# Vibro-acoustic radiation from a fluid-filled viscoelastic cylindrical shell with internal turbulent flow

код для вставкиСкачатьThe Pennsylvania State University The Graduate School THE PREDICTION OF BROADBAND SHOCK-ASSOCIATED NOISE USING REYNOLDS-AVERAGED NAVIER-STOKES SOLUTIONS A Dissertation in Aerospace Engineering by Steven Arthur Eric Miller c 2009 Steven Arthur Eric Miller Submitted in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy December 2009 UMI Number: 3399683 All rights reserved INFORMATION TO ALL USERS The quality of this reproduction is dependent upon the quality of the copy submitted. In the unlikely event that the author did not send a complete manuscript and there are missing pages, these will be noted. Also, if material had to be removed, a note will indicate the deletion. UMI 3399683 Copyright 2010 by ProQuest LLC. All rights reserved. This edition of the work is protected against unauthorized copying under Title 17, United States Code. ProQuest LLC 789 East Eisenhower Parkway P.O. Box 1346 Ann Arbor, MI 48106-1346 The dissertation of Steven Arthur Eric Miller was reviewed and approved? by the following: Philip J. Morris Boeing/A.D. Welliver Professor of Aerospace Engineering Dissertation Advisor, Chair of Committee Dennis K. McLaughlin Professor of Aerospace Engineering Kenneth S. Brentner Professor of Aerospace Engineering Daniel C. Haworth Professor of Mechanical Engineering George A. Lesieutre Professor of Aerospace Engineering Head of the Department of Aerospace Engineering ? Signatures are on file in the Graduate School. Abstract Broadband shock-associated noise is a component of jet noise generated by supersonic jets operating off-design. It is characterized by multiple broadband peaks and dominates the total noise at large angles to the jet downstream axis. A new model is introduced for the prediction of broadband shock-associated noise that uses the solution of the Reynolds averaged Navier-Stokes equations. The noise model is an acoustic analogy based on the linearized Euler equations. The equivalent source terms depend on the product of the fluctuations associated with the shock cell structure and the turbulent velocity fluctuations in the jet shear layer. The former are deterministic and are obtained from the Reynolds averaged Navier-Stokes solution. A statistical model is introduced to describe the properties of the turbulence. Only the geometry and operating conditions of the nozzle need to be known to make noise predictions. Unlike other models, the developed broadband shockassociated noise model is a true prediction scheme and not calibrated for a finite range of operating conditions. The broadband shock-associated noise model developed represents the only prediction method in existence that has no restrictions on nozzle geometry or jet operating conditions. This overcomes the limitations and empiricism present in previous broadband shock-associated noise models. Extensive validation of the Reynolds averaged Navier-Stokes solution is performed using experimental data. Validation efforts of the Reynolds averaged Navier-Stokes solutions include comparisons of Pitot and static probe measurements and schlieren visualization. These validations show both the strengths and deficiencies for modeling strategies of supersonic jets operating off-design using Reynolds averaged Navier-Stokes equations and associated turbulence closure schemes. Predictions for various nozzle and operating conditions are compared with experimental noise measurements of the associated jets to validate the broadband shock-associated noise model. The operating conditions include under-expanded and over-expanded jets with and without heating. iii Table of Contents List of Figures vi List of Tables xiii List of Symbols xiv Acknowledgments xviii Chapter 1 Introduction 1.1 Supersonic Jets . . . . . . . . . . . . . . . . . . . 1.1.1 Structure of the Flow Field . . . . . . . . 1.1.2 Three Components of Supersonic Jet Noise 1.1.2.1 Mixing Noise . . . . . . . . . . . 1.1.2.2 Screech . . . . . . . . . . . . . . 1.1.2.3 BBSAN . . . . . . . . . . . . . . 1.2 Current BBSAN Prediction Methodology . . . . . 1.2.1 Harper-Bourne and Fisher?s Model . . . . 1.2.2 Tam?s Model . . . . . . . . . . . . . . . . 1.2.3 Current Model Limitations . . . . . . . . . Chapter 2 Computational Fluid Dynamics 2.1 Introduction to CFD . . . . . . 2.1.1 Governing Equations . . 2.1.2 Grid Generation . . . . . 2.1.3 Solver . . . . . . . . . . 2.1.4 Post Processing . . . . . iv . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 5 8 11 14 16 18 19 20 24 26 . . . . . 31 32 32 36 40 42 2.2 CFD Results . . . . . . . . . . . . . . . . . . . 2.2.1 Computational Grids . . . . . . . . . . . 2.2.2 General CFD Results . . . . . . . . . . . 2.2.3 Menter SST and Chien K- Comparisons 2.2.4 Pitot Probe Comparisons . . . . . . . . . 2.2.5 Schlieren Comparisons . . . . . . . . . . 2.2.6 Off-Design Study . . . . . . . . . . . . . 2.2.7 Laminar and Turbulent Flow Nozzle . . 2.2.8 Helium and Hot Jet Comparisons . . . . Chapter 3 Broadband Shock-Associated Noise 3.1 Model Development . . . . . . . . . . . . . 3.2 Implementation . . . . . . . . . . . . . . . 3.3 Parametric Studies . . . . . . . . . . . . . 3.4 Single Stream Axisymmetric Jets . . . . . 3.5 Dual Stream Axisymmetric Jets . . . . . . 3.6 Three Dimensional Jets . . . . . . . . . . . 3.7 The Effect of Laminar Flow in the Nozzle 3.8 Helium / Air Mixture and Hot Air . . . . 3.9 Turbulent Scale Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 45 52 56 59 76 84 86 90 . . . . . . . . . 93 93 104 115 118 140 145 156 158 160 Chapter 4 Conclusion 163 4.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 4.2 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168 4.3 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 Appendix A Correlation and Sound Pressure Level 172 Appendix B Model Integrations 177 B.1 Integration of I?? . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 B.2 Integration of I?? . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178 Appendix C Green?s Function 180 Bibliography 184 v List of Figures 1.1 1.2 1.3 1.4 1.5 2.1 2.2 2.3 Possible operating conditions of converging and converging-diverging nozzles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The three possible types of operating conditions in a supersonic on- and off-design supersonic jet. Numerical contours of ??/?y are gradients of density in the cross-stream direction representative of a schlieren image. a) An over-expanded jet. b) An on-design jet. c) An under-expanded jet. . . . . . . . . . . . . . . . . . . . . . . . SP L per unit St of a Md = 1.00, Mj = 1.50, N P R = 3.67, T T R = 1.00, D = 0.0127 m, jet at R/D = 100 and various observer locations, ?. The experimental data is courtesy of NASA. . . The Harper-Bourne and Fisher model prediction compared with experimental results of a Md = 1.00, Mj = 1.42, D = 0.025m jet at 118 D, and ? = 135 degrees. The experimental data is digitized from Harper-Bourne and Fisher [1]. . . . . . . . . . . . . . . . . . . Tam?s model prediction compared with experimental results of a Md = 1.00, Mj = 1.50, N P R = 3.67, T T R = 1.00, D = 0.0127 m jet at R/D = 100 and ? = 130. The experimental data is courtesy of NASA. Note that the prediction has been translated by -5 dB to better match experiment. . . . . . . . . . . . . . . . . . . . . . . . . Example placement of boundary conditions for CFD of an axisymmetric jet simulation. . . . . . . . . . . . . . . . . . . . . . . . . . . Nozzle profiles and closeup of the computational grid near the nozzle exit plane. a) Contour of the converging nozzle. b) Closeup of the nozzle exit of the converging nozzle. c) Contour of the convergingdiverging nozzle. d) Closeup of the nozzle exit for the convergingdiverging nozzle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dual stream nozzle profile and closeup view of computational grid near the jet exit planes. . . . . . . . . . . . . . . . . . . . . . . . . vi 7 10 13 23 27 39 46 48 2.4 2.5 2.6 2.7 2.8 2.9 2.10 2.11 2.12 2.13 2.14 2.15 The computational domain of the rectangular nozzle with Md = 1.50. The domain extends 75 equivalent diameters downstream and 50 equivalent diameters in the cross-stream directions from the nozzle exit center plane. The width of the nozzle exit is w = 0.0208 m and the height is h = 0.0119 m. . . . . . . . . . . . . . . . . . . . An enlarged view of the rectangular nozzle computational grid. . . . Contours of the velocity component, u m/s, for a Md = 1.50, Mj = 1.30, T T R = 1.00, D = 0.0127 m converging-diverging axisymmetric jet. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Contours of the Mach number, M , for a Md = 1.50, Mj = 1.30, T T R = 1.00, D = 0.0127 m converging-diverging axisymmetric jet. Contours of the turbulent kinetic energy, K m2 /s2 , for a Md = 1.50, Mj = 1.30, T T R = 1.00, D = 0.0127 m converging-diverging axisymmetric jet. . . . . . . . . . . . . . . . . . . . . . . . . . . . . Contours of the dissipation, m2 /s3 , for a Md = 1.50, Mj = 1.30, T T R = 1.00, D = 0.0127 m converging-diverging axisymmetric jet. Contours of the shock pressure, ps Pa, for a Md = 1.50, Mj = 1.30, T T R = 1.00, D = 0.0127 m converging-diverging axisymmetric jet. Extracted values along y/D = 0.50 of a converging conical nozzle operating at Mj = 1.50 and T T R = 1.00 or T T R = 2.20 using the Menter SST or Chien K- turbulence models. . . . . . . . . . . . . . Extracted values along the cross stream directions of a converging conical nozzle operating at Mj = 1.50 and T T R = 1.00 or T T R = 2.20 using the Menter SST or Chien K- turbulence models. a) Transverse at x/D = 5.00 b) Transverse at x/D = 10.00. . . . . . . Comparison between the experimental (dots) and numerical (lines) po1 /po of the Md = 1.00, Mj = 1.50, converging nozzle case. Each set of data is separated by x/D = 0.20 starting at x/D = 0.0 at the left and stopping at x/D = 2.0 on the right. . . . . . . . . . . . Comparison between the experimental (dots) and numerical (lines) p/(1/2?u2j ) of the Md = 1.00, Mj = 1.50, converging nozzle case. Each set of data is separated by x/D = 0.20 starting at x/D = 0.0 at the left and stopping at x/D = 2.0 on the right. . . . . . . . . . Comparison between the experimental (dots) and numerical (lines) M of the Md = 1.00, Mj = 1.50, converging nozzle case. Each set of data is separated by x/D = 0.20 starting at x/D = 0.0 at the left and stopping at x/D = 2.0 on the right. . . . . . . . . . . . . . vii 50 51 54 54 54 55 55 57 58 62 63 63 2.16 Comparison between the experimental (dots) and numerical (lines) po1 /po of the Md = 1.50, Mj = 1.30, converging-diverging nozzle case. Each set of data is separated by x/D = 0.20 starting at x/D = 0.0 at the left and stopping at x/D = 2.0 on the right. . . 2.17 Comparison between the experimental (dots) and numerical (lines) p/(1/2?u2j ) of the Md = 1.50, Mj = 1.30, converging-diverging nozzle case. Each set of data is separated by x/D = 0.20 starting at x/D = 0.0 at the left and stopping at x/D = 2.0 on the right. . 2.18 Comparison between the experimental (dots) and numerical (lines) M of the Md = 1.50, Mj = 1.30, converging-diverging nozzle case. Each set of data is separated by x/D = 0.20 starting at x/D = 0.0 at the left and stopping at x/D = 2.0 on the right. . . . . . . . . 2.19 Pitot probe comparisons between the experimental (dots) and numerical (lines) of po1 /po of the Md = 1.50, Mj = 1.30, De = 0.01776 m jet along the major axis plane. . . . . . . . . . . . . . . . . . . 2.20 Pitot probe comparisons between the experimental (dots) and numerical (lines) of po1 /po of the Md = 1.50, Mj = 1.30, De = 0.01776 m jet along the minor axis plane. . . . . . . . . . . . . . . . . . . 2.21 Pitot probe comparisons between the experimental (dots) and numerical (lines) of po1 /po of the Md = 1.50, Mj = 1.70, De = 0.01776 m jet along the major axis plane. . . . . . . . . . . . . . . . . . . 2.22 Pitot probe comparisons between the experimental (dots) and numerical (lines) of po1 /po of the Md = 1.50, Mj = 1.70, De = 0.01776 m jet along the minor axis plane. . . . . . . . . . . . . . . . . . . 2.23 Pitot probe comparisons between the experimental (dots) and numerical (lines) of M of the Md = 1.50, Mj = 1.30, De = 0.01776 m jet along the major axis plane. . . . . . . . . . . . . . . . . . . . . 2.24 Pitot probe comparisons between the experimental (dots) and numerical (lines) of M of the Md = 1.50, Mj = 1.30, De = 0.01776 m jet along the minor axis plane. . . . . . . . . . . . . . . . . . . . . 2.25 Pitot probe comparisons between the experimental (dots) and numerical (lines) of M of the Md = 1.50, Mj = 1.70, De = 0.01776 m jet along the major axis plane. . . . . . . . . . . . . . . . . . . . . 2.26 Pitot probe comparisons between the experimental (dots) and numerical (lines) of M of the Md = 1.50, Mj = 1.70, De = 0.01776 m jet along the minor axis plane. . . . . . . . . . . . . . . . . . . . . viii . 64 . 64 . 65 . 68 . 69 . 70 . 71 . 72 . 73 . 74 . 75 2.27 Comparison between the experimental and numerical schlieren of a) the converging nozzle case Md = 1.00, Mj = 1.50, b) the convergingdiverging nozzle case Md = 1.50, Mj = 1.30. The nozzle exit is at x/D = 0.0 and the flow moves from x/D = 0.0 to the right. The nozzle centerline is at y/D = 0.0 and the nozzle lips at y/D = 0.5. 2.28 Comparison between the experimental and numerical schlieren of the converging nozzle case Md = 1.00, Mj = 1.50 a) T T R = 2.20 air, schlieren on top and numerical schlieren on bottom, b) top: schlieren of T T R = 2.20 air. bottom: numerical schlieren of helium / air mixture. The nozzle exit is at x/D = 0.0 and the flow moves from x/D = 0.0 to the right. The nozzle centerline is at y/D = 0.0 and the nozzle lips are at y/D = 0.5. . . . . . . . . . . . . . . . . 2.29 Comparison between the experimental and numerical schlieren of the rectangular nozzle case Md = 1.50, Mj = 1.30, and T T R = 1.00. a) The minor axis plane top: schlieren, bottom: numerical schlieren. b) The major axis plane, top: schlieren, bottom: numerical schlieren. The nozzle exit is at x/De = 0.0 and the flow moves from x/De = 0.0 to the right. . . . . . . . . . . . . . . . . . . . . 2.30 Comparison between the experimental and numerical schlieren of the rectangular nozzle case Md = 1.50, Mj = 1.70, and T T R = 1.00. a) The minor axis plane top: schlieren, bottom: numerical schlieren. b) The major axis plane, top: schlieren, bottom: numerical schlieren. The nozzle exit is at x/De = 0.0 and the flow moves from x/De = 0.0 to the right. . . . . . . . . . . . . . . . . . . . . 2.31 Comparison between the experimental and numerical schlieren of the rectangular nozzle case Md = 1.50 and T T R = 2.20. The experimental results use a helium / air mixture and the simulations use heated air. a) Mj = 1.30 Major axis plane top: schlieren, bottom: numerical schlieren. b) Mj = 1.70 major axis plane. top: schlieren. bottom: numerical schlieren. The nozzle exit is at x/De = 0.0 and the flow moves from x/De = 0.0 to the right. . . . . . . . . . . . . 2.32 Variation of the logarithm of shock strength, K?p2 (dots), compared with the off-design parameter ? 4 (line) offset slightly. The logarithm of ? 4 has a nearly identical slope with K?p2 . . . . . . . 2.33 Comparison of Mach number contours for the Md = 1.50, Mj = 1.30, T T R = 2.20, and D = 0.0127 m jet. Top: Fully turbulent flow. Bottom: Laminar flow inside the nozzle and turbulent flow outside the nozzle. . . . . . . . . . . . . . . . . . . . . . . . . . . ix . 78 . 80 . 82 . 82 . 83 . 85 . 87 2.34 Centerline values for a Md = 1.50, Mj = 1.30, T T R = 2.20, D = 0.0127 m jet. Top: Variation of Mach number. Bottom: Variation of pressure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.35 Lipline values (y/D = 0.50) for a Md = 1.50, Mj = 1.30, T T R = 2.20, D = 0.0127 m jet. Top: Variation of turbulent kinetic energy. Bottom: Variation of streamwise velocity. . . . . . . . . . . . . . . 2.36 Comparison of Mach number contours for Md = 1.00, Mj = 1.50, T T R = 2.20 air or T T R = 1.00 helium/air, D = 0.0127 m jet. Top: Contours of M of hot air. Bottom: Contours of M of the helium / air mixture. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.37 Centerline values for a Md = 1.50, Mj = 1.30, T T R = 2.20 or T T R = 1.00 helium / air mixture, D = 0.0127 m jet. Top: Variation of density, ?. Bottom: Variation of centerline velocity, u. . . . 3.1 3.2 3.3 . 88 . 89 . 90 . 92 Flowchart of the BBSAN prediction code. . . . . . . . . . . . . . . 105 Flowchart of the BBSAN integration subroutine. . . . . . . . . . . . 107 Interpolation of u m/s onto the integration region from the CFD solution of a Md = 1.00, Mj = 1.50, T T R = 1.00, D = 0.0127 m jet. The red line encloses of the integration region. . . . . . . . . . . 110 3.4 The magnitude of the Fourier transform of the shock cell pressure p?s . 112 3.5 a) Tcf verses T T R for a Md = 1.00 and Mj = 1.50 jet. b) The associated change in dB when heating a Md = 1.00 and Mj = 1.50 jet. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 3.6 Comparisons of BBSAN predictions with experiments for Md = 1.00, Mj = 1.50, T T R = 1.00, R/D = 100. . . . . . . . . . . . . . . 125 3.7 The total BBSAN prediction and the accompanying contributions from selective integrations over contributing wavenumbers of p?s representing different waveguide modes of the shock cell structure. Md = 1.00, Mj = 1.50, T T R = 1.00, R/D = 100, ? = 120.0. . . . . . 126 3.8 Various plots of the flow-field region for Md = 1.00, Mj = 1.50, T T R = 1.00, R/D = 100, ? = 90.0. a) contours of ps . b) contours of K. c) The spatially distributed source of the BBSAN at the peak frequency of fp = 20241 Hz or St = 0.653. . . . . . . . . . . . . . . 127 3.9 Comparisons of BBSAN predictions with experiments for Md = 1.00, Mj = 1.50, T T R = 2.20, and R/D = 100. . . . . . . . . . . . . 128 3.10 Comparisons of BBSAN predictions with experiments for Md = 1.00, Mj = 1.22, T T R = 1.00, and R/D = 100. . . . . . . . . . . . . 129 3.11 Comparisons of BBSAN predictions with experiments for Md = 1.00, Mj = 1.22, T T R = 2.20, and R/D = 100. . . . . . . . . . . . . 130 x 3.12 Comparisons of BBSAN predictions with experiments for Md = 1.00, Mj = 1.22, T T R = 3.20, and R/D = 100. . . . . . . . . . . . . 3.13 Comparisons of BBSAN predictions with experiments for Md = 1.50, Mj = 1.30, T T R = 1.00, and R/D = 100. . . . . . . . . . . . . 3.14 Comparisons of BBSAN predictions with experiments for Md = 1.50, Mj = 1.30, T T R = 2.20, and R/D = 100. . . . . . . . . . . . . 3.15 Comparisons of BBSAN predictions with experiments for Md = 1.50, Mj = 1.40, T T R = 1.00, and R/D = 100. . . . . . . . . . . . . 3.16 Comparisons of BBSAN predictions with experiments for Md = 1.50, Mj = 1.60, T T R = 1.00, and R/D = 100. . . . . . . . . . . . . 3.17 Comparisons of BBSAN predictions with experiments for Md = 1.00, Mj = 1.47, T T R = 3.20, and R/D = 100. . . . . . . . . . . . . 3.18 Comparisons of BBSAN predictions with experiments for Md = 1.00, Mj = 1.56, T T R = 3.20, and R/D = 100. . . . . . . . . . . . . 3.19 Comparisons of BBSAN predictions with experiments for Md = 1.50, Mj = 1.70, T T R = 1.00, and R/D = 100. . . . . . . . . . . . . 3.20 Comparisons of BBSAN predictions with experiments for Md = 1.50, Mj = 1.70, T T R = 2.20, and R/D = 100. . . . . . . . . . . . . 3.21 Integration regions for the BBSAN calculation of the dualstream jet Mdp = 1.00, Mjp = 1.19, Mds = 1.00, Mjs = 1.04, T T Rp = 2.70. . 3.22 Comparisons of BBSAN predictions with experiments for the dualstream jet Mdp = 1.00, Mjp = 1.19, Mds = 1.00, Mjs = 1.04, T T Rp = 2.70, R/D = 100. . . . . . . . . . . . . . . . . . . . . . . . 3.23 Comparisons of BBSAN predictions with experiments for the dualstream jet Mdp = 1.00, Mjp = 1.19, Mds = 1.00, Mjs = 0.96, T T Rp = 2.70, R/D = 100. . . . . . . . . . . . . . . . . . . . . . . . 3.24 Comparisons of BBSAN predictions with experiments for the rectangular jet Md = 1.50, Mj = 1.30, T T R = 1.00, R/De = 100 in the major axis direction. . . . . . . . . . . . . . . . . . . . . . . . . 3.25 Comparisons of BBSAN predictions with experiments for the rectangular jet Md = 1.50, Mj = 1.30, T T R = 1.00, R/De = 100 in the minor axis direction. . . . . . . . . . . . . . . . . . . . . . . . . 3.26 Comparisons of BBSAN predictions with experiments for the rectangular jet Md = 1.50, Mj = 1.70, T T R = 1.00, R/De = 100 in the major axis direction. . . . . . . . . . . . . . . . . . . . . . . . . 3.27 Comparisons of BBSAN predictions with experiments for the rectangular jet Md = 1.50, Mj = 1.70, T T R = 1.00, R/De = 100 in the minor axis direction. . . . . . . . . . . . . . . . . . . . . . . . . xi 131 132 133 134 135 136 137 138 139 142 143 144 148 149 150 151 3.28 Comparisons of BBSAN predictions with experiments for the rectangular jet Md = 1.50, Mj = 1.30, T T R = 2.20, R/De = 100 in the major axis direction. . . . . . . . . . . . . . . . . . . . . . . . 3.29 Comparisons of BBSAN predictions with experiments for the rectangular jet Md = 1.50, Mj = 1.30, T T R = 2.20, R/De = 100 in the minor axis direction. . . . . . . . . . . . . . . . . . . . . . . . 3.30 Comparisons of BBSAN predictions with experiments for the rectangular jet Md = 1.50, Mj = 1.70, T T R = 2.20, R/De = 100 in the major axis direction. . . . . . . . . . . . . . . . . . . . . . . . 3.31 Comparisons of BBSAN predictions with experiments for the rectangular jet Md = 1.50, Mj = 1.70, T T R = 2.20, R/De = 100 in the minor axis direction. . . . . . . . . . . . . . . . . . . . . . . . 3.32 Comparisons of BBSAN predictions for the laminar / RANS and fully RANS simulation with experiments for a Md = 1.50, Mj = 1.30, T T R = 2.20, jet at R/D = 100. . . . . . . . . . . . . . . . . 3.33 Comparisons of BBSAN predictions for heated air and helium / air simulated jets with experiments using heated air and helium / air mixtures. The jet conditions are Md = 1.00, Mj = 1.50, T T R = 2.20, R/D = 100. . . . . . . . . . . . . . . . . . . . . . . 3.34 Comparisons of BBSAN predictions using turbulence coefficients that are optimized for a non-screeching jet with experiments for Md = 1.50, Mj = 1.70, T T R = 2.20, R/D = 100. . . . . . . . . . . xii . 152 . 153 . 154 . 155 . 157 . 159 . 161 List of Tables 3.1 3.2 3.3 3.4 3.5 3.6 Summary of the parametric study of the integration and wavenumber regions for the circular converging jet operating at Mj = 1.50, T T R = 1.00, D = 0.0127 m. . . . . . . . . . . . . . . . . . . . . . Recommended integration and index ranges based on the integration study of Mj = 1.50, T T R = 1.00, D = 0.0127 m. . . . . . . . Recommended non-dimensional integration values based on D and index ranges. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Jet operating conditions of the RANS CFD and BBSAN predictions for axisymmetric single stream jets. . . . . . . . . . . . . . . . . . Jet operating conditions for the RANS CFD and BBSAN predictions of the dual stream jets. . . . . . . . . . . . . . . . . . . . . . Jet operating conditions for the RANS CFD and BBSAN predictions of the 3D calculations. . . . . . . . . . . . . . . . . . . . . . xiii . 116 . 117 . 117 . 119 . 142 . 145 List of Symbols a Local speed of sound cl Coefficient of the characteristic streamwise coherent large length scale of the convecting turbulence cp Specific heat at constant pressure c? Coefficient of the characteristic coherent large time scale of the convecting turbulence c? Coefficient of the characteristic cross-stream coherent large length scale of the convecting turbulence c? Speed of sound in the far-field D Exit diameter of a nozzle De Equivalent exit diameter of a non-circular nozzle Dj Fully expanded diameter of a nozzle Dje Equivalent fully expanded diameter of a non-circular nozzle D? The diameter at the throat of the nozzle e Specific internal energy f Frequency fc Characteristic frequency fp Peak Broadband Shock-Associated Noise frequency fs Screech frequency xiv fia Unsteady force per unit volume related to the interaction of fluctuations in the sound speed (or temperature), caused by the turbulence and the shock cells, and the associated pressure gradients fiv Unsteady force per unit volume associated with interactions between the turbulent velocity fluctuations and the velocity perturbations associated with the shock cells fxv Component of fiv in the direction of the observer K Turbulent kinetic energy k Wavenumber l Characteristic streamwise coherent large length scale of the convecting turbulence l? Characteristic cross-stream coherent large length scale of the convecting turbulence M Mach number Mc Convective Mach number Md Design Mach number of the nozzle Mj Fully expanded Mach number of the jet N P R Nozzle pressure ratio N P Rp Nozzle pressure ratio of the primary jet N P Rs Nozzle pressure ratio of the secondary jet p Thermodynamic pressure pref po po1 Reference acoustic pressure Stagnation pressure Stagnation pressure or total pressure at Pitot probe ps Shock pressure p?s Fourier transform in the streamwise direction of the jet of the shock pressure xv p? Ambient pressure R Gas constant or distance from nozzle exit to observer Rv v Rnm v Rxx r Two-point cross correlation function of the turbulent velocity fluctuations in the observer direction Two-point cross correlation function of fnv Two-point cross correlation function of the components of fnv in the direction of the observer Radial direction Re Reynolds number S Sv Spectral density Cross spectral density of the two-point cross correlation function of the turbulent velocity fluctuations in the observer direction St Strouhal number based on the characteristic frequency SP L Sound Pressure Level T Tcf To T? Thermodynamic static temperature Tam temperature correction factor Stagnation temperature Thermodynamic static temperature in the far-field T T R Total temperature ratio t Time uc Convective velocity uj Fully expanded jet velocity vg Component of the vector Green?s function x Vector from the center of the nozzle exit to the observer y Vector from the center of the nozzle exit to a source point in the jet near-field xvi ?f Frequency bandwidth ? Off-design parameter ?? ? Heat expansion coefficient Ratio of specific heats ?() Dirac delta function ?ij Kronecker delta function Turbulent dissipation ? Vector between two points in the source region ? Observer angle from the downstream axis, [degrees], or alternatively the dilatation rate ? Thermal conductivity ? Kinematic viscosity ?t Eddy viscosity ? The local density ?j Fully expanded density of a jet ? Logarithmic expression of pressure ?g Component of the vector Green?s function ? Retarded time ?s Characteristic coherent large time scale of the convecting turbulence ?ij Shear stress tensor ? Specific dissipation ? Radial frequency (? = 2?f ) xvii Acknowledgments This research is sponsored by the NASA Cooperative Agreement NNX07AC88A entitled, ?A Comprehensive Model for the Prediction of Supersonic Jet Noise,? a multiyear project involving professors, students, industry, and government. The Technical Monitor of the Cooperative Agreement is Dr. Milo Dahl of NASA. Dr. Nick Georgiadis of NASA provided helpful discussions regarding the Wind-US CFD solver. Experimental data is provided mainly by the author?s colleague and experimental counterpart Dr. Jeremy Veltin of The Pennsylvania State University. Additional acoustic data is courtesy of Mr. Ching-Wen Kuo of The Pennsylvania State University, Dr. K. Viswanathan of the Boeing Company, and also various NASA sources. All experimental data presented in this dissertation is published with permission. The author?s parents, Mr. Arthur F. Miller and Mrs. Beth A. Miller, and his sister, Dr. Tiffany A. Fidler, are acknowledged for encouraging studies in Aerospace Engineering. Their continuous love and support is invaluable. The author?s former advisor at Michigan State University, Professor of Aerospace Engineering Mei Zhuang of Ohio State University, is acknowledged for encouraging studies in Aeroacoustics. The Ph.D. committee, Professor Dennis K. McLaughlin, Professor Kenneth S. Brentner, Professor Daniel C. Haworth, and former member Professor Vigor Yang, are acknowledged for committing their time to providing insight into the project. Most importantly, Boeing/A.D. Welliver Professor of Aerospace Engineering Philip J. Morris, is acknowledged for acting as the author?s adviser and committee chair. His many years of guidance and support for this and many other research projects is invaluable. He provided the best graduate and research opportunities possible. Finally, his work ethic and dedication to scientific investigations will be used as a guide for the author?s research career. xviii Dedication Dedicated to Pierre-Simon, Marquis de Laplace. ?We ought then to regard the present state of the universe as the effect of its anterior state and as the cause of the one which is to follow. Given for one instant an intelligence which could comprehend all the forces by which nature is animated and the respective situation of the beings who compose it - an intelligence sufficiently vast to submit these data to analysis - it would embrace in the same formula the movements of the greatest bodies of the universe and those of the lightest atom; for it, nothing would be uncertain and the future, as the past, would be present to its eyes.? - Laplace 1820.1 ?The curve described by a simple molecule of air or vapor is regulated in a manner just as certain as the planetary orbits; the only difference between them is that which comes from our ignorance.? - Laplace 1820. ?Were she then to watch me live through it, she might smile condescendingly, as one who watches a marionette dance to the tugs of strings that it knows nothing about.?2 1 Laplace, P., 1820, Essai Philosophique sur les Probabilits, Paris: V Courcier; repr. F.W. Truscott and F.L. Emory (trans.), A Philosophical Essay on Probabilities, New York: Dover, 1951. 2 Causal Determinism, The Stanford Encyclopedia of Philosophy (Winter 2008 Edition), Edward N. Zalta (ed.). xix Chapter 1 Introduction This dissertation is a component of a larger research project for the development of a comprehensive prediction method for noise generated by supersonic jets. The project consists of a tightly combined effort between universities, industry, and government using analytical, experimental, and numerical methods to develop noise models. The project uses some existing prediction methods, however, the focus of this dissertation is the development of a new broadband shock-associated noise (BBSAN) prediction mathematical model, which is based primarily on first principles and is free of the restrictions of past empirical and semi-empirical models. However, much physical insight and mathematical approaches were adapted from past efforts. The BBSAN model developed only uses the operating conditions and geometry of the nozzle to form predictions from a single closed formula consisting of integrations over the region containing noise sources in the flow-field of the jet plume. This approach removes a large number of restrictions and forms a unique and new model for BBSAN. BBSAN is one of three components of supersonic jet noise and is always the dominant component at certain frequencies and angles to the jet axis. Past prediction methods of BBSAN are based in part on empirical correlations of radiated noise and jet flow measurements. These methods are restricted to circular single and dual stream jet nozzle geometries. These BBSAN models were mainly developed based on observations with cold jets. However, BBSAN remains very important in hot jets also, considering that engine exhaust is always heated. In heated jets the mixing noise dominates in the downstream quadrant with respect 2 to the jet exit. However, the BBSAN is always higher in intensity than the mixing noise at larger angles to the jet downstream axis. Since typical engine exhaust is hot and has nozzles that may be non-circular in practice, it is difficult to apply current BBSAN models in these situations. Gathering the acoustic spectra of shock noise from experiments in jet facilities is an alternative to prediction. However, performing experiments using model nozzles in small or medium scale facilities using heated air is relatively expensive compared to calculations. Furthermore, numerical calculations may be faster to obtain than experiments, which increases the number of points in the design space when creating new aircraft configurations. Various governments and industries pursuing new designs of supersonic business and military planes using supersonic jets are also in need of improved prediction capability. Furthermore, community noise issues related to military fighter aircraft are becoming a serious concern for residents living near military facilities. Unfortunately, no present capability exists to make reliable predictions of noise produced by either existing jet engines or new nozzle designs for future aircraft. The model presented herein provides industry and government aircraft and engine developers with a low cost method of predicting BBSAN. Many of the current approaches to jet noise reduction are based on modifying the jet flow field, which is mainly achieved through changes to the nozzle exit geometry. However, there is no method that can relate the changes in the flow field to the radiated noise. Therefore, there is a need to develop a model that predicts BBSAN over a wide range of operating conditions, both hot and cold, and takes into account the nozzle geometry, which essentially controls the development of the flow field. This will yield more understanding between the jet flow field predictions, experimental measurements, and BBSAN prediction and measurement. Burns [2] states that since the beginning of civil and military air transportation there has been great interest in reducing the noise emitted from the jets. This is mainly due to community noise around military facilities and civilian airports. When aircraft take off the noise from the jet is dominant relative to other sources and also problematic during cruise. During aircraft landing jet noise is less of a concern relative to take-off and cruise. However, the intensities of jet noise are still a major component of the total noise spectrum. In 1987 a renewed interest 3 was generated in supersonic jet noise research instituted by the United States Government as it recognized its need to remain competitive in aerospace research [3], which occurred long after the termination of supersonic transport programs [4]. To guide noise reduction efforts, the Federal Aviation Administration [5] creates long term noise suppression goals for noise reduction from jets. The BBSAN model developed is designed to be compatible with next generation aircraft noise prediction tools as described by Golub and Posey [6] and will be implemented in future versions of the Aircraft Noise Prediction code of NASA. Early BBSAN prediction methods, such as those available in the NASA Aircraft Noise Prediction Program (ANOPP) [7] and the SAE ARP876 [8], are based on the work of Harper-Bourne and Fisher [1]. These noise prediction methods involve master spectra and correlation functions based on experimental observations of supersonic jets from a single stream convergent nozzle. While such schemes are useful, they are only reliable within the range of jet conditions and geometries included in the existing database. Prediction methodologies for BBSAN based on Large Eddy Simulations (LES) or similar large scale computations such as Direct Numerical Simulation (DNS) are able to predict BBSAN. Recent research at Penn State by Paliath and Morris [9], as well as related computations by Shur et al. [10], demonstrated the capabilities of this prediction methodology. Complex geometries, including a beveled nozzle and simulated chevrons were studied. The results are very encouraging and indicate the quality of noise predictions that can be achieved through a judicious use of discretization methods, grid distribution, and turbulence modeling. Unfortunately, the computations are expensive and time-consuming. These methods are so computationally expensive that they will remain a research based tool for the foreseeable future. The BBSAN model developed in this thesis uses Reynolds Averaged Navier-Stokes (RANS) solutions as input. These are generated knowing only the geometry of the nozzle and operating conditions, and so are computationally more efficient than unsteady simulation-based methods such as LES, DNS, or another computationally expensive scheme. The BBSAN model development makes extensive use of experimental measurements of two-point statistical properties of turbulent fluctuations in supersonic jets. As will be shown they are a key component of the BBSAN prediction. The far-field 4 measurements of fluctuation pressure are also key, and their breakdown into different components of noise is also helpful in development of a model. The coupling of experimental methods with model development is a most effective approach of research, more so than either methods of research individually as there are strengths and weaknesses associated with each. There are two original contributions to science presented in this dissertation. First is the validation of RANS solutions for off-design supersonic jets operating over a wide range operating conditions using various nozzle geometries. These comparisons show both the strengths and deficiencies of current state-of-the-art modeling strategies for supersonic jets operating off-design. The second major contribution is the development of a BBSAN model that is based on first principles and has no restrictions regarding nozzle geometry and operating conditions. Only a precise RANS solution of the jet flow-field is needed for a BBSAN prediction. Unlike other models, the developed BBSAN prediction methodology is a true prediction scheme and not calibrated for a finite range of operating conditions. The BBSAN model developed in this dissertation represents the only prediction method in existence that has no restrictions. This dissertation is presented in the same order as the BBSAN model was developed and is divided into two main parts. The first is the computational fluid dynamics (CFD) development of the RANS solutions, which is described in Chapter 2. The CFD results presented are compared with experimental data. Since these RANS results are the input to the BBSAN model their accuracy is extremely important. Furthermore, RANS solutions that are compared with various experiments give validity to the numerical output of the CFD code and associated equations that model fluid. Chapter 2 is divided into two parts. The first part introduces the CFD and associated equations and turbulence models. The second part presents various CFD results and compares them with experimental data. Chapter 3 shows the BBSAN development mathematically and then predictions based on the model are compared to far-field spectra for various jet conditions. Multiple studies of the effect of the variation of parameters for the BBSAN predictions are made to help establish the sensitivity of the model. Results for axisymmetric single and dual stream, and rectangular jets are compared with experiment. Finally, conclusions and suggestions for future research are provided in Chapter 4. 5 Before the methods and results of the RANS solutions are presented, the remaining part of this introductory chapter examines the parameters that describe off-design supersonic jets and reviews the past BBSAN modeling efforts. 1.1 Supersonic Jets Before the aerodynamic and acoustic properties of the flow-field of supersonic jets are discussed, a few important parameters and conditions that are related to BBSAN are presented. First, the nozzle pressure ratio is the ratio of the stagnation pressure inside the jet plenum to the ambient pressure, N P R = po /p? . The total temperature ratio, T T R, is related to the stagnation temperature, To in the plenum, divided by the ambient temperature, T? , T T R = To /T? . The stagnation or total conditions are found through simple isentropic gas dynamic theory. These two parameters completely specify the operating conditions of a nozzle. The design Mach number, Md , of a nozzle is only dependent on the geometry of the nozzle. Specifically, the ratio of the exit diameter, D, to the choked throat, D? , forms a relation for Md , (?+1)/(2?2?) 1 + ??1 D2 2 = (?+1)/(2?2?) ??1 D?2 Md 1 + 2 Md2 (1.1) where ? is the ratio of specific heats for an ideal gas. This equation is implicit for Md and must be found by a simple iterative method once the ratio of the diameters are given. Note that if the ratio of D/D? = 1.00 then Md = 1.00. The fully expanded jet diameter, Dj , is defined as the diameter of the jet to which an off-design jet adjusts downstream of the nozzle. It is given by, " 1+ Dj = D 1+ (??1) Mj2 2 (??1) Md2 2 #(?+1)/(4??4) Md Mj 1/2 (1.2) where Mj is the fully expanded Mach number. The fully expanded Mach number, Mj is related only to the N P R by, 1/2 ??1 2 Mj = NP R ? ? 1 ??1 (1.3) 6 If the N P R is needed to produce a Mach number of Md then Md can be substituted into the above equation for Mj . If Md 6= Mj then the jet is said to be operating off-design. The Reynolds number, introduced by Osborne Reynolds [11] [12], is the ratio of the dynamic forces to the viscous forces. In general the Reynolds number, Re, is defined by, Re = uj D/? (1.4) Figure 1.1 shows all possible operating conditions of a jet operating both subsonically and supersonically on- and off-design. The left upper diagram shows the cross-section of a converging-diverging nozzle where Md > 1.00. The upper right diagram shows the cross-section of a converging nozzle where Md = 1.00. The high pressure (and possibly high temperature) fluid moves from inside the nozzle through the throat to the exit where it forms a plume in the ambient environment. Corresponding graphs below each nozzle show possible static pressure, p, distributions normalized by the stagnation pressure, po , along the centerline of the nozzle. Depending on the ratio of po to p? , which forms the N P R, various flow-fields inside and outside the nozzle exit can be obtained. If the N P R is small then a subsonic exit flow results as seen in the static pressure distributions of lines A or G. If the stagnation pressure is raised, thus increasing the N P R, eventually the flow will reach M = 1.0 at the throat of the nozzle but return to a subsonic condition as seen in line B or H. Notice that line H also corresponds to the exit of the converging nozzle where M = Md = 1.00 and is said to be operating ondesign. In the on-design case the pressure at the exit of the nozzle exactly matches the ambient pressure, p = p? . For the flow to reach M = 1.0 which is deemed choked flow, the ratio of the static pressure in the throat of the nozzle relative to the stagnation pressure in the plenum is 0.528 for ? = 1.4. If the N P R is increased further in the converging-diverging case then a normal shock will form in the diverging portion of the nozzle. As the ratio of po /p? grows, the normal shock travels downstream and eventually will exit the interior of the nozzle?s diverging section. Once this occurs there are only three possible operating conditions for the nozzle. The first is on-design, where the nozzle is perfectly expanded and the fluid exits the converging-diverging nozzle at a supersonic speed, as seen in line 7 E. In this case p? exactly matches p at the exit of the nozzle. Since there is no pressure mismatch the nozzle is considered to be operating on-design. This is also the case in line H of the converging nozzle except that the exit M = Md = 1.00. If po /p? does not allow for the static pressure to match the ambient pressure and Mj 6= Md then the nozzle operates off-design. This situation is shown by lines D or F of the converging-diverging nozzle and line I of the converging nozzle. The cases of interest of study in this dissertation focus on when the values of N P R are such that the operating conditions of the converging-diverging or converging nozzles operate off-design and supersonically as shown in lines D, F, or I. These lines of static pressure in particular are drawn very jagged outside the nozzle as the pressure mismatch causes a very unique flow-field to develop. For a more thorough introduction to the subject, Anderson [13] provides an overview with a historical perspective. Figure 1.1. Possible operating conditions of converging and converging-diverging nozzles. 8 1.1.1 Structure of the Flow Field The previous section described parameters necessary to specify the boundary conditions of the fluid dynamics problem and also some of the parameters needed to quantify the flow. Now, the structure of the flow-field or jet plume is discussed. This is accomplished by showing plots of numerical simulations of supersonic onand off-design jets. A single circular nozzle with Md = 1.50 (converging-diverging nozzle) and T T R = 1.00 is used for this task. The details of the simulations and their validation are given in Chapter 2. Figure 1.2 shows the three types of supersonic jets that are of interest to this work, over-expanded a), on-design b), and under-expanded c). The three cases are illustrative of those shown in the converging-diverging nozzle of Figure 1.1 cases D, E, and F respectively. The converging nozzle operating off-design is not shown because it is has the same characteristics as the under-expanded converging-diverging nozzle. The over-expanded jet is operating with a N P R of 2.77 and Mj of 1.30. The on-design jet is operating with a N P R of 3.67 and a Mj of 1.50. Finally, the under-expanded jet is operating with a N P R of 4.94 and a Mj of 1.70. In each of the contour plots the scale has been normalized by the exit diameter of the jet, D. The x-axis is in the downstream direction and the y-axis is in the cross-stream direction. x/D = 0.0 is at the nozzle exit plane and y/D = 0.0 is the center axis of the jet. The nozzle lips are at x/D = 0.0 and y/D = ▒0.5. The large white region in the vicinity of x/D < 0.0 is the interior of the nozzle. High pressure (and in aircraft engines heated, T T R > 1.00) fluid exits the nozzle supersonically. In this figure the flow-field of the jet is axisymmetric and three dimensional, but only a slice of the flow-field is shown through the center line of the nozzle. The contours of Figure 1.2 represent the slope of the density in the y direction, which simulates a numerical Z-type schlieren. This is discussed in detail in Chapter 2. Region A represents the inside of the nozzle and is the converging section of the nozzle. In this area the Mach number of the fluid is all less than one. In Region B the fluid accelerates past M = 1.00 and becomes supersonic exiting the nozzle, in this particular case at approximately M = 1.50. Note that distribution of M in the y direction is quadratic, as found by Meyer [14]. Waves in region B emit from the wall because the nozzle is not perfectly expanding the flow by eliminating the characteristic waves when they impinge on the opposite wall. 9 In the over-expanded case, a), conical oblique shock waves may be seen at point C. This oblique shock wave always originates from the nozzle lip in overexpanded jets, and it happens that this particular nozzle is over-expanded enough that the oblique shock waves terminate in a barrel shock or a Mach disc, which is characterized by the vertical line near x/D = 0.25. Mach discs do not form in slightly off-design jets. Oblique shock waves can also be found in the underexpanded jet, but they do not originate from the nozzle lip, and can be see also at point C in case c). Region D in case a) and c) are two locations of PrandtlMeyer [14] expansion waves. In these regions the flow accelerates and the pressure gradient is less than one in the streamwise direction. In case b) there are no strong oblique shock waves, only relatively weak ones in comparison. This is because the static pressure in the jet closely matches the ambient pressure at location E. In the off-design jet cases either a Prandtl-Meyer expansion fan, region D of case c), or an oblique shock wave must occur to match the ambient pressure. The repeating pattern of oblique shock and Prantdl-Meyer waves form shock cells. This is the so called shock cell structure1 and is present in some form in all supersonic offdesign jets. In 1950, Pack [15] gave an excellent analytical method for predicting the pressure fluctuations in the shock cell structure, but it is not accurate for jets operating highly off-design. The pattern of shock waves and Prandtl-Meyer expansion waves repeat and become weaker as they interact with the turbulent shear layer (initially at x/D = 0.0 and y/D ▒ 0.5) and finally dissipate in the turbulent mixing region far-downstream (x/D ? 10). The turbulent shear layer develops in all jets, subsonic and supersonic, overexpanded, under-expanded, or on-design, due to Kelvin-Helmholtz [16] instabilities that arise in shear layers of jets. These instabilities form a toroid (or another threedimensional shape due to a non-circular nozzle) vortex. These toroid vortices are also unstable and Widnall [17] instabilities form that eventually cause the shear layer to become fully turbulent. It is the growth of the turbulent shear layer that acts like a waveguide which contains the shock cell structure. The shear layer eventually dissipates the shock cell structure due to turbulent dissipation. The multiple oblique shock wave interactions with convecting turbulence in the shear layer is the cause of BBSAN. Details of how the shock cell structure and turbulent 1 Some researchers refer to this structure in off-design supersonic jets as shock diamonds. 10 shear layer are predicted are given in Chapter 2. These results are illustrative of the physics of the problems and it will be shown that almost identical plots can be produced from experimental methods. Figure 1.2. The three possible types of operating conditions in a supersonic on- and off-design supersonic jet. Numerical contours of ??/?y are gradients of density in the cross-stream direction representative of a schlieren image. a) An over-expanded jet. b) An on-design jet. c) An under-expanded jet. For supersonic jets operating off-design, one of the most important parameters used to link the operating conditions of the jet with the BBSAN intensity is the off-design parameter. The off-design parameter was first used by Harper-Bourne and Fisher [1], ?= q Mj2 ? 1 where ? is the off-design parameter. This definition of ? works very well for 11 converging nozzles where Md = 1.00. However, it needs to take into account the situations where converging-diverging nozzles are present. For Md > 1, the traditional definition of Harper-Bourne and Fisher is modified as, ?= q |Mj2 ? Md2 | (1.5) This definition of ? is used throughout the dissertation and is equivalent to the original definition for converging nozzles. If ? = 0 then the jet is operating ondesign and as ? increases the jet operates further off-design. Harper-Bourne and Fisher made many observations of the effect of ? as it varies for converging nozzles, Md = 1.00, and found that the intensity of sound from an off-design supersonic jet increases as ? 4 until a Mach disc or barrel shock forms in the core of the jet near ? ? 1.00. Once the barrel shock forms, then the noise intensity no longer increases in the same fashion. This observation is confirmed with numerical predictions in Chapter 2. Another important observation regarding the off-design parameter is that the shock-cell spacing is linearly proportional to ? for slightly or moderately off-design jets. 1.1.2 Three Components of Supersonic Jet Noise The flow-fields of supersonic off-design jets have been described in the previous section. This section describes the acoustic properties associated with these flowfields. Excellent overviews of the subject are given by Ffowcs Williams [18][19], Goldstein [20], and for supersonic jet noise, Tam [21]. Supersonic jet noise from off-design jets is often shown as acoustic spectra. One such plot may be seen in Figure 1.3. The x-axis of the figure is the St (Strouhal) number and is based on the characteristic frequency, fc of the jet. The characteristic frequency of a jet is simply the fully expanded velocity uj , divided by the fully expanded diameter Dj , fc = uj /Dj . The St number is the non-dimensional frequency, in this case the St = f /fc , where f is the frequency of the sound in Hz. The y-axis represents the Sound Pressure Level (SP L) per unit St where the bandwidth of the spectra is one unit St. All the results in the present dissertation are presented as SP L per unit St. SP L found experimentally is calculated from the spectral density of the time history of the acoustic signal and is discussed in detail in Appendix A. The 12 jet conditions in Figure 1.3 are Md = 1.00, Mj = 1.50, N P R = 3.67, T T R = 1.00, D = 0.0127 m, R = 100D, and various observer locations labeled as ?. R is the distance from the nozzle exit to the observer and ? is the angle to the observer measured from the downstream axis. The spectrum at each observer location is labeled in the plot at various observer angles. The three kinds of supersonic jet noise are mixing, screech, and BBSAN and are labeled in Figure 1.3 at ? = 70.0 degrees. These different components of jet noise are also apparent at the other observer angles. Additional strong directional waves are also created by some supersonic jets. This is called Mach wave radiation and is classified as a type of mixing noise and will be discussed later. The mixing noise, in both subsonic and supersonic jets, is broadband in nature and contributes to all frequencies of the spectrum. It is always the dominant component of noise in the downstream arc of the jet, especially so in hot jets. Notice that as ? is decreased from 70.0 to 60.0 degrees that the peak of the mixing noise grows. The screech, which is the discrete component of shock associated noise, is the most obvious noise component as it is located at discrete St and is often more intense than the other noise components for cold jets. As ? varies, the screech frequency does not change, and in this case remains at St = 0.30. The screech tones located at values of St = 0.60 are harmonics, and sometimes as many as five or six harmonics can be seen in experimental data. Screech is more intense in the upstream direction to the jet. Finally, screech is suppressed when the jet is heated. Finally, the BBSAN noise consists of multiple broadband peaks and dominates the mixing noise for high values of ?. The peaks of the BBSAN are closer together for high ? and spread out greatly for low ?. The BBSAN makes very little contribution to frequencies lower than its first broadband peak, only the screech and mixing noise contribute to these frequencies. More details of research performed for each of the types of noise observed in the acoustic spectra are given in the following sections, along with some of the methods of prediction, giving insight into the physical noise generation mechanisms. 13 Figure 1.3. SP L per unit St of a Md = 1.00, Mj = 1.50, N P R = 3.67, T T R = 1.00, D = 0.0127 m, jet at R/D = 100 and various observer locations, ?. The experimental data is courtesy of NASA. 14 1.1.2.1 Mixing Noise Turbulent mixing noise is present in all types of jets, at all jet velocities uj , and is associated with the turbulence in the jet shear layer and the downstream mixing region. Extensive experimental measurements of the acoustic spectra from these jets operating supersonically and on-design, as off-design jet spectra are contaminated by shock noise, were performed by Yu and Dosanjh [22] and cold and hot jets by Seiner et al. [23]. Mixing noise is divided into two components, at least by the majority of the aeroacoustics community, as fine scale and large scale. The fine scale mixing noise is found at larger observer angles from the downstream axis while the large scale mixing noise is more dominant at relatively low angles to the downstream axis. The large and small scale mixing noise has nothing to do with its frequency content and is broadband in nature. The large scale mixing noise originates from the large turbulent eddies convecting in the jet while the small scale mixing noise originates from the small scale turbulence. The fine scale mixing noise is believed to create a very broad range of contributions to the spectrum, while the dominant mixing noise intensity is due to the large scale mixing noise. All mixing noise depends primarily on the T T R and Mj . As Mj increases, the relative intensities between the small and large scale mixing noise increase. If one holds Mj constant while increasing the T T R of the jet then the mixing noise spectrum broadens and the overall sound pressure increases. Fine scale turbulent mixing noise occurs due to the turbulent mixing of the fluid in the shear layer, transition region, and fully turbulent regions of the jet. It may be predicted by various methods. A revolutionary method for jet noise prediction was proposed by Lighthill [24][25] where the Navier-Stokes equations are reorganized into a inhomogeneous wave equation. The inhomogeneous wave equation can be solved by analytical methods such as with Green?s functions. However, the inhomogeneous right hand sides are not easily assigned values because the spatial and temporal variations of the flow-field of the jet are not known exactly. The method, and many based on it, are known as acoustic analogies because the sources of jet noise on the right hand side of the modified equations are analogous acoustic sources. Generally, the right hand side of the equation is split up into analogous acoustic sources that are like a monopole, dipole, and quadrapole. The quadrapole term is analogous to the turbulent source term of fine scale mixing 15 noise. However, the right hand terms of acoustic analogies are only representative of the source of flow induced noise and give no insight into the actual physical mechanism of noise generation. A more recent method of predicting mixing noise can be performed computationally with DNS or LES but these simulations at realistic Re are extremely expensive. DNS and LES simulations are essentially a numerical experiment and give no additional insight into the actual physics of jets. Finally, predictions of mixing noise can be found based on RANS CFD solutions such as those by Raizada and Morris [26] for high speed subsonic jets. This method could be extended to the supersonic regime. Traditional RANS turbulence models and acoustic analogies do not give any indication of the large scales associated with large scale mixing noise. Instead, models for the large scale structures were pioneered by Plaschko [27][28] and greatly improved upon by Morris et al. [29] and then by Liou[30]. Spreading rates of supersonic jets are much smaller than subsonic jets because of the large associated Re and the turbulent statistics vary slowly in the streamwise direction. This implies that the large scale turbulent structures in the jet may be modeled by a linear superposition of the modes of instability waves. The modes of the instability waves reference a Fourier decomposition with respect to the azimuthal angle. This yields a general equation for flow-field variables in the jet. The pressure associated with the instability wave can be written as, for example, p (x, r, ?, t) = ? Z X n=?? ?w an (?) (eigenfunction) exp [in? ? ?w t] d?w (1.6) ??w where an is the amplitude of the instability wave and is considered to be a stochastic function, n is the azimuthal mode number, and ? is the angular frequency of the instability wave. Statistically, the large turbulent eddies and the instability wave model of the jet are identical. This model assumes that the turbulent structures are instigated by ?scientifically random? fluctuations at the nozzle lip. Tam and Burton [31] used matched asymptotic expansions to construct a prediction scheme to connect the far-field spectra to the near-field model of the instability waves. A wavy wall analogy suggests that the instability waves act as a wavy-wall, which has the 16 same characteristics of the instability wave such as wavelength and speed. If the instability waves move faster than the ambient speed of sound, then Mach wave radiation results and is characteristic of the large scale mixing noise from supersonic jets in which the large scale turbulent structures move faster than the ambient speed of sound. The directivity of the Mach wave radiation is arccos (c? /C), where C is the wave speed of the mode of the instability wave. McLaughlin et al. [32] confirmed this theory with their helium / air jet experiments by comparing the largest instability wave modes predictions with peak sound intensity frequencies. Since various instability waves have different speeds, C, then they radiate Mach wave radiation in slightly different directions and have the affect of broadening the peak of the large scale mixing noise. However, when the convection velocities are subsonic, relative to the ambient speed of sound, the analogy of a wavy wall is less applicable. Recently, Morris [33] connected the far field spectral densities of pressure fluctuations from subsonic and supersonic jets to wavenumber / frequency spectrum of pressure fluctuations in the jet shear layer. It is shown that for subsonic instability waves, C < 1, the noise in peak directions is from the large turbulent structures in the streamwise direction and not due to small scale mixing as previously believed. Tam [34] demonstrates that the growth and decay of the large turbulent structures needs to be taken into consideration for subsonically traveling waves. Tam showed that the subsonically traveling large turbulent structures may generate Mach wave radiation via broadening of the wavenumber spectrum of the traveling waves. 1.1.2.2 Screech The discrete component of shock-associated noise, the screech tone, was first observed by Powell [35][36] in his studies of choked jets. Powell observed that a powerful screech tone has an associated wavelength that is related to the shock cell spacing in the jet. Powell then proposed an empirical formula for its frequency based the jet diameter and the nozzle pressure ratio. Indeed, he was the first to attribute screech tones to a feedback loop of acoustic waves traveling upstream in the subsonic portion of the shear layer and impinging on the jet lipline, causing instability waves to propagate downstream in the jet. Many more experimental studies follow the work of Powell. Davies et al. [37] 17 proposed the mechanism of the screech tones in rectangular jets. Sherman et al. [38] found the screech frequency is inversely proportional to the length of the first shock cell by varying the N P R of an axisymmetric vertical jet of air. Norum [39] studied screech from underexpanded off-design jets, found the maximum amplitude of screech occurs near ? = 160 degrees, and used various methods of suppression by changing the jet exit geometry. Rosfjord and Toms [40] studied the temperature dependence of screech from an axisymmetric sonic jet with Powell?s empirical formula. Fisher and Morfey [41] proposed an improvement to the model of Powell based on a phased array of acoustic sources. Tam et al. [42] made further improvements to screech tone prediction with a semi-empirical relation for the peak screech frequency, " #?1 ?1/2 ?1/2 0.67uj To 1/2 ??1 2 2 fs = Mj Mj ? 1 1 + 0.7Mj 1 + Dj 2 T? (1.7) where fs is the screech frequency, Mj is the fully expanded Mach number of the jet, To is the stagnation temperature in the plenum, T? is the ambient static temperature, and uj is the fully expanded velocity of the jet. Panda [43] found with his experimental observations that another length scale may be present that is more representative of the average length scales than those proposed by Powell [35], Fisher and Morfey [41], or Tam [42]. Panda used the standing wavelength Lsw to predict a more precise screech frequency, fs = uc Lsw (1 + Mc ) (1.8) where Lsw is the standing wavelength length scale. The standing wavelength length scale may be larger or smaller than the average shock cell spacing. Unfortunately, Lsw can not be found analytically based on the jet operating conditions. It can be shown that by using Equation 1.8, and assuming an average length scale for the shocks along with Tam?s empirical approximations, Equation 1.7 may be obtained. Unfortunately, no empirical or analytical model exists that predicts the amplitude of screech. The screech phenomenon itself makes the jet?s flow-field structure oscillate. The different modes of oscillation present makes the problem of predict- 18 ing amplitude even more difficult. Various factors affect the intensity of screech. Larger lip thicknesses of the nozzle greatly increase screech. Increasing D or Mj also increases the amplitude of screech. Surfaces near the jet may also increase the intensity as they work like an additional acoustic reflector. However, as T T R is increased, then the intensity of screech is greatly reduced to a point where other sources of noise are dominant or the mechanism that produces screech is suppressed. Since jet engines operate with a high T T R then screech is often not a concern. Furthermore, screech in cold jets may be eliminated by disrupting the feedback loop via a single tab protruding into the flow from the nozzle lip. 1.1.2.3 BBSAN BBSAN is observable in acoustic spectra in the far-field as shown in Figure 1.3. The first acoustic measurements that were conducted to specifically study BBSAN were performed by Harper-Bourne and Fisher [1] using converging round nozzles with unheated air. Tanna [44] made more extensive measurements of shock related noise using convergent round nozzles. Seiner and Norum [45] [46] examined plumes of various jets performing cross correlations of the turbulence and shock positions. Norum and Seiner [47] [48] studied a wide range of conical and contoured nozzles operating over a wide range of pressure ratios to study the mean static pressure and far-field acoustic spectra. This greatly extended the available experimental database of shock related noise. Tanna [49] performed additional experiments on a Md = 1.67 nozzle and formed a basic peak frequency dependence formula. Seiner [50], Seiner and Yu [51], and Yamamoto et al. [52] carried out additional extensive studies of BBSAN. Viswanathan [53] has shown that turbulent mixing noise can be of the same level as BBSAN at high frequencies and care must be exercised in trying to extract the contributions from each noise source for the total spectrum. Generally BBSAN is omnidirectional except for very low observer angles. Recently, Bridges [54] showed that as the T T R ratio of the jet increases then the BBSAN will increase slightly. However, as T T R continues to increase then the BBSAN remains constant. This is most likely because increasing the T T R in supersonic jets stabilizes the magnitude of the instability waves or large scale structures in the shear layer. 19 1.2 Current BBSAN Prediction Methodology The first experimental investigation and semi-empirical BBSAN model was developed by Harper-Bourne and Fisher [1]. Because of the importance of this model, it is discussed further in the following section. Their work explained some of the fundamental physics that causes BBSAN. Following Harper-Bourne and Fisher, an extremely simple empirical model developed by Mani et al. [55] that includes some of the basic concepts of Harper-Bourne and Fisher. The model by Mani et al. [55] yields only the peak sound pressure level at the peak frequency and the peak frequency as a function of observer position, and consist of two simple algebraic formulae and uses heavy empiricism. A simple model such as this is important for rough calculations but isn?t useful in predictions outside the model?s range of operating conditions. These types of purely empirical models give no insight into the physics of the problem and only serve to replace interpolations of tables of experimental data. More recently, Stone [56] developed an empirical BBSAN model based on correlations of data from conical nozzle geometries. This model further includes some of the changes in the shock cell structure when Mach disks form when the off-design parameter becomes large. A direct formula is given by Stone for the Overall SPL (OASPL) and also the sound pressure level in 1/3 octave bands. Unfortunately, this model relies heavily on a sizable table of values of measured OASPL and SPL and therefore contains the same limitations as the model by Mani. Finally, Tam [57] developed a stochastic BBSAN model for circular nozzles operating slightly off design and unheated, which is discussed further in the following sections because of its importance for BBSAN prediction. Tam [58] extended this model for moderately off-design jets operating with slight heating. This is perhaps the most widely used BBSAN model. This list of empirical and semi-empirical models is incomplete, however, it is sufficient to describe the range of models thus developed. Comparisons of these notable models for various nozzle and operating conditions are given by Kim et al. [59]. 20 1.2.1 Harper-Bourne and Fisher?s Model Harper-Bourne and Fisher were the first to propose a model for BBSAN that included both physical modeling as well as empirical data. They performed a combination of experimental and analytical studies to make the first accurate BBSAN predictions. Their research methods should be replicated by other groups for other studies in fluid dynamics. This section is based on their research and will only briefly reproduce the important development and conclusions of their model. Many details are intentionally left out, but may be found in their paper [1]. They first observed that when converging nozzles choke Mj > 1.00, significant increases in noise occur above ? > 90. They postulated, based on experimental observations, that this additional noise source is relatively unrelated to the jet stagnation temperature To , and the angle of the observer to the jet. Harper-Bourne and Fisher extended Powell?s model for discrete components of shock noise. The model employs an array of sources on the nozzle lip line that are equally spaced by a constant shock cell spacing, L. It was subsequently assumed that each subsequent shock cell after the first one decreased by 6% in relative strength. Another assumption is that as the turbulent eddies convect on the lip line of the nozzle they interact with the tip of each shock cell and emit acoustic waves. Experiments show that the turbulent eddies between successive shock cells have correlation coefficients of 0.60, and two shock cell regions apart have correlation coefficients of 0.20. This reduces greatly as correlations are made further apart between different shock cell structures. Also, the time it takes for a turbulent eddy to travel between successive shocks in the shear layer is the distance between successive shock cell structures divided by the convection velocity of the turbulent eddies, u»c . Another important observation that was taken into account while developing the model, is that as the N P R increases, the shock cell spacing increases in a linear fashion and that as the shear layer growth increases, that the turbulent structures decay faster [60]. Harper-Bourne and Fisher first stated that each source contributes to the farfield acoustic pressure from a random source fluctuation evaluated at a retarded time. The sound intensity, p2 (x, t) can be formed by summing the contributions from each source, taking the square, and then the time average, 21 N X N X 1 r r m n p2 (x, t) = 2 Fm t ? Fn t ? x m=1 n=1 a? a? where Fm is a random source fluctuation, N is the number of sources, and rm is the distance from each source location to the observer. If an assumption is made that the fluctuations of the acoustic pressure are statistically stationary and that the turbulent convection speed controls the phase between the sources then with some algebraic detail omitted Harper-Bourne and Fisher state, ?(xn ? xm ) rm rn (1 ? Mc cos ?) Fm t ? Fn t ? = Gmn (?)cos a? a? uc where Gmn is the source cross spectral density between the sources Fm of shocks at positions xm and xn . Inserting the cross correlation between sources into the equation for the sound intensity immediately yields an equation for the spectral density of the pressure time history at the observer position, N N 1 XX ?(r1n ? r1m ) S (x, ?) = 2 (1 ? Mc cos ?) Gmn (?)cos x m=1 n=1 uc (1.9) S (x, ?) is the spectral density per Hz, and r1n is the distance downstream from the nozzle to the nth shock cell. If the shock cells decrease a small amount after the previous shock cell then the nth shock cell may be approximated by Ln = L1 ? (n ? 1)/?L. If this relation is inserted into Equation 1.9 then, N N n?1 X 1 XX ?L1 S (x, ?) = 2 Gmn (1 ? Mc cos ?) |n ? m| ? ?L/L1 K x m=1 n=1 uc m ! (1.10) where L1 is the length of the first shock cell, and K is an integer from m to n ? 1. Equation 1.10 is the Harper-Bourne and Fisher model for BBSAN [1]. All that is left to do is to model Gmn . Equation 1.9 may be expanded in terms of group source spectral densities, Sn , 22 ?L S (x, ?) = So (x, ?) + S1 (x, ?) cos (1 ? Mc cos(?) u?c 2?L +S2 (x, ?) cos (1 ? Mc cos(?) u?c 2?L (1 ? Mc cos(?) + ... +S3 (x, ?) cos u?c (1.11) where So , represents the summation of the source spectral densities from a single shock, S1 is the summation of the source spectral densities from all the adjacent shocks, S2 represents the source spectral densities of shocks separated by 2L, and so on. Si , for i > 0 is representative of the source spectral densities that are either destructive (negative summations) or constructive (positive summations) to the final far-field spectral density and are the source of the peaks of BBSAN above the peak frequency, fp . Harper-Bourne and Fisher show that values of Si may be related to values of So recursively through Si (x, ?) = 2(N ? i)/ci (?)So (x, ?) /N . ci is the group cross-correlation coefficient for shocks separated by iL and is Smn / (Smm Snn ). It is relatively independent of frequency, but at very high frequencies it drops off. This is not surprising as small scale fluctuations decorrelate before they convect to downstream shocks. Finally So , Z? So (x, ?) d? ? D2 ? 4 x2 (1.12) 0 where ? is the off-design parameter. Essentially, Harper-Bourne and Fisher found So through an empirical fit by a method of least squares and dimensional arguments for various values of ?, D, and x, and experimental data. Example results from Equation 1.10 appear in Figure 1.4. They have been generated for a Md = 1.00 converging nozzle with a N P R = 3.25, T T R = 1.00, Mj = 1.42, D = 0.025m, and uj = 410.0m/s, fc = 16, 404 measured at R = 118D with off-design parameter ? = 1.00. The number of shock cells taken into account is N = 8. There is a 6% decrease in shock cell length per shock cell as observed from experiment, and a convective Mach number of uc = 0.7uj is assumed. Using Equation 1.9 and assuming that the shock cell spacing in an off-design supersonic jet is constant, then the maximum spectral density, S (x, ?), will occur when the cosine terms have an argument of zero or an integer multiple of 2?. When 23 Figure 1.4. The Harper-Bourne and Fisher model prediction compared with experimental results of a Md = 1.00, Mj = 1.42, D = 0.025m jet at 118 D, and ? = 135 degrees. The experimental data is digitized from Harper-Bourne and Fisher [1]. the argument of the cosine is zero then the peak of the BBSAN may be found for (n 6= m), fp = uc L (1 ? Mc cos ?) (1.13) where fp is the peak frequency of the BBSAN. It is a combination of the convective speed of the turbulence in the shear layer, the shock cell spacing, and the angle to the observer. At fp the sound radiation from each BBSAN source is constructive and at other frequencies there is at least some destructive interference or purely destructive interference where lower spectral densities are expected. This behavior 24 yields the characteristic broadband peaks of BBSAN. 1.2.2 Tam?s Model The stochastic models of Tam [57] [58] are summarized here. Because of the complexity and length of the development of the model, only a summary of its development is given. For full details about the model, including some of the notation and definitions of the constants, the reader is directed to the original source. In the first prediction method, Tam [57] models large turbulent structures in the jet shear layer as instability wave modes of the mean flow set in a cylindrical coordinate system. These instability waves interact with the quasi-periodic shock cell structure and this interaction is considered the source of BBSAN. The form of the stochastic instability waves used in the model are based on the work of Tam and Chen [61]. The solution of the shock values and magnitudes of the instability waves are matched with the solutions of Pack [15]. The model development starts with the non-linear Euler equations consisting of budgets of mass, momentum, and energy. Field variables are split into a contribution of the mean flow, a perturbation due to the shock cell structure, a perturbation due to the turbulence, and a final term that is a perturbation due to the interaction of the turbulence and shock cells. This final term is arranged on the left hand side of the governing equations and the other terms are arranged on the right hand side as source terms. It is important to note that both the shock cell structure and the turbulence alone satisfy the linearized Euler equations (LEE). The system of equations now forms the LEE with source terms on the right hand side. Tam and Tanna [49] observed that only source terms with a supersonic phase velocity radiate sound while sources with subsonic phase velocity are inefficient acoustic sources. The LEE with corresponding source terms are then expanded as a Fourier series and the method of matched asymptotic expansions is used to solve for the field variable pressure using the method of Van Dyke [62]. This process forms an equation for pressure due to the mth waveguide mode and the turbulence in the shear layer in cylindrical coordinates, 25 ? Z X pm = n=?? ? ? Z ?? an (?)gnm Hn(1) exp [in? ? i?t + ix? + i(n + 1)?/2] d?d? ?? (1.14) where an is a stochastic amplitude function, Hn1 is the nth order Hankel function of the first kind, and 1 gnm (?, ?) = 2? Z ? Anm (x, ?) exp [i (?n ? ?m ) /] dx (1.15) ?? where Anm is an amplitude function, and is a small valued constant that characterizes the rate of spreading of the jet. The autocorrelation of the pressure is found and then the spectral density is formed after some simplification by performing the inverse Fourier transform. Anm (x, ?) exp [i (?n ? ?m ) /] remains to be modeled and could be calculated numerically but this involves extensive computational cost. Instead, the function is approximated by an analytical model of a Gaussian curve assuming that the shock cells are steady, Anm (x, ?) exp [i (?n ? ?m ) /] uj 1/2 ? ? exp ? ln 2 (?x/uj ? Xm )2 /L2 + i (kt ? km ) x a? (1.16) where L is the half-width of the Gaussian function, and Xm is representative of the maximum location of the instability wave. Using this analytical approximation and performing some simplification, a model for the far-field spectral density is obtained, S (R, ?, f ) = CL2 Mj2 ? Md2 2 Aj ?2? a2? u2j 2 1 + ??1 Md2 R2 f 2 ? X exp ? (fm /f ? 1)2 (1 + Mc cos ?)2 / (uc /uj )2 (L2 /2 ln 2) О 2 ?m m=1 (1.17) where C, L, fm , are constants or are given by simple formulas relating the physics of the jet to the model, ? is the observer angle from the upstream axis of the jet, Aj is the fully expanded jet area based on Dj , and ?m is the mth zero of the Bessel 26 function of the first kind. Full details of the constants and the model are given in Tam [57], along with a formula for the spectral density in the near-field. To take into account the effect of slightly heating the jet or having the jet operate moderately off-design some extensions were made to the model of Tam [58]. In this modified model a slight change is made to the instability wave spectrum of Equation 1.16 to allow for high frequency predictions. A semi-empirical model is also used to scale the shock cell strength depending on whether the jet is operating over- or under-expanded. Finally, to account for heating effects, a simple multiplication factor to the spectra density is introduced, given by, ?j ?? ??1 2 1+ Mj 2 ?1 (1.18) where ?j is the fully expanded density of the jet. The paper published by Tam [58] shows no comparisons with heated jets, however, results of the model compare well with experimental data for moderately heated and moderately off-design jets. Example results of the model by Tam [57] are shown in Figure 1.5 evaluated from Equation 1.17. They have been generated for a Md = 1.00 converging nozzle with a N P R = 3.67, T T R = 1.00, Mj = 1.50, D = 0.0127 m, fc = 16, 404 measured at R/D = 100 and ? = 1.12 with the following additional conditions, c? = 343.20 m/s, R = 287.00, ? = 1.40, To = 293.15 K, Tj = 202.17 K, T? = 293.15 K. The number of shock cells taken into account is N = 8. The experimental results are courtesy of NASA and are represented by a thick black line. The prediction itself is the orange line and components of the prediction are the red lines underneath the total spectrum. In the figure, the total prediction is a summation of individual BBSAN predictions corresponding to the mth waveguide mode of the shock cells. 1.2.3 Current Model Limitations Existing BBSAN models are based on empirical correlations of the measured aerodynamic characteristics of the jet plume and radiated noise; further, they are restricted to circular geometries. This is due to the ease of creating semi-empirical models and the abundance of past experimental investigations of cold axisymmetric jets. Perhaps the biggest shortcoming is that they were developed predominantly 27 Figure 1.5. Tam?s model prediction compared with experimental results of a Md = 1.00, Mj = 1.50, N P R = 3.67, T T R = 1.00, D = 0.0127 m jet at R/D = 100 and ? = 130. The experimental data is courtesy of NASA. Note that the prediction has been translated by -5 dB to better match experiment. for cold jets, for which the shock-associated noise dominates the jet mixing noise. The models for BBSAN of Harper-Bourne and Fisher [1] and Tam [58] use a number of assumptions in their development that restrict their application to certain nozzle geometries and operating conditions. However, these two models are much more advanced than their empirical counterparts. Harper-Bourne and Fisher restricted their model development and experiments in their study to converging nozzles operating with a T T R = 1.00. Although no data shown supporting heated jets, their model does support slightly heated conditions with some success. Furthermore, their model is developed only for 28 circular nozzle geometries. The model is mainly empirical in that it uses a master spectrum that is formed with a least squares analysis based on measured data. BBSAN is observed to be lower in intensity at small observer angles with respect to the downstream axis of the jet. Unfortunately, the Harper-Bourne and Fisher BBSAN model has a relatively constant amplitude of its spectral density with respect to observer angle. This constant BBSAN amplitude is not observed in experimental data. With these restrictions of the model considered, the model development of Harper-Bourne and Fisher is an excellent example of a combined experimental and theoretical approach that should be followed for other studies in science. The first stochastic BBSAN model of Tam [57] supports both converging and converging-diverging nozzles operating at over- and under-expanded conditions. Nozzle geometries are required to be circular and the temperature ratios are also required to be T T R = 1.00 or slightly heated. The off-design parameter is required to be small so the amount that the nozzle is operating off-design must be small. These restrictions are partially overcome by Tam [58] by updating the approximation of the shock cell structure and using an empirical temperature correction factor to multiply the final spectral density. This allows prediction of jets from circular converging-diverging nozzles operating at over- and under-expanded conditions while moderately off-design and heated. However, no results were presented for heated jets, and only comparisons with Norum and Seiner [47] are shown. The BBSAN models developed thus far have either been entirely empirical or semi-empirical with restrictions on having circular nozzles, mainly unheated or slightly heated flow, and operating slightly or moderately off-design. In the present dissertation a BBSAN model is developed that overcomes the limitations of past modeling efforts by removing the restrictions on the jet operating conditions and the shape of the nozzle. Any shape of nozzle may be considered, operating at any N P R or T T R, and is applicable to a large range of ?. Only the nozzle geometry and operating conditions need to be specified to make a prediction. This is accomplished by drawing heavily on the modeling efforts of Harper-Bourne and Fisher and Tam in conjunction with validation between the new predictions, models of Tam, and experiments conducted at the Pennsylvania State University (PSU) and the Boeing Company. The BBSAN model developed draws on the steady fluid 29 dynamic flow-field produced by a steady RANS simulation using any CFD solver. The BBSAN model developed here consists of an integration over the physical sources of BBSAN and its corresponding wavenumber spectrum of perturbation pressure due to the shock waves in the jet. The turbulent statistics of the large scale coherent eddies in the jet shear layer are modeled with an analytic function based on traditional descriptions of turbulent scales from the turbulent kinetic energy, K, and the turbulent dissipation, . A list has been constructed to summarize the accomplished goals of this research project that will be presented in the proceeding chapters. Each item in the list represents a goal and accomplishment that has been achieved. ? Develop the first BBSAN model that... - is only dependent on nozzle geometry and operating conditions. - uses RANS solutions of the flowfield. - can make predictions from any nozzle geometry. - can use any off-design parameter. - is able to plot the spatial distribution of the source strength for a given frequency and observer angle. ? Perform converging and converging-diverging circular jet BBSAN predictions. ? Perform the first dual stream jet BBSAN predictions. ? Perform the first non-circular jet BBSAN predictions. ? Validate off-design... - circular cold jet RANS solutions with Pitot measurements. - circular cold and hot jet RANS solutions with schlieren. - rectangular cold jet RANS solutions with Pitot measurements. - rectangular cold and hot jet RANS solutions with schlieren. ? Validate the first helium / air mixture RANS solutions for off-design jets and compare with experiment. 30 ? Compare heated air and helium / air mixture BBSAN predictions with helium / air and heated experimental measurement. ? Obtain RANS solutions for a fully turbulent and laminar flow nozzle and make corresponding BBSAN predictions. The developed BBSAN prediction scheme enables supersonic jet noise to be predicted based only on the nozzle geometry and operating conditions. In this dissertation, the connection to the jet mean flow is made, enabling BBSAN to be predicted for general geometries based on knowledge of the jet operating conditions alone. This provides a significant advance in the state-of-the-art in supersonic jet noise predictions. Since the developed model relies heavily on accurate CFD input, the next chapter of this dissertation explains in detail how the RANS solutions are obtained and validated closely with experiments in the near-field. The third chapter gives details of the BBSAN model development, and its validation with experimental data and comparisons with other models for both circular and rectangular jets at many different operating conditions. Finally, conclusions are drawn for the project along with suggestions for future improvements. Chapter 2 Computational Fluid Dynamics This chapter contains two parts. The first section gives an overview of CFD and how it is applied to the BBSAN prediction. This brief introduction is meant to give the reader the necessary background into the subject. The second part, section 2.2, shows various CFD calculations. These calculations are validated against experimental results in both a qualitative and quantitative sense with experimental data from the PSU and Boeing. Validation of the CFD with experimental measurement is important because these results are used as the input for the BBSAN model. Fluids deform continuously under any force. Generally systems of partial differential equations model fluids. Unfortunately, the vast majority of mathematical descriptions of fluids contain no analytical or general solution. The most popular method to obtain an approximate solution to a system of partial differential equations is to use a discretization method which approximates the derivatives of the equations and forms a set of algebraic equations that are discrete. These discrete algebraic equations can be solved by a digital computer on a computational grid. Discretization schemes vary widely and are an area of intense mathematical research. The quality of the discretization scheme chosen will affect the quality of the solution. The process of solving systems of partial differential equations which model the physics of various kinds of fluid dynamic situations is termed CFD. A good entry level text that covers many of the ideas and algorithms for the vast field of CFD is given by Ferziger and Peric [63]. 32 2.1 Introduction to CFD In general there are well defined methods for finding solutions to CFD problems. First, equations need to be developed to model the phenomena of interest. These are most often the Navier-Stokes equations or a simplified version of them. The equations that describe the fluid dynamics of the off-design supersonic jets are introduced in section 2.1.1. These equations are discretized by a CFD system onto computational grids described in section 2.1.2. The CFD system used to generate solutions in this dissertation is the Wind-US solver [64]. 2.1.1 Governing Equations The equations of motion that model fluid motion are the Navier-Stokes equations, named after Claude-Louis Navier and George Gabriel Stokes. These equations are conservative in the sense that they preserve the amount of mass, momentum, and energy in a system. They can be used to model many different fluids and flow types. One of these flow types is the off-design supersonic jet. The Navier-Stokes equations usually refer to a statement of the conservation of momentum of a fluid dynamic system. However, without additional supporting equations, the system of equations is not closed. The equation for the conservation of mass is, ?? ??ui =0 + ?t ?xi (2.1) where ? is the density, and ui is the velocity vector. Einstein summation is implied over repeated indices. The compressible Navier-Stokes equations, stating that momentum is conserved in orthogonal directions is, ??ui ??ui uj ?p ??ij + =? + ?t ?xj ?xi ?xj (2.2) where p is the static pressure, and ?ij is the shear stress tensor. A final partial differential equation is needed. It is the energy equation, 33 ??uj cp T ??cp T + = ?T ?t ?xj ?p ?p + uj ?t ?xj ?ui ? + ?ij + ?xj ?xj ?T ? ?xj (2.3) where cp is the specific heat at constant pressure, T is the temperature, ? is the heat expansion coefficient, and ? is the thermal conductivity. The ideal gas equation is usually used to close the system of equations. An excellent reference for the development of these equations and their description is found in the famous work by Schlichting [65]. Unfortunately, solutions of coupled partial differential equations are difficult or impossible to find and analytic solutions to the compressible NavierStokes equations remain unknown. Not only is a general solution to the NavierStokes equations unknown but it is also unknown if such a solution exists or is unique. However, with the advent of modern computers, numerical solutions of the Navier-Stokes equations can be found for many realistic problems. It is possible to simply solve the discretized Navier-Stokes equations with appropriate boundary conditions for an off-design supersonic jet and find the associated BBSAN using a computer. Directly solving the Navier-Stokes equations in this fashion with computers is called DNS. The amount of computing power needed to perform a DNS simulation of a high Reynolds number supersonic jet operating off-design is not available for the foreseeable future. A very conservative estimate of the computer power needed to solve the NavierStokes equations can be made. As stated in the introduction, the Reynolds number of a jet is Re = uj D/?. As an approximation, the integral length scale l of the turbulence in the supersonic off-design jet is the exit diameter of the jet, therefore D = l with associated Rel = ul/?. If the integral scale of the velocity in the jet is approximated as the square root of the turbulent kinetic energy, ? K 1/2 , and ?3/4 the space between each grid point, ?x, behaves as ?x/l ? Rel Kolomogorov [66], then u?t/l ? obtain a solution goes as Re3l . ?3/4 Rel . according to Therefore, the computer time required to As an example of the number of grid points needed for a DNS simulation, let u ? 0.2uj , l ? D, then Rej = 5 О 105 for a very simple jet experiment in a laboratory environment, the number of grid points in l3 , Rel ? 9/4 5 О 105 , is Rel ? 2 О 1011 , for DNS. If a Large Eddy Simulation is conducted, 34 ?1/2 then ?/l = Rel is the number of grid points needed which equates roughly to 32 million. Therefore, it is necessary to model the turbulence by modifying the Navier-Stokes equations because DNS is currently impossible to perform for realistic jet Re and LES is extremely expensive. There are various approaches that have been used to model the turbulence. However, the approach used in this dissertation makes use of Reynolds averaging of the Navier-Stokes equations (RANS). Tannehill et al. [67] show the well known development of the Reynolds Averaged Navier-Stokes equations. The RANS solutions in this work are found by numerically by solving the density weighted RANS equations as presented by Wilcox [68] as, ? ? ?? + (??u?i ) = 0 ?t ?xi (2.4) ? ? ??P ? (??u?i ) + (??u?j u?i ) = + ??ij ? ?u00j u00i ?t ?xj ?xi ?xj (2.5) ????ij = ??u00i u00j (2.6) where, and, " # ? u?i u?i ?u00i u00i ? u?i u?i 00 00 ?? e? + + + ??u?j h? + + u?j ?ui ui ?t 2 2 ?xj 2 i ? h = ?qL ? ?u00j h00 + ?ij u00i ? 1/2?u00j u00i u00i ?xj ? + u?i ?ij ? ?u00i u00j ?xj (2.7) where the tilde represents a mass average, the bar represents an ensemble average, the double prime is a fluctuation associated with the Favre averaging, and ? is the Kronecker delta function. Once the mass weighted averaging is performed, a turbulence model is needed (additional equations) to close the modified equations because of additional terms such as the Reynolds stresses that appear. There are various ways to close the modified system of equations. A thorough review of two 35 equation turbulence models, their development, and their implementation is given by Wilcox. [68] To find the appropriate turbulence statistics and mean flow quantities required for the BBSAN model the Menter Shear-Stress Transport (Menter SST) model is used. Menter used an empirical approach to combine the traditional strengths of the K ? and K ? ? models. The Menter SST turbulence model uses the traditional K ? ? of Wilcox in the inner region of the boundary layer. Additional information may be found on the traditional K ? ? model in Wilcox [69]. The Menter SST model automatically switches to a traditional K ? in the outer region of the boundary layer and in free shear flows. This is done because the K ? ? model is much better than the K ? model in the inner region of the boundary layer. Another additional modification in the Menter SST turbulence model is made to the definition of the eddy-viscosity which better accounts for the effects of transport of turbulent shear stress. Along with the Reynolds Averaged NavierStokes equations the following two transport equations are used, which define the Menter SST turbulence model, D?k ? ?k ?ui ? ? = ?ij ? ? ?? k + (х + ?k хt ) Dt ?xj ?xj ?xj (2.8) ?k ?ui ? D?? ? 1 ?k ?? ? = ?ij ? ??? ?2 + [(х + ?? хt )] + 2 (1 ? F1 ) ??w2 ? (2.9) Dt vt ?xj ?xj ? ?xj ?xj The constants1 in the Menter SST turbulence model, ? ? , ?k , ?k , ?, ?? , F1 , ??2 are defined by Menter [70]. The eddy viscosity is defined as, vt = 0.31k max (0.31?, |? О ~u|F ) (2.10) where ? is the magnitude of the vorticity, and F is a function of K, ?, and 1 Note that the constants represented here for the Menter SST turbulence model share notation found elsewhere in this dissertation. However, this is the notation chosen by Menter and is kept here only for consistency. 36 the distance to the wall. The distance to the wall has been set to a very large value. Both K and ? are very important parameters for modeling turbulence in the BBSAN model and these spatially varying quantities can only be provided by solving the RANS. The K ? turbulence model of Chien [71] is also available and works in the same fashion as the Menter SST turbulence model. However, the Menter SST turbulence model is used because it has been found to be more robust in obtaining RANS solutions. 2.1.2 Grid Generation The governing equations that model fluid flow behavior need to be discretized. Since these equations can not be solved analytically they need to be solved with a computer. The vast majority of modern computers are digital, meaning that values must consist of finite discrete quantities. These quantities are found in both space and time. A set of points are referred to as grid points and fill a solution space of the domain where the solution is desired to be found. Typically these points are connected in some fashion with connectivity data. There are generally two systems that connect the grid points in the domain. These are referred to as unstructured and structured grids. Structured grids represent an orderly implied connection between grid points. Unstructured grids have grid points connected in a non-implied orderly fashion, but have more variety in the ways they may be connected. The computational grids used in this work were created with the grid generation software package, Gridgen version 15. Gridgen was first developed in 1984 and started to be commercially available for purchase in 1994. Gridgen runs on a number of platforms including Windows, Mac, Linux, and Unix. Gridgen supports the creation of both structured, unstructured, and hybrid meshes in two and three dimensions. It has the capability to import a large number of computer aided design files or simple coordinates in a text file. More importantly, it has the ability to directly export grids into binary file formats for many commercial, industrial, and government CFD programs. Some of the more notable formats include Fluent, Star-CD, and Wind-US. The computational grids used in this research are all structured. This is ad- 37 vantageous for the CFD in that the finite difference methods can be used. The computational grids used in this research are either two or three dimensional. Two dimensional meshes consist of a list of x and y points while three dimensional meshes are lists of nodal coordinates in the x, y, and z directions. Connectivity information is assumed from the order in which the points are written in files for structured grids. Unstructured computational grids could also be used, without limitation in the BBSAN model or for finding RANS solutions, however the CFD solver chosen has much more flexibility with structured grids. Many numerical techniques and turbulence models are not implemented in the unstructured portion of the code. The computational grids are created by importing the nozzle geometries into Gridgen, which subsequently is used to create the computational domain inside the nozzle and downstream in the jet plume. Gridgen also has the ability to export files into formats for CFD codes to read in their desired format. It also has the ability to prescribe elementary information to help guide the CFD solver. Wind-US is the chosen CFD RANS solver for the present work and it has the capability to set boundary condition information based on Gridgen output. This saves a large amount of time when setting up the simulations because there is no other step necessary before solving the RANS equations. For supersonic jet simulations a number of boundary conditions can be prescribed. The first is an inflow boundary condition which prescribes the total pressure and total temperature in the plenum of the nozzle. These values correspond directly to the operating conditions of the nozzle, N P R and T T R. An initial Mach number must also be set, however, this can vary as only the total pressure and total temperature must remain constant. Freestream boundary conditions are applied in the far-field of the domain. Far-field boundary conditions require the specification of an ambient pressure p? , ambient temperature T? , and freestream Mach number M? . These may fluctuate and are not held constant, although they are not expected to change greatly as flow gradients near these boundary conditions are expected to be small. The freestream Mach number must never be zero for stability purposes and is set to a very small value of 0.0001. A freestream boundary condition is never expected to be used where a large mass flux of fluid will be moving across it. Regions with large masses and gradients of fluid exiting the computational domain cannot be handled by the 38 freestream boundary condition and are handled by an outflow boundary condition. A subsonic outflow boundary condition requires that a single quantity be specified, the back pressure, which in the present simulations is set to the ambient pressure p? . To simulate the walls of the nozzle there needs to be a boundary in the computational domain between the fluid and the solid wall. There are typically two kinds of boundary conditions for walls in traditional CFD. The viscous or no-slip wall boundary condition represents an adiabatic wall where the fluid at the wall must have a velocity of zero. The inviscid, Eulerian, or slip wall boundary condition has the same parameters as the viscous wall boundary condition except that it allows the velocity at the wall to have a component tangent to the wall. The velocity normal to the wall must be zero. This implies that the flow may ?slip? while not allowing mass flux to move through the wall. The final boundary condition used is the interface. The interface boundary condition allows the fluid flow to be continuous across various sub domains. The computational domain, since it is structured, is made of multiple zones or smaller sub domains. These zones need to exchange information and an interface boundary condition performs this task. Figure 2.1 shows the location of the boundary conditions for a two dimensional computational grid designed for axisymmetric jet RANS calculations. The sub domains of the computational grid are separated by black lines. The zone in the lower left hand corner represents the interior of a convergent divergent nozzle. The zone just above it represents the area outside the nozzle surface. Finally, the largest zone represents the area into which the nozzle will exhaust and the jet plume forms. Different boundary conditions are labeled on all the black lines which were described in the previous paragraph. The lower inviscid wall represents the centerline of the jet. Even though the domain is in two dimensions, the equations being solved are in an axisymmetric form. Gridlines from the computational domain have been removed to better illustrate the sub zone interfaces and the locations of the boundary conditions. 39 Figure 2.1. Example placement of boundary conditions for CFD of an axisymmetric jet simulation. 40 2.1.3 Solver The CFD solver finds solutions to either the Navier-Stokes equations or the RANS equations closed by the Menter SST or Chien turbulence models discussed in the previous sections of this Chapter. It can approximate the solution to the NavierStokes equations or the Menter SST equations in the entire computational domain or a combination of the sub domains in different physical regions of the computational grid. The Navier-Stokes or RANS simulations are conducted using the Wind-US 2.0 code [64]. Wind-US is a product of the National Project for Application-oriented Research in CFD (NPARC) Alliance, a partnership between the NASA Glenn Research Center and the USAF Arnold Engineering Development Center. Wind-US is the merger of four different CFD systems, NASTD, NPARC, NXAIR, and ICAT. The code has been extensively tested and numerous validation cases for benchmark flow regimes have been established. It can find solutions to the Euler and Navier-Stokes equations including turbulence models and chemically reacting flows [72]. In relation to the present research, the code is used to predict the development of single and dual stream supersonic jets operating off-design for both axisymmetric and non-axisymmetric jets. Wind-US can solve the field equations for perfect gases, real gases, or flows undergoing chemical reactions. The solutions are found by iterative schemes for time integration, including Runge-Kutta [73], Jacobi iteration, Gauss-Seidel iteration, MacCormack method, or Eulerian. Solutions in this work used a combination of time dependent explicit Euler iterations with final implicit Euler iterations to find the final steady solution. In the Spatial differencing for structured grids can be either central or upwind. Wind-US is written almost entirely in Fortran 90. The code is compiled with an Intel Fortran compiler and the GNU compilers. For the simulations in the present work, Roe second order upwind differencing is used for spacial discretization and a basic iterative solver is used to march the solutions in time until a steady state solution is reached. The Total Variational Diminishing (TVD) method is also used in some solutions for stability, both in the computational domain and at boundaries. The Wind-US solver has the capability to run in parallel using the Message Passing Interface (MPI) or the Parallel Virtual Machine (PVM). MPI is by far the most popular method for parallel processing today on distributed computers. The 41 most complete reference for MPI is the recent standards publication [74]. However, due to technical limitations of the PSU development Linux cluster2 system, Wind-US can not run in MPI mode, unlike many other programs developed at the PSU. PVM can run on the development Linux cluster and alternatively a heterogeneous computer environment connected by a simple network. The PVM system automatically distributes the load of the problem to different computer architectures if necessary to achieve the fastest possible solution time. Sunderam [75] gives an overview of PVM and its framework. The capabilities of PVM to run on a non-homogeneous computer network are not needed as the cluster consists of a homogeneous computing environment. PVM is used for the Wind-US calculations which are too large for a single computational processor on the development Linux cluster or to lower the time required to obtain a solution. In this work, PVM is only applied to the threedimensional cases that simulate rectangular jets. Wind-US unfortunately has no automatic grid decomposition such as METIS for unstructured grids or even an automatic algebraic grid decomposition for structured grids. For this reason, when a PVM or MPI Wind-US simulation is performed, the grid must be manually divided into separate sub domains, each assigned its own processor. Unfortunately, Wind-US 2.0 has a maximum number of indices that are allowed in any direction for a structured grid. This artificial limitation is set at 1000 grid points in an indice direction. Fortunately, the NPARC Alliance has included a utility called ?Decompose? to help divide larger domains into ones that are smaller for either PVM or for bypassing the artificial restriction on zone size. Still, with the program Decompose the user has to manually specify which zone to split up, which index to split, and into how many sub domains. This is often time consuming with multiple large zones in large three-dimensional simulations. Besides some of these extra limitations imposed by the code, PVM has proved to be very versatile as a library for distributed computing. The primary environment for the Wind-US solver used in this dissertation is a Linux cluster. The server of the cluster consists of 4 Intel Xeon 1.5 Gigahertz (Ghz) processors with 4 Gigabytes (Gb) of RAM. Each of the 64 nodes of the cluster contains dual 2.4 Ghz Intel Xeon processors and 2 Gb of RAM. Gigabit and Fast 2 The name of the computer cluster is Cocoa3. 42 Ethernet are used for connectivity between nodes. Serial jobs are performed to find the steady solutions of axisymmetric jets and the parallel jobs using PVM are performed for the three-dimensional cases. When performing an axisymmetric solution using approximately 500,000 grid points, solutions on a single processor are obtained in approximately five days. The three dimensional simulations are more expensive. A typical three dimensional solution is found using three to four processors over the course of 20 days. This is taking into account that only one fourth of the solution is calculated in three dimensions by taking advantage of the planes of symmetry of a rectangular jet. 2.1.4 Post Processing Post processing is a term to describe the analysis of the solution after Wind-US completes or partially completes calculating a solution to the RANS or NavierStokes equations. When the Wind-US solver terminates it produces solution files containing all the CFD information on the computational grid points. A serial or parallel run will produce two files, one is the CFL file containing the solution and the other is the CGD file containing the computational grid. These files are used to view the CFD solution and also as input to the BBSAN prediction code. More details about how the CFL and CGD file are integrated with the BBSAN code are given in the next chapter. For the CFD solutions of this section they are used for post-processing. CFPOST is a post-processing program that reads common grid and common flow files. It can produce listings, results, reports, files for other post-processing programs, and many other miscellaneous functions [76]. CFPOST is a command line driven program. Each time it is run the user must specify commands just as at a command prompt to instruct CFPOST to perform various options. Since CFPOST has been used so much in the process of producing results in this dissertation, various script files were developed to automatically generate results from CFD simulations. By using scripts for producing different kinds of plots a large amount of time is saved. CFPOST, even though it has a large amount of capability, has only been used in the scope of this dissertation to produce PLOT3D files for interpretation by the Tecplot program, which is used to calculate other 43 variables and produce figures. PLOT3D files, just as CFL and CGD files, are often split up as a solution and grid file. A single grid file may be used for several solution files. PLOT3D files can only contain structured computational grids and do not support unstructured computational grids. The files themselves may be formatted, unformatted, or in ASCII formats. PLOT3D has been in use for the storage of numerical simulation results, and not just for CFD simulations, for a considerable time and is an industry standard. More information about the development of PLOT3D, its format, and application can be found in Walatka et al. [77]. PLOT3D files in this dissertation are used for interpretation of the CFD results, as input to another post-processor, and are not used to transfer data to the BBSAN prediction code. The CFD results and later the BBSAN results, used for comparison with experimental measurement in this dissertation have been plotted with Tecplot. Tecplot has the ability to read a multitude of government and industrial CFD solutions. 2.2 CFD Results Since the ultimate goal of the BBSAN model is to make predictions for both axisymmetric and three dimensional off-design supersonic jets, CFD cases have been developed to validate these jets. In particular two axisymmetric jets operating off-design at a single condition and one rectangular jet operating off-design at two different conditions are chosen for validation. Two methods for validation are performed for these jets. The first is a quantitative approach comparing various flow quantities in the jet with experimental data. The second is to further validate the CFD, a study of how the shock wave strength scales with the off-design parameter changes is conducted. This is done to help validate the CFD with respect to Harper-Bourne and Fisher?s observations [1]. Sometimes the flow transitions to turbulence inside or outside the nozzle when the Reynolds number is low. A single study has been produced where the flow in the nozzle is laminar and outside is turbulent. Finally, since some of the experimental measurements for low density jets have been conducted with helium / air mixtures to simulate hot jets, CFD comparisons are made between a helium / air jet and a hot air jet. Axisymmetric simulations are run until both convergence is achieved and a 44 visual inspection of the solution yields satisfactory results. Convergence is achieved when the convergence residual of the system of RANS equations level reaches 10?15 universally or the residual of the various equations being solved becomes steady as the number of iterations increase. Grid sequencing is used in both cases to speed up convergence. Sequencing, in terms of Wind-US, means systematically skipping grid points generating a coarsened grid for the initial iterations, until later iterations where all the grid points are used. The transition from a sequenced domain to an un-sequenced domain uses linear interpolation so initial iterations on the unsequenced domain require a temporary low CFL number. Sequencing is also used to study the grid independence of the axisymmetric solutions. By sequencing the grid in each index direction and comparing the solution with non-sequenced solutions, grid independence is proven. This is performed for both axisymmetric validation cases and it is shown that the solutions are independent of the computational grid. Axisymmetric simulations use approximately 20,000 iterations until convergence if there are no stability problems. Axisymmetric simulations are run in serial mode because the benefit from parallelism would be small due to the relatively low number of grid points. For the initial iterations a global CFL number is utilized with a value of 0.75 in conjunction with a basic explicit Euler method. This drives the solution towards a steady state. After the initial disturbance waves exit the domain then the CFL number is not used and a constant time step is used at each computational grid point. This constant time step is chosen so that the CFL number at any grid point is less than one and is used in conjunction with the same explicit Euler method. This process creates a very smooth steady solution which demonstrates a constant residual with increasing time step. To validate the axisymmetric CFD simulations of off-design supersonic jets, two sets of CFD simulations were compared with experimental results. These included one converging nozzle, Md = 1.00, and one converging-diverging nozzle, Md = 1.50. The converging nozzle operated at N P R = 3.67 and the convergingdiverging nozzle operated at N P R = 2.77. Both of the nozzle operating conditions used T T R = 1.00. This implies that the total temperature specified in the plenum is the ambient temperature. The exit diameter of both nozzles is D = 0.0127m. The nozzle walls are contoured to reduce the number of Prandtl-Meyer waves in the supersonic region of the nozzles. 45 Validation of the three dimensional CFD results from Wind-US are conducted with a rectangular jet. The design Mach number of the rectangular jet is Md = 1.50. The inside walls of the rectangular jet are contoured to minimize the number of Prandtl-Meyer waves originating from the wall of the nozzle in the supersonic region. The rectangular jet can not be modeled with a two dimensional grid as it is not axisymmetric. However, there are two planes of symmetry in a rectangular jet, one through the major axis and one through the minor axis, parallel to the streamwise direction of the jet. Therefore, the computational domain can be represented by one fourth of the entire domain. When creating the computational grids, the two symmetric planes are used by specifying them as symmetric boundary conditions or slip-boundary conditions. Many CFD simulations have been conducted to assess the capabilities of the Wind-US system, to produce more input for the BBSAN prediction code, and for other research projects. However, this chapter focuses on the CFD results used only for validation. The creation of the additional CFD results that are not presented, but noted here and later in the BBSAN prediction chapter, follow the same procedure. Results are also presented for axisymmetric dual stream calculations, however, there is no experimental data of the flow-field available, but an overview of the CFD of these jets is presented. 2.2.1 Computational Grids For the simulations of the axisymmetric jets in the present research, a two dimensional grid is created. For the converging and converging-diverging nozzles with design Mach numbers of Md = 1.00 and Md = 1.50, the y + distance is set to one and the corresponding distance from the first grid point to the wall is then found. The total number of grid points for the Md = 1.00 and Md = 1.50 axisymmetric nozzles are 759, 609 and 485, 323 points, respectively. The number of grid points used is quite large for axisymmetric supersonic jet simulations. This is in part needed to make sure that there are enough grid points to fully resolve the shock cell region of the jet. This is discussed more in the next chapter, where studies are performed regarding the needed grid density for precise noise prediction. The computational domain downstream from the nozzle exit extends 75 nozzle diameters. 46 The range of the computational domain in the cross-stream direction is 50 nozzle diameters from the centerline axis. Figure 2.2 shows the computational grids near the nozzle exit for the two axisymmetric benchmark cases. The centerline is along y/D = 0 and the nozzle exit in both cases is at x/D = 0. Part a) shows the computational grid around the Md = 1.00 nozzle itself and the body of the nozzle can be seen faintly. Part b) shows a close-up view of the computational grid in the region of the nozzle Md = 1.00 exit. Part c) shows the converging-diverging nozzle computational grid. The white region in the middle of part c) is the structure of the nozzle. Finally, in part d) a closeup of the computational grid near the nozzle exit can be seen for the Md = 1.50 nozzle. The lines in the computational grid in the cross-stream direction do not vary in the streamwise direction. This makes extracting cross-stream profiles of different field variables of the RANS solution simple and avoids interpolation. Figure 2.2. Nozzle profiles and closeup of the computational grid near the nozzle exit plane. a) Contour of the converging nozzle. b) Closeup of the nozzle exit of the converging nozzle. c) Contour of the converging-diverging nozzle. d) Closeup of the nozzle exit for the converging-diverging nozzle. 47 An additional axisymmetric computational grid needs to be mentioned because it has additional important features not present for single stream jet calculations. A dual stream nozzle contains a primary nozzle, just like a single stream jet, but with an additional secondary annular jet stream around the primary jet. Figure 2.3 illustrates this geometry by showing a computational grid that has been created for RANS CFD calculations. The entire computational domain has been divided into various sub domains that have been colored for identification. The red region represents the inside of the primary nozzle. The orange region represents the interior of the secondary nozzle. The green color is an exterior region outside the nozzle that is upstream from the exit of the primary nozzle and the black region is downstream from the exit of the primary nozzle. Once again, the xaxis represents the centerline of the nozzle and the entire domain is considered to be axisymmetric about it. There are 15,390 grid points in the interior of the primary nozzle and 12,150 grid points inside the secondary nozzle. There are 228,943 exterior grid points present and the total number of grid points in the computational domain is 256,483. The y + distance at the exit of the nozzle is set to one and the corresponding wall distance from the first grid point to the wall is found. Both the primary and secondary nozzle are converging, so the design Mach numbers are 1.00 for both jets. Note that the two nozzle exits are off-set. The secondary jet exit is close to x/D = ?1.6 while the primary jet exit is at x/D = 0. This is often the case with commercial aircraft engines. The geometry for the dual stream nozzle and associated experimental data is courtesy of Dr. K. Viswanathan of the Boeing Company. The rectangular jet with design Mach number Md = 1.50 has an exit width of the nozzle exit of w = 0.0208 m and height of h = 0.0119 m. For the validation case a three dimensional grid is created. Noting the advantage of two planes of symmetry, one through the major axis and one through the minor axis of the rectangular exit, only one quadrant of the flow-field is modeled. Taking advantage of the planes of symmetry would not be possible if the simulation were unsteady. This is highly advantageous because it reduces the run time of the CFD solver by approximately four or more and saves a large amount of memory and PVM communication. The number of grid points in the computational domain of the rectangular jet is 19, 768, 058. Since the nozzle is highly three-dimensional and the 48 Figure 2.3. Dual stream nozzle profile and closeup view of computational grid near the jet exit planes. number of grid points so large, the domain needs to be further broken down into additional sub domains. Six different zones, or separate ordered pairs of i, j, k, indices are created. The number of grid points in the jet plume region is 501 by 190 by 190. 501 points are used in the streamwise direction. This value is still lower than the axisymmetric case, but increasing it adds unreasonable runtimes when calculating solutions with the available Linux development cluster. It is advantageous for noise calculations to have more grid points in the plume region, compared with the interior of the nozzle and the freestream region of the flow. Therefore, 91.41% of the grid points are located downstream of the nozzle. The number of grid points allocated for the three-dimensional calculations is much greater than needed. Figure 2.4 shows the entire computational grid used for the rectangular jet calculations. The exterior of the computational domain consists of six planes but only three are visible. The x direction is from the left to the right and represents the streamwise direction of the jet. The other two Cartesian directions are cross-stream directions. The z direction corresponds to the major axis and the y direction corresponds to the minor axis of the rectangular nozzle. The computational grid has 49 been colored to represent different regions of the flow for clarity. The red region represents the grid points that are inside the nozzle. The black region represents grid points that are downstream of the nozzle exit. The green points represent the grid points that are upstream of the nozzle exit, but not inside the nozzle. The planes have been labeled as symmetry and freestream. The symmetric planes represent where cuts have been made in a much larger fictitious computational domain. These symmetric planes are locations of symmetric boundary conditions. The freestream label only applies to the green domain as these grid points of this plane are freestream boundary conditions. The red grid points on the freestream plane are representative of an inflow boundary condition. The planes on the other side of the computational domain that are not seen are freestream boundary conditions and an outflow boundary condition in the y ? z plane. To obtain the entire flow-field, the RANS solution is mirrored across the planes of symmetry to fill in the other quadrants. Another view of the rectangular grid is shown in Figure 2.5. The representative coloring of the computational grid remains the same and additional thick black lines have been placed to separate each of the sub domains of the computational domain. Various grid clustering can be seen near these thick black boundaries. There are many grid points near the walls of the nozzle and these extend downstream from the nozzle exit into the far-field both in the cross-stream and streamwise directions. This is very beneficial in the near-field of the jet to properly resolve the three dimensional shock waves that form. At the inflow boundary there is no grid clustering, as it is not needed because this region of the flow-field is solved with no-slip walls: essentially solving the Euler equations (inviscid equations of motion). This is done for the stability of the calculations. Note that the grid spacing near the center line of the rectangular jet is also very small. This is to ensure that the oblique shock waves in the jet core do not artificially become Mach disks at the centerline. 50 Figure 2.4. The computational domain of the rectangular nozzle with Md = 1.50. The domain extends 75 equivalent diameters downstream and 50 equivalent diameters in the cross-stream directions from the nozzle exit center plane. The width of the nozzle exit is w = 0.0208 m and the height is h = 0.0119 m. 51 Figure 2.5. An enlarged view of the rectangular nozzle computational grid. 52 2.2.2 General CFD Results In chapter 3 it will be shown that the BBSAN noise model is based on an integration over space and wavenumber. Therefore, before detailed comparisons are made between experiment and numerical prediction of the CFD results, contour plots of variables that the BBSAN prediction model will use are presented. This should give the reader a better physical understanding for the values of the variables in the BBSAN model when it is developed later in this dissertation. This task is performed by presenting contour plots of the converging-diverging axisymmetric validation case. This nozzle operates at Md = 1.50, Mj = 1.30, T T R = 1.00, and D = 0.0127 m. Distances have been normalized by the exit nozzle diameter. Since the BBSAN model is dimensional, the variables have been left in their dimensional form for these plots only, to better facilitate understanding of the quantities involved. The simulation is axisymmetric, so for a whole plane of the flow-field to be shown, the results are mirrored across the x-axis by the operation ymirror = ?y. The RANS solution variables in the BBSAN noise model are the average velocity of the fluid in the streamwise direction u, the local Mach number M , the turbulent kinetic energy K, the viscous dissipation rate , and the shock pressure ps . All these values are dependent on space alone y = (x, y, z), and not time t, as the RANS solution is steady. Figure 2.6 shows the velocity in the streamwise direction, u m/s. The centerline of the jet is located at y/D = 0 and the jet exit plane is located at x/D = 0. The lip of the nozzle is located at x/D = 0 and y/D = ▒0.50. The high pressure fluid (air) is expanded out of the nozzle in the positive x direction, moving from the left of the page to the right. One can see immediately that in the over-expanded case of Mj = 1.30 a conical oblique shock originates at the lip of the nozzle and terminates in a barrel shock at x/D ? ?0.25. Prandtl-Meyer expansion waves originate in the second shock cell and again form another conical oblique shock. The slip lines downstream of the barrel shock propagate though the entire shock cell structure. As expected, u drops significantly after shocks and accelerates through Prandtl-Meyer expansion waves. Clearly, the values of u near the nozzle exit and above the lipline are close to zero, however there is some entrainment of the fluid. Figure 2.7 shows the same simulation and spatial ranges of the plot of u. In this case the local contours of M are plotted. The local Mach number is found from 53 the ratio of the magnitude of the velocity vector and the local speed of sound. The local speed of sound is a function of the local static temperature. For the BBSAN model input, this is done beforehand by extracting the static temperature from the common flow library solution file. The contours of M are consistent with respect to the shock waves, both conical oblique and the barrel shock. Figures 2.8 and 2.9 show contour plots of the turbulent kinetic energy K, and viscous dissipation rate, . The turbulent kinetic energy contour plot shows that it grows steadily from the lips of the nozzle in the downstream direction. The maximum value of K is at approximately x/D = 4.00. Further downstream the fully turbulent mixing region is present. There the annular regions of K meet and subsequently disappear very far downstream from the nozzle exit. There are slight waves in the contours of K in the core of the jet which are due to the expansions and shock waves of the shock cell structure. The viscous dissipation rate, , shown in Figure 2.9, is not a field variable that may be extracted from the Menter SST solution. Instead, the specific dissipation ? is converted to viscous dissipation rate as described by Wilcox [68] or Menter [70]. The relationship between and ? used in this dissertation is = 0.09K?. A large amount of dissipation is seen close to the nozzle lips. As the nozzle lips in all the CFD simulations are finite, small recirculation regions occur here. The same wavy contours of dissipation are also apparent like those of K because of the shock cell structure. Finally, Figure 2.10 shows contours of shock pressure, ps . The shock pressure perturbation is the thermodynamic static pressure minus the ambient static pressure, p? . The shock pressure perturbation is not close to zero at the exit of the nozzle because the jet is operating off-design. The regions where the shocks and expansions occur are very apparent in this plot because it removes the shear layer present in the other plots. Clearly, the first few shock shear layer interactions are very strong relative to ones further downstream in the jet. This is apparent from the large values of ps and because the spacing between the contour lines is small. 54 Figure 2.6. Contours of the velocity component, u m/s, for a Md = 1.50, Mj = 1.30, T T R = 1.00, D = 0.0127 m converging-diverging axisymmetric jet. Figure 2.7. Contours of the Mach number, M , for a Md = 1.50, Mj = 1.30, T T R = 1.00, D = 0.0127 m converging-diverging axisymmetric jet. Figure 2.8. Contours of the turbulent kinetic energy, K m2 /s2 , for a Md = 1.50, Mj = 1.30, T T R = 1.00, D = 0.0127 m converging-diverging axisymmetric jet. 55 Figure 2.9. Contours of the dissipation, m2 /s3 , for a Md = 1.50, Mj = 1.30, T T R = 1.00, D = 0.0127 m converging-diverging axisymmetric jet. Figure 2.10. Contours of the shock pressure, ps Pa, for a Md = 1.50, Mj = 1.30, T T R = 1.00, D = 0.0127 m converging-diverging axisymmetric jet. 56 2.2.3 Menter SST and Chien K- Comparisons The scaling of u and K with T T R for jets controls the scaling of BBSAN intensity. There is some skepticism regarding the ability of the Menter SST turbulence model to correctly predict correct K and values of a jet when unheated and especially when heated. Georgiadis et al. [78] compared various turbulence model implementations in Wind-US, the CFD RANS solver of choice for this work, for subsonic jets. In general, the Chien K- turbulence model displayed slightly better spreading rates and values of K closer to experimental data than the Menter SST turbulence model. This is mainly because the implementation of the Chien K ? turbulence model contains the ?PAB? temperature corrections introduced by Abdol-Hamid et al. [79] in the PAB3D finite volume code. The PAB temperature correction essentially implements a Sarkar compressibility correction and modifies a closure coefficient based on gradients of total temperature in the jet. The PAB3D code shows superior mean flow prediction capabilities relative to standard K ? RANS solutions for supersonic heated jets. Because of the improvements of predicted values of K from the Chien K ? turbulence model relative to experimental jet data presented by Georgiadis, K comparisons are made of the Menter SST and Chien K ? turbulence models for the same jet with two temperature ratios. The converging benchmark nozzle of Md = 1.00 and D = 0.0127 m is utilized to compare how K varies with temperature for both turbulence models. Four total simulations are conducted with a Mj = 1.50 and T T R = 1.00 or T T R = 2.20 using either the Menter SST or Chien K- turbulence models. Unfortunately, the Chien K- with heating at T T R = 2.20 used a coarsened computational grid relative to the other three simulations. This was necessary for the Chien K- model to converge to a steady solution. Since the BBSAN sources in an axisymmetric jet are located spatially along the lipline of the jet, lipline values of u and K are shown in Figure 2.11. Note that there are multiple y axis in this figure representing two variables separately. Clearly, the shock cell structure prediction from the Chien model is stronger in the velocity plot than the Menter model. This is because the shear layer growth in the Chien model is greater than the Menter model. Cross stream plots of the values of u and K show more detail of both models capability. Figure 2.12 shows cross stream values at x/D = 5.00 in a) and x/D = 10.00 in b). The values near x/D = 5.00 are located in a region with strong BBSAN 57 sources for this particular nozzle and operating conditions. The maximum values of u at x/D = 5.00 are 477.0, 671.0, 461.0, and 646.0 m/s for the Menter SST model at T T R = 1.00/2.20 and Chien model T T R = 1.00/2.20 respectively. The maximum values of K at x/D = 5.00 are 5527, 11547, 4249, and 8567 m2 /s2 for the Menter SST model at T T R = 1.00/2.20 and Chien model T T R = 1.00/2.20 respectively. Ratios of the maximum velocity from T T R = 1.00 to T T R = 2.20 are 1.406 and 1.401 for the Menter and Chien models. Ratios of the maximum K from T T R = 1.00 to T T R = 2.20 are 2.09 and 2.02 for the Menter and Chien models. Similar ratios are present at other x/D locations. Predicted values of u are in close agreement between the two models operating at different temperature ratios, although the spreading rate is slightly different. Even though the predicted magnitude of K is different between the two models, the scaling of K at various streamwise and cross-stream locations is exactly the same. This scaling is the same even though the Chien model has superior prediction capabilities of K. Since the scaling of u, K, or other field variables is the same for supersonic jets operating off-design, and the Menter SST turbulence model implementation in Wind-US is extremely robust relative to the Chien K-, the Menter SST RANS CFD solutions are utilized for all other CFD simulations and BBSAN predictions. Figure 2.11. Extracted values along y/D = 0.50 of a converging conical nozzle operating at Mj = 1.50 and T T R = 1.00 or T T R = 2.20 using the Menter SST or Chien K- turbulence models. 58 Figure 2.12. Extracted values along the cross stream directions of a converging conical nozzle operating at Mj = 1.50 and T T R = 1.00 or T T R = 2.20 using the Menter SST or Chien K- turbulence models. a) Transverse at x/D = 5.00 b) Transverse at x/D = 10.00. 59 2.2.4 Pitot Probe Comparisons In this section, comparisons of the RANS CFD results produced from the Wind-US calculations introduced in this chapter are compared directly with experimental results. Specifically, these experimental results are produced with Pitot and static pressure probes. The experimental data was collected in The Pennslvania State University High Speed Jet Facility by Dr. Jeremy Veltin. For more details on this facility and its capabilities see Doty and McLaughin [80]. The static pressure probe and the Pitot pressure probe are separate. Also, some experimental data was collected with a Pitot rake. The experimental data produced from these probes provides values of total pressure, static pressure, and using oblique shock tables and isentropic theory, Mach number. These values can be directly or indirectly extracted at the same locations where the probe resides from the CFD simulations. The experimental results are compared at various downstream and cross-stream locations near the jet exit in the region where the BBSAN sources reside. Predicted results are directly compared with experimental results at various downstream locations. Comparisons are made for the axisymmetric jets of Md = 1.00, Mj = 1.50, T T R = 1.00, and Md = 1.50, Mj = 1.30, T T R = 1.00. Rectangular jet pitot probe comparisons are made at the following conditions: Md = 1.50, Mj = 1.30, T T R = 1.00, and Md = 1.50, Mj = 1.70, T T R = 1.00. These CFD results have been discussed in the previous sections of this chapter. Dual stream comparisons are not made because experimental data are not available. In general, the predictions and experimental comparisons are made in the first few diameters of the jet. This is not to imply that comparisons far downstream are not important, but the source of BBSAN is in the shear layer close to the exit of the nozzle relative to the fully turbulent region of the jet. Before various results are presented, a short discussion of the analysis needed to obtain them is necessary for their full understanding. There are some choices made in the combination of equations used to evaluate the flow parameters (such as the local Mach number). In these choices, the Pitot pressure is always used. For this reason the simulation data are manipulated to produce profiles of local stagnation pressure at a point that would be measured directly behind a hypothetical Pitot probe. This is done because when a probe is physically put into a supersonic flow it creates a normal shock in front of it, effectively lowering the total pressure. This 60 manipulation involves using the Mach number and local static pressure ahead of the hypothetical shock that would be formed in the presence of the probe. For the full details of how the various comparisons are conducted see Miller et al. [81]. The total pressure measured by a Pitot probe, po1 , resides behind a normal shock wave which is created by the probe interfering with the supersonic portion of the flow. Alternatively, the Pitot probe measures the total pressure in a subsonic region of the jet flow where no normal shock exists. This quantity, po1 is compared directly with the experimental Pitot pressure as additional evidence of the quality of the computations. Figure 2.13 shows a comparison of the ratio of the Pitot pressure and the plenum pressure, po1 /po , data for the axisymmetric converging nozzle, Md = 1.0. The uncertainty estimate shown graphically on the figure is approximately 6% of the mid range static pressure measurement, but does not include errors caused by the distortion of the flow in a non-uniform flow field. Such an effect is evident in specific regions of the flow including the region near where the oblique shock wave originates at approximately x/D = 1.0. The data in this region near the centerline of the jet shows larger differences between the experimental and numerical solutions (larger than the pressure transducer uncertainty). It is likely that the discrepancies observed can be attributed to the interaction of the flow within the oblique shock created by the tip of the probe with other flow gradients (such as the mixing layer or the jet shock waves) that reflects back to the probe body upstream of the static pressure hole. This can produce an error difficult to predict due to the complex shape of the mixing layer and the three dimensionality of the problem. Figure 2.14 shows the next detailed profile comparison, namely a plot of the ratio of static pressure and plenum pressure, p/po data for the converging nozzle case. The static pressure at almost all locations shows very good agreement. However, the region near where the oblique shock wave originates at approximately x/D = 1.10 shows some difference between the numerical and experimental solutions when the probe is near the centerline of the jet. This is again due to the oblique shock wave interacting with the probe inside the flow. As before, as the probe moves away from this region, the numerical results again agree with experiment. Figure 2.15 shows the local Mach number, M , calculated as described above 61 and presented at the same locations as the previous figures. Some discrepancies appear near the jet centerline starting at x/D = 1.2 due to the presence of the normal shock observed in the experiment but not in the numerical predictions. Some discrepancies are also apparent in the outer part of the mixing layer, where the experimental results exhibit a less smooth appearance. At these locations, the computed velocities are very low. These local speeds correspond to very low pressure values measured by the Pitot probe pressure transducers. As discussed earlier, the uncertainty is estimated at 4% and 6% of the mid range value respectively for the static and the Pitot pressure. Therefore, these low speed measurements are below the precision achievable with the experimental setup. An additional problem appears near the nozzle exit where the static pressure probe cannot traverse to the nozzle lip-line where it would encounter the nozzle. Therefore, the M experimental plot at x/D = 0.0 shows a smaller radial range than its numerical counterpart. Furthermore, the numerical results at this location show that the velocity is zero on the nozzle lip; this accounts for the large gradient in the region near y/D = 0.5 compared to the experimental data. The spreading rate of the jet, shown by the lower gradients of pressure in the shear layer, is consistent with the numerical results for the range of measured values. The results of the probe traverses for the converging-diverging nozzle case are shown in Figure 2.16. These predicted Pitot pressure profiles show overall good agreement with the experiment. Figure 2.17 shows the corresponding static pressure measurements. Note that the static pressure has been non-dimensionalized with respect to (1/2)?u2j for a clearer representation. These also have good agreement. However, as in the convergent nozzle case, there is one small disagreement in the static pressure data. This is at approximately x/D = 0.2, which is the region near the bottom of a barrel shock. The reasons for this discrepancy are probably due to the same ones related to the converging nozzle case. The final profile comparison, of the local Mach number in the convergingdiverging case, is shown in Figure 2.18. These Mach number estimates use the data from the static pressure probe with the Rayleigh pitot formula to produce quite good agreement between computations and experiment. However, the error due to the static pressure probe interaction with the barrel shock at x/D = 0.2 is apparent. The static pressure measurement has altered the solution for the Mach 62 number at this point. By inspection of Figure 2.18, the predicted growth rate of the shear layer is lower than the experimental data. This is most likely a problem with the turbulence model in the simulations. Figure 2.13. Comparison between the experimental (dots) and numerical (lines) po1 /po of the Md = 1.00, Mj = 1.50, converging nozzle case. Each set of data is separated by x/D = 0.20 starting at x/D = 0.0 at the left and stopping at x/D = 2.0 on the right. 63 Figure 2.14. Comparison between the experimental (dots) and numerical (lines) p/(1/2?u2j ) of the Md = 1.00, Mj = 1.50, converging nozzle case. Each set of data is separated by x/D = 0.20 starting at x/D = 0.0 at the left and stopping at x/D = 2.0 on the right. Figure 2.15. Comparison between the experimental (dots) and numerical (lines) M of the Md = 1.00, Mj = 1.50, converging nozzle case. Each set of data is separated by x/D = 0.20 starting at x/D = 0.0 at the left and stopping at x/D = 2.0 on the right. 64 Figure 2.16. Comparison between the experimental (dots) and numerical (lines) po1 /po of the Md = 1.50, Mj = 1.30, converging-diverging nozzle case. Each set of data is separated by x/D = 0.20 starting at x/D = 0.0 at the left and stopping at x/D = 2.0 on the right. Figure 2.17. Comparison between the experimental (dots) and numerical (lines) p/(1/2?u2j ) of the Md = 1.50, Mj = 1.30, converging-diverging nozzle case. Each set of data is separated by x/D = 0.20 starting at x/D = 0.0 at the left and stopping at x/D = 2.0 on the right. 65 Figure 2.18. Comparison between the experimental (dots) and numerical (lines) M of the Md = 1.50, Mj = 1.30, converging-diverging nozzle case. Each set of data is separated by x/D = 0.20 starting at x/D = 0.0 at the left and stopping at x/D = 2.0 on the right. 66 Pitot probe measurements have also been conducted for the rectangular jet. A five probe Pitot static rake is used to gather total and static pressure at various downstream and cross-stream locations in the rectangular jet just as in the circular jet comparisons. The rake probe has five probes spaced evenly on a single device allowing for multiple data to be collected at each downstream location. More details about the rake Pitot probe can be found in Veltin and McLaughlin [82]. The exit geometry of the rectangular nozzle is described by the height h and width w of its geometry. In the circular calculations the downstream and cross-stream directions have been normalized by the diameter, D. A length scale is chosen for the rectangular jet as the equivalent diameter, De , which is formed by matching p the exit area of the rectangular nozzle to a fictitious circular one, De = 4wh/?. The fully expanded diameter of the rectangular nozzle can also be found as before by replacing D by De . For the rectangular jet tested, De = 0.01776 m and the aspect ratio of the nozzle is 1.75. Comparisons between the simulations and experiment are made for two rectangular jets with Md = 1.50, T T R = 1.00, and Mj = 1.30 or Mj = 1.70. The variation of po1 and M are determined at various downstream x/De locations in the minor and major axes. The major and minor axes are in the z and y directions respectively. Just as in the axisymmetric comparisons, the numerical results use a simulated normal shock, if the local M is greater than one, to correspond to the experimental results. The major and minor axis Pitot pressure traverse results for the rectangular jets are shown in Figures 2.19 through 2.22 for both the underand over-expanded cases. Each of the figures has a corresponding downstream location labeled as x/De and an integer n is used to separate the data on the axis. The dots represent the experimental measurements while the lines represent the numerical predictions. Generally, the agreement in the Mj = 1.30 case is excellent for the Pitot pressures except farther downstream where the spreading rate of the experiment is higher. The same good agreement can also be seen in the Mj = 1.70 case. However, on the minor axis there is more discrepancy in the Pitot comparisons than in the other three figures. Comparisons of M at various x/De locations for the same rectangular jet operating conditions are found in Figures 2.23 through 2.26. Generally the agreement between the predictions and measurements for the Mach number are good near the nozzle exit. Comparisons downstream 67 demonstrate less agreement. Recall that the experimental results use a combined approach of analytical and experimental measurement to find the M data and this needs to be considered when comparing results. The direct comparison of the Pitot pressure is a much better metric for comparing experimental and numerical predictions because it does not include assumptions for the jet static pressure variation across the jet. Overall the rectangular pitot probe comparisons are consistent with the circular jet comparisons. This is encouraging because both simulations used the same simulation parameters and it infers a degree of quantitative confidence in the RANS results. Figure 2.19. Pitot probe comparisons between the experimental (dots) and numerical (lines) of po1 /po of the Md = 1.50, Mj = 1.30, De = 0.01776 m jet along the major axis plane. 68 Figure 2.20. Pitot probe comparisons between the experimental (dots) and numerical (lines) of po1 /po of the Md = 1.50, Mj = 1.30, De = 0.01776 m jet along the minor axis plane. 69 Figure 2.21. Pitot probe comparisons between the experimental (dots) and numerical (lines) of po1 /po of the Md = 1.50, Mj = 1.70, De = 0.01776 m jet along the major axis plane. 70 Figure 2.22. Pitot probe comparisons between the experimental (dots) and numerical (lines) of po1 /po of the Md = 1.50, Mj = 1.70, De = 0.01776 m jet along the minor axis plane. 71 Figure 2.23. Pitot probe comparisons between the experimental (dots) and numerical (lines) of M of the Md = 1.50, Mj = 1.30, De = 0.01776 m jet along the major axis plane. 72 Figure 2.24. Pitot probe comparisons between the experimental (dots) and numerical (lines) of M of the Md = 1.50, Mj = 1.30, De = 0.01776 m jet along the minor axis plane. 73 Figure 2.25. Pitot probe comparisons between the experimental (dots) and numerical (lines) of M of the Md = 1.50, Mj = 1.70, De = 0.01776 m jet along the major axis plane. 74 Figure 2.26. Pitot probe comparisons between the experimental (dots) and numerical (lines) of M of the Md = 1.50, Mj = 1.70, De = 0.01776 m jet along the minor axis plane. 75 76 2.2.5 Schlieren Comparisons Schlieren3 visualization techniques are widely used in the Aerospace industry for flow visualization in all flow regimes and were invented by Robert Hooke. An excellent introductory overview of visualization techniques for fluid dynamics using schlieren techniques, with a historical perspective, including other visualization techniques and their strengths is presented by Settles. [83] Schlieren techniques are characterized by producing parallel rays of visible light. This light then travels through a flow-field that has density gradients (compressibility). The density gradients bend the beams of parallel lights due to refraction. A mirror focuses the resultant altered beams and some of them are removed with a ?knife edge? because their trajectory has been changed due to the density gradients. The light rays that do not collide with the knife edge are captured with a visualization device such as a camera. For the schlieren images shown below a Z-Type schlieren system is used. The specific techniques that produce schlieren for these comparisons, and a detailed overview and setup of the experiments themselves, are given by Veltin and McLaughlin. [84] The schlieren shown in this dissertation were produced by Dr. Jeremy Veltin. For a qualitative comparison of the experimental schlieren images it is necessary to produce the same images from the Wind-US CFD solutions. The Z-type schlieren essentially produces density gradients in the cross-stream direction of the fluid flow. Therefore, it is a simple process to take the derivative of the density in the cross-stream direction from the numerical solutions. Essentially, if y is the cross-stream direction of the fluid flow then the data for an artificial or numerical schlieren are produced by performing the simple operation of d?/dy using the Tecplot CFD analysis package. The numerical solutions, which are either axisymmetric or have planes of symmetry, are mirrored on the center axis or a plane of symmetry so that the entire flow-field is visible for a comparison. This allows for plotting of contours across the centerline or the plane of symmetry. The sign of the gradient of density below the centerline or plane of symmetry is then changed. This is done to simulate the inversion of the light gradients observed in the experiments. The numerical contours of d?/dy are plotted over a very small range of negative 3 Schlieren from the German word schliere, often confused as the inventors name. It is the plural of schliere meaning streak. 77 one to one with one hundred intervals. This is performed to emphasize the smallest features of the density gradients and to magnify any possible imperfections in the numerical simulation. The converging axisymmetric jet case of Md = 1.00, Mj = 1.50, and T T R = 1.00, is shown in Figure 2.27 part a). The top image in the figure represents the experimental schlieren result and the bottom image shows the numerical schlieren. The axes of the figures are non-dimensionalized by the diameter of the nozzle D. The flow moves from left to right in the positive x direction. The lips of the nozzle at y/D = ▒0.5 are evident by the shear layer spreading from these locations. The sharp demarcation between dark and light regions emanating from the nozzle lip and ending on the jet centerline indicates the end of the Prandtl-Meyer expansion fan and the initiation of compression waves caused by the interaction of the expansion waves with the free shear layer. Additionally, the shock waves and their corresponding angles can be seen as strong gradients of d?/dy. The first conical oblique shock originates at the same location in both experiment and simulations at approximately x/D = 1.1 near the centerline. The second oblique shock wave occurs at approximately x/D = 2.5. In both the experimental and numerical schlieren, the origin of the shock waves are the same. By inspection, the angles of both the experimental and numerical oblique shock waves agree. This shows that the calculated and experimental shock wave strengths are equal. However, a slight discrepancy occurs at approximately x/D = 1.1, at the jet center. The experimental visualization shows a very short normal shock, prior to the oblique shock waves. This normal shock creates a small subsonic region downstream, delimited by slip lines. This phenomenon can also be referred to as a barrel shock. Any increase in the nozzle pressure ratio shows an increase of this subsonic region. The numerical results only start showing this phenomenon when the simulated nozzle pressure ratio is increased slightly to produce Mj = 1.53. This value was found by incrementing the N P R until a normal shock was noticeable. It could also be created artificially by increasing the grid spacing along the centerline. The sensitivity to grid spacing near the centerline is found to be as important as the lipline grid density for this reason. Figure 2.27 part b), shows a comparison between the experimental and numerical schlierens for the converging-diverging axisymmetric jet of Md = 1.50, 78 Mj = 1.50, and T T R = 1.00. The arrangements of the shocks are very different because, unlike the converging nozzle case, an oblique shock wave originates at the nozzle lip and terminates as a barrel shock. The appearance of the more typical conical oblique shocks downstream ensues. A qualitative comparison of the position and angles of the shock waves show good agreement. In particular, the location of the normal shock just downstream of the exit of the nozzle are in the same position. The slip stream downstream of the normal shock is seen clearly in the simulations. Figure 2.27. Comparison between the experimental and numerical schlieren of a) the converging nozzle case Md = 1.00, Mj = 1.50, b) the converging-diverging nozzle case Md = 1.50, Mj = 1.30. The nozzle exit is at x/D = 0.0 and the flow moves from x/D = 0.0 to the right. The nozzle centerline is at y/D = 0.0 and the nozzle lips at y/D = 0.5. The PSU small scale jet anechoic facility does not have the ability to heat the air in the plenum that subsequently emerges from the nozzle. Instead, helium is mixed with compressed air in the plenum before it exits from the nozzle. This creates a lower density mixture of helium and air, which simulates heated jets, where T T R > 1.00. To help validate the acoustic results of heat-simulated jets from this facility, and to ensure that they compare favorably to hot jet simulations, CFD 79 calculations have been carried out between a Md = 1.00, Mj = 1.47, N P R = 3.52, T T R = 2.20 and a Md = 1.00, Mj = 1.47, T T R = 1.00 helium / air mixture. This helium / air simulation for supersonic jets operating off-design is the first ever conducted. The specific values of the helium / air mixture jet are p? = 98.00 kP a, T? = 294.3K, R = 703.5 m2 K/s2 , ? = 1.55, and the density of the mixture is ? = 0.756 kg/m3 . Both of these simulations used the converging computational grid discussed in section 2.1.2. Note that the helium / air mixture is used to simulate a T T R = 2.20 but the T T R specified in the numerical simulation is 1.00. In the heated air simulation only hot air exits from the converging nozzle, while in the helium / air mixture simulation the mass fraction of helium to air in the nozzle is 0.2326. Figure 2.28 shows schlieren comparisons of these two numerical simulations against a Z-type schlieren of the helium / air experimental jet. The top of a) and b) are the same image: the helium / air simulated jet. The bottom image of a) is a numerical schlieren of the hot air jet while the bottom image of b) is the numerical schlieren of the helium / air jet. Clearly, the numerical result of the helium / air mixture jet compares extremely well with that of the experimental schlieren. There is a slightly different shock cell structure in the hot air simulations relative to the helium / air numerical result and the schlierens. The shock cell structure in the helium / air mixture jet has been shortened compared to the hot air results. However, in section 2.2.8 it will be shown that more subtle differences occur according to contour plots of M and plots of various quantities. Z-type schlieren comparisons are made for the rectangular jets with Md = 1.50 that correspond to the same conditions described for Pitot probe comparisons. In addition, a comparison is made through the major axis for the same overand under-expanded cases with a T T R = 2.20. These schlierens are conducted through the highly three-dimensional flow-fields of the rectangular jets while the numerically generated schlierens are generated from the major and minor axis plane data alone. This causes some discrepancy in the qualitative comparisons because the numerical schlierens are not able to capture the effect of the density gradients of the main major and minor axis. It is not possible to plot contributions from multiple planes without transparency. The numerical schlieren in the major axis plane is generated by taking density gradients in the z direction and numerical schlieren in the minor axis plane is formed by taking density gradients in the y 80 Figure 2.28. Comparison between the experimental and numerical schlieren of the converging nozzle case Md = 1.00, Mj = 1.50 a) T T R = 2.20 air, schlieren on top and numerical schlieren on bottom, b) top: schlieren of T T R = 2.20 air. bottom: numerical schlieren of helium / air mixture. The nozzle exit is at x/D = 0.0 and the flow moves from x/D = 0.0 to the right. The nozzle centerline is at y/D = 0.0 and the nozzle lips are at y/D = 0.5. direction. These directions should conform to those of the experimental schlieren. Numerical data is mirrored across the centerline axis to form an entire plane and then the density gradient in the lower half of the corresponding plane is multiplied by negative one for the gradients of density to conform to experimental ones. The major and minor axis experimental and numerical schlieren of the rectangular Md = 1.50, Mj = 1.30, T T R = 1.00 jet is shown in Figure 2.29. Part a) shows the minor axis plane with the numerical schlieren under the experimental one and part b) shows the major axis plane with the same configuration. The major axis plane shows excellent qualitative agreement of spreading rate, shock position, and shock angles with respect to the experimental result. Less agreement can be seen in the minor axis plane. There is significant interference in the picture of density gradients of the minor axis plane in the z direction. Figure 2.30 shows the rectangular Md = 1.50, Mj = 1.70, T T R = 1.00 schlieren comparisons 81 in the same layout as Figure 2.29. Once again the major axis plane in part b) shows overall excellent agreement in terms of the spreading rate, shock position, and shock angles. Comparisons for the minor axis plane in part a) show much better agreement than the corresponding over-expanded cold rectangular jet. In particular, the first expansion waves and corresponding downstream oblique shocks through both the major and minor axis compare almost perfectly. Finally, comparisons are shown for experimental and numerical schlieren of the helium / air simulated T T R = 2.20 rectangular jets operating at Md = 1.50, Mj = 1.30 and Mj = 1.70. Numerical schlierens are produced only for the major axis because experimental data is only available for this plane. Figure 2.31 shows both the over- and under-expanded qualitative comparisons of the helium / air simulations. Part a) shows the Mj = 1.30 rectangular jet major axis and part b) shows the Mj = 1.70 rectangular jet major axis. Note that the simulations are performed with an actual T T R = 2.20, unlike the experiments, which are heat simulated with the helium / air mixture. Unfortunately, the experimental plots in the top half of the figure are heavily populated by the turbulent eddies as long time averaging of the helium / air jets is not possible due to the relative expense of helium / air experiments. Very little can be observed of the jet shock cell structure by comparing the plots in part a). However, in part b) the first oblique shock wave is easily discernable in the experiment and the numerical result and compares favorably. 82 Figure 2.29. Comparison between the experimental and numerical schlieren of the rectangular nozzle case Md = 1.50, Mj = 1.30, and T T R = 1.00. a) The minor axis plane top: schlieren, bottom: numerical schlieren. b) The major axis plane, top: schlieren, bottom: numerical schlieren. The nozzle exit is at x/De = 0.0 and the flow moves from x/De = 0.0 to the right. Figure 2.30. Comparison between the experimental and numerical schlieren of the rectangular nozzle case Md = 1.50, Mj = 1.70, and T T R = 1.00. a) The minor axis plane top: schlieren, bottom: numerical schlieren. b) The major axis plane, top: schlieren, bottom: numerical schlieren. The nozzle exit is at x/De = 0.0 and the flow moves from x/De = 0.0 to the right. 83 Figure 2.31. Comparison between the experimental and numerical schlieren of the rectangular nozzle case Md = 1.50 and T T R = 2.20. The experimental results use a helium / air mixture and the simulations use heated air. a) Mj = 1.30 Major axis plane top: schlieren, bottom: numerical schlieren. b) Mj = 1.70 major axis plane. top: schlieren. bottom: numerical schlieren. The nozzle exit is at x/De = 0.0 and the flow moves from x/De = 0.0 to the right. 84 2.2.6 Off-Design Study To further validate the CFD solutions for the BBSAN prediction model, a shock strength study has been conducted. Using the same geometry as the convergingdiverging validation case, simulations are conducted at off-design conditions from Mj = 1.60 to Mj = 2.50 in 0.1 increments, creating a total of 10 new cases for study. Besides a different pressure specification at the inlet boundary condition seen in Figure 2.1, which corresponds to the appropriate NPR, all other simulation parameters are the same. By varying Mj , the off-design parameter ? changes accordingly as Md = 1.50 remains constant. As in all off-design cases of supersonic jets oblique shock waves originate from inside the core of the jet. When Mj is larger than Md an expansion originates at the nozzle exit followed by an oblique shock. The pressure difference across this first oblique shock can easily be found when post-processing the CFD results. A simple model for shock associated noise, such as the model of Harper-Bourne and Fisher [1], suggests that the noise should scale depending on both the shock cell strength, where it intersects the shear layer, and the typical velocity fluctuation in the shear layer. The resulting intensity would depend on the square of this product. Using the pressure difference across the first oblique shock and the turbulent kinetic energy in the shear layer, K, the strength of the shock-associated noise source can be estimated as K?p2 . If the logarithm of K?p2 at the first oblique shock/shear layer intersection is compared with the logarithm of the variation of the off-design parameter then a nearly linear relationship results. Figure 2.32 shows the logarithm of K?p2 as a function of ? for the validation converging-diverging nozzle. The slope of this relationship is almost exactly 4.0 as Harper-Bourne and Fisher [1] and Tam and Tanna [49] have shown for cold jets. 85 Figure 2.32. Variation of the logarithm of shock strength, K?p2 (dots), compared with the off-design parameter ? 4 (line) offset slightly. The logarithm of ? 4 has a nearly identical slope with K?p2 . 86 2.2.7 Laminar and Turbulent Flow Nozzle Some of the conditions at which the nozzles operate have a low Reynolds number because of their small size and inflow conditions. This implies that the flow inside the nozzle or exiting the nozzle may not be turbulent but laminar. One such nozzle and condition that has been tested is the convergent divergent nozzle with Md = 1.00, Mj = 1.30, T T R = 2.20, and D = 0.0127 m. Under these conditions the Reynolds number of the flow is Re = 337, 000. This low Reynolds number can allow for laminar flow to exist fully inside the nozzle and to transition to turbulence downstream of the nozzle exit due to the Kelvin-Helmholtz instability. Since it is not known exactly where the flow transitions from laminar to turbulent, it is assumed as a test case that transition occurs at the nozzle exit. If a flow is laminar then it is modeled by the Navier-Stokes equations. If a flow transitions from laminar to turbulent at the nozzle exit, the need arises to solve both the RANS and Navier-Stokes equations in different regions of the flow. In this case, the laminar equations of motion are solved inside the nozzle and the RANS equations are solved everywhere outside the nozzle. Figure 2.33 shows contours of Mach number for the jet with transition at the nozzle exit. The top contour plot shows the solution given by RANS in the entire domain and the bottom contour plot shows the laminar / RANS jet. The most obvious difference between the two numerical solutions is that the oblique shock wave in the RANS solution is outside the nozzle and formed at the nozzle lip. In the laminar / RANS case characteristic waves are able to coalesce and form an oblique shock inside the nozzle at approximately x/D = ?0.1. This difference in origin of the first oblique shock changes the entire shock cell structure. The laminar / RANS case has a separation or recirculating region after the oblique shock forms on the inside wall of the nozzle near the exit. This is in contrast to the RANS simulation, which has an attached turbulent boundary layer along the entire nozzle wall. The slipstream in the fully RANS case, that is created by the barrel shock, is much stronger than in the laminar / RANS case. This observation is due to the lower effective Dj in the transition case. By carefully comparing both cases, it can be observed that the shock cell structure itself is very similar but shifted by a small amount x/D in the negative streamwise direction for the RANS case relative to the laminar / RANS case. Interestingly, nozzle exit velocities of the laminar / 87 RANS case are over 140 m/s lower than the RANS case on the centerline. Figure 2.33. Comparison of Mach number contours for the Md = 1.50, Mj = 1.30, T T R = 2.20, and D = 0.0127 m jet. Top: Fully turbulent flow. Bottom: Laminar flow inside the nozzle and turbulent flow outside the nozzle. Figure 2.34 shows two plots along the centerline of the jet. The top plot is M and the bottom is p/po . In each case both the fully turbulent RANS and laminar / RANS solution are shown. Note, that the exit of the nozzle is at x/D = 0.0, so part of each plot is inside the nozzle itself. The qualitative observation made in Figure 2.33 that the shock cell structures appear very similar is even more apparent in the plots of M and p/po . The very sharp vertical lines of M and p/po inside the nozzle of the laminar / RANS jet and outside the nozzle of the RANS jet are due to shock waves. As expected, in the first plot the Mach number is greater than one before a shock and reduces to a value lower than one after the shock, and a corresponding jump in static pressure also occurs at the same x/D location. If either variable, or another flow-field variable of the laminar / RANS case, is shifted by a value of x/D = 0.30 then the two solutions, at least in terms of the shock cell structure, would be almost identical. This would exclude very small changes in amplitudes between the two cases. Both sets of data show that the solutions inside 88 the nozzle are identical up until x/D = ?0.1. Not coincidentally, this is where the waves coalesce in the laminar nozzle. The spacing of the shocks is generally unchanged between the laminar and turbulent simulations on the centerline. The values of turbulent kinetic energy and streamwise velocity on the lipline are important for BBSAN intensity scaling. Figure 2.35 shows values of K and u on the lipline of the nozzle, y/D = 0.50. Both K and u have identical trends just after the nozzle exit but with the same shift as seen on the centerline. However, the turbulent nozzle simulation shows higher K and u at the exit of the nozzle. This is due to the recirculating region just inside the nozzle lip at the nozzle wall of the laminar nozzle and the existence of the oblique shock at the nozzle lip of the fully RANS jet. Figure 2.34. Centerline values for a Md = 1.50, Mj = 1.30, T T R = 2.20, D = 0.0127 m jet. Top: Variation of Mach number. Bottom: Variation of pressure. 89 Figure 2.35. Lipline values (y/D = 0.50) for a Md = 1.50, Mj = 1.30, T T R = 2.20, D = 0.0127 m jet. Top: Variation of turbulent kinetic energy. Bottom: Variation of streamwise velocity. 90 2.2.8 Helium and Hot Jet Comparisons Additional comparisons are made of the helium / air mixture jet and hot air jet of T T R = 2.20. The parameters and details of this simulation were presented in section 2.2.5. Here, additional quantitative comparisons are made between the numerical simulations for the two jets. Figure 2.36 shows contours of Mach number of the two jets. The top half plane above y/D = 0.0 represents the contours of M of the hot air jet and the bottom plane below y/D = 0.0 shows the helium / air mixture contours of M . The contours are almost identical near the nozzle exit, especially at the centerline. Far downstream the two numerical simulations show some subtle differences in spreading rate and positions of the shock waves on the center line. This is most likely due to a slight variation in the rates of diffusion between a helium / air mixture and a hot air jet. Figure 2.36. Comparison of Mach number contours for Md = 1.00, Mj = 1.50, T T R = 2.20 air or T T R = 1.00 helium/air, D = 0.0127 m jet. Top: Contours of M of hot air. Bottom: Contours of M of the helium / air mixture. Centerline data has been extracted from the two numerical simulations of the helium / air and heated air jets. Two important variables in particular have been extracted from the centerline and are shown in Figure 2.37. The top plot shows the density variation in the jet and the bottom plot shows the variation of the streamwise velocity component, u. These values have not been normalized and remain dimensional. The raw values in this case help to illustrate how very similar these jets are. Data produced from these simulations are used to perform a prediction of BBSAN to illustrate the ability to use helium / air jets to simulate 91 hot jets for aeroacoustic applications. This chapter has presented various equations of motion that model steady fluid flow of off-design supersonic jets. These equations have been solved by Wind-US and subsequently validated against experimental results from various sources. The validation involved comparisons of predicted values of pressures and Mach number with experimental data found from Pitot measurements. Qualitative comparisons of predicted and Z-type schlieren were also shown for various jets. The jets examined included heated and unheated, helium / air, and transitional. Also, both rectangular and circular nozzles were validated. In the next chapter the development, implementation, and calibration of the BBSAN model is shown. Predictions based on the validated RANS solutions of this chapter, along with additional RANS solutions for other jets, are used by the developed BBSAN model. These predictions are compared with experimental data from various sources and the predictions of Tam for a wide range of jet conditions and nozzle geometries. 92 Figure 2.37. Centerline values for a Md = 1.50, Mj = 1.30, T T R = 2.20 or T T R = 1.00 helium / air mixture, D = 0.0127 m jet. Top: Variation of density, ?. Bottom: Variation of centerline velocity, u. Chapter 3 Broadband Shock-Associated Noise Thus far an overview of the current prediction models of BBSAN and validation for the RANS CFD of supersonic jets operating off-design have been presented. This chapter is presented in three main parts. The first part of the chapter is the mathematical development of the BBSAN model where the prediction method is developed into a single mathematical formula for the spectral density in the farfield. The second part discusses the implementation of the mathematical model and subsequent parametric studies. Finally, in the third part, the prediction results over a wide range of jet conditions are presented both for the CFD validation cases developed in the previous chapter and also using additional RANS CFD solutions. These BBSAN predictions are compared with the prediction methods of Tam and with experimental data. Conditions are chosen to include a wide range of nozzle shapes, pressure ratios, and temperature ratios to test the capability of the model. 3.1 Model Development The BBSAN model builds on the analysis developed by Tam [57]. Tam?s analysis is considerably simplified if the following form of the inviscid compressible equations of motion are used. D? ?vi + =0 Dt ?xi (3.1) 94 Dvi ?? + a2 =0 Dt ?xi (3.2) D ? ? = + vi Dt ?t ?xi (3.3) where, where a is the local speed of sound, t is time, and vi are the velocity components in the xi directions of a Cartesian coordinate system. ? is related to the logarithm of the pressure, ?= 1 ln (p/p? ) ? (3.4) where p is the pressure, p? is the ambient pressure, and ? is the ratio of specific heats of an ideal gas. Following Tam [57], the instantaneous flow-field properties are separated into four components. That is, ? ? ? ? ? ?? + ?s + ?t + ? 0 ? ?=? ? 0 vi v?i + vsi + vti + vi (3.5) where the overbar denotes the long time averaged value, the subscript s denotes the perturbations associated with the shock cell structure, the subscript t denotes the fluctuations associated with the turbulence, and the primes denote the fluctuations generated by the interaction of the turbulence and the shock cell structure. In addition, it is assumed that the unsteady linearized version of these equations is also satisfied by the turbulent velocity fluctuations. This is justified if the important components of the turbulence, so far as the BBSAN is concerned, are coherent over relatively large axial distances. These components are described well by a linear instability wave model. It will be assumed that the shock cell structure satisfies the steady linearized version of Equations 3.1 and 3.2. That is, ??s ?vsi + =0 ?xj ?xi (3.6) ?v?i ?vsi ??s + v?j + a?2 =0 ?xj ?xj ?xi (3.7) v?j vsj 95 The shock cell structure is assumed to be steady in the model even though the shock cell structure oscillates slightly in the jet. This could be improved later if necessary. It is also assumed that the mean static pressure is constant throughout the flow-field. The mean pressure in the jet is the sum of the ambient pressure, p? , and the steady perturbations, ?s . Furthermore, it is assumed that to linear order ?? = 0. Finally, it is assumed that the turbulence responsible for the generation of BBSAN consists only of the large-scale structures, as BBSAN requires significant coherence lengths, and they also satisfy the linearized equations. This assumption is consistent with the ?instability wave? model of the turbulence. Making these assumptions yields, ??t ??t ?vti + v?j + =0 ?t ?xj ?xi (3.8) ?vti ?vti ?v?i ??t + v?j + vtj + a?2 =0 ?t ?xj ?xj ?xi (3.9) Next the decomposition given by Equation 3.5 is substituted in Equations 3.1 and 3.2. Use is also made of Equations 3.6 through 3.9. This gives, ?v 0 ?? 0 ?? 0 ??t ??s + v?j + i = ?vsj ? vtj ?t ?xj ?xi ?xj ?xj ?vi0 ?v 0 ?v?i ?? 0 ?vti ?vsi ??t ??s + v?j i + vj0 + a?2 = ?vsj ? vtj ? a2s ? a2t ?t ?xj ?xj ?xi ?xj ?xj ?xi ?xi (3.10) (3.11) The additional terms contribute to the development of the turbulence and the mean flow, which are found from the RANS CFD solutions. Clearly, this also requires the inclusion of viscous terms in the mean flow calculation though they are not crucial for the BBSAN generation. The operators on the left hand side of Equation 3.10 and 3.11 are the linearized Euler equations. The following definitions are introduced, ?vsj ??t ??s ? vtj =? ?xj ?xj (3.12) 96 ?vsj ?vti ?vsi ? vtj = fiv ?xj ?xj (3.13) ?a2s ??s ??t ? a2t = fia ?xi ?xi (3.14) Making these definitions, the inhomogeneous equations for the fluctuations associated with the interaction of the turbulence with the shock cells can be written, ?? 0 ?? 0 ?v 0 + v?j + i =? ?t ?xj ?xi (3.15) ?vi0 ?v 0 ?v?i ?? 0 + v?j i + vj0 + a?2 = fiv + fia ?t ?xj ?xj ?xi (3.16) where ? is a dilatation rate generated by the interaction between the pressure gradients and the turbulent velocity perturbations and the shock cells. fiv is the unsteady force per unit volume associated with interactions between the turbulent velocity fluctuations and the velocity perturbations associated with the shock cells. Finally, fia is the unsteady force per unit volume related to the interaction of fluctuations in the sound speed (or temperature), caused by the turbulence and the shock cells, and the associated pressure gradients. In traditional approaches to turbulence mixing noise models these equivalent sources have been treated separately and the same assumption will be made here. The solution to Equations 3.15 and 3.16, can be written in terms of the vector Green?s function that satisfies the equations, n ??gn ??gn ?vgi + v?j + = ? x ? y ? (t ? ? ) ?0n ?t ?xj ?xi (3.17) n n ?vgi ?vgi ??gn n ?v?i 2 + v?j + vgj + a? = ? x ? y ? (t ? ? ) ?in (3.18) ?t ?xj ?xj ?xi n n where ?gn = ?gn x, y, t ? ? and vgi = vgi x, y, t ? ? are the components of the vector Green?s function, x denotes the observer position, y denotes the source location, ?( ) is the Dirac delta function, and ? is the source emission time. ?ij is the Kronecker delta function. In the far-field, 97 1 ? = ln ? p? + p0 p? (3.19) since ?s and ?t vanish for the jet. Also, if p0 << p? , p0 p0 = ?p? ?? a2? ?0 ' (3.20) Then the solution for p0 (x, t) in the far-field is given by, 0 p (x, t) = ?? a2? Z ? Z ... ?? + 3 X ? ?gn ?g0 x, y, t ? ? ? y, ? ?? x, y, t ? ? (3.21) [fnv + fna ] y, ? d? dy n=1 Let the periodic Green?s function be defined by, ?gn 1 x, y, t ? ? = 2? ?gn Z Z ? ?gn x, y, ? exp [?i? (t ? ? )] d? (3.22) ?? ? x, y, ? = ?gn x, y, t ? ? exp [i? (t ? ? )] dt (3.23) ?? From this point, only the source term associated with the velocity perturbations will be considered. It is expected that the scaling of the other source terms would be similar. The exception would be the source term associated with the temperature fluctuations. However, the importance of this term remains the subject of debate in the prediction of turbulent mixing noise in heated jets. So, for the moment, this term will not be considered further. Then, the far-field pressure is given by, ? Z? X 3 X 3 2 4 Z ? a ? ? ?gn x, y, ?1 ?gm (x, z, ?2 ) p0 (x, t) p0 (x, t + ? ? ) = ... 2 (2?) ?? ?? n=1 m=1 v (z, ? ) exp [?i? (t ? ? ) ? i? (t ? ? ) ? i?? ? ] Оfnv y, ?1 fm 2 1 1 2 2 Оd?1 d?2 d?1 d?2 dydz The spectral density is related to the autocorrelation by, (3.24) 98 Z ? p0 (x, t) p0 (x, t + ? ? ) exp [i?? ? ] d? ? S (x, ?) = (3.25) ?? The spectral density is found from the autocorrelation. Making use of the integral, Z ? exp [i (? ? ?2 ) ? ? ] d? ? = 2?? (? ? ?2 ) (3.26) ?? yields the spectral density, ?2 a4 S (x, ?) = ? ? 2? Z ? Z ? ... ?? 3 X 3 X ?gn x, y, ?1 ?gm (x, z, ?2 ) ?? n=1 m=1 Оfnv y, ?1 fnv (z, ?2 )? (? ? ?2 ) exp [?i (?1 + ?2 ) t] (3.27) О exp [i?1 ?1 + i?2 ?2 ] d?1 d?2 d?1 d?2 dydz fnv is dependent on the strength of the shock cells and the turbulent fluctuations and its product is significant in regions where the shocks and expansions intersect with the turbulent shear layer. That is, if there is no turbulence present or pressure perturbation due to shock cells, then the term is small. Furthermore, the amplitude of fnv is proportional to the shock cell pressure perturbations and the turbulent velocity fluctuations. The two-point cross correlation function of the BBSAN source term is given by, v v (z, ? ) Rnm (y, ?, ? ) = fnv y, ?1 fm 2 (3.28) where ? is a vector between two source positions, ? = z ? y, and ? = ?2 ? ?1 . This is consistent with the statistics of the turbulence as a local function of the separation distance and time delay between two source locations. This yields, ?2 a4 S (x, ?) = ? ? 2? Z ? Z ... ?? ? 3 X 3 X ?gn x, y, ?1 ?gm (x, z, ?2 ) ?? n=1 m=1 v ОRnm y, ?, ? ? (? ? ?2 ) exp [?i (?1 + ?2 ) t] О exp [i (?1 + ?2 ) ?1 + i?2 ? ] d?1 d?2 d?1 d?2 dydz (3.29) 99 Now, the integrations with respect to ?1 , ?1 , and, ?2 can be performed using, S (x, ?) = ?2? a4? Z ? Z ? ... ?? 3 X 3 X ?gn x, y, ?? ?gm x, y + ?, ? ?? n=1 m=1 (3.30) v ОRnm y, ?, ? exp [i?? ] d?dy BBSAN is known to radiate at larger angles to the jet downstream axis. In those directions the propagation of the sound through the jet flow is minimally affected by the mean velocity and temperature gradients. In addition, the regions of strongest interaction between the turbulence and the shocks cells will occur at the center of the jet shear layer. The Green?s functions could be calculated numerically for a given mean flow. This could involve a locally parallel approximation or the full diverging flow. Also, the problem could be formulated in terms of the adjoint Green?s function for the linearized Euler equations as described by Tam and Auriault [85]. However, BBSAN is radiated predominantly at large angles to the jet downstream axis where the refractive effects of the mean flow would be small or absent. In view of this, the Green?s function is approximated with the absence of a mean flow as shown in Appendix C. The components of the vector Green?s function are readily related to the Green?s function of the Helmholtz equation. This gives, ?gn x, y, ? = i? xn x ? y /a? exp i? 4?a3? x x (3.31) where xn is the nth component of x, and x is the magnitude of x. Also, if y << |x|, ?gm x, y + ?, ? = ?gm ?x x, y, ? exp ?i и? a? x (3.32) With the use of these approximate forms for the far field Green?s functions, the spectral density can be written as, ?2? ? 2 S (x, ?) = 16? 2 a2? x2 ? xn xm v ... R y, ?, ? nm x2 ?? ?? ?x и ? d? d?dy О exp i?? ? i co x Z ? Z (3.33) 100 In Equation 3.33 the summation is implied over the indices n and m. v . From the continuity equations for the shock All that remains to model is Rnm cell structure Equation 3.6, v?j ??s ?vsi + =0 ?xj ?xi (3.34) Thus, it is assumed that the velocity perturbations associated with the shock cell structure are proportional to the local mean velocity multiplied by the perturbation in ?. That is, u? ?s vti l fiv ? (3.35) where l is the characteristic length scale, taken to be the same for the shock cell structure and the turbulence. From the form of fiv given by Equation 3.13, and on dimensional grounds, it is assumed to scale as, fiv ? ps v t ? ? a? l (3.36) where ps represents the shock cell strength, and vt is a characteristic turbulent velocity fluctuation. These variables will be determined from the RANS CFD solution. Also, for simplicity, use is made of the Proudman form for the cross correlation. That is, v where Rxx x n xm v v R y, ?, ? = R y, ?, ? (3.37) nm xx x2 y, ?, ? = fxv y, t fxv y + ?, t + ? , and fxv is the component of fiv in the direction of the observer. This simplification amounts to assuming that the source term is isotropic. Since the BBSAN radiation is dominant at large angles to the jet axis, this is a reasonable assumption. But it could be relaxed in the future with other models for the source statistics. Use of these relationships gives, v Rxx y, ?, ? = where, 1 ps ?2? a2? l2 y ps y + ? Rv y, ?, ? (3.38) 101 Rv (y, ?, ? ) = vx y, t vx y + ?, t + ? (3.39) is the two-point cross correlation function of the turbulent velocity fluctuations in the observer direction. Then the far field spectral density can be written as, Z? ?2 S (x, ?) = 16? 2 a4? x2 Z? 1 p y + ? p y s s l2 ?? ?? xи? v ОR (y, ?, ? ) exp i? ? ? d? d?dy xa? иии (3.40) It is convenient to introduce the cross spectral density of the turbulent velocity fluctuations. This enables the turbulent velocity statistics to be characterized either in terms of the cross correlation or cross spectral density. Both have been used in the modeling of turbulent mixing noise. The cross spectral density is given by, S v Z? y, ?, ? = Rv y, ?, ? exp (i?? ) d? (3.41) ?? with, R v 1 y, ?, ? = 2? Z? ? S v y, ?, ? ? e?i? ? d? ? (3.42) ?? Then, ?2 S (x, ?) = 32? 3 a4? x2 Z? Z? 1 ps y ps y + ? S v (y, ?, ? ? ) 2 l ?? ?? ?i?x ? О exp [?i? ? ] exp [i?? ] exp и ? d? ? d? d?dy a? x ... (3.43) The integration with respect to ? and then ? ? can be performed as before since, 102 Z? exp [?i (? ? ? ?) ? ]d? = 2?? (? ? ? ? ) (3.44) ?? So that, ?2 S (x, ?) = 16? 2 a4? x2 Z? Z? 1 ps y ps y + ? 2 l ?? ?? ?i?x ? y, ?, ? exp и ? d?dy a? x ОS v ... (3.45) In order to emphasize the quasi-periodic nature of the shock cell structure and to assist in the implementation of the model, the axial spatial Fourier transform of the shock cell?s pressure perturbation is defined. It is given by, 1 ps y = 2? Z? p?s (k1 , y2 , y3 ) exp [ik1 y1 ] dk1 (3.46) ?? with Z? p?s (k1 , y2 , y3 ) = p?s y exp [?ik1 y1 ] dy1 (3.47) ?? where k1 is the wavenumber in the axial direction, y1 . These relationships can be substituted into Equation 3.40. It should be noted that the axial Fourier transform of the shock cell pressure perturbation is only applied to one of the two terms in the integrand. Performing only one transform of the shock pressure results in the evaluation of the BBSAN to be more convenient in terms of computational efficiency. After some simplification, the spectral density is found to be given by, ?2 S (x, ?) = 32? 2 a4? x2 Z? Z? иии ?? ?? ОS v 1 ps y p?s (k1 , y2 , y3 ) exp [ik1 (y1 + ?)] 2 l xи? y, ?, ? exp i? ? ? dk1 d?dy xa? with, ? = (?, ?, ?). A model is now proposed for Rv y, ?, ? in the form, (3.48) 103 Rv y, ?, ? = K exp [? |? | /?s ] exp ? (? ? u?c ? )2 /l2 h 2 2 i О exp ? ? 2 + ? 2 /l? (3.49) where ?s is the turbulent time scale, l? is the turbulent length scale in the crossstream direction, and K is the turbulent kinetic energy. The scales, ?s , l, l? , are found directly from the CFD RANS solution. Then, ?2 S (x, ?) = 32? 3 a4? x2 Z? Z? K p y p?s (x, y, k) exp [ikz + ik? + i?? ] s l2 ?? ?? h 2i 2 2 2 2 2 О exp ? |? | /?s ? (? ? u?c ? ) /l ? ? + ? /l? ?i?x v ОS y, ?, ? exp и ? dkd?dy a? x ... (3.50) If x is written x/x = (sin ? cos ?, sin ? sin ?, cos ?) then the following integral needs to be evaluated, I?? Z? Z? i? 2 2 = sin ? cos ? exp ?? /l? ? a? ?? ?? i? 2 2 О exp ?? /l? ? sin ? sin ?? d?d? a? (3.51) The evaluation of I?? is shown in Appendix B. Making the substitution of the evaluated integral I?? yields, Z? Z? 2 Kl? ?2 S (x, ?) = ... ps y p?s (x, y, k) exp [ikz] 2 4 2 2 32? a? x l ?? ?? " #) ? |? | (? ? u?c ? )2 i? cos ?? О exp ? + i?? ? + ik? dkd? d?dy ?s l2 a? (3.52) The integral over ? and ? now needs to be evaluated. This integral is defined as I?? and is shown in Appendix B. Once this integration is performed and substituted into the above equation the final model for the spectral density is found, 104 Z? Z? 2 1 Kl? ps y p?s (k1 , y2 , y3 ) exp (ik1 y1 ) S (x, ?) = ? 4 2 иии l?s ? ?a? x ?? ?? ) 2 ? 2 ?s2 exp ?l2 (k1 ? ? cos ?/a? )2 /4 ? ? 2 l? sin2 ?/4a2? О dk1 dy 1 + (1 ? Mc cos ? + u?c k1 /?)2 ? 2 ?s2 (3.53) Equation 3.53 provides the prediction formula for the BBSAN. All of the parameters can be determined from a RANS CFD solution. The equations developed thus far can be applied to three-dimensional flow fields. In the case of an axisymmetric jet, the integrations with respect to the cross stream direction can be reduced to a single integration in the radial direction. The implementation of this prediction model is described in the next section. 3.2 Implementation The BBSAN integration model, shown in Equation 3.53, is implemented in a Fortran 90 program and linked with the common flow libraries. The Fortran 90 code conforms to the Fortran 95 standard ISO 1539-1997 [86] outlined by the J3 Fortran committee. The program is compiled with the Intel Fortran compiler for 32-bit applications, version 9.1 build 20060323z. The BBSAN code is a stand-alone program that is designed to load Wind-US structured grid solutions in their native format (or from another CFD program that provides output format in common flow libraries as structured grids). It outputs values of SP L per unit St (dB/[20хP a2 ]) at each prescribed observer location. Figure 3.1 shows an overview of the BBSAN program structure. The BBSAN program starts with a main function that calls the various subroutines depending on if the jet is axisymmetric or three-dimensional. Immediately the main program prints a comment card to the standard output detailing the version of the code and its associated library versions. This also lists the primary contact for the BBSAN program. Next the BBSAN program calls a subroutine that reads in the parameter input file. The parameter input file is unique to each jet simulation and contains information the program needs to successfully find spec- 105 Figure 3.1. Flowchart of the BBSAN prediction code. tral densities. It contains a flag that defines if the solution is three dimensional or axisymmetric, if the CFD solution is a dual stream jet, the names of the CFD grid and solution files, the number of the sub-zones to load from the CFD simulation, the ranges of the integration regions, the number of indices to be used in each direction of the integration regions, the jet or jet?s D, Md , Mj , and To , the number of observers and observer angles from the nozzle exit from the downstream axis, the number and ranges of frequencies to calculate, and an origin for the observer positions. Each CFD solution for each calculation is stored in sub-directories. This allows for a unique parameter input file to be stored in each sub-directory where the corresponding common flow library solutions are stored for a specific jet. This eliminates the need to enter the simulation parameters more than once and yields sub-directories with self contained executables and case files. The sub-zones of the CFD solution are specified as much of the solution is not needed for a successful calculation. Only the sub-zones that contain the shock waves and the shear layers are loaded. This saves a large amount of computer memory when examining large flow-field solutions from three dimensional jets. In fact, three dimensional integrations would not be possible if the ability to load subzones were not present in the serial implementation of the BBSAN model. Ranges of the interpolated integration regions are specified so that the BBSAN code only integrates over a small area of the loaded sub-zones. The sources of BBSAN are compact relative to the entire flow-field. Integration studies and reasons that an 106 interpolated integration regions are used instead of directly integrating over the CFD computational grid are discussed in the following sections. The diameter and operating conditions of the off-design supersonic jet are specified only for plotting purposes and are not used in the actual integration procedure. The BBSAN model evaluation is completely independent of these parameters. The BBSAN program only makes one assumption about the CFD solution: the positive axial streamwise direction of the jet is in the positive x direction and that the cross-stream direction is either r or y and z. This assumption could easily be removed. After the subroutine that reads the parameter input file finishes, the main function calls a subroutine that reads the common flow library grid and solution files. This subroutine, as seen in the third step of Figure 3.1, reads in specific subzones of the computational grid and solution files using the common flow library subroutines. It automatically detects if the variables written by Wind-US are single or double precision and stores them in double precision Fortran arrays. Also, the far-field quantities such as p? , T? , and ?? are read automatically. The field data and computational grid are immediately converted to the SI system as the common flow libraries store data in a non-dimensional format. This conversion is performed because the BBSAN program is written in dimensional form. Since the majority of the CFD simulations use the Menter SST K ? ? turbulence model and the BBSAN model requires the viscous dissipation rate, ? needs to be converted to . The relation between the two variables is = 0.09?K as specified by Wilcox [68] or Menter [70]. If a K ? model is being read from the common flow libraries no conversion is necessary. After the subroutine reads the common flow libraries it echoes each sub-zone?s dimensional data to individual Tecplot files for inspection to ensure that they are stored in memory correctly and that the solution is reasonable. These files can be compared directly to those produced by the Wind-US post processing program CFPOST. CFPOST can also perform the conversion of ? to and convert the non-dimensional solution to dimensional SI units. If the echoed files from the developed subroutine and CFPOST agree perfectly, this demonstrates that the subroutines that read the solution files are implemented correctly. The fourth step in Figure 3.1 involves calculating the metrics of the computational subdomains from the common flow libraries. These metrics were originally used to calculate derivatives of flow-field quantities. However, no derivatives are needed in 107 the current model. Finally, the main function calls either the axisymmetric single and dual stream BBSAN subroutine or the three dimensional BBSAN subroutine which calculate the spectral densities. The general form of the axisymmetric or three-dimensional BBSAN spectral density subroutine is shown in Figure 3.2. Figure 3.2. Flowchart of the BBSAN integration subroutine. The BBSAN subroutine immediately calculates the values of Tj , uj , and Dj based on D, Md , Mj , and To . These values are used to plot the results. The equations for this task are based on isentropic theory and are, Tj = 1+ uj = Mj To ??1 Mj2 2 p ?RTj (3.54) (3.55) 108 1 + (? ? 1)Mj2 /2 Dj = D 1 + (? ? 1)Md2 /2 (?+1)/(4??4) Md Mj 1/2 (3.56) The integration of the prediction equation is performed on structured grids that are derived from the CFD solution with constant spacing between grid points. This is done for two reasons. First, this method allows grid independence studies using the highly-resolved CFD solution databases. Second, it simplifies the calculation of the Fourier transform of ps since the grid spacing is constant with respect to x, r or y, and z. Thus a standard discrete Fourier transform or fast Fourier transform library can be used. Also, this ensures that the radial locations of p?s and the other field variables of the integration region are the same. Arrays that hold interpolated values in the integration regions are allocated based on the values in the parameter input file. These arrays include values in the integration region of K, , p, x, r or y, z, area, u, ps , and xk . These arrays hold the values in the integration regions, which are created and interpolated to based on the CFD solution. There are two interpolated regions that correspond to a jet. The first is the integration region and consists of a structured computational grid in two- or three-dimensions over the sources of BBSAN. The spacing between grid points in each direction is constant and the grid lines are parallel to the x-, y-, r-, or z-axes. The second interpolated region is the wavenumber region and it also has constant grid point spacing with its grid lines parallel to the x-, y-, r-, or z-axes. The spatial coordinates of the integration and wavenumber regions are found based on the parameter input file integration ranges and the respective number of indices. The range of the wavenumber region in the axial direction is based on the minimum distance between points in the axial direction. The interpolation scheme selected for calculating data on the integration and wavenumber regions is the inverse weighted distance interpolation algorithm of Shepard [87]. This algorithm is not a high order interpolation algorithm. It was found that higher order interpolation algorithms such as those based on radial basis functions of Li [88] are not needed and do not affect the accuracy of the BBSAN calculations. These high order interpolation algorithms create an additional unneeded computational cost. The inverse weighted distance formula for u, for example, 109 N P u(y) = wn (y)un n=0 N P (3.57) wn (y) n=0 where N is the maximum number of points close to the interpolated point, n is an integer representing the nth closest point, and wn is a weighting function given by, wn (y) = (x ? xn )2 1 + (y ? yn )2 + (z ? zn )2 (3.58) Clearly, the interpolation scheme is simple to implement and inexpensive. However, it requires finding the nearest N points, which can be an expensive operation. For axisymmetric BBSAN calculations N = 3 and for three-dimensional BBSAN calculations N = 4. An algorithm is implemented that calculates the distances between the interpolation location and each of the computational grid points on the CFD solution. The nearest N points can easily be identified in this fashion, although in three-dimensions it is very expensive because the search space is extremely large, even if it is restricted to one of the sub-zones. Conveniently, the structured computational domains used in the CFD are ordered. Therefore, some amount of computational logic is used in the three-dimensional subroutine to overcome the limitations of the brute force algorithm. This is accomplished by restricting the search space to planes in the cross-stream direction that are always parallel to the nozzle exit. The search algorithm for the nearest N points could be generalized by sorting the points of the CFD solution. Entire computational libraries exist for this purpose alone. However, the search algorithm is sufficient for structured CFD data but would need improvement if unstructured CFD data were being used as input to the BBSAN program. The variables that are interpolated onto the integration region are K, , p, and u. Figure 3.3 shows the integration region superimposed on the CFD solution with contours of u for the axisymmetric jet with Md = 1.00, Mj = 1.50, T T R = 1.00, and D = 0.0127 m. The CFD solution is calculated only in the top half plane. However, it has been mirrored in the bottom half plane to illustrate the accuracy of the interpolation scheme. Other variables show the same agreement. The interpolation from the CFD solution onto the wavenumber region uses the 110 Figure 3.3. Interpolation of u m/s onto the integration region from the CFD solution of a Md = 1.00, Mj = 1.50, T T R = 1.00, D = 0.0127 m jet. The red line encloses of the integration region. same interpolation scheme. However, the wavenumber region generally extends two core lengths downstream relative to the integration region. Note the integration region, in both the axisymmetric calculations and the three-dimensional calculations is chosen to enclose only the BBSAN sources. This extends from the nozzle exit to the end of the core or beyond and in the cross-stream direction from the centerline to roughly a single jet diameter. The wavenumber region is chosen to extend downstream of the integration region and shock cell structure by approximately one to two core lengths. The grid points of the wavenumber region in the y or r and z directions correspond to the same y or r and z grid points of the integration region. This is because in the BBSAN model equation the wavenumber spectrum must be known at the same cross-stream locations as the other field variables. The locations of the interpolated points of the wavenumber region in the streamwise direction do not correspond to streamwise points of the integration region. This is because the streamwise direction of the wavenumber integration region is transformed from x (streamwise direction) to wavenumber k. The index values of the wavenumber spectrum are now k, y or r, and z. When interpolating using inverse weighted distance interpolation only the interpolated values of p are evaluated on the wavenumber region. The shock pressure in the wavenumber region is then found by subtracting the ambient 111 pressure, p? . The variation of ps is constructed as an even function by mirroring the values across the nozzle exit plane. Now, values of shock pressure are even across x/D = 0.0 and the Hanning window [89], wn , is applied at each crossstream location in the stream-wise direction to ps , 1 2?n wn = 1 ? cos 2 N ?1 (3.59) where N is the maximum number of points in the streamwise direction of the wavenumber region and n is an integer from zero to N . Performing this operation helps to smooth the final wavenumber spectrum. After the Hanning window has been applied, the Fast Fourier Transform (FFT) or the Discrete Fourier Transform (DFT) is applied in the streamwise direction at each cross-stream location. Three different Fourier transform schemes have been implemented in the code and all produce the same result. The first is a simple discrete Fourier transform programmed directly from the definition of the Fourier transform. The second is the FFT written by Ferziger [90]. Since the Ferziger implementation is an FFT, it requires that the number of points in the streamwise direction be an integer power of two. Finally, the scheme normally used in the BBSAN code is the Single Precision Complex Fast Fourier Transform (SPCFFT) written by Ferguson [91]. The subroutine is further modified to use double precision values and also to use dynamic arrays that are available to Fortran 90. SPCFFT is unique because it reverts automatically to an FFT if the number of points in the streamwise direction is an integer power of two and works as a DFT otherwise. Therefore, there is no need to ensure that the number of points chosen in the streamwise direction is a power of two when specifying the indices in the parameter input file. SPCFFT has been benchmarked extensively after its modification by transforming common functions such as cos and exp in the forward and backward directions. After the DFT of the shock pressure has been calculated, the correction for the Hanning window is applied by multiplying the result by 8/3. The Fourier transform of the shock pressure in the axial direction, p?s , has now been constructed. An example result of p?s is shown in Figure 3.4 for Md = 1.00, Mj = 1.50, and T T R = 1.00. However, the absolute value of the quantity is shown to make the contours more easily identifiable. 112 Figure 3.4. The magnitude of the Fourier transform of the shock cell pressure p?s . Each of the peaks of the wavenumber spectrum, p?s , contribute to a peak in the predicted BBSAN spectrum. It is desirable to have a very large axial range for the shock pressure relative to the core length of the jet so that the spacing between wavenumber points in the spectrum is small. This helps resolve each peak?s contribution to the BBSAN. Note the values of p?s are real, symmetric about k = 0, and are a function of space only in the cross-stream directions. In the three dimensional BBSAN implementation p?s is a function of k, y, and z, and is also three-dimensional. Once p?s is calculated, files are created that are readable by Tecplot that contain all the interpolated data for the integration and wavenumber regions. The BBSAN program now has all the values ready to integrate as seen in Figure 3.2 using the developed model. In the evaluation of Equation 3.53 there are essentially three or four integrations to be performed over x, r or y, z, and k. These integrals are performed in a group of computational ?do? loops as seen in Figure 3.2. The outside do-loop is over the observer angle ? and the inside do-loop is over wavenumber, k. In the axisymmetric version of the BBSAN subroutine the fourth do-loop is skipped because the data is in two-dimensions. In the outer do-loop, ?, the observer angle is calculated from the parameter input file and a vector from the nozzle exit to the observer is also calculated. Inside the integration number do-loop the radian frequency ? is calculated. Inside the spatial do-loops of x, r or y, and z the three turbulence coefficients ? , l, and l? are found based on the values in the 113 integration region. These loops and corresponding inner loops use the Fortran cycle command if K < 0.10Kmax . Cycle essentially skips the evaluation of the do-loop at a particular value under certain conditions. This saves a great amount of time evaluating the spectral density because spatial regions that do not possess sufficient K will not contribute to the spectral density. Finally, the innermost do-loop is over the wavenumber k. By examining Figure 3.4 it is clear that certain ranges of wavenumber have very small values of p?s that do not contribute to the spectral density. Therefore, cycling is used in the innermost do-loop when p?s is not in the range of wavenumbers of interest to BBSAN. Performing a full wavenumber integration is more expensive and no difference in the final solutions are apparent. The integrand of Equation 3.53 is in the center of the seven do-loops. After the loops finish, the spectral densities at each observer location are multiplied by the term in front of the integrals of Equation 3.53 and also by a prefactor that controls the amplitude of the spectral densities, Pf . This is discussed in a later section. After the integral is multiplied by the prefactor, an adjustment to the spectral density is made if the jet is heated. This is necessary because of the deficiency in scaling of K with T T R as discussed in Chapter 2 of the RANS CFD solutions. To make the adjustment for heated jets the fully expanded jet density is calculated, po ?j = RTo ??1 2 Mj 1+ 2 ?1 ??1 (3.60) and the correction to account for a heated jet is, Tcf ?j = ?? ??1 2 1+ Mj 2 ?1 (3.61) where Tcf is the temperature correction factor proposed by Tam [58]. Tcf = 1.00 if T T R = 1.00 and is less than one if T T R > 1.00. The variation of Tcf with T T R is shown in Figure 3.5 part a) for a Md = 1.00 and Mj = 1.50 jet. Part b) of the figure shows the associated correction in dB applied to the SP L for the same jet. Because the Menter SST turbulence model is used to close the RANS equations the Tcf is applied to all the BBSAN calculations shown in this dissertation. By using Tcf better agreement is found with heated jets in terms of magnitude, although, the spectrum shapes are not affected by either Tcf or Pf . 114 Figure 3.5. a) Tcf verses T T R for a Md = 1.00 and Mj = 1.50 jet. b) The associated change in dB when heating a Md = 1.00 and Mj = 1.50 jet. Once the spectral density is found using Equation 3.53 and appropriate correction factors applied, it is written to files corresponding to each observer location as an ordered list of St, SP L, and f . Since Equation 3.53 uses radian frequency, the radian frequency is converted to St as, St = ?Dj 2?Uj (3.62) and the SP L is written to the file as, ? SP L = 20 log10 S 2 О 10?5 ! + 10 log10 (Uj /Dj ) (3.63) where S is the summation of the spectral density contributions from each integration region. The predictions can now be compared directly with experimental data. See Appendix A for additional details regarding the evaluation and scaling of the spectral density. 115 3.3 Parametric Studies A parametric study has been performed for the various computed jet flow-fields to determine the effects of changing the integration range and the grid point densities of the interpolated regions. Table 3.1 summarizes this work. The first column is the case number that is used for identification. The last column identifies if the case is a base case, or if no change, a minor change, or a very large change to the BBSAN is observed from the preceding base case. The bold values are changes in the specification of the interpolation regions relative to the base case. An integration region is defined by two points for both the three dimensional and axisymmetric cases. For the axisymmetric cases, the spatial point is (xmin , ymin ) and the second is (xmax , ymax ), forming a rectangle with sides parallel to the x- and r-axis. The three dimensional cases have an associated zmin and zmax value. As described in the previous section, the integration region is divided by a number of indices in the x- and r-axis directions that correspond to iindex and jindex respectively. The range of the wavenumber region always starts with the center of the nozzle exit plane and forms a rectangle with the second point at kmax and ymax . The number of indices in the x- and r-axis directions for the wavenumber region is jindex and kindex respectively. The first base case in Table 3.1 matches the experimental SPL well. Cases 2 through 8 vary the range and number of indices in the wavenumber region, thereby finding a range and grid density that is independent of the BBSAN solution. A new base case is constructed as number 8 and the ranges and indices are varied in the streamwise direction for the integration region. Cases 14 through 24 are conducted to examine the affects of varying the grid density and ranges of the integration region in the cross-stream direction. Finally, cases 25 through 29 are conducted partly to determine how close the integration region needs to be to the nozzle lip. Table 3.2 presents similar results to Table 3.1 but the ?case? column is replaced. The first row, ?Fast,? shows recommended dimensional values to obtain a BBSAN solution using the least number of recommended grid points in the integration and wavenumber regions. The last row uses the least number of grid points to get the most accurate solution from the model and achieve grid independence. 116 Case 01 02 03 04 05 06 07 08 09 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 xmin 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0100 0.0195 ymin 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0020 0.0040 0.0001 0.0001 0.0001 0.0001 0.0001 xmax 0.1137 0.1137 0.1137 0.1137 0.1137 0.1137 0.1137 0.1137 0.1421 0.1421 0.1421 0.1421 0.1421 0.1421 0.1421 0.1421 0.1137 0.1421 0.1421 0.1421 0.1421 0.1421 0.1421 0.1421 0.1421 0.0102 0.0722 0.1421 0.1421 ymax 0.0200 0.0200 0.0200 0.0200 0.0200 0.0200 0.0200 0.0200 0.0200 0.0200 0.0200 0.0200 0.0200 0.0200 0.0100 0.0200 0.0200 0.0100 0.0100 0.0200 0.0127 0.0100 0.0100 0.0100 0.0127 0.0127 0.0127 0.0127 0.0127 iind. 400 400 400 400 400 400 400 400 100 300 200 100 150 300 300 300 300 300 300 300 300 300 300 300 300 300 300 300 300 jind. 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 50 100 200 50 100 50 75 75 75 75 75 75 75 75 kmax 0.4000 0.4000 0.6000 0.4000 0.3000 0.4000 0.4000 0.4000 0.4000 0.4000 0.4000 0.4000 0.4000 0.4000 0.4000 0.4000 0.4000 0.4000 0.4000 0.4000 0.4000 0.4000 0.4000 0.4000 0.4000 0.4000 0.4000 0.4000 0.4000 kind. 300 500 300 400 300 200 300 400 400 400 400 400 400 400 400 400 400 400 400 400 400 400 400 400 400 400 400 400 400 Change Base Minor,1 Minor,2 No,2 No,4 Yes,5 Minor,5 Base No,8 Minor,9 Minor,9 Minor,9 Minor,9 Base Yes,14 Yes,14 No,16 Base Minor,18 Minor,18 Minor,18 Minor,18 Minor,22 Yes,22 Base Minor,25 Minor,25 Yes,25 Yes,25 Table 3.1. Summary of the parametric study of the integration and wavenumber regions for the circular converging jet operating at Mj = 1.50, T T R = 1.00, D = 0.0127 m. Finally, the ?Fast/Accurate? row shows a trade off between the two extremes of Fast and Precise. This middle row represents recommended values that allow fast solutions without sacrificing any accuracy of the BBSAN solutions, while allowing small deviations from the precise solution at very high and very low frequencies, well away from the BBSAN peaks. Clearly, the number of indices needed for a BBSAN solution is much less than those in the computational grids used in the 117 CFD solutions. Table 3.3 shows the ranges of the grid in a non-dimensional format based on the exit diameter of the nozzle. The ranges in the downstream direction roughly correspond to just more than the core length of the jet for the integration region and roughly three times the core length for the wavenumber region. The cross-stream direction range is recommended to be a single jet diameter and easily encompasses the BBSAN sources. In general, the chosen ranges and number of indices for the BBSAN solutions shown later in this chapter are based on the exit or equivalent diameter of the nozzle geometries based on the ?Fast/Accurate? row of Table 3.3. Case Fast Fast/Accurate Precise xmin 0.0001 0.0001 0.0001 ymin 0.0001 0.0001 0.0001 xmax 0.1421 0.1421 0.1421 ymax 0.0127 0.0127 0.0127 iindex 150 300 500 jindex 75 75 100 kmax 0.4000 0.4000 0.4000 kindex 300 300 400 Table 3.2. Recommended integration and index ranges based on the integration study of Mj = 1.50, T T R = 1.00, D = 0.0127 m. Case Fast Fast/Accurate Precise xmin 0.0001 0.0001 0.0001 ymin 0.0001 0.0001 0.0001 xmax 11.189 11.189 11.189 ymax 1.00 1.00 1.00 iindex 150 300 500 jindex 75 75 100 kmax 31.5 31.5 31.5 kindex 300 300 400 Table 3.3. Recommended non-dimensional integration values based on D and index ranges. The scaling coefficients in the BBSAN model control the shape of the spectrum and in part its magnitude. c? is set to 1.25 and controls the relative magnitude of the spectrum and is the primary means to control the sharpness of each BBSAN peak. cl is set to 3.25 and determines the turbulent length scale in the streamwise direction. It controls to some degree the width of the peaks, the smoothness of the spectra, and the relative magnitude of the BBSAN as a function of observer angle. Increasing cl smooths the BBSAN peaks, increases the width of each peak in the BBSAN spectrum, and lowers the relative magnitude between the peaks and 118 troughs. It should be noted that the value of cl of 3.25 is much larger than that used by Tam and Auriault [85] in their model of fine-scale turbulent mixing noise. This reflects the fact that BBSAN is controlled by the large scale structures in the jet shear layer that are coherent over relatively large axial distances. c? is set according to experimental observations, which show that the cross-stream length scale is approximately 30% of the streamwise length scale and is therefore set to 0.30. c? controls the rate at which the high frequency predictions decay. These values were chosen based on the spectrum of a jet at ? = 100 and operating at Md = 1.00, Mj = 1.50, and T T R = 1.00. Finally, Pf is chosen by matching the SPL at various frequencies of the same jet at ? = 100. The same values of c? , cl , c? , and Pf are used for every calculated spectral density. 3.4 Single Stream Axisymmetric Jets BBSAN predictions have been made for the jet conditions shown in Table 3.4 based on corresponding Wind-US CFD solutions using the Menter SST turbulence model. The cases selected include both unheated and heated jets operating at over(Mj < Md ) and under-expanded (Mj > Md ) conditions for two different nozzle geometries with design Mach numbers, Md = 1.00 and Md = 1.50. The CFD simulations for all the cases use a jet exit diameter, D = 0.0127 m. However, other CFD simulations for the same operating conditions but with different diameters have been performed and results are similar for the BBSAN prediction. The BBSAN predictions are made at various angles ? from the downstream jet axis at a radial polar distance of 100 D from the center of the jet exit. Predictions are lossless and corresponding experimental data in each case has had humidity and atmospheric absorption corrections applied so that the presented SPL are also lossless. The experimental data provided by Boeing was measured at 97.5 D from the nozzle exit while the Pennsylvania State University (PSU) data was measured at 150 D. Both sets of data have been corrected to 100 D. The PSU experimental data has its origin centered at x/D = 5.0 or 0.0635 m, but this is of less importance for BBSAN since the source region is relatively close to the jet exit and the dominant radiation is at large angles to the jet downstream axis. The 119 Md 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.50 1.50 1.50 1.50 1.50 1.50 Mj 1.22 1.22 1.22 1.47 1.56 1.50 1.50 1.30 1.30 1.40 1.60 1.70 1.70 NP R 2.50 2.50 2.50 3.50 4.50 3.67 3.67 2.77 2.77 3.18 4.25 4.94 4.94 TTR 1.00 2.20 3.20 3.20 3.20 1.00 2.20 1.00 2.20 1.00 1.00 1.00 2.20 D (m) 0.0508 0.0508 0.0508 0.0508 0.0508 0.0127 0.0127 0.0127 0.0127 0.0127 0.0127 0.0127 0.0127 Dj (m) 0.05176 0.05176 0.05176 0.05460 0.05604 0.01377 0.01366 0.01209 0.01209 0.01236 0.01309 0.01354 0.01209 fc 07115 10552 12728 13791 14004 31040 45784 31897 47191 32936 34106 34294 56826 ? 0.700 0.700 0.700 1.077 1.077 1.118 1.118 0.748 0.748 0.539 0.557 0.800 0.800 Tj (K) 225.6 496.4 722.1 655.8 631.5 202.2 450.3 219.1 479.6 210.6 193.9 185.8 406.6 uj (m/s) 368.3 546.2 658.8 753.0 784.8 427.5 625.3 385.7 570.6 407.2 446.6 464.5 687.2 Table 3.4. Jet operating conditions of the RANS CFD and BBSAN predictions for axisymmetric single stream jets. Boeing experiments were performed with heated air while the PSU experiments were performed with helium / air mixtures to simulate heated jets. The diameter of the convergent nozzle in the Boeing experiments is 0.0622 m while the diameter of the PSU nozzle is 0.0127 m. Experimental data, measured in the Small Hot Jet Aeroacoustic Research facility at NASA Glenn Research Center, have also been used for evaluation of the predictions. The SHJAR jets have a 0.0508 m diameter nozzle with microphones on a polar arc at R/D = 50 centered at the center of the nozzle exit. This data has also been extended to R/D = 100. The integration regions chosen for the converging jets with D = 0.0127 m are 0.0001 ? x ? 0.1421 m and 0.0001 ? r ? 0.0127 m, with 300 and 75 grid points in the x and r directions respectively. The spacing between these grid points are constant in both directions. These parameters were chosen based on the integration studies. The shock pressure used for the Fourier transform was taken from 0.0001 ? x ? 0.4000 m using 512 grid points. Since the data is mirrored about x/D = 0.0, the total number of points in the Fourier transform is 1024. This yields a wavenumber spacing, ?k, of approximately 7.86 m?1 . The integration regions for the convergent-divergent jets are 0.0001 ? x ? 0.1421 m and 0.0001 ? r ? 0.0127 m with 400 and 100 grid points in the x and y directions 120 respectively. The wavenumber integration range remains the same. More grid points are used because the source regions are more compact relative to the underexpanded cases. The model developed in the present dissertation is only for the BBSAN, which is only one of the components of off-design supersonic jet noise. Thus, it is useful to separate the individual noise components from the total spectrum. Viswanathan [53] has developed a framework that separates the individual components of the total noise spectrum into mixing noise and BBSAN. The Boeing experimental data presented for the converging nozzle predictions shows this breakdown to help illustrate the capability of the prediction scheme. Specifically, the proposed model for BBSAN is compared directly against the extracted shock component from the total noise for the two convergent nozzle cases. The BBSAN predictions are presented in terms of SP L per unit St. The case that has been used to fix the coefficients for the turbulence scales is the underexpanded converging jet with Mj = 1.50 and T T R = 1.00 at ? = 100 degrees. Results of the model prediction in this case, in addition to various experiments, are shown in Figure 3.6. There are eight comparisons of the predictions with experimental data for different observer angles ? with respect to the downstream jet axis. Each comparison is labeled with its corresponding observer angle and the maximum SPL level of the experiments of Boeing. The screech tones are not used to find the maximum value. Experimental data from Boeing, NASA, and PSU are shown for each observer angle. Though there are minor differences between the different sets of experimental data, the overall agreement is good. Also shown are predictions based on the BBSAN prediction formulas provided by Tam [57] or Tam [58] where appropriate. The breakdown of the total Boeing spectra into the BBSAN and mixing noise components is also shown. At the different observer angles the predictions are in good agreement with respect to all the experiments. The predictions capture the multiple peaks in the BBSAN spectra as well as the broadening in the spectral shape with decreasing angle to the jet downstream axis. For these operating conditions the amplitude of the primary peak in the BBSAN is relatively insensitive to the observer angle. This is captured by the predictions. It is important to emphasize that the predictions at every angle use the same scaling coefficients. As the angle to the jet downstream axis decreases, the primary peak 121 in the BBSAN spectrum moves to higher Strouhal numbers. This is also predicted in the BBSAN models of Harper-Bourne and Fisher [1] and Tam [57]. The increase in the relative importance of the mixing noise at lower Strouhal numbers is evident in the experiments. Before showing predictions for other jet operating conditions it is interesting to show how the predicted BBSAN spectrum is constructed from various wavenumber components. The wavenumber integration in Equation 3.53 can be limited to contributions associated with individual components of the shock cell?s wavenumber spectrum. This is equivalent to examining the contributions of the interactions of the turbulence with the Fourier modes in a waveguide model of the shock cell structure. Figure 3.7 shows these contributions and the corresponding peaks they generate in the BBSAN spectrum. This selection of wavenumber ranges replaces the summation over Fourier components of the waveguide model for the shock cell structure in Tam?s model [57]. However, in the present model, the spectral width is controlled by both the spectrum shape associated with the turbulent velocity fluctuations and the finite bandwidth of the dominant lines in the Fourier transform representation of the shock cell structure. The source regions of the BBSAN according to the model equation can be specifically identified. It was postulated that the regions of BBSAN occur where there are large values of |ps | and large values of turbulent kinetic energy, K. Figure 3.8 shows contour plots of the Md = 1.00, Mj = 1.50, T T R = 1.00 converging nozzle case. The spatial ranges of these contour plots do not correspond to the entire integration region and the values in the contour plots are of interpolated values from the RANS CFD solution. Figure 3.8 part a) shows a contour plot of the shock pressure, the axisymmetric shock cell structure is readily visible. Part b) shows contours of K that are strongest in the spatially growing turbulent shear layer of the jet. By evaluating the integrand of the model equation at ? = 90.0 degrees, St = 0.653, f = 20241 Hz, and R/D = 100 and plotting values spatially over x/D and r/D, the source distribution of the BBSAN is shown in part c). This observer angle and frequency are chosen because they correspond to the first dominant BBSAN peak and have the smallest effect due to the shear layer on the predicted frequency. Part c) shows that the sources of BBSAN are strong peaks that are relatively compact compared to the flow-field of the jet and are located where shock 122 waves interact with the turbulent shear layer. The dominant BBSAN source for this angle and frequency occurs at the first oblique shock wave at x/D = 1.50, the second dominant source is at x/D = 3.00, the third at x/D = 4.50, etc. Clearly, the largest contribution to the BBSAN occurs at x/D = 1.50 and the sources are relatively equally spaced. Since the shock cell structure is confined by the shear layer, the shocks themselves terminate at lower r/D locations as x/D increases. BBSAN sources also follow this trend at a relatively constant rate. That is, as x/D increases the positions of the BBSAN sources decrease at a relatively constant rate towards r/D = 0.0. The strength of the BBSAN sources far downstream becomes smaller as the shock pressure decreases. Finally, there is no BBSAN source contribution near the lip of the nozzle or a positive contribution in regions where the Prandtl-Meyer expansion fans interact with the shear layer. Only weak negative contributions exist in this region. The remaining BBSAN prediction results are presented in order of increasing Mj and then associated T T R as shown in Table 3.4. Using the same N P R as the previous case, heating is added to the converging nozzle flow at a T T R = 2.20. The BBSAN predictions are shown in Figure 3.9. Overall, the spectral shape of the predictions is excellent and the peaks of each BBSAN component are captured. However, the predicted BBSAN magnitude at all frequencies is too high by 6 dB without applying Tcf . Tam [58] argued that as the jet temperature increases, for supersonic conditions, the growth rate and amplitude of the large scale turbulent structures should decrease. This is supported by instability theory. Thus it is expected that the turbulence levels should decrease as the jet temperature increases. This is not found in either the K ? or K ? ? RANS simulations. In the absence of an improved RANS turbulence model for the effect of jet heating, Tam?s correction factor has been adopted here as an empirical correction for the effects of jet heating. Tcf is applied to every prediction in this dissertation. Turbulence modeling may be one of the issues that contributes to poor scaling of BBSAN intensity with increasing T T R. It could also be attributed to a failure to capture this phenomena with the acoustics model. Unfortunately, the reason why the BBSAN intensity does not continue to increase with increasing T T R is not understood by the aeroacoustics community and is an area of current research. To explore the effect of heating the jets further, three over-expanded cases 123 are conducted with the converging nozzle and T T R = 1.00, T T R = 2.20, and T T R = 3.20. The noise predictions in these three simulations are shown in Figures 3.10, 3.11, and 3.12. In each case the predictions and experiment agree very well both in magnitude and peak frequencies. The higher frequency BBSAN peaks are captured more accurately as the T T R is increased. However, if the factor Tcf is not applied to the predicted spectral densities, the higher values of T T R will correspond to very high predicted BBSAN magnitudes relative to experiment. The Tcf corrected dB for the three cases is 0.0, -3.45, and -5.07 dB, respectively. As T T R increases even more, the overshoot of the BBSAN and the associated corrected dB due to heating increases even further. Two over-expanded cases of Md = 1.50, Mj = 1.30, and T T R = 1.00 and T T R = 2.20 are shown in Figures 3.13 and 3.14 respectively. In the unheated case the peak levels are underpredicted and the spectral peaks are somewhat narrower than in the experiments and the BBSAN is evident at all angles to the jet axis. However, in the heated case, the mixing noise is clearly dominant at angles less than 80 degrees to the jet downstream axis. There is also close agreement between the present predictions and Tam?s model in the heated case. In the unheated case, Tam?s model overpredicts the levels, whereas the present model underpredicts them. Two slightly off-design cases have been predicted with over- and under-expanded jets for Md = 1.50 and T T R = 1.00. This enables an important illustration of the differences between Tam?s model and the current one to be shown. Predictions for jets with Mj = 1.40 and Mj = 1.60 are presented in Figures 3.15 and 3.16 respectively with SHJAR data at Mj = 1.38 and Mj = 1.61 also shown. The general shape of the dominant BBSAN peak agrees well between the two prediction schemes, with the current implementation having a slightly narrower peak. When the observer angle is less than ? = 90, the present prediction scheme gives levels that are lower than the peaks predicted by Tam?s model. However, the overall amplitude of Tam?s model at low observer angles is generally higher than the experimental data while the present predictions are lower. These two cases illustrate that computationally the model is more robust as the off-design parameter is increased. Two final BBSAN predictions from converging nozzles are conducted with mod- 124 erately off-design parameters and jet heating of T T R = 3.20. These are shown in Figures 3.17 and 3.18. The fully expanded Mach numbers are very close to one another and the temperature correction factor applied to both cases is -5.07 dB. The small change in Mj between the two cases increases both the experimental data and predictions by approximately the same amount of 2 to 3 dB. At lower angles the peak frequencies are slightly higher than experiment and the magnitudes increase too much as the observer angle increases. However, the higher frequency peaks of the BBSAN are captured extremely well. BBSAN predictions are now made for Md = 1.50, Mj = 1.70 with T T R = 1.00 and T T R = 2.20. The unheated and heated predictions with accompanying experimental data are shown in Figures 3.19 and 3.20 respectively. In the unheated case both prediction schemes give similarly good agreement with the experimental data. The multiple peaks in the BBSAN predictions are more evident in Tam?s model and the spectral shape given by the present model is closer to the measured spectrum. In the heated case, the peak BBSAN levels are slightly overpredicted in the present model and again corrected by Tcf . One interesting feature of the effect of jet temperature on the BBSAN can be seen in Figures 3.19 and 3.20. From the measured spectra it is clear that the peak amplitude of the BBSAN is only slightly higher in the heated case. However, the fully expanded jet velocity has increased by 48%. Since the amplitude of the BBSAN depends on the turbulence levels as well as shock cell strength, a larger increase in levels would be expected. A close examination of the prediction formula 3.53 suggests a dependence on jet velocity of u3j . In the present case, assuming that the shock cell strength is relatively independent of T T R, this would give an increase of 5-6 dB. This increase is consistent with the predictions and also with the predicted effects of heating in the other cases shown previously, but it is not evidenced in the experiments. 125 Figure 3.6. Comparisons of BBSAN predictions with experiments for Md = 1.00, Mj = 1.50, T T R = 1.00, R/D = 100. 126 Figure 3.7. The total BBSAN prediction and the accompanying contributions from selective integrations over contributing wavenumbers of p?s representing different waveguide modes of the shock cell structure. Md = 1.00, Mj = 1.50, T T R = 1.00, R/D = 100, ? = 120.0. Figure 3.8. Various plots of the flow-field region for Md = 1.00, Mj = 1.50, T T R = 1.00, R/D = 100, ? = 90.0. a) contours of ps . b) contours of K. c) The spatially distributed source of the BBSAN at the peak frequency of fp = 20241 Hz or St = 0.653. 127 128 Figure 3.9. Comparisons of BBSAN predictions with experiments for Md = 1.00, Mj = 1.50, T T R = 2.20, and R/D = 100. 129 Figure 3.10. Comparisons of BBSAN predictions with experiments for Md = 1.00, Mj = 1.22, T T R = 1.00, and R/D = 100. 130 Figure 3.11. Comparisons of BBSAN predictions with experiments for Md = 1.00, Mj = 1.22, T T R = 2.20, and R/D = 100. 131 Figure 3.12. Comparisons of BBSAN predictions with experiments for Md = 1.00, Mj = 1.22, T T R = 3.20, and R/D = 100. 132 Figure 3.13. Comparisons of BBSAN predictions with experiments for Md = 1.50, Mj = 1.30, T T R = 1.00, and R/D = 100. 133 Figure 3.14. Comparisons of BBSAN predictions with experiments for Md = 1.50, Mj = 1.30, T T R = 2.20, and R/D = 100. 134 Figure 3.15. Comparisons of BBSAN predictions with experiments for Md = 1.50, Mj = 1.40, T T R = 1.00, and R/D = 100. 135 Figure 3.16. Comparisons of BBSAN predictions with experiments for Md = 1.50, Mj = 1.60, T T R = 1.00, and R/D = 100. 136 Figure 3.17. Comparisons of BBSAN predictions with experiments for Md = 1.00, Mj = 1.47, T T R = 3.20, and R/D = 100. 137 Figure 3.18. Comparisons of BBSAN predictions with experiments for Md = 1.00, Mj = 1.56, T T R = 3.20, and R/D = 100. 138 Figure 3.19. Comparisons of BBSAN predictions with experiments for Md = 1.50, Mj = 1.70, T T R = 1.00, and R/D = 100. 139 Figure 3.20. Comparisons of BBSAN predictions with experiments for Md = 1.50, Mj = 1.70, T T R = 2.20, and R/D = 100. 140 3.5 Dual Stream Axisymmetric Jets The axisymmetric implementation of the BBSAN model can also be applied to axisymmetric dual stream jets. The only physical difference between the single stream and dual stream axisymmetric jets is the addition of a secondary stream surrounding the core flow. The dual stream jets of interest could have a supersonic primary and subsonic secondary flow, subsonic primary or supersonic secondary flow, or both supersonic flows operating off-design. The case where the core of the jet is operating supersonically and the secondary flow operates subsonically is essentially a problem similar to the single stream jet but with a high speed co-flow. If the secondary flow is supersonic while the primary flow is subsonic then the integration regions need only be constructed over the shock cell structures that interact with the turbulent shear layers. Finally, the most difficult case occurs when both streams are supersonic, as two separate integration and wavenumber regions must be placed over the regions of corresponding shock wave shear layer interactions. Table 3.5 shows the dual stream experimental conditions available to the author. The characteristic frequency, fcp , is based on the primary jet only. The method to obtain the RANS CFD solution for these cases was outlined in the previous chapter. Clearly, both the primary and secondary nozzle are converging which yields Md = 1.00. The secondary flow is cold and the primary flow is heated. The characteristic frequency is based on the fully expanded diameter and velocity of the primary jet. The off-design parameters for both flows are listed. The offdesign parameter does not apply to the subsonic flow of one of the cases and is listed as not applicable. For the dual stream case with two supersonic flows, multiple integration regions must be used to calculate the BBSAN contribution from each shock cell / shear layer interaction in the primary and secondary stream. Figure 3.21 shows the shock pressure contours and the integration regions. Note that the axes of the figure have been normalized by the primary nozzle diameter, Dp . The integration regions? interpolated data is overlaid on the flow-field of the RANS CFD solution and the associated boundaries are represented by red lines. The first integration region covers the primary stream while the second integration region covers the 141 secondary stream. There are two wavenumber regions that correspond to both of the integration regions and to the respective radial locations of the interpolated data of the integration regions. The spectral densities calculated from each individual integration and wavenumber region are summed to give a total BBSAN prediction. In one case, the off-design parameter of the secondary stream is relatively weak and the BBSAN is dominated by the primary stream. Both integration regions extend from just after the nozzle lips at x/Dp = 0.0 and x/Dp = ?1.36, as there is no BBSAN source at the nozzle lips in under-expanded jets, to 9.64 Dp . The wavenumber regions extend in the streamwise direction from their respective nozzle lips to 19.28 Dp . Figures 3.22 and 3.23 show BBSAN predictions for the dual stream jets with conditions shown in Table 3.5. The addition of the supersonic secondary stream increases the maximum SPL at all angles by 3 to 4 dB. This is reflected in the predictions as the maximum BBSAN at all observer angles is also increased by approximately 3 to 4 dB. Peak frequencies agree with experiment best at ? = 90.0 degrees for both cases. This implies that the high speed secondary flow, both subsonic or supersonic, has the effect of lowering the peak BBSAN frequencies. Mismatch of predicted frequency becomes quite extreme at high and low observer angles. The peak frequencies of the predictions could be improved by using a better approximation of the vector Green?s function to include secondary flow effects. The BBSAN predicted amplitudes compare very well with experiment. At high observer angles of Figure 3.22 there are regions where no prediction is present near St = 0.50. This is in contrast with the results of Figure 3.23 where there is a BBSAN contribution across the same St range. The higher speed secondary stream of Figure 3.22 causes a stronger Doppler shift to the predicted BBSAN and shifts the peaks to lower frequency. The prediction could be improved by lowering c? which would have the effect of broadening the BBSAN peaks, thus eliminating the region where no BBSAN is predicted. 142 Mdp 1.00 1.00 Mjp 1.19 1.19 Mds 1.00 1.00 Mjs 0.96 1.04 T T Rp 2.70 2.70 T T Rs 1.00 1.00 Dp (m) 0.0622 0.0622 fcp 9398 9398 ?p 0.645 0.645 ?s N/A 0.578 Table 3.5. Jet operating conditions for the RANS CFD and BBSAN predictions of the dual stream jets. Figure 3.21. Integration regions for the BBSAN calculation of the dualstream jet Mdp = 1.00, Mjp = 1.19, Mds = 1.00, Mjs = 1.04, T T Rp = 2.70. 143 Figure 3.22. Comparisons of BBSAN predictions with experiments for the dualstream jet Mdp = 1.00, Mjp = 1.19, Mds = 1.00, Mjs = 1.04, T T Rp = 2.70, R/D = 100. 144 Figure 3.23. Comparisons of BBSAN predictions with experiments for the dualstream jet Mdp = 1.00, Mjp = 1.19, Mds = 1.00, Mjs = 0.96, T T Rp = 2.70, R/D = 100. 145 3.6 Three Dimensional Jets Thus far axisymmetric single and dual stream CFD RANS solutions have been used to perform BBSAN predictions. These used the axisymmetric form of the BBSAN model equation and its implementation. The flow-fields produced by rectangular or non-axisymmetric jets are highly three-dimensional and are not applicable to the axisymmetric BBSAN model. Here, the BBSAN model is evaluated with the three dimensional implementation of the BBSAN code using the rectangular jet RANS solutions validated for rectangular jets in Chapter 2. Both cold rectangular jets with Md = 1.50 were validated with Pitot and schlieren comparisons. The two off-design rectangular jet simulations with T T R = 2.20 were validated by comparing schlieren of a simulated hot jet using a helium / air mixture. Table 3.6 shows the rectangular nozzle operating parameters that are used for the corresponding BBSAN predictions. The rectangular nozzle chosen has Md = 1.50, and operates either over- or under-expanded at Mj = 1.30 or Mj = 1.70 and T T R = 1.00 or T T R = 2.20. The experimental results were provided by Veltin and the experimental details are documented in Goss et al. [92]. Microphone positions were placed at 1.905 meters from the jet at various ? in the minor and major axis planes and at an azimuthal angle of 45 degrees. The experimental data was extended to 100 De to be consistent with the predictions presented in the previous sections. Only predictions are presented in the major and minor axis planes because the aspect ratio of the rectangular jet is relatively small, therefore the intensity of the BBSAN will vary only a little with respect to azimuthal angle. The characteristic frequencies of the rectangular jets are based on the fully expanded equivalent diameter found from the De and the N P R. Md 1.50 1.50 1.50 1.50 Mj 1.30 1.70 1.30 1.70 NP R 2.77 4.94 2.77 4.94 TTR 1.00 1.00 2.20 2.20 De (m) 0.01778 0.01778 0.01778 0.01778 Dej (m) 0.01693 0.01896 0.01693 0.01896 fc 22888 24499 33792 36334 ? 0.748 0.800 0.748 0.800 Tj (K) 219.1 185.8 482.0 408.7 uj (m/s) 385.7 464.5 572.1 688.9 Table 3.6. Jet operating conditions for the RANS CFD and BBSAN predictions of the 3D calculations. 146 Unlike the axisymmetric version of the code, the three-dimensional BBSAN implementation evaluates the integrals over the three dimensional integration region and wavenumber region. Since there is an additional integral that needs to be evaluated in the z-axis direction, the run time of the computer code can increase greatly. To minimize the amount of time needed to make a three-dimensional BBSAN prediction, careful choices are made of the range and number of indices of the integration and wavenumber regions. Based on the conclusions for the integration studies of the axisymmetric jet shown in Table 3.3, integration ranges for the four rectangular cases are chosen as 0.001m < x < 0.1437m, 0.001m < y < 0.011m, and 0.001m < z < 0.021m with corresponding index values of 300, 75, and 75 in the streamwise, minor-axis, and major-axis directions respectively. Since the RANS CFD simulations uses planes of symmetry of the nozzle, the implemented BBSAN code automatically integrates across the planes. There is no restriction in the implemented version of the BBSAN model regarding the flow-field, even though planes of symmetry were used to construct the RANS solution. The same implementation of BBSAN may be applied to any three-dimensional jet flow-field. The range of the wavenumber region is chosen as 0.001m < xk < 0.250 m with 512 indices. Summation of the integration region is selective by only including contributions from 10% of K using cycling. This cycling technique saves a large amount of computational time just as it had in the axisymmetric calculations. Also, the solution is found relatively quickly by specifying a limited range for the integrations over wavenumber. The wavenumber integration range is carefully chosen to only encompass peak values of the wavenumber spectrum. For example, in the first rectangular BBSAN condition of Table 3.6, the spacing between each value of wavenumber in the wavenumber spectrum is ?k = 12.57 and only wavenumber contributions are used in the range of ?25 < k < ?166 and 25 < k < 166. BBSAN predictions of the over-expanded cold rectangular jet in the major and minor axis planes are shown in Figures 3.24 and 3.25. By comparing the BBSAN predictions with the experimental data, it is seen that only a small 2 dB difference is apparent between the BBSAN peaks in the major and minor axis. Additional strong screech tones are present in the minor axis direction. The predictions in both figures are the same as the azimuthal angle is not included in the current prediction formula. This azimuthal dependance could easily be included if necessary. The 147 peak BBSAN at all observer locations matches well with respect to frequency and magnitude. As in the circular nozzle case operating at Md = 1.50 and Mj = 1.30, the overall prediction is slightly low. The width of the BBSAN peaks is also too narrow relative to experimental measurements. Figures 3.26 and 3.27 show the under-expanded unheated rectangular jet operating at Mj = 1.70. In this case, the variation of the magnitude of the BBSAN is invariant with azimuthal angle, unlike the previous over-expanded case. The predictions at all observer angles agree extremely well with the experimental data. The first BBSAN peak and subsequent minor peaks at higher frequencies generally align with those of experiment and the amount of fall-off in the spectrum is not significant. The small high frequency peaks of the BBSAN are more pronounced in the experimental data in the major axis plane. The predictions could be slightly improved for this jet if c? is lowered to flatten out the strong dominant BBSAN peak. BBSAN predictions are performed for an over-expanded rectangular jet operating at Mj = 1.30 and T T R = 2.20. These predictions are shown in Figures 3.28 and 3.29 in the major and minor axes planes respectively. Variation of maximum dB in the experiment with azimuthal angle only varies by up to 2 dB. Like the heated axisymmetric case, the predictions are slightly lower than the experimental data. The magnitude of BBSAN matches the experiment at high observer angles, and also matches the peak frequencies. At ? = 90.0 degrees and lower the mixing noise dominates the BBSAN. The width of the peaks, as with the axisymmetric jet, are too narrow. This is especially true for low St when the prediction has extreme fall-off. Note that for this heated prediction, along with all the other rectangular jet predictions, the temperature correction factor has been applied. The final rectangular predictions are shown in Figures 3.30 and 3.31 for the major and minor axis directions respectively, for the Mj = 1.70 and T T R = 2.20 jet. The peak frequencies match those of the experiment very well at most angles except for large values. BBSAN magnitudes align well perpendicular to the jet, however, are over-predicting at high observer angles. At low observer angles the BBSAN is dominated by the mixing noise because the jet is heated. The predicted values of BBSAN do not contribute to the total spectra just as in the corresponding heated circular jet case. The variation of the peak BBSAN magnitude is partly 148 controlled by cl . If cl is smaller, then the variation of BBSAN magnitude with angle would better match the experiment. Figure 3.24. Comparisons of BBSAN predictions with experiments for the rectangular jet Md = 1.50, Mj = 1.30, T T R = 1.00, R/De = 100 in the major axis direction. 149 Figure 3.25. Comparisons of BBSAN predictions with experiments for the rectangular jet Md = 1.50, Mj = 1.30, T T R = 1.00, R/De = 100 in the minor axis direction. 150 Figure 3.26. Comparisons of BBSAN predictions with experiments for the rectangular jet Md = 1.50, Mj = 1.70, T T R = 1.00, R/De = 100 in the major axis direction. 151 Figure 3.27. Comparisons of BBSAN predictions with experiments for the rectangular jet Md = 1.50, Mj = 1.70, T T R = 1.00, R/De = 100 in the minor axis direction. 152 Figure 3.28. Comparisons of BBSAN predictions with experiments for the rectangular jet Md = 1.50, Mj = 1.30, T T R = 2.20, R/De = 100 in the major axis direction. 153 Figure 3.29. Comparisons of BBSAN predictions with experiments for the rectangular jet Md = 1.50, Mj = 1.30, T T R = 2.20, R/De = 100 in the minor axis direction. 154 Figure 3.30. Comparisons of BBSAN predictions with experiments for the rectangular jet Md = 1.50, Mj = 1.70, T T R = 2.20, R/De = 100 in the major axis direction. 155 Figure 3.31. Comparisons of BBSAN predictions with experiments for the rectangular jet Md = 1.50, Mj = 1.70, T T R = 2.20, R/De = 100 in the minor axis direction. 156 3.7 The Effect of Laminar Flow in the Nozzle BBSAN predictions have been made for the laminar / RANS and fully RANS jet flows and compared to experimental data. This required the two CFD simulations that were conducted to examine the effect of laminar or turbulent flow inside the nozzle for a Md = 1.50, Mj = 1.30, T T R = 2.20 jet. The first CFD simulation solved the RANS equations in the entire domain, and the second CFD simulations solved the Navier-Stokes equations in the interior of the nozzle and the RANS equations in the exterior. The second simulation thereby forces the flow to be laminar inside the nozzle and turbulent outside the nozzle. In the laminar / RANS case, a conical oblique shock originates from inside the nozzle. This is more similar to an under-expanded jet than an over-expanded jet. The boundary layer in the laminar nozzle separates due to the oblique shock originating inside the nozzle. This is in contrast to the boundary layer in the RANS simulation where it remains fully attached to the nozzle wall. BBSAN predictions based on the model formula are conducted using no modification to the BBSAN code, as turbulence values are available in the exterior region in both cases. Results are found by performing the same spatial and full wavenumber integration of the model as previously discussed. The spectral densities of both solutions are multiplied by Tcf . Figure 3.32 shows the measured SPL at various microphone positions from the PSU anechoic jet chamber, the prediction with Tam?s [58] model, and the BBSAN predictions based on the two RANS solutions. The laminar nozzle case shows slightly lower peak frequencies of BBSAN than the fully turbulent case, and the SPL is 2 to 3 dB higher at all angles. Also, the BBSAN fall-off in the laminar nozzle case is much steeper at high frequencies. These differences are due to a lower effective exit diameter of the laminar / RANS jet due to separation of the boundary layer. This lower effective diameter lowers the initial speed of the turbulent shear layer of the jet thus lowering the predicted frequencies. In addition, there are initially higher values of K and the existence of an oblique shock wave at the nozzle lip of the fully RANS solution. This is in contrast where K near the nozzle lip of the laminar / RANS jet is almost zero and no shock wave exists. This explains the magnitude difference between the two simulations as the shock wave turbulence interaction is the extra source of BBSAN in the fully RANS simulation. 157 Figure 3.32. Comparisons of BBSAN predictions for the laminar / RANS and fully RANS simulation with experiments for a Md = 1.50, Mj = 1.30, T T R = 2.20, jet at R/D = 100. 158 3.8 Helium / Air Mixture and Hot Air BBSAN predictions were performed for helium / air mixtures and hot air using the two RANS solutions previously presented in Chapter 2 Section 2.2.8. The two RANS solutions were constructed from a hot air jet operating at Md = 1.00, Mj = 1.50, and T T R = 2.20 and a helium / air mixture operating at T T R = 1.00. The helium / air jet simulates a jet with a T T R = 2.20. The local densities and Mach numbers of the two simulations are very similar. BBSAN predictions based on the model formula are conducted here with no modification to the implementation. The integration, and wavenumber ranges and indices are the same in both cases. Experimental data from Boeing used heated air at T T R = 2.20 while PSU used a helium / air mixture to simulate heating. Plots of BBSAN predictions for the jets with operating conditions of the PSU and Boeing experiments are shown in Figure 3.33. Tcf is applied to the heated air and the helium / air mixture predictions. Even though the helium / air mixture is not heated, Tcf must be applied to the simulations to be consistent because the mixture in the plenum is low density and Tcf will not be unity. The overall amplitude of both predictions, including that of the second model of Tam [58], are too high relative to the Boeing experimental data and less so relative to the PSU data. However, the magnitudes of the peaks of all three predictions are relatively the same, though higher than the experimental data. Peak frequencies of the heated prediction agree better with experimental data at observer angles perpendicular to the jet, however, the helium / air mixture prediction shows much better agreement in peak frequencies at high observer angles. Experimental data from Boeing, which is heated, shows a lower peak frequency amplitudes than the helium / air mixture experimental data of PSU. This is also reflected in the corresponding predictions. The helium / air prediction also shows slower fall-off compared to the heated case. Differences between the two predictions are partially due to the use of the Menter SST turbulence model which does not have the PAB temperature correction of the K ? model of Chien. Also, the hot air simulation uses an ideal gas model while the helium / air simulation uses a three species model of helium, nitrogen, and oxygen that does not account for other species. Furthermore, even though the two simulations have similar streamwise velocities and Mach numbers as seen in 159 Chapter 2, the maximum K in the hot air simulation is 12,847 m2 /s2 while the maximum K in the helium / air simulation is 15,047 m2 /s2 . This is primarily the reason that the BBSAN magnitudes are slightly higher for the predictions. Figure 3.33. Comparisons of BBSAN predictions for heated air and helium / air simulated jets with experiments using heated air and helium / air mixtures. The jet conditions are Md = 1.00, Mj = 1.50, T T R = 2.20, R/D = 100. 160 3.9 Turbulent Scale Coefficients The coefficients, c? , cl , and c? , have been calibrated for a Md = 1.00, Mj = 1.50, T T R = 1.00 jet at ? = 100.0 degrees. These coefficients, and associated scaling factors Pf and Tcf , have been used to make all predictions thus far in this dissertation. The same coefficients are applied to each observer angle, nozzle geometry, and jet conditions creating a true prediction scheme. The calibration jet spectral density possesses screech tones. A screech tone changes the velocity cross-correlation. Cross-correlations of screeching jets are very different from crosscorrelations of non-screeching jets. Screech is not often found when jets operate with T T R > 1.00 or on large scale engines. Since the axisymmetric Md = 1.50, Mj = 1.70, T T R = 2.20 jet does not possess a screech tone, the coefficients chosen are not applicable because of the large difference in the cross-correlation of the velocities in the shear layer. The coefficients connecting the various length and time scales to the RANS solutions for the Md = 1.50, Mj = 1.70, T T R = 2.20 jet have been recalibrated to illustrate this point. The angle chosen for calibration is ? = 100 degrees as before. Coefficients for a jet without screech are recommended to be c? = 0.85, cl = 3.00, c? = 0.30, Pf = 101.3 . These coefficients yield good predictions for a wide range of jets without screech. Figure 3.34 shows a Md = 1.50, Mj = 1.70, and T T R = 2.20 jet at R/D = 100. Experimental data from the PSU, Tam?s model, and two BBSAN predictions using the developed model are shown. The black line shows BBSAN predictions using the coefficients, c? = 1.25, cl = 3.25, c? = 0.30, that were calibrated for the screeching jet operating at Md = 1.00, Mj = 1.50, and T T R = 1.00. The predictions using the new coefficients are shown in red. Since the jet is heated the temperature correction factor has been applied to both BBSAN predictions. Using the new coefficients to predict the BBSAN shows that more precise scaling of BBSAN is captured with increasing ?, the peak BBSAN magnitudes are captured successfully, and the width of the primary and sometimes secondary BBSAN peaks matches the experimental data successfully. 161 Figure 3.34. Comparisons of BBSAN predictions using turbulence coefficients that are optimized for a non-screeching jet with experiments for Md = 1.50, Mj = 1.70, T T R = 2.20, R/D = 100. 162 In this chapter a model for the BBSAN from off-design jets has been developed. This model overcomes many of the restrictions of the models developed by HarperBourne and Fisher [1] and Tam [57] [58]. The BBSAN model has been implemented in a Fortran 90 code that directly reads the RANS solutions produced by Wind-US. These RANS solutions were validated with experiment in Chapter 2. The BBSAN program is used to calibrate the coefficients connecting the RANS solutions with the turbulent scales in the model. Various parametric studies were performed to find the optimal integration and wavenumber region ranges. These studies were performed for circular jets and applied successfully to dual stream and rectangular jets. Most importantly, the BBSAN model coefficients were only calibrated against one jet condition and observer angle and applied to many other jets as a true prediction methodology. BBSAN predictions were compared with the two models of Tam for circular, dual stream, and rectangular nozzles. All the nozzles had varying diameters. The operating conditions of the nozzles varied greatly, both in terms of N P R and T T R. The model was also used to make predictions of helium / air jets and a laminar nozzle. These results agreed favorably with corresponding heated and fully RANS simulations. Finally, modeling coefficients were suggested for jets operating without a screech tone, although, some jets without a discrete shock tone had BBSAN predictions that compared very well with experimental data. The next chapter will review the methods and results presented in this dissertation. Future developments of the BBSAN and code will be discussed. Chapter 4 Conclusion This chapter is divided into three sections. The first gives a summary of the research presented in the previous chapters. The second section draws conclusions from the findings based upon the RANS solutions and BBSAN model. The final section discusses future investigations to be completed after the publication of this dissertation. 4.1 Summary Convergent or convergent divergent nozzles that operate supersonically and offdesign contain a shock cell structure that is confined by the jet shear layer. This flow-field can be very complicated as it is often produced by nozzles with complex geometries. Three different types of noise are created by the flow-field of an offdesign supersonic jet: Mixing noise, screech, and broadband shock-associated noise (BBSAN). Mixing noise occurs due to the presence of the large and small scale turbulent structures in the shear layer, transitional region, and mixing region of the jet. Noise due to shocks consists of the discrete screech tone that is caused by a feedback loop of acoustic waves propagating upstream in the subsonic region of the shear layer and turbulence convecting downstream in the jet shear layer. This feedback loop of screech is easily disturbed and is generally not supported in hot jets. Finally, the BBSAN is due to the turbulence in the jet shear layer interacting with the shock cell structure and causes constructive and destructive sound wave interference in the far-field. It is characterized by multiple broad peaks in the 164 spectral density in the far-field. BBSAN is the dominant noise source in cold and hot jets, operating off-design, and with a wide variety of nozzle geometries, mainly in the upstream quadrant of the jet. The noise due to the presence of shocks is a cause of concern for both military personnel and civilian residents in the vicinity of an airport and for passengers in an aircraft at cruise. Various models, both empirical and semi-empirical, have been developed for BBSAN prediction. Harper-Bourne and Fisher [1] performed the first study dedicated to BBSAN and constructed the first prediction methodology in a combined experimental and numerical investigation. Their work was built on by Tam [57], who formed a stochastic model for BBSAN from circular nozzle jets operating moderately off-design and slightly heated. The models of Harper-Bourne and Fisher and Tam have significant restrictions on their application. These restrictions include the need to have cold or moderate total temperature ratios, circular and more recently dual stream axisymmetric nozzles, and operation at slightly or moderately off-design conditions. Alternatively, a LES or DNS could be used to perform a numerical simulation of the jet. Unfortunately, these methods are impractical for the foreseeable future due to their extremely high computational demands, especially when modeling jets with realistic Reynolds numbers. The need for an approach using a three-dimensional steady RANS solution is apparent so that these limitations can be overcome without expending large amounts of computer resources. Steady RANS CFD solutions obtained from many commercial, government, or university CFD solvers are fast to obtain relative to LES or DNS solutions, but still allow for any geometry and operating conditions of the nozzle to be specified. The Wind-US solver developed by the NPARC alliance is used to obtain steady flow-field solutions using the RANS equations closed by the Menter SST turbulence model. The RANS solution is the only input that the developed BBSAN model requires, thus verification of the accuracy of the RANS solutions with experiment is extremely important. Extensive Pitot probe measurements have provided experimental data for static and total pressure. Axisymmetric and rectangular jet RANS simulations at various operating conditions have been compared with these measurements. Furthermore, Mach numbers derived from the simulations were directly compared with experiment. Overall the agreement between the predicted and measured Mach number, static pressure, and total pressure measured is good. 165 The total pressure comparisons agree most favorably in all cases because no assumptions are made. This is unlike the calculation of the Mach number, where an estimation of the static pressure is required. The static pressure simulations often diverge from experiments. This is due to the shock waves generated by the probe tip reflecting off the shear layer and interfering with the downstream static pressure port. Furthermore, simulated helium / air mixtures and simulated hot air schlieren are compared directly with experimental schlieren. Generally, the spreading rate in the jet shear layer and the shock cell structure near the jet exit agreed very well with the experiment. Farther downstream from the nozzle exit the experimental and numerical data diverged to a greater degree. This is most likely due to an error in the positioning of the experimental probes far from the nozzle exit and also due to a lack of accurate turbulence models for cold turbulent shock containing jets. When jets are heated the prediction capability of the turbulence models in CFD codes are less satisfactory for calculating turbulent kinetic energy or dissipation. The Chien K ? and Menter SST K ? ? turbulence models have been used to examine the effects of the scaling of K with increasing T T R both in an overand an under-expanded jet. Even though the Chien model contains temperature corrections for jets and predicts levels of K more precisely than that of the Menter SST model, the scaling of K and u with temperature remains the same in both models. The coefficients of the RANS models available have been optimized for wall-based flows and not three-dimensional shear layers. Scaling of the turbulent kinetic energy with total temperature ratio for both the Menter SST and Chien K ? turbulence models is not appropriate for BBSAN prediction. This suggests development of a turbulence model that can improve the prediction of the flow-field of heated supersonic jets. When the Reynolds number of the jet is low, it is possible that the flow will remain laminar in the nozzle and transition to turbulent flow in the region downstream from the nozzle exit due to the Kelvin-Helmholtz and Widnall instabilities. Since this location is often unknown and difficult to predict it has been assumed for practical purposes to occur at the nozzle exit. A single simulation has been performed where the Navier-Stokes equations have been solved inside the nozzle and the RANS equations were solved outside the nozzle, thus simulating the transition. 166 Also, helium / air jets have been used to simulate heated jets. A RANS solution has been constructed that simulates both the hot air and helium / air jet for a total temperature ratio of 2.20. Theoretically, both of these CFD results should yield the same BBSAN. Finally, Harper-Bourne and Fisher observed that BBSAN intensity scales with the 4th power of the off-design parameter for cold jets. An off-design study has been performed for a series of jets using a converging-diverging nozzle by varying the off-design parameter, while keeping the total temperature ratio constant at 1.00. It has been shown that the BBSAN sources scale with the off-design parameter as observed by Harper-Bourne and Fisher. A BBSAN prediction model has been developed that requires only the specification of the operating conditions and geometry of the nozzle. The RANS solution is the only input that the BBSAN model implementation requires. A vector Green?s function is used to find the acoustic pressure in the far-field from the compressible linearized Euler equations. The final prediction formula involves a spatial and wavenumber integration over the BBSAN sources. It does not possess many of the restrictions of the Harper-Bourne and Fisher and Tam models. Namely, it can be applied to any kind of jet flow-field, nozzle shape, off-design parameter, or temperature ratio. An exponential / Gaussian based model for the two-point cross-correlation of the velocity perturbation of the turbulence has been used. The model does require the knowledge of the turbulent length and time scales as a function of spatial position. These values are linked to the RANS CFD simulation by simple formulas and scaling coefficients. The BBSAN model has been implemented in a Fortran 90 computer program. The program has been used to conduct parametric studies of the integration and wavenumber regions, along with grid density studies of these regions, to find grid independent solutions. The model has been calibrated by adjusting the scaling coefficients that represent the relevant turbulent length and times scales at one operating condition only. Furthermore, many simulations have been performed to find the coefficients relating the turbulence scales for the model from the Menter SST turbulence model. This has yielded accurate results for a wide range of experimental conditions. These coefficients are only calibrated against the single operating condition of a cold under-expanded jet. The coefficients of the turbulence scales have not been optimized for a wide range of jet conditions. This optimization 167 has not been performed in order to help better illustrate the model?s capability. A wide range of calculations of single stream jets with various Md , Mj , and T T R have been chosen to test the developed model and corresponding implementation. Selective integration over small ranges of the wavenumber region has shown how the peaks of the wavenumber spectrum contribute to different peaks in the predicted BBSAN spectral density. The dominant peak of the wavenumber spectrum corresponds in frequency to the dominant broadband peak of the BBSAN spectrum adjusted by the Doppler factor. The predicted absolute levels for unheated under-expanded jets are slightly low in the calculations. The effect of jet heating is to increase the noise levels, for the same N P R, by 5-6 dB for a T T R = 2.20. This can be associated with a jet velocity scaling given by u3j , as seen in the noise prediction formula. However, the measured experimental changes are of the order of 1 dB. A temperature correction factor suggested by Tam [58] has been implemented for all predictions, heated and unheated. This adjustment does not change the spectral density shape. A limited number of dual stream calculations have been performed. This is partly due to the limited amount of experimental data and only a single set of dual stream nozzle geometry available to the author. However, the single stream axisymmetric implementation of the BBSAN model applied to the dual stream cases with multiple integration and wavenumber regions is shown to give initially satisfactory results. However, the high speed co-flow can lower the predicted BBSAN peak frequencies below those observed in the experiment. This could be corrected by numerically evaluating the Green?s functions based on the RANS solution to account for sound propagation through the high speed secondary stream. The general form of the BBSAN model developed is fully three dimensional with wavenumber integration in the streamwise direction only. This three-dimensional version of the model, like the axisymmetric one, supports multiple integration regions. The RANS solutions of the rectangular jets have been conducted by taking advantage of nozzle symmetry about the major- and minor-axes. Therefore, the integration region of the BBSAN model in the predictions presented here only uses one fourth of the flow-field. The integration and wavenumber regions may have ranges across the entire BBSAN source region or exist only in one fourth of the flowfield. The rectangular jet BBSAN predictions compare very favorably in both the 168 over- and under-expanded cold jet calculations. However, as for the axisymmetric jet predictions, the over-expanded BBSAN peaks are slightly low relative to the under-expanded counterparts. When the same rectangular jet conditions are used but with a heating of T T R = 2.20, the same over-prediction of the magnitudes is apparent as in the axisymmetric cases. However, with the temperature correction factor, the magnitudes and spectra are in-line with experimental spectra for the hot three-dimensional calculations. 4.2 Conclusion The large number of comparisons of the RANS solutions with experimental data for off-design supersonic jets showed good agreement overall. Agreement of the results are better in the near field region of the jet as turbulent mixing plays a smaller role relative to the convective terms of the equations of motion. Far downstream from the nozzle exit the prediction capability is not as accurate, specifically with respect to the spreading rates and field variables of the equations. Fortunately, sources of BBSAN are located in the shear layer where strong shocks exist near the nozzle exit. However, the predicted RANS solutions were sufficient to develop and create a BBSAN model that is accurate. As the T T R increases from unity to larger values the BBSAN intensity increases slightly then ceases to increase. The reason that BBSAN ceases to increases with increasing T T R is unknown to the aeroacoustics community. This may be due to a possible limitation of the instability waves ability to scale with a linearly increasing T T R or the onset of the barrel shock. Nonetheless, additional experimental and numerical investigations need to be conducted to determine why the BBSAN intensity is limited with increasing T T R. The developed BBSAN model and the model developed by Tam [58] have similar BBSAN intensity scaling with increasing T T R. Unfortunately, both models display different BBSAN intensity scaling with T T R than observed in experiment. This deficiency in both models was corrected by an empirical temperature correction factor, Tcf , proposed by Tam. The developed BBSAN model has better prediction capabilities than those of Harper-Bourne and Fisher [1] and Tam [57]. Harper-Bourne and Fisher?s model does not scale in intensity with observer angle, and only applies to convergent cir- 169 cular nozzles, and with unheated under-expanded jets. Tam?s model only makes predictions for convergent or convergent divergent circular nozzles and moderately heated and moderately off-design jets. The developed BBSAN model has been shown to make accurate predictions for convergent or convergent divergent circular nozzles that operate highly off-design and with or without heating. In addition, accurate predictions were made for dual stream circular nozzles operating with heating. Rectangular convergent divergent nozzles producing over- and under-expanded jets with highly three dimensional flow fields produced accurate BBSAN predictions. Tam?s model usually underpredicted the BBSAN contribution between dominant BBSAN peaks. The developed model makes better predictions relative to experimental data between these BBSAN peaks compared to Tam?s model. The new model has no restriction on the type of nozzle geometry or associated operating conditions. Because of the lack of limitations of nozzle geometry and operating conditions, relative to the existing BBSAN models, the newly developed BBSAN model represents a major improvement of prediction capabilities. Unlike other models the developed BBSAN prediction methodology is a true prediction scheme and is not calibrated for a finite range of operating conditions. The developed BBSAN model is almost non-empirical and has no restrictions regarding nozzle geometry and operating conditions. The BBSAN model developed in this dissertation represents the only prediction method in existence that has no restrictions. 4.3 Future Work The RANS CFD solutions could be improved slightly with the use of a shock capturing scheme such as the Weighted Essentially Non-Oscillatory scheme for spatial discretization. Currently, Wind-US does not support such a scheme and the current solutions are generated with a second order upwind Roe flux vector splitting algorithm. This scheme does not preserve the discontinuities as well as a shock capturing scheme would and tends to dissipate the strength of the shock waves prematurely. Unfortunately, a turbulence model has not been developed by the scientific community to properly simulate supersonic jets, in particular, heated supersonic jets operating off-design. Since the scaling of the BBSAN amplitude with 170 respect to increasing T T R is too large, and based on the investigation comparing the PAB temperature corrections in the K ? model with no such corrections in the Menter SST turbulence model, the scaling of K is wrong. This scaling with temperature needs to be corrected in the RANS CFD solutions so that the temperature correction factor Tcf is not needed. Furthermore, improved modeling techniques will yield results that are in better agreement with the experimental flow-field Pitot and schlieren data. The effects of heating jets on BBSAN is clearly a topic for future research. It is possible that the predictions could be improved by adding a frequency dependence to the characteristic turbulent length scale l. Experimental data shows that the length scale decreases with increasing frequency while it remains constant in the model. In addition, the assumed form for the two-point cross correlation function of the turbulent velocity fluctuations is exponential / Gaussian in nature and is based on similar forms used in the prediction of turbulent mixing noise. However, the components of the turbulence that are most important for BBSAN are the large scale coherent structures. The statistical properties of these structures would be expected to show significant positive and negative peaks in their cross correlations. The effects of changes in the model for the turbulent velocity cross correlation should be examined. Furthermore, the cross-stream length scale has been assumed to be independent of azimuthal angle. The prediction formula could be further improved by allowing for a dependence on azimuthal angle, thus improving predictions for highly three-dimensional flow-fields with cross-stream turbulent length scales that vary in the azimuthal direction. This could improve the predicted magnitudes of BBSAN in rectangular jets or jets with other noncircular geometries. The order of the computational loops that evaluate the model could also be rearranged so as to speed up the process of prediction. However, this would only lower the wall-clock time for a prediction and would not improve the accuracy. A near field prediction formula should also be developed so that the model can be implemented in noise prediction codes that take into account scattering by the aircraft structures. Near field noise is also important for structural loading estimates as well as sound transmission studies for both civilian and military aircraft. A vector Green?s functions could be evaluated numerically rather than assuming 171 that propagation effects through the mean flow are negligible. This is likely to be most important for dual stream jets where the BBSAN from the primary jet must propagate through the secondary stream. This would greatly increase the prediction capability for dual stream jets. Appendix A Correlation and Sound Pressure Level A correlation measures the similarity between two signals or functions that vary with a dependent or independent variable. Finding correlation involves a spatial or temporal analysis that may be useful for finding propagation times of quantities or frequency content of fluid dynamic problems of interest. The correlation between two signals x and y can be found by finding the average value of the product of the signals and dividing by the number of products. This can be written as, Z T /2 hx (t) y (t + ? )i = lim T ?? x (t) y (t + ? ) dt (A.1) ?T /2 The correlation will be higher between two signals if they are more similar. The auto-correlation is a measure between the same signals offset by a variable amount ? , hx (t) x (t + ? )i (A.2) while the cross-correlation involves two different signals obtained at different physical locations or times. If ? = 0 then two identical signals are entirely correlated and the autocorrelation evaluates to the mean square value of the signal. Furthermore, the autocorrelation is always symmetric about the origin. If the signal or signals are scientifically random in nature then there is usually only a noticeable 173 correlation near ? = 0. Finally, if the signal or signals are periodic then the autocorrelation or cross-correlation will be periodic. In this work there is interest in both auto and cross-correlations both for modeling turbulence and finding spectral density. The mean square value of a periodic scientifically random signal may be represented by the real part of a Fourier series as, " x (t) = Re ? X # ? Cn exp (in?t) = n=1 1X (Cn exp (in?t) + Cn? exp (?in?t)) (A.3) 2 n=1 where Cn are the Fourier coefficients of the signal, n is an integer, ? is the radial frequency, and t is an independent variable such as time. The mean square value of x(t) is, 1 hx i = lim 4 T ?? 2 ZT X ? 0 ? (Cn exp (in?t) + Cn? n=1 1X Cn Cn? exp (?in?t)) dt = 2 n=1 2 (A.4) The contribution to the mean square in the frequency bandwidth, ?f , is defined as, 1 G (fn ) = Cn Cn? 2 (A.5) where G is the power spectrum of the signal. The power spectral density, S, is defined as the power spectrum divided by a frequency bandwidth ?f . Substituting the definitions of G into the relations for the mean square of the signal yields, ? 1X hx i = S (fn ) ?f 2 n=1 2 (A.6) If the limit of the frequency bandwidth goes to zero then the mean square value of the signal is, 2 Z hx i = ? S (f ) df (A.7) 0 The advantage of using the spectral density instead of the power spectrum 174 is that the value of the spectral density is the same as the frequency bandwidth varies. The spectral density can also be easily written as a summation of its Fourier coefficients, ? 1X S (?i ) = Cn Cn? ? (? ? ?i ) 2 n=1 (A.8) It is useful in the development of the BBSAN model to express the spectral density in terms of the autocorrelation of a signal. The inverse Fourier transform is, ? Z X (f ) exp (2?if (t + ? )) df x (t + ? ) = (A.9) ?? and the autocorrelation is, 1 R (? ) =< x (t) x (t + ? ) >= lim T ?? T Z 1 R (? ) = lim T ?? T ? Z R (? ) = ?? x (t) x (t + ? ) dt (A.10) ?? X (f ) exp (2?if (t + ? )) df dt (A.11) ?? ?? ? ? ? x (t) Z Z 1 ? lim X (f ) X (f ) exp (2?if t) df T ?? T (A.12) because, 1 hx i = lim T ?? T 2 Z T /2 Z 2 x (t) dt = ?T /2 ? 1 X (f ) X ? (f ) df T ?? T ?? lim (A.13) we obtain, 1 X (f ) X ? (f ) T ?? T S (f ) = lim (A.14) This expression can be substituted directly into the auto correlation R (? ) to obtain, Z ? R (t) = S (f ) exp (2?if t) df (A.15) ?? This Equation shows that the inverse Fourier transform of the Spectral density is the auto correlation. The inverse of this operation shows that the spectral 175 density is the forward Fourier transform of the autocorrelation, Z ? R (t) exp (?2?if t) dt S (f ) = (A.16) ?? The equations which relate the Spectral density and the auto-correlation are called the Wiener-Khintchine equations. These also hold for cross-correlations. If the signal is the acoustic pressure in Pascals, then the dimensions of the autocorrelation are Pascal squared. If the forward Fourier transform is performed and the variable of integration is time, then the power spectral density obtains units of Pascals squared per second. So, the dimensions of the power spectral density is mean square pressure per frequency bandwidth. The frequency bandwidth in this dissertation is one Hertz. Therefore, the dimensions of power spectral density is mean square pressure per Hertz. In this work and those of Harper-Bourne and Fisher and Tam, the ultimate goal of a BBSAN noise model is to produce an equation that is in terms of spectral density per unit Hz. Often, aeroacoustic results are presented in Sound Pressure level, as those that are shown in Chapter 3. To convert spectral density, S(x, ?) [p2 /Hz] SPL per unit Hz, SP Lf = 20 log10 ! p S(x, ?) pref (A.17) where SP Lf is the sound pressure level in decibels per Hz, and pref = 2.0 О 10?5 is the reference pressure in pascals. Results produced in Chapter 3 are shown in SP L per unit St, which requires an additional term applied to the previous equation to preserve the energy in the spectral density, SP L = 20 log10 ! p S(x, ?) + 10log10 (fc ) pref (A.18) where fc is the characteristic frequency of the jet, uj /Dj . This also requires the frequency to be changed to St, St = f /fc (A.19) now, plots can be made of SP L per unit St by placing St on the x-axis and SP L on 176 the y-axis. To convert SP Lf to a sound pressure level other than unit frequency, SP Lf b = SP Lf + 10 log10 (?f ) where ?f is the frequency bandwidth. (A.20) Appendix B Model Integrations B.1 Integration of I?? The integral that needs to be evaluated is, I?? Z? Z? i? 2 2 sin ? cos ? = exp ?? /l? ? a? ?? ?? i? 2 2 О exp ?? /l? ? sin ? sin ?? d?d? a? (B.1) First, integrating over ? with the known integral and taking into account the limits of ?? to ?, Z? ? 2 2 exp ?p x ▒ qx dx = 2 ? q exp p 4p2 (B.2) ?? yields an integral only over ? that may be evaluated as, Z? exp ?? ?? 2 2 /l? 2 ? i? ?? 2 sin2 ? cos2 ?l? ? sin ? cos ?? d? = l? ? exp a? 4a2? and the integral with corresponding limits of I?? yields, (B.3) 178 I?? = B.2 2 ?l? 2 ?? 2 sin2 ?l? exp 4a2? (B.4) Integration of I?? The integral that needs to be evaluated is, Z? Z? I?? = ?? ?? " # ? |? | (? ? u?c ? )2 i? cos ?? exp ? + ik? d?d? + i?? ? ?s l2 a? (B.5) Expanding the integrand, Z? Z? I?? = ?? ?? i? cos ?? ? 2 2u?c ? ? u?2c ? 2 |? | exp ik? ? ? 2 + 2 ? 2 + i?? ? d? d? (B.6) a? l l l ?s Performing the integration with respect to ? first with the known integral (integration tables), Z? ? 2 2 exp ?p x ▒ qx dx = 2 ? q exp p 4p2 (B.7) ?? So, Z? i? cos ?? ? 2 2u?c ? ? exp ik? ? ? 2 + 2 d? a? l l ?? " 2 # ? l2 ? cos ? 2u?c ? = ?l exp i k? + 2 4 a? l " # 2 u? ? 2 ? ?l2 ? cos ? ? cos ? c = ?l exp k? + iu?c k ? ?+ 4 a? a? l (B.8) Integration of ? is now complete and now ? must be integrated after substituting the result from the integration of ?, 179 Z? |? | ? cos ? +? ? ? d? exp i u? k ? a? ?s ?? Z0 = ?? Z? + ? ? cos ? exp + i u?c k ? + ? ? d? ?s a? ?? ? cos ? exp + i u?c k ? + ? ? d? ?s a? 0 = + = (B.9) 1 1 ?s h h + i u?c k ? ? cos ? a? ? cos ? a? +? i +? i 1 1 ?s ? i u?c k ? 1 1 ?s2 h + u?c k ? ? cos ? a? +? i2 Thus, ?s2 ? ?l exp ?l2 4 h 1 + u?c k ? k? ? cos ? a? ? cos ? a? +? 2 i2 (B.10) ?s2 Finally, simplifying the equation and reducing u?c /a? to M yields, ? 2 ?l?s exp I?? = ?l2 4 k? ? cos ? a? 2 1 + [1 ? M cos ? + u?c k/?]2 ? 2 ?s2 (B.11) Appendix C Green?s Function Vector Green?s functions must be developed to allow the formation of the solution for the time variation of acoustic pressure in the far-field relative to the jet plume. It is assumed that the Green?s function is well approximated by the no flow solution. This is a relatively good assumption because BBSAN emits primarily in the crossstream direction relative to the jet and the BBSAN sources are location roughly in the middle of the shear layer, so only half of the shear layer has an effect on its emission direction. Starting with, ?i??gn + n ?vgi = ? x ? y ?0n ?xi (C.1) ??gn = ? x ? y ?in ?xi (C.2) n ?vgi = ?i?? x ? y ?0n ?xi (C.3) n ?i?vgi + a2? is, ?? 2 ?gn ? i? and ?i? n ?vgi ? 2 ?gn ? + a2? = ? x ? y ?in 2 ?xi ?xi ?xi If Equations C.3 and C.4 are combined then, (C.4) 181 a2? ? 2 ?gn ? 2 n + ? ? = i?? x ? x ? y ? + ? y ?in 0n g ?x2i ?xi (C.5) or ? 2 ?gn ? 2 ?gn 1 ? i? + 2 = 2 ? x ? y ?0n + 2 ? x ? y ?in = R x, y, ? 2 ?xi a? a? a? ?xi (C.6) Now the solution to the problem, ?2 ? 2 g (x, z, ?) + g (x, z, ?) = ? (x ? z) ?x2i a2? (C.7) Is given by, g (x, z, ?) = ? exp [i? |x ? z| /a? ] 4? |x ? z| (C.8) ?gn may now be formed and simplified as, ?gn Z ? Z ? Z ? g (x, z, ?)R x, y, ? dz ?? ?? ?? Z Z Z i? ? ? ? = 2 g (x, z, ?)? z ? y ?0n dz a ?? ?? ?? Z ??Z ? Z ? 1 ? + 2 ? z ? y ?in dz g (x, z, ?) a? ?? ?? ?? ?zi Z ?Z ?Z ? i? 1 ? = 2 g (x, z, ?) ?0n 2 [g (x, z, ?)] ? z ? y ?in dz a? a? ?? ?? ?? ?zi x, y, ? = i? 1 ? i?gn x, y, ? = 2 g x, y, ? ?0n ? 2 g x, y, ? ?in a? a? ?yi (C.9) (C.10) consider, ? ? exp [i?R/a? ] g x, y, ? = ? ?yi ?yi 4?R p R = x ? y = (xi ? yi ) (xi ? yi ) So, (C.11) 182 ?R ? (xi ? yi ) = ?yi R (C.12) then, ? g x, y, ? = ?yi ?1 i? + 2 4?R a? 4?R (xi ? yi ) exp [i?R/a? ] R (C.13) In the far-field, for this gives, i? xi ? g x, y, ? = exp [i?R/a? ] ?yi 4?a? R R (C.14) Thus, ?i? exp [i?R/a? ] 4?a2? R (C.15) i? xi exp [i?R/a? ] 4?a3? R R (C.16) ?g0 x, y, ? = ?gn x, y, ? = Using these Green?s function results, ?gn x, y, ? = i? xn x ? y /a? exp i? 4?a3? x x (C.17) From definition, ?gn x, y, ?? = ?gn? x, y, ? (C.18) and, p p x ? y = (xi ? yi ) (xi ? yi ) = xi xi ? 2xi yi + yi yi r xi y i y i y i =x 1?2 2 + 2 x x xиy 'x? x Following a similar analysis, xиy xи? x ? y ? ? ' x ? ? x x Yields, (C.19) (C.20) 183 ?gm x, y + ?, ? = ?gm ?x и? x, y, ? exp ?i co x (C.21) ?gn? x, y, ? ?gm x, y, ? may now be written as, ?gn? x, y, ? ?gm x, y, ? i? xm ?i? xn x ? y /a? x ? y /a? exp ?i? exp i? = 4?a3? x x 4?a3? x x xn xm ?2 = 2 6 2 16? a? x x2 and substituted in for the vector Green?s functions. (C.22) Bibliography [1] Harper-Bourne, M. and M. J. Fisher (1973) ?The noise from shockwaves in supersonic jets,? AGARD, 131, pp. 1?13. [2] Burns, W. (1973) Noise and Man, 2 ed., Lippincott, Philadelphia. [3] Darden, C. M., E. D. Olson, and E. W. Shields (1993) ?Elements of NASA?s High-Speed Research Program,? AIAA Journal. [4] Stitt, L. E. (1990) ?Exhaust Nozzle for Propulsion Systems with Emphasis on Supersonic Cruise Aircraft,? NASA RP-1235. [5] FAA (1990) ?DOT/FAA Noise Standards: Aircraft Type and Airworthiness Certifcation,? FAR Part 39. [6] Golub, R. A. and J. W. Posey (2003) ?Aircraft System Noise Prediction: Past, Present and Future,? Journal of the Acoustical Society of America, 114(4), pp. 2244?2245. [7] Zorumski, W. E. (1982) ?Aircraft noise prediction program. Theoretical manual. Parts 1 and 2,? NASA TM 83199. [8] SAE (1994) ?SAE ARP876 Revision D, Gas turbine jet exhaust noise prediction,? SAE International. [9] Paliath, U. and P. J. Morris (2005-3096) ?Prediction of Jet Noise from Circular Beveled Nozzles,? AIAA. [10] Shur, M. L., P. R. Spalart, M. K. Strelets, and A. V. Garbaruk (2006-0485) ?Further Steps in LES-Based Noise Prediction for Complex Jets,? AIAA. [11] Reynolds, O. (1883) ?An experimental investigation of the circumstances which determine whether the motion of water in parallel channels shall be 185 direct or sinuous and of the law of resistance in parallel channels,? Philos. Trans. R. Soc, 174, p. 935 82. [12] ??? (1895) ?On the dynamical theory of incompressible viscous fluids and the determination of the criterion,? Philos. Trans. R. Soc, 186, p. 123 164. [13] Anderson, J. D. (1990) Modern Compressible Flow, 2 ed., McGraw-Hill Publishing Company. [14] Meyer, T. (1908) ?ber Zweidimensionale Bewegungsvorgnge in einen Gas, das mit berschallgeschwindigkeit strmt (on Two-dimensional motion in a gas that is flowing at supersonic speed),? VDI-Heft, 62. [15] Pack, D. C. (1950) ?A note on Prandtl?s Formula for the Wavelength of a Supersonic Gas Jet,? Quarterly Journal Mech. Applied Mathematics, pp. 173?181. [16] Kelvin, L. (1871) ?Hydrokinetic Solutions and Observations,? Philosophical Magazine, 4(42), pp. 362?377. [17] Widnall, S. E. and J. P. Sullivan (1973) ?On the Stability of Vortex Rings,? Proc. R. Soc. Lond. A., 332, pp. 335?353. [18] Ffowcs-Williams, J. E. (1969) ?Hydrodynamic Noise,? Annual Review of Fluid Mechanics, 1, pp. 197?222. [19] ??? (1977) ?Aeroacoustics,? Annual Review of Fluid Mechanics, 9, pp. 447?68. [20] Goldstein, M. E. (1984) ?Aeroacoustics of Turbulent Shear Flows,? Annual Review of Fluid Mechanics, 16, pp. 263?285. [21] Tam, C. K. W. (1995) ?Supersonic Jet Noise,? Annual Review of Fluid Mechanics, 27, pp. 17?43. [22] Yu, J. C. and D. S. Dosanjh (1972) ?Noise Field of a Supersonic Mach 1.5 Cold Model Jet,? Journal of the Acoustical Society of America, 51, pp. 1400?1410. [23] Seiner, J. M., M. K. Ponton, B. J. Jansen, and N. T. Lagen (1992) ?The Effect of Temperature on Supersonic Jet Noise Emission,? AIAA 92-02046. [24] Lighthill, M. J. (1952) ?On Sound generated Aerodynamically I General Theory,? Processions of the Royal Society of London Series A, 211, pp. 564? 587. 186 [25] ??? (1954) ?On Sound Generated Aerodynamically II Turbulence as a Source of Sound,? Processions of the Royal Society of London Series A, 222, pp. 1?32. [26] Raizada, N. and P. J. Morris (2006) ?Prediction of Noise from High Speed Subsonic Jets using an Acoustic Analogy,? AIAA 2006-2596. [27] Plaschko, P. (1981) ?Stochastic Model Theory for Coherent Turbulent Structures in Circular Jets,? Physics of Fluids, 24, pp. 187?193. [28] ??? (1983) ?Axial Coherence Functions of Circular Turbulent Jets Based on an Inviscid Calculation of Damped Modes,? Physics of Fluids, 26, pp. 2368?2872. [29] Morris, P. J., M. G. Giridharan, and G. M. Lilley (1990) ?The Turbulent Mixing of Compressible Free Shear Layers,? Procession of the Royal Society of London, pp. 219?243. [30] Liou, W. W. and P. J. Morris (1992) ?Weakly Non-Linear Models for Turbulent Mixing in a Plane Mixing Layer,? Physics of Fluids, 4(2798-2808). [31] Tam, C. K. W. and D. E. Burton (1984) ?Sound generated by Instability Waves of Supersonic Flows. Part 1 Two-Dimensional Mixing Layers; Part 2 Axisymmetric Jets,? Journal of Fluid Mechanics, 138, pp. 249?295. [32] McLaughlin, D. K., W. D. Barron, and A. R. Vaddempudo (1992) ?Acoustic Properties of Supersonic Helium/Air Jets at Low Reynolds Number,? AIAA 92-02-047. [33] Morris, P. J. (2009) ?A Note on Noise Generation by Large Scale Turbulent Structures in Subsonic and Supersonic jets,? International Journal of Aeroacoustics, 8(4), pp. 301?315. [34] Tam, C. K. W. (2009) ?Mach Wave Radiation from High-Speed Jets,? AIAA Paper 2009-13. [35] Powell, A. (1953) ?On the Mechanism of Chocked Jet Noise,? Proc. Phys. Soc. London, 66, pp. 1039?1056. [36] ??? (1953) ?The Noise of Chocked Jets,? Journal of the Acoustical Society of America, 25, pp. 385?389. [37] Davies, M. G. and D. E. S. Oldfield (1962) ?Tones from a Chocked Axisymmetric Jet II: The Self-Excited Loop and Mode of Oscillation,? Acustica, 12, pp. 267?277. 187 [38] Sherman, P. M. and D. R. Glass (1976) ?Jet Flow Field During Screech,? Applied Scientific Research, 32(3), pp. 283?303. [39] Norum, T. D. (1983) ?Screech Suppression in Supersonic Jets,? AIAA Journal, 21, pp. 235?240. [40] Rosfjord, T. J. and H. L. Toms (1975) ?Recent Observations Including Temperature Dependence of Axisymmetricjet Screech,? AIAA Journal, 13(10), pp. 1384?138. [41] Fisher, M. J. and C. L. Morfey (1976) ?Jet noise.? AGARD Lecture Series, 80. [42] Tam, C. K. W., J. M. Seiner, and J. C. Yu (1986) ?Proposed Relationship between Broadband Shock Associated Noise and Screech Tones,? Journal of Sound and Vibration, 110, pp. 309?321. [43] Panda, J. (1999) ?An Experimental Investigation of Screech Noise Generation,? Journal of Fluid Mechanics, 378, pp. 71?96. [44] Tanna, H. K. (1977) ?An Experimental Study of Jet Noise. Part 2: Shock Associated Noise,? Journal of Sound and Vibration, 50(3), pp. 429?444. [45] Seiner, J. M. and T. D. Norum (1979) ?Experiments on Shock Associated Noise of Supersonic Jets,? AIAA Paper 79-1526. [46] ??? (1980) ?Aerodynamic Aspects of Shock-Containing Plumes,? AIAA Paper 80-0965. [47] Norum, T. D. and J. M. Seiner (1982) ?Measurements of Static Pressure and Far Field Acoustics of Shock-Containing Supersonic Jets,? NASA TM 84521. [48] ??? (1982) ?Broadband Shock Noise from Supersonic Jets,? AIAA Journal, 20, pp. 68?73. [49] Tam, C. K. W. and H. K. Tanna (1982) ?Shock-Associated Noise of Supersonic Jets from Convergent-Divergent Nozzles,? J. Sound Vib., 81(3), p. 337358. [50] Seiner, J. M. (1984) ?Advances in High Speed Jet Aeroacoustics,? AIAA Paper 84-2275. [51] Seiner, J. M. and J. C. Yu (1984) ?Acoustic Near-Field Properties Associated with Broadband Shock Noise,? AIAA Journal, 22, pp. 1207?1215. 188 [52] Yamamoto, K., J. F. Brausch, B. A. Janardan, D. J. Hoerst, A. O. Price, and P. R. Knott (1984) ?Experimental Investigation of Shock-Cell Noise Reduction for Single-Stream Nozzles in Simulated Flight,? Test Nozzles and Acoustic Data, Comprehensive Data Report, Vol. 1, NACA CR-168234. [53] Viswanathan, K. (2006) ?Scaling Laws and a Method for Identifying Components of Jet Noise,? AIAA Journal, 44(10), pp. 2274?2285. [54] Bridges, J. (2009) ?Broadband Shock Noise in Internally-Mixed DualStream Jets,? AIAA Paper 2009-3210. [55] Mani, R., T. F. Balsa, P. R. Gliebe, R. A. Kantola, and E. J. Stringas (1977) ?High Velocity Jet Noise Source Location and Reduction. Task 2. Theoretical Developments and Basic Experiments,? FAA-RD-76-79-2. [56] Stone, J. R. (2004) ?Jet Noise Modeling for Supersonic Business Jet Application,? NASA CR-2004-212984. [57] Tam, C. K. W. (1987) ?Stochastic Model Theory of Broadband Shock Associated Noise from Supersonic Jets,? Journal of Sound and Vibration, 116(2), pp. 265?302. [58] ??? (1990) ?Broadband Shock-Associated Noise of Moderatley Imperfectly Expanded Supersonic Jets,? Journal of Sound and Vibration, 140(1), pp. 55? 71. [59] Kim, C. M., E. A. Krejsa, and A. Khavaran (1992) ?A Survey of the Broadband Shock Associated Noise Prediction Methods,? NASA Technical Memorandum 105365. [60] Davies, P. O. A. L., M. J. Fisher, and M. J. Barratt (1963) ?The Characteristics of the Turbulence in the Mixing Region of a Round Jet,? Journal of Fluid Mechanics, 15(3), pp. 33?367. [61] Tam, C. K. W. and K. C. Chen (1979) ?A Statistical Model of Turbulence in Two-Dimensional Mixing Layers,? Journal of Fluid Mechanics, 92, pp. 303?326. [62] Van Dyke, M. (1975) Perturbation Methods in Fluid Mechanics, Parabolic Press. [63] Ferziger, J. H. and M. Peric (2002) Computational Methods for Fluid Dynamics, Springer. [64] Nelson, C. and G. Power (2001) ?The NPARC Alliance Flow Simulation System,? AIAA Paper 2001-0594. 189 [65] Schlichting, H. and K. Gersten (2000) Boundary Layer Theory, 8 ed., Springer. [66] Kolmogorov, A. N. (1941) ?The local structure of turbulence in incompressible viscous fluid for very large Reynolds numbers,? Proceedings of the Royal Society of London, Series A: Mathematical and Physical Sciences, 434, pp. 9?17. [67] Tannehill, J. C., D. A. Anderson, and R. H. Pletcher (1997) Computational Fluid Mechanics and Heat Transfer, Taylor and Francis. [68] Wilcox, D. C. (2006) A Comprehensive Introduction to Turbulence Modeling for CFD, DCW Industries. [69] ??? (1988) ?Reassessment of the Scale-Determining Equation for Advanced Turbulence Models,? AIAA Journal, 26(11), pp. 1299 ? 1310. [70] Menter, F. R. (1994) ?Two-Equation Eddy-Viscosity Turbulence Models for Engineering Applications,? AIAA Journal, 32(8), pp. 1598 ? 1605. [71] Chien, K. Y. (1982) ?Predictions of Channel and Boundary-Layer Flows with a Low Reynolds Number Turbulence Model,? AIAA Journal, 20(1), pp. 33?38. [72] NPARC Alliance (2008) Wind-US User?s Guide. [73] Runge, C. (1895) Math. Ann., 46(167). [74] Message Passing Interface Forum (2008) MPI: A Message Passing Interface Standard, 2.1 ed. [75] Sunderam, V. S. (1990) ?PVM: A Framework for Parallel Distributed Computing,? Concurrency: Practice and Experience, 2(4), pp. 315?339. [76] The NPARC Alliance (2008) CFPOST User?s Guide. [77] Walatka, P., P. G. Buning, L. Pierce, and P. A. Elson (1990) ?PLOT3D User?s Manual,? NASA Technical Memorandum 101067. [78] Georgiadis, N. J., D. A. Yoder, and W. A. Engblom (2006) ?Evaluation of Modified Two-Equation Turbulence Models for Jet Flow Predictions,? 44th AIAA Aerospace Sciences Meeting and Exhibit. [79] Abdol-Hamid, K., S. Pao, S. Massey, and A. Elmiligui (2003) ?Temperature Corrected Turbulence Model for High Temperature Jet Flow,? AIAA Paper 2003-4070. 190 [80] Doty, M. J. and D. K. McLaughlin (2003) ?Acoustic and Mean Flow Measurements of High-Speed, Helium/Air Mixture Jets,? International Journal of Aeroacoustics, 2(2), pp. 293?334. [81] Miller, S. A. E., J. Veltin, P. J. Morris, and D. K. McLaughlin (2008) ?Validation of Computational Fluid Dynamics for Supersonic Shock Containing Jets,? AIAA/CEAS Aeroacoustics Conference and Exhibit. [82] Veltin, J. and D. K. McLaughlin (2009) ?Flow Field and Acoustic Measurements of Rectangular Supersonic Jets,? AIAA Aerosciences Meeting and Aerospace Exposition 2009-19. [83] Settles, G. S. (2001) Schlieren and shadowgraph techniques: Visualizing phenomena in transparent media, Springer-Verlag. [84] Veltin, J. and D. K. McLaughlin (2008) ?Noise mechanisms investigation in shock containing screeching jets using optical deflectometry,? AIAA Paper 2008-2889. [85] Tam, C. K. W. and L. Auriault (1998) ?Mean flow refraction effects on sound radiated from localized sources in a jet,? Journal of Fluid Mechanics, 370, pp. 149?174. [86] ISO/IEC 1539-1:1997. Information technology Programming languages Fortran Part 1: Base language. [87] Shepard, D. (1968) ?A Two-Dimensional Interpolation Function for Irregularly-Spaced Data,? in Proceedings of the 1968 23rd ACM national conference, ACM, New York, NY, USA, pp. 517?524. [88] Li, J. and C. S. Chen (2002) ?A Simple Efficient Algorithm for Interpolation Between Different Grids in Both 2D and 3D,? Mathematics and Computers in Simulation, 58(2), pp. 125?132. [89] Blackman, R. B. and J. Tukey (1959) The Measurement of Power Spectra, From the Point of View of Communications Engineering, Dover. [90] Ferziger, J. H. (1998) Numerical Methods for Engineering Applications, 2 ed., Wiley-Interscience. [91] Ferguson, W. (1979) ?A Simple Derivation of Glassmans?s General N Fast Fourier Transform,? MRC Tech. Summ. Rept. 2029. [92] Goss, A. E., J. Veltin, J. Lee, and D. K. McLaughlin (2009) ?Acoustic Measurements of High-Speed Jets from Rectangular Nozzle with Thrust Vectoring,? AIAA Journal, 47(6), pp. 1482?1506. Vita Steven Arthur Eric Miller Steven Arthur Eric Miller attended Michigan State University, studied Mechanical Engineering, and subsequently earned his Bachelor of Science in December 2003. While at Michigan State University he performed undergraduate research in Aerodynamics and Aeroacoustics under Professor Mei Zhuang. He began graduate studies in Aerospace Engineering at The Pennsylvania State University in January 2004 under the guidance of Boeing/A.D. Welliver Professor Philip J. Morris. He was awarded his Master of Science degree in May 2006 with a thesis entitled, ?The Aerodynamics of Wind Turbine Airfoils.? Immediately after earning his Master of Science he began work on his Doctor of Philosophy in Aerospace Engineering under the same adviser. In addition to research activities at The Pennsylvania State University he acted as a teaching assistant or an instructor for a number of Aerospace Engineering courses. After completion, he will accept a research position at NASA Langley and continue research in the field of Aeroacoustics. ecomes quite extreme at high and low observer angles. The peak frequencies of the predictions could be improved by using a better approximation of the vector Green?s function to include secondary flow effects. The BBSAN predicted amplitudes compare very well with experiment. At high observer angles of Figure 3.22 there are regions where no prediction is present near St = 0.50. This is in contrast with the results of Figure 3.23 where there is a BBSAN contribution across the same St range. The higher speed secondary stream of Figure 3.22 causes a stronger Doppler shift to the predicted BBSAN and shifts the peaks to lower frequency. The prediction could be improved by lowering c? which would have the effect of broadening the BBSAN peaks, thus eliminating the region where no BBSAN is predicted. 142 Mdp 1.00 1.00 Mjp 1.19 1.19 Mds 1.00 1.00 Mjs 0.96 1.04 T T Rp 2.70 2.70 T T Rs 1.00 1.00 Dp (m) 0.0622 0.0622 fcp 9398 9398 ?p 0.645 0.645 ?s N/A 0.578 Table 3.5. Jet operating conditions for the RANS CFD and BBSAN predictions of the dual stream jets. Figure 3.21. Integration regions for the BBSAN calculation of the dualstream jet Mdp = 1.00, Mjp = 1.19, Mds = 1.00, Mjs = 1.04, T T Rp = 2.70. 143 Figure 3.22. Comparisons of BBSAN predictions with experiments for the dualstream jet Mdp = 1.00, Mjp = 1.19, Mds = 1.00, Mjs = 1.04, T T Rp = 2.70, R/D = 100. 144 Figure 3.23. Comparisons of BBSAN predictions with experiments for the dualstream jet Mdp = 1.00, Mjp = 1.19, Mds = 1.00, Mjs = 0.96, T T Rp = 2.70, R/D = 100. 145 3.6 Three Dimensional Jets Thus far axisymmetric single and dual stream CFD RANS solutions have been used to perform BBSAN predictions. These used the axisymmetric form of the BBSAN model equation and its implementation. The flow-fields produced by rectangular or non-axisymmetric jets are highly three-dimensional and are not applicable to the axisymmetric BBSAN model. Here, the BBSAN model is evaluated with the three dimensional implementation of the BBSAN code using the rectangular jet RANS solutions validated for rectangular jets in Chapter 2. Both cold rectangular jets with Md = 1.50 were validated with Pitot and schlieren comparisons. The two off-design rectangular jet simulations with T T R = 2.20 were validated by comparing schlieren of a simulated hot jet using a helium / air mixture. Table 3.6 shows the rectangular nozzle operating parameters that are used for the corresponding BBSAN predictions. The rectangular nozzle chosen has Md = 1.50, and operates either over- or under-expanded at Mj = 1.30 or Mj = 1.70 and T T R = 1.00 or T T R = 2.20. The experimental results were provided by Veltin and the experimental details are documented in Goss et al. [92]. Microphone positions were placed at 1.905 meters from the jet at various ? in the minor and major axis planes and at an azimuthal angle of 45 degrees. The experimental data was extended to 100 De to be consistent with the predictions presented in the previous sections. Only predictions are presented in the major and minor axis planes because the aspect ratio of the rectangular jet is relatively small, therefore the intensity of the BBSAN will vary only a little with respect to azimuthal angle. The characteristic frequencies of the rectangular jets are based on the fully expanded equivalent diameter found from the De and the N P R. Md 1.50 1.50 1.50 1.50 Mj 1.30 1.70 1.30 1.70 NP R 2.77 4.94 2.77 4.94 TTR 1.00 1.00 2.20 2.20 De (m) 0.01778 0.01778 0.01778 0.01778 Dej (m) 0.01693 0.01896 0.01693 0.01896 fc 22888 24499 33792 36334 ? 0.748 0.800 0.748 0.800 Tj (K) 219.1 185.8 482.0 408.7 uj (m/s) 385.7 464.5 572.1 688.9 Table 3.6. Jet operating conditions for the RANS CFD and BBSAN predictions of the 3D calculations. 146 Unlike the axisymmetric version of the code, the three-dimensional BBSAN implementation evaluates the integrals over the three dimensional integration region and wavenumber region. Since there is an additional integral that needs to be evaluated in the z-axis direction, the run time of the computer code can increase greatly. To minimize the amount of time needed to make a three-dimensional BBSAN prediction, careful choices are made of the range and number of indices of the integration and wavenumber regions. Based on the conclusions for the integration studies of the axisymmetric jet shown in Table 3.3, integration ranges for the four rectangular cases are chosen as 0.001m < x < 0.1437m, 0.001m < y < 0.011m, and 0.001m < z < 0.021m with corresponding index values of 300, 75, and 75 in the streamwise, minor-axis, and major-axis directions respectively. Since the RANS CFD simulations uses planes of symmetry of the nozzle, the implemented BBSAN code automatically integrates across the planes. There is no restriction in the implemented version of the BBSAN model regarding the flow-field, even though planes of symmetry were used to construct the RANS solution. The same implementation of BBSAN may be applied to any three-dimensional jet flow-field. The range of the wavenumber region is chosen as 0.001m < xk < 0.250 m with 512 indices. Summation of the integration region is selective by only including contributions from 10% of K using cycling. This cycling technique saves a large amount of computational time just as it had in the axisymmetric calculations. Also, the solution is found relatively quickly by specifying a limited range for the integrations over wavenumber. The wavenumber integration range is carefully chosen to only encompass peak values of the wavenumber spectrum. For example, in the first rectangular BBSAN condition of Table 3.6, the spacing between each value of wavenumber in the wavenumber spectrum is ?k = 12.57 and only wavenumber contributions are used in the range of ?25 < k < ?166 and 25 < k < 166. BBSAN predictions of the over-expanded cold rectangular jet in the major and minor axis planes are shown in Figures 3.24 and 3.25. By comparing the BBSAN predictions with the experimental data, it is seen that only a small 2 dB difference is apparent between the BBSAN peaks in the major and minor axis. Additional strong screech tones are present in the minor axis direction. The predictions in both figures are the same as the azimuthal angle is not included in the current prediction formula. This azimuthal dependance could easily be included if necessary. The 147 peak BBSAN at all observer locations matches well with respect to frequency and magnitude. As in the circular nozzle case operating at Md = 1.50 and Mj = 1.30, the overall prediction is slightly low. The width of the BBSAN peaks is also too narrow relative to experimental measurements. Figures 3.26 and 3.27 show the under-expanded unheated rectangular jet operating at Mj = 1.70. In this case, the variation of the magnitude of the BBSAN is invariant with azimuthal angle, unlike the previous over-expanded case. The predictions at all observer angles agree extremely well with the experimental data. The first BBSAN peak and subsequent minor peaks at higher frequencies generally align with those of experiment and the amount of fall-off in the spectrum is not significant. The small high frequency peaks of the BBSAN are more pronounced in the experimental data in the major axis plane. The predictions could be slightly improved for this jet if c? is lowered to flatten out the strong dominant BBSAN peak. BBSAN predictions are performed for an over-expanded rectangular jet operating at Mj = 1.30 and T T R = 2.20. These predictions are shown in Figures 3.28 and 3.29 in the major and minor axes planes respectively. Variation of maximum dB in the experiment with azimuthal angle only varies by up to 2 dB. Like the heated axisymmetric case, the predictions are slightly lower than the experimental data. The magnitude of BBSAN matches the experiment at high observer angles, and also matches the peak frequencies. At ? = 90.0 degrees and lower the mixing noise dominates the BBSAN. The width of the peaks, as with the axisymmetric jet, are too narrow. This is especially true for low St when the prediction has extreme fall-off. Note that for this heated prediction, along with all the other rectangular jet predictions, the temperature correction factor has been applied. The final rectangular predictions are shown in Figures 3.30 and 3.31 for the major and minor axis directions respectively, for the Mj = 1.70 and T T R = 2.20 jet. The peak frequencies match those of the experiment very well at most angles except for large values. BBSAN magnitudes align well perpendicular to the jet, however, are over-predicting at high observer angles. At low observer angles the BBSAN is dominated by the mixing noise because the jet is heated. The predicted values of BBSAN do not contribute to the total spectra just as in the corresponding heated circular jet case. The variation of the peak BBSAN magnitude is partly 148 controlled by cl . If cl is smaller, then the variation of BBSAN magnitude with angle would better match the experiment. Figure 3.24. Comparisons of BBSAN predictions with experiments for the rectangular jet Md = 1.50, Mj = 1.30, T T R = 1.00, R/De = 100 in the major axis direction. 149 Figure 3.25. Comparisons of BBSAN predictions with experiments for the rectangular jet Md = 1.50, Mj = 1.30, T T R = 1.00, R/De = 100 in the minor axis direction. 150 Figure 3.26. Comparisons of BBSAN predictions with experiments for the rectangular jet Md = 1.50, Mj = 1.70, T T R = 1.00, R/De = 100 in the major axis direction. 151 Figure 3.27. Comparisons of BBSAN predictions with experiments for the rectangular jet Md = 1.50, Mj = 1.70, T T R = 1.00, R/De = 100 in the minor axis direction. 152 Figure 3.28. Comparisons of BBSAN predictions with experiments for the rectangular jet Md = 1.50, Mj = 1.30, T T R = 2.20, R/De = 100 in the major axis direction. 153 Figure 3.29. Comparisons of BBSAN predictions with experiments for the rectangular jet Md = 1.50, Mj = 1.30, T T R = 2.20, R/De = 100 in the minor axis direction. 154 Figure 3.30. Comparisons of BBSAN predictions with experiments for the rectangular jet Md = 1.50, Mj = 1.70, T T R = 2.20, R/De = 100 in the major axis direction. 155 Figure 3.31. Comparisons of BBSAN predictions with experiments for the rectangular jet Md = 1.50, Mj = 1.70, T T R = 2.20, R/De = 100 in the minor axis direction. 156 3.7 The Effect of Laminar Flow in the Nozzle BBSAN predictions have been made for the laminar / RANS and fully RANS jet flows and compared to experimental data. This required the two CFD simulations that were conducted to examine the effect of laminar or turbulent flow inside the nozzle for a Md = 1.50, Mj = 1.30, T T R = 2.20 jet. The first CFD simulation solved the RANS equations in the entire domain, and the second CFD simulations solved the Navier-Stokes equations in the interior of the nozzle and the RANS equations in the exterior. The second simulation thereby forces the flow to be laminar inside the nozzle and turbulent outside the nozzle. In the laminar / RANS case, a conical oblique shock originates from inside the nozzle. This is more similar to an under-expanded jet than an over-expanded jet. The boundary layer in the laminar nozzle separates due to the oblique shock originating inside the nozzle. This is in contrast to the boundary layer in the RANS simulation where it remains fully attached to the nozzle wall. BBSAN predictions based on the model formula are conducted using no modification to the BBSAN code, as turbulence values are available in the exterior region in both cases. Results are found by performing the same spatial and full wavenumber integration of the model as previously discussed. The spectral densities of both solutions are multiplied by Tcf . Figure 3.32 shows the measured SPL at various microphone positions from the PSU anechoic jet chamber, the prediction with Tam?s [58] model, and the BBSAN predictions based on the two RANS solutions. The laminar nozzle case shows slightly lower peak frequencies of BBSAN than the fully turbulent case, and the SPL is 2 to 3 dB higher at all angles. Also, the BBSAN fall-off in the laminar nozzle case is much steeper at high frequencies. These differences are due to a lower effective exit diameter of the laminar / RANS jet due to separation of the boundary layer. This lower effective diameter lowers the initial speed of the turbulent shear layer of the jet thus lowering the predicted frequencies. In addition, there are initially higher values of K and the existence of an oblique shock wave at the nozzle lip of the fully RANS solution. This is in contrast where K near the nozzle lip of the laminar / RANS jet is almost zero and no shock wave exists. This explains the magnitude difference between the two simulations as the shock wave turbulence interaction is the extra source of BBSAN in the fully RANS simulation. 157 Figure 3.32. Comparisons of BBSAN predictions for the laminar / RANS and fully RANS simulation with experiments for a Md = 1.50, Mj = 1.30, T T R = 2.20, jet at R/D = 100. 158 3.8 Helium / Air Mixture and Hot Air BBSAN predictions were performed for helium / air mixtures and hot air using the two RANS solutions previously presented in Chapter 2 Section 2.2.8. The two RANS solutions were constructed from a hot air jet operating at Md = 1.00, Mj = 1.50, and T T R = 2.20 and a helium / air mixture operating at T T R = 1.00. The helium / air jet simulates a jet with a T T R = 2.20. The local densities and Mach numbers of the two simulations are very similar. BBSAN predictions based on the model formula are conducted here with no modification to the implementation. The integration, and wavenumber ranges and indices are the same in both cases. Experimental data from Boeing used heated air at T T R = 2.20 while PSU used a helium / air mixture to simulate heating. Plots of BBSAN predictions for the jets with operating conditions of the PSU and Boeing experiments are shown in Figure 3.33. Tcf is applied to the heated air and the helium / air mixture predictions. Even though the helium / air mixture is not heated, Tcf must be applied to the simulations to be consistent because the mixture in the plenum is low density and Tcf will not be unity. The overall amplitude of both predictions, including that of the second model of Tam [58], are too high relative to the Boeing experimental data and less so relative to the PSU data. However, the magnitudes of the peaks of all three predictions are relatively the same, though higher than the experimental data. Peak frequencies of the heated prediction agree better with experimental data at observer angles perpendicular to the jet, however, the helium / air mixture prediction shows much better agreement in peak frequencies at high observer angles. Experimental data from Boeing, which is heated, shows a lower peak frequency amplitudes than the helium / air mixture experimental data of PSU. This is also reflected in the corresponding predictions. The helium / air prediction also shows slower fall-off compared to the heated case. Differences between the two predictions are partially due to the use of the Menter SST turbulence model which does not have the PAB temperature correction of the K ? model of Chien. Also, the hot air simulation uses an ideal gas model while the helium / air simulation uses a three species model of helium, nitrogen, and oxygen that does not account for other species. Furthermore, even though the two simulations have similar streamwise velocities and Mach numbers as seen in 159 Chapter 2, the maximum K in the hot air simulation is 12,847 m2 /s2 while the maximum K in the helium / air simulation is 15,047 m2 /s2 . This is primarily the reason that the BBSAN magnitudes are slightly higher for the predictions. Figure 3.33. Comparisons of BBSAN predictions for heated air and helium / air simulated jets with experiments using heated air and helium / air mixtures. The jet conditions are Md = 1.00, Mj = 1.50, T T R = 2.20, R/D = 100. 160 3.9 Turbulent Scale Coefficients The coefficients, c? , cl , and c? , have been calibrated for a Md = 1.00, Mj = 1.50, T T R = 1.00 jet at ? = 100.0 degrees. These coefficients, and associated scaling factors Pf and Tcf , have been used to make all predictions thus far in this dissertation. The same coefficients are applied to each observer angle, nozzle geometry, and jet conditions creating a true prediction scheme. The calibration jet spectral density possesses screech tones. A screech tone changes the velocity cross-correlation. Cross-correlations of screeching jets are very diffe

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