Dev. Chem. Eng. Mineral Process., I1 (5/6), pp. 465-479, 2003. Direct Numerical Simulation of Open Channel Flow with Buoyancy Itsuro Hondal, Naotaka Kanagawa and Yosuke Kawashima Dept. of Chemical Engineering, Himeji Institute of Technology, 21 67 Syosha, Himeji, Hyogo, Japan Free surface flow is important in chemical reactions, heat exchange, atmospheric science and industrial technology. In terms of fundamental research, turbulence near a free surface is important in transport phenomenon. However, free surface effects exhibit some nonlinear behavior, so that numerical methods are usually employed instead of analytical approaches. In this investigation, direct numerical simulation is applied to turbulent channel flow with a free surface. The Navier-Stokes equations are solved using a finite difference method, and an interpolation scheme is applied to the convective term. Grid resolution is 128x65~128and Reynolds number based on the friction velocity and channel elevation is 150. Slip wall approximations have been used extensively. This means that the free surface has no stress, velocity or deformation. In order to solve such problems, low Froude number approximations have recently been presented and these are used to calculate the free surface flow. In turbulent flow, dissipation and production are essential qualities, Reynolds-stresses balances and pressurestrain correlations are examined numerically. It is shown that the main difference between free surface and slip wall approximations is in pressurestrain correlations near the surface. *Authorfor correspondence (pitot@mech. eng.himeji-tech.ac.jp). 465 I. Honda,N.Kanagawa and Y Kawashima Introduction Free surface flows are important in thermal science, mechanical engineering and industrial technology. Most of these flows are turbulent, and have surface waves that play an important role in the interaction of materials across the surface by means of turbulent air-water interfaces and wind-wave interactions. For vertical flows near a free surface, an important interaction mechanism is when vortex lines disconnect and subsequently terminate at the free surface. In terms of fundamental research, turbulence near a free surface is important in transport phenomena. However, current knowledge of transport mechanisms near free surfaces is neither complete nor satisfactory. This is because measurement of fluctuations of turbulent quantities in a layer near a free surface is very difficult. In a various equipment, control of thermo-fluid phenomena near free surfaces is crucial in maintaining high quality operation. Since free surface effects are dominated by nonlinear behavior, numerical methods are usually employed instead of analytical approaches. In the early 1980s, advances in computer architecture made direct numerical simulation [ 13 of turbulent flows possible. Many numerical studies have investigated wall turbulence. Moin and Kim [ 11 calculated turbulent channel flow by large eddy simulations. In their calculations, 64 and 128 uniform spacing grid points were used in the streamwise and spanwise directions respectively, while 63 non-uniform spacing grid points were used in the direction normal to the wall. Most experimentally observed wall-layer structures were reproduced using this setup. Kim et al. [2] calculated the channel flow by direct numerical simulation using 192x129~160grid points. Their results have been used to clearly explain wall turbulent phenomena and reproduce statistical turbulent properties with a high degree of accuracy. Lam and Banerjee [3] treated free surfaces as a flat slip wall, giving a Froude number of zero. This method has been widely used. However, the lack of variations in the surface shape precludes accurate computation of turbulence statistics. To address this problem, Komori et al. [4] considered variations in the free surface height, giving a non-zero Froude number. They applied an upwind scheme to the convective term in the NavierStokes equations. However, adding numerical viscosity may have a significant impact on turbulence statistics. Nagaosa [5] applied a consistent scheme developed by Kajishima to the convective term to avoid this effect, but also employed the slip wall approximation. 466 Direct Numerical Simulation of Open Channel Flow with Buoyancy In this paper we study open channel flow with buoyancy using numerical techniques and the nonzero Froude number approximation proposed by Borue et al. [ 6 ] .The purpose of this investigation is to find the effects of buoyancy and Froude number on statistical characteristics of free surface flow. Therefore, we perform direct numerical simulations employing a consistent scheme and allowing free surface fluctuations. Numerical Method Channel flow can be treated as an incompressible Newtonian fluid. Governing equations are the equation of continuity and Navier-Stokes equations with Boussinesq approximation: O = 'au. h i The equations are shown in non-dimensional form. Variables in these equations are normalized to H , u , and A T . Cartesian axis, being either streamwise, normal to wall or spanwise while; S is Kronecker's delta. Nondimensional parameters in the above equations are defined as follows: The flow geometry and coordinate system of the channel is shown in Figure1 . The two walls are kept at different but constant temperatures, without fluctuations. The Reynolds number ( R e ) was set to 150. The Prandtl number (Pr ) was assumed to be 1, and the Richardson number ( Ri ) was set to 10. This value for the Prandtl number is much smaller than that for water. If the Prandtl number is larger than 1, then a large number of grid points are necessary for DNS computation. It is thought that Prandtl number effects are not significant in large-scale motion. 467 I. Honda, N. Kanagawa and I! Kawashima Figure 1. Low Froude number flow. Spatial terms excluding convective terms are approximated by a secondorder central finite difference scheme. Convective terms in the Navier-Stokes and scalar transport equations are discretized using the second-order interpolation scheme proposed by Kajishima [7] on a staggered grid: where Sxi denotes central difference and denotes mean value. Since this scheme accurately reproduces the continuity equation, results are in good agreement with those of the spectral method [B]. For integration over time, second-order Adams-Bashforth and Crank-Nicolson methods are applied to the nonlinear and viscous terms respectively [9]. 468 Direct Numerical Simulation of Open Channel Flow with Buoyancy pn+l= P“ 1 + Q - -AttS,, 2 Re s,, Q In the above equations, Q is a scalar potential and Ai is a convective term. The dynamic boundary conditions on the free surface are: h p=-+-Fr2 ah at 2 du, Re ax2 ah axl -+u,-+u3-= ah dx, u2 Grid Coupling Algorithm Time Advancement Spatial Scheme I Boundary Conditions Boundary Conditions (Temperature) Staggered Grid Fractional Step Method 2nd-OrderAdams Bashforth Method (Convective Term) 2nd-OrderCrank Nicolson Method (Viscous Term) 2nd-OrderConsistent Scheme (Convective Term) 2nd-OrderCentral Scheme (Other Terms) Periodic ( x , ,x , direction) Slip Wall ( x2direction, x2 = H ) Zero Froude ( x2direction, x 2 = H ) Low Froude ( x2 direction, x2 = H ) (I) x2 = O / T = O , x , = l / T = l (11’) x, =OIT=l,x, = 1 / T = O Fr is the Froude number, defined as Fr = u,/(gh)”’, and h is the height of the free surface above the mean height. Equation ( 1 1 ) represents a type of hyperbolic. Since the equations need to be solved to a high degree of accuracy, a third-order Adams-Bashforth method with sixth-order central discretization 469 I. Honda, N. Kanagawa and X Kawashima is applied to spatial derivatives in this study. The case of a slip-wall approximation is the Fr = 0 limit of the present problem. Boundary conditions are shown in Tablel, a fully developed flow field was assumed so that periodic boundary conditions could be applied in both streamwise and spanwise directions. Time increments in this investigation were O . O 7 5 ~ l ( u , ) ~Results . were from ~ time steps) ensemble averages over space and time taken 7 5 0 0 v l ( ~ , )(100,000 after the flow fields were judged to have reached a fully developed state. Computations were carried out on a personal computer with Pentium 4 CPU, which took approximately 4 seconds of CPU time per time-step. Results and Discussion First, channel flow was computed by applying the non-zero Froude number approximation without buoyancy. Figure 2 shows mean flow velocity as a function of vertical position compared with Nagaosa’s data, which was computed using slip-wall approximations. Figure 3 shows root mean square (r.m.s.) fluctuations in the three velocity components (u, v and w represent velocities in the streamwise xl, vertical x2 and crosswise x3 directions, respectively) as a function of vertical position. The results are in good agreement with Nagaosa’s data near the wall, but show differences near the free surface arising from the different boundary conditions. The vertical component of turbulent intensity at the free surface is zero because the boundary condition forces the vertical component of velocity (v) to be zero at the surface in both results. Figure 4 shows the flatness factor as a function of vertical position in the flow. A flatness factor of 3 means that velocity fluctuations are distributed normally. Factors greater than 3 are indicative of highly intermittent phenomena. Flatness factors of the vertical component of velocity near the free surface were larger than those found by Nagaosa. These large flatness factors seem to be the result of highly intermittent phenomena near the free surface which are allowed by the non-zero Froude number approximation. Flatness of the vertical component was found to decrease rapidly near the free surface, but in slip-wall calculations remained large at the free surface. 4 70 Direct Numerical Simulation of Open Channel Flow with Buoyancy X., Figure 2. Mean velocity. 3 +$ 1 3 0 X., Figure 3. Turbulent intensity. 471 I, Honda, N.Kanagawa and I! Kawashima 0 0 50 A 100 150 Figure 4. Flatness factors. Figure 5 shows skewness of the flow. Nagaosa found large positive skewness at the free surface in slip-wall calculations, indicating large vertical velocities. However, skewness of vertical velocity in non-zero Froude number calculations decreased sharply near the free surface, indicating that the distribution of vertical velocity fluctuations becomes the normal distribution at the free surface. Velocity fluctuations found at the surface in non-zero Froude number calculations are lower than in slip-wall calculations, because the shape of the free surface is allowed to change. Vertical skewness factors confirm highly intermittent phenomena in the slip-wall case. These results seem to indicate that streamwise vortices organize coherent structures near the surface. These vortices are transported from the lower wall. Figure 6 shows pressure-strain correlations. The streamwise component was negative throughout the channel, but the other two components were positive, indicating that streamwise turbulent energy is redistributed into vertical and spanwise turbulence through pressure-strain effects, with more energy transferred to spanwise turbulence than to vertical turbulence. While the results of non-zero Froude number calculations agree with slip-wall calculations near the free surface, vertical components differ at the surface. 4 72 Direct Numerical Simulation of Open Channel Flow with Buoyancy Figure 5. Skewness. 0.c . 0 .r .r 'Q -0.02 t t Nagaosa Present 411 422 -0-041 -0.061 0 4 33 I I I 50 - ----__... I I + 100 I I O A 0 I I 1 1 I 150 Figure 6. Pressure-Strain correlation. 4 73 I. Honda. N. Kanagawa and L Kawashima 20 'I 5 3 0 -Nagaosa 5 9 0.5 1 X1 Figure 7. Mean velocity on heated plane. Figure 8. Turbulent intensity. 4 74 + 5 10 50 100 Direct Numerical Simulation of Open Channel Flow with Buoyancy l o A, I, , 0 0 1 I I I I 50 , 1 f , I 100 1 , , , , 150 X? Figure 9. Flatness on the heated plane. We concluded that free surface deformation causes a decrease in energy of the vertical component of pressure-strain. Figure 7 shown mean velocity profiles at two different boundary temperature conditions, lower surface heating and upper surface heating, as compared to Nagaosa's results. It should be noted that the velocity profile for a heated lower surface is much larger than for a heated upper surface. The velocity profile heated under the bottom wall is smaller in the region from x; = 55 to 150 than Nagaosa's result. In the case of a heated lower surface, the friction velocity at that surface does not change. It is clear that the turbulent boundary layer becomes relaminarized. Figure 8 shows r.m.s. velocity fluctuation components for the two different surface heating schemes compared to Nagaosa's results. Components calculated across the flow in the region from x2+ = 90 to the free surface are greater than that in slip-wall calculations. It is thought that longitudinal vortices become stronger with respect to surface splatting. This tendency is also seen in flatness and skewness, as shown in Figure 9 and 10 respectively. Vertical -components of flatness decrease sharply near the surface, but are larger between x2+ = 100 and the surface. 4 75 I. Honda. N Kanagawa and X Kawashima ~ ~~~~~ Figure 10. Skewness on the heatedplane. Streamwise and vertical components of skewness in the heated lower wall case are very different from those of the slip-wall case. Calculated streamwise skewness was smaller than in slip-wall calculations, suggesting that velocity fluctuations are lower. It is thought that a low velocity core will break up the wall by buoyancy. The v component is larger than the Nagaosa result in the region from x2+ = 80 to the free surface. This is because the energy of buoyancy is transferred from the energy of u component to the v component. Figure 11 shows pressure-strain correlations. Calculated pressure-strain correlation components $ 1 1and $22 were smaller than those of channel flow. For channel flow, $ 1 1 9 2 2 are zero at the free surface, meaning that no energy redistributions exist at the surface. Streamwise pressure-strain correlation is negative overall, indicating that streamwise energy is redistributed to vertical and spanwise components. Vertical component $22 decreases rapidly near the free surface, with its sign changing from positive to negative, while spanwise component $33 increases near the surface. These distributions suggest that vertical turbulent energy is transferred into spanwise turbulence. Surface deformation derived from the pressure gradient hastens the redistribution of energy. - 4 76 Direct Numerical Simulation of Open Channel Flow with Buoyancy 0.04 Llr 'Q a - -0.06 , 0 a 'r, a - a 411 a I Nagaosa Present , 50 , - 422 ----- 433 , , - - -I - - + 100 " - I A m A a (upper)(lower) , , , , 150 Figure 11. Pressure-Strain correlation. Conclusions We have performed a series of numerical simulations of channel flow using slip-wall and low Froude number approximations. The relation between the turbulent structures and energy redistributions in open channel flows with buoyancy were investigated. Vertical velocity fluctuations of slip-wall calculations are lower than those found using low Froude number approximations at the free surface, while vertical flatness factors computed by low Froude number calculations are smaller than those for slip-wall. Simulations of two boundary temperature conditions were performed. These indicate that the flow is relaminarized by free surface heating. In the case of a heated lower wall, longitudinal vortices seem to become strong with respect to surface splatting, while energy redistributions become larger near the free surface. Furthermore, it appears from numerical results that a very thin layer near the free surface prevents fluctuations from increasing in the non-zero Froude number calculations. These results appear to be necessary to explain surface energy redistribution mechanisms. 477 I. Honda, N. Kanagowa and L Kawashima Courant number Froude number gravitational acceleration free surface height reference length space index pressure Prandtl number Reynolds number Richardson number time time increment temperature xI(x)component of velocity wall friction velocity x2 (y)component of velocity xj (w) component of velocity velocity vector Cartesian reference length divisions thermal diffusion bulk modulus of elasticity Kronecker’s delta scalar potential viscosity kinematic viscosity density shear stress References [ l ] Moin, P. and Kim, J. 1982. Numerical investigation of turbulent channel flow, J. Fluid Mech.. 118, 341-377. [2] Kim, J. et a]. 1978. Turbulent statistics in fully developed channel flow at low Reynolds number, J. Fluid Mech., 117, 133-166. [3] Lam. K. and Banerjee, S. 1992. On the condition of streak formulation in bounded turbulent flow, Phys. Fluids, 4(2), 306-320. [4] Komori, S. et al. 1993. Direct numerical simulation of three-dimensional open-channel flow with zero-shear gas-liquid interface, Phys. Fluids, 5(1). 115-125. 4 78 Direct Numerical Simulation of Open Channel Flow with Buoyancy [ 5 ] Nagaosa, R. and Saito, T. 1997. Turbulence Structure and Scalar Transfer in Stratified Free-Surface Flows, AICnE J., 43(10), 2393-2404. (61 Borue, V. et al. 1995. Interaction of surface waves with turbulence: direct numerical simulations of turbulent open-channel flow, J. Fluid Mech., 286, 1-23. [7] Kajishima. T., 1994. Conservation Properties of Finite Difference Method for Convection (in Japanese), Trans. JSME, 60(574), 2058-2063. [S] Honda, I. et al. 2002, A Study on Turbulent Channel Flow With Buoyancy, Proceeding of JSME, 24-1.31 (In Japanese) [9] Kim, J. and Moin, P. 1985. Application of a fractional-step method to incompressible Navier-Stokes equations, J. Comput. Phys., 59,308-323. 479

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