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Stochastic Forcing of the Batchelor Vortex
Z. W. Guo, C. Chen, and D. J. Sun
Citation: AIP Conference Proceedings 1376, 302 (2011);
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Published by the American Institute of Physics
Proceedings of the Sixth International Conference on Fluid Mechanics
AIP Conf. Proc. Vol 1376, 302-304 (2011)
American Institute of Physics 978-0-7354-0936-1/$30.00
Stochastic Forcing of the Batchelor Vortex
Z. W. Guo∗ , C. Chen, D. J. Sun
Department of Modern Mechanics, University of Science and Technology of China, Hefei 230027, China
Abstract The dynamics of the high-Reynolds-number vortices with axial flow continuously forced by a random excitation is investigated. Large response is possible, especially for the helical modes (with azimuthal wavenumbers |n|
= 1). The general trends are complicated by a number of issues, including a long-wavelength effect and a resonance
effect, both of which were recently discovered for optimal perturbation and are found here to be present for stochastic
forcing of swirling flow. Overall, the results suggest that strong response energy are present in the moderate- to highswirl regime of practical interest (swirl number q ≥ 2). For axisymmetric (n = 0) and higher azimuthal wavenumbers
|n| > 1, the axial flow contributes little to the response energy.
Key words: random excitation, stochastic forcing
Swirling flows are known to be highly unstable and susceptible to breakdown at high Reynolds number. Transient
growth for swirling flows has been investigated recently [1-3]. Antkowiak and Brancher firstly calculated such growth
for the Lamb-Oseen vortex and found evidence of a core contamination mechanism combining Orr effects [1]. Then,
Pradeep and Hussian(referred to as PH06) found similar transient amplifications and explained the physical mechanism of the energy growth [2]. Much attention has been paid to the response of the shear flows [4]. Farrell and Ioannou
[5] investigated the variance maintained by stochastic forcing in two-dimensional unbounded constant shear and deformation flows. It was found that the variance in pure deformation flows increases without bound under stochastic
excitation for any value of the viscosity. The first detailed investigations of stochastic forcing of swirling flow have
only recently appeared. Fontane, Brancher and Fabre(referred to as FBF) discussed the response of the system excited
by stochastic forcing for the Lamb-Oseen vortex, and they presented the effects of noises from the background [6]. The
emergence of the displacement wave under continuous external perturbations is a good candidate for explaining the
vortex meandering (or wandering) phenomenon.
The swirl number q is a very important parameter for the Batchelor vortex. We focus on q ≥ 2, where the linear operator
is stable. What is more, whether eigenmodes can be activated naturally by the background noise present in uncontrolled
conditions is studied.
The cylindrical (r, θ , z) coordinates with corresponding velocity components (u, v, w) is adopted here. Scaling velocities
and lengths with the maximum swirl velocity Vs and the vortex core radius rs . The Reynolds number based on these
characteristic scales is Re = Vs rs /ν , where the ν denotes the kinematic viscosity. The mean flow is the Batchelor vortex
[3], given by
e −r
U = 0,V = (1 − e −r ), W =
where the q is swirl number, which is related to the ratio of the maximum swirl velocity to the maximum axial velocity
defect (or excess). When q → ∞, the base flow can be treated as the Lamb-Oseen vortex.
The perturbation is Fourier-decomposed along θ and z
{u, v, w, p} = { i F(r,t), G(r,t), H(r,t), P(r,t)} e i(kz+nθ )
The use of the energy-based inner product requires a coordinate transform [7]. The numerical codes used here were
validated in three stages. First, The eigenvalues obtained for the swirling flows added the basic axial flow agreed to at
least eight significant digits with the results of [8]. Second, the transient growth is the same to that in [2]. Third, the
response energy when q is close to infinity, is the same to that of the Lamb-Oseen vortex.
The energy amplification for stochastic forcing of the Lamb-Oseen vortex has been discussed by FBF. We now present
the detailed results of our growth calculations for stochastic forcing of Batchelor vortex. Figure 1 shows the quantity
G vs. k for −2 ≤ n ≤ 2 and Re = 1 000 as q increases to infinity. The first comment to make about the plots in Figure 1
is that for q → ∞ the curves of each plot recover the results of FBF (see their Figures 1, 5 and 8). The Batchelor vortex
show strong response effects, especially for helical modes (n = ±1).
Figure 1: Energy amplification G vs. axial wavenumber k for various swirl numbers q, Re = 1 000. (a) n = −2, (b) n = −1, (c) n = 0,
(d) n = 1, (e) n = 2.
We now examine the results in Figure 1 in detail. The energy amplification in the case of the Lamb-Oseen vortex is the
same for n and −n (FBF 2008), but we immediately see that for the Batchelor vortex, the non-zero axial flow breaks
this symmetry and the results are different. For |n| = 2 (Figure 1(a) and 1(e)), the energy growth changes little as q
increases. The results for |n| > 2 is similar to the Case |n| = 2.
The results for helical modes( |n| = 1) are different. Figure 1(b) and 1(d) shows the large increase of the gain for small
k is observed and the peaks emerge. For left-helical mode (n = −1), the peak moves forward as q increase, but for righthelical mode, the process is reverse, the peak moves backward as q increases. When q is close to infinity, both of them
is around k = 1.5, which is agreement with that of the Lamb-Oseen vortex. This phenomenon is same to that in optimal
perturbation in PH06 (Figure 7(b) and 7(d)) with q ≥ 2, the physically important regime in which strong exponential
instabilities do not immediately explain breakdown. The peaks are not very prominent meaning that the dominant
structure does not emerge strongly in the vortex response for this range of wavenumbers. We investigated the case of
Re = 1 000, n = −1 and q = 2.8 to show the structures of forcing and response (Figure 2). The dominant structure
does not emerge strongly in the vortex response like that in the Lamb-Oseen vortex since the first output structure
only contributes 30% of the sustained variance for k = 1. This is because the coexistence of several perturbations
participate in the energy amplification when the system is stochastically forced. The rest emerging output structures
contributes less for k = 1 (e.g., the second output structure contributes 11%), so the vortex response to the forcing will
be dominated by the first output structure. From Figure 2(a), we see the forcing structure is composed of the a pair
of left-handed spiralling vorticity sheets similar to that of the Lamb-Oseen vortex (FBF) and the optimal perturbation
found by Antkowiak and Brancher. The two folded vorticity layers located in r ≈ 3.04 that is the resonant radii
discussed detailedly in PH06. This means Orr mechanism is still at play. The modes for the Batchelor vortex can also
be divided like that in the Lamb-Oseen vortex (Fabre et al. 2006), and the response structure (Figure 2(b)) corresponds
to the first critical-layer mode. Similarly, other kinds of waves got from the stability can also be found with different k.
In conclusion, due to different the swirl numbers, the only changes of the energy amplifications happen at moderate
wavenumbers for helical modes. Besides, the eigenmodes of the Batchelor vortex can be excited naturally by the
background noise present in uncontrolled conditions.
Figure 2: Isocontours of azimuthal vorticity for n = −1, q = 2.8 and Re = 1 000. Equally spaced levels are displayed
and dashed contours correspond to negative values. First input structure for (a) k = 1.0 accounting for 30% of the
energy amplification. First output structure for (b) k = 1.0 and 30%
The supports of the National Natural Science Foundation of China (10772172) and the 111 Project of China (B07033)
are gratefully acknowledged.
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