Large-Eddy Simulation of acoustic propagation in a turbulent channel flow Pierre Comte1 , Marie Haberkorn2 , Gilles Bouchet2 , Vincent Pagneux3, and Yves Aur´egan3 1
2 3
LEA, 43 rue de l’A´erodrome, F86000 Poitiers
[email protected] IMFS, 2 rue Boussingault, F67000 Strasbourg
[email protected] LAUM, 2 avenue Olivier Messiaen, 72085 Le Mans Cedex 9
[email protected]
Summary. Large-Eddy Simulation of a turbulent pulsating channel flow is performed in order to investigate the origin of a critical Strouhal-number-range, in which the modulus of the wall-shear impedance (that can be interpreted in terms of sound attenuation) is lower than in the laminar regime. Comparisons with linearized calculations, in which a passive scalar oscillating at the walls is advected by a non-pulsating flow, confirm the non-linear origin of this critical range.
1 Introduction As shown experimentally in [7], sound propagation in turbulent ducted flows can be investigated either by introducing a long-wave pulsation of the mean flow, or by making the wall oscillate without pulsating the mean flow. Results obtained with both techniques collapse quite well, relatively independently of the forcing amplitude. The most relevant quantity to sound propagation τ) is the (non-dimensional) attenuation factor ατ = Re(Z co ρL , in which the wall shear impedance Zτ is the ratio of the complex amplitudes of the wall shear and the forced velocity. In general, attenuation is larger in a turbulent flow than in a laminar flow (ατ /ατStokes > 1) except in a rather small range of pulsations (Strouhal number ω + = ων/u2τ ∈ [5 10−3 , 3 10−2 ]) the origin of which has remained unclear. Two types of explanations have been proposed, the second of which is essentially linear: a) energy feedback from turbulence to the deterministic pulsating part, due to the synchronization of the bursting events in the boundary layers [9] b) diffraction of the shear wave due to the wall-normal gradient of eddyviscosity (in a RANS or phase-averaged sense)[7]. The suitability of Large-Eddy Simulation to tackle this problem of identification has been demonstrated in Ref. [8], in which incompressible pulsating
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P. Comte, M. Haberkorn, G. Bouchet, V. Pagneux, Y. Aur´egan
channel-flow simulations were performed for a wide range of frequencies of the driving pressure gradient, with thorough comparison between LES and DNS. Among the results, the Stokes regime was retrieved beyond ω + = 0.1 for the AC component of the flow, extracted by phase averaging. At lower forcing frequencies, the non-deterministic part (denoted above with a double prime superscript) was found to develop unsteadiness, with quasi-total cyclic relaminarisation. The critical zone was retrieved, and an eddy-viscosity model was proposed for the ”turbulent Stokes length”, namely, the penetration thickness of the shear waves into the boundary layers. This model yields lt+ ' lS+ for + + + 2 high frequencies p (κlS /2 1), and lt ' κ(lS ) at lower frequencies. Here, κ and ls = 2ν/ω denote the K´ arm´ an constant and laminar Stokes length, respectively, and we have ω + = ων/u2τ = 2/(lS+)2 . Our motivation here is to evaluate the possibilities of improving upon existing diffusion models in use in the acoustic propagation community, by means of lower cost LES and within a fully-compressible framework capable of taking into account arbitrarily large density and temperature gradients. Because of the complexity of the physics involved, we will nevertheless restrict ourselves to low Mach numbers and nearly isothermal situations. We proceed as follows: 1. we validate our results against [8] and the associated experimental background. 2. we reduce the forcing amplitude from A0 = 70% to A0 = 20% to get closer to acoustically-forced configurations, and check whether cyclic relaminarisation is needed to observe the critical zone. This is not so easy, because the effet to capture is rather subtle. We also check whether the presence of the critical zone is accompanied by overall drag reduction. 3. we check whether the critical zone can be retrieved in a linearized framework, either by solving for the advection, by a non-pulsating flow, of a passive scalar oscillating at the walls, or by solving a diffusion model with prescribed profiles of eddy-viscosity.
2 Methodology The filtered compressible Navier-Stokes equations are solved in the macrotemperature [3] closed form, by means of an explicit McCormack-type method which solves the convective terms with 4th-order accuracy, and the Filtered Structure-Function subgrid-scale model is used, as in Ref. [1]. With respect to [8], the time-averaged Reynolds number has been reduced from Reτ = 395 to Reτ = 180, to minimize the influence of the SGS model. The domain size is (Lx, Ly , Lz ) = (2πh, 2h, πh) with (Nx , Ny , Nz ) = (64, 109, 64). The mesh spacing is uniform in the streamwise and spanwise directions (hereafter denoted x and z, respectively), with ∆x+ = 19 and ∆z + = 10. Hyperbolictangent stretching is used in the wall-normal y direction, with the first mesh line away from the wall located at y + = 0.2.
LES of acoustic propagation in a turbulent channel flow
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The forcing term Q(t) = Q0 [1 − A0 sin(ωt)] prescribes the flow rate, which enables us to investigate the influence of the pulsation on the time-averaged drag, in contrast with the prescribed-shear forcing used in [8]. Two different forcing amplitudes have been considerered, as in the experiments [9], namely, 70% and 20% (relative to the centerline velocity of the laminar Poiseuille profile of same flowrate as the mean flow). Up to the difference in Reynolds numbers, the first case has been treated in [8]. To the best of our knowledge, the second case at lower amplitude, and therefore closer to acoustic propagation, has not been simulated yet. As in Ref. [8], flow decomposition is performed, after the spin-up of each simulation, in terms of phase average at the forcing pulsation: for each variable f , this reads f (x, y, z, t) = hf iΦ (y, t0 ) + f ”(x, y, z, t) with Nz Nx X N X X 1 hf iΦ (y, t0 ) = f (x, y, z, t0 + nT ) , t0 ∈ [0, 2π/ω] (1) N Nx Nz n=1 i=1 k=1
This indeed corresponds to a triple decomposition, into mean, cylic and nondeterministic parts, as introduced in [2]: e t) +f ”(x, y, z, t) . f (x, y, z, t) = hf i(y) + f(y, {z } | hf iΦ (y,t0 )
The wall friction is thus decomposed, yielding τw = µw
(2)
∂u ∂y w
= hτw i +
Aτ cos(ωt + Φτ ) + τw ”, in which Aτ and Φτ minimize τw ”. We then have Ac (τw ) τ i(Φτ −Φu ) Zτ = A , in which Φu denotes the phase shift of the cen= A Ao e c (uw ) terline velocity wrt Q(t). p Amplitude Ao is eliminated using the laminar Stokes solution AτStokes = ων µw Ao , which yields ατ ∝ Aτ Aτ cos (Φτ − Φu ). Stokes
3 Pulsed simulations The parametres of the LES undertaken are summarized in Table 1 below, together with, in italics, those of cases considered in [8]. N stands for the number of forcing periods during which the statistics have been considered, using phase and spanwise averaging. The corresponding CPU time, on a NEC SX-5 vector supercomputer, is also mentioned for information. A steady simulation has also been performed, not only for validation purposes, but also for the simulation of the advection of several passive scalars, forced harmonically at the walls with different frequencies, as detailed in section 4. As in Ref. [8], it is found that, for A0 = 70%, the non-deterministic part u00 exhibits unsteadiness when the forcing frequency is sufficiently low: Fig. 4 shows the time evolution of the vortex structure during a cycle, starting, from top to bottom, at the beginning of the acceleration phase. In contrast, unsteadiness is not visible at ω + ∼ 0.01. Figure 1 shows Aτ Aτ and (Φτ − Φu ) for the whole set of numerical and Stokes experimental results we know of. The solid line corresponds to the experiments
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P. Comte, M. Haberkorn, G. Bouchet, V. Pagneux, Y. Aur´egan Ao 0.7 0.7 Ref. [8] 0.7 0.7 Ref. [8] 0.7 0.7 Ref. [8] 0.2 0.2
ω ω+ 0.054 0.00604 0.0016 0.108 0.0107 0.01 1.08 0.093 0.1 0.108 0.0097 1.08 0.0941
ls+ 18.2 35 13.68 14 4.63 4.4 14.37 4.61
lt+ 138.25 504.7 79.14 63.16 10.78 9.89 87.03 10.69
N CPU hours 6 700 15
880
20
120
7 48
410 280
Table 1. Parametres 2.5
50 40
u
Φ −Φ
1.5
τ
Aτ/Aτ
Stokes
2
1
20 10 0
0.5
0 0
30
−10 0.02
0.04
0.06
+
ω
0.08
0.1
0.12
0
0.1
ω+
0.2
Fig. 1. Amplitude (left) and phase shift (right) of deterministic wall shear: —— [7] A < 5% ; • [8] A = 70% ; + [9] A = 70% ; + [9] A = 20% ; • present LES, A = 70% ; ◦ present LES, A = 20%.
of [7], which were performed in a pipe flow (as [5] and [6]), whereas the other ones were in plane channel flows. This might have been thought to be a reason for the dispersion observed, but the channel-flow numerical results of [8] fall right on the those of [7], which is not so clear for ours. However, the trends are retrieved for both forcing amplitudes, namely, the asymptotic Stokes-like at low ω + , due to behaviour at high ω + , the ω −1/2 behaviour of Aτ Aτ Stokes
the fact that AτStokes ∼ ω 1/2 when Aτ goes to a quasi-steady finite limit at vanishing ω + , and the critical zone in which Aτ Aτ < 1. Stokes Figure 2, analogous to Fig. 15 in [8], confirms that cyclic relaminarisation occurs at forcing amplitude 70%, but not really at 20%, at least for ω + ≥ 10−2 . This suggests quite convincingly that cyclic relaminarisation is not required for the presence of the critical region. On the other hand, the relative indifference of this critical region to the forcing amplitude is reminiscent of a linear process, in which the shear wave would be diffused and maybe refracted by the turbulent flow, without the latter being necessarily affected.
LES of acoustic propagation in a turbulent channel flow
ω+∼0.1
+
ω ∼0.01
160
160
140
140
140
120
120
120
100
100
100
80
80
80
60
60
60
40
40
40
20
20
20
ω+∼0.006
2
Φ
τ
/
160
5
0
0
0
10
0
0
10
y
+
0
10
Fig. 2. Phase-averaged evolution of the trace of the Reynolds stress tensor. Forcing amplitude 70% (black), 20% (medium grey) and 0% (pale grey). Profiles are at T /8 apart and offset by 20 units in the wall-normal direction. The dash-dotted and dashed lines are at y + = ls+ and y + = 2lt+ , resp. 22 20
/
18 16 14 12 10 8 6 4 2 0
0
10
10
y+
1
10
2
ω+ 0.006 0.01 ” 0.1 ”
Ao 0.7 0.7 0.2 0.7 0.2
Rew 27200 13600 1100 1400 100
h/ls Cf ref. 9.9 -11.7% · · · 13.16 -6.2% −− ” -1.5% −− 40.9 0% — ” ” —
Fig. 3. Mean velocity profiles. That at zero forcing amplitude is also plotted. It is collapsed with − − −.
Note also that significant drag reduction is observed, for sufficiently low forcing frequency and sufficiently high amplitude, as shown in Figure 3. Although not unanimously observed, this is consistent with the experimental findings of Ref. [4]
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P. Comte, M. Haberkorn, G. Bouchet, V. Pagneux, Y. Aur´egan
z/h
3 2 1
1
2
3
4
5
6
4
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6
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5
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5
6
x/h
z/h
3 2 1
1
2
3 x/h
z/h
3 2 1
1
2
3 x/h
z/h
3 2 1
1
2
3 x/h
z/h
3 2 1
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2
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3 2 1
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x/h
z/h
3 2 1
1
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3 x/h
Fig. 4. Evolution of the coherent structures during a cycle, for ω + = 0.006. Left: 00 00 contours of u at y¯+ = 10. Right: isosurface Q = 0.6(ub /h)2 .
LES of acoustic propagation in a turbulent channel flow
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4 Linearized calculations and eddy-viscosity arguments To check on the linear origin of the critical zone, simulations of the advection of a passive scalar c, forced harmonically at the walls and advected by a nonpulsed turbulent flow, have been performed. The results have been compared with those of a semi-analytical diffusion model, with a prescribed profile of 00 00 v iΦ effective eddy-diffusivity obtained from κt (y) = −hc ∂hciΦ /∂y . 1.4
50 45
Φτ−Φu
1.1
τ
A /A
τ
Stokes
40 35 30 25 20 15 0.8 0
0.05
10 0
0.1
+
ω
0.05
0.1
ω+
Fig. 5. Amplitude (left) and phase shift (right) of deterministic wall shear: —— [7] A < 5% ; passive scalar LES ; ∗ linear model κt (y)
and (Φτ − Φu ) are shown in Figure 5. The same trends Plots of Aτ Aτ Stokes are observed for both calculations, namely, the recovery of the asymptotic regimes at low and high ω + , but the critical region is lost. 0.16
0.1
0.14
ν /(h u ) t
τ
νt/(h uτ)
0.12 0.1 0.08
0.05
0.06 0.04 0.02 0 0
20
40
60
80
100
+
120
140
160
180
0 0
20
40
60
80
100
+
120
140
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180
200
y
y
Fig. 6. Profiles of eddy-viscosity in time and phase averages, for ω + ' 0.006 (left) −hu0 v 0 i and ω + ' 0.1 (right). Black crosses: hνt i(y) = ∂hui/∂y in steady LES. Grey solid line: same in unsteady calculations Other symbols: hνt iΦ (y, t) = 0
spaced phase angles t . Portion y
+
−hu”v”i ∂huiφ /∂y
for 8 equally-
≥ 140 irrelevant due to 0/0-type numerical errors.
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P. Comte, M. Haberkorn, G. Bouchet, V. Pagneux, Y. Aur´egan
Refs. [7, 8] strongly support the idea that the effect of turbulence on the shear waves can be modelled by means of an eddy-viscosity, in a RANS sense. Figure 6 shows strong dependance of νt on the phase angle at low frequency.
5 Conclusions LES of a pulsating plane channel flow have been performed to shed light on the critical frequency range where sound attenuation is lower than in the laminar regime. This has been retrieved in a reasonable agreement with the LES of Ref. [8] and the experimental data of Refs. [7, 9]. Up to 11% drag reduction has been found, as in Ref. [4]. The results at a forcing amplitude of 20%, which are original to the best of our knowledge, showed that cyclic relaminarization was not needed to observe the critical zone. The asymptotic behaviours of the shear-wave impedance at low and high forcing frequencies have been retrieved in LES of the advection by a steady flow of a passive scalar oscillating at the walls, but not the critical zone. Finally, eddy-viscosity profiles have been computed, showing strong variation with the phase angle at low frequencies, and phase-dependent analytical models are in progress, in order to check whether the critical zone can be retrieved with more realistic eddy-viscosity profiles than the rigid-wall model proposed in Ref [7], which is in use in current sound-propagation codes in industrial configurations. Acknowledgement: most of the CPU time used was allocated free by IDRIS, the French CNRS supercomputing center.
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