PAMM · Proc. Appl. Math. Mech. 8, 10763 – 10764 (2008) / DOI 10.1002/pamm.200810763
DNS of stochastically forced laminar plane Couette flow G. Khujadze∗1,2 , M. Oberlack 1 , and G. Chagelishvili 2 1 2
Department of Mechanics, Group of Fluid Dynamics, Hochschulstr. 1, Darmstadt, Germany Abastumani Astrophysical Observatory, Kazbegi Ave. 2A, Tbilisi, Georgia
Background of three dimensional hydrodynamic/vortical fluctuations in a stochastically forced, laminar, incompressible, plane Couette flow is simulated by direct numerical simulations (DNS). It was found that the fluctuating field has well pronounced peculiarities: (i) The hydrodynamic fluctuations exhibits non-exponential, transient growth; (ii) Streamwise non-constant fluctuations with the characteristic length scale of the order of the channel width are predominant in the fluctuating spectrum; (iii) Existence of coherent structures in the fluctuating background; (iv) Stochastic forcing breaks the spanwise reflection symmetry (inherent to the linear and full Navier-stokes equations in a case of the Couette flow) and inputs an asymmetry on dynamical processes. c 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
The resolution of the paradox of a subcritical transition in shear flows (plane Couette or pipe Poisuille flows for example) is a long-standing problem in fluid mechanics. It is well-known that certain laminar flows are linearly stable at all Reynolds numbers but in practice always become turbulent at sufficiently large Reynolds number [1]. On the other hand the stochastic forcing is inherent for the environmental and engineering flow systems. During the past two decades it has become evident that thermally excited fluctuations in fluids in nonequilibrium steady states, even in the absence of convection or turbulence are always spatially long ranged. By recent studies it has been shown that the relatively small stochastic forcing of the laminar Couette flow in the linear regime leads to the high level of fluctuation energy production (see [2] and references therein). Our aim was to calculate the fluctuating background of stochastically forced plane Couette flow by DNS, to define its characteristic length scales and to verify or refute the predominance of streamwise constant vortices. We consider intrinsic stochastic forcing – Gaussian random stress tensor satisfying the fluctuation-dissipation theory relation [3]. We study 3D, incompressible, plane Couette flow (U0 (Ax2 , 0, 0)) with shear parameter A, channel half-width L and Reynolds number Re based on the mean centerline velocity and the half-width of the channel (Re ≡ AL2 /ν). In the laminar case (Re < 350) the Couette flow is slightly nonequilibrium and the fluctuations can be neglected beyond the linear order. Consequently, the linearized Navier-Stokes equations for small stochastically forced fluctuations has the following form: ∂ui (r, t) = 0, ∂xi ∂ ∂ 1 ∂p(r, t) ∂sij (r, t) + ν∆ui (r, t) + , ui (r, t) + Au2 (r, t)δi1 = − + Ax2 ∂t ∂x1 ρ0 ∂xi ∂xj
(1) i, j = 1, 2, 3
(2)
were ρ0 is uniform flow density; ui (r, t) and p(r, t) – components of velocity and pressure fluctuations respectively; ∆ – Laplacian; sij (r, t) – spontaneous strain tensor. The last term containing sij (r, t) defines stochastic forcing of the system. Statistical properties of the spontaneous strain tensor are modelled in accordance with the fluctuation-dissipation theory [3]: 2T ν 2 (3) δik δjl + δil δkj − δij δkl δ(r − r )δ(t − t ). sij (r, t)skl (r , t ) = ρ0 3 For the numerical purposes we have to define stochastic forcing in spectral space. So, using Fourier transform sij (r, t) = dkSij (k, t)exp(ıkr) we can introduce the following expression for Sij (k, t) to satisfy the statistic characteristics of Eq.3: ⎧
⎫ 4 ⎪ ⎪ cos[2πφ (k, t)] cos[2πφ (k, t)] cos[2πφ (k, t)] ⎪ ⎪ 1 2 3 ⎪ ⎪ 3 ⎨ ⎬
4 2 , (4) Sij (k, t) ≡ cos[2πφ (k, t) + π] cos[2πφ (k, t)] cos[2πφ2 (k, t)] 1 4 3 3 ⎪ ⎪
⎪ ⎪ ⎪ ⎪ 4 4 ⎩ cos[2πφ (k, t)] ⎭ cos[2πφ4 (k, t)] 3 3 cos[2πφ1 (k, t) + 3 π] where ≡ 8T ν/ρ0 is the measure of the stochastic forcing, φ1 (k, t), φ2 (k, t), φ3 (k, t), and φ4 (k, t) are random numbers in the range [0, 1] different for different k and t. Originally the code for the DNS was developed at KTH, Stockholm (for details see [4]) using a spectral method with Fourier decomposition in the horizontal directions and Chebyshev discretization in the wall normal direction. Time integration is performed using a third order Runge-Kutta scheme for the advective and Crank-Nicolson for the viscous terms. Simulations ∗
Corresponding author: e-mail:
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c 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
10764
Sessions of Short Communications 14: Applied Stochastics
were performed on the IBM supercomputer (Regatta-H) at Technische Universit¨at Darmstadt, Germany. DNS of linear NavierStokes equations with stochastic forcing was performed using the simulation box (Lx = 6π)×(Ly = 2)×(L z = 2π) and the number of modes 256 × 217 × 128. DNS was done for the following parameters: [A] = 1, [L] = 1, [ν] = 1/300, i.e. for Re = 300. The only perturbations in the system are that ones due to the stochastic forcing. The performed linear simulations show that the variance reaches a finite statistically stationary level at [t] > 150. This level greatly exceed the variance levels resulting from the balance between energy accumulated from stochastic forcing and energy dissipated by the normal modes. This fact is in agreement with the recent studies of a stochastically forced shear flows (see referencies in [2]). The reason of this is a transient extraction of the background shear energy by eddy fluctuations that is a consequence of the non-normality of channel flows linear dynamics.
Fig. 1 (a) Streamwise, (b) wall-normal and (c) spanwise veloctity fluctuations; (d) Wall-normal vorticity. Fluctuation energy: coherent (e) and incoherent parts (white noise) (f).
The results of the DNS are presented in the Fig. 1. Predominance of the streamwise non-constant structural regularities with the characteristic length-scale of the order of the channel width in the fluctuating background are observed. The figure shows the velocity components, wall-normal vorticity and energy of fluctuations in three dinensional domain. One of the main result of the presented work is the different configurations of the different components of the fluctuation velocity field (Fig. 1, plots (a,b,c)) due to the non-normality of the linear dynamics of the shear flows. Different component of velocity perturbation extract the energy from the mean flow not by the classical/exponential but by the algebraic laws different for each component. This circumstance leads to the different characteristic configurations and scales of the hydrodynamic fluctuation background. The plots (d,e,f) in this figure show the wall-normal vorticity, coherent and incoherent parts (white noise) of the fluctuation energy respectively. Let’s summarize the DNS of Navier-Stokes equations at small amplitude (linear) stochastic forcing. The performed simulations revealed evident peculiarities of statistically stationary fluctuating field of a laminar plane Couette flow: The hydrodynamic fluctuations exhibit non-exponential, transient growth; An anisotropy of the fluctuating velocity field increasing with the shear rate; Existence of the streamwise structural regularities (coherent structures) with the characteristic length-scale of the order of the channel width; Appearance of the nonzero cross-correlations of velocity components; Spanwise reflection symmetry breaking of dynamical processes due to the stochastic forcing. It was found that the hydrodynamic fluctuation background in a laminar, plane Couette flow is anisotropic and has well pronounced peculiarities due to the non-normality of the dynamical processes. It was proved that the high level of vortical perturbations can be produced in the system by the intrinsic stochastic forcing (hydrodynamics fluctuations) of the flow. Acknowledgements The authors are grateful to Prof. Henningson (KTH, Stockholm) who kindly gave us the code for our simulations; One of the author (G.Ch.), thanks the support from DFG, Project GZ 436.
References [1] [2] [3] [4]
P. J. Schmid, Annu. Rev. Fluid Mech., 39, 129, (2007); G. Khujadze, M. Oberlack and G. Chagelishvili, Phys. Rev. Letters, 97, 034501-1, (2006). L. D. Landau and E. M. Lifshitz, Statistical Physics, Pergamon Press, (1980). M. Skote, Studies of turbulent boundary layer flow through direct numerical simulation, KTH, Stockholm, Sweden, (2001).
c 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
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