Eddy viscosity of cellular flows by upscaling

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Abstract

The eddy viscosity is the tensor in the equation that governs the transport of the large-scale (modulational) perturbations of small-scale stationary flows. As an approximation to eddy viscosity the effective tensor, that arises in the limit as the ratio between the scales ε→0, can be considered. We are interested here in the accuracy of this approximation. We present results of computational investigation of eddy viscosity, when the small-scale flows are cellular, special periodic stationary flows with the stream function φ=siny1siny2+δcosy1cosy2, y=x/ε, 0⩽δ⩽1. For small ε we used a numerical upscaling method. We designed this method so that it captures the modulational perturbations for any ε with O(ε2) accuracy and independent of ε complexity.

Section snippets

Formulation

For incompressible highly oscillatory small-scale flow v subject to forcing periodic in space and time the eddy viscosity, which controls the large-scale transport of momentum, can be determined in a systematic way by multiscale analysis (see e.g. [3], [12], [13], [14]). More specifically, assume that v is modeled as a stationary solution of the Navier–Stokes equations with some auxiliary forcingv·∇v=νΔv−∇p+f,∇·v=0.The role of the forcing is to sustain the flow pattern of v, and its detailed

Upscaling

Our Eq. (3) is very similar to the convection–diffusion of a passive scalar∂tωε(x,t)+1εvxε·∇ωε(x,t)=1PeΔωε(x,t),∇·v=0,where the Péclet numberPe=max|v|σ,σ is the diffusivity of the passive scalar ωε.

The convection–diffusion equation of a passive scalar for large Péclet number has been studied for cellular flows in [5] (for random flows see e.g. [4] and references therein). It was shown in [5], that if the passive scalar ωε is decomposed into a smooth mean part and a highly oscillatory mean-zero

Numerical implementation

In the numerical implementation of upscaling MATLAB is used primarily with the exception of the solution of the linear system (21), where C was used. Time-stepping in (18) for 〈ωε〉 is performed by Backward Euler method with variable time-step Δt=.001/max(1,ln|aεoo|). The four Fourier coefficients of ω̄ε are determined in terms of sums and differences (19) in order to keep track of symmetries (20). The elliptic problem (12) for ω̃ε is solved by LU decomposition of (21).

If we say that the result

Results

The linearity and scaling arguments (see e.g. [9]) imply that it is sufficient to solve (3) with initial conditions (15), assuming that the wave-vector is normalized to have length 1: m=(cosθ,sinθ). Our primary interest is the single mean flow Fourier coefficient aεoo and its dependence on ε, m, Re, δ. Therefore we present the numerical results for this coefficient only.

The first example was done by matrix exponentiation, because ε=0.25. In Fig. 2 we plot the value of aεoo(m,t), and the

Conclusions

The behavior of modulational perturbations (eddy viscosity) of cellular flows was studied numerically for Re∈[1,100], ε=.1,.01,.001. The effective equations for modulational perturbations predict accurately the solution when the ratio between scales ε is sufficiently small or the Reynolds number Re is not large. More specifically, a priori error estimates from [9] guarantee that the error is expected to be small for our numerical experiments if εRe⩽C, where C≪1, numerically we observe that the

Acknowledgements

We are grateful to H.-M. Zhou for reading a draft of this paper and making remarks about it. We would also like to thank anonymous referees for their valuable comments.

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Present address: Department of Mathematics, Penn State University, University Park, PA 16802, USA.

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