Stokes eigenmodes in square domain and the stream function–vorticity correlation

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Abstract

The Stokes eigenmodes in the square are numerically determined and their symmetry properties are identified. The spectra evolution laws are in excellent qualitative agreement with the theoretical asymptotic predictions proposed by Constantin and Foias (in “Navier–Stokes equations”, University of Chicago Press, 1988), λk≃k+O(k). The slopes are reported here and are found to be specific to the eigenmodes symmetry family. The dynamic equilibria are analyzed and show a linear relationship between the vorticity and the stream function in the core of the eigenmodes. These features of the Stokes eigenmodes confined in the square are shared by the fully periodic Stokes eigenmodes.

Introduction

Why to study the Stokes eigenmodes, whereas nonlinear dynamics is the most attractive part of the Navier–Stokes equations? The dynamical behaviors governed by the Navier–Stokes equations result from the way this nonlinear dynamics is controlled by diffusion. For instance, in isothermal mono-component fluid, turbulence can be primarily seen as resulting from an imbalance between the (u·∇)u and Δu contributions. No internal length scale can be found to put these terms in equilibrium. Understanding then the intrinsic dynamics of the diffusion part of Navier–Stokes can supply an interesting point of view on turbulence.

Since the early work of Taylor [35] on the leading eigenmode of the buckling load problem, only a few attempts have been made to compute few Stokes eigenvalues and/or eigenmodes in fully confined geometries [5], [6], [7], [8], [9], [10], [11], [13], [36], [37], [38]. Apart from the theoretical predictions proposed in [14] on the asymptotic behavior of the eigenvalues in 2D or 3D confined domains, there is as yet no global view of the Stokes eigenspace in the square. This is because Stokes eigenmodes in confined geometries are not easily accessible.

Numerically determining the confined Stokes eigenmodes is not an easy task. The choice made for uncoupling the pressure from the velocity fields is crucial for the consistency and computational complexity of the scheme. For instance, in [5] a Uzawa uncoupling method is applied for computing the eigenmodes of the generalized singular Stokes problem in order to provide a basis for the channel and grooved channel flows. This approach – already expensive in 2D – practically excludes access to 3D eigenmodes. To the authors' point of view, the projection-diffusion (PrDi) uncoupling [20], which is also consistent with the continuous problem [22] but optimal in computation cost, allows the 2D/3D eigenmodes to be computed.

The scope of the present work is to provide the first deep insight into the Stokes eigenspace in the square. The eigenvalues and eigenmodes are accurately computed by two different means, namely, a Chebyshev PrDi solver and a Galerkin–Reid–Harris (RH) expansion of the stream function. The symmetries which underlie the eigenmode patterns are also identified. The evolution laws of the spectra were fitted. They qualitatively agree with the theoretical estimations proposed in [14]. An analysis of the peculiar dynamics of the eigenmodes is also performed. Each symmetry family exhibits its own spectrum slopes. Furthermore, a linear relationship between the vorticity and the stream function is inferred from the dynamic equilibria at the core of each eigenmode. Frequent references are made to the fully periodic Stokes eigenspace features.

Confined 2D flows can thus be characterized by functional relationships between the vorticity and the stream function not only in the inviscid regions at high Reynolds number [4].

Section snippets

Outline

The paper is organized as follows. The governing equations are first presented in Section 3 for the various Stokes formulations. In view of further comparisons, Section 4 recalls the analytical expressions of the Stokes eigenmodes which are periodic in all, and in all but one, space directions. Section 5 presents the symmetry families of the Stokes eigenmodes in the square. Section 6 presents the solvers. Since corner eddies are expected to be part of each eigenmode, Section 7 makes a brief

Governing equations

Let us write the dimensionless unsteady (time t) 2D Stokes equations, with primitive variables, in the open domain Ω=]−1,1[2 with coordinates x=(x,y) and T a real positive number:ut=Δup+ffor(x,t)∈Ω×]0,T[,·u=0for(x,t)∈Ω×]0,T[,where u=(u,v) is the velocity field, p the pressure and f a source term. We denote the closure of Ω by Ω and the boundary by Ω. Dirichlet boundary conditions are imposed on the velocityu=Vfor(x,t)∈Ω×]0,T[,and compatible initial conditions are givenu(t=0)=V0forx∈Ω.

The presently known Stokes eigenmodes

This paper concerns the Stokes eigenmodes in the square. To the authors' knowledge, the Cartesian Stokes eigenmodes are not analytically known except when they are periodic in all, or in all but one, space directions. If they are indeed constrained to satisfy velocity no-slip conditions on a closed boundary they can only be determined by a numerical approach. For future reference a brief survey of the analytically known Cartesian eigenmodes is provided in this section.

Symmetries

Apart from translation, which is not of interest here, two planar isometries will be considered [32]. Firstly, the θ-rotation. Secondly, the (D)-symmetry which is the reflection about the straight line (D) making an angle α/2 with the unit horizontal axis êx (Fig. 1). Let us successively denote R(θ) and S(α) as the operators which describe these transformations, and |Ψ(M)〉 a state (defined below) known as a scalar function Ψ(x,y). One will have |Ψ(M)〉=R(θ)|Ψ(M)〉 and |Ψ(M″)〉=S(α)|Ψ(M)〉 the new

Solvers

Computing the Stokes eigenmodes can be made from either their (velocity–pressure) primitive variables or stream function formulation, but the choice of the scheme is particularly relevant. For instance, one of the problems raised by the accurate numerical determination of the eigenmodes regards the possible requirement of enforcing the numerical velocity to be divergence free in order to obtain relevant and convergent results.

With the former formulation, the well-known Stokes solvers are either

About the singular corner eddies

As is well known since Moffatt's work [24], the Stokes eigenmodes contain an infinite sequence of similar corner eddies, singular at each of the square four corners, verifying Δ2ψ=0 with ψ=∂ψ/∂n=0 on the boundaries. They are not specific of the Stokes eigenmodes but of their symmetry: the corner eddies are even about the square diagonals for the families |1,1,1〉 and |1,−1,−1〉 (Fig. 2(a)), odd for |1,−1,1〉 and |1,1,−1〉 (Fig. 2(b)), and without diagonal symmetry for the last families |−1,/,±1〉 (

Results

The complete L spectrum is computed (MATLAB software, [1]) with N=16, 32, 48, and only the 200 leading eigenmodes (ARPACK Library, [21]) for N=64, 96. The N=96 results are taken as reference (cf. [23]). The RH eigenvalue systems are solved using the Mathematica software [40]. A significant number of Stokes eigenmodes is computed, in each symmetry family, with the cut-off I=15, 31, 63.

Conclusions

Two different solvers have been used to compute the Stokes eigenmodes in the square, a (velocity, pressure) PrDi solver based on a Chebyshev collocation scheme and a Galerkin–RH expansion for the Stokes stream function formulation. The former solver supplies genuine Stokes eigenmodes whose divergence cancels asymptotically with the grid refinement, distributed among many eigenmodes of another system, the divergence diffusion problem. From both solvers comes the complete identification of the

Acknowledgements

The authors thank Prof. M.O. Deville for helpful discussions. The second author gratefully acknowledges the financial support from the ERCOFTAC visitor programm sponsored by the L. Euler Pilot Center (Switzerland) at EPFL, and the FSTI-EPFL for the Invited Professor Fellowship. The computing resources were made available by CSCS, Manno, Switzerland. The authors thank Dr. N. Borhani (LMF-ISE-FSTI-EPFL) for proof reading the manuscript.

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    On leave from Université Paris-Sud, LIMSI-CNRS, BP 133, 91403 Orsay Cedex, France.

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