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{\large\bf NUMERICAL SIMULATIONS OF SELF-ORGANIZATION IN
2-D CIRCULAR SHEAR FLOWS\\}
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\noindent
K. Bergeron$^1$, E.A. Coutsias$^1$,
J.P. Lynov, \underline{A.H. Nielsen.}\\

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\noindent
Association EURATOM - Ris\o$\;$National Laboratory,
Optics and Fluid Dynamics Department, \\
DK-4000 Roskilde, Denmark 

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\noindent
$^1$Department of Mathematics and Statistics,
University of New Mexico, Albuquerque,\\ NM 87131, USA

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\noindent
A fluid submitted to a velocity shear can be unstable, and may form a line
of vortices which is in itself unstable and evolves through vortex-pairing
processes. Such a shear flow is studied numerically in an annulus geometry.
We have performed direct numerical simulations of the experiment of 
Rabaud-Couder$^1$ (and more resently Chomaz et al.$^2$). Here a flow is
enclosed in a vertical cylinder of radius $R_{2}$ and very small height e. 
A disk of radius $R_{1}$ at the centre of each end, rotates at angular velocity 
$\Omega_{1}$, while the rest of the cell rotates at $\Omega_{2}$.
Viscous friction on the containing walls forces the fluid into solid rotations
at $\Omega_{1}$ in the central part and $\Omega_{2}$ in the external annulus
respectively, separated by a shear zone. Chomaz et al. have demonstrated 
that for low differential rotation (i.e. low $\Delta \Omega = 
\mid \Omega_{2} - \Omega_{1} \mid$) the flow will be stable while for
increasing $\Delta \Omega$ 
a Pitchfork bifurcation point is reached where the flow becomes unstable and 
a necklace of vortices is observed. By increasing $\Delta \Omega$, the size 
of the vortices will grow and a series of transitions towards patterns with 
even smaller numbers of vortices will take place. Arguing that the Ekman 
layers at the top and bottom bounding walls dominate the flow, they proposed 
a model in which the flow at the mid-plane is essentially two dimensional, 
with the motion of the vessel manifesting itself as a body force.

We have modeled the experiments by the Navier-Stokes equation:
\begin{equation}
\frac{\partial \omega}{\partial t} + \vec{u} \cdot \nabla \omega  = 
\nu \nabla^2 \omega + \nabla \times \vec{F} \; ,\label{eq:eq1}
\end{equation}
combined with the Poisson equation
\begin{equation}
\nabla^{2} \psi  =  - \omega  \; , \label{eq:eq2}
\end{equation}
where $\vec{u}(x,y) = \nabla \psi \times \vec{z}$ is the velocity, 
$\omega (x,y)$ the scalar vorticity, $\psi (x,y)$ the stream function, $\nu$ the viscosity, and $\vec{F}$
represent the body force due to the friction on horizontal walls. 

The Navier-Stokes equation, given by eqn.~\ref{eq:eq1} - ~\ref{eq:eq2}, are solved numerically in an annulus geometry employing a fully dealiased, spectral scheme based on Fourier-Chebyshev expansions. This implies very high resolution of boundary layers. The flow field is subject to no-slip conditions, so that it matches the wall velocity at the boundaries. Precomputed solvability constrains are enforced on each Fourier mode of $\omega$. A more detailed description of how these solvability constrains are implemented in the numerical scheme is given by Coutsias and Lynov$^3$. 


The forcing term has to be modeled in order to obtain a closed set of equations
in two space dimensions.
Assuming a parabolic vertical dependence characteristic of a Poiseuille flow 
between two planes we obtain:
\begin{equation}
\vec{F} \approx \frac{8 \nu}{e^2} ( \vec{u}^* - \vec{u} ) , \label{eq:eq3}
\end{equation}
where $\vec{u}^*$ is the boundary velocity on top or bottom walls. In the 
experiments this velocity is a discontinuous function of the radial distance 
to rotation axis. Such a discontinuous function is, however, not feasible 
to use in spectral numerical simulations, and it does not correctly describe 
the effect of the walls at the mid-plane.   
Rabaud and Couder$^2$ have shown that the typical length of the shear is of the 
order of the cell height e. We have thus chosen a 
$\omega^* \vec{z} = \nabla \times \vec{u}^*$ which is constant in two radial
regions with a sharp transition region of the scale length e. In order to obtain a steady state solution correcsponding to this forcing we insert the 
prescribed $\omega^*$ in equation ~\ref{eq:eq1} and we solve the resulting
Helmholtz equation:
\begin{equation}
(1 - \frac{e^2}{8} \nabla^{2} )\bar{\omega} = \omega^* \; .
\end{equation}
This $\bar{\omega}$ is used as an initial condition for the code.
To determine the Reynolds number (defined here as $Re = \Delta \Omega R_1 e/
\nu$) at which the cylindrically symmetric steady state is marginally stable
to perturbations of various wavenumbers $k$, we carried out a linear stability
analysis. The method used involved the expression of the linearized equations 
for the flow about the symmetric steady state in terms of Chebyshev polynomial
expansions. The stability analysis of these equations results in 
a generalized eigenvalue problem which is solved using standard algorithms.
We find the critical Reynolds numbers for various transitions and our 
predictions are within a few percent above the experimentally observed values,
in agreement with the expectations of Chomaz et al. according to whom their
experiment ought to be underpredicting the values of $Re_c$.
Since the flow was modeled by the two dimensional approximation described
above, the agreement between the predictions of the model and the experiment
lends further credibility to the two dimensional approximation.

 Guided by the results of the linear analysis, we were able to reproduce the
results on transitions among various vortex states as reported, while our
predictions are also quantitatively in agreement with the experiments.


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\noindent
1. M. Rabaud and Y. Couder, J. Fluid Mech., {\bf 136} (1983) 291-319.

\noindent
2. J.M. Chomaz, R. Rabaud, C. Basdevant and Y. Couder, J. Fluid Mech.,
{\bf 187} (1988) 115-140.

\noindent
3. E.A. Coutsias and J.P. Lynov, Physica D., {\bf 51} (1991) 482-497.


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