Oscillatory eddy structure in a container

Abstract

We consider the periodic, two-dimensional motion of a viscous, incompressible liquid which fills a rectangular container. The motion is due to the periodic motion of the lid which moves in its own plane. If the velocities are sufficiently small the motion will be governed by the linearized Navier–Stokes equations and consequently the dimensionless stream function $\Psi(x,z,t)\,{=}\,\psi(x,z){\rm e}^{{\rm i}t}$ will satisfy the equation $ \nabla^4 \psi - {\rm i}\hbox{\it Re} \nabla^2 \psi\,{=}\,0$, where ${\it Re}$ is the Reynolds number. If we then seek separable solutions for $\psi (x,z)$ that satisfy the no-slip conditions on the sidewalls, it is easy to show that the problem reduces to the eigenvalue problem $$\lambda \tan{\textstyle\frac{1}{2}}\lambda\,{=}\,\sqrt{\lambda^2 - {\rm i} \hbox{\it Re}} \tan {\textstyle{1 \over 2}} \sqrt{\lambda^2 - {\rm i} \hbox{\it Re}}$$ where $\lambda$ is the eigenvalue. A detailed analysis is made of this eigenvalue problem. All the eigenvalues are complex; all eigenvalues with positive real part either belong to a set $\{\lambda_n^u \}$ in the upper half-plane or to another $\{\lambda_n^l\}$ in the lower half-plane. They satisfy the important relationship $\lambda_n^l\,{=}\,\sqrt{\overline{\lambda_n^u}^2 + {\rm i} \hbox{\it Re}}.$ We show by an asymptotic analysis that while the $\lambda_n^l$ move to the neighbourhood of the real axis as $\hbox{\it Re}\rightarrow \infty$, the $\lambda_n^u$ move away from the origin and approach the line $\lambda_i\,{=}\,\lambda_r$ in the complex-$\lambda$-plane. This fact has an important bearing on the damping of gravity waves at high Reynolds numbers. The eigenfunctions derived above are used to write down a formal expansion for the stream function and the coefficients are determined from the boundary conditions using a least-squares procedure. An examination of the resulting streamline patterns reveals interesting inertial effects even at low Reynolds numbers. In particular we examine the mechanism by which the flow field reverses its direction when the lid stops and reverses its direction of motion. If inertial effects are completely negelected, as has been done till now, one would infer an immediate response of the fluid to the changes in the lid motion; for example, one would conclude, wrongly, that when the lid is at rest so is the fluid. Our analysis shows, in fact, a very intricate and beautiful mechanism, involving among other things an apparent engulfing of the corner eddy by the new primary eddy, by which the direction of the circulation is reversed in the fluid. These results should be of importance in the analysis of mixing, where such effects appear to have been ignored till now.