On the steady flow in a rectangular cavity at large Reynolds numbers: A numerical and analytical study

Steady flow in a rectangular cavity at high Reynolds numbers is numerically and analytically investigated. Numerical simulations are reported up to a maximum Reynolds number, Re, value of 15000 for deep cavities and 20000 for shallow cavities using a compact fourth-order accurate central difference scheme and a stream function-vorticity formulation. At high Reynolds numbers, the eddy structure in shallow cavities consists of counter-rotating primary eddies, with each eddy behaving as an inviscid core with uniform vorticity. For deep cavities, the increase in Reynolds number results in the growth and eventually merger of the corner eddies into new primary eddies. Two merger patterns are identified, a symmetric pattern and an asymmetric pattern depending on a local Reynolds number based on the properties of the bottom primary eddy. A cavity with effectively infinite depth, D=10, is also numerically investigated up to a maximum Re value of 10000. Numerical results indicate that for an infinitely deep cavity and at a large Reynolds number, inertia effects would dominate near the upper moving wall, while Stokes flow behavior would dominate away from the moving wall. An overlap region would exist, in which both inertia and viscous effects are of comparable magnitude. Finally, an analytical solution is developed for the steady flow in a rectangular cavity at large Reynolds numbers. Results from the analytical model are compared to numerical solutions obtained from the full Navier-Stokes equations for both one-sided and four-sided driven cavity configurations.