Numerical study of transitions in lid-driven flows in shallow cavities

Abstract

In this article, three dimensional (3D) lid-driven flow in shallow cavities with a unit square base are studied. The numerical solution of the Navier–Stokes equations modeling incompressible viscous fluid flow in a cavity is obtained via a methodology combining a first order accurate operator-splitting scheme, a $L^2$-projection Stokes solver, a wave-like equation treatment of the advection and finite element space approximations. Numerical results of a lid-driven flow in a cubic cavity show a good agreement with those reported in literature. The critical Reynolds numbers ($\mathit{Re}_{\mathrm{cr}}$) for having flow with increasing of oscillating amplitude (a Hopf bifurcation) in different shallow cavities are obtained and associated oscillating modes are studied.

Keywords

• Lid driven cavity flow
• Shallow cavity
• Taylor–Görtler-like vortices
• Hopf bifurcation
• Projection method