Nonlinear three-dimensional flow in the lid-driven square cavity

Abstract

The three-dimensional flow in a lid-driven cuboid is investigated numerically. The geometry is an extension to three dimensions of the lid-driven square cavity by translating the two-dimensional lid-driven cavity parallel to the third orthogonal direction. The incompressible Navier–Stokes equations are discretized by a pseudospectral Chebyshev-collocation method. The singularities caused by the discontinuous velocity boundary conditions are reduced by including asymptotic analytical solutions in the solution ansatz. The flow is computed for Reynolds numbers above the critical onset of Taylor–Görtler vortices. Nonlinear Taylor–Görtler cells are calculated for periodic and for realistic no-slip endwall conditions. For periodic boundary conditions the bifurcation is either sub- or supercritical, depending on the wavenumber. The limiting tricritical case arises near the critical wavenumber of the linear-stability problem. On the other hand, no-slip endwall conditions have a significant effect on the supercritical three-dimensional flow. In agreement with recent experimental results we find that Taylor–Görtler vortices are suppressed near no-slip endwalls.