A numerical study of the transition to oscillatory flow in 3D lid-driven   cubic cavity flows
  In this article, three dimensional (3D) lid-driven cubic cavity flows have been studied numerically for various values of Reynolds number ($Re$). The numerical solution of the Navier-Stokes equations modeling incompressible viscous fluid flow in a cubic cavity is obtained via a methodology combining a first order accurate operator-splitting, $L^2$-projection Stokes solver, a wave-like equation treatment of the advection and finite element methods. The numerical results obtained for Re$=$400, 1000, and 3200 show a good agreement with available numerical and experimental results in literature. Simulation results predict that the critical Re$_{cr}$ for the transition from steady flow to oscillatory (a Hopf bifurcation) is somewhere in [1870, 1875] for the mesh size $h=1/96$. Via studying the flow field distortion of fluid flow at Re before and after Re$_{cr}$, the occurrence of the first pair of Taylor-G\"ortler-like vortices is connected to the flow field distortion at the transition from steady flow to oscillatory flow in 3D lid-driven cubic cavity flows for Re $< 2000$. 
Shang-Huan Chiu, Tsorng-Whay Pan, Jiwen He, Aixia Guo, and Roland   Glowinski
physics.flu-dyn