Numerical investigations of the nature of the first bifurcation for the flow in an annular rotor/stator cavity

Direct numerical simulation is performed to study a transitional flow in an annual rotating cavity of different aspect ratios $L=4.0$, 5.0 and different curvature parameters $R_m=(R_1+R_0)/(R_1-R_0)=1.5-6.0$. This paper reports on the influence of the curvature parameter $R_m$ and end-wall boundary layers on the nature of the first bifurcation to unsteadiness and on instability structures in the rotor and stator boundary layers. For all considered end-wall boundary conditions, we have observed clearly supercritical transition to unsteadiness for larger $R_m$ and direct transition from a steady flow to chaotic one for small $R_m$. A spectral collocation method based on the Chebyshev polynomial is used for solving the incompressible Navier-Stokes equations. The time scheme is semi-implicit and second-order accurate; it corresponds to a combination of the second-order backward differentiation formula for the viscous diffusion terms and the Adams-Bashforth scheme for the non-linear terms. The method uses a projection scheme to maintain the incompressibility constraint.