Existence regime of symmetric and asymmetric Taylor vortices in wide-gap spherical Couette flow

Abstract

We study the existence regime of symmetric and asymmetric Taylor vortices in wide-gap spherical Couette flow by time marching the three-dimensional incompressible Navier–Stokes equations numerically. Three wide-gap clearance ratios, $\beta =\left( R_{2}-R_{1}\right) /R_{1}=0.33$, 0.38 and 0.42 are investigated for a range of Reynolds numbers respectively. Using the 1-vortex flow for clearance ratio $\beta =0.18$ at Reynolds number ${Re}=700$ as the initial conditions and suddenly increasing $\beta$ to the target value, we can compute Taylor vortices for the three wide gaps. For $\beta =0.33$, Taylor vortices exist in the range $450\le {Re}\le 2050$. With increasing Re the steady symmetric 1-vortex flow becomes steady asymmetric at ${Re}=1850$, and then become periodic at ${Re}=2000$. When ${Re}>2050$ the flow returns back to the steady basic flow state with no Taylor vortices. For $\beta =0.38$, Taylor vortices can exist in the range $500\le {Re}\le 1400$. With increasing Re, the steady symmetric 1-vortex flow become steady asymmetric at ${Re}=1200$, and then the flow evolves into the steady basic flow for ${Re}>1400$. For $\beta =0.42$, Taylor vortices can exist in the range $650\le {Re}\le 1300$. With increasing Re, steady asymmetric Taylor vortices occur at ${Re}=1150$, and then the flow evolves into the steady basic flow for ${Re}>1300$. The present numerical results are in good agreement with available numerical and experimental results. Furthermore, the existence regime of Taylor vortices in the $(\beta ,{Re})$ plane for $\beta \ge 0.33$ and the three-dimensional transition process from periodic asymmetric vortex flow to steady basic flow with increasing Re are presented for the first time.