Numerical simulation of the Taylor-Couette flow between eccentric cylinders
Abstract
We present the numerical simulation of the Taylor-Couette (TC) flow between eccentric cylinders with only the inner cylinder rotating at low Reynolds numbers. A fourth-order finite difference method was used with a bipolar mesh for cylinders with two radius ratios of 0.5 and 0.9, inner Reynolds numbers of 50 and 100, and varying eccentricity $e$ ($0\le~e\le0.8$). These conditions ensured a complete description of all three flow regimes: the circular Couette flow (CCF), separation flow (SF), and eccentric Taylor vortex flow (TVF). In numerical verifications, the velocity profiles were compared with those in the case of a concentric cylinder, and the forces were compared with those in the case of eccentric cylinders with a radius ratio of 0.5. The present results emphasize on the flow regimes transition and the forces exerted on the inner cylinder. With regard to the description of flow regimes, we first elucidated the characters of three states: the CCF with two-dimensional (2D) streamlines attached to both the inner and outer cylinders, SF with streamlines attached to the inner cylinder while separation/reattachment occurs at the outer cylinder, and TVF with separatrices on both the inner and outer cylinders. For a transition from the TVF to the SF state, first, each separatrix denoting separation degenerates into a source point and saddle point; then, each separatrix representing reattachment degenerates into a sink point and saddle point. This two-step transition implies a transition from the Taylor vortex to a three-dimensional (3D) separation state. This 3D separation then quickly degenerates into 2D separation, as shown in the SF regime. Discussion of the forces on the inner cylinder showed that forces in the $x$ direction (perpendicular to the line connecting the two centers) increase with the increase in $e$, whereas the forces in the $y$ direction (along the line connecting the two centers) first increase and then decrease. When the regression of the 3D TC flow to a 2D SF is examined, forces in the $x$ direction around the critical eccentricity exhibit a plateau. For pressure and viscous forces, the ratio of the pressure and resultant force in the $x$ direction increases with $e$, whereas in the $y$ direction, this ratio is close to unity. Furthermore, when $e$ approaches zero, the ratio in the $x$ direction is a function of the radius ratio: it is close to unity for large radius ratios and decreases quickly as the radius ratio decreases (in the present study, the ratio of pressure and resultant force in the $x$ direction is 0.8 for a radius ratio of 0.5). For 3D TC flow cases, this ratio is even smaller.
