偏心圆柱Taylor-Couette流动的数值模拟研究

Numerical simulation of the Taylor-Couette flow between eccentric cylinders

本文采用四阶有限差分的数值模拟方法研究了低雷诺数下偏心双圆柱间的Taylor-Couette流动,通过选取两个半径比(0.5和0.9),两个内圆柱雷诺数(50和100)并改变偏心度e (0≤e≤0.8),详细讨论了随着偏心度的增长,流动形态的变化和内圆柱受力的变化.模拟结果完整地描述了层流、分离流和偏心泰勒涡流动三种不同流动形态.在流态变化方面,通过流线图分析发现,三维泰勒涡流转变为三维分离流,然后退化为二维分离流的过程.在内圆柱受力方面,从层流到分离流的过程,发现在与圆心连线相垂直的方向即x方向,合力绝对值因不断增长的挤压效应持续增大;在圆心连线方向即y方向,合力绝对值由于惯性影响呈现先增大后减小趋势.此外,在三维泰勒涡流退化为二维分离流的过程中,临界偏心度附近内圆柱在x方向受力形成平台,变化很小. x方向的压力占比随偏心度持续增大.在偏心度趋于零的情况下, x方向的压力占比随半径比而不同:大半径比层流的压力占比接近1(半径比为0.9时压力占比约为0.97),小半径比层流的压力占比减小很多(半径比为0.5时压力占比约为0.8),在三维泰勒涡流的情况下的压力占比则更小.

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 ≤ e ≤0.8). 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.