Numerical Simulation Of Eccentric Taylor-Couette Flow At Low Reynolds Number

We present numerical simulation of Taylor-Couette(TC)flow between eccentric cylinders with only inner cylinder rotating at low Reynolds number.A fourth-order finite difference method is employed on bipolar mesh for cylinders with two radius ratios of 0.5 and 0.9.Numerical verifications have been implemented in three aspects.First,we com-pare the distribution of normalized angular velocity along the radial direction at high Reynolds number.And then,we show the grid convergence at low Reynolds number.Finally the influence of the asepct ratio,which is the ratio of the axial height to the dif-ference between the inner radius and the outer radius,is carefully studied.It is found that for the cases with radius ratio of 0.5,the height ratio of ? is appropriate,while for the cases with radius ratio of 0.9,the height ratio should be set to 3?.In description of flow regimes,we first show the character of four states:circular Couette flow(CCF)with 2D attached streamlines on both inner and outer cylinder,sep-aration flow(SF)with attached lines on inner cylinder while separation/reattachment on outer cylinder.Then eccentric Taylor vortex flow(TVF)and wavy vortex flow(WVF)with separatrices on both inner and outer cylinders.The Q value and streamline dia-grams of the outer cylinder with radius ratio of 0.5 and 0.9 are studied when the three-dimensional flow pattern changes to two-dimensional flow pattern with eccentricity at different Reynolds number.The qualitative analysis of flow regime transition process with different radius ratio is given.It is found that in TVF,there is no wave along the angular direction,while in WVF,the wave number in the angular direction is 6.For transition from TVF and WVF to SF state,first each separatrix denoting separation degenerates into a source point and a saddle point,then each separatrix representing reattachment degenerates into a sink point and a saddle point.This implies a transi-tion from Taylor vortex to 3D separation flow state.This 3D separation then quickly degenerate into 2D separation as shown in SF regime.Discussion of forces on inner cylinder shows that,with inner Reynolds numbers of 50 and 100 and varying eccentricity e(0?e?0.8).Those setups ensure a complete description of all three flow regimes:CCF,SF and TVF.The forces were compared with those in the case of eccentric cylinders with a radius ratio of 0.5.Present results emphasize on the discussion the forces exerted on the inner cylinder.Forces on x direc-tion(perpendicular to the line connecting two centers)increases with e,while forces on y direction(along the line connecting two centers)first increase,then decrease.When regression of 3D TC flow to 2D separation flow is discussed,forces on x direction at cases around the critical eccentricity shows a plateau.For pressure force and viscous force,the ratio between pressure and resultant force in x direction increases with e,while this ratio in y direction is close to unit.Furthermore,when e goes to zero,ratio in x direction differs with radius ratio:it is close to unity for large radius ratio while decrease quickly with decreasing radius ratio(in present study is 0.8 for radius ratio of 0.5).We note that for 3D TC flow cases,this ratio is even smaller.The angular momen-tum transport of eccentric TVF and WVF will appear an approximate plateau region at the wide side,but the variation trend is almost the same at the narrow side.For the torque of the inner cylinder,when the flow state is two-dimensional,cases at both the radius ratio 0.5 and 0.9 agree well with the previous theoretical results.In the case of radius ratio 0.5,when TVF occurs,the torque is almost constant with eccen-tricity,and the torque of different Reynolds number can be normalized by relation of Rei1.5.In the case of radius ratio 0.9,the appearance of WVF makes the analyze more complicate.