V形空腔内Moffatt涡旋的数值模拟

Numerical Simulation of Moffatt Vortices in a V-shaped Notch

应用边界元方法 ,对低雷诺数粘性不可压流体在V形空腔内的流动问题进行数值分析 ,数值模拟V形空腔内流体的流场表明 :流场内出现Moffatt涡 ,所形成的两个较大涡旋不但大小不等、强度差别很大而且旋转方向相反 ,其中大涡的涡心坐标为 y1=4 .4cm ,小涡的涡心坐标为 y2 =2 .2cm ,第一个涡的分离点坐标为Y1=4 .8cm ,两涡的分离点坐标为Y2 =2 .4cm ,亦即 :y2 =y1/ 2 ,Y2 =Y1/ 2 ,数值计算结果与Taneda的实验结果比较完全一致。特别是我们还发现V形空腔内不仅仅有两个涡旋 ,而是应有强度急剧减弱的n个涡旋 ,各涡的涡心坐标 yn 以及相邻两涡的分离点坐标Yn 满足如下条件 :yn +1=yn/ 2 ,Yn +1=Yn/ 2。数值计算结果表明在V形空腔内的第四个涡旋的强度已经非常弱 ,其数量级已经是 10 - 4,因此我们有理由认为对θ =2 8.5°的V形空腔有nmax=4。

The principal intent of this research is to observe and investigate Moffatt vortices in the wedge-shaped region,and so a numerical study was presented for low-Reynolds-number incompressible viscous fluid flow near a V-shapes notch by boundary element method.The numerical simulation of velocity field show that the Moffatt vortices formed by rolling a cylinder over the V-shaped notch.We can see that a largest vortex produced by the rotating cylinder and a next smaller one induced by it .The direction of rotation of the small vortex is opposite to that of the large vortex,and the distance between the centre of the small vortex and the corner point is 0.5 times the distance between the centre of the large vortex and the corner point.We also found that the distance between the dividing streamline between two vortices and the corner point is 0.5 times the distance between the dividing streamline between the large vortex and the cylinder and the corner point.By comparison,the numerical simulation of the velocity fields is in good agreement with Taneda’s experiment.The numerical results indicate that the fluid flow in the wedge-shaped region consists of and infinite sequence of vortices of rapidly diminishing strength,and also we find that y n+1 =y n/2,Y n+1 =Y n/2,where y n is the centre coordinate of the vortex ,Y n is the coordinate of dividing streamline between two vortices,but for the V-shaped notch with θ=28.5° the maximum n is 4 thus n max =4,because of the strength of fourth vortex is 10 -4 .In particular,the numerical invetigation show that the method is very effective in resseaching and solving the problem of the fluid flow near a V-shaped notch with variable θ and h,furthermore,finding the condition of appearance Moffatt vortices by using the method.The question of especial importance is the method has some advantages of computational convenience,the surperior occuracy and the decreasing of the numerical computational work.