Spectral Galerkin Approximation of Couette-Taylor Flow
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摘要: 利用谱方法对轴对称的旋转圆柱间的Couette-Taulor流进行数值模拟．首先给出Navier-Stokes方程流函数形式，利用Couette流把边界条件齐次化．其次给出Stokes算子的特征函数的解析表达式，证明其正交性，并对特征值进行估计．最后利用Stokes算子的特征函数作为逼近子空间的基函数，给出谱Galerkin逼近方程的表达式．证明了Navier-Stokes方程非奇异解的谱Galerkin逼近的存在性、唯一性和收敛性，给出了解谱Galerkin逼近的误差估计，并展示了数值计算结果．
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关键词:
- Navier-Stokes方程 /
- Coutte-Taylor流 /
- 谱逼近 /
- Stokes算子
Abstract: Axisymmetric Couette-Taylor flow between two concentric rotating cylinders was simulated numerically by the spectral method.First,stream function form of the Navier-Stokes equations which homogeneous boundary condition was given by introducing Couette flow.Second,the analytical expressions of the eigenfunction of the Stokes operator in the cylindrical gap region were given and its orthogonality was proved.The estimates of growth rate of the eigenvalue were presented.Finally, spectral Galerkin approximation of Couette-Taylor flow was discussed by introducing eigenfunctions of Stokes operator as basis of finite dimensional approximate subspaces.The existence,uniquence and convergence of spectral Galerkin approximation of nonsingular solution for the steady-state NavierStokes equations are proved.Moreover,the error estimates are given.Numerical result is presented.-
Key words:
- Navier-Stokes equation /
- Couette-Taylor flow /
- spectral approximation /
- Stokes operator
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[1] Anderreck C D,Liu S S,Swinney H L.Flow regimes in a circular Couette system with independent rotating cylinders[J].J Fluid Mech,1986,164(1):155—183. doi: 10.1017/S0022112086002513 [2] Yasusi Takada.Quasi-periodic state and transition to turbulence in a rotating Couette system[J].J Fluid Mech,1999,389(1):81—99. doi: 10.1017/S0022112099005091 [3] Coughlin K T,Marcus P S,Tagg R P,et al.Distinct quasiperiodic modes with like symmetry in a rotating fluid[J].Phys Rev Lett,1991,66(1):1161—1164. doi: 10.1103/PhysRevLett.66.1161 [4] Meincke O,Egbers C.Routes into chaos in small and wide gap Taylor-Coutte flow[J].Phys Chem Earth B,1999,24(5):467—471. [5] Takeda Y,Fischer W E,Sakakibara J.Messurement of energy spectral density of a flow in a rotating Couette system[J].Phys Rev Lett,1993,70(1):3569—3571. doi: 10.1103/PhysRevLett.70.3569 [6] Wimmer M.Experiments on the stability of viscous flow between two concentric rotating spheres[J].J Fluid Mech,1980,103(1):117—131. [7] Taylor G I.Stability of a viscous liquid contained between two rotating cylinders[J].Phil Trans Roy Soc London A,1923,223(1):289—343. doi: 10.1098/rsta.1923.0008 [8] 李开泰，黄艾香.有限元方法及其应用——发展及应用[M].西安：西安交通大学出版社，1988，320—321. [9] 刘式达，刘式适.特殊函数[M].北京：气象出版社，1988，202—203. [10] 杨应辰.数学物理方程 特殊函数[M].北京：国防工业出版社，1991，160—162. [11] 萧树铁，居余马，李海中.代数与几何[M].北京：高等教育出版社，2000，198—199. [12] 王立周，李开泰.Navier-Stokes方程非奇异解分支的谱Galerkin逼近[J].计算数学，2002，24(1)：39—52. [13] Brezzi F,Rappaz J,Raviart P A.Finite dimensional approximation of nonlinear problem—Part Ⅰ:branches of nonsingular solution[J].Numer Math,1980,36(1):1—25. doi: 10.1007/BF01395985 -
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