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 Navier-Stokes equations are proved. Moreover, the error estimates are given. Numerical result is presented.
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References
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.
Yasusi Takada. Quasi-periodic state and transition to turbulence in a rotating Couette system[J],J Fluid Mech, 1999,389(1):81–99.
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.
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.
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.
Wimmer M. Experiments on the stability of viscous flow between two concentric rotating spheres [J].J Fluid Mech, 1980,103(1):117–131.
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.
LI Kai-tai, HUANG Ai-xiang.The Finite Element Method and Its Application[M]. Xi'an: Xi'an Jiaotong University Press, 1988, 320–321. (in Chinese)
LIU Shi-da, LIU Shi-kuo.The Special Function[M]. Beijing: China Meteorological Press, 1988, 202–203. (in Chinese)
YANG Ying-chen.Mathematical Physics Equation and Special Function[M]. Beijing: National Defence Industry Press, 1991,160–162. (in Chinese)
XIAO Shu-tie, JU Yu-ma, LI Hai-zhong.Algebra and Geometry[M]. Beijing: Higher Education Publishing House, 2000,198–199. (in Chinese)
WANG Li-zhou, LI Kai-tai. Spectral Galerkin approximation for nonsingular solution branches of the Navier-Stokes equations[J].Journal of Computation Mathematics, 2002,24(1):39–52. (in Chinese)
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.
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(Communicated by ZHOU Xian-chu, Original Member of Editorial Committee, AMM)
Foundation items: the National Key Basic Research Special Foundation of China (G1999032801–07); the National Natural Science Foundation of China (10101020)
Biography: WANG He-yuan (1963≈)
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He-yuan, W., Kai-tai, L. Spectral galerkin approximation of couette-taylor flow. Appl Math Mech 25, 1184–1193 (2004). https://doi.org/10.1007/BF02439871
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DOI: https://doi.org/10.1007/BF02439871