Direct solution of the vorticity-stream function ordinary differential equations by a Chebyshev approximation

Abstract A numerical method for solving the coupled vorticity-stream function equations in one dimension with an exact noniterative determination of the vorticity boundary values is presented. The one-dimensional form considered can represent the Fourier modes of a two-dimensional problem. The equations are separated by replacing the derivative specifications for the stream function at the boundary points with equivalent conditions of an integral type for the vorticity. A spectral approximation by means of Chebyshev polynomials is considered. The numerical properties of the algorithm are investigated against a few analytical examples which demonstrate the accuracy of the proposed method.

[1]  V. A. Patel.  Time-dependent solutions of the viscous incompressible flow past a circular cylinder by the method of series truncation , 1976 .

[2]  D. Gottlieb,et al.  Numerical analysis of spectral methods : theory and applications , 1977 .

[3]  L. Quartapelle,et al.  Vorticity conditioning in the computation of two-dimensional viscous flows , 1981 .

[4]  V. A. Patel.  Kármán vortex street behind a circular cylinder by the series truncation method , 1978 .

[5]  U. Schumann,et al.  Treatment of incompressibility and boundary conditions in 3-D numerical spectral simulations of plane channel flows , 1980 .

[6]  G. S. Patterson,et al.  Numerical Simulation of Three-Dimensional Homogeneous Isotropic Turbulence , 1972 .

[7]  Parviz Moin,et al.  On the numerical solution of time-dependent viscous incompressible fluid flows involving solid boundaries , 1980 .

[8]  Steven A. Orszag,et al.  Transition to turbulence in plane Poiseuille and plane Couette flow , 1980, Journal of Fluid Mechanics.

[9]  S. Orszag.  Spectral methods for problems in complex geometries , 1980 .

[10]  R. Glowinski,et al.  Numerical methods for the first biharmonic equation and for the two-dimensional Stokes problem , 1977 .

[11]  S. Dennis,et al.  Calculation of the steady flow past a sphere at low and moderate Reynolds numbers , 1971, Journal of Fluid Mechanics.

[12]  H. B. Keller,et al.  Viscous flow past circular cylinders , 1973 .

[13]  L. Fox,et al.  Chebyshev polynomials in numerical analysis , 1970 .

[14]  D. Ingham,et al.  The steady flow due to a rotating sphere at low and moderate Reynolds numbers , 1980, Journal of Fluid Mechanics.

[15]  Comparison of spectral methods for flows on spheres for flows on spheres , 1979 .

[16]  Dale B. Haidvogel,et al.  The Accurate Solution of Poisson's Equation by Expansion in Chebyshev Polynomials , 1979 .

[17]  Bengt Fornberg,et al.  A numerical study of 2-D turbulence , 1977 .