Numerical solution of steady, symmetric and laminar flow around a circular cylinder

Abstract

A numerical integration of the Navier-Stokes equations is given for the steady, symmetric flow around a circular cylinder. The problem is formulated in terms of a streamfunction and the vorticity. The method used is the semi-analytical one of series truncation, in which the streamfunction and the vorticity are expanded in a finite Fourier sine series with argument β, the polar angle. Substitution of the truncated series into the Navier- Stokes equations yields a system of non linear, coupled, ordinary differential equations for the Fourier coefficients which is subjected to boundary conditions on the cylinder and at infinity. In order to be able to do a numerical calculation the free stream conditions at infinity are replaced by the asymptotic Oseen conditions at a finite distance from the cylinder. The resulting two point boundary value problem for the system of differential equations is solved numerically by a finite difference method. This method gives rise to a non linear system of algebraic difference equations. Four different iteration methods are discussed to solve this algebraic system. The most efficient iteration method seems to be Newton's method, which needs only about three iterations to converge to a solution of the difference equations. The approximation of the solution of the finite difference equations to the exact solution of the differential equations is improved by performing a Richardson extrapolation. It can be concluded that a very efficient scheme has been obtained to solve the system of ordinary differential equations which follow from the application of the method of series truncation. It has been found however that the number of terms in the Fourier series needed to describe the flow adequately and correspondingly the computation time increase considerably with the Reynolds number. Nevertheless, it is believed that the method developed here is much more efficient than previous ones. Calculations have been done for R = 0.5, 2.0, 3.5 and 5.0 where R = Ua/ γ (a is the radius of the cylinder). The results compare reasonably well with previous numerical calculations of Keller-Takami and Dennis-Chang.