Analysis of the Modified Oseen Equations Using Orthogonal Function Expansions

CANADIAN APPLIED MATHEMATICS QUARTERLY Volume 1, Number 4, Fall 1993

ANALYSIS OF THE MODIFIED OSEEN EQUATIONS USING ORTHOGONAL FUNCTION EXPANSIONS

S. KOCABIYIK

ABSTRACT. The t-dimensional Navier-Stokes equations, describing a steady flow of a viscous incompressible fluid past a circular cylinder, are analyzed by using an orthogonal function expansion in an angular variable on the basis of the modified Oseen approximation due to Southwell and Squire. The analysis is developed in terms of the scalar vorticity and stream function. An asymptotic solution to the vorticity is obtained separately from the stream function using modified Mathieu functions, the determination of which can be effected by suc- ceasive approximations in terms of the low Reynolds number. The usefulness of the modified Oseen type of successive approximation in dealing with the Navier-Stokes equations is confirmed by estimating the critical value of the Reynolds number, and some suggestions for extension to the full Navier- Stokes equations are made.

1. Introduction. It is well known that the steady two-dimensional flow of a viscous fluid past a cylindrical body at small Reynolds number can be discussed on the basis of the standard Oseen approximation. Expansion formulas in terms of Reynolds number were first obtained for a circular cylinder by Lamb [ll]in 1911, later by Bairstow et al. [I],Harrison [lo], Filon [8], Fax& [7], Griffith [Q], Sidrak [14], To- motika and Aoi [17, 181, Yamada [19], Skinner [15], Ranger [13]and, more recently, Dennis and Kocabiyik [6]. Southwell and Squire [16] have proposed a modification of the Oseen equations in which the fist approximation to the modified Oseen solution agrees with the first approximation to the standard Oseen solution. In the modified Oseen the- ory, the equations are linearized by introducing a perturbation stream function. Unliie the standard Oseen linearization from the external stream, in which the perturbation stream function satisfies none of the boundary conditions at the cylinder surface, this linearization satisfies one of the conditions of no slip. Southwell and Squire solved the modified equations by approximate methods (largely graphical) for the case

Received by the editors on August 17, 1992, and in revised form on February 18, 1993. Copyright 01993 Rocky Mountain Mathematics Consortium 476 S. KOCABIMK of the flow around a circular cylinder. The method adopted in their work was that of adding a number of separate solutions of the modified vorticity equation and combining them so that the boundary conditions are satisfied at a finite number of points. Southwell and Squire obtained five particular solutions of the vorticity equation and satisfied the boundary conditions at five points only. In this paper the problem of modified Oseen flow past a circular cylinder is analyzed. A prior step in the analysis is the use of a set of orthogonal functions in an angular variable which enables us to formu- late an expansion for the scalar vorticity. The basic governing equation for the orthogonal functions is obtained by applying a technique which is similar to the separation of variables and which is suggested by a perturbation from the potential flow. By means of these orthogonal functions, related to the Mathieu functions, the modified Oseen linearized equation for the vorticity can be reduced to an infinite set of ordinary differential equations in one space variable. The solutions of these equations are determined so that the vorticity will have the correct form in some asymptotic limit. In the present case the asymp totic limit is chosen as the solution valid in the limiting domain at large enough distances from the body. The method is based on the satisfaction of necessary conditions involving integrals of the vorticity throughout the flow region. This is considered as a very important part of the solution procedure, since the integral conditions ensure both the correct decay of the vorticity at large distances from the cylinder and satisfaction of the physically essential results for the existence of the flow. These conditions are derived by consideration of the Poisson equation governing the stream function together with its boundary conditions, but without explicitly obtaining a solution. Also, according to our solution for the modified Oseen problem, the critical value of the Reynolds number at which the standing vortex-pair formed is found to be in qualitative agreement with the numerical solutions (see, for example, Dennis and Chang [5] of the full Navier-Stokes equations).

2. Governing equations and boundary conditions. The origin is fixed at the center of the cylinder, and modified polar coordinates (t,8) are used where ( = log(r/a) and a is the radius of the cylinder, so that 8 = 0 coincides with the axis of the cylinder in the downstream direction.