# CAIMS Prize Award

Department of Applied Mathematics, University of Western Ontario

Supervisor: S. C. R. Dennis

Thesis title: Steady Asymmetric Flow of a Viscous Fluid Past a Cylinder.

Abstract:

The steady flow of viscous incompressible fluid past a cylinder is considered. In particular, asymmetric flows, which involves a lift force in addition to the drag, are investigated. The Navier-Stokes equations and the equation of continuity are formulated in terms of the stream function and vorticity. The work may be divided into two parts, an asymptotic solution of the governing equations of motion at far distances from the cylinder, and a numerical solution of these equations near the cylinder. An asymptotic solution to the vorticity is obtained in the form of a Hermite polynomial expansion. This solution is in perfect agreement with Imai (1953). The asymptotic solution involves unknown constants (namely the lift and drag coefficients) which must be obtained from a numerical solution valid near the cylinder. Leading terms of this solution are then used as the boundary condition at far distances in the numerical calculations.

A study of the asymptotic solution illustrates the difficulty in obtaining a numerical solution in the far wake. This is especially prevalent in asymmetric flow calculations as an errors occurring here may not be damped out. To ensure that the numerical solution properly matches the asymptotic solution the is transformed and the grid size in the angular direction is reduced in the far wake. This s is done in a manner so as to minimize the total number of grid points but allowing a fine enough angular grid size in the far wake to be used. A numerical solution is then obtained from which the asymptotic boundary condition is continuously updated.

Numerical results are given for the flows past a rotating circular cylinder and an inclined elliptic cylinder for Reynolds number 5 and 20. The influence of the asymptotic solution and grid size upon the numerical solutions is investigated. In the case of the rotating circular cylinder, results are given that are in agreement with unsteady results that are taken to a limit in large time (Badr et al., (1998)). The flow past an elliptic cylinder with ratio of minot to major axes of 0.2 is investigated for various angles of incidence. Results of the symmetric problem are compared to Dennis and Chang (1969). The dependence of the life, drag and moment upon the Reynolds number and angle of incidence is studied. For both types of cylinder, surface pressure and vorticity distributions given. In addition, plots of streamlines and lines of equivorticity for the flow past the cylinder are presented.