Relaxation Methods Applied to Determine the Motion, in Two Dimensions, of a Viscous Fluid Past a Fixed Cylinder

In this paper relaxational techniques are applied to the general case of steady laminar motion of an incompressible viscous fluid past a stationary cylinder: that is, to motion at speeds such that neither inertia nor viscosity can be neglected. The governing equation is υδ/δx+νδ/δy−νΔ2ζ = 0, where Δ2 = δ2/δx2+δy2/δ2, υ and ν are the component velocities, ν is the kinematio viscosity, and ζ (the vorticity) = δν/δx−δυ/δy. It must be solved in conjunction with the equation of continuity δυ/δx+δ/δy = 0, which permits the introduction of a stream-function π such that υ = δπ/δy, ν = −δπ/δx, ζ =−Δ2π. The numerical computations relate to a circular cylinder, but the methods are applicable to any shape (an initial conformal transformation changes the independent variables from x and y to α and β, the irrotational velocity-potential and stream-function for flow past the specified cylinder). The flow-patterns (contours of π and ζ) change as the ‘Reynolds number’ R increases; but an introduction of variables involving R makes the change relatively slow, and thereby (e.g.) the accepted solution for R = 10 is made a good starting assumption for R = 100. Fig. 6 relates computed values of the total drag with experimental and other theoretical estimates.