The prediction of time-dependent, separated flow around cylindrical bodies has important applications in both aero and marine engineering Numerical methods may be broadly divided into Eulerian and Lagrangian schemes. The Lagrangian vortex method has several advantages over Eulerian schemes It is less prone to instability and there is no mesh-generated artificial viscosity. It is thus well suited to high Reynolds number flows Furthermore, it is automatically adaptive, with the greatest density of discrete vortices forming in regions of physical interest.
In modelling separated flows by an inviscid vortex method, a common approach is to discretize only the vortiaty in the wake, where viscous effects are neglected New discrete vortices are generated at the boundary-layer separation points, which are fixed either at sharp edges (Stansby, 1985), or at positions determined from an empirical boundary-layer calculation (Sarpkaya and Shoaff, 1979) These methods have the disadvantage that vorticity, which is created and entrained into the flow downstream of separation, is neglected Spalart et al (1983) incorporated a finite-difference boundary-layer calculation, which was interfaced with an inviscid vortex method at a small distance from the wall A more general approach is that of Chorin (1973) viscous effects are incorporated into the vortex method itself, which may then be used to represent both the wake and the boundary layer. In this fractional-step scheme, vortices are created along the cylinder surface in order to satisfy the dynamic boundary condition The process of vorticity diffusion is simulated by adding a random walk to the position of each vortex Finally, the vortices are convected in their mutually induced velocity field Accurate flow simulations require a very large number of vortices, and their convection may be handled efficiently by the vortex-in-cell method (Christiansen, 1973).
The use of a surface-fitted mesh in the vortex-in-cell calculation enables the zero velocity boundary condition to be satisfied simply and accurately Smith (1986) computed the flow around a circular cylinder using an exponentially expanding polar mesh, defined over an annular domain external to the cylinder. This gives fine resolution in the boundary layer where it is required and coarse definition in the far field In this chapter, we extend the use of the polar mesh to the simulation of flows around cylinders of more general shape The computations are performed in a transformed complex plane, the region outside the cylinder in the physical plane is mapped conformally on to the region outside the cylinder in the transformed plane. The vorticity field is modelled, in the physical plane, as a distribution of vortices outside the cylinder and, in the transformed plane, as a distribution of vortices, with unchanged circulations, at corresponding points outside the unit circle The use of conformal mapping requires modifications to the expressions for the random perturbation and convection of the vortices.
Smith and Stansby (1987) used a numerically generated transformation to compute the flows around cylinders of several geometries, but were restricted to shapes without sharp edges.