1. Introduction

A number of engineering problems involve flows of gases or liquids over solid bodies. For example: air flows over cars and aeroplanes; wind blowing over bridges and buildings; sea waves slashing against the supporting columns of an off-shore oil rig and many more. Often these flows do not follow the contour of the solid surface completely, but separate from it, creating a wake such as behind a ship. Such separated flows are difficult to handle by conventional numerical schemes. The objective of this work is to develop a numerical procedure to solve such flows. It is based on determination of the vorticity, defined as the curl of the flow velocity. According to Stokes [225] vorticity is the twice the local angular velocity of the fluid if it moves as a solid body.

The main reason to base the numerical method on the vorticity is that, typically, only a small portion of the flow contains vorticity. This can lead to significant savings in storage and computational effort. Our numerical method will compute the evolution of vorticity using a Lagrangian approach, in which the computational points follow the motion of the fluid. Such a method is commonly called a vortex method, and the computational points are the vortices.

Vortex methods can offer significant advantages for the computation of separated flows:

• We need to describe only the small portion of the flow region where vorticity occurs. Typical conventional methods must resolve the whole flow.

• In our implementation no grid is required. This makes it easier to handle flows around complicated geometries.

• If the vorticity separates from the solid surfaces, the vortices follow that motion, so that numerical resolution is maintained.

• In a conventional computation, the motion of the vorticity must be found by numerical approximation on a fixed grid, which can induce significant numerical dissipation. A vortex method avoids this since the vortices follow the motion of the fluid.

• In the computation of the velocity field from the vorticity, mass conservation can be satisfied exactly; see, for example, Van Dommelen & Rundensteiner [233]. Gresho [98,101] has shown that mass conservation can be a cause of difficulty in other computations.

• In external flows, most methods need to restrict the computational domain artificially to a finite size. But as long as the vorticity remains limited to a finite region, vortex methods need not.

These advantages are most often critical for high Reynolds number flows, which are commonly separated. An important concern at high Reynolds numbers is that the numerical dissipation should not overwhelm the natural viscous diffusion process and destroy the small scale features. Such requirements make vortex methods a natural choice. Yet, the implementation of vortex methods is not simple. The two main physical processes that must be represented numerically are convection of the vorticity by the velocity field and diffusion due to viscosity. Each has its difficulties.

For convection of the vortices, the velocity field can in principle be found from the Biot-Savart law [14]. Such a velocity field implicitly satisfies mass conservation. However, the computational effort required to evaluate it directly is high; it is proportional to the square of the number of vortices. Fast algorithms have been developed to do it with much less effort. Van Dommelen & Rundensteiner [233,240] developed the first `solution adaptive' fast method that could efficiently handle the sparse and complex vorticity distributions of high Reynolds number separated flows. An earlier non-adaptive routine was given by Greengard & Rokhlin [97]. Even faster algorithms are available now [2,5,10,36,74].

However, representing diffusion is a more difficult problem; one of the main difficulties is the chaotic distribution of the vortices. It is this difficulty that is the main topic of this thesis. In this thesis, we have developed an accurate mesh-free procedure to overcome that difficulty. We next discuss the importance the mesh-free property in Lagrangian methods.