In numerical simulations it is desirable to use numerical methods that are well suited to the physics of the problem at hand. As the dominant physics of a flow can vary in different parts of the domain, it is often advantageous to implement hybrid numerical schemes.

In this chapter we discuss hybrid numerical methods that combine, to various extents, vortex methods with Eulerian grid-based schemes. In these hybrid schemes, Lagrangian vortex methods and Eulerian schemes may be combined in the same part of the domain, in which each method is used in order to discretize different parts of the governing equations. Alternatively, vortex methods and grid-based methods can be combined in the same flow solver, in which each scheme resolves different parts of the domain. In this case we will discuss domain-decomposition formulations. Finally we consider the case of using different formulations of the governing equations in different parts of the domain. In that context we discuss the combination of the velocity–pressure formulation (along with grid-based methods) and the velocity–vorticity formulation (along with vortex methods) for the governing Navier–Stokes equations.

For simplicity, we often use in this chapter the terminology of finite-difference methods but it must be clear that in most cases the ideas can readily be extended to other Eulerian methods, such as finite-element or spectral methods.

One of the attractive features of vortex methods is the replacement of the nonlinear advection terms with a set of ordinary differential equations for the trajectories of the Lagrangian elements, resulting in robust schemes with minimal numerical dissipation.

Vortex methods were originally conceived as a tool to model the evolution of unsteady, incompressible, high Reynolds number flows of engineering interest. Examples include bluff-body flows and turbulent mixing layers. Vortex methods simulate flows of this type by discretizing only the vorticity-carrying regions and tracking the computational elements in a Lagrangian frame. They provide automatic grid adaptivity and devote little computational effort to regions devoid of vorticity. Moreover the particle treatment of the convective terms is free of numerical dissipation.

Thirty years ago simulations using inviscid vortex methods predicted the linear growth in the mixing layer and were able to predict the Strouhal frequency in a variety of bluff-body flow simulations. In three dimensions, we have seen that inviscid calculations using the method of vortex filaments have provided us with insight into the evolution of jet and wake flows. However, the inviscid approximation of high Reynolds number flows has its limitations. In bluff-body flows viscous effects are responsible for the generation of vorticity at the boundaries, and a consistent approximation of viscous effects, including diffusion, is necessary at least in the neighborhood of the body. In three-dimensional flows, vortex stretching and the resultant transfer of energy to small scales produce complex patterns of vortex lines. The complexity increases with time, and viscous effects provide the only limit in the increase of complexity and the appropriate mechanism for energy dissipation. In this chapter we discuss the simulation of diffusion effects in the context of vortex methods.

The role of the theory of flow of an inviscid fluid

We have completed a study of the general effects of the viscosity of the fluid, and are now in a position to take advantage of the fact that the viscosities of the common fluids air and water are quite small. The Reynolds number ρLU/µ (in the notation of §4.7) is usually a measure of the ratio of the representative magnitude of inertia forces to that of viscous forces; and, when this Reynolds number is large compared with unity, viscous forces frequently play a negligible part in the equation of motion over nearly all the flow field. In many cases in which separation of the boundary layer from a rigid boundary does not occur, the flow field tends to the form appropriate to an inviscid fluid, as ρLU/µ → ∞, over the whole of the region occupied by the fluid, and the fact that viscous forces remain significant in certain thin layers in the fluids however large the Reynolds number may be, is of little consequence for many purposes. However, in cases in which the boundary layer does separate from a rigid boundary, the limit is a singular one and, although the region of fluid in which viscous forces are significant may decrease in size to zero as ρLU/µ ∞, the limiting form of the flow field is not the same as that appropriate to a completely inviscid fluid.

While teaching fluid dynamics to students preparing for the various Parts of the Mathematical Tripos at Cambridge I have found difficulty over the choice of textbooks to accompany the lectures. There appear to be many books intended for use by a student approaching fluid dynamics with a view to its application in various fields of engineering, but relatively few which cater for a student coming to the subject as an applied mathematician and none which in my view does so satisfactorily. The trouble is that the great strides made in our understanding of many aspects of fluid dynamics during the last 50 years or so have not yet been absorbed into the educational texts for students of applied mathematics. A teacher is therefore obliged to do without textbooks for large parts of his course, or to tailor his lectures to the existing books. This latter alternative tends to emphasize unduly the classical analytical aspects of the subject, and the mathematical theory of irrotational flow in particular, with the probable consequence that the students remain unaware of the vitally important physical aspects of fluid dynamics. Students, and teachers too, are apt to derive their ideas of the content of a subject from the topics treated in the textbooks they can lay hands on, and it is undesirable that so many of the books on fluid dynamics for applied mathematicians should be about problems which are mathematically solvable but not necessarily related to what happens in real fluids.

Specification of the flow field

The continuum hypothesis enables us to use the simple concept of local velocity of the fluid, and we must now consider how the whole field of flow may be specified as an aggregate of such local velocities. Two distinct alternative kinds of specification are possible. The first, usually called the Eulerian type, is like the specification of an electromagnetic field in that the flow quantities are defined as functions of position in space (x) and time (t), The primary flow quantity is the (vector) velocity of the fluid, which is thus written as u(x, t). This Eulerian specification can be thought of as providing a picture of the spatial distribution of fluid velocity (and of other flow quantities such as density and pressure) at each instant during the motion.

The second, or Lagrangian type of specification, makes use of the fact that, as in particle mechanics, some of the dynamical or physical quantities refer not only to certain positions in space but also (and more fundamentally) to identifiable pieces of matter. The flow quantities are here defined as functions of time and of the choice of a material element of fluid, and describe the dynamical history of this selected fluid element.

Solidss liquids and gases

The defining property of fluids, embracing both liquids and gases, lies in the ease with which they may be deformed. A piece of solid material has a definite shape, and that shape changes only when there is a change in the external conditions. A portion of fluid, on the other hand, does not have a preferred shape, and different elements of a homogeneous fluid may be rearranged freely without affecting the macroscopic properties of the portion of fluid. The fact that relative motion of different elements of a portion of fluid can, and in general does, occur when forces act on the fluid gives rise to the science of fluid dynamics.

The distinction between solids and fluids is not a sharp one, since there are many materials which in some respects behave like a solid and in other respects like a fluid. A ‘simple’ solid might be regarded as a material of which the shape, and the relative positions of the constituent elements, change by a small amount only, when there is a small change in the forces acting on it. Correspondingly, a ‘simple’ fluid (there is no one term in general use) might be defined as a material such that the relative positions of the elements of the material change by an amount which is not small when suitably chosen forces, however small in magnitude, are applied to the material.

Introduction

In this chapter the discussion of the flow of a viscous incompressible fluid of uniform density will be continued.

The values of the kinematic viscosity for air and water are so small that the Reynolds numbers for most of the flow systems of importance, whether in nature or in technology or in the laboratory, are very much larger than unity. A Reynolds number of 103 is attained in air at 20°C when UL has the very modest value 150 cm2/sec, and in water when UL is only 10 cm2/sec, where U and L are representative values for the velocity variations and the distances over which they occur in the flow system concerned. Such small values of the product UL are so readily and so often exceeded that flow at large Reynolds number must be regarded as the standard case.

The largeness of R = UL/v has its implications for the relative importance of the various terms in the equation of motion, as was seen in §4.7. Provided the non-dimensional quantities |Du′/Dt′| and |∇2u′| are both of order unity over most of the flow field (which would of course exclude some simple flow fields, such as steady unidirectional flow in a tube, in which the fluid acceleration is zero everywhere), R is a measure of the ratio of the magnitudes of inertia and viscous forces acting on the fluid; and a flow field for which R ≫ 1 is presumably one in which inertia forces are much greater than viscous forces over most of the field.