Vortex jumps

It has been supposed so far that the velocity and vorticity fields are continuous and have continuous derivatives. In a real viscous fluid, these assumptions are generally regarded as appropriate, and the velocity field is assumed analytic everywhere, except possibly at an initial instant o, when the speed of boundaries changes in a non-analytic manner. However, if we go to the the limit of vanishingly small viscosity and consider ideal fluids, to which the Helmholtz laws apply, the mathematics allows non-analytic behaviour and we cannot assert on physical grounds that only continuous fields should be considered. Velocity or vorticity fields with singularities or discontinuities are indeed of considerable importance. The singularities are, of course, not arbitrary and must be consistent with integral forms of the Euler equations or equivalently with the conservation of mass, momentum and energy. In particular, the dynamical constraint that pressure is continuous across a surface of discontinuity must be satisfied unless there are also singularities in the external force fields. We suppose in this chapter that the density is uniform and put it equal to unity unless explicitly stated otherwise. Also, external forces are supposed conservative unless non-conservative forces are explicitly introduced.

In the past three decades, the study of vortices and vortex motions – which originated in Helmholtz's great paper of 1858, ‘Uber Integrale der hydrodynamischen Gleichungen welche den Wirbelbewegungen entsprechen’ (translated by Tait [1867]), and continued in the brilliant work of Lord Kelvin and others in the nineteenth century, and Prandtl and his Göttingen school in the first half of this century – has received continuing impetus from problems arising in physics, engineering and mathematics. As aptly remarked by Küchemann [1965], vortices are the ‘sinews and muscles of fluid motions’. Aerodynamic problems of stability, control, delta wing aerodynamics, high lift devices, the jumbo jet wake hazard phenomenon, among other concerns, have led to a myriad of studies. Smith [1986] reviews some of this work. The realisation that many problems involving interfacial motion can be cast in the form of vortex sheet dynamics has stimulated much interest. The discovery (rediscovery?) of coherent structures in turbulence has fostered the hope that the study of vortices will lead to models and an understanding of turbulent flow, thereby solving or at least making less mysterious one of the great unsolved problems of classical physics. Vortex dynamics is a natural paradigm for the field of chaotic motion and modern dynamical system theory. It is perhaps not well known that the father of modern dynamics and chaos wrote a monograph on vorticity (Poincaré [1893]).

In this chapter we consider stresses in a fluid. We start by setting the forces resulting from these stresses in their proper perspective, i.e., in relation to body forces, together with which they raise accelerations. This results in a set of momentum equations, which are needed later.

We then consider the relations between the various stress components and proceed to inspect stress in fluids at rest and in moving fluids.

The Momentum Equations

In this section we establish relations between body forces, stresses and their corresponding surface forces and accelerations. Newton's second law of motion is used, and the results are the general momentum equations for fluid flow.

Consider a system consisting of a small cube of fluid, as shown in Fig. 2.1. A system is defined in classical thermodynamics as a given amount of matter with well-defined boundaries. The system always contains the same matter and none may flow through its boundaries.

As a rule a fluid system does not retain its shape, unless, of course, the fluid is at rest. This does not prevent the choice of a system with a certain particular shape, e.g., a cube. The choice means that imaginary surfaces are defined inside the fluid such that at the considered moment they enclose a system of fluid with a given shape. A moment later the system may have a different shape, because the shape is not a property of the fluid or of the location.

Differential Representation

Two conditions must be satisfied to make the integral analysis considered in the previous chapter useful: The sought information must be such that no details are required; and enough information must be known a priori to supply numerical values for all the symbols in the equations of the integral analysis. In many situations details are meaningful and necessary, and in very many cases, particularly engineering ones, no a priori information is given.

The difficulty mentioned in Chapter 4, that of identification of fluid particles, cannot be alleviated any more by an integral approach and the concept of the field must be introduced.

The field point of view, which is traditionally called the Eulerian approach, uses the same properties associated with fluid particles, e.g., density, velocity, temperature, etc., but assigns them to the field. One then considers the field density, i.e., the density at a particular point (x, y, z) and at a particular time, or one considers the field velocity or the field temperature. There is then a flow field, with all its properties at each point and for any time, and the problem of identification seems to have been successfully circumvented.

There is still, however, an intermediate step necessary before field analysis can be applied: the basic laws dealing with mass and momentum, which are formulated for well-defined, and therefore identifiable, thermodynamic systems, must be translated into expressions which contain field concepts only. An analogous step has been taken in going from thermodynamic systems to control volumes in Chapter 4.

This textbook is intended to be read by undergraduate students enrolled in engineering or engineering science curricula who wish to study fluid mechanics on an intermediate level. No previous knowledge of fluid mechanics is assumed, but the students who use this book are expected to have had two years of engineering education which include mathematics, physics, engineering mechanics and thermodynamics.

This book has been written with the intention to direct the readers to think in clear and correct terms of fluid mechanics, to make them understand the basic principles of the subject, to induce them to develop some intuitive grasp of flow phenomena and to show them some of the beauty of the subject.

Care has been taken in this book to strictly observe the chain of logic. When a concept evolves from a previous one, the connection is shown, and when a new start is made, the reason for it is clearly stated, e.g., as where the turbulent boundary layer equations are derived anew. As related later, there may be programs of study in which certain parts of this book may be skipped. However, from a pedagogical point of view it is important that these skipped parts are there and that the student feels that he or she has a complete presentation even though it is not required to read some particular section.

It is believed that an important objective of engineering education is to induce the students to think in clear and exact terms.

The nonlinear terms on the left-hand side of the Navier–Stokes equations result from the acceleration of the fluid. We already know that these terms contribute much to the difficulties one encounters in the solution of the equations. Indeed, we have noted the relative ease with which exact solution have been obtained in Chapter 6 for fully developed flows, where those nonlinear terms vanish. As we now look for approximations, obtained by the solutions of approximate forms of the Navier–Stokes equations, the first idea that comes to mind is to remove the terms which cause the greatest difficulty, i.e., the nonlinear acceleration terms.

Our subject here is fluid mechanics, and in this context we must ask if there exist real flows in which the acceleration terms are negligible; and if they exist, how do we identify them? The answer to this enquiry is that there are at least two such families of flows: flows in narrow gaps and creeping flows.

Flow in Narrow Gaps

Consider the two-dimensional flow of an incompressible fluid in the narrow gap between the two plates shown in Fig. 9.1. For simplicity we assume the plates to be flat. The results obtained in this analysis may be extended to cases where the plates are curved, as long as the gap is much smaller than the radius of curvature. Another possible extension is to three-dimensional flows, where a w-component of the velocity also exists. Here we present the simplest examples of gap flows and limit the analysis to two-dimensional flows between flat plates.

The Field of Fluid Mechanics

Fluid mechanics extends the ideas developed in mechanics and thermodynamics to the study of motion and equilibrium of fluids, namely of liquids and gases.

The beginner in the study of fluid mechanics may have some intuitive notion as to the nature of a fluid, a notion that centers around the idea of a fluid not having a fixed shape. This idea indicates at once that the field of fluid mechanics is more complex than that of solid mechanics. Fluid mechanics has to deal with the mechanics of bodies that continuously change their shape, or deform. Similarly, the ideas developed in classical equilibrium thermodynamics have to be extended to allow for the additional complexity of properties which vary continuously with space and time, normally encountered in fluid mechanics.

Fluid mechanics bases its description of a fluid on the concept of a continuum, with properties which have to be understood in a certain manner. So far, neither the idea of a continuum nor that of a fluid have been properly defined. We, therefore, begin with the explanation of what a continuum is and how local properties are defined. We then describe the various forces that act in a continuum leading to the definition of the concept of stress at a point. The concepts of stress and continuum are then used to define a fluid.

Moving fluids are subject to the same laws of physics as are moving rigid bodies or fluids at rest. The problem of identification of a fluid particle in a moving fluid is, however, much more difficult than the identification of a solid body or of a fluid particle in a static fluid. When the fluid properties, e.g., its velocity, also vary from point to point in the field, the analysis of these properties may become quite elaborate.

There are cases where such an identification of a particle is made, and then this particle is followed and the change of its properties is investigated. This is known as the Lagrangian approach. In most cases one tries to avoid the need for such an identification and presents the phenomena in some field equations, i.e., equations that describe what takes place at each point at all times. This differential analysis, which is called the Eulerian approach, is the subject of the next chapter.

There exist, however, quite a few cases where there are sufficient restrictions on the flow field, such that while the individual fluid particles still elude identification, whole chunks of fluid can be identified. When this is the case, some useful engineering results may be obtained by integrating the relevant properties over the appropriate chunk. This approach is known as integral analysis.

In most cases where integral representation is possible, there are walls impermeable to the fluid.