Governing equations of fluid mechanics

Preamble

This chapter is the first of two which set out the fundamental fluid dynamics required for the further development of aerodynamics. In it the study of air in motion starts with the physics and mathematics of one-dimensional fluid motion. Many of the physical phenomena evident in all stages of aerodynamics are most readily approached by considering the one-dimensional mode, without prejudice to the wider analysis of two – and three-dimensional motions.

The laws governing the changes in the physical properties of air are first covered and the relevant mathematics introduced. These laws are applied to the accelerating gas as it moves out of the low-speed ( incompressible) regime and into the transonic and supersonic regimes where the abrupt changes in properties are manifest.

2.1 Introduction

The physical laws that govern fluid flow are deceptively simple. Paramount among them is Newton’s second law of motion which states that:

Mass x Acceleration = Applied force

In fluid mechanics we prefer to use the equivalent form of

Rate of change of momentum = Applied force

Apart from the principles of conservation of mass and, where appropriate, conserva­tion of energy, the remaining physical laws required relate solely to determining the forces involved. For a wide range of applications in aerodynamics the only forces involved are the body forces due to the action of gravity* (which, of course, requires the use of Newton’s theory of gravity; but only in a very simple way); pressure forces (these are found by applying Newton’s laws of motion and require no further physical laws or principles); and viscous forces. To determine the viscous forces we

Body forces are commonly neglected in aerodynamics.

need to supplement Newton’s laws of motion with a constitutive law. For pure homogeneous fluids (such as air and water) this constitutive law is provided by the Newtonian fluid model, which as the name suggests also originated with Newton. In simple terms the constitutive law for a Newtonian fluid states that:

Viscous stress oc Rate of strain

At a fundamental level these simple physical laws are, of course, merely theoretical models. The principal theoretical assumption is that the fluid consists of continuous matter – the so-called continuum model. At a deeper level we are, of course, aware that the fluid is not a continuum, but is better considered as consisting of myriads of individual molecules. In most engineering applications even a tiny volume of fluid (measuring, say, 1 pm3) contains a large number of molecules. Equivalently, a typical molecule travels on average a very short distance (known as the mean free path) before colliding with another. In typical aerodynamics applications the m. f.p. is less than 100 nm, which is very much smaller than any relevant scale characterizing quantities of engineering significance. Owing to this disparity between the m. f.p. and relevant length scales, we may expect the equations of fluid motion, based on the continuum model, to be obeyed to great precision by the fluid flows found in almost all engineering applications. This expectation is supported by experience. It also has to be admitted that the continuum model also reflects our everyday experience of the real world where air and water appear to our senses to be continuous substances.

There are exceptional applications in modem engineering where the continuum model breaks down and ceases to be a good approximation. These may involve very small – scale motions, e. g. nanotechnology and Micro-Electro-Mechanical Systems (MEMS) technology,[4] where the relevant scales can be comparable to the m. f.p. Another example is rarefied gas dynamics (e. g. re-entry vehicles) where there are so few mole­cules present that the m. f.p. becomes comparable to the dimensions of the vehicle.

We first show in Section 2.2 how the principles of conservation of mass, momen­tum and energy can be applied to one-dimensional flows to give the governing equations of fluid motion. For this rather special case the flow variables, velocity and pressure, only vary at most with one spatial coordinate. Real fluid flows are invariably three-dimensional to a greater or lesser degree. Nevertheless, in order to understand how the conservation principles lead to equations of motion in the form of partial differential equations, it is sufficient to see how this is done for a two­dimensional flow. So this is the approach we will take in Sections 2.4-2.8. It is usually straightforward, although significantly more complicated, to extend the principles and methods to three dimensions. However, for the most part, we will be content to carry out any derivations in two dimensions and to merely quote the final result for three-dimensional flows.