Theory of shock waves and introduction to gas dynamics/Introduction

by Yakov Zeldovich

THEORY OF SHOCK WAVES AND INTRODUCTION TO GAS DYNAMICS

By

Ya. B. Zeldovich

Introduction

Gas dynamics is a component part of hydrodynamics, the science of fluids, liquids and gases.

A particular feature of gas dynamics is the need to keep account of the compressibility of the medium. Liquids may be considered incompressible under normal circumstances, whereas gases change their volume considerably even under a slight variation in pressure.

It is obvious that specific formulas and laws of gas dynamics have to be applied to gases only insofar as we are dealing with pressure changes of great magnitude.

In the case of small velocities, the motion of gas can be regarded in the same way as the motion of a liquid, i. e., ignoring the change of volume and compressibility.

Depending upon the condition, the order of magnitude of pressure differentials arising in a flow changes from $\rho u^{2}/2$ the value of dynamic Impact according to Bernoulli's formula, to $\rho uc$, where $c$ is the speed of sound, $u$ to the speed of motion and p$\rho$ is gas density. Gas pressure is approximately equal to $\rho c^{2}$.

If we juxtapose the expressions, we see that at subsonic velocities the pressure differentials are small as compared with pressure proper and, consequently, we may therefore, as a rule, ignore the compressibility of the medium.

Following is a definition of the scope of gas dynamics. Gas dynamics is the science of motion at great pressure differentials and high velocities, velocity being measured in terms of the speed of sound.

In similarity theory we have the following ratio between motion and speed of sound:

$u/c=M$

where $M$ is known as the Mach number.

Gas dynamics studies motion and $M$ values close to unity. If $M$ is considerably smaller than 1, the general equation of gas dynamics becomes those of hydrodynamics of an incompressible liquid.

It will be assumed in the following that laws of hydrodynamics of an incompressible liquid are known, and we shall therefore not dwell on the derivation of the corresponding formulas.

To take account of compressibility means that one also has to take account of the change in the state of the medium. In hydrodynamics the action of dissipative forces (viscosity) leads to a release of heat In the liquid and to a change in its temperature, but it does not lead to a change in volume: the changes within the liquid have no inverse effect on the nature of the flow and have little importance for the phenomena investigated in hydrodynamics.

In gas dynamics, instead, we shall continuously deal with changes in the state of medium in the flow proper. This aspect of gas dynamics requires that any and all phenomena be also investigated from a thermal dynamics point of view; thus, thermodynamics is totally indispensable for the study of gas dynamics.

In the present book we shall deal only with specific phenomena of gas dynamics, i.e., such that have no analogies in the mechanics of an incompressible liquid. We shall not dwell on those subjects in which gas dynamics and the consideration of compressibility give only slight correction for the conventional formulas of hydrodynamics of an incompressible liquid. The emphasis in the present book will be on the careful definition of the fundamentals of gas dynamics, of the fundamental laws, and of the methods for solving the simplest problems, rather than on the computational methods of gas dynamics, the methods of numerical integration of complex two- and three-dimensional flows, etc. We shall proceed here from the simple to the complex, rather than from general problems to particular ones. Instead of writing first the equations of gas dynamics in their most general form (taking into consideration all the factors), searching for general solutions and then, by simplifying these solutions, going on to the particular solution of simple cases, we shall solve simple, elementary problems that describe certain aspects of some phenomena, and then, by means of these individual partial solutions piece together the solution of more complex problems.

We can outline the following, fundamental fields of application of gas dynamics. The first, which today is the better known and more developed one, comprises problems of flow around bodies moving at great speeds. This involves, first of all, the corrections in ordinary formulas of resistance and lift for bodies moving at subsonic speeds, i.e., corrections that are already applicable to contemporary aviation. A radical change in flow around bodies occurs when we deal with velocities exceeding the speed of sound. These speeds are involved in ballistics, i.e., the science of the motion of missiles and projectiles, and also in the study of rocket aircraft of the near future.

This application of gas dynamics to the problem of the motion of a body in a gas at speeds of the order of the speed of sound or exceeding it is dealt with in detail in text books, hence we shall deal with it only marginally here.

The second, extremely important field is that of the motion of a gas in ducts, such as nozzles and pipes. Again, gas dynamics becomes indispensable if and when the velocity of the gas attains or exceeds the speed of sound. In this field, the nature of the flow, and the dependence of velocity and flow rate on pressure drop, are subject to qualitative changes. This group of problems is of great significance for the theory of turbines, jet engines and missiles.

A peculiar field of gas dynamics based on the consideration of the compressibility of the moving medium is the teaching on sound - acoustics. The velocity of the medium and the amplitude of pressure changes under the effect of sound are very small. Nevertheless, consideration of compressibility becomes indispensable when studying the initial stages of any motion, and when studying rapidly changing, especially periodical motion.

Shock waves are of particular interest from various points of view, and they will be one of the main subjects of the present book. On the one hand, wherever the attempts of integrating equations without introducing discontinuities (i.e., shock waves) lead to paradoxes which make it impossible to solve these equations, the theory of shock waves eliminates the paradoxes and makes it possible to design a regime of motion under any conditions.

On the other hand, the shock waves themselves are a paradoxical phenomenon. They are paradoxical in that, without introducing any assumptions regarding dissipative forces (viscosity and thermal conductivity), from elementary considerations we can derive shock wave laws which include the increase in entropy, i.e., laws which include the irreversibility of the processes occurring in shock waves.

From this point of view shock waves afford a considerable logical and scientific interest, irrespective of their application.

It is worth noting that all basic relations and fundamental concepts have been established from the study of the general equations of gas dynamics some 50 years ago, at a time, that is, when there existed no experimental material, and long before shock waves were investigated by researchers.

As Emile Jouguet once said in a very poignant figure of speech, "the shock waves first Appeared on the point of the pen of a theoretician."

We cannot but marvel at the keen analysis and theoretizing power of the great minds of the past century, first of all of the German mathematician Bernhard Riemann, the English physicist Rankine and the French artillerist Hugoniot; from different approaches and independently of one another they have created the theory of shock waves which, to this day, has not lost its significance.

Finally, the interest in shock waves has increased over recent years in connection with the problem of the destructive effect of explosions and the propagation of the explosion on the explosive substance (capable of chemical reaction). It is necessary to know exactly the condition of the substance compressed by the shock wave, the rate of compression and similar properties of the wave. The present book is an introduction to the theory of explosions.

It is the author's pleasant duty to express his gratitude to Prof. N. N. Andreyev, B. P. Konstantinov, L. D. Landau, A. A. Sadovskiy, 0. M. Todes and Yu. B. Khariton for going over his manuscript and giving valuable advice.

Literature: Popular introduction to hydrodynamics [22];^[1] some general manuals on gas dynamics [4, 23, 23, 27, 39, 106].

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1. Figures in brackets correspond to the numbers of the bibliography.