Theory of shock waves and introduction to gas dynamics/Chapter 1

by Yakov Zeldovich

Chapter 1

Gas Dynamlcs Equations

We set up gas dynamics equations and neglect the effect of the force of gravity and also (see below) that of viscosity and thermal conductivity. For the sake of simplicity we shall write the equation for the one-dimensional case; generalization to two and three-dimensional cases will then not be difficult.

We begin with the continuity equation, i.e., the equation that expresses the law of conservation of matter. We denote, as usual, by $d/dt$ the substantial derivative in time, i.e., the derivative taken for the given particle along its path, and by $\partial /\partial t$ the local derivative in time which characterizes the change of the studied quantities at the given point in space, and write

${\frac {d\rho }{dt}}={\frac {\partial \rho }{\partial t}}+u{\frac {\partial \rho }{\partial x}}=-\rho {\frac {\partial u}{\partial x}},$

(I-1)

or

${\frac {\partial \rho }{\partial t}}=-{\frac {\partial (\rho u)}{\partial x}}=-\rho {\frac {\partial u}{\partial x}}-u{\frac {\partial \rho }{\partial x}}.$

(I-2)

Both formulas are, of course, completely equivalent. To derive the first formula we observe the motion of the layer of matter that comprises a constant amount of that matter. The second formula is derived by observing the change in density at the given point in space.

The equation of motion does not differ from the equation of motion for incompressible fluids:

$\rho {\frac {du}{dt}}=\rho {\frac {\partial u}{\partial t}}+\rho u{\frac {\partial u}{\partial x}}=-{\frac {\partial p}{\partial x}}.$

(I-3)

Finally, the third equation is substantially new; it represents a characteristic feature of gas dynamics. This is the equation of the change of state.

In the hydromechanics of Incompressible fluids we added the incompressibility equation $\rho =$ const to the first two equations. How do we find the relation between density and pressure in a compressible fluid? Density, pressure and temperature of a fluid are connected by &n equation known as the equation of state. If we know the thermal capacity, we can connect temperature with energy. To determine the connection between density and pressure, we must set up another equation - the equation of energy of a fluid in motion. In the absence of dissipative forces (viscosity and thermal conductivity) we have

$dE=-pdv;\quad {\frac {dE}{dt}}=-p{\frac {dv}{dt}}=-p{\frac {d(1/\rho )}{dt}}={\frac {p}{\rho ^{2}}}{\frac {d\rho }{dt}},$

(I-4)

where $v$ is specific volume, a quantity inverse to density $\rho$.

The energy of any element of matter under investigation can only change on account of the work of compression that is being performed on it by the surrounding volumes of the fluid (gas).

Bearing in mind the fundamental thermodynamics equation

$dE=TdS-pdv,$

(I-5)

from the energy equation we readily obtain for the studied case of the absence of dissipative forces the natural conclusion

$TdS=0;\quad {\frac {dS}{dt}}=0.$

(I-6)

In other words, the state of matter changes according to the adiabatic curve, it changes with constant entropy.

As is known, for an ideal gas with constant thermal capacity, the adiabatic equation is

$p=A\rho ^{k},$

(I-7)

where $k=c_{p}/c_{v}$, $k=$const. It can also be found without considering entropy, and it was found that way in 1818 by Poisson who integrated Eq. (1-4), in which for an ideal gas we substitute Clapeyron's law

$E=c_{v}T={\frac {c_{v}}{R}}RT={\frac {c_{v}}{R}}pv;\quad dE={\frac {c_{v}}{R}}pdv+{\frac {c_{v}}{R}}vdp.$

(I-8)

Which are the conditions of applicability of the above equations$^{3}$ in which the effect of viscosity and thermal conductivity was disregarded? It is obvious, in the first place, that in order to apply these equations the Reynolds and Peclet numbers must be high. "As is known from similarity theory and hydrodynamics of an incompressible fluid, the Reynolds number characterizes the relation of inertia and viscosity. The Peclet number plays an analogous role in that it characterizes the relation of molar heat transfer of a flowing fluid and the heat flows transferred by molecular thermal conductivity.

Thus, a high Reynolds number means that one may disregard viscosity in gas dynamics equations. A high Peclet number means that thermal conductivity may be ignored; it means that along the flow line motion takes place virtually adiabatically.

From the molecular-kinetic theory it follows that in gases the ratio of thermal conduction to volume thermal capacity (known as thermal diffusivity) is approximately equal to the viscosity to density ratio (known as kinematic viscosity). For this reason in a gas flow the Reynolds number is quite close to the Peclet number, and both conditions (namely, a high Reynolds number and a high Peclet number) coincide.

Following Karman we can give a different formulation to the condition of a high Reynolds number. We use the molecular expression for the viscosity coefficient

$\eta =\nu \rho ={\frac {1}{2}}\rho c'l,$

(I-9)

where $l$ is the length of the free path of the molecules in the gas, $c'$ is the velocity of molecules, a quantity equal in magnitude to the speed of sound, and <math\nu</math> is kinematic viscosity (cm$^{2}/$sec).

If we substitute the expression for viscosity into the Reynolds number formula, we get

$Re={\frac {U\rho d}{\eta }}={\frac {Ud}{\nu }}=3{\frac {Ud}{c'l}}\approx M{\frac {d}{l}}$

(I-10)

where $d$ is the characteristic size, $U$ is the characteristic velocity of the motion investigated.

The relation between the speed of motion and the speed of sound is known as the Mach number

${\frac {u}{c}}=M.$

(I-11)

In the field of gas dynamics interesting us, where the speed of motion is of the same order of magnitude as the speed of sound $M\sim 1$, the Reynolds number turns out to be of the same order of magnitude as the ratio of the dimensions of system $d$ to the length of the molecule path $l$.

The condition stated above according to which $Re\gg 1$, and according to which it is possible to ignore dissipation forces (viscosity and thermal conduction), leads to the requirement that the dimensions of the system be considerably greater than the length of the free path of molecules. We see further, however, that the fulfillment of that condition, i.e., a system of large size, does in reality not always ensure small dissipation forces and the possibility of studying adiabatic processes only. We shall see in the following that in the presence of shock waves in a flow there occur exceedingly large gradients of all the quantities studied; the magnitude of these gradients does no longer depend upon the dimension of the system, and also does not drop as the dimensions of the system increase. In these cases, we will have to consider the possibility of changing entropy no matter how large the Reynolds number is.

Generally speaking, the possibility of an increase in entropy does, in principle, depend upon the dissipation forces; all the observed large-size properties of the flow, however, and, specifically, the numerical value of entropy increase in a shock wave, do not depend upon the magnitude of viscosity and thermal conductivity (they are self-modeling with respect to thermal conductivity and viscosity); the laws of the change of state in a shock wave can thus be derived without investigating the structure of its front from the equations of conservation of matter, the amount of motion and energy, applied to the states prior and after the passage of the wave.

In the case of high Reynolds numbers, we could expect a considerable effect of turbulence. In matter of fact, however, studies of the simultaneous effect of turbulence and extremely high (of the order of the speed of sound) velocities are very few. To some extent, this lack appears to be due to the complexity of such a comparatively far-out field. On the other hand, in most typical problems of gas dynamics we are faced with short pipes and nozzles, short bodies to be flowed around; in a short pipe turbulence has no time to develop, even if the $Re$ number is high. Finally, in the hydrodynamics of small velocities, with $M<1$, the formation of eddies and turbulence is the only resistance mechanism for $Re\gg 1$; their consideration is absolutely necessary for studying the forces affecting a body moving in a fluid. In the case of supersonic speeds there occurs what is known as wave resistance and the possibility of irreversible dissipation of energy in steady-state shock waves; a resistance different from $0$ may be found also without studying turbulence.

Appendix

In order to determine the applicability of Eq. (I-1) - (I-6), let us take the general form of gas dynamic equations (see, for instance, [23, 27]).

The equation of motion, has the form:

$\rho {\frac {du_{x}}{dt}}=\rho X-{\frac {\partial p}{\partial x}}-{\frac {\partial T_{xx}}{\partial x}}-{\frac {\partial T_{xy}}{\partial y}}-{\frac {\partial T_{xz}}{\partial z}},$

(I-12)

where the quantities $X$, $Y$, $Z$ are components of volumetric force applied to a unit of mass, and the quantities $T_{xx}$, $T_{xy}$, and so forth, are components of the tensor of stresses due to the effect of viscosity. The effect of viscosity depends on the relative motion of neighboring fluid particles. From the conditions of tensor symmetry, confining ourselves to terms proportional to the first derivatives of velocity with respect to the coordinate, taking the invariant sum of normal stresses on three mutually perpendicular platforms to be equalled to the three-fold pressure, and isolating pressure from the stress tensor, as this already has been done in formula (I-12), we arrive at the following expression for the stress tensor:

$T_{xx}={\frac {2}{3}}\eta \left({\frac {\partial u_{x}}{\partial x}}+{\frac {\partial u_{y}}{\partial y}}+{\frac {\partial u_{z}}{\partial z}}\right)-2\eta {\frac {\partial u_{x}}{\partial x}};\quad T_{xz}=-\eta \left({\frac {\partial u_{z}}{\partial x}}+{\frac {\partial u_{x}}{\partial z}}\right);\quad T_{xy}=-\eta \left({\frac {\partial u_{y}}{\partial x}}+{\frac {\partial u_{x}}{\partial x}}\right).$

(I-13)

The equations of motion with respect to the two other coordinates are found from (I-12) and (I-13) by a cyclic shifting of indices.

The coefficients in (I-13) have been chosen such that

$T_{xx}+T_{yy}+T_{zz}=0.$

A11

In the one-dimensional case

$u_{x}=u(x),\,u_{y}=u_{z}=0,\,{\frac {\partial u_{x}}{\partial y}}={\frac {\partial u_{x}}{\partial z}}=0$

B11

and the equation of motion (1-12) can be simplified to

$\rho {\frac {du_{x}}{dt}}=\rho X-{\frac {\partial p}{\partial x}}+{\frac {4}{3}}{\frac {\partial }{\partial x}}\eta \left({\frac {\partial u_{x}}{\partial x}}\right).$

(I-14)

If viscosity and thermal conduction are taken into consideration, additional terms appear also in the equation of energy: In the general case of three-dimensional motion ($\lambda$ is thermal conduction)

${\begin{aligned}\rho {\frac {d}{dt}}\left(E+{\frac {u_{x}^{2}+u_{y}^{2}+u_{z}^{2}}{2}}\right)&=\rho (u_{x}X+u_{y}Y+u_{z}Z)-{\frac {\partial }{\partial x}}[u_{x}(p+T_{xx}+u_{y}(p+T_{yy})+u_{z}(p+T_{zz})]-{\frac {\partial }{\partial y}}[u_{y}(\dots )+\dots ]\\&-{\frac {\partial }{\partial z}}[u_{z}(\dots )+\dots ]+{\frac {\partial }{\partial x}}\lambda {\frac {\partial T}{\partial x}}+{\frac {\partial }{\partial y}}\lambda {\frac {\partial T}{\partial y}}+{\frac {\partial }{\partial z}}\lambda {\frac {\partial T}{\partial z}}.\end{aligned}}$

(I-15)

We remind the reader that $T$ without indices is absolute temperature. By using the continuity equation, the equations of motion in the form (I-12) and the thermodynamic relation $dE=-pdv+TdS$, we can transform (I-15) to the following form:

${\begin{aligned}\rho T{\frac {dS}{dt}}=&-T_{xx}{\frac {\partial u_{x}}{\partial -}}-T_{xy}{\frac {\partial u_{x}}{\partial x}}-T_{xz}{\frac {\partial u_{x}}{\partial x}}\\&+{\frac {\partial }{\partial x}}\lambda {\frac {\partial T}{\partial x}}+{\frac {\partial }{\partial y}}\lambda {\frac {\partial T}{\partial y}}+{\frac {\partial }{\partial z}}\lambda {\frac {\partial T}{\partial z}}.\end{aligned}}$

(1-16)

By substituting the expressions (I-13) of the components of the tensor of viscous stresses, we reduce the expression for the work performed by viscosity, irreversibly transforming itself into heat in (I-16), to a form which shows that this quantity is essentially positive:

${\begin{aligned}\rho T{\frac {dS}{dt}}&=\eta \left\{\left({\frac {\partial u_{x}}{\partial y}}+{\frac {\partial u_{y}}{\partial x}}\right)^{2}+\left({\frac {\partial u_{x}}{\partial z}}+{\frac {\partial u_{z}}{\partial x}}\right)^{2}+\left({\frac {\partial u_{y}}{\partial z}}+{\frac {\partial u_{z}}{\partial y}}\right)^{2}\right.\\&\left.+{\frac {4}{3}}\eta \left[\left({\frac {\partial u_{x}}{\partial x}}-{\frac {\partial u_{y}}{\partial y}}\right)^{2}+\left({\frac {\partial u_{y}}{\partial y}}-{\frac {\partial u_{z}}{\partial z}}\right)^{2}+\left({\frac {\partial u_{z}}{\partial z}}-{\frac {\partial u_{x}}{\partial x}}\right)^{2}\right]\right\}\\&+{\frac {\partial }{\partial x}}\lambda {\frac {\partial T}{\partial x}}+{\frac {\partial }{\partial y}}\lambda {\frac {\partial T}{\partial y}}+{\frac {\partial }{\partial z}}\lambda {\frac {\partial T}{\partial z}}.\end{aligned}}$

(I-17)

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