§13 Shock waves in hydrodynamics

One of the most important physical phenomena that occur in supersonic flow is the existence of discontinuities in the different hydrodynamical quantities describing the flow. In order for the flow to possess such discontinuities, certain conditions have to be satisfied at the boundaries between the media. Mathematically, these boundaries are treated as infinitesimal, so that they may be assumed to be surfaces. For a steady flow, an element of area on a surface of discontinuity can be considered to be a plane and it can be assumed to be at rest by an appropriate choice of the system of reference. If the flow is not steady, then the argument remains valid for a short interval of time. In order to simplify the calculations, and without loss of generality, the surface of discontinuity can be chosen to be a plane which is parallel to the $Oyz$ plane, so that the unit vector $\hat{ \boldsymbol{e} }_x$ in the positive $x$ direction is normal to it.

Let us consider a closed three dimensional timelike cylinder $S$ that intersects the surface of discontinuity. The axis of the cylinder is such that it is parallel to the normal to the plane of the surface of discontinuity, in the direction of the $x$ axis. Integrating eq.(8.1) and eq.(9.1) along the volume enclosed by this hypersurface and using Gauss's theorem we obtain:

$$\oint{ \mathcal{T}^{i\alpha} \mathrm{d} f_\alpha = 0 },$$ (13.1)

$$\oint{ n u^\alpha \mathrm{d} f_\alpha = 0 }.$$ (13.2)

with $\mathrm{d} f_\alpha$ the area element along the surface $S$. Taking the limit when the volume enclosed by the area $S$ tends to zero gives:

$$[ \mathcal{T}^{i1} ] = 0, \qquad [ n u^1 ] = 0.$$ (13.3)

where the difference between the values on either side of the discontinuity (sides 1 and 2) are represented by $q_1 - q_2 \equiv [ q ]$ for any quantity $q$. As we saw in section §8, $\ensuremath{\mathsf{c}}\mathcal{T}^{0\alpha}$ represents the energy flux vector, $\mathrm{diag} ( \mathcal{T}_{\alpha\beta} )$ is the 3-momentum flux density vector and $n u^k$ is the particle flux 4-vector. From this it follows that the particle flux, the energy flux and the momentum flux vectors are conserved across the surface of discontinuity according to eq.(13.3):

$$[ n u^x ] = 0, \qquad [ \mathcal{T}^{xx} ] = [ w (u^x)^2 + p ] = ... ...= [ \omega u^x u^y ] = 0, \qquad [ \mathcal{T}^{zx} ] = [ \omega u^x u^z ] = 0.$$ (13.4)

where $x, y,$    and $z$ are Cartesian coordinates. From eqs.(13.4)-(13.5) it follows that it is possible to define two types of discontinuities. In the first place, those in which there is no particle flux through the surface of discontinuity. That is, $n_1 u_1^x = n_2 u_2^x = 0$. Since the particle number densities on both sides of the discontinuity are non-zero it follows that the velocities $u_1^x = u_2^x = 0$. This satisfies identically all relations in eq.(13.5) as well as the first and third of eq.(13.4). The second relation in eq.(13.4) implies $[ p ] = 0$. That is, a discontinuity for which the mass flux through its surface is zero is such that its normal velocity components are zero and the pressure is continuous across it:

$$\ensuremath{v}_{x_1} = \ensuremath{v}_{x_2} = 0, \qquad [ p ] = 0.$$ (13.5)

The values of the other hydrodynamical quantities can take any value across this surface of discontinuity. A discontinuity of this kind is called a tangential discontinuity.

The second type of discontinuity occur when the particle flux through the surface is non-zero. According to eqs.(13.4)-(13.5), this implies that the tangential component of the velocity is preserved across the surface of discontinuity:

$$[ \ensuremath{v}_x ] = 0, \qquad [ \ensuremath{v}_y ] = 0.$$ (13.6)

Such discontinuities are called shock waves. Substitution of the 4-velocity components in eq.(13.4) gives:

$$\ensuremath{v}_1 \gamma_1 / \mathprm{V}_1 = \ensuremath{v}_2 \gamma_2 / \mathprm{V}_2 \equiv \mathprm{j},$$ (13.7)

$$\omega_1 \ensuremath{v}_1^2 \gamma_1^2 / \ensuremath{\mathsf{c}}^... ...p_1 = \omega_2 \ensuremath{v}_2^2 \gamma_2^2 / \ensuremath{\mathsf{c}}^2 + p_2,$$ (13.8)

$$\omega_1 \ensuremath{v}_1 \gamma_1^2 = \omega_2 \ensuremath{v}_2 \gamma_2^2,$$ (13.9)

in which $\mathprm{V} \! \equiv \! 1 / n$ is the proper volume per particle number. From eq.(13.8) and eq.(13.10) it follows that the particle number flux $j$ is given by:

$$\mathprm{j}^2 = ( p_2 - p_1 ) \ensuremath{\mathsf{c}}^2 / ( \omega_1 \mathprm{V}_1^2 - \omega_2 \mathprm{V}_2^2 ).$$ (13.10)

Algebraic manipulation of eqs.(13.8)-(13.10) imply that (Taub, 1948; Landau & Lifshitz, 1995):

$$\omega_1^2 \mathprm{V}_1 - \omega_2 \mathprm{V}_2 + ( p_2 - p_1 ) ( \omega_1 \mathprm{V}_1^2 + \omega_2 \mathprm{V}_2^2 ) = 0,$$ (13.11)

which is called the relativistic shock adiabatic relation or Taub adiabatic. For a given $p_1, \mathprm{V}_1$, the shock adiabatic gives a relation between $p_2, \mathprm{V}_2$.

Writing $\ensuremath{v}/ \ensuremath{\mathsf{c}}\! = \! \tanh \varphi,$    so that $\gamma = \! \cosh \varphi$, the velocities of the gas on either side of the discontinuity can be easily shown to be:

$$\frac{ \ensuremath{v}_1 }{ \ensuremath{\mathsf{c}}} = \sqrt{ \fra... ...= \sqrt{ \frac{ ( p_2 - p_1 ) ( e_1 + p_2 ) }{ ( e_2 - e_1 ) ( e_2 + p_1 ) } },$$ (13.12)
while their relative velocity is

$$\ensuremath{v}_{12} = \frac{ \ensuremath{v}_1 -\ensuremath{v}_2 }... ...{c}}\sqrt{ \frac{ ( p_2 - p_1 ) ( e_2 - e_1 ) }{ ( e_1 + p_2 ) ( e_2 + p_1) } }$$ (13.13)

according to the relativistic rule for addition of velocities. The entropy density, as any other thermodynamic quantity, is discontinuous across a shock wave. From the law of the increase of the entropy it follows that the entropy can only increase across a shock wave. It is possible to show under very general arguments (Thorne, 1973; Landau & Lifshitz, 1995) that the shock wave is a compression wave, that is $p_2 \! > p_1$, if

$$\left( \partial^2 ( \omega \mathprm{V}^2 ) / \partial p^2 \right)_{\sigma \mathprm{V}} \! > \! 0.$$ (13.14)

When $p_2 \! > \! p_1$ it follows from eqs.(13.8)-(13.10) that $\mathprm{V}_1 \! > \! \mathprm{V}_2$. Using the definition of the particle number flux in eq.(13.8), this implies that $\ensuremath{v}_1 \! > \! \ensuremath{v}_2$. In other words, provided that the inequality in eq.(13.15) is satisfied, then a shock wave satisfies:

$$p_2 > p_1, \qquad \mathprm{V}_1 > \mathprm{V}_2 \quad \ensuremath{v}_1 > \ensuremath{v}_2.$$ (13.15)

Very general arguments about the stability of shock waves (Landau & Lifshitz, 1995) show that, for any shock wave, whatever the thermodynamic conditions of the gas:

$$\ensuremath{v}_1 > a_1 \quad \ensuremath{v}_2 < a_2,$$ (13.16)

for a gas with sound speed $a$.

In order to derive the classical expressions of the relations written above, we take the limit $\ensuremath{\mathsf{c}}\! \rightarrow \! \infty$ and use eq.(10.1). It is common practice in classical hydrodynamics to use the mass flux density $j$ as opposed to the particle number density $\mathprm{j}$, and the inverse of the mass density, the volume per unit mass $\ensuremath{\mathit{V}}$, instead of the volume per unit particle $\mathprm{V}$.

The mass density flux $j$ is then given by:

$$j^2 = ( p_2 - p_1 ) / ( \ensuremath{\mathit{V}}_1 - \ensuremath{\mathit{V}}_2 ) = 0.$$ (13.17)

The shock adiabatic relation, also called Hugoniot adiabatic in classical fluid dynamics, is:

$$w_1 - w_2 + \frac{ 1 }{ 2 } ( \ensuremath{\mathit{V}}_1 + \ensuremath{\mathit{V}}_2 )( p_2 - p_1 ) = 0.$$ (13.18)

The velocity difference in eq.(13.14) gives:

$$\ensuremath{v}_1 - \ensuremath{v}_2 = \sqrt{ ( p_2 - p_1 ) ( \ensuremath{\mathit{V}}_1 - V_2 ) }.$$ (13.19)

All the inequalities in eqs.(13.16)-(13.17) remain valid and that in eq.(13.15) becomes $( \partial^2 \ensuremath{\mathit{V}}/ \partial p^2 )_s \! > \! 0$.

Footnotes

... discontinuity.

In classical hydrodynamics tangential discontinuities are shown to be unstable and spread out to form a turbulent region. However, this stability arguments do not apply for the particular case in which the velocity of the flow is continuous across the surface of discontinuity. A tangential discontinuity with this properties is called a contact discontinuity.