Rankine-Hugoniot conditions for a perfect gas

In the rest frame of a one-dimensional shock, states with velocities $w_0,w_1$ satisfyand

These are the Rankine-Hugoniot conditions specialized to the mass, momentum and energy fluxes of a perfect gas.

For a stationary normal shock in a polytropic perfect gas, the upstream and downstream normal speeds obey $u_1{u}_2=c_{\mathrm{cr}}^2$, where $c_{\mathrm{cr}}$ is the sonic speed on their common Bernoulli streamline.

When the upstream pressure is negligible, a perfect-gas shock has density ratio $(\gamma +1)/(\gamma -1)$. For $\gamma =5/3$, the downstream density is four times the upstream density.

A strong, nonradiative shock in a monatomic ideal gas gives $p_{\mathrm{d}}=3{\rho }_{\mathrm{u}}{v}_{\mathrm{u}}^2/4$ and ${\rho }_{\mathrm{d}}=4{\rho }_{\mathrm{u}}$. The ideal gas law therefore gives the temperature in the title, with $\mu$ the mean particle mass. If the upstream speed is a Newtonian free-fall speed, $k_{\mathrm{B}}{T}_{\mathrm{d}}=3\mu GM/(8R)$. Mass alone does not determine this temperature without a radius and composition; electron-ion disequilibrium and radiation can also make the emergent spectrum differ from the ideal gas temperature.

In the rest frame of a strong planar shock wave, a perfect gas has ${\rho }_2/{\rho }_1=(\gamma +1)/(\gamma -1)$, $u_2/u_1=(\gamma -1)/(\gamma +1)$ and $p_2=2{\rho }_1{u}_1^2/(\gamma +1)$. Its downstream specific internal energy is ${ϵ}_2=p_2/[(\gamma -1){\rho }_2]=2u_1^2/(\gamma +1{)}^2$. Multiplying by mass flux $j={\rho }_1{u}_1$ gives the internal-energy flux in the title. The lost bulk kinetic energy flux instead equals $j(u_1^2-u_2^2)/2=\gamma F_{\mathrm{int}}$: this includes downstream pressure work and is the increase of specific enthalpy flux in the cold-upstream limit. Neither quantity is a temperature change per unit time of a fixed target mass without further information about that target.

The shock compression ratio is the downstream-to-upstream density ratio. For a strong perfect-gas shock,In the upstream rest frame, the downstream velocity is $2V_s/(\gamma +1)$ and its pressure is $2{\rho }_0{V}_s^2/(\gamma +1)$.

A plane shock moving at speed $V_s$ through initially stationary gas has upstream speed $u_1=V_s$ and downstream speed $u_2=V_s/\chi$ in the shock frame, where $\chi$ is the shock compression ratio. Transforming back to the initial slab frame gives $v_2=V_s-u_2$. At the breakout time $L/V_s$, a uniform downstream slab has thickness $L-v_{2}L/V_s=L/\chi$. The strong monatomic limit has $\chi =4$, hence $L_2=L/4$ and $v_2=3V_s/4$. This uses a maintained planar post-shock state; boundary rarefactions and variations of shock speed can change the actual thickness of a freely evolving finite cloud.

For a strong monatomic shock, the planar shock compression of a finite slab gives bulk speed $3V_s/4$ and kinetic energy $E_{\mathrm{bulk}}=9M_c{V}_s^2/32$. The initial internal energy is $E_{\mathrm{th}}=3M_{c}p/(2\rho )$. A bulk-ballistic escape estimate $E_{\mathrm{bulk}}\gtrsim W$ uses positive binding-energy magnitude $W$, not negative gravitational potential energy. With adiabatic sound speed $c_{\mathrm{ad}}^2=5p/(3\rho )$ it yields the first displayed coefficient. Using the isothermal sound speed $c_T^2=p/\rho$ instead yields $16/3\simeq 5.3$. The definitions obey ${\mathcal{M}}_T^2=(5/3){\mathcal{M}}_{\mathrm{ad}}^2$. Heating, subsequent expansion and energy loss can alter the physical escape threshold, so the coefficient is not independent of the energy budget and Mach convention.

If a piston moves at speed $V$ into stationary perfect gas and drives a shock at speed $U$, then the shock-frame upstream and downstream speeds are $U$ and $U-V$. If $p_2-p_1=\beta p_1$, the jump conditions give

If a shock travels into stationary gas at speed $V$ while the downstream piston and gas travel at $V/3$, then for $\gamma =3$,

Let $P=p_2/p_1$ and $D={\rho }_2/{\rho }_1$ be the pressure and mass density ratios across a normal shock wave in a perfect gas with specific-heat ratio $\gamma$. Eliminating the velocities from the Rankine-Hugoniot conditions for a perfect gas givesAs $P\to \infty$, the ratio tends to the shock compression ratio $(\gamma +1)/(\gamma -1)$.

For the pressure-density Hugoniot relation for a perfect gas, the change in specific entropy is $[s]/c_v=\log P-\gamma \log D(P)$, with $c_v$ the specific heat capacity at constant volume. Its derivative isIt is positive for a compressive normal shock wave with $P>1$. For a weak shock with $P=1+\delta$, integrating the leading term givesThus the entropy production is cubic in the small pressure jump, even though the mass density and temperature changes already appear at first order.

For $N$ equal weak shocks with individual pressure ratio $1+\delta$ and total ratio $P_s=(1+\delta {)}^N$, entropy production in a perfect-gas shock givesAt fixed $P_s>1$, this tends to zero as the compression is divided into increasingly many smaller weak shocks. A single finite normal shock wave instead gives the positive value $\log P_s-\gamma \log D(P_s)$. Gradual compression can consequently approach reversible isentropic flow while an abrupt compression generates finite entropy.