Normal shock wave

A normal shock is a discontinuity perpendicular to the flow direction across which mass, momentum, and total energy fluxes are conserved.

For a moving level surface, $n=\nabla S/|\nabla S|$, $v_n=-S_t/|\nabla S|$, and the surface delta distribution is ${\delta }_s=|\nabla S|\delta (S)$. Piecewise classical conservation laws acquire a distributional flux defect $[b]\cdot n-v_n[a]$. Vanishing of this defect is the Rankine-Hugoniot condition.

The shock frame is an inertial reference frame in which the shock is stationary. A travelling shock then becomes a steady flow through a fixed discontinuity.

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

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.

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 $2{\rho }_0{V}_s^2/(\gamma +1)$.

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.

A weak shock has a small relative pressure jump. Its density jump agrees with the reversible adiabatic prediction through second order in the pressure jump, while entropy production first appears at third order.

An oblique shock is inclined relative to the upstream velocity. In an inviscid perfect gas, the normal velocity satisfies the normal-shock jump conditions while the tangential velocity is continuous.

If $p_2/p_1=1+ϵ$ with $0<ϵ\ll 1$ and the upstream Mach number is $M_1>1$, the shock angle is the Mach angle to leading order and the flow deflection isThe normal speed decreases, so the downstream flow turns toward the shock front.

The Mach angle of a supersonic flow with Mach number $M>1$ is $\mu =\arcsin (M^{-1})$. Infinitesimal pressure disturbances lie on the corresponding Mach cone, and a weak oblique shock approaches this angle as its strength tends to zero.