Rankine-Hugoniot Relation

Passing through a shock front moving with a speed $V_s$, the physical variables $\rho$, $p$, and $u$ change abruptly. Since the basic equations of hydrodynamics is unchanged after chosing a system moving $V_s$, the continuity equation

\frac{\partial \rho}{\partial t}+\frac{\partial \rho u}{\partial x}=0
\end{displaymath} (A.26)

gives an equation for a steady state as
\frac{\partial \rho u}{\partial x}=0.
\end{displaymath} (A.27)

Considering a region containing the shock front at $x_s$ which extends from $x=x_s-\Delta x$ to $x=x_s+\Delta x$ and integrating the above equation, we get
\int_{x_s-\Delta x}^{x_s+\Delta x}\frac{d \rho u}{d x}dx=
\left[\rho u\right]_{x_s-\Delta x}^{x_s+\Delta x}.
\end{displaymath} (A.28)

Thus, we obtain the jump condition conserning the mass coservation as
\rho_1 u_1=\rho_2 u_2,
\end{displaymath} (A.29)

where the quatities with suffix 1 are for preshock and those with suffix 2 are for postshock.

Equation of motion for steady state

\rho u\frac{\partial u}{\partial x} =-\frac{\partial p}{\partial x},
\end{displaymath} (A.30)

p_1+\rho_1 u_1^2=p_2+\rho_2 u_2^2,
\end{displaymath} (A.31)

where we used equation(A.29).

Kohji Tomisaka 2012-10-03