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Shocks form adiabatic discontinuities that ALWAYS take an upstream relative supersonic Mach number $M^- > 0$ and decelerates it to a downstream subsonic Mach number $M^+ <0$. It is useful to think of shocks as walls. Whereas in incompressible flow, we deal with all values of $P$, $T$, $u$, etc as a continuum through space, a shock is modeled as a boundary where these values can discontinuously jump.
Expansion fans are the inverse of shocks. Expansion fans are not discontinuities, but rather isentropic gradients that take an upstream relative subsonic Mach number $M^- <0$ and accelerates it to a downstream supersonic Mach number $M^+ >0$.
A useful Rankine-Hugoniot relation online calculator developed by the Virginia Tech Aerospace Dept exists here:
For a normal shock (1D problem), the Rankine Hugoniot relations tell us how thermodynamic values jump across the shock discontinuity. Notice that all you need is the upstream Mach number $M_1$ and specific heat ratio $\gamma$ and the entire relationship between the upstream and downstream states are defined.
\[M_2^2 = \frac{2 + (\gamma-1)M_1^2}{2\gamma M_1^2 - (\gamma-1)}\] \[\frac{\rho_2}{\rho_1} = \frac{u_1}{u_2} = \frac{(\gamma+1)M_1^2}{2 + (\gamma-1)M_1^2}\] \[\frac{P_2}{P_1} = 1 + \frac{2\gamma}{\gamma + 1}(M_1^2-1)\] \[\frac{T_2}{T_1} = \frac{P_2}{P_1}\frac{\rho_1}{\rho_2} = [1 + \frac{2\gamma}{\gamma + 1}(M_1^2-1)][\frac{2 + (\gamma-1)M_1^2}{(\gamma+1)M_1^2}]\] \[\Delta s = s_1 - s_2 = c_p \ln(\frac{T_2}{T_1}) - R\ln(\frac{P_2}{P_1})\]Note that there are a few caveats/things to remember regarding the usage of the R-H relations.
The general equation for thrust (of any propulsion system air or fuel-breathing) is
\[T = \dot{m_a}(U_e - U_0) + (P_e - P_0)A_e + \dot{m_f}U_e\]Notice that the three terms here are essentially