Rayleigh flow

Example

Rayleigh flow is the steady, one-dimensional, frictionless flow of an ideal gas through a duct of constant cross-section. Heat is added or removed through the walls. Let $\gamma$ be the ratio of specific heats and $M$ the Mach number. The mass flux $\rho u$ and the momentum flux $p + \rho u^2$ are constant along the duct, and the stagnation temperature $T_0$ changes with the heat added. With the index $*$ for the state at $M = 1$ on the same flow, the relations are $$\frac{p}{p^*} = \frac{1 + \gamma}{1 + \gamma M^2}, \qquad \frac{T}{T^*} = \frac{(1 + \gamma)^2 M^2}{(1 + \gamma M^2)^2}, \qquad \frac{T_0}{T_0^*} = \frac{2(\gamma + 1) M^2 \left(1 + \frac{\gamma - 1}{2} M^2\right)}{(1 + \gamma M^2)^2}.$$ Check at $M = 1$: each right side equals $1$; for the last one, $2(\gamma + 1) \cdot \frac{\gamma + 1}{2} = (\gamma + 1)^2$. The function $T_0 / T_0^*$ increases on $0 < M < 1$, decreases for $M > 1$, and has its maximum $1$ at $M = 1$.