Fanno flow

Example

Fanno flow is the steady, one-dimensional, adiabatic flow of an ideal gas through a duct of constant cross-section with wall friction. Let $D$ be the hydraulic diameter of the duct and $f$ the Fanning friction factor, assumed constant. Let $\gamma$ be the ratio of specific heats and $M$ the Mach number. The mass flux $\rho u$ and the stagnation temperature are constant along the duct, while friction raises the entropy. Let $L^*$ be the length of duct needed to bring the flow from Mach number $M$ to $M = 1$. Then $$\frac{4 f L^*}{D} = \frac{1 - M^2}{\gamma M^2} + \frac{\gamma + 1}{2\gamma} \ln\frac{(\gamma + 1) M^2}{2 + (\gamma - 1) M^2}.$$ Check at $M = 1$: both terms vanish, so $L^* = 0$, as it should. For $M \to 0$ the right side tends to $+\infty$, so a very slow flow needs a very long duct to reach sonic speed. For $M \to \infty$ it tends to $-\frac{1}{\gamma} + \frac{\gamma + 1}{2\gamma}\ln\frac{\gamma + 1}{\gamma - 1}$, which is about $0.82$ for $\gamma = 1.4$. So a supersonic Fanno flow can only cover a bounded length before it becomes sonic.