Adiabatic nozzle terminal speed ($u_{\infty }=c_{\ast }\sqrt{(\gamma +1)/(\gamma -1)}$)

For a lossless polytropic flow with $\gamma >1$, negligible gravity and a sonic throat, the Bernoulli function is $u^2/2+c_s^2/(\gamma -1)=(\gamma +1)c_{\ast }^2/[2(\gamma -1)]$. On an expanding supersonic branch for which $\rho \to 0$, the specific enthalpy vanishes and the limiting velocity is the displayed value. If area grows as $C{r}^2$, mass conservation further gives $\rho \sim \dot{M}/(C{u}_{\infty }{r}^2)$. A subsonic pressure-confined branch need not have this asymptotic behavior.