Nozzle throat velocity

Question

I have the following question regarding nozzles; It is known that in a CD nozzle there can be only one of the following outcomes;

1. a flow fully subsonic in the convergent, subsonic at the throat and then subsonic in the divergent area; That's in the case that the pressure ratio is below the critical pressure ratio.
2. a flow fully subsonic in the convergent, sonic at the throat and then supersonic in the divergent (no shock) or supersonic and then subsonic (with a shock).

So far so good; my question now about the second case. From a physical point of view, why can't the sonic transition take place before the nozzle throat? For the sake of simplicity in exercises we know that the nozzle has a CD shape, so we assume a priori that if there is a sonic transition it will be right at the throat. But physically speaking, the flow is being accelerated and "it does not know" what's happening ahead and if the cross section is going to be even smaller later. So what prevents it from going sonic before the nozzle throat? Can someone help with this thought? Am I missing something fundamental or approaching it in a completely wrong way?

Thank you and cheers, ndjojo

Answer 1

Excellent question, which took me a moment to figure out. But I'll walk you through the end results.

The equation of mass conservation can be written as $$(1)\;\;\; \rho v A=const. ,$$ with the gas mass density $\rho$, the gas velocity $v$ and the cross-sectional area $A$. For our purposes we can recast this into $$(2)\;\;\;\nabla \ln \rho + \nabla \ln v + \nabla \ln A=0.$$

The Navier-Stokes Equation(s) in 1-D with all simplifications and an isothermal equations of state $\partial P/\partial \rho = c_s^2$ can with without much ado be written as $$(3)\;\;\; \frac{\nabla \ln \rho}{\nabla \ln v} = -\frac{u^2}{c^2_s}. $$

So from this fundamental equations we see already that something interesting happens at the sonic transition: When $u^2/c^2_s=1,$ according to (3) we get a change in the dominant contribution to the differentials that are necessary to keep the mass flux constant in (2). Now combining (3) and (2) and elimiating the density differential gives you the equation, which you need to answer your questions:

$$ (4)\;\;\;(M^2-1)\nabla \ln v = \nabla \ln A. $$ Now Eqn. (4) shows that, if you are supersonic ($M^2>1$) in the convergent part of the nozzle ($\nabla\ln A < 0$) then this implies $\nabla \ln v < 0$ i.e. you decelerate. On the other hand, once you are subsonic you accelerate again, so this would imply that you have either a flow that finds an equilibrium at $M=1$ or there could be oscillations around $M=1$ until you hit the throat.

I am not sure what happens next. Depending if you get oscillations around $M=1$, then you have a chance of hitting the throat while being supersonic. But when you hit the throat while being supersonic, you will decelerate to subsonic speeds and the rocket fails. If there are no oscillations however, and your flow reaches the throat with a stable $M=1$, then the usual chocking conditions should apply and you successfully accelerate to supersonic speeds.

Answer 2

Signals (like pressure change) can propagate upstream in subsonic flow so the flow ahead of the throat will sense the throat ahead. That is my best explanation. From an equation standpoint I remember from engineering thermodynamics class (I looked the details up): dA/A = -(1-M^2) dV/V where A is crosssectional area, M is Mach number and V is velocity. So this shows us that for dV positive and M < 1 the area decreases (dA negative) and at M = 1 the area does not change.
The length of the nozzle from where speed is low does not make a difference (within reason, there can be greater losses), it is the minimum area that matters.