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When I am reading lectures or exercises about Convergent-Divergent nozzle, many times the throat is supposed to be in sonic condition. But I don't see how scientists asses that sonic condition is really happening for a known nozzle geometry, and by knowing also the inlet condition and ambient pressure (which can be different from exit pressure of the exhaust).

For instance :

a gas is arriving at the inlet of the nozzle at (known data)

$T_0$ , $T_{s_0}$ respectively stagnation and static temperatures $P_0$ , $P_{s_0}$ respectively stagnation and static pressure $Q_0$ mass flow rate $M_O$ mach number ... and so on

we know the ambient pressure $P_{amb}$ outside of the exhaust

and we know the geometry (cross-section) of the nozzle : $A_0$ , $A_t$ (throat) , $A_e$ (exit) But we don't know yet the critical cross section $A_*$

With these data, how can we asses that the throat is in a sonic state ?

When I read this website it seems it is just by comparing $\frac{P_0}{P_{amb}}$ with $(\frac{\gamma + 1}{2})^{\frac{\gamma}{\gamma +1}}$

But I feel it is too easy to tell whether the throat is sonic.

Or do we have to do some assumptions, then check if the results with these assumptions are physically possible ?

Thank you

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Experimental rocket nozzles are extensively instrumented with pressure taps all along their length- convergent to throat and throat to divergent. Pressure readings taken from these during test firings establish whether or not shocks are present and if so, where they are forming.

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