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Here is the mass flow rate equation from the NASA site(enter link description here). enter image description here enter image description here According to this the maximum mass flow occurs at Mach number M = 1, which I can understand. But I can substitute Mach values greater than one (M>1) in the equation, then I get a lower mass flow rate. enter image description here I know that in a convergent-divergent nozzle, the Mach number grows greater than 1 in the divergent section as the back pressure at the exit of the nozzle continues to reduce. But here the mass flow rate continues to remain constant and equal to chocked mass flow rate at the nozzle throat. Is the equation correct in suggesting that the flow can accelerate past Mach 1 as long as the mass flow is reducing?

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The vertical axis is labeled Normalised Mass Flow Rate ( and the brackets [-] indicates no units). It is the value calculated on the right side of the equation divided by the value of the right side with M = 1. In a convergent-divergent nozzle, the stagnation pressure and temperature ($p_t$ and $T_t$) remain constant (at least for the ideal case) and the mass flow rate remains constant. Therefore, the decrease of the curve above Mach = 1 shows area has to increase to get higher Mach number. If you try to add or remove flow you have to do a whole bunch of other calculations (see Shapiro's The Dynamics and Thermodynamics of Compressible Fluid Flow (Volume 1)for example.

It took looking in three books to find the equation I wanted to share that clearly shows why the diverging section has supersonic flow (to increase flow speed).

$\frac{dA}{A}=-(1-M^2)\frac{dV}{V}$

where $dA$ is the increase of cross-sectional area, $M$ the Mach number, $dV$ the increase in flow velocity. Found in Fundamentals of Gas Dynamics by Owczarek (c. 1964 International Textbook Company), page 201. If you want a derivation I can add it.

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  • $\begingroup$ "Therefore, the decrease of the curve above Mach = 1 shows AREA has to increase to get higher Mach number". If the flow has attained the sonic condition (M=1) at a particular area A then increasing that area will indeed increase the mass flow rate but the increase of area will not increase the Mach number at that area as the flow will be chocked. My question is how to explain the behavior eluded to in the equation when we input M>1. $\endgroup$
    – GRANZER
    Aug 5, 2022 at 12:04
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As long as the choked flow rate (maximum $\dot{m}$) is achieved at minimum flow area, i.e. the sonic condition is achieved at the nozzle throat, then no downstream changes will alter the fact that the mass flow rate remains constant (steady-state). This is because the flow is supersonic past the nozzle throat and downstream pressure changes will not alter the mass flow rate past the choking point. This is an intrinsic property associated with transonic flow, where the quasilinear PDE changes from elliptic (for $M<1$ ) to hyperbolic ($M>1$), so that downstream pressure perturbations cannot propagate upstream and change the mass flow rate (unchoke the flow).

If flow choking is never attained (i.e. $M =1$ at minimum flow area is not attained), then it is true that as the Mach number increases, the mass flow rate decreases (e.g. supersonic inlet where you vary the inlet Mach number).

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