[Friendly]

Hello everyone,


I wonder if there are any analytical methods to estimate the angle of a supersonic free stream e. g. out of a pipe with given p, T and Ma inside the pipe.


Does anyone know if there is a possibility? Unfortunately I cant find one.

[Friendly]

I am a bit confused by this reply.

[agd]

I think you got hit by a spam reply. Regarding your question, it is somewhat ill-posed. Flow won't just go supersonic inside a pipe, and how the flow reaches the state it possesses at the exit may play a role in further analysis.



But assuming that the flow is supersonic at a duct exit you have three cases to consider. First, the exit pressure at the duct matches the pressure of the atmosphere. In this case the jet should leave fairly smoothly since there will be no need for compression or expansion. There will be a strong shear layer between the jet and the freestream that will go unsteady.


If the pressure at the duct exit is less than the freestream you may get a normal shock at the duct exit that will jump the pressure up to freestream, and the flow will effectively leave the duct subsonically. If the pressure difference is not too large the pressure adjustment may occur through oblique shocks and expansions that will work to correct the pressure and adjust the velocity angles.


If the pressure in the duct is higher than the freestream, the flow will continue to expand upon exit through a series of expansion fans and possible weak oblique shocks.


The classic example of these behaviors is the converging-diverging nozzle operation. I suggest you look there and see where that leads you.

[Friendly]

Thanks for your reply!


I know all that about the nozzle.


The stream inside the pipe is subsonic and then reaches Ma = 1 in the narrowest cross section, where is also the end of the pipe. After that the stream leaves the pipe while expanding (Prandtl-Meyer Expansion).


I want to know if there is a possibility to determine the angle of the outflow for e.g. given pressure ratios without using CFD.

[agd]

You might find something through the method of characteristics.

[FMDenaro]

Originally Posted by Friendly

Thanks for your reply!


I know all that about the nozzle.


The stream inside the pipe is subsonic and then reaches Ma = 1 in the narrowest cross section, where is also the end of the pipe. After that the stream leaves the pipe while expanding (Prandtl-Meyer Expansion).


I want to know if there is a possibility to determine the angle of the outflow for e.g. given pressure ratios without using CFD.

Have a look to the textbook of Zucrow