[mkhm]

Hi,



I am working on the design of supersonic nozzles and I am using the method of characteristics. In this method, we should find the Mach number by inversion of the of the Prandtle-Meyer relation.


For low Mach number, one can use the Newton-Raphson method. and for air, there is the very well know receipt of : Hall, I. M. "Inversion of the prandtl-meyer relation." Aeronautical Journal 79 (1975): 417.


In my case, I have high Mach numbers. So, if I use Newton-Raphson, the residual is very high. And I treat a case with CO2 molecules and therefore gamma different from 1.4.



My search through internet was not successful. I am wondering if one supersonic expert could help me with that.



Thanks in advance

[FMDenaro]

Please, provide details about what are you doing in solving numerically the equation for determining the angle

[mkhm]

Originally Posted by FMDenaro

Please, provide details about what are you doing in solving numerically the equation for determining the angle

I want to find the Mach number through a known angle (nu) with respect to which the flow turns isentropically. Please find the prandtl-meyer in the attached file.

This relation gives nu as a function of Mach number. What I want is to find Mach number for a given angle nu. This implies to inverse the above equation. I know two methods. First: We can do it by using the Hall relation. However, hall relation is for gamma equal to 1.4. Second, doing Newton-Raphson where residual is very high for me (high mach number and low gamma) and I dont reach convergence by the mentioned method. So, I am wondering if there are other methods which can give me Mach number as a function of nu. I looked if there is analytical expressions, I did not find any thing for gamma equal to 1.29. Or even tabulated table would be nice. But my search was not successful. Do you have any suggestion ?

[FMDenaro]

And does your fixed angle fulfill the max deviation limit?

[mkhm]

Originally Posted by FMDenaro

And does your fixed angle fulfill the max deviation limit?

Yes, and lets say that this equation is not aware of the fact that there is a maximum deviation fixed by tetha maximum at the diverging part of the sharp nozzle and should find a Mach number for a given nu, whatever is the value of nu.

[FMDenaro]

Have you plotted the function along the Mach number to see it graphically?

[mkhm]

Originally Posted by FMDenaro

Have you plotted the function along the Mach number to see it graphically?

I know the trend should be something like the photo being attached.
Do you suggest me to make a tabulated table and use the table afterwards to identify a Mach number for a given nu ?