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I am doing a nondimensional Navier-Stokes simulation, but it's not exactly clear what values to use for the various gas constants: $R$, $C_v$, and $C_p$. Obviously, once I know one of these and $\gamma$, I can define the others. A colleague of mine is using the formula $$C_v=\frac{1}{\gamma*(\gamma-1)M_\infty^2}$$ but it's not clear where this comes from and what value to use for $M_\infty$, especially in the context of shock-tube problems. For lack of a better option, I've just been setting $M_\infty=1$, but I'm not convinced this is defensible. What do other people do in this context?

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  • $\begingroup$ What is $M_{\infty}$ here? Dimensional analysis suggests it must have units of reciprocal square root heat capacity, which would be unusual. $\endgroup$ – Geoff Oxberry Oct 9 '14 at 18:30
  • $\begingroup$ @GeoffOxberry, for external flows, $M_\infty$ is the free stream Mach number, typically. I think there's something funny with this formula. $\endgroup$ – Bill Barth Oct 9 '14 at 19:26
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If you are using an ideal gas equation of state, only the ratio of specific heats should appear in the nondimensional governing compressible Navier-Stokes equations. Knowing that you want to simulate (say) air, which provides gamma, is sufficient.

If this is your setting and my answer unclear, please let me know and I can point you to a write up.

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  • $\begingroup$ Rhys, your writeup cleared everything up. Thanks. $\endgroup$ – Truman Ellis Oct 10 '14 at 19:44
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It would probably be instructional if you went through the process of non-dimensionalization yourself, starting from the dimensional equations for which it is always clear which constant to use where. This way, you will discover which combinations of constants appear where.

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  • $\begingroup$ I just went through the process and it clarified thing considerably. Thanks. $\endgroup$ – Truman Ellis Oct 10 '14 at 19:43

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