Hegedus Aerodynamics

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Opening Statement

The non-dimensional values were obtained from reference (1).

Values with tildes (~) are dimensional while non accented values are non-dimensional.

Within CFD literature, it seems the variable used to represent total (stagnation) energy in the Q vector differs.  Sometimes one sees $\rho e$, sometimes $e$ or sometimes $e_0$.  To clarify what $e_0$ refers to in Aero Troll CFD, I'll show some equations below.  Some equations have been adapted from reference (2). $${\tilde{e}}_{\mathrm{total}}={\tilde{e}}_t=\tilde{e}={\tilde{e}}_0=\widehat{u}+\frac{1}{2}{\tilde{V}}^2+\tilde{g}\tilde{z}$$ At the moment, Aero Troll CFD neglects gravity, therefore: $${\tilde{e}}_0=\widehat{u}+\frac{1}{2}{\tilde{V}}^2$$ Assuming the perfect-gas law and constant specific heats: $$\hat{u}={\tilde{c}}_v\tilde{T}$$ Therefore: $${\tilde{e}}_0={\tilde{c}}_v\tilde{T}+\frac{1}{2}{\tilde{V}}^2$$ Next, the pressure equation will be derived using the perfect gas law and constant specific heats.  For a perfect gas, the following is true: $\tilde{R}={\tilde{c}}_p-{\tilde{c}}_v=\left(\gamma -1\right){\tilde{c}}_v$

Starting with the perfect gas law: $$\tilde{p}=\tilde{\rho }\tilde{R}\tilde{T}$$ $$\tilde{p}=\left(\gamma -1\right)\tilde{\rho }{\tilde{c}}_v\tilde{T}$$ $$\tilde{p}=\left(\gamma -1\right)\tilde{\rho }\left({\tilde{e}}_0-\frac{1}{2}{\tilde{V}}^2\right)$$

Non-Dimensionalized Values

$$\rho =\frac{\tilde{\rho }}{{\tilde{\rho }}_{\infty }}$$ $$V=U=\frac{\tilde{U}}{{\tilde{a}}_{\infty }}$$ $$u=\frac{\tilde{u}}{{\tilde{a}}_{\infty }}$$ $$v=\frac{\tilde{v}}{{\tilde{a}}_{\infty }}$$ $$w=\frac{\tilde{w}}{{\tilde{a}}_{\infty }}$$ $$a=\frac{\tilde{a}}{{\tilde{a}}_{\infty }}$$ $$p=\frac{\tilde{p}}{{\tilde{\rho }}_{\infty }{\tilde{a}}_{\infty }^2}=\frac{\tilde{p}}{{\tilde{\gamma }}_{\infty }{\tilde{p}}_{\infty }}$$ $$T=\frac{\tilde{T}}{{\tilde{T}}_{\infty }}=a^2$$ $$e_0=\frac{{\tilde{e}}_0}{{\tilde{a}}^2}$$ $$R=\frac{\tilde{R}}{{\gamma }_{\infty }{\tilde{R}}_{\infty }}$$ $$\mu =\frac{\tilde{\mu }}{{\tilde{\mu }}_{\infty }}$$

Non-Dimensionalized Equations

Perfect-gas law:

$$p=\rho RT$$ Speed of sound:

$$a^2=\gamma RT=\frac{\gamma p}{\rho }$$ Pressure:

$$p=\left(\gamma -1\right)\left(\rho e_0-\frac{1}{2}\rho \left(u^2+v^2+w^2\right)\right)$$ Total Enthalpy:

$$H_0=\frac{{\tilde{H}}_0}{{\tilde{a}}_{\infty }^2}=\frac{a^2}{\gamma -1}+\frac{1}{2}\overrightarrow{V}·\overrightarrow{V}=\frac{\gamma }{\gamma -1}\frac{p}{\rho }+\frac{1}{2}\overrightarrow{V}·\overrightarrow{V}=\frac{1}{\rho }\left(\rho e_0+p\right)$$ Entropy: $$s-s_{\infty }=\frac{\gamma -1}{\tilde{R}}\left(\tilde{s}-{\tilde{s}}_{\infty }\right)=\ln \left(\gamma p\right)-\gamma \ln \left(\rho \right)$$

References

1) Krist, S. L., Biedron, R. T., and Rumsey, C. L., "CFL3D User's Manual (Version 5.0)," NASA TM 1998-208444, June 1998.
2) White, F. R., "Fluid Mechanics, 2nd Edition," McGraw Hill, 1986.