Hegedus Aerodynamics

Non-Dimensionalization for Aero Troll CFD

Note: the equations below require MathML to be displayed.

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). $e ~ total = e ~ t = e ~ = e ~ 0 = u ^ + 1 2 V ~ 2 + g ~ z ~$ At the moment, Aero Troll CFD neglects gravity, therefore: $e ~ 0 = u ^ + 1 2 V ~ 2$ Assuming the perfect-gas law and constant specific heats: $u ^ = c ~ v T ~$ Therefore: $e ~ 0 = c ~ v T ~ + 1 2 V ~ 2$ Next, the pressure equation will be derived using the perfect gas law and constant specific heats.  For a perfect gas, the following is true: $\stackrel{~}{R}={\stackrel{~}{c}}_{p}-{\stackrel{~}{c}}_{v}=\left(\gamma -1\right){\stackrel{~}{c}}_{v}$

Starting with the perfect gas law: $p ~ = ρ ~ R ~ T ~$ $p ~ = γ - 1 ρ ~ c ~ v T ~$ $p ~ = γ - 1 ρ ~ e ~ 0 - 1 2 V ~ 2$

Non-Dimensionalized Values

$ρ = ρ ~ ρ ~ ∞$ $V = U = U ~ a ~ ∞$ $u = u ~ a ~ ∞$ $v = v ~ a ~ ∞$ $w = w ~ a ~ ∞$ $a = a ~ a ~ ∞$ $p = p ~ ρ ~ ∞ a ~ ∞ 2 = p ~ γ ~ ∞ p ~ ∞$ $T = T ~ T ~ ∞ = a 2$ $e 0 = e ~ 0 a ~ 2$ $R = R ~ γ ∞ R ~ ∞$ $μ = μ ~ μ ~ ∞$

Non-Dimensionalized Equations

Perfect-gas law:

$p = ρ R T$ Speed of sound:

$a 2 = γ R T = γ p ρ$ Pressure:

$p = γ - 1 ρ e 0 - 1 2 ρ u 2 + v 2 + w 2$ Total Enthalpy:

$H 0 = H ~ 0 a ~ ∞ 2 = a 2 γ - 1 + 1 2 V → · V → = γ γ - 1 p ρ + 1 2 V → · V → = 1 ρ ρ e 0 + p$ Entropy: $s - s ∞ = γ - 1 R ~ s ~ - s ~ ∞ = ln⁡ γ p - γ ln⁡ ρ$

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.