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Old   June 3, 2012, 09:37
Question Mach-number-independence Oswatitsch
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for my studies, I've to develop a solution method for the mach-number-independence principle which is valid for very high mach-numbers. In my case I assume
- ideal gas,
- inviscid flow,
- flow over a blunt body,
- 2D flow,
- steady flow (at the end of the iteration),
- shock fitting.

In nondimensional form the systems consists of Crocco's theorem (a combination of the momentum equation and the first and second law of thermodynamics) with two equations for {Ma\rightarrow\infty}

{\left( {\begin{array}{*{20}{c}}u\\v\end{array}} \right)_t} = \dfrac{{1}}{{2\kappa }}\left( {1 - V^2} \right)\left( {\begin{array}{*{20}{c}}{s{_x}}\\{s{_y}}\end{array}} \right) + ({v_x} - {u_y})\left( {\begin{array}{*{20}{c}}v\\{ - u}\end{array}} \right)

and the gas dynamic equation which is a combination of the momentum and the coninuity equation and yields for

    (u u_t + v v_t  - \frac{1}{\rho}\frac{\kappa-1}{2}(1-V^2) \rho_t)&+ 
    (u^2 - \frac{\kappa  - 1}{2}(1-V^2))u_x + \\
(v^2 - \frac{\kappa  - 1}{2}(1-V^2))v_y &+
    u v (u_y + v_x) = 0

with entropy s, x-component of velocity u, y-component of velocity v, density \rho and V^2 = u^2 + v^2 with its spatial x and y and time derivatives t. I complement the sytems with the equation of state and the entropy equation which can be calculated in a seperate way.

I know that the system is elliptic and hyperbolic in space depending on the local velocity. So a time-dependent method must be applied. I already used MacCormack (without artificial viscosity) for a very simple case which was a 1-D streamline but the system didn't converged and became unstable.

Can anybody help me to find the mistake or has anybody even already implemented these equations???
Kathrin is offline   Reply With Quote


hypersonic, maccormack, oswatitsch, time dependent

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