Computational Aerodynamics Assignment 4 — Flux Jacobian and Eigenvalues
 Question #1
Starting from the Euler equations: $$U=\left[\begin{array}{c}\rho \\ \rho u\\ \rho E \end{array} \right] ~~~~ F=\left[\begin{array}{c}\rho u\\ \rho u^2+P\\ \rho u H \end{array}\right]$$ Show that the vectors $U$ and $F$ can be expressed as: $$U=\left[\begin{array}{c}\rho \\ \rho u\\ \frac{1}{\gamma-1}\rho d + \frac{1}{2}\rho u^2\end{array} \right] ~~ \textrm{and} ~~ F=\left[\begin{array}{c}\rho u \\ \rho u^2+\rho d\\ \frac{\gamma}{\gamma-1}\rho u d + \frac{1}{2}\rho u^3\end{array} \right]$$ with $d\equiv RT$ and $\gamma\equiv c_p/c_v$. Outline clearly your assumptions.
 04.04.17
 Question #2
Starting from $F$ and $U$ found in Question #1, prove that the flux Jacobian $A\equiv\partial F/\partial U$ is equal to: $$A=\left[ \begin{array}{ccc} 0 & 1 & 0\\ \frac{\gamma-3}{2}u^2 & (3-\gamma)u & \gamma-1 \\ \frac{\gamma}{1-\gamma}ud +\frac{\gamma-2}{2}u^3 & \frac{\gamma}{\gamma-1}d + \frac{3-2\gamma}{2}u^2 & \gamma u \end{array} \right]$$ Note: you only need to prove the terms on the third row of $A$. Do not derive the terms on the first and second rows.
 Question #3
Starting from the flux Jacobian obtained in the previous question, show that $u$ is a valid wavespeed of the Euler equations.
 04.06.17
 Question #4
Consider the following system of equations: $$\frac{\partial U}{\partial t} + \frac{\partial F}{\partial x} =0$$ with $$U=\left[ \begin{array}{c} \rho_1 \\ \rho_2 \\ \rho u \\ \rho E \end{array} \right] =\left[ \begin{array}{c} U_1 \\ U_2 \\ U_3 \\ U_4 \end{array} \right] ~~~~{\rm and} ~~~~ F=\left[ \begin{array}{c} \rho_1 u \\ \rho_2 u\\ \rho u^2 +P \\ \rho u H \end{array} \right] =\left[ \begin{array}{c} F_1 \\ F_2 \\ F_3 \\ F_4 \end{array} \right]$$ with $$E = \frac{\rho_1}{\rho} c_{v1} T + \frac{\rho_2}{\rho} c_{v2} T + \frac{u^2}{2}$$ $$H = \frac{\rho_1}{\rho} c_{p1} T + \frac{\rho_2}{\rho} c_{p2} T + \frac{u^2}{2}$$ $$P = \left( \rho_1 R_1 + \rho_2 R_2 \right) T$$ $$\rho=\rho_1+\rho_2$$ and with $c_{v1}$, $c_{v2}$, $c_{p1}$, $c_{p2}$, $R_1$, $R_2$ some constants. Using the method of your choice find the following elements within the flux Jacobian:
 (a) Find $\partial F_1/\partial U_1$ (b) Find $\partial F_3/\partial U_4$
Hint: $\rho H=\rho E+P$.
 04.05.18
 Question #5
Consider the following system of equations: $$\frac{\partial U}{\partial t} + \frac{\partial F}{\partial x} =0$$ with $$U=\left[ \begin{array}{c} \rho_1 \\ \rho_2 \\ \rho u \\ \rho E \end{array} \right] =\left[ \begin{array}{c} U_1 \\ U_2 \\ U_3 \\ U_4 \end{array} \right] ~~~~{\rm and} ~~~~ F=\left[ \begin{array}{c} \rho_1 u \\ \rho_2 u\\ \rho u^2 +P \\ \rho u H \end{array} \right] =\left[ \begin{array}{c} F_1 \\ F_2 \\ F_3 \\ F_4 \end{array} \right]$$ with $$E = \frac{\rho_1}{\rho} e_1 + \frac{\rho_2}{\rho} e_2 + \frac{u^2}{2}$$ $$P = \left( \rho_1 R_1 + \rho_2 R_2 \right) T$$ $$\rho=\rho_1+\rho_2$$ $$H=E+\frac{P}{\rho}$$ $$e_1=\xi_1 + \xi_2 T + \xi_3 T^2 +\xi_4 T^3 + \xi_5 T^4$$ $$e_2=\xi_6 T$$ and with $\xi_1$, $\xi_2$, $\xi_3$, $\xi_4$, $\xi_5$, $\xi_6$, $R_1$, $R_2$ some constants. Find $\partial F_3/\partial U_4$ within the flux Jacobian.
 04.04.19
 4. $u\left(1-\frac{\rho_1}{\rho}\right)$, $\frac{\rho_1 R_1+ \rho_2 R_2 }{c_{v1}\rho_1 + c_{v2}\rho_2}$. 5. $\frac{\rho_1 R_1 + \rho_2 R_2}{\rho_1 (\xi_2 + 2 \xi_3 T+ 3 \xi_4 T^2+4\xi_5 T^3)+\rho_2 \xi_6}$
 $\pi$