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
Answers
4.  $u\left(1-\frac{\rho_1}{\rho}\right)$, $\frac{\rho_1 R_1+ \rho_2 R_2 }{c_{v1}\rho_1 + c_{v2}\rho_2}$.
Due on Thursday April 12th at 16:30. Do Questions 2, 3, and 4.
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