Introduction to CFD 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.
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.
Due on Thursday April 13th at 16:30. Solve the 3 Problems.
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