2019 Computational Aerodynamics Midterm Exam
April 25th 2019
16:30 — 18:30

NO NOTES OR BOOKS; USE COMPUTATIONAL AERODYNAMICS TABLES THAT WERE DISTRIBUTED; ANSWER ALL 4 QUESTIONS; ALL QUESTIONS HAVE EQUAL VALUE.
 04.03.19
 Question #1
Starting from the imposed dependencies on the generalized coordinates $\tau$, $\xi$, and $\eta$:
 Cartesian Coordinates Generalized Coordinates $t=t(\tau)$ $\tau=\tau(t)$ $x=x(\xi,\eta,\tau)$ $\xi=\xi(x,y,t)$ $y=y(\xi,\eta,\tau)$ $\eta=\eta(x,y,t)$
Demonstrate that the metrics of the generalized coordinates correspond to: $$\xi_t=\frac{\Gamma}{\Omega}\left(y_\tau x_\eta - x_\tau y_\eta \right),~~~~ \xi_x = \frac{y_\eta}{\Omega},~~~~ \xi_y=-\frac{x_\eta}{\Omega}$$ and $$\eta_t=\frac{\Gamma}{\Omega}\left(x_\tau y_\xi - x_\xi y_\tau \right),~~~~ \eta_x = -\frac{y_\xi}{\Omega},~~~~ \eta_y=\frac{x_\xi}{\Omega}$$ with $\Gamma\equiv \tau_t$ and $\Omega$ the inverse of the metrics Jacobian defined in 2D as: $$\Omega \equiv x_\xi y_\eta - y_\xi x_\eta$$
 Question #2
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.
 Question #3
Consider the following grid that you wish to reproduce using CFDWARP:
Do the following:
 (a) Outline the block-by-block strategy. (b) Write down the code needed within the Grid() module.
 Question #4
Consider the following nodes on a mesh:
The position of every node along with the density and temperature correspond to:
 Node $x$, m $y$, m $T$, K $\rho$, kg/m$^3$ 1 0.01 0 317 0.133 2 0.02 0.0066 321 0.300 3 0.03 0.0131 345 0.285 4 0.04 0.0195 358 0.271 5 0.05 0.0258 360 0.257 6 0.06 0.0318 356 0.245 7 0.01 0.01 300 0.119 8 0.02 0.0165 317 0.133 9 0.03 0.0229 337 0.153 10 0.04 0.0291 350 0.168 11 0.05 0.0352 355 0.174 12 0.06 0.0411 355 0.174 13 0.01 0.02 300 0.119 14 0.02 0.0263 302 0.121 15 0.03 0.0326 310 0.128 16 0.04 0.0387 325 0.143 17 0.05 0.0447 342 0.162 18 0.06 0.0505 351 0.172 19 0.01 0.03 300 0.119 20 0.02 0.0362 300 0.119 21 0.03 0.0423 301 0.120 22 0.04 0.0484 304 0.123 23 0.05 0.0542 316 0.135
Using generalized coordinates and second-order accurate stencils if possible and first-order accurate stencils otherwise, find $$\frac{\partial}{\partial x}\left(\sqrt{T} \frac{\partial \rho}{\partial x} \right) ~~\textrm{at node 9}$$ For simplicity, you can assume that the inverse of the metric Jacobian $\Omega$ is equal to $10^{-4}$ m$^2$ for all nodes.
 $\pi$