2017 Introduction to CFD Midterm Exam
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Poll ended at 3:43 pm on Tuesday April 25th 2017. Total votes: 19. Total voters: 9.
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 04.09.17
Thursday April 20th 2017
18:00 — 20:00

NO NOTES OR BOOKS; USE CFD TABLES THAT WERE DISTRIBUTED; ANSWER ALL 4 QUESTIONS; ALL QUESTIONS HAVE EQUAL VALUE.
 04.14.17
 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
Starting from the principle of conservation of mass, show that the mass conservation equation for a fluid corresponds to: $$\frac{\partial \rho}{\partial t} + \frac{\partial \rho u}{\partial x} + \frac{\partial \rho v}{\partial y} + \frac{\partial \rho w}{\partial z} =0$$ with $\rho$ the mass density, and $u,v,w$ the $x,y,z$ components of the velocity vector.

 Question #3
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: $\rho H=\rho E+P$.

 Question #4
Consider the following grid:
with $R=1$ m, $L_1=1.5$ m, $L_2=1.2$ m, $L_3=0.5$ m, and $ds_3=10^{-4}$ m. Note that $ds_1$ and $ds_2$ are uniform in the range $0^\circ \le \theta \le 90^\circ$ (but $ds_1 \neq ds_2$) and that the mesh should have 123 grid lines along $i$ and 56 grid lines along $j$. Do the following:
 (a) Outline the gridding strategy (b) Write the code that would generate this grid. The grid should be such that there is no sudden change in mesh spacing at any location.
 3 (a) $u\frac{\rho_2}{\rho}$, (b) $\frac{\rho_1 R_1+\rho_2 R_2}{c_{v1}\rho_1 + c_{v2}\rho_2}$
 $\pi$