2018 Introduction to CFD Midterm Exam  
Tuesday April 24th, 2018
16:30 to 18:30


NO NOTES OR BOOKS; USE INTRODUCTION TO CFD TABLES THAT WERE DISTRIBUTED; ANSWER ALL 4 QUESTIONS; ALL QUESTIONS HAVE EQUAL VALUE.
04.18.18
Question #1
Starting from Newton's law $\vec{F}_y=m\frac{dv}{dt}$ and the mass conservation equation show that the $y$-component of the momentum transport equation for a fluid corresponds to: $$ \frac{\partial \rho v}{\partial t} + \frac{\partial \rho u v}{\partial x} + \frac{\partial \rho v^2}{\partial y} + \frac{\partial \rho w v}{\partial z} = -\frac{\partial P}{\partial y} $$ with $P$ the pressure.
Question #2
Starting from the Euler equations $$ \frac{\partial U}{\partial t} + \frac{\partial F_x}{\partial x} + \frac{\partial F_y}{\partial y}=0 $$ and the metrics $\eta_x$, $\xi_y$, $\Omega$, etc outlined in the tables, show that the Euler equations can be written in generalized coordinates in strong conservative form as follows: $$ \frac{\partial Q}{\partial \tau} + \frac{\partial G_\xi}{\partial \xi} + \frac{\partial G_\eta}{\partial \eta}=0 $$ with $$ Q\equiv \Omega \Gamma U $$ $$ G_\xi\equiv \Omega(\xi_x F_x + \xi_y F_y) $$ $$ G_\eta\equiv \Omega(\eta_x F_x + \eta_y F_y) $$ Outline clearly your assumptions.
Question #3
Create a grid for the following problem
Q3.png  ./download/file.php?id=4130&sid=3a5126efe9587686283908db733e3c19  ./download/file.php?id=4130&t=1&sid=3a5126efe9587686283908db733e3c19
given the dimensions $dw=10^{-4}$ m, $R_1=0.5$ m, $R_2=1$ m, $x_{\rm A}=0$, $y_{\rm A}=R_1$, $x_{\rm B}=0.5$ m, $y_{\rm B}=0.9 R_1$, $x_{\rm C}=1$ m, $y_{\rm C}=0.7R_1$, $x_{\rm D}=2$ m, $y_{\rm D}=0$, $H=R_2$, and $\theta=20^\circ$. Notes:
(a)  Outline clearly the strategy used.
(b)  Make sure that the grid spacing does not vary abruptly at any location.
(c)  Points A, B, C, D are not in a straight line and should be joined using a smooth curve.
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 $$ E = \frac{\rho_1}{\rho} e_1 + \frac{\rho_2}{\rho} e_2 + \frac{u^2}{2} $$ $$ P = \left( \rho_1 R_1 + \rho_2 R_2 \right) T $$ $$ \rho=\rho_1+\rho_2 $$ $$ H=E+\frac{P}{\rho} $$ $$ e_1=\xi_1 + \xi_2 T + \xi_3 T^2 $$ $$ e_2=\xi_4 T $$ and with $\xi_1$, $\xi_2$, $\xi_3$, $\xi_4$, $R_1$, $R_2$ some constants. Find $\partial F_3/\partial U_4$ within the flux Jacobian.
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