2019 Computational Aerodynamics Midterm Exam


B. Parent • AE68714 Computational Aerodynamics

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:

M2019Q3.png ./download/file.php?id=5107&sid=06977d22c7116cf2600e41845d563665 ./download/file.php?id=5107&t=1&sid=06977d22c7116cf2600e41845d563665

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:

Q4.png ./download/file.php?id=5119&sid=06977d22c7116cf2600e41845d563665 ./download/file.php?id=5119&t=1&sid=06977d22c7116cf2600e41845d563665

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.

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