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 CoordinatesGeneralized 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 $$ and with $\xi_1$, $\xi_2$, $\xi_3$, $\xi_4$, $\xi_5$, $\xi_6$, $R_1$, $R_2$ some constants. Find $\partial F_3/\partial U_4$ within the flux Jacobian.
Question #3
Consider the following grid that you wish to reproduce using CFDWARP:
M2019Q3.png  ./download/file.php?id=5107&sid=af7344590bcceecb0f2bc7e349d2cfd2  ./download/file.php?id=5107&t=1&sid=af7344590bcceecb0f2bc7e349d2cfd2
Do the following:
(a)  Outline the block-by-block strategy.
(b)  Write down the code needed within the Grid() module.
Question #4
Consider the following nodes on a mesh:
Q4.png  ./download/file.php?id=5119&sid=af7344590bcceecb0f2bc7e349d2cfd2  ./download/file.php?id=5119&t=1&sid=af7344590bcceecb0f2bc7e349d2cfd2
The position of every node along with the density and temperature correspond to:
Node$x$, m$y$, m$T$, K$\rho$, kg/m$^3$
10.0103170.133
20.020.00663210.300
30.030.01313450.285
40.040.01953580.271
50.050.02583600.257
60.060.03183560.245
70.010.013000.119
80.020.01653170.133
90.030.02293370.153
100.040.02913500.168
110.050.03523550.174
120.060.04113550.174
130.010.023000.119
140.020.02633020.121
150.030.03263100.128
160.040.03873250.143
170.050.04473420.162
180.060.05053510.172
190.010.033000.119
200.020.03623000.119
210.030.04233010.120
220.040.04843040.123
230.050.05423160.135
Using generalized coordinates and second-order accurate stencils if possible and first-order accurate stencils otherwise, find $$\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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