Introduction to CFD Assignment 3 — Generalized Coordinates  
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 $$
03.21.17
Question #2
Starting from the Euler equations $$ \frac{\partial U}{\partial t} + \frac{\partial F_x}{\partial x} + \frac{\partial F_y}{\partial y}=0 $$ and the metrics $\eta_x$, $\xi_y$, $\Omega$, etc derived above in Question #1, show that the Euler equations can be written in generalized coordinates in strong conservative form as follows: $$ \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) $$ Outline clearly your assumptions.
03.22.17
Question #3
Consider the following nodes in the $x$-$y$ plane:
Q3.png
with the following associated properties:
Node$x$, mm$y$, mm$\rho$, kg/m$^3$
1530-901.0
2400-2101.05
3570-2201.05
4750-2001.1
5220-3601.05
6380-3801.1
7550-4001.15
8730-4101.2
9900-4201.25
10320-5401.15
11500-5801.20
12650-6301.25
13410-7001.30
Using the latter, and knowing that $$ F_x=F_y=\rho $$ and with second-order accurate stencils for the metrics and the derivatives do the following:
(a)  Find $G_\eta$ at node 3.
(b)  Find $G_\eta$ at node 11.
(c)  Find $\partial G_\eta/\partial\eta$ at node 7.
Question #4
For the nodes shown in Question #3 above, do the following:
(a)  Find $\Omega$ at node 7 using second-order accurate stencils for the metrics.
(b)  Find the cell area at node 7 using a method of your choice and compare it with $\Omega$ found in (a).
03.24.17
Due on Thursday March 30th at 16:30. Do Questions #2, #3, and #4 only.
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