Introduction to CFD Assignment 2 — Generalized Coordinates
 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: $$\Omega \equiv x_\xi y_\eta - y_\xi x_\eta$$
 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:
with the following associated properties:
 Node $x$, mm $y$, mm $\rho$, kg/m$^3$ 1 530 -90 1.0 2 400 -210 1.05 3 570 -220 1.05 4 750 -200 1.1 5 220 -360 1.05 6 380 -380 1.1 7 550 -400 1.15 8 730 -410 1.2 9 900 -420 1.25 10 320 -540 1.15 11 500 -580 1.20 12 650 -630 1.25 13 410 -700 1.30
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 29th at 16:30. Do Questions 2, 3, and 4 only.
 03.22.18
 $\pi$