Computational Aerodynamics Assignment 2 — Generalized Coordinates


B. Parent • AE68714 Computational Aerodynamics

Computational Aerodynamics 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: $$ \Omega \equiv x_\xi y_\eta - y_\xi x_\eta $$

03.20.17

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.

03.21.17

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.

(d)		Find $\partial^2\rho/\partial x^2$ 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 28th at 16:30. Do Questions #2 and #3 only.

03.22.19

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