Grid transformation to computational space

mech_eng_gr · May 31, 2020, 13:17

Hello,

So I've got this problem that I need to solve the governing equations (x,y momentum and continuity with pressure correction (SIMPLE)) over a computational plane. But, I dont know how will I procceed from the physical plane to the computational one. I've been trying to figure this out from John D. Anderson's book but I can't find the solution.
The physical space is a duct with a 90degrees turn at some distance. I have the original grid with its nodes so I know the position of the nodes at the first place. The grid is compressed near the wall also.
How could I transform it into a computational space with η, ξ ? How can I transform the governing equations correctly with η and ξ ?

Thanks.

Eifoehn4 · May 31, 2020, 22:17

Dear mech_eng_gr,

are you familiar with the chain rule?

Consider a field variable f as a function of $f(x(\xi,\eta),y(\xi,\eta))$ . Now you can start from the physical side

$\frac{ \partial f}{ \partial x} = \frac{ \partial f}{ \partial \xi} \frac{ \partial \xi}{ \partial x} + \frac{ \partial f}{ \partial \eta} \frac{ \partial \eta}{ \partial x},$
$\frac{ \partial f}{ \partial y} = \frac{ \partial f}{ \partial \xi} \frac{ \partial \xi}{ \partial y} + \frac{ \partial f}{ \partial \eta} \frac{ \partial \eta}{ \partial y},$

resulting in

$\begin{pmatrix} f_x \\ f_y \end{pmatrix}= \begin{pmatrix} \xi_x & \eta_x \\ \xi_y & \eta_y \end{pmatrix} \begin{pmatrix} f_{\xi} \\ f_{\eta} \end{pmatrix}.$

Or you start from the other side with

$\frac{ \partial f}{ \partial \xi} = \frac{ \partial f}{ \partial x} \frac{ \partial x}{ \partial \xi} + \frac{ \partial f}{ \partial y} \frac{ \partial y}{ \partial \xi}$
$\frac{ \partial f}{ \partial \eta} = \frac{ \partial f}{ \partial x} \frac{ \partial x}{ \partial \eta} + \frac{ \partial f}{ \partial y} \frac{ \partial y}{ \partial \eta}$

resulting in

$\begin{pmatrix} f_{\xi} \\ f_{\eta} \end{pmatrix}= \begin{pmatrix} x_{\xi} & y_{\xi} \\ x_{\eta} & y_{\eta} \end{pmatrix} \begin{pmatrix} f_{x} \\ f_{y} \end{pmatrix}.$

Now since both sides must be identical it holds

$\begin{pmatrix} \xi_x & \eta_x \\ \xi_y & \eta_y \end{pmatrix} = \begin{pmatrix} x_{\xi} & y_{\xi} \\ x_{\eta} & y_{\eta} \end{pmatrix}^{-1} .$

Here you may use the Cramer's rule for the right side.

Now all you have to do is simply to replace your derivative expressions with the relations above before discretization. Moreover after that you also have to discretize the covariant metric terms $x_{\xi}, y_{\xi}, x_{\eta}, y_{\eta}$ .

Regards

May 31, 2020, 13:17	Grid transformation to computational space	#1
mech_eng_gr New Member Join Date: May 2020 Posts: 1 Rep Power: 0	Hello, So I've got this problem that I need to solve the governing equations (x,y momentum and continuity with pressure correction (SIMPLE)) over a computational plane. But, I dont know how will I procceed from the physical plane to the computational one. I've been trying to figure this out from John D. Anderson's book but I can't find the solution. The physical space is a duct with a 90degrees turn at some distance. I have the original grid with its nodes so I know the position of the nodes at the first place. The grid is compressed near the wall also. How could I transform it into a computational space with η, ξ ? How can I transform the governing equations correctly with η and ξ ? Thanks.

May 31, 2020, 22:17		#2
Eifoehn4 Senior Member - Join Date: Jul 2012 Location: Germany Posts: 184 Rep Power: 14	Dear mech_eng_gr, are you familiar with the chain rule? Consider a field variable f as a function of $f(x(\xi,\eta),y(\xi,\eta))$ . Now you can start from the physical side $\frac{ \partial f}{ \partial x} = \frac{ \partial f}{ \partial \xi} \frac{ \partial \xi}{ \partial x} + \frac{ \partial f}{ \partial \eta} \frac{ \partial \eta}{ \partial x},$ $\frac{ \partial f}{ \partial y} = \frac{ \partial f}{ \partial \xi} \frac{ \partial \xi}{ \partial y} + \frac{ \partial f}{ \partial \eta} \frac{ \partial \eta}{ \partial y},$ resulting in $\begin{pmatrix} f_x \\ f_y \end{pmatrix}= \begin{pmatrix} \xi_x & \eta_x \\ \xi_y & \eta_y \end{pmatrix} \begin{pmatrix} f_{\xi} \\ f_{\eta} \end{pmatrix}.$ Or you start from the other side with $\frac{ \partial f}{ \partial \xi} = \frac{ \partial f}{ \partial x} \frac{ \partial x}{ \partial \xi} + \frac{ \partial f}{ \partial y} \frac{ \partial y}{ \partial \xi}$ $\frac{ \partial f}{ \partial \eta} = \frac{ \partial f}{ \partial x} \frac{ \partial x}{ \partial \eta} + \frac{ \partial f}{ \partial y} \frac{ \partial y}{ \partial \eta}$ resulting in $\begin{pmatrix} f_{\xi} \\ f_{\eta} \end{pmatrix}= \begin{pmatrix} x_{\xi} & y_{\xi} \\ x_{\eta} & y_{\eta} \end{pmatrix} \begin{pmatrix} f_{x} \\ f_{y} \end{pmatrix}.$ Now since both sides must be identical it holds $\begin{pmatrix} \xi_x & \eta_x \\ \xi_y & \eta_y \end{pmatrix} = \begin{pmatrix} x_{\xi} & y_{\xi} \\ x_{\eta} & y_{\eta} \end{pmatrix}^{-1} .$ Here you may use the Cramer's rule for the right side. Now all you have to do is simply to replace your derivative expressions with the relations above before discretization. Moreover after that you also have to discretize the covariant metric terms $x_{\xi}, y_{\xi}, x_{\eta}, y_{\eta}$ . Regards Last edited by Eifoehn4; June 1, 2020 at 08:34.

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