CFD Online Logo CFD Online URL
www.cfd-online.com
[Sponsors]
Home > Forums > Main CFD Forum

Roe matrix for Finite volume scheme.

Register Blogs Members List Search Today's Posts Mark Forums Read

Like Tree1Likes
  • 1 Post By ripperjack

Reply
 
LinkBack Thread Tools Display Modes
Old   December 6, 2012, 00:41
Default Roe matrix for Finite volume scheme.
  #1
Member
 
Dokeun, Hwang
Join Date: Apr 2010
Posts: 71
Rep Power: 7
dokeun is on a distinguished road
Hello

From J.Blazek's, intercell flux for roe is expressed as a production of roe matrix and difference of conservation variables between left & right

So, I'm trying to derive eigenvalues, eigenvectors, wave strength from roe matrix written for 2D Euler Finite volume scheme. But I confused whether I'm doing right.

Many references explain the conservation lows and roe flux in serperated coordinates. From Toro, there is a method to express these flux terms as one augmented 1D flux.

Where can I find a whole process to derive roe matrix for the augmented 1D flux!?

Any kind of advice will be a help. Thank you in advance.
dokeun is offline   Reply With Quote

Old   December 6, 2012, 13:52
Default
  #2
New Member
 
Vasiliy
Join Date: Feb 2011
Posts: 9
Rep Power: 6
Vasiliy is on a distinguished road
Try to find Weiss and Smith paper "Preconditioning Applied to Variable and Constant Density Time-Accurate Flows on Unstructured Meshes".
May by it can be useful for you.
Vasiliy is offline   Reply With Quote

Old   December 11, 2012, 09:00
Default
  #3
Member
 
Ping
Join Date: Dec 2011
Posts: 63
Rep Power: 5
ripperjack is on a distinguished road
Quote:
Originally Posted by dokeun View Post
Hello

From J.Blazek's, intercell flux for roe is expressed as a production of roe matrix and difference of conservation variables between left & right

So, I'm trying to derive eigenvalues, eigenvectors, wave strength from roe matrix written for 2D Euler Finite volume scheme. But I confused whether I'm doing right.

Many references explain the conservation lows and roe flux in serperated coordinates. From Toro, there is a method to express these flux terms as one augmented 1D flux.

Where can I find a whole process to derive roe matrix for the augmented 1D flux!?

Any kind of advice will be a help. Thank you in advance.


To find more details about Roe scheme, the chapter 11 from "Riemann Solvers and Numerical Methods for Fluid Dynamics A Practical Introduction" by Toro, E. F. will be very helpful!
ripperjack is offline   Reply With Quote

Old   December 12, 2012, 00:58
Default
  #4
Member
 
Dokeun, Hwang
Join Date: Apr 2010
Posts: 71
Rep Power: 7
dokeun is on a distinguished road
Dear Vasiliy, ripperjack

Thank you for your advices. I just read the paper and the took.

But I think they didn't resolve my concern. Maybe I didn't describe the exact point I stuck.

So, I'd like to change my question.

The discretized integral form of conservation laws(2d euler) for unstructured scheme is

U_{t} = - \frac{1}{\Omega_I} \sum^{N}_{m=1} \vec{F_c} \Delta S_m.

In order to find inter cell flux by roe shceme, I needed to fined \tilde{B}(\tilde{Q}) and \tilde{C}(\tilde{Q}) which satisfy following relations,

\Delta U = \tilde{B} \Delta \tilde{Q}, \Delta F_{c} = \tilde{C} \Delta \tilde{Q},

Paramether vector \tilde{Q} = \frac{1}{2}\left( \sqrt{\rho_{R}}+\sqrt{\rho_{L}} ~~~ \sqrt{\rho_{R}}u_{R}+\sqrt{\rho_{L}}u_{L} ~~~ \sqrt{\rho_{R}}v_{R}+\sqrt{\rho_{L}}v_{L} ~~~ \sqrt{\rho_{R}}H_{R}+\sqrt{\rho_{L}}H_{L} \right)^T = \left( \tilde{q}_1 ~~~ \tilde{q}_2 ~~~ \tilde{q}_3 ~~~ \tilde{q}_4 \right)^T,

And \Delta \tilde{Q} = \left( \sqrt{\rho_{R}}-\sqrt{\rho_{L}} ~~~ \sqrt{\rho_{R}}u_{R}-\sqrt{\rho_{L}}u_{L} ~~~ \sqrt{\rho_{R}}v_{R}-\sqrt{\rho_{L}}v_{L} ~~~ \sqrt{\rho_{R}}H_{R}-\sqrt{\rho_{L}}H_{L} \right)^T

from these I can find roe matrix \tilde{A} = \tilde{C} \tilde{B}^{-1}.

But, the problem is that it's too complicate to find \tilde{C} from flux \vec{F}_c.

Here, \vec{F}_c = \left( \rho V ~~~ \rho u V+n_{x}P ~~~ \rho v V+n_{y}P ~~~ \rho H V \right)^{T}.

I'd like to know that obtaining \tilde{C} from above relation is straightforward despite of its complexity.

For the 1st row of \tilde{C}, I found \left(n_{x}\tilde{q}_{2}+n_{y}\tilde{q}_{3} ~~~ n_{x}\tilde{q}_{1} ~~~ n_{y}\tilde{q}_{1} ~~~ 0 \right) but I couldn't for the other rows. It's too complicated.

If the fluxs are splitted x, y direction(structured scheme) I have no problem with it.

I'd like to know if I'm going a right way.

Thank you.
dokeun is offline   Reply With Quote

Old   December 12, 2012, 11:09
Default
  #5
Member
 
Ping
Join Date: Dec 2011
Posts: 63
Rep Power: 5
ripperjack is on a distinguished road
Quote:
Originally Posted by dokeun View Post
Dear Vasiliy, ripperjack

Thank you for your advices. I just read the paper and the took.

But I think they didn't resolve my concern. Maybe I didn't describe the exact point I stuck.

So, I'd like to change my question.

The discretized integral form of conservation laws(2d euler) for unstructured scheme is

U_{t} = - \frac{1}{\Omega_I} \sum^{N}_{m=1} \vec{F_c} \Delta S_m.

In order to find inter cell flux by roe shceme, I needed to fined \tilde{B}(\tilde{Q}) and \tilde{C}(\tilde{Q}) which satisfy following relations,

\Delta U = \tilde{B} \Delta \tilde{Q}, \Delta F_{c} = \tilde{C} \Delta \tilde{Q},

Paramether vector \tilde{Q} = \frac{1}{2}\left( \sqrt{\rho_{R}}+\sqrt{\rho_{L}} ~~~ \sqrt{\rho_{R}}u_{R}+\sqrt{\rho_{L}}u_{L} ~~~ \sqrt{\rho_{R}}v_{R}+\sqrt{\rho_{L}}v_{L} ~~~ \sqrt{\rho_{R}}H_{R}+\sqrt{\rho_{L}}H_{L} \right)^T = \left( \tilde{q}_1 ~~~ \tilde{q}_2 ~~~ \tilde{q}_3 ~~~ \tilde{q}_4 \right)^T,

And \Delta \tilde{Q} = \left( \sqrt{\rho_{R}}-\sqrt{\rho_{L}} ~~~ \sqrt{\rho_{R}}u_{R}-\sqrt{\rho_{L}}u_{L} ~~~ \sqrt{\rho_{R}}v_{R}-\sqrt{\rho_{L}}v_{L} ~~~ \sqrt{\rho_{R}}H_{R}-\sqrt{\rho_{L}}H_{L} \right)^T

from these I can find roe matrix \tilde{A} = \tilde{C} \tilde{B}^{-1}.

But, the problem is that it's too complicate to find \tilde{C} from flux \vec{F}_c.

Here, \vec{F}_c = \left( \rho V ~~~ \rho u V+n_{x}P ~~~ \rho v V+n_{y}P ~~~ \rho H V \right)^{T}.

I'd like to know that obtaining \tilde{C} from above relation is straightforward despite of its complexity.

For the 1st row of \tilde{C}, I found \left(n_{x}\tilde{q}_{2}+n_{y}\tilde{q}_{3} ~~~ n_{x}\tilde{q}_{1} ~~~ n_{y}\tilde{q}_{1} ~~~ 0 \right) but I couldn't for the other rows. It's too complicated.

If the fluxs are splitted x, y direction(structured scheme) I have no problem with it.

I'd like to know if I'm going a right way.

Thank you.
Dear Dokeun,

It seems that you are trying to derive Roe matrix \tilde{A}, and calculate the flux at the cell surfaces.
I did not know the details of derivation of Roe matrix \tilde{A}, I just used it.
And \tilde{A} is exact the same as the original convective Jacobian A in 2D Euler equation, except that the flow variables are replaced with the Roe averaged variables (see J.Blazek's book, p108).
And the flux at the cell surface can be calculate by:
F_{i+1/2}=0.5*(F(U_L)+F(U_R))-0.5*\sum^{N}_{m=1} {\alpha_m} \left| {\lambda_m}\right| T_m
The T_m is the mth right eigenvector based on the Roe averaged matrix \tilde{A}, T_m=(r_1,r_2,r_3,r_4)^T
The {\lambda_m} is the mth eigenvalue also based on the \tilde{A}
The wave strength {\alpha_m} is defined by:
\Delta \vec{U}=U_R-U_L= \sum^{N}_{m=1} {\alpha_m} T_m=\vec{T}\vec{\alpha}
so the wave strength {\alpha_m} can be calculated by:
\vec{\alpha}=\vec{T}^{-1}\Delta \vec{U}=({\alpha_1},{\alpha_2},{\alpha_3},{\alpha_4})^T
and \vec{T}^{-1} is the left eigenvector based on the \tilde{A}
The left and right eigenvector can be find in J.Blazek's book A.11
In addition, if you just want to use Roe scheme, the Eq. 4.91-4.95 in J.Blazek's book can be used directly. I used that equations and they worked good.
That is all I can do, hope that would help!
dokeun likes this.
ripperjack is offline   Reply With Quote

Reply

Thread Tools
Display Modes

Posting Rules
You may not post new threads
You may not post replies
You may not post attachments
You may not edit your posts

BB code is On
Smilies are On
[IMG] code is On
HTML code is Off
Trackbacks are On
Pingbacks are On
Refbacks are On


Similar Threads
Thread Thread Starter Forum Replies Last Post
On the damBreak4phaseFine cases paean OpenFOAM Running, Solving & CFD 0 November 14, 2008 22:14
Roe scheme for general equation of state zouchu Main CFD Forum 3 August 10, 2000 16:46
roe scheme Jian Xia Main CFD Forum 7 August 9, 2000 01:18
Roe FDS scheme applied to backward facing Mohammad Kermani Main CFD Forum 6 December 29, 1999 12:11
Roe Scheme; Shock Boundary layer Interaction Mohammad Kermani Main CFD Forum 5 December 20, 1999 16:44


All times are GMT -4. The time now is 19:46.