discretization of convective terms of Spalart Allmaras equations

Abolfazl_cfd · March 28, 2019, 07:34

Hi everyone.

I'm about to discretize the spalart allaras equation and add it into my compressible code based on Roe scheme.

I was wondering can i use a simple first or second upwind for discretization of convective terms of spalart allmaras like incompressible flow or I should use the method of Roe!?

thanks a lot

sbaffini · March 28, 2019, 08:47

How would you define the Roe method for a scalar equation as the one for SA and how an upwind scheme?

They are the same for, say, unstructured schemes

Abolfazl_cfd · March 29, 2019, 02:53

Quote:

Originally Posted by sbaffini

How would you define the Roe method for a scalar equation as the one for SA and how an upwind scheme?

They are the same for, say, unstructured schemes

Hi sbaffini

Actually I don't know. I know how i can use forward and backward discretization for convective term of spalart and this is all I know. I have no idea how I can use Roe for spalart!

actually maybe I'm not asking the right question. I'm not well familiar with turbulence models.

sbaffini · March 29, 2019, 06:00

Well, I tougth that, as you have a Roe scheme in your compressible code, you kind of knew it. For a general vector equation of the form:

$\frac{\partial \textbf{W}}{\partial t} + \frac{\partial \textbf{F}\left(\textbf{W}\right)}{\partial \textbf{x}}=0$

your Roe scheme for the flux $\textbf{F}$ at the interface between cells L and R typically is:

$\textbf{F} = \frac{1}{2}\left[\textbf{F}\left(\textbf{W}_L\right)+\textbf{F}\left(\textbf{W}_R\right)-\textbf{M}\left|\Lambda\right|\textbf{M}^{-1}\left(\textbf{W}_R-\textbf{W}_L\right)\right]$

where:

$\frac{\partial \textbf{F}}{\partial \textbf{W}} = \textbf{M}\left|\Lambda\right|\textbf{M}^{-1}$

is the Jacobian of the flux (upwind altered by taking the absolute eigenvalues), the surface normal is assumed to point toward the R cell and the L and R states can be, in general, cell values (1st order) or suitably reconstructed values (2nd or higher order).

Now, if you apply this to a scalar equation for $\phi$ with assigned mass flux, say

, which is the first component of $\textbf{F}$ above, you get as flux:

$F = \frac{1}{2}\left[m\left(\phi_L+\phi_R\right)-\left|m\right|\left(\phi_R-\phi_L\right)\right]$

which actually is the upwind scheme in its most common version. In layman's terms, the point is that, for a scalar equation, you don't have a wave structure anymore, just a single eigenvalue,

.

sbaffini · March 29, 2019, 06:23

Note that I assumed that you are solving the SA equation as decoupled from the main equations, as otherwise you have no option for the scheme.

Abolfazl_cfd · April 7, 2019, 09:48

Quote:

Originally Posted by sbaffini

Note that I assumed that you are solving the SA equation as decoupled from the main equations, as otherwise you have no option for the scheme.

Dear sbaffini

thank you for your kind reply.
I checked the "computational fluid dynamics Vo.II" written by Hoffmann.
I found that I can discrete the convective terms using a simple upwind scheme.

but now I have another question.
there is a term in Spalart Allmaras model: $\sqrt{2S_{ij} S_{ij} }$ ?
I know $S_{ij}$ is $S_{ij}=\frac{1}{2}(\frac{u_i}{x_j}-\frac{u_j}{x_i})$ but I don't get it how i can expand $\sqrt{2S_{ij} S_{ij} }$ !

is the expansion for 2D like this? $\sqrt{2S_{ij} S_{ij} }=\sqrt{2*(\frac{1}{2}(V_x-U_y))^2 }$ ?
How like is the expansion for 3D?

thanks a lot!

FMDenaro · April 7, 2019, 12:46

Quote:

Originally Posted by Abolfazl_cfd

Dear sbaffini

thank you for your kind reply.
I checked the "computational fluid dynamics Vo.II" written by Hoffmann.
I found that I can discrete the convective terms using a simple upwind scheme.

but now I have another question.
there is a term in Spalart Allmaras model: $\sqrt{2S_{ij} S_{ij} }$ ?
I know $S_{ij}$ is the strain rate tensor and $S_{ij}=\frac{1}{2}(\frac{u_i}{x_j}-\frac{u_j}{x_i})$ but I don't get it how i can expand $\sqrt{2S_{ij} S_{ij} }$ !

is the expansion for 2D like this? $\sqrt{2S_{ij} S_{ij} }=\sqrt{2*(\frac{1}{2}(V_x-U_y))^2 }$ ?
How like is the expansion for 3D?

thanks a lot!

No, Sij is the symmetric part of the velocity gradient. But what is more, stands in that fact the a turbulence model for incompressible flows does not necessarily can be extended as it is to compressible flows.
Have a look here http://iccfd.org/iccfd7/assets/pdf/p...1902_paper.pdf

selig5576 · April 7, 2019, 16:29

Unless you absolutely need to use the Roe scheme, I would suggest the AUSM+ scheme (or one of the variants.) It's easier to implement, cheaper in computational cost, and can be more accurate than the Roe scheme (I can't think of a case of where it's not.)

Abolfazl_cfd · April 8, 2019, 03:54

Quote:

Originally Posted by FMDenaro

No, Sij is the symmetric part of the velocity gradient. But what is more, stands in that fact the a turbulence model for incompressible flows does not necessarily can be extended as it is to compressible flows.
Have a look here http://iccfd.org/iccfd7/assets/pdf/p...1902_paper.pdf

Dear FMDenaro
Thank you for correcting my mistake.
I toke a look into the paper and found it really helpful. specially the part about negative values of source term in Spalart Allmaras.
Thank U.

Abolfazl_cfd · April 8, 2019, 03:59

Quote:

Originally Posted by selig5576

Unless you absolutely need to use the Roe scheme, I would suggest the AUSM+ scheme (or one of the variants.) It's easier to implement, cheaper in computational cost, and can be more accurate than the Roe scheme (I can't think of a case of where it's not.)

Dear selig5576

Thank you for your precious advice.
Actually I am working on immersed boundary method.
years ago I tried to add my immersed code to a AUSM solver but some instabilities near the immersed boundary started to grow and after several attempts I gave up upon AUSM. But for Roe scheme it works properly.
Thanks a lot.

Abolfazl_cfd · April 9, 2019, 04:47

Quote:

Originally Posted by Abolfazl_cfd

Hi everyone.

I'm about to discretize the spalart allaras equation and add it into my compressible code based on Roe scheme.

I was wondering can i use a simple first or second upwind for discretization of convective terms of spalart allmaras like incompressible flow or I should use the method of Roe!?

thanks a lot

Thank U all.

finally I could write the spalart allmaras code. Although instead of Roe scheme, I used E-CUSP scheme, but the results in channel flow in Re=10000 are satisfactory.
If anyone need the code, can send me an email. I would happy to share it and I hope it would help.
moosavi.abolfazl@gmail.com

March 28, 2019, 07:34	discretization of convective terms of Spalart Allmaras equations	#1
Abolfazl_cfd New Member Abolfazl Join Date: Oct 2016 Posts: 28 Rep Power: 10	Hi everyone. I'm about to discretize the spalart allaras equation and add it into my compressible code based on Roe scheme. I was wondering can i use a simple first or second upwind for discretization of convective terms of spalart allmaras like incompressible flow or I should use the method of Roe!? thanks a lot

March 29, 2019, 06:00		#4
sbaffini Senior Member Paolo Lampitella Join Date: Mar 2009 Location: Italy Posts: 2,192 Blog Entries: 29 Rep Power: 39	Well, I tougth that, as you have a Roe scheme in your compressible code, you kind of knew it. For a general vector equation of the form: $\frac{\partial \textbf{W}}{\partial t} + \frac{\partial \textbf{F}\left(\textbf{W}\right)}{\partial \textbf{x}}=0$ your Roe scheme for the flux $\textbf{F}$ at the interface between cells L and R typically is: $\textbf{F} = \frac{1}{2}\left[\textbf{F}\left(\textbf{W}_L\right)+\textbf{F}\left(\textbf{W}_R\right)-\textbf{M}\left\|\Lambda\right\|\textbf{M}^{-1}\left(\textbf{W}_R-\textbf{W}_L\right)\right]$ where: $\frac{\partial \textbf{F}}{\partial \textbf{W}} = \textbf{M}\left\|\Lambda\right\|\textbf{M}^{-1}$ is the Jacobian of the flux (upwind altered by taking the absolute eigenvalues), the surface normal is assumed to point toward the R cell and the L and R states can be, in general, cell values (1st order) or suitably reconstructed values (2nd or higher order). Now, if you apply this to a scalar equation for $\phi$ with assigned mass flux, say , which is the first component of $\textbf{F}$ above, you get as flux: $F = \frac{1}{2}\left[m\left(\phi_L+\phi_R\right)-\left\|m\right\|\left(\phi_R-\phi_L\right)\right]$ which actually is the upwind scheme in its most common version. In layman's terms, the point is that, for a scalar equation, you don't have a wave structure anymore, just a single eigenvalue, . Abolfazl_cfd likes this.

March 29, 2019, 06:23		#5
sbaffini Senior Member Paolo Lampitella Join Date: Mar 2009 Location: Italy Posts: 2,192 Blog Entries: 29 Rep Power: 39	Note that I assumed that you are solving the SA equation as decoupled from the main equations, as otherwise you have no option for the scheme. Abolfazl_cfd likes this.

April 7, 2019, 16:29	Discretization	#8
selig5576 Senior Member Selig Join Date: Jul 2016 Posts: 213 Rep Power: 11	Unless you absolutely need to use the Roe scheme, I would suggest the AUSM+ scheme (or one of the variants.) It's easier to implement, cheaper in computational cost, and can be more accurate than the Roe scheme (I can't think of a case of where it's not.) Abolfazl_cfd likes this.

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March 28, 2019, 08:47		#2
sbaffini Senior Member Paolo Lampitella Join Date: Mar 2009 Location: Italy Posts: 2,192 Blog Entries: 29 Rep Power: 39	How would you define the Roe method for a scalar equation as the one for SA and how an upwind scheme? They are the same for, say, unstructured schemes