[srinath]

Hello

I was reading Hrv's thesis and have a couple of doubts on the interpolation routine in OpenFoam

For the NVD schemes, there is an expression on page 109, which gives
\phi c \tilda = 1 - (gradphi)f.d/(2*grad phi)c.d

And a nice algorithm to give the value of phi on the face, using blended/gamma differencing.

But how do we calculate (grad phi)c?
Suppose i had rho at cell centers and wanted to obtain a limited value of rho on the surface, what would i do?
Should i use a least squared approach to find (grad phi)c?

Also i can't seem to find a formulation of TVD using slope limiters on unstructured grids.

Could someone please help me on this?

Thanks
Srinath

[hjasak]

Yes. Both TVD and NVD are calculated the way I described in the original Gamma paper:

@Article{Jasak:GAMMAPAPER,
author = {Jasak, H. and Weller, H.G. and Gosman, A.D.},
title = {High resolution NVD differencing scheme for
arbitrarily unstructured meshes},
journal = {Int. J. Numer. Meth. Fluids},
year = 1999,
volume = 31,
pages = {431-449}
}


Here's the code (below). As you can see, it uses the cell gradient and a face gradient (actually a difference across the face). For the cell gradient, you can use whatever you like: Gauss gradient will do. The code implements phict for the NVD and r for TVD.

Enjoy,

Hrv


scalar phict
(
const scalar faceFlux,
const vector& phiP,
const vector& phiN,
const tensor& gradcP,
const tensor& gradcN,
const vector& d
) const
{
vector gradfV = phiN - phiP;
scalar gradf = gradfV & gradfV;

scalar gradcf;

if (faceFlux > 0)
{
gradcf = gradfV & (d & gradcP);

else
{
gradcf = gradfV & (d & gradcN);
}

// Stabilise for division
gradcf = stabilise(gradcf, VSMALL);

return 1 - 0.5*gradf/gradcf;
}


scalar r
(
const scalar faceFlux,
const vector& phiP,
const vector& phiN,
const tensor& gradcP,
const tensor& gradcN,
const vector& d
) const
{
vector gradfV = phiN - phiP;
scalar gradf = gradfV & gradfV;

scalar gradcf;

if (faceFlux > 0)
{
gradcf = gradfV & (d & gradcP);
}
else
{
gradcf = gradfV & (d & gradcN);
}

// Stabilise for division
gradf = stabilise(gradf, VSMALL);

return 2*(gradcf/gradf) - 1;
}
}

[srinath]

Thanks for the reference Professor Jasak, this scheme seems so much better computationally than something like ENO.

For the cell gradient, when you say Gauss gradient, do you mean
integral(grad (phi)dV) = \sigma dS * phi_face
Where dS is the outward pointing Area vector?

But in that case how do we get phi_face?
Is it ok to use the cell centre averages of the 2 cells sharing that face?

Regards
Srinath

[hjasak]

Yes, correct: you interpolate from the cell centre values. Remember how the scheme says Gauss linear - it is the linear that tells you how to interpolate (=linear interpolation}. You could of course do Gauss harmonic as well, you can guess what that does.

Enjoy,

Hrv

[srinath]

Thanks Professor Jasak

That clears up this issue totally

Regards
Srinath

[ville]

Hi,

I'm testing discretization schemes for LES and have come up with the following picture on a
scalar pulse that is originally:

c = 1, when 0.2m < x < 0.7m
c = 0, elsewhere

and the pulse is advected to the right with velocity 1m/s. The Courant number is 0.1 and time integration is backward. The Gamma scheme is
the Gamma01 scheme.


A couple of questions about Gamma schemes in
OpenFOAM in general:

1) I've noted that the smaller I make the value
0<= psi <= 1, the larger overshoots I make.
What is the connection between psi and the parameter beta given in the above-mentioned
paper (Jasak et al.)? I seached the code
(as given above and related files) but
could not find a connection... The paper tells
us that typically beta = 0.1 is the lower limit.

2) What is the difference between the GammaV
scheme and the Gamma scheme? I can use both
for velocity.. So, where does the "V" come into
play?

Regards,
Ville

[Tushar@cfd]

Originally Posted by ville

Hi,

A couple of questions about Gamma schemes in
OpenFOAM in general:

1) I've noted that the smaller I make the value
0<= psi <= 1, the larger overshoots I make.
What is the connection between psi and the parameter beta given in the above-mentioned
paper (Jasak et al.)? I seached the code
(as given above and related files) but
could not find a connection... The paper tells
us that typically beta = 0.1 is the lower limit.

2) What is the difference between the GammaV
scheme and the Gamma scheme? I can use both
for velocity.. So, where does the "V" come into
play?

Regards,
Ville

Do you got the answer?

[s.v]

Hi Everyone:

I know this is an old thread but this might be useful for others.

Regarding Ville's first question -- psi = 2*Beta (as far as I understand).

So to avoid the "switching" instability discussed by Jasak et al. in their original Gamma scheme paper (https://citeseerx.ist.psu.edu/viewdo...=rep1&type=pdf) we would need to have psi>=0.2.

This is discussed in my PhD thesis (https://drum.lib.umd.edu/handle/1903/21883) on page 32. In my thesis psi is called "k".

Cheers ....