[Funkymouton]

Hello,

I would like to implement a 3rd order WENO scheme in OpenFOAM. This scheme would be used with a 2D structured grid.

Regarding the validation part, I will be running simulations on basic test cases such as : flow around a cylinder, flow around the NACA0012.

I am new to OpenFOAM and I am not sure on how to deal with such a scheme. Hence, I am looking for advices. Has anyone done it before ?



I have been studying different references :

Liu XD, Osher S, Chan T: Weighted Essentially Nonoscillatory Schemes, JOURNAL OF COMPUTATIONAL PHYSICS 115 (1): 200-212 NOV 1994

Jiang GS, Shu CW: Efficient implementation of weighted ENO schemes, JOURNAL OF COMPUTATIONAL PHYSICS 126 (1): 202-228 JUN 1996.

Shu CW: Essentially Non-Oscillatory and Weighted Essentially Non-Oscillatory Schemes for Hyperbolic Conservation Laws, ICASE-1997-65.

[Luca Cornolti]

Hi,

some times ago, I tried to implement the weno scheme in OpenFOAM, then I found out that the problem was quite complex and that I didn't have enougth time for it.
Anyway, I can share with you what I found out and some suggestions.

There are two main problems related to this topic:

1) Adaptation of high order schemes for 3D unstructured meshes are not straightforward and only few articles are avaiable in the literature about this topic. For this reason, the general validity of the approaches proposed in these articles should be proven with a wider set of test cases.

2) OpenFOAM was made to deal with general 3D unstructured meshes about 20 years ago. Therefore, it was programmed and optimazed to work only with two cells stencil. As a conseguence, from a programming point of view, it is not easy to change what is already avaiable to implement the WENO scheme.

ideas/suggestions:

in 2008 H. Weller actually implemented higher order schemes.
These are explained in the article "Voronoi, Delaunay, and Block-Structured Mesh Refinement for Solution of the Shallow-Water Equations on the Sphere" by Hillary and Henry Weller.
They are quite different from a WENO scheme. In short, they use third and fourth order polynomial to interpolate the face value of a field, using the values of the cells which share the face and their neighbours. These schemes are only technically high order, as the interpolated value is computed as the sum of an implicit linear term with a higher order explicit correction term (they are not fully implicit or fully explicit as theory would require).

You can find these schemes in this folder: ... /src/finiteVolume/interpolation/surfaceInterpolation/schemes.
The are called quadraticLinear... quadraticUpwind...

The useful part of the code of these schemes, for your purpose, is the one which builds the large stencils that these schemes use. You should pay attention that the word "upwind" used in these schemes has nothing to do with the upwind direction of WENO schemes, as it is the owner cell direction which does not correspond to the velocity direction.


If you will work further on this topic and find something more, please let me know, as I'm still interest in the implmentation of WENO shcemes in OpenFOAM.