# OpenFOAM guide/Reconstruction

## 1 Introduction

Reconstruction operation is used in OpenFOAM e.g. in BuoyantBoussinesqPisoFoam to reconstruct "...a volume field from a face flux field introducing pseudo-staggered grid setup... to avoid checker-board pressure oscillations that may occur on co-located grids".

## 2 Method description

Let’s consider a reconstruction fvc::reconstruct(…) operator for the case of the cell centered values recovery of the velocity $\vec{u}$ vector based on the face flux f f S r r j = u × of the control volume.

Surface normal vectors is

(1)

which can be represented as the scalar product of the face normal unity vector f n and the aread of the face f f A S r = :

According to the fvcReconstruct.C file:


00042 template<class Type>
00043 tmp
00044 <
00045 GeometricField
00046 <
00047 typename outerProduct<vector,Type>::type, fvPatchField, volMesh
00048 >
00049 >
00050 reconstruct
00051 (
00052 const GeometricField<Type, fvsPatchField, surfaceMesh>& ssf
00053 )
00054 {
00055 typedef typename outerProduct<vector, Type>::type GradType;
00056
00057 const fvMesh& mesh = ssf.mesh();
00058
00059 tmp<GeometricField<GradType, fvPatchField, volMesh> > treconField
00060 (
00062 (
00063 IOobject
00064 (
00065 "volIntegrate("+ssf.name()+')',
00066 ssf.instance(),
00067 mesh,
00069 IOobject::NO_WRITE
00070 ),
00071 inv(surfaceSum(sqr(mesh.Sf())/mesh.magSf()))
00072 & surfaceSum((mesh.Sf()/mesh.magSf())*ssf),
00074 )
00075 );
00076
00077 treconField().correctBoundaryConditions();
00078
00079 return treconField;
00080 }

Thereby, the reconstructed velocity values * u r are calculated as:

(3)

Here one should remember that in the OpenFOAM the square operator (...)2 performed on the vector argument produces a symmetric tensor (it is not the scalar product of a vector with itself) using an outer product of the vector:

(4)

The obtained 2nd rank tensor is normalized by the face area f A and summarized (using operator Σ f ... ) over all faces of control volume:

(5)

The inversion of the tensor W is used to represent the normalization (is this correct?). According to the Eq. (2), the face normal component of velocity vector f n u r is calculated as

(6)

thus

(7)

Finally the Eq. (3) can be rewritten as the weighted sum (with face areas as the weighted coefficients) averaged using normalization tensor W:

(8)

--Makaveli lcf 01:12, 23 October 2011 (CEST)