[mrishi]

My question is, is it accurate to use Subtracting 1/3 * trace(A)*I from matrix A, and call it deviatoric part of the matrix?


The solution to UEqn.H leads to turbulence->divDevRhoReff (or divDevReff depending on solver). Here, the deviatoric part of stress tensor is calculated using the dev() or dev2() operators on gradU. These are defined in OpenFOAM/primitives/TensorI.H where it makes use of trace.

//- Return the deviatoric part of a tensor

template<class Cmpt>
inline Tensor<Cmpt> dev(const Tensor<Cmpt>& t)
{
    return t - SphericalTensor<Cmpt>::oneThirdI*tr(t);
}


//- Return the deviatoric part of a tensor
template<class Cmpt>
inline Tensor<Cmpt> dev2(const Tensor<Cmpt>& t)
{
    return t - SphericalTensor<Cmpt>::twoThirdsI*tr(t);
}

If I understand correctly, the definition for i,j th element of deviatoric matrix should be . (I know technically there is another term, namely which comes from gradU vs Transpose(gradU) but it gets absorbed as laplacian and we are considering the Transpose(gradU) as matrix A here.)

However, since there is a divergence operator subsequent to all of this, the final result, after summation over indices, becomes equivalent to 1/3*tr(gradU). But is it mathematically correct to have the definition of diagonal matrix as 1/3*trace(A)*I?


Or in a simpler question of equivalence

Is the matrix
1 0 0
0 5 0
0 0 6

equivalent to
4 0 0
0 4 0
0 0 4



I would really appreciate if someone with mathematical, or openfoam knowledge can help me understand this.