Diffusion term

The diffusion term amounts to

\int_{A}^{} t_{ij}n_j da, (605)

which amounts to, taking into account Equations () and ():

\int_{A}^{} \mu^T v_{i,j}n_j da + \int_{A}^{} \mu^Tv_{j,i}n_j da - \frac{2}{3} \int_{A}^{} [\mu^T v_{k,k}+\rho k] \delta_{ij}n_j da, (606)

where \mu^T:=\mu+\mu_t is the total dynamic viscosity. The first term contains the gradient in normal direction. For face e between elements P and E it is approximated by:

\int_{A_e}^{} \mu^T v_{i,j}n_j da \approx \mu^{T(m-1)}_e A_e\frac{(v_i)_E^{(m)}-(v_i)_P^{(m)}}{l_{PE}} + \mu^{T(m-1)}_e A_e (\nablav_i)_e^{(m-1)} \cdot (\boldsymbol{n}_e-\boldsymbol{j}_e). (607)

This amounts to the following approximation:

\nabla_n^{(m)} \approx \nabla_j^{(m)} + (\nabla_n^{(m-1)}-\nabla_j^{(m-1)}). (608)

This amounts to a deferred correction for the gradient. Terms 2 and 3 of Equation () are computed from iteration (m-1):

\mu^{T(m-1)} (v_{j,i})_f^{(m-1)} (n_j)_f A_f - \frac{2}{3} \left[ \mu^{T(m-1)} (v_{k,k})_f^{(m-1)} + \rho_f^{(m-1)} k_f^{(m-1)} \right ] \delta_{ij} n_jA_f (609)

The boundary conditions for the diffusion term deserve special attention. The following cases are distinguished: