Diffusion term

The diffusion term amounts to

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

which amounts to, taking into account Equations (591) and (592):

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

$\displaystyle \int_{A_e}^{} \mu^T v_{i,j}n_j da \approx \mu^{T(m-1)}_e A_e \fra...
...T(m-1)}_e A_e (\nabla v_i)_e^{(m-1)} \cdot (\boldsymbol{n}_e-\boldsymbol{j}_e).$ (607)

This amounts to the following approximation:

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

This amounts to a deferred correction for the gradient. Terms 2 and 3 of Equation (606) are computed from iteration $ (m-1)$:

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

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



Subsections