Convection term

\int_{A}^{} c_v T \rho \boldsymbol{v} \cdot \boldsymbol{n} da \approx \sum_{f}^{} \dot{m} _f ^{(m)} c_v ^{(m-1)} \overrightarrow{T}_f ^{(m)}. (700)

Notice that the corrected mass flow (calculated in iteration (m)) is taken! \overrightarrow{T}_f ^{(m)} is approximated by (cf. the exposure in the section on the conservation of momentum):

\overrightarrow{T}_f ^{(m)} \approx \overrightarrow{T}_f ^{UD(m)} + \left [ \overrightarrow{T}_f ^{(m-1)} - \overrightarrow{T}_f^{UD(m-1)} \right ]. (701)

The boundary conditions amount to:

• Inlet: \overrightarrow{T}_f ^{(m)} is given
• Outlet:

  \overrightarrow{T}_f ^{(m)} = \overrightarrow{T}_f ^{UD(m)} = T_P ^{(m)} (702)

  Consequently, the facial value is identical to the upstream element center value.

• wall and sliding conditions: \dot{m}_f ^{(m)} =0, so there is no contribution.

For the convective interpolation of T the modified smart algorithm has not shown any advantages, therefore, the upwind difference scheme is always used.