Diffusion term

For turbulent flow \lambda in the energy equation has to be replaced by \lambda + \lambda_t where

\lambda_t = \frac{c_p \mu_t}{Pr_t}, (703)

where Pr_t \approx 0.9 is the turbulent Prandl number. Therefore, one now arrives at:

\int_{A}^{} \lambda^T \frac{\partial T}{\partial n} da \approx \sum_{f}^{} \lambda_f^{T(m-1)} A_f \left [ \frac{T_F ^{(m)} - T_P ^{(m)} }{l_{PF}} + \nabla T_f ^{(m-1)} \cdot (\boldsymbol{n}_f - \boldsymbol{j}_f) \right ], (704)

where \lambda^T := \lambda + \lambda_t and

(\lambda_t)_f ^{(m-1)} = \frac{c_p ^{(m-1)} \rho ^{(m-1)} k ^{(m-1)} }{Pr_t \; \omega ^{(m-1)} }. (705)

\omega is the turbulence frequency. The dynamic turbulent viscosity \mu_t can be written as \mu_t = \rho k / \omega.

The boundary conditions for the diffusion term amount to:

• Inlet: the temperature T_f ^{(m)} is known, therefore the term reduces to:

  \lambda_f^{T(m-1)} A_f \left [ \frac{T_f ^{(m)} - T_P ^{(m)} }{l_{Pf}} + \nabla T_f ^{(m-1)} \cdot (\boldsymbol{n}_f - \boldsymbol{j}_f) \right ] (706)

• Outlet: the temperature gradient in \boldsymbol{ j}-direction is assumed to be zero and the term now reads:

  \lambda_f^{T(m-1)} A_f \left [ \nabla T_f ^{(m-1)} \cdot (\boldsymbol{n}_f - \boldsymbol{j}_f) \right ] (707)

• wall or sliding conditions.

  Either the temperature is given or the heat flux is given (may be zero as for adiabatic conditions). If the temperature is given the treatment is analogous to the case of an inlet, if the heat flux is given the solution is trivial since the diffusion term for an external surface is nothing else than the value of the heat flux through this surface.