# Incompressible Viscous Turbulent Heat Transfer Energy Equation

Generally, incompressible viscous turbulent heat transfer energy equation is
$\dfrac{\partial T}{\partial t} + \nabla\cdot(\boldsymbol{U} T) = \alpha \nabla^2 T + \dfrac{\nu}{c_\mathrm{p}} [\nabla \boldsymbol{U} +(\nabla \boldsymbol{U})^\mathrm{T}]:\nabla \boldsymbol{U}$

where $\alpha = \dfrac{k}{\rho c_\mathrm{p}}$

When turbulent stress considered, the equation becomes
$\dfrac{\partial T}{\partial t} + \nabla\cdot(\boldsymbol{U} T) = \alpha_\mathrm{eff}\nabla^2 T + \dfrac{\nu_\mathrm{eff}}{c_\mathrm{p}} [\nabla \boldsymbol{U} +(\nabla \boldsymbol{U})^\mathrm{T}]:\nabla \boldsymbol{U}$

where $\alpha_\mathrm{eff}= \nu/Pr + \nu_t/Pr_t$, $latex Pr$ is Prandl number, $latex \nu_t$ is turbulent viscosity and $latex Pr_t$ is turbulent Prandl number.

With OpenFOAM, the incompressible viscous turbulence energy equation is written


volScalarField alphaEff
(
"alphaEff",
turbulence->nu()/Pr + turbulence->nut()/Prt
);

{
}