Incompressible Viscous Turbulent Heat Transfer Energy Equation

Generally, incompressible viscous turbulent heat transfer energy equation is
[latex]\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}
[/latex]
ddt(T) + div(U T) = alpha laplacian(T) + ((nu/cp (grad(U) + (grad(U)).T)) && grad(U))

where [latex]\alpha = \dfrac{k}{\rho c_\mathrm{p}}[/latex]

When turbulent stress considered, the equation becomes
[latex]\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}
[/latex]

where [latex]\alpha_\mathrm{eff}= \nu/Pr + \nu_t/Pr_t[/latex], $latex Pr$ 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
    );

    volTensorField gradU = fvc::grad(U);

    volTensorField tauEff = turbulence->nuEff() / cp * (gradU + gradU.T());

    {
        // Add energy equation
        fvScalarMatrix TEqn
        (
            fvm::ddt(T)
          + fvm::div(phi, T)
          - fvm::laplacian(alphaEff, T)
          - (tauEff && gradU)
        );

        TEqn.solve();
    }

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