EquationsToSolve

Conservation of mass: \begin{align} \underbrace{\frac{\partial \rho}{\partial t} + \mathbf{v} \cdot \nabla \rho}_{\displaystyle =\dfrac{D \rho}{D t}} = -\rho \left( \nabla ^{T}\mathbf{v} \right) \end{align} Conservation of momentum: \begin{align} \underbrace{\frac{\partial \mathbf{v}}{\partial t} + (\mathbf{v} \cdot \nabla) \mathbf{v}}_{\displaystyle =\dfrac{D \mathbf{v}}{D t}} + \frac{1}{\rho }\nabla p=\frac{1}{\rho }\left( \nabla ^{T}\mathbf{S}_{s} \right)^{T}+\frac{1}{\rho }\left( \nabla ^{T}\mathbf{S}_{v}\left( \mathbf{v} \right) \right)^{T}+\mathbf{g}-\beta \cdot \left( \mathbf{v} - \mathbf{v}_{\beta} \right) \end{align} Conservation of energy \begin{align} \rho c_{v} \underbrace{\left( \frac{\partial T}{\partial t} + \mathbf{v} \cdot \nabla T \right)}_{\displaystyle =\dfrac{D T}{D t}} = \nabla ^{T}\left( \lambda \cdot \nabla T \right) - p \, (\nabla ^{T}\mathbf{v}) + q \end{align} The variables are

• \( \nabla\) => gradient operator
• \( \rho\) => density, see %ind_r%
• \( \mathbf{v} = \left( u,v,w \right)^T\) => velocity, see %ind_v(1)% , %ind_v(2)% , %ind_v(3)%
• \( p\) => pressure, see %ind_p% and %ind_p_dyn%
• \( \mathbf{g}\) => gravity / body forces, see %ind_g(1)% , %ind_g(2)% , %ind_g(3)%
• \( T\) => temperature, see %ind_T%
• \( c_v\) => specific heat capacity, see %ind_CV%
• \( \lambda\) => heat conductivity, see %ind_LAM%
• \( q\) => heat sources, given by external heat sources and internal processes (viscous heating), see %ind_diss%
• \( \eta_{\text{eff}}\) => effective viscosity, might consist of laminar and turbulent partitions, see %ind_ETA_eff% , %ind_ETA% , %ind_ETA_sm% .
• \( \mathbf{S}_v \left( \mathbf{v} \right) = \eta_{\text{eff}} \cdot \left( \left( \nabla \mathbf{v}^T \right) + \left( \nabla \mathbf{v}^T \right)^T - \frac{2}{3} \nabla^T \mathbf{v} \cdot \mathbf{I} \right)\) => viscous stress tensor
• \( \mathbf{S}_s\) => solid stress tensor, refer to StressTensorAlgorithm, see also %ind_Sxx% , %ind_Sxy% , %ind_Sxz% , %ind_Syy% , %ind_Syz% , %ind_Szz%
• \( \beta\) => The Darcy / Brinkman constant \( \beta = \frac{\tilde{\beta}}{\rho}\) , see DarcyConstant
• \( \mathbf{v}_{\beta}\) => basis velocity of porous material, see DarcyBasisVelocity
• \( - p \, (\nabla ^{T}\mathbf{v})\) => by default this term is ignored, but the user can control it with COEFF_p_divV

For the solution algorithm it is important to be aware of the following two remarks: DeriveDivergenceOfVelocity DerivePoissonEquationForPressure