ForchheimerConstant

While the constant defined in DarcyConstant represents the classical Darcy relation for porous media of the form \begin{align} -\nabla p = \tilde{\beta}_{D}{\vb u}\end{align} one may further extend this by an inertial contribution of Forchheimer type by defining the constant \( \tilde{\beta}_{F}\) in \begin{align} -\nabla p = \tilde{\beta}_{D}{\vb u} + \tilde{\beta}_{F} \|{\vb u}\|{\vb u}\end{align} via In case this constant is defined, \( \beta\) in EquationsToSolve is given by \( \beta = \frac{1}{\rho}\left(\tilde{\beta}_{D} + \tilde{\beta}_{F}\|\mathbf{v} - \mathbf{v}_{\beta}\|\right)\).

Isotropic materials

If in the RightHandSideExpression one argument is given, e.g. then the porous material is assumed to be isotropic. Thus, \( \beta\) in EquationsToSolve can be viewed as a scalar quantity.

Anisotropic materials

If DarcyConstant is specified for three perpendicular directions, three arguments can be supplied to ForchheimerConstant, e.g. Then, the constant &bx&, &by&, &bz& in DarcyConstant are modified in the sense that \( &bx& = &bx& + &Fx&\|\mathbf{v} - \mathbf{v}_{\beta}\|\).

Notes

• The behavior of ForchheimerConstant replicates the behavior of specifying the Forchheimer term via an equation in DarcyConstant which uses %ind_v_0(1:3)% (the main purpose of ForchheimerConstant is thus to simplify inputs)
• In particular, the relative velocity norm within the Forchheimer term is based on \( \mathbf{v}^{n}\) and DarcyBasisVelocity
• Despite the naming convention, %ind_betaDarcy% will not store \( \tilde{\beta}_{D}\), but the above \( \beta\) (in case of a non-zero ForchheimerConstant)
• The case of non-scalar DarcyConstant but scalar ForchheimerConstant will be treated as if 3 identical values (&Fx&=&Fy&=&Fz&) were supplied to ForchheimerConstant.
• ForchheimerConstant is only available for points in LIQUID chambers.