Commit 5b3969bb authored by Hao Wu's avatar Hao Wu
Browse files

[doxygen,handbook,naming]

doxygen
- added zeroeq and multidomain models
- add model and problem descriptions

handbook
- changes in the model description of models which do not use a free
  flow are just related to changes in doxygen
- added equation subsections for multidomain and zeroeq models

naming
- applied naming convention to stokes, zeroeq, and multidomain
  models/problems

reviewed by fetzer



git-svn-id: svn://svn.iws.uni-stuttgart.de/DUMUX/dumux/trunk@15023 2fb0f335-1f38-0410-981e-8018bf24f1b0
parent acf53d21
......@@ -207,6 +207,42 @@
*
* \copydetails Dumux::StokesncniModel
*/
/*!
* \ingroup ImplicitModels
* \defgroup BoxZeroEqModel ZeroEq
*
* \copydetails Dumux::ZeroEqModel
*/
/*!
* \ingroup ImplicitModels
* \defgroup BoxZeroEqncModel N-component ZeroEq
*
* \copydetails Dumux::ZeroEqncModel
*/
/*!
* \ingroup ImplicitModels
* \defgroup BoxZeroEqncniModel Non-isothermal N-component ZeroEq
*
* \copydetails Dumux::ZeroEqncniModel
*/
/*!
* \ingroup ImplicitModels
* \defgroup BoxZeroEqModel ZeroEq
*
* \copydetails Dumux::ZeroEqModel
*/
/*!
* \ingroup ImplicitModels
* \defgroup BoxZeroEqncModel N-component ZeroEq
*
* \copydetails Dumux::ZeroEqncModel
*/
/*!
* \ingroup ImplicitModels
* \defgroup BoxZeroEqncniModel Non-isothermal N-component ZeroEq
*
* \copydetails Dumux::ZeroEqncniModel
*/
/*!
* \ingroup ImplicitModels
* \defgroup ElasticBoxModel Linear elastic
......@@ -233,16 +269,42 @@
* \ingroup ImplicitModels
* \defgroup TwoPTwoCStokesTwoCModel Two-component, Stokes-Darcy
*
* \copydetails Dumux::TwoCStokesTwoPTwoCLocalOperator
* <br><br><br>
* \copydetails Dumux::TwoPTwoCModel
* <br>
* \copydetails Dumux::StokesncModel
*/
/*!
* \ingroup ImplicitModels
* \defgroup TwoPTwoCNIStokesTwoCNIModel Non-isothermal, two-component, Stokes-Darcy
*
* \copydetails Dumux::TwoCNIStokesTwoPTwoCNILocalOperator
* <br><br><br>
* \copydetails Dumux::TwoPTwoCNIModel
* <br>
* \copydetails Dumux::StokesncniModel
*/
/*!
* \ingroup ImplicitModels
* \defgroup TwoPTwoCZeroEqTwoCModel Two-component, ZeroEq turbulence-Darcy
*
* \copydetails Dumux::TwoCStokesTwoPTwoCLocalOperator
* <br><br><br>
* \copydetails Dumux::TwoPTwoCModel
* <br>
* \copydetails Dumux::ZeroEqncModel
*/
/*!
* \ingroup ImplicitModels
* \defgroup TwoPTwoCNIZeroEqTwoCNIModel Non-isothermal, two-component, ZeroEq turbulence-Darcy
*
* \copydetails Dumux::TwoCNIStokesTwoPTwoCNILocalOperator
* <br><br><br>
* \copydetails Dumux::TwoPTwoCNIModel
* <br>
* \copydetails Dumux::ZeroEqncniModel
*/
/*!
* \ingroup ImplicitModel
* \defgroup ImplicitBaseProblems Base Problems
......
......@@ -20,6 +20,10 @@ set(TEX_INPUTS
tutorial-coupled.tex
tutorial-decoupled.tex
tutorial.tex
ModelDescriptions/2cstokes2p2cmodel.tex
ModelDescriptions/2cnistokes2p2cnimodel.tex
ModelDescriptions/2czeroeq2p2cmodel.tex
ModelDescriptions/2cnizeroeq2p2cnimodel.tex
ModelDescriptions/1p2cimplicitmodel.tex
ModelDescriptions/1pdecoupledmodel.tex
ModelDescriptions/1pimplicitmodel.tex
......@@ -42,6 +46,9 @@ set(TEX_INPUTS
ModelDescriptions/stokesimplicitmodel.tex
ModelDescriptions/stokesncimplicitmodel.tex
ModelDescriptions/stokesncniimplicitmodel.tex
ModelDescriptions/zeroeqimplicitmodel.tex
ModelDescriptions/zeroeqncimplicitmodel.tex
ModelDescriptions/zeroeqncniimplicitmodel.tex
../../tutorial/tutorial_coupled.cc
../../tutorial/tutorial_coupled.input
../../tutorial/tutorialproblem_coupled.hh
......
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% This file has been autogenerated from the LaTeX part of the %
% doxygen documentation; DO NOT EDIT IT! Change the model's .hh %
% file instead!! %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
This model implements the coupling between a free-\/flow model and a porous-\/medium flow model under non-\/isothermal conditions. Here the coupling conditions for the individual balance are presented\-:
The total mass balance equation\-: \[ \left[ \left( \varrho_\textrm{g} {\boldsymbol{v}}_\textrm{g} \right) \cdot \boldsymbol{n} \right]^\textrm{ff} = -\left[ \left( \varrho_\textrm{g} \boldsymbol{v}_\textrm{g} + \varrho_\textrm{l} \boldsymbol{v}_\textrm{l} \right) \cdot \boldsymbol{n} \right]^\textrm{pm} \] in which $n$ represents a vector normal to the interface pointing outside of the specified subdomain.
The momentum balance (tangential), which corresponds to the Beavers-\/\-Jospeh Saffman condition\-: \[ \left[ \left( {\boldsymbol{v}}_\textrm{g} + \frac{\sqrt{\left(\boldsymbol{K} \boldsymbol{t}_i \right) \cdot \boldsymbol{t}_i}} {\alpha_\textrm{BJ} \mu_\textrm{g}} \boldsymbol{{\tau}}_\textrm{t} \boldsymbol{n} \right) \cdot \boldsymbol{t}_i \right]^\textrm{ff} = 0 \] with $ \boldsymbol{{\tau}_\textrm{t}} = \left[ \mu_\textrm{g} + \mu_\textrm{g,t} \right] \nabla \left( \boldsymbol{v}_\textrm{g} + \boldsymbol{v}_\textrm{g}^\intercal \right) $ in which the eddy viscosity $ \mu_\textrm{g,t} = 0 $ for the Stokes equation.
The momentum balance (normal)\-: \[ \left[ \left( \left\lbrace \varrho_\textrm{g} {\boldsymbol{v}}_\textrm{g} {\boldsymbol{v}}_\textrm{g}^\intercal - \boldsymbol{{\tau}}_\textrm{t} + {p}_\textrm{g} \boldsymbol{I} \right\rbrace \boldsymbol{n} \right) \cdot \boldsymbol{n} \right]^\textrm{ff} = p_\textrm{g}^\textrm{pm} \]
The component mass balance equation (continuity of fluxes)\-: \[ \left[ \left( \varrho_\textrm{g} {X}^\kappa_\textrm{g} {\boldsymbol{v}}_\textrm{g} - {\boldsymbol{j}}^\kappa_\textrm{g,ff,t,diff} \right) \cdot \boldsymbol{n} \right]^\textrm{ff} = -\left[ \left( \varrho_\textrm{g} X^\kappa_\textrm{g} \boldsymbol{v}_\textrm{g} - \boldsymbol{j}^\kappa_\textrm{g,pm,diff} + \varrho_\textrm{l} \boldsymbol{v}_\textrm{l} X^\kappa_\textrm{l} - \boldsymbol{j}^\kappa_\textrm{l,pm,diff} \right) \cdot \boldsymbol{n} \right]^\textrm{pm} = 0 \] in which the diffusive fluxes $ j_\textrm{diff} $ are the diffusive fluxes as they are implemented in the individual subdomain models.
The component mass balance equation (continuity of mass/ mole fractions)\-: \[ \left[ {X}^{\kappa}_\textrm{g} \right]^\textrm{ff} = \left[ X^{\kappa}_\textrm{g} \right]^\textrm{pm} \]
The energy balance equation (continuity of fluxes)\-: \[ \left[ \left( \varrho_\textrm{g} {h}_\textrm{g} {\boldsymbol{v}}_\textrm{g} - {h}^\textrm{a}_\textrm{g} {\boldsymbol{j}}^\textrm{a}_\textrm{g,ff,t,diff} - {h}^\textrm{w}_\textrm{g} {\boldsymbol{j}}^\textrm{w}_\textrm{g,ff,t,diff} - \left( \lambda_\textrm{g} + \lambda_\textrm{g,t} \right) \nabla {T} \right) \cdot \boldsymbol{n} \right]^\textrm{ff} = -\left[ \left( \varrho_\textrm{g} h_\textrm{g} \boldsymbol{v}_\textrm{g} + \varrho_\textrm{l} h_\textrm{l} \boldsymbol{v}_\textrm{l} - \lambda_\textrm{pm} \nabla T \right) \cdot \boldsymbol{n} \right]^\textrm{pm} \]
The energy balance equation (continuity of temperature)\-: \[ \left[ {T} \right]^\textrm{ff} = \left[ T \right]^\textrm{pm} \]
This is discretized by a fully-\/coupled vertex-\/centered finite volume (box) scheme in space and by the implicit Euler method in time.
This model implements the same coupling conditions as the model
concepts, which couples Darcy and Stokes flow presented \hyperref[sc_2cnistokes2p2cni]{above}.
The only difference is that the eddy coefficients $\mu_\textrm{g,t}$,
$D_\textrm{g,t}$, and $\lambda_\textrm{g,t}$ are not necessarily non-zero
at the coupling interface.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% This file has been autogenerated from the LaTeX part of the %
% doxygen documentation; DO NOT EDIT IT! Change the model's .hh %
% file instead!! %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
This model implements the coupling between a free-\/flow model and a porous-\/medium flow model under isothermal conditions. Here the coupling conditions for the individual balance are presented\-:
The total mass balance equation\-: \[ \left[ \left( \varrho_\textrm{g} {\boldsymbol{v}}_\textrm{g} \right) \cdot \boldsymbol{n} \right]^\textrm{ff} = -\left[ \left( \varrho_\textrm{g} \boldsymbol{v}_\textrm{g} + \varrho_\textrm{l} \boldsymbol{v}_\textrm{l} \right) \cdot \boldsymbol{n} \right]^\textrm{pm} \] in which $n$ represents a vector normal to the interface pointing outside of the specified subdomain.
The momentum balance (tangential), which corresponds to the Beavers-\/\-Jospeh Saffman condition\-: \[ \left[ \left( {\boldsymbol{v}}_\textrm{g} + \frac{\sqrt{\left(\boldsymbol{K} \boldsymbol{t}_i \right) \cdot \boldsymbol{t}_i}} {\alpha_\textrm{BJ} \mu_\textrm{g}} \boldsymbol{{\tau}}_\textrm{t} \boldsymbol{n} \right) \cdot \boldsymbol{t}_i \right]^\textrm{ff} = 0 \] with $ \boldsymbol{{\tau}_\textrm{t}} = \left[ \mu_\textrm{g} + \mu_\textrm{g,t} \right] \nabla \left( \boldsymbol{v}_\textrm{g} + \boldsymbol{v}_\textrm{g}^\intercal \right) $ in which the eddy viscosity $ \mu_\textrm{g,t} = 0 $ for the Stokes equation.
The momentum balance (normal)\-: \[ \left[ \left( \left\lbrace \varrho_\textrm{g} {\boldsymbol{v}}_\textrm{g} {\boldsymbol{v}}_\textrm{g}^\intercal - \boldsymbol{{\tau}}_\textrm{t} + {p}_\textrm{g} \boldsymbol{I} \right\rbrace \boldsymbol{n} \right) \cdot \boldsymbol{n} \right]^\textrm{ff} = p_\textrm{g}^\textrm{pm} \]
The component mass balance equation (continuity of fluxes)\-: \[ \left[ \left( \varrho_\textrm{g} {X}^\kappa_\textrm{g} {\boldsymbol{v}}_\textrm{g} - {\boldsymbol{j}}^\kappa_\textrm{g,ff,t,diff} \right) \cdot \boldsymbol{n} \right]^\textrm{ff} = -\left[ \left( \varrho_\textrm{g} X^\kappa_\textrm{g} \boldsymbol{v}_\textrm{g} - \boldsymbol{j}^\kappa_\textrm{g,pm,diff} + \varrho_\textrm{l} \boldsymbol{v}_\textrm{l} X^\kappa_\textrm{l} - \boldsymbol{j}^\kappa_\textrm{l,pm,diff} \right) \cdot \boldsymbol{n} \right]^\textrm{pm} = 0 \] in which the diffusive fluxes $ j_\textrm{diff} $ are the diffusive fluxes as they are implemented in the individual subdomain models.
The component mass balance equation (continuity of mass/ mole fractions)\-: \[ \left[ {X}^{\kappa}_\textrm{g} \right]^\textrm{ff} = \left[ X^{\kappa}_\textrm{g} \right]^\textrm{pm} \]
This is discretized by a fully-\/coupled vertex-\/centered finite volume (box) scheme in space and by the implicit Euler method in time.
This model implements the same coupling conditions as the model
concepts, which couples Darcy and Stokes flow presented \hyperref[sc_2cstokes2p2c]{above}.
The only difference is that the eddy coefficients $\mu_\textrm{g,t}$
and $D_\textrm{g,t}$ are not necessarily non-zero at the coupling
interface.
......@@ -8,7 +8,7 @@ This model solves equations of the form \[ \phi \left( \rho_w \frac{\partial S_w
\[ - \text{div}\, \left[\lambda \boldsymbol K \left(\textbf{grad}\, p_w + f_n \textbf{grad}\, p_c + \sum f_\alpha \rho_\alpha \, g \, \textbf{grad}\, z\right)\right] = q, \]
a non-\/wetting ( $ n $) phase pressure yields \[ - \text{div}\, \left[\lambda \boldsymbol K \left(\textbf{grad}\, p_n - f_w \textbf{grad}\, p_c + \sum f_\alpha \rho_\alpha \, g \, \textbf{grad}\, z\right)\right] = q, \] and a global pressure leads to \[ - \text{div}\, \left[\lambda \boldsymbol K \left(\textbf{grad}\, p_{global} + \sum f_\alpha \rho_\alpha \, g \, \textbf{grad}\, z\right)\right] = q. \] Here, $ p_\alpha $ is a phase pressure, $ p_ {global} $ the global pressure of a classical fractional flow formulation (see e.\-g. P. Binning and M. A. Celia, ''Practical implementation of the fractional flow approach to multi-\/phase flow simulation'', Advances in water resources, vol. 22, no. 5, pp. 461-\/478, 1999.), $ p_c = p_n - p_w $ is the capillary pressure, $ \boldsymbol K $ the absolute permeability, $ \lambda = \lambda_w + \lambda_n $ the total mobility depending on the saturation ( $ \lambda_\alpha = k_{r_\alpha} / \mu_\alpha $), $ f_\alpha = \lambda_\alpha / \lambda $ the fractional flow function of a phase, $ \rho_\alpha $ a phase density, $ g $ the gravity constant and $ q $ the source term.
a non-\/wetting ( $ n $) phase pressure yields \[ - \text{div}\, \left[\lambda \boldsymbol K \left(\textbf{grad}\, p_n - f_w \textbf{grad}\, p_c + \sum f_\alpha \rho_\alpha \, g \, \textbf{grad}\, z\right)\right] = q, \] and a global pressure leads to \[ - \text{div}\, \left[\lambda \boldsymbol K \left(\textbf{grad}\, p_{global} + \sum f_\alpha \rho_\alpha \, g \, \textbf{grad}\, z\right)\right] = q. \] Here, $ p_\alpha $ is a phase pressure, $ p_ {global} $ the global pressure of a classical fractional flow formulation (see e.\-g. P. Binning and M. A. Celia, \textquotesingle{}\textquotesingle{}Practical implementation of the fractional flow approach to multi-\/phase flow simulation\textquotesingle{}\textquotesingle{}, Advances in water resources, vol. 22, no. 5, pp. 461-\/478, 1999.), $ p_c = p_n - p_w $ is the capillary pressure, $ \boldsymbol K $ the absolute permeability, $ \lambda = \lambda_w + \lambda_n $ the total mobility depending on the saturation ( $ \lambda_\alpha = k_{r_\alpha} / \mu_\alpha $), $ f_\alpha = \lambda_\alpha / \lambda $ the fractional flow function of a phase, $ \rho_\alpha $ a phase density, $ g $ the gravity constant and $ q $ the source term.
For all cases, $ p = p_D $ on $ \Gamma_{Dirichlet} $, and $ \boldsymbol v_{total} \cdot \boldsymbol n = q_N $ on $ \Gamma_{Neumann} $.
......@@ -18,7 +18,7 @@ In the I\-M\-P\-E\-S models the default setting is\-:
\begin{itemize}
\item formulation\-: $ p_w-S_w $ (Property\-: {\itshape Formulation} defined as {\itshape \hyperlink{a00099_a601a847774d6e1b2e2a2b469f70c3f22}{Decoupled\-Two\-P\-Common\-Indices\-::pwsw}})
\item formulation\-: $ p_w-S_w $ (Property\-: {\itshape Formulation} defined as {\itshape \hyperlink{a00095_a601a847774d6e1b2e2a2b469f70c3f22}{Decoupled\-Two\-P\-Common\-Indices\-::pwsw}})
\item compressibility\-: disabled (Property\-: {\itshape Enable\-Compressibility} set to {\itshape false})
\end{itemize}
......
......@@ -16,7 +16,7 @@ The total velocity formulation is only implemented for incompressible fluids and
In the I\-M\-P\-E\-S models the default setting is\-:
formulation\-: $ p_w $ -\/ $ S_w $ (Property\-: {\itshape Formulation} defined as {\itshape \hyperlink{a00099_a601a847774d6e1b2e2a2b469f70c3f22}{Decoupled\-Two\-P\-Common\-Indices\-::pwsw}})
formulation\-: $ p_w $ -\/ $ S_w $ (Property\-: {\itshape Formulation} defined as {\itshape \hyperlink{a00095_a601a847774d6e1b2e2a2b469f70c3f22}{Decoupled\-Two\-P\-Common\-Indices\-::pwsw}})
compressibility\-: disabled (Property\-: {\itshape Enable\-Compressibility} set to {\itshape false})
......
......@@ -4,5 +4,5 @@
% file instead!! %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
See \hyperlink{a00615}{Two\-P\-Two\-C\-Model} for reference to the equations used. The \hyperlink{a00075}{C\-O2} model is derived from the 2p2c model. In the \hyperlink{a00075}{C\-O2} model the phase switch criterion is different from the 2p2c model. The phase switch occurs when the equilibrium concentration of a component in a phase is exceeded, instead of the sum of the components in the virtual phase (the phase which is not present) being greater that unity as done in the 2p2c model. The \hyperlink{a00082}{C\-O2\-Volume\-Variables} do not use a constraint solver for calculating the mole fractions as is the case in the 2p2c model. Instead mole fractions are calculated in the Fluid\-System with a given temperature, pressurem and salinity. The model is able to use either mole or mass fractions. The property use\-Moles can be set to either true or false in the problem file. Make sure that the according units are used in the problem setup. use\-Moles is set to false by default.
See \hyperlink{a00633}{Two\-P\-Two\-C\-Model} for reference to the equations used. The \hyperlink{a00074}{C\-O2} model is derived from the 2p2c model. In the \hyperlink{a00074}{C\-O2} model the phase switch criterion is different from the 2p2c model. The phase switch occurs when the equilibrium concentration of a component in a phase is exceeded, instead of the sum of the components in the virtual phase (the phase which is not present) being greater that unity as done in the 2p2c model. The \hyperlink{a00078}{C\-O2\-Volume\-Variables} do not use a constraint solver for calculating the mole fractions as is the case in the 2p2c model. Instead mole fractions are calculated in the Fluid\-System with a given temperature, pressurem and salinity. The model is able to use either mole or mass fractions. The property use\-Moles can be set to either true or false in the problem file. Make sure that the according units are used in the problem setup. use\-Moles is set to false by default.
......@@ -4,9 +4,9 @@
% file instead!! %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
This model implements a one-\/phase flow of an incompressible fluid, that consists of two components. The deformation of the solid matrix is described with a quasi-\/stationary momentum balance equation. The influence of the pore fluid is accounted for through the effective stress concept (Biot 1941). The total stress acting on a rock is partially supported by the rock matrix and partially supported by the pore fluid. The effective stress represents the share of the total stress which is supported by the solid rock matrix and can be determined as a function of the strain according to Hooke's law.
This model implements a one-\/phase flow of an incompressible fluid, that consists of two components. The deformation of the solid matrix is described with a quasi-\/stationary momentum balance equation. The influence of the pore fluid is accounted for through the effective stress concept (Biot 1941). The total stress acting on a rock is partially supported by the rock matrix and partially supported by the pore fluid. The effective stress represents the share of the total stress which is supported by the solid rock matrix and can be determined as a function of the strain according to Hooke\textquotesingle{}s law.
As an equation for the conservation of momentum within the fluid phase Darcy's approach is used\-: \[ v = - \frac{\textbf K}{\mu} \left(\textbf{grad}\, p - \varrho_w {\textbf g} \right) \]
As an equation for the conservation of momentum within the fluid phase Darcy\textquotesingle{}s approach is used\-: \[ v = - \frac{\textbf K}{\mu} \left(\textbf{grad}\, p - \varrho_w {\textbf g} \right) \]
Gravity can be enabled or disabled via the property system. By inserting this into the volume balance of the solid-\/fluid mixture, one gets \[ \frac{\partial \text{div} \textbf{u}}{\partial t} - \text{div} \left\{ \frac{\textbf K}{\mu} \left(\textbf{grad}\, p - \varrho_w {\textbf g} \right)\right\} = q \;, \]
......
......@@ -4,9 +4,9 @@
% file instead!! %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
This model implements a two-\/phase flow of compressible immiscible fluids $\alpha \in \{ w, n \}$. The deformation of the solid matrix is described with a quasi-\/stationary momentum balance equation. The influence of the pore fluid is accounted for through the effective stress concept (Biot 1941). The total stress acting on a rock is partially supported by the rock matrix and partially supported by the pore fluid. The effective stress represents the share of the total stress which is supported by the solid rock matrix and can be determined as a function of the strain according to Hooke's law.
This model implements a two-\/phase flow of compressible immiscible fluids $\alpha \in \{ w, n \}$. The deformation of the solid matrix is described with a quasi-\/stationary momentum balance equation. The influence of the pore fluid is accounted for through the effective stress concept (Biot 1941). The total stress acting on a rock is partially supported by the rock matrix and partially supported by the pore fluid. The effective stress represents the share of the total stress which is supported by the solid rock matrix and can be determined as a function of the strain according to Hooke\textquotesingle{}s law.
As an equation for the conservation of momentum within the fluid phases the standard multiphase Darcy's approach is used\-: \[ v_\alpha = - \frac{k_{r\alpha}}{\mu_\alpha} \textbf{K} \left(\textbf{grad}\, p_\alpha - \varrho_{\alpha} {\textbf g} \right) \]
As an equation for the conservation of momentum within the fluid phases the standard multiphase Darcy\textquotesingle{}s approach is used\-: \[ v_\alpha = - \frac{k_{r\alpha}}{\mu_\alpha} \textbf{K} \left(\textbf{grad}\, p_\alpha - \varrho_{\alpha} {\textbf g} \right) \]
Gravity can be enabled or disabled via the property system. By inserting this into the continuity equation, one gets \[ \frac{\partial \phi_{eff} \varrho_\alpha S_\alpha}{\partial t} - \text{div} \left\{ \varrho_\alpha \frac{k_{r\alpha}}{\mu_\alpha} \mathbf{K}_\text{eff} \left(\textbf{grad}\, p_\alpha - \varrho_{\alpha} \mathbf{g} \right) - \phi_{eff} \varrho_\alpha S_\alpha \frac{\partial \mathbf{u}}{\partial t} \right\} - q_\alpha = 0 \;, \]
......
......@@ -4,7 +4,7 @@
% file instead!! %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
This model implements a linear elastic solid using Hooke's law as stress-\/strain relation and a quasi-\/stationary momentum balance equation\-: \[ \boldsymbol{\sigma} = 2\,G\,\boldsymbol{\epsilon} + \lambda \,\text{tr} (\boldsymbol{\epsilon}) \, \boldsymbol{I}. \]
This model implements a linear elastic solid using Hooke\textquotesingle{}s law as stress-\/strain relation and a quasi-\/stationary momentum balance equation\-: \[ \boldsymbol{\sigma} = 2\,G\,\boldsymbol{\epsilon} + \lambda \,\text{tr} (\boldsymbol{\epsilon}) \, \boldsymbol{I}. \]
with the strain tensor $\boldsymbol{\epsilon}$ as a function of the solid displacement gradient $\textbf{grad} \boldsymbol{u}$\-: \[ \boldsymbol{\epsilon} = \frac{1}{2} \, (\textbf{grad} \boldsymbol{u} + \textbf{grad}^T \boldsymbol{u}). \]
......
......@@ -6,7 +6,7 @@
This model implements a $M$-\/phase flow of a fluid mixture composed of $N$ chemical species. The phases are denoted by lower index $\alpha \in \{ 1, \dots, M \}$. All fluid phases are mixtures of $N \geq M - 1$ chemical species which are denoted by the upper index $\kappa \in \{ 1, \dots, N \} $.
The momentum approximation can be selected via \char`\"{}\-Base\-Flux\-Variables\char`\"{}\-: Darcy (\hyperlink{a00278}{Implicit\-Darcy\-Flux\-Variables}) and Forchheimer (\hyperlink{a00279}{Implicit\-Forchheimer\-Flux\-Variables}) relations are available for all Box models.
The momentum approximation can be selected via \char`\"{}\-Base\-Flux\-Variables\char`\"{}\-: Darcy (\hyperlink{a00280}{Implicit\-Darcy\-Flux\-Variables}) and Forchheimer (\hyperlink{a00281}{Implicit\-Forchheimer\-Flux\-Variables}) relations are available for all Box models.
By inserting this into the equations for the conservation of the mass of each component, one gets one mass-\/continuity equation for each component $\kappa$ \[ \sum_{\kappa} \left( \phi \frac{\partial \left(\varrho_\alpha x_\alpha^\kappa S_\alpha\right)}{\partial t} + \mathrm{div}\; \left\{ v_\alpha \frac{\varrho_\alpha}{\overline M_\alpha} x_\alpha^\kappa \right\} \right) = q^\kappa \] with $\overline M_\alpha$ being the average molar mass of the phase $\alpha$\-: \[ \overline M_\alpha = \sum_\kappa M^\kappa \; x_\alpha^\kappa \]
......@@ -20,7 +20,7 @@ These three equations constitute a non-\/linear complementarity problem, which c
Several non-\/linear complementarity functions have been suggested, e.\-g. the Fischer-\/\-Burmeister function \[ \Phi(a,b) = a + b - \sqrt{a^2 + b^2} \;. \] This model uses \[ \Phi(a,b) = \min \{a, b \}\;, \] because of its piecewise linearity.
These equations are then discretized using a fully-\/implicit vertex centered finite volume scheme (often known as 'box'-\/scheme) for spatial discretization and the implicit Euler method as temporal discretization.
These equations are then discretized using a fully-\/implicit vertex centered finite volume scheme (often known as \textquotesingle{}box\textquotesingle{}-\/scheme) for spatial discretization and the implicit Euler method as temporal discretization.
The model assumes local thermodynamic equilibrium and uses the following primary variables\-:
\begin{itemize}
......
......@@ -4,9 +4,9 @@
% file instead!! %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
In the unsaturated zone, Richards' equation \[ \frac{\partial\;\phi S_w \varrho_w}{\partial t} - \text{div} \left\lbrace \varrho_w \frac{k_{rw}}{\mu_w} \; \mathbf{K} \; \left( \text{\textbf{grad}} p_w - \varrho_w \textbf{g} \right) \right\rbrace = q_w, \] is frequently used to approximate the water distribution above the groundwater level.
In the unsaturated zone, Richards\textquotesingle{} equation \[ \frac{\partial\;\phi S_w \varrho_w}{\partial t} - \text{div} \left\lbrace \varrho_w \frac{k_{rw}}{\mu_w} \; \mathbf{K} \; \left( \text{\textbf{grad}} p_w - \varrho_w \textbf{g} \right) \right\rbrace = q_w, \] is frequently used to approximate the water distribution above the groundwater level.
It can be derived from the two-\/phase equations, i.\-e. \[ \phi\frac{\partial S_\alpha \varrho_\alpha}{\partial t} - \text{div} \left\lbrace \varrho_\alpha \frac{k_{r\alpha}}{\mu_\alpha}\; \mathbf{K} \; \left( \text{\textbf{grad}} p_\alpha - \varrho_\alpha \textbf{g} \right) \right\rbrace = q_\alpha, \] where $\alpha \in \{w, n\}$ is the fluid phase, $\kappa \in \{ w, a \}$ are the components, $\rho_\alpha$ is the fluid density, $S_\alpha$ is the fluid saturation, $\phi$ is the porosity of the soil, $k_{r\alpha}$ is the relative permeability for the fluid, $\mu_\alpha$ is the fluid's dynamic viscosity, $\mathbf{K}$ is the intrinsic permeability, $p_\alpha$ is the fluid pressure and $g$ is the potential of the gravity field.
It can be derived from the two-\/phase equations, i.\-e. \[ \phi\frac{\partial S_\alpha \varrho_\alpha}{\partial t} - \text{div} \left\lbrace \varrho_\alpha \frac{k_{r\alpha}}{\mu_\alpha}\; \mathbf{K} \; \left( \text{\textbf{grad}} p_\alpha - \varrho_\alpha \textbf{g} \right) \right\rbrace = q_\alpha, \] where $\alpha \in \{w, n\}$ is the fluid phase, $\kappa \in \{ w, a \}$ are the components, $\rho_\alpha$ is the fluid density, $S_\alpha$ is the fluid saturation, $\phi$ is the porosity of the soil, $k_{r\alpha}$ is the relative permeability for the fluid, $\mu_\alpha$ is the fluid\textquotesingle{}s dynamic viscosity, $\mathbf{K}$ is the intrinsic permeability, $p_\alpha$ is the fluid pressure and $g$ is the potential of the gravity field.
In contrast to the full two-\/phase model, the Richards model assumes gas as the non-\/wetting fluid and that it exhibits a much lower viscosity than the (liquid) wetting phase. (For example at atmospheric pressure and at room temperature, the viscosity of air is only about $1\%$ of the viscosity of liquid water.) As a consequence, the $\frac{k_{r\alpha}}{\mu_\alpha}$ term typically is much larger for the gas phase than for the wetting phase. For this reason, the Richards model assumes that $\frac{k_{rn}}{\mu_n}$ is infinitly large. This implies that the pressure of the gas phase is equivalent to the static pressure distribution and that therefore, mass conservation only needs to be considered for the wetting phase.
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......@@ -10,7 +10,7 @@ Momentum Balance\-: \[ \frac{\partial \left(\varrho_g {\boldsymbol{v}}_g\right)}
Mass balance equation\-: \[ \frac{\partial \varrho_g}{\partial t} + \boldsymbol{\nabla}\boldsymbol{\cdot}\left(\varrho_g {\boldsymbol{v}}_g\right) - q_g = 0 \]
\hyperlink{a00088}{Component} mass balance equations\-: \[ \frac{\partial \left(\varrho_g X_g^\kappa\right)}{\partial t} + \boldsymbol{\nabla} \boldsymbol{\cdot} \left( \varrho_g {\boldsymbol{v}}_g X_g^\kappa - D^\kappa_g \varrho_g \frac{M^\kappa}{M_g} \boldsymbol{\nabla} X_g^\kappa \right) - q_g^\kappa = 0 \]
\hyperlink{a00084}{Component} mass balance equations\-: \[ \frac{\partial \left(\varrho_g X_g^\kappa\right)}{\partial t} + \boldsymbol{\nabla} \boldsymbol{\cdot} \left( \varrho_g {\boldsymbol{v}}_g X_g^\kappa - D^\kappa_g \varrho_g \frac{M^\kappa}{M_g} \boldsymbol{\nabla} X_g^\kappa \right) - q_g^\kappa = 0 \]
This is discretized using a fully-\/coupled vertex centered finite volume (box) scheme as spatial and the implicit Euler method in time.
......@@ -10,7 +10,7 @@ Momentum Balance\-: \[ \frac{\partial \left(\varrho_g {\boldsymbol{v}}_g\right)}
Mass balance equation\-: \[ \frac{\partial \varrho_g}{\partial t} + \boldsymbol{\nabla}\boldsymbol{\cdot}\left(\varrho_g {\boldsymbol{v}}_g\right) - q_g = 0 \]
\hyperlink{a00088}{Component} mass balance equation\-: \[ \frac{\partial \left(\varrho_g X_g^\kappa\right)}{\partial t} + \boldsymbol{\nabla} \boldsymbol{\cdot} \left( \varrho_g {\boldsymbol{v}}_g X_g^\kappa - D^\kappa_g \varrho_g \frac{M^\kappa}{M_g} \boldsymbol{\nabla} x_g^\kappa \right) - q_g^\kappa = 0 \]
\hyperlink{a00084}{Component} mass balance equation\-: \[ \frac{\partial \left(\varrho_g X_g^\kappa\right)}{\partial t} + \boldsymbol{\nabla} \boldsymbol{\cdot} \left( \varrho_g {\boldsymbol{v}}_g X_g^\kappa - D^\kappa_g \varrho_g \frac{M^\kappa}{M_g} \boldsymbol{\nabla} x_g^\kappa \right) - q_g^\kappa = 0 \]
Energy balance equation\-: \[ \frac{\partial (\varrho_g u_g)}{\partial t} + \boldsymbol{\nabla} \boldsymbol{\cdot} \left( \varrho_g h_g {\boldsymbol{v}}_g - \sum_\kappa \left[ h^\kappa_g D^\kappa_g \varrho_g \frac{M^\kappa}{M_g} \nabla x^\kappa_g \right] - \lambda_g \boldsymbol{\nabla} T \right) - q_T = 0 \]
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This model implements an single-\/phase isothermal free flow solving the mass and the momentum balance. For the momentum balance the Reynolds-\/averaged Navier-\/\-Stokes (R\-A\-N\-S) equation with zero equation (algebraic) turbulence model is used.
Mass balance\-: \[ \frac{\partial \varrho_\textrm{g}}{\partial t} + \boldsymbol{\nabla}\boldsymbol{\cdot}\left(\varrho_\textrm{g} {\boldsymbol{v}}_\textrm{g}\right) - q_\textrm{g} = 0 \]
Momentum Balance\-: \[ \frac{\partial \left(\varrho_\textrm{g} {\boldsymbol{v}}_\textrm{g}\right)}{\partial t} + \boldsymbol{\nabla} \boldsymbol{\cdot} \left( \varrho_\textrm{g} {\boldsymbol{v}_\textrm{g} {\boldsymbol{v}}_\textrm{g}} - \left[ \mu_\textrm{g} + \mu_\textrm{g,t} \right] \left(\boldsymbol{\nabla} \boldsymbol{v}_\textrm{g} + \boldsymbol{\nabla} \boldsymbol{v}_\textrm{g}^T \right) \right) + \left(p_\textrm{g} {\bf {I}} \right) - \varrho_\textrm{g} {\bf g} = 0, \]
This is discretized by a fully-\/coupled vertex-\/centered finite volume (box) scheme in space and by the implicit Euler method in time.
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This model implements an single-\/phase isothermal compositional free flow solving the mass and the momentum balance. For the momentum balance the Reynolds-\/averaged Navier-\/\-Stokes (R\-A\-N\-S) equation with zero equation (algebraic) turbulence model is used.
Mass balance\-: \[ \frac{\partial \varrho_\textrm{g}}{\partial t} + \boldsymbol{\nabla}\boldsymbol{\cdot}\left(\varrho_\textrm{g} {\boldsymbol{v}}_\textrm{g}\right) - q_\textrm{g} = 0 \]
Momentum Balance\-: \[ \frac{\partial \left(\varrho_\textrm{g} {\boldsymbol{v}}_\textrm{g}\right)}{\partial t} + \boldsymbol{\nabla} \boldsymbol{\cdot} \left( \varrho_\textrm{g} {\boldsymbol{v}_\textrm{g} {\boldsymbol{v}}_\textrm{g}} - \left[ \mu_\textrm{g} + \mu_\textrm{g,t} \right] \left(\boldsymbol{\nabla} \boldsymbol{v}_\textrm{g} + \boldsymbol{\nabla} \boldsymbol{v}_\textrm{g}^T \right) \right) + \left(p_\textrm{g} {\bf {I}} \right) - \varrho_\textrm{g} {\bf g} = 0, \]
\hyperlink{a00084}{Component} mass balance equations\-: \[ \frac{\partial \left(\varrho_\textrm{g} X_\textrm{g}^\kappa\right)}{\partial t} + \boldsymbol{\nabla} \boldsymbol{\cdot} \left( \varrho_\textrm{g} {\boldsymbol{v}}_\textrm{g} X_\textrm{g}^\kappa - \left[ D^\kappa_\textrm{g} + D^\kappa_\textrm{g,t} \right] \varrho_\textrm{g} \frac{M^\kappa}{M_\textrm{g}} \boldsymbol{\nabla} x_\textrm{g}^\kappa \right) - q_\textrm{g}^\kappa = 0 \]
This is discretized by a fully-\/coupled vertex-\/centered finite volume (box) scheme in space and by the implicit Euler method in time.
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% doxygen documentation; DO NOT EDIT IT! Change the model's .hh %
% file instead!! %
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This model implements an single-\/phase non-\/isothermal compositional free flow solving the mass and the momentum balance. For the momentum balance the Reynolds-\/averaged Navier-\/\-Stokes (R\-A\-N\-S) equation with zero equation (algebraic) turbulence model is used.
Mass balance\-: \[ \frac{\partial \varrho_\textrm{g}}{\partial t} + \boldsymbol{\nabla}\boldsymbol{\cdot}\left(\varrho_\textrm{g} {\boldsymbol{v}}_\textrm{g}\right) - q_\textrm{g} = 0 \]
Momentum Balance\-: \[ \frac{\partial \left(\varrho_\textrm{g} {\boldsymbol{v}}_\textrm{g}\right)}{\partial t} + \boldsymbol{\nabla} \boldsymbol{\cdot} \left( \varrho_\textrm{g} {\boldsymbol{v}_\textrm{g} {\boldsymbol{v}}_\textrm{g}} - \left[ \mu_\textrm{g} + \mu_\textrm{g,t} \right] \left(\boldsymbol{\nabla} \boldsymbol{v}_\textrm{g} + \boldsymbol{\nabla} \boldsymbol{v}_\textrm{g}^T \right) \right) + \left(p_\textrm{g} {\bf {I}} \right) - \varrho_\textrm{g} {\bf g} = 0, \]
\hyperlink{a00084}{Component} mass balance equations\-: \[ \frac{\partial \left(\varrho_\textrm{g} X_\textrm{g}^\kappa\right)}{\partial t} + \boldsymbol{\nabla} \boldsymbol{\cdot} \left( \varrho_\textrm{g} {\boldsymbol{v}}_\textrm{g} X_\textrm{g}^\kappa - \left[ D^\kappa_\textrm{g} + D^\kappa_\textrm{g,t} \right] \varrho_\textrm{g} \frac{M^\kappa}{M_\textrm{g}} \boldsymbol{\nabla} x_\textrm{g}^\kappa \right) - q_\textrm{g}^\kappa = 0 \]
Energy balance equation\-: \[ \frac{\partial (\varrho_\textrm{g} u_\textrm{g})}{\partial t} + \boldsymbol{\nabla} \boldsymbol{\cdot} \left( \varrho_\textrm{g} h_\textrm{g} {\boldsymbol{v}}_\textrm{g} - \sum_\kappa \left( h^\kappa_\textrm{g} \left[ D^\kappa_\textrm{g} + D^\kappa_\textrm{g,t} \right] \varrho_\textrm{g} \frac{M^\kappa}{M_\textrm{g}} \nabla x^\kappa_\textrm{g} \right) - \left[ \lambda_\textrm{g} + \lambda_\textrm{g,t} \right] \boldsymbol{\nabla} T \right) - q_\textrm{T} = 0 \]
This is discretized by a fully-\/coupled vertex-\/centered finite volume (box) scheme in space and by the implicit Euler method in time.
......@@ -20,6 +20,7 @@
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