diff --git a/slides/coupled_ff-pm.md b/slides/coupled_ff-pm.md
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@@ -2,114 +2,245 @@
title: Coupled Freeflow and Porous Media Flow Models in DuMu<sup>X</sup>
---
-# Coupled Freeflow and Porous Media Flow Models in DuMu<sup>X</sup>
+# Motivation
+
+## Environmental and Agricultural Issues
+
+{style="width: 60%; margin: auto; "}
+<figcaption align = "center">
+<font size = "2">
+Fig.1 - Evaporation of soil water (Heck et al. (2020))[^3]
+</font>
+</figcaption>
+
+* Evaporation of soil water
+* Soil salinization
+* Underground storage (e.g. CO2, atmoic waste)
+
+## Technical Issues
+
+{style="width: 45%; align: left;"}
+<figcaption align = "center">
+<font size = "2">
+Fig.2 - Filter (Schneider et al. (2023))[^1]
+</font>
+</figcaption>
+
+* Fuel cells
+* Filters (e.g. air)
+* Heat exchangers (e.g. CPU cooling)
+
+## Biological Issues
+
+{style="width: 25%; align: left;"}
+<figcaption align = "center">
+<font size = "2">
+Fig.3 - Brain tissue (Koch et al. (2020))[^2]
+</font>
+</figcaption>
+
+* Brain tissue
+* Leaf structure
+
+# Model Overview
+
+## Conceptual Physical Model
+<img src=img/FFPM-PhysicalModelOverview.png width="80%">
+<figcaption align = "center">
+<font size = "2">
+Fig.4 - Coupled dynamics at the soil-atmosphere interface (Photo: Edward Coltman)
+</font>
+</figcaption>
+
+## Conceptual Physical Model
+{style="width: 80%; align: left;"}
+<figcaption align = "center">
+<font size = "2">
+Fig.5 - Exchange processes at the free-flow porous-medium interface at different scales (Photo: Martin Schneider)
+</font>
+</figcaption>
-## Coupled Freeflow and Porous Media Flow Systems?
+## Mathematical Model: Overview
-<img src=img/FFPM-SoilWaterEvapField.png width="100%">
-<img src=img/FFPM-SaltPrecip.png width="100%">
-<img src=FFPM-FuelCellsSim.png width="100%">
+{style="width: 15%; margin: auto; float: left;"}
-[ETHZurich](https://emeritus.step.ethz.ch/the-step-group.html)
-[EOS-SoilSalinization](https://eos.com/blog/soil-salinization/)
-[EllerEtAl2011](https://iopscience.iop.org/article/10.1149/1.3596556#artAbst)
+<font size = "6">
-## Conceptual Physical model
-<img src=img/FFPM-PhysicalModelOverview.png width="100%">
+**Freeflow:**
-## Mathematical Model: Overview
-<img src=img/FFPM-ModelConceptColumn.png width="100%">
+* Stokes / Navier-Stokes / RANS
+* 1-phase, n-components, non-equilibrium
+
+**Interface condtions:**
+
+* no thickness, no storage
+* local thermodynamic equilibrium
+* continuity of fluxes
+* continuity of state variables
+
+**Porous media:**
-Freeflow: NS/RANS Equations, Non-isothermal, multi-component
-Porous Medium: Multi-phase Darcy, Non-isothermal, multi-component
-Coupling Conditions: Local Thermodynamic Equilibrium, continuity of fluxes
+* Darcy/ Forchheimer / Richards
+* m-phases, n-components, non-isothermal
+
+</font>
+
+## Mathematical Model: Freeflow
+<img src=img/FFPM-freeflowsymbol.png width="40%">
## Mathematical Model: Freeflow
-<img src=img/FFPM-freeflowsymbol.png width="100%">
-\begin{equation}
-\frac{\partial \rho_g \textbf{v}_g}{\partial t} + \nabla \cdot (\rho_g \textbf{v}_g \textbf{v}_g^T) - \DIV(\bm{\tau}_g + \bm{\tau}_{g,t}) +\DIV (p_g\textbf{I})- \rho_g \textbf{g} = 0\, .
-\end{equation}
+* Total mass balance
+$$
+\frac{\partial \rho_g \textbf{v}_g}{\partial t} + \nabla \cdot (\rho_g \textbf{v}_g \textbf{v}_g^T) - \nabla \cdot (\mathbf{\tau}_g + \mathbf{\tau}_{g,t}) +\nabla \cdot (p_g\textbf{I})- \rho_g \textbf{g} = 0
+$$
-\begin{equation}
-\frac{\partial \left(\rho_g X^\kappa_g\right)}{\partial t}
-+ \nabla \cdot \left( \rho_g \textbf{v}_g X^\kappa_g
-- \mathbf{j}_{\text{diff}}^\kappa\right)
-- q^\kappa = 0\, .
-\end{equation}
+* Momentum balance
+$$
+\frac{\partial \left(\rho_g X^\kappa_g\right)}{\partial t} + \nabla \cdot \left( \rho_g \textbf{v}_g X^\kappa_g - \mathbf{j}_{\text{diff}}^\kappa\right) - q^\kappa = 0
+$$
-\begin{equation}
-\frac{\partial (\rho_g u_g) }{\partial t} + \DIV (\rho_g h_g \textbf{v}_g) + \sum_{i} {\DIV (h_g^\kappa \textbf{j}_{\text{diff},t}^\kappa)} - \DIV ( (\lambda_{g} + \lambda_{t})) \grad T) = 0\, ,
-\end{equation}
+* Component mass balance
+$$
+\frac{\partial (\rho_g u_g) }{\partial t} + \nabla \cdot (\rho_g h_g \textbf{v}_g) + \sum_{i} {\nabla \cdot (h_g^\kappa \textbf{j}_{\text{diff},t}^\kappa)} - \nabla \cdot ( (\lambda_{g} + \lambda_{t}) \nabla T) = 0
+$$
+
+## Mathematical Model: Porous Medium Flow
+<img src=img/FFPM-pmfsymbol.png width="40%">
## Mathematical Model: Porous Medium Flow
-<img src=img/FFPM-pmfsymbol.png width="100%">
-\begin{equation}
- \sum\limits_{\alpha \in \{\text{l, g} \}}
- \left(\phi \frac{\partial \left(\rho_\alpha S_\alpha X_\alpha^\kappa\right)}{\partial t } + \nabla \cdot \textbf{v}_\alpha\rho_\alpha X_\alpha^\kappa + \sum_\kappa \nabla \cdot \left( \textbf{D}_{pm,\alpha}^\kappa\rho_\alpha\nabla X_\alpha^\kappa \right)\right) = 0 ,
-\end{equation}
+* Component mass balance
+$$
+\sum\limits_{\alpha \in \{\text{l, g} \}} \left(\phi \frac{\partial \left(\rho_\alpha S_\alpha X_\alpha^\kappa\right)}{\partial t } + \nabla \cdot \textbf{v}_\alpha\rho_\alpha X_\alpha^\kappa + \sum_\kappa \nabla \cdot \left( \textbf{D}_{pm,\alpha}^\kappa\rho_\alpha\nabla X_\alpha^\kappa \right)\right) = 0
+$$
-\begin{equation}\label{eq:darcy}
- \textbf{v}_\alpha = - \frac{k_{r,\alpha}}{\mu_\alpha} K \left(\nabla p_\alpha - \rho_\alpha \textbf{g}\right) .
-\end{equation}
+* Darcy velocity
+$$
+\textbf{v}_\alpha = - \frac{k_{r,\alpha}}{\mu_\alpha} K \left(\nabla p_\alpha - \rho_\alpha \textbf{g}\right)
+$$
+
+* Energy balance
+$$
+\sum\limits_{\alpha \in \{\text{l, g} \}}\left(\phi\frac{\partial \left(\rho_\alpha S_\alpha u_\alpha\right)}{\partial t} + \nabla \cdot \left(\rho_\alpha h_\alpha \textbf{v}_\alpha \right)\right) + \left(1-\phi\right) \frac{\partial \left(\rho_s c_{p,s}T\right)}{\partial t} - \nabla\cdot \left(\lambda_{pm} \nabla T \right) = 0
+$$
-\begin{equation}
- \sum\limits_{\alpha \in \{\text{l, g} \}}
- \left(\phi\frac{\partial \left(\rho_\alpha S_\alpha u_\alpha\right)}{\partial t}
- + \nabla \cdot \left(\rho_\alpha h_\alpha \textbf{v}_\alpha \right)\right)
- + \left(1- \phi\right) \frac{\partial \left(\rho_s c_{p,s}T\right)}{\partial t}
- - \nabla\cdot \left(\lambda_{pm} \nabla T \right) = 0 .
-\end{equation}
## Mathematical Model: Coupling Conditions
-<img src=img/FFPM-couplingsymbol.png width="100%">
+<img src=img/FFPM-couplingsymbol.png width="30%">
-\begin{equation}
-[(\rho_g \textbf{v}_g) \cdot \textbf{n}]^{\text{ff}} = - [(\rho_g \textbf{v}_g + \rho_w \textbf{v}_w) \cdot \textbf{n}]^{\text{pm}}\, .
-\end{equation}
+* Total mass condition
+$$
+[(\rho_g \textbf{v}_g) \cdot \textbf{n}]^{\text{ff}} = - [(\rho_g \textbf{v}_g + \rho_w \textbf{v}_w) \cdot \textbf{n}]^{\text{pm}}
+$$
-<img src=img/FFPM-BJS.png width="100%">
+## Mathematical Model: Coupling Conditions
+<img src=img/FFPM-BJS.png width="30%">
-\begin{equation}
-\left[\left(- \textbf{v}_g - \frac{\sqrt{(\textbf{K}\textbf{t}_i)\cdot \textbf{t}_i}}{\alpha_{BJ}} (\nabla \textbf{v}_g + \nabla \textbf{v}_g^T)\textbf{n} \right) \cdot \textbf{t}_i \right]^{\text{ff}} = 0\, , \quad i \in \{1, .. ,\, d-1\}\, .
-\end{equation}
+* Momentum (tangential)condition
+$$
+\left[\left(- \textbf{v}_g - \frac{\sqrt{(\textbf{K}\textbf{t}_i)\cdot \textbf{t}_i}}{\alpha_{BJ}} (\nabla \textbf{v}_g + \nabla \textbf{v}_g^T)\textbf{n} \right) \cdot \textbf{t}_i \right]^{\text{ff}} = 0\, , \quad i \in \{1, .. ,\, d-1\}\,
+$$
-\begin{equation}
-[((\rho_g \textbf{v}_g \textbf{v}_g^T - (\bm{\tau}_g + \bm{\tau}_{g,t}) + p_g\textbf{I}) \textbf{n} )]^{\text{ff}} = [(p_g\textbf{I})\textbf{n}]^{\text{pm}}\, .
-\end{equation}
+## Mathematical Model: Coupling Conditions
-\begin{equation}
-[(\rho_g X_g^\kappa \textbf{v}_g + \textbf{j}_{\text{diff}, t}) \cdot \textbf{n}]^{\text{ff}} = - \left[\left( \sum_{\alpha} (\rho_{\alpha} X_{\alpha}^\kappa \textbf{v}_\alpha + \textbf{j}^\kappa_{\text{diff}, \alpha})\right) \cdot \textbf{n}\right]^{\text{pm}}\, .
-\end{equation}
+* Momentum (normal) condition
+$$
+[((\rho_g \textbf{v}_g \textbf{v}_g^T - (\mathbf{\tau}_g + \mathbf{\tau}_{g,t}) + p_g\textbf{I}) \textbf{n} )]^{\text{ff}} = [(p_g\textbf{I})\textbf{n}]^{\text{pm}}\,
+$$
-\begin{equation}
-\left[\left(\rho_g h_g \textbf{v}_g + \sum_i h_g^\kappa \textbf{j}_{\text{diff},g}^\kappa + \lambda_{g}\grad T\right)\cdot \textbf{n}\right]^{\text{ff}} = - \left[\left( \sum_\alpha (\rho_\alpha h_\alpha \textbf{v}_\alpha + \sum_i h_\alpha^\kappa \textbf{j}_{\text{diff},\alpha}^\kappa) - \lambda_{\text{pm}}\grad T\right)\cdot \textbf{n}\right]^{\text{pm}}\, .
-\end{equation}
+* Component mass condition
+$$
+[(\rho_g X_g^\kappa \textbf{v}_g + \textbf{j}_{\text{diff}, t}) \cdot \textbf{n}]^{\text{ff}} = - \left[\left( \sum_{\alpha} (\rho_{\alpha} X_{\alpha}^\kappa \textbf{v}_\alpha + \textbf{j}^\kappa_{\text{diff}, \alpha})\right) \cdot \textbf{n}\right]^{\text{pm}}\,
+$$
+
+* Energy condition
+$$
+\left[\left(\rho_g h_g \textbf{v}_g + \sum_i h_g^\kappa \textbf{j}_{\text{diff},g}^\kappa + \lambda_{g}\nabla T\right)\cdot \textbf{n}\right]^{\text{ff}} = - \left[\left( \sum_\alpha (\rho_\alpha h_\alpha \textbf{v}_\alpha + \sum_i h_\alpha^\kappa \textbf{j}_{\text{diff},\alpha}^\kappa) - \lambda_{\text{pm}}\nabla T\right)\cdot \textbf{n}\right]^{\text{pm}}\,
+$$
## Numerical Model: Coupled Model
-<img src=img/FFPM-numericalmodel.png width="100%">
+<img src=img/FFPM-numericalmodel.png width="25%">
+<figcaption align = "center">
+<font size = "2">
+Fig.6 - Discretization scheme (Fetzer, 2018)[^5]
+</font>
+</figcaption>
+
+# Example: Soil-Water Evaporation
+
+## Soil-Water Evaporation: Soil-Water Evaporation
+
+<img src=img/FFPM-TurbulentBoundaryLayer.png width="40%">
+
+## Example: Soil-Water Evaporation
+<img src=img/FFPM-SoilWaterEvapField.png width="40%">
+
+<figcaption align = "center">
+<font size = "2">
+Fig.7 - Evaporation in the water cycle (Photo: ETHZ)[^6]
+</font>
+</figcaption>
-## Soil-Water Evaporation: Further Concepts
+## Example: Soil-Water Evaporation
-<img src=img/FFPM-TurbulentBoundaryLayer.png width="100%">
+<img src=img/FFPM-evapStages.png width="60%">
-<img src=img/FFPM-evapStages.png width="100%">
-[OrEtAl2013](https://doi.org/10.2136/vzj2012.0163)
+<figcaption align = "center">
+<font size = "2">
+Fig.8 - Different evaporation stages (Or et al., 2013)[^4]
+</font>
+</figcaption>
-<img src=img/FFPM-evapReynoldsNum.png width="100%">
+## Example: Simple Evaporation Setup
-# Exercises:
+{style="width: 60%; margin: auto; float: left;"}
+
+<font size = "2">
+ Tab1: Input parameter
+</font>
+
+<font size = "5">
+
+| Parameter | Value |
+|:----------------------------|--------------:|
+| $\textbf{v}_g^{ff}$ [m/s] | (3.5,0)$^T$ |
+| $p_g^{ff}$ [Pa] | 1e5 |
+| $X_g^{w,ff}$ [-] | 0.008 |
+| $T^{ff}$ [K] | 298.15 |
+| $p_g^{pm}$ [Pa] | 1e5 |
+| $S_l^{pm}$ [-] | 0.98 |
+| $T^{pm}$ [K] | 298.15 |
+
+</font>
+
+<figcaption align = "left">
+<font size = "2">
+Fig.9 - Model setup (Fetzer, 2018)[^5]
+</font>
+</figcaption>
+
+
+## Example: Results
+
+
+<figcaption align = "center">
+<font size = "2">
+Fig.10 - Results: Evaporation from a simple setup (Fetzer, 2018)[^5]
+</font>
+</figcaption>
+
+# Exercises
## Exercise: Interface
_Tasks_
-- Change flow direction for a tangetial flow as opposed to the original normal flow.
-- Introduce the beavers joseph tangential flow interface condition.
-- Redevelop the grid and introduce an undulating interface.
-- Change the inflow boundary condition to a velocity profile.
+- Change flow direction for a tangetial flow as opposed to the original normal flow
+- Introduce the beavers joseph tangential flow interface condition
+- Redevelop the grid and introduce an undulating interface
+- Change the inflow boundary condition to a velocity profile
## Exercise: Models
@@ -125,4 +256,39 @@ _Tasks_
- Introduce a Turbulence model to the free-flow domain
- Reduce the free-flow domain by using a symmetry condition at the upper domain boundary
-- Vary grid resolution and perform a qualitative grid convergence test.
+- Vary grid resolution and perform a qualitative grid convergence test
+
+
+
+[^1]:
+<font size = "2">
+ Schneider, M., Gläser, D., Weishaupt, K., Coltman, E., Flemisch, B., Helmig, R., Coupling staggered-grid and vertex-centered finite-volume methods for coupled porous-medium free-flow problems. Journal of Computational Physics. 2023; 112042. https://doi.org/10.1016/j.jcp.2023.112042.
+</font>
+
+[^2]:
+<font size = "2">
+Koch, T, Flemisch, B, Helmig, R, Wiest, R, Obrist, D. A multiscale subvoxel perfusion model to estimate diffusive capillary wall conductivity in multiple sclerosis lesions from perfusion MRI data. Int J Numer Meth Biomed Engng. 2020; 36:e3298. https://doi.org/10.1002/cnm.
+</font>
+
+[^3]:
+<font size = "2">
+ Heck, K., Coltman, E., Schneider, J., & Helmig, R. (2020). Influence of radiation on evaporation rates: A numerical analysis. Water Resources Research, 56, e2020WR027332. https://doi.org/10.1029/2020WR027332
+</font>
+
+[^4]:
+<font size = "2">
+Or, D., Lehmann, P., Shahraeeni, E. and Shokri, N. (2013), Advances in Soil Evaporation Physics—A Review. Vadose Zone Journal, 12: 1-16 vzj2012.0163. https://doi.org/10.2136/vzj2012.0163
+</font>
+
+[^5]:
+<font size = "2">
+Fetzer, Thomas:
+Coupled Free and Porous-Medium Flow Processes Affected by Turbulence and
+Roughness – Models, Concepts and Analysis, Universität Stuttgart. - Stuttgart: Institut für Wasser- und Umweltsystemmodellierung, 2018
+</font>
+
+
+[^6]:
+<font size = "2">
+Or, D. (2023, 31. March). https://emeritus.step.ethz.ch/the-step-group.html
+</font>
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