The aim of this exercise is to get familiar with the set up of coupled free flow/porous medium flow problems.
## Problem set-up
The model domain consists of two non-overlapping subdomains.
Free flow is modeled in the upper subdomain, while the lower subdomain models a flow in a porous medium.
Both single-phase flow and two-phase flow will be considered in the porous domain.

### 0. Getting familiar with the code
* Navigate to the directory `exercises/exercise-coupling-ff-pm`
There are three sub folders: `interface`, `models` and `turbulence`.
The problem-related files for this exercise are:
* Three __main files__ for the three sub-tasks :`ex_interface_coupling_ff-pm.cc`, `ex2_coupling_ff-pm.cc`, `ex2_coupling_ff-pm.cc`,
* Three __free flow problem files__: `ex_interface_ffproblem.hh`, `ex2_ffproblem.hh`, `ex3_ffproblem.hh`
* Three __porous medium flow problem files__: `ex_interface_pmproblem.hh`, `ex2_pmproblem.hh`, `ex3_pmproblem.hh`
* The __input files__: ...
* The __spatial parameters files__: `1pspatialparams.hh`, `2pspatialparams.hh`
In the main file, `TypeTags` for both submodels are defined. The same applies for types such as `GridManager`, `FVGridGeometry`, `Problem`, etc.. Since we use a monolithic coupling scheme, there is only one `Assembler` and one
`NewtonSolver`.
The problem files very much look like "regular", uncoupled ones with the exception that they hold a pointer to the `CouplingManager` which allows to evaluate the coupling conditions and to exchange information between the coupled models. The coupling conditions are realized technically in terms of boundary condition. For instance, in lines 178 and 179
in `ex_interface_ffproblem.hh`, `couplingNeumann` boundary conditions are set, which means that the free flow models evaluates the
mass and momentum fluxes coming from the porous domain and uses these values as boundary conditions at the interface.
Note the certain checks are performed when combining different models, e.g., the fluid system has to be the same for both domains.
We will use a staggered grid for the free flow and a cell-centered finite volume method for the porous medium.
Keep in mind that the staggered grid implementation distinguishes between face variables (velocity components) and
cell center variables (all other variables), therefore in some cases either the `stokesCellCenterIdx`
or the `stokesFaceIdx` is used respectively, while for the porous medium all variables can be accessed with `darcyIdx`.
The main part of the Stokes/Darcy coupling is implemented in the folder `dumux/multidomain/boundary/stokesdarcy`.
__Task__:
Take a closer look at the Stokes/Darcy coupling files and try to answer the following questions before moving to the next part of the exercise:
* Where are the elements at the coupling interface paired with their respective coupling partner elements of the other domain?
* Where are the coupling conditions implemented and evaluated?
* You will find four implementations of the `massCouplingCondition`. What are the differences between them?
* Which data is stored and updated regularly such that it is accessible from the respective other domain?
* How can we access volume variables like density and pressure from the respective other domain?
* ...
### 1. Changing the interface
In this part of the exercise, a simple coupled system consisting of a one-phase (1p) free flow and a one-phase flow in a porous medium is set up. Both subproblems have no-flow boundaries at the sides.
A velocity profile is set on the upper free flow boundary, which leads to a vertical flow into the porous medium.
* We will first change the flow direction such that the free flow is parallel to the porous medium.
* Afterwards, the Beavers-Joseph-Saffman condition will be used as an interface condition for the tangential momentum transfer.
* Last, we change the flat interface between the two domains to a wave-shaped one.
__Tasks__:
Open the file `ex_interface_ffproblem.hh` and navigate to line 155, where the types of boundary condition are set.
Instead of applying a fixed velocity profile at the top of the domain, we want to use fixed pressure boundary conditions
at the left and right side of the free flow domain, while the top represents an impermeable wall.
__Change the flow direction__
Set a Dirichlet boundary condition for the pressure at the left and ride side of the domain:
at the position where the coupling boundary conditions are set in `ex_interface_ffproblem.hh`.
To check if the simulation behaves as expected, we can compare the velocity profile $`v_x(y)`$ with the analytical solution provided by [Beavers and Joseph (1967)](https://doi.org/10.1017/S0022112067001375).
in `ex_interface_coupling_ff-pm.cc`. After re-compiling and re-running the executable, we should be able to see also
the analytical solution of $`v_x`$ on the free flow domain. Play around with the grid resolution to see how that affects the velocity profile.
__Cange shape of interface__
Now we want to include a non-flat interface between the two domains. We use `dune-subgrid` to construct
two grids for the two domains from one common host grid. Comment out lines 93-103 in `ex_interface_coupling_ff-pm.cc` and comment lines 112-147 in the same file. This will instantiate a host grid and define two helper lambda functions that are used to choose elements from to host grid for the respective sub grid. In the given case,
the domain is split in two haves, separated by a sinusoidal interface.