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@@ -26,6 +26,85 @@ Key reasons for using the lid-driven cavity (referred to as "cavity" in the foll
 
 ### 3D Lid-driven cavity benchmark
 
+The first test case chosen is the 3-D version of the [Lid-driven cavity flow tutorial](https://www.openfoam.com/documentation/tutorial-guide/tutorialse2.php).
+
+This test case has simple geometry and boundary conditions, involving transient, isothermal, incompressible laminar flow in a three-dimensional box domain. The _icoFoam_ solver is used in such test-case.
+
+It is intended to stress test the linear algebra solver, most of the time being spent in the pressure equation.
+
+**_Definition of geometrical and physical properties_**
+
+In this simple geometry, all the boundaries of the box are walls. The top wall moves in the x-direction at the speed of 1 m/s while the other five are stationary. Three different sizes have been selected:
+
+- Small (S)
+- Medium (M)
+- Extra-Large (XL)
+- Extra-extra Large (XXL)
+
+The following table shows the geometrical and physical properties of the different test cases, which have an increasing number of cells: 1 million (m) (S), 8 m (M) and 64 m (XL) of cells, obtained by halving ∆x when moving from the smaller to the bigger test-case. The Courant number Co=(U ∆t)/∆x is kept under the stability limit, and it is halved when moving to bigger test cases. The time step ∆t is reduced proportionally to Co and ∆x, by 4 times. The physical time to reach a steady state in laminar flow is T= 0.5.
+| Parameters / Test-case | **S** | **M** | **XL** |
+|------------------------|-------|-------|--------|
+| ∆x (m.) | 0.001 | 0.0005 | 0.00025 |
+| N. of cells tot. (m) | 1 | 8 | 64 |
+| N. of cells lin. (m) | 100 | 200 | 400 |
+| kinematic viscosity ν (m^2/s) | 0.01 | 0.01 | 0.01 |
+| d (m) | 0.1 | 0.1 | 0.1 |
+| Co | 1 | 0.5 | 0.25 |
+| ∆t (s) | 0.001 | 0.00025 | 0.0000625 |
+| Reynolds number | 10 | 10 | 10 |
+| U (m/s) | 1 | 1 | 1 |
+| n. of iter. | 500 | 2000 | 8000 (100) |
+| End physical Time (s.) | 0.5 | 0.5 | 0.5 (0.00625) |
+
+The test-case XL is used with a number of iterations equal to 100 to reduce the computational time.
+
+**_Set-up of linear algebra solvers_**
+
+The setup for the linear algebra solvers comprises the following:
+
+Pressure:
+
+- FOAM-DIC-PCG: is an OpenFOAM iterative solver that combines a Diagonal-based Incomplete-Cholesky (DIC) preconditioner with a Conjugate Gradient (PCG) solver. DIC is a common preconditioner in OpenFOAM for its easy configuration and high efficiency
+- FOAM-GAMG-PCG: is an OpenFOAM iterative solver that uses a Conjugate Gradient (CG) accelerated by a generalized geometric-algebraic multigrid (GAMG) method that supports both geometrical and algebraic multigrid.
+- PETSc-ICC-CG(\*): is the PETSc counterpart of the FOAM-DIC-PCG method. ICC is the Incomplete Cholesky factorization implementation of PETSc
+- PETSc-AMG-CG(\*): is the PETSc counterpart of the FOAM-GAMG-PCG method. In this solver, we use an algebraic multigrid method (**_BoomerAMG_**), provided by the third-party library of PETSc named Hypre.
+- PETSc-AMG-CG(\*) + caching: as above but the matrix is converted only at the beginning and cached together with the preconditioner for all the time steps. In this particular case of the constant-coefficients matrix, the numerical solution is equivalent to the case without caching.
+
+(\*) For the PETSc solvers, the \[PETSc4FOAM library\]\[readme petscfoam\] is used for embedding PETSc and its external dependencies (i.e. Hypre) into arbitrary OpenFOAM simulations.
+
+Velocity:
+
+- The solver/preconditioner pair for the momentum equation is kept fixed by using a DILU-PBiCGStab that is a combination of Diagonal Incomplete LU (asymmetric) factorization for the preconditioner with a Stabilized Preconditioned (bi-)conjugate gradient solver.
+
+The following Table summarizes the different preconditioner/solver pairs used to solve the pressure equation
+| Method | Preconditioner | Solver |
+|--------|----------------|--------|
+| FOAM-DIC-PCG | Diagonal-based incomplete-Cholesky | Conjugate Gradient |
+| FOAM-GAMG-PCG | Geometric-algebraic multigrid | Conjugate Gradient |
+| PETSc-ICC-CG | Incomplete-Cholesky | Conjugate Gradient |
+| PETSc-AMG-CG | Classic algebraic multigrid (BoomerAMG) | Conjugate Gradient |
+
+The iterative method can be run using two different convergence criteria:
+
+1. `fixedITER`: In this case, the computational load is fixed, when running with the given solver. It is NOT representative of the real setup. Used only for preliminary tests in the development phase or stress HPC architectures. This case is useful for comparing different hardware configurations by keeping constant the computational load.
+2. `fixedNORM`: in this case, the exit norm is fixed.
+
+The two different options are selected by choosing one of the two configurations files, where the keyword is selected according to the previous Table:
+
+1. `system/fvSolution.<Method>.fixedITER`, with a constant number of iterations per time step for the pressure solver, different according to the test-case size. The velocity solver is set to 5 iterations per time step for the different test cases.
+2. `system/fvSolution.<Method>.fixedNORM`, with a fixed exit norm value of 10^{-4} for the pressure solver.
+
+To use one of the selected methods copy/rename the chosen file in `fvSolution` or use the symbolic link of the unix environment. For example,
+
+```
+cd system
+ln -s fvSolution.<Method>.fixedITER fvSolution
+```
+
+![Figure](Lid_driven_cavity-3d/assets/time_3d.png "total time for PISO solver, XL Test case")
+
+This figure shows the total time for solving PISO with different preconditioner/solver pairs reported in the Table above, for the XL test-case, with fixed exit norm configuration.
+
 ### 3D Lid-driven cavity micro-benchmark
 
 1. credits to @Serge.