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**_Definition of geometrical and physical properties_**
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</summary>
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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:
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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:
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- Small (S)
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- Medium (M)
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- Extra-extra Large (XXL)
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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.
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| Parameters / Test-case | **S** | **M** | **XL** |
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|------------------------|-------|-------|--------|
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| ∆x (m.) | 0.001 | 0.0005 | 0.00025 |
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| End physical Time (s.) | 0.5 | 0.5 | 0.5 (0.00625) |
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The test-case XL is used with a number of iterations equal to 100 to reduce the computational time.
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</details>
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<details>
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<summary>
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**_Set-up of linear algebra solvers_**
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</summary>
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The setup for the linear algebra solvers comprises the following:Pressure:
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- 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
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- 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.
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The following Table summarizes the different preconditioner/solver pairs used to solve the pressure equation
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| Method | Preconditioner | Solver |
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|--------|----------------|--------|
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| FOAM-DIC-PCG | Diagonal-based incomplete-Cholesky | Conjugate Gradient |
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![Figure](Lid_driven_cavity-3d/assets/time_3d.png "total time for PISO solver, XL Test case")
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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.
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</details>
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### 3D Lid-driven cavity micro-benchmark
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