# Backward Step

We describe the benchmark proposed in [Armaly],[Erturk] and [Stefano]

## 1. Description

• Problem summary :

Let us consider the backward-facing step benchmark illustrated in Figure 1, which is an example of an inflow/outflow problem. The inflow is at $x=-1$ and the outflow is at $x=5$ for $Re=10$ and $Re=100$, at $x=10$ for $Re=200$ and at $x=20$ for $Re=400$.

Figure 1. Step geometry: computational domain
• Associated EDP

$-\nu\Delta\mathbf{u} +\rho (\mathbf{u} \cdot \nabla \mathbf{u}) \mathbf{u} +\nabla p = \mathbf{0}, \mbox{ in } \Omega \\ \nabla \cdot \mathbf{u} = 0, \mbox{ in } \Omega$

We choose an implicit treatment of the convective term and a non symmetric formulation of the deformation tensor. We will deal with the nonlinear system arising from the discrete Navier-Stokes equations by using Picard iterations.

### 1.1. Boundary conditions

• Boundary conditions formulation

• a no-flow condition is imposed on the wall

• a Newmann condition is applied at the outflow boundary

• A Poiseuille flow profile is imposed on the inflow boundary. The 2D and 3D Poiseuille profiles are defined respectively as follow:

u_x = 6y(1-y)\\ u_y=0

and

$u_x=24y(1-y)z(1-z)\\ u_y=0\\ u_z=0$

### 1.2. Initial conditions

• Initial condition: The initial iterate (\mathbf{u}_0, p_0) is obtained by solving the corresponding discrete Stokes problem.

## 2. Inputs

• Parameter set definition

 Name Description Nominal Value $D$ height of the step 2 $L$ length of the step { 5, 10, 20 } $\rho$ density of the fluid 1 $\nu$ kinematic viscosity { 0.2, 0.1, 0.01, 0.005 } Re Reynolds number $\quad \quad \frac{2}{\nu}$ { 10, 100, 200, 400 }
• Solver and preconditioner used:

• Gmsh: mesh generation

• Metis: partitioner

• Paraview: post process

• PCD: preconditioner (GAMG for A_p and M_p sub-problems, as for F_u we coupled Fieldsplit with block Jacobi. For each components of F_u we applied a GAMG preconditioner for Re=10, 100 and Re=200. As for Re=400 we used the DD method GASM with LU in the subdomains for the components of F_u sub-matrix. (We used a relative tolerance of 10^{-6} for each sub-problem).

• GCR: solver

The stopping criterion of the nonlinear iteration is when the vector Euclidean norm of the nonlinear residual reaches a relative error of 10^{-6}, that is

$\| \begin{pmatrix} \mathbf{f} - \left( F_\mathbf{u}(\mathbf{u}^{m})\mathbf{u}^{(m)} + B^Tp^{(m)}\right)\\ \mathbf{g} - B\mathbf{u}^{(m)}\end{pmatrix} \| \leq 10^{-6} \| \begin{pmatrix} \mathbf{f}\\ \mathbf{g}\end{pmatrix}\|$

As for the starting vector for the linearized iteration it is set to zero and the stopping criterion is

$\|\mathbf{r}^{(k)}\| \leq 10^{-6}\|S^{(m)}\| ,$

where $\mathbf{r}^{(k)}$ is the residual of the linear system and $S^{(m)}$ is the left-hand side residual associated with the final nonlinear system.

## 3. Discretization

The geometry was carried out using Gmsh, and the partitioning was done using Metis. The mesh characteristics and the total number of DOF per configuration is reported in table 2

Figure 2. Total number of DOF for the 2D and 3D step geometry for $L=5$ , $L=10$ and $L=20$ with $\mathbb{P}_2\mathbb{P}_1$, $\mathbb{P}_3\mathbb{P}_2$ and a $\mathbb{P}_4\mathbb{P}_3$ configurations.

## 6. Bibliography

• [Armaly] Bassem F Armaly, F Durst, JCF Pereira, and B Schönung. Experimental and theoretical investigation of backward-facing step flow. Journal of Fluid Mechanics, 127:473–496, 1983.

• [Stefano] G De Stefano, FM Denaro, and G Riccardi. Analysis of 3 d backward-facing step incompressible flows via a local average-based numerical procedure. International journal for numerical methods in fluids, 28(7):1073–1091, 1998.

• [Erturk] Ercan Erturk. Numerical solutions of 2D steady incompressible flow over a backward-facing step,part i: High reynolds number solutions. Computers & Fluids, 37(6):633–655, 2008.