Backward Step
1. Description

Problem summary :
Let us consider the backwardfacing 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 \(Re=400\).

Associated EDP
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 NavierStokes equations by using Picard iterations.
1.1. Boundary conditions

Boundary conditions formulation

a noflow 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:

and
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 subproblems, 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 submatrix. (We used a relative tolerance of 10^{6} for each subproblem).

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
As for the starting vector for the linearized iteration it is set to zero and the stopping criterion is
where \(\mathbf{r}^{(k)}\) is the residual of the linear system and \(S^{(m)}\) is the lefthand 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
6. Bibliography

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

[Stefano] G De Stefano, FM Denaro, and G Riccardi. Analysis of 3 d backwardfacing step incompressible flows via a local averagebased 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 backwardfacing step,part i: High reynolds number solutions. Computers & Fluids, 37(6):633–655, 2008.