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. 2D Geometry

The mathematical model reads as follows:

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

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

Table 1. Fixed and Variable Input Parameters
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 }

Mesh generation: Gmsh
Partitioner: Metis
PostProcessing: Paraview
Preconditioner: PCD (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).
Solver: GCR

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

4. Results

5. Conclusion

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.