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 \(Re=400\).

FDA
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

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 }

  • 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

FDA
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.

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.