Validation of the backward facing step test case.

Configuration

The backward facing step flow is a well known benchmark largely studied in the literature. We will compare our work to the ones of Gartling [1], Kim & Moin [2], Sohn [3], and to the experience of Armaly [4], for \( Re=100 \) to \( Re=800 \).

The height of the domain is \( H_d=2m \), the length is \( L=40m \). A step of \( l=4m \) length and \( H=1m \) height is placed on the left bottom corner of the domain. Walls are set at the bottom and top boundaries. Neumann boundary condition is set at the exit. A parabolic poiseuille flow is set at the inlet (see Figure 1).

At \(Re=200\), we set the mean inlet velocity to \(1 m/s\), density to \(1 kg/m^3\).

Figure 1: Backward facing step configuration and schematic flow

Runtime parameters

Regular mesh size varies from \( 200 \times 50 \) to \( 1600 \times 400 \). The second order centered advection scheme is used. First order volume penalization immersed boundary method is used.

Test case list

Label
backward_facing_step_2D.nts

Comments

As noticed by Armaly [5] and verified numerically by Williams, from Re=400 the flow is 3D. This is the reason why discrepancies exist between experimental and 2D numerical values. Results are provided for \(Re=200\) only.

Results

At \(Re=200\), experimental value of \(X_s/H\) is equal to 5, to compare to the numerical value of 4.96.

Figure 2: Backward facing step at Re=200: streamlines and horizontal velocity field

The next table shows a spatial convergence order of one for several domain-averaged quantities.

Mesh	mean kinetic energy	Order	mean_pressure	Order	mean_velocity_u	Order
100	1.88996258e-01	n/a	2.34924172e-01	n/a	5.28397116e-01	n/a
200	1.88342258e-01	n/a	2.38531649e-01	n/a	5.28106612e-01	n/a
400	1.88585374e-01	n/a	2.38116483e-01	n/a	5.27207647e-01	-1.6297
800	1.88712759e-01	+0.9325	2.37951589e-01	+1.3321	5.26762944e-01	+1.0154

Mesh	mean_velocity_v	Order	mean_velocity_magnitude	Order
100	-6.89835746e-03	n/a	5.31635498e-01	n/a
200	-6.72797904e-03	n/a	5.32195204e-01	n/a
400	-6.64242566e-03	+0.9938	5.31877883e-01	n/a
800	-6.60286005e-03	+1.1126	5.31742497e-01	+1.2289

References

[1] Gartling D.K., A test problem for outflow boundary conditions – flow over a backward- facing step, International Journal of Numerical Methods in Fluids, 11, 953-967, 1990.

[2] Kim J. & Moin P., Application of a fractional-step method to incompressible Navier- Stokes equations, Journal of Computational Physics, 59, 308-323, 1985.

[3] Sohn J., Evaluation of FIDAP on some classical laminar and turbulent benchmarks, International Journal of Numerical Methods in Fluids, 8, 1469-1490, 1988.

[4] Armaly P.H., Durst F., Pereira J.C.F. & Schonung F., Experimental and theoretical investigation of backward-facing step flow, Jounal of Fluid Mechanics ,127, 473-496, 1983.

[5] Williams P.T. & Baker A.J., Numerical simulations of laminar flow over a 3D backward- facing step, International Journal for Numerical Methods in Fluids, 24, 1159-11831 1997.