# UFR 3-15 Test Case

# 2D flow over backward facing step

Underlying Flow Regime 3-15 © copyright ERCOFTAC 2004

# Test Case

## Brief description of the study test case

The geometry of the problem is shown in Figure 1.

*Figure 1: geometric configuration (not to scale)*

The first case correspond to the experimental measures performed by Jovic and Driver (1995) and there is also the case calculated by DNS by Le and Moin (1997). The Reynolds number based on the step height (h) is 5000, and the expansion ratio is 1.2. The domain height is 5h and 6h before and after the expansion respectively. This gives the expansion ratio of 1.2. The domain length in the experiment is given in the following section, and in the DNS was taken to be from 10h prior to the step to 20h after the expansion. The inlet velocity profile was that of a flow over a constant pressure boundary layer. This profile was calculated by Spalart (1988) and can be found in the
ERCOFTAC Classic database
here. The results found by Jovic and Driver (1995) and Le et al. (1997) include the velocity and turbulent quantities profiles at *x/*h = -3.0, 4.0, 6.0, 10.0, 15.0 and 19.0 as well as the friction and pressure coefficients along the bottom wall.

The second case corresponds to the experiment of Driver and Seegmiller (1985). The Reynolds number (based on the step height) is 37500, and the domain is 8h prior to the step and 9h after it so the expansion ratio is 1.125. The inlet was chosen to be fully turbulent with a developed boundary layer at a momentum thickness Reynolds number of 5000. The results presented by Driver and Seegmiller (1985) include the velocity and turbulent quantities profiles at *x/*h = -4.0, -2.0, -1.0, 1.0, 1.5, 2.0, 2.5, 3.0, 4.0, 5.0, 5.5, 6.0, 6.5, 7.0, 8.0, 10.0, 12.0, 14.0, 16.0, 20.0 and 32.0 as well as the friction and pressure coefficients along the bottom wall.

## Test Case Experiments

In the experiments performed by Jovic and Driver (1995), they have measured the flow over a BFS with a fully developed turbulent boundary layer as inlet. The test section is symmetrical with respect to the centre line forming a double-sided expansion. The channel height upstream is 96 mm and downstream 115 mm. The channel aspect ratio was 31 to avoid 3D effects.

*Figure 2: experimental set-up of Jovic and Driver (1995)*

A wind tunnel with the fan at the exit was used to generate a fully developed turbulent boundary layer over a flat plate to a backward facing step. The opposite wall was a mirror image plate and step to form a plane of symmetry at the tunnel centreline and an expansion ratio of 1.2. Figure 2 sketches the situation. As mentioned before, the experiment was made at Re = 5000 calculated using the centre line velocity ahead of the step which was approximately 7.7 m/s. The main parameters of the inflow boundary layer at x/h=-3.05 are:

δ = 11.5 mm ; δ* = 1.7 mm ; θ = 1.2 mm ; C_{f} = 0.0049 ; R_{q} = 610

where δ is the boundary layer thickness δ* the displacement thickness and θ the momentum thickness, C_{f} the skin friction coefficient, and R_{q} the Reynolds number based on the momentum thickness. The measured quantities were the mean flow velocities as well as the mean turbulent stresses. Also the local wall shear stress and the surface pressures were measured. The instrumentation included a LDV with frequency shifting for directional resolution, a laser interferometer for oil flow measurement of skin friction and surface pressure taps. The uncertainties were 2% for the mean velocities and 15% for the Reynolds stresses.

Driver and Seegmiller (1985) performed measurements of the flow over a BFS with a turbulent boundary layer at the inlet. It was generated using a low speed wind tunnel. The wall opposite to the step had an inclination of 0 or 6 degrees as shown in Figure 3. The experiment was performed with a velocity of 44.2 m/s at the centre of the channel giving the Reynolds number of 37500.

The inlet boundary layer parameters at x/h=-4.0 were:

δ = 19 mm ; δ* = 2.7 mm ; θ = 1.9 mm; C_{f} = 0.0029 ; R_{q} = 5000

*Figure 3: experimental setup of Driver and Seegmiller (1985)*

The measured quantities were the mean flow velocities as well as the mean turbulent stress, local wall shear stress and the surface pressures. The instrumentation is similar to the JD experiment. The uncertainties were estimated to be 1.5 % for the mean velocities and 12% for the Reynolds stresses.

## CFD Methods

As mentioned before, there exist many studies of this problem. Most of them have taken the experiments and DNS results for developing RANS turbulence models.

Driver and Seegmiller (1985) presented, together with the experiments, CFD calculations that were carried out with k-ε and ASM models. They also tested a modification in the production term of the P equation proposed by Sindir and Launder. They studied also the dependence of the flow with respect to the top wall deflection angle.

Rodi (1991) collected the experience with two layer models, combining k-ε with a one equation model near the wall. Different criteria for matching the models, in order to avoid the need of matching at fixed and preselected grid points, were analysed together with damping functions for the viscosity and length scale prescription. The predictions were compared to the results of Driver and Seegmiller (1985).

Menter (1994) developed the BSL and SST models and tested them, comparing the predictions against the DS experimental results. The performance of these models (based on the k-ω one) was compared to that of the standard k-ω and to the JL k-ε model. In another work, Menter (1996), also developed a transformation from the k-ε model to a one equation model for the turbulent viscosity.

Non linear equation models performance were reported in Apsley and Leschziner (1998) where the quadratic k-ε model of Speziale, the cubic k-ε models of Lien et al. and their own model, were compared against the standard k-ε.

Based on the fact that the velocity scale provided by the k-ε model (given by k) is not appropriate for turbulent transport towards the wall, Durbin (1995) proposed the k-ε-χ^{2} model based on an elliptic relaxation equation that allows the derivation of low Reynolds number models from high Reynolds ones. Its performance was tested comparing to the experimental results of both, JD and DS. This model was then implemented in an industrial CFD code by Manceau and Parnaix (1999).

LES calculations were performed by Akselvol and Moin (1993) using the dynamic model of Germano and the modifications proposed by Lilly (using a time dependent Smagorinsky constant). They used a staggered grid with second order finite difference operators in space solving the system with a fractional step method. The grid used was stretched in the normal-to-the-wall direction (and uniform in the others) using a non slip condition in the lower wall. The results were compared to DNS and the JD experimental results (although those experimental results were previous ones, not those found in the databases, with some difference in the aspect ratio, that causes some change in the recirculation length).

Also Chan and Mittal (1996) calculated the flow corresponding to the JD case, using the standard Smagorinsky model with a Van Driest damping function for the subgrid modelling. They implemented this model on a spectral element solver.

Second moment closure modelling is presented by Hanjalic and Jakirlic (1998). A second moment low Re model was developed and compared to the LRRG and SSG models. They also included a modification to the e equation: an invariant reformulation of a productive term originally proposed by Yap (1987). Extensive comparisons were performed using six different experimental (including JD, DS) and DNS results. They used a finite volume code treating convection by a second order linear upwind differencing scheme and also tested first order upwind obtaining similar results.

The DNS simulation of Le, Moin and Kim (1997), corresponding to the JD test case, was used in Parneix and Durbin (1996) and Parneix et al. (1996) to develop second moment turbulence models. They calibrated models performing differential tests, which consist in introducing the DNS data into the equations, evaluating the differences, a technique introduced by Rodi and Mansour (1993), and also in solving for an individual variable while keeping the others to the DNS value.

Other studies comparing to these results (JD, DS or the DNS) are those of Michelassi et al. (1996), Perot and Moin (1996), Poroseva and Iaccarino (2001), Manceau (2000), and Ross and Larock (1997).

Cruz and Silva Freire (2002) calculated the heat transfer using two equation models and wall laws for the velocity and also for the temperature (they proposed a correction that depends on the pressure gradient at the wall), but they did not compare them against experimental data.

© copyright ERCOFTAC 2004

Contributors: Arnau Duran - CIMNE