Flow over a backward-facing step

The second test case contains a turbulent separating flow over a backward-facing step which was studied experimentally by Jovic and Driver (1994). The height (h) of the step is 0.98 cm. The Reynolds number used in the experiment was Reh = 5000 based on the step height and free-stream velocity, Uo = 7.72 m/s. The computational domain layout is shown on Fig. 2.10. It consists of a streamwise length Lx = 30h, vertical height Ly = 6h, and spanwise width Lz = 4h. The length of the upstream inlet channel is chosen to be 10h

in order to limit the effect of inlet turbulence on the separated region downstream of the sudden expansion.

Large Eddy Simulations were performed using 400 x 140 x 40 grid points along the streamwise, vertical, and spanwise directions, respectively. A pressure outlet (convective condition) was imposed at the downstream location. The flow in the spanwise direction is assumed to be statistically homogeneous (Hungle et al., 1997) and therefore, a periodic boundary condition was used. A no-stress (free slip) wall was applied at the top boundary of the computational domain and a no-slip condition was used for the bottom walls. The inflow velocity field is taken from an instantaneous simulation of a fully developed plane channel flow (Barri et al., 2009) of finite length where periodic boundary conditions were used in the streamwise direction.

Figure 2.10: The computational domain of the backward-facing step (Hungle et al., 1997)

The simulations were started from a stationary flow field and were allowed to evolve to a statistical steady state for a total computational time of ∼ 1600h/Uo. The time step of each simulation was adjusted by keeping the CFL number (Courant et al., 1928) below 0.5. Statistical averaging of the individual flow fields were performed from 1600h/Uo to 4000h/Uo which is approximately 53 ‘flow-through’ times.

Comparisons of the time averaged streamwise velocity profiles at four different locations (x/h = -3 in the entrance, x = 4 in the recirculation, x/h = 6 close to the flow reattachement

Figure 2.11: Time averaged streamwise velocity profiles

point, and x/h = 10 in the recovery region) between the LES results and the experimental data of Jovic and Driver (1994) are given on Fig. 2.11. Overall, all LES schemes give velocity profiles which are in a good agreement with the experimental data. In the re- attachment region (x/h = 6), it is observed that both the SM, and SgsKEM schemes are not able to capture the mean streamwise velocities close to the bottom wall which is mainly due to the limitations of a constant model coefficient in these two schemes. A transverse flow with negative streamwise velocity is also observed from the profiles of both SM and SgsKEM at x/h = 6 which is an indication of overestimations in the re-attachment lengths. The mean turbulence fields in terms of the velocity fluctuations for the streamwise (< u0u0 >) profiles are given on Fig. 2.12. The comparisons were performed at the same four

Figure 2.12: Mean streamwise turbulent profiles

locations. The numerical < u0u0 > predictions from the DSM are in good agreement with the profiles observed with the experimental data both in the re-circulation and recovery regions. Downstream of the sudden expansion, the DMM is found to be dissipative by underpredicting the streamwise velocity fluctuations, however in the entrance region (x/h = -3), both the DMM and DSM are shown to be better for the < u0u0 > profiles. The peak < u0u0 > values predicted by the SM and SgsKEM before the sudden expansion (x/h = -3) are lower than the values reported by the experimental data which is also consistent to the values found for these two LES models in the previous section for the wall bounded open channel flow. Downstream of the step, a high rate of velocity damping is

observed for the constant model coefficient LES models (SM and SgsKEM). This high rate of velocity damping gives strong and long re-circulation regions with an overestimation of the streamwise turbulence fields (Fig. 2.12).

Figure 2.13: Mean wall normal turbulent profiles

Figures 2.13, and 2.14 show the mean representation of the wall normal (< v0v0 >), and cross-stress (< u0v0 >) turbulence profiles at the four locations respectively. There are some minor differences in the < v0v0 > and < u0v0 > profiles from the DSM scheme, but overall it reproduces good turbulent statistics as the experimental data. A slighter underestimation is observed for the two turbulent quantities from the DMM both in the re-circulation and recovery regions. The SM and SgsKEM schemes overpredict the peak values in regions

below the sudden expansion. A similar overprediction is also observed for the streamwise turbulent intensities for these two schemes as noted earlier. These comparisons clearly show the advantageous of dynamic model coefficients, and limitations of a constant model coefficient SGS models for resolving complex turbulent flows. It is important to note that using an inlet bounday condition from a periodic boundary simulation of amplitude length is quite adequate to get good predictions of mean flow and turbulent fiields both upstream and downstream of the step. According to Aider and Danet (2006), replication of the inlet boundary condition is essential to get good profiles of the hydrodynamic variables below the sudden expansion.

Table 2.2: Comparisons of reattachment lengths (Xr/h)

Method Exp. SgsKEM SM DSM DMM

Uw = 0 6 7 6.7 6.1 5.6

Streamline 6 6.8 6.6 5.9 5.4

Figure 2.15: Streamline and instantaneous vorticity contors from SM and SgsKEM

Another important parameter for evaluating the ability of various turbulence models in simulating complex flows with re-circulation and flow separation is the flow reattachment location (Ghosal et al., 1995; Hungle et al., 1997). The reattachment length (Xr) is the distance from the step to the point of zero wall shear stress or streamwise velocity. It is known to be one of the key parameters to test the numerical accuracy of SGS models in addition to the mean flow and turbulent fields. We have used two methods to calculate the flow re-attachment location as reported by Hungle et al. (1997); i) the longitudinal distance where the mean streamwise velocity is zero at the first grid point normal to the wall; and ii) the location at which mean streamlines touch the lower wall after the sudden expansion (Fig. 2.15 and 2.16). Tab. 2.2 shows the comparisons of the flow reattachment locations obtained from different LES models. Jovic and Driver (1994) reported a reattachment length of 6 step heights downstream of the step which is close to the values estimated in the current simulation by the DSM scheme. As it can be seen from the table, and the streamline plots,

Figure 2.16: Streamline and instantaneous vorticity contors from DSM and DMM the SM, and SgsKEM schemes tend to overestimate the flow reattachment location which can also explained by the turbulence fields from the two models as discussed previously. An increase in the reattachment lengths and delay in the transition of the shear layer can possibily be due to the absence of longitudinal turbulent vortices in the recirculation region (Fig. 2.15).