Baseline Simulations

First, the flow is simulated without excitation. In a very simular case Franke et al. [13] could obtain convincing results by using the LLR $k$-$\omega$ model of Rung and Thiele [11] that are significantly better compared to standard $k$-$\varepsilon$ and $k$-$\omega$ formulations. Their simulation results strongly depend on the transition position on the flap as well as on the turbulence model. In the fully turbulent case, the flow remains attached to the flap surface whereas tripped transition leads to flow separation.

Figure: Pressure distribution on main airfoil and flap for different turbulence models without excitation

Figure: $u$-velocity field of the separated flow around the flap without excitation

Figure: Spectrum of lift coefficient for different models without excitation

In the present investigation the flow is characterized by massive separation which is well predicted by all turbulence models. The prediction of pressure is also in fairly good agreement with the experimental data. Most promising results were obtained by using the $k$-$\omega$ models (Fig. ). The separation point for this case is located in the laminar part of the flap boundary-layer slightly upstream of the excitation slot (Fig. ), such that a large reverse flow region forms downstream. The two-equation models provide a correct prediction of the flap separation. In the case of the Spalart-Allmaras model however, the flap separation position is located too close to the leading edge, resulting in a slightly underpredicted pressure in the reverse flow region. Here the flow is almost steady and compared to the lift coefficient spectra of the two-equation models, no dominant frequencies, that might indicate vortex-shedding, occur (Fig. ). Results of the two-equation models, however, show a strong amplitude for a non-dimensional frequency of $F^+=F\frac{c_k}{u_\infty} \approx 0.5$. These results are in agreement with those obtained by Bunge [14] reporting a very low unsteadyness in the case of the SA-model for oscillating airfoil flows. In all following investigations the LLR $k$-$\omega$ model is used.

Compared to the experimental results the suction peak is overpredicted in the numerical simulation. This is caused by the direction of the flow-vector behind the main airfoil trailing edge. In the experiments strong three-dimensional effects appear that are neglected in the simulation.

Figure: Pressure distribution for different time stepping

To correctly capture all unsteady flow features a proper time-resolution is required and the simulation has to last at least several periods to avoid launching effects. In the present case, all simulations consider an unsteady flow field with a non-dimensional time stepping of $\Delta t^+=\frac{\Delta t \ u_{\infty}}{c_k}=0.01$. One period of vortex-shedding is resolved by 230 time steps in this case. Results of a finer time-stepping with 595 steps per period ( $\Delta t^+=0.004$) do not show significant differences to the present results (Fig. ).

Results of numerical simulations often strongly depend on the intensity of free-stream turbulence. In the present study the influence of $Tu_{in}$ in the range of $0.2\%<Tu_{in}<2\%$ (wind-tunnel quantities) is of minor importance.