VKI Turbine

Unsteady RANS simulations have been achieved for the VKI turbine case. A coarse grid (18 000 nodes, referenced as stp2) was firstly used, but in order to resolve the shocks appropriately, a finer grid has been also designed (72 000 nodes, referenced as stp1). The sonic region and the Von Karman street can be seen on the instantaneous map of the computed Mach number contours on Fig. 1-left.

The time averaged isentropic Mach number distributions along the pressure and the suction sides are shown on Fig. 2. The computational results (lin­ear model of Wilcox, 1993b, and non linear model of Craft et al., 1996) are

Figure 2. VKI turbine. Time averaged isentropic Mach number distribution on the blade

very close to the experimental results. The prediction is very good on the pressure side, and a good agreement is noticeable on the suction side where the numerical distribution is above the experimental points, probably a time – integration effect of the experimental data. The unsteadiness due to the vortex shedding modify mainly the area located downstream of the nearly sonic area (x/c > 0.5), on the suction side.

The stabilization on one specific frequency is difficult to obtain with the non­linear k — и model. It seems that unsteady shock boundary layer interactions was not completely stabilized. All simulated Strouhal numbers are quite close to the experimental value of 0.219, as shown in Table 1.

Table 1. VKI turbine. Strouhal number RANS Iin. RANS non-Iin. Fine grid (stpl) 0.253 0.225 Coarse grid (stp2) 0.231

Results are better on the coarse grid than on the fine grid. Consequently, one needs to be careful when analyzing the numerical results. k — и models give a frequency a little too high, which suggests a too small value of the turbulent kinetic energy.

Figure 3. VKI turbine. Left: Comparison of vortex location. Right: Comparison of vortex density evolution at the center of vortex (fine grid).

Figure 1-right shows the density iso-contours for the experiment and for the linear and non-linear k — и models. The vortex locations and the shock wave structures are close, but the intensity of the numerical results seems higher than the experimental results. The vortex location obtained by experiment is well compared to linear and non-linear k — и model (Fig. 3-left). For the numerical results, the distance between the lower and upper vortex is a little too wide suggesting that the amplitude of the flictuations is too strong near the trailing edge.

The density at the vortex center is compared to the extrema of VKI exper­imental data for linear and non-linear k — и model (Fig. 3-right). The exper­imental results show a symmetric vortex shedding. The linear and non-linear k — и models increase the deficit of density which shows that the fhctuation level is over-predicted as was suggested on Fig. 1-right. For the linear model, the modification is non-symmetric and affects mainly the pressure side of the trailing edge. As for the non-linear k — и model, it predicts correctly the lower vortex intensity, but over-estimates the upper vortex intensity.

Time averaged static pressure at different angular locations around the trail­ing edge is presented in Fig. 4-left. With the linear model, the agreement is reasonable on the pressure side, whereas more discrepancies are visible on the suction side. Otherwise, the non-linear model gives better results on the suc­tion side except at 80 degrees where the shock wave appears. It seems that the computation is not completely stabilized (locally). It will be noticed that the experimental values are almost symmetric, the numerical solution appearing more influenced by the small supersonic regions upstream of the trailing edge. More investigation will be necessary on spatial schemes and limiters. An in­teresting point is to observe the comparison between the time averaged value

Figure 4. VKI turbine. Left: Comparison of the trailing edge static pressure distribution from pressure side (left hand) to suction side (right hand) (fine grid). Right: Comparison of unsteady static pressure fhctuation between numerical results and phase-locked Fourier analysis (solid line) or wavelet experimental analysis (dots).

Figure 5. VKI turbine. Time averaged stagnation temperature (left) and stagnation pressure (right) atX/D=2.5.

and the steady value. The higher level on steady static pressure confirms that the vortex shedding increases the losses.

The excessive amplitude of the deficit given by the linear k — и model is also visible looking the instantaneous static pressure flictuations due to vor­tex shedding in Fig. 4-right. The curve shapes are similar, but the amplitude is higher for the finer grid, particularly on the suction side where the increase of flrctuations represents 40% of the total flrctuation amplitude. This is con­sistent with a sharper description of local gradients. Two decompositions of raw experimental data were done at VKI using phase-locked Fourier analysis (solid line) or wavelet analysis (dots). The amplitude measured using wavelet analysis is in better agreement with the numerical results. This could be ex­plained by a slight random frequency variation during the experimental vortex shedding to which the Fourier analysis is more sensitive.

Total temperature and total pressure flictuations were recorded 2.5 diame­ters downstream the trailing edge (Fig. 5). Once again, the curve shapes are similar, particularly on the pressure side, the amplitude being slightly over pre­dicted.