Rod test-case

The flow past a circular rod is an interesting academic study about vortex shedding. It will be used to pursue a deeper analysis, taking advantage of the extensive experimental database. Three URANS k — и turbulence models will be compared, in order to estimate the infhence of this parameter. Also, a large-eddy simulation (LES) has been carried-out. It enables an interesting comparison and creates an outlook.

4 Description of the confi guration. In the present study, the ft>w past the rod is characterized by a Reynolds number of Red = 48,000 [the infbw velocity is U ж = 72 m/s and the rod diameter: d = 0.01 m]. Accord­ing to the experiments, the vortex shedding occurs at the Strouhal number: St = fd/Ucv ~ 0.2, and the transition to turbulence takes place in the shear layer, just after the separation (sub-critical regime). As a consequence, the wake is fully turbulent, though dominated by the von Karman vortices. This ft>w topology is sufficiently complex to be compared to practical applications.

The results analyzed in this paper are extracted from a larger configuration previously used to study the emission of broadband noise. This configuration involves a symmetric airfoil (chord length: c = 10 d) located in the wake of a rod, one chord length downstream. We are here concerned with the vortex shedding, so only the flow past the rod will be analyzed. The studies have shown that the airfoil has nearly no influence on the rod because of the distance between the two bodies. Information about the study on the whole rod-airfoil configuration can be found in references Jacob et al., 2002 and Boudet, 2003.

5 Numerical characteristics. Figure 6 presents the overall rod – airfoil computational domain and a view of the mesh near the rod. The URANS mesh is 2D and involves 197 points around the circumference, with the first grid line located at У + < 8 from the wall. A 3D mesh is not necessary be­cause URANS only features the mean components of the ft>w, which are bi­dimensional in this case.

The LES mesh is the same, but duplicated along span to cover 3 diameters (31 points), in order to simulate the tri-dimensionality induced by the turbu­lence. In the turbulent region, the resolution is characterized by y + < 3 per-

Figure 6. Rod test-case. Left: Rod-airfoil computational domain (7 blocks). R/ght: mesh near the rod (1pt/2).

pendicularly to the wall, x+ < 50 tangentially to the curvature of the wall, and z+ < 200 along the span.

Three URANS computations were carried-out, using respectively three k — и models: the linear model of Wilcox, 1993b, the low-Reynolds number model of Wilcox, 1993a, and the non-linear model of Shih et al., 1995. A no-slip condition is imposed at the walls, and a non-refecting condition reduces the spurious confinement at the outer boundaries. The computation was led to convergence for 10 aerodynamic cycles, then it was saved 100 times per cycle for 18 cycles.

The LES computation used the auto-adaptive model of Casalino et al., 2003, to estimate the subgrid scale viscosity. As for URANS, no-slip and non­refecting conditions are imposed at the relevant boundaries, and a slip condi­tion is applied to the limiting plane in the spanwise direction. The computation was led to convergence for 6 aerodynamic cycles (initial field: URANS), then it was recorded 100 times per cycle for 18 cycles.

4.0.6 Results. The vortex shedding occurs naturally for all the computa­tions. However, as mentioned in table 2, URANS overestimates the frequency: the Strouhal number is St = 0.24 for the three k — и models, whereas the experimental value is St = 0.20. LES performs better, it predicts the experi­mental value.

Table 2 also presents the integral forces on the rod: the mean drag (< CD >), the fhctuating drag (C D), and the fhctuating lift (C L). The lin­ear URANS model underestimates these forces. As a comparison, the low – Reynolds number model performs slightly better. The non-linear model is fi-

Table 2. Rod test-case. Vortex shedding Strouhal number and aerodynamic forces on the rod – (measurements: Gerrard, 1961, Achenbach, 1968, Cantwell and Coles, 1983, Szepessy and Bearman, 1992). RANS lin. RANS low-Re RANS non-lin. LES Measurements St 0.24 0.24 0.24 0.19 ~ 0.20 < Cd > 0.79 0.86 1.03 1.17 [1.0, 1.35] C’D 0.012 0.021 0.079 0.12 [0.08,0.16] C’l 0.34 0.48 0.76 0.57 [0.4, 0.8]

Figure 7. Rod test-case. Left: Mean pressure on the wall. Right: Mean friction on the wall. [x x: measurements (Szepessy and Bearman, 1992, Achenbach, 1968),———— : RANS lin., — • —: RANS low-Re,………… : RANS non-lin.,———– : LES]. Angle = Odeg is the forward stagnation point.

nally better than the two other URANS computations. It performs as well as LES, they both exhibit results in agreement with the experimental ranges.

Next, Fig. 7 presents the mean pressure and the mean friction on the wall. Concerning the mean pressure, we first notice a discrepancy in the recirculation region, represented by the plateau near the angle 180 deg. This turbulent recir­culating region is the most sensitive. However, the non-linear URANS and the LES exhibit better results. The region near the separation is more interesting. There, all the URANS computations fail, exhibiting similar curves. They em­phasize the minimum, and this can be related to a delaying of the separation. As a comparison, LES achieves to predict the experimental curve.

The mean friction confirms the preceding comments. The separation, lo­cated by Cf = 0, is delayed by the URANS computations, whereas the LES features the experimental location.

The overestimate of the Strouhal number by the URANS can be related to the delaying of the separation. The mean recirculation bubble is thiner than

Figure 8. Rod test-case. Left: Mean velocity. Right: Fluctuating velocity, x/d = 7.5. [x x: measurements (Jacob et al., 2002),———- : RANS lin., — • —: RANS low-Re,………….. : RANS non-lin.,———- : LES].

in the experiments, and the vortex shedding frequency is consequently higher. Also, the good estimate of the shedding frequency by LES is consistent with the simulation of the separation, located in agreement with the measurements.

Figure 8 presents the mean and flictuating velocities in a section perpen­dicular to the wake, at x/d = 7.5 downstream of the rod axis. This location is influenced by the airfoil, but computations and experiments are consistent, carried-out for the same configuration (rod + airfoil). We notice similar results for the three URANS computations, but level differences appear. The URANS wake is thiner than in the experiment, in agreement with the delaying of the separation. Also, the two peaks on each side of the axis show that the von Karman vortices are convected along the same path from one cycle to another. This is not true for the experiment and the LES, because they take the turbulent dispersion of the trajectories into account.

Finally, Fig. 9 shows views of the instantaneous vortex identification func­tion, Г2, designed by Graftieaux et al., 2001. The experiment (PIV), the linear URANS and the LES are presented. We notice a good agreement between LES and PIV, many small vortical scales agglomerates to shape the major eddies. On the contrary, URANS only exhibits large von Karman vortices, with a small inter-vortices distance due to the higher frequency.

Figure 10 presents the same quantity, but reconstructed using only the mean field and the two major POD modes (POD: Proper Orthogonal Decomposi­tion). The agreement is good between the three images. It shows that URANS is able to feature the major modes, in agreement with PIV and LES.

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2. Conclusion

The present studies focused on the abilities of the numerical approaches to predict vortex shedding. Firstly, various U-RANS results has been presented on the VKI test-case. Then, an academic study was pursued on an isolated rod undergoing a sub-critical vortex street. One observes that RANS is able to predict the main features of the fbw, qualitatively. However, differences are noticeable between the various turbulence models used. As an outlook, LES performs better because of its direct description of the flow, including the diversity of the scales. For example, boundary layer separation has an important influence on the vortex shedding, and it is accurately featured by LES.