Kármán vortex and turbulent wake generation by wind park piles

Abstract

Observational evidence of turbulent wakes behind wind parks’ piles motivated a series of numerical experiments, aiming to identify the dynamic regimes associated with wakes’ generation in tidal basins. We demonstrate that the obstacles such as piles of wind parks give rise to vortices similar to the known Kármán vortices which affect substantially the turbulent kinetic energy. The latter can be considered as the agent enhancing sediment remobilization from the ocean bottom, thus making wakes well visible in satellite data. The temporal and spatial variability of studied processes is analyzed under stationary and nonstationary conditions. The dependence of a vortex generation and evolution upon the environmental conditions is also studied, which demonstrates a large variety of appearances of turbulent wakes. The comparison between simulations using a suspended sediment model and satellite images demonstrated that the model is capable to realistically simulate sediment wakes observed in remote sensing data.

Appendices

Appendix 1: The laboratory experiments of Lloyd and Stansby (1997a, b) and numerical simulations

The model used here has been applied for estuaries, coastal waters, and regional simulations with a resolution of hundred to thousand meters (Pein et al. 2014; Zhang et al. 2016a). Before using this model for the simulations of wakes in the coastal ocean, it was necessary to test its capability to resolve scales from meters to kilometers. However, it was also necessary to have a reference from other independent observations or simulations. Therefore, below, we will refer to the research of Lloyd and Stansby (1997a, b), who conducted a series of experiments to investigate a shallow-water flow in the wakes of conical islands with gently sloping sides. They measured flow velocity and carried out numerical simulations with a depth-averaged and 3D hydrostatic numerical model. Their flume has horizontal dimensions of 9.75 m by 1.52 m, and the experiments were conducted for flumes with different basin depths varying between 1.3 and 13.6 cm. Measurements have been done for Re ranging between 1300 and 7040, and the Froude number was less than 0.3. Figure 12a shows the surface velocity field as presented by of Lloyd and Stansby (1997a).

Joseph Zhang (personal communication) repeated the simulations of Lloyd and Stansby (1997b) with the same scales demonstrating that SCHISM replicates well the reported by these authors’ simulations and laboratory observations. This motivated us to rescale the experiment of these authors to realistic dimensions of a wind park turbine pile (about 100 times larger scales). The respective surface velocity is shown in Fig. 12b.

Appendix 2: Sensitivity to channel length and horizontal resolution

Long-channel experiment

In the experiment presented in Sect. 3, the Kármán street reaches the downstream model boundary, which could question the validity of results because of nonadequate boundary conditions prescribed. In order to demonstrate the sensitivity of simulations to the channel length, we increase the grid cells’ side length in the x-direction by factor of 3. The surface velocity in Fig. 13a demonstrates that the area where sea surface velocity behind the pile is lower than in the surrounding fluid extends up to about 3 km. The wavy-like pattern ends after about 1000 m behind the obstacle. Surface vorticity (Fig. 13b) demonstrates that Kármán vortices are well developed also beyond 1000 m; at 3 km, the values for vorticity become very low. Thus, this experiment demonstrates that in order to adequately simulate the entire process of formation and disappearance of wakes, the model area has to be larger than 3–4 km.

Sensitivity to horizontal resolution

In order to illustrate the effect of different resolutions in the basic experiment and the one presented in this appendix, we show in Fig. 14 the vorticity field in the elongated channel plotted over the area covered in the basic experiment. The comparison with the simulations with finer resolution (Fig. 2b) shows small differences, but the overall result does not change much. It appears that the Kármán vortices in the basic experiment are sharper and develop closer to the bump, which is characteristic for a higher Reynolds number (see, e.g., Rajani et al. 2009).

Differences between the long-channel experiment and simulation presented in Sect. 3

The time-averaged characteristics in the long-channel experiment show also a similar behavior as in the basic case. Here, the surface velocity increases until about 900 m behind the obstacle, then decreases and reaches a stable state until about 1750 m and then increases again. Noteworthy is that velocity starts to decrease at the area where a maximum of TKE (Fig. 15a, b) has been reached. In comparison to Fig. 4b, the distance of the maximum of turbulent energy to the bump is higher.

The major difference between Figs. 4 and 15 is the position of the maximum of the surface velocity and the surface velocity fluctuation. The locations of the extremes are moved away from the obstacle. This may be a consequence of the change of the grid resolution which results in a different representation of the bathymetric bump and also affects the numerical mixing in horizontal direction and thus changing the Reynolds number like it has already been mentioned above. The detailed analysis of the influence of the grid resolution on the resulting dynamics will be subject to future studies.

Appendix 3: Reynolds number and depth dependence

Sensitivity to flow magnitude

We repeat the basic experiment with a two times higher flow. The surface vorticity (Fig. 16a) increases correspondingly (compare with Fig. 2b); Kármán vortices develop at a larger distance behind the bump. In an experiment with two times lower velocities than in the basic one (Fig. 16b), the overall structure is very similar to the one shown in Fig. 2b, but the values of the vorticity inside the Kármán vortex street decrease very strongly at the end of the model domain. It is noteworthy that the pattern of Kármán vortices is similar to the ones in Fig. 6 before and after a stable Kármán vortex street has been formed.

Sensitivity to depth

Another test was carried out to study the sensitivity of Kármán vortices to the depth (Fig. 16c). In this experiment, the depth is increased by a factor of 4 and correspondingly, the volume flow at the right boundary was increased by a factor of 4. In this case, the Kármán vortices behind the bump become very unstable. This is consistent with the theory that bottom friction stabilizes the wake behind an obstacle (Lloyd and Stansby 1997a).

Fig. 12

Snapshots of surface velocity a from the laboratory experiment (Lloyd and Stansby 1997a with permission from ASCE, © American Society of Civil Engineers) and b from the numerical simulation of Sect. 3

Fig. 13

Snapshots of surface velocity SV (a) and vorticity Ω (b) 35,050 s after initialization in the experiment with three times coarser resolution in x-direction. For comparison, see the basic experiment of Sect. 3 (Fig. 2)

Fig. 14

The same as Fig. 13b plotted in the model area of the basic experiment of Sect. 3

Fig. 15

The same as Fig. 4 but plotted for the sensitivity experiment presented in Appendix 2

Fig. 16

As in Fig. 2b, but with two times higher velocity (a), two times lower velocity (b), and four times higher depth (c)