Effect of wind turbulence on wave generation over a viscous liquid

When wind blows over the surface of a viscous liquid, a clear transition from irregular small-amplitude streamwise-oriented wrinkles to well-defined nearly two-dimensional regular waves is observed at a critical wind velocity. We examine how free-stream turbulence in the air influences the growth of wrinkles and regular waves, as well as the transition between these two regimes. Experiments are carried out in a wind tunnel, in which air is blown over a tank filled with silicone oil whose viscosity is fifty times higher than that of water. The free-stream turbulence is enhanced using upstream grids, achieving relative turbulence intensities up to 8%. Surface deformations are measured using Free-Surface Synthetic Schlieren with micrometer accuracy. Velocity measurements are performed using hot-wire anemometry above the interface and particle image velocimetry in the liquid. Results reveal two primary effects of grid-enhanced free-stream turbulence: an increase in the wrinkle amplitude, and a reduction in the critical wind speed at the onset of regular waves. Nevertheless, the wrinkle-wave transition still corresponds to an approximately constant friction velocity. Similar to a classical boundary layer over a flat plate, the friction velocity is found to decrease with fetch. From a wave energy balance, we develop a qualitative model explaining why, with the highly viscous liquid considered here, this decrease in the friction velocity results in a non-monotonic variation of the wave amplitude with the fetch.

💡 Research Summary

The paper investigates how free‑stream turbulence in the air influences the early stages of wind‑generated surface deformations on a highly viscous liquid. The authors use a closed‑loop wind tunnel equipped with a 50‑mm‑deep tank filled with silicone oil whose kinematic viscosity is 50 × 10⁻⁶ m² s⁻¹ (about fifty times that of water). Three inlet conditions are examined: (i) no grid (NG), (ii) a grid with 31.6 mm mesh and 6 mm bars (G32), and (iii) a grid with 64.1 mm mesh and 12 mm bars (G64). The grids are placed 0.2 m upstream of the liquid surface and generate free‑stream turbulence intensities (Tu) up to 8 %, compared with ~0.6 % for the NG case.

Velocity profiles and turbulence statistics in the air are measured with a hot‑wire probe (20 kHz sampling). The presence of a grid produces a classic “overshoot” in the mean velocity profile and raises the turbulence intensity in the outer part of the boundary layer while leaving the near‑wall peak essentially unchanged. The turbulent boundary‑layer thickness remains of order 20 mm, but the mean streamwise velocity near the wall can be 10‑20 % larger downstream of the grid.

The liquid flow is characterised by particle‑image velocimetry (PIV) in the vertical mid‑plane, allowing a direct determination of the friction velocity u* at the air–liquid interface. Surface deformations are captured with Free‑Surface Synthetic Schlieren (FS‑SS), which provides micrometre‑scale vertical resolution over a horizontal field of view of about 0.19 m × 0.15 m. From the time‑averaged displacement field of a random dot pattern placed under the tank, the surface elevation ζ(x,y) is reconstructed with a vertical accuracy of ~0.1 µm.

Two distinct regimes are observed as wind speed increases. At low wind speeds (≈1–2 m s⁻¹) the surface exhibits streamwise‑oriented “wrinkles” of very small amplitude. The root‑mean‑square amplitude ζrms follows the scaling proposed by Phillips and later refined by Perrard et al.:

 ζrms δ ∝ ρa ρℓ u*³⁄² (g νℓ)¹⁄²  (1)

where δ is the boundary‑layer thickness, ρa and ρℓ are the densities of air and liquid, g is gravity, and νℓ is the liquid kinematic viscosity. The experiments confirm that ζrms grows with increasing free‑stream turbulence: the grids raise ζrms by a factor of 1.5–2 relative to the NG case, consistent with a larger pressure‑fluctuation amplitude injected by the turbulent air.

When the wind speed reaches a critical value Uc, a sudden transition to regular, nearly two‑dimensional waves occurs. In the NG configuration Uc≈3.2 m s⁻¹, whereas with G32 and G64 the critical wind speed is reduced to ≈2.8 m s⁻¹ and ≈2.6 m s⁻¹ respectively. Despite this shift, the friction velocity at the transition remains essentially constant (u*≈0.25 m s⁻¹) for all three cases. This observation supports the Miles mechanism, which predicts that wave growth becomes unstable when the air‑side shear stress (represented by u*) exceeds a threshold, independent of the mean wind speed.

A further important finding is that u* decreases with fetch (the distance downstream from the grid). This behaviour mirrors that of a classical turbulent boundary layer over a flat plate, where the shear stress decays as the flow develops. By writing a wave‑energy balance (input from wind stress minus viscous dissipation in the liquid), the authors argue that the decreasing u* reduces the growth rate of the regular waves. Because the liquid is highly viscous, the dissipation term is large, leading to a non‑monotonic evolution of ζrms with fetch: the amplitude first rises (as wrinkles are amplified) and then falls as the diminishing shear can no longer sustain the wave energy.

In summary, the study demonstrates three key effects of grid‑generated free‑stream turbulence on wind‑wave generation over a viscous liquid:

1. Amplification of wrinkle amplitude – turbulence raises the pressure‑fluctuation forcing, increasing ζrms in the linear regime.
2. Lowering of the critical wind speed for wave onset – stronger turbulent fluctuations trigger the Miles instability earlier, but the transition still occurs at an almost invariant friction velocity.
3. Fetch‑dependent friction velocity leading to non‑monotonic wave amplitudes – as u* decays downstream, the balance between wind input and viscous dissipation produces a peak in wave amplitude followed by decay.

These results reconcile the large scatter of critical wind speeds reported in the literature for the air–water system by attributing part of the variability to differences in incoming turbulence intensity. Moreover, the qualitative model based on the wave‑energy balance provides a useful framework for predicting wave growth on highly viscous liquids, which is relevant to industrial processes such as oil recovery, coating flows, and the design of wind‑driven mixers. The work also highlights the importance of measuring and controlling free‑stream turbulence in laboratory wind‑wave experiments, as it can significantly modify both the linear and nonlinear stages of wave development.