Effect of wind turbulence on wave generation over a viscous liquid

TL;DR

This study investigates how free-stream turbulence in the air modifies wind-induced surface deformations over a viscous liquid. Using grid-generated turbulence in a wind tunnel and measurements of wrinkles and waves via FS-SS, hot-wire, and PIV, the authors show that turbulence intensities raise the friction velocity $u^*$ and reduce the wrinkle-to-wave transition wind speed $U_{ac}$, while the critical friction velocity at transition remains near $u^*_c \approx 0.33$ ms$^{-1}$. A simple wave-energy balance explains the observed non-monotonic dependence of the wave amplitude with fetch, as $u^*(x)$ decays downstream and crosses $u^*_c$, with a stronger grid causing a larger early-amplitude response and an earlier drop. The results highlight the role of boundary-layer turbulence structure in wind–wave onset and motivate extending the study to higher turbulence levels with active grids to test the universality of the constant $u^*_c$ criterion.

Abstract

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.

Figures (12)

• Figure 1: Experimental setup. (a) Wind tunnel, with the 50 mm-deep tank filled with silicone oil and the three measurement techniques. (b) and (c) Grids G32 and G64, respectively. The edges of the grids (in brown) are recessed into the channel walls.
• Figure 2: Normalized mean velocity profiles (a) and turbulence intensity profiles (b) in the air at $y=0\,$mm and $x=430\,$mm, i.e., $X=x-x_g = 630\,$mm downstream of the grid location, for the free-stream velocity $U_a=$ 3.2ms. The horizontal dashed lines correspond to the boundary layer thickness for the no-grid case (NG, blue line), and to the location of the center of the first horizontal rod for grids G32 and G64 (red and yellow dashed lines, respectively).
• Figure 3: Streamwise variation of the free-stream turbulence intensity for $U_a=$ 3.2ms (hot wire measurements). (a) Linear plot versus the distance to the grid. The light grey area indicates the region where the FS-SS measurements of the free-surface displacements are performed. (b) log-log plot against $(X-X_0)/M$, with $X_0/M=1$. The corresponding Reynolds numbers are $Re_M= 7500$ for G32 and 14500 for G64. The dashed blue line represents the uniform free-stream turbulence level (0.6%) in the NG case.
• Figure 4: Mean velocity profile in the liquid determined by PIV at fetch $x=500\,$mm, in the absence (a) or in the presence (b,c) of a grid. Arrows indicate increasing wind speeds $U_a$. Filled symbols correspond to wind speeds below the onset of waves ($U_a=3.2$ and 5.1 ms for the three inlet conditions) while open symbols refer to wind speeds beyond the onset of waves ($U_a=8.3$, 6.1 and 5.6 ms for NG, G32 and G64, respectively).
• Figure 5: Longitudinal variation of the friction velocity $u^*$ deduced from PIV measurements in the liquid. (a) in the absence of a grid; (b) with the G32 grid; (c) with the G64 grid. Dashed lines correspond to fits based on the Schlichting law \ref{['eq:tauvsx']}, with $C=0.029$ and and $x_0=-1530\,$mm (no-grid case), $C=0.032$ and $x_0=0\,$mm (G32 case) and $C=0.038$ and $x_0=150\,$mm (G64 case). The horizontal dotted line $u^* = 0.35~\mathrm{m~s^{-1}}$ corresponds to the critical friction velocity associated with the wrinkle-wave transition. The light grey area indicates the FS-SS measurement window.