Wind-sustained viscous solitons

When wind blows at the surface of a liquid of sufficiently high viscosity, a wave packet of small amplitude is first generated, which sporadically forms large-amplitude fluid bumps that rapidly propagate downstream. These nonlinear structures, first observed by Francis [J. R. D. Francis, Philos. Mag. 42, 695 (1954)], have an almost vertical rear facing the wind and a weak slope at the front. We call them viscous solitons. We investigate their dynamics in a wind-tunnel experiment using silicon oil of kinematic viscosity $1000{\mathrm{mm}}^2{\mathrm{s}}^{-1}$ by means of laser sheet profilometry and particle image velocimetry. We give evidence of their subcritical nature: They are emitted in a region of large shear stress but, once formed, they are sustained by the wind and propagate in a region of lower stress. Their propagation velocity is given by the balance between aerodynamic drag in the air and viscous drag in the liquid. The stable soliton branch of the subcritical bifurcation diagram is reconstructed from the measured soliton amplitude at various wind velocities and distances along the channel. At large wind velocity, the emission frequency of solitons increases, resulting in a long-range sheltering of downstream mature solitons by newly formed upstream solitons, which limits their course.

Figure 1

Viscous soliton propagating from left to right, generated by a wind of velocity $U_a=9.63\mathrm{m}{\mathrm{s}}^{-1}$ on a silicon oil bath of viscosity ${\nu }_{\ell }=1000{\mathrm{mm}}^2{\mathrm{s}}^{-1}$ and depth $h=35\mathrm{mm}$. The size of the grid pattern is 1.3 cm.

Figure 2

Sketch of the wind tunnel and of the liquid tank: (a) side view and (b) front view, looking downstream. Measurements are performed by laser sheet profilometry and particle image velocimetry. The wave amplitude is exaggerated for visibility.

Figure 3

Free surface measurement using LSP showing three viscous solitons propagating downstream, for $U_a=9.95\mathrm{m}{\mathrm{s}}^{-1}$ and $h=35\mathrm{mm}$. The grayscale image shows the raw image of the tilted laser sheet penetrating the oil; the solid red line is the reconstructed interface $\zeta \left(x\right)$. Note that the vertical scale is magnified for visibility.

Figure 4

Flow in a vertical plane measured by PIV (a) below the onset of solitons, $U_a=7.4\mathrm{m}{\mathrm{s}}^{-1}$, and (b) showing the formation of a soliton, $U_a=10.1\mathrm{m}{\mathrm{s}}^{-1}$. Color codes the norm of the velocity. Because of the camera incidence, the 5 mm at the bottom of the tank are not resolved. In (b) a soliton measured by LSP is also shown at the same location as in the PIV field (PIV and LSP measurements are not simultaneous).

Figure 5

(a) Shear stress $\tau$ as a function of the distance $x$ for three wind velocities $U_a$, obtained from the drift velocity measurement using Eq. (2). (b) Normalized shear stress (skin friction coefficient) as a function of the Reynolds number based on the total streamwise distance $x+x_0$, where $x_0$ is the length of the flat plate before the edge of the tank. In both figures the lines show the empirical fit [Eq. (3)].

Figure 6

Spatiotemporal diagrams of the surface elevation $\xi (x,t)$ during 50 s for a liquid depth $h=35\mathrm{mm}$ at three wind velocities (a) $U_a=9.15\text{m}{\text{s}}^{-1}$, (b) $U_a=9.31\text{m}{\text{s}}^{-1}$, and (c) $U_a=9.95\text{m}{\text{s}}^{-1}$, showing the emission, propagation, and decay of viscous solitons. The dashed box in (a), magnified in (d), shows the interval of $x$ where the initial wave packet is present. The horizontal dashed arrows and the circles highlight the influence of new upstream solitons on mature downstream solitons: strong sheltering at moderate wind velocity, leading to a marked slowdown sometimes followed by a decay in (b), and weak sheltering at larger wind velocity in (c).

Figure 7

Stacked plots of the surface elevation $\xi \left(x\right)$ for $U_a=9.15\mathrm{m}{\mathrm{s}}^{-1}$ taken every $\mathrm{\Delta }t=0.1$ s, showing the propagation and growth of the initial wave packet and the generation of a soliton. The profiles are vertically shifted by $0.1\mathrm{mm}$ at each time step.

Figure 8

Sketch illustrating the subcritical formation of viscous solitons. (a) Decrease of the shear stress along the channel, $\tau \sim {(x+x_0)}^{-0.2}$ [Eq. (3)], and the two thresholds ${\tau }_1>{\tau }_2$ defining the two distances $x_1<x_2$. (b) Bifurcation diagram showing the primary wave branch, unstable for $\tau >{\tau }_1$, and the soliton branch, stable for $\tau >{\tau }_2$ (stable branches in solid lines, unstable branches in dashed lines). The vertical black arrows illustrate the linear growth rate ${\sigma }_i$. The red trajectory illustrates the evolution of the system when governed by a time-varying shear stress $\tau$ close to the onset ${\tau }_1$ (see Sec. 4).

Figure 9

Determination of the critical shear stress ${\tau }_1$ for the onset of solitons, obtained by measuring the critical wind velocity $U_{ac}$ as a function of the length $L_m$ of a floating rigid membrane that extends the length of the rigid-wall boundary layer (red dashed line in the sketch). From this measurement, the critical shear stress ${\tau }_1\simeq 0.25$ Pa is obtained using the decay law (3), shown in the inset. The liquid depth $h=35\mathrm{mm}$.

Figure 10

(a) Amplitude of the initial wave packet and soliton crest as a function of the relative time $\mathrm{\Delta }t=t-t_s$, for $U_a=9.15\mathrm{m}{\mathrm{s}}^{-1}$ and $h=35\mathrm{mm}$. The time of formation of a soliton $t_s$ is defined such that $A=0.5\mathrm{mm}$ (see the arrow). For each event (different symbols), the data at $\mathrm{\Delta }t<0$ show the amplitude of the wave envelope at the center of the wave packet, $x_c=50\mathrm{mm}$, while the data at $\mathrm{\Delta }t>0$ show the amplitude following the soliton crest. The growth of the initial wave packet ($\mathrm{\Delta }t<0$) is irregular, while the soliton growth ($\mathrm{\Delta }t>0$) is reproducible, with a well-defined growth rate ${\sigma }_s$. (b) Mean growth rate of the initial wave packet ${\sigma }_i$ just before the transition to the soliton and growth rate of the soliton ${\sigma }_s$ as functions of the wind velocity $U_a$.

Figure 11

(a) Emission frequency of solitons for the three liquid depths. (b) Fluctuation rate of the waiting time $T_e$ between two solitons for $h=35\mathrm{mm}$. The dashed line, ${(U_a-U_{ac})}^{-1}$ with $U_{ac}=9.25\mathrm{m}{\mathrm{s}}^{-1}$ the critical wind velocity, is shown as a guide to the eye.

Figure 12

(a) Spatial evolution of the soliton amplitude $A$ for a set of $N=12$ solitons, for a wind velocity $U_a=9.15\mathrm{m}{\mathrm{s}}^{-1}$ and liquid depth $h=35\mathrm{mm}$. (b) Soliton trajectories, in the plane amplitude $A$ vs velocity $V_s$. Each color represents a single soliton. (1) shows the unstable primary wave, (2) the growth phase, (3) the slowly decreasing mature state, and (4) the rapid decay. The horizontal dashed line represents the mean phase velocity of the regular waves, $c\simeq 19\mathrm{mm}{\mathrm{s}}^{-1}$.

Figure 13

Shape of mature solitons at different wind velocities $U_a$ for a liquid depth $h=35\mathrm{mm}$. Each curve is an average over a set of solitons at constant $U_a$, with the center $X_s$ defined such that $\xi \left(X_s\right)=0$. (a) Profiles $\xi$ in physical units. (b) Profiles rescaled by the soliton amplitude $A=max\left(\xi \right)$. Note that the measurements at $x\simeq X_s$ are not reliable, where the true slope may become negative (overhanging profile).

Figure 14

Flow below a soliton for different liquid depths (a) $h=20$ mm, (b) $h=35$ mm, and (c) $h=50$ mm (wind velocity $U_a=10.1\mathrm{m}{\mathrm{s}}^{-1}$). The colormaps show the norm of the velocity $\left|\mathbf{u}\right|={(u_x^2+u_z^2)}^{1/2}$ (left column) and the vorticity $\omega$ (right column). The selected fields are shown just after the emission, at the beginning of the mature phase (note that for small $h$ the emission occurs at a larger fetch).

Figure 15

(a) Mean velocity $V_s$, and (b) mean amplitude $A$ of solitons in the quasistationary phase as a function of the wind velocity for various liquid depths $h$.

Figure 16

Coefficient $\alpha$ characterizing the soliton velocity in Eq. (8) as a function of the wind velocity $U_a$ for the three liquid depths $h$. The inset shows the mean value of $\alpha$ as a function of $h$, with a best fit $\alpha \sim h^{0.3}$.

Figure 17

Amplitude of viscous solitons as a function of the local shear stress $\tau$ for various wind velocities, obtained from the distance $x$ along the channel using the decay law (3). The distance $x$ increases from right to left in this diagram. The critical shear stresses ${\tau }_1$ and ${\tau }_2$ are from Eqs. (4) and (5). The dashed line is a guide to the eye representing the hypothetical unstable branch of the subcritical bifurcation diagram of Fig. 8.