Wave Turbulence on the Surface of a Fluid in a High-Gravity Environment

We report on the observation of gravity-capillary wave turbulence on the surface of a fluid in a high-gravity environment. By using a large-diameter centrifuge, the effective gravity acceleration is tuned up to 20 times Earth’s gravity. The transition frequency between the gravity and capillary regimes is thus increased up to one decade as predicted theoretically. A frequency power-law wave spectrum is observed in each regime and is found to be independent of the gravity level and of the wave steepness. While the timescale separation required by weak turbulence is well verified experimentally regardless of the gravity level, the nonlinear and dissipation timescales are found to be independent of the scale, as a result of the finite size effects of the system (large-scale container modes) that are not taken currently into account theoretically.

Figure 1

Experimental setup. Top: schema of the large diameter centrifuge showing the generation of an apparent gravity normal to the fluid surface. Bottom: experimental setup inside the gondola. See movies in Supplemental Material [20].

Figure 2

Power spectrum density (PSD) of the wave height $\eta (t)$ for $1g$ and $19.4g$. Dashed lines display best power-law fits for (blue) gravity and (red) capillary regimes. Vertical dash-dotted lines correspond to $f_{gc}(g^{*})$ of Eq. (1). ${\sigma }_A=15.5\mathrm{mm}$. Inset: PSD for different vibration amplitudes ${\sigma }_A=0$, 3.7 and 15.5 mm (bottom to top). $g^{*}=19.4g$. Vertical solid (dashed) lines are the first axisymmetrical $m=0$ (nonaxisymmetrical $m=1$) basin mode solutions of $J_m^{\prime }(k_{n}D/2)=0$, with $J_m^{\prime }(·)$ the derivative of the $m$th order Bessel function of the first kind [21].

Figure 3

Transition frequency $f_{gc}$ between gravity and capillary wave turbulence regimes versus $g^{*}$. Vibration amplitude: ${\sigma }_A=3.7$ (open circles) and 15.5 mm (solid diamonds). Solid line is the prediction of Eq. (1). Inset: Frequency power-law exponents $\alpha$ of the wave spectrum for capillary (red) and gravity (blue) regimes. Dotted lines: mean values $⟨{\alpha }_c{⟩}_{g^{*}}$ and $⟨{\alpha }_g{⟩}_{g^{*}}$. Solid lines: weak turbulence predictions (see text).

Figure 4

Temporal decay of the wave spectrum $S_{\eta }(f,t)$ for $15g$. Inset: semilogy plot of the decay of wave energy ${\sigma }_{\eta }^2$ ($+$), and of the Fourier modes $S_{\eta }(f^{*},t)\delta f$ at different frequencies $f^{*}$ corresponding either to the main container mode (open circles), the gravity regime (blue symbols), or the capillary regime (red symbols). Dashed vertical lines delimit the fast power-law decay (solid-line fits) and then the slow exponential decay (dashed-line fits). ${\sigma }_A=15.5\mathrm{mm}$.

Figure 5

Wave turbulence timescale separation for $g^{*}=15g$. Solid line: linear timescale ${\tau }_{\mathrm{lin}}=1/f$. Nonlinear timescale ${\tau }_{\mathrm{nl}}$ estimated from the fast decay power-law fit of Fig. 4 (*) or from the phenomenological model of Eq. (2) $1/b$ (solid square) and $1/c$ (open square). Dashed red lines are predictions of ${\tau }_{\mathrm{nl}}$ in the capillary and gravity regimes with constants chosen to $c_g=c_c=1$. Open circle indicates dissipation timescale, ${\tau }_{\text{diss}}$, estimated from the slow decay of Fig. 4. ${\sigma }_A=15.5\mathrm{mm}$.