Mach-like capillary-gravity wakes

We determine experimentally the angle $\alpha$ of maximum wave amplitude in the far-field wake behind a vertical surface-piercing cylinder translated at constant velocity $U$ for Bond numbers ${\mathrm{Bo}}_D=D/{\lambda }_c$ ranging between 0.1 and 4.2, where $D$ is the cylinder diameter and ${\lambda }_c$ the capillary length. In all cases the wake angle is found to follow a Mach-like law at large velocity, $\alpha \sim U^{-1}$, but with different prefactors depending on the value of ${\mathrm{Bo}}_D$. For small ${\mathrm{Bo}}_D$ (large capillary effects), the wake angle approximately follows the law $\alpha \simeq c_{\mathrm{g},\min }/U$, where $c_{\mathrm{g},\min }$ is the minimum group velocity of capillary-gravity waves. For larger ${\mathrm{Bo}}_D$ (weak capillary effects), we recover a law $\alpha \sim \sqrt{gD}/U$ similar to that found for ship wakes at large velocity [Rabaud and Moisy, Phys. Rev. Lett. 110, 214503 (2013)]. Using the general property of dispersive waves that the characteristic wavelength of the wave packet emitted by a disturbance is of order of the disturbance size, we propose a simple model that describes the transition between these two Mach-like regimes as the Bond number is varied. We show that the new capillary law $\alpha \simeq c_{\mathrm{g},\min }/U$ originates from the presence of a capillary cusp angle (distinct from the usual gravity cusp angle), along which the energy radiated by the disturbance accumulates for Bond numbers of order of unity. This model, complemented by numerical simulations of the surface elevation induced by a moving Gaussian pressure disturbance, is in qualitative agreement with experimental measurements.

Figure 1

Wake patterns in the small-scale towing tank experiments, for a cylinder of diameter $D=1.5$ mm (Bond number ${\mathrm{Bo}}_D=0.10$), at velocity $U=0.60,0.80$, and $1.80 \mathrm{m} \mathrm{s}{}^{-1}$. The tank is 2 m long, and only the last 0.85 m are shown. The waves are visualized by shadowgraphy on the bottom of the water tank.

Figure 2

Wake patterns in the swimming pool experiments, for a cylinder of diameter $D=16$ mm (Bond number ${\mathrm{Bo}}_D=1.0$), at velocity $U=0.63$ and $2.5 \mathrm{m} \mathrm{s}{}^{-1}$. The wake angle is determined from the intersection (shown by the vertical arrow) of the dashed line, going through the waves of maximum amplitude, to the back edge of the pool.

Figure 3

Wake angle as a function of the cylinder velocity $U$, normalized by $c_{\min }$ in (a) and by $\sqrt{gD}$ in (b). Open symbols (blue): Small-scale experiments; filled symbols (red): swimming-pool experiments. The solid lines show best fits of the data at large velocity: (a) $\alpha =0.85c_{\min }/U$ for ${\mathrm{Bo}}_D=0.10$ and 0.34 and (b) $\alpha =0.5/{\mathrm{Fr}}_D$ for ${\mathrm{Bo}}_D=2.0$ and 4.2 (numerical values are given for $\alpha$ in radians).

Figure 4

Construction of the stationary wake pattern. The disturbance is at point O at time 0, and we consider a wave vector $\mathbf{k}$ emitted at point M at time $-t$. In the frame of the disturbance, its energy propagates along the radiation angle $\alpha \left(k\right)$. This is the direction of the relative group velocity ${\mathbf{c}}_g^{\prime }\left(k\right)={\mathbf{c}}_g\left(k\right)-\mathbf{U}$.

Figure 5

Phase velocity $c_{\phi }$ and group velocity $c_g$, normalized by the minimum phase velocity $c_{\min }$, as a function of the normalized wave number $k/\kappa$, with $\kappa ={(\rho g/\gamma )}^{1/2}$. The minimum group velocity is $c_{\mathrm{g},\min }/c_{\min }\simeq 0.77$, at $k/\kappa \simeq 0.39$. A disturbance velocity $\mathcal{U}=U/c_{\min }>1$ selects a range $[k_1,k_2]$ such that the stationary condition (3) is satisfied.

Figure 6

Radiation angle $\alpha \left(k\right)$ as a function of the wave number $k$ normalized by the minimum wave number $k_1$ given by Eq. (4). The bold curve shows the pure gravity case $\left(\mathcal{U}\to \infty \right)$, and the thin curves show capillary-gravity cases for $\mathcal{U}$ between 1 and 8. The dashed lines show the locus of the upper and lower extrema of $\alpha \left(k\right)$, corresponding to the gravity and capillary cusp angles, ${\alpha }_g^{\mathrm{cusp}}$ ($--$, red) and ${\alpha }_c^{\mathrm{cusp}}$ ($-·-$, blue). $\circ$: Onset of existence of the two cusps for ${\mathcal{U}}^{*}=1.938$, at ${\alpha }^{*}\simeq 22.{06}^{\mathrm{o}}$. $\square$: Asymptotic gravity cusp angle for $\mathcal{U}\gg 1$, which corresponds to the Kelvin angle ${sin}^{-1}(1/3)\simeq 19.{47}^{\mathrm{o}}$.

Figure 7

Patterns of isophase lines (e.g., crest lines) for six velocity ratios $\mathcal{U}$. For each velocity, only two crest lines are shown, corresponding to the capillary branch (long dashed line) and the gravity branch (continuous line). Note that a phase shift of $\pi /2$ appears at each cusp point. The filled circles (red) show the gravity cusp and the empty circles (blue) show the capillary cusp, present for $\mathcal{U}>{\mathcal{U}}^{*}=1.938$ [the two cusps merge at ${\mathcal{U}}^{*}$ (b)]. The short dashed lines show the cusp angles ${\alpha }_g^{\mathrm{cusp}}$ (red) and ${\alpha }_c^{\mathrm{cusp}}$ (blue), along which the energy radiated from the source accumulates.

Figure 8

(a) Gravity and capillary cusp angles ${\alpha }_g^{\mathrm{cusp}}$ and ${\alpha }_c^{\mathrm{cusp}}$ and (b) normalized cusp wave numbers $k_g^{\mathrm{cusp}}/\kappa$ and $k_c^{\mathrm{cusp}}/\kappa$, as a function of the normalized velocity $\mathcal{U}$. The circle indicates the onset of the cusp at ${\mathcal{U}}^{*}\simeq 1.938$ (${\alpha }^{*}\simeq 22.{06}^{\mathrm{o}}$ and $k^{*}/\kappa \simeq 0.275$). The line $-·-$ shows the cusp precursor (angle such that $\partial \alpha /\partial k$ is minimum), where a preferential accumulation of energy may take place even before the apparition of the cusps.

Figure 9

Radiation angle $\alpha \left(k\right)$ in the case $\mathcal{U}=6$. Regions in gray represent the range of wave numbers centered on $k_f\simeq L^{-1}$ and of characteristic width $\Delta k\simeq k_f$ present in the wave packet radiated by the disturbance. See text for the definition of the regimes 1–4.

Figure 10

Wake pattern of a Gaussian pressure disturbance at Bond number $\mathrm{Bo}=1$, for increasing velocity $\mathcal{U}$ between 1.2 and 6. The computation domain $L_{\mathrm{box}}$ is $200L$, and only a subdomain of size $L_{\mathrm{box}}/4$ is shown here. The arrows and dashed lines show the cusp precursor angle [panels (a) and (b) for $\mathcal{U}<{\mathcal{U}}^{*}$], and the gravity ${\alpha }_g^{\mathrm{cusp}}$ and capillary ${\alpha }_g^{\mathrm{cusp}}$ cusp angles [panels (c)–(f)]. The black contours show the isoenergy level given by 0.3 times the maximum energy.

Figure 11

Wake pattern at constant velocity $\mathcal{U}=6$, for increasing disturbance size: $\mathrm{Bo}=1$ to 32. Same line patterns as in Fig. 10. As the disturbance size increases, the angle of maximum wave amplitude drifts from the capillary cusp angle ${\alpha }_c^{\mathrm{cusp}}$ (blue dashed line) to the gravity cusp angle ${\alpha }_g^{\mathrm{cusp}}\simeq {sin}^{-1}(1/3)$ (red dashed line).

Figure 12

(a) Map of the isovalues of the angle of maximum wave amplitude $\alpha$ in the plane $(\mathrm{Bo},\mathcal{U})$ for the Gaussian pressure disturbance defined by Eq. (9). The three wake regimes are as follows: (1) Kelvin regime, $\alpha \simeq {sin}^{-1}(1/3)$; (2) Mach regime for gravity waves, $\alpha \simeq a/\mathrm{Fr}$; (3) Mach regime for capillary waves, $\alpha \simeq c_{\mathrm{g},\min }/U$. The numbers in boxes indicate the angle $\alpha$ in degrees. (b) Plot of $\alpha$ as a function of $\mathcal{U}$, for different Bond numbers $\mathrm{Bo}$. The solid line is $\alpha =c_{\mathrm{g},\min }/U=0.77/\mathcal{U}$ (regime 3).

Figure 13

(a) Map of the isovalues of the angle of maximum wave amplitude $\alpha$ in the plane (Bo, Fr). The lower border $\mathcal{U}=1$ is given by $\mathrm{Fr}=1/\sqrt{\pi \mathrm{Bo}}$. (b) Plot of $\alpha$ as a function of $\mathrm{Fr}$ for different values of $\mathrm{Bo}$. The solid line shows the law $\alpha =1/\left({40}^{1/4}{\pi }^{1/2}\mathrm{Fr}\right)$ of Darmon et al. [11] (regime 2).

Figure 14

Compensated angle $\alpha \mathrm{Fr}$ as a function of $\mathrm{Fr}$ for a Gaussian pressure disturbance using various definitions for the wake angle. The dash-dotted curve shows the Kelvin regime ${\alpha }_K\mathrm{Fr}$, with ${\alpha }_K={sin}^{-1}(1/3)$. The reference definition $(*)$, corresponding to the angle of maximum wave amplitude $\zeta$, is compared to the exact value $a=1/\left({40}^{1/4}{\pi }^{1/2}\right)\simeq 0.224$ (horizontal line). The four alternate definitions, based on maximum wave slope ($\circ ,▵$, and $\square$) and curvature $(\diamond )$, give smaller values of $a$.