Long-time behavior of three-dimensional gravity-capillary solitary waves on deep water generated by a moving air-blowing forcing: Numerical study

Long-time simulations are conducted on a forced three-dimensional (3D) nonlinear viscous gravity-capillary wave equation that describes the surface wave pattern when the forcing moves on the surface of deep water with speeds less than the linear phase speed $c_{\min }=23\mathrm{cm}/\mathrm{s}$. Three different states are identified according to forcing speeds $U$ below $c_{\min }$. At relatively low speeds below a certain speed ($c_1$), a steady circular dimple is observed below the moving forcing. At relatively high speeds above a certain speed ($c_2$), “symmetric” shedding phenomena of 3D depressions are observed behind the moving forcing. At intermediate speeds ($c_1\le U\le c_2$), steady 3D gravity-capillary solitary waves are generated behind the moving forcing and are maintained for some time. After long-time simulations, however, those gravity-capillary solitary waves break up and 3D local depressions are shed asymmetrically behind the moving forcing. In more detail, when the forcing speed ($U$) is very close to $c_1$, the asymmetric shedding is “almost regular” and when the forcing speed ($U$) is very close to $c_2$, the asymmetric shedding is “regular antisymmetric,” after a transient period of an “irregular” asymmetric shedding from the steady state of 3D gravity-capillary solitary waves. On the contrary, for the remaining cases of the entire forcing speeds ($c_1<U<c_2$), the asymmetric shedding is “irregular.”

Figure 1

Time histories of the maximum depressions of the solutions to Eq. (1) with the initial still-water condition according to forcing speeds α.

Figure 2

Steady wave solution corresponding to Fig. 1 ($\alpha =0.9$). The left moving forcing is located at the origin ($x=0, y=0$). A steady dimple is formed just below the moving forcing. (a) Slanted view, (b) Cross-sectional view (solid line: profile on the $y=0$ plane, dashed line: profile on the plane which passes through the minimum of the profile on the $y=0$ plane).

Figure 3

Steady wave solutions from Eq. (13). (a) $|{\eta }_{\min }|$ vs α diagram, (b) magnified version of Fig. 3 near $\alpha =1$. (c)–(h) Cross-sectional view (solid line: profile on the $y=0$ plane, dashed line: profile on the plane which passes through the minimum of the profile on the $y=0$ plane) of steady solitary-wave solutions for $\alpha =0.9$, 0.91, 0.92, 0.95, 0.96, and 0.97 corresponding to the dot points (1)–(6) in (a) and (b). These solutions are the same as steady wave solutions in Figs. 1–1.

Figure 4

Steady wave solution corresponding to Fig. 1 ($\alpha =0.97, t=4\sim 13\mathrm{s}$). The left moving forcing is located at the origin ($x=0, y=0$). Steady solitary waves are formed behind the moving forcing. (a) Slanted view, (b) cross-sectional view (solid line: profile on the $y=0$ plane, dashed line: profile on the plane which passes through the minimum of the profile on the $y=0$ plane).

Figure 5

Irregular asymmetric shedding of 3D local depressions corresponding to Fig. 1 ($\alpha =0.97, t=13\sim 35\mathrm{s}$). The left moving forcing is located at the origin ($x=0, y=0$).

Figure 6

Regular antisymmetric shedding of 3D local depressions corresponding to Fig. 1 ($\alpha =0.97, t>35\mathrm{s}$). The left-moving forcing is located at the origin ($x=0, y=0$). Figures 6–6 are the same as Figs. 6–6, respectively, and the shedding period is about 1.3 s.

Figure 7

Regular symmetric shedding of 3D local depressions corresponding to Fig. 1 ($\alpha =0.99$). The left-moving forcing is located at the origin ($x=0, y=0$).

Figure 8

Steady wave solution corresponding to Fig. 1 ($\alpha =1.01$). The left-moving forcing is located at the origin ($x=0, y=0$). Steady “V”-shaped surface waves are formed behind the moving forcing.

Figure 9

Time evolution of the energy $E={\int }_{-\infty }^{\infty }{\int }_{-\infty }^{\infty }{{\eta }^{\prime }}^{2}dxdy$ ($t$ vs ln $E$), for the (a) symmetric initial condition ${\eta }^{\prime }(x,y,t=0)={\overline{\eta }}_x$ and (b) antisymmetric initial condition ${\eta }^{\prime }(x,y,t=0)={\overline{\eta }}_y$.

Figure 10

Time histories of the maximum depressions of the solutions to Eq. (1) for the (a) symmetric initial condition ${\eta }^{\prime }(x,y,t=0)=\overline{\eta }+{\overline{\eta }}_x$ and (b) antisymmetric initial condition ${\eta }^{\prime }(x,y,t=0)=\overline{\eta }+{\overline{\eta }}_y$.