Self-consistent theory of capillary-gravity-wave generation by small moving objects

We investigate theoretically the onset of capillary-gravity waves created by a small object moving at the water-air interface. It is well established that, for straight uniform motion, no steady waves appear at velocities below the minimum phase velocity $c_{\text{min}}=23\text{cm}/\text{s}$. At higher velocities, the emission of capillary-gravity waves creates an additional drag force. The behavior of this force near the critical velocity is still poorly understood. A linear-response theory where the object is replaced by an effective pressure source predicts a singular behavior for the wave drag. However, experimental data tend to indicate a more continuous transition. In this paper, we show that a proper treatment of the flow equations around the obstacle can regularize wave emission, even in the linear wave approximation, thereby ensuring a continuous behavior of the drag force.

Figure 1

Geometry of the submerged cylinder and of the free surface above. The far-field velocity is indicated by the arrows $\mathbf{V}$.

Figure 2

Overview of the variable transformations of the full theory in Secs. 2, 3 (upper panel) and of the dipolar model in Sec. 5 (lower panel).

Figure 3

(Color online) Numerical solution of Eq. (15). Dashed lines represent the discretization with linear finite elements. Parameters: $H\kappa =0.5$, $R\kappa =0.3$, and $V=1.3c_{\text{min}}$.

Figure 4

(Color online) Visualization of the flow velocity around the cylinder. The velocities are represented by arrows the size of which is proportional to the velocity magnitude. The color code is yellow/gray for large magnitudes and blue/black of small ones. Parameters are as in Fig. 3.

Figure 5

(Color online) Wave drag as a function of the far-field velocity $V$ for the parameters for $H\kappa =0.5$ and $R\kappa =0.3$. The results of three different models are presented: dashed curve corresponds to a linear-response calculation where $V_C=V\text{cos}\theta$, red (gray) curve corresponds to a self-consistent calculation using Eq. (15), and black curve corresponds to the dipolar model explained in Sec. 5.