Hysteretic wave drag in shallow water

During motion from deep to shallow water, multiple equilibria may emerge, each with identical drag—a phenomenon that can be explained by a localized amplification of the wave drag near the shallow wave speed. The implication of this is the emergence of several previously unstudied bifurcation patterns and hysteresis routes. Here, we address these nonlinear dynamics by considering the quasisteady motion of a body between deep and shallow water, where the depth is slowly varying. We survey several theoretical models for the drag, compare these against our tow-tank experimental measurements, and then use the validated theory to explore the bifurcation patterns using two parameters: the depth of motion and the forcing. In particular, using a case study of a lake with a sinusoidal depth profile, we illustrate that hysteresis effects can play a significant role on the speed of motion and journey time, presenting interesting implications for naval and racing applications.

Figure 1

Labeled photograph of our experimental setup: A hull is pulled through the water at constant velocity by a linear motor and connected via supporting bars to a force sensor. The water depth for each experiment is uniform.

Figure 2

(a)–(c) Comparison between experimental measurements and theoretical predictions of the drag force for different depths and Froude numbers, and an illustration of each depth case above. Theoretical $R_w$ predictions are summed with wind tunnel measurements of $R_f+R_s$ for comparison with tow-tank measurements of $R_d$ (see discussion in the main text). The critical depth-based Froude number ${\text{Fr}}_h=u/\sqrt{gh}=1$ is indicated in each plot, and areas of accentuated drag are illustrated with shading. An illustration of the calculated boundary layer profile for our hull shape, which we use in our modification to Sretensky's theory, is given in (d) (see the discussion towards the end of Sec. 2).

Figure 3

Bifurcation diagrams, showing all of the existing branches of the quasisteady equilibrium solutions to (1) and (2), for the two bifurcation parameters $F/mg$ and $h/L$. We also illustrate the different subcritical, critical, and supercritical regimes, the critical forcing $F=F_c$ (see the discussion in Sec. 4), and possible hysteresis routes.

Figure 4

Phase plane analysis of the case study of a boat moving in a lake with sinusoidal depth (16), where the rate of change of depth is sufficiently small that the quasisteady approximation is appropriate. Trajectories for different values of the forcing term $F/mg$ are illustrated in both (a) the depth-Froude plane and (b) the distance-Froude plane. (c) Times to complete the first half of the race $0\le x\le nL/2$ for the different solution branches.