Steady free-surface flow at the stern of a ship

Abstract

New solutions for steady two-dimensional free-surface flow past a curved plate are considered here. They can be interpreted as approximations to the flow locally at the stern of a ship. Weakly nonlinear solutions are derived analytically and nonlinear solutions are computed by boundary integral equation methods. Analysis in the phase plane provides a way to determine the geometries of hulls that give rise to wave-free stern flows. These waveless flows are desirable as they reduce ship-drag.

Steady Two-Dimensional Free-Surface Flow Past Disturbances in an Open Channel: Solutions of the Korteweg–De Vries Equation and Analysis of the Weakly Nonlinear Phase Space

“…For example, Binder et al [30] considered flow with a curved plate and in the phase plane this gives a curved jump between trajectories and fixed points instead of the horizontal jump that we have seen for a flat plate. A similar approach has also been used to examine the waves in subcritical flow at the stern of a ship [11,85]. In these problems, the stern consists of a curved portion of the hull that is connected to a semi-infinite horizontal plate, which is assumed to have a constant pressure, P, applied to the plate as x → ∞.…”

“…Note that Equation 14(and Equation 11in the the full problem) immediately rule out the possibility of solution types VI, VIII, X, XI when σ(±∞) = 0. The comparison between exact solutions of the full problem, Equations (10) and (11), and values of the weakly nonlinear analysis, Equations (6) and (14), is presented in Figure 4b,c, and illustrate that only four of the eight basic flow types that are wave-free in the far-field are possible with the constraints considered so far. The results also show that there is good quantitative agreement between the two theories when 0.8 < F < 1.2 for flow type V. Furthermore, it is important to recognise that the analysis does not guarantee the existence of solutions for a given type of disturbance, or forcing and this needs to be established on a case-by-case basis.…”

“…Note that Equations (31) and (32) relax to those of Equations (10) and (11) when h = 0. The comparison between weakly nonlinear, Equations (27) and (30), and fully nonlinear, Equations (31) and (32), results for h = ±0.05 is presented in Figure 8, and qualitatively similar results are also found for other non-zero values of the step height, h = 0.…”

“…It is understood that free-surface flow occurs over channel beds within tidal estuaries, streams, irrigation channels, rivers and hydraulic infrastructures [1][2][3][4][5][6], and that when the speed of the base flow is close to the critical speed gH, where H is the depth of the channel, surface waves generated by disturbances in a channel can damage both the integrity of waterway banks and hydraulic infrastructures. The study of free-surface flow has also explored energy lost in surface waves as an important component of the overall drag on the disturbance (e.g., the steady wave pattern and drag at the stern of a ship) [7][8][9][10][11][12][13]. The nature of research into free-surface flow has led to experimental studies predominantly in rectangular open channels [14][15][16][17][18][19] (e.g., see Figure 1).…”

Free surface flow past topography: A beyond-all-orders approach

“…Other early examples of similar observations are given for flows past submerged bodies and pressure distributions by Dagan [13] and Doctors & Dagan [15]. Note that these are in contrast to certain other configurations, such as flow past a semi-infinite plate [2,26,28,29,31,37], for which solutions with downstream waves do not exist for sufficiently small Froude numbers, and so there is no corresponding small Froude number nonuniformity.…”