Comparison of nonlinear wave-resistance theories for a two-dimensional pressure distribution

Abstract

The wave resistance of a two-dimensional pressure distribution which moves steadily over water of finite depth is computed with the aid of four approximate methods: (i) consistent small-amplitude perturbation expansion up to third order; (ii) continuous mapping by Guilloton's displacements; (iii) small-Froude-number Baba & Takekuma's approximation; and (iv) Ursell's theory of wave propagation as applied by Inui & Kajitani (1977). The results are compared, for three fixed Froude numbers, with the numerical computations of von Kerczek & Salvesen for a given smooth pressure patch. Nonlinear effects are quite large and it is found that (i) yields accurate results, that (ii) acts in the right direction, but quantitatively is not entirely satisfactory, that (iii) yields poor results and (iv) is quite accurate. The wave resistance is subsequently computed by (i)-(iv) for a broad range of Froude numbers. The perturbation theory is shown to break down at low Froude numbers for a blunter pressure profile. The Inui-Kajitani method is shown to be equivalent to a continuous mapping with a horizontal displacement roughly twice Guilloton's. The free-surface nonlinear effect results in an apparent shift of the first-order resistance curve, i.e. in a systematic change of the effective Froude number.

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

“…The finite-depth versions of these problems are treated in McCue & Forbes [27] and Mekias & VandenBroeck [30], for example, where the main goal was to solve the problems numerically for all Froude numbers; in these cases, numerical results strongly suggest the amplitude of the waves is exponentially small in the Froude number as F → 0. 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.…”

On reduced models for gravity waves generated by moving bodies

“…As we have reviewed in §1.3, others have proposed integral formulations of the low-Froude problem (see e.g. the collection of models in Doctors & Dagan 1980), but such models typically depended on ad-hoc linearizations of the two-dimensional potential flow equations.…”

“…Reviews of these and other models were presented by Doctors & Dagan (1980) and Miloh & Dagan (1985), and many others. However, what distinguishes Tulin and Tuck's work is the shifted focus towards the analytic continuation of the flow problem into the complex domain; indeed, Tulin notes his strong motivation by the work of Davies (1951) in his 2005 review.…”

“…Origin of the low-Froude paradox Ogilvie (1968) Two-dimensional and three-dimensional linearizations Dagan & Tulin (1972), Keller (1979), Ogilvie & Chen (1982), Doctors & Dagan (1980), Tulin (1984), Brandsma & Hermans (1985) On numerical solutions Vanden-Broeck & Tuck (1977), Vanden-Broeck et al (1978), Madurasinghe & Tuck (1986), Farrow & Tuck (1995) On exponential asymptotics applied to water waves Chapman & Vanden-Broeck (2002, 2006, Trinh et al (2011), Trinh & Chapman (2013a, Lustri et al (2013); Lustri & Chapman (2014), Trinh & Chapman (2014) Review articles Tuck (1991a), Tulin (2005) …”

“…The two-term truncation is the closest analogy to the Tulin (1982) model (see §9 for an extended discussion), as well as various other models (c.f. the review by Doctors & Dagan 1980). However the fact that such two-term approximations do not accurately predict the pre-factor A is not well established in the literature.…”