Boundary-integral representation sans waterline integral for flows around ships steadily advancing in calm water

doi:10.1016/j.euromechflu.2021.06.002

European Journal of Mechanics - B/Fluids

Volume 89, September–October 2021, Pages 259-266

https://doi.org/10.1016/j.euromechflu.2021.06.002 Get rights and content

Highlights

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Neumann–Kelvin theory of potential flow around a ship steadily advancing in calm water is reconsidered.
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New boundary-integral flow representation, which does not involve a waterline integral, is given.
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This exact flow representation solves a 50-year old puzzle and extends the Neumann–Michell theory.

Abstract

The boundary-integral flow-representation associated with the boundary-value problem, commonly called Neumann–Kelvin (NK) problem, that corresponds to linear potential flow around a ship steadily advancing in calm water involves an integral around the mean waterline of the ship. This ‘waterline integral’ is a notorious source of numerical difficulties and has been extensively studied. The waterline integral in the NK theory is largely – but not fully – eliminated in the modification, called Neumann–Michell (NM) theory, of the NK theory. Specifically, the NM theory includes a residual waterline-distribution of weak Rankine singularities, ignored in practical applications. A crucial element of the NM theory is a mathematical transformation that is based on a vector Green function, which is associated with the common scalar Green function used in the NK theory. This transformation is revisited in the present study. A rigorous analysis yields an answer to a fifty-year old puzzle: an exact boundary-integral flow representation that does not include a waterline integral. A remarkable feature of this new flow representation, which is a modification of the NM flow representation given previously, is that it explicitly determines the flow potential and the flow velocity at a ship hull surface in terms of the flow velocity at the hull surface, rather than in terms of the hull-surface potential as in usual boundary-integral flow representations obtained via Green’s classical identity.

Introduction

The wave drag experienced by a ship steadily advancing in calm water is a major element of ship design, and accordingly has been considered in a huge literature since a theoretical method for predicting the wave drag of a ship was first proposed by Michell [1] in 1898. A partial review of this vast literature may be found in e.g. [2], [3], and in studies listed further on. This literature includes approximate analytical theories based on Michell’s thin-ship assumption and similar flat-ship or slender-ship assumptions, the Neumann–Kelvin (NK) theory and the related Neumann–Michell (NM) theory, which are based on applications of the method of Green functions and boundary-integral flow representations, and CFD methods based on the Euler or RANS flow equations.

Applications to ship design, especially hull-form optimization and early-stage design, require methods that satisfy a dual, largely conflicting, requirement. On one hand, methods that account for the dominant flow physics and are sufficiently accurate are required to design modern energy-saving ships. On the other hand, design methods based on hull-form optimization require computational methods that are practical and robust.

Most ships are streamlined slender bodies operating at high Reynolds number of order $1 0^{9}$ . Viscous effects are then confined to a thin boundary layer that only has a significant influence in the vicinity of the ship stern, and potential-flow theory is a realistic flow model. Moreover, the boundary condition at the free surface can realistically be linearized about the uniform speed that opposes the ship speed, in the manner first used by Kelvin [4] in 1887 and Michell [1] in 1898. The Kelvin–Michell linear free-surface boundary condition, given further on by (3c), is examined in [3], [5]. In fact, nonlinearities associated with the free-surface boundary condition can be appreciable [6] and have a significant local influence, notably at a ship bow [5], [7], [8]. Nonlinearities also have a large influence on short divergent waves, which can be too steep to exist [9], [10], [11] and must be filtered [11], [12], [13], [14]. In short, the NK linear potential-flow theory corresponds to an approximate yet realistic flow model. An important feature of this flow model is that the boundary condition at the ship hull surface is enforced exactly, at the precise location of the hull surface, which is an essential requirement for effective hull-form optimization.

Accordingly, the NK theory has been extensively studied for about 50 years [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28]. Unfortunately, the boundary integral associated with the NK boundary-value problem and the Green function that satisfies the Kelvin–Michell free-surface boundary condition involves a troublesome line integral around the mean waterline of the ship. The well-documented difficulties related to this waterline integral are well summarized in [26] as “the Neumann–Kelvin theory, as it is currently understood, does not give satisfactory wave resistance results for realistic ship hull forms. These results lead to, and reinforce, the suggestion that the problem lies with the waterline integral term; a new treatment of this term may substantially increase the applicability of Neumann–Kelvin theory”. Indeed, [28] provides strong numerical evidence that the NK waterline integral results in an ill-conditioned matrix of influence coefficients.

An important modification of the NK (Neumann–Kelvin) theory, called NM (Neumann–Michell) theory, is proposed in [2], [3], [29]. The waterline integral in the NK theory is mostly (but not entirely) eliminated in the NM theory by means of a physical argument and a mathematical transformation. Briefly, the waterline integral in the NK theory involves the terms $G φ_{ξ}$ and $φ G_{ξ}$ where $G$ and $φ$ denote the Green function and the flow potential and $ξ$ is the coordinate along the ship length.

The term $G φ_{ξ}$ is shown in [2], [3], and in Section 5 of the present study, to correspond to an inconsistent linear flow model; thus, this term does not appear in a consistent flow model.

The term $φ G_{ξ}$ is eliminated by means of a mathematical transformation based on a vector Green function $G$ associated with the scalar Green function $G$ via the relation (12b) given in Section 6 of this study.

More precisely, the mathematical transformation related to the vector and scalar Green functions $G$ and $G$ in (12b) is applied to the Fourier components $G^{F}$ and $G^{F}$ in the Rankine–Fourier decomposition $4 π G = G^{R} + G^{F} and 4 π G = G^{R} + G^{F}$ of the Green functions $G$ and $G$ in [3], [28], and is only applied to the wave components $G^{W}$ and $G^{W}$ in the waves and local-flow decompositions $4 π G = G^{R} + G^{L} + G^{W} and 4 π G = G^{R} + G^{L} + G^{W}$ of $G$ and $G$ in [2]. As a result, the boundary-integral formulations of the NM theory given in [2], [3], [28] involve distributions of weak Rankine singularities, which are related to the Rankine component $G^{R}$ , around the ship waterline. Thus, the NM theories given in these previous studies are approximate theories that involve line integrals around the ship waterline, although these waterline integrals are ignored in the practical applications of the NM theories reported in [5], [29], [30], [31], [32], [33], [34], [35].

These studies show that various numerical implementations of the NM theory expounded in [2], [3], [29] yield realistic predictions of the drag, the sinkage and the trim experienced by a ship, and the wave profile along a ship hull, that are in good overall agreement with experimental measurements and are sufficiently accurate for practical purposes, notably for hull-form optimization, within a wide range of Froude numbers. In particular, [34], [35] show that the influence of sinkage and trim on the drag of a freely-floating monohull ship can easily be taken into account in the NM theory, and [5] presents simple post-processing nonlinear corrections (which do not require additional flow computations) of the NM linear theory that account for dominant nonlinear effects, notably the substantial decrease in the wave drag that occurs for a ship with a large bulbous bow. Furthermore, flow predictions obtained via the NM theory or via far more complicated CFD methods compare well, as is illustrated in [3]. An important feature of the NM theory is that it is a simple and practical theory. Indeed, the flow around a ship hull can be evaluated in about 1 s via a PC. The theory is then well suited for routine applications to hydrodynamic optimization, and indeed has been widely used for hull-form optimization [36], [37], [38], [39], [40], [41], [42], [43], [44].

Thus, the NM theory is practical and useful. Nevertheless, the waterline integral in the NK theory is mostly (for practical purposes) but not fully removed in the NM theory given in [2], [3]. This puzzling theoretical blemish is removed in the present study, which shows that the waterline integral in the NK theory can be completely removed, and that this removal can be accomplished via a straightforward rigorous analysis. The refined ‘NM theory sans waterline integral’ expounded in the study therefore provides a resolution of a fifty-year old puzzle related to the appearance of a waterline integral in the NK theory, which has been considered in numerous studies since it was first proposed in [15], [16].

Section snippets

Flow around a ship steadily advancing in calm water

Thus, this study considers potential flow around a ship, of length denoted as $L$ , that advances at a constant speed $V$ in calm water of effectively infinite depth and horizontal extent. The flow around the ship is observed in a Galilean frame of reference attached to the moving ship and a related system of Cartesian coordinates $X \equiv (X, Y, Z)$ . As is shown in Fig. 1, the undisturbed free surface, denoted as $Σ^{F}$ , is chosen as the plane $Z = 0$ and the $Z$ axis points upward. The $X$ axis is taken along the path

Neumann-Kelvin (NK) boundary-value problem and related Green function

The velocity potential $φ (ξ)$ of the flow created by a ship steadily advancing in calm water satisfies the far-field boundary condition $\nabla_{ξ} φ \approx 0 at Σ^{\infty},$ the Laplace equation $\nabla_{ξ}^{2} φ \equiv \partial_{ξ}^{2} φ + \partial_{η}^{2} φ + \partial_{ζ}^{2} φ = 0 in D,$ the Kelvin–Michell linear free-surface boundary condition $\partial_{ζ} φ + F^{2} \partial_{ξ}^{2} φ - ϵ F \partial_{ξ} φ = F \partial_{ξ} p^{F} - q^{F} at Σ^{F}$ and the Neumann hull-surface boundary condition $n \cdot \nabla_{ξ} φ = F n^{x} + σ^{H} at Σ^{H} .$ One has $ϵ = + 0$ in the free-surface boundary condition (3c), in accordance with a flow that slowly grows from a state of rest at time $t = - \infty$ . The time-growth

Basic boundary-integral representation

[2], [3] show that application of a basic Green’s identity to the velocity potential $φ \equiv φ (ξ)$ and the Green function $G \equiv G ($ $ξ$ $, x)$ associated with the boundary-value problems (3) or (6) yield the basic boundary-integral flow representation $ϕ = \int_{Σ^{F}} d ξ d η G (q^{F} - F \partial_{ξ} p^{F}) + F^{2} \int_{Γ} d η [G \partial_{ξ} φ - (φ - ϕ) \partial_{ξ} G] + \int_{Σ^{H}} d a [G (F n^{x} + σ^{H}) - (φ - ϕ) n \cdot \nabla_{ξ} G]$ where $p^{F} \equiv p^{F} (ξ, η)$ and $q^{F} \equiv q^{F} (ξ, η)$ denote the pressure or the flux at a point $(ξ, η, 0)$ of the free surface $Σ^{F}$ , as was already noted, and $ϕ \equiv ϕ (x)$ . Moreover, $d a \equiv d a ($ $ξ$ ) and $σ^{H} \equiv σ^{H} ($ $ξ$ ) are the

Consistent linear flow model and partial elimination of the waterline integral

An interesting and important modification of the Neumann–Kelvin flow representation (9) is given in [2]. This modification is based on a short elementary ‘physical’ argument, which is related to a basic flow-modeling consideration. The argument is repeated here for completeness.

Expression (4) shows that the linear approximation to the free-surface elevation is given by $ζ^{F} (ξ, η) = F \partial_{ξ} φ$ in the common case of flows around ships, for which one has $p^{F} = 0$ . The contribution of the band of water bounded by

A mathematical transformation and total elimination of the waterline integral

The boundary-integral representation (11) involves a surface integral over the ship hull surface $Σ^{H}$ and a line integral around the waterline $Γ$ . An alternative boundary-integral representation that only involves a surface integral over $Σ^{H}$ can be obtained via a mathematical transformation based on a vector Green function $G$ that is related to the scalar Green function $G$ as $G \equiv (0, \partial_{ζ}^{ξ} G, - \partial_{η}^{ξ} G)$ where $\partial^{ξ}$ means integration with respect to $ξ$ . Expression (12a) yields $\nabla_{ξ} G = \nabla_{ξ} \times G + T where T \equiv (\partial^{ξ} \nabla_{ξ}^{2} G, 0, 0) .$ One has $m \cdot$

Velocity representation

Another notable difference between the boundary-integral flow representation (17) and the flow representations (9), (11) is that the representation (17) determines the flow potential $ϕ (x)$ at $x \in Σ^{H}$ in terms of the flow velocity at the ship hull surface $Σ^{H}$ , whereas the representations (9), (11) determine $ϕ$ in terms of the potential $φ$ at $Σ^{H}$ .

Indeed, the flow velocity $n \times \nabla_{ξ} φ$ at a point $ξ$ of the ship hull surface $Σ^{H}$ is tangent to $Σ^{H}$ and can be expressed as $n \times \nabla_{ξ} φ = n \times (σ \partial_{σ} φ + τ \partial_{τ} φ)$ where $n$ is the unit vector

Approximate theories

The boundary-integral flow representation (19) corresponds to the boundary-value problem (3), where the boundary condition at the ship hull surface $Σ^{H}$ is satisfied exactly. Useful approximations can be associated with the exact (within the context of linear potential flow theory) flow representation (19).

In particular, the approximations $\partial_{τ} φ \approx 0$ and $\partial_{σ} φ \approx 0$ on the right sides of the flow representation (19a) yield the flow approximation $ϕ \approx ϕ_{H}$ where $4 π ϕ_{H} \equiv \int_{Σ^{H}} d a (G^{R} + G^{L} + G^{W}) (F n^{x} + σ^{H})$ and (2) was used. The

Scalar and vector Green functions

The Green function $G$ determined by Eqs. (6) can be expressed as $4 π G = G^{R} + G^{F}$ where $G^{R}$ and $G^{F}$ represent Rankine and Fourier components defined in terms of elementary (free-space) Rankine sources or a double Fourier superposition of elementary wave functions. The Rankine–Fourier decomposition (23a) of the Green function $G$ and expression (12a) for the corresponding vector Green function $G$ yield $4 π G = G^{R} + G^{F} where$ $G^{R} \equiv (0, \partial_{ζ}^{ξ} G^{R}, - \partial_{η}^{ξ} G^{R}) and G^{F} \equiv (0, \partial_{ζ}^{ξ} G^{F}, - \partial_{η}^{ξ} G^{F})$ represent Rankine and Fourier components.

The

Conclusion

In summary, a new boundary-integral flow representation associated with the Neumann–Kelvin boundary-value problem for linear potential flows around ships steadily advancing in calm water has been given. Unlike the alternative boundary integrals given in the literature, this new flow representation, given by (19) or (21), does not involve a line integral around the ship waterline.

The mathematical transformation (12) is a crucial element of the flow representation (21). This mathematical

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

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The dominant computational task involved in the methods, widely called panel methods, currently used to compute 3D potential flows around large floating bodies such as ships and offshore structures (and other classes of dispersive waves such as flexural-gravity waves in very large floating structures or ice sheets) consists in evaluating $N^{2}$ coefficients, called influence coefficients. These coefficients are associated with the flows created by distributions of singularities (sources, dipoles) over the $N$ panels that approximate the surface of the body (ship, offshore structure) at every panel of the body surface. In contrast to the $O (N^{2})$ computations required in existing panel methods, the dominant part of the computations of influence coefficients are independent of $N$ in the Fourier–Kochin (FK) method if the Kochin functions in the FK flow representation are approximated by means of Fourier series and Chebyshev polynomials. The numerical analysis reported in this study shows that Fourier–Chebyshev approximations to the Kochin functions are feasible and indeed practical. In particular, the analysis shows that the number $N^{F}$ of basic Fourier integrals (the major and most difficult computational task) that must be evaluated within the approach to the numerical implementation of the FK method considered in the study is given by $N^{F} \approx {(600)}^{2},$ i.e. is smaller than $N^{2}$ if $600 < N .$ Thus, one has $N^{F} ≪ N^{2}$ for typical panel numbers $N = O (1 0^{4}),$ and the FK approach opens the way for a class of panel methods in which the dominant computations are independent of the number $N$ of panels instead of $O (N^{2})$ in existing panel methods. Another major advantageous feature of the FK method is that this approach circumvents the notorious difficulties involved in the numerical evaluation and the panel-surface-integration of the complicated Green functions associated with ship and offshore hydrodynamics.
Three alternative boundary integral flow representations for wave diffraction-radiation by offshore structures
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Citation Excerpt :
Specifically, the boundary integral flow representation based on the free waterplane linear flow model contains a line integral around the ship waterline, whereas the rigid waterplane flow model applied in He et al. (2022b) yields a boundary integral representation that does not contain a waterline integral. The notorious difficulties associated with the waterline integral in the usual linear flow model, documented in Marr (1996), Diebold (2007), Liang et al. (2018), Noblesse et al. (2013), He et al. (2018, 2021a) and Chen et al. (2018) and other studies, are then moot in the rigid waterplane flow model, which also settles a fifty-year old puzzle related to the waterline integral in the usual Neumann–Kelvin problem of flows around ships advancing in calm water or in waves as is noted in He et al. (2021a, 2022b) . Thus, the formulation of a linear flow model and of a related boundary integral flow representation – which evidently is the first, and arguably most important, step in the development of a panel method – involves nontrivial fundamental issues that require careful consideration, although these basic issues are largely ignored in the extensive applications of the method of Green function and boundary integral (panel method) reported in the marine hydrodynamics literature.
Three alternatives to the classical boundary integral representation of diffraction-radiation of regular waves by large stationary bodies, such as offshore structures and moored ships, that underlies existing panel methods are defined. These three alternative flow representations are associated with two alternative linear flow models, called ‘free waterplane flow model’ and ‘rigid waterplane flow model’, of potential flow around an offshore structure in regular waves. As was shown previously, these two alternative linear flow models of diffraction-radiation of regular waves by stationary bodies yield identical boundary integral flow representations and are then consistent, although the rigid waterplane flow model precludes irregular frequencies. Moreover, this flow model – and mathematical transformations – yield two other boundary integral flow representations, so that three alternatives to the classical boundary integral flow representation used in existing panel methods are defined. These three alternative flow representations are weakly singular. The first of these flow representations, given previously, involves a surface integral over the waterplane inside the body, while the second flow representation involves a line integral around the waterline of the body, i.e. the intersection curve between the body and the undisturbed free surface. The third flow representation is of particular interest because it involves neither a surface integral over the waterplane nor a line integral around the waterline. Moreover, this boundary integral flow representation does not involve a distribution of dipoles over the body surface; indeed, it expresses the flow potential in terms of the normal and tangential components of the flow velocity at the surface of the body. It is also notable that this boundary integral flow representation is identical to the boundary integral representation previously obtained for flows around ships advancing at a constant speed in calm water or in regular waves, and thus provides a common basis for the analysis of three major classes of flows in ship and offshore hydrodynamics.
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Thus, the ultimate value of a linear flow model cannot be predicted a priori. However, the linear flow models considered in this study and in [24,40–42] arguably are interesting and reasonable alternatives to the free-waterplane linear flow model that are worth considering. Moreover, the fact that the rigid-waterplane flow model applied in the present study and the different linear flow model considered in the NM theory expounded in [24,40–42] yield identical boundary integral flow representations, and the fact that the NM theory has been found to yield realistic predictions and to be useful for hull-form optimization in [43–57], suggest that the rigid-waterplane flow model may well provide a useful basis for the analysis of flows around ships advancing in regular waves.
The basic issue of formulating a linear flow model for the analysis of potential flow around a ship advancing at a constant speed in regular waves is considered. Specifically, the ‘rigid-waterplane flow model’, previously considered for diffraction–radiation of regular waves by offshore structures, is applied to analyze flows around ships advancing in regular waves. A straightforward application of Green’s fundamental identity to the rigid-waterplane flow model yields a remarkable new boundary integral flow representation. Indeed, this flow representation does not involve a line integral around the mean waterline of the ship hull surface, is weakly singular, does not involve a distribution of dipoles over the ship-hull surface, and is free from irregular frequencies. In the particular case of flow around a ship steadily advancing in calm water, the boundary integral flow representation obtained in the study via an analysis based on the rigid-waterplane flow model is identical to the flow representation associated with the Neumann–Michell theory, which is based on a different linear flow model that only considers the flow region strictly outside the ship but consistently accounts for the linear contribution of the thin band of water between the wave profile along the ship hull surface and the undisturbed free surface. This result strongly suggests that the classical application of Green’s basic identity to analyze linear potential flow around a ship advancing in regular waves or in calm water, which yields a notoriously troublesome line integral around the ship waterline, may be questionable due to possible incompatibilities between the Neumann boundary condition at the ship hull surface and the linear free surface boundary condition associated with flows around ships advancing in calm water or in regular waves. This fundamental difficulty may be alleviated in the rigid-waterplane flow model, which imposes a compatibility condition between the Neumann boundary condition at the rigid lid that artificially closes the open ship hull surface and the linear boundary condition at the free surface above the waterplane-lid. The boundary integral flow representation given in the study also holds for diffraction–radiation of regular waves by offshore structures. Thus, this new flow representation provides a common mathematical basis, which fundamentally differs from the basis steadfastly adopted over the past fifty years, for the analysis of three basic classes of flows in ship and offshore hydrodynamics, and indeed can be applied more generally to other boundary-value problems such as those associated with flexural-gravity waves for ice sheets or very large floating structures.
Optimal Fourier–Kochin flow representations in ship and offshore hydrodynamics: Theory
2022, European Journal of Mechanics, B/Fluids
This study considers the core issue of evaluating flows created by general distributions of sources and dipoles over panels (typically flat/curved triangles/quadrilaterals) used to approximate the surface of a body (ship or offshore structure) in usual implementations of the Green function and boundary integral method in marine hydrodynamics. This crucial basic issue is considered – within the framework of the Fourier–Kochin (FK) method – for four classes of flows in deep-water ship and offshore hydrodynamics: diffraction-radiation of regular waves by an offshore structure, and flow around a ship that advances at a constant speed $V$ in calm water or in regular waves of (encounter) frequency $ω$ in the regimes $τ \equiv ω V / g < 1 / 4$ or $0.3 \leq τ$ where $g$ is the acceleration of gravity; diffraction-radiation of regular waves by an offshore structure in water of uniform finite depth is also considered. Two notable features of the Green functions $G$ used in the study of these five classes of flows are that (i) they are based on optimal decompositions into Rankine and Fourier components $G^{R}$ and $G^{F},$ and that (ii) these Green functions are consistent. In particular, a new representation of the Green function for flows around ships advancing in regular waves at $τ < 1 / 4$ that is consistent with the Green functions for the special cases $V = 0$ or $ω = 0$ is given. The study also provides a complete and largely self-contained account of the FK method for evaluating the Fourier component $ϕ^{F}$ in the Rankine–Fourier decomposition $ϕ^{R} + ϕ^{F}$ of the velocity potential $ϕ$ (and the related velocity) of the flow created by a general distribution of singularities. An essential element of this FK theory is an optimal waves/local-effects (WL) decomposition $ϕ^{F} = ϕ^{W} + ϕ^{L},$ given for a general dispersion relation associated with a broad class of dispersive waves, in which the component $ϕ^{W}$ represents the (far-field and near-field) waves contained in $ϕ^{F}$ and the component $ϕ^{L}$ corresponds to a non-oscillatory local disturbance that is mostly significant in a small near-field region. Applications of this general WL decomposition to the five classes of flows in marine hydrodynamics of primary interest in the study yield simple analytical flow representations – for general compact distributions of singularities – that offer two notable advantages over the classical direct Green function method: (i) Integration over hull-surface panels within the FK method only involves smooth ordinary functions, which are incomparably simpler than the highly singular functions $G^{F}$ and $\nabla G^{F},$ and (ii) The computations related to $G^{F}$ and $\nabla G^{F}$ in the FK theory – implemented in the manner expounded in the study – are proportional to the number $N$ of panels that approximate the body surface, whereas common panel methods require $O (N^{2})$ computations. Thus, the Fourier–Kochin method and the optimal Rankine–Fourier decompositions and optimal waves/local-flow decompositions given in the study lay the foundation of a new type of computational methods, which offers two compelling advantages over the classical ‘direct Green function method’ steadfastly applied in the past fifty years.

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