Interference effects on the Kelvin wake of a monohull ship represented via a continuous distribution of sources

European Journal of Mechanics B/Fluids 51 (2015) 27–36

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Interference effects on the Kelvin wake of a monohull ship represented via a continuous distribution of sources Chenliang Zhang, Jiayi He, Yi Zhu, Chen-Jun Yang, Wei Li, Yifei Zhu, Muyuan Lin, Francis Noblesse ∗ State Key Laboratory of Ocean Engineering, School of Naval Architecture, Ocean & Civil Engineering, Shanghai Jiao Tong University, Shanghai, China

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Article history: Received 22 July 2014 Received in revised form 24 December 2014 Accepted 27 December 2014 Available online 6 January 2015 Keywords: Ship waves Wave signature Kelvin wake Interference effects

abstract Wave-interference effects on the wave pattern of a monohull ship that advances at constant speed along a straight path in calm water are considered. The flow around the ship is represented via a continuous distribution of sources over the hull surface, rather than via a point source at the bow and a point sink at the stern. A practical and realistic numerical method for determining the apparent wake angle ψmax , where the largest waves are found at high Froude numbers, for general ship hulls (including multihulls and/or distributions of pressure at the free surface) is considered. The method only involves elementary numerical computations that can be performed simply and efficiently. The method is therefore used here to perform systematic computations for seven ship hulls that correspond to broad ranges of main hull-shape parameters (beam/length, draft/length, beam/draft, waterline entrance angle) at ten Froude numbers. This numerical study shows that the apparent wake angle ψmax is only weakly influenced by the shape of the ship hull. A useful practical consequence of this numerical finding is that the wake angle ψmax can be estimated for general monohull ships via simple analytical relations. These relations provide relatively-accurate practical estimates (without computations) of the apparent wake angle ψmax (F ) for arbitrary ship hull forms and Froude numbers. The explicit approximations for the wake angle ψmax (F ) obtained here account for both longitudinal interference between the divergent waves created by the fore and aft regions of a monohull ship and lateral interference between the divergent waves created by the port and starboard sides of the hull. As expected, these relations yield an estimate of the wake angle ψmax (F ) for a monohull ship that is more precise than the analytical estimate given by an elementary analysis of interference between the dominant waves created by the bow and the stern of a ship, which corresponds to a 2-point wavemaker approximation of the continuous hull-surface distribution of sources considered here. © 2015 Elsevier Masson SAS. All rights reserved.

1. Introduction 1.1. Kelvin’s analysis and related literature The simplest analysis of the wave pattern of a ship that advances at constant speed along a straight path, in calm water of large depth, was given by Kelvin in 1887 [1]. A ship is approximated quite simply, as a 1-point wavemaker, in this classical analysis. Although particularly simple, the 1-point wavemaker approximation is sufficient to determine major features of the far-field waves created by a ship. In particular, Kelvin showed that ship waves can only be found inside a wedge |ψ| ≤ ψK with ψK ≈ 19°28′ . Kelvin’s

∗

Corresponding author. E-mail address: [email protected] (F. Noblesse).

http://dx.doi.org/10.1016/j.euromechflu.2014.12.006 0997-7546/© 2015 Elsevier Masson SAS. All rights reserved.

analysis also shows that the pattern of transverse and divergent waves created by a ship does not depend on the length L or the shape of the ship and only depends on the ship speed V , specifically on (X , Y )g /V 2 where g denotes the acceleration of gravity and the origin (0, 0) of the horizontal coordinates (X , Y ) is taken at the centroid of the ship waterplane. Numerous experimental observations, analytical predictions and numerical computations show that the largest waves created by a ship are found in the vicinity of the cusp lines ψ = ±ψK of the Kelvin wave pattern, in agreement with Kelvin’s analysis. However, it has long been observed [2–8] that the largest waves created by a ship are found along ray angles ψ = ±ψmax that are inside Kelvin’s cusp lines ψ = ±ψK at high Froude numbers FK ≤ F with FK ≈ 0.6. In particular, [8] reports 37 observations of ship wakes for the wide range of Froude numbers 0.1 < F < 1.7. The wake angles for the 12 observations reported for 0.6 < F are consistently and significantly smaller than the Kelvin cusp angle ψK .

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C. Zhang et al. / European Journal of Mechanics B/Fluids 51 (2015) 27–36

Several theoretical explanations of these seemingly unexpected observations of ship wakes have been offered. These explanations include [5,7,9,10] where nonlinear, unsteady, wind, or finite waterdepth effects are invoked, and [8,11,12] that only involve linear steady potential-flow hydrodynamics. [8] appears to have motivated a series of studies, including [11–20], that consider various issues related to the Kelvin wake of a ship, although in fact most of these studies are only remotely related to Kelvin’s practical purpose of explaining the wave patterns created by common ships that advance at constant speed in calm water. Indeed, [8,11,15–18] consider the far-field waves created by a Gaussian distribution of pressure at the free surface, hardly a very realistic model of the flow around a common ship hull; [14] considers the waves created by a submerged point disturbance (source or dipole); [15] extends [11] to include the effect of a shear current; and [16] extends [8] to include surface-tension effects, negligible for typical ships [12]. Moreover, [8,11,14,16–18] mostly focus on the asymptotic behavior, in the limit F → ∞, of the apparent wake angle ψmax that corresponds to the largest waves. However, the high Froude number limit 1 ≪ F is unrealistic within the context of the classical theory of water waves considered in [8,11–20]. Indeed, this theory is woefully inadequate at very high Froude numbers, where wave-breaking and spray have a predominant influence. The wake angle ψmax is determined in [11,15,17] via a straightforward analysis, used in [21,22] nearly thirty years earlier, that is based on the numerical determination of the peak of the amplitude function in the Fourier–Kochin representation of far-field ship waves (for the special case of a Gaussian pressure distribution) and the method of stationary phase. The asymptotic (analytical and numerical) studies reported in [8,11,14,16–18] show that the wake angle ψmax decreases like 1/F as F → ∞. In fact, this result is implied by the relations (4a)–(4b) and (5) that follow from the basic relations (2a) and (3a) obtained in [12] via an elementary analysis of interference effects. Although mathematically correct, the conclusion ψmax = O(1/F ) as F → ∞ is shown here to hold only at very high (unrealistic) Froude numbers for a monohull ship. Indeed, the numerical computations reported here for simple (mathematically-defined) but realistic monohull ships (rather than Gaussian distributions of pressure at the free surface) show that the wake angle ψmax decreases approximately like 1/F 2 within the range FK ≤ F ≤ FX with FK ≈ 0.57 and FX ≈ 0.85, and like 1/F for FY ≤ F with FY ≈ 6. The transition between the 1/F 2 and 1/F regimes occurs gradually within the wide range FX < F < FY . The far-field waves created by ship hulls at realistic high Froude numbers 0.6 < F < 2 in deep or shallow water are considered in [12,19] and the present study. [8,16,18] consider numerical computations of far-field waves (for Gaussian pressure distributions) and also offer phenomenological models to explain their observations and computations of narrow wakes. These models ultimately amount to assumptions and restrictions with respect to the wavelength of the waves that are created by a ship. In particular, [8] relies on the key assumption that a ship does not create significant waves that are longer than the ship length. This assumption is not obvious for a study of the Kelvin wakes of ships at high Froude numbers 0.5 < F because the wavelength 2π F 2 of the longest waves created by a ship is larger than the ship length for 0.4 ≤ F . Moreover, this basic assumption is associated with flawed logic because the relations λ < λcut < 4π F 2 /3 and ψ < ψ cut < 19°28′ are mathematically equivalent for divergent waves, as noted in [19]. Thus, a model that invokes a cutoff-wavelength assumption λ < λcut , as in [8], cannot explain observations of ship wakes that are narrower than Kelvin’s wake unless the assumed choice of cutoff-wavelength λcut is rationally justified. No justification is given in [8], and that study therefore does not explain observed narrow ship wakes. Arguments based on the Cauchy–Poisson initial-value problem are belatedly offered in [18]. However, the benefits of considering the Cauchy–Poisson problem to analyze steady-state ship waves are not a priori obvi-

ous because the initial-value problem and the steady-state problem are based on identical basic flow physics, associated with linear inviscid waves. In particular, the correspondence between effects of wave-interference on the steady Kelvin wake of a ship and the assumption of a cutoff wavelength associated with the Cauchy–Poisson initial-value problem is not a priori evident. Within the context of an analysis that assumes linear inviscid waves, as considered here and in [8,11–13,15–19], waves can neither grow nor be attenuated due to nonlinear or dissipative effects. Within this context, the models and related assumptions offered in [8,16,18] are unnecessary, as is shown by the analysis given in [11] and here, and moreover are shown in [19] to be unconvincing. Indeed, the arguments based on the Cauchy–Poisson problem offered in [18] might be an indirect way of interpreting results, but these arguments do not provide a simple direct explanation of observed narrow ship wakes, and moreover do not yield a quantitative prediction of the apparent wake angle for common ships. In fact, these phenomenological models obfuscate the elementary flow physics, wave interference, that govern far-field ship waves (as well as numerous other classes of problems that involve waves, e.g. in optics or electromagnetism). In short, the observations of narrow ship wakes reported in [4–8] are merely the unsurprising result of interference effects between the dominant divergent waves created by the bow and the stern of a ship, as noted in [12], and more precisely are the result of longitudinal and lateral interference between the waves created by a continuous distribution of sources over the fore and aft regions of a ship hull or its port and starboard sides, as noted in [19] and shown here. Indeed, within the context of a theory of linear inviscid ship waves (and other dispersive waves), wave interference is the only flow physics that can explain (observed or computed) ship wave patterns, as already noted, and the sensational reference to a Mach-like wake angle made in [8,16] is then misleading. 1.2. Elementary analysis of wave interference Within the context of a linear potential-flow analysis, the flow around a ship hull can be represented via a continuous distribution of sources over the ship hull surface. The 1-point wavemaker approximation used in Kelvin’s analysis cannot account for interference effects that occur between the waves created by the sources distributed over a ship hull surface. Wave-interference effects are important. In particular, interference between the transverse waves created by the bow and the stern of a ship is an essential element of the design of common displacement ships, and interference between divergent waves is shown in [12] to result in the appearance of narrow wave patterns at high Froude numbers. Indeed, [12] shows how interference effects can be approximately taken into account via a trivial refinement of Kelvin’s classical analysis, based on a 1-point wavemaker approximation of a ship as already noted. Specifically, a monohull ship is approximated in [12] as a 2-point wavemaker, in accordance with the well-known property that a steadily-advancing ship creates two dominant waves that originate at the ship bow and stern (where the hull geometry varies most rapidly). The superposition of two basic Kelvin wakes with origins at the bow and the stern of a ship, although a trivial modification of Kelvin’s classical wave pattern, introduces an important additional parameter, the ship length L, that controls the occurrence of constructive or destructive interference between the two basic ‘bow and stern Kelvin wakes’. Thus, the superposition of two Kelvin wakes associated with the dominant waves created by the bow and the stern of a monohull ship, or by the bows (or sterns) of the twin hulls of a catamaran, introduces the Froude numbers F ≡ V / gL or Fs ≡ V / gS





(1)

where S denotes the lateral separation distance between the two hulls of a catamaran, and incorporates important additional flow physics associated with wave-interference effects. The amplitudes

C. Zhang et al. / European Journal of Mechanics B/Fluids 51 (2015) 27–36

of the dominant bow and/or stern waves created by a ship are not considered in the elementary analysis of wave-interference effects given in [12]. This analysis is then a ‘geometrical’ (kinematic) analysis, like Kelvin’s analysis. [12] shows that longitudinal (x) interference between two basic Kelvin wakes associated with the dominant bow and stern waves created by a monohull ship yields largest waves along rays ψ = x ±ψmax that are inside the cusp lines ψ = ±ψK of the Kelvin wake x for Froude numbers F x < F . The apparent wake angle ψmax and the x related Froude number F are given by the analytical relations

    π 2 F 4 /ℓ2 − 1 0.16 ℓ ψ ≈ arctan ≈ arctan (2a) 2π 2 F 4 /ℓ2 − 1 F2  √ and F x = (3/2)1/4 ℓ/π ≈ 0.62 ℓ. (2b) Here, ℓ denotes the nondimensional distance (related to the dimensional distance ℓL) between the effective origins of the bow x max

and stern waves, assumed to be located slightly aft of the bow or slightly ahead of the stern. For lack of better knowledge, ℓ is taken as ℓ = 0.9 in [12], as commonly used by naval architects in the analysis of interference between the transverse waves created by a ship bow and stern and the selection of a ship length L that avoids unfavorable interference effects (and related humps of the wave resistance curve). The choice ℓ = 0.9 in (2a)–(2b) yields ψmax ≈ arctan(0.14/F 2 ) and F x ≈ 0.59. [12] also shows that lateral (y) interference between two Kelvin wakes associated with the dominant waves created by the twin bows (or the twin sterns) of the two hulls of a catamaran similarly y yields largest waves along rays ψ = ±ψmax that are inside the cusp y lines ψ = ±ψK of the Kelvin wake for Froude numbers Fs < Fs y with Fs defined by (1). The wake angle ψmax and the Froude number y Fs are given by

  ψ

y max

≈ arctan 

1 + 16π 2 Fs4 − 1

2(1 + 16π 2 Fs4 )



   ≈ arctan 0.2 Fs

(3a)

√

and Fsy ≡ 31/4 /(2 π ) ≈ 0.37.

(3b)

The Froude √ numbers F that correspond to Fs = 0.37 are given by F = Fs S /L ≈ 0.17, 0.26, 0.33 for S /L = 0.2, 0.5, 0.8. The basic relations (2a) and (3a) show that the apparent wake angle ψmax decreases like 1/F 2 for longitudinal (x) interference or like 1/F for lateral (y) interference. These relations are based on a highly-simplified analysis that essentially approximates a continuous distribution of sources over a ship hull surface by means of a point source and a point sink (for a monohull ship) or two point sources (for a catamaran), i.e. as a 2-point wavemaker. This relatively crude approximation has the merit of providing insight into wave-interference effects, ignored in Kelvin’s classical analysis. Indeed, the elementary longitudinal (x) interference and lateral (y) interference analysis considered in [12] are basic elements of the more complicated wave-interference effects that are associated with a distribution of sources over a ship hull surface. Another merit of this elementary analysis is that it yields the simple analytical relations (2a) and (3a). These relations provide a realistic practical estimate of the apparent wake angle of a ship without computations. However, the analytical estimates (2a) and (3a) are based on a relatively crude 2-point wavemaker approximation, and therefore cannot be expected to be very accurate (obviously). The far-field waves created by a ship are influenced by both longitudinal interference between the waves created by sources and sinks distributed over the fore and aft regions of a ship hull and lateral interference between sources (or sinks) distributed over the port and starboard sides of the hull, as noted in [19]. The relations y x (2) and (3) show that the apparent wake angles ψmax and ψmax associated with longitudinal (x) or lateral (y) interference for a monohull ship can be expressed as

x ψmax ≈ arctan(0.16 ℓe /F 2 )  y ψmax ≈ arctan(0.2 be /F )

29

(4a) (4b)

where ℓe and be stand for an effective ship length or an effective ship beam. The relations (4a) and (4b) yield

 x y ψmax < ψmax if Fxy ≡ 0.8 ℓe / be < F .

(5)

E.g., (5) yields Fxy ≈ 1.4 for ℓe = 0.8 and be = 0.2. The foregoing elementary considerations suggest a gradual transition from a ‘low-speed regime’ to a ‘high-speed regime’, with wake angles ψmax given by (4a) or (4b), that are dominated by longitudinal or lateral interference. 1.3. Numerical analysis of wave interference The primary purpose and main merit of the highly-simplified analysis of wave-interference effects given in [12] are to explain how ‘narrow ship wakes’ can occur, and to obtain an analytical estimate of the apparent wake angle ψmax related to the largest waves created by a ship. A more precise analysis, and a more accurate estimate of ψmax than the estimate given by the analytical relations (2) and (3), requires computations. The far-field waves created by a ship hull, of length L, that advances at constant speed V along a straight path in calm water of large depth are then considered here, within the framework of linear potential flow theory. This theoretical framework is realistic, and the only practical option, to analyze far-field waves of interest here (and indeed is also realistic and useful for near-field ship waves). The flow due to the ship is observed from a Galilean frame of reference and a related system of orthogonal coordinates X ≡ (X , Y , Z ) attached to the moving ship, and thus appears steady with flow velocity given by the sum of an apparent uniform current (−V , 0, 0) opposing the ship speed V and the (disturbance) flow velocity given by the gradient (ΦX , ΦY , ΦZ ) of the flow potential Φ (X). The undisturbed free surface is chosen as the plane Z = 0 with the Z axis directed upward, and the X axis is taken along the ship path and directed toward the ship bow. The length L and the speed V of the ship are used to define nondimensional coordinates x ≡ X/L, flow potential φ ≡ Φ /(VL) and flow velocity ∇x φ ≡ (φx , φy , φz ) ≡ (ΦX , ΦY , ΦZ )/V . Within the framework of the theory of linear inviscid waves considered here, the flow around a ship hull can be represented by means of a distribution of singularities (sources and sinks) over the ship hull surface, as already noted. These elementary singularities correspond to a fundamental solution (Green function) associated with the boundary condition at the free surface. This fundamental solution, commonly called ‘free-surface Green function’, has been widely considered in the ship-hydrodynamics literature and is given in e.g. [23]. The density of the distribution of sources over the ship hull surface in the linear potential flow theory of ship waves is explicitly specified in terms of the hull geometry in the approximate theories associated with the classical Hogner approximation [24] or the similar slender-ship approximation given in [25]. Alternatively, the source density can be determined numerically via the Neumann–Kelvin theory [26,27] or the related Neumann–Michell theory [28]. The Hogner approximation [24] and the approximation given in [25] are shown in [28,25] to correspond to approximate solutions of the Neumann–Michell or Neumann–Kelvin theories given in [28] or [26,27], respectively. The slender-ship approximations given in [24,25] are also closely related to Michell’s classical thin-ship approximation [29], and indeed can be regarded as extensions of Michell’s approximation. The slender-ship approximations given in [24,25] are of practical interest because they are particularly simple yet are realistic, although (obviously) not highly accurate. In particular, numerical

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C. Zhang et al. / European Journal of Mechanics B/Fluids 51 (2015) 27–36

predictions (of the sinkage, trim, and drag experienced by several ship hulls, and of wave profiles along the hulls, for a range of Froude numbers) based on the Hogner approximation are found in [28,30] to be consistent with experimental measurements as well as numerical predictions given by the more accurate Neumann–Michell theory. The slender-ship approximations given in [24,25] and the related Michell thin-ship approximation [29] are considerably simpler than the Neumann–Michell or Neumann–Kelvin theories because they explicitly define the flow created by a ship in terms of the Froude number F (i.e. the speed and the length of the ship) and the hull shape via a distribution of sources with density nx equal to the x-component of the unit vector n ≡ (nx , ny , nz ) normal to the hull surface. An important major consequence of this property is that the far-field waves created by a ship (and the related wave drag of the ship) can be determined very simply, without having to compute the near-field flow around the ship hull, as is well known and noted in e.g. [25,28]. The Hogner approximation [24] is considered here. Thus, the flow around a ship hull is represented via a distribution of sources with density nx , in accordance with [24,28,30]. A well-known fundamental property, emphasized in e.g. [23,31], of the flow u ≡ ∇φ created by a distribution of sources is that the flow velocity u can be formally decomposed into waves uW that decay slowly away from the source distribution (ship hull L surface), and a non-oscillatory local flow u that decays rapidly. Specifically, uW decays like 1/ x2 + y2 and uL decays like 1/(x2 + y2 + z 2 )3/2 as is well known. The wave component uW therefore is dominant in the far field (indeed, uW is also dominant in the near field). An important consequence of the slow rate of decay of ship waves is that the waves, notably the far-field waves, created by a ship are greatly influenced by interference effects. In particular, constructive and destructive interference between the divergent waves generated by sources distributed over a ship hull surface result in largest waves along ray angles ψ = ±ψmax that, at sufficiently high Froude numbers F , are inside the classical Kelvin wedge ψ = ±ψK with ψK ≈ 19°28′ . The approach adopted here to determine the apparent wake angle ψmax of a ship represented via a distribution of sources over a ship hull surface closely follows [21,22]. A difference between [21, 22] and the present study is that the effect of a particular geometrical feature of a ship hull (the bow flare) on the apparent wake angle ψmax is of primary interest in [21,22], whereas the farfield waves created by an entire ship hull (rather than the ship bow) are considered here. Moreover, the slender-ship approximation given in [25] and Michell’s thin-ship approximation are considered in [21,22] instead of the Hogner approximation, used here. As already noted, the Hogner approximation fully determines far-field ship waves in terms of the Froude number F and the hull geometry, and is realistic and well suited to determine far-field ship waves [28,30]. Indeed, this approximation yields a practical numerical method to compute the apparent wake angle ψmax for arbitrary ship hulls and/or distributions of pressure at the free surface. This method, given in the next section, only involves elementary numerical computations that can be performed simply and efficiently. The method is used in Section 3 to compute the wake angle ψmax for seven hull forms at ten Froude numbers. These numerical computations yield insight into the relative importance of longitudinal and lateral interference effects for monohull ships. The systematic computations considered here show that the apparent wake angle ψmax is only weakly influenced by the hull shape. A useful practical consequence of this numerical finding is that the wake angle ψmax can be estimated for general monohull ships via simple analytical relations. These relations provide relativelyaccurate practical estimates (without computations) of the apparent wake angle ψmax (F ) for arbitrary ship hull forms and Froude numbers. The explicit approximations for the wake angle ψmax (F ) obtained here account for both longitudinal interference between

the waves created by the fore and aft regions of a monohull ship and lateral interference between the waves created by the port and starboard sides of the hull. As expected, these relations yield a practical estimate of the wake angle ψmax (F ) for a monohull ship that is more precise than the (fully-analytical) estimate (2a)–(2b) obtained via the simple analysis of interference between the dominant waves created by the bow and the stern of a ship (a 2-point wavemaker approximation of the hull-surface distribution of sources considered here) considered in [12]. 2. Far-field waves due to source and pressure distributions 2.1. Basic expression for the flow potential The flow created by a distribution of sources with density π H over a ship hull surface Σ H , and/or a distribution of pressure π F over a near-field region Σ F of the mean free-surface plane z = 0 where the boundary condition φz + F 2 φxx = π F holds for (x, y) ∈ Σ F , is a classical ship-hydrodynamics topic that has been widely considered; e.g. in [31,28]. The general case of an arbitrary distribution of sources π H and/or pressure π F is considered here, as in [28]. The velocity potential φ( x) at a flow-field point  x ≡ ( x, y, z ) is given by

φ( x) =

 ΣH

G( x, x)π H (x)da(x)

 − ΣF

G( x, x, y, 0) π F (x, y)dxdy

(6)

where da(x) denotes the differential element of area at a point x of the hull surface Σ H , and the Green function G ≡ G( x, x) represents the velocity potential of the flow that is created by a unit source located at a point x and is observed at a flow-field point  x. The Hogner approximation for the flow around the hull Σ H of a displacement ship corresponds to π H = nx and π F = 0. The pressure distribution π F over a finite region Σ F of the free surface is useful for hovercrafts, planing hulls and some hybrid ships. 2.2. The Green function An important basic property of the Green function G is that it can be expressed as the sum of a wave component W and a nonoscillatory local flow component L. The decomposition G = W + L is not unique. In particular, three alternative representations and decompositions are considered in [23]. The representation of G recommended in [23] is used here. The wave component W ( x, x) in this representation is given in e.g. [23,28,30] as W ( x, x) =

H (x − x)

πF2



q∞

Im

(q, Λ x) E (q, x)E (q, x)dq.

(7)

−q ∞

Here, H (·) is the usual Heaviside unit-step function, F is the Froude number (1), Im means that the imaginary part is considered,  E ≡  E (q, x) and E ≡ E (q, x) are the elementary wave functions

√ 2 2 2  2  E ≡ e (1+q )z /F + i 1+q (x+q y)/F √ 2 2 2 2 E ≡ e (1+q )z /F − i 1+q (x+qy)/F

(8) (9)

and q denotes the wavenumber. The finite limits of integration (q, ±q∞ and the function Λ x) in (7) filter unrealistic short waves, e.g. as in [32]. 2.3. Fourier–Kochin representation of waves The decomposition G = W + L yields the corresponding decomposition φ( x) = φ W ( x) + φ L ( x) where the wave potential φ W and the local-flow potential φ L are given by (6) with G taken as W or L, respectively. The wave potential φ W is dominant in the far field

C. Zhang et al. / European Journal of Mechanics B/Fluids 51 (2015) 27–36

(indeed, φ W is also dominant in the vicinity of the ship) and one then has φ ≈ φ W . Behind a ship, one has  x < x and H (x − x) = 1. Expressions (6) and (7) then define the ‘far-field’ wave potential φ W ( x) as the Fourier superposition

φ W ( x) =

F2

q∞



(q, Λ x)A(q) E (q, x)dq

Im

π

(10)

−q∞

of elementary waves  E (q, x) with amplitude A(q) given by the distribution A(q) ≡



1 F4

−

ΣH

1

π (x)E (q, x) da(x) H



F4

π (x, y)E0 (q, x, y) dxdy

(11)

of elementary waves E (q, x) given by (9) or E0 (q, x, y) defined as

√ E0 (q, x, y) ≡ e

−i

1+q2 (x+qy)/F 2

.

(12)

The free-surface elevation ζ ( x, y) = F ∂φ( x)/∂ x with  z = 0 is determined by (10) and (8) as 2

ζ ( x, y) =

F2

π



√   1 + q2 A(q)e i 1+q2 (x+qy)/F 2 dq Λ

q∞

Re

where  C x,  Sx,  C y and  S y are defined as

  S x ≡ sin  ϕ x where  ϕ x ≡ 1 + q2  x/F 2 (17a)  S y ≡ sin  ϕy  where  ϕ y ≡ q 1 + q2  y/F 2 (17b) x x y y  C , S ,  C are even functions of q, and  S is an odd function of q.  C x ≡ cos  ϕ x,  C y ≡ cos  ϕ y,

The wave potential (10) then yields

φ ( x) = W

(13)

−q ∞

where Re means that the real part is considered. The Fourier–Kochin representation (8)–(13) of the far-field waves created by a distribution of sources over a ship hull surface Σ H and a pressure distribution over a free-surface patch Σ F is a straightforward consequence of well-known basic results, notably the integral representation of the Green function given in [23], and is given in e.g. [28,30]. For the common case of a ship hull with port and starboard symmetry that advances straight ahead, the source density π H (x) and the free-surface pressure π F (x, y) are even functions of y, and the ranges of integration in the Kochin representation (11) of the amplitude function A(q) and in the Fourier integrals (10) and (13) can be halved. These expressions for port and starboard symmetry are given in [28,30] for the Neumann–Michell theory, and are also given below for the particular case of the source and pressure distribution (6). 2.4. Port and starboard symmetry

ζ ( x, y) =

E=e

E0

where E0 ≡ (C x − i S x )(C y − i S y ).

(14)

Here, C x , S x , C y and S y are defined as



1 + q2 x/F 2

S x ≡ sin ϕ x

where ϕ x ≡

C y ≡ cos ϕ y ,

S y ≡ sin ϕ y

where ϕ y ≡ q 1 + q2 y/F 2 . (15b)

(15a)



For the common case of a ship hull with port and starboard symmetry that advances straight ahead, the source density π H and the free-surface pressure π F are even functions of y, as already noted. Expressions (11) and (14) then yield 2



F4

−

2 )z /F 2

H Σ+

2 F4

π H e (1+q

 ΣF

C y (C x − i S x ) da

π F C y (C x − i S x ) dxdy ≡ Are + i Aim

(16)

+

H F where Σ+ and Σ+ denote the positive halves 0 ≤ y of Σ H or Σ F . The amplitude function (16) is an even function of q. Expression (8) for the function  E yields

 E=e

(1+q2 ) z /F 2

( C x + i S x )( C y + i Sy)

e (1+q2 )z /F 2 Λ C y ( S x Are +  C x Aim ) dq.

(18)

0

2F 2

π

q∞



  1 + q2  Λ C y ( C x Are −  S x Aim ) dq.

(19)

0

2.5. Stationary-phase approximation Only the free-surface elevation ζ ( x, y) is now considered. Expression (13) is taken as starting point, i.e. Zg V2

=

1

π

q∞

 Re



1 + q2 Ae ihϕ dq.

(20)

−q ∞

 was ignored, the amplitude Here, the short-wave filter function Λ function A is given by (16), h denotes the nondimensional speedscaled horizontal distance h ≡ Hg /V 2 ≡

 ξ 2 + η2 where

(21)

(ξ , η) ≡ ( x, y)/F ≡ ( X, Y )g /V 2 (22)  2 and ϕ ≡ 1 + q (ξ + q η)/h. The phase function ϕ and its derivatives ϕ ′ ≡ dϕ/dq and ϕ ′′ ≡ d2 ϕ/dq2 are then given by  (23a) ϕ ≡ (q sin ψ − cos ψ) 1 + q2 2

ϕ′ ≡ ϕ ′′ ≡

C x ≡ cos ϕ x ,

A ≡

π

q∞



The Fourier–Kochin representation (15)–(19) determines the farfield wave potential φ W ( x) and the related free-surface elevation ζ ( x, y) for the waves created by a source and pressure distribution (6) that is symmetric about the centerplane y = 0. These basic analytical representations of φ W and ζ are simplified further if the method of stationary phase is used to approximate the Fourier integrals (18) and (19). This classical analytical approximation is now given.

Expressions (9) and (12) yield (1+q2 )z /F 2

2F 2

The free-surface elevation is then given by

F

ΣF

31

(1 + 2q2 ) sin ψ − q cos ψ  1 + q2

(23b)

q(3 + 2q2 ) sin ψ − cos ψ

(23c)

(1 + q2 )3/2

where ψ is defined as tan ψ ≡

 η Y ≡ . −ξ − X

(24)

For large values of h ≡ ξ 2 + η2 , the dominant contributions to the wave integral (20) stem from the points where the phase ϕ of the trigonometric function in (20) is stationary, as is well known. These points of stationary phase are then determined by the roots of the equation ϕ ′ = 0, i.e. the equation



(1 + 2q2 ) tan ψ − q = 0.

(25)

Inside the Kelvin wedge, i.e. for −ψK < ψ < ψK with ψK ≈ 19°28′ , the stationary-phase relation (25) has two distinct roots qT and qD that are given by 2 tan ψ

qT ≡ 1+



1 − 8 tan2 ψ

and

qD ≡

1+



1 − 8 tan2 ψ

4 tan ψ

(26)

32

C. Zhang et al. / European Journal of Mechanics B/Fluids 51 (2015) 27–36

Fig. 1. Amplitude ζ D of the divergent waves, defined by (32), created by the hull (33) and predicted by the Hogner approximation, for 2°30′ ≤ ψ < 19°28′ and six Froude numbers F within the range 0.65 ≤ F ≤ 1.5. The apparent wake angle ψmax where the largest waves are found is determined by the first peak, marked by a dot and a dashed vertical line, of the amplitude function ζ D .

and correspond to transverse and divergent waves. The roots qT and qD satisfy the relations qT qD =

1 2

−1 1 1 √ ≤ qT ≤ √ , √ < |qD |.

,

2

2

and qD = ±qC

√

where qC ≡ 1/ 2 ≈ 0.7.

(28)

−ϕ =



1 − 8 tan2 ψ

√

8 2

cos ψ ∆ψ | tan ψ|

(29a)

√  ϕD′′ = 2 2 1 − 8 tan2 ψ | sin ψ|/∆ψ   where ∆ψ ≡ 1 + 4 tan2 ψ + 1 − 8 tan2 ψ.

(29b)

−ϕ T =



1 − 8 tan2 ψ

√

2

ϕT′′ = − cos2 ψ

(30a)

2∆ψ



1 − 8 tan ψ 2

2

∆ψ .

(30b)

V2

≈

2/π h

Im(eD + eT ) where

1 + (qT )2 i(hϕ T −π /4) e . ′′

(31c)

−ϕT

Here, AD and AT denote the values of the amplitude function A at q = qD or q = qT with qD and qT given by (26). The asymptotic approximation (31) is not valid in the vicinity of the cusp of the Kelvin wake, as is well known, because the second derivatives ϕD′′ and ϕT′′ vanish there as already noted. The transverse wave eT in the far-field stationary-phase approximation (31a) is ignored hereafter. This simplification is justified to analyze interference effects between divergent waves, dominant for ray angles that are well inside the cusp lines ψ = ±ψK , of primary interest here. In the vicinity of the cusp lines, transverse waves are important and should then be considered for an accurate analysis. 2.6. Largest waves and apparent wake angle Fig. 1 depicts the amplitude function ζ D ≡

ζD ≡





2/π F 2 |AD | [1 + (qD )2 ]/ϕD′′

z2 b y = ± (1 − 16x4 ) 1 − 2 2 d



Expressions (29b) and (30b) show that 0 < ϕD and ϕT < 0. At the cusp lines ψ = ±ψK , one has tan2 ψ = 1/8 and the second derivatives ϕD′′ and ϕT′′ vanish. In the far field 1 ≪ h, the wave integral (20) can be evaluated analytically via the classical stationary-phase approximation



 T

√

2/π F 2 |eD |, i.e. (32)

where AD , qD and ϕD′′ are given by (16), (26), (29b) and (29c), for the simple hull form ′′

Zg

(31b)

(29c)

Similarly, the phase ϕ T and the corresponding second derivative ϕT′′ at the stationary point q = qT associated with transverse waves are given by 3+

1 + (qD )2 i(hϕ D +π /4) e ′′

ϕD

e ≡A

The values ϕ D and ϕD′′ of the phase ϕ and its second derivative ϕ at the stationary point q = qD associated with divergent waves are determined by (26), (23a) and (23c) as 3−

e ≡A

T

′′

D

D

(27)

2

At the cusp lines ψ = ±ψK , qT and qD are equal and given by qT = ±qC

 D

′′

(31a)

 (33)

with beam/length ratio b = 0.15 and draft/length ratio d = 0.05. The function (32) is depicted in Fig. 1 for ray angles ψ within the range 2°30′ ≤ ψ ≤ 19°28′ and six Froude numbers F = 0.65, 0.7, 0.9, 1.1, 1.3 and 1.5. The ray angle ψ = ψmax where the largest waves are found is determined by the highest peak of the amplitude function (32).

C. Zhang et al. / European Journal of Mechanics B/Fluids 51 (2015) 27–36

33

Table 1 Main hull-shape parameters associated with the seven hulls considered in the computations. B/L

D/L

B/D

n

2α

0.15 0.15 0.15 0.1 0.15 0.2 0.25

0.05 0.05 0.05 0.1 0.075 0.05 0.025

3 3 3 1 2 4 10

1 2 3 2 2 2 2

33°24′ 61°55′ 83°59′ 43°36′ 61°55′ 77°19′ 90°

Fig. 3. Values of the effective length ℓe in the relation (4a), associated with longitudinal interference between the dominant waves created by a ship bow and stern, that correspond to the values of ψmax computed via the Hogner approximation and depicted in Fig. 2 for seven hulls and ten Froude numbers F within the range 0.6 < F ≤ 1.5.

Fig. 2. Apparent wake angle ψmax , determined numerically from the first peak of the amplitude function (32) computed via the Hogner approximation, for seven hulls and ten Froude numbers within the range 0.6 < F ≤ 1.5. The wake angle ψmax defined by the approximate relations (39) as well as the Kelvin angle ψK and x the analytical approximation ψmax obtained in [12] via an elementary analysis of interference between the dominant waves created by the bow and the stern of a ship are also shown.

This peak is the ‘first’ peak, located closest to the cusp angle, and it is marked by a dot and a dashed vertical line in Fig. 1. As already noted, the second derivative ϕD′′ vanishes at the cusp angle ψ ≈ 19°28′ , where the function (32) is then unbounded. Fig. 1 suggests that the value of the wake angle ψmax , determined from the location of the first peak as just noted, is not appreciably influenced by the weak singularity of the amplitude function (32) at the cusp angle, except possibly for F = 0.65. Fig. 1 also shows that the first peak is sharp and significantly higher than all the other peaks for Froude numbers 0.9 ≤ F , but is broad and not as dominant for F ≤ 0.7. 3. Numerical computations The amplitude function (32) is now considered for seven simple hulls. Three hulls are defined by y=±

b 2



[1 − (2x) ] 1 − 2n

z2 d2

 (34)

with (b, d) = (0.15, 0.05) and n = 1, 2, 3. The waterline entrance angles 2α for the hulls that correspond to n = 1, 2, 3 are approximately equal to 33°, 62°, 84°, and these three hulls are identified as fine, medium and blunt in Figs. 2 and 3, considered further on. Four hulls are defined by (34) with n = 2 and beam/length ratio b and draft/length ratio d chosen as

(b, d) = (0.1, 0.1), (0.15, 0.075), (0.2, 0.05), (0.25, 0.025). (35) The beam/length and draft/length ratios for these four hulls vary within the broad ranges 0.1 ≤ b ≤ 0.25 and 0.025 ≤ d ≤ 0.1, and the beam/draft ratio is equal to 1, 2, 4 and 10. The waterline

entrance angles 2α for the four hulls are approximately equal to 44° for b = 0.1, 62° for b = 0.15, 77° for b = 0.2, and 90° for b = 0.25. The values of the beam/length ratio B/L, the draft/length ratio D/L, the beam/draft ratio B/D and the waterline entrance angle 2α for the seven hulls considered in the computations are listed in Table 1. This table shows that the seven hulls considered here are characterized by the broad ranges 0.1 ≤ 1≤

B

B L

≤ 0.25,

≤ 10,

0.025 ≤

D L

≤ 0.1, (36)

33° ≤ 2 α ≤ 90°

D of the main hull-form parameters B/L, D/L, B/D and 2α . Fig. 2 depicts the apparent wake angle ψmax , determined numerically from the first peak of the amplitude function (32) as already explained, for the seven hulls and ten Froude numbers F ≈ 0.65 and F = 0.7, 0.8, . . . , 1.5. This figure shows that the apparent wake angle ψmax is only weakly influenced by the hull shape. The effective lengths ℓe in the relation (4a), associated with longitudinal interference between the dominant waves created by a ship bow and stern, that correspond to the values of ψmax computed in the manner just explained and depicted in Fig. 2 are determined from (4a) as

ℓe = 6.28F 2 tan ψmax .

(37)

These values of ℓe are depicted in Fig. 3 for the seven hulls and the ten Froude numbers considered in Fig. 2. Fig. 3 shows that the effective length ℓe in the relations (4a) and (37) associated with strict longitudinal interference is significantly affected by the shape of a ship hull. However, Fig. 2 shows that the wake angle ψmax is only weakly influenced by the hull shape, as already noted. A rough fit to the numerical values of ℓe depicted in Fig. 3 is then sufficient to obtain a useful practical approximation for the wake angle ψmax . The fit

ℓe ≈ 0.725 for F ≤ 0.85 ℓe ≈ 0.3 + 0.5F for 0.85 ≤ F

(38a) (38b)

is used here. Fig. 3 shows that the influence of dominant hull-shape parameters, notably the beam/length ratio, on the effective length ℓe could be taken into account via a more precise fit. The relation (38b) yields 1 ≤ ℓe if 1.4 ≤ F , and Fig. 3 indeed includes several computed values of ℓe that are greater than 1. These seemingly-surprising numerical results in fact are consistent with the property that the wave signature of a monohull ship is not only

34

C. Zhang et al. / European Journal of Mechanics B/Fluids 51 (2015) 27–36

Fig. 4. Free-surface elevation Z /L for the hull defined by (33) at four Froude numbers F = 0.65, 0.8, 1 and 1.4. The upper and lower halves of the wave patterns are computed via the Hogner approximation or the more precise Neumann–Michell theory given in [28,30]. The figure also shows three systems of rays with origins at the bow of the hull. x These rays are the Kelvin rays ψ = ±19°28′ and the rays ψ = ±ψmax with ψmax given by (39) or by the analytical approximation ψmax obtained in [12] via an elementary analysis of interference between the dominant waves created by the bow and the stern of a ship. In accordance with Fig. 2, the rays ψ = ±ψmax with ψmax given by (39) are located inside the rays defined in [12] for F = 0.65, 0.8 and 1, but the opposite holds for F = 1.4. The rays ψ = ±ψmax defined by (39) or in [12] are very close for F = 1 and F = 1.4, also in accordance with Fig. 2. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

influenced by longitudinal interference, as assumed in (4a) and the related simple analysis considered in [12], but is also significantly influenced by lateral interference. Moreover, Fig. 3 and (38b) show that the importance of lateral interference grows with speed. The relations (38) and (2a) yield the simple approximate expressions tan ψmax ≈ 0.116/F 2

for F ≤ 0.85

tan ψmax ≈ 0.08(1 + 0.6/F )/F

for 0.85 ≤ F

(39a) (39b)

for the apparent wake angle ψmax . The wake angle ψmax defined by the relations (39a)–(39b) is shown as a solid line in Fig. 2. The x Kelvin angle ψK and the analytical approximation ψmax given by (2a)–(2b), obtained in [12] via an elementary analysis of interference between the dominant waves created by the bow and the stern of a ship, are also depicted in Fig. 2. This figure shows that the simple approximation ψmax (F ) given by (39a)–(39b) is in relatively good agreement with the numerical computations of ψmax for ten Froude numbers and seven hulls that correspond to a wide range of main hull-form parameters, and that the apparent wake angle ψmax is only weakly influenced by the hull shape as already noted. Fig. 4 shows the waves created by the hull defined by (33) at four Froude numbers F = 0.65, 0.8, 1 and 1.4. The free-surface elevation Z /L is depicted within the regions

(−9 ≤ x ≤ 1, −3 ≤ y ≤ 3) for F = 0.65 (−14 ≤ x ≤ 1, −4 ≤ y ≤ 4) for F = 0.8 and F = 1 (−19 ≤ x ≤ 1, −6 ≤ y ≤ 6) for F = 1.4 of the free-surface plane (x, y) ≡ (X , Y )/L. The upper and lower halves of the wave patterns shown in Fig. 4 are computed via the Hogner approximation or the more precise Neumann–Michell theory given in [28,30]. The Hogner and Neumann–Michell wave patterns shown in Fig. 4 are consistent. This figure and the comparisons of experimental measurements and numerical predictions (of the sinkage, trim, and drag, and of wave profiles along the hulls)

based on the Hogner approximation and the Neumann–Michell theory that are reported in [28,30] for several ship hulls and a range of Froude numbers show that the Hogner approximation is realistic (as well as practical). Fig. 4 also shows three systems of rays with origins at the bow (0.5, 0) of the hull. These rays consist of the Kelvin rays ψ = x ±19°28′ , the rays (black solid lines) ψ = ±ψmax defined by (2a) and obtained in [12] via an elementary analysis of interference between the dominant waves created by the bow and the stern of a ship, and the rays (pink solid lines) ψ = ±ψmax defined by the relations (39a)–(39b) and determined here from the first peak of the amplitude function (32) computed via the Hogner approximax tion. Fig. 4 shows that differences between the ray angle ψmax obtained from the highly simplified analysis given in [12] and the ray angle ψmax defined by the relations (39) obtained here from numerical computations are small for F = 1 and F = 1.4. Indeed, Fig. 2 shows that differences between these two wake angles do not exceed 1° for 1 ≤ F . Differences are larger, although not very large, for 0.6 ≤ F < 1. Fig. 4 also shows that the rays ψ = ±ψmax with ψmax given by (39a)–(39b) are located inside the rays defined in [12] for F = 0.65, 0.8 and 1, but the opposite holds for F = 1.4, in accordance with Fig. 2. As shown in Fig. 1 and already noted, the first peak of the amplitude function (32) is sharp and significantly higher than the other peaks of the function (32) at high Froude numbers 0.9 ≤ F , but is broad and not as dominant at the lower Froude numbers F = 0.7 and (especially) F = 0.65. Accordingly, the waves depicted in Fig. 4 are significantly larger (dominant) within a fairly narrow wedge in the vicinity of the rays ψ = ±ψmax for F = 1.4 and (to a lesser extent) F = 1. However, waves are significant within a relatively broad region around the rays ψ = ±ψmax for F = 0.8 and 0.65. This result means that scatter can be expected in observations of the apparent wake angle ψmax of a ship at Froude numbers F smaller than about 1. The far-field waves depicted in Fig. 4 are consistent with the amplitude function (32) depicted in Fig. 1, as expected.

C. Zhang et al. / European Journal of Mechanics B/Fluids 51 (2015) 27–36

4. Conclusion A main contribution of this study is a practical and realistic method for determining the apparent wake angle ψmax , where the largest waves are found, for arbitrary ship hulls and/or distributions of pressure at the free surface. The method only involves elementary numerical computations that can be performed simply and efficiently. Specifically, the method mostly requires the computation of the amplitude function A defined by the distribution (16) of elementary waves (15) over a ship hull surface Σ H and/or a free-surface patch Σ F . This method can be applied to realistic ship hulls (including multihulls) of arbitrary shape as well as general pressure distributions over the free surface. The method is then considerably more general than the special cases of axisymmetric or elliptical Gaussian pressure distributions considered in [8, 11,17,18]. The method closely follows [21,22] and is a straightforward application of basic results, notably the expression for the wave component in the Green function associated with the linearized free-surface boundary condition given in e.g. [23], that are widely known in the ship-hydrodynamics literature. This basic ship-hydrodynamics literature is ignored in [8,11,17,18] where unrealistic Gaussian distributions of pressure at the free surface are used to model the flow around a ship hull, and the widely-known fundamental decomposition of the ‘free-surface Green function’ into a wave component given by a single Fourier integral and a local-flow component is ignored. The computations reported here (for seven simple hull forms) show that the far-field waves created by a monohull ship are significantly influenced by longitudinal interference between the waves created by sources and sinks distributed over the fore and aft regions of a ship hull, as well as lateral interference between the waves created by sources (or sinks) distributed over the port and starboard sides of the hull. Thus, interference effects between the divergent waves created by a monohull ship are seemingly complex. The elementary considerations given in [12] and the computations reported here show that two main regimes can be defined for the wave signature of a monohull ship. For Froude numbers F < FK with FK ≈ 0.6, interference effects between transverse waves are dominant, and the Kelvin cusp angle ψK ≈ 19°28′ is a realistic practical estimate of the wake angle ψmax where the largest waves can be expected. However, interference effects between divergent waves are dominant for high Froude numbers FK < F . Furthermore, the ‘divergent-wave-interference’ regime FK < F can be subdivided into three regimes FK < F < FX , FX < F < FY , FY < F with FX ≈ 0.85 and 1 ≪ FY . Longitudinal interference between the divergent waves created by sources and sinks distributed over the fore and aft regions of a monohull ship is dominant in the ‘low-speed’ regime FK < F < FX , and the apparent wake angle ψmax decreases approximately like 1/F 2 in this regime in accordance with (4a). The ‘high-speed’ regime FY < F is predominantly influenced by lateral interference between the divergent waves created by sources (or sinks) distributed over the port and starboard sides of the hull, and ψmax decreases approximately like 1/F in this regime, in accordance with (4b). The computations reported here show that the transition between the predominantly ‘longitudinal’ interference regime FK < F < FX and the predominantly ‘lateral’ interference regime FY < F is gradual rather than abrupt, and indeed occurs within the wide range of Froude numbers FX < F < FY . Most displacement ships correspond to the Kelvin ‘low-speed transverse-wave interference regime’ F ≤ FK ≈ 0.6 or the ‘lowspeed divergent-wave interference regime’ FK ≤ F ≤ FX ≈ 0.85. These two ‘low-speed’ regimes are predominantly influenced by longitudinal interference between the transverse or divergent waves that are mostly generated by the fore and aft regions of a ship hull. Realistic practical estimates (that do not require computations) of the wake angle ψmax in these two ‘low-speed’

35

regimes are given by

ψmax ≈ ψK ≈ 19°28′ for F ≤ 0.573 (40a) and tan ψmax ≈ 0.16 ℓe /F 2 for FK ≤ F ≤ FX . The numerical predictions, depicted in Fig. 3, of the distance ℓe between the effective origins of the dominant bow and stern waves suggest that ℓe should be chosen somewhat smaller than the value ℓe = 0.9 used in [12]. Fig. 3 shows that ℓe ≈ 0.725 is a more precise choice. This choice yields the practical estimate

ψmax ≈ arctan(0.116/F 2 ) for 0.573 ≤ F ≤ 0.85 (40b) instead of the estimate ψmax ≈ arctan(0.14/F 2 ) given in [12]. In the ‘high-speed divergent-wave interference regime’ FX ≤ F , the numerical computations reported here yield the practical estimate

ψmax ≈ arctan[0.08(1 + 0.6/F )/F ] for 0.85 ≤ F .

(40c)

This relation yields

ψmax ≈ arctan(0.08/F ) for 6 < F .

(40d)

This ‘very-high-Froude-number’ approximation is identical√ to the y lateral interference relation given by (4b) as tan ψmax ≈ 0.2 be /F with the effective beam be taken as be ≈ 0.16. The regime (40d) only occurs for very large Froude numbers and indeed cannot exist in practice because the Froude number FY ≈ 6 is extremely large, and beyond the scope of linear potential flow theory. Fig. 3 shows that large variations in the shape of a ship hull have a significant influence on the length ℓe between the effective origins of the bow and stern waves in the ‘longitudinal-interference relation’ tan ψmax ≈ 0.16 ℓe /F 2 . However, Fig. 2 shows that the apparent wake angle ψmax is only weakly influenced by the shape of a ship hull. Indeed, the simple estimate given by (40b)–(40c) agrees well, for practical purposes, with the numerical computations of ψmax reported here for broad ranges of basic hull-form parameters (beam/length, draft/length, beam/draft, waterline entrance angle). x Moreover, even the analytical estimate ψmax given by (2a)–(2b) obtained in [12] via an elementary analysis of interference between the dominant waves created by the bow and the stern of a ship (a 2-point wavemaker approximation to the distribution of sources over a ship hull surface considered here) is realistic, although the estimate (40b)–(40c) is more precise as expected. Thus, expressions (40a)–(40c) provide a simple estimate (without computations) of the apparent wake angle ψmax for a general monohull ship (of arbitrary shape), and this estimate indeed is the main practical result of the computations reported here. The far-field waves created by an arbitrary distribution of sources over a ship hull surface and/or an arbitrary distribution of pressure at the free surface are considered for the sake of generality and completeness in (11) and (16). However, numerical computations are only reported here for distributions of sources over ship hull surfaces because a distribution of pressure at the free surface is not a realistic model of the flow around a ship hull (with the possible exception of high-speed planing hulls), as emphasized in [12,19]. Moreover, far-field ship waves created by free-surface pressure distributions have already been extensively considered in [8,11,17,18], although for the special cases of axisymmetric [8,11] or elliptical [17,18] Gaussian distributions. [8,11] show that the wake angle ψmax decreases like 1/F as F → ∞. The numerical computations reported here show that, for monohull ships, this conclusion is only valid for very large Froude numbers (outside the scope of linear potential flow theory) and is then a mathematical result with limited practical relevance. The analysis considered in [8,11] for an axisymmetric Gaussian pressure distribution is extended in [17,18] to an elliptical Gaussian pressure distribution. The asymptotic analysis given in [17] shows that ψmax decreases like 1/F as F → ∞, as already found in [8,11]. The numerical computations reported in [18] show that ψmax is O(1/F 2 ) for intermediate values of F and O(1/F ) for large values of F , as found here for a distribution of sources over a hull surface.

36

C. Zhang et al. / European Journal of Mechanics B/Fluids 51 (2015) 27–36

The numerical results reported in [8,11,17,18] and here are fully consistent with the wave-interference analysis considered in [12]. Indeed, expression (11) shows that a distribution of pressure π F over a free-surface patch Σ F and a distribution of sources π H over a ship hull surface Σ H involve both longitudinal and lateral wave interference between the elementary waves defined by the wave functions E0 or E (except in the special case when Σ F and Σ H are line segments parallel to the x axis, for which only longitudinal interference effects occur). The relations (4a)–(4b) and (5) then show that the wake angle ψmax ultimately vanishes like 1/F as F → ∞. Depending on the distributions π F and π H , notably the shape of Σ F and Σ H , this asymptotic behavior may occur quickly (for a catamaran, or even an axisymmetric Gaussian pressure distribution) or slowly (for a monohull ship as shown here, or a pressure distribution over a slender patch as considered in [17,18]). An axisymmetric distribution of pressure π F → ∞ over a patch Σ F → 0 yields the special case of a singular pressure point, as considered by Kelvin, and corresponds to F → ∞ and ψmax → 0. This special case however is a strictly mathematical result with no practical relevance because very small values of ψ correspond to very short waves that are affected by surface tension, viscosity and breaking, and thus are outside the scope of a linear potential flow analysis. The practical method given here to determine the apparent wake angle ψmax can readily be applied to catamarans. Systematic computations for seven hull forms that correspond to a broad range of main hull-shape parameters (beam/length, draft/length, beam/draft, waterline entrance angle), separated by a nondimensional lateral distance s ≡ S /L that varies within √ the range 0.2 ≤ s ≤ 0.8, at Froude numbers 0.4 ≤ Fs ≡ F / gS ≤ 3.5 show that the largest waves are found at a wake angle ψmax that is only weakly affected by the shape of the hulls of the catamaran, as found here for monohull ships. Moreover, the computations show that the wake angle ψmax (F , s) can be estimated via simple analytical relations, similar to the relations (40a)–(40c) for monohull ships. In particular, the computations show that lateral interference effects are dominant if the separation distance s between the twin hulls of the catamaran is large enough and/or if F is sufficiently large, i.e. for wide and/or fast catamarans. Moreover, the wake angle in this regime is well predicted by the relation ψmax = arctan(0.2/Fs ) in accordance with (3a). However, longitudinal interference effects √cannot be ignored for narrow slow catamarans, specifically if F s < 0.39 + 0.13s, i.e. if F < 0.93 for s = 0.2, F < 0.64 for s = 0.5, F < 0.55 if s = 0.8. This practical application to catamarans will be reported elsewhere. References [1] W. Thomson, On ship waves, Proc. Inst. Mech. Eng. 38 (1887) 409–434. [2] D.W. Taylor, Resistance of Ships and Screw Propulsion, Macmillan, 1910.

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