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Comparison of three simple models of Kelvin’s ship wake
European Journal of Mechanics B/Fluids 49 (2015) 12–19
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Comparison of three simple models of Kelvin’s ship wake Jiayi He, Chenliang Zhang, Yi Zhu, Huiyu Wu, Chen-Jun Yang, Francis Noblesse ∗ , Xiechong Gu, Wei Li 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 23 March 2014 Received in revised form 18 July 2014 Accepted 18 July 2014 Available online 24 July 2014 Keywords: Kelvin wake Wave interference Cutoff wavelength Gaussian pressure distribution Monohull Catamaran
abstract A theoretical explanation of observations of high-Froude-number ship wakes that are narrower than the classical Kelvin 39◦ angle was recently offered by Rabaud and Moisy. The explanation relies on the assumption that a ship hull does not create waves longer than its length. A validation of this theoretical model has also been given. The validation is based on the approximation of the flow created by a ship hull by means of a Gaussian distribution of pressure at the free surface. These two flow models predict a wake angle ψmax that decreases like 1/F as the Froude number F increases beyond F ≈ 0.5 . A third theoretical explanation was recently proposed by the authors. This theoretical explanation assumes that the wave pattern of a ship mostly consists of dominant waves that are created by the ship bow and stern, and is mostly determined by interference effects between these dominant waves. The analysis of interference effects√ on the Kelvin wake of a ship predicts a wake angle ψmax ≈ 0.14/F 2 for a monohull ship, or ψmax ≈ 0.2 b/F for a catamaran with beam/length ratio b. The ‘flow models’ underlying these three alternative theoretical explanations of narrow ship wakes are examined, and the corresponding theoretical predictions are compared to the 37 observations of ship wakes reported by Rabaud and Moisy for Froude numbers F within the wide range 0.1 < F < 1.7. The wake observations are found to be consistent with the predictions given by an analysis of interference between the bow and stern waves of a monohull ship, or a catamaran with beam/length ratio b within the range 0.4 ≤ b ≤ 0.8. Indeed, agreement is consistently strong for the 35 wake observations within the range 0.1 < F < 1.4. This range of Froude numbers includes the range F < 0.6, where interference between transverse bow and stern waves is important, and corresponds to the vast majority of ships. The predictions given by the Rabaud–Moisy ‘cutoff-wavelength model’ and the ‘Gaussian pressure distribution model’ are in close agreement with two wake observations for 1.6 < F < 1.7 and may also be consistent with several wake observations for 0.6 < F < 1.4, but are not consistent with most observations. This finding and a critical examination of the assumptions underlying the Rabaud–Moisy model and the Gaussian pressure distribution model suggest that these theoretical models may not be realistic for most ships. This conclusion is further validated by numerical computations of wave patterns for F = 1. The computed waves are largest along a ray angle that agrees with the prediction of the bow and stern waves interference model, but is noticeably smaller than predicted by the Gaussian pressure distribution model. © 2014 Elsevier Masson SAS. All rights reserved.
1. Introduction The far-field waves generated by a ship hull, of length Ls , that advances at constant speed Vs along a straight path in calm water of large depth are considered. Main features of far-field ship waves, commonly called the Kelvin wake, have been explained by Kelvin and are well known. A main result of Kelvin’s classical far-field analysis is that ship waves cannot exist outside a wedge, with
∗
Corresponding author. E-mail address: [email protected] (F. Noblesse).
http://dx.doi.org/10.1016/j.euromechflu.2014.07.006 0997-7546/© 2014 Elsevier Masson SAS. All rights reserved.
half angle
ψK ≈ 19°28′
(1)
from a ship track, that trails a ship. This angle is independent of the hull shape or the Froude number F ≡ Vs / gLs
(2)
where g denotes the acceleration of gravity. However, numerous observations of ship wakes that are significantly narrower than the wake angle ψK expected from Kelvin’s analysis have long been observed: e.g. [1–5]. This experimental fact is clear from the observations reported by Rabaud and Moisy in [5]. These
J. He et al. / European Journal of Mechanics B/Fluids 49 (2015) 12–19
Wake angle (°)
20
15
10
5
0
0
0.2
0.4
0.6
0.8 1 1.2 Froude number
1.4
1.6
1.8
Fig. 1. Observations (Exp.) of ship wake angles ψmax reported by Rabaud and Moisy in [5] and predictions given by three simple theoretical approaches. Specifically, the figure depicts the angles ψmax predicted by the ‘Gaussian pressure distribution model’ (Gauss), the Rabaud–Moisy ‘cutoff wavelength’ model with the cutoff wavelength λcut taken as the ship length (Ship length), and lateral interference between the divergent waves created by the twin hulls of a catamaran (Catamaran) with beam/length ratio b within the range 0.4 ≤ b ≤ 0.8. The figure also shows the ray angles ψn with n = 1, 2 and 6 (the curves that correspond to n = 3, 4, 5 are not shown because they are closely packed between the n = 2 and n = 6 curves) along which the waves are largest due to constructive longitudinal interference between the transverse and divergent waves created by the bow and the stern of a monohull ship (Monohull).
observations, reproduced here in Fig. 1, are within the range 13° ≤ ψmax ≤ 21° and mostly located around the Kelvin angle ψK for Froude numbers F < 0.6, but are consistently and significantly smaller than ψK for 0.6 < F . Several alternative theoretical explanations of these wake observations in apparent variance with Kelvin’s classical result have been proposed: e.g. [4–8]. The explanations offered in [5,8,9] are based on a linear potential-flow analysis of steady ship waves, unlike the considerably more complex explanations proposed in [2,4,6,7] that invoke effects of ambient waves, nonlinearities or finite water depth. These more complex theoretical explanations are not considered here; i.e. only the three alternative ‘simple theories’ recently offered in [5,8,9] are compared to one another and confronted to the 37 wake observations reported in [5]. These three theories have in common the fact that they each rely on a key assumption that greatly simplifies the analysis of the Kelvin wake of a ship, but differ in that the key assumptions underlying the alternative theories offered in [5,8,9] are markedly different. The explanation proposed in [5] relies on the assumption that waves with wavelengths Λ greater than the ship length Ls can be ignored, i.e. on the restriction λ ≡ Λ/Ls ≤ 1. This ‘cutoffwavelength model’ is also justified in [5] via the ‘Gaussian pressure distribution model’, which is based on approximating the flow created by a ship hull by means of a Gaussian distribution of pressure at the free surface. This ‘Gaussian pressure distribution model’ is further considered in [8,10]. Although based on different assumptions, the Rabaud–Moisy cutoff-wavelength model and the Gaussian pressure distribution model yield similar predictions that the wake angle ψmax of a ship decreases like 1/F as the Froude number F increases beyond F ≈ 0.5. The theoretical explanation proposed in [9] relies on the assumption that the wave pattern of a ship mostly consists of dominant waves that are created by the ship bow and stern, and is mostly determined by interference effects between these dominant waves. Specifically, the explanation offered in [9] relies on an elementary ‘geometrical’ analysis of interference between the dominant waves created by the bow and the stern of a monohull ship (longitudinal interference) or by the bows of the twin hulls of
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a catamaran (lateral interference). The ‘dominant waves interference model’ considered in [9] does not involve the amplitudes of the bow and stern waves, and is particularly simple. Although comparisons with the observations of ship wakes reported in [5] are given in [9], this previous study does not include comparisons of the basic assumptions underlying the three alternative flow models considered in [5,8,9], i.e. the ‘cutoff-wavelength model’ offered in [5], the ‘Gaussian pressure distribution model’ considered in [5,8], and the ‘dominant waves interference model’ proposed in [9]. Moreover, no comparison of the theoretical predictions given by the alternative theories proposed in [5,8,9] is included in these studies. It is then interesting and useful – indeed necessary – to compare the assumptions underlying the three theoretical models proposed in [5,8,9] and to compare the theoretical predictions given by these alternative explanations to the 37 observations of ship wakes reported in [5] for the broad range of Froude numbers 0.1 < F < 1.7. This comparison of underlying assumptions and predictions, not previously considered as already noted, is then considered here. As already noted, the theoretical explanation offered in [5] is based on the assumption that waves with wavelengths Λ larger than the ship length Ls can be ignored. This key assumption is justified in [5] via the consideration of the waves created by a Gaussian distribution of pressure at the free surface, also considered in [8] as noted earlier. A Gaussian distribution of pressure at the free surface may be a reasonable model of the flow due to a highspeed planing hull. However, a free-surface pressure distribution that has a single peak may not realistically account for the strong interference between the dominant waves created by the bow and the stern of a ship, and therefore may not be a realistic model of the flow around the hull of a typical displacement ship (notably, monohull ships or catamarans). Moreover, the Gaussian distribution of pressure considered in [5,8] is smooth (infinitely differentiable), whereas the flow created by the bow and the stern of a ship (especially a fine bow or stern) varies very rapidly and indeed is not smooth at a sharp bow and stern, where the hull geometry varies abruptly. The difference is mathematically important because farfield waves are strongly influenced by near-field singularities, as shown in e.g. [11]. Indeed, this basic property of Fourier transforms provides a mathematical interpretation (explanation) of the common observation that a ship mostly creates two dominant waves that originate from points where the hull geometry varies abruptly, i.e. the ship bow and stern. This well-known feature of the wavemaking of a ship – the key approximation underlying the dominant waves interference model considered in [9] – is clearly apparent from Fig. 2, which depicts the wave patterns created by the Wigley hull and the Series 60 model at a Froude number F = 0.3. These wave patterns were determined via the Neumann–Michell theory given in [12,13]. Thus, the smooth (Gaussian) free-surface pressure-distribution, with a single peak, considered in [5,8] may not be a realistic model of the flow around a ship hull, except for high-speed planing hulls, and consequently may not provide an adequate justification of the key assumption that underlies the ‘cutoff wavelength’ model of Kelvin’s wake proposed by Rabaud and Moisy in [5]. It is then necessary to test if the assumption λ ≡ Λ/Ls ≤ 1 that underlies the Rabaud–Moisy flow-model can be rationalized, notably in the light of the wave-interference analysis given in [9]. Indeed, it may be argued that the analysis of interference between the dominant waves created by the bow and the stern of a ship given in [9] provides a reasonable definition of a cutoff wavelength λcut ; specifically, this definition is related to the fact that interference between dominant divergent waves results in the apparent effective elimination of wavelengths λcut < λ via destructive interference.
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J. He et al. / European Journal of Mechanics B/Fluids 49 (2015) 12–19
Fig. 2. Elevation z = e(x, y) with e ≡ Eg /Vs2 of the free surface (computed by means of the Neumann–Michell theory) in the region (−3.5 ≤ x ≡ X /Ls ≤ 1, −2 ≤ y ≡ Y /Ls ≤ 2) for the Wigley hull (left side) or the Series 60 model (right) at a Froude number F = 0.3. The figure illustrates the well-known property that the dominant waves created by a ship hull originate at the ship bow and stern, where the hull geometry varies abruptly. This basic feature of the wavemaking of a ship underlies the analysis of wave-interference effects on the Kelvin wave pattern given in [9].
2. Wake angles predicted by three simple flow models The angles of the wakes created by monohull ships, catamarans, and planing hulls predicted by the three theoretical models considered in [5,8,9] are now summarized as they are required for purposes of comparison. [9] shows that constructive longitudinal interference between the (transverse or divergent) waves created by the bow and the stern of a monohull ship yields dominant waves along rays at angles ψ = ±ψn with 1 ≤ n that are defined in terms of the Froude number F as
(2n − 1)2 π 2 F 4 /ℓ2 − 1 tan ψn = 2(2n − 1)2 π 2 F 4 /ℓ2 − 1
√ ℓ/π (2/3)1/4 FK ≈ 0.59. (3a) with FnT ≡ √ ≤ F and FK ≡ (2/3)1/4 2n − 1 Here, ℓ ≈ 0.9 denotes the (nondimensional) distance between the effective origins of the bow and stern waves. The largest transverse and divergent waves are found along √ the ray angles √ ψ = ±ψn defined by (3a) with FnT ≤ F ≤ FK / 2n − 1 or FK / 2n − 1 ≤ F , respectively. The ray angles ψn are depicted in Fig. 1 for n = 1, 2 and 6. For longitudinal interference between the divergent bow and stern waves and FK ≤ F , one has ψn ≤ ψ1 ≡ ψmax with
π 2 F 4 /ℓ2 − 1 ℓ/2 0.14 ≈ ≈ 2 as F → ∞. 2π 2 F 4 /ℓ2 − 1 πF2 F
tan ψmax =
(3b)
Ref. [9] also shows that lateral interference between the divergent waves created by the bows and sterns of the twin hulls of a catamaran similarly results in dominant waves along rays at angles ψ = ±ψmax that are defined in terms of the Froude number
√
Fb ≡ Vs / gB ≡ F / B/Ls ≡ F / b
(4)
based on the lateral separation distance B between the twin hulls of the catamaran as
1 + 16π 2 F 4 /b2 − 1
tan ψmax = with FKcat ≡
2 + 32π 2 F 4 /b2 1 2
√
F 3/π ≈ 0.37 < √ b
√
and
1 tan ψmax ≈ √ 8π
b F
(5a)
√
≈ 0.2
b F
The wake angles ψ = ±ψmax given in [5,8] using the assumption λ ≤ λcut = 1 and/or the consideration of a Gaussian distribution of pressure at the free surface are given by
tan ψmax =
and
2π F 2 /λcut − 1
4π F 2 /λcut − 1
1 tan ψmax ≈ √ 8π
√ λcut F
with FKcut ≡
≈ 0.2
√ λcut F
1 cut 3λ /π ≤ F (6a) 2 as F → ∞.
(6b)
The wake angle ψmax given in [5] corresponds to the cutoff wavelength λcut and the related Froude number F cut
λcut = 1 and FKcut ≡
3/π /2 ≈ 0.49.
(6c)
The high-Froude-number wake created by a Gaussian distribution of pressure at the free surface considered in [8] corresponds to
λcut =
8/5 ≈ 1.265
and FKcut ≡
3/π /101/4 ≈ 0.55.
(6d)
The lower bound FKcut in (6a), (6c) and (6d) and the lower bounds FK and FKcat in (3a) and (5a) are the values of the Froude number
F for which the wake angle ψmax is equal to the Kelvin angle ψK , i.e. the value of F that yields ψmax = ψK . Thus, expressions (3a), (5a), (6a), (6c), (6d) predict that the largest waves created by a ship hull are found at an angle ψmax that is smaller than the Kelvin wake angle ψK if the Froude number F is larger than FK ≈ 0.59, FKcat ≈
√
0.37 b, FKcut ≈ 0.49 or FKcut ≈ 0.55. Interference between the waves created by the bow and the stern of a monohull yields ψmax = O(1/F 2 ), i.e. predicts a wake angle ψmax that decreases like 1/F 2 as F → ∞, whereas both expression (5b) for a catamaran and expression (6b) for a Gaussian free-surface pressure-distribution yield ψmax = O(1/F ), i.e. predict wake angles ψmax that decrease like 1/F as F → ∞. Indeed, the high-speed approximations (5b) and (6b) are identical except for the fact that the beam/length ratio b of a catamaran or the cutoff wavelength λcut for a Gaussian pressure-distribution appears in (5b) or (6b). Thus, these expressions, with the substitution b ↔ λcut , predict nearly identical ship wake angles. 3. Comparison with observations and computations 3.1. The Rabaud–Moisy observations of ship wakes
as F → ∞.
(5b)
Here, b ≡ B/Ls where Ls is the length of the catamaran, and B denotes the lateral separation distance between the twin hulls of the catamaran (beam/length ratio) as already noted.
Fig. 1 depicts the 37 observations of ship wake angles ψmax reported in [5] for the broad range of Froude numbers 0.1 < F < 1.7 and ray angles predicted by the three simple theoretical approaches considered here. Specifically, Fig. 1 depicts the wake
J. He et al. / European Journal of Mechanics B/Fluids 49 (2015) 12–19
angles ψmax predicted by the ‘Gaussian pressure distribution model’ (Gauss), the Rabaud–Moisy ‘cutoff wavelength’ model with the cutoff wavelength λcut taken as the ship length (Ship length), and lateral interference between the divergent waves created by the twin hulls of a catamaran (Catamaran) with beam/length ratio b within the range 0.4 ≤ b ≤ 0.8. In addition, Fig. 1 shows the ray angles ψn with n = 1, 2 and 6 along which the waves are largest due to constructive longitudinal interference between the transverse and divergent waves created by the bow and the stern of a monohull ship (Monohull). Fig. 1 shows that the four curves that correspond to expression (5a) associated with wave-interference effects for a catamaran, with b = 0.4 or b = 0.8, and expressions (6a), (6c) and (6d) related to a Gaussian free-surface pressure distribution or the Rabaud–Moisy cutoff-wavelength model are nearly parallel, in accordance with the high-speed approximations (5b) and (6b). The ten wake observations within the range 0.6 < F < 1.4 shown in Fig. 1 are fully consistent with the ray angles ψmax associated with longitudinal or lateral interference between the divergent waves created by the bow and the stern of a monohull ship (Monohull) or a catamaran (Catamaran) with beam/length ratio b within the range 0.4 ≤ b ≤ 0.8. The two wake observations within the high-speed range 1.6 < F < 1.7 are in close agreement with the predictions given by the Rabaud–Moisy cutoff wavelength model (Ship length) and the Gaussian pressure distribution model (Gauss) and may also be consistent with wave-interference effects for a catamaran with a large beam/length ratio b. The two curves marked ‘Gauss’ or ‘Ship length’ may also be consistent with several of the ten wake observations shown in Fig. 1 for 0.6 < F < 1.4, but are not consistent with most wake observations. Most of the observations (specifically 25 out of a total of 37 observations, i.e. nearly 68%) shown in Fig. 1 correspond to Froude numbers within the range F ≤ FK ≈ 0.59. The figure shows that the 25 observations for F ≤ 0.59 are within the range 13° ≤ ψmax ≤ 21° and are mostly located around the Kelvin angle ψK ≈ 19°28′ . However, three observations of wake angles shown in Fig. 1 for F < 0.59 are smaller than 15°, i.e. significantly smaller than ψK . Fig. 1 shows that these three observations, as well as the 22 observations of ship wake angles that are roughly equal to the Kelvin angle ψK for F < 0.6, are fully consistent with the ‘interference curves’ that depict three angles ψn for 1 ≤ n ≤ 6 along which the waves are largest due to constructive interference between the transverse or divergent waves created by a ship bow and stern [9]. The comparison of observations of ship wake angles and theoretical predictions shown in Fig. 1 provides solid evidence that interference between the bow and stern waves of a monohull ship or a catamaran is a highly probable, although remarkably simple, explanation for the observations of wake angles reported in [5] for the broad range of Froude numbers 0.1 < F < 1.7. This speed range includes the range F < 0.6 where interference between transverse bow and stern waves is important. The predictions given by an analysis of interference effects on the Kelvin wave pattern are in particularly good agreement with the wake observations within the range of Froude numbers 0.1 < F < 1.4. This range of Froude numbers includes 35 of the 37 observations (nearly 95%) shown in Fig. 1 and indeed corresponds to the vast majority of ships. 3.2. Numerical computations The wake angles ψmax predicted by the Gaussian pressure distribution model and the dominant waves interference model are now compared to numerical computations of the waves created by the classical Wigley parabolic hull at a Froude number F = 1. Fig. 3 depicts the waves created within the near-field region −3.5 ≤ x ≡ X /Ls ≤ 1, −1 ≤ y ≡ Y /Ls ≤ 1. The waves depicted above and below the centerline y = 0 are computed via the classical Hogner approximation, which is based on the Green
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Fig. 3. Waves created within the region −3.5 ≤ x ≤ 1, −1 ≤ y ≤ 1 by the Wigley hull at F = 1. The waves above and below the centerline y = 0 are computed via the Hogner approximation or a Rankine source panel method, respectively. The figure also shows the Kelvin rays ψ = ±19°28′ with origin at the bow of the hull, and the rays ψ ≈ ±8°8′ that are predicted by the interference relation (3b) and originate at the bow (0.5, 0) or the stern (−0.5, 0) of the Wigley hull.
Fig. 4. Waves created within the region −14 ≤ x ≤ 1, −5 ≤ y ≤ 5 by the Wigley hull at F = 1. The waves are computed via the Hogner approximation. The figure also shows the Kelvin cusp lines ψ ≈ ±19°28′ and the rays ψ ≈ ±12°39′ and ψ ≈ ±8°8′ predicted by the Gaussian pressure distribution model or the wave interference model, respectively.
function that satisfies the Kelvin–Michell linearized boundary condition at the free surface and is related to the Neumann–Michell theory given in [12,13] or correspond to a numerical solution of the Neumann–Kelvin problem (Laplace equation and Kelvin–Michell linear free-surface boundary condition) obtained via a panel method based on Rankine singularities in the manner explained in [14]. Although differences can be observed between the near-field waves depicted in the upper and lower halves of Fig. 3, these two wave patterns (determined by means of significantly different numerical methods as noted) are largely consistent. Fig. 3 also shows the two Kelvin rays ψ = ±19°28′ with origin at the bow (0.5, 0) of the Wigley hull and the four rays ψ = ±8°8′ that are predicted by the dominant waves interference relation (3b) and originate at the bow (0.5, 0) or the stern (−0.5, 0) of the hull. Both numerical computations show that the largest waves are approximated located along the rays ψ = ±8°8′ predicted by the dominant waves interference model. Fig. 4 depicts the waves, computed via the Hogner approximation, created by the Wigley hull at F = 1 within the larger region −14 ≤ x ≤ 1, −5 ≤ y ≤ 5. This figure shows that although waves exist inside the wedge bounded by the rays ψ = ±19°28′ that correspond to the cusp lines (marked as dashed lines) of the Kelvin wake, the largest waves are found well inside the Kelvin wake, approximately along the rays (marked as solid lines) ψ = ±8°8′ predicted by the dominant waves interference model. Fig. 4 also shows that the rays ψ = ±12°39′ (marked as short dashed lines) predicted by the Gaussian pressure distribution model are noticeably outside the region where the largest waves are found. Thus, the computed waves depicted in Fig. 4 are largest along ray angles that agree well with the predictions given by the bow and stern waves interference model, but are noticeably smaller
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J. He et al. / European Journal of Mechanics B/Fluids 49 (2015) 12–19
The wavelength at the cusp ψ = ψK of the Kelvin wake is given by
λcusp = 2 λmax /3 = 4π F 2 /3.
(12)
The is smaller than the ship length if F < √ cusp wavelength λ 3/π /2 ≈ 0.49 but is larger than the ship length if 0.49 < F . The cusp wavelength λcusp can be significantly larger than the ship length λs ≡ 1; e.g. one has λcusp ≈ 4.2 for F = 1 and λcusp ≈ 16.8 for F = 2. The range 0 ≤ λ ≤ λcusp that corresponds to divergent waves is considered hereafter. The wavelength λD of the divergent waves along a ray angle ψ is given by (11) as cusp
0≤
Fig. 5. Variation of the Kelvin angle ψK for 0 ≤ λ/λmax ≤ 1. Divergent waves (solid line) and transverse waves (dashed line) correspond to 0 ≤ λ/λmax ≤ 2/3 and 2/3 ≤ λ/λmax ≤ 1, respectively.
than the ray angles predicted by the Gaussian pressure distribution model. This finding provides strong evidence that constructive interference between the dominant bow and stern waves created by a ship, as illustrated in Fig. 2, results in the appearance of narrow ship wakes at high Froude numbers. 4. The cutoff-wavelength model
with F k = 1/ cos γ ≡ 1 + tan γ 2
2
and − π /2 < γ < π /2.
(7)
λ = 2π F 2 cos2 γ ≤ 2π F 2 ≡ λmax .
(8)
Thus, the wavelength λmax of the longest waves created by √a ship is smaller than the ship length for Froude numbers F < 1/ 2π ≈ 0.4 but is larger than the ship length if 0.4 < F , as well known. In the far-field limit h ≡ x2 + y2 → ∞, significant contributions to a superposition of elementary waves (7) only stem from values of γ for which the phase
θ ≡ (x cos γ + y sin γ )(1 + tan2 γ )/F 2
(9)
is stationary, i.e. from roots of the ‘stationary–phase relation’ dθ/dγ = 0. This equation yields y
−x
=
Within the divergent-wave range 0 ≤ λ ≤ λcusp , the function ψ(λ) defined by (11) increases, as already noted and shown in Fig. 5. The inequalities
λ ≤ λcut ≤ λcusp ≡ 2λmax /3 ≡ 4π F 2 /3 and √ max cut λ /λ − 1 cut −1 ≤ ψK ≈ 19°28′ ψ ≤ ψ ≡ tan 2λmax /λcut − 1
tan γ 1 + 2 tan2 γ
.
tan ψ
cut
(10)
The stationary–phase relation (10) and the dispersion relation (8) relate the ray angle ψ and the wavelength λ as
√ ′ √ max λ /λ − 1 λ (1 − λ′ ) λ tan ψ = ≡ with λ′ ≡ max . (11) max ′ 2λ /λ − 1 2−λ λ The ray angle ψ defined by (11) is real for 0 ≤ λ′ ≤ 1, increases from 0 to the Kelvin angle ψK for 0 ≤ λ′ ≤ 2/3 and decreases from ψK to 0 for 2/3 ≤ λ′ ≤ 1, as depicted in Fig. 5. The ‘short’ waves 0 ≤ λ′ ≤ 2/3 and the ‘long’ waves 2/3 ≤ λ′ ≤ 1 correspond to the divergent waves and the transverse waves, respectively, in the Kelvin wave pattern.
√ λcut λcut 1 ≈ √ ≡ √ max F 8π 2 λ √ λcut ≈ 0.2 with λcut ≪ λmax ≡ 2π F 2 . F
2
The dispersion relation F 2 k = 1/ cos2 γ defines the wavelength λ ≡ 2π/k as
tan ψ ≡
4.2. Two mathematically-equivalent inequalities
(14) (15)
√
At some distance behind a ship, the waves created by the ship can be represented as a linear superposition of elementary waves that travel at angles −π /2 < γ < π /2 from the path of the ship (the x axis). These elementary waves are given by e
(13)
therefore are mathematically equivalent. Expression (15) for ψ cut yields the approximation
4.1. Basic relations
kz +ik(x cos γ +y sin γ )
2 λD 8 tan2 ψ ≤ . ≡ max 2 2 λ 3 1 + 4 tan ψ + 1 − 8 tan ψ
(16)
The fact that the inequalities (14) and (15) are mathematically equivalent means that an assumed restriction λ ≤ λcut < λcusp has the automatic effect of reducing the wake angle ψ below the Kelvin angle ψK and indeed is tantamount to imposing that the wake angle ψ is smaller than the angle ψ cut < ψK . Clearly then, a model that invokes a cutoff-wavelength assumption λ ≤ λcut , as in [5], cannot explain observations of ship wakes that are narrower than Kelvin’s wake unless the assumed choice of cutoff-wavelength λcut can be rationally justified. A reasonable definition of a cutoff wavelength λcut , based on the analysis of interference between the bow wave and the stern wave of a monohull or between the twin bow waves of a catamaran summarized in (3a) and (5a), is then considered here. Specifically, expressions (3a) and (5a) for the wake angles ψmax predicted by an analysis of interference between the bow and stern waves of a monohull, or between the twin bow waves of a catamaran, are now used in expression (13) with ψ = ψmax to determine the cutoff wavelength λcut that corresponds to ψmax and a wave-interference analysis. This definition of a cutoff wavelength λcut is then
λcut ≡
16 π F 2 tan2 ψmax 1 + 4 tan2 ψmax +
1 − 8 tan2 ψmax
≈ 8π F 2 tan2 ψmax
(17)
where (8) was used. 4.3. Cutoff wavelengths related to wave interference The cutoff wavelength λcut that corresponds to interference between the bow and stern waves of a ship and the wake angle ψmax
J. He et al. / European Journal of Mechanics B/Fluids 49 (2015) 12–19
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the Froude number F is then reasonable for catamarans, as well as for planing hulls as shown in [5,8]. Furthermore, the cutoff wavelengths λcut given by (19b), (6c) and (6d) with 0.7 ≤ b ≤ 1.3 do not differ widely. The cutoff wavelengths λcut given by (6c) and (6d) are also consistent with the cutoff wavelength (18a) associated with interference between the bow and stern waves of a monohull for a very narrow range of Froude numbers F , roughly for 0.65 < F < 0.75. However, the cutoff wavelength (18a) related to interference between the bow and stern waves of a monohull is a rapidly decreasing function of F that cannot realistically be assumed constant. Thus, the key assumption λ ≤ 1 invoked in [5] cannot be rationalized by an analysis of interference between the bow and stern waves of a ship hull, and for the wide range of Froude numbers F < 1.2. 4.4. The cutoff-wavelength and wave-interference models Fig. 6. Variations of the cutoff wavelengths λcut related to longitudinal interference between the divergent waves created by the bow and the stern of a monohull ship (Monohull) for FK ≈ 0.59 < F ≤ 1.6 or to lateral interference between √ the divergent waves created by the twin hulls of a catamaran (catamaran) for FKcat b ≈
√
0.37 b ≤ F ≤ 1.6 with b = 0.1, 0.5, 1 and 1.5. The cutoff wavelengths λcut given by (6c) or (6d) and associated with the ship length or a Gaussian distribution of pressure at the free surface, as in [5] or [8], are also shown. The figure shows that the cutoff wavelength λcut related to longitudinal interference for a monohull ship (Monohull) is a rapidly decreasing function of F that cannot realistically be assumed constant, in contradiction with the key assumption used in [5].
given by (3b) is determined by (17) and (3b). One then has
λ
4
cusp K
≤λ
2
≡ π =2 ℓ ≈ 1.47 3 3 with FK ≡ ℓ 3/2/π ≈ 0.59 < F
cut
and λ
cut
≈
2 ℓ2
πF2
FK2
≈
0.516 F2
(18a)
as F → ∞.
(18b)
Thus, λcut decays like 1/F 2 as F → ∞. Within the Froude-number range FK ≈ 0.59 ≤ F ≤ 1.2, the cutoff wavelength λcut varies within the approximate range 1.47 ≥ λcut ≥ 0.36. The cutoff wavelength λcut that corresponds to interference between the twin bow waves of a catamaran and the wake angle ψmax given by (5a) is similarly determined by (17) and (5a). One then has √ = λcusp F =F cat b K
with FKcat
4 3
b
π (FKcat )2 b = √ ≈ 0.58b ≤ λcut ≤ b
√ b≡
3
1/4
√
√
b
2 π
and λcut ≈ b ≡ B/Ls
3
√ ≈ 0.37 b ≤ F
(19a)
as F → ∞.
(19b)
Thus, λ → b as F → ∞. The cutoff wavelength λ varies cut within √ the range 0.58 b < λ < b as F varies within the range 0.37 b ≤ F < ∞. Fig. 6 depicts the cutoff wavelengths λcut given by (18a) or (19a) with b = 0.1, 0.5, 1 and 1.5. As already explained, these wavelengths are determined from an analysis of interference between the bow and stern waves of a monohull, or between the twin bow waves of a catamaran with beam/length ratios b. The cutoff wavelengths given in [5,8] and by (6d) and the cusp wavelength λcusp given by (12) are also shown in Fig. 6. For Froude numbers F larger than Fhigh with Fhigh ≈ 1.2, the cutoff wavelength λcut given by (19a) and associated with interference between the twin bow waves of a catamaran does not vary significantly with F and is given by λcut ≈ b as shown in (19b). The assumption of a cutoff wavelength λcut that is independent of cut
The approaches used in [5,9] differ in several other important respects as well, as now noted. In particular, expression (3) fully determines the wake angle of a monohull ship in terms of the Froude number F . Similarly, (5) fully determines the wake√ angle of a catamaran in terms of the Froude number Fb ≡ F / B/Ls based on the lateral separation distance B between the twin bows of the catamaran. Indeed, expressions (3) and (5) are based on the classical theories of ship waves and wave interference, and accordingly do not involve any a-priori unknown parameter, whereas the flow-model used in [5] involves an a-priori unknown cutoff wavelength λcut that must be chosen somehow. Furthermore, the flow-model used in [5] assumes that the wavelength λ can be restricted as
λ ≤ λcut or H (λcut − λ) (20) where H (·) denotes the usual Heaviside step function. The waveinterference analysis given in [9], on the other hand, is based on the interference relations
ℓ cos γ = (2n − 1)λ/2 or b sin γ = nλ with 1 ≤ n
and
λ ≡ 2π F 2 cos2 γ
(21)
for a monohull or a catamaran. Here, ℓ cos γ represents the projection of the distance ℓ between the effective origins of the dominant waves created by the bow and the stern of a monohull ship, and b sin γ is the projection of the lateral separation distance b between the twin bows of a catamaran, upon the direction of propagation (cos γ , sin γ ) of the waves in the Kelvin wake. The interference relations (21) state that these projections yield constructive interference (and therefore largest waves) between the bow and stern waves of a monohull ship or between the bow waves of a catamaran. These interference relations essentially restrict the wavelength λ via the Dirac function
(2n − 1)λ δ ℓ cos γ − 2
cut
or δ(b sin γ − nλ)
with 1 ≤ n
(22)
instead of the Heaviside function in (20) that is used in [5]. The relations (22), where 1 ≤ n, define a series of angles ψn along which large waves can be found due to constructive interference between divergent waves as well as transverse waves, unlike the relations (20) that only correspond to divergent waves with n = 1. Indeed, the flow model based on the assumption of a cutoff wavelength λcut cannot be applied within the Froudenumber ranges
ℓ ≈ 0.59 or π √ √ √ 31/4 b F < FKcat b = ≈ 0.37 b √ 2 π 1/4
F < FK = (3/2)
(23)
18
J. He et al. / European Journal of Mechanics B/Fluids 49 (2015) 12–19
for a monohull or a catamaran, respectively. Thus, this flow model cannot explain the observations of ship wake angles that are significantly smaller than the Kelvin angle ψK reported in [5] and shown in Fig. 1 for Froude numbers F < 0.6, whereas interference effects between the divergent or transverse waves created by the bow and the stern of a monohull ship are consistent with these wake observations. Moreover, Fig. 1 shows that ship wake angles that are roughly equal to the Kelvin angle ψK for F < 0.6 are also consistent with interference between the bow and stern waves of a monohull ship. 5. Conclusion Fig. 1 shows that the theoretical predictions given by the Rabaud–Moisy ‘cutoff-wavelength model’ proposed in [5] and the ‘Gaussian pressure distribution model’ considered in [5,8] are in close agreement with two wake observations within the range 1.6 < F < 1.7 and may possibly also be consistent with several observations within the range 0.6 < F < 1.4, but are not consistent with most of the 37 observations reported in Fig. 1. This finding suggests that the ‘cutoff-wavelength model’ and the ‘Gaussian pressure distribution model’ may not be realistic for most ships, as suggested also by a critical examination of the assumptions underlying these two flow models and the comparisons with numerical computations of wave patterns given in Fig. 4. In particular, Fig. 6 demonstrates that the assumption of a cutoff wavelength λcut that is independent of the Froude number and approximately equal to the ship length invoked in the cutoff-wavelength model cannot be rationalized via the ‘dominant-wave-interference analysis’ given in [9] for monohull ships in the most common speed range 0.1 < F < 1.2. Moreover, the smooth distribution of free-surface pressure, with a single peak, considered in the Gaussian pressure distribution model is not a realistic model of the flow around the hull of a typical displacement ship, which creates two dominant (bow and stern) waves as illustrated in Fig. 2. Fig. 1 shows that the 35 observations of ship wakes reported in [5] for Froude numbers within the range 0.1 < F < 1.4 are fully consistent with the ray angles ψn and the wake angles ψmax predicted by an analysis of longitudinal or lateral interference between the dominant waves created by the bow and the stern of a monohull ship or the twin hulls of a catamaran with beam/length ratio b within the range 0.4 ≤ b ≤ 0.8. This range of Froude numbers includes 35 of the 37 observations (nearly 95%) shown in Fig. 1 and indeed corresponds to the vast majority of ships. The two wake observations reported in Fig. 1 within the range 1.6 < F < 1.7 may possibly be consistent with wave interference for a high beam/length ratio catamaran. Nearly 68% of the 37 wake observations reported in [5] and in Fig. 1 correspond to the range F < 0.6 where interference between transverse bow and stern waves is important. Fig. 1 shows that the ray angles ψn along which the waves are largest due to constructive interference between the transverse or divergent waves created by the bow and the stern of a monohull ship are fully consistent with the 25 observations of ship wakes in the ‘transverse-wave range’ F < 0.6. These 25 wake observations include 22 wake angles that are roughly equal to the Kelvin angle ψK and three wake angles significantly smaller than ψK . Thus, the comparison of observations of ship wake angles and theoretical predictions given in Fig. 1 provides strong evidence that interference between the bow and stern waves of a monohull ship or a catamaran is a highly probable, although remarkably simple, explanation for the observations of ship wakes reported in [5] for a broad range of Froude numbers. Furthermore, the numerical computations of wave patterns depicted in Fig. 4 show that the computed waves are largest along ray angles that agree well with the predictions of the bow and stern waves interference model. This finding provides solid additional evidence for the conclusion that interference between the dominant bow and stern waves created by a ship, as illustrated in Fig. 2, results in the appearance of narrow ship wakes at high Froude numbers.
However, elementary theoretical considerations and Fig. 1 suggest that the bow and stern waves interference model may not be realistic in a high Froude number range Fhigh < F . Indeed, an obvious limitation of this simple flow model is that the basic theoretical framework of linear potential flow theory may be unrealistic at very high Froude numbers, notably because a high-speed ship typically creates a large spray sheet and related wave breaking wake that may be expected to have a significant influence on the appearance of the ship wake [15]. Another obvious limitation of the model is that high-speed ships in the planing-hull regime do not create distinct bow and stern waves like common displacement ships, and a pressure distribution model may be a reasonable flow model for high-speed planing hulls (the two wake observations for 1.6 < F < 1.7 in Fig. 1 may possibly be due to planing hulls). Moreover, even for a ship that creates distinct bow and stern waves at high speed, interference effects are more complex than assumed in the elementary geometrical analysis considered in [9]. Indeed, potential flow around a ship hull (a slender body) can be represented by means of a distribution of sources and sinks over the bow and stern regions of the ship hull, e.g. as given by the Hogner approximation [12,13]. Thus, lateral interference occurs between the sources (or sinks) distributed over the port and starboard sides of the bow (or stern) regions of the ship hull, and longitudinal interference occurs between the waves created by the sources and the sinks distributed over the bow and stern regions. The flow around a monohull ship is therefore not fundamentally different from the flow created by a narrow catamaran with small beam/length ratio. [9] shows that the wake angle ψmax decreases like 1/F or 1/F 2 for lateral or longitudinal interference, respectively, and the behavior of ψmax for a high-speed monohull ship can then be expected to depend on both F and the hull shape, notably the beam/length ratio. Indeed, as shown in [16], combined longitudinal and lateral interference is significantly more complex than either the pure longitudinal interference or the pure lateral interference considered in [9]. The simplified analysis given in [9] nevertheless provides valuable insight. √In particular, the wake angles ψmax ≈ 0.14/F 2 and ψmax ≈ 0.2 b/F associated with √longitudinal or lateral interference are equal if F = F∗ ≈ 0.7/ b. The ‘crossover’ Froude number F∗ is given by F∗ ≈ 2.2 for b = 0.1 and F∗ ≈ 1.6 for b = 0.2. These large values of F∗ suggest that the approximation ψmax ≈ 0.14/F 2 may in fact be adequate for most monohull ships encountered in practice. Although the differences between the cutoff wavelength model, the Gaussian pressure distribution model, and the dominant bow and stern waves interference model proposed in [5,8,9] are mostly emphasized in the foregoing, these three flow models share an important similarity that also deserves to be emphasized. Specifically, the three models consider the simple theoretical framework of steady, deep-water, linear potential-flow hydrodynamics – unlike the considerably more complex theories considered in [2,4,6,7] that involve effects of ambient waves, nonlinearities or finite water depth – and moreover are based on a simple key assumption or approximation. The highly simplified flow models considered in [5,8,9] illustrate, yet again, the necessity and the value of considering analytical models based on simplifying approximations to understand seemingly complicated flows. It is also appropriate to note that credit is due to Rabaud and Moisy [5] for first proposing a simple flow model of the Kelvin ship wake to seek an explanation of narrow ship wakes, as well as for providing invaluable observations of ship wakes over the broad range of Froude numbers 0.1 < F < 1.7. References [1] W.H. Munk, P. Scully-Power, F. Zachariasen, Ships from space, Proc. R. Soc. Lond. Ser. A 412 (1987) 231–254. [2] E.D. Brown, S.B. Buchsbaum, R.E. Hall, J.P. Penhune, K.F. Schmitt, K.M. Watson, D.C. Wyatt, Observations of a nonlinear solitary wave packet in the Kelvin wake of a ship, J. Fluid Mech. 204 (1989) 263–293.
J. He et al. / European Journal of Mechanics B/Fluids 49 (2015) 12–19 [3] A.M. Reed, J.H. Milgram, Ship wakes and their radar images, Annu. Rev. Fluid Mech. 34 (2002) 469–502. [4] M.C. Fang, R.Y. Yang, I.V. Shugan, Kelvin ship wake in the wind waves field and on the finite sea depth, J. Mech. 27 (2011) 71–77. [5] M. Rabaud, F. Moisy, Ship wakes: Kelvin or Mach angle? Phys. Rev. Lett. 110 (2013) 214503. [6] C.C. Mei, M. Naciri, Note on ship oscillations and wake solitons, Proc. R. Soc. Lond. Ser. A 432 (1991) 535–546. [7] Q. Zhu, Y. Liu, D.K.P. Yue, Resonant interactions between Kelvin ship waves and ambient waves, J. Fluid Mech. 597 (2008) 171–197. [8] A. Darmon, M. Benzaquen, E. Raphael, Kelvin wake pattern at large Froude numbers, J. Fluid Mech. (2014) 738. [9] F. Noblesse, J. He, Y. Zhu, L. Hong, C. Zhang, R. Zhu, C. Yang, Why can ship wakes appear narrower than Kelvin’s angle? Eur. J. Mech. B Fluids 46 (2014) 164–171. [10] F. Dias, Ship waves and Kelvin, J. Fluid Mech. 746 (2014) 1–4.
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[11] M.J. Lighthill, An Introduction to Fourier Analysis and Generalised Functions, Cambridge University Press, 1958. [12] F. Noblesse, F. Huang, C. Yang, The Neumann–Michell theory of ship waves, J. Engrg. Math. 79 (2013) 51–71. [13] F. Huang, C. Yang, F. Noblesse, Numerical implementation and validation of the Neumann–Michell theory of ship waves, Eur. J. Mech. B Fluids 42 (2013) 47–68. [14] S. Bal, Prediction of wave pattern and wave resistance of surface piercing bodies by a boundary element method, Internat. J. Numer. Methods Fluids 56 (2008) 305–329. [15] F. Noblesse, G. Delhommeau, P. Queutey, C. Yang, An elementary analytical theory of overturning ship bow waves, Eur. J. Mech. B Fluids 48 (2014) 193–209. [16] J. He, C. Zhang, Y. Zhu, H. Wu, F. Noblesse, D. Wan, L. Zou, W. Li, Interference effects on the Kelvin wake of a catamaran, Eur. J. Mech. B Fluids (2014) submitted.
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European Journal of Mechanics B/Fluids journal homepage: www.elsevier.com/locate/ejmflu
Comparison of three simple models of Kelvin’s ship wake Jiayi He, Chenliang Zhang, Yi Zhu, Huiyu Wu, Chen-Jun Yang, Francis Noblesse ∗ , Xiechong Gu, Wei Li State Key Laboratory of Ocean Engineering, School of Naval Architecture, Ocean & Civil Engineering, Shanghai Jiao Tong University, Shanghai, China
article
info
Article history: Received 23 March 2014 Received in revised form 18 July 2014 Accepted 18 July 2014 Available online 24 July 2014 Keywords: Kelvin wake Wave interference Cutoff wavelength Gaussian pressure distribution Monohull Catamaran
abstract A theoretical explanation of observations of high-Froude-number ship wakes that are narrower than the classical Kelvin 39◦ angle was recently offered by Rabaud and Moisy. The explanation relies on the assumption that a ship hull does not create waves longer than its length. A validation of this theoretical model has also been given. The validation is based on the approximation of the flow created by a ship hull by means of a Gaussian distribution of pressure at the free surface. These two flow models predict a wake angle ψmax that decreases like 1/F as the Froude number F increases beyond F ≈ 0.5 . A third theoretical explanation was recently proposed by the authors. This theoretical explanation assumes that the wave pattern of a ship mostly consists of dominant waves that are created by the ship bow and stern, and is mostly determined by interference effects between these dominant waves. The analysis of interference effects√ on the Kelvin wake of a ship predicts a wake angle ψmax ≈ 0.14/F 2 for a monohull ship, or ψmax ≈ 0.2 b/F for a catamaran with beam/length ratio b. The ‘flow models’ underlying these three alternative theoretical explanations of narrow ship wakes are examined, and the corresponding theoretical predictions are compared to the 37 observations of ship wakes reported by Rabaud and Moisy for Froude numbers F within the wide range 0.1 < F < 1.7. The wake observations are found to be consistent with the predictions given by an analysis of interference between the bow and stern waves of a monohull ship, or a catamaran with beam/length ratio b within the range 0.4 ≤ b ≤ 0.8. Indeed, agreement is consistently strong for the 35 wake observations within the range 0.1 < F < 1.4. This range of Froude numbers includes the range F < 0.6, where interference between transverse bow and stern waves is important, and corresponds to the vast majority of ships. The predictions given by the Rabaud–Moisy ‘cutoff-wavelength model’ and the ‘Gaussian pressure distribution model’ are in close agreement with two wake observations for 1.6 < F < 1.7 and may also be consistent with several wake observations for 0.6 < F < 1.4, but are not consistent with most observations. This finding and a critical examination of the assumptions underlying the Rabaud–Moisy model and the Gaussian pressure distribution model suggest that these theoretical models may not be realistic for most ships. This conclusion is further validated by numerical computations of wave patterns for F = 1. The computed waves are largest along a ray angle that agrees with the prediction of the bow and stern waves interference model, but is noticeably smaller than predicted by the Gaussian pressure distribution model. © 2014 Elsevier Masson SAS. All rights reserved.
1. Introduction The far-field waves generated by a ship hull, of length Ls , that advances at constant speed Vs along a straight path in calm water of large depth are considered. Main features of far-field ship waves, commonly called the Kelvin wake, have been explained by Kelvin and are well known. A main result of Kelvin’s classical far-field analysis is that ship waves cannot exist outside a wedge, with
∗
Corresponding author. E-mail address: [email protected] (F. Noblesse).
http://dx.doi.org/10.1016/j.euromechflu.2014.07.006 0997-7546/© 2014 Elsevier Masson SAS. All rights reserved.
half angle
ψK ≈ 19°28′
(1)
from a ship track, that trails a ship. This angle is independent of the hull shape or the Froude number F ≡ Vs / gLs
(2)
where g denotes the acceleration of gravity. However, numerous observations of ship wakes that are significantly narrower than the wake angle ψK expected from Kelvin’s analysis have long been observed: e.g. [1–5]. This experimental fact is clear from the observations reported by Rabaud and Moisy in [5]. These
J. He et al. / European Journal of Mechanics B/Fluids 49 (2015) 12–19
Wake angle (°)
20
15
10
5
0
0
0.2
0.4
0.6
0.8 1 1.2 Froude number
1.4
1.6
1.8
Fig. 1. Observations (Exp.) of ship wake angles ψmax reported by Rabaud and Moisy in [5] and predictions given by three simple theoretical approaches. Specifically, the figure depicts the angles ψmax predicted by the ‘Gaussian pressure distribution model’ (Gauss), the Rabaud–Moisy ‘cutoff wavelength’ model with the cutoff wavelength λcut taken as the ship length (Ship length), and lateral interference between the divergent waves created by the twin hulls of a catamaran (Catamaran) with beam/length ratio b within the range 0.4 ≤ b ≤ 0.8. The figure also shows the ray angles ψn with n = 1, 2 and 6 (the curves that correspond to n = 3, 4, 5 are not shown because they are closely packed between the n = 2 and n = 6 curves) along which the waves are largest due to constructive longitudinal interference between the transverse and divergent waves created by the bow and the stern of a monohull ship (Monohull).
observations, reproduced here in Fig. 1, are within the range 13° ≤ ψmax ≤ 21° and mostly located around the Kelvin angle ψK for Froude numbers F < 0.6, but are consistently and significantly smaller than ψK for 0.6 < F . Several alternative theoretical explanations of these wake observations in apparent variance with Kelvin’s classical result have been proposed: e.g. [4–8]. The explanations offered in [5,8,9] are based on a linear potential-flow analysis of steady ship waves, unlike the considerably more complex explanations proposed in [2,4,6,7] that invoke effects of ambient waves, nonlinearities or finite water depth. These more complex theoretical explanations are not considered here; i.e. only the three alternative ‘simple theories’ recently offered in [5,8,9] are compared to one another and confronted to the 37 wake observations reported in [5]. These three theories have in common the fact that they each rely on a key assumption that greatly simplifies the analysis of the Kelvin wake of a ship, but differ in that the key assumptions underlying the alternative theories offered in [5,8,9] are markedly different. The explanation proposed in [5] relies on the assumption that waves with wavelengths Λ greater than the ship length Ls can be ignored, i.e. on the restriction λ ≡ Λ/Ls ≤ 1. This ‘cutoffwavelength model’ is also justified in [5] via the ‘Gaussian pressure distribution model’, which is based on approximating the flow created by a ship hull by means of a Gaussian distribution of pressure at the free surface. This ‘Gaussian pressure distribution model’ is further considered in [8,10]. Although based on different assumptions, the Rabaud–Moisy cutoff-wavelength model and the Gaussian pressure distribution model yield similar predictions that the wake angle ψmax of a ship decreases like 1/F as the Froude number F increases beyond F ≈ 0.5. The theoretical explanation proposed in [9] relies on the assumption that the wave pattern of a ship mostly consists of dominant waves that are created by the ship bow and stern, and is mostly determined by interference effects between these dominant waves. Specifically, the explanation offered in [9] relies on an elementary ‘geometrical’ analysis of interference between the dominant waves created by the bow and the stern of a monohull ship (longitudinal interference) or by the bows of the twin hulls of
13
a catamaran (lateral interference). The ‘dominant waves interference model’ considered in [9] does not involve the amplitudes of the bow and stern waves, and is particularly simple. Although comparisons with the observations of ship wakes reported in [5] are given in [9], this previous study does not include comparisons of the basic assumptions underlying the three alternative flow models considered in [5,8,9], i.e. the ‘cutoff-wavelength model’ offered in [5], the ‘Gaussian pressure distribution model’ considered in [5,8], and the ‘dominant waves interference model’ proposed in [9]. Moreover, no comparison of the theoretical predictions given by the alternative theories proposed in [5,8,9] is included in these studies. It is then interesting and useful – indeed necessary – to compare the assumptions underlying the three theoretical models proposed in [5,8,9] and to compare the theoretical predictions given by these alternative explanations to the 37 observations of ship wakes reported in [5] for the broad range of Froude numbers 0.1 < F < 1.7. This comparison of underlying assumptions and predictions, not previously considered as already noted, is then considered here. As already noted, the theoretical explanation offered in [5] is based on the assumption that waves with wavelengths Λ larger than the ship length Ls can be ignored. This key assumption is justified in [5] via the consideration of the waves created by a Gaussian distribution of pressure at the free surface, also considered in [8] as noted earlier. A Gaussian distribution of pressure at the free surface may be a reasonable model of the flow due to a highspeed planing hull. However, a free-surface pressure distribution that has a single peak may not realistically account for the strong interference between the dominant waves created by the bow and the stern of a ship, and therefore may not be a realistic model of the flow around the hull of a typical displacement ship (notably, monohull ships or catamarans). Moreover, the Gaussian distribution of pressure considered in [5,8] is smooth (infinitely differentiable), whereas the flow created by the bow and the stern of a ship (especially a fine bow or stern) varies very rapidly and indeed is not smooth at a sharp bow and stern, where the hull geometry varies abruptly. The difference is mathematically important because farfield waves are strongly influenced by near-field singularities, as shown in e.g. [11]. Indeed, this basic property of Fourier transforms provides a mathematical interpretation (explanation) of the common observation that a ship mostly creates two dominant waves that originate from points where the hull geometry varies abruptly, i.e. the ship bow and stern. This well-known feature of the wavemaking of a ship – the key approximation underlying the dominant waves interference model considered in [9] – is clearly apparent from Fig. 2, which depicts the wave patterns created by the Wigley hull and the Series 60 model at a Froude number F = 0.3. These wave patterns were determined via the Neumann–Michell theory given in [12,13]. Thus, the smooth (Gaussian) free-surface pressure-distribution, with a single peak, considered in [5,8] may not be a realistic model of the flow around a ship hull, except for high-speed planing hulls, and consequently may not provide an adequate justification of the key assumption that underlies the ‘cutoff wavelength’ model of Kelvin’s wake proposed by Rabaud and Moisy in [5]. It is then necessary to test if the assumption λ ≡ Λ/Ls ≤ 1 that underlies the Rabaud–Moisy flow-model can be rationalized, notably in the light of the wave-interference analysis given in [9]. Indeed, it may be argued that the analysis of interference between the dominant waves created by the bow and the stern of a ship given in [9] provides a reasonable definition of a cutoff wavelength λcut ; specifically, this definition is related to the fact that interference between dominant divergent waves results in the apparent effective elimination of wavelengths λcut < λ via destructive interference.
14
J. He et al. / European Journal of Mechanics B/Fluids 49 (2015) 12–19
Fig. 2. Elevation z = e(x, y) with e ≡ Eg /Vs2 of the free surface (computed by means of the Neumann–Michell theory) in the region (−3.5 ≤ x ≡ X /Ls ≤ 1, −2 ≤ y ≡ Y /Ls ≤ 2) for the Wigley hull (left side) or the Series 60 model (right) at a Froude number F = 0.3. The figure illustrates the well-known property that the dominant waves created by a ship hull originate at the ship bow and stern, where the hull geometry varies abruptly. This basic feature of the wavemaking of a ship underlies the analysis of wave-interference effects on the Kelvin wave pattern given in [9].
2. Wake angles predicted by three simple flow models The angles of the wakes created by monohull ships, catamarans, and planing hulls predicted by the three theoretical models considered in [5,8,9] are now summarized as they are required for purposes of comparison. [9] shows that constructive longitudinal interference between the (transverse or divergent) waves created by the bow and the stern of a monohull ship yields dominant waves along rays at angles ψ = ±ψn with 1 ≤ n that are defined in terms of the Froude number F as
(2n − 1)2 π 2 F 4 /ℓ2 − 1 tan ψn = 2(2n − 1)2 π 2 F 4 /ℓ2 − 1
√ ℓ/π (2/3)1/4 FK ≈ 0.59. (3a) with FnT ≡ √ ≤ F and FK ≡ (2/3)1/4 2n − 1 Here, ℓ ≈ 0.9 denotes the (nondimensional) distance between the effective origins of the bow and stern waves. The largest transverse and divergent waves are found along √ the ray angles √ ψ = ±ψn defined by (3a) with FnT ≤ F ≤ FK / 2n − 1 or FK / 2n − 1 ≤ F , respectively. The ray angles ψn are depicted in Fig. 1 for n = 1, 2 and 6. For longitudinal interference between the divergent bow and stern waves and FK ≤ F , one has ψn ≤ ψ1 ≡ ψmax with
π 2 F 4 /ℓ2 − 1 ℓ/2 0.14 ≈ ≈ 2 as F → ∞. 2π 2 F 4 /ℓ2 − 1 πF2 F
tan ψmax =
(3b)
Ref. [9] also shows that lateral interference between the divergent waves created by the bows and sterns of the twin hulls of a catamaran similarly results in dominant waves along rays at angles ψ = ±ψmax that are defined in terms of the Froude number
√
Fb ≡ Vs / gB ≡ F / B/Ls ≡ F / b
(4)
based on the lateral separation distance B between the twin hulls of the catamaran as
1 + 16π 2 F 4 /b2 − 1
tan ψmax = with FKcat ≡
2 + 32π 2 F 4 /b2 1 2
√
F 3/π ≈ 0.37 < √ b
√
and
1 tan ψmax ≈ √ 8π
b F
(5a)
√
≈ 0.2
b F
The wake angles ψ = ±ψmax given in [5,8] using the assumption λ ≤ λcut = 1 and/or the consideration of a Gaussian distribution of pressure at the free surface are given by
tan ψmax =
and
2π F 2 /λcut − 1
4π F 2 /λcut − 1
1 tan ψmax ≈ √ 8π
√ λcut F
with FKcut ≡
≈ 0.2
√ λcut F
1 cut 3λ /π ≤ F (6a) 2 as F → ∞.
(6b)
The wake angle ψmax given in [5] corresponds to the cutoff wavelength λcut and the related Froude number F cut
λcut = 1 and FKcut ≡
3/π /2 ≈ 0.49.
(6c)
The high-Froude-number wake created by a Gaussian distribution of pressure at the free surface considered in [8] corresponds to
λcut =
8/5 ≈ 1.265
and FKcut ≡
3/π /101/4 ≈ 0.55.
(6d)
The lower bound FKcut in (6a), (6c) and (6d) and the lower bounds FK and FKcat in (3a) and (5a) are the values of the Froude number
F for which the wake angle ψmax is equal to the Kelvin angle ψK , i.e. the value of F that yields ψmax = ψK . Thus, expressions (3a), (5a), (6a), (6c), (6d) predict that the largest waves created by a ship hull are found at an angle ψmax that is smaller than the Kelvin wake angle ψK if the Froude number F is larger than FK ≈ 0.59, FKcat ≈
√
0.37 b, FKcut ≈ 0.49 or FKcut ≈ 0.55. Interference between the waves created by the bow and the stern of a monohull yields ψmax = O(1/F 2 ), i.e. predicts a wake angle ψmax that decreases like 1/F 2 as F → ∞, whereas both expression (5b) for a catamaran and expression (6b) for a Gaussian free-surface pressure-distribution yield ψmax = O(1/F ), i.e. predict wake angles ψmax that decrease like 1/F as F → ∞. Indeed, the high-speed approximations (5b) and (6b) are identical except for the fact that the beam/length ratio b of a catamaran or the cutoff wavelength λcut for a Gaussian pressure-distribution appears in (5b) or (6b). Thus, these expressions, with the substitution b ↔ λcut , predict nearly identical ship wake angles. 3. Comparison with observations and computations 3.1. The Rabaud–Moisy observations of ship wakes
as F → ∞.
(5b)
Here, b ≡ B/Ls where Ls is the length of the catamaran, and B denotes the lateral separation distance between the twin hulls of the catamaran (beam/length ratio) as already noted.
Fig. 1 depicts the 37 observations of ship wake angles ψmax reported in [5] for the broad range of Froude numbers 0.1 < F < 1.7 and ray angles predicted by the three simple theoretical approaches considered here. Specifically, Fig. 1 depicts the wake
J. He et al. / European Journal of Mechanics B/Fluids 49 (2015) 12–19
angles ψmax predicted by the ‘Gaussian pressure distribution model’ (Gauss), the Rabaud–Moisy ‘cutoff wavelength’ model with the cutoff wavelength λcut taken as the ship length (Ship length), and lateral interference between the divergent waves created by the twin hulls of a catamaran (Catamaran) with beam/length ratio b within the range 0.4 ≤ b ≤ 0.8. In addition, Fig. 1 shows the ray angles ψn with n = 1, 2 and 6 along which the waves are largest due to constructive longitudinal interference between the transverse and divergent waves created by the bow and the stern of a monohull ship (Monohull). Fig. 1 shows that the four curves that correspond to expression (5a) associated with wave-interference effects for a catamaran, with b = 0.4 or b = 0.8, and expressions (6a), (6c) and (6d) related to a Gaussian free-surface pressure distribution or the Rabaud–Moisy cutoff-wavelength model are nearly parallel, in accordance with the high-speed approximations (5b) and (6b). The ten wake observations within the range 0.6 < F < 1.4 shown in Fig. 1 are fully consistent with the ray angles ψmax associated with longitudinal or lateral interference between the divergent waves created by the bow and the stern of a monohull ship (Monohull) or a catamaran (Catamaran) with beam/length ratio b within the range 0.4 ≤ b ≤ 0.8. The two wake observations within the high-speed range 1.6 < F < 1.7 are in close agreement with the predictions given by the Rabaud–Moisy cutoff wavelength model (Ship length) and the Gaussian pressure distribution model (Gauss) and may also be consistent with wave-interference effects for a catamaran with a large beam/length ratio b. The two curves marked ‘Gauss’ or ‘Ship length’ may also be consistent with several of the ten wake observations shown in Fig. 1 for 0.6 < F < 1.4, but are not consistent with most wake observations. Most of the observations (specifically 25 out of a total of 37 observations, i.e. nearly 68%) shown in Fig. 1 correspond to Froude numbers within the range F ≤ FK ≈ 0.59. The figure shows that the 25 observations for F ≤ 0.59 are within the range 13° ≤ ψmax ≤ 21° and are mostly located around the Kelvin angle ψK ≈ 19°28′ . However, three observations of wake angles shown in Fig. 1 for F < 0.59 are smaller than 15°, i.e. significantly smaller than ψK . Fig. 1 shows that these three observations, as well as the 22 observations of ship wake angles that are roughly equal to the Kelvin angle ψK for F < 0.6, are fully consistent with the ‘interference curves’ that depict three angles ψn for 1 ≤ n ≤ 6 along which the waves are largest due to constructive interference between the transverse or divergent waves created by a ship bow and stern [9]. The comparison of observations of ship wake angles and theoretical predictions shown in Fig. 1 provides solid evidence that interference between the bow and stern waves of a monohull ship or a catamaran is a highly probable, although remarkably simple, explanation for the observations of wake angles reported in [5] for the broad range of Froude numbers 0.1 < F < 1.7. This speed range includes the range F < 0.6 where interference between transverse bow and stern waves is important. The predictions given by an analysis of interference effects on the Kelvin wave pattern are in particularly good agreement with the wake observations within the range of Froude numbers 0.1 < F < 1.4. This range of Froude numbers includes 35 of the 37 observations (nearly 95%) shown in Fig. 1 and indeed corresponds to the vast majority of ships. 3.2. Numerical computations The wake angles ψmax predicted by the Gaussian pressure distribution model and the dominant waves interference model are now compared to numerical computations of the waves created by the classical Wigley parabolic hull at a Froude number F = 1. Fig. 3 depicts the waves created within the near-field region −3.5 ≤ x ≡ X /Ls ≤ 1, −1 ≤ y ≡ Y /Ls ≤ 1. The waves depicted above and below the centerline y = 0 are computed via the classical Hogner approximation, which is based on the Green
15
Fig. 3. Waves created within the region −3.5 ≤ x ≤ 1, −1 ≤ y ≤ 1 by the Wigley hull at F = 1. The waves above and below the centerline y = 0 are computed via the Hogner approximation or a Rankine source panel method, respectively. The figure also shows the Kelvin rays ψ = ±19°28′ with origin at the bow of the hull, and the rays ψ ≈ ±8°8′ that are predicted by the interference relation (3b) and originate at the bow (0.5, 0) or the stern (−0.5, 0) of the Wigley hull.
Fig. 4. Waves created within the region −14 ≤ x ≤ 1, −5 ≤ y ≤ 5 by the Wigley hull at F = 1. The waves are computed via the Hogner approximation. The figure also shows the Kelvin cusp lines ψ ≈ ±19°28′ and the rays ψ ≈ ±12°39′ and ψ ≈ ±8°8′ predicted by the Gaussian pressure distribution model or the wave interference model, respectively.
function that satisfies the Kelvin–Michell linearized boundary condition at the free surface and is related to the Neumann–Michell theory given in [12,13] or correspond to a numerical solution of the Neumann–Kelvin problem (Laplace equation and Kelvin–Michell linear free-surface boundary condition) obtained via a panel method based on Rankine singularities in the manner explained in [14]. Although differences can be observed between the near-field waves depicted in the upper and lower halves of Fig. 3, these two wave patterns (determined by means of significantly different numerical methods as noted) are largely consistent. Fig. 3 also shows the two Kelvin rays ψ = ±19°28′ with origin at the bow (0.5, 0) of the Wigley hull and the four rays ψ = ±8°8′ that are predicted by the dominant waves interference relation (3b) and originate at the bow (0.5, 0) or the stern (−0.5, 0) of the hull. Both numerical computations show that the largest waves are approximated located along the rays ψ = ±8°8′ predicted by the dominant waves interference model. Fig. 4 depicts the waves, computed via the Hogner approximation, created by the Wigley hull at F = 1 within the larger region −14 ≤ x ≤ 1, −5 ≤ y ≤ 5. This figure shows that although waves exist inside the wedge bounded by the rays ψ = ±19°28′ that correspond to the cusp lines (marked as dashed lines) of the Kelvin wake, the largest waves are found well inside the Kelvin wake, approximately along the rays (marked as solid lines) ψ = ±8°8′ predicted by the dominant waves interference model. Fig. 4 also shows that the rays ψ = ±12°39′ (marked as short dashed lines) predicted by the Gaussian pressure distribution model are noticeably outside the region where the largest waves are found. Thus, the computed waves depicted in Fig. 4 are largest along ray angles that agree well with the predictions given by the bow and stern waves interference model, but are noticeably smaller
16
J. He et al. / European Journal of Mechanics B/Fluids 49 (2015) 12–19
The wavelength at the cusp ψ = ψK of the Kelvin wake is given by
λcusp = 2 λmax /3 = 4π F 2 /3.
(12)
The is smaller than the ship length if F < √ cusp wavelength λ 3/π /2 ≈ 0.49 but is larger than the ship length if 0.49 < F . The cusp wavelength λcusp can be significantly larger than the ship length λs ≡ 1; e.g. one has λcusp ≈ 4.2 for F = 1 and λcusp ≈ 16.8 for F = 2. The range 0 ≤ λ ≤ λcusp that corresponds to divergent waves is considered hereafter. The wavelength λD of the divergent waves along a ray angle ψ is given by (11) as cusp
0≤
Fig. 5. Variation of the Kelvin angle ψK for 0 ≤ λ/λmax ≤ 1. Divergent waves (solid line) and transverse waves (dashed line) correspond to 0 ≤ λ/λmax ≤ 2/3 and 2/3 ≤ λ/λmax ≤ 1, respectively.
than the ray angles predicted by the Gaussian pressure distribution model. This finding provides strong evidence that constructive interference between the dominant bow and stern waves created by a ship, as illustrated in Fig. 2, results in the appearance of narrow ship wakes at high Froude numbers. 4. The cutoff-wavelength model
with F k = 1/ cos γ ≡ 1 + tan γ 2
2
and − π /2 < γ < π /2.
(7)
λ = 2π F 2 cos2 γ ≤ 2π F 2 ≡ λmax .
(8)
Thus, the wavelength λmax of the longest waves created by √a ship is smaller than the ship length for Froude numbers F < 1/ 2π ≈ 0.4 but is larger than the ship length if 0.4 < F , as well known. In the far-field limit h ≡ x2 + y2 → ∞, significant contributions to a superposition of elementary waves (7) only stem from values of γ for which the phase
θ ≡ (x cos γ + y sin γ )(1 + tan2 γ )/F 2
(9)
is stationary, i.e. from roots of the ‘stationary–phase relation’ dθ/dγ = 0. This equation yields y
−x
=
Within the divergent-wave range 0 ≤ λ ≤ λcusp , the function ψ(λ) defined by (11) increases, as already noted and shown in Fig. 5. The inequalities
λ ≤ λcut ≤ λcusp ≡ 2λmax /3 ≡ 4π F 2 /3 and √ max cut λ /λ − 1 cut −1 ≤ ψK ≈ 19°28′ ψ ≤ ψ ≡ tan 2λmax /λcut − 1
tan γ 1 + 2 tan2 γ
.
tan ψ
cut
(10)
The stationary–phase relation (10) and the dispersion relation (8) relate the ray angle ψ and the wavelength λ as
√ ′ √ max λ /λ − 1 λ (1 − λ′ ) λ tan ψ = ≡ with λ′ ≡ max . (11) max ′ 2λ /λ − 1 2−λ λ The ray angle ψ defined by (11) is real for 0 ≤ λ′ ≤ 1, increases from 0 to the Kelvin angle ψK for 0 ≤ λ′ ≤ 2/3 and decreases from ψK to 0 for 2/3 ≤ λ′ ≤ 1, as depicted in Fig. 5. The ‘short’ waves 0 ≤ λ′ ≤ 2/3 and the ‘long’ waves 2/3 ≤ λ′ ≤ 1 correspond to the divergent waves and the transverse waves, respectively, in the Kelvin wave pattern.
√ λcut λcut 1 ≈ √ ≡ √ max F 8π 2 λ √ λcut ≈ 0.2 with λcut ≪ λmax ≡ 2π F 2 . F
2
The dispersion relation F 2 k = 1/ cos2 γ defines the wavelength λ ≡ 2π/k as
tan ψ ≡
4.2. Two mathematically-equivalent inequalities
(14) (15)
√
At some distance behind a ship, the waves created by the ship can be represented as a linear superposition of elementary waves that travel at angles −π /2 < γ < π /2 from the path of the ship (the x axis). These elementary waves are given by e
(13)
therefore are mathematically equivalent. Expression (15) for ψ cut yields the approximation
4.1. Basic relations
kz +ik(x cos γ +y sin γ )
2 λD 8 tan2 ψ ≤ . ≡ max 2 2 λ 3 1 + 4 tan ψ + 1 − 8 tan ψ
(16)
The fact that the inequalities (14) and (15) are mathematically equivalent means that an assumed restriction λ ≤ λcut < λcusp has the automatic effect of reducing the wake angle ψ below the Kelvin angle ψK and indeed is tantamount to imposing that the wake angle ψ is smaller than the angle ψ cut < ψK . Clearly then, a model that invokes a cutoff-wavelength assumption λ ≤ λcut , as in [5], cannot explain observations of ship wakes that are narrower than Kelvin’s wake unless the assumed choice of cutoff-wavelength λcut can be rationally justified. A reasonable definition of a cutoff wavelength λcut , based on the analysis of interference between the bow wave and the stern wave of a monohull or between the twin bow waves of a catamaran summarized in (3a) and (5a), is then considered here. Specifically, expressions (3a) and (5a) for the wake angles ψmax predicted by an analysis of interference between the bow and stern waves of a monohull, or between the twin bow waves of a catamaran, are now used in expression (13) with ψ = ψmax to determine the cutoff wavelength λcut that corresponds to ψmax and a wave-interference analysis. This definition of a cutoff wavelength λcut is then
λcut ≡
16 π F 2 tan2 ψmax 1 + 4 tan2 ψmax +
1 − 8 tan2 ψmax
≈ 8π F 2 tan2 ψmax
(17)
where (8) was used. 4.3. Cutoff wavelengths related to wave interference The cutoff wavelength λcut that corresponds to interference between the bow and stern waves of a ship and the wake angle ψmax
J. He et al. / European Journal of Mechanics B/Fluids 49 (2015) 12–19
17
the Froude number F is then reasonable for catamarans, as well as for planing hulls as shown in [5,8]. Furthermore, the cutoff wavelengths λcut given by (19b), (6c) and (6d) with 0.7 ≤ b ≤ 1.3 do not differ widely. The cutoff wavelengths λcut given by (6c) and (6d) are also consistent with the cutoff wavelength (18a) associated with interference between the bow and stern waves of a monohull for a very narrow range of Froude numbers F , roughly for 0.65 < F < 0.75. However, the cutoff wavelength (18a) related to interference between the bow and stern waves of a monohull is a rapidly decreasing function of F that cannot realistically be assumed constant. Thus, the key assumption λ ≤ 1 invoked in [5] cannot be rationalized by an analysis of interference between the bow and stern waves of a ship hull, and for the wide range of Froude numbers F < 1.2. 4.4. The cutoff-wavelength and wave-interference models Fig. 6. Variations of the cutoff wavelengths λcut related to longitudinal interference between the divergent waves created by the bow and the stern of a monohull ship (Monohull) for FK ≈ 0.59 < F ≤ 1.6 or to lateral interference between √ the divergent waves created by the twin hulls of a catamaran (catamaran) for FKcat b ≈
√
0.37 b ≤ F ≤ 1.6 with b = 0.1, 0.5, 1 and 1.5. The cutoff wavelengths λcut given by (6c) or (6d) and associated with the ship length or a Gaussian distribution of pressure at the free surface, as in [5] or [8], are also shown. The figure shows that the cutoff wavelength λcut related to longitudinal interference for a monohull ship (Monohull) is a rapidly decreasing function of F that cannot realistically be assumed constant, in contradiction with the key assumption used in [5].
given by (3b) is determined by (17) and (3b). One then has
λ
4
cusp K
≤λ
2
≡ π =2 ℓ ≈ 1.47 3 3 with FK ≡ ℓ 3/2/π ≈ 0.59 < F
cut
and λ
cut
≈
2 ℓ2
πF2
FK2
≈
0.516 F2
(18a)
as F → ∞.
(18b)
Thus, λcut decays like 1/F 2 as F → ∞. Within the Froude-number range FK ≈ 0.59 ≤ F ≤ 1.2, the cutoff wavelength λcut varies within the approximate range 1.47 ≥ λcut ≥ 0.36. The cutoff wavelength λcut that corresponds to interference between the twin bow waves of a catamaran and the wake angle ψmax given by (5a) is similarly determined by (17) and (5a). One then has √ = λcusp F =F cat b K
with FKcat
4 3
b
π (FKcat )2 b = √ ≈ 0.58b ≤ λcut ≤ b
√ b≡
3
1/4
√
√
b
2 π
and λcut ≈ b ≡ B/Ls
3
√ ≈ 0.37 b ≤ F
(19a)
as F → ∞.
(19b)
Thus, λ → b as F → ∞. The cutoff wavelength λ varies cut within √ the range 0.58 b < λ < b as F varies within the range 0.37 b ≤ F < ∞. Fig. 6 depicts the cutoff wavelengths λcut given by (18a) or (19a) with b = 0.1, 0.5, 1 and 1.5. As already explained, these wavelengths are determined from an analysis of interference between the bow and stern waves of a monohull, or between the twin bow waves of a catamaran with beam/length ratios b. The cutoff wavelengths given in [5,8] and by (6d) and the cusp wavelength λcusp given by (12) are also shown in Fig. 6. For Froude numbers F larger than Fhigh with Fhigh ≈ 1.2, the cutoff wavelength λcut given by (19a) and associated with interference between the twin bow waves of a catamaran does not vary significantly with F and is given by λcut ≈ b as shown in (19b). The assumption of a cutoff wavelength λcut that is independent of cut
The approaches used in [5,9] differ in several other important respects as well, as now noted. In particular, expression (3) fully determines the wake angle of a monohull ship in terms of the Froude number F . Similarly, (5) fully determines the wake√ angle of a catamaran in terms of the Froude number Fb ≡ F / B/Ls based on the lateral separation distance B between the twin bows of the catamaran. Indeed, expressions (3) and (5) are based on the classical theories of ship waves and wave interference, and accordingly do not involve any a-priori unknown parameter, whereas the flow-model used in [5] involves an a-priori unknown cutoff wavelength λcut that must be chosen somehow. Furthermore, the flow-model used in [5] assumes that the wavelength λ can be restricted as
λ ≤ λcut or H (λcut − λ) (20) where H (·) denotes the usual Heaviside step function. The waveinterference analysis given in [9], on the other hand, is based on the interference relations
ℓ cos γ = (2n − 1)λ/2 or b sin γ = nλ with 1 ≤ n
and
λ ≡ 2π F 2 cos2 γ
(21)
for a monohull or a catamaran. Here, ℓ cos γ represents the projection of the distance ℓ between the effective origins of the dominant waves created by the bow and the stern of a monohull ship, and b sin γ is the projection of the lateral separation distance b between the twin bows of a catamaran, upon the direction of propagation (cos γ , sin γ ) of the waves in the Kelvin wake. The interference relations (21) state that these projections yield constructive interference (and therefore largest waves) between the bow and stern waves of a monohull ship or between the bow waves of a catamaran. These interference relations essentially restrict the wavelength λ via the Dirac function
(2n − 1)λ δ ℓ cos γ − 2
cut
or δ(b sin γ − nλ)
with 1 ≤ n
(22)
instead of the Heaviside function in (20) that is used in [5]. The relations (22), where 1 ≤ n, define a series of angles ψn along which large waves can be found due to constructive interference between divergent waves as well as transverse waves, unlike the relations (20) that only correspond to divergent waves with n = 1. Indeed, the flow model based on the assumption of a cutoff wavelength λcut cannot be applied within the Froudenumber ranges
ℓ ≈ 0.59 or π √ √ √ 31/4 b F < FKcat b = ≈ 0.37 b √ 2 π 1/4
F < FK = (3/2)
(23)
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J. He et al. / European Journal of Mechanics B/Fluids 49 (2015) 12–19
for a monohull or a catamaran, respectively. Thus, this flow model cannot explain the observations of ship wake angles that are significantly smaller than the Kelvin angle ψK reported in [5] and shown in Fig. 1 for Froude numbers F < 0.6, whereas interference effects between the divergent or transverse waves created by the bow and the stern of a monohull ship are consistent with these wake observations. Moreover, Fig. 1 shows that ship wake angles that are roughly equal to the Kelvin angle ψK for F < 0.6 are also consistent with interference between the bow and stern waves of a monohull ship. 5. Conclusion Fig. 1 shows that the theoretical predictions given by the Rabaud–Moisy ‘cutoff-wavelength model’ proposed in [5] and the ‘Gaussian pressure distribution model’ considered in [5,8] are in close agreement with two wake observations within the range 1.6 < F < 1.7 and may possibly also be consistent with several observations within the range 0.6 < F < 1.4, but are not consistent with most of the 37 observations reported in Fig. 1. This finding suggests that the ‘cutoff-wavelength model’ and the ‘Gaussian pressure distribution model’ may not be realistic for most ships, as suggested also by a critical examination of the assumptions underlying these two flow models and the comparisons with numerical computations of wave patterns given in Fig. 4. In particular, Fig. 6 demonstrates that the assumption of a cutoff wavelength λcut that is independent of the Froude number and approximately equal to the ship length invoked in the cutoff-wavelength model cannot be rationalized via the ‘dominant-wave-interference analysis’ given in [9] for monohull ships in the most common speed range 0.1 < F < 1.2. Moreover, the smooth distribution of free-surface pressure, with a single peak, considered in the Gaussian pressure distribution model is not a realistic model of the flow around the hull of a typical displacement ship, which creates two dominant (bow and stern) waves as illustrated in Fig. 2. Fig. 1 shows that the 35 observations of ship wakes reported in [5] for Froude numbers within the range 0.1 < F < 1.4 are fully consistent with the ray angles ψn and the wake angles ψmax predicted by an analysis of longitudinal or lateral interference between the dominant waves created by the bow and the stern of a monohull ship or the twin hulls of a catamaran with beam/length ratio b within the range 0.4 ≤ b ≤ 0.8. This range of Froude numbers includes 35 of the 37 observations (nearly 95%) shown in Fig. 1 and indeed corresponds to the vast majority of ships. The two wake observations reported in Fig. 1 within the range 1.6 < F < 1.7 may possibly be consistent with wave interference for a high beam/length ratio catamaran. Nearly 68% of the 37 wake observations reported in [5] and in Fig. 1 correspond to the range F < 0.6 where interference between transverse bow and stern waves is important. Fig. 1 shows that the ray angles ψn along which the waves are largest due to constructive interference between the transverse or divergent waves created by the bow and the stern of a monohull ship are fully consistent with the 25 observations of ship wakes in the ‘transverse-wave range’ F < 0.6. These 25 wake observations include 22 wake angles that are roughly equal to the Kelvin angle ψK and three wake angles significantly smaller than ψK . Thus, the comparison of observations of ship wake angles and theoretical predictions given in Fig. 1 provides strong evidence that interference between the bow and stern waves of a monohull ship or a catamaran is a highly probable, although remarkably simple, explanation for the observations of ship wakes reported in [5] for a broad range of Froude numbers. Furthermore, the numerical computations of wave patterns depicted in Fig. 4 show that the computed waves are largest along ray angles that agree well with the predictions of the bow and stern waves interference model. This finding provides solid additional evidence for the conclusion that interference between the dominant bow and stern waves created by a ship, as illustrated in Fig. 2, results in the appearance of narrow ship wakes at high Froude numbers.
However, elementary theoretical considerations and Fig. 1 suggest that the bow and stern waves interference model may not be realistic in a high Froude number range Fhigh < F . Indeed, an obvious limitation of this simple flow model is that the basic theoretical framework of linear potential flow theory may be unrealistic at very high Froude numbers, notably because a high-speed ship typically creates a large spray sheet and related wave breaking wake that may be expected to have a significant influence on the appearance of the ship wake [15]. Another obvious limitation of the model is that high-speed ships in the planing-hull regime do not create distinct bow and stern waves like common displacement ships, and a pressure distribution model may be a reasonable flow model for high-speed planing hulls (the two wake observations for 1.6 < F < 1.7 in Fig. 1 may possibly be due to planing hulls). Moreover, even for a ship that creates distinct bow and stern waves at high speed, interference effects are more complex than assumed in the elementary geometrical analysis considered in [9]. Indeed, potential flow around a ship hull (a slender body) can be represented by means of a distribution of sources and sinks over the bow and stern regions of the ship hull, e.g. as given by the Hogner approximation [12,13]. Thus, lateral interference occurs between the sources (or sinks) distributed over the port and starboard sides of the bow (or stern) regions of the ship hull, and longitudinal interference occurs between the waves created by the sources and the sinks distributed over the bow and stern regions. The flow around a monohull ship is therefore not fundamentally different from the flow created by a narrow catamaran with small beam/length ratio. [9] shows that the wake angle ψmax decreases like 1/F or 1/F 2 for lateral or longitudinal interference, respectively, and the behavior of ψmax for a high-speed monohull ship can then be expected to depend on both F and the hull shape, notably the beam/length ratio. Indeed, as shown in [16], combined longitudinal and lateral interference is significantly more complex than either the pure longitudinal interference or the pure lateral interference considered in [9]. The simplified analysis given in [9] nevertheless provides valuable insight. √In particular, the wake angles ψmax ≈ 0.14/F 2 and ψmax ≈ 0.2 b/F associated with √longitudinal or lateral interference are equal if F = F∗ ≈ 0.7/ b. The ‘crossover’ Froude number F∗ is given by F∗ ≈ 2.2 for b = 0.1 and F∗ ≈ 1.6 for b = 0.2. These large values of F∗ suggest that the approximation ψmax ≈ 0.14/F 2 may in fact be adequate for most monohull ships encountered in practice. Although the differences between the cutoff wavelength model, the Gaussian pressure distribution model, and the dominant bow and stern waves interference model proposed in [5,8,9] are mostly emphasized in the foregoing, these three flow models share an important similarity that also deserves to be emphasized. Specifically, the three models consider the simple theoretical framework of steady, deep-water, linear potential-flow hydrodynamics – unlike the considerably more complex theories considered in [2,4,6,7] that involve effects of ambient waves, nonlinearities or finite water depth – and moreover are based on a simple key assumption or approximation. The highly simplified flow models considered in [5,8,9] illustrate, yet again, the necessity and the value of considering analytical models based on simplifying approximations to understand seemingly complicated flows. It is also appropriate to note that credit is due to Rabaud and Moisy [5] for first proposing a simple flow model of the Kelvin ship wake to seek an explanation of narrow ship wakes, as well as for providing invaluable observations of ship wakes over the broad range of Froude numbers 0.1 < F < 1.7. References [1] W.H. Munk, P. Scully-Power, F. Zachariasen, Ships from space, Proc. R. Soc. Lond. Ser. A 412 (1987) 231–254. [2] E.D. Brown, S.B. Buchsbaum, R.E. Hall, J.P. Penhune, K.F. Schmitt, K.M. Watson, D.C. Wyatt, Observations of a nonlinear solitary wave packet in the Kelvin wake of a ship, J. Fluid Mech. 204 (1989) 263–293.
J. He et al. / European Journal of Mechanics B/Fluids 49 (2015) 12–19 [3] A.M. Reed, J.H. Milgram, Ship wakes and their radar images, Annu. Rev. Fluid Mech. 34 (2002) 469–502. [4] M.C. Fang, R.Y. Yang, I.V. Shugan, Kelvin ship wake in the wind waves field and on the finite sea depth, J. Mech. 27 (2011) 71–77. [5] M. Rabaud, F. Moisy, Ship wakes: Kelvin or Mach angle? Phys. Rev. Lett. 110 (2013) 214503. [6] C.C. Mei, M. Naciri, Note on ship oscillations and wake solitons, Proc. R. Soc. Lond. Ser. A 432 (1991) 535–546. [7] Q. Zhu, Y. Liu, D.K.P. Yue, Resonant interactions between Kelvin ship waves and ambient waves, J. Fluid Mech. 597 (2008) 171–197. [8] A. Darmon, M. Benzaquen, E. Raphael, Kelvin wake pattern at large Froude numbers, J. Fluid Mech. (2014) 738. [9] F. Noblesse, J. He, Y. Zhu, L. Hong, C. Zhang, R. Zhu, C. Yang, Why can ship wakes appear narrower than Kelvin’s angle? Eur. J. Mech. B Fluids 46 (2014) 164–171. [10] F. Dias, Ship waves and Kelvin, J. Fluid Mech. 746 (2014) 1–4.
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[11] M.J. Lighthill, An Introduction to Fourier Analysis and Generalised Functions, Cambridge University Press, 1958. [12] F. Noblesse, F. Huang, C. Yang, The Neumann–Michell theory of ship waves, J. Engrg. Math. 79 (2013) 51–71. [13] F. Huang, C. Yang, F. Noblesse, Numerical implementation and validation of the Neumann–Michell theory of ship waves, Eur. J. Mech. B Fluids 42 (2013) 47–68. [14] S. Bal, Prediction of wave pattern and wave resistance of surface piercing bodies by a boundary element method, Internat. J. Numer. Methods Fluids 56 (2008) 305–329. [15] F. Noblesse, G. Delhommeau, P. Queutey, C. Yang, An elementary analytical theory of overturning ship bow waves, Eur. J. Mech. B Fluids 48 (2014) 193–209. [16] J. He, C. Zhang, Y. Zhu, H. Wu, F. Noblesse, D. Wan, L. Zou, W. Li, Interference effects on the Kelvin wake of a catamaran, Eur. J. Mech. B Fluids (2014) submitted.