Elementary ship models and farfield waves

Accepted Manuscript Elementary ship models and farfield waves Yi Zhu, Jiayi He, Huiyu Wu, Wei Li, Francis Noblesse, Gerard Delhommeau

PII: DOI: Reference:

S0997-7546(17)30158-9 https://doi.org/10.1016/j.euromechflu.2017.09.013 EJMFLU 3215

To appear in:

European Journal of Mechanics / B Fluids

Received date : 25 March 2017 Revised date : 18 September 2017 Accepted date : 20 September 2017 Please cite this article as: Y. Zhu, J. He, H. Wu, W. Li, F. Noblesse, G. Delhommeau, Elementary ship models and farfield waves, European Journal of Mechanics / B Fluids (2017), https://doi.org/10.1016/j.euromechflu.2017.09.013 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Elementary ship models and farfield waves Yi Zhua , Jiayi Heb , Huiyu Wua , Wei Lia , Francis Noblessea,∗, Gerard Delhommeauc a State

Key Laboratory of Ocean Engineering, School of Naval Architecture, Ocean & Civil Engineering a Collaborative Innovation Center for Advanced Ship and Deep-Sea Exploration a Shanghai Jiao Tong University, Shanghai, China b Shanghai Key Laboratory of Ship Engineering, Marine Design and Research Institute of China, Shanghai, China c Ecole Centrale, Nantes, France

Abstract Kelvin’s classical 1-point ship model, the basic 2-point models of a monohull ship and a catamaran previously studied via elementary geometrical considerations, optimal versions of these two basic 2-point models, and a 4-point model of a catamaran (not considered previously) are analyzed within a common theoretical approach based on potential flow theory, the Green function associated with the Kelvin-Michell linear boundary condition at the free surface, the related Fourier representation of farfield ship waves, and the farfield stationary-phase approximation. The analysis shows that the apparent wake angle associated with the highest waves found inside the cusps of the Kelvin wake as a result of interferences among the divergent waves created by a ship at a high Froude number can be realistically predicted via elementary ship models. In particular, the apparent wake angles for monohull ships and catamarans can be predicted very well via a 2-point model or a 4-point model. Moreover, these models provide basic insight into main features of the farfield waves created by fast ships. Notably, differences between the amplitudes of the bow and stern waves are shown to have no influence on the occurrence of constructive or destructive interferences, although they affect the intensity of wave-interference effects (strong if the amplitudes of the bow and stern waves are commensurate, weak otherwise). Another notable conclusion is that lateral interference effects become more important as the Froude number increases. Indeed, lateral interferences between the twin hulls of a catamaran dominate longitudinal interferences between the fore and aft of the ship at Froude numbers greater than about 1. However, longitudinal interferences between the bow and the stern waves remain dominant for common fast monohull ships. Keywords: Ship models, farfield ship waves, wave interferences, wake angle, Green function 1. Introduction The farfield waves due to a monohull ship or a catamaran of length L that travels at a constant speed V along a straight path, taken as the X axis, in calm water of large depth and horizontal extent are considered. The Froude number F is defined as p F = V/ gL (1)

where g is the acceleration of gravity. The waves created by the ship are observed from a Cartesian system of coordinates (X, Y, Z) attached to the ship. The undisturbed free surface is taken as the plane Z = 0, and the Z axis is vertical and points upward. The X axis points toward the ship bow. The farfield ship waves are analyzed within the usual framework of linear potential flow theory, as in numerous studies since Kelvin’s classical analysis [1]. This basic flow model is realistic, and the only available practical option, to analyze farfield ship waves. Within this theoretical framework, the flow created by the ship can be represented in terms of a distribution of sources (and possibly dipoles) over the mean wetted surface of the ship hull, ∗ Corresponding

author Email address: [email protected] (Francis Noblesse)

Preprint submitted to Elsevier

as is well known; e.g. [2]. Far behind the ship, this hull-surface distribution of sources can be modeled as a point source (or point pressure) located at the centroid of the ship waterplane. This 1-point wavemaker model of farfield ship waves, used by Kelvin in his analysis, yields important fundamental insight into dominant features of farfield ship waves. In particular, Kelvin’s elementary 1-point ship model shows that farfield ship waves consist of two sets of waves, called transverse and divergent waves, that exist inside a wedge with angle 2 ψK ≈ 39◦ aft of the ship. The 1-point ship model evidently cannot account for interferences among the elementary waves (contained within the Green function G associated with the boundary condition at the free surface) that radiate from every point of the ship hull surface, in accordance with the classical representation of potential flows in terms of surface distributions of sources given in e.g. [2]. Wave-interference effects can be approximately taken into account via the 2-point ship models considered in [3] for deep water, and in [4, 5] for the more general (and considerably more complicated) case of uniform finite water depth. These 2-point models of farfield ship waves consist of a point source at the bow and a point sink at the stern of a monohull ship, or a pair of point sources (or sinks) at the twin bows (or sterns) of a catamaran with identical twin hulls, as is illustrated on the left and September 17, 2017

Y

Y

Y

V

V

V S

S X

X

X

L

L

L

Figure 1: The figure on the left depicts a top view of a monohull ship of length L that travels in calm water at a constant speed V along the x axis.

This figure also shows the 1-point ship model (red bullet) used in Kelvin’s classical analysis, and the 2-point ship model (two blue crosses) used to analyze interferences between the waves created by the bow and the stern of the ship. The figures in the center and on the right similarly depict top views of a catamaran with identical twin hulls of length L, separated by a distance S , and the 1-point ship model (red bullet). In addition, these two figures show the 2-point ship model (two blue crosses) used to analyze interferences between the waves created by the twin bows of the catamaran, and the more precise 4-point ship model (four blue crosses) that also accounts for longitudinal interferences between the waves created by the bows and the sterns of the catamaran.

in the center of Fig.1. The 2-point models of a monohull ship and a catamaran considered in [3] are analyzed again here via a different, more formal, approach. Specifically, the analysis given in [3] is based on elementary geometrical interference relations that only rely on the dispersion relation (like Kelvin’s classical analysis for the 1-point ship model). Indeed, the elementary analysis given in [3] does not involve the wave component W in the Green function G, and moreover does not consider the amplitudes of the waves created by the bow and the stern of a ship. The more precise analysis considered in the present study accounts for the amplitudes of the bow and stern waves, and involves the wave component W in G and the farfield stationary-phase approximation. This more elaborate analysis corroborates and refines the results of the elementary analysis previously considered in [3], and provides further insight. In particular, the analysis given here shows that differences between the amplitudes of the bow and stern waves have no influence on the occurrence of constructive or destructive interferences, although they affect the intensity of wave-interference effects (strong if the amplitudes of the bow and stern waves are commensurate, weak otherwise). The extensive numerical studies, reported in [6–8], of the farfield waves created by monohull ships and catamarans, modeled by means of realistic hull-surface distributions of sources in accordance with Hogner’s classical theory [9], show that the two elementary 2-point ship models depicted on the left and in the center of Fig.1 yield realistic predictions of apparent wake angles associated with the highest waves that result from constructive interferences of divergent waves at high Froude numbers. The apparent wake angles predicted by these two basic 2-point ship models are also shown in [8, 10] to be consistent with the observations of narrow ship wakes reported in the literature (a list of references to these observations of apparent narrow ship wakes may be found in [3] and the other previously-noted studies of wave-interference effects on farfield ship waves). However, the elementary 2-point model of a catamaran used in [3] evidently cannot account for the longitudinal interferences between the waves created by the bows and the sterns of the catamaran and is not accurate at low Froude numbers, for which longitudinal interference effects are important. In particular, [11] shows that the 2-point model of a catamaran poorly predicts the wavelength of the highest waves created by

the ship at F < 1. The analysis of the basic 2-point model of a catamaran depicted in the center of Fig.1 and considered in [3] is then extended here. Specifically, a generalized 2-point model of a catamaran in which the distance between the pair of point sources is allowed to differ from the distance between the twin bows of the catamaran is considered. A generalized 2-point model of a monohull ship in which the distance between the point source and the point sink can significantly differ from the distance between the bow and the stern of the ship is considered as well. Optimal 2-point models that yield apparent wake angles, where the highest divergent waves are found, that agree with the apparent wake angles predicted in [6] and [7] via the Hogner sourcedistribution model are then considered. However, a realistic model of a catamaran requires two point sources near the bows and two point sinks near the sterns of the twin hulls of the catamaran, as is illustrated on the right of Fig.1. This more realistic 4-point model of a catamaran, not considered previously, is analyzed in the present study. Thus, elementary ship models — the basic 2-point models of a monohull ship and a catamaran previously considered in [3] via an elementary geometrical analysis, optimal versions of these two basic 2-point models, and the 4-point model of a catamaran — are analyzed within the classical framework of potential flow theory, the Green function associated with the linear Kelvin-Michell boundary condition at the free surface, the related Fourier representation of farfield ship waves, and the farfield stationary-phase approximation. The analysis shows that these elementary ship models can realistically predict the highest waves found inside the cusps of the Kelvin wake as a result of interferences among the divergent waves created at the bow and the stern of a fast ship. The 1-point, 2-point and 4-point ship models that are considered here are the most basic and simplest models that can be used to analyze farfield ship waves. Other simple flow models can provide further insight and have been used. In particular, a ship is modeled via a distribution of pressure over the free surface in [13, 14], or via a distribution of sources over the ship hull surface in [6, 7]. This hull-surface source-distribution model, due to Hogner, arguably is the most realistic of the simple flow models that can be used to analyze farfield ship waves. References to numerous other studies of farfield ship waves can be found in [7, 6, 11, 10]. 2

e defined by (4b) and (4c) represents an elemenThe function E tary wave that travels at an angle γ from the x-axis given by q (cosγ, sinγ) = (1, q)/ 1 + q2 (5a)

2. Farfield waves due to a point source The waves created by a ship are observed at flow-field points e Y, e Z) e g/V 2 e x ≡ (e x,e y,e z) ≡ (X,

(2a)

The Fourier variable q in (4a)-(4c) and (5a) corresponds to the wavenumber k = 1 + q2 and the wavelength

The Cartesian coordinates e x and e y can be expressed in the polar form

e x = −h cosψ and e y = h sinψ

λ≡

(2b)

The finite limits of integration ±q∞ in (4a) filter unrealistic short waves that can be influenced by surface tension and viscosity. The relation (5b) shows that waves with wavelengths λ < λ∞ are eliminated if q∞ in (4a) is chosen as q∞ = λ0 /λ∞ − 1. E.g., waves shorter than 10%, 5%, 2% or 1% of the longest ship waves are eliminated if q∞ is chosen equal to 3, 4.4, 7 or 10. The simplest model of the farfield waves created by a ship is Kelvin’s 1-point wavemaker model, in which the ship is modeled as a single source (or pressure) point, which may be taken at the centroid (0, 0, 0) of the ship waterplane. Expression (6) then shows that the amplitude function A(q) in expression (4a) associated with this elementary ship model is given by

The ray ψ = 0 corresponds to the track e x < 0, e y = 0, e z = 0 of the ship. 2.1. Green function and related wave component

Within the linear potential flow analysis considered here, the flow around a ship hull is represented via a Green function G that satisfies the Kelvin-Michell linearized boundary condition at the free surface Z = 0. The Green function G(e x, x) represents the (nondimensional) velocity potential of the flow created at a flow-field point e x by a unit source located at a source point

(3)

A1 = 1

The coordinates (2a) and (3) of a flow-field point e x and a source point x are nondimensional with respect to the ship speed V or length L, as is appropriate for an analysis of farfield ship waves. The classical Green function G associated with the KelvinMichell free-surface boundary condition consists of waves W and a local flow L, which is ignored in this analysis of farfield waves. Thus, the waves created by a point source are defined in terms of the wave component W in the basic decomposition G = L + W of the Green function G into waves and a local flow. The elevation of the free surface behind a point source located at x is then given by

2.2. Farfield stationary-phase approximation Far behind a ship, i.e. for 1  h, the dominant contributions to the wave integral (4a) stem from points where the phase ϕ of the trigonometric function e i h ϕ in (4b) is stationary, as is well known. The relations (2b) show that the phase function ϕ defined by (4c) and its derivatives are given by q ϕ ≡ 1 + q2 (q sinψ − cosψ) (8a)

as is well known; e.g. [2, 12]. Thus, the free-surface elevation e ζ is determined by the Fourier superposition of elementary plane waves Z q∞ q e ζg 1 e dq ≈ Re 1 + q2 A E (4a) V2 π −q∞

dϕ (1 + 2q2 ) sinψ − q cosψ ≡ p dq 1 + q2

(8b)

d2 ϕ q (3 + 2q2 ) sinψ − cosψ ≡ dq2 (1 + q2 )3/2

(8c)

The points of stationary phase are determined by the roots of the equation dϕ/dq = 0, i.e. the equation

e represents Here, Re means that the real part is considered, and E the elementary wave where the phase h ϕ is given by q x + qe y) h ϕ = 1 + q2 (e

(7)

This model provides important basic information about farfield ship waves, most notably the classical Kelvin pattern of transverse and divergent waves, but cannot account for interference effects among the waves that are radiated by the sources distributed over the ship hull surface.

e ζ g/V 2 ≈ ∂W(e x, x)/∂e x with e z=0

e ≡ e (1+q2 )ez + i h ϕ E

(5b)

where λ0 is the wavelength of the longest waves created by a ship along its track. The wave-amplitude function A in the Fourier representation (4a) of W(e x, x) is given by A = E where √ 2 2 2 2 (6) E ≡ e (1+q ) z/F − i 1+q ( x+q y)/F

where the nondimensional horizontal distance h ≡ Hg/V 2 and the ray angle ψ are defined as q x2 + e y2 and tanψ ≡ e y/(−e x) (2c) h≡ e

x ≡ (x, y, z) ≡ (X, Y, Z)/L

Λg 2π 2π = ≤ 2π ≡ λ0 = 2 k V 1 + q2

tanψ = q/(1 + 2 q2 )

(4b)

(9)

as readily follows from (8b). The stationary-phase relation (9) yields

(4c) 3

cosψ = p

1 + 2 q2 (1 +

q2 )(1 +

4q2 )

and sinψ = p

q (1 +

q2 )(1 +

4q2 )

Here, expression (10) for ϕ00 (q) was used. For short divergent waves, i.e. in the limit q → ∞, one has 1  Q = O(q3/2 ). Expression (6) shows that the wave amplitude function A in the Fourier representation (4a) is O(1) as q → ∞ for the special case of point sources located at the free-surface plane z = 0 considered hereafter. However, the function A vanishes rapidly as q → ∞ for a distribution of sources over a ship hull surface. The farfield approximation (14) is not valid in the vicinity of the cusps ψ = ±ψK of the Kelvin wake, where (10) and (13) show that the second derivative ϕ00 of the phase function ϕ vanishes. Specifically, the amplitude function a defined by (14b) is weakly singular (like 1/|ψ ∓ ψK |1/4 ) at ψ = ±ψK , as can be seen in Fig.2, Fig.4 and Fig.5 considered further on. The influence of this weak singularity is negligible for ray angles ψ that are not very close to the cusp angles ±ψK , as is considered here.

These two relations and expressions (8a) and (8c) show that the phase function ϕ and its second derivative at a point of stationary phase are given by d2 ϕ −(1 + q2 ) 2q2 − 1 and ϕ= p = p dq2 1 + 4 q2 (1 + q2 ) 1 + 4 q2

(10)

The stationary-phase relation (9) has two roots

qT =

2 tanψ 1 and qD ≡ p 2 2 qT 1 + 1 − 8 tan ψ

(11)

These roots are real and correspond to two sets of waves, called transverse and divergent waves, if tan2 ψ < 1/8, i.e. for ray angles ψ within the Kelvin wedge −ψK < ψ < ψK where ψK ≡ arcsin(1/3) ≈ 19◦ 280

(12)

3. Two-point ship models

At the cusps ψ = ±ψK of the Kelvin wake, the roots qT and qD are √ equal, and given by qT = ± qC and qD = ± qC where C q = 1/ 2. The transverse and divergent waves correspond to √ (13) | qT | ≤ qC = 1/ 2 ≈ 0.7 ≤ |qD |

3.1. Two-point model of a monohull ship An elementary model of wave-interference effects on the farfield waves created by a monohull ship is considered in [3]. This model only accounts for interferences between the dominant waves created by the bow and the stern of the ship. The model yields a simple analytical interference relation, obtained in [3] via an elementary geometrical consideration that (like Kelvin’s classical analysis of the 1-point ship model) only requires the dispersion relation. Indeed, expressions (4) and (6) associated with the waves contained in the Green function G are not used in the geometrical analysis considered in [3]. A more formal analysis of the 2-point model of a monohull ship is now considered. In this model, a monohull ship is represented as a point source at (` x /2, 0, 0) and a point sink at (−` x /2, 0, 0), as is depicted on the left of Fig.1. The distance ` x between the point source and the point sink is taken as ` x = 0.9 in [3] and in Fig.2 considered further on. The amplitude function A in (4a) for this elementary 2-point ship model is given by A = ab A2x where 0 < ab denotes the strength of the bow source and √ 2 √ 2 2 2 (16) A2x ≡ e− i 1+ q `x /(2F ) − a sb e i 1+ q `x /(2F )

These inequalities show that the second derivative ϕ00 ≡ d2 ϕ/dq2 defined by (10) is negative for transverse waves and positive for divergent waves, i.e. one has ϕ00T < 0 and 0 < ϕ00D . In the far field 1  h, the wave integral (4a) can be evaluated analytically via the classical stationary-phase approximation T D e ζ g Re[aT e i h (ϕ −π/4) + aD e i h (ϕ +π/4) ] ≈ √ √ V2 π/2 h

(14a)

Here, ϕT and ϕD denote the phase function ϕ(q) defined by (10) for the transverse or divergent waves q = qT or q = qD . The amplitudes aT and aD of the transverse or divergent waves are defined in terms of the amplitude function A in (4a) as q p a(q) ≡ A(q) 1 + q2 / | ϕ00 (q)| (14b) where q = qT or q = qD . The relations (11) show that the phase function ϕ(q) in (14a) and its second derivative ϕ00 (q) in (14b), as well as the amplitude function A(q) in (14b), are functions of the ray angle ψ with −ψK ≤ ψ ≤ ψK . Thus, as is well known, farfield ship waves consist of transverse and divergent waves with amplitudes that slowly decay, at a rate inversely proportional to the square root of the horizontal distance h aft of the ship, and vary across the Kelvin wake −ψK < ψ < ψK according to the function q |a(q)| = Q(q) A2r (q) + A2i (q) (15a)

where q is taken equal to the functions qT (ψ) or qD (ψ) defined by (11), Ar and Ai denote the real and imaginary parts of the function A(q), and q Q(q) ≡ (1 + q2 )(1 + 4 q2 )1/4 / |1 − 2 q2 | (15b)

as follows from (6). In (16), a sb ≡ a s /ab where 0 < a s denotes the strength of the stern sink. Expression (16) yields r q |A2x | =

1 + a2sb − 2 a sb cos( 1 + q2 ` x /F 2 )

(17)

where 0 < a sb . This relation shows that constructive interferences associated with the largest values of |A2x | occur if q 1 + q2 ` x /F 2 = (2n − 1) π where 1 ≤ n (18a)

Although expression (17) involves the ratio a sb ≡ a s /ab of the amplitudes a s and ab of the stern and bow waves, the interference relation (18a) is independent of a sb . This fact means that the amplitudes of the bow and stern waves have no influence on 4

7

|Ax2| and |ax2|

x

|A2| x |a2|

Monohull ship, F = 0.7, asb=1

6

Monohull ship, F = 1, asb=1

x

|A2| x |a2|

Monohull ship, F = 1.3, asb=1

x

|A2| x |a2|

5 4 3 2 1 0 7

|Ax2| and |ax2|

|Ax2| x |a2|

Monohull ship, F = 0.7, asb=0.6

6

Monohull ship, F = 1, asb=0.6

|Ax2| x |a2|

Monohull ship, F = 1.3, asb=0.6

|Ax2| x |a2|

5 4 3 2 1 0 7

y

|A2|

|Ay2| and |ay2|

6

y

|A2|

Catamaran, F = 0.7, ly=0.2

y

y

|A2|

Catamaran, F = 1, ly=0.2

y

|a2|

Catamaran, F = 1.3, ly=0.2

y

|a2|

|a2|

5 4 3 2 1 0 7

y

|A2| y |a2|

|Ay2| and |ay2|

6

y

|A2| y |a2|

Catamaran, F = 0.7, ly=0.5

y

|A2| y |a2|

Catamaran, F = 1, ly=0.5

Catamaran, F = 1.3, ly=0.5

5 4 3 2 1 0 0

2

4

6

8

10 ψ

12

14

16

18

0

2

4

6

8

10 ψ

12

14

16

18

0

2

4

6

8

10 ψ

12

14

16

18

Figure 2: The top half depicts the amplitude functions |A2x | and |a2x | given by (17) and (19) associated with the divergent waves created by the 2-point model of a monohull ship with a sb = 1 or 0.6 at Froude numbers F = 0.7, 1 or 1.3. The primary peaks of the functions |A2x | and |a2x | are marked via open or solid circles. The bottom half depicts the amplitude functions |Ay2 | and |ay2 | given by (22b) and (24) associated with the divergent waves created by the 2-point model of a catamaran with `y = 0.2 or 0.5 at Froude numbers F = 0.7, 1 or 1.3. The primary peaks of the functions |Ay2 | and |ay2 | are marked via open or solid squares.

the occurrence of interferences, although they affect their intensity. Indeed, wave interferences are strongest if a sb = 1 and negligible if a sb  1 or if 1  a sb . The relation (18a) can be expressed in the equivalent form ` x = (2n − 1)πF 2 cosγ where 1 ≤ n

as follows from (13). The main practical application of the interference relation (18a) therefore is that it defines a series of unfavorable Froude numbers p √ Fn = F1 / 2 n − 1 where F1 ≡ ` x /π

(18b)

as follows from (5a). The relation (18b) is identical to the constructive interference relation obtained in [3] via an elementary geometrical consideration. This geometrical analysis is then corroborated by the mathematical analysis of the 2-point model of a monohull ship given above. The interference relations (18) are valid for both transverse and divergent waves. Transverse waves correspond to the fairly narrow range q p 1 ≤ 1 + q2 ≤ 3/2 ≈ 1.2

that correspond to peaks of the wave drag and are then avoided in ship design, as is well known. One has F1 ≈ 0.5 for ` x = 0.9. The low-speed regime F < F1 is dominated by interferences between the transverse waves created by the bow and the stern of the ship. This well-known regime is not considered here. Thus, only the high-speed regime dominated by interferences between divergent waves is considered hereafter. The function |a(q)| that corresponds to the function |A2x | given by (17) is denoted as |a2x | and is defined by (15a) as 5

|a2x | = Q |A2x |

(19)

where Q is given by (15b). The amplitude functions |A2x | and |a2x | defined by (17) and (19), where q is taken as q = qD in accordance with (11), are depicted in the upper half of Fig.2 for three Froude numbers F = 0.7, 1 or 1.3, and for two values of the relative amplitude a sb of the stern and bow waves equal to 1 or 0.6, i.e. for a s = ab or a s = 0.6 ab . The roots q (20) qnx = (2n − 1)2 π2 F 4/`2x − 1 where 1 ≤ n

The finding that a sb has no influence on the apparent wake angle associated with constructive interferences is consistent with the conclusion, established via a numerical analysis based on the Hogner source-distribution model, stated in [6] that the hull shape only has a relatively small influence on the apparent wake angle. Indeed, the fact that a sb has no influence on the apparent wake angle actually extends the conclusion given in [6], where only hull forms with fore and aft symmetry (which corresponds to a s = ab and a sb = 1) are considered.

of the constructive-interference relation (18a) define a series of values of q that correspond to peaks of the amplitude function |A2x | . One has q1x ≤ qnx . The relations (11) then show that one has ψnx ≤ ψ1x for the ray angles ψnx of the divergent waves that correspond to the roots qnx . Thus, the largest ray angle ψ associated with the highest divergent waves defined by the interference relation (18a) occurs for n = 1, i.e. for primary interferences, and is given by ψ1x . This ray angle can be regarded as an ‘apparent wake angle’ that may explain the observations of narrow ship wakes reported in the literature, as is suggested in [3] and related studies of wave interference effects on farfield ship waves. The ray angle ψ1x associated with the primary peak of the amplitude function |A2x | is given by p π2 F 4/`2x − 1 x (21a) tan ψ1 = 2π2 F 4/`2x − 1

3.2. Two-point model of a catamaran An elementary model of wave-interference effects on the farfield waves created by a catamaran with identical twin hulls is considered in [3]. This model only accounts for interferences between the dominant waves created by the bows (or the sterns) of the twin hulls of the catamaran. The model yields a simple analytical interference relation, obtained in [3] via an elementary geometrical consideration. A more formal analysis of the 2-point model of a catamaran is now considered. In this model, a catamaran with two identical hulls is modeled via two point sources at the twin bows of the catamaran, as is depicted in the center of Fig.1. Thus, the lateral separation distance `y between the two point sources at (0, ±`y /2, 0) is equal to the distance s ≡ S /L between the twin bows of the catamaran. The amplitude function A in expression (4a) for this elementary 2-point model of a catamaran is given by A = ab Ay2 where ab denotes the strength of the twin bow sources and q (22a) Ay2 ≡ 2 cos[q 1 + q2 `y /(2F 2 )]

as follows from (9) and (20) with n = 1. The related high Froude number approximation tan ψ1x ≈

` x /F 2 as F → ∞ 2π

(21b)

as follows from (6). One then has q |Ay2 | ≡ 2 | cos[q 1 + q2 `y /(2F 2 )] |

shows that ψ1x decreases like 1/F 2 as F increases. The ray angle ψ1x is equal to the Kelvin cusp angle ψK ≈ 19◦ 280 for p F 2/` x = 3/2/π (21c)

(22b)

Expression (22b) shows that constructive interferences related to the largest values of |Ay2 | occur if q q 1 + q2 `y /(2F 2 ) = nπ where 1 ≤ n (23a)

i.e. for F ≈ 0.6 if ` x = 0.9. Fig.2 shows that the ray angle ψ1x , which corresponds to the first peak of the amplitude function |A2x |, and the ray angle associated with the corresponding peak of the amplitude function |a2x | differ by less than 1◦ for F = 1 and 1.3. For F = 0.7, the peak of the amplitude function |a2x | cannot be identified, and the peak of the function A2x is not as clearly apparent as for the higher Froude numbers F = 1 and 1.3. Moreover, this peak occurs at a ray angle ψ ≈ 16◦ that is close to the Kelvin cusp angle ψK . Fig.2 illustrates the fact that a sb ≡ a s /ab has no influence on the occurrence of constructive interferences between the divergent waves created by the 2-point ship model. Indeed, the peaks of the amplitude functions |A2x | and |a2x | occur at the same ray angles for a sb = 1 and a sb = 0.6. However, a sb affects the intensity of wave interferences. Indeed, the amplitude functions |A2x | and |a2x | vanish for a sb = 1 at a succession of ray angles, associated with destructive interferences, but do not vanish for a sb = 0.6. The peaks of |A2x | and |a2x |, associated with constructive interferences, similarly are higher for a sb = 1 than for a sb = 0.6.

The relations (5a) show that (23a) can be expressed in the equivalent form `y sinγ = 2 nπF 2 cos2 γ where 1 ≤ n

(23b)

The relation (23b) is identical to the constructive interference relation obtained in [3] via an elementary geometrical consideration. This geometrical analysis is then corroborated by the mathematical analysis given above. The function |a(q)| that corresponds to the function |Ay2 | given by (22b) is denoted as |ay2 | and is defined by (15a) as |ay2 | = Q |Ay2 |

(24)

where Q is given by (15b). The amplitude functions |Ay2 | and |ay2 | defined by (22b) and (24), where q is taken as q = qD in accordance with (11), are depicted in the bottom half of Fig.2 for three Froude numbers F = 0.7, 1 or 1.3, and for distances `y = s between the pair of point sources equal to 0.2 or 0.5. 6

√

The roots rq

2 qyn ≡

1 + 16 n2 π2 F 4 /`y2 − 1 where 1 ≤ n

of the ‘optimal distance’ ` x between the bow source and the stern sink in the 2-point model of a monohull ship are depicted in the top left corner of Fig.3 together with the fit

(25)

` x = 0.7 + 0.025(1 + 3F 2 )F 4 for F ≤ 1

of the constructive-interference relation (23a) define a series of values of q that correspond to peaks of the amplitude function |Ay2 | . One has qy1 ≤ qyn , and consequently ψyn ≤ ψy1 for the ray angles ψyn of the divergent waves that correspond to the roots qyn . The largest ray angle ψ associated with the highest divergent waves defined by the interference relation (23a) then occurs for n = 1, i.e. for primary interferences, and is given by ψy1 . This ray angle can be regarded as an apparent wake angle that may result in the appearance of narrow ship wakes, as was noted earlier for monohull ships. Expressions (9) and (25) define the ray angles ψyn associated with the peaks of the amplitude function |Ay2 | as tan ψyn

=

v u u tq

1 + 16 n2 π2 F 4 /`y2 − 1

2 (1 + 16 n2 π2 F 4 /`y2 )

` x = 0.25 + 0.55 F for 1 ≤ F

(27a) (27b)

The smooth fit (27) is more precise than the fit given by equations (38) and depicted in Fig.3 of [6]. Expression (27b) shows that ` x < 1 if F < 1.36. This result suggests that dominant features of the farfield waves created by a monohull ship can be well represented via a 2-point model of the ship if F < 1.36, for which lateral wave interferences between the sources and sinks distributed over the port and starboard sides of the ship hull are not very important. Conversely, the fact that (27b) yields 1 < ` x if 1.36 < F suggests that the 2point model is less realistic at high Froude numbers, for which lateral interferences are important and the 2-point model struggles to account for lateral wave-interference effects by means of longitudinal wave interferences. The apparent wake angle defined by (21a), where ` x is taken as given by (27), is smaller than the Kelvin cusp angle ψK for F K < F with F K ≈ 0.52. Expressions (21a) and (27) define the apparent wake angle ψ1x in terms of the Froude number F. Indeed, the numerical analysis reported in [6] shows that the hull shape only has a relatively small influence on the apparent wake angle, as is illustrated in the bottom left corner of Fig.3. This figure also shows that the apparent wake angle defined by (21a) and (27) closely fits the results of the numerical analysis, based on the Hogner flow model, considered in [6]. The optimal 2-point model, with the distance ` x given by (27), is then a realistic model for the purpose of predicting the apparent wake angle associated with the highest waves created by a fast monohull ship.

(26a)

where 1 ≤ n. The related high Froude number approximation p `y /(2F) tan ψyn ≈ √ √ as F → ∞ (26b) 2π n shows that the ray angles ψyn decrease like 1/F as F increases. The angles ψyn are equal to the Kelvin cusp angle ψK ≈ 19◦ 280 for q√ q √ 3/π/ n (26c) 2F/ `y =

p i.e. for F ≈ 0.37 `y /n. The ray angle ψy1 , which corresponds to the first peak of the amplitude function |Ay2 |, and the ray angle associated with the corresponding peak of the amplitude function |ay2 | can hardly be distinguished in Fig.2. This figure shows that the Froude number F and the hull spacing s both have a significant influence on wave interferences. In particular, the apparent wake angle associated with the peaks of the amplitude functions |Ay2 | and |ay2 | decreases as the Froude number F increases or as the distance `y = s between the pair of point sources decreases.

4.2. Catamarans The extensive numerical study, reported in [7], of the farfield waves created by 42 catamarans, modeled via Hogner’s distribution of sources over the twin hull surfaces of the catamarans, that correspond to six hull-separation distances within the range 0.2 ≤ s ≡ S /L ≤ 0.8 and seven hulls associated with a broad range of main hull-shape√parameters, for Froude numbers within the range 0.4 ≤ F/ s ≤ 3.5, shows that the 2point model of a catamaran is quite realistic for Froude numbers 1 < F. More generally, the 2-point model of a catamaran is shown in [7] to be realistic for high Froude numbers F and/or large hull spacings s, i.e. for fast and/or wide catamarans. [7] also defines two notable wake angles ψi and ψo , called inner and outer wake angles, that correspond to the ray angles ψ = ±ψi and ψ = ±ψo where the highest divergent waves due to a catamaran are found inside the Kelvin wake −ψK ≤ ψ ≤ ψK . The hull form is found in [7] to have a relatively minor influence on the wake angles ψi and ψo , which can then approximated via simple analytical relations valid for all hull forms. Specifically, the inner wake angle ψi is approximated in [7]

4. Optimal 2-point models 4.1. Monohull ships The numerical analysis, reported in [6], of the farfield waves created by a monohull ship, modeled via the classical Hogner distribution of sources over the ship hull surface, for seven hull forms characterized by broad ranges of main hull-shape parameters (beam/length, draft/length, beam/draft, waterline entrance angle) shows that the 2-point model of a monohull ship is realistic for Froude numbers within the usual range 0.6 ≤ F ≤ 1.5. The numerical analysis given in [6] also yields values of ` x for which the 2-point model is approximately equivalent to the Hogner source-distribution model. These numerical estimates 7

1.4

4

1.3

3.5 Optimal ly / s

Optimal lx

1.2 1.1 1 0.9

3

2

0.7

1

0.6 20

0.5

18 16 Optimal ψy1(°)

14 12

x

i

FK i F*

2.5

1.5

0.8

Optimal ψ1(°)

s=0.2 s=0.35 s=0.5

10 8 6 4 2 0 0

0.2

0.4

0.6

0.8

1 F

1.2

1.4

1.6

1.8

2

20 18 16 14 12 10 8 6 4 2 0

s=0.2 s=0.35 s=0.5 FiK Fi*

0

0.2

0.4

0.6

0.8

1 F

1.2

1.4

1.6

1.8

2

Figure 3: Top left corner: Numerical estimates obtained in [6] and related fit (27) of the optimal distance ` x between the point source and the point

sink in the 2-point model of a monohull ship. Bottom left corner: Numerical estimates of the apparent wake angle, where the highest divergent waves created by a monohull ship are found, obtained in [6], and the wake angle ψ1x given by (21) and (27). Bottom right corner: Numerical estimates, obtained in [7] and given by (28), of the inner wake angle ψi for catamarans with hull-spacings s ≡ S /L = 0.2, 0.35 and 0.5. The Froude numbers FKi and F∗i that define the Kelvin 1-point model regime and the basic 2-point model regime are also marked. Top right corner: Optimal distance `y (F, s) for which the apparent wake angle ψy1 predicted by the 2-point model of a catamaran is equal to the inner wake angle ψi obtained in [7] via a numerical analysis based on the Hogner source-distribution model.

Expressions (28a), (28c) and (26a) show that the angle ψi∗ is equal to the angle ψy1 if the distance `y between the twin point sources in the 2-point model of a catamaran in (26a) is equal to the separation distance s between the twin bows of the catamaran, as is depicted in Fig.1. The basic 2-point model of a catamaran, with `y = s, is then accurate for F∗i < F. This elementary model becomes more accurate as F increases and/or as s increases, and is realistic for fast and/or wide catamarans as was already noted. The relations (29) yield F∗i = FKi for s ≈ 1.55, and therefore predict that the 2-point model with `y = s is accurate within the whole speed range FKi < F for very wide catamarans with hull spacings 1.55 < s. This prediction however√is uncertain because expressions (29) are only validated for F/ s ≤ 3.5 in [7]. The distance `y between the twin point sources in the 2point model of a catamaran can be chosen so that the wake angle ψy1 given by (26a) is equal to the inner wake angle ψi given by (28) for FKi ≤ F. The optimal source-separation distance `y (F, s) determined by the relation ψy1 = ψi and expressions (26a) and (28) is depicted in the top right corner of Fig.3 for FKi ≤ F ≤ 2 and three hull-spacings s ≡ S /L = 0.2, 0.35 and 0.5. As is expected, the optimal distance `y (F, s) is equal to the hull-spacing s for F∗i (s) ≤ F where lateral wave interferences between the twin hulls of the catamaran are dominant and the 2-point model is realistic. However, the optimal source-

as ψi ≈ ψi∗ for F∗i ≤ F i

ψ ≈

ψi∗

+

50 (F∗i

(28a) 2

− F) s for

FKi

≤F≤

F∗i

(28b)

where (28b) presumes that the ray angles ψi and ψi∗ are expressed in degrees. Moreover, ψi∗ is given by (26a) where n is taken as n = 1 and the distance `y between the two point sources is taken as the distance s ≡ S /L between the twin bows of the catamaran, as is depicted in Fig.1, i.e. sp 1 + 16 π2 F 4/s2 − 1 (28c) tan ψi∗ ≡ 2 (1 + 16 π2 F 4/s2 ) Finally, the Froude numbers F∗i and FKi in (28a) and (28b) are defined as √ (29a) F∗i ≈ (0.13 + 0.47/s) s √ √ i FK ≈ 0.35 s + (0.135 − 0.01/s)/ s (29b) The inner wake angle ψi is smaller than the Kelvin cusp angle ψK for Froude numbers FKi < F, and is equal to the ray angle ψi∗ for F∗i < F. The inner wake angle ψi defined by (28) is depicted in the bottom right corner of Fig.3 for s = 0.2, 0.35 and 0.5. The Froude numbers FKi (s) and F∗i (s) are also marked in this figure. 8

and the inner rays ψ = ±ψi already identified in the analysis of the 2-point model of a catamaran. The angles ψo and ψi that correspond to the outer peak and the primary inner peak of the amplitude function |A4 | can be regarded as outer and inner apparent wake angles, as is shown in [7] for the Hogner model and below in Fig.4 for the 4-point model. The numerical analysis of the farfield waves created by catamarans, modeled via Hogner’s distribution of sources over the twin hull surfaces of the catamarans, reported in [7] shows that the inner wake angle ψi is approximately given by the relations (28) and (29), as was already noted, and that the outer wake angle ψo can similarly be approximated as √ ψo ≈ ψo∗ ≡ arctan(0.37 s/F ) + 0.02(0.64/s2 − 1)

separation distance `y (F, s) differs from the distance s between the twin bows of the catamaran in the regime FKi ≤ F < F∗i for which longitudinal interferences between the waves created at the fore and aft of the hulls of the catamaran are significant and the 2-point model is dubious. The large increase of the optimal source-separation distance `y (F, s) depicted in the figure for Froude numbers F in the vicinity of FKi is uncertain because the approximation (28b) is not accurate for ray angles ψi near the cusp angle ψK , as is explained in [7]. 5. Four-point model of a catamaran The foregoing analysis shows that the basic 2-point model of a catamaran, where the distance `y between the pair of point sources is equal to the distance s between the twin bows of the catamaran as is shown in the center of Fig.1, realistically accounts for lateral interferences between the divergent waves created by the ship at Froude numbers 1 < F, for which lateral interferences are dominant. Moreover, the optimal 2-point model of a monohull ship with the distance ` x between the source-sink pair chosen according to (27) realistically accounts for longitudinal interferences between the divergent waves created by the bow and the stern of the ship. These two results suggest that a 4-point model of a catamaran that consists of two point sources at (` x /2, ±s/2, 0) and two point sinks at (−` x /2, ±s/2, 0), with ` x chosen as in (27), may be expected to be realistic. This 4-point model of a catamaran, illustrated on the right side of Fig.1, is now considered. The amplitude function A in expression (4a) for this 4-point model of a catamaran is given by A = ab A4 where ab denotes the strength of the twin bow sources, A4 ≡ A2x Ay2

o

ψ ≈

(30a)

(30b)

(30c)

where Q is given by (15b). The relation (30b) suggests that the 4-point model of a catamaran is considerably more complex than the 2-point models of a monohull ship or a catamaran, especially for narrow catamarans at low Froude numbers for which neither lateral nor longitudinal interferences are dominant. The complexities of this ‘slow and narrow’ catamaran regime are noted in [7]. In particular, the 4-point model analyzed here and the more precise Hogner source-distribution model considered in [7] show the existence of a peak, called outer peak in [7] and hereinafter, of the amplitude function |A| in the Fourier-Kochin representation (4a) of farfield waves and the related stationary-phase approximation (14). This outer peak is found at ray angles ψ = ±ψo that are located between the Kelvin cusps ψ = ±ψK

for FKo ≤ F ≤ F∗o

(31a)

(31b)

The outer wake angle ψo is smaller than the Kelvin cusp angle ψK for Froude numbers FKo < F. The amplitude functions |A4 | and |a4 | defined by (30), where q is taken as q = qD in accordance with (11), that correspond to three catamarans with hull-spacings s = 0.2, 0.35 or 0.5 at four Froude numbers F = 0.6, 0.8, 1.1 and 1.4 are depicted in Fig.4. The relative amplitude a sb ≡ a s /ab of the stern and bow waves is taken equal to 0.8, i.e. a s = 0.8 ab , for the three catamarans considered in this figure. Fig.4 shows that outer peaks of the amplitude function |A4 | exist for the three catamarans at F = 1.5, 1.1 and 0.8, but only exist at the lower Froude number F = 0.6 for the narrow catamaran s = 0.2; i.e., no outer peak can be observed at F = 0.6 for the wider catamarans s = 0.35 and s = 0.5. Moreover, the outer peaks for 0.6, s = 0.2 and for F = 0.8, s = 0.5 occur at ray angles ψ that are extremely close to the Kelvin cusp angle ψK . Outer peaks of the amplitude function |a4 | can only be identified in Fig.4 at F = 1.1 for s = 0.2, and at F = 1.5 for s = 0.2 and s = 0.35. The first inner peaks are higher than, or approximately as high as, the second inner peaks for every Froude number F and every hull-spacing s, except for the case F = 0.6, s = 0.2 that corresponds to the slowest and narrowest catamaran. Moreover, the groups of waves that are centered at the first inner peaks are wider than the groups of waves centered at the second inner peaks in every case, i.e. for every Froude number and every hull-spacing, considered in Fig.4, except for one case that corresponds to F = 0.6, s = 0.2. The first inner peak then seems more likely than the second inner peak to stand out as an apparent wake angle. The sharper first peaks also seem more likely to stand out than the broader outer peaks in Fig.4. Thus, this figure suggests that the first inner peak of the amplitude function |A4 | is likely to stand out as an apparent wake angle for most catamarans.

where |A2x | and |Ay2 | are given by (17) and (22b). The function |a(q)| that corresponds to the function |A4 | given by (30b) is denoted as |a4 | and is defined by (15a) as |a4 | = Q |A4 |

for F∗o ≤ F √ + 22[ 0.6 + (11.4 − 10F/ s) s2 ]3

where the angles ψo and ψo∗ are expressed in degrees. Moreover, the Froude numbers F∗o and FKo in (31) are defined as √ (32a) F∗o ≈ (1.14 + 0.06/s2 ) s √ o FK ≈ (1.1 + 0.04/s) s (32b)

and A2x and Ay2 are given by (16) and (22a). One then has |A4 | ≡ |A2x | |Ay2 |

ψo∗

9

10

|A4| and |a4|

8

|A4| |a4|

F=0.6, s=0.2

|A4| |a4|

F=0.8, s=0.2

|A4| |a4|

F=1.1, s=0.2

|A4| |a4|

F=1.5, s=0.2

|A4| |a4|

F=0.6, s=0.35

|A4| |a4|

F=0.8, s=0.35

|A4| |a4|

F=1.1, s=0.35

|A4| |a4|

F=1.5, s=0.35

|A4| |a4|

F=0.6, s=0.5

|A4| |a4|

F=0.8, s=0.5

|A4| |a4|

F=1.1, s=0.5

|A4| |a4|

F=1.5, s=0.5

6 4 2

0 10

|A4| and |a4|

8 6 4 2 0 10

|A4| and |a4|

8 6 4 2 0 0

2

4

6

8

10 12 14 16 18 ψ

0

2

4

6

8

10 12 14 16 18 ψ

0

2

4

6

8

10 12 14 16 18 ψ

0

2

4

6

8

10 12 14 16 18 ψ

Figure 4: Amplitude functions |A4 | and |a4 | defined by (30b) and (30c) for three catamarans with a sb = 0.8 and hull-spacings s = 0.2 (top row), 0.35

(center row) or 0.5 (bottom row) at four Froude numbers F = 0.6 (left column), 0.8 (second column), 1.1 (third column) or 1.5 (right column). The outer peaks and the first and second inner peaks of the functions |A4 | and |a4 | are marked via solid or open diamonds (outer peaks), triangles (first inner peaks) or pentagons second inner peaks).

Fig.4 also shows that the first and second inner peaks, as well as the outer peaks, of the amplitude function |A4 | occur at ray angles ψ that decrease as the Foude number F increases or as the hull-spacing s decreases. Thus, the apparent wake angle that is associated with the first inner peak (or possibly with the outer peak or the second inner peak) is narrower for a fast narrow catamaran than for a slow wide catamaran, in agreement with the results of the numerical analysis considered in [7] and the computations of wave patterns reported in [8]. As was already noted, the second inner peak can be higher than the first inner peak, as is the case in Fig.4 for s = 0.2 at F = 0.6 and is also illustrated in Fig.3 of [7] for the Hogner source-distribution model of a catamaran. The 4-point model considered here amplifies the height of the second peak in comparison to the height of the first peak because the function Q in (15) increases as q increases, i.e. as the ray angle ψ decreases. Specifically, the function Q is O(q3/2 ) as q → ∞ for the 4-point model of a catamaran, whereas Q vanishes rapidly as q → ∞ for a more realistic source-distribution model such as the Hogner model considered in [7]. Fig.5 illustrates the influence of the relative amplitude a sb ≡ a s /ab of the stern and bow waves, chosen as a sb = 1 or 0.6, for a catamaran with hull-spacing s = 0.35 at four Froude numbers F = 0.6, 0.8, 1.1 or 1.5. The peaks of the amplitude functions |A4 | and |a4 | are smaller for a sb = 0.6 than for a sb = 1, but the peaks occur at the same ray angles ψ. These results show that the relative amplitude a sb of the stern and bow waves

does not influence the occurrence of constructive wave interferences, and only affects the intensity of interferences, which are strongest for a sb = 1, as was already observed for the 2-point model of a monohull ship. Fig.6 depicts the outer wake angle and the first and second inner wake angles that are predicted by the 4-point model of a catamaran via identification of the outer peak and the first and second inner peaks of the amplitude function |A4 |, as is illustrated in Fig.4. The figure shows that the wake angle associated with the first inner peak is quite close to the inner wake angle ψi predicted by the Hogner source-distribution model and given by (28), and that the wake angle associated with the outer peak is somewhat larger (by nearly 2◦ for some Froude numbers) than the outer wake angle ψo predicted by the Hogner source-distribution model and given by (31). This result shows that longitudinal and lateral interference effects on the farfield waves created by a catamaran can be realistically represented via a simple 4-point model, i.e. via a pair of point sources and a pair of point sinks separated by a lateral distance `y equal to the distance s between the twin bows of the catamaran and a longitudinal distance ` x equal to the optimal separation distance given by (27). Fig.6 also shows that the first and second inner wake angles predicted by the 4-point model of a catamaran are very close to the inner wake angles ψy1 and ψy2 predicted by the 2-point model of a catamaran and given by (26a) at Froude numbers F greater than about 0.9, 0.7 or 0.6 for hull spacings s equal to 0.2, 0.35 10

10

|A4| and |a4|

8

|A4| |a4|

F=0.6, asb=1

|A4| |a4|

F=0.8, asb=1

|A4| |a4|

F=1.1, asb=1

|A4| |a4|

F=1.5, asb=1

|A4| |a4|

F=0.6, asb=0.6

|A4| |a4|

F=0.8, asb=0.6

|A4| |a4|

F=1.1, asb=0.6

|A4| |a4|

F=1.5, asb=0.6

6 4 2 0

10

|A4| and |a4|

8 6 4 2 0 0

2

4

6

8

10 12 14 16 18 ψ

0

2

4

6

8

10 12 14 16 18 ψ

0

2

4

6

8

10 12 14 16 18 ψ

0

2

4

6

8

10 12 14 16 18 ψ

Figure 5: Amplitude functions |A4 | and |a4 | defined by (30b) and (30c) for two catamarans with hull-spacing s = 0.35 and a sb = 1 (top row) or 0.6 (bottom) at four Froude numbers F = 0.6 (left column), 0.8 (second column), 1.1 (third column) or 1.5 (right column). The outer peaks and the first and second inner peaks of the functions |A4 | and |a4 | are marked via solid or open diamonds (outer peaks), triangles (first inner peaks) or pentagons (second inner peaks). Outer, 4 point Outer, Hogner

First Inner, 4 point First Inner, 2 point First Inner, Hogner

Second Inner, 4 point Second Inner, 2 point

20 18 16 14

ψ(°)

12 10 8 6 4 2

s = 0.2

s = 0.35

s = 0.5

0 0

0.2

0.4

0.6

0.8

1 F

1.2

1.4

1.6

1.8

2

0

0.2

0.4

0.6

0.8

1 F

1.2

1.4

1.6

1.8

2

0

0.2

0.4

0.6

0.8

1 F

1.2

1.4

1.6

1.8

2

Figure 6: The figure depicts the outer and inner wake angles ψo and ψi predicted by the Hogner source-distribution model and given by (31) and

(28), the inner wake angles ψy1 and ψy2 predicted by the 2-point model of a catamaran and given by (26a), and the outer wake angle and the first and second inner wake angles predicted by the 4-point model.

or 0.5. This result means that lateral interferences between the twin hulls of a catamaran dominate longitudinal interferences between the bow and the stern of the catamaran even at relatively low Froude numbers, and that the 2-point model of a catamaran in fact is realistic over a fairly broad range of Froude numbers. The vertical lines at F = 0.52 in Fig.6 provide a reminder of the fact that transverse waves, ignored in the analysis considered here, can be important for Froude numbers F < 0.52. Thus, the analysis of interferences between the divergent waves created by a catamaran considered here and in [7] cannot be expected to be realistic in this low Froude number regime where transverse waves also need to be considered.

point models of a monohull ship and a catamaran (already considered in [3] via an elementary geometrical consideration), optimal versions of these two basic 2-point models, and the 4point model of a catamaran (not considered previously) — that is reported in this study shows that major features of the farfield waves created by a ship, notably the highest waves that exist inside the cusps of the Kelvin wake as a result of interferences among the divergent waves created at the bow and the stern of a fast ship, can be well predicted by very simple ship models. These elementary ship models are considered in the study via a common theoretical approach based on potential flow theory, the Green function associated with the Kelvin-Michell linear boundary condition at the free surface, the related Fourier representation of farfield ship waves, and the farfield stationaryphase approximation. This mathematical analysis yields interference relations for longitudinal and lateral wave interferences that are identical to the relations previously obtained in [3] via an elementary ge-

6. Conclusion The analysis of the elementary ship models depicted in Fig.1 — namely, Kelvin’s classical 1-point ship model, the basic 211

ometrical consideration that does not involve the wave component W in the Green function G, and does not consider the amplitudes of the waves created by the bow and the stern of a ship. This geometrical analysis is then corroborated by the mathematical analysis considered in the study, which takes into account the amplitudes of the bow and stern waves and involves the wave component W in G and the farfield stationary-phase approximation. Moreover, the more precise analysis considered in the study provides further insight into interference effects on farfield ship waves. In particular, differences between the amplitudes of the bow and stern waves are shown to have no influence on the occurrence of constructive or destructive interferences; specifically, the highest and smallest waves associated with these interferences are found along ray angles that do not depend on the amplitudes of the bow and stern waves. This conclusion is consistent with the fact that the numerical studies (based on the Hogner source-distribution model) reported in [6] and [7] for monohull ships or catamarans show that the hull geometry only has a relatively minor influence on the apparent wake angle associated with the highest divergent waves created by a ship. This conclusion — based on a numerical analysis for several hull forms that have fore and aft symmetry, and therefore create bow and stern waves of equal amplitude — is extended here by the finding that the occurrence of constructive or destructive interferences is not affected by differences between the amplitudes of the bow and stern waves, i.e. is also valid for ship hulls that do not have fore and aft symmetry. Indeed differences between the amplitudes of the bow and stern waves only affect the intensity of wave interference effects, which are strong if the amplitudes of the bow and stern waves are commensurate or weak otherwise. The analysis considered in the study shows that wave interference effects on the farfield waves of a fast monohull ship can be well explained via a 2-point model that consists of a point source near the bow and a point sink near the stern of the ship if the distance ` x between the point source and the point sink is chosen according to the relation (27), which is more precise than the relation given in [6]. The apparent wake angle associated with the highest divergent waves created by the optimal 2-point model of a monohull ship defined by (27) is in excellent agreement with the predictions obtained in [6] via a numerical analysis based on the Hogner source-distribution model, and this optimal 2-point model is then realistic for a broad range of monohull ships and Froude numbers. The 4-point model of a catamaran that consists of two point sources at (` x /2, ±s/2, 0) and two point sinks at (−` x /2, ±s/2, 0), where ` x is chosen according to (27) and s ≡ S /L is the distance between the twin bows of the catamaran, is also realistic. Indeed, the apparent wake angles associated with the highest divergent waves created by a catamaran and the corresponding main peaks of the related farfield wave amplitude function are close to the wake angles predicted in [7] via a numerical analysis based on the Hogner source-distribution model. Specifically, the inner wake angle that corresponds to the first inner peak of the wave amplitude function predicted by the elementary 4-point model and the Hogner source-distribution model

analyzed in [7] are in close agreement, and the outer wake angle associated with the outer peak is somewhat larger (by nearly 2◦ for some Froude numbers) than the outer wake angle predicted by the Hogner model. A notable result of the analysis is that the first and second inner wake angles associated with the first and second inner peaks of the farfield wave amplitude function for the 4-point model of a catamaran are in close agreement with the corresponding first and second inner wake angles for the 2-point model considered in [3] at Froude numbers F greater than about 0.9, 0.7 or 0.6 for hull spacings s equal to 0.2, 0.35 or 0.5. This result means that lateral interferences between the twin hulls of a catamaran dominate longitudinal interferences between the bow and the stern of the catamaran even at relatively low Froude numbers. Thus, the 2-point model of a catamaran is realistic over a surprisingly broad range of Froude numbers. References [1] W. Thomson, On ship waves, Proc. Inst. Mech. Eng. 38 (1887) 409–434. [2] F. Noblesse, F. Huang, C. Yang, The Neumann-Michell theory of ship waves, J. Eng. Math. 79 (2013) 51–71 [3] 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. [4] Y. Zhu, J. He, C. Zhang, H. Wu, D. Wan, R. Zhu, F. Noblesse, Farfield waves created by a monohull ship in shallow water, Eur. J. Mech. B Fluids 49 (2015) 226–234. [5] Y. Zhu, C. Ma, H. Wu, J. He, C. Zhang, W. Li, F. Noblesse, Farfield waves created by a catamaran in shallow water, Eur. J. Mech. B Fluids 59 (2016) 197–204. [6] C. Zhang, J. He, Y. Zhu, C-J. Yang, W. Li, Y. Zhu, M. Lin, F. Noblesse, Interference effects on the Kelvin wake of a monohull ship represented via a continuous distribution of sources, Eur. J. Mech. B Fluids 51 (2015) 27–38. [7] J. He, C. Zhang, Y. Zhu, L. Zou, W. Li, F. Noblesse, Interference effects on the Kelvin wake of a catamaran represented via a hull-surface distribution of sources, Eur. J. Mech. B Fluids 56 (2016) 1–12. [8] F. Noblesse, C. Zhang, J. He, Y. Zhu, C.J. Yang, W. Li, Observations and computations of narrow Kelvin ship wakes, J. Ocean Eng. & Science, 1 (2016) 52–65. [9] E. Hogner, Hydromech, Probl. d. Schiffsantriebs, Herausgeg. v. Kempf u. E. Foerster, Hamburg (1932) 99–114. [10] J. He, C. Zhang, Y. Zhu, H. Wu, C.J. Yang, F. Noblesse, X. Gu, W. Li, Comparison of three simple models of Kelvin’s ship wake, European J Mech. B/Fluids, 49 (2015) 12-19. [11] C. Ma, Y. Zhu, J. He, C. Zhang, H. Wu, W. Li, F. Noblesse, Wavelengths of the highest waves created by fast monohull ships or catamarans, J. Ocean Engineering, 113 (2016) 208-214. [12] F. Noblesse, Alternative integral representations for the Green function of the theory of ship wave resistance, J. Engineering Mathematics, 15 (1981) 241-265. [13] A. Darmon, M. Benzaquen, E. Rapha¨el, Kelvin wake pattern at large Froude numbers, J. Fluid Mech. 738 (2014) R3. [14] M. Benzaquen, A. Darmon, E. Rapha¨el, Wake pattern and wave resistance for anisotropic moving disturbances, Phys. Fluids. 26 (2014) 092106.

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