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Equation for ship wave crests in the entire range of water depths
Coastal Engineering 153 (2019) 103542
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Coastal Engineering journal homepage: http://www.elsevier.com/locate/coastaleng
Equation for ship wave crests in the entire range of water depths Byeong Wook Lee a, Changhoon Lee b, * a b
Coastal Development and Ocean Energy Research Center, Korea Institute of Ocean Science & Technology, 385 Haeyang-ro, Busan, 49111, Republic of Korea Department of Civil and Environmental Engineering, Sejong University, 209 Neungdong-ro, Gwangjin-gu, Seoul, 05006, Republic of Korea
A R T I C L E I N F O
A B S T R A C T
Keywords: Ship wave crests Cusp locus angle Entire range of water depths Theoretical solution Numerical experiment
An equation for ship wave crests y =x in the entire range of water depths is developed using the linear dispersion relation. In deep water, the developed equation is reduced to the equation of Kelvin (1906). The locations of ship wave crests in the x - and y -directions are obtained using a dimensionless constant C. The wave ray angle θc at the cusp locus is determined using the condition that θc is maximal at the cusp locus and the cusp locus angle is determined as αc ¼ tan 1 ðy=xÞmax . Numerical experiments are conducted using the FLOW-3D to simulate ship wave propagation. The cusp locus angles of the FLOW-3D are similar to both those of the present theory and Havelock (1908) theory in the entire range of the Froude number. Both the present theory and the FLOW-3D yield that, with the increase of ship speed, the Froude number increases and does the wavelength. For the Froude number equal to or greater than unity, the wavelength becomes infinitely large and the transverse waves disappear. The wavelengths of the FLOW-3D are slightly smaller than those of the present theory because the FLOW-3D considers the decrease of wavelength due to energy dissipation which happens because of viscosity of water and turbulence of high-speed particle velocities.
1. Introduction When a ship navigates in a channel or ocean, it creates waves propagating outward in all directions from the source point, which is moving with the ship speed. The ship waves can be categorized into transverse and diverging waves. The transverse waves propagate behind the ship with narrower wave ray angles from the ship trajectory line. The diverging waves propagate behind the ship with wider wave ray angles. pffiffiffiffiffi When the Froude number (i.e., Fr ¼ Us = gh where Us is the ship speed and h is the water depth) is less than unity, the crestlines connecting these ship waves make a concave triangle in which the transverse waves make a bottom line and the diverging waves make right and left side lines. The transverse and diverging waves meet with the same wave ray angles at a point called the cusp locus. When the Froude number is equal to or greater than unity, the transverse waves disappear and the diverging waves exist. Ship wave crest patterns have been studied since the late 1800’s. Kelvin (1887) first found empirically that diverging and transverse wave crests in deep water meet at the cusp locus with a so-called cusp locus angle of αc ¼ 19:47� . Kelvin (1906) also developed an equation for the ship wave crest using the linear dispersion relation in deep water and also the concept of ship wave propagation with a group velocity. Newman (1970) also developed the same equation for the ship
wave crest using the concept of the wave phase being stationary in deep water. As ship speed increases, the relative water depth becomes shal lower and the Kelvin’s equation cannot be applied accurately. Havelock (1908) found that the cusp locus angles vary depending on the Froude number in the entire range of water depths. Recently, Fang et al. (2011) suggested models of ship waves in the wind wave field and also on in termediate water depth using the methods of kinematics wave theory (Lighthill and Whitham, 1955). However, they could not show a detailed crest pattern of the diverging and transverse waves as Kelvin (1906) did. The heights of ship waves depend on ship speed, shape, draft, and water depth, etc. (Stoker, 1957; Johnson, 1958; Newman, 1977). When propagating in open and deep oceans, the ship waves may not affect mooring boats nor damage coastal structures. However, the ship waves may cause safety problems in a narrow channel or when the ship moves in high speed. Therefore, ship waves need to be studied in the entire range of water depths, from deep to shallow waters, in order to protect coastal area from erosion problem and keep harbor calmness and safety. Lee et al. (2011) predicted ship wave crest pattern in water of inter mediate depth by extending the approach of Kelvin (1906) using the recursive relation of the dispersion relationship in shallower water. Recently, Lee et al. (2013) developed an equation of ship wave crest on varying water depth in water of intermediate depth. However, they
* Corresponding author. E-mail addresses: [email protected] (B.W. Lee), [email protected] (C. Lee). https://doi.org/10.1016/j.coastaleng.2019.103542 Received 15 February 2019; Received in revised form 28 June 2019; Accepted 2 September 2019 Available online 4 September 2019 0378-3839/© 2019 Elsevier B.V. All rights reserved.
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could not predict ship wave crest in shallow water where the Froude number is greater than unity. Some researchers used horizontally two-dimensional numerical wave models to simulate ship wave gener ation and propagation. In order to generate ship waves, they included a moving-ship induced pressure gradient in the momentum equations (Akylas, 1984; Ertekin et al., 1986; Wu and Wu, 1982; Ersan and Beji, 2013; David et al., 2017; Shi et al., 2018) or used a slender body theory (Tuck, 1966; Chen and Sharma, 1995). In order to simulate ship wave propagation, they used horizontally two-dimensional wave models such as the Boussinesq equations (Wu and Wu, 1982; Ersan and Beji, 2013; David et al., 2017; Shi et al., 2018), Koerteweg-de Vries (KdV) equation (Akylas, 1984), Green-Naghdi equations (Ertekin et al., 1986), or Kadomtsev-Petviashvilli (KP) equations (Chen and Sharma, 1995). Most researchers were concerned about the ship waves moving near the critical speed in which condition solitary waves appear in front of the ship and the forces on the ship become significantly large. Recently, Ersan and Beji (2013) investigated ship wave crest patterns using the Boussinesq equations. They found that, in the entire range of water depths, their measured cusp locus angles are close to those predicted by Havelock (1908). However, they did not show detailed wave crest pat terns. The ship waves have components of different periods with a ship speed. The ship wave components are shorter (i.e., relatively deeper) especially near the ship in the diverging waves and are longer (i.e., relatively shallower) in the transverse waves. The whole wave crest pattern cannot be simulated with any of the equations to simulate ship wave propagation because these equations cannot be applied in very deep water. Recently, some researchers simulated ship wave generation and propagation using the three-dimensional Navier-Stokes equations with the moving pressure condition at the ship boundary (Kang et al., 2008; Hur et al., 2011). Johnson (1958) and Sorensen (1967, 1969) conducted hydraulic experiments to measure heights and angles of the wave ray with various conditions of ship speed and draft. Based on the experimental data, Sorensen and Weggel (1984) suggested an interim model to predict ship-generated wave heights. In addition, some re searchers used radar images to study the Kelvin ship waves (Shemdin, 1990; Hennings et al., 1999; Reed and Milgram, 2002). To date, there has been no information about the ship wave crest pattern in the entire range of water depths from deep to shallow waters. In this study we develop an equation to predict ship wave crests in the entire range of water depths using the linear dispersion relation and we predict cusp locus angles. The predicted wave crest pattern is verified with numerical results of the FLOW-3D. In Section 2, we suggest an equation of ship wave crests in the entire range of water depths and the integration constant C is determined by the distance between adjacent ship wave crests for the wave ray angle equal to zero. In Section 3, we simulate ship wave propagation using the FLOW-3D on the condition of Johnson (1958) hydraulic experiment and compare the cusp locus an gles of the present theory, Kelvin (1887) theory, Havelock (1908) theory and the experimental data with the solutions of the FLOW-3D. Also, we conduct numerical experiments in various conditions to verify the pre sent theory of ship wave crests. In Section 4, we summarize present studies and suggest future research topics.
Fig. 1. Ship wave crests composed of diverging and transverse waves.
and the diverging waves have angles in a range of θc � θ � 90� . When pffiffiffiffiffi the Froude number Fr ð¼ Us = gh Þ is greater than unity, the transverse waves disappear and the range of existing wave ray angles decreases. In the figure, α is the angle of the line connecting the ship and the wave crest point measured from the ship trajectory. The angle at the cusp locus is called as the cusp locus angle αc . At the crest points, the diverging waves have come from the source point nearer than the transverse waves. In actual sea, viscosity of water affects the propagation of ship waves around a ship. However, its effect would be negligibly small if ship waves propagate away from the ship. Therefore, we can express ship wave crests using the velocity potential. As shown in Fig. 2, ðX; Y; ZÞ and ðx; y; zÞ are the coordinates in a fixed frame and a ship-moving frame, respectively. The velocity potential of linear waves can be defined in a fixed frame as ϕ¼i
ag coshkðh þ ZÞ i½ωt e cosh kh ω
kðX cos θþY sin θÞ �
(1)
where a is the amplitude of the water surface elevation, k is the wave number of ship waves, h is the still water depth, g is the acceleration of gravity, and ω is the angular frequency. The coordinates ðX; Y; ZÞ and ðx; y; zÞ have relations with X ¼ x þ Us t, Y ¼ y and Z ¼ z. Eq. (1) can be expressed in the ship-moving frame as ϕ¼i
ag cosh kðh þ zÞ i½ðω e cosh kh ω
kUs cos θÞt kðx cos θþy sin θÞ �
(2)
The velocity potential is stationary in the ship-moving frame, thus we get the following relation: (3)
ω ¼ kUs cos θ
Substituting Eq. (3) into the linear dispersion of waves yields the linear dispersion of ship waves as k¼
g ðUs cos θÞ2
tanh kh
(4)
Considering the propagation of a group wave, we use the coordinate ðx’ ; y’ Þ which is normal to the wave crests in the fixed frame. The ship waves travel in groups (i.e., Cg ¼ x’ =t ¼ dω=dk), which can be expressed
2. Development of ship wave crest equations 2.1. Ship wave crest pattern Ship wave crests are composed of several components arriving from different source points at different times. The wave crests are catego rized as diverging and transverse waves. The diverging wave crests are located more away from the ship trajectory. The transverse waves are located more away from the ship. Fig. 1 shows how each component of ship waves propagates from the source to the crest line with different angles. θ is the wave ray angle measured from the ship trajectory. The diverging and transverse waves meet at the cusp locus with an angle θc . Therefore, the transverse waves have angles in a range of 0� � θ � θc
Fig. 2. Coordinates expressing ship wave crests. 2
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as d ðkx’ dk
The second relation in Eq. (12) can be obtained using the linear dispersion relation given by Eq. (4), which implies that the Froude number depends on the wave ray angle θ as well as the relative water depth kh. Fig. 3(a) shows the variation of relative water depths of ship waves with the wave ray angle θ when the Froude number Fr is given. As the angle of wave ray becomes smaller for given values of the Froude number, the relative water depth becomes shallower. And, the larger the angle of wave ray, the deeper the relative water depth. For example, in the case of the Froude number being equal to 0.5, the entire range of angles of wave ray exist from θ ¼ 0� to θ ¼ 90� while the relative water depths exist only in deep water (i.e., kh > π). On the other hand, in the case in which the Froude number is equal to 2, the relative water depths exist in the entire range of water depths from deep to shallow waters while the angles of wave ray exist only from θ ¼ 60� to θ ¼ 90� . For the Froude number greater than unity, the transverse waves disappear. As the ship speed Us increases on a constant water depth, the Froude number increases and the relative water depth of the ship waves with given wave ray angle θ deceases and thus the wavelength λ increases. Fig. 3(b) shows the variation of relative water depths with the Froude number Fr when the wave ray angle θ is given. In the figure, there is nonexistent depth region on the left side from the line with θ ¼ 0� . For example, for the Froude number equal to 0.5, there are no relative water depths less than kh ¼ π . For the Froude number equal to 2, there exists an entire range of relative water depths and the angles of wave ray are always greater than 60� . Fig. 3(c) shows the variation of relative water depths with the angle α when the Froude number Fr is given. The diverging and transverse waves become deeper and shallower, respectively, from the water with the cusp locus angle αc . As the Froude number increases, the water depth range of both the diverging and transverse waves becomes shallower. In the case of the Froude number being less than or equal to 0.5, the angle α is in the range between α ¼ 0� and α ¼ αc where the cusp locus angle αc is constant as αc ¼ 19:47� . As the Froude number increases from 0.5 to unity, the cusp locus angle αc increases from αc ¼ 19:47� to αc ¼ 90� . As the Froude number increases from unity, no transverse waves exist and the cusp locus angle αc decreases from αc ¼ 90� to zero. Integrating Eq. (6) in terms of θ and using the linear dispersion relation, i.e., Eq. (4), yield the following relation
(5)
ωtÞ ¼ 0
where x’ ¼ X cos θ þ Y sin θ ¼ x cos θ þ y sin θ þ Us t cos θ. Using Eq. (4) and applying the chain rule for kðθÞ in Eq. (5) yield the following relation: d ½kðx cos θ þ y sin θÞ� ¼ 0 dθ
(6)
Expressing Eq. (6) in terms of y =x yields the following relation dk cos θ y k sin θ dθ ¼ dk x k cos θ þ dθ sin θ
(7)
Using the linear dispersion relation, i.e., Eq. (4), we can get the following relation: dk 2g sin θ ¼ dθ U 2s cos3 θ 1
tanh kh gh sech2 kh U 2s cos2 θ
(8)
Substituting Eqs. (4) and (8) into Eq. (7) gives the equation of ship wave crests as � �2 sin θ cos θ þ sechkh tan θ F r y ¼ ¼ tan α (9) �2 � x sechkh 1 þ sin2 θ Fr In deep water (i.e., kh � π) and shallow water (i.e., kh � 0:1π ), Eq. (9) is reduced, respectively, to y ffi x
sin θ cos θ 1 þ sin2 θ
y ffi x
sin θ cos θ þ F12 tan θ
(10)
r
1 þ sin2 θ
1 F2r
(11)
Eq. (10) was first developed by Kelvin (1906) who used the linear
dispersion relation in deep water k ¼ g=ðUs cos θÞ2 . This implies that Kelvin’s theory can be applied only in deep water. To the contrary, the present theory given by Eq. (9) can be applied in the entire range of water depths from shallow to deep waters. The Froude number Fr can be expressed as rffiffiffiffiffiffiffiffiffiffiffiffiffiffi Us 1 tanh kh (12) Fr ¼ pffiffiffiffiffi ¼ kh gh cos θ
g tanh kh ðx cos θ þ y sin θÞ ¼ U 2s cos2 θ
C
(13)
where C is a dimensionless positive value which is related to the distance between neighboring ship wave crests. Substituting Eq. (9) into Eq. (13)
Fig. 3. Relations among relative water depth, Froude number, wave ray angle of ship waves: (a) Variation of relative water depths with the wave ray angle when the Froude number is given, (b) Variation of relative water depths with the Froude number when the wave ray angle is given, (c) Variation of relative water depths with the angle α when the Froude number is given. 3
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yields the ship-wave crest position in the x -axis as �2 � sechkh 1 þ sin2 θ 2 F r U cos θ x¼ C s � �2 g tanh kh sechkh 1 sec θ Fr
2.2. Cusp locus angle Havelock (1908) derived the cusp locus angle analytically. He considered an impulse point moving with a ship speed as shown in Fig. 4 where Us is the ship speed, cg is the group velocity, dð ¼ OPÞ is the cusp locus line, t is travelling time from the source point A to the ship position O, and P is the cusp locus in which the transverse and diverging waves meet. Then, he derived the cusp locus angle αc using the first and second laws of cosines with the geometry of the ship, the source, and the cups locus as � 8 � 2kh > >8 1 > > sinh 2 kh > > >� � 2 ; Fr < 1 > < 2kh 3 cos2 αc ¼ (23) sinh 2 kh > > > > > > 1 > > : 1 ; Fr > 1 F2r
(14)
Substituting Eq. (14) into Eq. (13) yields the ship-wave crest position in the y -axis as � �2 sin θ cos θ þ sechkh tan θ 2 F r U cos θ y¼C s (15) � �2 g tanh kh sechkh 1 sec θ Fr In deep water, Eqs. (14) and (15) can be expressed, respectively, as � CU 2s cos θ 1 þ sin2 θ ¼ g
x ffi
y ffi
1 CU 2s ð5 cos θ 4 g
CU 2s 1 CU 2s ðsin θ þ sin 3 θÞ sin cos2 θ ¼ 4 g g
cos 3 θÞ
(16)
where the subscript c means the value at the cusp locus. Havelock (1908) also found that, at the cusp locus, the Froude number does not depend on the wave ray angle but on the relative water depth as ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi� ffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi� tanh kh 2kh Fr ¼ 3 (24) 2kh sinh 2 kh
(17)
Kelvin (1906) and Lamb (1945) derived the second relation in Eqs. (16) and (17) using a constant number of að ¼ CU2s =gÞ. Their x -axis is positive backward from the ship, while the present x -axis is positive forward from the ship. The constant value of a is in length scale, while C used in the present study is a distance normalized by the wave number k. In shallow water, Eqs. (14) and (15) can be expressed, respectively, as x ffi
y ffi C
C
2 1 U 2s cos θ 1 þ sin θ F2r � �2 gkh sec θ 1 Fr
In a condition of the ship speed Us and the water depth h, the wave number k can be obtained using Eq. (4). Then, the cusp locus angle αc can be obtained using Eq. (23), and the Froude number Fr at the cusp locus can be obtained using Eq. (24). Also, the angle of the wave ray θc at the cusp locus point can be obtained by substituting the obtained kh and Fr into Eq. (12). Alternatively, we can find the cusp locus angle αc using the presently developed Eq. (9). That is, we can find θ ¼ θc which yields the maximum value of y =x as ðy=xÞmax and then the cusp locus angle can be determined as αc ¼ tan 1 ðy=xÞmax . From Eq. (9), the derivative of y =x with respect to the wave angle θ can be found as 8 þ at 0� � θ < θc d �y� < ¼ 0 at θ ¼ θc (25) : dθ x at θc < θ � 90�
(18)
tan θ U 2s cos θ sin θ cos θ þ F2r � �2 gkh sec θ 1 Fr
(19)
Eqs. (14) and (15) of the present theory can be applied to the entire depths of water from deep to shallow, while Eqs. (16) and (17) of Kel vin’s theory can be applied only in deep water. Use of Eqs. (4) and (13) � with θ ¼ 0 (i.e., a component of the transverse wave propagating on the ship trajectory) yields the integration constant C as C¼
g x 2 tanh kh ¼ Us
In deep water, the derivative of y =x with respect to θ can be explicitly expressed from Eq. (10) as
(20)
k0 x
d �y� ¼ dθ x
The x -directional distance Δdn between neighboring ship waves (i.e., (n 1)-th and n -th ship waves where n is the order of wave crests starting from the ship) is the wavelength λ0 when the wave ray angle is zero (i.e., Δdn ¼ λ0 ). Thus, ΔCn , which determines the distance of neighboring ship wave crests, can be expressed as
(26)
Thus, the condition of dðy=xÞ =dθ ¼ 0 gives the angle of wave ray at cusp locus point and the cusp locus angle as pffiffiffiffiffiffiffiffi θc ¼ sin 1 1=3 ¼ 35:26� and αc ¼ tan 1 ðy=xÞmax ¼ 19:47� , respec tively, which was first found by Kelvin (1887). the
(21)
ΔCn ¼ k0 Δdn ¼ 2π
3 sin2 θ �2 1 þ sin2 θ
1
In Eqs. (20) and (21), the subscript 0 means the wave number of ship � waves with θ ¼ 0 . The difference of C (i.e., ΔC), which is equal to 2π , was also found by Lamb (1945). From Eq. (21), the coefficient CN cor responding to the N -th ship wave crest is given by N X
CN ¼ C 1 þ
ΔCn ¼ C1 þ 2πðN
1Þ
(22)
n¼2
where C1 is the value of C with the wave crest nearest to the ship. The position of the nearest wave crest may not be constant depending on the ship size and speed since it is close to the ship. Theoretical solutions of the ship wave crest positions given by Eqs. (14) and (15) are compared to numerical solutions with the software FLOW-3D in Section 3.2. Fig. 4. Locations of cusp locus point P, ship position O, and source point A and cusp locus angles αc and θc in Havelock (1908) theory. 4
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3. Numerical experiments to simulate the ship wave crest pattern
However, the experimental data are smaller than the solutions for the Froude number greater than unity. Johnson’s data would have non-negligible errors in finding the cusp locus angles because he determined the cusp locus angles from coarse gridded wave gages at distances of y ¼ 0.457m, 0.914m, 1.524m, 2.435m, 3.353m (see Fig. 6). The comparison implies that the present theory accurately predicts the cusp locus angles in entire range of the Froude number. Also, the FLOW-3D can be applied to simulate the wave crest pattern in entire range of the Froude number.
3.1. Comparison of cusp locus angles among theories, solutions, and hydraulic experiments Johnson (1958) conducted hydraulic experiments to measure maximum wave amplitudes at distances from the ship trajectory and found the cusp locus angles. In the conditions of Johnson’s hydraulic experiment, we compare the cusp locus angles of the present theory, Kelvin (1906) theory, Havelock (1908) theory, numerical solutions of the FLOW-3D, and the hydraulic experimental data altogether. The ship dimensions used in the hydraulic experiment are 102:91cm �28:65cm� 3:96cm in (length, width, draft) and the water depth is h ¼ 15:85cm. In the FLOW-3D simulation, the computational domain is 42:67m �12:19m � 0:3048m in ðx; y; zÞ, the grid size is Δx ¼ Δy ¼ Δ z ¼ 3:048cm, and the total time is t ¼ 100sec. The Froude numbers are chosen as Fr ¼ 0.65, 0.75, 0.85, 0.95, 1.0, 1.2, 1.5, 1.8, 2.1, 2.4, 2.7, 3.0. The ship speed is determined when the Froude number and the water depth are given. In the numerical simulation, ship moves along the x -axis and over the free surface boundary and a symmetry boundary condition is applied along the ship trajectory (i.e., y ¼ 0) to reduce computational time. Ship wave amplitudes would be maximal near the source point and inversely proportional to the square root of the distance from the source point. Thus, amplitudes of ship waves with the wave ray angle of θ ¼ 90� would be maximal and decrease with the decrease of the wave ray angle. In the case that the Froude number is less than unity, at the cusp locus, both the diverging and transverse waves meet with same ray angles, and thus the amplitudes would be locally maximal. In the FLOW-3D solution, we find the cusp loci of the first, second, third wave crests, etc. where the wave amplitudes are locally maximal. Then, we find the cusp locus angle of the line connecting the cusp loci. In the case of the Froude number equal to or greater than unity, only diverging waves exist, and thus we find the cusp locus angles of the outer line of the diverging wave crests. Fig. 5 shows the variation of the cusp locus angles with the Froude number. The present theory, Havelock’s theory, and the FLOW-3D so lution yield similar values of the cusp locus angle in entire range of the Froude number. When the Froude number is less than 0.4, the cusp locus angle is αc ¼ 19:47� . As the Froude number increases from Fr ¼ 0:4 to Fr ¼ 1, the cusp locus angle increases from αc ¼ 19:47� to αc ¼ 90� . As the Froude number increases to greater than unity, the cusp locus angle decreases from αc ¼ 90� to αc ¼ 0� . The present theory gives the same angles as Havelock’s theory for the Froude number less than 0.4 or greater than unity. However, the present theory gives the cusp locus angles slightly greater than Havelock’s theory for the Froude number between 0.4 and unity. Kelvin’s theory incorrectly gives constant value of the cusp locus angle αc ¼ 19:47� for any value of the Froude number. Johnson (1958) hydraulic experimental data are around the theoretical and numerical solutions for the Froude number less than unity.
3.2. Comparison of wave crest patterns among theories and solutions In this section, we conduct numerical experiments using the FLOW3D with a yacht of typical size and compare numerical solutions with the present theory and Kelvin (1906) theory in constant water depth. The tested ship dimensions are 17:1m �5:5m � 2:0m in (length, width, draft). We test 56 cases for water depths of h ¼ 8m; 9m; 10m; 11m, the ship speeds in the range of Us ¼ 5:5e9m=s with 0:5m =s interval and in the range of Us ¼ 9e15m=s with 1m =s interval. The computational do mains in the x -, y -, z -directions are from 700m �200m � 13m to 2000m �1000m � 16m depending on the ship speed and the water depth. The grid size is Δx ¼ Δy ¼ Δz ¼ 1m, and the total simulation time is t ¼ 100sec. First, we compare wavelengths λ0 of the θ ¼ 0� transverse ship waves among the present theory and the Kelvin theory, and the FLOW-3D so lution in Fig. 7. For the FLOW-3D solution, average value of the five wavelengths is plotted in the figure. The wavelength λ0 is normalized by the wavelength λ0K of the Kelvin theory. When the ship speed increases such that the Froude number approaches unity, the transverse waves disappear and numerical solutions cannot be obtained for θ ¼ 0� . Therefore, analytical solutions of the normalized distances are compared with the numerical solutions in the range of the Froude number Fr ¼ 0.53–0.91. The figure shows that, as the Froude number Fr increases from zero to unity, the wavelength λ0 increases infinitely large. For the Froude number Fr greater than or equal to unity, the transverse waves no longer exist. These phenomena are properly considered in both the present theory and the FLOW-3D solution. However, the Kelvin theory which was developed based on a deep water assumption does not consider these phenomena properly. The figure also shows that, there exists energy dissipation due to viscosity of water and turbulence of high-speed particle velocities, which results in the decrease of wave length. These phenomena are considered in the FLOW-3D. However, both the present and Kelvin theories neglect the decrease of wavelength due to energy loss. Therefore, the FLOW-3D solutions yield the wave lengths λ0 always less than the present theory. Second, we compare the first to fifth ship wave crests of the present theory, the Kelvin theory, and the FLOW-3D solution. Fig. 8(a) shows the ship wave crests for Fr ¼ 0:66 (Us ¼ 6:5m=s, kh � 0:724π), which are mostly in deep water. This figure shows that both the present and Kelvin theories give longer ship wave crests compared to the FLOW-3D solution
Fig. 5. Variation of cusp locus angles with the Froude number.
Fig. 6. Configuration of Johnson (1958) hydraulic experiment. 5
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Fig. 7. Variation of wavelengths of θ ¼ 0� transverse ship waves with the Froude number.
because the theories neglect the energy dissipation. The Kelvin theory does not properly consider the phenomenon of longer ship waves in shallower water depth and thus yields the ship wave crests closer to the FLOW-3D compared to the present theory. Fig. 8(b) shows the ship wave crests for Fr ¼ 0:86 (Us ¼ 8:5m= s, kh � 0:342π ) which are mostly in intermediate-depth water. Again, the figure shows that both the present and Kelvin theories give longer ship wave crests compared to the FLOW-3D solution because the theories neglect the energy dissipation. The Kelvin theory does not properly consider the phenomenon of longer ship waves in shallower water depth and thus yields the ship wave crests much smaller than both the present theory and the FLOW-3D solution. Fig. 8(c) shows the ship wave crests for Fr ¼ 1:21 (Us ¼ 12:0m= s, kh � 0:003π ) which are mostly in shallow water. Both the present theory and the FLOW-3D solution yield infinitely large wavelengths, and thus the transverse wave crests have disappeared. The present theory gives longer ship wave crests compared to the FLOW-3D solution because the theory neglects the energy dissipation. However, the difference is not so significant because the phenomenon of longer ship waves in shallower water is more dominant than the phenomenon of shorter ship waves due to energy dissipation. The Kelvin theory incorrectly yields finite values of wavelength and shows transverse wave crests. The FLOW-3D solution shows one crest of waves starting at the bow of the ship which would be the shock waves. The shock waves would occur when the ship speed Us is pffiffiffiffiffi equal to or greater than the long wave speed gh. Third, we measure numerical solutions of distances of the adjacent ship wave crests for angle θ ¼ 0� . In section 2.2, we found that the x -directional distance Δdn between two adjacent ship waves is the wavelength λ0 with angle θ ¼ 0� . The coefficient Cn can be calculated analytically using Eq. (22). Fig. 9 shows the variation of adjacent distances of the θ ¼ 0� trans verse ship waves with the Froude number. The distance is normalized by the wave length λ0 , which is calculated using Eq. (4) with θ ¼ 0� . We compare numerical solutions of the FLOW-3D with theoretical solutions of the distances given by Eq. (22). The positions of the first wave crest may vary depending on both the ship size and the ship speed since it is closest to the ship. Therefore, we analyze the first crest separately in Fig. 9(a) and the other crests in Fig. 9(b). For all cases, the numerical solutions are smaller than the theoretical solutions. When the ship moves, there would occur energy dissipation due to viscosity of water
Fig. 8. Comparison of ship wave crest patterns: (a) Fr ¼ 0:66 (Us ¼ 6:5m=s, kh � 0:724π), (b) Fr ¼ 0:86 (Us ¼ 8:5m=s, kh � 0:342π), (c) Fr ¼ 1:21 (Us ¼ 12:0m=s, kh � 0:003π). Line definition: red solid line ¼ present theory; yellow dashed line ¼ Kelvin theory; white dot ¼ FLOW-3D solution. (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.)
especially near the ship and also turbulence of high-speed particle ve locities. The FLOW-3D considers the energy dissipation but the theory neglects it. That is why the FLOW-3D solutions are always smaller than the theoretical solutions. More energy dissipation would occur near the ship and thus numerical solutions of the first crests are smaller than the other crests. We obtain regression curves to find any trend of Δdn =λ0 with different Froude numbers. The second-order polynomial is used to best fit to the numerical solutions. Mean squared errors of the regression curves are 4:06 � 10 3 and 1:63 � 10 3 for the first and the other crests, respectively. In the case of the other crests, we use mean value of the distances for each Froude number. The fitted regression curves and mean values of the normalized distances are given by � � Δd1 Δd1 ¼ 0:734 (27) ¼ 2:76 5:70Fr þ 3:92F 2r ; λ0 λ0
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Fig. 9. Variation of distances between adjacent transverse ship wave (θ ¼ 0� ) crests with the Froude number: (a) 1st ship wave crest, (b) Mean value excluding 1st ship wave crest.
Δdn ¼ 0:31 þ 1:59Fr λ0
1:10F 2r ;
� � Δdn ¼ 0:874 λ0
(28)
The trend of the adjacent distances with different Froude numbers is not clearly seen. Around Fr ¼ 0.73, the adjacent distance is minimal for the first crests while it is maximal for the other crests. Also, the variation with the Froude number is not so significant. Fourth, we quantitatively compare theoretical and numerical solu tions of wave ray distances for all the 56 cases. In order to compare the theoretical solution and the FLOW-3D solution, we calculate RMSE (i.e., root-mean-squared error) of the wave ray distances, which are measured from the source to the crest at different angles θ. The RMSE is defined as vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u i u1 X RMSE ¼ t (29) ðrT rN Þ2k i k¼1 where rN and rT are the distances of the wave ray from the source to the crest obtained by the FLOW-3D numerical solution and the present theory, respectively, and i is the total number of the numerical solutions of the FLOW-3D. Fig. 10 shows the distances rN and rT and a relation of r ¼ Y=sin θ. Fig. 10(a) and (b) show the ray distances for the transverse and diverging waves, respectively. For each component of the wave crests, i.e., the first, second crests, etc., we find optimal values of C, which give minimal values of root-mean-squared error of the wave ray distances between the theory and the FLOW-3D solution. These ship wave crests with optimal values of C are regarded as those of the FLOW3D solution for comparing ray distances between the theories and the FLOW-3D solution. Figs. 11–13 show the variation of the ray distances rN and rT with the wave angle θ. The water depth is h ¼ 10m. The figure shows five ship wave crests. In figures (a) and (b), the ray distances rT are for the present theory and the Kelvin theory, respectively. Fig. 11 which is for the Froude number Fr ¼ 0:66 shows the FLOW-3D solutions for the trans verse waves only because the diverging wave amplitudes are negligibly small (see Fig. 8(a)). Fig. 12 which is for the Froude number Fr ¼ 0:86 shows the FLOW-3D solutions for both the transverse and diverging waves because the amplitudes of both the transverse and diverging waves can be measured (see Fig. 8(b)). Fig. 13 which is for the Froude number Fr ¼ 1:21 shows the FLOW-3D solutions for the diverging waves only because the transverse waves have disappeared (see Fig. 8(c)). The ray distances are minimal at the first crests, and they increase with the increase of the crest order to the second, third, etc. Also, the ray
Fig. 10. Distances of the wave ray rN and rT from source to crest: (a) transverse wave, (b) diverging wave.
distances are minimal at θ ¼ 90� , and they increase with the decrease of θ up to θ ¼ 0� . As the Froude number increases, the degree of the in crease of ray distance with the decrease of wave angle becomes more significant. When the Froude number is greater than or equal to unity, the ray distances become infinitely large near the cusp locus angle θ ¼ θc . These phenomena are well predicted by the present theory and the FLOW-3D solution. Also, the present theory analytically predicts the variation of the cusp locus angles with the Froude number. The present theory always predicts the ray distances greater than the FLOW-3D so lution because the present theory neglects the decrease of wavelength due to energy loss. The Kevin theory incorrectly predicts a constant value of the wave ray angle at the cusp locus, i.e., θc ¼ 35:26� for any value of the Froude number and it improperly predict the phenomena of the increase of the ray distance with the decrease of wave angle. At the Froude number Fr ¼ 1:12; the Kelvin theory incorrectly predicts finite values of the ray distance at wave angles between θ ¼ 0� and θ ¼ θc .
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Fig. 13. Variation of distances of wave ray rN and rT with the wave ray angle θ for Fr ¼ 1:21, h ¼ 10m: (a) present theory, (b) Kelvin theory.
Fig. 11. Variation of distances of wave ray rN and rT with the wave ray angle θ for Fr ¼ 0:66, h ¼ 10m: (a) present theory, (b) Kelvin theory.
cusp locus angle was determined analytically as αc ¼ tan 1 ðy=xÞmax ¼ 19:47� . We conducted numerical experiments using the FLOW-3D to simu late ship wave propagation and found that the cusp locus angles of nu merical solutions are similar to both the present theory and Havelock (1908) theory in the entire range of the Froude number. We further conducted numerical experiments to see ship wave patterns in detail. Both the present theory and the FLOW-3D solution show that, with the increase of ship speed, the Froude number increases and the wavelength increases. These also show that, when the Froude number is equal to or greater than unity, the wavelength becomes infinitely large and the transverse waves disappear. The wavelengths are minimal at the wave ray angle of θ ¼ 90� and they increase with the decrease of wave ray angle to θ ¼ 0� . The degree of ship wavelength decrease with the decrease of wave ray angle becomes more significant as the Froude number becomes larger. The wavelengths are minimal at the first wave crests and increase with the crest order to the second, third, etc. Wavelengths of the FLOW-3D are a little smaller than those of the pre sent theory because the FLOW-3D considers the decrease of wavelength due to energy dissipation which happens because of viscosity of water and turbulence of high-speed particle velocities. However, the present theory neglects the energy dissipation. The Kelvin theory improperly yields the increase of wavelength with the increase of ship speed. For the Froude number equal to or greater than unity, the Kelvin theory incor rectly yields finite values of wavelength and predict transverse waves. For a ship moving in a river or in ocean with tidal current, the location of ship wave crests will be different depending on the direction and velocity of the current and the ship. In the future, an equation can be developed for the location of crests of ship waves in a current by using the linear dispersion relation for waves in a current.
Fig. 12. Variation of distances of wave ray rN and rT with the wave ray angle θ for Fr ¼ 0:86, h ¼ 10m: (a) present theory, (b) Kelvin theory.
4. Conclusions We developed an equation for ship wave crests in the entire range of water depths, that is, the ratio of ship wave crest locations y =x using the characteristics that the velocity potential is stationary in a ship-moving frame and the ship waves propagate in group velocity, and using the linear dispersion relation. In deep water, the developed equation is reduced to the equation of Kelvin (1906), who used the linear dispersion in deep water. The locations of ship wave crests in the x - and y -di rections were obtained using a dimensionless constant C. The constant C was determined using the characteristics in which neighboring ship wave crests are at a distance of wavelength on the ship trajectory. Further, we determined the wave ray angle θc at the cusp locus using the condition in which θc is maximal at the cusp locus and also determined the so-called cusp locus angle αc ¼ tan 1 ðy=xÞmax . In deep water, the
Acknowledgements This research was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (No. NRF-2018 R1D1A1B07048606). References Akylas, T.R., 1984. On the excitation of long nonlinear water waves by a moving pressure distribution. J. Fluid Mech. 141, 455–466.
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Coastal Engineering 153 (2019) 103542 Lee, C., Lee, B.W., Kim, Y.J., Ko, K.O., 2011. Ship wave crests in intermediate-depth water. In: Proceedings of the 6th International Conference on Asian and Pacific Coasts, pp. 1818–1825. Lee, B.W., Lee, C., Kim, Y.J., Ko, K.O., 2013. Prediction of ship wave crests on varying water depths and verification by FLOW-3D. J. Korean Soc. Civil Eng. 33 (4), 1447–1454 (in Korean). Lighthill, M.J., Whitham, G.B., 1955. On kinematic waves: I. Flood movement in long rivers; II. Theory of traffic flow on long crowded roads. Proc. R. Soc. A 229, 281–345. Newman, J.N., 1970. Recent research on ship waves. In: Proceedings of the 8th Symposium on Naval Hydrodynamics, pp. 519–545. Newman, J.N., 1977. Marine Hydrodynamics. The MIT Press. Reed, A.M., Milgram, J.H., 2002. Ship wakes and their radar images. Annu. Rev. Fluid Mech. 34, 469–502. Shemdin, O.H., 1990. Synthetic aperture radar imaging of ship wakes in the Gulf of Alaska. J. Geophys. Res. 95 (C9), 16319–16338. Shi, F., Malej, M., Smith, J.M., Kirby, J.T., 2018. Breaking of ship bores in a Boussinesqtype ship-wake model. Cost Eng. 132, 1–12. Sorensen, R.M., 1967. Investigation of ship-generated waves. J. Waterw. Harb. Div. 85–99. ASCE. Sorensen, R.M., 1969. Waves generated by model ship hull. J. Waterw. Harb. Div. 513–538. ASCE. Sorensen, R.M., Weggel, J.R., 1984. Development of ship wave design information. In: Proceedings of the 19th Conference on Coastal Engineering, pp. 3227–3243. ASCE. Stoker, J.J., 1957. Water Waves: the Mathematical Theory with Applications. Interscience Publishers. Tuck, E.O., 1966. Shallow-water flows past slender bodies. J. Fluid Mech. 26, 81–95. Wu, D.M., Wu, T.Y., 1982. Three-dimensional nonlinear long waves due to moving surface pressure. In: Proceedings of the 14th Symposium on Naval Hydrodynamics, pp. 103–129.
Chen, X.N., Sharma, S.D., 1995. A slender ship moving at a near-critical speed in a shallow channel. J. Fluid Mech. 291, 263–285. David, C.G., Roeber, V., Goseberg, N., Schlurmann, T., 2017. Generation and propagation of ship-borne waves – solutions from a Boussinesq-type model. Cost Eng. 127, 170–187. Ersan, D.B., Beji, S., 2013. Numerical simulation of waves generated by a moving pressure field. Ocean Eng. 59, 231–239. Ertekin, R.C., Webster, W.C., Wehausen, J.V., 1986. Waves caused by a moving disturbance in a shallow channel of finite width. J. Fluid Mech. 169, 275–292. Fang, M.-C., Yang, R.-Y., Shugan, I.V., 2011. Kelvin ship wake in the wind waves field and on the finite sea depth. J. Mech. 27 (1), 71–77. Havelock, T.H., 1908. The propagation of groups of waves in dispersive media with application to waves on water produced by a travelling disturbance. Proc. Royal Soc. London Ser. A 398–430. Hennings, I., Romeiser, R., Alpers, W., Viola, A., 1999. Radar imaging of Kelvin arms of ship wakes. Int. J. Remote Sens. 20 (13), 2519–2543. Hur, D.S., Lee, J., Choi, D.S., Lee, H.W., 2011. On run-up characteristics of revetment under interaction among ocean wave, current and ship induced wave in the canal. In: Proceedings of the 37th Conference on the Korean Society of Civil Engineers, pp. 588–591 (in Korean). Johnson, J.W., 1958. Ship waves in navigation channels. In: Proceedings of the 6th Conference on Coastal Engineering, pp. 666–690. Kang, Y.S., Kim, P.J., Hyun, S.K., Sung, H.K., 2008. Numerical simulation of ship-induced wave using FLOW-3D. J. Korean Soc. Coast. Ocean Eng. 20 (3), 255–267 (in Korean). Kelvin, 1887. On ship waves. In: Proceedings of the Institution of Mechanical Engineering, pp. 409–433. Kelvin, 1906. Deep sea ship-waves. Proc. R. Soc. Edinb. 25 (2), 1060–1084. Lamb, H., 1945. Hydrodynamics. Dover Publications.
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Coastal Engineering journal homepage: http://www.elsevier.com/locate/coastaleng
Equation for ship wave crests in the entire range of water depths Byeong Wook Lee a, Changhoon Lee b, * a b
Coastal Development and Ocean Energy Research Center, Korea Institute of Ocean Science & Technology, 385 Haeyang-ro, Busan, 49111, Republic of Korea Department of Civil and Environmental Engineering, Sejong University, 209 Neungdong-ro, Gwangjin-gu, Seoul, 05006, Republic of Korea
A R T I C L E I N F O
A B S T R A C T
Keywords: Ship wave crests Cusp locus angle Entire range of water depths Theoretical solution Numerical experiment
An equation for ship wave crests y =x in the entire range of water depths is developed using the linear dispersion relation. In deep water, the developed equation is reduced to the equation of Kelvin (1906). The locations of ship wave crests in the x - and y -directions are obtained using a dimensionless constant C. The wave ray angle θc at the cusp locus is determined using the condition that θc is maximal at the cusp locus and the cusp locus angle is determined as αc ¼ tan 1 ðy=xÞmax . Numerical experiments are conducted using the FLOW-3D to simulate ship wave propagation. The cusp locus angles of the FLOW-3D are similar to both those of the present theory and Havelock (1908) theory in the entire range of the Froude number. Both the present theory and the FLOW-3D yield that, with the increase of ship speed, the Froude number increases and does the wavelength. For the Froude number equal to or greater than unity, the wavelength becomes infinitely large and the transverse waves disappear. The wavelengths of the FLOW-3D are slightly smaller than those of the present theory because the FLOW-3D considers the decrease of wavelength due to energy dissipation which happens because of viscosity of water and turbulence of high-speed particle velocities.
1. Introduction When a ship navigates in a channel or ocean, it creates waves propagating outward in all directions from the source point, which is moving with the ship speed. The ship waves can be categorized into transverse and diverging waves. The transverse waves propagate behind the ship with narrower wave ray angles from the ship trajectory line. The diverging waves propagate behind the ship with wider wave ray angles. pffiffiffiffiffi When the Froude number (i.e., Fr ¼ Us = gh where Us is the ship speed and h is the water depth) is less than unity, the crestlines connecting these ship waves make a concave triangle in which the transverse waves make a bottom line and the diverging waves make right and left side lines. The transverse and diverging waves meet with the same wave ray angles at a point called the cusp locus. When the Froude number is equal to or greater than unity, the transverse waves disappear and the diverging waves exist. Ship wave crest patterns have been studied since the late 1800’s. Kelvin (1887) first found empirically that diverging and transverse wave crests in deep water meet at the cusp locus with a so-called cusp locus angle of αc ¼ 19:47� . Kelvin (1906) also developed an equation for the ship wave crest using the linear dispersion relation in deep water and also the concept of ship wave propagation with a group velocity. Newman (1970) also developed the same equation for the ship
wave crest using the concept of the wave phase being stationary in deep water. As ship speed increases, the relative water depth becomes shal lower and the Kelvin’s equation cannot be applied accurately. Havelock (1908) found that the cusp locus angles vary depending on the Froude number in the entire range of water depths. Recently, Fang et al. (2011) suggested models of ship waves in the wind wave field and also on in termediate water depth using the methods of kinematics wave theory (Lighthill and Whitham, 1955). However, they could not show a detailed crest pattern of the diverging and transverse waves as Kelvin (1906) did. The heights of ship waves depend on ship speed, shape, draft, and water depth, etc. (Stoker, 1957; Johnson, 1958; Newman, 1977). When propagating in open and deep oceans, the ship waves may not affect mooring boats nor damage coastal structures. However, the ship waves may cause safety problems in a narrow channel or when the ship moves in high speed. Therefore, ship waves need to be studied in the entire range of water depths, from deep to shallow waters, in order to protect coastal area from erosion problem and keep harbor calmness and safety. Lee et al. (2011) predicted ship wave crest pattern in water of inter mediate depth by extending the approach of Kelvin (1906) using the recursive relation of the dispersion relationship in shallower water. Recently, Lee et al. (2013) developed an equation of ship wave crest on varying water depth in water of intermediate depth. However, they
* Corresponding author. E-mail addresses: [email protected] (B.W. Lee), [email protected] (C. Lee). https://doi.org/10.1016/j.coastaleng.2019.103542 Received 15 February 2019; Received in revised form 28 June 2019; Accepted 2 September 2019 Available online 4 September 2019 0378-3839/© 2019 Elsevier B.V. All rights reserved.
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could not predict ship wave crest in shallow water where the Froude number is greater than unity. Some researchers used horizontally two-dimensional numerical wave models to simulate ship wave gener ation and propagation. In order to generate ship waves, they included a moving-ship induced pressure gradient in the momentum equations (Akylas, 1984; Ertekin et al., 1986; Wu and Wu, 1982; Ersan and Beji, 2013; David et al., 2017; Shi et al., 2018) or used a slender body theory (Tuck, 1966; Chen and Sharma, 1995). In order to simulate ship wave propagation, they used horizontally two-dimensional wave models such as the Boussinesq equations (Wu and Wu, 1982; Ersan and Beji, 2013; David et al., 2017; Shi et al., 2018), Koerteweg-de Vries (KdV) equation (Akylas, 1984), Green-Naghdi equations (Ertekin et al., 1986), or Kadomtsev-Petviashvilli (KP) equations (Chen and Sharma, 1995). Most researchers were concerned about the ship waves moving near the critical speed in which condition solitary waves appear in front of the ship and the forces on the ship become significantly large. Recently, Ersan and Beji (2013) investigated ship wave crest patterns using the Boussinesq equations. They found that, in the entire range of water depths, their measured cusp locus angles are close to those predicted by Havelock (1908). However, they did not show detailed wave crest pat terns. The ship waves have components of different periods with a ship speed. The ship wave components are shorter (i.e., relatively deeper) especially near the ship in the diverging waves and are longer (i.e., relatively shallower) in the transverse waves. The whole wave crest pattern cannot be simulated with any of the equations to simulate ship wave propagation because these equations cannot be applied in very deep water. Recently, some researchers simulated ship wave generation and propagation using the three-dimensional Navier-Stokes equations with the moving pressure condition at the ship boundary (Kang et al., 2008; Hur et al., 2011). Johnson (1958) and Sorensen (1967, 1969) conducted hydraulic experiments to measure heights and angles of the wave ray with various conditions of ship speed and draft. Based on the experimental data, Sorensen and Weggel (1984) suggested an interim model to predict ship-generated wave heights. In addition, some re searchers used radar images to study the Kelvin ship waves (Shemdin, 1990; Hennings et al., 1999; Reed and Milgram, 2002). To date, there has been no information about the ship wave crest pattern in the entire range of water depths from deep to shallow waters. In this study we develop an equation to predict ship wave crests in the entire range of water depths using the linear dispersion relation and we predict cusp locus angles. The predicted wave crest pattern is verified with numerical results of the FLOW-3D. In Section 2, we suggest an equation of ship wave crests in the entire range of water depths and the integration constant C is determined by the distance between adjacent ship wave crests for the wave ray angle equal to zero. In Section 3, we simulate ship wave propagation using the FLOW-3D on the condition of Johnson (1958) hydraulic experiment and compare the cusp locus an gles of the present theory, Kelvin (1887) theory, Havelock (1908) theory and the experimental data with the solutions of the FLOW-3D. Also, we conduct numerical experiments in various conditions to verify the pre sent theory of ship wave crests. In Section 4, we summarize present studies and suggest future research topics.
Fig. 1. Ship wave crests composed of diverging and transverse waves.
and the diverging waves have angles in a range of θc � θ � 90� . When pffiffiffiffiffi the Froude number Fr ð¼ Us = gh Þ is greater than unity, the transverse waves disappear and the range of existing wave ray angles decreases. In the figure, α is the angle of the line connecting the ship and the wave crest point measured from the ship trajectory. The angle at the cusp locus is called as the cusp locus angle αc . At the crest points, the diverging waves have come from the source point nearer than the transverse waves. In actual sea, viscosity of water affects the propagation of ship waves around a ship. However, its effect would be negligibly small if ship waves propagate away from the ship. Therefore, we can express ship wave crests using the velocity potential. As shown in Fig. 2, ðX; Y; ZÞ and ðx; y; zÞ are the coordinates in a fixed frame and a ship-moving frame, respectively. The velocity potential of linear waves can be defined in a fixed frame as ϕ¼i
ag coshkðh þ ZÞ i½ωt e cosh kh ω
kðX cos θþY sin θÞ �
(1)
where a is the amplitude of the water surface elevation, k is the wave number of ship waves, h is the still water depth, g is the acceleration of gravity, and ω is the angular frequency. The coordinates ðX; Y; ZÞ and ðx; y; zÞ have relations with X ¼ x þ Us t, Y ¼ y and Z ¼ z. Eq. (1) can be expressed in the ship-moving frame as ϕ¼i
ag cosh kðh þ zÞ i½ðω e cosh kh ω
kUs cos θÞt kðx cos θþy sin θÞ �
(2)
The velocity potential is stationary in the ship-moving frame, thus we get the following relation: (3)
ω ¼ kUs cos θ
Substituting Eq. (3) into the linear dispersion of waves yields the linear dispersion of ship waves as k¼
g ðUs cos θÞ2
tanh kh
(4)
Considering the propagation of a group wave, we use the coordinate ðx’ ; y’ Þ which is normal to the wave crests in the fixed frame. The ship waves travel in groups (i.e., Cg ¼ x’ =t ¼ dω=dk), which can be expressed
2. Development of ship wave crest equations 2.1. Ship wave crest pattern Ship wave crests are composed of several components arriving from different source points at different times. The wave crests are catego rized as diverging and transverse waves. The diverging wave crests are located more away from the ship trajectory. The transverse waves are located more away from the ship. Fig. 1 shows how each component of ship waves propagates from the source to the crest line with different angles. θ is the wave ray angle measured from the ship trajectory. The diverging and transverse waves meet at the cusp locus with an angle θc . Therefore, the transverse waves have angles in a range of 0� � θ � θc
Fig. 2. Coordinates expressing ship wave crests. 2
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as d ðkx’ dk
The second relation in Eq. (12) can be obtained using the linear dispersion relation given by Eq. (4), which implies that the Froude number depends on the wave ray angle θ as well as the relative water depth kh. Fig. 3(a) shows the variation of relative water depths of ship waves with the wave ray angle θ when the Froude number Fr is given. As the angle of wave ray becomes smaller for given values of the Froude number, the relative water depth becomes shallower. And, the larger the angle of wave ray, the deeper the relative water depth. For example, in the case of the Froude number being equal to 0.5, the entire range of angles of wave ray exist from θ ¼ 0� to θ ¼ 90� while the relative water depths exist only in deep water (i.e., kh > π). On the other hand, in the case in which the Froude number is equal to 2, the relative water depths exist in the entire range of water depths from deep to shallow waters while the angles of wave ray exist only from θ ¼ 60� to θ ¼ 90� . For the Froude number greater than unity, the transverse waves disappear. As the ship speed Us increases on a constant water depth, the Froude number increases and the relative water depth of the ship waves with given wave ray angle θ deceases and thus the wavelength λ increases. Fig. 3(b) shows the variation of relative water depths with the Froude number Fr when the wave ray angle θ is given. In the figure, there is nonexistent depth region on the left side from the line with θ ¼ 0� . For example, for the Froude number equal to 0.5, there are no relative water depths less than kh ¼ π . For the Froude number equal to 2, there exists an entire range of relative water depths and the angles of wave ray are always greater than 60� . Fig. 3(c) shows the variation of relative water depths with the angle α when the Froude number Fr is given. The diverging and transverse waves become deeper and shallower, respectively, from the water with the cusp locus angle αc . As the Froude number increases, the water depth range of both the diverging and transverse waves becomes shallower. In the case of the Froude number being less than or equal to 0.5, the angle α is in the range between α ¼ 0� and α ¼ αc where the cusp locus angle αc is constant as αc ¼ 19:47� . As the Froude number increases from 0.5 to unity, the cusp locus angle αc increases from αc ¼ 19:47� to αc ¼ 90� . As the Froude number increases from unity, no transverse waves exist and the cusp locus angle αc decreases from αc ¼ 90� to zero. Integrating Eq. (6) in terms of θ and using the linear dispersion relation, i.e., Eq. (4), yield the following relation
(5)
ωtÞ ¼ 0
where x’ ¼ X cos θ þ Y sin θ ¼ x cos θ þ y sin θ þ Us t cos θ. Using Eq. (4) and applying the chain rule for kðθÞ in Eq. (5) yield the following relation: d ½kðx cos θ þ y sin θÞ� ¼ 0 dθ
(6)
Expressing Eq. (6) in terms of y =x yields the following relation dk cos θ y k sin θ dθ ¼ dk x k cos θ þ dθ sin θ
(7)
Using the linear dispersion relation, i.e., Eq. (4), we can get the following relation: dk 2g sin θ ¼ dθ U 2s cos3 θ 1
tanh kh gh sech2 kh U 2s cos2 θ
(8)
Substituting Eqs. (4) and (8) into Eq. (7) gives the equation of ship wave crests as � �2 sin θ cos θ þ sechkh tan θ F r y ¼ ¼ tan α (9) �2 � x sechkh 1 þ sin2 θ Fr In deep water (i.e., kh � π) and shallow water (i.e., kh � 0:1π ), Eq. (9) is reduced, respectively, to y ffi x
sin θ cos θ 1 þ sin2 θ
y ffi x
sin θ cos θ þ F12 tan θ
(10)
r
1 þ sin2 θ
1 F2r
(11)
Eq. (10) was first developed by Kelvin (1906) who used the linear
dispersion relation in deep water k ¼ g=ðUs cos θÞ2 . This implies that Kelvin’s theory can be applied only in deep water. To the contrary, the present theory given by Eq. (9) can be applied in the entire range of water depths from shallow to deep waters. The Froude number Fr can be expressed as rffiffiffiffiffiffiffiffiffiffiffiffiffiffi Us 1 tanh kh (12) Fr ¼ pffiffiffiffiffi ¼ kh gh cos θ
g tanh kh ðx cos θ þ y sin θÞ ¼ U 2s cos2 θ
C
(13)
where C is a dimensionless positive value which is related to the distance between neighboring ship wave crests. Substituting Eq. (9) into Eq. (13)
Fig. 3. Relations among relative water depth, Froude number, wave ray angle of ship waves: (a) Variation of relative water depths with the wave ray angle when the Froude number is given, (b) Variation of relative water depths with the Froude number when the wave ray angle is given, (c) Variation of relative water depths with the angle α when the Froude number is given. 3
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yields the ship-wave crest position in the x -axis as �2 � sechkh 1 þ sin2 θ 2 F r U cos θ x¼ C s � �2 g tanh kh sechkh 1 sec θ Fr
2.2. Cusp locus angle Havelock (1908) derived the cusp locus angle analytically. He considered an impulse point moving with a ship speed as shown in Fig. 4 where Us is the ship speed, cg is the group velocity, dð ¼ OPÞ is the cusp locus line, t is travelling time from the source point A to the ship position O, and P is the cusp locus in which the transverse and diverging waves meet. Then, he derived the cusp locus angle αc using the first and second laws of cosines with the geometry of the ship, the source, and the cups locus as � 8 � 2kh > >8 1 > > sinh 2 kh > > >� � 2 ; Fr < 1 > < 2kh 3 cos2 αc ¼ (23) sinh 2 kh > > > > > > 1 > > : 1 ; Fr > 1 F2r
(14)
Substituting Eq. (14) into Eq. (13) yields the ship-wave crest position in the y -axis as � �2 sin θ cos θ þ sechkh tan θ 2 F r U cos θ y¼C s (15) � �2 g tanh kh sechkh 1 sec θ Fr In deep water, Eqs. (14) and (15) can be expressed, respectively, as � CU 2s cos θ 1 þ sin2 θ ¼ g
x ffi
y ffi
1 CU 2s ð5 cos θ 4 g
CU 2s 1 CU 2s ðsin θ þ sin 3 θÞ sin cos2 θ ¼ 4 g g
cos 3 θÞ
(16)
where the subscript c means the value at the cusp locus. Havelock (1908) also found that, at the cusp locus, the Froude number does not depend on the wave ray angle but on the relative water depth as ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi� ffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi� tanh kh 2kh Fr ¼ 3 (24) 2kh sinh 2 kh
(17)
Kelvin (1906) and Lamb (1945) derived the second relation in Eqs. (16) and (17) using a constant number of að ¼ CU2s =gÞ. Their x -axis is positive backward from the ship, while the present x -axis is positive forward from the ship. The constant value of a is in length scale, while C used in the present study is a distance normalized by the wave number k. In shallow water, Eqs. (14) and (15) can be expressed, respectively, as x ffi
y ffi C
C
2 1 U 2s cos θ 1 þ sin θ F2r � �2 gkh sec θ 1 Fr
In a condition of the ship speed Us and the water depth h, the wave number k can be obtained using Eq. (4). Then, the cusp locus angle αc can be obtained using Eq. (23), and the Froude number Fr at the cusp locus can be obtained using Eq. (24). Also, the angle of the wave ray θc at the cusp locus point can be obtained by substituting the obtained kh and Fr into Eq. (12). Alternatively, we can find the cusp locus angle αc using the presently developed Eq. (9). That is, we can find θ ¼ θc which yields the maximum value of y =x as ðy=xÞmax and then the cusp locus angle can be determined as αc ¼ tan 1 ðy=xÞmax . From Eq. (9), the derivative of y =x with respect to the wave angle θ can be found as 8 þ at 0� � θ < θc d �y� < ¼ 0 at θ ¼ θc (25) : dθ x at θc < θ � 90�
(18)
tan θ U 2s cos θ sin θ cos θ þ F2r � �2 gkh sec θ 1 Fr
(19)
Eqs. (14) and (15) of the present theory can be applied to the entire depths of water from deep to shallow, while Eqs. (16) and (17) of Kel vin’s theory can be applied only in deep water. Use of Eqs. (4) and (13) � with θ ¼ 0 (i.e., a component of the transverse wave propagating on the ship trajectory) yields the integration constant C as C¼
g x 2 tanh kh ¼ Us
In deep water, the derivative of y =x with respect to θ can be explicitly expressed from Eq. (10) as
(20)
k0 x
d �y� ¼ dθ x
The x -directional distance Δdn between neighboring ship waves (i.e., (n 1)-th and n -th ship waves where n is the order of wave crests starting from the ship) is the wavelength λ0 when the wave ray angle is zero (i.e., Δdn ¼ λ0 ). Thus, ΔCn , which determines the distance of neighboring ship wave crests, can be expressed as
(26)
Thus, the condition of dðy=xÞ =dθ ¼ 0 gives the angle of wave ray at cusp locus point and the cusp locus angle as pffiffiffiffiffiffiffiffi θc ¼ sin 1 1=3 ¼ 35:26� and αc ¼ tan 1 ðy=xÞmax ¼ 19:47� , respec tively, which was first found by Kelvin (1887). the
(21)
ΔCn ¼ k0 Δdn ¼ 2π
3 sin2 θ �2 1 þ sin2 θ
1
In Eqs. (20) and (21), the subscript 0 means the wave number of ship � waves with θ ¼ 0 . The difference of C (i.e., ΔC), which is equal to 2π , was also found by Lamb (1945). From Eq. (21), the coefficient CN cor responding to the N -th ship wave crest is given by N X
CN ¼ C 1 þ
ΔCn ¼ C1 þ 2πðN
1Þ
(22)
n¼2
where C1 is the value of C with the wave crest nearest to the ship. The position of the nearest wave crest may not be constant depending on the ship size and speed since it is close to the ship. Theoretical solutions of the ship wave crest positions given by Eqs. (14) and (15) are compared to numerical solutions with the software FLOW-3D in Section 3.2. Fig. 4. Locations of cusp locus point P, ship position O, and source point A and cusp locus angles αc and θc in Havelock (1908) theory. 4
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3. Numerical experiments to simulate the ship wave crest pattern
However, the experimental data are smaller than the solutions for the Froude number greater than unity. Johnson’s data would have non-negligible errors in finding the cusp locus angles because he determined the cusp locus angles from coarse gridded wave gages at distances of y ¼ 0.457m, 0.914m, 1.524m, 2.435m, 3.353m (see Fig. 6). The comparison implies that the present theory accurately predicts the cusp locus angles in entire range of the Froude number. Also, the FLOW-3D can be applied to simulate the wave crest pattern in entire range of the Froude number.
3.1. Comparison of cusp locus angles among theories, solutions, and hydraulic experiments Johnson (1958) conducted hydraulic experiments to measure maximum wave amplitudes at distances from the ship trajectory and found the cusp locus angles. In the conditions of Johnson’s hydraulic experiment, we compare the cusp locus angles of the present theory, Kelvin (1906) theory, Havelock (1908) theory, numerical solutions of the FLOW-3D, and the hydraulic experimental data altogether. The ship dimensions used in the hydraulic experiment are 102:91cm �28:65cm� 3:96cm in (length, width, draft) and the water depth is h ¼ 15:85cm. In the FLOW-3D simulation, the computational domain is 42:67m �12:19m � 0:3048m in ðx; y; zÞ, the grid size is Δx ¼ Δy ¼ Δ z ¼ 3:048cm, and the total time is t ¼ 100sec. The Froude numbers are chosen as Fr ¼ 0.65, 0.75, 0.85, 0.95, 1.0, 1.2, 1.5, 1.8, 2.1, 2.4, 2.7, 3.0. The ship speed is determined when the Froude number and the water depth are given. In the numerical simulation, ship moves along the x -axis and over the free surface boundary and a symmetry boundary condition is applied along the ship trajectory (i.e., y ¼ 0) to reduce computational time. Ship wave amplitudes would be maximal near the source point and inversely proportional to the square root of the distance from the source point. Thus, amplitudes of ship waves with the wave ray angle of θ ¼ 90� would be maximal and decrease with the decrease of the wave ray angle. In the case that the Froude number is less than unity, at the cusp locus, both the diverging and transverse waves meet with same ray angles, and thus the amplitudes would be locally maximal. In the FLOW-3D solution, we find the cusp loci of the first, second, third wave crests, etc. where the wave amplitudes are locally maximal. Then, we find the cusp locus angle of the line connecting the cusp loci. In the case of the Froude number equal to or greater than unity, only diverging waves exist, and thus we find the cusp locus angles of the outer line of the diverging wave crests. Fig. 5 shows the variation of the cusp locus angles with the Froude number. The present theory, Havelock’s theory, and the FLOW-3D so lution yield similar values of the cusp locus angle in entire range of the Froude number. When the Froude number is less than 0.4, the cusp locus angle is αc ¼ 19:47� . As the Froude number increases from Fr ¼ 0:4 to Fr ¼ 1, the cusp locus angle increases from αc ¼ 19:47� to αc ¼ 90� . As the Froude number increases to greater than unity, the cusp locus angle decreases from αc ¼ 90� to αc ¼ 0� . The present theory gives the same angles as Havelock’s theory for the Froude number less than 0.4 or greater than unity. However, the present theory gives the cusp locus angles slightly greater than Havelock’s theory for the Froude number between 0.4 and unity. Kelvin’s theory incorrectly gives constant value of the cusp locus angle αc ¼ 19:47� for any value of the Froude number. Johnson (1958) hydraulic experimental data are around the theoretical and numerical solutions for the Froude number less than unity.
3.2. Comparison of wave crest patterns among theories and solutions In this section, we conduct numerical experiments using the FLOW3D with a yacht of typical size and compare numerical solutions with the present theory and Kelvin (1906) theory in constant water depth. The tested ship dimensions are 17:1m �5:5m � 2:0m in (length, width, draft). We test 56 cases for water depths of h ¼ 8m; 9m; 10m; 11m, the ship speeds in the range of Us ¼ 5:5e9m=s with 0:5m =s interval and in the range of Us ¼ 9e15m=s with 1m =s interval. The computational do mains in the x -, y -, z -directions are from 700m �200m � 13m to 2000m �1000m � 16m depending on the ship speed and the water depth. The grid size is Δx ¼ Δy ¼ Δz ¼ 1m, and the total simulation time is t ¼ 100sec. First, we compare wavelengths λ0 of the θ ¼ 0� transverse ship waves among the present theory and the Kelvin theory, and the FLOW-3D so lution in Fig. 7. For the FLOW-3D solution, average value of the five wavelengths is plotted in the figure. The wavelength λ0 is normalized by the wavelength λ0K of the Kelvin theory. When the ship speed increases such that the Froude number approaches unity, the transverse waves disappear and numerical solutions cannot be obtained for θ ¼ 0� . Therefore, analytical solutions of the normalized distances are compared with the numerical solutions in the range of the Froude number Fr ¼ 0.53–0.91. The figure shows that, as the Froude number Fr increases from zero to unity, the wavelength λ0 increases infinitely large. For the Froude number Fr greater than or equal to unity, the transverse waves no longer exist. These phenomena are properly considered in both the present theory and the FLOW-3D solution. However, the Kelvin theory which was developed based on a deep water assumption does not consider these phenomena properly. The figure also shows that, there exists energy dissipation due to viscosity of water and turbulence of high-speed particle velocities, which results in the decrease of wave length. These phenomena are considered in the FLOW-3D. However, both the present and Kelvin theories neglect the decrease of wavelength due to energy loss. Therefore, the FLOW-3D solutions yield the wave lengths λ0 always less than the present theory. Second, we compare the first to fifth ship wave crests of the present theory, the Kelvin theory, and the FLOW-3D solution. Fig. 8(a) shows the ship wave crests for Fr ¼ 0:66 (Us ¼ 6:5m=s, kh � 0:724π), which are mostly in deep water. This figure shows that both the present and Kelvin theories give longer ship wave crests compared to the FLOW-3D solution
Fig. 5. Variation of cusp locus angles with the Froude number.
Fig. 6. Configuration of Johnson (1958) hydraulic experiment. 5
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Fig. 7. Variation of wavelengths of θ ¼ 0� transverse ship waves with the Froude number.
because the theories neglect the energy dissipation. The Kelvin theory does not properly consider the phenomenon of longer ship waves in shallower water depth and thus yields the ship wave crests closer to the FLOW-3D compared to the present theory. Fig. 8(b) shows the ship wave crests for Fr ¼ 0:86 (Us ¼ 8:5m= s, kh � 0:342π ) which are mostly in intermediate-depth water. Again, the figure shows that both the present and Kelvin theories give longer ship wave crests compared to the FLOW-3D solution because the theories neglect the energy dissipation. The Kelvin theory does not properly consider the phenomenon of longer ship waves in shallower water depth and thus yields the ship wave crests much smaller than both the present theory and the FLOW-3D solution. Fig. 8(c) shows the ship wave crests for Fr ¼ 1:21 (Us ¼ 12:0m= s, kh � 0:003π ) which are mostly in shallow water. Both the present theory and the FLOW-3D solution yield infinitely large wavelengths, and thus the transverse wave crests have disappeared. The present theory gives longer ship wave crests compared to the FLOW-3D solution because the theory neglects the energy dissipation. However, the difference is not so significant because the phenomenon of longer ship waves in shallower water is more dominant than the phenomenon of shorter ship waves due to energy dissipation. The Kelvin theory incorrectly yields finite values of wavelength and shows transverse wave crests. The FLOW-3D solution shows one crest of waves starting at the bow of the ship which would be the shock waves. The shock waves would occur when the ship speed Us is pffiffiffiffiffi equal to or greater than the long wave speed gh. Third, we measure numerical solutions of distances of the adjacent ship wave crests for angle θ ¼ 0� . In section 2.2, we found that the x -directional distance Δdn between two adjacent ship waves is the wavelength λ0 with angle θ ¼ 0� . The coefficient Cn can be calculated analytically using Eq. (22). Fig. 9 shows the variation of adjacent distances of the θ ¼ 0� trans verse ship waves with the Froude number. The distance is normalized by the wave length λ0 , which is calculated using Eq. (4) with θ ¼ 0� . We compare numerical solutions of the FLOW-3D with theoretical solutions of the distances given by Eq. (22). The positions of the first wave crest may vary depending on both the ship size and the ship speed since it is closest to the ship. Therefore, we analyze the first crest separately in Fig. 9(a) and the other crests in Fig. 9(b). For all cases, the numerical solutions are smaller than the theoretical solutions. When the ship moves, there would occur energy dissipation due to viscosity of water
Fig. 8. Comparison of ship wave crest patterns: (a) Fr ¼ 0:66 (Us ¼ 6:5m=s, kh � 0:724π), (b) Fr ¼ 0:86 (Us ¼ 8:5m=s, kh � 0:342π), (c) Fr ¼ 1:21 (Us ¼ 12:0m=s, kh � 0:003π). Line definition: red solid line ¼ present theory; yellow dashed line ¼ Kelvin theory; white dot ¼ FLOW-3D solution. (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.)
especially near the ship and also turbulence of high-speed particle ve locities. The FLOW-3D considers the energy dissipation but the theory neglects it. That is why the FLOW-3D solutions are always smaller than the theoretical solutions. More energy dissipation would occur near the ship and thus numerical solutions of the first crests are smaller than the other crests. We obtain regression curves to find any trend of Δdn =λ0 with different Froude numbers. The second-order polynomial is used to best fit to the numerical solutions. Mean squared errors of the regression curves are 4:06 � 10 3 and 1:63 � 10 3 for the first and the other crests, respectively. In the case of the other crests, we use mean value of the distances for each Froude number. The fitted regression curves and mean values of the normalized distances are given by � � Δd1 Δd1 ¼ 0:734 (27) ¼ 2:76 5:70Fr þ 3:92F 2r ; λ0 λ0
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Fig. 9. Variation of distances between adjacent transverse ship wave (θ ¼ 0� ) crests with the Froude number: (a) 1st ship wave crest, (b) Mean value excluding 1st ship wave crest.
Δdn ¼ 0:31 þ 1:59Fr λ0
1:10F 2r ;
� � Δdn ¼ 0:874 λ0
(28)
The trend of the adjacent distances with different Froude numbers is not clearly seen. Around Fr ¼ 0.73, the adjacent distance is minimal for the first crests while it is maximal for the other crests. Also, the variation with the Froude number is not so significant. Fourth, we quantitatively compare theoretical and numerical solu tions of wave ray distances for all the 56 cases. In order to compare the theoretical solution and the FLOW-3D solution, we calculate RMSE (i.e., root-mean-squared error) of the wave ray distances, which are measured from the source to the crest at different angles θ. The RMSE is defined as vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u i u1 X RMSE ¼ t (29) ðrT rN Þ2k i k¼1 where rN and rT are the distances of the wave ray from the source to the crest obtained by the FLOW-3D numerical solution and the present theory, respectively, and i is the total number of the numerical solutions of the FLOW-3D. Fig. 10 shows the distances rN and rT and a relation of r ¼ Y=sin θ. Fig. 10(a) and (b) show the ray distances for the transverse and diverging waves, respectively. For each component of the wave crests, i.e., the first, second crests, etc., we find optimal values of C, which give minimal values of root-mean-squared error of the wave ray distances between the theory and the FLOW-3D solution. These ship wave crests with optimal values of C are regarded as those of the FLOW3D solution for comparing ray distances between the theories and the FLOW-3D solution. Figs. 11–13 show the variation of the ray distances rN and rT with the wave angle θ. The water depth is h ¼ 10m. The figure shows five ship wave crests. In figures (a) and (b), the ray distances rT are for the present theory and the Kelvin theory, respectively. Fig. 11 which is for the Froude number Fr ¼ 0:66 shows the FLOW-3D solutions for the trans verse waves only because the diverging wave amplitudes are negligibly small (see Fig. 8(a)). Fig. 12 which is for the Froude number Fr ¼ 0:86 shows the FLOW-3D solutions for both the transverse and diverging waves because the amplitudes of both the transverse and diverging waves can be measured (see Fig. 8(b)). Fig. 13 which is for the Froude number Fr ¼ 1:21 shows the FLOW-3D solutions for the diverging waves only because the transverse waves have disappeared (see Fig. 8(c)). The ray distances are minimal at the first crests, and they increase with the increase of the crest order to the second, third, etc. Also, the ray
Fig. 10. Distances of the wave ray rN and rT from source to crest: (a) transverse wave, (b) diverging wave.
distances are minimal at θ ¼ 90� , and they increase with the decrease of θ up to θ ¼ 0� . As the Froude number increases, the degree of the in crease of ray distance with the decrease of wave angle becomes more significant. When the Froude number is greater than or equal to unity, the ray distances become infinitely large near the cusp locus angle θ ¼ θc . These phenomena are well predicted by the present theory and the FLOW-3D solution. Also, the present theory analytically predicts the variation of the cusp locus angles with the Froude number. The present theory always predicts the ray distances greater than the FLOW-3D so lution because the present theory neglects the decrease of wavelength due to energy loss. The Kevin theory incorrectly predicts a constant value of the wave ray angle at the cusp locus, i.e., θc ¼ 35:26� for any value of the Froude number and it improperly predict the phenomena of the increase of the ray distance with the decrease of wave angle. At the Froude number Fr ¼ 1:12; the Kelvin theory incorrectly predicts finite values of the ray distance at wave angles between θ ¼ 0� and θ ¼ θc .
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Fig. 13. Variation of distances of wave ray rN and rT with the wave ray angle θ for Fr ¼ 1:21, h ¼ 10m: (a) present theory, (b) Kelvin theory.
Fig. 11. Variation of distances of wave ray rN and rT with the wave ray angle θ for Fr ¼ 0:66, h ¼ 10m: (a) present theory, (b) Kelvin theory.
cusp locus angle was determined analytically as αc ¼ tan 1 ðy=xÞmax ¼ 19:47� . We conducted numerical experiments using the FLOW-3D to simu late ship wave propagation and found that the cusp locus angles of nu merical solutions are similar to both the present theory and Havelock (1908) theory in the entire range of the Froude number. We further conducted numerical experiments to see ship wave patterns in detail. Both the present theory and the FLOW-3D solution show that, with the increase of ship speed, the Froude number increases and the wavelength increases. These also show that, when the Froude number is equal to or greater than unity, the wavelength becomes infinitely large and the transverse waves disappear. The wavelengths are minimal at the wave ray angle of θ ¼ 90� and they increase with the decrease of wave ray angle to θ ¼ 0� . The degree of ship wavelength decrease with the decrease of wave ray angle becomes more significant as the Froude number becomes larger. The wavelengths are minimal at the first wave crests and increase with the crest order to the second, third, etc. Wavelengths of the FLOW-3D are a little smaller than those of the pre sent theory because the FLOW-3D considers the decrease of wavelength due to energy dissipation which happens because of viscosity of water and turbulence of high-speed particle velocities. However, the present theory neglects the energy dissipation. The Kelvin theory improperly yields the increase of wavelength with the increase of ship speed. For the Froude number equal to or greater than unity, the Kelvin theory incor rectly yields finite values of wavelength and predict transverse waves. For a ship moving in a river or in ocean with tidal current, the location of ship wave crests will be different depending on the direction and velocity of the current and the ship. In the future, an equation can be developed for the location of crests of ship waves in a current by using the linear dispersion relation for waves in a current.
Fig. 12. Variation of distances of wave ray rN and rT with the wave ray angle θ for Fr ¼ 0:86, h ¼ 10m: (a) present theory, (b) Kelvin theory.
4. Conclusions We developed an equation for ship wave crests in the entire range of water depths, that is, the ratio of ship wave crest locations y =x using the characteristics that the velocity potential is stationary in a ship-moving frame and the ship waves propagate in group velocity, and using the linear dispersion relation. In deep water, the developed equation is reduced to the equation of Kelvin (1906), who used the linear dispersion in deep water. The locations of ship wave crests in the x - and y -di rections were obtained using a dimensionless constant C. The constant C was determined using the characteristics in which neighboring ship wave crests are at a distance of wavelength on the ship trajectory. Further, we determined the wave ray angle θc at the cusp locus using the condition in which θc is maximal at the cusp locus and also determined the so-called cusp locus angle αc ¼ tan 1 ðy=xÞmax . In deep water, the
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