Journal of Marine Science and Engineering
Article Kinematics of the Ship’s Wake in the Presence of a Shear Flow
Igor Shugan 1,2,* and Yang-Yih Chen 1
1 Department of Marine Environment and Engineering, National Sun Yat-Sen University, Kaoshiung 804, Taiwan; [email protected] 2 Laboratory of Shelf and Sea Coasts, Shirshov Institute of Oceanology, Russian Academy of Sciences, 117997 Moscow, Russia * Correspondence: [email protected]
Abstract: We present the kinematic model of the ship wake in the presence of horizontal subsurface current linearly varying with the depth of water. An extension of the Whitham–Lighthill theory for calm water is developed. It has been established that the structure of ship waves under the action of a shear flow can radically differ from the classical Kelvin ship wake model. Co propagating ship and shear current lead to increasing the total wedge angle up to full one 180◦ and decreases for the counter shear current. At relatively large unidirectional values of the shear current, cusp waves in the vicinity of the wedge boundary are represented by transverse waves and, conversely, by diverging waves directed almost perpendicular to the ship track for the opposite shear current. The presence of a shear flow crossing the direction of the ship’s movement gives a strong asymmetry of the wake. An increase in the perpendicular shear flow leads to an increase in the difference between the angles of the wake arms. The limiting value of the shear current corresponds to one or both arms angles equal to 90◦. Transverse and divergent edge waves for this limiting case coincide.
Keywords: ship wake; wake angle; transverse and divergent waves; shear current
1. Introduction The structure of waves on the water surface behind a ship, called a wake, was first Citation: Shugan, I.; Chen, Y.-Y. Kine- examined by Lord Kelvin (Thomson) in 1887 [1]. Starting at approximately several lengths matics of the Ship’s Wake in the Pres- of the ship behind it, the wake geometry and wave pattern are quite universal and inde- ence of a Shear Flow. J. Mar. Sci. Eng. pendent of the ship’s speed and shape in deep water conditions. The wake consists of 2021, 9, 7. https://dx.doi.org/10.3390/ two sorts of waves: transverse waves with crests across the ship’s path and longitudinal jmse9010007 waves, moving outward. They are placed in a wedge region behind the ship, and the angle of the wake is approximately 39◦. The wake’s edges with the maximum amplitude Received: 15 November 2020 of the longitudinal waves are called Kelvin arms, and the waves in this area are called Accepted: 21 December 2020 the cusp waves. The Kelvin solution’s main features were reproduced by Lighthill and Published: 23 December 2020 Whitham [2,3], who suggested the kinematics of the ship wave’s model.
Publisher’s Note: MDPI stays neu- Comparison of these theoretical results with observations revealed that, although tral with regard to jurisdictional claims there is generally acceptable qualitative agreement, there are essential differences between in published maps and institutional the predicted and measured geometry of the Kelvin diagram in the ship’s wake. The main affiliations. difference in the ship’s wake structure is the significant variability of the wake angles [4–6]. In the wake, the half Kelvin angle can be as narrow as 9◦ or much wider, up to full 90◦ [6,7]. The wake angle statistics observed by the synthetic aperture radar (SAR) method [7] show that, in general, wake angles are narrower than the Kelvin results. Another feature Copyright: © 2020 by the authors. Li- is the asymmetry of the ship’s wake’s observed shape–very often, only one of the Kelvin censee MDPI, Basel, Switzerland. This arms is visible on SAR images [7–10]. article is an open access article distributed SAR techniques used to detect ships have greatly contributed to the study of ship under the terms and conditions of the tracks over the past few decades. Ships may not even be visible on radar images due to their Creative Commons Attribution (CC BY) small size, but the ship’s trail extends over kilometers and can provide essential information license (https://creativecommons.org/ about the ship, its length, speed, direction of movement, etc. [11,12]. Speed estimation from licenses/by/4.0/).
J. Mar. Sci. Eng. 2021, 9, 7. https://dx.doi.org/10.3390/jmse9010007 https://www.mdpi.com/journal/jmse J. Mar. Sci. Eng. 2021, 9, x FOR PEER REVIEW 2 of 15
ured geometry of the Kelvin diagram in the ship's wake. The main difference in the ship’s wake structure is the significant variability of the wake angles [4–6]. In the wake, the half Kelvin angle can be as narrow as 9o or much wider, up to full 90o [6,7]. The wake angle statistics observed by the synthetic aperture radar (SAR) method [7] show that, in general, wake angles are narrower than the Kelvin results. Another feature is the asymmetry of the ship’s wake’s observed shape – very often, only one of the Kelvin arms is visible on SAR images [7–10]. SAR techniques used to detect ships have greatly contributed to the study of ship tracks over the past few decades. Ships may not even be visible on radar images due to their small size, but the ship’s trail extends over kilometers and can provide essential information about the ship, its length, speed, direction of movement, etc. [11,12]. Speed estimation from radio images helps to track vessels, which is an im- portant part of shipping management. In recent years, several meth- ods for recognizing radar images have been proposed to analyze ship
J. Mar. Sci. Eng. 2021, 9, 7 tracks. The direction of movement of the vessel and its speed can be 2 of 12 estimated both by the wake’s angle and by the length of waves in the cusp region of the wake [12–15]. Thus, a quantitative assessment of the geometry of Kelvin arms is one of the practically demanded tasks. Muchradio imagesof the ship helps’s energy to track is vessels, lost within which the is ship's an important wake [16 part–18], of so shipping the management. propertiesIn recent years, of ship several waves methods and wave for recognizing resistance are radar particularly images have im- been proposed to portantanalyze in ship shipbuilding. tracks. The direction of movement of the vessel and its speed can be estimated bothThe by the difference wake’s anglebetween and the by theKelvin length scheme of waves and in the the ship cusp track region ob- of the wake [12–15]. servationsThus, a quantitative can be attributed assessment to several of the geometryreasons arising of Kelvin from arms the m isod- one of the practically eldemanded’s basic assumptions. tasks. Much Modeling of the ship’s a ship's energy wake is lost in withincalm water the ship’s and in wake [16–18], so the deepproperties water of approaching ship waves and is one wave of resistance the main reasonsare particularly for this important discrep- in shipbuilding. ancy.The Wind difference waves and between a finite the sea Kelvin depth scheme approach and canthe ship significantly track observations can be alterattributed a ship to’s severalwake [19,20]. reasons The arising wake from angle the and model’s the basiccusp assumptions.waves' char- Modeling a ship’s wake in calm water and in deep water approaching is one of the main reasons for this acteristics in this case strongly depend on the ship's speed and the discrepancy. Wind waves and a finite sea depth approach can significantly alter a ship’s depth of the water. The use of the theory of linear waves on water wake [19,20]. The wake angle and the cusp waves’ characteristics in this case strongly significantly simplified the task [21,22] and, in addition to nonlineari- depend on the ship’s speed and the depth of the water. The use of the theory of linear ty, did not consider all the effects of the hull shape, the interaction of waves on water significantly simplified the task [21,22] and, in addition to nonlinearity, bow and stern waves, which does not allow adequately simulating the did not consider all the effects of the hull shape, the interaction of bow and stern waves, flow field near the ship. Modern ideas about the structure of the ship’s which does not allow adequately simulating the flow field near the ship. Modern ideas wake are presented in Figure 1 [19]. about the structure of the ship’s wake are presented in Figure1[19].
Figure 1. Structure of the ship wake.
The ship waves’ structure in the presence of horizontal shear current that changes with water depth can be strongly different from the classical one and exhibit some conceptually new properties. Depth sheared currents are often observed in coastal waters due to wind produced drift, tides, and underwater currents [23,24]. A long list of works is devoted to propagating surface waves in the presence of a horizontal flow that changes with the depth of the fluid. Most of them consider two- dimensional interaction in the approximation of constant flow vorticity [23–27]. This pro- vides a significant simplification of the problem because the theory of potential waves is applicable in this case. The three-dimensional problem of the interaction of the transverse current with waves was mainly developed in the last decade [28,29]. The dispersion rela- tion for waves indicates a significant difference in the characteristics of waves depending on their directions with respect to the direction of the shear current. One of the iconic three-dimensional tasks is studying the structure of the ship’s wake on a shear flow. A recent study [30] showed a strong dependence of the wake structure on the shear flow characteristics. The classical Kelvin model, based on consideration of the ship as a moving pressure distribution, was re-analyzed, and the stationary phase method was applied to calculate the far-field wake. It was shown that the angle of the ship’s wake could be either large or smaller, and the wake can lose its symmetry due to J. Mar. Sci. Eng. 2021, 9, 7 3 of 12
three-dimensional interaction with the shear current. The significant effects of finite water depth were included in consideration of this problem in [17]. Another possibility of analyzing the influence of horizontal shear current on the ship’s wake is developing an extended kinematic model of the wake, originally proposed in [2,3]. This is the purpose of this study. A model of propagating ship waves in the presence of a constant vorticity flow for an infinite water depth is considered. The advantage of the presented kinematic model is its relative simplicity, which makes it possible to adequately describe the main features of the wake on the shear current. Explicit formulas describe the geometry of the ship’s wave structure depending on the magnitude of the shear current’s vorticity and its direction relative to the ship’s path. The structure of the paper is as follows. Section2 presents the basic kinematic model of the ship wake in the presence of horizontal shear current, describes the method of solving, and presents the results. Section3 describes the results of waves modeling for three variants of the mutual ship path and current directions: - Collinear (Section 3.1), - cross (Section 3.2), - inclined (Section 3.3). Conclusions are made in Section4.
2. Kinematic Model of the Ship Wake → Let us consider plane surface waves with a vector of wavenumber k = (kX, kY) = (−k Cosψ, k Sinψ) (see Figure2), forming the ship’s wake, in the presence of a horizon- → → → 0 tal shear flow Uc = U0 + U 0z, linearly varying with the water depth z. The moving frame of reference has the ship fixed at its origin (P) with the uniform surface current → → → → U = −(Uship + U0) moving in the positive direction along the X-axis. Uship is the vector of → → → 0 0 0 the ship’s speed relative to the water surface, U 0 = (U 0X, U 0Y) is the constant horizontal vector of a shear flow gradient. Our study aims to analyze the structure of stationary ship → waves in the presence of a given shear current Uc. The kinematics model for a ship wave’s wake in calm water first was suggested by Lighthill and Whitham [2,3]. We extend the theory for the wave’s propagation in the pres- → 0 ence of the shear current with constant vertical shear U 0. The deepwater approximation kh 1 is assumed, which is acceptable for the ship wave’s pattern with the length of the ship and waves much less than the depth of the fluid h. We will also take the approximation of an incompressible and inviscid fluid. The linear dispersion equation for surface gravity waves in the problem under consid- eration includes the Doppler frequency shift and the effect of the shear flow and has the well-known form [29,30]:
→ → 0 → → 2 → → → ( k ·U0 ) G(kX, kY) = (ω − k ·U) + (ω − k ·U) − k g = 0 (1) → k
where ω is the frequency of waves, g is gravity acceleration. J. Mar. Sci. Eng. 2021, 9, 7 4 of 12
J. Mar. Sci. Eng. 2021, 9, x FOR PEER REVIEW 5 of 15
→ → FigureFigure 2. 2.The The geometry geometry of of waves waves behind behind the the ship. ship.U Uis the is flow the flow velocity, velocity,k —wave k number vector of ∂G —wave number vector of surface waves, (, )—ray and wave number vec- = surface waves, (ξ, ψ )—ray and wave number vector angles, respectively; vector → GkX , GkY G ∂ k tor angles, respectively; vector = ( GG, ) express( es) the ray direction expresses the ray direction tan ξ GkY kXkk, kY /G kX kX, kY . k XY tan G ( k ,k)/ G ( k ,k) . The kinematicskYX X Y of wave k X motion Y is described by the law of conservation of the number of waves [31]: → ∂ k The kinematics of wave motion is + described∇ω = 0 by the law of con- (2) servation of the number of waves [31]:∂T and consistency equation for the wavenumber function: k 0 (2) T∂k ∂k X = Y (3) ∂Y ∂X and consistency equation for the wavenumber function: We consider a stationary problemkk with zero frequency of the field of surface waves in the selected coordinate system; therefore,XY Equation (2) is identically valid. In(3 polar)
YX → → coordinates, we have (Figure1): kX = −Cos(ψ) k , kY = Sin(ψ) k , −π/2 < ψ < π/2. We consider a stationary problem with zero frequency of the (k k ) fieldFinally, of surface we have waves the pair in the of Equations selected coordinate (1) and (3) forsystem; two unknown therefore, functions X, Y . The dependence k = f (k ) can be found from the dispersion relation (1), and the quasi- Equation (2) is identicallyX Y valid. In polar coordinates, we have (Figure linear Equation (3) can be solved by the method of characteristics. 1): kXY Cos( ) k , k Sin ( ) k , / 2 / 2 . Equation (3) for kY function may be rewritten in the following form: Finally, we have the pair of Equations (1) and (3) for two un- ∂k ∂k known functions (,)kk . The dependenceY = f 0(k ) kY f() k can be XY ∂X Y ∂YXY found from the dispersion relation (1), and the quasi-linear Equation The(3) cansystem be solved of characteristics by the method for this of characteristics. equation is: Equation (3) for kY functiondY may be rewrittendk in the following = − f 0(k ), Y = 0 (4) form: dX Y dX kk For the point source P, we have theYY central fk() wave and characteristics—straight rays with a constant wavenumber along theXY directionY of ray ξ: The system of characteristics for this equation is:( ) 0 GkY kX, kY tan ξ = Y/X = − f (kY) = ; kY, kX = const (5) Gk dk(kX, kY) dY ，X Y fk()Y 0 (4) where ξ is the ray angle (see FiguredX 1). dX AFor complete the point picture source of P, the we wave have pattern the central behind wave the and ship characteris- can also be drawn with lines withtics— thestraight same waverays with phase a andconstant with wavenumber crest lines. The along expression the direction for the phaseof function may be found from its definition: ray : → X Z → → θ(X, Y)Gk =k( Xk ,k·dX Y ) tan Y / X f ( k ) Y ； k, k const (5) Y Gk0( ,k ) YX kX X Y J. Mar. Sci. Eng. 2021, 9, 7 5 of 12
When calculating the integral, you can use any trajectory, since the wave number field → is irrotational. The rays ξ = const are convenient for calculation, since the wave number k along them is constant, and we have:
→ θ(X, Y) = r k Cos(µ), µ = π − ψ − ξ, (6)
where r—is the distance from the origin, µ—is the angle between the vectors of the ray and the wavenumber (see Figure1).
3. Results 0 0 3.1. Collinear Propagation of the Ship and Shear Flow (U0X 6= 0, U0Y = 0) We start with the simplest case of collinear ship propagation and shear flow along the X-axis. For steady-state waves in the chosen moving frame of reference with zero frequency ω = 0, the dispersion relation (1) takes the form:
2 → 0 2 U WX UU0X G(kx, ky) = kX( + ) − k = 0, WX = − , (7) g → g k
where U is the speed of surface flow, WX is a dimensionless parameter that characterizes the vorticity of the underlying flow. The ray Equation (5) takes the form:
tan ψ + WX 2 tan ψ Gk (kX, kY) 1 + tan ψ tan ξ = Y/X = Y = (8) Gk (kX, kY) 1 2 X −WX + 1 + 2 tan ψ 1 + tan2 ψ
The directions of ship waves ψ: −kY/kX = tan(ψ) depending on the ray angle tan ξ are shown in Figure3 for unidirectional with the ship ( WX > 0) and oncoming (WX < 0) shear current. → Because k = − k Cos(ψ) < 0, only the range −π/2 < ψ < π/2 is permissible. X The dependence (8) is evidently symmetrical concerning the X-axes, so we present only the plane’s upper half 0 < ψ < π/2. Two types of waves are represented within the ship’s wake: transverse waves (crests intersect the ship’s trajectory) and divergent waves (crests are approximately parallel to J. Mar. Sci. Eng. 2021, 9, x theFOR ship’sPEER REVIEW trajectory and move outward), which is consistent with Kelvin’s results7 of 15 [1].
Figure 3. DirectionsFigure 3. Directions of ship waves of shipψ: waves−kY/k X=: tankkYX(ψ/) depending tan( ) ondepending the ray angle on thetan rayξ for angle differenttan values for of shear current: (a) Unidirectional with the ship (WX > 0) shear current; (b) Oncoming (WX < 0) shear current. different values of shear current: (a) Unidirectional with the ship (WX 0 ) shear current; (b) On-
coming (WX 0 ) shearThe current. ship’s wake structure drastically depends on the magnitude of the shear current. The ship wake’s total wedge angle increases for a unidirectional with ship shear current Two types of waves are represented within the ship’s wake: transverse waves (crests intersect the ship's trajectory) and divergent waves (crests are approximately parallel to the ship’s trajectory and move outward), which is consistent with Kelvin’s results [1]. The ship’s wake structure drastically depends on the magnitude of the shear current. The ship wake’s total wedge angle increases for a o unidirectional with ship shear current up to 180 for WX 1 (Figure 3a) and decreases for the counter shear current (Figure 3b). At rela- tively large positive values of the shear current, cusp ship waves with
the angle m in the wedge boundary’s vicinity are mainly repre- sented by transverse waves and, conversely, by divergent waves for the counter shear current. The cusp wavelength 1 21U 2 increases with increasing unidirectional mX2 W g Cos m shear current up to infinity. The lines of the constant phase (,)X Y const in polar coordi- nates (,)r may be presented in the parametric form in terms of :
U 221 tan2 2(1 tan ) tan r 22; tan ; g (1 tan WXX ) W (1 tan ) tan W tan (9) X 1 tan2 tan YX / . 1 W 1 2 tan2 X 1 tan2 The value of is negative. Phase lines can also be represented in Cartesian coordinates: J. Mar. Sci. Eng. 2021, 9, 7 6 of 12
◦ up to 180 for WX = 1 (Figure3a) and decreases for the counter shear current (Figure3b) . At relatively large positive values of the shear current, cusp ship waves with the angle ψm in the wedge boundary’s vicinity are mainly represented by transverse waves and, conversely, by divergent waves for the counter shear current. The cusp wavelength −1 2πU2 1 λm = 2 − WX increases with increasing unidirectional shear current up to g Cos ψm infinity. The lines of the constant phase θ(X, Y) = const in polar coordinates (r, ξ) may be presented in the parametric form in terms of ψ: √ θU2 1+tan2 µ 2(1+tan2 ψ) tan ψ r = − 2 ; tan µ = 2 ; g (1+tan ψ−WX ) WX −(1+tan ψ) tan ψ WX +tan ψ 1 + tan2 ψ (9) tan ξ = Y/X = . 1 J. Mar. Sci. Eng. 2021, 9, x FOR PEER REVIEW 2 8 of 15 −WX +1+2 tan ψ 1 + tan2 ψ
The value of θ is2 negative. Phase lines can also be represented in Cartesian coordinates: WX 2 12 v 4 tan 221 tan u !2 U u WX Cos X t 1− +4 tan; 2 ψ 2 2 2 g W 2 1 + tan1ψ tan Cos2ψ X = − θUX √ ; 1 g 2 !2 2 1 tan WX 1+tan ξ 1− (10) 1 + tan2 ψ v2 (10) W u !2 tan tan ψ X u W2 X W tanW 1+tan ψ 2 t 1−4 tan +4 tan2 ψ 22X 2 X 2 2 U 1 tan 2 1 + tanψ1 tan 1 + tan ψCos Cos2ψ Y Y = − θU . √ g 2 !2 2 . g 1 2 1 WX 2 1+tan ξ W 1− W 2 tan +1+2 tan2 Wψ X 1 tan X 2 X 2 1 1− 1 tan 1 + tan ψ 2 + 2 1 tan 1 tan ψ
Typical examplesTypical of examples wave patterns of wave are shown patterns in Figure are shown4 for unidirectional in Figure 4 ( forWX = 0.75)
and oppositeunidirectional (WX = −0.75) (WX shear 0.75 current.) and opposite (WX 0.75 ) shear current.
Figure 4. TheFigure lines 4. of The the lines constant of the phase constantθ(X, Yphase) = const (,)Xfor Y different const values for different of shear values current: of shear (a) Unidirectional current: (a) with the
ship (WX =Unidirectional0.75) shear current; with the (b) ship Opposite (WX (W0.75X = −) shear0.75) shearcurrent; current. (b) Opposite (WX 0.75 ) shear current.
The phaseThe lines phase are symmetrical lines are symmetrical about the X-axis. about The the wavenumber X-axis. The wave-vectors at each point are perpendicular to the phase line. All phase lines have a cusp that separates number vectors at each point are perpendicular to the phase line. All transverse and divergent waves. The phase lines’ geometry is quite similar to that pre- phase lines have a cusp that separates transverse and divergent waves. The phase lines’ geometry is quite similar to that presented in [30], which can serve as a test and clearly demonstrates the applicability of the presented model.
3.2. Cross Direction of the Shear Flow Next, we analyze the effect of the shear current perpendicular to
the ship’s speed (WWYX0, 0 ). For steady-state waves in the chosen moving frame of reference with zero frequency 0 , the dispersion relation (1) has the form:
2 2 U WY UU0Y ' G(kx , k y ) k X kXY k k 0, WY . (11) ggk
The ray Equation (5) is the following: J. Mar. Sci. Eng. 2021, 9, 7 7 of 12
sented in [30], which can serve as a test and clearly demonstrates the applicability of the presented model.
3.2. Cross Direction of the Shear Flow Next, we analyze the effect of the shear current perpendicular to the ship’s speed (WY 6= 0, WX = 0). For steady-state waves in the chosen moving frame of reference with zero frequency ω = 0, the dispersion relation (1) has the form:
2 → 0 J. Mar. Sci. Eng. 2021, 9, x FOR PEER REVIEW 2 U WY UU0Y 9 of 15 G(kx, ky) = kX + kXkY − k = 0, WY = − . (11) g → g k
W Y tan The ray Equation (5) is the following: 1 tan2 tan YX / (12) W 1 2 tan2 W tanY + tan ψ 1 2 tan2 Y1 + tan2 ψ 2 = = 1 tan tan ξ Y/X 2 (12) 1 + 2 tan ψ 2 Equation (12) is WY invarianttan ψ under + 1 the+ 2 tan transformation:ψ 1 + tan2 ψ WWYY ,,, therefore, we consider only positive Equation (12) is invariant under the transformation: WY → −WY, ψ → −ψ, ξ → −ξ, values of WY 0 . The directions of ship waves kkYX/ tan( ) therefore, we consider only positive values of WY > 0. The directions of ship waves depending on the ray angle are shown in Figure 5a. Dependence −kY/kX = tan(ψ) depending on the ray angle ξ are shown in Figure5a. Dependence (12) is clearly asymmetric(12) is clearly about asymmetric the X-axis; about therefore, the X-axis; we presented therefore, the we entire presented variation range −π/2 < ψthe< entireπ/2. A variation typical sketch range of the/ 2 phase lines / 2 for. A the typical shear sketch value of (W theY = 0.75) is shown in Figure5b. phase lines for the shear value (WY 0.75) is shown in Figure 5b.
Figure 5. Cross interaction of waves and shear current: (a) Directions of ship waves Figure 5. Cross interaction of waves and shear current: (a) Directions of ship waves −kY/kX = tan(ψ) depending on the kk/ tan( ) depending on the ray angle for different values W 0. ; (b) Phase lines ray angle ξ forYX different values WY > 0; (b) Phase lines θ(X, Y) = const for the current shearY value (WY = 0.75).
(,)X Y const for the current shear value (WY 0.75). Transverse waves have a negative direction (ψ < 0, ξ = 0) along the trajectory of the ship. TwoTransverse wake arms waves have have different a negative absolute direction values ( of angles0, 0 (see) along Figure 5a,b). The positive wake arm angle ξm increases with the flow vorticity WY and reaches the value ◦ the trajectory of the ship. Two wake arms have different absolute val- 45 for WY = 1, while the () value limits the negative arm angle. Figure5b of the constant phase linesues demonstrates of angles (see the Figure strong 5a,b). asymmetry The positive of the wake ship wakearm angle in the presencem in- of the cross-shear flow. o creases with the flow vorticity WY and reaches the value 45 for Cusp waves are placed along the arms of the wake. The angle of inclination of the W 1, while the () value limits the negative arm angle. Figure 5b of cusp waves Yψm in depending on WY is shown in Figure6. It decreases monotonically with WY and isthe equal constant to zero phase for W Y lines= 1—transverse demonstrates cusp the waves strong propagate asymmetry along of the the direction of the shipship motion. wake Ain further the presence increase of inthe shear cross-shear flow leads flow. to the negative direction of the arm angles’ cuspCusp waves waves and are values placedξm along> π/ the4. The arms limit of the value wake. of the The shear angle flow of WY = 2 corresponds to arms angles ξ = ±90◦ and zeroing the group velocity of the cusp waves. inclination of the cuspm waves m in depending on is shown in The transverse and divergent edge waves for that limiting case coincide and have an angle Figure 6. It decreases monotonically with and is equal to zero for
WY 1—transverse cusp waves propagate along the direction of the ship motion. A further increase in shear flow leads to the negative di-
rection of the arm angles' cusp waves and values m /4. The
limit value of the shear flow WY 2 corresponds to arms angles o m 90 and zeroing the group velocity of the cusp waves. The transverse and divergent edge waves for that limiting case coincide o and have an angle m 45 . The resulting limit value WY coin- cides with that given in [30], which further validates our model. J. Mar. Sci. Eng. 2021, 9, 7 8 of 12
◦ ψm = −45 . The resulting limit value WY coincides with that given in [30], which further J. Mar. Sci. Eng. 2021, 9, x FOR PEERvalidates REVIEW our model. 10 of 15
FigureFigure 6. 6.The The angle angle of of inclination inclination of of the the cusp cusp waves wavesψ m inm dependingin depending on shearon flow WY.
3.3.shear Inclined flow W DirectionY . of the Shear Flow In the general case, it is necessary to investigate the shear flow’s oblique direction (W3.3.Y 6=Inclined0, WX Direction6= 0). of the Shear Flow DispersionIn the general relation case, (1) it hasis necessary the form: to investigate the shear flow’s
oblique direction2 (WWYX0, 0 ). → 0 0 2 U 2 WX WY UU0Y UU0Y G(kxDispersion, ky) = kX relation+ kX (1) has+ k XthekY form: − k = 0, WY = − , WX = − . (13) g → → g g k k 2 22U WWXY UU00YY'' UU G(kx , k y ) k X kX kXY k k 0, WY ,WX . (13) g The raykk direction Equation (5) is: g g
tan ψ WY The ray direction Equation (5) is: WX + + tan ψ 1 + tan2 ψ 1 + tan2 ψ tan ξ = Y/X = (14) tan W 2 Y 1 1 + 2 tan ψ 2 WX 22−WX +tanWY tan ψ + 1 + 2 tan ψ 1 tan 11 tan+ tan2 ψ 1 + tan2 ψ tan YX / (14) 1 1 2 tan2 Equation (14) is obviously invariant under the transformation:2 W → −W , ψ → −ψ, WWXY22 tan1 2 tan Y Y ξ → −ξ,1 therefore, tan we consider1 only tan positive values of WY > 0. Examples of ship waves’ directions −k /k = tan(ψ) depending on the ray angle ξ Equation (14) is obviously invariant underY X the transformation: are shown in Figure7 for positive and negative X-components of shear flow WX = ±WY. ForWWYY each ray,,, angle ξ , we still have therefore, a pair of weship consider waves: only transverse positive and divergent; the
wakevalues wave’s of WY pattern 0 . is asymmetric about the X-axis. For each value of the shear current, there is a cusp on the ray ξ = ξm with the wave’s direction ψ = ψm. For the positive Examples of ship waves' direction√ s kkYX/ tan( ) depend- shear components WX = WY = 1/ 2 (see curve (IV) in Figure7a), the edge wave’s ing on the ray angle are shown in Figure 7 for positive and nega- ◦ angles are equal to ψm = 0, which means that the waves on the cusp ξm = 67 will be tive X-components of shear flow WW . For each ray angle , strictly parallel to the√ ship’s movement.XY For opposite signs of shear flow components WweX =still− WhaveY = a− pair1/ of2 (seeship curvewaves: (IV transverse) in Figure and7b), divergent; the corresponding the wake cusp angle will be ◦ ξwave’sm = 22 pat. Thetern shear is asymmetric current for about this case the satisfies X-axis. theFor relationship: each value of the
shear current, there is a cusp on the ray m with the wave’s di- W 2 + W 2 = 1 X Y rection m . For the positive shear components WWXY1/√ 2 √ (see A curve sketch (IV of) thein Figure phase lines 7a), the for theedge shear wave's currents anglesW Y are= equal1/ 2 ,toW X = ±1/ 2 are presented in Figure8a,b. o m 0 , which means that the waves on the cusp m 67 will be strictly parallel to the ship's movement. For opposite signs of shear
flow components WWXY 1/ 2 (see curve (IV) in Figure 7b), o the corresponding cusp angle will be m 22 . The shear current for this case satisfies the relationship: J. Mar. Sci. Eng. 2021, 9, x FOR PEER REVIEW 11 of 15
J. Mar. Sci. Eng. 2021, 9, x FOR PEER REVIEW 11 of 15 22 WWXY1 22 A sketch of the phase WW linesXY for the1 shear currents J. Mar. Sci. Eng. 2021, 9, 7 WW1/ 2, 1/ 2 9 of 12 YXA sketch of the are phase presented lines in for Figure the 8a,b. shear currents
WWYX1/ 2, 1/ 2 are presented in Figure 8a,b.
Figure 7. Directions of ship waves kYX/() k tg depending on the ray angle : (a) (I)—( WW0 ), (II)—(WW0.4 ), (III)—(WW0.6 ), (IV)—(WW1/ 2 ); (b) (I)—( FigureXY 7. Directions of XY ship− waves= kYX( /()) k tgXY depending on theXY ray angle : (a=) (I)—=( Figure 7. Directions of ship waves kY/kX tg ψ depending√ on the ray angle ξ :(a)(I)—(WX WY 0), (II)— (W = W = 0.4), (III)—(W), = W (II)—=(0.6WW), (XYIV)—( W = 0.2W =),1 / (2III); ()b—)((I)—(WWWXY= W = 00.45), (II)—(),W (=IV−)—W( = −0.2), X YWWXY0 ), (IIX)—(WWYXY0.4 ), (IIIX )—(√WWYXY0.6 ), (IV)—X(WWXYY 1/ 2 ); (Xb) (I)—(Y (III)—(WX WW= −W Y = − 0.45), 1/ (IV 2 ).)—( WX = −WY = −1/ 2). XY), (II)—( WWXY 0.2 ), (III)—( WWXY 0.45 ), (IV)—(
WWXY 1/ 2 ).
Figure 8. The lines of the constant( phase) = (,)X Y const for different values of shear current: (a) Figure 8. The lines√ of the constant phase θ X, Y const√for different values of shear current: (a) positive shear components (WX = WYFigurepositive= 1/ 8. 2);shear The (b) lines Oppositecomponents of the (W constantX (WW=XY−W phaseY = − 1/1/(,)X 22)); Y shear(b) constOpposite current for components. (differentWWXY values 1/ of shear 2 ) shear current: current (a) components. positive shear components (WW1/ 2 ); (b) Opposite (WW 1/ 2 ) shear current A furtherXY increase in shear flow leads toXY a negative direction of cusp waves ψm < 0 ◦ components. and arm anglesA ξfurtherm > 68 increase. The directions in shear offlow ship leads waves to −a knegativeY/kX = tandirection(ψ) depending of on the ray angle ξ are shown√ in Figure9 for positive and negativeo X-components of shear flow cusp waves m 0 and arm angles m 68 . The directions of WX = ±WY, WAY further> 1/ 2increase. A sketch in ofshear the flow phase leads lines to for a thenegative shear currentsdirection components of ship waves kk/ tan( ) depending on o the ray angle are WY = 0.82,cuspWX waves= ±0.82 aremYX presented0 and arm in Figure angles 10 a,b.m The68 upper. The limit directions value of theof positive shear flow W = W = 0.92 corresponds to the arm angle ξ = 90◦ and zeroing the group shipshownX waves in Y Figure kk / 9 for positive tan( ) and depending negative on X -components them ray angle of shear are YX( ) = velocity of the cusp waves GkX kX, kY 0. Transverse and divergent edge waves for shownflow WWW inXYY Figure , 9 for positive 1/ 2 . andA sketch negative of theX◦ -components phase lines of for shear the this limiting case coincide and have an angle ψm = −10 . For the limiting negative shear shear currents components WW0.82, 0.82 are presented◦ in current componentflow WWWXYYW X = ,−WY = 1/−2.47 2 . YX,A the sketch arms angles of the are phaseξm = lines±90 for, cusp the waves are ◦ ◦ ψm = −44shearFigure, and currents the10a,b. total Thecomponents wake upper angle limitisWW 180 value0.82,. of the 0.82 positive are presented shear flow in WW0.92 YX o FigureXY 10a,b. The corresponds upper limit to valuethe arm of angle the positivem 90 shearand zero- flow o WWXY0.92 corresponds to the arm angle m 90 and zero- J. Mar. Sci. Eng. 2021, 9, x FOR PEER REVIEW 12 of 15
J. Mar. Sci. Eng. 2021, 9, x FOR PEER ingREVIEW the group velocity of the cusp waves G( k , k ) 0. Trans-12 of 15 kX X Y verse and divergent edge waves for this limiting case coincide and ing the group velocity of the cusp waves Gk( k X , k Y ) 0. Trans- have an angle 10o . For the limiting negativeX shear current verse and divergentm edge waves for this limiting case coincide and o o component WWXY 2.47 , the arms angles are m 90 , have an angle m 10 . For the limiting negative shear current o o o cusp waves are m 44 , and the total wake angle is 180 . J. Mar. Sci. Eng. 2021, 9, 7 component WWXY 2.47 , the arms angles are m 90 , 10 of 12 o o cusp waves are m 44 , and the total wake angle is 180 .
Figure 9. Directions of ship waves kkYX/ tan( ) depending on the ray angle : (a) (I)—( WW1/ 2 ), (II)—(WW0.8 ), (III)—(WW0.85 ), (IV)—(WW0.9 ); (b) √ FigureXY 9. Directions of ship− XY waves= kk(/) tan(XY ) depending on the ray XY angle : (a=) (I)—( = Figure 9. Directions of ship waves kY/kX tanYXψ depending on the ray angle ξ :(a)(I)—(WX WY √ 1/ 2), (II)—(WX (=I)—W(YWW= 0.8 ), (III)—( W1/X = 2W),Y (II=)—0.85( WW), (IV )—(W X =1.25WY), = (III0.9)—);( (WWb)(I)—( WX = 1.75−WY), = (IV−)—1/( 2), (II)— WWXYXY1/ 2 ), (II)—(WWXY0.8XY), (III)—(WWXY0.85 ), (XYIV)—(WWXY0.9 ); (b) (WX = −WY = −1.25), (III)—(WX = −WY = −1.75), (IV)—(WX = −WY = −2.25). WWXY 2.25 ). (I)—( WWXY 1/ 2 ), (II)—( WWXY 1.25 ), (III)—( WWXY 1.75 ), (IV)—(
WWXY 2.25 ).
Figure 10. The lines of the constant phase (,)X Y const : (a) positive shear components ( Figure 10. The lines of the constant phase θ(X, Y) = const:(a) positive shear components (WX = WY = 0.82); (b) opposite (W = −WWW=−0.82)0.82 shear); current (b) opposite components. (WW 0.82 ) shear current components. X FigureY XY 10. The lines of the constantXY phase (,)X Y const : (a) positive shear components ( WW0.82); (b) opposite (WW 0.82 ) shear current components. XY 4. Conclusions4. ConclusionsXY The structureThe structure of ship waves of ship in thewaves presence in the of presence a horizontal of a shearhorizontal flow, whichshear changes 4. Conclusions with waterflow, depth, which can changes radically with differ water from depth, the classical can radically Kelvin differ ship from wake the model and possess severalclassicalThe strikingly Kelvinstructure ship new of wakeship properties. waves model in Theand the presentedpossess presence several kinematicof a horizontal strikingly model shear new of the ship’s wake developsproperties.flow, whichthe well-known The changes presented withWhitham–Lighthill kinematic water depth, model canmodel of radically the and ship can differ’s adequately wake from devel- the describe the main featuresopsclassical the of the well-knownKelvin stationary ship wake pictureWhitham model of– theLighthill and ship’s possess wake. model several and canstrikingly adequately new Thedescribeproperties. ship’s collinear the The main presented propagation features ofkinematic the and stationary the model shear picture flowof the result ofship the’ ins shipwake a symmetrical’s wake.devel- wake structureops concerning theThe well-known ship the’s ship’s collinear path. Whitham propagation There–Lighthill are two and types model the of shear andwaves flow can in theadequately result wake in of a the ship: transversesymmetricaldescribe and divergent, the mainwake whichfeatures structure is consistentof concerning the stationary with the the pictureship classical’s path. of the results. There ship’ sare Cowake. two propagating ◦ ship andtypes shearThe current of wavesship leads’s collinear in to the increasing wake propagation of the the total ship: and wedge transversethe shear angle up flow and to resultfull divergent, one in180 a for the no dimensionalwhichsymmetrical is magnitude consistent wake ofstructurewith shear theW classicalconcerningX = 1 and results. decreasesthe ship Co ’propagatings forpath. the There counter ship are shearandtwo current. At relativelysheartypes large current of unidirectional waves leads in theto increasing values wake of the the sheartotal ship: wedgecurrent, transverse angle cusp and wavesup to divergent, full in the one vicinity of the wedgewhich boundary is consistent are represented with the byclassical transverse results. waves Co propagating and, conversely, ship byand diverging waves directedshear current almost leads perpendicular to increasing to thethe shiptotal trackwedge for angle the oppositeup to full shear one current. The cusp wavelength increases with increasing unidirectional shear current. The presence of a shear flow crossing the direction of the ship’s movement gives a strong asymmetry of the wake. The positive wake arm angle increases with the positive ◦ value of shear WY and reaches the value 45 for WY = 1, while the negative arm angle is limited by the (−22.3◦) value. The inclination of the cusp waves decreases monotonically with WY and is equal to zero for WY = 1—transverse cusp waves propagate along the direction of the ship motion. A further increase in the shear flow leads to a negative J. Mar. Sci. Eng. 2021, 9, 7 11 of 12
◦ direction of cusp waves and values of the arm angles ξm > 45 . The limiting value of ◦ the shear current WY = 2 corresponds to arms angles ξm = ±90 and zeroing of the cusp waves’ group velocity. Transverse and divergent edge waves for this limiting case coincide ◦ and have an angle ψm = −45 . In the general case, the shear flow’s oblique direction to the trajectory of the ship also gives the wake asymmetry. The positive wake arm angle increases with the positive values 2 2 of the shear components, and for WX + WY = 1 the edge waves will be strictly parallel to the movement of the ship. A further increase in shear flow leads to a negative direction of cusp waves. The upper limit value of the positive shear flow WX = WY = 0.92 corresponds ◦ to the arm angle ξm = 90 and zeroing the group velocity of the cusp waves. Transverse ◦ and divergent edge waves for this limiting case coincide and have an angle ψm = −10 . For the limiting negative shear current component WX = −WY = −2.47, the arms angles ◦ ◦ are ξm = ±90 , cusp waves angles are ψm = −44 . The presented model has a vastly different approach from the classical one, based on consideration of the ship as moving pressure distribution on the sea surface. Asymptotic nontrivial numerical integration for every specific oceanographic situation in the solution’s traditional scheme needs much effort and has its restrictions. Despite its relative simplicity, our kinematic model reproduces some results similar to those obtained according to the classical scheme [30], which, in turn, can serve as a test of the model’s applicability. Here we can cite the typical pictures of the phase lines of ship waves, the critical values of the shear flow gradient during cross interaction, etc. As an advantage of the presented model, it can be noted that its analytical solution makes it possible to explicitly describe the structure of ship waves in the entire range of changes in the defining parameters of the problem, and not only its typical behavior and limiting cases.
Author Contributions: Conceptualization, I.S. and Y.-Y.C.; methodology I.S. and Y.-Y.C.; software, I.S.; validation, I.S.; formal analysis, I.S.; investigation, I.S.; resources, I.S. and Y.-Y.C.; writing— original draft preparation, I.S.; writing—review and editing I.S. and Y.-Y.C.; visualization, I.S.; supervision, Y.-Y.C.; project administration, Y.-Y.C.; funding acquisition, Y.-Y.C. All authors have read and agreed to the publication of the manuscript. Funding: The reported study was funding by RFBR and TUBITAK according to the research project 20-55-46005. The study was also funded by the Ministry of Science and Higher Education of the Russian Federation, theme no. 0149-2019-0005. Acknowledgments: The authors acknowledge the financial support from the Ministry of Science and Technology of Taiwan, under Grant Number MOST 106-2221-E-110-036-MY3 for this study. Conflicts of Interest: The authors declare no conflict of interest. The funders had no role in the study’s design; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results.
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