Coherent structures in phase space, governing the nonlinear surge motions of ships in steep waves

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Ocean Engineering 120 (2016) 339-345

E L S E V f f i R

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Ocean Engineering

j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / o c e a n e n g

Coherent structures in phase space, governing the nonlinear surge

motions of ships in steep waves

loannis Kontolefas, Kostas J. Spyrou*

School of Naval Architecture and tviarine Engineering, National Technical University of Athens, Greece

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A R T I C L E I N F O

Article history:

Received 12 Novennber 2015 Accepted 5 February 2016 Available online 18 February 2016 Keywords:

Ship Motions Surf-riding Waves

Lagrangian Coherent Structures Feature Flow Field

A B S T R A C T

In steep multi-chromatic seas, ship surge dynamics can become intricate and the full variety of exhibited motions is u n k n o w n . This accrues, partly, from the nonlinear nature of surge motion; a n d partly because, for multi-frequency waves, the phase-space flow of the dynamical system becomes time-dependent. Accordingly, conventional concepts that w e r e applied in the past for analyzing stationary phase-space flows a r c rendered incapable to support in-depth expluralion of ship dynamics. Towards overcoming this limitation, use of the concept of hyperbolic Lagrangiatt Coherent Structures (LCSs) is proposed. T h e s e phase-space objects can be regarded as the "finite-time" generalizations of the stable and unstable manifolds of hyperbolic fixed points defined in "time-invariant" dynamical systems. T h e y can be described as, locally, the strongest repelling or attracting material surfaces (curves in the case of 2-dimensional systems) advected w i t h the phase fiow. We have identified hyperbolic LCSs that are innate to the phase-fiow associated w i t h the surge motion of a ship in astern seas. To the global approach of LCS identification, a supplementary computational scheme is incorporated, aiming to track, in space-time, local "features" of the flow, connected w i t h surf-riding. T h e emerging toolset can enhance current efforts towards a rigorous assessment of ship d y n a m i c stability in following seas.

1. Introduction

The mectianisms generating s u r f - r i d i n g behaviour f o r a ship o p e r a t i n g i n regular f o l l o w i n g seas have been extensively studied i n the past (e.g. Kan, 1990; Spyrou, 1996). However, g a i n i n g u n d e r s t a n d i n g beyond t h e context o f h a r m o n i c waves has been considered d a u n t i n g ; because the m u l t i - f r e q u e n c y wave f i e l d brings-in new qualitative features i n ship response, by-and-large as yet unrecognised, descending f r o m the t i m e - d e p e n d e n t nature of t h e system's phase f l o w .

For "regular sea" scenarios, i t is w e l l k n o w n that s u r f - r i d i n g can be i d e n t i f i e d as an e q u i l i b r i u m s o l u t i o n o f the surge equation o f m o t i o n . Such solutions m a y appear i n coexistence w i t h the o r d i n a r y periodic type o f ship surge response, or they m a y even e n t i r e l y d o m i n a t e the surge behaviour o f a ship. A detailed account o f progress on u n d e r s t a n d i n g t h e nonlinear surging and s u r f - r i d i n g o f a ship i n h a r m o n i c waves can be f o u n d i n Spyrou (2006). Consideration, t h o u g h , o f m o r e general w a v e f o r m s introduces p r o f o u n d complications. For i r r e g u l a r seas, the key d e s c r i p t i o n o f s u r f - r i d i n g as a stationary state needs to be reap-praised, since one cannot reasonably assume t h a t the u n d e r l y i n g

* Corresponding author: TeL: + 3 0 210 7 721418; fax: +30 210 7721408. E-mail address: k.spyrou@central.ntua.gr (K.J. Spyrou).

n o n - a u t o n o m o u s dynamical system w i l l a d m i t constant solutions. Hence, a broader d e f i n i t i o n o f s u r f - r i d i n g is entailed.

W i t h these difficulties recognized, a phenomenological approach to s u r f - r i d i n g i n irregular seas has been proposed recently, expanding u p o n the n o t i o n o f wave celerity and its role i n signalling a ship's capture to surfriding (Spyrou et al., 2012, 2014a). I n p a r t i -cular, d e f i n i t i o n and methods f o r the calculation o f wave celerity f o r a n irregular seaway w e r e proposed and their relevance to the p r o -b l e m o f s u r f - r i d i n g was evaluated. The appeal o f such an approach is that i t enables a straightforward statistical a p p r o x i m a t i o n o f t h e probability o f surf-riding i n irregular seas, by setting up a direct c o u n t i n g scheme o f velocity threshold exceedances.

In another recent w o r k , Belenky et al. (2012) endeavoured t o gain i n s i g h t i n t o the surge dynamics i n m u l t i - c h r o m a t i c f o l l o w i n g waves t h r o u g h the i d e n t i f i c a t i o n o f the points o f the wave p r o f i l e w h e r e , e q u i l i b r i u m o f forces along ship's l o n g i t u d i n a l d i r e c t i o n is instantaneously satisfied. For the calculation o f such points, cel-e r i t y o f irrcel-egular wavcel-es, w h cel-e r cel-e a b o u t ship's p o s i t i o n , m u s t bcel-e k n o w n . This technique could be useful, i n some instances, as a n a p p r o x i m a t e calculation scheme.

Also, Spyrou et al. (2014b) examined the p o s s i b i l i t y o f e x t r a c t i n g and t r a c k i n g "features" related to the surge dynamics i n irregular seas. The w o r d "feature" is attached here to any o b j e c t ostensibly relevant t o the realisation o f s u r f r i d i n g . I t was d i s -covered that, such features can be i d e n t i f i e d a m o n g the elements

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340

o f the zero set o f the "acceleration f i e l d " i.e., points o f the phase-plane w h e r e the acceleration and its t i m e derivative attain, instantly, zero values.

In the current w o r k , m e t h o d s having potential to yield insight i n t o the dynamics o f the surge m o t i o n in m u l t i - c h r o m a t i c astern seas are employed. In particular, the concept o f hyperbolic Lagrangian Coherent Stiuctures is applied and its capacity i n u n v e i l i n g the c h a n g i n g - i n - t i m e organization o f system's phase f l o w is evaluated. T h r o u g h their organizing role, these structures can be considered as analogues o f the stable and unstable m a n i -folds of hyperbolic f i x e d points d e f i n e d i n autonomous d y n a m i c a l systems. For t h e i r i d e n t i f i c a t i o n , one can choose a m o n g a n u m b e r o f d i f f e r e n t methods. Here, a popular n u m e r i c a l scheme is selec-ted, based on the calculation o f the spatial d i s t r i b u t i o n o f the largest f i n i t e - t i m e Lyapunov exponent. Furthermore, a scheme a i m i n g to the t r a c k i n g o f critical points o f the vector f i e l d , d e f i n e d by the value o f acceleration along the surge d i r e c t i o n and o f its material derivative, is applied. This supplementary n u m e r i c a l scheme is based on the Feature Flow Field concept, w h i c h addresses the p r o b l e m o f feature t r a c k i n g in non-stationary f l o w fields (Theisel and Seidel, 2003).

W i t h the c u r r e n t w o r k i t is aimed to propose concepts and toolsets w h i c h can enable deeper u n d e r s t a n d i n g o f the s u r f - r i d i n g and broaching-to behaviour i n irregular seas. Such p h e n o m e n a o f extreme ship behaviour c u r r e n t l y receive detailed a t t e n t i o n at IMO and, i t is expected t h a t t h e y w i l l be covered i n f u t u r e legislation as additional dynamic s t a b i l i t y requirements (Peters et al., 2011).

2. Concepts and computational tools 2.1. Lagrangian Coherent Structures

The concept o f Lagrangian Coherent Structures seems to have emerged as result o f the interbi^eeding o f ideas o r i g i n a t i n g f r o m the fields o f d y n a m i c a l systems t h e o r y and f l u i d dynamics. A l t h o u g h the t e r m was f i r s t l y i n t r o d u c e d by Haller and Yuan (2000), m a n y people have c o n t r i b u t e d i n the d e v e l o p m e n t o f c o m p u t a t i o n a l strategies f o r their i d e n t i f i c a t i o n - f o r a short review see Shadden (2011). LCSs have been extensively used d u r i n g the last years i n a w i d e range o f applications concerning physical and biological f l o w s , w h i l e the theory, as w e l l as e f f i c i e n t calculation methods, are still developing.

Although one can select among d i f f e r e n t schemes f o r the iden-tification of hyperbolic LCSs [such as the finite-size Lyapunov Exponent (FSLE) approach, or the variational theory o f hyperbolic LCSs developed recently by Haller (2011) that enables a more r i g -orous computation] f o r the needs o f the current study w e w i l l consider a w i d e l y used computational procedure involving the cal-culation o f the largest finite-time Lyapunov exponent (FILE) field. Let us consider the f o l l o w i n g d y n a m i c a l system t h a t defines a flow on the plane,

x=f(x,t), x e D c R ^ t e [ t - , t + ] c R (1) A trajectory o f system (1) at t i m e t, s t a r t i n g f r o m the i n i t i a l

c o n d i t i o n Xo at to, w i l l be denoted by x(t; to, Xo). W e can w r i t e f o r the flow map F[ (Xo) o f (1),

Xo^x{t;to,Xo) (2)

T h r o u g h (2), the phase-particle passing f r o m Xo at t i m e to is associated w i t h its p o s i t i o n at t i m e t. We, f u r t h e r m o r e , consider t w o i n f i n i t e s i m a l l y close phase-paiticles, located a t x o and Xo+6o at t i m e to. The m a g n i t u d e o f the linearized p e r t u r b a t i o n at t i m e

f o + T is g i v e n by (see, e.g. Shadden (2011)),

ll^ll = ||VF[° + ^(Xo)<5o|| = ll<5o II ^ e ; [ V F J " + ^(XO)] VFj° + ^ ( X o ) e o (3) In the above, eo is the u n i t vector along the d i r e c t i o n o f So, denotes the transpose o f A, w h i l e V F [ ° + ' ' ( X o ) is the d e f o r m a t i o n

g r a d i e n t and c E j + ' ' ( X o ) = [vFj°'*"''(Xo)] V F [ » ' ^ ' ' ( X O ) is t h e r i g h t

Cau-chy-Green d e f o r m a t i o n tensor, b o t h evaluated at X Q . C [ ° ' ^ ' " ( X O ) is a real s y m m e t r i c positive d e f i n i t e tensor and, as such, i t has real positive eigenvalues, Ai, i = 1,2. Moreover, the corresponding eigenvectors, e,-, / = 1,2, f o r m an o r t h o n o r m a l basis.

The Cauchy-Green d e f o r m a t i o n tensor provides a measure o f h o w l i n e elements i n the n e i g h b o u r h o o d o f Xo d e f o r m under the flow; i.e., h o w the lengths and the angles b e t w e e n l i n e elements change, w h e n considering the c o n f i g u r a t i o n i n the close v i c i n i t y o f x(t; t o , X Q ) at times to and to -i- r . A circular b l o b o f i n i t i a l conditions centred at Xo w i l l evolve i n t o an ellipse, w i t h the m a j o r ( m i n o r ) axis aligned w i t h the d i r e c t i o n o f the eigenvector 62 ( e i ) . The coefficients o f expansion along these directions w i l l be given by v / ^ , i = l , 2 .

The finite-time Lyapunov exponents are d e f i n e d as f o l l o w s ,

A ( = T l n \ / i ; , 1 = 1,2 (4)

The largest FTLE, A2, is usually referred to as "FTLE" w i t h o u t d i s t i n c t i o n . By v i r t u e o f (4), A 2 can be regarded as a time-averaged measure o f stretching and therefore, as a ( r o u g h ) measure o f a trajectory's hyperbolicity. Yet, as noted b y Haller (2011) and Shadden (2011), this does n o t hold in general.

T h r o u g h the calculation o f the spatial FTLE d i s t r i b u t i o n , the i d e n t i f i c a t i o n o f LCSs is made possible. The latter w i l l appear as local m a x i m i z i n g curves o f the FTLE field. Typically, the calculation of the field is p e r f o r m e d on the basis o f a structured g r i d o f i n i t i a l conditions spanning a considered d o m a i n at a given t i m e t o . The g r i d is integrated over a specified t i m e i n t e r v a l T , using a n u m e r -ical i n t e g r a t i o n a l g o r i t h m . Once the final p o s i t i o n o f each g r i d p o i n t is calculated, the d e f o r m a t i o n g r a d i e n t is o b t a i n e d b y i m p l e m e n t i n g a finite difference scheme o n the nodes o f the i n i t i a l g r i d . I n the final step o f the procedure, the largest eigenvalue o f the d e f o r m a t i o n gradient is c o m p u t e d and the FTLE field is cal-culated d i r e c t l y f r o m expression (4). The l o c a t i o n o f r e p e l l i n g / attracting LCSs can be i d e n t i f i e d as ridges o f the FTLE field w h e n f o r w a r d / b a c k w a r d i n t e g r a t i o n times are considered.

2.2. Feature tracking

A c c o r d i n g to Spyrou et al. (2014b), "features" relevant to the p r o b l e m o f s u r f - r i d i n g i n m u l t i - c h r o m a t i c waves can be i d e n t i f i e d on the basis o f critical points o f a planar vector field, w i t h coor-dinates t h a t correspond to the acceleration, along the surge direction, and its m a t e r i a l derivative. The paths o f such features can be calculated using the Feature Flow Field (FFF) m e t h o d . W i t h respect to the latter, and given a vector valued f u n c t i o n o f the f o r m ,

a ( x i , X 2 , t) = (a, ( X i , X 2 , t), 02 ( X , , X 2 , t)) (5) a t h r e e - d i m e n s i o n a l vector field w ( X i , X 2 , t ) is constructed, b y

d e m a n d i n g t h a t vector w points t o w a r d the d i r e c t i o n o f m i n i m a l change o f a i n a first order a p p r o x i m a t i o n . This d i r e c t i o n is d e f i n e d by the intersection o f the planes perpendicular to V a i and Va2, w h e r e the V operator is related to the t h r e e - d i m e n s i o n a l Euclidian space w i t h coordinates ( X i , X 2 , t ) (Theisel and Seidel, 2 0 0 3 ) . Thus,

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Fig. 1. Harmonic excitation, (i) and (ii) The graph of the forward FTLE field over a ( X i . X j ) domain, (iii) Loci of points where the field surpasses a selected threshold.

Once the p o s i t i o n o f some critical p o i n t o f the vector valued 4. Application f u n c t i o n a is d e t e r m i n e d , its p a t h can be o b t a i n e d as an integral

line o f w . emanating f r o m the p o i n t u n d e r consideration. 4.7. Verification

3. Mathematical m o d e l of surge m o t i o n

Consider the f o l l o w i n g u n i d i r e c t i o n a l w a v e f o r m comprised o f N propagating, h a r m o n i c wave components.

f ( x , t ) = cos kiX-cOit+ef^ (7)

In the above, x is the distance f r o m an E a r t h - f i x e d p o i n t o f reference, w h i l e A,-, kj and w, are the a m p l i t u d e , wave n u m b e r and frequency, respectively, o f the d i s t i n c t w a v e c o m p o n e n t corre-s p o n d i n g to the index /; e j " denotecorre-s the r a n d o m phacorre-se o f the latter, u n i f o r m l y d i s t r i b u t e d i n the range [ 0 , 2 ^ ) .

We, f u r t h e r m o r e , consider an e l e m e n t a r y m a t h e m a t i c a l m o d e l t h a t can reproduce a s y m m e t r i c s u r g i n g and s u r f - r i d i n g occur-rences i n f o l l o w i n g waves o f the f o r m ( 7 ) , see f o r example Spyrou e t a l . (2012, 2014a), = (Ton^ ( m - X ü ) f = iTor, ^ V V inertia ^— -tin^ r - ^ A / R A O ; sin fci^-ffl,t-i-e!'^+e/i (8) waveforce

In the equation above, ^ is the p o s i t i o n o f a s h i p - f i x e d p o i n t o f reference w i t h respect to the E a r t h - f i x e d o r i g i n , w h i l e RAO, and £ƒ, denote t h e response-amplitude-operator and phase, respectively, o f the surging force c o r r e s p o n d i n g t o the w a v e c o m p o n e n t i ; m,Xu is the mass and surge added mass o f the ship, n corresponds to the propeller revolutions, w h i l e 7 , , / = 0 , 1 , 2 and r i , i = 1,2,3 are p o l y n o m i a l coefficients. The o v e r d o t denotes d i f f e r e n t i a t i o n w i t h respect to t i m e t. Setting, X i = f , X 2 = ^ Eq. (8) can be w r i t t e n i n n o r m a l f o r m , X , = X 2 (9) X2 = ( m - X , - , ) ' | r o n ^ - l - s i n [k,x, - «jt+£('"'-i-£j-,-- (r, «jt+£('"'-i-£j-,-- T , n)X2 «jt+£('"'-i-£j-,-- ( r 2 «jt+£('"'-i-£j-,-- «jt+£('"'-i-£j-,--r2)xi «jt+£('"'-i-£j-,-- r^xl

w h e r e , i n the above, ƒ,• = A j • RAO,.

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W e select, as a case study, the t u m b l e h o m e h u l l f r o m the ONR topside series (Bishop et al., 2 0 0 5 ) w i t h L = 1 5 4 m ( l e n g t h ) , B = 18.8 m ( b r e a d t h ) and r = 5 . 5 m ( d r a u g h t ) . To v e r i f y the applic-a b i l i t y o f the LCS i d e n t i f i c applic-a t i o n scheme i n the p r o b l e m considered, w e w r i t e f i r s t l y system (10) f o r N = 1 (regular waves),

X i = X 2

X 2 = (m - X +f sin [ / « I - t a t - i - t y ]

-(r^ - i - i n ) X 2 - ( r 2 - T 2 ) x ^ - r 3 x i _{( 1 1 )}

W e set the w a v e l e n g t h and steepness values to X = L a n d H/X = 0.04, respectively, w h e r e H denotes the wave height. Deep w a t e r is assumed, w h i l e the selected value o f n corresponds to a " n o m i n a l " ( c a l m w a t e r ) speed o f 12 m / s .

For the calculation o f the FTLE field, a g r i d is considered at to = 3 0 0 s on a ( x , , X 2 ) d o m a i n . A f t e r some e x p e r i m e n t a t i o n , i n t e -g r a t i o n t i m e is set to T = 55 s. The -graph o f the r e s u l t i n -g field, n o r m a l i z e d i n the range [0, 1], can be seen i n Fig. 1(1) a n d ( i i ) . I n Fig. l ( i i i ) , w e visualize loci o f p o i n t s w h e r e the field surpasses a selected t h r e s h o l d . The emergent curves indicate t h e p o s i t i o n o f r e p e l l i n g LCSs at t i m e to.

To f u r t h e r examine the relevance o f these structures w i t h t h e stable and unstable m a n i f o l d s o f h y p e r b o l i c fixed points arising i n t h e c o n t e x t o f s u r f - r i d i n g i n regular waves, w e c o n v e r t (11) t o a u t o n o m o u s f o r m by considering the f o l l o w i n g t r a n s f o r m a t i o n s ,

X i = x ' / - f - C t , X 2 = X ^ ' + C (12)

In the above, x^" and x^^ is the l o n g i t u d i n a l p o s i t i o n and velocity, respectively, o f t h e ship w i t h respect to a f r a m e located at a w a v e crest, translating w i t h the wave celerity c.

A p p l y i n g expressions (12) to system (11) w e o b t a i n , a f t e r rearranging, the f o l l o w i n g set o f equations.

x'^ = ( m - X , - , ) - ' { g ( c , n)+f sin [ / < +ef]

+ [ r , n - r , H - 2 c ( T 2 - r 2 ) - 3 r 3 c 2 ] x ^ ^ - K ( r 2 - r 2 - 3 r 3 C ) ( x 5 ' ) ' - r 3 ( x ^ " ) ' }

( 1 3 )

w h e r e ,

g(c, n) = T o n ^ + ( r i n - r i ) c - i - ( T 2 - r 2 ) c 2 - r 3 c 3 (14) It can be seen that, system (13) does n o t depend explicitly o n t i m e . Stationary solutions can be obtained by setting the r i g h t h a n d side to be equal to zero and solving w i t h respect to x ' / and x ^ . I n

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xl |in 10''21 x l (m 10^2]

0.0 0.2 0.4 0.6 0.8 1.0

Fig. 2. (i) and (ii) Manifolds of hyperbolic (saddle-type) points of (13). (iii) and (iv) LCSs of (11) revealed via forward (red) and backward (blue) FTLE fields. Wave steepness values, s = 0.02 |(i) and (iii)|, s = 0.04 [(ii) and (iv)]. (For interpretation of the references to color in this figure, the reader is referred to the web version of this article.)

Fig. 2(i) and (ii), a n u m b e r o f saddle points are identified and the unstable/stable manifolds are " g r o w n " b y integrating perturbed ( w i t h respect to the f i x e d points and along the eigendirections) initial conditions, f o r w a r d / b a c k w a r d i n t i m e , respectively. Wave length and n o m i n a l speed are set to X = L and Unom = 1 2 . 5 m / s respecdvely. Fig. 2(1) and ( i i ) correspond t o wave steepness values s = 0.02 and s = 0.04, respectively.

We, consequently, consider system (11) and calculate, f o r the same settings, the f o r w a r d ( r = 420 s) and bacl<ward ( r = - 240 s) FTLE f i e l d , f o r to = 0 s. LCSs are i d e n t i f i e d as i n the case o f Fig. l ( i i i ) . Results are presented i n Fig. 2 ( i i i ) and ( i v ) ; r e d (blue) lines corre-spond to repelling ( a t t r a c t i n g ) LCSs. W e notice t h a t the

arrangement o f the structures revealed is, substantially, identical to the arrangement o f manifolds integrated f r o m the saddle points. The o n l y difference is that the f o r m e r are translating w i t h the wave celerity (this difference w o u l d become visible i n the pictures i f w e m o n i t o r e d LCSs e v o l u t i o n i n t i m e , rather t h a n observing t h e i r arrangement f o r one t i m e instant). This is o w e d to the fact that, the t w o calculation schemes w e r e applied w i t h respect to d i f f e r e n t reference frames [as recalled, system (11) is expressed w i t h respect to an E a r t h - f i x e d f r a m e ) .

4.2. Multi-frequency wave excitation

We n o w i n t r o d u c e a second wave c o m p o n e n t i.e., system (10) is considered w i t h N = 2. The length a n d steepness o f the reference wave are set to ; i i = L and S i = 0.035. The parameters o f the second wave c o m p o n e n t are fixed, such that, 0)2/co^ = 0.9 and S 2 / S 1 = 0.3. N o m i n a l speed is set to iinom = 1 2 m / s . Instantaneous FTLE fields are calculated at 50 and 76 s (Fig. 3). As noticed, t h e LCSs seem t o persist.

The same procedure is repeated f o r the case o f a JONSWAP spectrum w i t h a peak period and significant height o f Tp = 9.93 s and H s = 4.5 m , respectively. A frequency range o f ( « ( o , v . ® f i / g ( . ) =

(0.4,1.27) is considered and a 6 9 - c o m p o n e n t w a v e is produced (frequency values i n rad/s). N o m i n a l speed is again set to u„om ^ 1 2 m / s . Results are displayed in Fig. 4 . As i t can be seen, the i d e n t i f i e d structures appear to be f a i r l y complicated.

R e t u r n i n g to t h e b i - c h r o m a t i c scenario, w e attempt, this t i m e , to ascertain the o r g a n i z i n g role o f t h e i d e n t i f i e d LCSs o n the t i m e -v a r y i n g phase-flow. W e set 1^ =L, S i = 0 . 0 2 , Ö ; 2 / ' » I = 0 . 7 , S 2 / S 1 =

0.3 and u„om = 1 2 m / s . I n Fig. 5, a parcel o f particles is integrated, these particles corresponding to d i f f e r e n t i n i t i a l conditions o f the ship. The e v o l u t i o n o f the parcel u n d e r the flow reveals d i f f e r e n t " l o n g - t e r m " behaviour o f particle trajectories, as the parcel, after some t i m e , splits i n t w o sub-parcels. Some particles seem t o respond i n a surging-like manner [Fig. 5(iv), g r o u p o f particles o n the l e f t ] w h i l e others seem to be engaged to s u r f - r i d i n g [Fig. 5(iv), r i g h t p a r t ] .

W e keep the same setting and calculate the FTLE field at the t i m e instant t = 258 s, o n a d o m a i n c o n t a i n i n g the previous parcel of i n i t i a l conditions. As revealed, responsible f o r the situation depicted above are the r e p e l l i n g LCSs o f the phase flow. Specifi-cally, the r e p e l l i n g LCS associating w i t h the h y p e r b o l i c trajectory passing near (7.7,15) at time t = 258 s [Fig. 6(1) a n d ( v ) | , acts as a separatrix b e t w e e n regions o f the flow w i t h d i s t i n c t dynamics. I n

xl [m 10'^2]

0.0 0.2 0.4 0.6

Fig. 3. Bi-chromatic excitation; attracting (blue) and repelling (red) LCSs. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

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(. Kontolefas, K.J. Spyrou / Ocean Engineering 120 (2016) 339-345 3 4 3

0.0 0.2 0.4 0.6 0.8 1.0

Fig. 4 . Attracting (blue) and repelling (red) LCSs for the case of a JONSWAP spectrum (69 wave components). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

m o v i n g along paths that "resemble" to solutions o f (10), seem to m a r k regions i n the extended phase-space w h e r e ensembles o f trajectories are engaged to s u r f - r i d i n g .

In Fig. 7, a s i m u l a t i o n c o r r e s p o n d i n g to the a f o r e m e n t i o n e d scenario can be seen (black line). Distance is measured f r o m amidships (x-axis) w h i l e ii-axis refers to velocity as measured by a n on-shore observer. Three critical points o f the acceleration f i e l d have been detected at a r o u n d t = 240 s (these have been selected as they are related to the calculated trajectory; one could f i n d m o r e critical points at d i f f e r e n t space-time intervals). Their paths (blue and red lines) have been c o m p u t e d using the FFF m e t h o d . I n t h e same figure, w e have i n c l u d e d sections d e p i c t i n g LCSs t h a t have been i d e n t i f i e d on phase-space w i n d o w s a r o u n d the ship at selected t i m e instants. There seems to be a strong correspondence between the paths o f t w o critical points (denoted w i t h red lines) and hyperbolic trajectories revealed via t h e FTLE fields. The t h i r d critical p o i n t , o n the o t h e r hand, appears near the core o f an attracting LCS, i n a region o f the phase f l o w w h e r e a s u r f - r i d i n g state can be revealed. It is noted that, f o r the considered arrangement, this w o u l d be a periodic t r a j e c t o r y w i t h an a t t r a c t i n g character.

5. C o n c l u d i n g r e m a r k s

M o d e r n m e t h o d s f o r g a i n i n g i n s i g h t i n t o the dynamics o f ship surge m o t i o n i n astern m u l t i - c h r o m a t i c seas are i n t r o d u c e d . Spe-cifically, an i d e n t i f i c a t i o n m e t h o d f o r hyperbolic Lagrangian Coherent Structures (LCSs) is applied o n the phase flow d e f i n e d by the surge equation o f m o t i o n . I t is based o n a w e l l - k n o w n calcu-l a t i o n scheme w i t h a w i d e range o f appcalcu-lications i n t h e calcu-literature, w h i c h involves the calculation o f the spatial d i s t r i b u t i o n o f the largest finite-time Lyapunov exponent (FTLE). T h r o u g h the FTLE field, LCSs, i.e., i n f l u e n t i a l m a t e r i a l lines shaping the p a t t e r n o f the t i m e - d e p e n d e n t flow, w e r e obtained. T h e i r role as p h a s e - f l o w o r g a n i z i n g structures was e x a m i n e d . It was f o u n d that, f o r the case o f a b i - c h r o m a t i c scenario, the i d e n t i f i c a t i o n o f h y p e r b o l i c LCSs can explain the e v o l u t i o n o f ensembles o f i n i t i a l c o n d i t i o n s , by p r o v i d i n g the location o f t r a n s p o r t barriers, as w e l l as the final destinations o f particle trajectories.

Furthermore, the Feature Flow Field m e t h o d was i m p l e m e n t e d f o r the t r a c k i n g o f features, c o r r e s p o n d i n g to elements o f the zero set o f the acceleration field d e f i n e d b y the surge equation o f

0.5 0.7 0.9 2^ 0.9 I . I 1.3 1 = 258. s t = 283.s 20 . 20 M. -15 s g SI 10 10 (i) (ii) 5 5 xl | m l 0 ^ 3 | x l |ml0'>3| 1.7 1.9 2.1 3.25 3.5 3.75 4 t = 333.s l = 458.s - 20 - 20 (iii) (iv) 5 5 x l Im 10"3| x l \m \0^3\

Fig. 5. Bi-chromatic excitation: Integration of a dense patch of initial conditions (black) reveals qualitatively different "long-term" behaviour of particle trajectories.

fact, particles travel along this r e p e l l i n g structure towards the h y p e r b o l i c trajectory, w h e r e t h e y are redirected towards d i f f e r e n t branches o f the a t t r a c t i n g LCSs correlating w i t h the same trajec-t o r y [Fig. 6 ( i i ) - ( i v ) and ( v i ) - ( v i i i ) ] .

Lastly, the eariier described "feature" t r a c k i n g scheme is i m p l e m e n t e d , f o r a b i - c h r o m a t i c wave scenario. The frequency and steepness ratio o f t h e t w o wave components are set to co2/co\ = 0.93 and S 2 / S 1 = 0 . 4 5 , respectively. The reference wave has been chosen such that X-[=L and si = 1 / 3 5 , w h i l e n o m i n a l speed is set to u„om = 1 2 m / s .

W e d i f f e r e n t i a t e (10) w i t h respect to time to o b t a i n t h e accel-e r a t i o n fiaccel-eld a = ( x i , X 2 ) - thaccel-e usaccel-e o f this t accel-e r m is j u s t i f i accel-e d f r o m thaccel-e fact t h a t one can i n t e r p r e t (10) as a velocity field o n the phase plane. Our objective is to track critical points o f a i.e., points w h e r e the acceleration vector vanishes. I n Spyrou et al. (2014b) i t has been c o n j e c t u r e d t h a t certain critical points o f this field correlate w i t h s u r f - r i d i n g events. Furthermore, the critical points o f a,

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344 ;. Kontolefas. IQ. Spyrou / Ocean Engineering 120 (2016) 339-345

0.0 0.2 0.4 0,6 0.8 1.0

Fig. 6. Same setting as in Fig. 5. Advection of two adjacent pfiase-particle parcels, (i)-(iv) and (v)-(viii) Integration of 65,000 and 85,000 (approx.) initial conditions (shown in black), respectively.

m o t i o n . Results obtained f r o m the t r a c k i n g o f such features and t h e LCS i d e n t i f i c a t i o n procedure w e r e c o m b i n e d . I t has been s h o w n that the paths o f certain features correlate to hyperbolic trajectories o f the surge equation; w h i l e others, to trajectories w i t h a t t r a c t i n g character that seem t o evolve i n the core o f specific branches o f a t t r a c t i n g LCSs.

It is believed that, the application o f the above methods can lead to enhanced understanding o f s u r f - r i d i n g and broaching-to behaviour o f ships i n irregular seas.

Aclmowledgements

The calculation and t e s t i n g o f LCSs f o r surging and s u r f - r i d i n g behaviour described i n this paper have been f u n d e d by the Greek General Secretariat o f Research and Technology (Greek M i n i s t r y o f Education, Research and Religious A f f a i r s ) - project category Excellence ("Aristeia") - 1, p r o j e c t t i t l e HOMSHIP (ID code 252), contract reference n u m b e r GSRT-252. The w o r k regarding the feature t r a c k i n g m e t h o d applied i n Section 4 has been f u n d e d by

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/. Kontolefas. K.]. Spyrou / Ocean Engineering 120 (2016) 339-345 345

the Office o f Naval Research, ONRG grant n u m b e r N62909-13-1-7, under Dr. Ki-Han K i m , Dn Tom Fu and D r W o e i - M i n Lin.

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