Source: Physics of Fluids Vol 6 No pp 6-64 4 DOI: 6/64445 IMPACT OF AN OIQUE REAKING WAVE ON A WA Jian-Jun SHU School of Mechanical & Aerospace Engineering Nanyang Technological Universiy 5 Nanyang Avenue Singapore 69798 E-mail: mjjshu@nuedusg Absrac The inenion of his paper is o sudy impac force of an oblique-angled slamming ave acing on a rigid all In he presen sudy he analyical approach is pursued based on a echnique proposed by he auhor A nonlinear heory in he conex of poenial flo is presened for deermining accuraely he free-surface profiles immediaely afer an oblique breaking ave impingemen on he rigid verical all ha suddenly sars from res The small-ime expansion is aken as far as necessary o include he acceleraing effec The analyical soluions for he free-surface elevaion are derived up o he hird order The resuls derived in his paper are of paricular ineres o he marine and offshore engineering indusries hich ill find he informaion useful for he design of ships coasal and offshore Keyords: Oblique plunging ave impulsive pressure Inroducion One of he mos devasaing forces of naure is ha of breaking aves The desrucive force of breaking aves is economically and physically derimenal and faal Hence a considerable amoun of research has been devoed o he sudy of he impac forces of breaking aves paricularly ha of breaking aves impacing on a rigid all hich is suddenly sared from res and made o move oards a fluid aper Such sudies can yield useful resuls ha ould benefi designers of dams ships oil-rigs and oher coasal and offshore srucures hich are direcly subjeced o he impac forces of breaking aves When a breaking ave srikes a all he impac produced is of shor duraion bu considerable inensiy This direc collision generaes an impulsive pressure on he all hich is similar o he problem of iniial-sage aer impac Hoever exising ave heories based on small and finie ampliude assumpions canno accuraely model he breaking ave force on a all due o he highly nonlinear and ransien naure of he problem In revieing he previous sudies one of he mos imporan and unresolved quesions is ho he iniial sage of he breaking ave impingemen on he all can be properly characerized and simulaed Cumberbach [] considered he case of symmeric normal impac of a aer edge on a all and Zhang e al [] ook i furher by sudying an oblique impac These o orks assumed prescribed funcions of he free surface profiles close o he all: in Cumberbach [] a linear funcion as assumed hile in Zhang e al [] an exponenial funcion as assumed These o orks ere semmed from an ad hoc assumpion on he free-surface profiles close o he all In Shu & Chang [] an analyical approach as aken o solve he breaking ave problem for a normal ave ihou prescribed funcions I has been found ha he free-surface profile close o all is neiher linear in Cumberbach's assumpion [] nor exponenial in Zhang e al's assumpion [] This paper aims o ake he same analyical approach bu insead of a normal ave e shall derive and solve he impac problem due o an oblique angled ave
In he presen sudy e do no assume any prescribed funcions for he free surface profiles Effecs of graviy viscosiy and surface ension can be negleced since ineria forces are dominan during he small-ime impac process The essenial mechanism involved in he impac process can be described by he heoreical reamen of poenial flo A small porion of he breaker ip is iniially cu off o produce a finie eed area on he all and a high spike in he consequen impac resuls from an acceleraion of aer oards he all We are ineresed in he shor ime successive riggering of he non-linear effecs using a small-ime expansion of he full non-linear iniial/boundary value problem The leading small-ime expansion is aken o include he acceleraing effec Governing equaions We consider a rigid horizonal all being suddenly sared from res and made o move verically ih a cos oards a consan acceleraion a o-dimensional fluid aper ih semi-angle ( / A definiion skech of he flo is shon in Fig The axis of he fluid aper is a an angle ( / o he verical / e us nondimensionalize ime by ( / a disance ( x y by velociy ( u v by / ( a pressure p by a here is he righ-side eed all semi-lengh hen he breaking ave jus ouches he all a ime and is he densiy of he fluid A mahemaical saemen of he above problem can no be rien as u v p y on x yan ( u v p y on x ( y yan ( ( y (5 here he lef-side eed all semi-lengh can be expressed in he erms of angles and as follos: cos cos (6 On he all surfaces he normal velociy of fluid paricles mus be he same as ha of he all a all ime v a on y a / (7 The pressure vanishes a infiniy p as y (8 Mahemaical analysis The full nonlinear iniial/boundary value problem consiss of equaions (-( ih condiions (4-(8 These equaions are solved analyically by employing a small-ime expansion We assume ha u v x y u u u p u v x y x v v v p u v x y y ( ( ( u( y u ( y O( v( y v ( y O( p( y p( y O( ( y ( y ( y ( y (9 ( O( O( ( For negaive ime everyhing is a res u v for (4 here and are he free surface ``elevaions'' in he x direcion beyond he undisurbed surface On he surfaces he kinemaic and dynamic boundary condiions require The leading-order equaions are u v p u x y x subjec o he condiions p v y (
u u p on x yan ( on p x yan ( ( (4 v a on y (5 p as y (6 I is clear ha pressure p saisfies he aplace equaion p x p y (7 Inroducing a complex-conjugae funcion q ih respec o p e can consruc an analyic funcion f( z p iq z x iy (8 As shon in Fig he conformal mapping ( z ( d (9 given by he Scharz-Chrisoffel ransformaion maps he upper half of he plane ( i ono he region occupied by he fluid Here is he incomplee ea funcion defined by ( x x ( d ( ( Funcion f is also analyic in he ransformed variable On he free surfaces hich correspond o and p on he posiive real axis vanishes On he all surface hich corresponds o he line segmen e ake p / n a plane e have Therefore along he real axis in he Re( on f ( f Re a on n ( Re( f on (4 If s ( measures he disance from poin C in Fig o any poin on he all surface he Cauchy- Riemann equaions give Re( on f (5 Im( f a on (6 Re( f on (7 here he disance s ( is given by (9 as ( ( d on If e inroduce a ne analyic funcion g ( by g ( (8 / ( / f ( (9 he boundary condiions for g ( are unmixed Im( g on ( / / Im( g a ( on ( Im( g on ( The analyic funcion g ( ha is regular in he upper half plane and vanishes a infiniy can be obained from he Scharz inegral formula
Im( g ( d g ( Subsiuing (9 -- ( ino ( e have a f ( / / ( d / ( ( / From boundary condiions ( o (6 e have Im( f / ( on Im( z / Im( f / ( on Im( z / Afer some mahemaical manipulaion e obain a ( ( sin ( ( ( ( ( ( a sin on on (4 (5 (6 (7 (8 Impac free-surface profiles for differen and are shon in Figs and 4 I has been found ha he free-surface profiles ( and ( y y / close o he all are proporional o y and / y respecively hich are neiher linear in Cumberbach's assumpion [] nor exponenial in Zhang e al's assumpion [] Conclusions The problems of oblique breaking ave impingemen on he all and he free-surface profiles have been solved analyically by using a small-ime expansion The resuls obained sho ha he free-surface profiles can be deermined if he angles and acceleraion of he oblique breaking ave are given The resuls of his paper agree ih he resuls of he case of a normal impac (Shu & Chang [] ih angle In addiion e have furher confirmed ha he free surface profile close o he all is neiher linear in Cumberbach s assumpion [] nor exponenial in Zhang e al s assumpion [] References [] Cumberbach E 96 The Impac of a Waer Wedge on a Wall Journal of Fluid Mechanics 7 pp 5-74 [] Zhang S Yue DKP and Tanizaa K 996 Simulaion of Plunging Wave Impac on a Verical Wall Journal of Fluid Mechanics 7 pp -54 [] Shu J-J and Chang AT Hydrodynamic Force on a Verical Wall due o reaking Waves Proceedings of he Tenh Inernaional Offshore and Polar Engineering Conference III pp 77-8 A A S a (y D S y (y C z - plane Fig : Fluid body in physical z -plane Re (fo = Im (fo = ao C - plane Re (fo = Fig : Physical z plane is conformally mapped ono he upper half of he plane C D x
4 5 6 7 8 9 Fig : Impac free-surface profile ( / a for / 6 5 5 5 Fig 4: Impac free-surface profile ( / a for / 6 / 4