HYDROELASTICITY OF QUARTER-INFINITE PLATE ON WATER OF FINITE DEPTH

Transcription

1 HYDOELASTIITY OF QUATE-INFINITE LATE ON WATE OF FINITE DETH Alex I. Adriaov, Aad J. Hermas Delft Uiversit of Techolog Facult Iformatio Techolog ad Sstems Departmet of Applied Mathematical Aalsis Mekelweg 4, 2628 D Delft, The Netherlads A.Adriaov@ITS.TUDelft.Nl, A.J.Hermas@ITS.TUDelft.Nl Summar Diffractio of surface waves b a ver large floatig platform is studied for the case of fiite water depth. I our paper large platform is the flexible thi quarter-ifiite plate. usig of ra method we describe the propagatio of wave modes iside of the platform area sequetiall. Kewords: diffractio of surface waves, VLF, deflectio, wave modes, ra method, itegral-differetial equatio. 1 Itroductio ecetl the problems of the behavior of floatig flexible thi plates o waves obtaied great iterest. This sectio of hdroelasticit is importat ad well studied b umerical approaches while remai some difficulties with aaltical ivestigatio especiall for the case of fiite depth. The thickess of the ver large floatig platform (VLF) compared to horizotal parameters is small ad the are modeled as thi elastic plates. The dimesios of these objects ca be i order of 5 i legth ad 1 i width while thickess is aroud 1. I HEMANS [1], [2], ANDIANOV ad HEMANS [3], [4] solutios for the deflectio ad the trasmissio ad reflectio coefficiets are obtaied for ifiite, fiite ad shallow water depth for such forms of the platform: the strip of the ifiite legth ad the semi-ifiite plate. The results were preseted at the previous IWWWF i [3]. Here is theor developed further for the quarter-ifiite plate. HEMANS [1] derived a exact itegral-differetial equatio for the deflectio of a VLF at deep water. The equatio was solved umericall b meas of a boudar elemet method ad a mode expasio. Later HEMANS [2] used this formulatio to derive boudar coditios to appl the ra method for short wave diffractio. Some other approaches ca be used for the solutio of this class of problems, for istace parabolic approximatio method, see TAKAI [5], OHKUSU ad NAMA [6]. Here we stud the diffractio of surface waves b VLF i the form of quarter-ifiite plate (QI) floatig o the surface of a fluid of fiite depth. It is reasoable to split up the problem o two cases: the first with oblique icidet waves ad the secod with perpedicular waves. We use a ra method for the solutio which cosists of 3 parts. The platform is idealized as a plate with elastic properties of zero thickess. I this paper we igore the effect of corer poit. A aaltical stud is preseted for the deflectio i all parts of the solutio. The first part of the solutio is based o the approach ([2], [3]) for the problem of a semi-ifiite platform. There a itegral-differetial equatio, a ree s fuctio, boudar ad edge coditios are used. Oe travelig ad some evaescet wave modes are cosidered. A specialt of preset approach is i the cosideratio of the propagatio of travelig mode (mai ra) of the solutio ad calculatio of its reflectio o other edge of the platform (as QI is cosidered, platform has 2 edges perpedicular each other). Later a special matchig coditio itroduced alog the lie which split up the platform area o part where the ier reflectio is exist ad where is ot. This coditio is valid i all area of the plate. 2 Formulatio of the roblem We cosider a floatig flexible thi plate which covers a quarter (x, ) of the surface of a ideal icompressible fluid of depth h. z is the positive upwards coordiate. We assume waves i still water ad itroduce the velocit potetial Φ x z t V x z t where Φ x z t is a solutio of the Laplace equatio Φ (1) i the fluid (z ) together with the coditio at the bottom (z h) Φ z ad surface coditios at z Φ z w Φ, x ; t z 1 g Φ, x (2) t 2 where w x z t deotes the free surface elevatio uder the platform, is the ope fluid area ( x x ) ad is the platform area. The dividig surfaces are defied as (x ) ad as x ( x ). The fluid regio where the icidet field ad reflected waves from coexist is defied as 1 ad the regio i which icidet waves trasmittig x as 2. Icomig short waves propagate from the ope fluid i the directio which makes a agle to x axis (it is show i figure 1). The platform draft d is shallow ad the platform is assumed to be a thi laer at the free surface. VLF is modeled the as a elastic plate with zero thickess. To describe the deflectio of the platform w we appl the thi plate theor, which leads to a differetial equatio of the vertical

2 111 Sfrag replacemets ic. wave!! "" ## $ $$ %%% & && ' '' ((( ))) ** +++,, /// QI :: ;; << == AA Figure 1: eometr of the problem displacemet of the platform: D x D w m 2 w t 2 (3) at z i the platform area, where m is the mass of uit area of the platform, D is its equivalet flexural rigidit, x z t is the liearized pressure ρ Φ t ρgw, (4) here ρ is the desit of the water. After applig the operator E t to (3) ad usig (2) ad (4) we arrive at the followig equatio for Φ at z : FH x D µ 2 t 2 1 I Φ z 1 g x Φ, (5) t 2 where parameters are DE ρg ad µ me ρg. The edges of the platform are free of momet ad shear force, the edge coditios at (x ) are: w x 2 ad at x ( ): w 2 ν 2 w 2 ; 3 w x 3 2 ν ν 2 w x 2 ; 3 w 3 2 ν 3 w x 2 (6) 3 w, (7) x2 where ν is oisso s ratio. The harmoic wave ca be writte i the form Φ x z t J φ x z iωt. The we reduce time-depedece ad cosider waves of a sigle frequec ω ad obtai at z : where K φ ic x D µ φ 1 D Kφ (8) z ω 2 E g. For fiite water icidet waves equals coshk z h coshk h ga iω eik L xcosm sin, (9) here A is the wave height ad the wave umber k obes the water dispersio relatio k tahk h K. Legth of icomig waves is λ 2πE k. 3 Solutio (latform Area) We cosider the case of oblique waves icomig from the field (x ) with agle of icidece, πe 2. As the effect of corer poit (x ) is ot cosidered, we ma use ra method for this geometr of icidet field. A itegral-differetial formulatio derived i HE- MANS [2] ad ANDIANOV ad HEMANS [4] for the case of fiite depth. The deflectio of the platform due to the propagatio of mai wave modes represeted as a superpositio of expoetial fuctio i the followig form w 1 x O a e iκ xm ik si, (1) where a are the amplitudes of wave modes ad κ are reduced wave umbers. w 1 is the largest part of the deflectio correspoded to the situatio whe ras trasmit but ot reach x et. Wave fuctios κ defied as κ 2 κl N 2 k 2 si 2, (11) here κl N are roots of the dispersio relatio κ 4 µ 1Q κtahκh k. (12) Number N 1 deotes the umber of roots takig ito accout (oe real ad N imagiar) lead us to fidig N 3 ukow amplitudes a. We itroduce the ree s fuctio after splittig up the fluid domai ad usig the thi plate theor for the determiatio of w 1. If the itegral-differetial formulatio, the surface coditios ad ree s theorem are used to the potetials i ad respectivel, we obtai such itegraldifferetial equatio: K 4π S x D µ 1 D w 1 x O ξ 2 2 η 2 D µ D w 1 ξ η UTV x ;ξ η dξdη Ae ik L xcosm sin, (13) ree s fuctio has the followig form: TV x ;ξ η WẌ 2 YJZ k coshkh k sihkh K coshkh J kr dk (14) at z, where []\ is cotour of itegratio i the complex k- Sfrag plae replacemets from to udereath the sigularit k k, chose for fulfillig the radiatio coditio, J kr - essel s fuctio ad r 2 ^ x ξ 2 η 2. k [ \ Figure 2: otour of itegratio We cosider the zeros of dispersio relatio for water surface ad the replacig of (1) ito (13) result i a _ κl N 4 µq κ k cos cos A. (15) k

3 «ªª ³²± _ žž gg kk lllmmm ss ww Here k K K 1 Kh k 2 hq. To complete the sstem itroducig of these coditios, the costructio of the of N 3 equatios, two equatios ma be obtaied from solutio for whole area of the platform will be completed. edge coditios (6). Now amplitude fuctios a ad fuctio Istead of the fuctio w 2 will used ew deflectio fuc- w 1 ca be received. tio, which exist i both zoes. We ivestigate the propagatio of mai ra further. After For the case of oblique waves we use a straight-forward the reflectio of travelig wave mode (mai ra) o x ra method. For a perpedicular icidet field OHKUSU with a agle θ (that is show i figure 3) the QI get the ad NAMA [6] obtaied a solutio for platform which floats added deflectio w 2 geerated b the vibratio of the plate o shallow water b usig a parabolic approximatio ad edge x: matchig of differet zoes. w 2 x O a ǹ e iκa M iκ1x We use the geeral formulatio for the diffractio of, (16) waves b QI ad appl the ra method. The solutio of the where a are the amplitudes after the reflectio at Laplace equatio (1) is Φ ǹ x ad x t ad the potetial of udisturbed icidet wave φ correspoded wave umbers are 2 κ` κl N 2 κ 2 1. this is give b (9). wa κ`1 k si ad the largest term of reflected travelig wave is a`1 ik sim iκ 1 x Œ Ž. I mathematical pla a ǹ š š Ÿ Ÿ Sfrag replacemets œ œ are the solutios of the sstem depedig from a 1 κ 1 ad cosistig from itegral-differetial ad edge equatios at ic. wave QI. hhi ppq ffg jjk ooopp ttu xx x\ rrs vvw \ Sfrag replacemets bbc dde zz{ QI } θ θ ~~ µ µ µ µ µ µ µ µ µ µ µ µ µ µ µ µ µ x ic. wave ƒ Figure 4: eometr of curret aalsis θ ˆˆ x Let us cosider the ew sstem of coordiates x\ \ which is obtaied b rotatio agle πe 2 θ from x ad x\ is the Figure 3: eflectio of travelig wave mode (mai ra) at x-axis I geeral, the deflectio of the QI ca be writte i the followig form: w x O w 1 x w 2 x (17) where w 1 x ad w 2 x were described alread. The reflected part of the deflectio w 2 exists i the regio xe taθ. The agle of the ra reflectio θ ma be obtaied from taθ k E κ 1. To fid w 2 we appl a aalsis similar to the determiatio of w 1. Itegral-differetial formulatio leads us to the equatio for a ǹ : a ǹ _ κl N 4 µq κ k cosθ cosθ a 1 κ 1 Š (18) with the two edge coditios (7). Now deflectio fuctios w 1 ad w 2 are kow. We obtai the followig itermediate results for total deflectio w x i platform zoes: i 1 (x taθ) w w 1 ad i 2 (x taθ) w w 1 w 2. I ext sectio the preset approach will be improved b the itroducig a special matchig coditio alog the border betwee these zoes ad we will fid a ew part of total deflectio. 4 Matchig oditio Here we derive the matchig coditios alog the border x taθ which split up the platform area i 1 ad 2. border betwee 1 ad 2 ad its directio is coicidig with directio of ras after reflectio at. We assume that the potetial udereath the platform ca be writte as a superpositio of ra-mode solutios φ x O ϕ x e iκxm κz (19) where ϕ x is the amplitude fuctio of the th mode. We skip the primes. The deflectio is represeted b w s x O ia ωg φ z x (2) We assume that our approximatio is valid i whole area of the platform. Isertio of (19) to Laplace equatio (1) gives: ϕ 2κ iϕ x ϕ z º¹ K» (21) We write ` K 1¼ 2, κ Kr ad elimiate the z-derivative b the followig expressio ϕ z iϕ x ϕ a a 2r (22) Isertio of (19) ito (8) lead us after droppig the asterisk to: ad ¹ K : ¹ K : r 4 µ 1Q r 1 (23) r 4 µ 1QJ½ iϕ x ϕ 2r D 2r 3 ϕ 4ir 4 ϕ x Q¾ (24)

4 À (23) is the dispersio relatio at z. Now partial differetial equatio 2irϕ x ϕ (25) have to be cosidered. Two terms are equal i order of magitude whe x ¹ 1 ad ¹ K 1¼ 2. Use of Laplace trasform ψ s us to the followig equatio: ϕ x sx dx lead 2irsψ ψ 2irϕ (26) ad we require the iitial coditios, see MEI ad TUK [7], which differs with chage of sig of : ϕ Š A s, ; (27) ϕ O, (28) Value of the costat A s ma be obtaied from the secod part of our solutio b usig of the results for w 2. From geeral solutio of (26) we arrive to the equatios: ψk s O α 1 s eá 2irs A s s (29) ψ M s O 2 s Á 2irs (3) with special otatio of the amplitude fuctio ad its trasforms: ϕk ad ψk i regio ad ϕ M ad ψ M i regio. We ca determie the costats: α 1  2 A s E 2s. Applig of Laplace iverse trasform ad usig of trasitio coditios ϕ M ϕk ad ϕ M ϕk at lead us to the followig formulas for the amplitude fuctio: ϕk x»ẍ ϕ M x» A s à π A s à π λ 2 dλ λ 2 dλ Ä 2ir 2Ä x Ä 2ir 2Ä x λ 2 dλ D A s (31) λ 2 dλ D (32) After the calculatios of the itegral values of the potetial i the platform area ad respectivel of the deflectio w s ma be determied too. efereces [1] HEMANS A.J. (21). A eometrical-optics Approach for the Deflectio of a Floatig Flexible latform. roceedigs of the 16th Iteratioal Workshop o Water Waves ad Floatig odies, Hiroshima, Japa, pp [2] HEMANS A.J. (23). The a Method for the Deflectio of a Floatig Flexible latform i Short Waves. Joural of Fluids ad Structures, i press. [3] ANDIANOV A.I. ad HEMANS A.J. (22). A VLF o Ifiite, Fiite ad Shallow Water. roceedigs of the 17th Iteratioal Workshop o Water Waves ad Floatig odies, ambridge, Eglad, pp.1-4. [4] ANDIANOV A.I. ad HEMANS A.J. (23). The Ifluece of Water Depth o the Hdroelastic espose of a Ver Large Floatig latform. Marie Structures, i press. [5] TAKAI K. (21). arabolic Approximatio of the Hdro-Elastic ehavior of a Ver Large Floatig Structure i Oblique Waves. roceedigs of the 16th Iteratioal Workshop o Water Waves ad Floatig odies, Hiroshima, Japa, pp [6] OHKUSU M. ad NAMA Y. (23). Hdroelastic Aalsis of a Large Floatig Structure. Joural of Fluids ad Structures, i press. [7] MEI.. ad TUK E.O. (198). Forward Scatterig b Log Thi odies. SIAM Joural of Applied Mathematics, vol.39, o.3, pp [8] STOKE J.J. (1957). Water Waves. Itersciece ublishers, New York, 567p. [9] WEHAUSEN J.V. ad LAITONE E.V. (196). Surface Waves, Ecclopedia of hsics, vol.9, Spriger-Verlag, erli, pp esults & oclusios Fiall, the total deflectio of the quarter-ifiite platform is writte as the followig sum: w x W w 1 w s (33) i the whole area of the platform, where the first term is the mai part of the solutio ad the secod part represets the solutio alog the ras b stretchig the coordiates. This approach ad aalsis of the behavior of VLF will be developed further. The details ad obtaied results will be preseted ad discussed at the Workshop. Later we will exted the preseted approach to the case whe icidet waves propagate perpedicularl to the platform ad to the case whe icidet waves propagate from ( ). The same aalsis we are goig to exted to the problem for two other forms of the platform: the strip of semi-ifiite legth ad plate of fiite sizes. The latter is a more realistic case ad it is the mai goal of our stud.

5 18th IWWF Le roisic (Frace) 6-9 April 23 Questio b : T. Miloh Is it true that our solutio is valid ol awa from the origi ad is it possible to obtai a local solutio at the corer? Author s repl: Yes, ideed our solutio is ot valid ear the corer of the plate. Yes, it seems possible to obtai a solutio at the corer poit ad i shadow regio geerated b the presece of the corer. Hopefull, o the ext Workshop we will show these results. Questio b : M. Ohkuhu A problem ou treated with is almost the same as those studied i the papers of Takagi (22) ad Ohkusu (23) ou referred. It seems though more geeral, is somehow mathematicall related to the oes of those papers. I woder if ou would give a descriptio of the relatioship of them. Author s repl: Yes, i all three papers the quarter-ifiite plate is cosidered. ut i our paper, the geeral case - case of fiite depth is studied. The, b takig the limits, we ca solve the problems for ifiite ad shallow water, which is cosidered i the referred papers. The results we preseted are valid for restricted values of the agle of icidece. I future we will exted our approach to urestricted icidet agle. Questio b : D.V Evas Is our theor a ra theor, ol valid for short waves? Author s repl: I the derivatio of the direct ad reflected wave fields we have ot used small values of the wavelegth explicitl. However, to obtai smooth solutio we itroduced a coordiate stretchig to ed up with a parabolic equatio. So our approach is valid ol for short waves.