WAVE EQUATION MODEL FOR SHIP WAVES IN BOUNDED SHALLOW WATER

109Joumal of HydrodynamicsE fer. BE& *2000£0+09- 119China Ocean Press£ Beijing - Printed in ChinaWAVE EQUATION MODEL FOR SHIP WAVES IN BOUNDEDSHALLOW WATERT.S.LeeMechanical & Production Enginering Departmentf-National University of Singapore£-Singapore119620Wu Jian- kangf Xiong Chuan- guangMechanics Department of Huazhong University of Science & Technology E-Wuhan 430074E-P. R.ChinaC. ShuMechanical & Production Engineering Department- rNational University of Singapore£- Singaporef Received Dec. 18E-1999£OABSTRACT2 Ships were modelled as moving pressure disturbances on the free surface of a shal-ow water basin in the present paper. The moving- pressure generating waves were subjected to thereflection of land boundaries and the radiation of open boundaries. This paper proposed and exam-ined a wave equation modef" WEME@o solve the shallow water equations with moving surface pres-sures simulating ship waves in a bounded shallow water region. 'The Galerkin finite element methodwas used to solve a second order wave equation for the free surface elevations and the hydrodynamicpressure of the ship botom simultaneously. Horizontal velocities were obtained from the momentumequations. Numerical solutions of Series 60 Cp = 0. 6 ships moving with the depth Froude numberof Fn=0. 6E1. 0£1. 3 in a rectangular shallow water harbor were investigated. Three dimensional surfaceelevation profiles and the depth- averaged horizontal velocities were analysed. The numerical resultscharacterised very well the ship waves in shallow water. Strong boundary reflection waves werefound in the case of high depth Froude numbe£" Fn = 1. 32@ Waves generated by the interactionsof two ships moving in the same directions and in the opposite directions were also numerically in-vestigated in the present study .KEY WORDS ship wavesE- shallowv water equationsE FEM1. INTRODUCTIONIn the recent yearsE- waves in coastal shallow water£ generated by marine traffic and intensifiedby port expansion£ interaction with structures and reflection of land boundariesf Hhas become a cru-cial factor affecting water environments and engineering operation. In comparison with the windwaves and ocean swellsE-waves inside harbor ex中国煤化工1 certain areas. Thedominant ship waves are hardly dissipated in then with shorelinesf andmay cause wave resonance. This accumulatedMYHCNMHGevere increase of waveheight which causes damage to port facilities£- rdegradation of harbor infrastructures£- and excessivedisturbance to moored and moving vessels . The ship waves in riverfHakes£ Testuaries and coastal ar-eas are of great importance to environments£- engineering applications as well as ships.In shallow water areas£- the typical water depth is usually less than ship length. The wavelengthof ship waves aallow water is longer than that in deep water at the same speed. In many cases110the order of magnitude of the wavelength may be the same as ship length. In such situations the shipwaves in shallow water may be treated as shallow water waves£-i.e. k:h < < 1 2 and studied byshallow water equations. The nonlinearityE- energy dissipation and free surface disturbance may besignificant to shallow water waves£ 7and explicitly appear in the shallow water equations. The numer-ical solution of the Navier-Stokes equations are gaining popular in hydrodynamics in recent years .The shallow water equations are the depth-averaged Navier Stokes equations in the cases that the wa-ter depth is much less than horizontal flow scale. The shallow water is often bounded by shorelinesand varying seabed. The shoreline reflection makes ship waves even more complicated. The fielddata and experimental results in such situations are rarely available. Numerical model is of great in-terest in practical applications . The objective of this paper is to develop a finite element model to in-vestigate the characteristics of ship waves in bounded shallow water. In the present work ships areconsidered as moving pressure disturbance on the free surface£- rwhich is explicitly involved in theshallow water equations . This paper focuses the attention on the moving pressure on the free surface2-which generates waves in shallow water .The continuity equation is the first-order hyperbolic differential equation. It is known that thenumerical solution of the first-order hyperbolic differential equation has long been a difficult problemin computational fluid dynamics field . The numerical solutions of shallow water equations are easilyspoiled by numerical oscillationS 262EY. The wave equation modelE" WEME-first developed byLynch & Gra&U1E ntransforms the continuity equation and momentum equations to a second-orderwave equation2- then solves the wave equation for shallow water solutions£- instead of directly solvingcoupled continuity equation and momentum equations. The wave equation model eliminates numeri-cal oscillation effectively and decouples the continuity equation and momentum equations. Wave e-quation model has been sucssfully applied to many tidal computationS- 324252. The wave equationmodel and the Galerkin finite element method are employed to solve shallow water equations for in-vestigating ship waves in bounded shallow water in this paper. The numerical solutions of three-di-mensional free surface profiles and horizontal velocities are presented .2. WAVE EQUATION MODEL TO SOLVE SHALLOW WATER EQUATION WITHMOVING PRESSUREThe depth- averaged shallow water equations are given bydζ+ ViE" HVEO= 0f"120JHV+ Vi@"HVVE@+ gH°ζ + P£O+ tHV= 0f"20.pgwhere ζ is the surface elevation relative to still water£-V f" ufrfQs the depth- averaged horizontalvelocities£-H = h + ζ is the total water depthE-h is the still water depth£f" xEyEfGs the movingpressure on water surface£p is the water density£ g is the gravity acceleration£ tτ is the floor frictioncofficientE-und V =f" i. j EQs horizonl中国煤化工The present work is mainly concerned withMYHCNMH(Jing ship waves in shal-low water. The following assumptions have been made in the momentum equationsE" 22@ Lateral mo-mentum transfer and friction between ship and water surface are negligible. Floor friction is approxi-mately linearized by a constant cofficient τ . The ratio of water depth to wavelength is smallfi.e.f-hh < < 1 Eand the frequency dispersion is ignored . Propulsive flows are also not considered in thepresent st万a方数据.111Eq2"”1E@and" 2f@are the continuity equation and horizontal momentum equations respectively.The proposed wave equation model first transforms the continuity equations and the momentum equa-tions to a second-order wave equation£- then solves the wave equation for surface elevation and themomentum equations for horizontal velocitiesE bseparately . Differentiating the continuity equatior£" 120with respect to time£ taking divergence of momentum equatiorf£" 2f国and combining these two equa-tions yield a second-order wave equation2些+ r今- vjUe" HVV2C+ gH°ζ +⊥咽Y= 0"320)tNow a new variable is defined&2C=ζ+pf"4E0wherep = p/pg . Substituting Eqf" 4EGnto Eqf" 3E国-the Eqf" 3EGs rewitten as£22CdC2p. dp"5202 +τdi- Vi@0Vi" HVVEO+ gHV CEY=Jt22 +iHtThe variable C represent the surface elevations at the free surface because of zero pressure. Ifthe water depths of ship bottom are knownf the hydrodynamic pressure can be found from the vari-able C . The pressure terms on the right side represent pressure disturbance . Both pressure positionof the free surface and pressure magnitude are changed while the ships moving. The effect of movingpressure on the free surface is dominanted by the change of pressure position. For a still ship thepressure disturbance terms on the right side are zero. A zero flux boundary condition is specified atclosed land boundariesf the first-order non-eflection conditionf nindicating out-going wavesEris im-posed at open boundariesfi.e. f-dζ鄂+VgH张.=0["6E⑥where s is the propagation direction of out- going waves£which is not the same as the outward normaldirection of open boundaries generally . The direction s is determined as followsdζ/Jxs =E"s。E-s,2国- s, =- sighassai9 Vζ 1sy =- sigh"8iVζ1720The ships are assumed to start moving from still water with zero surface elevation and the flowvelocities are set to zero initial conditions .中国煤化工.3. FINITE ELEMENT SOLUTIONS OF SIMHCNMHGTIONS.Applying the Galerkin finite element method to the wave equatiorf" 5ECand the momentum e-quatiorf" 2E@yields,2(i2φ折数据;φ i4i'V i80v i HVVEO+ gHV C室φ ;iμ112i M≌∞;i4n" τtφ ;iμf"820.Jl2.日HV日Ep ;iH+i'Vi@" HVVEQp ;iH g HV CEp ;i4i”τHVEψ iμ= 0"9fOwherep&" i = 1E-2E-i- NE@are the global interpolation functions£-N is the total number of dis-cretized nodes in horizontal domainf-undi ' fiEf2il_ fi f2 dx dy fis integral over the entire 2-D domain. By integrating by parts and making use of the momentum equation&" 2E国-EqE" 8EGre-duces toi°qZfφ;iμi°τ能φ;iμ i' hV CE7Vφ ;iμ=iφ;i4iτ只φ;iμ.aHV,-i’BEVφ ;iH一f。. dt+ tHV,2@ ;dsf"10⑥where B = Vi闵" HVVEQ gζV C is the nonlinear terms which are moved to the right side of the e-quation and explicitly treated. At the closed boundaries the boundary integral in the Eqf" 10E@van-ishes. The Eq2" 10E@s replaced by non-reflection boundary conditiorf" 6E@at the open boundaries .The time derivative of the wave equatiorf" 10EGs approximated by finite differences£2"EO 2"2@+1 _ 2"2@ +&"2@-1。2"20 f"2@+1 f"2@-1Jt2Or2f anddt20tf"11EOwherf" EGlenote C orp . The superscripts n + 1f rnEn - 1 denote time levels. The same definitionis throughout this paper. The non-time derivative terms are weighted over three time levelsEi.e.亡20= a&"E+1+ a屹"烟+ a&~2O-'E- with a1+ a2+ a3= 1.0f"120where anE-a2fa3≥0 are time weighting factors. Substituting~ 11£82: 12E@ntcC" 10f⑥one obtainsa matrix wave equation29E0A9Ccm+2y= fQfyf 1320and the matrix coffcients are given asA; ="1+台0φ Lrp ;i4 a1gOrf°hVφf" 14E0中国煤化工FYHCNMHGQ; =i°1←毕p°+12φ ;i4-2 p°£φ;i4"1-n pp"-lEφ ;iμ2++F^ C"E→φ:il_°"1-台0Cn-1Epil- Of' B"EVφ ;iμ .113- g0eUak' hVC°&Vφ ;i4 ai"hVC" -12Vφ ;iaYf 15E0All variables in the previous time leves n£-n - 1 are known. The pressure at the new time lev-el n + 1f7p"+1 is unknown. p"+1 may be approximated by p”. The position of p”at the surface hasbeen moved forward Ox . With boudary conditions the variable C can be solved from the wave equa-tiorf" 13f@ When variable Cn+1 is solved free surface elevations and ship bottom pressure can be si-multaneously solved. The hydrodynamic pressure of the ship bottom are given asp"+l= Cn+1- h,f"162◎where h, are the depths of ship bottomE which are known. p"+1 could be used to solve the wave e-quationf" 132@one more time to improve the accuracy of solution. In this study the change of ship po-sitionf such as drafE- pitch angleL- ure ignored. With solved variable C + 1 the horizontal velocitiesare solved from the momentum equatiorE" 92@ The final matrix equations are written asEUMSE° Hvf@+1&y= £Q,2yf"17E⑥EUMSLr" Hu&@+12y= fQ,&yf" 18E0τOt,M;=f"1 +0 P f£φjiμ["19E⑥E" Q.0="1-台国HuA@E~p ;iLi&".2”Hu2E@ A"~ Huu2@+52园p ;iμJxdy一+HGU周piμf"20f⑥2ar"Q.sO=f"1-兮细Hr2OErxp ;iui;- Hu2@+2 Ho2fOQ.p ;iμ吟r Hp+12C+tac"f"21E@ay+ H" gy 4φiμZero normal flux condition is imposed at the closed boundaries .4. NUMERICAL RESULTS AND DISCUSSIONSFig. 1 shows a rectangular harbour E“840mE@<臣~ 100m£Qvith constant water depthh = 10m. It is descretised into 7000 rectangular elements of sizE" Ox = 6mE-Dy = 2mEO A Series 60 Cp= 0.6 ship moves into the harbour entrance. Thethredlimrmins h. dta of Series 60 Cp =0.6 ship is given in RefEU9f2-where the ship len中国煤化工nd drat=2.5m. Thedepth Froude number Fn = U/N gh are0. 6E-MHCNMH G= Ax/U f-where Uisship speed. The time weighting factor in Eq2" 10£Qa1 = a3 = .0.5E-a2 = ('"symmetric implicit scheme£O Linear floor friction cofficient τ = 0.01. Normal fluxis zero at the harbour walls. The first-order non- rflection conditionf" 42Cs imposed at harbour en-trance. Locallyfit jis the out-going plane waves . Surface elevations at the new time level at the har-bour entrance a整查ven by the integral.114ζ士~ xE-yLO=里x- Ot√gHsEy- Ot VgHs,EO"220Fig. 1 Skelch of harbourf: rmoving ship and finite element gridsFig.2" aL⑥The free surface elevations around the ship moving in harbour with Fn =0.6Fig.2" bE@ The wave patterm generated by a ship moving in harbour with Fn =0.6中国煤化工MHCNMH GFig. 2" LO The horizontal flows around a ship moving in harbour with Fi =0.6The诟有敞据surface elevations and velocities are assumed to be zero£ and the initial pressure115of ship bottom is hydrostaticf- given by ship hull data. When the ship moves to the middle of theharbourf- the free surface elevations£ wave patterns and horizontal velocities are shown in Fig.2 forFn =0.6E-Fig.3 for Fh =1.0f-and Fig.4 for Fh = 1.3. In these and the subsequent figures ofwave patterns2-the light colour region represents wave crest and the darker colour region representsthe wave trough . These results well characterise the most important physical features of ship waves inshallow water. A wave crest appears around the ship bowfwhere the free surface rises upf the wateris displaced away£- and the flow pattern is much like a point source. The free surface sets downaround the ship sternE- where the water is convergedE- and the flow patterm is like a point sink .The pressure on central line of ship bottom is shown in Fig.5 where p* = p/pg. It can beseen that the pressure is different from hydrostatic one. The pressure around the bow increases£ andthe pressure around the stern decreases. A wave-like pressure distribution is found on ship botom.The dynamic pressure is mainly induced by surface waves around the ship hull. The wavelength isfound to be about one third of the ship lengthE which is longer than water depth in this case.Fig.8" afO The free suface elevations around a ship moving in harbour with Fi = 1.0Fig.8" bEO The wave pattern generated by a ship moving in harbour with Fn = 1.0中国煤化工YHCNMHGFig.8" cEO The horizontal flows around a ship moving in harbour with Fn =1.0In the case of Fn =0.65 low speedE- subcritical flows&国-the waves are generated mainly bythe movings鬥数据the reflection waves from the side walls are weak. A strong reflection wave is116Fig.8" aL⑥The free surface elevations around the ship moving in harbour with Fh = 1.3Fig. bEO The wave patterm generated by a ship moving in harbour with Fh = 1.3Fig. A" c2⑥The horizontal flows around a ship moving in harbour withFn =1.3found in the case of Fn = 1.3~high speedE rsuperitical flows2@ Reflection flows from the harbourside walls generate complicated wake flow whichE in many casesf- significantly contributes to waveresistance of the ship. These generated waves are of great importance to water environments of a har-bor. They also have significant influences on the performances and movement of the ships .Fig. 6 shows two ships moving in the same directions. The dimensions of the harbour and the .ships are the same as in the previous case. When the ships move to the middle of the harbourf thefree surface elevations£ wave patterns and horizontal velocities are shown in Figs. 6~ af6-6" bf国6 cfOrespectively for Fn = 1.0. A strong wave interaction between the two ships is found with theS = 40m considered in this workE where S denotes the中国煤化工lines of the two ships.In present study the distance between the two shhe ship speed is 9. 9m/s. For the case considered here£ the ships expenMYHCNMHG"due to the wave inter-action. This will greatly affect the motion of the two shipS" 1. Fig. 7 shows the solution for two shipsmoving in the opposite directions to each other. The dimensions of the harbour and ships are thesame as in the previous case. The free surface elevationsE- wave patterns and horizontal velocities areshown in Figs. 早igfoA”bf国-A”c国for Fh = 1.0. The flows are observed to start in front of oneshipf- and conerngeto the sterm of another ship. A strong wave interaction of two ships is also found.117When the bow of one ship approaches to the stem of the other£ the ships experience a large attrac-tive side force if two ships are close enoughf" S < 40 mE0 An impact accident could thus occur.D.StemEowhydostaticF二1.3国F'r=0.'g- I.5-第-2.55拿-3-3. 5480100120ship Jength z (m)Fig.5 Hydrodynamic pressure on ship bottomFig.6" aLO The free surface elevations around two ship&" S = 40mE@noving prllellyl in harbourwith Fn =1.0中国煤化工Fig.6" lbE⑥ The wave pattern generated by two shipf"MYHCNMHGarbour5. CONCLUTIONA prp府数据ave Equation ModelC" WEMECand the Galerkin finite element method are em-.118Fig. 6" cL⑥'The horizontal flows around two shipsL" S = 40m2@moving parllelly in harbourwith Fr =1.0Fig.I" ac⑥The free suface elevations around two shipL" S = 40mE@noving in opposite directions in harbour withFn =1.0Fig.9" "bEO The wave pattern generated by two shipL" S = 40mE@noving in opposite directions in harbour with Fn= 1.0中国煤化工THCNMHGFig. 9" c2⑥The horizontal flows around two shipL" S = 40mE@noving in opposite directions in harbour with Fn =1.0ployed to月券数据°water equations with moving pressure for the waves generated by moving ships119in bounded shallow water. The ships are considered as moving pressure on free surface. In compari-son with potential theory the shallow water equations with moving pressure are more suitable to modelship waves in shallow waterf when the ratio of water depth to wavelength is small. The irrotationalcondition is not needed . The nonlinearity£- energy dissipation£ free surface disturbance and boundaryreflection can be directly treated in the shallow water equations . Numerical results obtained throughthis work simulate very well the characteristics of the ship waves in a shallow water harbour wherethe boundary reflection and the radiation of open boundary are considered.ACKNOWLEDGEMENTSThis work was completed during Professor Wu Jiankang' s attachment to the Department of Me-chanical and Production Engineering of the National University of SingaporL" NUSEO Professor Wu isgrateful to NUS for providing the financial support. The discussion with Dr. Chen Yongze is alsogratefully acknowledged.REFERENCES1. Lynch D. R. and Gray W. G. £+978E9A Wave Equation Model for Finite Element 'Tidal Computations£ -Computerand Fluid2- 72207-288.2. Lynch D. R. E-1983E9Progress in Hydrodynamics Modellingf Review of U. s. Contribution£ 1979- 1982E -Reviewof Geophysics and Space PhysicsE- 2”320-741-754.3. Francisco E. 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