NIVERSITYOSouthampton(3) Two-dimensional (2D)Hydrodynamics andStrip Theory.Examplesfrom:Hydroelasticity of Ships,R.E.D.Bishop&W.G.Price, cUP1979Hydrodynamics and HydroelasticityProf.P.Temarel,WUT,July2015JIVERSITYOStrip Theorv:FundamentalsSouthamptonPotential flowanalysisObjective:Estimateforceappliedbystripoffluid on corresponding strip of hull2Danalysis;infinitelylonguniformcylinderofarbitrarycrosssection;flowinthemiddle2D,i.e.noend effectsas in3DflowDoes not satisfy speed dependent linearised.freesurfaceconditionforoscillatinghullApplicable to low speeds and high.Verysuccessful intermsoffrequenciespractical applications fora range of mono-hulledFirst developed in late1950s using relative.Vessels.displacement conceptIgnores interactions betweenManyversions,improvements since;most·Strips along the ship;important:Salvesen,Tuck&Faltinsen,especiallyimportantforTrans.SNAME,197o;DerivationofStripmedium tohigh speedsTheoryfrom3Danalysis
1 (3) Two-dimensional (2D) Hydrodynamics and Strip Theory. Examples from: Hydroelasticity of Ships, R.E.D. Bishop & W.G. Price, CUP 1979 Hydrodynamics and Hydroelasticity Prof. P. Temarel, WUT, July 2015 2 Strip Theory: Fundamentals • Potential flow analysis • Objective: Estimate force applied by strip of fluid on corresponding strip of hull • 2D analysis; infinitely long uniform cylinder of arbitrary cross section; flow in the middle 2D, i.e. no end effects as in 3D flow • Does not satisfy speed dependent linearised free surface condition for oscillating hull • Applicable to low speeds and high frequencies • First developed in late 1950s using relative displacement concept • Many versions, improvements since; most important: Salvesen, Tuck & Faltinsen, Trans. SNAME, 1970; Derivation of Strip Theory from 3D analysis Very successful in terms of practical applications for a range of mono-hulled Vessels. Ignores interactions between Strips along the ship; especially important for medium to high speeds
NIVERSITYOFOtherhydrodynamictheoriesSouthampton(Table from Probabilistic theoryof Ship Dynamics, 1979,L: Lengthby R.E.D. Bishop & W.G. Price)B: BeamDmlH4aCT:DraughtFr=VeL:Wavelengtho:FrequencyD(p)0(1)0(1)0(1)zero or o(1)Thin shipFn:Froude Notheoryzero oro(1)g:Gravitational0(1)0(1)c(p)0(1)Flet shipAccelerationtheoryU: Ship speed0(p)op)0(1)0(1)zero oro(1)Slender shipβ<<1theory0(8-1/2)O(R)0(p)0(p)zero or o(1)Strip theoryVarious3Dlinearhydrodynamictheories(potentialflow).Use sourcedistributions on calm waterwettedsurface, and if required free surface,e.g.Rankinesource, pulsating source, translating-pulsating sourceetc.,usuallyreferred to as“Green'sfunction”3UNIVERSITYOFRadiationproblem:SimpleexplanationSouthampton?OscillationofhullinstillwaterF, :MechanicalexcitationTFeTwt(SHM)t212Cn :Restoring coefficientCnz=pVgV:Volume between WoLoWand OyaxisSoC pg Water Plane Areaassuming hull wall-sidedHull in equilibriumMechanical excitation of hull in still waterm2=-W+A-Cz+Fi(0)or mZ=-Czzz+Fi(t)instill waterignoring hydroddynamic effects△=Wm2=-mzzz-Nzzz-Czzz+Fi(t)assuminghydrodynamicpressurehascomponentsA=pvproportional to velocity (Nzzfluiddamping)W=mgand acceleration (mzz added mass).A2
2 3 Other hydrodynamic theories (Table from Probabilistic theory of Ship Dynamics, 1979, by R.E.D. Bishop & W.G. Price) • Various 3D linear hydrodynamic theories (potential flow). Use source distributions on calm water wetted surface, and if required free surface, e.g. Rankine source, pulsating source, translating-pulsating source etc., usually referred to as “Green’s function” L: Length B: Beam T: Draught λ: Wavelength ω: Frequency Fn: Froude No g: Gravitational Acceleration U: Ship speed β << 1 4 Radiation problem: Simple explanation • Oscillation of hull in still water W m g W in still water Hullin equilibrium assuming hull wall-sided So C : g Water Plane Area and Oy axis V :Volume between W L C z V g C :Restoring coefficient (SHM) F : Mechanical excitation zz 0 0 zz zz 1 and acceleration (m added mass). proportional to velocity (N fluid damping) assuming hydrodynamic pressure has components m z m z N z C z F (t) ignoring hydroddynamic effects m z W C z F (t) or m z C z F (t) Mechanical excitation of hull in still water zz zz zz zz zz 1 zz 1 zz 1
IVERSITYORelative displacement: Simple explanation SouthamptonOscillationofhnllinwavesa-C.(a-52)m=-W+-Cz (z-)=-Cz(z-)ignoring hydrodynamic effectsmz=-mzz (z-)-Nzz(2-)-Cz (z-L)assuming hydrodynamic pressure has componentsproportional to relative velocity (Nzz fluid damping)andrelativeacceleration(mzzaddedmass)G:assumed waveelevationat hull's centreline,ie.(x,t)NIVERSITYOFTotal Derivative:SimpleexplanationSouthamptonConsider function F=f(x,t)af.afAf--At+AXataxorAf_afatofAxAtatAtoxAtDistance in thespace-fixed AXoYoZo axesaf.afdxXA=Ut+xatoxdtwitho-0%d(XA)=U+datxdtdtAt timet= 0,Oand A coincide; henceTOTALDERIVATIVEdx=-U.D_a00dtDtatax3
3 5 Relative displacement: Simple explanation • Oscillation of hull in waves : assumed wave elevation at hull's centre line,i.e. (x, t) and relative acceleration (m added mass) proportional to relative velocity (N fluid damping) assuming hydrodynamic pressure has components m z m (z ) N (z ) C (z ) ignoring hydrodynamic effects m z W C (z ) C (z ) zz zz zz zz zz zz zz 6 Total Derivative: Simple explanation x f U t f t x x f t f t x x f t t t f t f x x f t t f f F f x t d d or Consider function ( , ) x U D t t D TOTAL DERIVATIVE . dt d At time 0,O and A coincide; hence . dt d ( ) dt d with X Distance in the space - fixed AX Y Z axes A 0 0 0 U x t x X U U t x A
VERSITYOFluid Force acting on a stripSouthamptonPuttingtogetherall thesimpleexplanations,fluidforceon a stripDD三(x,t)DE(x,t)F(x,t)=+ N(x)+pg B(x)z(x,tm(x)DtDtDtm(x):Addedmass(heave)perunit lengthN(x):Fluiddamping (heave)per unit lengthB(x):Breadth along calm water line三(x,t)=w(x,t)-5(x,t)w(x,t):Vertical displacementS(x,t):Regularwaveprofileathull'scentrelineDE(x,t)Note that m(x)representsfluidmomentumDtVERSITYOFluid Force acting on a stripSouthampton.Performing the operationsD?E(x,t)[N(a)-dm(DE(x,) F(x,t)=-m(x)pg B(x)三(x, t)dxDtDt2as the added mass pul m(x) is only a function of position along the shipWe can further break down the Fluid action into components of w(x,t)and (x,t):F(x,t)=-H(x,t)+ Z(x,t)whereD? w(x,t)udm()|Dw(x,)H(x,t)=m(x)N(x)-+ pg B(x)w(x,t)DtDt2dxD?(x,t)[N(x)-Udm(DS(x,)Z(x,t)=m(x)+pg B(x)s(x,t)D2dxDt
4 7 Fluid Force acting on a strip • Putting together all the simple explanations, fluid force on a strip representsfluid momentum ( , ) Note that ( ) ( , ):Regular wave profile at hull's centre line ( , ):Vertical displacement ( , ) ( , ) ( , ) ( ):Breadth along calm water line ( ):Fluid damping (heave) per unit length ( ): Added mass(heave) per unit length ( ) ( , ) ( , ) ( ) ( , ) ( , ) ( ) Dt D z x t m x x t w x t z x t w x t x t B x N x m x g B x z x t Dt D z x t N x Dt D z x t m x Dt D F x t 8 Fluid Force acting on a strip • Performing the operations ( ) ( , ) ( , ) d d ( ) ( ) ( , ) ( , ) ( ) ( ) ( , ) ( , ) d d ( ) ( ) ( , ) ( , ) ( ) where ( , ) ( , ) ( , ) We can further break down the Fluid action into components of ( , ) and ( , ): as the added mass pul ( ) is only a function of position along the ship. ( ) ( , ) ( , ) d d ( ) ( ) ( , ) ( , ) ( ) 2 2 2 2 2 2 g B x x t D t D x t x m x N x U D t D x t Z x t m x g B x w x t D t D w x t x m x N x U D t D w x t H x t m x F x t H x t Z x t w x t x t m x g B x z x t D t D z x t x m x N x U D t D z x t F x t m x
IVERSITYOFluid Force acting on a stripSouthampton.Incident waveforceon a stripUsing complex notation the regularwave profile(at centre line)(x,t)=a exp(kT)exp[i(krcos x-er)such thatReal[(x,t)]=aexp(-kT)cos(kxcos x-Oet)Continuing with the complexnotationD(%-0%)c(x,t)=-ieaexp(-kT)exp[i(kxcos x-oet)-C(x,t) =Dt(arax-iUkcos xaexp(-kT)exp[i(kxcos -@et)=-i(e+Uk cos x)aexp(-kT)exp[i(kxcos x-0et)=-io(x,t)D2D-[-i(x,)=(-i0)(-i0)(x,t)=-2(x,t)-C(x,t) =DtDt2Thus Z(x,t) becomes:Z(x t)= 2m(x)-io[N(x)-Um(x)+ pg B(x))5(x, )ERSITYOAddedmass&FluidDampingSouthamptonInfinite Fluid Domain- Ideal Fluid.Added mass determined for spheres, ellipsoids and cylindrical bodies (ofspecifiedcross section shape)using suitable velocitypotential.Easier in 2D case; more difficult in 3D case:Suitable velocity potentials difficult to find for prismatic bodies, e.g shipshapedsections:No fluid damping (inviscid & irrotational fluid)On the free surface-Ideal fluid.Added mass &Fluid damping:Analytical solutionsdifficult,evenfor simplegeometriesUsecombinationsof velocitypotentialsRepresentshapeofsection(2D)orship(3D)accurately105
5 9 Fluid Force acting on a strip • Incident wave force on a strip Z(x,t) ( ) ( ) ( ) ( ) ( , ) Thus ( , ) becomes: [ ] ( )( ) . ( cos ) exp( ) exp[ ( cos )] cos exp( ) exp[ ( cos )] exp( ) exp[ ( cos )] Continuing with the complex notation Real[ ] exp( ) cos( cos ) such that exp( ) exp[ ( cos )] Using complex notation the regular wave profile (at centre line) 2 2 2 2 m x i N x Um x g B x x t Z x t i ζ(x,t) i i ζ(x,t) ζ(x,t) Dt D ζ(x,t) Dt D i ζ(x,t) i Uk a kT i kx t iUk a kT i kx t ζ(x,t) i a kT i kx t x U t ζ(x,t) Dt D ζ(x,t) a kT kx t ζ(x,t) a kT i kx t e e e e e e e 10 Added mass & Fluid Damping Infinite Fluid Domain- Ideal Fluid • Added mass determined for spheres, ellipsoids and cylindrical bodies (of specified cross section shape) using suitable velocity potential • Easier in 2D case; more difficult in 3D case • Suitable velocity potentials difficult to find for prismatic bodies, e.g ship shaped sections • No fluid damping (inviscid & irrotational fluid) On the free surface – Ideal fluid • Added mass & Fluid damping • Analytical solutions difficult, even for simple geometries • Use combinations of velocity potentials • Represent shape of section (2D) or ship (3D) accurately
UNIVERSITYOFAdded mass in infinitefluid (2D)SouthamptonSimpleexample:Uniform circularcylinder ofradiusa=B/2,:infinitelylong,movingwithvelocityU(t)perpendiculartoitslongitudinal axis (i.e.xdirection);fluid stationary.a2Velocity potential @-U(t)"cOSOinpolarcoordinates (r,0)P"-02oadadVelocitycomponentsqr=cosO,U(sin O).q0=orroo(-r2)r2Kinetic Energy of fluid per unit length (2D):002元0.5pl (u?+w2)dxdz=0.5pJ了(qi+q)rdrdoaosince a2) (++)-2yThus kinetic energy of fluid (pul) becomes2元100<11Sovaf00rapU2a2mU250a0-2元=2.22Oa11UNIVERSITYOFAdded mass in infinite fluid (2D)SouthamptonKinetic energy of fluid + cylinder (pul)1(m+m)U2wherem:massof cylinderpulForce F acting on cylinder, in direction of motion,thenfromPower=RateofchangeofkineticEnergyd(m +m)U2FU(t) =2 = U(t)U(t) (m +m)dt22where :added mass pul(m + m): Virtual mass pul;m=p元a-ADDED MASS- an explanationTo setabodyinmotion itrequireskinetic energy.In addition the fluid which is displaced by the body's motion also has to be given kinetic energy.The work done in accelerating the body in fluid is thus greater than that for the body only.It isas if work is doneonthe"bodymass+addedmassOthemassofthebodyappearstobegreaterbyanamountof"addedmassPhysically speaking, the result of a body moving in infinite ideal fluid is hydrodynamic pressure12proportional tobody'sacceleration6
6 11 Added mass in infinite fluid (2D) • Simple example: Uniform circular cylinder of radius a=B/2, infinitely long, moving with velocity U(t) perpendicular to its longitudinal axis (i.e. x direction); fluid stationary. 2 2 2 2 2 4 2 0 4 2 4 4 4 2 2 2 2 2 2 2 2 0 2 2 2 2 2 2 2 2 1 2 1 2 2 1 2 1 2 1 Thus kinetic energy of fluid (pul) becomes since ( ) ( ) . 0.5 ( ) 0.5 ( ) Kinetic Energy of fluid per unit length (2D): cos ; ( sin ). ( ) Velocity components Velocity potential ( ) cos in polar coordinates ( , ) U a mU r dr d U a r r U a r a u w q q U u w dx dz q q r dr d r a U r q r a U r q r r a U t a a r r a r 12 Added mass in infinite fluid (2D) (m m):Virtual mass pul; m : added mass pul where (m m)U U(t)U(t) (m m) 2 1 d t d FU(t) then from Power Rate of change of kinetic Energy Force F acting on cylinder, in direction of motion, (m m)U where m mass of cylinder pul 2 1 Kinetic energy of fluid cylinder (pul) 2 2 2 a : ADDED MASS – an explanation To set a body in motion it requires kinetic energy. In addition the fluid which is displaced by the body’s motion also has to be given kinetic energy. The work done in accelerating the body in fluid is thus greater than that for the body only. It is as if work is done on the “body mass + added mass” or the mass of the body appears to be greater by an amount of “added mass”. Physically speaking, the result of a body moving in infinite ideal fluid is hydrodynamic pressure proportional to body’s acceleration
IVERSITYOAddedmassininfinitefluid (2D)SouthamptonAlternativeapproachesforthestudentstoattempt:Same velocity potential, in Cartesian coordinatesa"xΦ-U()-;with velocitycomponents u=-(a/ax), w=-(ab/az)(x2 +22)evaluationofvelocitycomponentsmoredifficult.Evaluate pressure on the cylinder (pul) and force (pul) in direction-ofmotion,usingp=-p(ob/at) and Fx=muUsealternativepotential wherethe cylinderis stationaryand thefluid has velocity U(t) perpendicular to the cylinders longitudinalaxisa@=U(t)r+cosO-BecarefulwiththemagnitudeoftheF,force;thinkFroude-Krylov!13UNIVERSITYOFAdded MassCoefficientSouthampton.Genericform:Addedmaspul of shipshapedsectionCAdded mass pul of comparable infinite cylinderVerticalmotion (i.e.heave)mybased on B= 2aCy=p元B2/8Horizontalmotion (i.e.sway)and rotational motion (i.e.roll)1CT =mHCH=PRT4p元T2/2147
7 13 Added mass in infinite fluid (2D) Alternative approaches for the students to attempt • Same velocity potential, in Cartesian coordinates evaluation of velocity components more difficult. • Evaluate pressure on the cylinder (pul) and force (pul) in direction of motion, using • Use alternative potential where the cylinder is stationary and the fluid has velocity U(t) perpendicular to the cylinders longitudinal axis Be careful with the magnitude of the Fx force; think Froude-Krylov! ; with velocity components ( / ), ( / ) ( ) ( ) 2 2 2 u x w z x z a x U t p t Fx mU ( / ) and ( ) cos 2 r a U t r 14 Added Mass Coefficient • Generic form: • Vertical motion (i.e. heave) • Horizontal motion (i.e. sway) and rotational motion (i.e. roll) B a B m C V V ; based on 2 / 8 2 Added mass pul of infinite cylinder Added mas pul of ship shaped section comparable C 4 T I CT 2 2 ρ π T / m C H H
UNIVERSITYOFConformal MappingSouthamptonGenericdefinition:Ashape inn-plane ismapped intoa shape inK-plane, i.e.K=f(n), through a coordinate transformation, generically2K=Zenn"R=1n=e/β=(cosβ+isinβ)K=Y+iZ=reie=r(cosa+isine)aBm4planeK-plane215UNIVERSITYOFConformal MappingSouthamptonIt's real advantage: Perform calculations on a simple shape, e.g.circle (semi-circular section atfree surface)and findresults for a complexshape (shipsection atfree surface)bymappingthe semi-circle onto the ship sectionFor a port-starboard symmetric section,thetransformation is (only n=-1,1,3,5.etc,and ao=c-,a,=c,/c-,a=cg/c.,etc)1171K=Y+iz=aon+ai=+3+as+a7-nn27for =1ao:ScalefactorandLewis developed a 2-parameter transformation,using sectional area, beam anddraughttoobtaintheparameters3-parametertransformationwiththeaddition of2ndmomentofareaMulti-parameter transformation,based on definingpoints along section contourandobtainingparametersfromleastsquaresolutionIn every case,resultant mapped section perpendicular atfree surfaceand centreline168
8 15 Conformal Mapping • Generic definition: A shape in η-plane is mapped into a shape in Kplane, i.e. K=f(η), through a coordinate transformation, generically (cos sin ) (cos sin ) 1 e i K Y i Z r e r i K c i i n n n 16 Conformal Mapping • It’s real advantage: Perform calculations on a simple shape, e.g. circle (semicircular section at free surface) and find results for a complex shape (ship section at free surface) by mapping the semi-circle onto the ship section • For a port-starboard symmetric section, the transformation is (only n=-1, 1, 3, 5 etc, and a0=c-1, a1=c1/c-1, a3=c3/c-1 etc) • Lewis developed a 2-parameter transformation, using sectional area, beam and draught to obtain the parameters • 3-parameter transformation with the addition of 2nd moment of area • Multi-parameter transformation, based on defining points along section contour and obtaining parameters from least square solution • In every case, resultant mapped section perpendicular at free surface and centre line for 1 and :Scale factor . 1 1 1 1 0 7 7 5 5 3 0 1 3 a K Y i Z a a a a a
VERSITYCConformal Mapping- Lewis method SouthamptonY= ao Real(eiβ + aje-iβ + ae-13β)=ao[(I+ai)cos β+ a3 cos3P]Z=ao Imag(eiβ + aje-iβ + a3e-i3β)=o[(I-a1)sin βa3 sin3β]with1anddY/dβ=0forβ=0dY I dβ=ao[-(1 + a)sin β-3a3 sin 3β]dz /dβ=ao[(1-ai)cosβ-3a3 cos3β]anddz/dβ=0forβ=元/2AlsoY(β=0)=ao(1+ai +a)=B/2Sectionhalf-breadthZ(β=元 /2)=ao(1-a1 +a3)=TSection draughtLewis Method CoefficientsB3-Co+ (9-2Co)(1+a3)(A-1)aoaja3(A +1)Co2(1+aj +a3)+ 4g + (4 -1)2(1- 4g)where Co=3+(A+1)2(元元17FRSITYOConformal Mapping- Lewis method Southampton.SectionalArea2S=[YdZ=[Y(dZIdB)dβ2元=a了 [(I+ai)cos β+a3 cos3β[(1-ai)cos β-3a cos3β]dβ0=a (1-a )cos? β+[-3a3(1+a1)+(1-a1)a3]cos βcos3β- 3ag cos? 3β)dβ= r(1-α2 -3a3)2242元T cos? 3βdβ[ cos’βdβ=π=sinceandcosβcos3βdβ=0000SectionalArea Coefficient =S/BTHalf-breadth/DraughtratioA=B/2T·Using relationsfor Half-breadth,oand A,the scale factoraandthetwoparameters a,and acan beuniquelydefined (see slide17)9
9 17 Conformal Mapping- Lewis method ( / 2) (1 ) Section draught ( 0) (1 ) / 2 Section half - breadth Also / [(1 ) cos 3 cos3 ] and / 0 for /2 / [ (1 )sin 3 sin 3 ] and / 0 for 0 with Imag( ) [(1 )sin sin 3 ] Real( ) [(1 ) cos cos 3 ] 0 1 3 0 1 3 0 1 3 0 1 3 0 1 3 3 0 1 3 0 1 3 3 0 1 3 Z a a a T Y a a a B dZ d a a a dZ d dY d a a a dY d Z a e a e a e a a a Y a e a e a e a a a i i i i i i 4 1 2 ( 1) 2 4 ( 1) where 0 3 0 3 0 (9 2 0) , 3 ( 1) (1 3)( 1) , 1 2(1 1 3) 0 Lewis Method Coefficients C C C C a a a a a B a 18 Conformal Mapping- Lewis method • Sectional Area • Using relations for Half-breadth, σ and Λ, the scale factor a0 and the two parameters a1 and a3 can be uniquely defined (see slide 17) since cos cos 3 and cos cos 3 0 (1 3 ) {(1 ) cos [ 3 (1 ) (1 ) ]cos cos 3 3 cos 3 } [(1 ) cos cos 3 ][(1 ) cos 3 cos 3 ] d 2 ( / ) 2 0 2 2 0 2 2 0 2 3 2 1 2 0 2 2 3 1 1 3 3 2 2 1 2 0 2 0 1 3 1 3 2 0 2 0 d d d a a a a a a a a a a d a a a a a S Y dZ Y dZ d d B T S BT Half - breadth/Draught ratio / 2 Sectional Area Coefficient /
IVERSITYOConformal Mapping- Lewis method Southampton.Tvnical Iewis Sertinnsand Restrictionson Iewisforms0.29452.0r0.48No Lewis form0.6926/4 (circulan)0.9405L36max (o)Tunnel bulbousbulbousxannel1.0FullerLewisfomsCeattnliirdformavailahtehullshullsD:0.29.A:0.5min (o,)No Lenror2:1.0A:0.5wlhoous o:1.15 A:0.52.01.51.00.5景19UNIVERSITYOFConformalMappingSouthamptonComparison of Lewis and Multi-parametermapping0.02.04.0直8.012.0E210
10 19 Conformal Mapping- Lewis method • Typical Lewis Sections and Restrictions on Lewis forms reentrant bulbous tunnel Tunnel bulbous 20 Conformal Mapping • Comparison of Lewis and Multi-parameter mapping