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