AIVERSITYOSouthampton(2) Waves: Regular waves.Hydrodynamics and HydroelasticityProf.P.Temarel,WUT,July2015JIVERSITYORegular(sinusoidal)waveSouthampton.Main particulars(X,t)=acos(kX -ot+α) (l)a:waveamplitude (m)h=2a:wave height (m)W=2元/T:wave frequency (rad/s)T:wave period (s)2:wavelength (m)Z=(X.Y.n)k=2元/:wavenumber (1/m)c= /T = /k : wave or phase velocity (m/s)d :water depth (m)Long-crestedregular wavearbitrary phase angle (rad)aα:β:waveslopedsβ==-ak sinkXandI βmax [= akdx
1 (2) Waves: Regular waves. Hydrodynamics and Hydroelasticity Prof. P. Temarel, WUT, July 2015 2 Regular (sinusoidal) wave • Main particulars (X ,t) acos(kX t ) (1) ak kX ak X d c λ/T ω/k k π/λ T ω π/T h a a max sin and d d : wave slope : arbitrary phase angle (rad) : water depth (m) : wave or phase velocity (m/s) 2 : wave number (1/m) : wave length (m) : wave period (s) 2 : wave frequency (rad/s) 2 : wave height (m) : wave amplitude (m) Long-crested regular wave
JIVERSITYOBernoulli'sEquation (2D)SouthamptonTo obtain the regular wave a velocity potential is used-ideal fluidBernoulli'sequationtakestheform:--(2+w)--gz=0 (2)at2padproportional to dynamic pressureatpressure (relative to atmospheric)p:fluid densityp:acceleration dueto gravityg:fluid particle velocity componentsu, w:adadu=W=azax3FRSITYOVelocity potential for regular wave Southampton(X,Z,1)=-a cosh(k(Z+dsi(X-01+a) (3)cosh(kd)satisfying Laplace's equationV@=0.Linearised dynamic free surface condition (from Bernoull's eqn.,neglecting highorderterms)1adS=on Z=0 (4)01-85=0 orgatKinematic free surface conditionas-W=0onZ==01a@adat:0on Z =0 (5)which combined with eq.(4) results in:ozgarBottom (no flow through) condition: oo=0 on Z= -d4az2
2 3 Bernoulli’s Equation (2D) • To obtain the regular wave a velocity potential is used – ideal fluid • Bernoulli’s equation takes the form: ( ) 0 (2) 2 1 2 2 gZ p u w t : fluid particle velocity components : acceleration due to gravity : fluid density : pressure (relative to atmospheric) : proportional to dynamic pressure u, w g p t X u Z w 4 Velocity potential for regular wave • satisfying Laplace’s equation • Linearised dynamic free surface condition (from Bernoulli’s eqn., neglecting high order terms) • Kinematic free surface condition which combined with eq.(4) results in: (5) • Bottom (no flow through) condition: sin( ) (3) cosh( ) cosh[ ( )] ( , , ) kX t kd g k Z d X Z t a 0 2 0 or g t on 0 (4) 1 Z g t 0 on 0 w Z t 0 on 0 1 2 2 Z g t Z Z d Z 0 on
VERSITYCRegular waves: dispersion relation SouthamptonExaminethedynamicfree surface condition on Z=o,i.e.eq.(5)1 ad-a(-の) sin(kX - t +α)gar?ad-ggk sinh[k(Z+d in(X - 01 +α)azcosh(kd)0adagktanh(kd)sin(kX-@t+α) forZ =0az01 a@dFromazg at?agk2 = kg tanh(kd) (6)tanh(kd)orao=0RSTYeRegular waves: dispersion relation Southammpton:Applying the dispersion relationship to the wave propagation velocityC=03 tanh(2元 g)gtanh(kd) = kVkVk2.ExaminetheinfluenceofwaterdepthShallow water: d / a→0 or kd →0 .tanh(kd)→kd .c=gdDeep water: d / a→oo or kd-→oo :.tanh(kd)-→1 .c=g / kIndepwaler: -%-是 .0 - g125kDeepwater:d/a>0.5.Noteifit is not deep water,itdoesn'tmean it is shallow,ie.betweendeepandshallowhavetousedispersionrelationship3
3 5 Regular waves: dispersion relation • Examine the dynamic free surface condition on Z=0, i.e. eq.(5) ( ) ( ) 1 2 2 a sin kX t g t tanh( ) sin( ) for 0 sin( ) cosh( ) sinh[ ( )] kd kX t Z gk a Z kX t kd gk k Z d a Z g t Z 2 2 1 From tanh( ) or tanh( ) (6) 2 kd kg kd agk a 6 Regular waves: dispersion relation • Applying the dispersion relationship to the wave propagation velocity • Examine the influence of water depth • Note if it is not deep water, it doesn’t mean it is shallow, i.e. between deep and shallow have to use dispersion relationship tanh( ) tanh(2 ) d k g kd k g k c Shallow water : d / 0 or kd 0 tanh(kd) kd c gd Deep water : d / or kd tanh(kd) 1 c g / k Deep water : 0 5 In deep water : 2 2 2 2 d / . kg k g k c
JIVERSITYORegularwave:PressureSouthamptonDependenceondistancefromfreesurface,indeepwatercosh[k(Z + d)) _ cosh(kZ)cosh(kd)+ sinh(kZ)sinh(kd)cosh(kd)cosh(kd)=cosh(kZ)+sinh(kZ)tanh(kd)=cosh(kZ)+sinh(kZ)indeepwater=ekzUsing this exponential variation and Bernoulli'seqn.,ignoring higherorder terms the pressure (relative to atmospheric) anywhere (i.e.X.Z)in the fluid is:adp=p gleks(X,1)-Z) in deep water (7)Ap=gzVERSITYORegular wave: Presence of moving ship Southampton
4 7 Regular wave: Pressure • Dependence on distance from free surface, in deep water • Using this exponential variation and Bernoulli’s eqn., ignoring higher order terms the pressure (relative to atmospheric) anywhere (i.e. X, Z) in the fluid is: cosh( ) cosh( )cosh( ) sinh( )sinh( ) cosh( ) cosh[ ( )] kd kZ kd kZ kd kd k Z d kZ kZ kZ kZ kZ kd e cosh( ) sinh( ) in deep water cosh( ) sinh( )tanh( ) gZ g[e (X ,t) - Z] in deep water (7) t p kZ 8 Regular wave: Presence of moving ship
SRegular wave: Presence of moving ship SouthamptonCoordinatetransformationX.HX8,X,= 0a +ab =(y sn p+x cospgi=ac-de=ycosp-xsinpALSO(he-fb)X=e-he=x,cosh-y,simry= eb+fg = x,smp +y,coshSYRegular wave: Presence of moving ship SouthamptonRegular waveeqn.with referencetoamovingship.Transformation ofcoordinatefor wave propagating atheading%,i.e.fromAXYZ to AX,Y.Z.(Z=Z.)axes systemsX = Xocosx +Yosinx.TransformationofcoordinatetoOxzX=x+Ut and y=Yo·Combiningboth(x, y,t) = acos(kxcosx +kysinx +kUcosx t-ot +α)=acos(kxcosx+kysinx-oet+α).Where ,is the encounter frequency, frequency of waveas seenbyanobserveronamoving ship105
5 9 Regular wave: Presence of moving ship • Coordinate transformation 10 Regular wave: Presence of moving ship Regular wave eqn. with reference to a moving ship • Transformation of coordinate for wave propagating at heading χ, i.e. from AXYZ to AX0Y0Z0 (Z=Z0) axes systems • Transformation of coordinate to Oxz • Combining both • Where ωe is the encounter frequency, frequency of wave as seen by an observer on a moving ship X X0cos Y0sin 0 Y0 X x U t and y cos( cos sin ) ( ) cos( cos sin cos ) a kx ky t x, y,t a kx ky kU t t e
VERSITYCRegular wave: Encounter frequency Southampton0。=0-kUcosx= k(c-Ucosx)OUindeepwater0.=0-cOSXgV(radlis)oUHead wxveHeadwavesx=180°..@=0+VasmisgQuartering(bow) waves,e.g. x=13500Beam waves x = 90° .. 0 = 0AIVERSITYORegular wave: Encounter frequency SouthamptonOUUFollowing waves x = 0° : 0。= @ --gg0。=0when 0=0-1 (ic. WhenU =c)Lradis)0e=0 when0FellowodU=5m1 (i.e.whenU>c)NesecskonthauzekoLSA0。 treated as+ve; following waves overtaken by shipU<1followingwavesovertakeshipFor0ghas a maximum valueQeado.=0=1-20-gwhenand the max. valueis40dogSamerelationship is also valid for any x<900Quartering(stern) waves,e.g. ×= 4506
6 11 Regular wave: Encounter frequency cos in deep water ( cos ) cos 2 g U k c U kU e e g U e 2 o Head waves 180 e o Beam waves 90 o Quartering (bow) waves,e.g. 135 12 Regular wave: Encounter frequency 4U g 0 1 2 and the max. value is d d when has a maximum value For 1 following waves overtake ship treated as ve; following waves overtaken by ship 0 when 1 (i.e. when ) 0 when 1 (i.e. when ) 0 when 0 Following waves 0 1 e e 2 o g U g U U c g U U c g U g U g U e e e e e o o Quartering (stern) waves,e.g. 45 Same relationship is also valid for any 90
ERSTYRegular wave: Encounter frequency SouthamptonExampleinfollowingwaves(=00):Forward speed9.81m/s0.6@ (rad/s)0.41,2k=0°/g (1/m)0.01630.03670.146842.8入=2元/k (m)171.2385.2c=0/k (m/s)8.1824.5216.35() 0.24。 (rad/s)0.240.2413NIVERSITYOIncidentwaveactionsonship:DynamiccaseSouthamptonFROUDE-KRYLOV hypothesisWhen evaluatingthefluid actionsduetoincidentwaves,it isassumed that the pressure distribution on the ship is that of theundisturbed incident wave.:Eq.(2)impliesthatpressurebeneaththewavecrest is smaller andbeneaththewavetroughlargerthanhydrostaticpressure(pressureproportional to distance from free surface).This is due to the orbitalmotion of fluid particles..Thisis called theSMITHCorrectionInthedynamiccase,toavoidintegration inthevertical direction z,ameandraughtvalueischosensuchthato<T<TThe wave elevation equation becomes:S(x, y,t)= a e-kT cos(krcosx + hysinx -0 1 +a)147
7 13 Regular wave: Encounter frequency • Example in following waves (χ=0o) • Forward speed 9.81m/s ω (rad/s) 0.4 0.6 1.2 k=ω2/g (1/m) 0.0163 0.0367 0.1468 λ=2π/k (m) 385.2 171.2 42.8 c=ω/k (m/s) 24.52 16.35 8.18 ωe (rad/s) 0.24 0.24 (-) 0.24 14 Incident wave actions on ship: Dynamic case FROUDE-KRYLOV hypothesis • When evaluating the fluid actions due to incident waves, it is assumed that the pressure distribution on the ship is that of the undisturbed incident wave. • Eq. (7) implies that pressure beneath the wave crest is smaller and beneath the wave trough larger than hydrostatic pressure (pressure proportional to distance from free surface). This is due to the orbital motion of fluid particles. • This is called the SMITH Correction • In the dynamic case, to avoid integration in the vertical direction z, a mean draught value is chosen such that • The wave elevation equation becomes: 0 T T ( ) e cos( cos sin ) x, y,t a kx ky t e kT
NIVERSITYOSouthamptonIncidentwaveactionsonship:Ouasi-staticcaseShipbalanced onawavelength equal toshiplength,for crest at amidships(hogging)andtrough atamidships (sagging).Waveheight typically ofthe orderof L/10 (small ships) or L/20(largeships)nRather than regularsinusoidal wave, trochoidalwave used.V=T+tSmith correction, impliesmore sectional areaforsections underatroughandlessforsectionsundercrest.The concept of EquivalentDesignWavesisbecomingtrochordalwaveprevalent.SinvsoidapwaveNIVERSITYOFRegularwaves:SummarySouthampton.Velocitypotentialfor sinusoidal wave:Waveencounterfrequency,withrespecttoa shiptravellingwithforwardspeedandaheadingtotheregularwave:Froude-Krylov hypothesisInfluenceofdynamic effectsonpressuresundertrough&crest : Smith effect.Quasi-static analysiswiththe shippoised on atrochoidalwaveoflength equal to ship'slength;sagging&hoggingconditions.Thequasi-staticanalysisonatrochoidal waveisreplacedbyEquivalent Design waves.8
8 15 Incident wave actions on ship: Quasi-static case • Ship balanced on a wave length equal to ship length, for crest at amidships (hogging) and trough at amidships (sagging). Wave height typically of the order of L/10 (small ships) or L/20 (large ships) • Rather than regular sinusoidal wave, trochoidal wave used. • Smith correction, implies more sectional area for sections under a trough and less for sections under crest. • The concept of Equivalent Design Waves is becoming prevalent. 16 Regular waves: Summary • Velocity potential for sinusoidal wave • Wave encounter frequency, with respect to a ship travelling with forward speed and a heading to the regular wave • Froude-Krylov hypothesis • Influence of dynamic effects on pressures under trough & crest : Smith effect • Quasi-static analysis with the ship poised on a trochoidal wave of length equal to ship’s length; sagging & hogging conditions. • The quasi-static analysis on a trochoidal wave is replaced by Equivalent Design waves