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