《船舶水动力学》课程教学课件（讲稿）第三章 二维水动力理论及切片理论 2D hydrodynamics Strip theory

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 