13.42 Lecture: Vortex Induced VibrationsClassic VIV CatastrophePotential FlowAxial Pressure ForceReynolds Number DependencyShear layer instability causes vortex roll-upWake InstabilityClassical Vortex SheddingVortex shedding dictated by the Strouhal numberAdditional VIV ParametersStrouhal Number vs. Reynolds NumberVortex Shedding Generates forces on CylinderVortex Induced ForcesForce Time TraceAlternate Vortex shedding causes oscillatory forces which induce structural vibrations“Lock-in”Equation of Cylinder Heave due to Vortex sheddingLift Force on a CylinderLift Force Components:Total Force:Coefficient of Lift in Phase with VelocityLift in phase with velocityAmplitude EstimationDrag AmplificationSlide 25Flexible CylindersSlide 27Wake Patterns Behind Heaving CylindersTransition in Shedding PatternsSlide 30End Force CorrelationVIV in the OceanOscillating Tapered CylinderSpanwise Vortex Shedding from 40:1 Tapered CylinderFlow Visualization Reveals: A Hybrid Shedding ModeDPIV of Tapered Cylinder WakeSlide 37VIV SuppressionVIV Suppression by Helical StrakesOscillating CylindersReynolds # vs. KC #Forced Oscillation in a CurrentWall ProximityGallopingLift Force, Y(a)Galloping motionInstability Criterionb is shape dependentInstability:Torsional GallopingGalloping vs. VIVReferences13.42 Lecture:Vortex Induced VibrationsProf. A. H. Techet18 March 2004Classic VIV Catastrophe If ignored, these vibrations can prove catastrophic to structures, as they did in the case of the Tacoma Narrows Bridge in 1940.Potential FlowU() = 2U sinP() = 1/2  U()2 = P + 1/2 U2Cp = {P() - P }/{1/2  U2}= 1 - 4sin2Axial Pressure Force i) Potential flow: -/w <  < /2ii) P ~ PB /2    3 /2(for LAMINAR flow)Base pressure(i) (ii)Reynolds Number DependencyRd < 55-15 < Rd < 4040 < Rd < 150150 < Rd < 300300 < Rd < 3*1053*105 < Rd < 3.5*1063.5*106 < RdTransition to turbulenceShear layer instability causes vortex roll-up•Flow speed outside wake is much higher than inside•Vorticity gathers at downcrossing points in upper layer•Vorticity gathers at upcrossings in lower layer•Induced velocities (due to vortices) causes this perturbation to amplifyWake InstabilityClassical Vortex SheddingVon Karman Vortex StreetlhAlternately shed opposite signed vorticesVortex shedding dictated by the Strouhal number St=fsd/Ufs is the shedding frequency, d is diameter and U inflow speed•Reynolds Number–subcritical (Re<105) (laminar boundary)•Reduced Velocity•Vortex Shedding Frequency–S0.2 for subcritical flowAdditional VIV ParametersDSUfseffects viscouseffects inertialRevUDDfUVnrnStrouhal Number vs. Reynolds NumberSt = 0.2Vortex Shedding Generates forces on CylinderFD(t)FL(t)UoBoth Lift and Drag forces persist on a cylinder in cross flow. Lift is perpendicular to the inflow velocity and drag is parallel.Due to the alternating vortex wake (“Karman street”) the oscillations in lift force occur at the vortex shedding frequency and oscillations in drag force occur at twice the vortex shedding frequency.Vortex Induced ForcesDue to unsteady flow, forces, X(t) and Y(t), vary with time.Force coefficients:Cx = Cy = D(t)1/2  U2 dL(t)1/2  U2 dForce Time TraceCxCyDRAGLIFTAvg. Drag ≠ 0Avg. Lift = 0Alternate Vortex shedding causes oscillatory forces which induce structural vibrationsRigid cylinder is now similar to a spring-mass system with a harmonic forcing term.LIFT = L(t) = Lo cos (st+)s = 2 fsDRAG = D(t) = Do cos (2st+ )Heave Motion z(t)2( ) cos( ) sin( ) cosoooz t z tz t z tz t z tww ww w==-=-&&&“Lock-in”A cylinder is said to be “locked in” when the frequency of oscillation is equal to the frequency of vortex shedding. In this region the largest amplitude oscillations occur. v = 2fv = 2St (U/d)n = km + maShedding frequencyNatural frequencyof oscillationEquation of Cylinder Heave due to Vortex sheddingAdded mass termDampingIf Lv > b system is UNSTABLEkbmz(t)( )mz bz kz L t+ + =&& &( ) ( ) ( )a vL t L z t L z t=- +&& &( ) ( ) ( ) ( ) ( )a vmz t bz t kz t L z t L z t+ + =- +&& & && &{( ) ( ) ( ) ( ) ( ) 0a vm L z t b L z t k z t+ + - + =&& &1 4 2 4 3 1 42 43Restoring forceLIFT FORCE: Lift Force on a Cylinder( ) cos( )o oL t L tw y= +vif w w<( ) cos cos sin sino o o oL t L t L tw y w y= -2cos sin( ) ( ) ( )o o o oo oL LL t z t z tz zy yw w-= +&& &where v is the frequency of vortex sheddingLift force is sinusoidal component and residual force. Filtering the recorded lift data will give the sinusoidal term which can be subtracted from the total force.Lift Force Components:Lift in phase with acceleration (added mass):Lift in-phase with velocity:2( , ) cosoa oLM aaw yw=sinov oLLayw=-Two components of lift can be analyzed:(a = zo is cylinder heave amplitude)Total lift:( ) (( , () () , ))a vL t z t L aM za tww=- +&& &Total Force:•If CLv > 0 then the fluid force amplifies the motion instead of opposing it. This is self-excited oscillation.•Cma, CLv are dependent on  and a.( ) (( , () () , ))a vL t z t L aM za tww=- +&& &( )( )24212( , ) ( )( , ) ( )maLvd C a z tdU C a z tpr wr w=-+&&&Coefficient of Lift in Phase with VelocityVortex Induced Vibrations areSELF LIMITEDIn air: air ~ small, zmax ~ 0.2 diameterIn water: water ~ large, zmax ~ 1 diameterLift in phase with velocityGopalkrishnan (1993)Amplitude Estimation = b2 k(m+ma*)ma* = V Cma; where Cma = 1.0Blevins (1990)a/d = 1.29/[1+0.43 SG]3.35~SG=2  fn2 2m (2d2; fn = fn/fs; m = m + ma*^^__Drag AmplificationGopalkrishnan (1993)Cd = 1.2 + 1.1(a/d)VIV tends to increase the effective drag coefficient. This increase has been investigated experimentally.Mean drag:Fluctuating Drag:Cd occurs at twice the shedding frequency.~321Cd|Cd|~0.1 0.2 0.3fdUad= 0.75Single Rigid Cylinder Resultsa) One-tenth highest transverse oscillation amplitude ratiob) Mean drag coefficientc) Fluctuating drag coefficientd) Ratio of transverse oscillation frequency to natural frequency of cylinder1.01.0Flexible CylindersMooring lines and towing cables act in similar fashion to rigid cylinders except that their motion is not spanwise uniform.tTension in the cable must be considered when determining equations of motionFlexible Cylinder Motion TrajectoriesLong flexible cylinders can move in two directions and tend to trace a figure-8 motion. The motion is dictated bythe tension in the cable and the speed of