113.42 Lecture:Vortex Induced VibrationsA. H. Techet13 March 2003Petrogras Rig Sinking off BrazilWhy study VIV?www.offshore-technology.comMany direct applications to Ocean Engineering:Cables, risers, platforms, islands in the wind2Classic VIV CatastropheIf ignored, these vibrations can prove catastrophic to structures, as they did in the case of the Tacoma Narrows Bridge in 1940.Classical Vortex SheddingVon Karman Vortex StreetlhPotential FlowU(θ) = 2U∞sinθP(θ) = 1/2 ρ U(θ)2 = P∞+ 1/2 ρ U∞2Cp = {P(θ) - P ∞}/{1/2 ρ U∞2}= 1 - 4sin2θ3Axial Pressure Force i) Potential flow:-π/w < θ < π/2ii) P ~ PBπ/2 ≤θ≤3π/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 amplify4Wake InstabilityAlternately Shed Opposite Sign VorticesImage: Professor T.T. Lim - University of MelbourneVortex shedding dictated by the Strouhal numberSt=fsd/Ufsis the shedding frequency, d is diameter and U inflow speed5• 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.6Alternate 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 cosωstωs= 2π fsDRAG = D(t) = Do cos (2ωst)LIFT FORCE: Y(t) = Yocos (ωt + ψo) if ω < ωv;Y(t) = Yocos ωt cos ψo-Yosin ωt sin ψo-Yocos ψoz(t) + Yosin ψoz(t) where ωv= freq of vortex sheddingzoω2zoωEquation for cylinder heavez = zocosωtz = -zoωsin ωtz = -zoω2 cosωt...Y(t) =.. .Equation of Cylinder Heave due to Vortex sheddingmz + bz + kz = Y(t)...Y(t) = -YAz(t) + YVz(t).. .mz + bz + kz = -YAz(t) + YVz(t)... ..... .(m + YA) z + (b-Yv) z + kz = 0Added mass termdampingIf Yv> b system is UNSTABLEkbmz(t)7Coefficient of Lift in Phase with VelocityVortex Induced Vibrations areSELF LIMITEDIn air: ρair~ small, zmax~ 0.2 diameterIn water: ρwater~ large, zmax~ 1 diameter1/2 ρ U2d“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.ωv= 2π fv= 2π St(U/d)ωn= km + maShedding frequencyNatural frequencyof oscillationSingle 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 cylinder8VIV in the Ocean• Non-uniform currents effect the spanwise vortex shedding on a cable or riser. • The frequency of shedding can be different along length.• This leads to “cells” of vortex shedding with some length, lc.Flexible 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 Trajectories9Freely Vibrating CylindersFreely vibrating cylinders closely follow the critical curve. Williamson and Roshko (1988)• Shedding patterns in the wake of oscillating cylinders are distinct and exist for a certain range of heave frequencies and amplitudes.• The different modes have a great impact on structural loading.Wake Patterns Behind Heaving Cylinders‘2S’‘2P’f , Af , AUUTransition in Shedding PatternsWilliamson and Roshko (1988)A/df* = fd/UVr = U/fd10Formation of ‘2P’ shedding patternEnd Force CorrelationUniform CylinderTapered CylinderHover, Techet, Triantafyllou (JFM
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