113.42 Lecture:Vortex Induced VibrationsProf. A. H. Techet18 March 2004Classic VIV CatastropheIf ignored, these vibrations can prove catastrophic to structures, as they did in the case of the Tacoma Narrows Bridge in 1940.2Potential 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 - 4sin2θAxial Pressure Force i) Potential flow:-π/w < θ < π/2ii) P ~ PBπ/2 ≤θ≤3π/2(for LAMINAR flow)Base pressure(i) (ii)3Reynolds 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 InstabilityClassical Vortex SheddingVon Karman Vortex StreetlhAlternately shed opposite signed vortices5Vortex shedding dictated by the Strouhal numberSt=fsd/Ufsis 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 =effectsviscouseffects inertialRe ≈=vUDDfUVnrn=6Strouhal 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.7Vortex Induced ForcesDue to unsteady flow, forces, X(t) and Y(t), vary with time.Force coefficients:Cx= Cy= D(t)1/2 ρ U2dL(t)1/2 ρ U2dForce Time TraceCxCyDRAGLIFTAvg. Drag ≠ 0Avg. Lift = 08Alternate 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 (ωst+ψ)ωs= 2π fsDRAG = D(t) = Do cos (2ωst+ ψ)Heave Motion z(t)2() cos() sin() cosooozt z tzt z tzt z tωωωωω==−=−“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 oscillation9Equation of Cylinder Heave due to Vortex sheddingAdded mass termDampingIf Lv> b system is UNSTABLEkbmz(t)()mz bz kz L t++= () () ()avLt L zt L zt=−+ () () () () ()avmz t bz t kz t L z t L z t++=− +   N( ) () ( ) () () 0avmLzt bLzt kzt++−+=  Restoring forceLIFT FORCE: Lift Force on a Cylinder() cos( )ooLt L tωψ=+vifωω<() cos cos sin sinooooLt L t L tωψωψ=−2cos sin() () ()oo ooooLLLt zt ztzzψψωω−=+ where ωvis the frequency of vortex sheddingLift force is sinusoidal component and residual force. Filteringthe recorded lift data will give the sinusoidal term which can be subtracted from the total force.10Lift Force Components:Lift in phase with acceleration (added mass):Lift in-phase with velocity:2(,) cosoaoLMaaωψω=sinovoLLaψω=−Two components of lift can be analyzed:(a = zois cylinder heave amplitude)Total lift:() ((, ()() , ))a vLt zt LaM za tωω=− + Total Force:• If CLv> 0 then the fluid force amplifies the motion instead of opposing it. This is self-excited oscillation.• Cma, CLvare dependent on ωand a.() ((, ()() , ))a vLt zt LaM za tωω=− + ()()24212(,)()(,)()maLvdC aztdU C a z tπρωρω=−+11Coefficient of Lift in Phase with VelocityVortex Induced Vibrations areSELF LIMITEDIn air: ρair~ small, zmax~ 0.2 diameterIn water: ρwater~ large, zmax~ 1 diameterLift in phase with velocityGopalkrishnan (1993)12Amplitude Estimationζ = b2 k(m+ma*)ma*= ρ V Cma; where Cma= 1.0Blevins (1990)a/d= 1.29/[1+0.43 SG]3.35~SG=2 π fn22m (2πζ)ρ 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:Cdoccurs at twice the shedding frequency.~321Cd|Cd|~0.1 0.2 0.3fdUad= 0.7513Single 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 motion14Flexible 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 towing.• 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 , AUU15Transition in Shedding PatternsWilliamson and Roshko (1988)A/df* = fd/UVr = U/fdFormation of ‘2P’ shedding pattern16End Force CorrelationUniform CylinderTapered CylinderHover, Techet, Triantafyllou (JFM 1998)VIV 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.17Strouhal Number for the tapered cylinder:St= fd / Uwhere d is the average cylinder diameter.Oscillating Tapered Cylinderxd(x)U(x) = UoSpanwise Vortex Shedding from 40:1 Tapered CylinderTechet, et al (JFM 1998)dmaxRd= 400; St = 0.198; A/d = 0.5Rd= 1500; St = 0.198; A/d = 0.5Rd= 1500; St = 0.198; A/d = 1.0dminNo Split: ‘2P’18Flow Visualization Reveals: A Hybrid Shedding Mode• ‘2P’ pattern results at the smaller end• ‘2S’ pattern at the larger end• This mode is seen to be repeatable over multiple cyclesTechet, et al (JFM 1998)DPIV of Tapered Cylinder Wake‘2S’‘2P’Digital particle image velocimetry (DPIV) in the