5Reynolds Number Dependency Transition to turbulence300 < Rd < 3*1053*105 < Rd < 3.5*1063.5*106 < Rd
6Shear layer instability causes vortex roll-up Flow speed outside wake is much higher than insideVorticity gathers at downcrossing points in upper layerVorticity gathers at upcrossings in lower layerInduced velocities (due to vortices) causes this perturbation to amplify
12Vortex Shedding Generates forces on Cylinder UoBoth Lift and Drag forces persist on a cylinder in cross flow. Lift is perpendicular to the inflow velocity and drag is parallel.FL(t)FD(t)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.
13Vortex Induced ForcesDue to unsteady flow, forces, X(t) and Y(t), vary with time.Force coefficients:D(t)1/2 r U2 dL(t)1/2 r U2 dCx =Cy =
14Force Time TraceDRAGCxAvg. Drag ≠ 0LIFTCyAvg. Lift = 0
15Alternate Vortex shedding causes oscillatory forces which induce structural vibrations Heave Motion z(t)LIFT = L(t) = Lo cos (wst+)DRAG = D(t) = Do cos (2wst+ )Rigid cylinder is now similar to a spring-mass system with a harmonic forcing term.ws = 2p fs
16“Lock-in” wv = 2p fv = 2p St (U/d) 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.Sheddingfrequencywv = 2p fv = 2p St (U/d)wn =km + maNatural frequencyof oscillation
17Equation of Cylinder Heave due to Vortex shedding z(t)mkbAdded mass termRestoring forceDampingIf Lv > b system isUNSTABLE
18Lift Force on a Cylinder Lift 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:where wv is the frequency of vortex shedding
19Lift Force Components: Two components of lift can be analyzed:Lift in phase with acceleration (added mass):Lift in-phase with velocity:Total lift:(a = zo is cylinder heave amplitude)
20Total 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 w and a.
21Coefficient of Lift in Phase with Velocity Vortex Induced Vibrations areSELF LIMITEDIn air: rair ~ small, zmax ~ 0.2 diameterIn water: rwater ~ large, zmax ~ 1 diameter
22Lift in phase with velocity Gopalkrishnan (1993)
23a/d = 1.29/[1+0.43 SG]3.35 Amplitude Estimation ~ Blevins (1990) _ _ 2m (2pz)r d2^^SG=2 p fn2; fn = fn/fs; m = m + ma*z =b2 k(m+ma*)ma* = r V Cma; where Cma = 1.0
24Drag Amplification ~ Cd = 1.2 + 1.1(a/d) VIV tends to increase the effective drag coefficient. This increase has been investigated experimentally.321Cd|Cd|~0.10.20.3fdUad= 0.75Gopalkrishnan (1993)Mean drag:Fluctuating Drag:~Cd = (a/d)Cd occurs at twice theshedding frequency.
25Single Rigid Cylinder Results 1.0One-tenth highest transverse oscillation amplitude ratioMean drag coefficientFluctuating drag coefficientRatio of transverse oscillation frequency to natural frequency of cylinder1.0
26Flexible 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 motion
27Flexible Cylinder Motion Trajectories Long flexible cylinders can move in two directions andtend to trace a figure-8 motion. The motion is dictated bythe tension in the cable and the speed of towing.
28Wake Patterns Behind Heaving Cylinders f , AUShedding 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.
29Transition in Shedding Patterns A/dWilliamson and Roshko (1988)Vr = U/fdf* = fd/U
32VIV in the OceanNon-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.
33Oscillating Tapered Cylinder Strouhal Number for the tapered cylinder:St = fd / Uwhere d is the averagecylinder diameter.xd(x)U(x) = Uo
35Flow Visualization Reveals: A Hybrid Shedding Mode ‘2P’ pattern results at the smaller end‘2S’ pattern at the larger endThis mode is seen to be repeatable over multiple cyclesTechet, et al (JFM 1998)
36DPIV of Tapered Cylinder Wake Digital particle imagevelocimetry (DPIV) in the horizontal plane leads to a clear picture of two distinct shedding modes along the cylinder.‘2S’z/d = 22.9z/d = 7.9‘2P’Rd = 1500; St = 0.198; A/d = 0.5
37Principal Investigator: Vortex Dislocations, Vortex Splits & Force Distribution in Flows past Bluff Bodies D. Lucor & G. E. KarniadakisObjectives:Confirm numerically the existence of a stable, periodic hybrid shedding mode 2S~2P in the wake of a straight, rigid, oscillating cylinderTechet, Hover and Triantafyllou (JFM 1998)Approach:DNS - Similar conditions as the MIT experiment (Triantafyllou et al.)Harmonically forced oscillating straight rigid cylinder in linear shear inflowAverage Reynolds number is 400VORTEX SPLITMethodology:Parallel simulations using spectral/hp methods implemented in the incompressible Navier- Stokes solver NEKTARNEKTAR-ALE SimulationsResults:Existence and periodicity of hybrid mode confirmed by near wake visualizations and spectral analysis of flow velocity in the cylinder wake and of hydrodynamic forcesPrincipal Investigator:Prof. George Em Karniadakis, Division of Applied Mathematics, Brown University
39VIV Suppression by Helical Strakes Helical strakes are acommon VIV suppresiondevice.
40Oscillating Cylinders Parameters:y(t)Re = Vm d / nReynolds #db = d2 / nTReducedfrequencyy(t) = a cos wty(t) = -aw sin(wt).KC = Vm T / dKeulegan-Carpenter #Vm = a wSt = fv d / VmStrouhal #n = m/ r ; T = 2p/w
41Reynolds # vs. KC # )( ( ) Re = KC * b Re = Vm d / n = wad/n = 2p a/d d /nT2)(()KC = Vm T / d = 2p a/dRe = KC * bb = d2 / nTAlso effected by roughness and ambient turbulence
42Forced Oscillation in a Current y(t) = a cos wtqw = 2 p f = 2p / TUParameters:a/d, r, n, qReduced velocity: Ur = U/fdMax. Velocity: Vm = U + aw cos qReynolds #: Re = Vm d / nRoughness and ambient turbulence
43Wall Proximity e + d/2 At e/d > 1 the wall effects are reduced. Cd, Cm increase as e/d < 0.5Vortex shedding is significantly effected by the wall presence.In the absence of viscosity these effects are effectively non-existent.
44Galloping is a result of a wake instability. my(t), y(t).Y(t)U-y(t)VaResultant velocity is a combination of the heave velocity and horizontal inflow.If wn << 2p fv then the wake is quasi-static.
45Lift Force, Y(a)Y(t)VaY(t)1/2 r U2 ApCy =CyStableaUnstable
46Galloping motion mz + bz + kz = L(t) .. z(t), z(t).L(t)U-z(t)Vamz + bz + kz = L(t)...aL(t) = 1/2 r U2 a Clv - ma y(t)..bk Cl (0) aCl(a) = Cl(0) ++ ...Assuming small angles, a:a ~ tan a = -zU.b = Cl (0) aV ~ U
47Instability Criterion ...bU(m+ma)z + (b + 1/2 r U2 a )z + kz = 0~bUIfb + 1/2 r U2 a< 0Then the motion is unstable!This is the criterion for galloping.
48b is shape dependent Shape Cl (0) a -2.7 U -3.0 -10 -0.66 1 1 2 2 U12-3.014-10-0.66
49Instability: ( ) U > b Cl (0) 1/2 r U a b = a < b 1/2 r a Critical speed for galloping:b1/2 r aU >( ) Cl (0) a
50Torsional Galloping Both torsional and lateral galloping are possible. FLUTTER occurs when the frequency of the torsionaland lateral vibrations are very close.
51Galloping vs. VIV Galloping is low frequency Galloping is NOT self-limitingOnce U > Ucritical then the instability occurs irregardless of frequencies.