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**13.42 Lecture: Vortex Induced Vibrations**

Prof. A. H. Techet 18 March 2004

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**Classic VIV Catastrophe**

If ignored, these vibrations can prove catastrophic to structures, as they did in the case of the Tacoma Narrows Bridge in 1940.

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**Potential Flow U(q) = 2U sinq P(q) = 1/2 r U(q)2 = P + 1/2 r U2**

Cp = {P(q) - P }/{1/2 r U2}= 1 - 4sin2q

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**Axial Pressure Force i) Potential flow: -p/w < q < p/2**

(ii) Base pressure i) Potential flow: -p/w < q < p/2 ii) P ~ PB p/2 q 3p/2 (for LAMINAR flow)

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**Reynolds Number Dependency**

Transition to turbulence 300 < Rd < 3*105 3*105 < Rd < 3.5*106 3.5*106 < Rd

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**Shear 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 amplify

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Wake Instability

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**Classical Vortex Shedding**

Von Karman Vortex Street Alternately shed opposite signed vortices

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**Vortex shedding dictated by the Strouhal number**

St=fsd/U fs is the shedding frequency, d is diameter and U inflow speed

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**Additional VIV Parameters**

Reynolds Number subcritical (Re<105) (laminar boundary) Reduced Velocity Vortex Shedding Frequency S0.2 for subcritical flow

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**Strouhal Number vs. Reynolds Number**

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**Vortex Shedding Generates forces on Cylinder**

Uo Both 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.

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Vortex Induced Forces Due to unsteady flow, forces, X(t) and Y(t), vary with time. Force coefficients: D(t) 1/2 r U2 d L(t) 1/2 r U2 d Cx = Cy =

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Force Time Trace DRAG Cx Avg. Drag ≠ 0 LIFT Cy Avg. Lift = 0

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**Alternate 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

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**“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. Shedding frequency wv = 2p fv = 2p St (U/d) wn = k m + ma Natural frequency of oscillation

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**Equation of Cylinder Heave due to Vortex shedding**

z(t) m k b Added mass term Restoring force Damping If Lv > b system is UNSTABLE

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**Lift 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

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**Lift 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)

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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 w and a.

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**Coefficient of Lift in Phase with Velocity**

Vortex Induced Vibrations are SELF LIMITED In air: rair ~ small, zmax ~ 0.2 diameter In water: rwater ~ large, zmax ~ 1 diameter

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**Lift in phase with velocity**

Gopalkrishnan (1993)

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**a/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 = b 2 k(m+ma*) ma* = r V Cma; where Cma = 1.0

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**Drag Amplification ~ Cd = 1.2 + 1.1(a/d)**

VIV tends to increase the effective drag coefficient. This increase has been investigated experimentally. 3 2 1 Cd |Cd| ~ 0.1 0.2 0.3 fd U a d = 0.75 Gopalkrishnan (1993) Mean drag: Fluctuating Drag: ~ Cd = (a/d) Cd occurs at twice the shedding frequency.

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**Single Rigid Cylinder Results**

1.0 One-tenth highest transverse oscillation amplitude ratio Mean drag coefficient Fluctuating drag coefficient Ratio of transverse oscillation frequency to natural frequency of cylinder 1.0

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Flexible Cylinders Mooring lines and towing cables act in similar fashion to rigid cylinders except that their motion is not spanwise uniform. t Tension in the cable must be considered when determining equations of motion

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**Flexible Cylinder Motion Trajectories**

Long flexible cylinders can move in two directions and tend to trace a figure-8 motion. The motion is dictated by the tension in the cable and the speed of towing.

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**Wake Patterns Behind Heaving Cylinders**

f , A U 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.

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**Transition in Shedding Patterns**

A/d Williamson and Roshko (1988) Vr = U/fd f* = fd/U

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**Formation of ‘2P’ shedding pattern**

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**End Force Correlation Uniform Cylinder Tapered Cylinder**

Hover, Techet, Triantafyllou (JFM 1998) Tapered Cylinder

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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.

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**Oscillating Tapered Cylinder**

Strouhal Number for the tapered cylinder: St = fd / U where d is the average cylinder diameter. x d(x) U(x) = Uo

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**Spanwise Vortex Shedding from 40:1 Tapered Cylinder**

Rd = 400; St = 0.198; A/d = 0.5 Rd = 1500; St = 0.198; A/d = 0.5 Rd = 1500; St = 0.198; A/d = 1.0 dmax Techet, et al (JFM 1998) No Split: ‘2P’ dmin

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**Flow 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 cycles Techet, et al (JFM 1998)

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**DPIV of Tapered Cylinder Wake**

Digital particle image velocimetry (DPIV) in the horizontal plane leads to a clear picture of two distinct shedding modes along the cylinder. ‘2S’ z/d = 22.9 z/d = 7.9 ‘2P’ Rd = 1500; St = 0.198; A/d = 0.5

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**Principal Investigator:**

Vortex Dislocations, Vortex Splits & Force Distribution in Flows past Bluff Bodies D. Lucor & G. E. Karniadakis Objectives: Confirm numerically the existence of a stable, periodic hybrid shedding mode 2S~2P in the wake of a straight, rigid, oscillating cylinder Techet, 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 inflow Average Reynolds number is 400 VORTEX SPLIT Methodology: Parallel simulations using spectral/hp methods implemented in the incompressible Navier- Stokes solver NEKTAR NEKTAR-ALE Simulations Results: Existence and periodicity of hybrid mode confirmed by near wake visualizations and spectral analysis of flow velocity in the cylinder wake and of hydrodynamic forces Principal Investigator: Prof. George Em Karniadakis, Division of Applied Mathematics, Brown University

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**VIV Suppression Helical strake Shroud Axial slats Streamlined fairing**

Splitter plate Ribboned cable Pivoted guiding vane Spoiler plates

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**VIV Suppression by Helical Strakes**

Helical strakes are a common VIV suppresion device.

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**Oscillating Cylinders**

Parameters: y(t) Re = Vm d / n Reynolds # d b = d2 / nT Reduced frequency y(t) = a cos wt y(t) = -aw sin(wt) . KC = Vm T / d Keulegan- Carpenter # Vm = a w St = fv d / Vm Strouhal # n = m/ r ; T = 2p/w

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**Reynolds # vs. KC # )( ( ) Re = KC * b**

Re = Vm d / n = wad/n = 2p a/d d /nT 2 )( ( ) KC = Vm T / d = 2p a/d Re = KC * b b = d2 / nT Also effected by roughness and ambient turbulence

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**Forced Oscillation in a Current**

y(t) = a cos wt q w = 2 p f = 2p / T U Parameters: a/d, r, n, q Reduced velocity: Ur = U/fd Max. Velocity: Vm = U + aw cos q Reynolds #: Re = Vm d / n Roughness and ambient turbulence

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**Wall Proximity e + d/2 At e/d > 1 the wall effects are reduced.**

Cd, Cm increase as e/d < 0.5 Vortex shedding is significantly effected by the wall presence. In the absence of viscosity these effects are effectively non-existent.

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**Galloping is a result of a wake instability.**

m y(t), y(t) . Y(t) U -y(t) V a Resultant velocity is a combination of the heave velocity and horizontal inflow. If wn << 2p fv then the wake is quasi-static.

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Lift Force, Y(a) Y(t) V a Y(t) 1/2 r U2 Ap Cy = Cy Stable a Unstable

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**Galloping motion mz + bz + kz = L(t) ..**

z(t), z(t) . L(t) U -z(t) V a mz + bz + kz = L(t) .. . a L(t) = 1/2 r U2 a Clv - ma y(t) .. b k Cl (0) a Cl(a) = Cl(0) + + ... Assuming small angles, a: a ~ tan a = - z U . b = Cl (0) a V ~ U

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**Instability Criterion**

.. . b U (m+ma)z + (b + 1/2 r U2 a )z + kz = 0 ~ b U If b + 1/2 r U2 a < 0 Then the motion is unstable! This is the criterion for galloping.

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**b is shape dependent Shape Cl (0) a -2.7 U -3.0 -10 -0.66 1 1 2 2**

U 1 2 -3.0 1 4 -10 -0.66

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**Instability: ( ) U > b Cl (0) 1/2 r U a b = a < b 1/2 r a**

Critical speed for galloping: b 1/2 r a U > ( ) Cl (0) a

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**Torsional Galloping Both torsional and lateral galloping are possible.**

FLUTTER occurs when the frequency of the torsional and lateral vibrations are very close.

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**Galloping vs. VIV Galloping is low frequency**

Galloping is NOT self-limiting Once U > Ucritical then the instability occurs irregardless of frequencies.

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References Blevins, (1990) Flow Induced Vibrations, Krieger Publishing Co., Florida.

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