Review on Vortex-Induced Vibration for Wave math.mit.edu/classes/18.376/TermPapers/18376_Term_Paper_Rao.pdf1 Review on Vortex-Induced Vibration for Wave Propagation Class By Zhibiao

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    Review on Vortex-Induced Vibration for Wave Propagation Class

    By Zhibiao Rao

    Whats Vortex-Induced Vibration?

    In fluid dynamics, vortex-induced vibrations (VIV) are motions induced on bodies interacting

    with an external fluid flow.

    The highly specialized subject of VIV is part of a number of disciplines, incorporating fluid

    mechanics, structural mechanics, vibrations, computational fluid dynamics, acoustics, wavelet

    transforms, complex demodulation analysis, statistics, and smart materials. They occur in many

    engineering situations, such as bridges, stacks, transmission lines, aircraft control surfaces,

    offshore structures, thermo wells, engines, heat exchangers, marine cables, towed cables, drilling

    and production risers in petroleum production, mooring cables, moored structures, tethered

    structures, buoyancy and spar hulls, pipelines, cable-laying, members of jacketed structures, and

    other hydrodynamic and hydro acoustic applications (Sarpkaya, 2004).

    How and Why does VIV Occur?

    When a flow passes through a circular cylinder in the direction perpendicular to its axis, the VIV

    will be seen. Thats because real fluids always present some viscosity, the flow around the

    cylinder will be slowed down while in contact with its surface, forming the so called boundary

    layer. At some points, however, this boundary layer can separate from the body because of its

    excessive curvature. Vortices are then formed changing the pressure distribution along the

    surface. These shedding vortices are called Von Karman Vortex Street. When the vortices are not

    formed symmetrically around the body, different lift forces develop on each side of the body,

    thus leading to motion transverse to the flow. This motion changes the nature of the vortex

    formation in such a way as to lead to limited motion amplitude.

    How to Model VIV of a Rigid Cylinder?

    The force induced by vortex is periodic. Thus an equation of motion is introduced to represent

    VIV of a cylinder oscillating in the transverse Y direction (normal to the flow) as follows:

    + + = (1)

    Where m is the structural mass, c the damping, k the constant spring stiffness, and F the fluid

    force in the transverse direction.

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    The principal assumption is that the cross-flow excitation is a steady state periodic force, which

    may be decomposed into a Fourier series. By the principle of superposition the response to each

    Fourier component may be computed individually. The hydrodynamic force shown on the right

    hand side of Equation (1) is in this case assumed to be the principal Fourier component of the

    cross-flow excitation. A good approximation to the force is given by

    = + (2)

    The steady state particular solution to this equation is given by

    = (3)

    Where is 2 and f the body oscillation frequency.

    Plug Equations (2) and (3) into the Equation (1), we can get two equations, in which the time

    dependent terms have cancelled out. Equation (4) establishes the equilibrium relationship

    between the component of the fluid excitation in phase with stiffness and inertial forces in the

    system.

    = (4)

    Equation (5) expresses the dynamic equilibrium that exists between the fluid lift force/length in

    phase with the cylinder velocity and , the force/length required to drive the dashpot.

    = (5)

    Both Equations (4) and (5) are valid at any steady state excitation frequency.

    An Example for a Typical VIV of Rigid Cylinder

    Feng (1968) conducted in air with a single-degree-of freedom flexible cylinder with a relatively

    large mass ratio and relatively large Reynolds number from 1x104

    to 5x104. Figure 1 is the

    classical self-excited oscillation plot showing the lock-in nature of the cylinder oscillation in

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    self-excited conditions. The two features of the plot show the reduced velocity (Vr=U/fairD)

    plotted against both the amplitude-to-diameter ration (A/D) and the ratio of vortex-shedding

    frequency to natural frequency (fex /f

    air). The plot shows the sudden increase in oscillation

    amplitude at Vr = 5. This lock-on continues to Vr=7.

    Figure 2 Lift coefficient is function of VIV amplitude (Vandiver, 2012)

    The Figure 1 shows that VIV is not a small perturbation superimposed on a mean steady motion.

    It is an inherently nonlinear, self-governed or self-regulated, multi-degree-of-freedom

    phenomenon. It presents unsteady flow characteristics manifested by the existence of two

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    unsteady shear layers and large-scale structures. In other words, the VIV amplitude has a limited

    value. However, from the Equation (2), lift coefficient seems be independent of amplitude. The

    corresponding lift coefficient model should be adjusted to capture the nonlinear, self-governed or

    self-regulated phenomenon. The real experimental data shows that the lift coefficient is the

    function of VIV amplitude, shown in Fig.2.

    What a Typical VIV Looks Like for Long Flexible Risers?

    In 2006, Prof. Vandiver at MIT conducted a large scale VIV experiment in the Gulf Stream,

    which is sponsored by DEEPSTAR. The set-up for the experiment is shown in Figure 3. The

    experiment was conducted on the Research Vessel F. G. Walton Smith from the University of

    Miami using a fiber glass composite pipe 500.4 feet in length and 1.43 inches in outer diameter.

    A railroad wheel weighing 805 lbs was attached to the bottom of the pipe to provided tension.

    Strain gauges were used to measure the VIV response of the pipe. Eight optical fibers containing

    thirty five strain gauges each were embedded in the outer layer of the composite pipe. The

    gauges had a resolution of 1 micro-strain. The R/V F. G. Walton Smith is equipped with

    Acoustic Doppler Current Profilers (ADCP). During the experiments, the ADCP was used to

    record the current velocity and direction along the length of the pipe.

    Figure 3 Setup for the Gulf Stream Experiments 2006.

    Prior to the Gulf Stream experiments, it was expected that the VIV response of the long flexible

    cylinder towed in sheared current would consist of multiple standing wave modes and

    frequencies responding simultaneously. However, once the data from the second Gulf Stream

    experiment was analyzed, it became evident that the response was not dominated by standing

    wave characteristics. It was found that the response is often characterized by the presence of

    traveling waves. The presence of traveling waves in the measured response can be inferred by

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    observing the frequency content of the PSD of strain at different locations and the spatial

    variation of the cross-flow strain. Figures 4 and 5 show the strong traveling waves in the strain

    contour plot. Figure 4 shows that waves were generated from the bottom of the riser and then

    propagated upwards. While Figure 5 shows that waves were generated from the region in the

    middle of the pipe and then traveled towards two ends of pipes.

    Figure 4 Strain contour plot of all sensor locations in quadrant 4 for case 20061023205557

    (Jaiswal, 2009).

    Figure 5 Strain contour plot of all sensor locations in quadrant 4 for case 20061022153003

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    (Jaiswal, 2009).

    How to Model VIV for Long Flexible Risers?

    For any simple-support flexible risers subjected to VIV, the equation of motion may be written

    as:

    ! "#$%,'"'# + (%"$%,'

    "' + )*"+$%,'

    "%+ ,"#$%,'

    "%# = -%, ' (6)

    Where m, EI, and T are the constant mass per unit length, bending stiffness and tension,

    respectively; y(x, t) is the displacement in the cross-flow direction; c(x) is the nonlinear damping

    per unit length; F(x, t) is the excitation force.

    The challenging work to predict the VIV response of a long flexible riser is to build reasonable

    mathematical models for both damping c(x) and lift force F(x, t).

    For a long flexible riser subjected to VIV, the whole riser may be split into several regions. The

    regions where the shedding of vortices is correlated with the structural responses are designated

    power-in regions. Regions on the pipe that are not a source of power are designated power-out

    regions. Some assumptions are made to build VIV prediction model.

    The hydrodynamic contribution to the damping term, c(x), is assumed to come only from the

    power-out region. The lift force per unit length F(x, t) is assumed to act only in the power-in

    region. Within power-in regions where the modal amplitude may be quite large, the lift

    coefficient may become negative, thus extracting power from the vibration (Rao, 2014).

    (1) Hydrodynamic Damping Model

    The Venugopal hydrodynamic damping model was widely used in VIV field to estimate the

    hydrodynamic damping in the power-out region. This model is based on the VIV

    experiments of rigid cylinders.

    Hydrodynamic damping model when reduced velocity ./ < 1

    ( = 234 + 5/6789 (7)

    234 = :;78#

    # 2?: + 534 @A8B

    #C (8)

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    Hydrodynamic damping model when reduced velocity ./ > 7

    ( = 5/F79#/: (9)

    Where HIJ is the still water contribution; KL is an empirical coefficient taken to be 0.18; HNO = OP

    Q , a vibration Reynolds number; IJ and KR are two other empirical

    coeff