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.
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
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
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.
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
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
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
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
Figure 5 Strain contour plot of all sensor locations in quadrant 4 for case 20061022153003
How to Model VIV for Long Flexible Risers?
For any simple-support flexible risers subjected to VIV, the equation of motion may be written
! "#$%,'"'# + (%"$%,'
"' + )*"+$%,'
"%# = -%, ' (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
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