IMECE2012

IMECE2012
Proceedings of ASME 2012 International Mechanical Engineering Congress and Exposition
IMECE 2012
November 9-15, 2012, Houston, Texas, USA
IMECE2012-93334
MASSIVELY PARALLEL SPECTRAL DIFFERENCE SOLVER FOR SIMULATING
VORTEX-INDUCED VIBRATIONS OF CIRCULAR CYLINDERS
Chunlei Liang∗, Andrew S. DeJong
Department of Mechanical and Aerospace Engineering
George Washington University
Washington, DC 20052
ABSTRACT
This paper presents high-fidelity simulations of the VortexInduced Vibration (VIV) phenomena using a new computational
model based on the high-order spectral difference (SD) method
on unstructured grids. The SD method has shown promise in the
past as a highly accurate, yet sufficiently fast method for solving
unsteady viscous compressible flows. A Riemann solver is used
to compute the inviscid fluxes at the cell interfaces, and the viscous fluxes are computed using an averaging mechanism based
on the fluxes from the two cells that share the interface. A thirdorder Runge-Kutta scheme is used to advance time. In this viscous, compressible flow solver, the displacement of an elastically
mounted bluff body has been coupled to the lift force created
by vortex shedding. The solver is validated by correlating the
lift and drag coefficients with previous published results of two
cases: single rigid cylinder, and single elastically mounted cylinder. The rigid cylinder case validates the accuracy of the fluid
solver. The elastically mounted case validates the fluid-structure
coupling. We simulate the phenomenon of wake galloping with
two cylinders after the solver has been validated using single
cylinder VIV phenomenon.
do f Degrees of Freedom
fN , f ∗ , FN Cylinder natural frequency, frequency ratio, and normalized frequency
k Dimensional spring constant
m, m∗ Dimensional and normalized cylinder mass
Re Reynolds number
SD Spectral difference
St Strouhal number
t, T dimensional and normalized time
u,U ∗ dimensional and normalized velocity
V IV Vortex-induced vibration
y,Y Dimensional and normalized cylinder displacement
µ Fluid viscosity
ρ Fluid density
INTRODUCTION
When a structure or bluff body is subjected to a wind load, it
is possible for it to undergo unintended vibrations due to aerodynamic instability phenomena. The main types of such phenomena are vortex-induced vibrations (VIV), galloping and fluttering, and wake galloping. VIV is the most classic type of aerodynamic instability phenomenon and is frequently used to generally
describe the other types. VIV occurs when fluid flow over bluff
bodies forms unsteady vortices on the trailing side of the body.
As the vortices shed off of the body, oscillatory forces are applied to the body. These oscillatory forces cause small amplitude
vibrations in a plane normal to the fluid flow. If the oscillatory
forces are applied at a frequency near the structure’s natural fre-
NOMENCLATURE
A∗ Normalized maximum cylinder displacement
c, ζ Dimensional and normalized damping coefficient
CL ,CD Lift and drag coefficients
d Cylinder diameter
∗ Corresponding
author. Email: [email protected]
1
c 2012 by ASME
Copyright FIGURE 1. Flutter aerodynamic instability phenomenon in an airfoil.
As the airfoil lifts, the torsional inertia forces the airfoil to twist downward, causing a plunging motion [2]
quency, resonance can severely increase the amplitude of vibration. Galloping and flutter are a special case of VIV where the
derivative of the steady state lift coefficient is negative. Thus,
as the structure oscillates in one direction, the lift forces generated by the body’s shape decrease and become negative to put
it back. These types of phenomena are frequently called diverging and self-regulating since the structure’s lift coefficient causes
increased divergence from its resting place and eventually cause
its return. In the case of flutter, this motion is accompanied by
a torsional twist in the body, causing a pitching and plunging
motion, as shown in Fig. 1. As the body lifts and twists, the torsional stiffness forces the body to twist back, causing a plunging
motion. This is the phenomenon witnessed during the Tacoma
Narrows Bridge accident in 1940 [1].
In 2008, a research group from the University of Michigan
proposed a new device called VIVACE [3] for harnessing renewable energy using VIV. Wake galloping occurs when multiple
tandem bodies undergo VIV. The downstream bluff body is affected by the shedding wake from the upstream body. This can
cause an increase in vibration amplitude in the downstream body.
If the distance between the two bodies is too great (about 6 times
the body diameter) or too small (about 2 times the body diameter) wake galloping may not occur [4].
If the body is not designed properly, these phenomena can
have disastrous effects. Historically, VIV (and its derivative
forms) have been intentionally designed out of systems with prejudice due to the extreme destructive power displayed during engineering failures involving VIV. This power has been seen in
simple cases such as water flowing past riser tubes bringing oil
up from the seabed, flow around heat exchanger tubes, and wind
flowing across launch vehicles, and in more famous cases such as
the structural failure of Ferrybridge power station cooling towers
in England which was resulted from VIV of turbulent flow [5].
While engineering practice seeks to design out the VIV phenomena, there has been considerable research into understanding the intense non-linear, unsteady, multi-dimensional flow that
causes VIV. The experiments conducted by Feng in 1963 are
generally considered the first modern study of VIV. Feng introduces many of the concerns and questions about VIV which have
FIGURE 2. X-Y plots for two studies of 2dof VIV where the cylinder
oscillated in a figure-8 (top) and half-circle (bottom) [12]
still not been answered. Even though theoretical, experimental,
and numerical research into VIV has been going on for decades,
the fluid behavior and structural responses cannot be fully explained, only empirically predicted. One of the main problems
is that VIV is extremely sensitive to the physical properties of
the system. Making simple assumptions, such as only allowing
vibrations in the plane normal to fluid flow, can cause wide divergence in results. Much of the research by Feng [6], Khalak
and Williamson [7], Pan et al. [8], Zhao and Cheng [9], Zhoa
et al. [10], and Ji et al. [11] has addressed a single cylinder in a
cross flow current undergoing one-degree of freedom (1dof) vibrations. In these studies, the cylinder is in some way restrained
from vibrating parallel to fluid flow. However, VIV is inherently
multi-dimensional and many situations in reality (such as suspension cables on bridges) do not display that constraint. Consequently, there have been 2-degree of freedom (2dof) studies
by Mittal and Kumar [12], Jeon and Gharib [13], Jauvtis and
Williamson [14], and Prasanth et al. [15] which address transverse and downstream vibrations of the body. These researchers
notice specific cases where the phase between the transverse and
downstream oscillations is offset such that the cylinder moves in
a circular, half-circular, or figure-8 motions, as shown in Fig. 2.
Even though the physical phenomena in the 1dof and 2dof
systems are not well understood, work has been done on study2
c 2012 by ASME
Copyright FN =
fN d
1
= ∗
u
U
(8)
with,
St = ( fshed d)/u
r
1 k
fN =
2π m
FL
CL = 1 2
ρu
d
2
ρud
Re =
µ
FIGURE 3. Schematic of wake galloping energy harvesting method
proposed by Jung and Lee [4]
ing VIV wake galloping effects on multiple cylinders within one
domain. This allows for anywhere between 0 to 4 dof to be investigated, although, in these cases the leading cylinder is usually
fixed and the downstream cylinder attached elastically some distance behind the first (measured in intervals of cylinder diameter,
d). The work done by Shimizu [16], Kim [1], Assi et al. [17], and
Jung and Lee [4] suggest that wake galloping can be used a form
of energy harvesting by converting the oscillations into electrical
current, as seen in Fig. 3. Understanding and predicting the wake
galloping response to the system’s properties is still quite difficult, yet Jung and Lee report a field test where the wake galloping
device is mounted on a bridge to harvest wind energy and possibly power remote sensor devices that would otherwise require
connection to the power grid.
In addition to the difficulty presented by the VIV sensitivity
to physical properties and assumptions, a second problem is the
wide range of variables to control. The above mentioned studies
have observed extremely non-linear VIV effects to varying free
stream velocities, cylinder sizes, spring and damper coefficients,
and Reynolds numbers. To help gather and compare data, a set
of non-dimensionalized variables are constructed in Eqn. 1-8 [15,
18].
y
d
tu
T=
d
4m
m∗ =
ρπd 2
c
ζ= √
2 km
u
∗
U =
fN d
ymax
A∗ =
d
f
shed
f∗ =
fN
Y=
(9)
(10)
(11)
(12)
where y is the cross-flow displacement, t is the time, u is the free
stream velocity, d is the cylinder diameter, m is the cylinder mass,
ρ is the fluid density, c is the structural damping coefficient, k is
the structural spring coefficient, fN is the structural natural frequency, ymax is the maximum amplitude of cross-flow vibration,
fshed is the vortex shedding frequency, St is the Strouhal number,
CL is the aerodynamic lift coefficient of the lift force FL , Re is the
Reynolds number, and µ is the fluid viscosity.
Even with these normalized values, it is difficult to create
a single chart, table, or medium to effectively communicate the
cylinder response of each experiment. This is partly due to the
nonlinear amplitude response. For example, a typical VIV amplitude response (A*) for a variety of normalized flow velocities (U*) can be seen in Fig. 4. There are clearly three nonlinear regimes, namely the initial branch, lower branch, and upper
branch. However, this only holds true if the mass-damping coefficient (m*) is sufficiently small, in the order of 101 [18]. When
m* increases in the order of 102 the regimes shift, as shown in
Fig. 5.
Although the non-linearity cannot necessarily be explained,
it has been well established empirically. This allows numerical
solutions to be compared with experimental results for validation
of their methods. The present work investigates 1dof transverse
vibrations in elastically mounted rigid cylinders using a massively parallel high-order spectral difference (SD) solver. The
code is built for moving and deforming unstructured grids [19]
which allows the solver to analyze a wide variety of bluff body
shapes and configurations, including square cylinders, tandem
cylinders, or even clusters of parallel cylinders. To validate the
solver, benchmark cases using a single elastically mounted cylinder have been compared to results from previous literature.
(1)
(2)
(3)
(4)
(5)
COMPUTATIONAL MODEL
Spectral Difference Method
The SD method has shown promise in the past as a highly
accurate, yet sufficiently fast method for solving unsteady viscous flows. More detailed derivation and analysis of the viscous,
(6)
(7)
3
c 2012 by ASME
Copyright FIGURE 6.
Diagram of mass-spring-damper system
Mass-Spring-Damper System
To allow the cylinder to vibrate, a mass-spring-damper system is modeled as shown in Fig. 6. The upstream cylinder is
rigid. The downstream cylinder is free to translate vertically. A
spring and damper connect the downstream cylinder to the rigid
ground. In some cases of this study, only a single cylinder is
used. In which case, the first cylinder is elastically mounted and
the second cylinder is removed.
The equation to describe the elastically mounted cylinder
can be written as:
FIGURE 4. Amplitude response regimes for varying normalized velocities with low mass-damping
mÿ + cẏ + ky = FL
(13)
Using the Newmark-β method, we can discretize Eqn. 13 in
the time domain into the following second order approximations.
ÿn+1 = (FL − kyn − cẏn )/m
(14)
ẏn+1 = ẏn + ∆t/2(ÿn + ÿn+1 )
(15)
yn+1 = yn + ∆t ẏn + (1/2 − β )∆t 2 ÿn + β ∆t 2 ÿn+1
(16)
where β is the Newmark variable, which is typically set to
β = 1/4, and n is the number of time steps ∆t. The Newmark-β
method is stable under the condition that [21]:
FIGURE 5. Amplitude response regimes for varying normalized velocities with high mass-damping
k
2
m ∆t
1 + mk ∆t 2 β
compressible SD method used in this solver can be found elsewhere [19,20]. A Riemann solver is used to compute the inviscid
fluxes at the cell interfaces, and the viscous fluxes are computed
using an averaging mechanism based on the fluxes from the two
cells that share the interface. An efficient and compact mathematical formulation of the spectral difference method on moving
and deforming grids can be found in [19]. For cases considered
in this paper, a third-order Runge-Kutta scheme is used for time
marching.
< 4.0
(17)
To easily compare results between different studies, it is useful to now non-dimensionalize the variables by normalizing them
with the system parameters in Eqn. 1-8. Based on Eqn. 1 and 2
it can be seen that,
ẏ = uẎ
u2
ÿ = Ÿ
d
4
(18)
(19)
c 2012 by ASME
Copyright Therefore, Eq. 13 can be rewritten,
Ÿ +
cd
kd 2
1 ρd 2
Ẏ + 2 Y =
mu
mu
2 mCL
(20)
By substituting Eq. 5 and 8,
Ÿ +
c
k 2
1 ρd 2
CL
FN Ẏ +
F
Y
=
m fN
2 m
m fN2 N
(21)
Lastly, by substituting Eq. 3, 4, and 7 and simplifying, the nondimensional form of Eq. 13 is:
Ÿ + 4πFN ζ Ẏ + (2πFN )2Y =
2CL
πm∗
FIGURE 7.
(22)
Domain coordinate system and boundaries
These non-dimensional variables can be used as a basis of
comparison between various studies using different dimensional
values.
Mesh Deformation and Domain Decomposition
The domain for this system was chosen to be a large rectangle, 60d wide and 40d high as shown in Fig. 7. Thus the
blockage, B, is 1/40 = 2.5%. The upstream cylinder is centered
at (0,0) and the downstream at (4d,0). The domain for one cylinder validation is identically shaped but without the downstream
cylinder. The inlet (left) and outlet (right) boundaries are constrained using Dirichlet conditions of flow speed and pressure.
The top and bottom boundaries have symmetric constraints applied to minimize the boundary interference. A close-up of the
mesh around the cylinders can be seen in Fig. 8. The single cylinder mesh uses 19,181 elements and the tandem cylinder mesh
uses 23,873 elements. These meshes were selected since they
are sufficiently fine to have converged on accurate solutions.
After the cylinder response was approximated using the
Newmark-β method, each vertex of the mesh is deformed at each
time step by some displacement yv where:
yv = (1 −Cde f orm )ycyl
Cde f orm =


0
r<0
10r3 − 15r4 + 6r5 0 ≤ r ≤ 1

1
r>1
r=
rv − rmin
rmax
FIGURE 8.
Close up of mesh around cylinder
where rmin is the distance of 0.6d and rv is the distance of the
vertex to the cylinder center. rmax is a prescribed outer limit of
mesh deformation. ycyl is the y displacement found using the
Newmark-β method. Thus the physical mesh is deformed based
on a blended analytical function such that each node’s deformation is based on its proximity to the deforming cylinder. Vertices
with rmin of the cylinder deform with the cylinder, and vertices
outside of rmax do not deform at all. These blended functions are
also used in [19, 22].
To allow for higher order solutions and finer meshes, the
domain has been decomposed into separate domains, each to be
solved on a separate processor. This essentially splits the problem into a group of parallel problems which can be solved si-
(23)
(24)
(25)
5
c 2012 by ASME
Copyright FIGURE 10. Lift and drag coefficients for the single rigid cylinder in
a flow field with Reynolds number 100
FIGURE 9. Domain decomposition onto 4 processors. Each shade
represents the mesh quadrant solved by a different processor. The processor interface is the boundary layer between each processors portion
of the domain
TABLE 1. Comparison of present results with other results for flow
over a single rigid cylinder at Reynolds number 100
multaneously on different processors within a cluster of servers.
A decomposed mesh using 4 processors can be seen in Fig. 9.
Since the fluid flows across the domain, and the solutions for the
cells on one side of a processor interface depend on the solutions
of the cells across the interface, some cross-processor communication and extra memory space is required for two portions of
the domain interact. Message Passing Interface (MPI) is used to
communicate values between the processors.
Investigator
Present
Liang [20]
Shar. [23]
Mene. [24]
Kang [25]
Re no.
100
100
100
100
100
Nodes
19393
33400
14441
13696
62127
Blockage
0.025
0.0312
0.02
0.047
–
CL
0.239
0.232
0.23
–
0.32
CD
0
1.385
1.365
1.33
–
–
CD
0.006
0.0086
0.0064
–
–
St no.
0.166
0.164
0.164
0.165
0.165
0
VALIDATION
In the following section two validation cases are tested to
ensure that the solver produces accurate results. The fist case is
a single rigid cylinder in a flow field. The second case is a single
elastically mounted cylinder in the same field.
[25]. The mean drag coefficient, CD , is slightly higher than those
0
reported by other authors, however the rms drag, CD , matches
that reported by Sharman et al. [23]. It is possible that the slight
compressibility of the present fluid may cause the slight increase
in mean drag coefficient, but the effect is minimal. Mittal and
Tezduyar [26] report a mean drag coefficient around 1.4 for the
same fluid flow conditions. Overall, this validation proves that
this solver correctly predicts VIV when the cylinder is fixed.
Single Rigid Cylinder
This test case uses a single cylinder in a flow field with
Re = 100. The freestream Mach number is set to 0.2. This
“rigid” single cylinder is actually an elastically mounted cylinder allowed to vibrate in the vertical direction connected by a
very stiff spring and damper (U ∗ = 0.063 and ζ = 50). This effectively reduces the maximum Y to the order of 10−6 . Figure 10
shows lift and drag coefficients for single “rigid” cylinder. Table
1 reports a comparison between the present compressible viscous
flow at Mach number 0.2 to other numerical and experimental
studies for incompressible viscous flow.
The reported lift and drag coefficients are very closely
matched with the previously published results. The rms lift co0
efficient, CL , is slightly higher than those reported by Liang et
al. [20] and Sharman et al. [23] yet below that reported by Kang
Single Elastically Mounted Cylinder
This test case uses a single elastically mounted cylinder with
ζ = 0.0, m∗ = 10. A variety of flow conditions have been used,
but in all cases U ∗ = 0.06Re. This condition effectively couples
the spring coefficient to the flow velocity. Doing so allows the response amplitude, A∗ , to be plotted for a variety of Re, as shown
in Fig. 11 which compares the present amplitudes to those predicted by Prasanth et al. [15, 27]. Similarly, Fig. 12 compares
6
c 2012 by ASME
Copyright the present lift coefficient amplitude to those predicted by Prasanth et al. The jump in vibration amplitude and lift coefficient
amplitude marks the synchronization, or “lock-in” region, where
the frequency ratio (Eqn. 7) approaches unity. It is this region
which contains the highest energy transfer from the fluid to the
structure.
more complicated cases, such as tandem cylinder wake galloping.
FIGURE 13. Nondimensional displacement of the elastically
mounted cylinder over nondimensional time
FIGURE 11. The maximum normalized vibration amplitude for a single elastically mounted cylinder
TANDEM CYLINDER WAKE GALLOPING
Figure 14 shows the vortex shedding of two synchronized
tandem cylinders at Re = 200, with the contour of ωd/u∞ ranging from -4 to 4. The first cylinder is rigid and the second is
elastically mounted with U ∗ = 6.6, m∗ = 10.0, and ζ = 0.0. The
mesh has 8,374 cells such that there are 30 cells around the periphery of each cylinder. The simulation was performed using a
third-order SD method. The cylinder walls are dealt by a highorder curved representation. The center-to-center distance between the cylinders is 4d. Figure 15 shows the non-dimensional
displacement, Y , of the second cylinder over non-dimensional
time.
Note that the amplitude of vibration for this tandem cylinder
wake galloping case is nearly twice that of the single elastically
mounted cylinder case shown in Fig. 11. The addition of the
upstream cylinder relative to the elastically mounted cylinder has
dramatically increased the vibrational amplitude. That is, more
energy is converted from the fluid into the kinetic energy of the
cylinder. In a system like that proposed by Jung and Lee [4]
in Fig. 3 the electricity generation is partially dependent on the
amplitude of vibration.
FIGURE 12. The maximum normalized lift coefficient for a single
elastically mounted cylinder
The amplitude of vibration and lift coefficients match very
well with those reported by Prasanth et al. [15, 27]. A plot of
the Y-displacement against time is shown in Fig. 13. In addition, these values are similar to those reported by Khalak and
Williamson [18]. This validation proves that this solver correctly
and accurately couples the fluid and structure responses together.
This fluid-structure interaction can now be used to investigate
CONCLUSION
A massively parallel, high-order spectral difference solver is
extended for simulating Vortex-Induced Vibrations in bluff bodies based on a robust and accurate scheme for moving and de7
c 2012 by ASME
Copyright wake galloping case is nearly twice that of the single elastically
mounted cylinder case. The addition of the upstream cylinder
relative to the elastically mounted cylinder has dramatically increased the vibrational amplitude. That is, more energy is converted from the fluid into the kinetic energy of the cylinder. Thus
more energy could be extracted from the vibrating cylinder and
converted into electricity. Further investigation of the wake galloping response to a variety of flow and structural conditions is
already underway.
ACKNOWLEDGMENT
Chunlei Liang and Andrew DeJong would like to acknowledge a research grant from ONR with award no.
N000141210500. They are also grateful to the intramural seed
funds awarded by the George Washington University.
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FIGURE 14. Vortex shedding of tandem cylinders. The upstream
(left) cylinder is fixed rigid while the downstream (right) cylinder is elastically mounted. The processor interface lines are shown to confirm that
there is no error in the inter-processor communication
FIGURE 15. Nondimensional displacement of the elastically
mounted cylinder over nondimensional time
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8
c 2012 by ASME
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9
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