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Y. Bazilevs

Department of Structural Engineering,

University of California–San Diego,

La Jolla, CA 92093

e-mail: yuri@ucsd.edu

A. Korobenko

Department of Structural Engineering,

University of California–San Diego,

La Jolla, CA 92093

X. Deng

Department of Structural Engineering,

University of California–San Diego,

La Jolla, CA 92093

J. Yan

Department of Structural Engineering,

University of California–San Diego,

La Jolla, CA 92093

M. Kinzel

Department of Aerospace Engineering,

Division of Engineering and Applied Science,

California Institute of Technology,

Pasadena, CA 91125

J. O. Dabiri

Department of Aerospace Engineering,

Division of Engineering and Applied Science,

California Institute of Technology,

Pasadena, CA 91125

Fluid–Structure Interaction

Modeling of Vertical-Axis

Wind Turbines

Full-scale, 3D, time-dependent aerodynamics and ﬂuid–structure interaction (FSI) simu-

lations of a Darrieus-type vertical-axis wind turbine (VAWT) are presented. A structural

model of the Windspire VAWT (Windspire energy, http://www.windspireenergy.com/)is

developed, which makes use of the recently proposed rotation-free Kirchhoff–Love shell

and beam/cable formulations. A moving-domain ﬁnite-element-based ALE-VMS (arbi-

trary Lagrangian–Eulerian-variational-multiscale) formulation is employed for the aero-

dynamics in combination with the sliding-interface formulation to handle the VAWT

mechanical components in relative motion. The sliding-interface formulation is aug-

mented to handle nonstationary cylindrical sliding interfaces, which are needed for the

FSI modeling of VAWTs. The computational results presented show good agreement with

the ﬁeld-test data. Additionally, several scenarios are considered to investigate the tran-

sient VAWT response and the issues related to self-starting. [DOI: 10.1115/1.4027466]

1 Introduction

In recent years, the wind-energy industry has been moving in

two main directions: off shore, where energy can be harvested

from stronger and more sustained winds, and urban areas, which

are closer to the direct consumer. In the offshore environments,

large-size horizontal-axis wind turbines (HAWTs) are at the lead-

ing edge. They are equipped with complicated pitch and yaw con-

trol mechanisms to keep the turbine in operation for wind

velocities of variable magnitude and direction, such as wind gusts.

The existing HAWT designs are currently more efﬁcient for

large-scale power production compared with the VAWT designs.

However, smaller-size VAWTs are more suitable for urban envi-

ronments and are currently employed for small-scale wind-energy

generation. Nevertheless, wind-energy technologies are maturing,

and several studies were recently initiated that involve placing

VAWTs off shore [2,3].

There are two main conﬁgurations of VAWTs, employing the

Savonius or Darrieus rotor types [4]. The Darrieus conﬁguration

is a lift-driven turbine. It is more efﬁcient than the Savonius con-

ﬁguration, which is a drag-type design. Recently, VAWTs resur-

faced as a good source of small-scale electric power for urban

areas. The main reason for this is their compact design. The gener-

ator and drive train components are located close to the ground,

which allows for easier installation, maintenance, and repair.

Another advantage of VAWTs is that they are omidirectional (i.e.,

they do not have to be oriented into the main wind direction),

which obviates the need to include expensive yaw control

mechanisms in their design. However, this brings up issues related

to self-starting. The ability of VAWTs to self-start depends on the

wind conditions as well as on airfoil designs employed [5]. Stud-

ies in Refs. [6,7] reported that a three-bladed H-type Darrieus

rotor using a symmetric airfoil is able to self-start. In Ref. [8], the

author showed that signiﬁcant atmospheric wind transients are

required to complete the self-starting process for a ﬁxed-blade

Darieus turbine when it is initially positioned in a dead-band

region deﬁned as the region with the tip-speed-ratio values that

result in negative net energy produced per cycle. Self-starting

remains an open issue for VAWTs, and an additional starting sys-

tem is often required for successful operation.

Due to increased recent emphasis on renewable energy, and, in

particular, wind energy, aerodynamics modeling and simulation

of HAWTs in 3D have become a popular research activity [9–17].

FSI modeling of HAWTs is less developed, although, recently,

several studies were reported showing validation at full-scale

against ﬁeld-test data for medium-size turbines [18], and demon-

strating feasibility for application to larger-size offshore wind-

turbine designs [10,19]. However, 3D aerodynamics modeling of

VAWTs is lagging behind. The majority of the computations for

VAWTs are reported in 2D [20–22], while a recent 3D simulation

in Ref. [23] employed a quasi-static representation of the air ﬂow

instead of solving the time-dependent problem. A detailed 3D

aerodynamics analysis of a VAWT used for laboratory testing was

recently performed by some of the authors of the present paper in

Ref. [24]. The studies included full 3D aerodynamic simulations,

validated using experimental data, and a simulation of two side-

by-site counter-rotating turbines.

The aerodynamics and FSI computational challenges in

VAWTs are different than in HAWTs due to the differences in

their aerodynamic and structural design. Because the rotation axis

Manuscript received March 25, 2014; ﬁnal manuscript received April 14, 2014;

accepted manuscript posted April 22, 2014; published online May 7, 2014. Assoc.

Editor: Kenji Takizawa.

Journal of Applied Mechanics AUGUST 2014, Vol. 81 / 081006-1Copyright V

C2014 by ASME

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is orthogonal to the wind direction, the wind-turbine blades expe-

rience rapid and large variations in the angle of attack resulting in

an air ﬂow that is constantly switching from being fully attached

to being fully separated. This, in turn, leads to high-frequency and

high-amplitude variations in the aerodynamic torque acting on the

rotor, requiring ﬁner mesh resolution and smaller time-step size

for accurate simulation [24]. VAWT blades are typically long and

slender by design. The ratio of cord length to blade height is very

low, requiring ﬁner mesh resolution also in the blade height direc-

tion in order to avoid using high-aspect-ratio surface elements,

and to better capture turbulent ﬂuctuations in the boundary layer.

When the FSI analysis of VAWTs is performed, the simulation

complexity is further increased. The ﬂexibility in VAWTs does

not come from the blades, which are practically rigid (although

blades deform at high rotational speeds), but rather from the tower

itself, and its connection to the rotor and ground. As a result, the

main FSI challenge is to be able to simulate a spinning rotor that

is mounted on a ﬂexible tower.

In the present paper, we focus on the following developments.

We propose a set of techniques that, for the ﬁrst time, enable FSI

simulations of VAWTs in 3D and at full-scale. We ﬁrst develop a

3D structural model of a Windspire VAWT [1], which makes use

of the recently proposed rotation-free Kirchhoff–Love shell and

beam/cable formulations, and their coupling. The model allows

for the rotor to spin freely and for the tower and blades to undergo

elastic deformations. We validate the aerodynamics of the Wind-

spire design using the ﬁeld data reported in Refs. [25–27]. For the

FSI computations, to accommodate the spinning rotor and deﬂect-

ing tower and blades, the FSI formulation from Ref. [19]is

enhanced to allow the cylindrical sliding interface to also move in

space. Finally, with these new techniques, we perform preliminary

FSI computations in an effort to better understand the self-starting

issues.

The paper is outlined as follows. In Sec. 2, we introduce the

FSI formulation and present the governing equations of aerody-

namics and structural mechanics. We also brieﬂy describe the

discretization techniques employed and the aforementioned

enhancement of the sliding-interface formulation. In Sec. 3,we

show the aerodynamic and FSI computations of the Windspire

VAWT and discuss start-up issues. In Sec. 4, we draw conclusions

and discuss possible future research directions.

2 Methods for Modeling and Simulation of VAWTs

2.1 Governing Equations at the Continuum Level. To per-

form the VAWT simulations, we adopt the FSI framework devel-

oped in Ref. [28]. The wind turbine aerodynamics is governed by

the Navier–Stokes equations of incompressible ﬂows. The

incompressible-ﬂow assumption is valid for the present applica-

tion because the Mach number is low (0:1). The Navier–Stokes

equations are posed on a moving spatial domain and are written in

the ALE frame [29] as follows:

q1

@u

@t^

x

þðu^

uÞ$uf1

$r1¼0(1)

$u¼0(2)

where q1is the ﬂuid density, f1is the external force per unit mass,

uand ^

uare velocities of the ﬂuid and ﬂuid mechanics domain,

respectively. The stress tensor r1is deﬁned as

r1u;pðÞ¼pIþ2leuðÞ (3)

where pis the pressure, Iis the identity tensor, lis the dynamic

viscosity, and euðÞis the strain-rate tensor given by

euðÞ¼

1

2$uþ$uT

(4)

In Eq. (1),j^

xdenotes the time derivative taken with respect to a

ﬁxed referential domain spatial coordinates ^

x. The spatial deriva-

tives in the above equations are taken with respect to the spatial

coordinates xof the current conﬁguration.

The governing equations of structural mechanics written in the

Lagrangian frame [30] consist of the local balance of linear

momentum, and are given by

q2

d2y

dt2f2

$r2¼0(5)

where q2is the structural density, f2is the body force per unit

mass, r2is the structural Cauchy stress, and yis the unknown

structural displacement vector.

At the ﬂuid–structure interface, compatibility of the kinematics

and tractions is enforced, namely

udy

dt¼0(6)

r1n1þr2n2¼0(7)

where n1and n2are the unit outward normal vectors to the ﬂuid

and structural mechanics domain at their interface. Note that

n1¼n2.

The above equations constitute the basic formulation of the FSI

problem at the continuous level. In what follows, we discuss the

discretization of the above system that is applicable to VAWT

modeling. For a variety of discretization options, FSI coupling

strategies, and applications to a large class of problems in engi-

neering, the reader is referred to the recent book on computational

FSI [31].

2.2 Discretization and Special FSI Techniques for

VAWTs. The aerodynamics formulation makes use of the FEM-

based ALE-VMS approach [29,32,33] augmented with weakly

enforced boundary conditions [34–36]. The former acts as a turbu-

lence model, while the latter relaxes the mesh size requirements in

the boundary layer without sacriﬁcing the solution accuracy. ALE-

VMS was successfully employed for the aerodynamics simulation

of HAWTs and VAWTs in Refs. [9,16,24,37,38], and ﬂuid–

structure interaction simulation of HAWTs in Refs. [10,18,19,28].

The structural mechanics of VAWTs are modeled using a com-

bination of the recently proposed displacement-based Kirchhoff–

Love shell [10,18,39] and beam/cable [40] formulations. Both are

discretized using NURBS-based isogeometric analysis (IGA)

[41,42].

The FSI modeling employed here makes use of nonmatching

discretization of the interface between the ﬂuid and structure sub-

domains. Nonmatching discretizations at the ﬂuid–structure inter-

face require the use of interpolation or projection of kinematic

and traction data between the nonmatching surface meshes (see,

for example, Refs. [28,31,32,43–52]), which is what we do here.

To handle the rotor motion in the aerodynamics problem, the

sliding-interface approach is employed. The sliding-interface for-

mulation was developed in Ref. [53] to handle ﬂows about objects

in relative motion, and used in the computation of HAWTs in Ref.

[16], including FSI coupling [19], and VAWTs in Ref. [24]. We

note that in application of the FEM to ﬂows with moving mechan-

ical components, alternatively to the sliding-interface approach,

the Shear–Slip Mesh Update Method [54–56] and its more general

versions [57,58] may also be used to handle objects in relative

motion. A recently developed set of space-time (ST) methods can

serve as a third alternative in dealing with objects in relative

motion. The components of this set include the ST/NURBS mesh

update method [17,59,60], ST interface tracking with topology

change [61], and ST computation technique with continuous

representation in time [62].

To accommodate the spinning motion of the rotor superposed

on the global elastic deformation of the VAWT, and to maintain a

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moving-mesh discretization with good boundary-layer resolution

critical for aerodynamics accuracy, the sliding-interface technique

is “upgraded” to handle more complex structural motions. While

at the ﬂuid–structure interface the ﬂuid mechanics mesh follows

the rotor motion, the outer boundary of the cylindrical domain

that encloses the rotor is only allowed to move as a rigid object.

The rigid-body motion part is extracted from the rotor structural

mechanics solution (see, e.g., Ref. [63]) and is applied directly to

the outer boundary of the cylindrical domain enclosing the rotor.

The inner boundary of the domain that encloses the cylindrical

subdomain also moves as a rigid object. It follows the motion of

the cylindrical subdomain, but with the spinning component of the

motion removed. The ﬂuid mechanics mesh motion in the interior

of the two subdomains is governed by the equations of elasto-

statics with Jacobian-based stiffening [43,57,64–67] to preserve

the aerodynamic mesh quality.

3 Computational Results

The computations presented in this section are performed for a

1.2 kW Windspire design [1], a three-bladed Darrieus VAWT.

The total height of the VAWT tower is 9.0 m and the rotor height

is 6.0 m. The rotor uses the DU06W200 airfoil proﬁle with the

chord length of 0.127 m, and is of the Giromill type with straight

vertical blade sections attached to the main shaft with horizontal

struts (see Fig. 1). The blades and struts are made of aluminum,

and the tower is made of steel. The material parameters and

masses of the main structural components are given in Tables 1

and 2.

The VAWT blades and part of the tower that spins with the

rotor are modeled using Kirchhoff–Love shells, while the struts

and main shaft are modeled as beams. The struts are connected to

the blades, tower shell, and main shaft, which gives a relatively

simple VAWT structural model that can represent the 3D mechan-

ics of a spinning, ﬂexible rotor mounted on a ﬂexible tower. See

Figs. 1(b)and 1(c)for more details of the VAWT geometry

description. The density of the tower shell is set to be very small

(see Table 1) so that most of tower mass is distributed evenly

along the beam. Quadratic NURBS are employed for both the

beam and shell discretizations. The total number of beam ele-

ments is 116, and total number of shell elements is 7029. We note

that all the aerodynamically important surfaces that are “seen” by

the ﬂuid mechanics discretization are modeled using shells.

The aerodynamics and FSI simulations are carried out at realis-

tic operating conditions reported in the ﬁeld-test experiments con-

ducted by the National Renewable Energy Laboratory [25] and

Caltech Field Laboratory for Optimized Wind Energy [26,27]. For

all cases, the air density and viscosity are set to 1.23 kg/m

3

and

1:78 105kg/ms, respectively.

The outer aerodynamics computational domain has the dimen-

sions of 50 m, 20 m, and 30 m in the streamwise, vertical, and

spanwise directions, respectively, and is shown in Fig. 2. The

VAWT centerline is located 15 m from the inﬂow and side boun-

daries. The radius and height of the inner cylindrical domain that

encloses the rotor are 1.6 m and 7 m, respectively.

At the inﬂow, a uniform wind velocity proﬁle is prescribed. On

the top, bottom, and side surfaces of the outer domain no-

penetration boundary conditions are prescribed, while zero

traction boundary conditions are set at the outﬂow.

The aerodynamics mesh has about 8 M elements, which are lin-

ear triangular prisms in the blade boundary layers, and linear tetra-

hedra elsewhere. The boundary-layer mesh is constructed using

18 layers of elements, with the size of the ﬁrst element in the

Fig. 1 Windspire VAWT structural model with dimensions

included: (a) full model using isogeometric NURBS-based

rotation-free shells and beams; (b) model cross section 1 show-

ing attachment of the struts to the blades and tower shell; (c)

model cross section 2 showing attachment of the struts and

tower shell

Table 1 Geometric and material properties of the main VAWT

structural components

Thickness/

radius

Young’s

modulus

Poisson’s

ratio Density

Part t/r(mm) E(GPa) q(kg/m

3

)

Blades 2/NA 70 0.35 2700

Strut 12.7/NA 70 0.35 2700

Tower shell 5/44.45 210 0.33 78

Tower beam NA/41.08 210 NA 5120.9

Table 2 Masses of the main VAWT structural components

Part Mass (kg)

Blades 26.3

Strut 14.1

Tower 243.4

Total 283.8

Fig. 2 The VAWT aerodynamics computational domain in the

reference conﬁguration, including the inner cylindrical region,

outer region, and sliding interface that is now allowed to move

in space as a rigid object

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wall-normal direction of 0.0003 m, and growth ratio of 1.1. A 2D

slice of the mesh near the rotor is shown in Fig. 3, while Fig. 4

shows the zoom on the boundary-layer mesh near one of the

blades. The mesh design employed in this simulation is based on a

reﬁnement study performed for a Darrieus-type experimental tur-

bine in Ref. [24].

All computations are carried out in a parallel computing envi-

ronment. The mesh is partitioned into subdomains using METIS

[68], and each subdomain is assigned to a compute core. The

parallel implementation of the methodology may be found in

Ref. [37]. The time-step is set to 1:0105s for the aerodynam-

ics computation and 2:0105s for the FSI analysis.

3.1 Aerodynamics Simulation of the Windspire VAWT.

We ﬁrst performed two pure aerodynamic simulations of the

Windspire VAWT, one using the wind speed of 8.0 m/s and rotor

speed of 32.7 rad/s, and another using the wind speed of 6.0 m/s

and rotor speed of 20.6 rad/s. The time history of the aerodynamic

torque for both cases is plotted in Fig. 5together with the experi-

mental values reported from ﬁeld-test experiments [25–27]. After

the rotor undergoes a full revolution, a nearly periodic solution is

attained in both cases. For 8.0 m/s wind, the predicted average tor-

que is 18.9 Nm, while its experimentally reported value is about

12.7 Nm. For 6.0 m/s wind, the predicted average torque is 9.5

Nm, while its experimentally reported value is about 4.8 Nm.

In both cases, the experimental value of the aerodynamic torque

is derived from the average power produced by the turbine at the

target rotor speed. The difference in the predicted and experimen-

tally reported aerodynamic torque is likely due to the mechanical

and electrical losses in the system, which are not reported. To esti-

mate those, we perform the following analysis. For simplicity, we

assume that the torque loss is proportional to the rotational speed

of the turbine, that is

Tloss ¼closs

_

h(8)

Here, Tloss is taken as the difference between the predicted and

reported torque values, _

his the rotation speed, and closs is the

“loss” constant that characterizes the turbine. The data for the

8.0 m/s wind give closs ¼0:19 kg m

2

/rad, while for 6.0 m/s wind

we ﬁnd that closs ¼0:23 kg m

2

/rad. The two values are reasonably

close, which suggests that the torque overestimation is consistent

with the loss model. In fact, this technique of combining experi-

mental measurements and advanced computation may be

employed to approximately estimate losses in wind turbines.

3.2 FSI Simulations of the Windspire VAWT. In this sec-

tion, we perform a preliminary investigation of the start-up issues

in VAWTs using the FSI methodology described earlier and the

structural model of the Windspire design. We ﬁx the inﬂow wind

speed at 11.4 m/s, and consider three initial rotor speeds: 0 rad/s,

4 rad/s, and 12 rad/s. Of interest is the transient response of the

system. In particular, we will focus on how the rotor angular

speed responds to the prescribed initial conditions, and what is the

range of the tower tip displacement during the VAWT operation.

The starting conﬁguration of the VAWT is shown in Fig. 3.

Blade 1 is placed parallel to the ﬂow with the airfoil leading edge

facing the wind. Blades 2 and 3 are placed at an angle to the ﬂow

Fig. 3 A 2D cross section of the computational mesh along the

rotor axis. The view is from the top of the turbine, and the

blades are numbered counterclockwise, which is the expected

direction of rotation. The sliding interface may be seen along a

circular curve where the mesh appears to be nonconforming.

Fig. 4 A 2D cross section of the blade boundary-layer mesh

consisting of triangular prisms

Fig. 5 Time history of the aerodynamic torque for the pure aerodynamics simulations. (a) 8.0 m/s wind with experimental data

from Ref. [25] and (b) 6.0 m/s wind with experimental data from Refs. [26,27].

081006-4 / Vol. 81, AUGUST 2014 Transactions of the ASME

with the trailing edge facing the wind. (Blade numbering is shown

in the ﬁgure.)

The time history of rotor speed is shown in Figs. 6–8. For the

0 rad/s case, the rotor speed begins to increase suggesting this con-

ﬁguration is favorable for self-starting. For the 4 rad/s case, the

rotor speed has a nearly linear acceleration region followed by a

plateau region. In Ref. [7], the plateau region is deﬁned as the re-

gime when the turbine operates at nearly constant (i.e., steady-

state like) rotational speed. From the angular position of the

blades in Fig. 7, it is evident that the plateau region occurs approx-

imately every 120 deg when one of the blades is in a stalled posi-

tion. It lasts until the blade clears the stalled region, and the lift

forces are sufﬁciently high for the rotational speed to start

increasing again. As the rotational speed increases, the angular ve-

locity is starting to exhibit local unsteady behavior in the plateau

region. While the overall growth of the angular velocity for the

4 rad/s case is promising for the VAWT to self-start, the situation

is different for the 12 rad/s case (see Fig. 8). Here, the rotor speed

has little dependence on the angular position and stays nearly con-

stant, close to its initial value. It is not likely that the rotor speed

will reach to the operational levels in these conditions without an

applied external torque, or a sudden change in wind speed, which

is consistent with the ﬁndings of Ref. [8].

Figure 9shows, for a full turbine, a snapshot of vorticity col-

ored by ﬂow speed for the 4 rad/s case. Figure 10 zooms on the

rotor and shows several ﬂow vorticity snapshots during the rota-

tion cycle. The ﬁgures indicate the complexity of the underlying

ﬂow phenomena and the associated computational challenges.

Note the presence of quasi-2D vortex tubes that are created due to

massive ﬂow separation, and that quickly disintegrate and turn

into ﬁne-grained 3D turbulence further downstream. For more dis-

cussion on the aerodynamics phenomena involved in VAWTs the

reader is referred to Ref. [24].

Figure 11 shows the turbine current conﬁguration at two time

instances during the cycle for the 4 rad/s case. The displacement is

mostly in the direction of the wind, however, lateral tower dis-

placements are also observed as a result of the rotor spinning

motion. The displacement amplitude is around 0.10–0.12 m,

which is also the case for the 0 rad/s and 12 rad/s cases.

Fig. 6 Time history of the rotor speed starting from 0 rad/s

Fig. 8 Time history of the rotor speed starting from 12 rad/s

Fig. 7 Time history of the rotor speed starting from 4 rad/s

Fig. 9 Vorticity isosurfaces at a time instant colored by veloc-

ity magnitude for the 4 rad/s case

Journal of Applied Mechanics AUGUST 2014, Vol. 81 / 081006-5

4 Conclusions and Future Work

In this paper, dynamic FSI modeling of VAWTs in 3D and at

full-scale was reported for the ﬁrst time. A structural model of a

Windspire wind turbine design was constructed and discretized

using the recently proposed isogeometric rotation-free shell and

beam formulations. This approach presents a good combination of

accuracy due to the structural geometry representation using

smooth, higher-order functions, and efﬁciency due to the fact that

only displacement degrees of freedom are employed in the formu-

lation. The ALE-VMS technique for aerodynamics modeling was

augmented with an improved version of the sliding-interface for-

mulation, which allows the interface to move in space as a rigid

object and accommodate the global turbine deﬂections in addition

to the rotor spinning motion. The pure aerodynamics computation

produced good agreement with the ﬁeld-test data for the Wind-

spire turbine, and the FSI simulations were performed to investi-

gate turbine start-up issues.

From the FSI computations, we see that for given wind condi-

tions, the rotor naturally accelerates at lower values of angular

speed. However, as the angular speed grows, the rotor may en-

counter a dead-band region. That is, the turbine self-starts, but

then it is trapped in a lower rotational speed than is required for

optimal performance, and some additional input (e.g., a wind gust

or applied external torque) is required to get the rotor to accelerate

further. There may be multiple dead-band regions that the turbine

needs to overcome, with external forcing applied before it reaches

the target rotational speed. In the future, to address some of these

issues, we plan to couple our FSI formulation with an appropriate

control strategy (see, e.g., Ref. [69]) to simulate more realistic

VAWT operation scenarios.

Acknowledgment

This work was supported through the NSF CAREER Award

No. 1055091. The computational resources of the Texas

Fig. 10 Vorticity isosurfaces of vorticity colored by velocity magnitude for the 4rad/s case. Zoom on the rotor. From left to

right: vorticity at 1.12 s, 1.24 s, 1.40 s, and 1.50 s.

Fig. 11 Turbine current conﬁguration at two time instances for the 4 rad/s case.

The tower centerline in the reference conﬁguration is shown using the dashed line

to illustrate the range of turbine motion during the cycle. The range of the tower tip

displacement during the cycle is about 0.10–0.12 m.

081006-6 / Vol. 81, AUGUST 2014 Transactions of the ASME

Advanced Computing Center (TACC) [70] were employed for the

simulations reported in this work. This support is gratefully

acknowledged.

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