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Accepted manuscript, 2018 – Transactions of the Canadian Society for Mechanical Engineering

Published version available online at: https://dx.doi.org/10.1139/tcsme-2017-0071

A Parametric Study of Energy Extraction from Vortex-Induced Vibrations

Olivier Paré Lambert and Mathieu Olivier

Laboratoire de Mécanique des Fluides Numérique, Département de Génie Mécanique,

Université Laval, Québec, Québec, Canada

ABSTRACT

This paper presents a parametric investigation of an oscillating-cylinder turbine concept based on the

mechanism of vortex-induced vibrations. The parametric space includes 4 parameters: the Reynolds

number, the mass ratio, the dimensionless stiffness, and the dimensionless damping. The damping-

stiffness space is explored for four different mass ratios at a fixed Reynolds number of 200. Also, the

influence of the parameters on the amplitude of the cylinder displacement and on the efficiency of

power harnessing is discussed. Vortex shedding patterns observed within the parametric space are

investigated. The 2S, 2P, and C(2S) wake modes are observed and related to the turbine performance.

Preliminary results show a maximum efficiency of 10.6%, which is obtained with low mass ratios.

1 INTRODUCTION

Every year, new advances are made in the field of renewable energies. In Canada, the most established

technology for clean energy production is hydropower. With 63% of its energy provided by water, Canada

is the third largest hydropower producer in the world (Canadian Hydropower Association (2017)). While

hydraulic power is generally harnessed with hydraulic turbines, new hydrokinetic turbine concepts are

currently receiving a lot of attention in literature (Güney & Kaygusuz (2010); Khan et al. (2009); Xiao &

Zhu (2014); Young, Lai & Platzer (2014)). Among these concepts, a novel approach for energy extraction

from water currents is the Vortex Induced Vibration Aquatic Clean Energy (VIVACE) converter, an

invention of Bernitsas and Raghavan (United States Patent No. 7,493,759 B2, 2009). This technology allows

hydrokinetic energy to be harnessed from the Vortex-Induced Vibration (VIV) phenomenon that occurs on

cylinders in a flowing fluid, even when facing slow water currents (Bernitsas et al. (2008)).

The main feature of the VIVACE converter is the passive cylinder oscillations caused by the fluid-structure

interaction between the water and the rigid body. As the flow passes around the cylinder, a so-called vortex

street caused by the unstable shear layers is formed. This vortex street produces unsteady lift and drag forces

on the cylinder, thereafter leading to its oscillation (Williamson (1996)). When the frequency of the shed

vortices is synchronized with the natural frequency of the structure, a lock-in regime appears, resulting in

larger vibrations which are generally damaging for structures. However, in a context of energy extraction,

these large vibrations are attractive since they allow power to be harnessed from the flow (Bernitsas et al.

(2008)). Therefore, in order to maximize energy extraction, it is important to choose the right parameters

that control the cylinder motion of such turbine. This can be achieved with experiments and numerical

simulations. In recent years, many studies on the VIV of elastically mounted cylinders were published.

These studies mostly present the effect of the parameters on the wake formation (Williamson and Roshko

(1987); Govardhan and Williamson (2000); Prasanth & Mittal (2008)) and on the amplitude of the cylinder

displacement (Bahmani and Akbari (2010); Klamo, et al. (2006)). Although fewer articles concerning the

power extraction from vortex-induced vibrations were published, the paper by Barrero-Gil et al. (2012)

presents the effects of different parameters influencing the energy extraction of a VIV-based turbine by

using a mathematical model built against experimental data. They mainly investigated the relation between

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the mass, the damping coefficient, and their combined effect on the extraction of energy from vortex-

induced vibrations with a one-degree-of-freedom model. They also examined the responses of varying the

Reynolds number. The parameters tested in their study exhibit an optimal configuration that reaches the

efficiency of about 0.17 at a Reynolds number of 3 800. By increasing the Reynolds number to 10 000, they

obtained efficiencies of about 0.24, which is promising.

This paper presents an oscillating-cylinder turbine model based on the mass-stiffness-damping equation

which governs the cylinder displacement in both the in-flow and transverse directions. The turbine model

is systematically studied in a dimensional analysis context so that only four parameters are involved: the

mass ratio, the dimensionless damping, the dimensionless stiffness, and the Reynolds number. The impact

of these parameters on both the efficiency and the amplitude of a two-degrees-of-freedom cylinder-based

turbine can thus be assessed. Additionally, the vortex modes are explored within the parametric space and

the widely used mass-damping parameter is employed to better understand the behavior of the proposed

turbine in terms of energy extraction and to further compare the results with other research works. The study

uses Computational Fluid Dynamics (CFD) to perform numerical simulations with a two-degree-of-freedom

solid-body solver combined in a fully implicit and strongly coupled Fluid-Structure Interaction (FSI)

algorithm.

In a first attempt to find the best parameters for harvesting energy from unconstrained flows with the

proposed turbine model, several numerical simulations have been carried out with numerous parameters for

a fixed Reynolds number of 200. The choice of using such a low Reynolds number is made to avoid the

added difficulty of proper turbulence modeling in obtaining a first insight of the problem. The results show

that the parameters adopted in this study exhibit optimal configurations that reach efficiencies up to about

10% in the damping-stiffness space for different mass ratios. The paper is divided as follows. In section 2,

the governing equations and the problem definition are introduced. Sections 3 and 4 present respectively

the numerical methods used in this study and the validation tests made to certify suitable numerical results.

In section 5, the results obtained with the different dimensionless parameters are presented and discussed.

2 PROBLEM DEFINITION AND GOVERNING EQUATIONS

This paper analyzes the two-dimensional interaction between a rigid cylinder mounted on two perpendicular

spring-damper systems and an incompressible fluid flow. Fig. 1 represents this interaction and shows the

variables that are important in this problem. 𝐷 and 𝑚 are, respectively, the diameter and the mass of the

cylinder, 𝑘 and 𝛾 are the stiffness and the damping of the system in both directions, and 𝑈∞ represents the

freestream velocity. The fluid flow is governed by the incompressible Navier-Stokes equations:

𝛻 ∙ 𝐮 = 0, (1)

Fig. 1: Problem setup: cylinder mounted on two perpendicular spring-damper systems in a flow field.

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𝜕𝐮

𝜕𝑡+ 𝐮 ∙ 𝛻𝐮 = −

1

𝜌𝛻𝑝 + 𝜈𝛻2𝐮, (2)

where 𝜌 is the fluid density, 𝜈 is the dynamic viscosity, 𝑝 is the pressure field and 𝐮 is the velocity field.

The displacement of the cylinder, which is represented by a mass-stiffness-damping system, is governed by

the following equation:

𝑚𝜕2𝐝

𝜕𝑡2+ 𝛾

𝜕𝐝

𝜕𝑡+ 𝑘𝐝 = 𝐅, (3)

where 𝐝 is the displacement vector of the rigid body and 𝐅 is the external fluid force vector. In order to

generalize the results of this study, dimensionless parameters are used and expressed below. These

parameters are the Reynolds number (𝑅𝑒), the mass ratio (𝑚∗), the dimensionless stiffness (𝑘∗) and the

dimensionless damping (𝛾∗), which are defined as:

𝑅𝑒 =𝑈∞𝐷

𝜈, 𝑚∗ =

4𝑚

𝜋𝜌𝐷2, 𝑘∗ =

2𝑘

𝜌𝑈∞2 , 𝛾∗ =

2𝛾

𝜌𝑉∞𝐷.

These non-dimensional parameters differ from those generally used in such studies, but they have the

advantage to remain definite in all the parametric space such as when the mass ratio is set to 0 (Shiels et al.

(2001)). Finally, setting 𝐝∗ = 𝐝/𝐷, the dimensionless harmonic equation used in this paper, in which CF is

a force coefficient vector, yields:

𝜋

2𝑚∗

𝜕2𝐝∗

𝜕𝑡2+ 𝛾∗

𝜕𝐝∗

𝜕𝑡+ 𝑘∗𝐝∗ = 𝐂𝐅. (4)

3 NUMERICAL METHODS

The coupled fluid-solid interaction model is solved using an in-house CFD simulation code based on

OpenFOAM version 3.0. The fluid flow is therefore solved with the second-order finite-volume method

with a collocated variable arrangement. The pressure-velocity coupling is handled with the so-called

PIMPLE algorithm, which is very analogous to an iterative PISO algorithm (or SIMPLE algorithm if the

amount of pressure correction steps is set to one). Second-order spatial discretization schemes are used along

with a second-order backward differencing scheme for the time integration. The same time integration

scheme is also used to discretize the solid equation so that numerical consistency is preserved at the fluid-

solid interface. Moreover, the conservation equations are implemented in an arbitrary Lagrangian-Eulerian

formulation and the mesh motion is handled by a Laplace differential equation. Further details on the fluid

discretization are provided by (Olivier & Dumas, 2016).

The fluid-solid coupling scheme is based on the partitioned approach and Broyden’s method (Press, 2007)

is used to accelerate the convergence rate, which is particularly beneficial for strongly coupled cases. This

coupling scheme, which thus belongs to the quasi-Newton family, is well suited for segregated fluid solver

algorithms and allows very strong fluid-solid interaction cases, such as those involving a massless and

damping-free oscillating cylinder, to be addressed efficiently. Once the solid body motion equation is

discretized, the resulting algebraic equation can be expressed as a residual:

𝐑 = 𝑎𝐝𝑛+1 + 𝐛 − 𝐅(𝐝𝑛+1), (5)

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in which 𝐑 is the residual itself, 𝑎 is a numerical parameter that depends on the temporal discretization as

well as on the mechanical properties 𝑚, 𝛾, and 𝑘, 𝐛 is the explicit contribution that depends on the previous

states of 𝐝, and 𝐅 is the fluid force acting on the cylinder, which depends on the position of the solid body.

This residual expression can be used to devise a Newton-Raphson scheme. However, since the derivatives

of 𝐅 with respect to 𝐝 cannot be obtained easily, an approximated Jacobian is used and updated with

Broyden’s formula. At a given time step, the coupling algorithm can be outlined as follows:

The solid displacement is estimated with a second-order Adams-Bashforth scheme based on the

solid velocity and, then, the fluid fields are updated accordingly with a fluid solver call. This allows

a first residual vector to be computed.

A very small perturbation is applied on the solid displacement vector and, again, the flow solver is

called. This allows a second residual vector to be computed.

A first diagonal matrix, which acts as an approximated Jacobian, can be computed by using finite

differences of the 2 residual vectors. Note that a full approximated Jacobian would require more

fluid solver calls which would increase the numerical cost. The solid displacement can be updated

with: 𝐝𝑖𝑛+1 = 𝐝𝑖−1

𝑛+1 − (𝐉𝑖)−1𝐑𝑖 and the fluid solver is called to obtain 𝐑𝑖+1

The previous step is repeated iteratively, but this time, the Jacobian is updated with Broyden’s

formula:

𝛿𝐝𝑖−1 = 𝐝𝑖−1 − 𝐝𝑖−2,

𝛿𝐑𝑖 = 𝐑𝑖 − 𝐑𝑖−1,

𝐉𝑖 = 𝐉𝑖−1 + (𝛿𝐑𝑖 − 𝐉𝑖−1 ⋅ 𝛿𝐝𝑖−1) ⨂ 𝛿𝐝𝑖−1

𝛿𝐝𝑖−1 ⋅ 𝛿𝐝𝑖−1.

(6)

4 VERIFICATION AND VALIDATION

All simulations proposed in this study are run with a 65065-cell mesh along with a time step set to

𝛥𝑡𝑈∞/𝐷 = 0.005. A typical numerical simulation has also been tested with coarser (15065 cells,

𝛥𝑡𝑈∞/𝐷 = 0.01) and finer (269140 cells, 𝛥𝑡𝑈∞/𝐷 = 0.0025) meshes and time steps. The proposed

intermediate mesh and time step then appeared to be a good compromise between precision and simulation

run time. The mesh is shown in Fig. 2 along with the domain dimensions. The coarse mesh is shown because

its lower density makes it easier to display. However, all three meshes are similar: a layer of quadrilateral

cells is extruded from the cylinder, a region with uniform unstructured polygonal cells is kept around the

cylinder and in its wake, and a growth rate is used to coarsen the mesh far from the cylinder and its wake.

For the medium mesh, the height of the first cell layer on the cylinder is 0.0036 𝐷, the growth ratio of the

layers is 1.026, and the characteristic cell length in the wake region is 0.031 𝐷. Characteristic lengths for

the coarser and finer meshes are respectively the double and the half of those of the medium mesh.

Moreover, the simulation results have been compared favorably with the results of Blackburn and

Karniadakis (1993) (see Fig. 3).

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Fig. 3: Coarse polyhedral mesh and domain dimensions.

Fig. 2: Cylinder trajectory of the last computed oscillation cycle with 𝑅𝑒 = 200, 𝑚∗ = 4/𝜋, 𝛾∗ = 2𝜋/125, and

𝑘∗ = 8𝜋2/25. Different meshes and time steps are compared with the data of Blackburn and Karniadakis (1993).

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5 RESULTS

In order to assess the performances of the turbine model against the dimensionless parameters, many

simulations have been performed with various values of 𝑚∗, 𝛾∗, and 𝑘∗. The purpose of these numerical

simulations is to determine the parameters that maximize the energy harnessing. All simulations were

carried out with a Reynolds number of 200. The simulations were run until a stable and periodic state was

obtained or until 𝑡𝑈∞/𝐷 = 500. In this study, almost all simulated cases have reached a periodic regime.

Nevertheless, the performance metrics defined below have typically been averaged over 50 cycles by

considering the dominant mode of the oscillation.

The performance of the turbine model is determined by its capability to extract power and by its efficiency.

The efficiency of the oscillating-cylinder turbine is calculated with the following equation:

𝜂 =�̅�

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𝜌𝑈∞3 𝐴1

, (7)

where �̅� is the cycle-averaged power per unit depth harvested by the supports, 𝜌 is the fluid density, 𝑈∞ is

the freestream velocity, and 𝐴1 is the area (length in 2D) swept by the cylinder during its oscillating motion.

The swept area and the harvested power are respectively given by:

𝐴1 = ( 𝑑𝑦,max − 𝑑𝑦,min) + 𝐷, (8)

�̅� =1

𝑛𝑇 ∫ 𝛾 (

𝜕𝐝

𝜕𝑡)

2

𝑑𝑡𝑡0+𝑛𝑇

𝑡0

, (9)

where 𝑑𝑦,max and 𝑑𝑦,min are the highest and the lowest points reached by the center of the cylinder in a

cycle, 𝑛 is the number of cycles considered for the averaging, and 𝑇 is the period of oscillations. All

simulation results are summarized in Figures 4 to 7 in which the black dots represent individual numerical

simulations. The contour plots in these figures represent the energy extraction efficiency for the four mass

ratios tested.

To better grasp the evolution of the power coefficient over a period of the cylinder displacement, Fig. 8

presents the trajectory of the cylinder for the simulation with the highest efficiency (𝑚∗ = 0, 𝑘∗ = 2.5, and

𝛾∗ = 2). The circle marker size represents the amount of instantaneous power extracted throughout the

cycle. The fact that the maximum power extraction occurs when the cylinder is in the vicinity of the

equilibrium position (whether in its upward or downward course) is not surprising since it is where the

cylinder has the highest velocity. This feature is represented by larger and more spaced circles in Fig. 8.

However, in the regions where the cylinder changes its direction, smaller and nearer circles indicate a much

lower power output. Nevertheless, it is interesting to note that the power coefficient is always positive,

which means that there is no interval in the oscillation cycle where the system needs to be provided with

external energy.

5.1 Influence of the mass ratio

Figures 4 to 7 show the results of the energy extraction efficiency for all mass ratios. An optimal

configuration is present in all four figures but is different depending on 𝑚∗. Indeed, for a mass ratio of 0

(Fig. 4), the highest efficiency obtained is of 10.6% with a dimensionless stiffness of 𝑘∗ = 2.5 and a

dimensionless damping of 𝛾∗ = 2. For the mass ratio of 𝑚∗ = 0.127 (Fig. 5), the maximum efficiency

reaches 10.45% at the point (2.5, 1.5) in the 𝑘∗ − 𝛾∗ space. In Fig. 6, for 𝑚∗ = 1, the optimal configuration

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is obtained at the point (5, 2), and results in an efficiency of 9.63%. Finally, Fig. 7 depicts the efficiency

contours of a cylinder with 𝑚∗ = 10. The highest reported efficiency is of 7.92%, which is of the same

order as the other mass ratios, but it is obtained with a much higher dimensionless stiffness, that is, 𝑘∗ =30. The corresponding dimensionless damping is 𝛾∗ = 2.5.

Throughout Figures 4 to 7, as the mass ratio increases, the optimum moves towards higher 𝑘∗ and the highest

efficiency decreases. Consequently, among the four studied mass ratios, the case with 𝑚∗ = 0 is the one

exhibiting the highest energy extraction efficiency. It is also the one featuring the largest amplitude ratio:

𝐴∗ = 0.89. It is evident from this data that the mass ratio has a great influence on the energy extraction.

Lastly, every figure also displays a rapid growth of the efficiency in the low dimensionless damping region

before reaching the optimal configuration. As all figures show, varying 𝛾∗ from 0 to 1 greatly increases the

efficiency as demonstrated by the closest contours on the figures. On the other hand, the influence of the

dimensionless damping variation appears to be less significant on the power extracted at higher values.

Indeed, increasing 𝛾∗ further, say in the range 1 to 3, does not change the efficiency value very much, which

means that the turbine is not very sensitive to damping variations when it operates near its best operation

point.

Fig. 7: Energy extraction efficiency (%) of the

oscillating cylinder for 𝑚∗ = 0 and 𝑅𝑒 = 200.

Fig. 7: Energy extraction efficiency (%) of the

oscillating cylinder for 𝑚∗ = 0.127 and 𝑅𝑒 = 200.

Fig. 7: Energy extraction efficiency (%) of the

oscillating cylinder for 𝑚∗ = 1 and 𝑅𝑒 = 200.

Fig. 7: Energy extraction efficiency (%) of the

oscillating cylinder for 𝑚∗ = 10 and 𝑅𝑒 = 200.

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5.2 Lock-in in the parametric space studied

Most simulated cases produce a periodic motion whose dominant mode frequency is synchronized with the

vortex shedding frequency 𝑓𝑠 such that 𝑓𝑠𝐷/𝑈∞ = 0.2. This implies that almost all cases are synchronized

with the frequency of the classical von Kármán vortex street, meaning that it is actually the flow that governs

the oscillation. Fig. 9 illustrates the correlation between the frequency ratio 𝑓∗ and the reduced velocity 𝑈∗,

which are respectively defined as:

𝑓∗ =𝑓

𝑓𝑛, (10)

𝑈∗ =𝑈

𝑓𝑛𝐷, (11)

where 𝑓 is the frequency of the cylinder displacement and 𝑓𝑛 is the structural natural frequency of the system.

Almost all simulated cases lie on the same line with a slope of 0.2, which corresponds to the Strouhal number

of the vortex street for a cylinder with 𝑅𝑒 = 200. A lock-in phenomenon appears between 𝑈∗ = 5 and 𝑈∗ =9 for 𝑚∗ = 1, where the cylinder oscillates at the structural natural frequency rather than at the vortex street

natural frequency 𝑓𝑠 (see the few red squares that lie horizontally at 𝑓∗ = 1 in Fig. 9). The vortex street

frequency then locks on the structural natural frequency. This phenomenon is obtained only with cases that

have a low dimensionless damping parameter for specific values of 𝑘∗ (or 𝑈∗), which is in good agreement

with the data reported in literature (Williamson & Govardhan, R. (2004); Jauvtis & Williamson (2004)).

However, these configurations involving lock-in, which is usually associated with branching and higher

amplitudes, lead to a significant decrease in the energy extraction efficiency since this turbine concept

strongly depends on the damping coefficient to extract energy (see Equations 7 and 9). Indeed, the damping

parameter 𝛾 represents the ability of the supports of the cylinder to extract energy from its motion, which is

itself induced by the vortices in the wake. Beyond the optimal point, decreasing its value further to achieve

higher amplitudes (and to eventually reach a locked-in configuration) would only reduce the amount of

Fig. 8: Displacement of the cylinder (left) and power extraction coefficient (right) of a whole cycle for the

simulation with the highest efficiency (𝑚∗ = 0, 𝑘∗ = 2.5, and 𝛾∗ = 2). The circle-shape markers size is

proportional to the power coefficient on the displacement curve. The “∗” blue markers indicate specific times.

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power extracted because the damping parameter is simply too low to extract significant power. Furthermore,

increasing the amplitude also hinders the efficiency because it increases the area swept by the cylinder 𝐴1

(see Equation 7).

5.3 Relation between the efficiency and the amplitude

An interesting way to illustrate the relation between the energy extraction efficiency and the motion

amplitude of the cylinder is to consider the so-called Griffin plot (Williamson & Govardhan, 2004), which

illustrates the maximum achievable amplitude at a given mass-damping parameter (𝑚∗𝜁, where 𝜁 is defined

below; see Fig. 10). This parameter was first formulated by Griffin to express the concept of the amplitude

limit as seen on the Griffin plot. This mass-damping parameter is directly proportional to the Skop-Griffin

parameter (SG), another form for the reduced damping, which is defined as (Khalak, Williamson (1996)):

𝑆𝐺 = 2𝜋3𝑆𝑡2(𝑚∗𝜁), (12)

where the variables composing the Skop-Griffin parameter are the mass-damping parameter (𝑚∗𝜁), which

itself depends on the damping ratio 𝜁, and the Strouhal number. These two parameters are defined as:

𝑆𝑡 =𝑓𝑠𝐷

𝑈∞, (13)

𝜁 =𝛾

2√𝑘𝑚. (14)

Since both the energy extraction and the motion amplitude strongly depend on the damping, it is important

to better understand the relation between the amplitude and the energy extraction. It is expected that the

optimal energy extraction configuration is likely not to correspond to a maximum amplitude configuration.

Fig. 9: Frequency ratio against the reduced velocity for all simulated cases. Note that all cases with 𝑚∗ = 0 are

located at the origin because their natural frequencies are undefined (𝑓𝑛 → ∞).

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This is effectively shown in Fig. 10 where the markers filled in red show where the highest energy extraction

efficiency is observed at a given mass ratio. In Fig. 10, 𝐴∗ is the amplitude of the oscillations normalized by

the cylinder diameter. In this case, the amplitude ratio corresponds to:

𝐴∗ =𝑑𝑦,max−𝑑𝑦,min

2𝐷. (15)

It is interesting to note that, effectively, these points do not generally correspond to the highest amplitude

achievable as reported by the Griffin plot. Indeed, for a mass ratio of 𝑚∗ = 0.127, the maximum efficiency

(10.45%) is obtained with a dimensionless amplitude of 𝐴∗ = 0.40, and for a mass ratio of 𝑚∗ = 0, the

maximum efficiency (10.56%) is obtained with a dimensionless amplitude of 𝐴∗ = 0.33. However, for the

mass ratios of 𝑚∗ = 1 and 𝑚∗ = 10, the best configuration for energy extraction is near the saturation limit

(the maximum amplitude at a given mass-damping parameter), implying that the optimal amount of

damping used to extract the energy has only a slight effect on the amplitude. The data shown for 𝑚∗𝜁 = 0

represents the physical limit of a zero-mass or a zero-damping system. It is also used to highlight the fact

that the saturation amplitude of the massless cylinder is higher than for all the other mass ratios.

Another interesting matter is the three distinct branches that are shown in Fig. 10 for 𝑚∗𝜁 < 0.1. Indeed,

for each individual mass ratio, the maximum reachable amplitude is different as seen on the griffin plot

where the height of the saturation limit for these three mass ratios is depicted. It is shown that as 𝑚∗ is

increased, the saturation limit decreases. This characteristic has been pointed out in several studies on

vortex-induced vibrations for other Reynolds numbers (Khalak & Williamson (1996), Newman &

Karniadakis (1997)). Also, even though there is no possible branch for the zero-mass ratio, Fig. 10 shows

that the highest amplitude is observed with this specific mass ratio (points at 𝑚∗𝜁 = 0). All these points

Fig. 10: Griffin plot including all simulations carried out with the mass ratios 𝑚∗ = 0 (Δ), 𝑚∗ = 0.127 (◊), 𝑚∗ =1 (), and 𝑚∗ = 10 () with 𝑅𝑒 = 200. The markers colored in red represent the cases with the best efficiency

at a given mass ratio. Four curves illustrate the saturation limit of the amplitude.

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have in common a small non-dimensional damping when compared to all other cases. The vortex streets of

almost all the simulations constituting the three branches display an early 2P mode. More information on

these vortex streets are presented in the next section. Interestingly, in the cases constituting the saturation

branch for 𝑚∗ = 10, the cylinder does not have a regular displacement pattern. Instead of a symmetrical

eight-shape form as normally seen and depicted in Fig. 3, the displacement is more irregular (although still

periodic) as displayed on Fig. 11 (a). Indeed, in this specific case, the cross-flow displacement frequency is

the same as the in-line displacement frequency. Consequently, the trajectory of the cylinder becomes a loop

rather than a symmetrical eight-shape. For most other cases, the frequency of the in-line displacement is

twice that of the cross-flow displacement. The irregularity of the trajectory comes from a slightly

asymmetric vortex mode (see Fig. 11 (b)). This feature has also been observed for high-density cylinders

with two degrees of freedom in other studies. For example, the paper of Marzouk (2010), which presents

the effect of the density ratios for both 1- and 2-DOF cylinders, also report this behavior. Lastly, it is worth

noting that the in-line motion remains much smaller than the cross-flow motion. Therefore, the irregular

motion is very subtle.

Fig. 12 shows the same griffin plot, but in this case, only some results from 𝑚∗ = 1 are shown. To better

illustrate the dependency of the amplitude with 𝑘∗ and 𝛾∗ and to compare the results with those of Fig 4,

curves for fixed dimensionless damping are drawn. The numbers that are presented on the figure are the

different dimensionless stiffness parameters. They are only shown on the first curve to indicate the trend

since the same pattern is reproduced along every curve. Finally, the maximum of all curves lies between

𝑘∗ = 3 and 𝑘∗ = 6.25.

This figure also clearly illustrates the effect of the damping parameter on the amplitude of the cylinder

displacement. Indeed, increasing this parameter increases the mass-damping parameter and decreases the

maximum amplitude attainable as predicted by the Griffin plot. The highest energy extraction efficiency is

obtained for 𝐴∗ = 0.308 and 𝑚∗𝜁 = 0.36. As mentioned before, both the energy harnessing and the

amplitude depend on the damping of the system. Imposing too much damping would result in small

amplitudes of oscillation and smaller efficiencies, which is against what is wanted in a context of energy

extraction. Additionally, setting a low damping will decrease the power that can be extracted from the

(a) (b)

Fig. 11: Irregular oscillation of a high amplitude case with 𝑚∗ = 10, 𝑘∗ = 30, and 𝛾∗ = 0.0001: (a) trajectory, (b)

two vorticity fields (magnitude) taken at a 𝑇/2 interval.

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cylinder displacement since this parameter is fundamental in the power equation (Equation 9). This figure

also shows that the amplitude seems very sensitive to the mass-damping parameter. Certainly, a small

variation of this parameter can drastically modify the magnitude of the cylinder displacement.

5.4 Vortex modes in the parametric space

Three patterns of vortex shedding are observed in the parametric space and are presented in Figures 13 to

15. The first mode reported is the 2S mode (Fig. 13), which is largely present over all the parametric space

for every mass ratio except for the lower-left corner where the dimensionless damping and stiffness are

smaller and where greater amplitudes are observed. In this mode, a vortex is shed into the wake of the

cylinder at each half cycle, which is similar to the von Kàrmàn vortex shedding (Williamson & Roshko

(1988)).

Fig. 12: Griffin plot for 𝑚∗ = 1 and 𝑅𝑒 = 200 with data connected by dimensionless damping. The “+” symbol

represents the highest efficiency obtained. The numbers on the curve 𝛾∗ = 0.25 are the 𝑘∗ values.

Fig. 13: Near-wake vortices of an oscillating cylinder with a 2S mode.

13

Another vortex shedding pattern visible is the early 2P mode displayed in Fig. 14. The 2P mode corresponds

to a pair of vortices that is formed at each half cycle. As it is shown in this figure, the second (smaller)

vortex is rapidly damped out, most likely because of the low Reynolds number used in the simulations. In

the presented results, this pattern is observed for the configuration with the greatest amplitudes for all mass

ratios. Mostly, the points constituting the three saturation limits in Fig. 10. As mentioned earlier, these points

represent the cases with low 𝛾∗. Although the 2P mode has been reported in literature for Reynolds numbers

as low as 200 (Han et al. (2015)), it is usually associated with a change from an initial branch of resonance

to an upper one at higher Reynolds (Yu, 2014)). However, such branching is generally not observed in the

laminar regime (Prasanth & Mittal (2008); Khalak & Williamson (1999)).

Lastly, Fig. 15 illustrates a C(2S) vortex shedding pattern. This pattern exhibits vortices that coalesce behind

the rigid body to create larger vortical structures in the wake of the cylinder, as presented in different studies

(Williamson and Roshko (1988); Prasanth & Mittal (2008)). This pattern is predominantly seen in the lower-

left corner of the parametric space for 𝑚∗ = 0, 0.127 and 1, around the points producing 2P modes. They

also are present for rather high amplitudes, but generally lower than those of the 2P mode. It is noted in

literature that a switch from the 2S to the C(2S) resonance mode can create a jump of the oscillation

amplitude (Yu (2014); Prasanth & Mittal (2008)). This feature can explain why these specific cases have a

larger amplitude than others.

Fig. 14: Near-wake vortices of an oscillating cylinder with an early 2P mode.

Fig. 15: Near-wake vortices of an oscillating cylinder with a C(2S) mode.

14

5.5 Relation between the efficiency and the mass-damping parameter

Another way to present the efficiency of vortex-induced vibrations is with the mass-damping parameter

(Barrero-Gil et al. (2012); Mackowski & Williamson(2013)). As shown in Fig. 12, a high or a low non-

dimensional damping results in a decrease of the amplitude and, hence, a decrease of the energy extraction

efficiency is expected. Indeed, lower amplitudes result in lower velocity amplitudes in the upward and

downward courses, which directly impairs the power output. Fig. 16 shows the efficiency of some turbine

configurations against the mass-damping parameter. The figure clearly illustrates that, with a given

dimensionless damping, better efficiencies are obtained with intermediate spring stiffness. Moreover, a

significant sensitivity of the mass-damping parameter is observed, especially before the optimal point where

a small variation of it greatly modifies the efficiency of the turbine. This was also observed by Barrero-Gil

et al. (2012) who show that the maximum efficiency varies from about 0.07 to 0.18 when the mass-damping

parameter 𝑚∗𝜁 is changed from about 0.04 to 0.25. However, they noted that, at higher Reynolds numbers,

the high-efficiency range is larger in terms of the mass-damping parameter. This means that the sensitivity

to the latter is less prominent, which is promising for an eventual application.

5.6 Perspectives of application

The maximum efficiency reported in this preliminary study is 10.6% and is obtained with the case where

the mass ratio is set to 0. For a mass ratio of 𝑚∗ = 0.127, the efficiency can reach 10.45%. This maximum

efficiency is similar to the one reported by Barrero-Gil et al. (2012) which is of 8% (based on the efficiency

definition of Equation 7). Moreover, while the efficiency definition used by Bernitsas et al. (2008) is

different, their experimental data suggests that they obtained efficiencies of the same order of magnitude if

the definition of Equation 7 is considered.

When compared to other technologies of hydrokinetic turbines such as axial-flow and cross-flow turbines

as well as oscillating wing turbines, the efficiency of the oscillating cylinder concept remains rather small

(Kinsey and Dumas (2012); Kinsey and Dumas (2017); Gosselin et al. (2016)). However, the simplicity of

the concept and its similarity with the oscillating wing concept make it interesting. Indeed, even if the

efficiency is moderate, the understanding of the flow mechanisms that maximize the efficiency may be

useful for the oscillating-wing turbines development. Finally, the model presented by Barrero-Gil et al.

(a) (b)

Fig. 16: Energy extraction efficiency (in %) of the oscillating cylinder turbine against the mass-damping parameter

with (a) 𝑚∗ = 0.127 and (b) 𝑚∗ = 10. The “+” symbol represents the maximum efficiency obtained. The numbers

on the curve 𝛾∗ are the 𝑘∗ values.

15

(2012) suggests that increasing the Reynolds number would produce higher efficiencies. Therefore, it is not

excluded that this concept may result in a viable turbine design.

6 CONCLUSION

This paper focuses on identifying the effects of 𝑚∗, 𝛾∗, and 𝑘∗ on the energy extraction efficiency of an

oscillating-cylinder turbine based on the mechanism of vortex-induced vibrations. As demonstrated in

Figures 4 to 7, optimal configurations can be observed in the 𝑘∗ − 𝛾∗ space and analyzed with the current

parameters. The results for small mass ratios exhibit the highest energy extraction potentials. Efficiencies

of 7.92%, 9.63%, 10.45% and 10.6% are obtained with 𝑚∗ = 10, 𝑚∗ = 1, 𝑚∗ = 0.127 and 𝑚∗ = 0,

respectively. Lastly, this paper helps in understanding the action of each parameter of the dimensionless

harmonic equation on the proposed turbine model. However, more investigation is required to optimize this

turbine concept and, eventually find the set of parameters that will maximize its efficiency. The next step

of the study will focus on a more fundamental perspective and try to connect the phenomena observed in

this study with the concept of oscillating foil turbines. The exploration of new parameters influencing the

energy efficiency such as the use of different stiffness and damping in the in-line and cross-flow directions

will be investigated. Non-linear springs will also be considered since it has been suggested by Mackowsky

and Williamson (2013) that those can be beneficial for energy extraction. Finally, tests at higher Reynolds

numbers and the implementation of an optimization algorithm that automates the search of optimal

configurations will be considered.

ACKNOWLEDGEMENTS

The authors would like to acknowledge the financial support from Université Laval as well as the computing

resources provided by Compute Canada.

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