View
4
Download
0
Category
Preview:
Citation preview
1
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
2
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.
3
𝜕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)
4
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).
5
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).
6
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 =�̅�
12
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
7
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.
8
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.
9
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 → ∞).
10
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.
11
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.
12
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.
REFERENCES
Bahmani, M. H., & Akbari, M. H. (2010). Effects of mass and damping ratios on VIV of a circular cylinder.
Ocean Engineering, 37, 511-519.
Barrero-Gil, A., Pindabo, S., & Avila, S. (2012). Extracting energy from Vortex-Induced Vibrations: A
parametric study. Applied Mathematical Modelling, 36, 3153-3160.
Bernitsas, M., & MacBain, J. (n.d.). VIVACE: A new concept for harnessing hydrokinetic energy. Retrieved
from National Institute of Standards and Technology:
https://www.nist.gov/sites/default/files/documents/tip/wp/pswp/87_vivace_a_new_concept_for_h
arnessing_hydrokenetic_energy.pdf
Bernitsas, M., & Raghavan, K. (2009). United States Patent No. 7,493,759 B2.
Bernitsas, M., Raghavan, K., Ben-Simon, Y., & Garcia, E. (2008, September). VIVACE (Vortex Induced
Vibration Aquatic Clean Energy): A New Concept in Generation of Clean and Renewable Energy
From Fluid Flow. Journal of Offshore Mechanics and Arctic Engineering - Transaction of the
ASME, 130(4).
Blackburn, H., & Karniadakis, G. (1993). Two-and-three-dimensional simulations of vortex-induced
vibration or a circular cylinder. In The Third International Offshore and Polar Engineering
Conference, vol. 3 (p. 715). Singapore: International Society of Offshore and Polar Engineers.
Canadian Hydropower Association. (2017). Retrieved 02 15, 2017, from https://canadahydro.ca
Gosselin, R., Dumas, G., & Boudreau, M. (2016). Parametric study of H-Darrieus vertical-axis turbines
using CFD simulations. Journal of Renewable and Sustainable Energy.
16
Govardhan, R., & Williamson, C. H. (2000). Modes of vortex formation and frequency response of a freely
vibrating cylinder. Journal of Fluid Mechanics, 420, 85-130.
Güney, M. S., & Kaygusuz, K. (2010). Hydrokinetic energy conversion systems: A technology status
review. Renewable and Sustainable Energy Reviews, 14(9), pp. 2996-3004.
Han, H., Lin, W., Tang, Y., Zhao, C., & Sammut, K. (2015). Effects of the natural frequency ratio on vortex-
induced vibration of a cylindrical structure. Computers & Fluids 110, 62-76.
Jauvtis, n., & Williamson, C. (2004). The effect of two degrees of freedom on vortex-induced vibration at
low mass and damping. Journal of Fluid Mechanics, 509, 23-62.
Khalak, A., & Williamson, C. (1996). Dynamics of a hydroelastic cylinder with very low mass and damping.
Journal of Fluids and Structures,10, 455-472.
Khalak, A., & Williamson, C. H. (1999). Motions, forces and mode transitions in vortex-induced vibrations
at low mass-damping. Journal of Fluids and Structures, 13, pp. 813-851.
Khan, M., Bhuyan, G., Iqbal, M., & Quaicoe, J. (2009). Hydrokinetic energy conversion systems and
assessment of horizontal and vertical axis turbines for river and tidal applications: A technology
status review. Applied Energy, 86(10), pp. 1823-1835.
Kinsey, T., & Dumas, G. (2012). Computational Fluid Dynamics Analysis of a Hydrokinetic Turbine Based
on Oscillation Hydrofoils. Journal of Fluids Engineering vol. 134, no 2, 21104.
Kinsey, T., & Dumas, G. (2017). Impact of channel blockage on the performance of axial and cross-flow
hydrokinetic turbines. Renewable Energy, 239-254.
Klamo, J. T., Leonard, A., & Roshko, A. (2006). The effects of damping on the amplitude and frequency
response of a freely vibrating cylinder in cross-flow. Journal of Fluids and Structures, 22, 845-856.
Mackowski, A. W., & Williamson, C. H. (2013). An experimental investigation of vortex-induced vibration
with nonlinear restoring. Physics of Fluids 25.
Marzouk, O. A. (2010). Characteristics of the Flow-Induced Vibration and Forces with 1- and 2-DOF
Vibrations and Limiting Solid-to-Fluid Density Ratios. Journal of Vibration and Acoustics,
0410139.
Newman, D. J., & Karniadakis, G. (1997). A direct numerical simulation study of flow past. J. Fluid Mech.
vol.344, 95-136.
Olivier, M., & Dumas, G. (2016). A parametric investigation of the propulsion of 2D chordwise-flexible
flapping wings at low Reynolds number using numerical simulations. Journal of Fluids and
Structures, 63, 210-237. doi:10.1016/j.jfluidstructs.2016.03.010
Prasanth, T., & Mittal, S. (2008). Vortex-induced vibrations of a circular cylinder at low Reynolds numbers.
Journal of Fluid Mechanics vol. 594, pp. 463-491.
Press, W. T. (2007). Numerical recipes. Cambridge University Press.
Shiels, D., Leonard, A., & Roshko, A. (2001). Flow-induced vibration of a circular cylinder at limiting
structural parameters. Journal of Fluids and Structures, 15, 3-21.
17
Williamson, C. H. (1996). Vortex dynamics in the cylinder wake. Annual Review of Fluid Mechanics, 477-
539.
Williamson, C. H., & Roshko, A. (1987, January 14). Vortex formation in the wake of an oscillating
cylinder. Journal of Fluids and Structures, 2, 355-381.
Williamson, C., & Govardhan, R. (2004). Vortex-Induced Vibrations. Annual Review of Fluid Mechanics,
36, 413-455.
Xiao, Q., & Zhu, Q. (2014). A review on flow energy harvesters based on flapping foils. Journal of Fluids
and Structures, 46, pp. 174-191.
Young, J., Lai, J. C., & Platzer, M. F. (2014). A review of progress and challenges in flapping foil power
generation. Progress in Aerospace Sciences, 67, pp. 2-28.
Yu, R. K. (2015). Flow-induced vibrations of in-line cylinder arrangement at low Reynolds number. Ph.D
thesis. École Polytechnique de Montréal.
Recommended