Multi-body interaction effect on the energy harvesting performance of a flapping hydrofoil

Accepted Manuscript Multi-body interaction effect on the energy harvesting performance of a flapping hydrofoil Mohsen Lahooti, Daegyoum Kim PII:

S0960-1481(18)30699-2

DOI:

10.1016/j.renene.2018.06.054

Reference:

RENE 10210

To appear in:

Renewable Energy

Received Date: 5 September 2017 Revised Date:

12 June 2018

Accepted Date: 13 June 2018

Please cite this article as: Lahooti M, Kim D, Multi-body interaction effect on the energy harvesting performance of a flapping hydrofoil, Renewable Energy (2018), doi: 10.1016/j.renene.2018.06.054. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

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Multi-body interaction effect on the energy harvesting performance of a flapping hydrofoil

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Mohsen Lahooti, Daegyoum Kim∗

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Department of Mechanical Engineering, KAIST, Daejeon 34141, Republic of Korea

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Abstract

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The effect of an upstream bluff body on energy harvesting performance of

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a heaving and pitching hydrofoil is investigated numerically using a two-

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dimensional immersed boundary method at Re = 1000. The presence of the

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upstream body changes flow structure around the hydrofoil and enhances

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efficiency significantly by two mechanisms. Mutual interaction of the vortex

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shed from the upstream body and the leading-edge vortex of the hydrofoil

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precipitates the separation of the leading-edge vortex from the hydrofoil and

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its streamwise transport. The incoming flow deflected by the upstream body

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changes the effective angle of attack for the hydrofoil. These phenomena

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significantly increase heaving force and pitching moment during stroke re-

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versal, and major contribution to efficiency enhancement is from the change

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in pitching moment. 30% increase in efficiency, relative to a hydrofoil without

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an upstream body, can be achieved for same kinematics. However, the up-

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stream body may be disadvantageous in some configurations. If the hydrofoil

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is placed closely to the body in transverse direction, the leading-edge vor-

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tex formation is suppressed after stroke reversal. When flapping frequency

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does not match with vortex shedding frequency of the upstream body, non-

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Corresponding author: D. Kim Emailsubmitted addresses: (Mohsen Lahooti), Preprint to [email protected] Renewable Energy [email protected] (Daegyoum Kim) ∗

June 14, 2018

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periodic flow structure formed around the hydrofoil can cause efficiency drop

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and irregular power generation.

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Keywords: Energy harvesting, Ocean energy, Hydrodynamics, Flapping

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hydrofoil, Bluff body, Leading-edge vortex

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1. Introduction

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Recently, a flapping hydrofoil has been widely studied as a novel method

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to extract energy from fluid flows. The flapping hydrofoil can perform with-

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out losing efficiency noticeably in unsteady flow environment. The hydrofoil

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is also better fitted for shallow water installation because of their rectangular

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sweeping area[1–3]. The idea of using the flapping hydrofoil was first intro-

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duced by McKinney and DeLaurier[4]. Motivated by their work, Jones and

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Platzer [5] showed that, for the hydrofoil of combined heaving and pitching

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motions, one could change from a propulsion regime to an energy harvesting

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regime if the pitching amplitude exceeds the heaving-based angle of attack.

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The basic mechanism of efficient energy extraction from a flapping foil

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(from now on the “hydrofoil” is referred to as “foil” for simplicity) can be

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explained with vortices generated by the foil. The heaving motion of the foil

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with an appropriate angle of attack forms a strong leading-edge vortex near

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the surface of the foil and produces vertical force due to the low pressure

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region inside the vortex. However, as the leading-edge vortex separates and

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washes away from the surface, the vertical force decreases. To maintain large

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power generation, the foil should rotate and change its heaving direction to

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form another leading-edge vortex near the opposite surface of the foil [1, 6–8].

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Many studies on the energy harvesting foil have addressed the role of 2

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motion parameters and foil geometry on power generation. Among the earli-

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est works is a numerical and experimental study conducted by Linsey [9] to

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determine the feasibility of energy extraction. Another comprehensive work

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was done by Dumas and Kinsey [10] and Kinsey and Dumas [7] in which they

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presented an efficiency map as a function of pitching amplitude and reduced

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frequency. Their map shows the efficiency as high as 34% can be reached for

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a single foil and the best performance is around f ∗ = 0.15 which is consis-

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tent with the analysis on optimal frequency for energy harvesting [8]. The

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foil shape is known to have minor impact on efficiency for thin foils [6, 7].

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It was also claimed that the increase in Reynolds number led to the better

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performance [7]. However, due to complication of turbulent flows, solid con-

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clusion on the Reynolds number effect cannot be established, and it needs

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more investigation [1]. The studies on a foil with finite span reported that

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three-dimensional effects caused the reduction in efficiency, compared to its

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two-dimensional counterpart [6, 11, 12].

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Several studies attempted non-sinusoidal pitching and heaving motions

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and achieved higher efficiency than that of sinusoidal motions for some spe-

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cific conditions [13, 14]. The flexible aero/hydrofoils motivated by the com-

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pliant propulsors of flying and swimming animals were also investigated, in

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which the flexible aero/hydrofoil showed efficiency improvement although

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the prescribed deformation of the aero/hydrofoil was considered instead of

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its passive deformation [15]. It is worth mentioning that the effect of a flex-

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ible foil on energy harvesting has not been fully understood, and in-depth

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research is required in this area [1].

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Another bio-inspired idea to improve the harvesting performance is from

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multi-body interaction. This idea was motivated by aquatic animals and

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birds travelling as a group to benefit from the flow induced by their neigh-

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bors [16–18]. One of the earliest studies on multiple foils for energy harvesting

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is the work by Jones et al. [19], in which they experimentally investigated

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the efficiency of twin foils in a tandem configuration with small clearance

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and phase lag of 90◦ between them. Since the idea of using tandem foils

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is to extract the remaining energy from the vortices generated by the front

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foil, both phase lag and relative distance between front and rear foils are im-

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portant parameters to strongly determine the overall performance [13, 20].

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The tandem configuration seems disadvantageous because the rear foil posi-

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tioned in the wake of the front foil is exposed to relatively lower flow velocity

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and fluid kinetic energy. Nevertheless, we are able to gain some benefit by

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properly positioning the rear foil for positive interaction with the vortices

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shed from the front foil [2]. Indeed, using the tandem configuration, higher

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efficiency can be achieved in some specific conditions (e.g., the phase lag of

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180◦ ) [21–23].

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In addition to the tandem arrangement, the parallel foil arrangement

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was investigated experimentally or numerically for both in-phase and out-of-

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phase modes [24–26]. For the in-phase mode, the per foil efficiency decreases

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as the distance between the two foils decreases. However, for the out-of-phase

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mode, the reduction in the gap distance produces larger overall output.

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Most, if not all, of previous works on multi-body effects focused on diverse

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arrangements of multiple foils with the same geometry. Instead of multiple

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foils, mutual interaction of an upstream bluff body and a downstream energy

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harvester is considered to improve the efficiency in our work. The stationary

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upstream body deflects uniform flow and increases the flow velocity encoun-

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tered by the downstream energy harvester, which is able to contribute to

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larger power generation. This idea was applied to a vertical-axis rotary tur-

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bine with an upstream bluff body, and the increase in efficiency was reported

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[27, 28]. In the application of this idea to the heaving and pitching foil, un-

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steady motion of the foil and its interaction with the vortex shed from an

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upstream body make flow dynamics too complicated to predict the overall

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effect based on available data reported in the literature. This difficulty moti-

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vates us to investigate the effect of an upstream body on the power generation

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of the downstream flapping foil and identify flow phenomena responsible for

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noticeable changes in power generation. We believe that this study will lead

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us to design an energy harvesting hydrofoil system with better performance.

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The rest of paper is organized as follows. In Sec. 2, a foil model and

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variables investigated in this study are described. The method of numerical

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simulation and the validation of our numerical code are explained in Sec. 3.

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Sec. 4 discusses our results on heaving force/pitching moment generation and

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overall efficiency, based on the physical interpretation of vortex dynamics,

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which is followed by concluding remarks in Sec. 5.

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2. Problem description

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Our model consists of a pitching and heaving foil and a stationary up-

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stream bluff body, a thin vertical plate, positioned upstream of the flapping

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foil (Fig. 1(a)). As a foil model, we used the symmetric Joukowski foil

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mathematically described as [8]: x=z+

λ2 + d − e, z−e 5

(1)

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U∞

(b)

c

αe

h0

θ

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h−Vˆ

Ly

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(a)

Û

1/3c

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y

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h

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c

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Figure 1: (a) Schematic of the flapping foil with an upstream body and (b) the Joukowski foil. In (a), the thick vertical line is the upstream body.

where x = x + iy and z = ξ + iζ are coordinates in (x, y) and (ξ, ζ)-

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planes respectively. The Joukowski foil is the mapping of a circle with radius

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r = λ + e + s from the (ξ, ζ)-plane to the (x, y)-plane. e and s are the

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parameters characterizing foil thickness and trailing-edge sharpness. d is

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defined as d = (λ+2e+s)+λ2 /(λ+2e+s)−(1/2+b). The parameters used for

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the Joukowski foil in this work are [b, λ, e, s] = [−0.167, 0.247, 0.027, 0.0074].

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With these parameters, the foil has a unit chord length with the maximum

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thickness of about 0.15 and the pitching axis located at one third of the chord

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from the leading edge (Fig. 1(b)). The kinematics of the foil are the combination of sinusoidal heaving and

pitching motions: h(t) = h0 sin(2πf t)

(2a)

θ(t) = −θ0 sin(2πf t + φ),

(2b)

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where h0 and θ0 are heaving and pitching amplitudes, f is flapping frequency, and φ = −π/2 is phase difference. Instantaneous heaving efficiency ηh (t), instantaneous pitching efficiency ηp (t), and time-averaged total efficiency η are defined as follows: h˙ 2Fy h˙ = C (t) h 3 A ρU∞ U∞ s ˙ ˙ 2M θ θc ηp (t) = = C (t) p 3 A ρU∞ U∞ s Z t+T 1 [ηh (t) + ηp (t)] dt, η= T t

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ηh (t) =

(3a)

(3b) (3c)

where As is the frontal area swept by the foil during its periodic motion

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2 2 and U∞ is free-stream velocity. Ch (= 2Fy /ρU∞ As ) and Cp (= 2M/ρU∞ cAs )

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are heaving force and pitching moment coefficients respectively. Pitching

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moment M and angle θ are positive in a clockwise direction (Fig. 1(b)). The

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averaged efficiency η was computed over ten cycles for all cases after initial

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transient effects disappeared. The efficiency indicates how much energy can

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be transformed to mechanical energy of the foil from fluid kinetic energy

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available for the swept area of the foil. In addition to these parameters, a

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parameter frequently used here is relative efficiency η ∗ defined as the ratio of

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the efficiency for a foil with an upstream body η to that of a foil without an

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upstream body (a single foil ) η0 ; η ∗ = η/η0 .

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In this study, non-dimensional variables we mainly considered are hor-

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izontal distance between the edge of the upstream plate and the pitching

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axis of the foil, L∗x ∈ [0.5, 0.75, 1.0], vertical distance between the top edge of

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the upstream plate and the middle of the sweeping area (the location of the

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pitching axis in the middle of a stroke), L∗y ∈ [0.5, 1.0, 1.5], and the height

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the upstream plate, H ∗ ∈ [1.0, 1.5]; L∗x = Lx /c, L∗y = Ly /c, and H ∗ = H/c 7

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(Fig. 1(a)).

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As mentioned in Sec. 1, this study focuses on finding the effects of an

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upstream body on energy harvesting performance and in particular physical

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mechanisms which cause the change in the performance, instead of obtaining

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optimal model conditions. Therefore, the heaving and pitching motions of the

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foil were fixed for all of our numerical simulations. The specific kinematics of

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the foil were chosen such that they produced high efficiency (not the highest,

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though) for a single foil without an upstream body. According to previous

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researches with the single foil, for high efficiency, the heaving and pitching

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amplitudes should be usually in the range of 0.5c < h0 < c and 70◦ < θ0 <

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80◦ with the reduced frequency f ∗ (= f c/U∞ ) around 0.15 [6–8]. In our

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simulations, the heaving and pitching amplitudes and the reduced frequency

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were fixed as h0 = 0.5c, θ0 = 76.33◦, and f ∗ = 0.14. The Reynolds number

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based on the free-stream velocity U∞ and the chord c is 1000.

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3. Numerical method

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Governing equations of a fluid domain are as follows for incompressible

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isothermal laminar flow:

∇·u = 0

(4a)

1 ∂u + u · ∇u = − ∇p + ν∇2 u, ∂t ρ

(4b)

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where ρ, ν, and p are fluid density, kinematic viscosity, and pressure, re-

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spectively, and u is a velocity vector. The simulation of the flow around a

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flapping foil is numerically a challenging problem due to the large-amplitude

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motion of the foil. Several numerical approaches have been reported in lit-

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erature including the panel method [9], the approach using moving meshes 8

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in combination with a translational reference frame [7, 12, 13, 22], the finite

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difference method using mapping between physical and computational spaces

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[8], the compressible flow solver with low Mach number [14], to name a few.

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In the present work, a different approach based on the Immersed Boundary

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Method (IBM) [29, 30] was employed, which enables us to arbitrarily move

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an object of any shape in stationary grids of a fluid domain and realize the

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relative motion of two objects, a moving foil and a fixed upstream body.

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An in-house IBM code of the finite different method was implemented

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for two-dimensional flow. A moving rigid foil was treated by the Hybrid

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Cartesian/Immersed Boundary (HCIB) method proposed by Glimanov and

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Sotiropoulos [31]. The HCIB method benefits from the sharp interface char-

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acteristic of Cartesian methods, rather than diffusing the interface over sev-

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eral cells. In addition, although the characteristics of the Cartesian methods

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area adopted, the HCIB method dose not require complex cell reconstruction

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and grid modification near immersed interfaces [31]. Therefore, it takes the

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advantages of sharp interface capturing capability of Cartesian methods and

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simpleness and robustness of immersed boundary methods in treating moving

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bodies inside flow, which makes it an appropriate choice for our problem. Fol-

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lowing Ge and Sotiropoulos [32], governing equations were discretized with

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the implicit second-order method. The QUICK scheme was used for the dis-

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cretization of convective terms, and viscous terms and pressure gradient was

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discretized using the standard central difference. For pressure-velocity cou-

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pling, the fractional step method of Van Kan [33] was employed. Summary

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of discretization schemes is provided in Table 1. The implicit second-order

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approach produces a nonlinear system of equations which can be solved with

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Discretization

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Table 1: Summary of discretization schemes

implicit second-order

convective term

QUICK method

pressure gradient term

second-order central difference

viscous term

second-order central difference

pressure-velocity coupling

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fractional step method of Van Kan [33]

the Newton-Krylov subspace methods such as GMRES. The linear system

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of equations is computed during nonlinear iterations, and linear system also

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arises for pressure equation. Convergence criteria for both nonlinear and

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linear systems are to reduce the relative L2 -norm of error by eight order of

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magnitude; i.e. ||ǫrel || < 10−8 where ||ǫrel || = ||ǫ||/||ǫ0 || and ||ǫ0 || is the error

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norm at start of solving the system of equations. For the complete solution

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procedure of the HCIB method, refer to Gilmanov and Sotiropoulos [31] and

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Ge and Sotiropoulos [32].

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For our simulations, a rectangular fluid domain of [−15c, 25c]×[−15c, 15c]

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size was constructed with the pitching axis in the middle of a stroke at origin.

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The domain was divided into non-uniform rectangular grids with the mini-

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mum spacing of dx = dy = 0.005c near the foil and upstream body. Since the

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foil moves through the domain, in order to have enough grid resolution in the

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vicinity of the foil and upstream body, uniform grids with dx = dy = 0.005c

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were used in the region with approximate size of [−1c, 1.2c] × [−2.5c, 1.2c]

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containing both upstream body and entire swept area of the foil. Outside

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the region of uniform grids, the grid size increases smoothly with the length

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ratio of r = 1.05 until the whole computational domain is covered. Accord-

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ingly, the total number of grids is Nx × Ny = 628 × 755. More details on

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the procedure of choosing grid size and grid convergence study are provided

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in Sec. 3.1. Time steps were determined by the Courant-Fredrisch-Levwy

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(CFL) criterion, which restricts a Lagrangian point on the foil to pass less

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than one grid of the fluid domain in each time step. The CFL number used

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in this work is CF L = 0.5 which results in the time steps size of typically

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2800 − 3000 steps per cycle (T /∆t = 2800). Since the Reynolds number of

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our model is 1000, actual flow is three-dimensional. However, in this study,

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based on two-dimensional simulation, we focus on obtaining insights into how

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the efficiency of the foil can be improved by placing an upstream body rather

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than simulating a realistic three-dimensional flow.

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3.1. Code validation

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To validate the accuracy of our in-house IBM code, two sets of tests

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were conducted. The classic problem of a uniform flow around a stationary

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circular cylinder with diameter d was first considered. A fluid domain is

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30d in both x and y directions, and the cylinder is placed in the middle

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of the domain. Non-uniform grids were used with the minimum spacing of

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0.01d near the cylinder; the total number of grids is Nx × Ny = 386 × 300.

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Drag and lift coefficients are compared with other references in Table 2 for

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Re = 200 and 1000. In addition, Fig. 2 compares Strouhal number St as a

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function of Re. For Re < 49, the flow around the cylinder is steady laminar

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with symmetric wake vortices, and, from Re = 50 to Re = 140 − 194, the

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around a circular cylinder.

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Table 2: Comparison of drag coefficient CD and lift coefficient CL for the uniform flow

CL

CD Re = 200

±0.67

1.34 ± 0.04

Linnick and Fasel [35]

1.34 ± 0.04

±0.69

Taria and Colonius [36]

1.36 ± 0.04

±0.69

1.49 ± 0.2

±1.36

Present work Henderson [37]

1.51

1.45 ± 0.2

±1.38

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flow is in laminar vortex-shedding regime. Increasing Re from 180 to 260,

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the flow is in three-dimensional wake-transition regime in which the near-

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cylinder wake develops vortices of smaller scale and the wake becomes three-

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dimensional. Increasing Re from 260 to 1000 leads to increase in disorder

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of fine-scale three-dimensionalities [34]. Although a flow over the circular

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cylinder becomes three-dimensional from about Re = 180, the purpose of

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this two-dimensional simulation is solely to validate the accuracy of our code

238

against available two-dimensional simulations reported in literature. The

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results of our simulations are found to be in excellent agreement with those

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of other references.

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Next, for the problem including a moving object, we compared our results

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of a single flapping foil to those of Kinsey and Dumas in order to validate our

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0.25

0.20

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St 0.15

0

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Present work Mittal et al. 2008 Williamson 1992 Zang et al. 1994

250

500

750

1000

Re

Figure 2: Strouhal number St as a function of Reynolds number Re for the circular cylinder. Filled circle: present work, filled triangle: Mittal et al.[38], hollow square:

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Williamson[39], and hollow diamond: Zang et al.[40].

code [7]; h0 = c, θ0 = 76.33◦, and f ∗ = 0.14 at Reynolds number Re = 1100.

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Since the Joukowski foil was used in our study, the foil geometry parameters

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were chosen such that it has the most resemblance to the NACA0015 foil used

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in Kinsey and Dumas. For this simulation, a fluid domain with 30c in both

247

x and y directions was discretized non-uniformly. To select appropriate grid

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size, a series of tests was conducted with different grid numbers. For all these

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meshes, grid size is distributed non-uniformly and grows smoothly to domain

250

boundaries. However, in the region covering the foil swept area, uniform grids

251

with the minimum grid size was used. Table 3 compares average efficiency in

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one period for these grids. As evident in Table 3, the solution converges in

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grid no. 3; the difference between no. 3 and no. 4 is negligible. Hence, the

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grid no. 3, i.e. the mesh with the minimum grid size of dx = dy = 0.005c

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Table 3: Comparison of time-averaged total efficiency η (Eq. 3c) for simulations with different grids.

No.

Nx × Ny

dxmin = dymin

η

1

235 × 255

0.02c

0.283

2

347 × 415

0.01c

3

547 × 755

0.005c

0.324

4

913 × 1262

0.0025c

0.325

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Table 4: Comparison of time-averaged total efficiency η (Eq. 3c) between our simulation and previous studies.

η

0.324

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0.337

Zhu [8]

0.315

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near the foil and the total grid number of Nx × Ny = 547 × 755, was chosen

256

for simulation of a single foil. Based on this grid convergence study, the same grid resolution, dx = dy = 0.005c near the foil and upstream plate, was

258

chosen for our simulations with an upstream body.

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Table 4 compares the averaged efficiency of our simulation with those of

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Kinsey and Dumas [7] and Zhu [8]. In addition, Fig. 3 compares drag and

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heaving force coefficients with those of Kinsey and Dumas[7]. Table 4 and

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Fig. 3 validate that our code produces reliable data in both time-averaged

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and time-resolved approaches.

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(b)

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Ch(t*) 0

Cx(t*) 2

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0 0

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0.75

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0.75

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(b) heaving force coefficient Ch (Eq. 3a) between our simulation (solid line) and Kinsey and Dumas[7] (circle dots). t∗ = t/T , and T is flapping period.

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4. Results and discussion

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4.1. Physical principles of efficiency improvement with an upstream body

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First, we describe the change in flow structure of a flapping foil with the

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presence of an upstream plate and address their effects on energy harvesting

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performance. In this section, we will consider only a single case of L∗x = 0.75,

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L∗y = 1.5, and H ∗ = 1.0 (Fig. 4). This particular case was chosen because

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the foil was located at the region of higher power output than the single-

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foil case and the characteristics of flow structure and force/moment trends

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shown in this case can be generalized to other cases which exhibit comparable

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improvement in efficiency over the sing-foil case.

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Figs. 4(a) and (b) show the time history of heaving force coefficient Ch

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and pitching moment coefficient Cp for one cycle. t∗ (= t/T ) = 0.0 (and

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1.0) and 0.5 correspond to the uppermost and lowermost positions of the

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foil respectively, where T is the flapping period; downstroke at 0 < t∗ < 0.5

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and upstroke at 0.5 < t∗ < 1.0. During downstroke motion, as long as the

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foil is far enough from the upstream plate (t∗ < 0.35), the influence of an

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upstream plate is negligible, and the heaving force and pitching moment are

281

almost identical to those of the single foil. When the foil moves upward

282

away from the upstream body (0.60 < t∗ < 1.0), the effect of the upstream

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body becomes insignificant again, and the heaving force and pitching moment

284

become similar to those of the single foil.

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However, at the late phase of the downstroke (0.35 < t∗ < 0.50) and

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at the early phase of the upstroke (0.50 < t∗ < 0.60), the heaving force

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and pitching moment increase noticeably compared to the single foil (Figs.

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4(a) and (b)). During the downstroke, the increase in the heaving force 16

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(a)

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(b) 1.5

4

1.0 2 0.5

Ch(t ) 0

Cp(t )

*

*

SC

0

−0.5

−2

−1.0

Foil with upstream body Single foil

0

0.25

−1.5

M AN U

−4

0.50

0.75

t*

(c)

1

0

0.25

0.50

0.75

1

0

0.25

0.50

0.75

1

t*

(d)

0.6

1.0

0.3

−0.3 −0.6

0.50

t*

0.75

−1.0 1

t*

(f) t*=1.0 (or 0.0) t*=0.75

AC C

1.2

0.25

0 −0.5

EP

0

(e)

ηp(t*)

D

0

TE

ηh(t*)

0.5

0.8 0.4

η(t ) *

Ly*=1.5

0

t*=0.5

−0.4

H*=1.0

−0.8 −1.2

0

0.25

0.50

t

*

0.75

1 Lx*=0.75

Figure 4: Time history of (a) Ch : heaving force coefficient, (b) Cp : pitching moment

17

coefficient, (c) ηh : heaving efficiency, (d) ηp : pitching efficiency, and (e) η(= ηh + ηp ): total efficiency for one cycle. (f): Schematic of the model. The foil with an upstream body of L∗x = 0.75, L∗y = 1.5, and H ∗ = 1.0 (solid line) and the single foil (dash-dot line). For the definition of output parameters, refer to Eqs. 3a−3c.

ACCEPTED MANUSCRIPT

is unfavorable because the heaving velocity h˙ is negative and the heaving

290

efficiency decreases accordingly (Eq. 3a); it is favorable at the early phase

291

of the upstroke, though. At 0.35 < t∗ < 0.50, heaving efficiency drops from

292

the single-foil case and reaches the minimum of −0.17 (Fig. 4(c)). Near the

293

end of the downstroke, the heaving velocity approaches to zero, and negative

294

effect of the heaving force on the heaving efficiency reduces.

SC

RI PT

289

Contrary to the heaving force that has both positive and negative effects

296

on the efficiency, the change in the pitching moment by the upstream plate

297

is almost favorable. During the stroke reversal (0.35 < t∗ < 0.60), the

298

pitching moment becomes higher than that of the single foil (Fig. 4(b)).

299

Since pitching angular velocity θ˙ is positive and reaches its maximum during

300

that time span, the augmentation of the pitching moment over the single foil

301

amplifies positive contribution to the efficiency (Fig. 4(d)). Near t∗ = 0.25

302

and 0.65, the pitching moment rather reduces from the single foil. However,

303

at these phases, the pitching velocity is near zero, which has little influence

304

on the pitching efficiency.

EP

TE

D

M AN U

295

The overall effects of the upstream body can be summarized as follows.

306

The noticeable rise in heaving force and pitching moment occur near the

AC C

305

307

stroke reversal from the downstroke to the upstroke. The change in heaving

308

force contributes positively to efficiency during the early upstroke. However,

309

its overall effect is to reduce the efficiency due to negative heaving velocity

310

during the downstroke. Meanwhile, in contrast to the heaving force, pitching

311

moment raises up the efficiency in both downstroke and upstroke. During

312

the stroke reversal, pitching angular velocity is near maximum while heaving

313

velocity is small. Therefore, major enhancement in total efficiency is due to

18

ACCEPTED MANUSCRIPT

the pitching moment (compare Figs. 4(c) and (d)). The combined effect of

315

both motions leads the gain in total efficiency at 0.4 < t∗ < 0.6 as evident

316

in Fig. 4(e). Consequently, with an upstream plate, the time-averaged total

317

efficiency η increases 31% from that of the single foil η0 ; i.e., η/η0 ≈ 1.3.

RI PT

314

To obtain more physical insight into the influence of the upstream body,

319

flow structures at several time steps are depicted in Fig. 5 for both the foils

320

without and with an upstream body. Compared to the single-foil case, the

321

local vorticity fields near the foil at t∗ = 0.15 (Figs. 5(a)-I and (b)-I) and

322

t∗ = 0.25 (Figs. 5(a)-II and (b)-II) show no significant difference, which was

323

expected since the foil is far away from the upstream body. Therefore, local

324

flow dynamics would be similar for both cases, so do the resulting force and

325

moment as discussed before. However, when the foil approaches more closely

326

to the body, the leading-edge vortex of the foil (red contour in Fig. 5) is

327

under the influence of the strong vortex created at the upper edge of the

328

upstream body (blue contour in Fig. 5). The flow induced by the vortex of

329

the upstream plate washes away the counter-rotating leading-edge vortex of

330

the foil in the x-direction, more quickly than the single foil (Figs. 5(a)-III∼V

331

and (b)-III∼V). At t∗ = 0.49 (Figs. 5 (a)-V and (b)-V), while the leading-

AC C

EP

TE

D

M AN U

SC

318

332

edge vortex is placed near the trailing edge for the single foil, it is already

333

behind the trailing edge for the foil with the body. Due to the separation

334

of the leading-edge vortex, the pressure near the lower surface of the foil

335

increases from the single-foil case, which results in the larger upward heaving

336

force. At the early phase of the upstroke (t∗ = 0.56), the leading-edge vortex

337

near the lower surface of the foil (red contour) becomes smaller and weaker

338

than that of the single foil due to the influence of the vortex of the upstream

19

ACCEPTED MANUSCRIPT

(II) t*=0.25

(V) t*=0.49

(VI) t*=0.56

(IV) t*=0.42

(III) t*=0.35

RI PT

(I) t*=0.15

SC

(a)

M AN U

(b)

(VIII) t*=0.75

AC C

(b)

EP

TE

D

(a)

(VII) t*=0.62

ωc/U∞

−20

−12

−4

4

12

20

Figure 5: Comparison of vorticity fields at several instances between (a) the foil with an upstream body (L∗x = 0.75, L∗y = 1.5, and H ∗ = 1.0) and (b) the single foil. Only the upper half of the upstream body is shown in each snapshot of (a).

20

ACCEPTED MANUSCRIPT

body (Figs. 5(a)-VI and (b)-VI), which keeps the upward heaving force

340

higher relative to the single foil. After t∗ = 0.56, the development of the

341

vortices around the foil show similar patterns between the two cases in spite

342

of some minor difference in vortex position and size (Figs. 5(a)-VII,VIII and

343

(b)-VII,VIII).

SC

RI PT

339

As described above, one of the mechanisms by which the upstream body

M AN U

affects the flow field near the foil is through the induced velocity caused by the strong vortex created at the edge of the upstream plate. To provide a clearer picture on how much this induced flow changes the location and shape of the leading-edge vortices, it is more convenient to compare directly the core and boundary of the vortices. Here the vortex identification method of Graftieaux et al. [41] is employed, in which two scalars Γ1 and Γ2 indicate the

TE

as follows:

D

core and boundary of the vortices, respectively. These scalars are computed

EP

Γ1 (xP ) =

Γ2 (xP ) =

1 X (xP M × uM ) · n N s |xP M | · |uM | 1 X [xP M × (uM − uP )] · n N

s

|xP M | · |(uM − uP )|

(5a) ,

(5b)

where xP is an arbitrary grid point in the computational domain for which

345

Γ1 and Γ2 are computed, and s is the area of 7 × 7 grid points, surrounding

346

xP , with a vector n normal to the area s. xP M is the vector from the point

347

xP to the point xM where xM presents one of the grid points lying on s.

348

uM is a velocity vector at xM , and uP is a velocity vector averaged on the

349

area s around xP . N is the number of points xM on s. Once Γ1 and Γ2 are

350

computed for the entire grids, their absolute values can be used to identify

351

vortex core and boundary respectively [41].

AC C

344

21

(b)

M AN U

SC

(a)

RI PT

ACCEPTED MANUSCRIPT

Figure 6: Comparison of vortex boundaries between the foil with an upstream body (blue contours) and the single foil (dashed red contours) at (a) t∗ = 0.42 and (b) t∗ = 0.56. In both (a) and (b), vortex centers (green contours) are shown as well.

Each subfigure of Fig. 6 superimposes vortex boundaries of the foils with

353

an upstream body (blue lines) and without an upstream body (red lines) in

354

downstroke (t∗ = 0.42) and upstroke (t∗ = 0.56). For the vortex boundary,

355

iso-contours of |Γ2 | = 0.65 (Eq. 5b) are plotted. For the identification of the

356

vortex center, the contours of |Γ1 | > 0.80 (green contours) are presented as

357

well (Eq. 5a). It should be noted that the vortex boundary may alter with

EP

TE

D

352

the change in the threshold value of |Γ2 |. Nevertheless, it is an acceptable

359

tool here to deliver how the upstream body would affect vortex dynamics.

AC C

358

360

At t∗ = 0.42, for the foil with an upstream body, the boundary of the

361

leading-edge vortex near the lower surface of the foil translates more in the

362

x-direction; its core is displaced away in the x-direction about ∆x ≈ 0.23c

363

relative to the single foil (Fig. 6(a)). In fact, at this moment, the leading-edge

364

vortex of the foil is located just above the detached vortex of the upstream

22

(a)

(I) t*=0.42

(II) t*=0.56

(c) 90 60

SC

30

RI PT

ACCEPTED MANUSCRIPT

αe

(b)

0

M AN U

−30 −60

−90 0 |u | *

Foil with upstream body Single foil

0.25

0 0.8 1.6 2.4

0.50

t

0.75

*

D

Figure 7: Effect of the upstream body on free-stream deflection and the effective angle of

TE

attack. Streamlines are colored by velocity magnitude at t∗ = 0.42 and 0.56 for (a) the foil with an upstream body and (b) the single foil. (c) The effective angle of attack over

EP

one cycle for the foil with a body (square) and the single foil (circle).

body, and their centers are aligned vertically, which enhances the induced

366

motion of the leading-edge vortex along the x-axis. Flow acceleration in-

AC C

365

367

duced by the two counter-rotating vortices can be more clearly observed by

368

comparing velocity magnitude (Fig. 7). The region of high velocity mag-

369

nitude |u∗ |(= |u|/U∞) (red color), which is approximately 2.5 times of the

370

free-stream velocity, exists between the boundaries of the two vortices for the

371

foil with a body (Fig. 7(a)).

372

Besides mutual interaction of the vortices, the upstream plate itself affects

373

the flow field by changing streamlines of the incoming flow as illustrated in 23

1

ACCEPTED MANUSCRIPT

Figs. 7(a) and (b). Due to the presence of the upstream plate, streamlines

375

are deflected upward in front of the sweeping area of the foil while, for the

376

single foil, streamlines remain horizontally until they reach the foil. This flow

377

deflection caused by the upstream body directly changes the effective angle

378

of attack αe that the foil actually encounters from the incoming flow. Here,

SC

we define the effective angle of attack as follows (Fig. 1(b)): ! ˙ − Vˆ h αe = θ − tan−1 , Uˆ

M AN U

379

RI PT

374

(6)

where Uˆ and Vˆ represent incoming flow velocities in x and y directions re-

381

spectively. For the single foil, we have Uˆ = U∞ and Vˆ = 0. However, for the

382

ˆ and foil with an upstream body, special attention is required to quantify U

383

Vˆ . In the study, at each time instance, Uˆ and Vˆ were obtained by averaging

384

u and v velocities over ten sample points distributed on the vertical line of

385

one chord length in front of the foil. The vertical line for the sample points

386

has the horizontal distance ∆x = 1.5c from the pitching axis.

TE

D

380

Fig. 7(c) depicts the variation of the effective angle of attack during one

388

cycle for the foils with an upstream body (square symbols) and without an

389

upstream body (circle symbols). The effective angle of attack for the foil

390

with an upstream body is different from that of the single foil almost for

391

the entire cycle. For early downstroke and late upstroke where the foil is

392

distanced from the body, αe is closer to that of the single foil. However,

393

the difference amplifies as the foil moves near the upstream body during

394

0.4 < t∗ < 0.6 and reaches its maximum value of ∆αe ≈ 15◦ shortly after the

395

foil changes its stroke.

AC C

EP

387

396

The change in the effective angle of attack, when the upstream body ex24

ACCEPTED MANUSCRIPT

ists, has a direct impact on flow structure near the foil. At the late phase of

398

downstroke, as shown in Fig. 6(a), small vortices (blue contours) near the

399

lower surface of the foil are slightly closer to the foil and stretched along the

400

foil. This is due to the smaller magnitude of the effective angle of attack

401

|αe | (Fig. 7(c)), which aligns the incoming flow toward the chord of the foil

402

(Fig. 7(a)-I). The influence of the upstream plate is also clearly identified on

403

pressure distribution near the foil. Because of the decrease in |αe |, pressure

404

magnitude on the upper surface (pressure surface) of the foil reduces, com-

405

pared to the single-foil case (Fig. 8). For the foil with the upstream body,

406

pressure also increases on the lower side (suction surface) of the foil mainly

407

because of the detached leading-edge vortex (Fig. 6(a)). Therefore, the in-

408

crease in pressure on the lower side of the foil together with the decrease in

409

pressure on the upper side (relative to single foil) can explain noticeable rise

410

in the upward heaving force at the late phase of the downstroke (Fig. 4(a)).

411

With the upstream body, the pressure increase on the lower surface is more

412

distinct near the leading edge than the trailing edge, which leads to the rise

413

in the clockwise pitching moment (Figs. 4(b) and 8).

SC

M AN U

D

TE

EP

On the other hand, at the early phase of upstroke, the magnitude of the

AC C

414

RI PT

397

415

effective angle of attack |αe | becomes larger for the foil with an upstream

416

body (Fig. 7(c)). The incoming flow hits the lower surface (pressure surface)

417

of the foil with larger angle, especially near the leading edge (Fig. 7(a)-II).

418

Moreover, for the foil with an upstream body, the vortex near the lower

419

surface share almost the same position but considerably smaller in size, com-

420

pared to the single-foil case (Fig. 6(b). These changes increase pressure on

421

the lower side of the foil compare to the single-foil case, which causes the rise

25

(b)

TE

D

M AN U

(a)

SC

RI PT

ACCEPTED MANUSCRIPT

p*

−6.2

−4.2

EP

−8.2

−2.2

−0.2

1.8

2 Figure 8: Comparison of non-dimensional pressure (p∗ = 2(p − p∞ )/ρU∞ ) fields at t∗ =

AC C

0.42. (a) The foil with an upstream body (L∗x = 0.75, L∗y = 1.5, and H ∗ = 1.0) and (b) the single foil.

26

ACCEPTED MANUSCRIPT

in upward heaving force and clockwise pitching moment at the early phase

423

of the upstroke although the rise is not as large as at the late phase of the

424

downstroke (Figs. 4(a) and (b)).

RI PT

422

Although we have discussed two mechanisms for efficiency improvement,

426

the change in the angle of attack and the interaction with the vortex are

427

not mutually exclusive. For example, the vortex shedding from the upstream

428

body will changes the angle of attack for the foil near the body. Thus, the

429

effect of each mechanism cannot be quantified separately, and the overall

430

effects of the upstream body have been addressed instead in this section.

431

4.2. Effect of relative position of a foil

M AN U

SC

425

In Sec. 4.1, general physical mechanisms on the effect of an upstream

433

body on energy harvesting were described. In this section, the effect of the

434

location of the foil relative to the upstream body will be discussed. Fig. 9

435

presents relative averaged efficiency (η ∗ = η/η0 ) for several values of L∗x and

436

L∗y . When the upstream body is vertically located far enough (L∗y = 1.0

437

and 1.5), both vertical and horizontal distances are not a critical factor to

438

change the performance. The efficiency is almost the same for all distances;

439

the gain is about 30%, relative to the single foil. For these cases, vorticity

440

distributions and time histories of the heaving force and the pitching mo-

441

ment are also similar with minor difference, and the principles of efficiency

442

improvement described in Sec. 4.1 can be applied to all of them. In this

443

study, we limited L∗y upto 1.5. By further increase in L∗y beyond 1.5, the

444

efficiency would eventually approach the efficiency of a single foil since the

445

effect of an upstream body on the foil vanishes away by increase in vertical

446

distance.

AC C

EP

TE

D

432

27

ACCEPTED MANUSCRIPT

1.0

η*

0.5

0.75

1.0

M AN U

0

SC

Ly*=0.5 Ly*=1.0 Ly*=1.5

0.5

RI PT

1.5

Lx*

Figure 9: Relative averaged efficiency η ∗ = η/η0 for the foil with an upstream body (H ∗ = 1.0).

However, when the foil is placed too closely to the upstream body in a

448

vertical direction (say, L∗y = 0.5), the fluid dynamics to cause the change in

449

energy harvesting performance are different from those mentioned in Sec 4.1,

450

and they generally result in the loss of efficiency (Fig. 9). We first consider

451

a situation with the upstream body located at (L∗x , L∗y ) = (0.75, 0.5), which

452

has the worst averaged efficiency in Fig. 9 and compare it with the foil at

AC C

EP

TE

D

447

453

(L∗x , L∗y ) = (0.75, 1.5). The discussion on this case can be generalized to the

454

other cases of L∗y = 0.5.

455

When the foil is located at L∗y = 0.5, even at the early part of downstroke

456

(t∗ < 0.25), the foil moves closely to the vortex of the upstream body. Due

457

to the larger incoming flow induced by the vortex of the body, the foil ex-

458

periences higher pressure at the upper surface of the foil, especially near the

459

leading edge (Fig. 10(c)-I). Thus, the foil at L∗y = 0.5 has larger (negative) 28

ACCEPTED MANUSCRIPT

heaving force and larger (negative) pitching moment than the foil at L∗y = 1.5

461

(Figs. 10(a) and (b)).

RI PT

460

After t∗ = 0.25, the foil starts to be submerged in the vortex of the

463

upstream body (Fig. 10(c)-II,III). The leading-edge vortex of the foil (red

464

contour) moves downstream more quickly, due to its interaction with the

465

vortex of the body, compared to its counterpart of L∗y = 1.5 (Fig. 5(a)-

466

II,III). As a result, the heaving force starts to increase sooner than that

467

of the foil at L∗y = 1.5 (Fig. 10(a)). The detached leading-edge vortex of

468

the foil moves upward closely to the foil and causes a low pressure region

469

near the trailing edge of the lower surface, which contributes to the increase

470

in the pitching moment near t∗ = 0.35 (Fig. 10(b)). The submerging foil

471

causes the increase in the heaving force and pitching moment earlier than

472

that of L∗y = 1.5 and experiences their peaks around t∗ = 0.35 instead of

473

around t∗ = 0.5. Compared to the foil of L∗y = 1.5, the earlier increase in the

474

heaving force has a negative effect on the efficiency because the magnitude

475

of negative heaving velocity is maximum near t∗ = 0.25. With a similar

476

argument, the earlier increase in the pitching moment is also undesirable

477

since the pitching angular velocity is maximum at t∗ = 0.5.

AC C

EP

TE

D

M AN U

SC

462

478

The most striking difference between L∗y = 0.5 and L∗y = 1.5 occurs

479

at the early phase of upstroke. Around t = 0.5, the foil is aligned in a

480

streamwise direction behind the upper edge of the upstream body. The

481

region just behind the upstream body has low pressure, especially below the

482

lower surface of the foil, and accordingly the heaving force becomes negative

483

(Fig. 10(a)). The leading-edge vortex generated during the downstroke (red

484

contour near the foil in Fig. 10(c)-III) sheds away from the trailing edge of

29

(a)

(b) 4

RI PT

ACCEPTED MANUSCRIPT

1.5 1.0

SC

2

Ch(t*) 0

Cp(t*) 0

−2

M AN U

0.5

−4

0

(I) t*=0.15

0.25

0.50

t*

(II) t*=0.25

0.75

−1.0 −1.5 0

1

(III) t*=0.35

0.25

(IV) t*=0.42

0.50

t*

0.75

(V) t*=0.49

AC C

EP

TE

D

(c)

−0.5

Ly*=0.5 Ly*=1.5

ωc/U∞

−20

−12

−4

4

12

20

Figure 10: (a,b) Comparison of the heaving force coefficient Ch and pitching moment coefficient Cp between the foils with an upstream body: (L∗x , L∗y ) = (0.75, 0.5) (dotted

line), (0.75, 1.5) (solid line). (c) Vorticity fields at several instances for (L∗x , L∗y ) = (0.75, 0.5).

30

1

(VI) t*=0.56

ACCEPTED MANUSCRIPT

the foil, and the leading-edge of the foil is under stronger influence of the

486

low pressure region just behind the upstream body. For these reasons, the

487

pitching moment also becomes negative near t = 0.5 (Fig. 10(b)). At the

488

start of the upstroke, since the incoming flow is blocked by the upstream

489

body, the formation of the leading-edge vortex on the upper surface of the

490

foil is delayed. Due to negative heaving force and pitching moment at t = 0.5

491

and delayed development of the leading-edge vortex, the force and moment

492

are noticeably reduced compared to the L∗y = 1.5 case.

M AN U

SC

RI PT

485

For L∗y = 1.0 and 1.5, main contribution to efficiency improvement is

494

from the increase in positive pitching moment near t∗ = 0.5 when positive

495

angular velocity is maximum (Sec. 4.1). However, when the upstream body is

496

positioned at L∗y = 0.5, the pitching moment is even negative near t∗ = 0.5.

497

This reduced pitching moment is the main reason that the averaged total

498

efficiency becomes much smaller than that of the foil at L∗y = 1.0 or L∗y = 1.5

499

and even smaller than that of the single foil (Fig. 9).

500

4.3. Regular and irregular formations of flow structure

EP

TE

D

493

In the previous section, it was shown that, if the height of the upstream

502

body is equal to the chord length (H ∗ = 1.0), averaged total efficiency η

503

is almost independent of the horizontal and vertical locations of the foil at

504

L∗y = 1.0 and 1.5. For H ∗ = 1.0, the dimensional vortex shedding frequency

505

of the upstream body is fv (= fv∗ U/H ≈ 0.14U/H) without a foil. This

506

vortex shedding frequency matches the flapping frequency of the foil f (=

507

f ∗ U/c = 0.14U/H) in our study. The match in the frequencies results in

508

the synchronization of the vortex shedding process of the upstream body

509

and the vortex formation of the foil and the production of periodic heaving

AC C

501

31

ACCEPTED MANUSCRIPT

(III) t*=0.35+2.00

(IV) t*=0.35+3.00

RI PT

(II) t*=0.35+1.00

M AN U

SC

(a)

(I) t*=0.35

TE

D

(b)

ωc/U∞

−12

−4

4

12

20

EP

−20

Figure 11: Vorticity fields of four successive cycles at the same phase of the foil. (a)

AC C

H ∗ = 1.0 and (b) H ∗ = 1.5. L∗x = 0.75 and L∗y = 1.5 for both (a) and (b).

510

force and pitching moment over cycles. This synchronization may also occur

511

even when the two frequencies have minor discrepancy, by tuning the vortex

512

shedding frequency fv to the flapping frequency f .

513

Then, a question arises whether this synchronization is possible even with

514

large discrepancy between the two frequencies. To answer this question, we

515

investigated a vortex formation pattern by increasing the height of the up-

516

stream body from H ∗ = 1.0 to H ∗ = 1.5 with the other parameters un32

ACCEPTED MANUSCRIPT

t*=0.56

t*=0.64

M AN U

SC

t*=0.50

RI PT

t*=0.43

t*=0.35

Figure 12: Vorticity contours for five time instances during the stroke reversal. L∗x = 0.75, L∗y = 1.5, and H ∗ = 1.5.

changed. Without a foil, the dimensional vortex shedding frequency fv in

518

H ∗ = 1.5 is reduced to 2/3 of fv in H ∗ = 1.0, from which we can expect

519

the mismatch of the vortex shedding frequency and the flapping frequency

520

in H ∗ = 1.5. Fig. 11 illustrates repeating and non-repeating patterns of

521

flow structure for both H ∗ = 1.0 and H ∗ = 1.5 respectively. For H ∗ = 1.0,

522

because of strong synchronization, vorticity distribution is almost the same

AC C

EP

TE

D

517

523

for all cycles (Fig. 11(a)). However, for H ∗ = 1.5, the synchronization is

524

lost, and vorticity distribution is not periodic at the same position of the foil

525

in each cycle (Fig. 11(b)). Especially the vortices detached from two edges

526

of the body exhibit stark difference in their size and position in each cycle.

527

For H ∗ = 1.5, the shear layers of the body become unstable and break into

528

small eddies whereas, for H ∗ = 1.0, periodic flow structure keeps the shear

529

layers from breaking into small eddies.

33

ACCEPTED MANUSCRIPT

As H ∗ increases from 1.0 to 1.5, the size of the vortex formed by the

531

body is also scaled with the height of the body. This implies that the body

532

vortex becomes more influential in the interaction with the foil. Due to

533

stronger interaction, the shear layers of the upstream body tend to become

534

more unstable in H ∗ = 1.5, and small eddies are formed more frequently, in

535

spite of the same kinematics and relative position of the foil as H ∗ = 1.0.

536

Fig. 12 provides the evolution of flow structure during stroke reversal. In

537

downstroke, the shear layer of the upper edge of the upstream body forms

538

smaller eddies (secondary vortices). Then, the small vortex near the edge is

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dragged up by the upward motion of the foil after t∗ = 0.50, and is extended

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to the lower surface of the foil. The dragged motion of the vortex toward the

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foil in the upstroke is not observed at H ∗ = 1.0 (refer to Fig. 5). Here we

542

emphasize again that the flow pattern depicted in Fig. 12 does not repeat

543

regularly over cycles although it is observed frequently.

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The non-periodic flow structure observed in H ∗ = 1.5 directly affects the

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heaving force and pitching moment (Fig. 13). While Ch and Cp clearly show

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regularly repeating patterns over cycles for H ∗ = 1.0 (Figs. 13(a) and (c)),

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the repeatability is absent for H ∗ = 1.5, and the peaks of Ch and Cp fluctuate

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every cycle (Figs. 13(b) and (d)). In addition to the case of Figs. 11−13

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(L∗x = 0.75, L∗y = 1.5, and H ∗ = 1.5), the aforementioned trend related with

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asynchronization between the foil motion and the vortex originated from the

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upstream plate is also observed at other relative positions of the foil with

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H ∗ = 1.5. This finding gives some insight into designing the system of

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an energy harvesting foil with an upstream body. To maintain stable and

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predictable behaviors of the system, we should avoid the configuration which

34

(a)

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4 2

Ch(t*) 0

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(b)

2

Ch(t*) 0 −2 −4 2

(c)

1

Cp(t ) 0

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(d)

1

Cp(t ) 0 *

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0

5

10

15

20

t*

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Figure 13: Time history of (a,b) heaving force coefficient Ch and (c,d) pitching moment coefficient Cp for H ∗ = 1.0 (dashed line) and H ∗ = 1.5 (solid line) over 20 cycles. L∗x = 0.75 and L∗y = 1.5.

35

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1.5

1.0

0.5

0.50

0.75

1.00

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η*

Lx*

Figure 14: Relative efficiency η ∗ for H ∗ = 1.0 (dashed lines) and H ∗ = 1.5 (solid lines). L∗y = 1.0 (square) and L∗y = 1.5 (circle).

causes irregular flow pattern and power generation.

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For H ∗ = 1.5, the averaged total efficiency η becomes smaller than that of

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H ∗ = 1.0 at the same relative positions, L∗x and L∗y , of the foil (Fig. 14). The

558

asynchronization of vortex formation not only causes the irregular generation

559

of power output, but also leads to the reduction in total efficiency. In spite of

560

irregular force and moment generation (Figs. 13(b) and (d)), in general, the

561

upstream body of H ∗ = 1.5 increases the heaving force particularly around

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t∗ ≈ 0.25, at which the magnitude of the negative heaving velocity is around

563

its maximum. This results in efficiency loss compared to H ∗ = 1.0. Moreover,

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pitching moment generally reduces from that of H ∗ = 1.0 during the stroke

565

reversal where pitching angular velocity is maximum, which is the major

566

reason of efficiency loss. Therefore, the upstream body with height greater

567

than chord length, i.e., H ∗ > 1.0, should be examined with extra care in

568

context of efficiency improvement as well as stable operation. 36

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5. Concluding remarks

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In this study, we have numerically investigated the effect of an upstream

571

bluff body on energy harvesting of an oscillating foil. Although we rely on

572

two-dimensional simulation at the Reynolds number much lower than that

573

of the actual application, we were able to find fluid-mechanical principles of

574

improving the efficiency significantly by using a simple additional structure.

575

The presence of the upstream body changes local flow structure around the

576

foil in two ways. First, it deflects an incoming flow into the sweeping area of

577

the foil, which results in the decrease in the effective angle of attack during

578

downstroke and its increase during upstroke. In addition, the vortex shed

579

from the upstream body induces faster separation of the leading-edge vor-

580

tex generated by the foil. These mechanisms change pressure distribution in

581

both surfaces of the foil and increase heaving force and pitching moment, es-

582

pecially during stroke reversal near the body, which contributes to improving

583

efficiency. By locating the foil at a proper location relative to the upstream

584

body, we are able to enhance overall power output by 30% from the single

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foil.

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The installation of the upstream body does not always provide positive

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results. Instead, it may deteriorate the performance of the foil in several con-

588

ditions. When the foil is close to the upstream body in a transverse direction

589

(e.g., L∗y = 0.5), the development of the leading-edge vortex responsible for

590

power generation is delayed due to the blockage of the incoming flow during

591

the stroke reversal. If the vortex shedding from the upstream body is not

592

synchronized with the formation of the leading-edge vortex of the foil (e.g.,

593

H ∗ = 1.5), the flow structure becomes unstable, producing irregular power 37

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output over cycles.

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The strategy to increase power output of an energy harvesting system by

596

manipulating the distribution of an incoming flow was originally suggested for

597

conventional rotary turbines. Here, we conducted a proof-of-concept study

598

to apply this strategy to a foil periodically moving in a uniform stream. In

599

particular we emphasized the change in flow dynamics by the presence of

600

the upstream body and its correlation with power generation. In future,

601

extensive study should be pursued in order to find the optimal configuration

602

of the system and confirm its applicability to actual three-dimensional pilot

603

devices in high Reynolds number.

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Acknowledgments

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This study was supported by the Basic Science Research Program through

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the National Research Foundation of Korea (NRF) funded by the Ministry

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of Science, ICT & Future Planning (NRF-2015R1C1A1A02037111) and by a

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grant [KCG-01-2016-04] through the Disaster and Safety Management Insti-

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tute funded by Korea Coast Guard of Korean government.

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Frontal swept area of a foil

αe

Effective angle of attack

c

Foil chord

Γ1

Scalar quantity indicating vortex core

Ch (t)

2 A ) Heaving force coefficient (2Fy (t)/ρU∞ s

Γ2

Scalar quantity indicating vortex boundary

Cp (t)

2 Pitching moment coefficient (2M (t)/ρU∞ cAs )

η

Average total efficiency

f

Oscillation frequency

η0

Average total efficiency of a foil

f∗

Reduced frequency (f c/U∞ )

η∗

Relative efficiency (η/η0 )

Fy

Heaving force

η(t)

Total efficiency (ηh (t) + ηp (t))

h(t)

Heaving position

ηh (t)

3 ˙ Heaving efficiency (2Fy (t)h(t)/ρU ∞ As )

h0

Heaving amplitude

ηp (t)

3 ˙ Pitching efficiency (2M (t)θ(t)/ρU ∞ As )

H∗

Non-dimensional height of an upstream plate (H/c)

θ0

Pitching amplitude

L∗ x

Non-dimensional horizontal distance between the edge of

θ(t)

Pitching angle

an upstream plate and the pitching axis of a foil (Lx /c)

φ

Phase difference between heaving and pitching motions

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Non-dimensional vertical distance between the top edge of an upstream plate and the middle of the swept area (Ly /c) Pitching moment

T

Oscillation period

U∞

Non-dimensional time (t/T ) Free-stream velocity

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t

∗

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As

L∗ y

Appendix A. Nomenclature

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Nomenclature

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• • •

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The effect of an upstream bluff body on the energy harvesting performance of a pitching and heaving hydrofoil is investigated. The upstream body can improve the efficiency of the hydrofoil by about 30%. Mutual interaction of the vortex shed from the upstream body and the leadingedge vortex of the hydrofoil is a main mechanism for efficiency improvement. The change in the effective angle of attack for the hydrofoil is another important mechanism for efficiency improvement. The upstream body placed in an improper position can cause efficiency drop and irregular power generation.

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•