Recent advancements in highly deformable smart materials have lead to increasing interest in small-scale energy harvesting research for powering low consumption electronic devices. One such recent experimental study by Goushcha et al. explored energy harvesting from a passing vortex ring by a cantilevered smart material plate oriented parallel to and offset from the path of the ring in an otherwise quiescent fluid. The present study focuses on modeling this experimental study using potential flow to facilitate optimization of the energy extraction from the passing ring to raise the energy harvesting potential of the device. The problem is modeled in two-dimensions with the vortex ring represented as a pair of counter-rotating free vortices. Vortex pair parameters are determined to match the convection speed of the ring in the experiments, as well as the imposed pressure loading on the plate. The plate is approximated as a Kirchhoff-Love plate and represented as a finite length vortex sheet in the fluid domain. The analytical model matches experimental measurements, including the tip displacement, the integrated force along the entire plate length as a function of vortex ring position, and the pressure along the plate. The potential flow solution is employed in a parametric study of the governing dimensionless parameters in an effort to guide the selection of plate properties for optimal energy harvesting performance. Results of the study indicate an optimal set of plate properties for a given vortex ring configuration, in which the time-scale of vortex advection matches that of the plate vibration.

In the past decade, considerable research efforts have been invested into the exploitation of electro-active materials to harvest energy from the surrounding environment, including human heat,1 ventricular wall motion,2 human/animal locomotion,3–5 and structural vibrations.6–10 A dynamically rich and exciting subset of the energy harvesting research field entails energy extraction directly from the motion of a fluid encompassing the harvester, see Refs. 11–14.

One of the first demonstrations of fluid energy harvesting using structural vibrations of an electro-active material utilized a time-varying pressure loading generated by the Kármán vortex street on a cantilevered strip behind a cylinder in a crossflow.11,15 The alternating pressure field caused by the vortices advecting past the harvester incited deformation of the material, which generated an electrical current through the device. Alternative harvesting configurations and modalities that have been subsequently demonstrated include the flutter instability of flapping flags,16,17 the flutter instability of a cantilevered plate with a T-shaped tip,18 and a cylindrical tip mass,19 as well as the mechanical buckling of a harvester plate caused by a Savonius rotor and a slider-crank mechanism.20 Energy harvesting from turbulent motion has been explored as well, including using arrays of harvesters placed in a turbulent crossflow21 and employing a cantilever plate to scavenge energy from a turbulent boundary layer.22 

Recently, the interaction and energy exchange between a single advecting vortical structure and a deformable solid has been studied. Direct impact of a vortex ring with a cantilever plate was studied experimentally in Ref. 23, while the problem was modeled analytically using a two-dimensional vortex pair representation in Ref. 24. Hu et al.25 studied the interaction of a vortex ring with a concentric annular plate. In the cantilevered plate configuration, the vortex ring is destroyed during the impact, which tends to transfer approximately 1% of its energy to the structure, while the annular impact produces a second vortex ring that advects away.

In a timely experimental study conducted by Goushcha et al.,14 the energy harvesting potential of single and multiple vortex rings passing by a cantilevered plate oriented parallel to and offset from the vortex ring trajectory was explored. The vortex ring was generated by a speaker/cylinder setup in otherwise stagnant air, and it interacted with a flexible cantilevered polycarbonate plate. The strain at the clamped edge was measured via a small strain gauge, and time-resolved particle image velocimetry (PIV) was used to obtain information about the flow kinematics. The pressure loading on the plate was subsequently inferred from the flow kinematics using both control volume and direct integration methods. The structural vibration of the cantilevered plate was found to be influenced by the pressure fluctuation caused by the passing vortex ring. Subsequent rings could be timed such that they either enhanced or mitigated the vibration amplitude of the beam, which is related to the energy harvesting potential of the configuration.

In this paper, we model the energy harvesting configuration introduced and explored experimentally by Goushcha et al.14 using a two-dimensional potential flow model.24 Specifically, the interaction between a single vortex pair and a cantilevered deformable plate in an otherwise quiescent ideal fluid is proposed as a template for performing optimization studies on the vortex-deformable structural energy transfer in which viscous fluid forces are negligible, in pursuit of maximizing the energy scavenger's harvesting potential. This paper is organized as follows: the problem is formulated and the potential flow model is introduced in Sec. II; a scheme to proxy a vortex ring through a vortex pair is developed in Sec. III; the model is validated against published experimental data in Sec. IV; optimization of the energy transfer based upon the analytical model is presented in Sec. V; and Sec. VI summarizes concluding remarks.

Here, we analytically explore the problem addressed experimentally by Goushcha et al.14 of a vortex ring passing over a cantilevered plate oriented parallel to and offset from the path of a vortex ring in an otherwise quiescent fluid, see Figure 1(a). The vortex ring has ring diameter ar, core diameter br, circulation Γr, initial distance of the ring center axis to the plane of the undeformed plate hr, and initial convection speed Vc. A Cartesian coordinate system is defined at the geometric center of the plate, with x oriented along the plate pointing from the fixed to the free end, y oriented normal to the plate in the undeformed state and pointing towards the vortex ring, and z forming a standard right-handed coordinate system, see Figure 1(a). The plate has length L, width W, and thickness T and is clamped at x=L/2.

FIG. 1.

(a) Schematic of the experiment performed by Goushcha et al.14 of a vortex ring passing over a cantilevered plate; and (b) two-dimensional vortex pair-based representation of the experiment.

FIG. 1.

(a) Schematic of the experiment performed by Goushcha et al.14 of a vortex ring passing over a cantilevered plate; and (b) two-dimensional vortex pair-based representation of the experiment.

Close modal

We assume that viscous dissipation of the vortex core while passing over the plate is negligible due to the short interaction time of the ring with the plate; thus, the circulation Γr is assumed to be constant throughout the interaction. We further assume that the vortex ring core diameter br is small in comparison with its overall diameter ar,26 and the plate and the vortex ring are sufficiently far away from one another such that the interaction between the vortex core and the induced vorticity on the plate is negligible. With these assumptions, the surrounding fluid is modeled as incompressible, inviscid, and irrotational, except in the core of the vortex ring. We hypothesize that the plate length is much larger than its thickness and that the plate vibration is of small amplitude and confined to the xy-plane. The effective Young's modulus of the plate and the mass density per unit volume are ϒ and ϱ, respectively. As such, the plate is modeled as a Kirchhoff-Love plate27 undergoing cylindrical bending in response to the applied fluid loading. The structural damping of the plate is captured using a distributed viscous damping model.28 

The governing equation for the plate dynamics is

Mδ¨(x,t)+Cδ̇(x,t)+BδIV(x,t)=p(x,t),
(1)

where δ is the plate deflection, B=ϒT3/12 is the plate bending stiffness, M=ϱT is the plate mass per unit surface area, C quantifies the structural damping, and p is the pressure difference between the top (positive y-plane) and bottom (negative y-plane) of the plate due to the fluid. The Roman numeral indicates the order of differentiation with respect to x, while a dot indicates differentiation with respect to time t. The initial conditions for the plate are δ(x,0)=δ̇(x,0)=0, while the fixed-free boundary conditions are δ(L/2,t)=δI(L/2,t)=δII(L/2,t)=δIII(L/2,t)=0. We note that Equation (1) only applies to weakly electromechanically coupled harvesters, such as ionic polymer metal composites.9,25,29 Materials with stronger coupling, such as piezoelectrics, would require consideration of bidirectional effects, whereby the electrical response directly influences the mechanical deformation, and vice versa.10,30

Employing L0=L/2 as the length scale, V0=Vc as the velocity scale, t0=L0/V0 as the time scale, and p0=ρV02 as the pressure scale, where ρ is the fluid density, Equation (1) can be expressed in dimensionless form as

μδ̂¨(x̂,t̂)+ηδ̂̇(x̂,t̂)+βδ̂IV(x̂,t̂)=p̂(x̂,t̂),
(2)

where β=B/(ρV02L03) is the ratio of the plate restoring force to the applied fluid loading, μ=M/(ρL0) is the plate mass to fluid mass ratio, and η=C/(ρV0) relates the structural damping to the applied fluid loading. A hat over a variable indicates that it is dimensionless. The dimensionless initial and boundary conditions are δ̂(x̂,0)=δ̂̇(x̂,0)=0 and δ̂(1,t̂)=δ̂I(1,t̂)=δ̂II(1,t̂)=δ̂III(1,t̂)=0, respectively.

In order to gain analytical traction on the vortex ring/plate interaction problem depicted in Figure 1(a), we recast the three-dimensional problem into a two-dimensional analog by assuming that the diameter of the vortex ring ar is much larger than the width of the plate W. Under this assumption, the vortex ring curvature in the vicinity of the plate is small. Furthermore, we neglect three-dimensional effects along the edges of the plate. A schematic of the two-dimensional model is presented in Figure 1(b). In two-dimensions, the vortex ring is represented by a pair of counter-rotating free vortices with initial separation, convection speed, and circulation of ap,Vc, and Γp, respectively. The midpoint between the pair is initially at a distance hp from the plane of the undeflected plate. We note that the convection speeds of both the vortex ring and pair are denoted by the same variable, Vc, which will be discussed in Sec. III. For the remainder of the manuscript, the subscript r will denote vortex ring properties, while a subscript p will be employed to denote properties of the vortex pair.

For convenience, we introduce the complex coordinate ζ=x+iy and the complex velocity w=uiv, where u and v are the velocity components in the respective x and y directions, and i=1. The positions of the two vortices in the vortex pair are given by ζ1 and ζ2, respectively, see Figure 1(b). Following Peterson and Porfiri,24 the velocity field at any location ζ in the domain at given time t in non-dimensional form is

ŵ(ζ̂,t̂)=i2π[2πΛ(1ζ̂ζ̂1(t̂)1ζ̂ζ̂2(t̂))11γ̂(α,t̂)ẑαdα],
(3)

where Λ=ap/L0. The first term on the right hand side of Equation (3) is the contribution from the vortex pair, while the second term is the velocity induced by a bounded vortex sheet whose vorticity distribution is given by γ̂(α,t̂), which represents the plate in the fluid domain. We note that the relationship between the vortex circulation and the pair convection speed, Vc=Γp/(2πap), is employed in the derivation and subsequent non-dimensionalization of Equation (3), see Ref. 24. The kinematics of each vortex in the pair is governed by the desingularized velocity31 at each respective vortex center as

ζ̂̇1(t̂)=Conj{i2π[2πΛζ̂1(t̂)ζ̂2(t̂)11γ̂(α,t̂)ζ̂1(t̂)αdα]},
(4a)
ζ̂̇2(t̂)=Conj{i2π[2πΛζ̂2(t̂)ζ̂1(t̂)11γ̂(α,t̂)ζ̂2(t̂)αdα]},
(4b)

where Conj(·) represents complex conjugation, and ζ̂̇1 and ζ̂̇2 are the velocities of vortex 1 and 2, respectively, see Figure 1(b). The initial vortex positions are ζ̂1(0)=x̂0+i(H+Λ/2) and ζ̂2(0)=x̂0+i(HΛ/2), where H=hp/L0, and x̂0 is the initial x location.

The vortex sheet has an initial vorticity distribution γ̂(x̂,0)=0, and according to Kelvin's circulation theorem,32 at all subsequent times the vorticity distribution must satisfy

11γ̂(α,t̂)dα=0.
(5)

The boundary conditions on the fluid are that it is at rest with zero pressure at infinity and the velocity component normal to the plate is equal to the plate velocity. The first boundary condition is automatically satisfied by Equation (3), while the second boundary condition24 yields

12π11γ̂(α,t̂)x̂αdα=δ̂̇(x̂,t̂)+Λ{1x̂ζ̂1(t̂)1x̂ζ̂2(t̂)},
(6)

where (·) is Cauchy's principal value integral and (·) is the real part of the complex variable. The pressure difference induced by the fluid across the plate can be computed using unsteady Bernoulli's equation33 as

p̂(x̂,t̂)=0x̂γ̂̇(α,t̂)dα+γ̂(x̂,t̂){Conj[ŵ(x̂,t̂)]}.
(7)

The fluid and structure models are solved simultaneously as outlined by Peterson and Porfiri.24 Briefly, the plate shape is projected onto a basis set of Chebyshev polynomials, which facilitate solution of Equation (6) for the plate vorticity distribution. Combining Equations (3) and (7), the pressure distribution is then substituted into Equation (2) to solve for the plate motion. The vortices are advected via Equation set (4). This system of equations is integrated in time to determine the overall system dynamics.

Due to the topological differences between a vortex ring, with its looping core, and a two-dimensional vortex pair, with infinitely long vortex line cores, the velocity and associated pressure fields induced by these fluid structures differ considerably. As such, a vortex ring cannot simply be represented by a vortex pair with identical geometric and kinematic parameters. To encapsulate the salient physics of the fluid-structure interaction, we propose a vortex ring to vortex pair conversion scheme that matches the pressure loading time scale and magnitude on the plate.

The pressure loading time scale is matched between the potential flow model and the experiment by ensuring that the vortex ring and the vortex pair pass the plate at the same rate; that is, the pair and ring must have equal initial convection speed Vc. Note that we are implicitly assuming that the ring and pair convection speeds do not vary significantly during the interaction with the plate, as it is not expected that the influence of the plate on the ring speed would necessarily be the same as that for the vortex pair. Since the convection speeds in both cases are equated, the non-dimensionalization scheme introduced in Sec. II applies to both the ring and the pair, with the plate dynamics parameters μ, β, and η remaining unaltered after the conversion from the experimental configuration to the two-dimensional potential flow representation. With the plate dynamics unaffected by the conversion, only the vortex pair geometric parameters Λ and H are impacted.

To match the pressure loading magnitude, we first simplify the problem by neglecting the influence of the plate on the fluid. This assumption effectively reduces the matching to a steady state relationship between the pressure distributions at the plane of the undeflected plate in the reference frame of the moving ring/pair. This also neglects the pressure at the back of the plate, which is primarily associated with the plate's movement. Since the plate vibrates in a similar manner as the ring/pair applies similar pressure loading, the differences in the pressure on the back of the plate should be small between the two cases; thus, the pressure loading on the back of the plate may be ignored as a first order approximation. For simplicity, we employ the same coordinate system as depicted in Figure 1(a) at the instant when the vortex ring/pair is passing through the xz-plane.

We begin with a model for an axisymmetric vortex ring with infinitesimal core thickness in an ideal fluid. The axial and radial velocity components ûr and v̂r at the plane of the plate (y = 0) induced by the ring can be expressed using the Biot-Savart law34 in dimensionless form as

ûr(x̂,ẑ)=ΛrΓ̂r8π[(Λr2+YAB)I2YBI1],
(8a)
v̂r(x̂,ẑ)=ΛrΓ̂r8π(x̂B)(I1AI2),
(8b)

where

Y=ẑ2+Hr2,
(8c)
A=x̂2+Y2+Λr2/4,
(8d)
B=ΛrY,
(8e)
C=x̂2+(Y+Λr/2)2,
(8f)
λ=2ΛrY/C2,
(8g)
I1=4CK(λ),
(8h)
I2=4C3E(λ)1λ,
(8i)

and Γ̂r=Γr/V0L0 is the dimensionless circulation for the vortex ring. K(λ) and E(λ) represent complete elliptic integrals of the first and second kind, respectively. Similarly, to obtain the steady state Cartesian velocity components ûp and v̂p at the plane of the plate for an analogous vortex pair, we rewrite Equation (3), without the vortex sheet, as

ûp(x̂)=Λp[Λp/2Hpx̂2+(Λp/2Hp)2+Λp/2+Hp(x̂2+Λp/2+Hp)2],
(9a)
v̂p(x̂)=Λp[x̂x̂2+(Λp/2Hp)2x̂x̂2+(Λp/2+Hp)2].
(9b)

We note that while ûr and v̂r correspond to cylindrical velocity components of the ring (azimuthal and radial components, respectively) and ûp and v̂p are Cartesian velocity components of the pair (in the x and y directions, respectively), the components are aligned when z = 0, corresponding to the centerline of the plate. In what follows, we often refer to these components as û and v̂ for both models.

The pressure field at the plane of the plate due to the vortex ring p̂r(x̂,ẑ) and pair p̂p(x̂) can be obtained from the steady state Bernoulli's equation

(10)
p̂r(x̂,ẑ)=12[ûr2(x̂,ẑ)+v̂r2(x̂,ẑ)],
(10a)
p̂p(x̂)=12[ûp2(x̂)+v̂p2(x̂)].
(10b)
Note that in obtaining Equation (10) we have assumed that the pressure and velocity of the fluid both vanish to zero in the far field. An exact match of the pressure profile is not expected; however, we find that the pressure profiles at the plane of the plate (along the x axis) have similar bell curve-like profiles for both the ring and the pair, which enables the determination of an optimal match by requiring the two profiles to have equal peak and integrated values along the plane the plate.

To match the peak pressure value, which occurs directly beneath the ring/pair at x̂=0, we set

p̂p(0)2Ŵ0Ŵ/2p̂r(0,ẑ)dẑ=0,
(11)

where Ŵ=W/L0. The integral is the average pressure across the width of the plate, which accounts for variations in the pressure due to the vortex ring curvature in the ẑ direction, see Figure 1(a). By noting that only the û component of the velocity exists at x̂=0 for both the ring and the pair, substitution of Equations (9a) and (8a) into Equation (10), and subsequent substitution into Equation (11) yields

(12)
Hp=(141U)12Λp,
(12a)
where

U=2Ŵ0Ŵ/2ûr(0,ẑ)dẑ.
(12b)

Equation (12) provides a relationship between the vortex pair geometric parameters (spacing between the vortices, Λp, and distance of the pair from the plate, Hp) and the vortex ring parameters, which are encapsulated in U.

An additional relationship is required to solve for Λp and Hp in terms of the vortex ring parameters. This is accomplished by matching the force induced by the ring and the pair at the plane containing the plate. Equating the integrated pressure for the ring and the pair yields

(13)
0ûp2(x̂)+v̂p2(x̂)dx̂F=0,
(13a)
where

F=2Ŵ0Ŵ/20ûr2(x̂,ẑ)+v̂r2(x̂,ẑ)dx̂dẑ.
(13b)

The bounds of the integrals above are from x̂=0 to x̂= since the pressure profile is symmetric about the xy-plane.

Equations (12) and (13) can be solved simultaneously for the two unknown vortex pair parameters Λp and Hp. We note that the vortex pair circulation Γp is a dependent parameters since it can be computed from the known vortex pair convection speed and pair geometry, as discussed previously. The vortex pair parameters Λp and Hp can be obtained by first calculating the vortex ring constants U and F, then substituting Equation (12) into Equation (13) to yield Λp. Finally, Hp can be found directly from Equation (12). The solution procedure enables efficient estimation of the vortex pair parameters that best match the pressure distributions for a given vortex ring/plate configuration without the need to apply a more computationally expensive optimization procedure to minimize the error between the vortex ring and the vortex pair pressure profiles at each point.

An example of the output of this conversion method is shown in Figure 2, which compares the pressure and the velocity components at the plate centerline (x-axis) of a vortex ring and its matched two-dimensional vortex pair for a plate of zero width (Ŵ=0). In this case, the integrals in Equations (11) through (13) with respect to ẑ are replaced by their integrands, that is, there is no averaging across the plate width. From Figure 2(a), we see that the vortex pair pressure profile is nearly identical to that of the vortex ring after matching. However, owing to the distinctly different topology of the vortex filaments in the ring and pair cases, there are small differences in the individual velocity components, as shown in Figures 2(b) and 2(c). These differences in the velocity components in the two cases can potentially lead to some error once the influence of the plate is included. For finite width plates Ŵ>0, the matched vortex pair pressure profile tends to under-predict the pressure near the centerline of the plate and over-predict the values nearer the edges of the plate due to the averaging effect. Overall, the matching enables representation of a vortex ring by a two-dimensional pair with nearly identical pressure loading distributions on a plate in the fluid, as well as identical interaction time scales due to matching the convection speeds of the ring and pair.

FIG. 2.

Comparison of the (a) pressure, (b) û velocity, and (c) v̂ velocity profiles between a vortex ring and its matched vortex pair for a plate of zero width. Solid lines and dash lines represent the vortex pair and the vortex ring, respectively.

FIG. 2.

Comparison of the (a) pressure, (b) û velocity, and (c) v̂ velocity profiles between a vortex ring and its matched vortex pair for a plate of zero width. Solid lines and dash lines represent the vortex pair and the vortex ring, respectively.

Close modal

To validate the analytical solution, model predictions are compared with the experimental results of Goushcha et al.14 Their experimental campaign employed a polycarbonate strip of dimensions L=0.1m,W=0.03m, and T=0.05mm. Reported vortex ring parameters are Γr=0.77m2/s and Vc=2.81m/s. Additionally, the ring diameter ar=0.14m and distance of the ring from the plate hr=0.096m used in their study were obtained through personal communication. The value of hr used in the analytical solution was slightly modified to hr=0.1011m to better match the reported pressure loading on the plate. This modest adjustment is, however, within the uncertainty limits of the experiments.

In addition to the plate geometry and vortex ring parameters, Goushcha et al. reported the first mode natural frequency to be 48.3rad/s. They further presented a time series of the measured strain at the clamped end of the plate in their Figure 3. Estimating the plate natural frequency from this time series yields a value of 40.0rad/s, which is within 20% of their reported value. Hypothesizing that the mass density per unit volume of the polycarbonate is ϱ=1200kg/m3,35–39 matching the fundamental resonance frequency to the reported value yields an effective Young's modulus ϒ=109GPa. Such an estimate of the Young's modulus is outside the range typical of polycarbonate, which is between 2.35 and 2.4GPa.35–39 It is tenable to hypothesize that the thickness was higher than the reported value, which we thus change to T=0.3mm to set the effective Young's modulus at ϒ=2.4GPa. We note that other possible contributors to the discrepancy include the impact of the strain gauge on the material stiffness and/or free vibration length, or incorrect material properties. A detailed study determined that these influences were relatively small. The damping factor C is selected to match the decay rate of the free vibration presented in their Figure 3.

FIG. 3.

Snapshots of the vortex pair/deformable plate interaction. The left column displays the position (circles) and the trajectories (dashed lines) of the vortices with respect to the plate (solid lines). The middle column displays the plate deflection, while the right column shows the pressure jump across the plate. Row 1: t̂=9.02; row 2: t̂=10.08; row 3: t̂=11.15; row 4: t̂=12.16; and row 5: t̂=13.20.

FIG. 3.

Snapshots of the vortex pair/deformable plate interaction. The left column displays the position (circles) and the trajectories (dashed lines) of the vortices with respect to the plate (solid lines). The middle column displays the plate deflection, while the right column shows the pressure jump across the plate. Row 1: t̂=9.02; row 2: t̂=10.08; row 3: t̂=11.15; row 4: t̂=12.16; and row 5: t̂=13.20.

Close modal

The dimensionless parameters used for the analytical solution are β=4.466,μ=5.878,η=0.419,Λ=1.004, and H = 1.335; the latter two parameters were obtained using the conversion method discussed in Sec. III. We only employ the first in-vacuum mode of the plate to simulate the plate dynamics in the analytical solution. This is in agreement with the experimental observations of Goushcha et al. that the plate vibrates along its fundamental mode shape. Including additional modes in the analysis does not significantly affect the results.

Figure 3 presents details of the vortex pair/plate interaction predicted from the analytical model based upon the experimentally derived governing dimensionless parameters. The left-most column shows the positions of the vortex pair (circles) and the plate (solid line), with the trajectories of the vortices depicted by dashed lines. The middle column shows the corresponding plate deflection, while the right column shows the differential pressure loading on the plate. Time is increasing from one row to the next. As the vortices approach the plate from the right, there is a positive pressure jump across the plate, that is, the pressure on the top of the plate is higher, which causes a negative plate deflection (row 1). As the vortices pass above the plate, a strong negative pressure on the plate develops due to the low pressure vortex core, which pulls on the plate (rows 2–4). In turn, the plate motion influences the trajectory of the vortex pair, drawing the pair in the negative ŷ direction. Once the vortex pair passes beyond the plate, there is again a positive pressure loading on the plate (row 5). We note that due to the inertia of the plate, its deflection is not necessarily in phase with the pressure loading. This process qualitatively agrees with the experimental observations of Goushcha et al. and a similar positive pressure jump is observed for a vortex advecting past a structure in a free stream.40 

A more quantitative validation of the model is presented in Figure 4, which compares the plate tip deflection δ̂(1,t̂), the pressure loading on the plate p̂ when the vortex ring/pair is positioned at x̂0.04, and the net force per unit width f̂ as a function of vortex ring/pair position between the analytical solution and the experimental data. The experimental data for the tip deflection shown in Figure 4(a) were not published in the original experimental paper, but were provided through personal communication and are reproduced herein with permission. The integrated force per unit width in Figure 4(c) is computed as

f̂(t̂)=11p̂(x̂,t̂)dx̂
(14)

with t̂ as a function of {ζ̂2}.

FIG. 4.

Comparison between model predictions and experimental measurements of (a) the plate tip deflection, (b) the differential pressure loading on the plate when the vortex ring/pair is located at x̂0.04, and (c) integrated force per unit width on the plate versus position of the bottom vortex {ζ̂2}. The analytical results are represented by solid lines. The experimental data are depicted by a dashed line in (a), and circles in (b) and (c).

FIG. 4.

Comparison between model predictions and experimental measurements of (a) the plate tip deflection, (b) the differential pressure loading on the plate when the vortex ring/pair is located at x̂0.04, and (c) integrated force per unit width on the plate versus position of the bottom vortex {ζ̂2}. The analytical results are represented by solid lines. The experimental data are depicted by a dashed line in (a), and circles in (b) and (c).

Close modal

The deflection (Figure 4(a)) shows generally good agreement between the present analytical solution and the experimental data. The analytical solution under-predicts the first positive peak in both cases by 20%, though subsequent positive peaks match very well.

The pressure distribution on the plate when the vortex ring/pair is located over the center of the plate is presented in Figure 4(b) for both the experimental data, obtained using PIV, and the analytical prediction. The minimum pressure across the plate is located at approximately the same x̂ position as the vortex ring/pair, which corresponds to t̂11 in Figure 4(a). The minimum pressure peak is matched in both magnitude and location between the experiment and the model, though the overall pressure distribution is slightly wider for the model prediction. In addition, the experimental data have a relatively pronounced positive peak in the pressure near the free end of the plate that is not captured by the potential flow solution.

The integrated force on the plate is plotted as a function of bottom vortex position {ζ̂2} (see Figure 1(b)) in Figure 4(c). Note, for the experimental data, “bottom vortex position” refers to the x position of the portion of the vortex ring core that is closest to the plate, see, for example, Figure 3 of Goushcha et al. In Figure 4(c), the experimental data, again obtained from PIV, are noisy, but agree well with the model prediction in terms of trend and magnitude.

The conversion procedure from vortex ring properties to a representative vortex pair implicitly assumes that the vortex ring/pair convection speed is not significantly influenced by the presence of the plate, as discussed in Sec. III. We observe in Figure 3 that the trajectories of the vortices constituting the vortex pair are modestly influenced by the plate. Given that the vortices have constant circulation, a relative change in trajectory of one vortex in the pair with respect to the other can result in an altered convection speed of the pair; that is, if the vortices move closer to one another their mutual induction speed will increase, for example. Figure 5 plots the distance between the vortices in the pair normalized by their initial separation, |ζ̂1ζ̂2|/Λ, as a function of the position of the bottom vortex, {ζ̂2}, as it passes over the plate. As the vortex pair approaches the plate, {ζ̂2} decreases from the initial value of x0, and the vortices begin to move apart slightly, indicating that the convection speed decreases. Once the pair reaches the free end of the plate ({ζ̂2}=1), the vortices start to move closer together as they are deflected downwards due to the interaction with the plate; see also the first column in Figure 3. The vortices continue to move closer together after advecting past the plate before stabilizing into a new configuration with a final separation distance that is 5% closer than their initial separation (Vc is increased by 5%). Over the course of the interaction, changes in the convection speed of the pair are very modest, as originally assumed for the vortex ring to pair conversion.

FIG. 5.

Change of vortices separation distance with respect to its initial condition Λ as a function of the bottom vortex location {ζ̂2}.

FIG. 5.

Change of vortices separation distance with respect to its initial condition Λ as a function of the bottom vortex location {ζ̂2}.

Close modal

As a final note on the applicability of the potential flow model, the ideal fluid assumption neglects vorticity generation, dissipation, and shedding along the plate.41 These viscous effects will introduce additional hydrodynamic loading onto the plate which the potential model does not capture,42–46 as well as a train of wake vortices convecting away from the tip of the plate.41,47 Based upon the agreement between experimental data and model predictions presented above, these viscous effects are expected to play a secondary role for a vortex pair sufficiently far from the plate during the initial interaction stage. Viscous effects are partially accounted for by the structural damping coefficient η, which is identified through in-air experiments rather than trials in-vacuum. As a result, η likely over-predicts the inherent structural damping, compensating for some of the unmodelled viscous effects.

However, if the initial vortex pair positions and trajectory result in the pair coming too close to the plate, then non-physical vortex trajectories can occur, as shown in Figure 6. In this case, the trajectory of vortex 2 is significantly altered, with the vortex actually passing around the bottom side of the plate before joining with vortex 1 again and advecting away from the plate. The proximity of vortex 2 to the plate would, in fact, lead to non-negligible viscous effects, namely, vorticity generation along the plate and subsequent interaction with the pair vorticity.13,48 Thus, the potential flow model behavior should be expected to deviate considerably from experimental observations of real fluids in such conditions. For the remainder of the document we restrict ourselves to cases in which viscous effects are small and the potential flow model as described in Sec. II is reasonable. Lastly, we call attention to the fact that the matching method is only valid for cases where the plate width is small compared to the vortex ring radius ar and the vortex ring to plate distance Hr.

FIG. 6.

Non-physical behavior of a vortex pair/plate interaction in an ideal fluid, wherein the bottom vortex is “trapped” by the plate. In this case, vorticity interaction between the plate and vortex pair would be non-negligible.

FIG. 6.

Non-physical behavior of a vortex pair/plate interaction in an ideal fluid, wherein the bottom vortex is “trapped” by the plate. In this case, vorticity interaction between the plate and vortex pair would be non-negligible.

Close modal

With the analytical model for the energy harvesting configuration shown in Figure 1(a) validated against the experimental data of Goushcha et al., we now focus on employing the model to optimize the energy exchange between the vortex structure and the plate; that is, we aim to optimize the plate properties for a given set of vortex parameters in order to maximize the energy transferred to the plate. We note that energy harvesting can be estimated through the addition of a damping term in the governing equations; however, we are principally concerned with the total energy transferred from the fluid to the plate, as the fluid is the sole energy source in this interaction. Details of the energy transduction through electromechanical coupling, which is material dependent, are not considered.

In this case, we consider maximizing the total energy transferred from the fluid to the plate as maximizing the total mechanical energy that is available for electrical conversion, which is proportional to the energy harvesting potential. This assumes, of course, weak electromechanical coupling. The nondimensional energy of the plate, Ê, is given by23 

Ê(t̂)=1211μ[δ̂̇(x̂,t̂)]2+β[δ̂II(x̂,t̂)]2dx̂,
(15)

where Ê=E/ρV02L02. The first term in Equation (15) corresponds to the plate kinetic energy and the second term represents the material strain potential energy. Both terms comprise a non-dimensional parameter that is a function of the physical properties of the plate coupled with the dynamic response of the plate. Note that we neglect damping (η = 0) for the energy transfer optimization, which, in turn, neglects all losses in the system in order to facilitate comparisons of the total plate energy acquired from the vortex pair. This assumes that viscosity does not play a significant role in the initial vortex pair-plate interaction, which was demonstrated in the previous section, and that structural damping is relatively weak.

The total plate energy as a function of time for two different plate/vortex pair parameter sets is presented in Figure 7. The solid line corresponds to the parameters representative of the experiments of Goushcha et al. in the absence of damping (μ=5.878,β=4.466, H = 1.335, Λ=1.004, and η = 0), while the dashed line has lower mass and stiffness ratios (μ=2.000 and β=1.000). In both cases, as the pair approaches the plate it deflects, resulting in an increase in the total plate energy. The plate then reaches a local maximum in energy, denoted as point A in Figure 7, when the vortex pair is near the free end of the plate. After point A, the plate energy decreases slightly before rapidly rising to point B as the vortex pair passes over the plate. Once the pair passes beyond the plate there is again a drop in energy to point C, after which the plate settles into a free vibration state, which is evident in the harmonic oscillation in the plate energy after point C. We note that the harmonic oscillation in the plate energy during free vibration is due to the exchange between the kinetic energy of the vortex sheet32 and the total energy of the plate via fluid-structural coupling. Shvydkoy49 demonstrated that a vortex sheet with zero total circulation has finite energy, which is conserved in the absence of the interactions; this condition is automatically satisfied for all time in the present model in Equation (5) and, therefore, the vortex sheet simply acts as another energy storage element in the system during steady state free vibration.

FIG. 7.

The energy of the plate as a function of time; the solid line represents dimensionless parameters matching the experiments of Goushcha et al. (μ=5.878,β=4.466, H = 1.335, Λ=1.004, η = 0); the dashed line represents an alternative parameter set with lower mass and stiffness ratios (μ=2.000,β=1.000, H = 1.335, Λ=1.004, η = 0).

FIG. 7.

The energy of the plate as a function of time; the solid line represents dimensionless parameters matching the experiments of Goushcha et al. (μ=5.878,β=4.466, H = 1.335, Λ=1.004, η = 0); the dashed line represents an alternative parameter set with lower mass and stiffness ratios (μ=2.000,β=1.000, H = 1.335, Λ=1.004, η = 0).

Close modal

As mentioned previously, the total energy available for harvesting is related to the total plate energy. As shown in Figure 7, the plate energy never settles to a fixed value, making assessment of the total energy transferred somewhat nebulous. Herein, we define the “steady state” plate energy Êss as the peak energy of the plate during its free vibration once the vortex pair has advected away; this point is denoted as D in Figure 7. Point D does not necessarily correspond to the maximum energy achieved by the plate due to the interaction with the vortex pair; in fact, point B is considerably larger than point D for the dashed line in Figure 7. However, the energy in the plate is quickly transferred back to the fluid in this case, with the free vibration phase having considerably less total energy than at point B. For the solid line, corresponding to the experimental conditions of Goushcha et al., point D is near the maximum energy obtained by the plate.

Though qualitatively similar in behavior, there are significant differences in the plate energy at the highlighted points in Figure 7 for the two cases; a 66% reduction in μ and 78% reduction in β results in a 154% increase at point A, a 4.7% increase at point B, a 25% reduction at point C, and a 32% reduction in Êss (point D). To further elucidate the physics associated with this behavior, we consider the integrated loading on the plate. Figure 8 shows the integrated force along the plate as a function of time, computed using Equation (14), for the two parameter sets presented in Figure 7. As the vortex pair approaches the plate there is a negative force on the plate that increases in magnitude as the distance of the pair from the plate free end decreases (t̂10). As the vortices pass over the plate (10<t̂12.5), there is a strong positive force due to the low pressures of the vortex cores pulling the plate towards the pair. The force again goes negative once the vortices are beyond the plate (12.5<t̂14). Oscillations in the force on the plate after the interaction t̂15 are due primarily to the plate free vibration. We note that in the free vibration phase, the plate energy oscillates at twice the frequency of the forcing, since maximum energy is achieved twice per period.

FIG. 8.

Integrated force on the plate with respect to time; the solid line represents dimensionless parameters matching the experiments of Goushcha et al. (μ=5.878,β=4.466, H = 1.335, Λ=1.004, and η = 0); the dashed line represents an alternative parameter set with lower mass and stiffness ratios (μ=2.000,β=1.000, H = 1.335, Λ=1.004, and η = 0).

FIG. 8.

Integrated force on the plate with respect to time; the solid line represents dimensionless parameters matching the experiments of Goushcha et al. (μ=5.878,β=4.466, H = 1.335, Λ=1.004, and η = 0); the dashed line represents an alternative parameter set with lower mass and stiffness ratios (μ=2.000,β=1.000, H = 1.335, Λ=1.004, and η = 0).

Close modal

As evidenced by Figure 8, there is an inherent time scale to the vortex pair/plate interaction, which is a function of the vortex parameters H and Λ. In both cases shown in the figure, the vortex pair positions and parameters are identical, and as such, the time scale of the interaction, denoted by the negative-positive-negative peaks around 9<t̂<13, are the same. There is a difference in the loading magnitude between the two cases, however, due to the differing degrees of coupling between the fluid and the structure associated with the differing mass and stiffness ratios. Specifically, for the dashed line, the mass and stiffness ratios are smaller than for the solid line, and thus the plate more readily deflects due to the incoming vortex pair, resulting in a lower peak loading. The plate natural frequency for the two cases differ, resulting in divergence of the loading in the free vibration phase (t̂15).

Figure 8 suggests a manner in which the plate dynamics may be optimized for energy transfer from the fluid; specifically, resonance will occur if the characteristic frequency of the pressure loading matches the plate natural frequency. This will, in turn, lead to larger plate deflection and an increase in the energy transfer capacity. Out of phase loading, however, results in the plate energy returning back to the fluid after the vortex interaction, as seen in point C of the dashed line in Figure 7. This is analogous to the frequency response of a single degree of freedom system.50 

To more clearly illustrate the coupling between the vortex pair convection and the plate response, a simplified analysis can be performed by representing the pressure loading of the vortex pair with a traveling point load with time-varying magnitude to mimic the negative-positive-negative loading observed during the interaction in Figure 8. In this case, the loading term in Equation (1) is

p(x,t)=P0sin(ωPt)δD(xx0+Vct),
(16)

where δD is the Dirac delta distribution. The point load is travelling at a constant velocity Vc, and has a harmonic magnitude with a frequency of ωP. We note that ωP represents the characteristic frequency of the pressure loading in this simplified analysis, which is a function of the advective time scale. Fryba51 demonstrated that in this simplified scenario, resonance occurs when

ωires=ωP±(λiL)VcL,
(17)

where ωi and (λiL) are the natural frequency and the characteristic constant for each mode, respectively. The simplified model neglects the loading distribution along the x-axis and the fluid-structure coupling, but it does highlight the important relationship between the vortex pair convection speed and the characteristic frequency of the loading, since both terms in Equation (17) are functions of Vc. We note that even when the point load has a constant magnitude, the resonance still occurs when ωires=Vc/L, see Ref. 51.

Based upon the above discussion, it is evident that the convection speed is key to achieve resonance; thus, we seek the appropriate plate natural frequency to loading time scale ratio that maximize the steady state energy, Ess. Using the nondimensionalization scheme discussed in Sec. II, the dimensionless natural radian frequency of the first plate vibration mode is defined as

Ω1=2ω1LVc(λ1L)2=βμ,
(18)

where ω1 is the plate fundamental natural radian frequency and λ1=1.8751/L is the corresponding eigenvalue. To investigate the optimal plate/vortex pair parameters in terms of energy transfer to the plate, we parametrically vary Ω1 and μ and evaluate the steady state plate energy Êss. Figure 9(a) plots Ess as a function of Ω1 and μ for the experimental configuration of Goushcha et al. in the absence in damping, that is H = 1.335, Λ=1.004, and η = 0.

FIG. 9.

(a) Parametric study of the steady state plate energy Êss for the experimental vortex pair parameters H = 1.335 and Λ=1.004 with zero damping η = 0; the dashed line indicates the resonant frequency. (b) Parametric study of the resonance frequency Ω1res with respect to changes in H. Solid lines indicated where Êss is maximized for H ranging from 1.2 to 2.2 with Λ=1.004 and η = 0.

FIG. 9.

(a) Parametric study of the steady state plate energy Êss for the experimental vortex pair parameters H = 1.335 and Λ=1.004 with zero damping η = 0; the dashed line indicates the resonant frequency. (b) Parametric study of the resonance frequency Ω1res with respect to changes in H. Solid lines indicated where Êss is maximized for H ranging from 1.2 to 2.2 with Λ=1.004 and η = 0.

Close modal

It is clear from the parametric study result in Figure 9(a) that there is an optimal dimensionless natural frequency that maximizes Êss at a given value of μ, which is indicated by the dashed line. This dashed line corresponds to the resonance condition alluded to in Equation (17), which results in a significant boost to the steady state plate energy. The resonance condition occurs at the frequency of Ω1res2.02 for μ = 1, and decreases as μ increases. The rate of change in Ω1res decreases as μ increases, suggesting that the coupling effect decreases as the mass ratio increases, which is in agreement with the study by Peterson and Porfiri24 for a vortex pair orthogonally approaching and passing around a deformable plate. Additionally, there is an overall increase in Ess as μ decreases, which suggests that one can improve the energy transferring by reducing the plate mass or, more practically, adjusting its thickness. However, one should be aware that the bending stiffness is sensitive to the plate thickness as well, which impacts the dimensionless natural frequency.

Figure 9(a) focuses on optimizing the plate parameters for the experimental geometry described by Goushcha et al., which has a fixed vortex ring height above the plate. That is, for a given vortex ring/plate configuration, Figure 9(a) provides insight into the best μ and Ω1 for maximum energy transfer to the plate. In terms of pressure loading on the plate, the closer the vortex ring/pair passes by the plate, the larger the pressure loading magnitude, and consequently the larger the plate deflection. Thus, it is important to examine how the resonance frequencies vary with changing H. We note that by changing H for the vortex pair, the corresponding vortex ring parameters change as well due to the conversion described in Sec. III. Figure 9(b) presents the shift in the resonance peaks for H ranging from 1.2 to 2.2 with Λ=1.004 and η = 0. As H increases, the dimensionless resonance frequency shifts towards lower values. In addition, the resonant frequencies are a weaker function of μ for larger values of H, demonstrating that coupling decreases as the pair is further away from the plate.

To understand the shift in resonance frequency as H changes, we compare the force loading on the plate at H = 1.2, H = 1.6, and H = 2.0 in Figure 10. In order to observe the differences in the loading characteristic frequency, the force is normalized by its maximum value for direct comparison. As shown in Figure 10, the primary positive peaks have similar profiles at different H values, but the negative peaks vary noticeably. The first negative peak becomes wider and has a small increase in the relative magnitude in comparison to the primary positive peak as H increases, while the second negative peak becomes much weaker. The significant drop in the relative magnitude of the second negative peak as H increases is due to the fluid-structure coupling reduction, which results in a smaller vortex pair path deflection towards the plate and leads to a drop in relative forcing.

FIG. 10.

Integrated force on the plate normalized by its maximum value with respect to time with H = 1.2, H = 1.6, and H = 2.0 represented by solid, long dashed, and short dashed lines respectively. (μ=5.878,β=4.466,Λ=1.004, and η = 0).

FIG. 10.

Integrated force on the plate normalized by its maximum value with respect to time with H = 1.2, H = 1.6, and H = 2.0 represented by solid, long dashed, and short dashed lines respectively. (μ=5.878,β=4.466,Λ=1.004, and η = 0).

Close modal

To aid in energy harvester selection, we present the dimensionless plate bending stiffness β that results in resonance for a given dimensionless plate mass μ and dimensionless vortex pair height above the plate H. This is obtained by fitting the parametric study results in Figure 9(b), yielding

(19)
β=4.264μe0.7314H+4.072H1.737.
(19a)
The coefficient of determination is R2=0.9998 for Equation (19a). The resonance frequency Ω1res can subsequently be found from Equation (18). For very large mass ratios, Equation (19a) can be simplified to

Ω1res=2.065e0.3657H,
(19b)

which suggests that the resonance frequency decreases exponentially as H increases for μ. We further note that for a given H, the values of μ and β that result in resonance follow a simple linear relationship β=aμ+b. Thus, for a given energy harvesting configuration featuring a cantilevered electroactive polymer strip and a passing vortex ring/pair, Equation (19) (in conjunction with the conversion strategy presented in Sec. III, as required) can be used to inform the harvester selection to maximize the energy transferred to the structure.

Herein, we have focused on cases where μ1, which is generally valid for in-air energy harvesting. Though not shown, cases where μ<1 follow the same trend as in Figure 9. However, due to the increased fluid-structural coupling effects as μ decreases, the computational time required for a converged solution increases.

In this article, we analytically examined the coupled interaction between a vortex ring passing over a cantilevered plate oriented parallel to and offset from the path of a vortex ring in an otherwise quiescent fluid, and compared results with a recent experimental study conducted by Goushcha et al.14 To gain analytical traction, we developed a method to convert the three-dimensional vortex ring/plate configuration into a two-dimensional vortex pair/plate representation, thus enabling a potential flow solution. The model predictions agree well with the experimental measurements of Goushcha et al., both in terms of plate kinematics and the pressure loading induced by the fluid on the structure. Additionally, the interaction between a single vortex element and a deformable structure is of fundamental interest for energy harvesters that utilize coherent fluid structures and smart materials; the current study provides a theoretical framework that informs the design for such energy harvesters. By extension, this work also offers insight into more complex vortex-structure interactions, taking place in broadband turbulence-driven energy transfer.

The analytical model was employed to perform a parametric study aimed at maximizing the energy transferred to the plate from a given vortex ring. Borrowing from the classical analysis of a point load moving along a plate, which shows that resonance can occur for specific ratios of the plate natural frequency to the speed of the point load, a nondimensional natural frequency for the plate was introduced based upon the vortex pair convective time scale. The parametric study presented the energy transfer to the plate as a function of the dimensionless plate resonant frequency for a given mass ratio. Both the overall energy transfer capacity and the resonant frequency decrease with increasing mass ratio, with the resonant frequency appearing to approach an asymptote for large values of the mass ratio.

The shift in resonant frequency with varying distance of the vortex pair from the plate was also explored. As the distance of the vortex pair from the plate increases, the resonant frequencies shift towards lower values. Empirical correlations were presented that relate the mass ratio, dimensionless bending stiffness, and distance of the vortex pair from the plate to the nondimensional plate natural frequency. These correlations can be employed to determine the optimal plate properties for maximizing energy transfer to the plate for a given vortex/harvester configuration. The overall analysis highlights the critical importance of the vortex ring/pair convection speed in relation to the plate natural frequency for optimal energy transfer.

This work was supported by the Natural Sciences and Engineering Research Council of Canada (NSERC) and the National Science Foundation (NSF) under Grant Nos. 386282-2010 and CBET-1332204, respectively.

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