The extraction of energy from vortical structures advecting through an ambient environment is a topic of interest due to the potential to power miniature in situ sensors and monitors. This work investigates the vortex dynamics and flow-induced vibrations of a flexible plate arising from a vortex ring passing tangentially over it. Experimental measurements of the flow field and plate dynamics are performed in tandem with a coupled potential flow/Kirchhoff-Love plate model in order to (i) elucidate the physics of the vortex-plate interactions in the specified orientation and relate the energy exchange between the ring and the plate to the attendant vortex dynamics; (ii) validate the potential flow model and provide any needed corrections to account for the simplifying assumptions; and (iii) provide empirical data for estimating energy harvesting capabilities in the specified orientation. The plate loading arises as a result of an initial down-wash, followed quickly by a region of reduced pressure as the vortex core passes over the plate. The fundamental physics of the interaction is discussed, identifying three regimes. When the centerline of the vortex ring is positioned greater than approximately 2 vortex ring radii away from the plate it can be considered to be in the far-field, and the resulting vibrations are well predicted through potential flow, once the plate dynamics are corrected for edge effects arising from a finite plate width. As the offset distance of the vortex ring is decreased, diffusion of induced vorticity on the plate into the flow field significantly alters the fluid dynamics, pressure loading, and the resultant plate dynamics, and dramatically increases the strain energy in comparison with the potential flow model predictions. A first-order correction to the potential flow model is proposed to account for the finite plate width, while empirical correlations are presented for the plate strain energy in cases where ring/induced vorticity interactions are significant.

Extraction of energy from ambient environments is a topic that has been of interest for decades.1,2 Recently, research efforts have focused on energy harvesting from fluidic environments, such as in the atmosphere,3–6 rivers and streams,7–10 and tidal currents,11–13 to name a few. Applications include both medium-scale power generation, such as charging of consumer electronics,14,15 and small-scale energy capture for powering low-consumption devices, as found in structural health monitors16,17 and aquatic tracking systems.10,18 Recent advancements in electro-active polymers, which can be used to extract fluidic energy,19 have boosted their efficiencies, making this a promising field of research.20,21

Energy harvesting from fluid-structure interactions has been studied for a variety of fluid flows. Examples include flow-induced vibrations of a plate: immersed in a vortical flow field, such as a von Kármán vortex street;22,23 in a shear flow;24 in the wake of a bluff body;6,25 and directly impacted by coherent structures.26,27 In all cases, the vortical structure(s) produce a time-varying pressure field that imparts energy to, and incites motion in, the structure.

The energy exchange between a fluid and a cantilevered plate placed in a von Kármán street, usually created in the wake of a bluff body, can by maximized by ensuring that the vortices pass periodically over the surface with a frequency equal to the natural frequency of the plate.2 However, there are situations where sub-optimal conditions may exist, such as a change in trajectory of the free-stream velocity that shifts the wake such that the plate is no longer centered in the vortex street, but rather is located off center. This orientation is also of interest in other research areas, such as voiced speech production, where vortices are generated and advect over the compliant surface of the vocal folds, impacting the energy exchange that drives self-sustained vocal fold oscillation and hence the production of voiced speech.28 In this scenario, vortices are believed to influence both the frequency and intensity of speech.29,30

While von Kármán vortex streets provide cyclic loading, in a more general case, the arrival time of an energy containing coherent fluid structure may be random and/or aperiodic. To this end, several studies have considered the interaction of single individual coherent structures with deformable surfaces. Peterson and Porfiri26 investigated the energy transfer arising from a vortex ring impacting the tip of a cantilevered smart material strip. They found that roughly 1% of the ring energy is translated to elastic energy in the strip. Zivkov et al.31 numerically studied the impact of a vortex dipole colliding with the tip of a flexible plate, observing the vortex dynamics as the dipole vorticity interacted with induced vorticity on the wall, causing it to eject and form smaller rebounding dipoles. These dipoles were found to either form smaller structures or decay through opposite-sign vorticity annihilation, depending on the Reynolds number. These numerical results were recently validated experimentally by Zivkov, Peterson, and Yarusevych.32 

More closely aligned with the orientation of the current study, Goushcha, Elvin, and Andreopoulos33 experimentally studied a tangential vortex-plate orientation, demonstrating that the pressure field produced by a vortex ring passing over one side of a plate is sufficient to set a cantilevered plate into motion. They utilized particle image velocimetry (PIV) data to estimate the pressure on a variety of plates with differing lengths, but did not explore the influence of the vortex ring to plate spacing. Hu, Porfiri, and Peterson34 developed a fluid-structure interaction model of Goushcha, Elvin, and Andreopoulos's33 configuration by coupling a potential flow model with Kirchhoff-Love plate theory,27 with the aim of optimizing the energy transfer to the plate. They were able to reproduce the plate motion observed in the experimental investigations of Goushcha, Elvin, and Andreopoulos,33 although there were insufficient data to fully validate the model. Using the potential flow model, the criterion for maximizing the energy of the plate as a function of the initial separation between the ring and the plate was determined. The use of a potential flow model, however, precluded the ability to capture diffusion of induced vorticity along the plate into the fluid and thus vorticity annihilation through cross-diffusion.

The primary objective of this study is to explore the dynamics of a vortex ring passing tangentially over a flexible plate as a function of the initial vortex ring/plate separation, thereby expanding the experimental study initially performed by Goushcha, Elvin, and Andreopoulos.33 This will aid in addressing several outstanding issues with this energy harvesting configuration, including:

  • A deeper exploration of the vortex dynamics, particularly in cases where fluid viscosity diffuses the induced plate vorticity into the flow field and significantly changes the interaction.

  • Direct assessment of the potential flow model of Hu, Porfiri, and Peterson,34 including possible corrections for neglected viscous and/or three-dimensional effects.

  • The prescription of empirical data for determining energy harvesting capacities in such cases.

The manuscript is outlined as follows: the problem formulation is introduced in Sec. II, the experimental methods are presented in Sec. III, the numerical approach is outlined in Sec. IV, results are discussed in Sec. V, and Sec. VI is left for the conclusions.

The problem of interest is a vortex ring passing tangentially over a cantilevered, flexible plate. The problem formulation, along with relevant parameters and coordinate system, is shown in Fig. 1. The dimensions L, W, and T are the plate length, width, and thickness, respectively. The distance between the plate surface and the centerline of the vortex ring is denoted by H. The vortex ring initially has a radius a, core radius b, and circulation Γ, while self-advecting over the plate surface from the free end (tip) to the fixed end (base) with an initial velocity U. The x-axis (see Fig. 1) is located on the initial ring centerline, with x = 0 corresponding to the tip of the plate. The plate vibrations are in the y-direction and perpendicular to gravity, which points in the negative z-direction; as such, body forces on the plate can be neglected.

FIG. 1.

Schematic of the vortex ring and cantilevered plate configuration.

FIG. 1.

Schematic of the vortex ring and cantilevered plate configuration.

Close modal

Since the plate length is large compared to its thickness, the plate dynamics can be modeled using Kirchhoff-Love plate theory, which for pure cylindrical bending is governed by35 

m s 2 δ t 2 + K 4 δ x 4 + C δ t = [ [ p ] ] ( x , t ) ,
(1)

where δ ( x , t ) is the plate deflection in the y-direction measured from its rest position (m), ms is the mass per unit area of the plate (kg⋅m−2), K is the plate flexural rigidity per unit width (N⋅m), C is the plate structural damping coefficient per unit area (kg⋅s−1⋅m−2), and [ [ p ] ] ( x , t ) is the spatially and temporally varying pressure difference across the upper (facing toward the vortex ring; –y plane) and lower (facing away from the vortex ring; + y plane) surfaces of the plate (Pa).

Equation (1) can be non-dimensionalized by using L as the length scale, U as the velocity scale, t0 =L/U as the time scale, and p 0 = ρ U 2 as the pressure scale, where ρ is the fluid density. This yields

Π m 2 δ * t * 2 + Π s 4 δ * x * 4 + Π d δ * t * = [ [ p * ] ] ( x * , t * ) ,
(2)

where Π m = m s / ρ L is a non-dimensional mass parameter, Π s = K / ρ U 2 L 3 is a non-dimensional stiffness parameter, and Π d = C / ρ U is a non-dimensional damping parameter. A superscript * indicates a variable that has been non-dimensionalized by the appropriate scale, e.g., δ * = δ / L . The pressure loading on the plate due to a passing vortex ring is a function of the geometric, kinematic, and dynamic parameters, such that

[ [ p * ] ] = f ( H * , a * , Γ * , Re , AR ) ,
(3)

where H * = H / L is the non-dimensional distance of the vortex ring from the plate, a * = a / L is the non-dimensional radius of the ring, Γ * = Γ / U L is the non-dimensional vortex ring circulation, Re is the vortex ring Reynolds number defined as Re = U L / ν , where ν is the kinematic viscosity of the fluid, and AR = W / L is the plate aspect ratio. While the beam length serves as a reasonable length scale for scaling the plate deflection and for the convective time scale, for the pressure loading, it is more meaningful to scale the separation between the plate and the ring by the ring radius. As such, the distance between the vortex ring and the plate will be referred to in terms of H * / a * = H / a when discussing the investigations.

Herein, the experimental facility and the methods for acquiring and processing the data are detailed.

The experimental facility utilized for the study is shown in Fig. 2, where the physical facility is shown in Figs. 2(a), and 2(b) is a schematic of the facility with the position and orientation of the measurement system included. A vortex ring is generated by movement of a baffled speaker cone. An 8 in Morel subwoofer with an effective cone diameter of 162 mm is housed in a cylindrical plenum and a d = 24.3 mm ID circular tube is connected to the end of the plenum such that forward movement of the speaker cone forces a slug of air out of the tube, which rolls up and forms a vortex ring of diameter 2 a = 36. mm . A variable sawtooth signal (short rising and long settling time) is produced in National Instruments (NI) LabVIEW software and amplified with a Kepco BOP-36 amplifier, before reaching the speaker. In this manner, the speaker cone motion can be adjusted to generate vortices of varying circulation strengths and advection velocities.

FIG. 2.

(a) The physical flow facility and (b) a schematic of the flow facility showing the location and orientation of the particle image velocimetry (PIV) system and the laser displacement sensor.

FIG. 2.

(a) The physical flow facility and (b) a schematic of the flow facility showing the location and orientation of the particle image velocimetry (PIV) system and the laser displacement sensor.

Close modal

For the current experiments, the speaker cone motion was specified to reach an amplitude of 2.1 mm in 30.0 ms. Based upon the volume of displaced air and the exit tube diameter, d, the vortex ring formation number,36F, was estimated to be F = L 0 / d = 4 , where L 0 is the length of the slug of displaced fluid. A formation number of 4 for the prescribed ramp-up time and tube exit conditions suggests that the vortex ring has achieved its maximum possible circulation.36 

A cantilevered polycarbonate plate with dimensions of L = 100 mm × W =25 mm × T = 0.25 mm was placed downstream of the tube. The plate material and dimensions were selected to be similar to those of Goushcha, Elvin, and Andreopoulos.33 The plate tip (free end) was placed at an axial distance of 1.5 d from the tube end. The offset distance of the plate from the vortex centerline, H, was adjusted by positioning the plate holder. The instantaneous deflection of the plate was acquired with a Wenglor PNBC005 laser displacement sensor with a sampling frequency of 10 kHz and a resolution of 1.5 μ m . The entire system (speaker, plenum, and plate) was housed in a closed acrylic box with a cross-section of 610 mm × 610 mm and a length of 1 220 mm . This isolated the flow field from ambient room eddies and contained the flow seeding particles, while being large enough such that the effect of the walls on the flow field and any flow reflections at the walls were negligible.

The mass per unit area of the plate, ms, was measured using a scale, and the mechanical properties were derived from several impulse response (flick) tests, the results of which are summarized in Table I. The data are presented along with their standard deviations obtained from 100 trials. In Table I, fn is the natural frequency of the plate. As can be seen, the mass and stiffness parameters are O ( 1 ) . The damping parameter, on the other hand, is much lower in magnitude, suggesting low structural damping. In common piezoelectric energy harvesting applications, O ( 10 3 ) < Π m < O ( 10 1 ) and O ( 10 2 ) < Π s < O ( 10 3 ) .8,10,22,26,27,33,34,37 The mechanical properties of the plate in the present study are well within these ranges.

TABLE I.

Mechanical properties of the cantilevered plate.

Description Variable Value (units)
Mass per unit area  m s   0.318 ± 0.003 (kg⋅m–2
Flexural rigidity per unit width  K  3.78 ± 0.08 (10–3 N⋅m) 
Natural frequency  f n   6.13 ± 0.02 (Hz) 
Mass parameter  Π m   2.44 ±0.03 
Stiffness parameter  Π s   1.09 ±0.02 
Damping parameter  Π d   0.03 ±0.01 
Description Variable Value (units)
Mass per unit area  m s   0.318 ± 0.003 (kg⋅m–2
Flexural rigidity per unit width  K  3.78 ± 0.08 (10–3 N⋅m) 
Natural frequency  f n   6.13 ± 0.02 (Hz) 
Mass parameter  Π m   2.44 ±0.03 
Stiffness parameter  Π s   1.09 ±0.02 
Damping parameter  Π d   0.03 ±0.01 

The velocity field was measured using a LaVision Flow Master Particle Image Velocimetry (PIV) system. Particle seeding was generated via a custom fabricated olive oil atomizer, with particle sizes in similar systems reported to be O ( 1 10 ) μ m .38 A 532 nm Nd:YAG (Litron Nano L) laser beam illuminated the viewing area. The laser beam was divided with a beam splitter with the resultant beams passing through mirror-cylindrical lens combinations, as shown in Fig. 2, thereby evenly illuminating both sides of the cantilevered plate. The laser sheet was aligned with the plate centerline bisecting the vortex ring. Care was taken in aligning the laser to minimize out-of-plane gradients that may arise from misalignment of the data plane with the vortex ring symmetry plane.

The laser timing was optimized (100 μ s) so that the maximum particle displacement in the vortex ring between image pairs was approximately one-quarter of the final interrogation region size of 32 × 32 pixels. As such, the uncertainty in the PIV velocity data in the undisturbed vortex ring was estimated to be 1.6%, which accounts for uncertainties arising from the calculation of particle displacement, PIV timing, particle sedimentation, and particle inertia.39 It is expected that velocity uncertainties are the highest near the plate surface and outer boundaries where particle displacements are low.

Image pairs of the vortex ring were acquired with a 2560 × 2160 pixel Image sCMOS camera at full resolution. The viewing area of the camera in the xy plane, as shown in Fig. 2, was 153 × 129 mm. Based on the camera resolution and the interrogation region size with 50% overlap, the spatial resolution was 0.923 mm, corresponding to a non-dimensional spatial resolution of Δ x * = 9.23 × 10 3 . A mean background subtraction was performed on each image pair to eliminate laser reflections along the cantilevered plate, and a geometric mask was applied to the plate and its holder before processing. All timing and processing were handled with the DaVis8 software package on a 2× quad-core XEON processor PC with 12 GB of RAM.

Phase-averaged PIV data were acquired at 13 instances in time as the vortex ring passed over the plate, ranging from t* = 1.43 to 2.61 with a non-dimensional time increment of 0.098, which corresponds to axial positions of the undisturbed vortex ring from x* = –0.1 to 1.1. Cycle-to-cycle changes in the vortex position were found to vary by less than 0.40 mm (x* < 0.004). Based on these observations, 50 vector fields were averaged at each instance in time, and the corresponding mean velocity fields were computed. This number was chosen as it was found to be sufficient to ensure convergence of the flow field to within 2.5% over the entire domain.

Timing of the speaker input signal (vortex generator), PIV trigger signal, and data acquisition from the laser displacement sensor was created, acquired, and synchronized with NI LabVIEW software. The non-dimensional time t* = 0 corresponds to the starting point of the output signal to the vortex generator, which is also the starting point of the input signal from the laser displacement sensor. The synchronized trigger signal was created with a NI USB-6001 DAQ card. Sufficient time (20 s) was allowed between each successive instance of data acquisition to ensure no transients from the previous instance of data acquisition persisted in either the plate oscillation or flow field. The input signal from the laser displacement sensor that captured the plate displacement was acquired with a NI 6320 DAQ card.

The properties of the undisturbed vortex ring were measured without the cantilevered plate in place and are reported in Table II. The vortex ring circulation was calculated by integrating the vorticity over the bounds of the vortex ring cores, obtained by finding the outer closed contour of vorticity near each core. Prior investigations, not shown here for brevity, determined that the vortex ring achieved its maximum circulation at x* = 0.30 (t* = 1.82), with the circulation gradually decreasing downstream. The circulation started to decay rapidly, and its trajectory altered after it passed x* = 1.2 (t* = 2.71) due to slight experimental perturbations. As such, x* = 0.0 was an optimum position for the plate tip due to experimental limitations. The difference between vortex ring circulation at x* = –0.10 (t* = 1.42) and its maximum value was ∼20%.

TABLE II.

Properties of the vortex ring.

Description Variable Value (units)
Circulation  Γ  0.12 ± 0.01 (m2⋅s–1
Advection velocity  U  1.64 ± 0.06 (m⋅s–1
Ring radius  a  18. ± 1. (mm) 
Plate width/ring diameter  W/2a  0.683 ±0.011 
Reynolds number  Re  10 300 ± 400 
Strouhal number  St  0.374 ± 0.014 
Description Variable Value (units)
Circulation  Γ  0.12 ± 0.01 (m2⋅s–1
Advection velocity  U  1.64 ± 0.06 (m⋅s–1
Ring radius  a  18. ± 1. (mm) 
Plate width/ring diameter  W/2a  0.683 ±0.011 
Reynolds number  Re  10 300 ± 400 
Strouhal number  St  0.374 ± 0.014 

The vortex ring circulation, advection velocity, and ring radius were computed by averaging the properties from 250 trials over all 13 instances in time. The resultant temporally and spatially averaged vortex ring properties are presented in Table II along with their standard deviations, revealing largely constant ring properties over the domain of interest.

The Strouhal number in Table II is defined as St = fnL/U and is the ratio of the ring convective timescale to the plate vibrational timescale. Also note that the ring diameter was larger than the plate width (W/2a < 1). The Reynolds number of the vortex ring utilized in the current study indicates a laminar vortex ring.40 

To estimate the load exerted on the plate arising from the passing vortex ring, the pressure field is calculated from the instantaneous velocity fields obtained from the PIV measurements. Several methods have been proposed to extract pressure fields from two-dimensional velocity fields. Line integration methods41–43 are most commonly employed, although there are some complications with these methods, such as the accumulation of PIV velocity errors and the resultant reduction of accuracy,44 difficulty in implementing the Neumann boundary condition of a moving solid boundary, and termination of integration lines at solid boundaries (e.g., the plate in the present study), thereby potentially obstructing the number of available paths for integration.

The method employed in this study is similar to that proposed by Shams, Jalalisendi, and Porfiri,45 where the pressure field is obtained from PIV velocity data by solving the pressure Poisson equation. Solving the pressure Poisson equation has the advantage of a uniform pressure distribution and connection of all nodes in the calculation process. However, the uncertainty is increased due to the implementation of second-order velocity derivatives. Nevertheless, it is shown by Charonko et al.44 that if the PIV errors are small, and if a conservative form of the Poisson equation is used, the accuracy of the results improves.

The PIV velocity field is two-dimensional in a plane cutting through the ring, but the flow field is three-dimensional. Consequently, any deviation of the laser sheet from the vortex mid-line will give rise to out-of-plane velocity components. Therefore, it is prudent to employ the conservative form of the pressure Poisson equation, with the out-of-plane velocity component included, given by

2 p x 2 + 2 p y 2 = ρ [ g 1 ( x , y , t ) + g 2 ( x , y , t ) + g 3 ( x , y , t ) ] ,
(4)

where the source terms are

g 1 ( x , y , t ) = ( u x ) 2 + 2 u y v x + ( v y ) 2 , g 2 ( x , y , t ) = t ( u x + v y ) + u x ( u x + v y ) + v y ( u x + v y ) , g 3 ( x , y , t ) = w z ( u x + v y ) + w x u z + w y v z ,
(5)

and u, v, and w are the flow velocity components in the x-, y-, and z-directions, respectively.

While viscosity plays a clear role in the evolution of the velocity field, ignoring the viscous terms when computing the pressure field is justified because of the negligible contribution of the viscous stresses in the flow to the pressure field. This was confirmed by computing the pressure field both with and without the viscous terms, resulting in no discernible difference; therefore, for simplicity, they are neglected.

The three-dimensionality of the flow, which arises due to curvature of the vortex ring, is captured by the terms in g2 and g3 in Eq. (4). All terms in g2 can be calculated from the two-dimensional PIV data, but the terms in g3 cannot be computed from the available data and are therefore neglected and considered a source of error.

Equation (4) is solved with a central difference scheme for the interior nodes. The transient terms are calculated using the successive PIV data acquisitions in time. The Immersed Boundary method with a three-point interpolation46 was used for the moving plate, where the normal pressure gradient was calculated from the known velocity and acceleration of the plate measured by the laser displacement sensor.

The analytical model employed to predict the movement of a flexible plate due to a passing vortex ring casts the three-dimensional viscous problem in a two-dimensional, inviscid framework. Specifically, the vortex ring is represented by a counter-rotating vortex pair, with the fluid assumed to be irrotational except within the infinitesimal vortex core; thus, the flow field can be obtained with potential flow theory. A brief introduction to the model is provided herein. The reader can find a complete description in previously published work.27,34

Due to the differences in vortex topology between a vortex ring (ring-shaped vortex lines) and a two-dimensional vortex pair (straight vortex lines), the dynamics and induced flow fields differ. Therefore, a ring-to-pair conversion scheme34 is necessary to match the pressure loading time scale and magnitude to ensure that the primary physics of the flow-induced vibrations of the plate are captured. The unknown parameters of the equivalent vortex pair ( H p * , a p * ), denoted by subscript p, can be obtained from the conversion scheme with the known vortex ring parameters ( Γ r * , H r * , a r * ), denoted by subscript r. Note that the vortex ring is approximated as an axisymmetric, infinitely thin core vortex ring; that is, the ratio b r / a r approaches zero. The closed-form expression for the induced flow field under these assumptions is known.47 

For the fluid-structure interaction, it is assumed that Γ r * remains constant throughout the interaction, since the viscous dissipation time scale is much longer than the convective time scale. The potential flow model of the fluid domain consists of a pair of point vortices representing the ring, and a bounded vortex sheet, which represents the plate.27 The flow field must satisfy the boundary conditions of zero pressure at infinity and normal fluid velocity equal to the plate velocity on its surface (i.e., no penetration), as well as the initial condition of zero total circulation.27 Being two-dimensional potential flow, it must also satisfy Kelvin's circulation theorem at all times.48 

The pressure jump across the plate can be solved with the unsteady Bernoulli equation.49 The kinematics of the vortex pair is found by determining the velocity at a given vortex center due to the contributions of all elements in the fluid domain except the vortex itself (i.e., desingularized flow field).50 In the structural domain, the plate dynamics are obtained by solving Eq. (2), assuming that the vibration magnitude is relatively small and undergoes purely cylindrical bending. Lastly, the coupled fluid and structural models are solved simultaneously with the procedures outlined by Peterson and Porfiri.27 

PIV measurements of the velocity field were captured at 5 different initial non-dimensional distance ratios, H * / a * = 3.00 , 2.46, 1.91, 1.64, and 1.37. Initial comparisons of the plate dynamics between the experimental results and the potential flow model revealed the importance of the three-dimensionality of the flow field on the resulting dynamics, which is not considered in the two-dimensional potential flow model. A correction that can be applied to the model to account for these effects is introduced in Sec. V A.

Results for the vortex ring/plate dynamics are grouped into three categories based upon the level of interactions between the core of the vortex ring and the vorticity fields induced along the flexible plate: a far-field region, where the pressure arising from the core of the vortex ring dominates the pressure loading (Sec. V B); a region where the effects of induced vorticity on the plate becomes appreciable (Sec. V C); and a region where complete breakdown of the vortex ring occurs (Sec. V D). As the distance ratio decreases and the vortex ring approaches the plate, vorticity diffusion and annihilation become more pronounced. A discussion on the strain energy in the plate as a function of the flow region and the implications on energy harvesting are provided in Sec. V E.

Figure 3 presents the non-dimensional plate tip deflection δ tip * = δ ( 0 , t ) / L at a distance ratio of H * / a * = 3.00 for both the experimental measures and the potential flow model as a function of time. In Fig. 3 and all subsequent plate tip deflection figures, the recorded experimental data and the numerical prediction are synchronized at the time when the vortex ring reaches x * = 0.20 , which is located at the left boundary of the camera field of view.

FIG. 3.

Non-dimensional plate deflection at H * / a * = 3.00 : numerical prediction () and experimental results (red square).

FIG. 3.

Non-dimensional plate deflection at H * / a * = 3.00 : numerical prediction () and experimental results (red square).

Close modal

It is clear from Fig. 3 that the potential flow model results differ from the experimental data in both amplitude and frequency. The amplitude predicted by the potential flow model is twice as large and the frequency of vibration is lower. Due to the far-field location of the vortex ring, the interaction between the induced vorticity on the plate and the ring/pair vorticity should be negligible, so good agreement between the two studies should be achievable. Discrepancies in the two methods arise due to the fact that the experimental plate has a finite width, in contrast with the potential flow model that solves the pressure loading for a two-dimensional, infinite width plate. In the real case, as the plate oscillates, edge vortices will roll up and alter the flow field.

The effects of these edge vortices are two-fold. First, they reduce the pressure difference between the top and bottom surfaces of the plate at the edges as flow moves from the high to low pressure side, thus producing a non-uniform pressure distribution along the span of the plate. This is analogous to the formation of wingtip vortices for a low-aspect ratio wing, where, in this case, the aspect ratio is given as AR = W/L (analogous to wing span to chord length for a rectangular wing). Helmbold's equation51 can be used to correct the lift coefficient for an infinite two-dimensional plate as a result of three-dimensional edge effects arising from edge vortices, given as

δ tip , 3 D * δ tip , 2 D * C L , 3 D C L , 2 D = 1 a 0 [ 1 a 0 ] 2 + [ 1 π AR ] 2 + 1 π AR ,
(6)

where CL is the total lift coefficient, a0 is the slope of the lift coefficient variation versus angle of attack for an infinite aspect ratio wing, and the subscripts 3D and 2D refer to the three- and two-dimensional results, respectively. It is assumed in this equation that the reduction in tip deflection of the three-dimensional plate is proportional to the reduction in total load.

The second effect of the edge vortices is to decrease the added mass associated with fluid dislocation as the plate moves. This has been discussed and quantified by Aureli, Basaran, and Porfiri52 and Falcucci et al.53 The change in the added mass will cause a change in the frequency and amplitude of vibration when compared to the two-dimensional case. These changes are estimated as

t 3 D * t 2 D * = f n , 2 D f n , 3 D m plate + m add , 3 D m plate + m add , 2 D
(7)

and

δ tip , 3 D * δ tip , 2 D * δ ¨ tip , 3 D * ( 0 , t ) δ ¨ tip , 2 D * ( 0 , t ) = m plate + m add , 2 D m plate + m add , 3 D ,
(8)

where mplate is the mass of the plate, madd is the added mass, and δ ¨ tip * is the plate tip acceleration. Again, it is assumed in Eq. (8) that a change in the plate tip deflection due to three-dimensionality is proportional to its tip acceleration.

As a first-order correction, the non-dimensional time histories of the two-dimensional plate oscillations are corrected by the ratio given in Eq. (7), and the non-dimensional tip deflections are corrected by the ratio given in Eqs. (6) and (8). These corrections are applied to all subsequent discussions of the plate dynamics in the numerical potential flow model.

The largest separation distance between the vortex ring and the plate considered herein is H * / a * = 3.00 . Based on the PIV velocity data, vorticity and pressure distributions are calculated and plotted at several instances in time in Fig. 4, with the top row displaying the vorticity contours, the middle row the pressure contours, and the bottom row the pressure difference across the surface of the plate. The data are presented at 6 instances in time as the vortex ring passes over the plate, t* = 1.53, 1.72, 1.92, 2.12, 2.31, and 2.51. These times correspond to the center of the vortex ring being positioned at x* = 0.0, 0.20, 0.40, 0.60, 0.80, and 1.0, respectively. In all of the vorticity plots, regions of negative vorticity, corresponding to clockwise (CW) rotation, are denoted by blue, and regions of positive vorticity, corresponding to counter-clockwise (CCW) rotation, are denoted by red.

FIG. 4.

Derived values from PIV data at a distance ratio of H * / a * = 3.00 at times (a) t * = 1.53 , (b) t * = 1.72 , (c) t * = 1.92 , (d) t * = 2.12 , (e) t * = 2.31 , and (f) t * = 2.51 . Top row - vorticity contours, middle row - pressure contours, and bottom row - resultant pressure distribution along the plate.

FIG. 4.

Derived values from PIV data at a distance ratio of H * / a * = 3.00 at times (a) t * = 1.53 , (b) t * = 1.72 , (c) t * = 1.92 , (d) t * = 2.12 , (e) t * = 2.31 , and (f) t * = 2.51 . Top row - vorticity contours, middle row - pressure contours, and bottom row - resultant pressure distribution along the plate.

Close modal

The reader is reminded that the exit of the vortex tube is located at a position of x * = 0.36 (1.5d before the tip of the plate), which is beyond the limits of the viewing plane. As the vortex ring propagates over the plate in Fig. 4 (top panel), it can be observed that the interaction between the vortex ring and the induced vorticity on the plate is minimal. As such, the vortex ring circulation and its trajectory are not significantly altered by interactions with the plate.

Observing the pressure field, it is seen that pressure minima are located at the cores of the vortex ring, which create a negative pressure difference across the plate in the vicinity of the core as the vortex ring passes over the plate surface (see Fig. 4, bottom panel). Regions of positive gauge pressure arise in front of and behind the vortex ring due to its advection through the domain. Although regions of negative gauge pressure are also found in the trailing vorticity, their influence on the plate dynamics is likely negligible because of their small magnitudes and greater distances from the plate surface.

The experimental non-dimensional time history of the plate tip deflection and non-dimensional moment in the z-direction about the base of the plate (fixed end) is displayed in Fig. 5 as red squares. The experimental time history data have been down-sampled for clarity. The non-dimensional moment per unit width at the base of the plate is calculated as

M z * = 0 1 [ [ p * ] ] ( 1 x * ) d x * .
(9)

A positive pressure difference across the plate produces a negative (CW) moment at the base of the plate, and vice versa, as defined by the coordinate system in Fig. 1.

Results from the potential flow model are superimposed as a solid black line, where the amplitude and frequency of the potential flow model have been corrected to account for the three-dimensionality of the experimental measures using Eqs. (6)–(8). In Fig. 5(a), vertical dashed lines indicate the span of time during which the vortex ring propagates over the plate, revealing that the loading behaves largely as an impulse; that is, the convective time scale of the ring is small in comparison with the fundamental period of oscillation. Because of the short interaction time scale, the vortex ring has a nearly constant circulation; thus, the applied moment at the base in Fig. 5(b) is nearly constant.

With corrections applied to account for three-dimensionality, Fig. 5(a) demonstrates very good agreement between the potential flow model and the experimental results. Similarly, good agreement is found when comparing the moment at the base of the plate, as shown in Fig. 5(b). This indicates that when the vortex ring is sufficiently far from the plate surface, the potential flow model, when corrected to consider three-dimensional edge effects and added mass differences, does an excellent job at predicting the plate dynamics. Clearly then, when the vortex ring is in the far-field, the vorticity induced on the plate is small, and the resultant diffusion of this vorticity into the flow field will have a negligible effect on any cross-sign interactions with the vortex ring. Similar agreement and observations are found when H * / a * = 2.46 , which are not shown here for brevity.

FIG. 5.

(a) Non-dimensional deflection and (b) non-dimensional moment for: potential flow model () and experimental results (red square) at a distance ratio of H * / a * = 3.00 . Vertical dashed lines in (a) indicate the non-dimensional time at which the vortex ring propagates over the plate.

FIG. 5.

(a) Non-dimensional deflection and (b) non-dimensional moment for: potential flow model () and experimental results (red square) at a distance ratio of H * / a * = 3.00 . Vertical dashed lines in (a) indicate the non-dimensional time at which the vortex ring propagates over the plate.

Close modal

The influence of induced vorticity interacting with the vortex ring becomes evident when the vortex ring-plate spacing is decreased below H * / a * = 1.91 , with significant interactions occurring at H * / a * = 1.64 , as shown in the vorticity and pressure contour plots of Fig. 6. When the vortex ring approaches the plate tip, as seen in Fig. 6(a), the CW rotation of the lower core creates a down-wash (in the + y-direction) that impinges on the top surface of the plate. The down-wash imposes a positive pressure on the plate, causing an initial downward deflection (positive δ * ). This is evident in the pressure distribution along the plate shown in the lower panel of Fig. 6(a), where a positive pressure peak is evident downstream of the maximum negative pressure peak. As the vortex ring passes over the plate, the positive pressure due to the down-wash is followed by a localized region of negative pressure difference across the plate that arises from the low pressure core, as seen in the second and third rows of Fig. 6.

FIG. 6.

Derived values from PIV data at a distance ratio of H * / a * = 1.64 at times (a) t * = 1.53 , (b) t * = 1.72 , (c) t * = 1.92 , (d) t * = 2.12 , (e) t * = 2.31 , and (f) t * = 2.51 . Top row - vorticity contours, middle row - pressure contours, and bottom row - resultant pressure distribution along the plate. Dashed boxes indicate the corresponding domains plotted in Fig. 7.

FIG. 6.

Derived values from PIV data at a distance ratio of H * / a * = 1.64 at times (a) t * = 1.53 , (b) t * = 1.72 , (c) t * = 1.92 , (d) t * = 2.12 , (e) t * = 2.31 , and (f) t * = 2.51 . Top row - vorticity contours, middle row - pressure contours, and bottom row - resultant pressure distribution along the plate. Dashed boxes indicate the corresponding domains plotted in Fig. 7.

Close modal

Magnified views of the vorticity contours in Fig. 6 are plotted in Fig. 7, with the corresponding domains of interest identified by dashed boxes in Fig. 6. The CW rotating lower core of the vortex ring is denoted as (i) in Fig. 7. As the down-wash impacts the plate, a CCW tip vortex, identified as (ii) in Fig. 7(a), is formed due to separation of the flow under the plate. As the plate is displaced downward due to the impinging flow, a second CW tip vortex is then also produced as can be observed in Fig. 7(b), denoted by (iii). The two tip vortices (ii) and (iii) eventually form a pair in Fig. 7(b) and move away from the top side of the plate due to mutual induction and entrainment into the wake of the vortex ring [see Fig. 7(c)].

FIG. 7.

Magnified view of the domains identified by the dashed boxes in Fig. 6 displaying vorticity contours for a vortex ring at a distance ratio of H * / a * = 1.64 from the plate at times t* = (a) 1.53, (b) 1.72, (c) 1.92, (d) 2.12, (e) 2.31, and (f) 2.51.

FIG. 7.

Magnified view of the domains identified by the dashed boxes in Fig. 6 displaying vorticity contours for a vortex ring at a distance ratio of H * / a * = 1.64 from the plate at times t* = (a) 1.53, (b) 1.72, (c) 1.92, (d) 2.12, (e) 2.31, and (f) 2.51.

Close modal

While the vortex ring is above the plate, a region of positive induced vorticity on the plate surface, located below the lower core, is also formed, as denoted by (iv) in Fig. 7. This induced vorticity is produced along the plate surface in tandem with the vortex ring advection, as seen in Figs. 7(b)–7(f). Another region of negative induced vorticity (denoted as “secondary” induced vorticity) is also observed to form in Figs. 7(d)–7(f), which is labeled as (v). If the vortex ring circulation is increased or the plate spacing is decreased, the regions of induced vorticity will concomitantly increase in size and strength.

Visible in Figs. 7(d)–7(f), the positive vorticity (iv) induced on the top of the plate interacts with the negative vorticity (i) in the lower core of the vortex ring, weakening it through opposite-sign vorticity annihilation. This, in turn, decreases the advection velocity and pulls the bottom of the ring closer to the plate surface. This skewing of the vortex trajectory is predicted by the potential flow model as well, since it captures the induced vorticity through the bound vortex sheet that represents the plate. This is demonstrated by observing that the vortex sheet (plate) initially has zero-circulation. As the vortex pair propagates over it, the vortex sheet must have non-zero circulation to satisfy the no fluid-penetration boundary condition. Therefore, opposite-sign induced vorticity is predicted on the plate boundary. However, due to the inviscid nature of the potential flow model, there is no mechanism that will diffuse vorticity into the flow field, thereby enabling opposite-sign vorticity annihilation with the vortex ring. As such, the effect of opposite-sign vorticity interactions is more pronounced in the experimental measures.

The non-dimensional deflection and moment time histories at H * / a * = 1.64 are displayed in Fig. 8. It is seen that the potential flow model underestimates the plate deflection amplitude by approximately 50%. Observing Fig. 8(b), a negative moment is initially imparted to the plate due to the down-wash of the vortex. Following this initial CW motion, the plate rebounds as the low pressure in the vortex core begins to pass over it, which imparts a positive moment to the plate, inciting a CCW plate rotation. Interestingly, the progression of the loading as the vortex ring propagates over the plate predicted by the potential flow model follows the experimental measurements until the vortex reaches the mid-point of the plate (t* = 1.92). At later times, the magnitude of the moment predicted by the potential flow model decreases, while the moment arising from the experimental loading remains relatively constant. The increased moment observed in the experimental investigations arises due to the decreased spacing between the vortex core and the plate, as previously discussed. In addition, viscosity diffuses the induced vorticity from the boundary into the flow field causing a pronounced decrease in the advection velocity as the two opposite-sign vorticity regions interact. This plays a role in producing a larger moment on the plate by increasing the effective moment arm (relative to instances of higher advection velocity) at which the pressure acts.

FIG. 8.

(a) Non-dimensional deflection and (b) non-dimensional moment for: potential flow model results () and experimental results (red square) at a distance ratio of H * / a * = 1.64 . Vertical dashed lines in (a) indicate the non-dimensional time during which the vortex ring propagates over the plate.

FIG. 8.

(a) Non-dimensional deflection and (b) non-dimensional moment for: potential flow model results () and experimental results (red square) at a distance ratio of H * / a * = 1.64 . Vertical dashed lines in (a) indicate the non-dimensional time during which the vortex ring propagates over the plate.

Close modal

To quantify the trajectory, the locations of the vortex ring cores in the experimental investigations are determined by computing the Q-criterion54 with a sub-pixel interpolation scheme55 to better locate the peak value. The experimental trajectory of the lower core (the closest core to the plate) is plotted in Fig. 9 as closed symbols for all of the distance ratios of interest. The ratio h l * / a * is introduced as the non-dimensional distance from the resting position of the end of the plate to the center of the lower vortex core. The trajectory of the lower core predicted by the potential flow model is also plotted in Fig. 9 as a solid line, with open symbols used to denote the same instances in time at which the experimental data were acquired. The symbol sizes are specified to be equal to the experimental positional uncertainty (approximately 0.50%).

FIG. 9.

Trajectories of the lower core of the vortex ring for the potential flow model (——) and the experimental investigations at initial non-dimensional distance ratios of h l * / a * = 2.00 ( H * / a * = 3.00 ) (red square), 1.46 ( H * / a * = 2.46 ) (red triangle), 0.91 ( H * / a * = 1.91 ) (red inverse triangle), 0.64 ( H * / a * = 1.64 ) (red diamond), and 0.37 ( H * / a * = 1.37 ) (red circle). The symbols denote the location of the vortex core at the successive times of t* =1.53, 1.63, 1.72, 1.82, 1.92, 2.02, 2.12, 2.21, 2.31, 2.41, and 2.51, with the open symbols pertaining to the corresponding times in the potential flow model results and the closed symbols pertaining to the experimental results.

FIG. 9.

Trajectories of the lower core of the vortex ring for the potential flow model (——) and the experimental investigations at initial non-dimensional distance ratios of h l * / a * = 2.00 ( H * / a * = 3.00 ) (red square), 1.46 ( H * / a * = 2.46 ) (red triangle), 0.91 ( H * / a * = 1.91 ) (red inverse triangle), 0.64 ( H * / a * = 1.64 ) (red diamond), and 0.37 ( H * / a * = 1.37 ) (red circle). The symbols denote the location of the vortex core at the successive times of t* =1.53, 1.63, 1.72, 1.82, 1.92, 2.02, 2.12, 2.21, 2.31, 2.41, and 2.51, with the open symbols pertaining to the corresponding times in the potential flow model results and the closed symbols pertaining to the experimental results.

Close modal

When the lower core is far from the plate ( h l * / a * = 2.00 and 1.46, corresponding to H * / a * = 3.00 and 2.46), the vortex core follows a path that is nearly parallel to the plate in both the potential flow model and the experimental results. At a vortex core distance of h l * / a * = 0.91 ( H * / a * = 1.91 ), the trajectory in both the experimental and potential flow results skews slightly towards the plate due to the effect of induced vorticity.

For a closer vortex core spacing of h l * / a * = 0.64 ( H * / a * = 1.64 ), the experimental trajectory diverges significantly from the potential flow model as the vortex progresses over the plate, due to circulation loss through vorticity annihilation, as previously discussed. This is clearly observed in Fig. 9, where at a distance of h l * / a * = 0.64 ( H * / a * = 1.64 ), the location of the experimental core in time (denoted by the markers) is significantly delayed relative to the same corresponding time in the potential flow model. This has the added effect of increasing the moment arm at which the pressure acts, thereby causing a further increase in the moment at the base, as is observed in Fig. 8(b), where good agreement is found up to the point where the trajectories diverge. The increased moment loading acts to reinforce the plate oscillations, imparting additional energy to it. For this reason, there is a significant increase in the experimental oscillation amplitude as well, as observed in Fig. 8(a).

In Fig. 9, at h l * / a * = 0.37 ( H * / a * = 1.37 ), the interaction predicted by the potential flow model between the vortex ring and the plate produces non-physical behavior. Therefore, for the parameters investigated herein, H * / a * = 1.91 is the limit of applicability of the potential flow model. At a distance ratio of H * / a * = 1.37 , primary and secondary regions of induced vorticity along the plate are diffused into the flow field due to viscosity, producing more complex vortex dynamics that lead to breakdown of the vortex ring. This distance ratio corresponds to the condition where the plate surface is initially tangential to the outer radius of the lower core of the vortex ring. The vorticity and pressure contours for this case are shown in Fig. 10. The magnified views of the vorticity contours are also plotted in Fig. 11, with the domains of the magnified regions of interest identified by dashed boxes in the corresponding plots of Fig. 10.

FIG. 10.

Derived values from PIV data at a distance ratio of H * / a * = 1.37 at times (a) t * = 1.53 , (b) t * = 1.72 , (c) t * = 1.92 , (d) t * = 2.12 , (e) t * = 2.31 , and (f) t * = 2.51 . Top row - vorticity contours, middle row - pressure contours, and bottom row - resultant pressure distribution along the plate. Dashed boxes indicate the corresponding domains plotted in Fig. 11.

FIG. 10.

Derived values from PIV data at a distance ratio of H * / a * = 1.37 at times (a) t * = 1.53 , (b) t * = 1.72 , (c) t * = 1.92 , (d) t * = 2.12 , (e) t * = 2.31 , and (f) t * = 2.51 . Top row - vorticity contours, middle row - pressure contours, and bottom row - resultant pressure distribution along the plate. Dashed boxes indicate the corresponding domains plotted in Fig. 11.

Close modal
FIG. 11.

Magnified view of the domains identified by the dashed boxes in Fig. 10 displaying vorticity contours for a vortex ring at a distance ratio of H * / a * = 1.37 from the plate at times t* = (a) 1.53, (b) 1.72, (c) 1.92, (d) 2.12, (e) 2.31, and (f) 2.51.

FIG. 11.

Magnified view of the domains identified by the dashed boxes in Fig. 10 displaying vorticity contours for a vortex ring at a distance ratio of H * / a * = 1.37 from the plate at times t* = (a) 1.53, (b) 1.72, (c) 1.92, (d) 2.12, (e) 2.31, and (f) 2.51.

Close modal

At this distance ratio, the CCW (ii) and CW (iii) tip vortices [Figs. 11(a) and 11(b)] are stronger and detach more rapidly than at a distance ratio of H * / a * = 1.64 [see Figs. 7(a) and 7(b)]. The region of positive induced vorticity (iv) under the ring is also significantly stronger in comparison with the greater distance ratio. Due to the tangential location of the CW lower vortex core (i) with the plate, it is quickly drawn toward the surface by the action of induced vorticity, as seen in Fig. 11(c). The induced positive vorticity (iv) on the plate becomes trapped downstream of the CW rotation of the lower core (i) and upstream of the CW tip vortex (iii) [see Fig. 11(c)]. This causes the induced vorticity to detach from the plate surface. Separation of the primary positive induced vorticity (iv) from the plate surface produces a secondary region of negative induced vorticity (v), between the separated primary vorticity and the plate, as shown in Fig. 11(c). At this time, the lower core (i) of the vortex ring is deformed due to interactions with the plate and the induced vortex structures (iv) and (v) and begins to break down. In Fig. 11(d), the positive induced vorticity, (iv), that was separated from the plate has dissipated due to interactions with the negative (CW) vorticity of the core (i) and the secondary negative vorticity (v). The merging of the regions (i) and (v) [Fig. 11(d)] produces an asymmetric distribution of vorticity that rapidly breaks down and ultimately creates a broad region of CCW vorticity (vi) that spans the majority of the plate length [see Figs. 11(e) and 11(f)]. During these interactions, several smaller and weaker regions of vorticity appear to be generated and then quickly dissipated. By t* = 2.51 [Fig. 11(f)], the vortex ring is completely annihilated.

Pressure contours and distributions in Fig. 10 show that a large pressure load initially acts on the plate tip [Figs. 10(a)–10(c)] due to the vortex core being near the plate surface and subsequently being pulled even closer due to the opposite-sign induced vorticity. Also, as shown in Fig. 9, the diffusion of the induced vorticity into the flow field influences the vortex dynamics by retarding the advection of the lower portion of the vortex ring, producing a longer relative moment arm with which the pressure acts on the plate. After this initial loading, the vortex quickly dissipates [Figs. 10(e) and 10(f)] and the pressure loading becomes negligible.

The time history of the plate tip deflection and the corresponding moment at H * / a * = 1.37 are shown in Fig. 12. Note that the moment scales on this figure are increased relative to the corresponding plots in Figs. 5 and 8. The potential flow model solution is not included because of the non-physical vortex motion predicted by it. The net moment is sharply increased in comparison to the larger distance ratios because the pressure loading is higher and acts for an extended period of time at a greater moment arm. This is observed in Fig. 9 where the location of the vortex core h l * / a * = 0.37 ( H * / a * = 1.37 ) is delayed relative to larger distance ratios at the same instant in time. After the vortex ring passes the mid-point of the plate, it quickly breaks down due to opposite-sign vorticity annihilation and is dissipated. However, because of the significantly higher moment that was achieved at the tip of the plate, it still produces an amplitude of oscillation that is almost double that at a distance ratio of H * / a * = 1.64 , and the maximum moment experienced by the plate is almost an order of magnitude larger. The maximum deflection amplitude at H * / a * = 1.37 is approximately 3% of the plate length and is the maximum observed plate deflection for all of the distance ratios studied.

FIG. 12.

(a) Non-dimensional deflection and (b) non-dimensional moment for the experimental results (red square) at a distance ratio of H * / a * = 1.37 . Vertical dashed lines in (a) indicate the non-dimensional time during which the vortex ring propagates over the plate.

FIG. 12.

(a) Non-dimensional deflection and (b) non-dimensional moment for the experimental results (red square) at a distance ratio of H * / a * = 1.37 . Vertical dashed lines in (a) indicate the non-dimensional time during which the vortex ring propagates over the plate.

Close modal

For distance ratios less than H * / a * = 1.37 (not shown here for brevity), the plate passes through the center of the vortex ring. This causes a decrease in the pressure loading on the plate due to the opposing load that develops on opposite sides of the plate and the rapid break down of the vortex ring. When H * / a * = 0 , the plate exactly bisects the vortex ring so that uniform loading is applied on both sides of the plate by the two cores, and the net pressure load is zero.

The impact of the aforementioned vortex ring-plate physics on energy harvesting can be determined by computing the strain energy produced in the cantilevered plate from the interaction. The stored strain energy E s is calculated as31 

E s = K W 2 L 0 1 ( 2 δ * x * 2 ) 2 d x * .
(10)

Because only the tip deflection is known from the laser displacement sensor, it is assumed that the spatial distribution of deflection is well approximated by the first mode of vibration,56 as observed herein. The percent of energy transferred to the plate can be estimated by assuming the vortex ring has a thin core, such that

E 0 = 1 4 ρ Γ 2 a [ ln ( 8 a b ) 2 ] .
(11)

This thin core assumption is admittedly strained in the present case, where b/a = 0.37; nevertheless, because the initial ring energy is constant for all of the experiments, it provides an acceptable reference for determining relative changes in energy exchange.

The maximum non-dimensional strain energy ( E s , max * = max { E s } / E 0 ) was computed for each distance ratio and is shown in Fig. 13. As mentioned in Secs. V A and V C, there are two types of interactions that govern the physics of this problem. The first arises due to the three-dimensionality of the flow, influencing both the pressure loading and the added mass that is displaced by the plate motion. As previously discussed, this can be easily accounted for by applying the corrections proposed in Eqs. (6)–(8). The impact on the plate oscillations is highlighted in Fig. 13 where the experimental measures are denoted by red markers, the long dashed black line indicates the uncorrected potential flow model results, the solid black line indicates the potential flow model results that include the correction for three-dimensional plate effects, and the short dashed gray lines are an empirical fit to the experimental results.

FIG. 13.

Maximum non-dimensional strain energy for: uncorrected potential flow model results (– –), corrected potential flow model results (), experimental results (red square), and empirical fits to the experimental results (- -).

FIG. 13.

Maximum non-dimensional strain energy for: uncorrected potential flow model results (– –), corrected potential flow model results (), experimental results (red square), and empirical fits to the experimental results (- -).

Close modal

It is clear from Fig. 13 that the three-dimensional plate effects are significant in this study, primarily arising from the relatively low aspect ratio of the plate. Three-dimensional effects may not play as important of a role in higher aspect ratio investigations, a situation that merits further investigation, though torsional modes may appear. For the corrected potential flow model, as has been previously discussed, very good agreement is found with the experimental measures at non-dimensional distance ratios that correspond to the vortex ring being in the far-field.

The impact of the vortex dynamics on the plate oscillation is elucidated by comparing the corrected strain energy predicted by the inviscid potential flow model, with the experimental measurements (see Fig. 13). As the distance ratio decreases below H * / a * = 1.91 , alteration of the flow field due to opposite-sign vorticity annihilation between the vortex ring and the induced plate vorticity becomes appreciable, and as previously mentioned, the potential flow solution severely under-predicts this behavior and consequently the strain energy. This, therefore, establishes the lower bound for which viscous diffusion can be neglected under the current parameter set.

For distance ratios less than H * / a * = 1.64 , the potential flow solution is unable to provide a physically realistic solution, as observed in Fig. 9. The marked increase in strain energy observed in the experimental investigations for these distance ratios arises from the skewing of the vortex core towards the plate and the concomitant retardation of the advection velocity that increases the moment arm of the load. Both of these phenomena occur due to opposite-sign vorticity interactions between the ring and the induced vorticity on the plate. It is interesting to note that the maximum strain energy occurs when the lower core of the vortex ring is aligned with the plate ( H * / a * = 1.37 ). As discussed earlier, even though this case leads to rapid break down and dissipation of the vortex ring, the increased loading at the plate tip offsets the loss in loading at later instances in time. The increase in stored energy during vortex breakdown was also observed by Peterson and Porfiri26 for collisions where the vortex ring trajectory is oriented normal to the plate surface.

As discussed in Sec. V D, for distance ratios of H * / a * < 1.37 , the plate passes through the vortex ring causing a marked decrease in strain energy due to the distribution of the pressure loading across both sides of the plate. At H * / a * = 0.00 , the plate is positioned at the centerline of the vortex ring and consequently there is no net load that acts on the plate.

The maximum strain energy captured by the plate in this study is < 1% of the total kinetic energy stored in the vortex ring, with the remaining energy either passing over the plate or being annihilated by opposite-sign interactions. This value is reasonable when compared to the achievable energy transfer of 0.5%–1.5% reported by Peterson and Porfiri26 for a vortex ring impacting a vertical plate and the maximum transferred energy of 5%, as reported by Zivkov et al.,31 for a vortex dipole impacting a vertical plate. This indicates that even in sub-optimal conditions, where the plate is not immersed in the wake of a bluff body, a significant amount of energy can be harnessed from vortical structures. Based on the results of Fig. 13, the captured energy would be minimal if the plate bisects the vortex ring, or is very far from it, but is maximized when the vortex core is tangent to the plate surface.

As a first-order estimation of the amount of energy transferred to the plate, an empirical correlation is provided for the non-dimensional stored strain energy, based on the experimental observations shown in Fig. 13. The non-dimensional stored energy as a function of the vortex ring distance ratio can be estimated as two, piece-wise functions corresponding to the regions where the vortex core is above the plate ( H * / a * > 1.37 ) and where it is bisected by the plate ( H * / a * < 1.37 ). This is given as

E s , max * = { 0.0041 ( H * a * ) 2.23 0 H * a * 1.37 0.0626 ( H * a * ) 6.37 1.37 H * a * 3.00 .
(12)

Although providing a general correlation requires a comprehensive study of the plate vibrations under a wide range of non-dimensional plate and vortex ring parameters, the provided correlation serves as a first estimation of the behavior, which could be used for comparison in future studies that utilize a similar vortex ring/plate orientation.

Results at only one plate aspect ratio have been reported herein. As previously discussed, it should be noted that corrections to the plate dynamics as a function of aspect ratio only consider first-order effects. The influence of the aspect ratio is, however, more complex, arising from the fact that the current experiments utilize a constant vortex ring diameter (2a =36.7 mm). It can be shown that higher plate aspect ratios decrease edge effects, but also decrease loading per unit width due to vortex ring curvature, while lower aspect ratios have the opposite effect. This trend is illustrated in Fig. 14, where experimental measures of the maximum plate deflection are presented at a distance ratio of H * / a * = 1.64 as a function of plate width to vortex ring diameter (W/2a), for a fixed plate length (L = 100.0 mm). As W increases, the plate aspect ratio increases and pressure losses due to edge effects decrease. For sufficiently low values of W, the constant diameter vortex ring is large relative to the plate width, and therefore, the loading per unit width is only minimally influenced by the increasing plate width. As a result, the maximum plate amplitude increases with increasing W. However, a critical value arises at approximately W/2a = 0.7 where the decrease in loading per unit width due to the increasing ratio of the plate width to vortex ring diameter becomes greater than the suppression of pressure losses due to edge effects. This manifests as a subsequent decrease in the maximum plate deflection. To maximize the plate displacements in the current experiments, a value of W/2a = 0.683 was utilized (see Table II).

FIG. 14.

Maximum plate tip deflection at a distance ratio of H * / a * = 1.64 as a function of the ratio of plate width to vortex ring diameter ( W / 2 a ).

FIG. 14.

Maximum plate tip deflection at a distance ratio of H * / a * = 1.64 as a function of the ratio of plate width to vortex ring diameter ( W / 2 a ).

Close modal

The dependence of the plate displacement on the aspect ratio also influences the energy exchange process as a lower amplitude of plate oscillation causes an increase in the relative distance between the vortex core and the surface of the plate. Investigations at different plate aspect ratios (not shown here for brevity) were found to exhibit the same physical interactions.

Vortex dynamics arising from a vortex ring passing tangentially over a cantilevered plate were investigated at different distance ratios between the plate surface and the ring centerline. Experimental results were compared with a two-dimensional potential flow model to highlight how vorticity diffusion, which produces opposite-sign vortex interactions in the flow field, significantly alters the vortex dynamics and the resultant beam dynamics. Regions of interaction were identified as a function of distance ratio.

Three-dimensionality of the plate was shown to have a significant impact on the plate dynamics due to the formation of edge vortices during oscillation. In comparison with a two-dimensional potential flow model, edge vortices decreased the pressure loading due to losses that arise from the edge effects and decreased the added mass of the fluid displaced by the plate motion. Simple corrections to the two-dimensional potential flow model were proposed to account for the three-dimensionality.

At large distance ratios, the induced vibration of the plate was unaffected by vorticity interactions and the vortex ring followed a nearly straight trajectory with a nearly uniform load applied to the plate. For these far field investigations, when corrected for three-dimensionality, the potential flow model provided very good agreement with the experimental results. As the vortex ring approached the plate, the induced vorticity on the plate surface, which was well predicted by the potential flow model, led to significant alteration of the vortex dynamics. Fluid viscosity led to diffusion of the induced vorticity into the flow field, and the subsequent opposite-sign interaction with the vortex ring caused the vortex ring to skew towards the plate surface and the resultant pressure loading to increase. Retardation of the advection velocity of the lower core of the vortex ring due to opposite-sign vorticity interactions also caused the moment arm at which the pressure loading acted to be larger. Hence, the moment, plate deflection, and maximum strain energy, were greatly increased as the distance ratio was decreased. Although the potential flow model was able to predict skewing of the vortex ring, it was unable to capture diffusion of the induced vorticity along the plate into the flow field and the resultant opposite-sign vorticity interactions that dominated the vortex dynamics for distance ratios of H * / a * 1.64 . At a distance ratio corresponding to where the lower core of the vortex ring was tangent to the plate ( H * / a * = 1.37 ), diffusion of the boundary vorticity gave rise to annihilation of the vortex ring due to opposite-sign vorticity interactions. In addition, primary and secondary regions of vorticity production and annihilation were observed during breakdown of the vortex ring.

Computing the strain energy stored in the plate showed that the distance ratio of H * / a * = 1.37 produced the highest observed deflection amplitude and highest extracted energy from the flow. An empirical correlation was provided for the observed maximum strain energy of the plate, which could be used for the prediction of energy capture in scenarios with similar parameters and orientation. The correlation can also be used as a correction to improve the predictive/optimization capabilities of potential flow models.

Future studies are planned to encompass a broader range of non-dimensional parameters for both the plate and vortex properties to investigate the entire domain of interest in vortex-plate interactions, hopefully yielding a more general empirical correlation for predicting energy extraction in this flow scenario. Also of interest is the passing of a periodic train of vortices over the plate at varying fundamental frequencies relative to the plate natural frequency.

This material is based upon work supported by the National Science Foundation (NSF) under Grant No. CBET-1511761 and the Natural Sciences and Engineering Research Council of Canada (NSERC), under Grant No. 05778-2015.

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