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Abstract

Flow-induced vibrations have proven to be a valuable power source for wireless sensors and MEMS devices. In this study, the energy harvesting aspects of an open cavity flow is explored and modelled as a single-degree-of-freedom system. Experiments with flow over a cavity of length-to-depth ratio of 3, with an incoming velocity of 30 m/s was taken as a case study to validate the model. It was observed in experiments that three distinct flow oscillation frequencies of significant amplitudes, corresponding to the different modes of oscillation were generated. To harvest this oscillatory power, a piezoelectric cantilever beam, mounted on the aft wall was subjected to unsteady pressure forces. To harvest efficiently, the natural frequency of the cantilever beam was tuned to lock in with the flow oscillation frequencies. The frequencies of cavity oscillations agreed well with Rossiter’s model in the experiments. A single-degree-of-freedom system assuming lumped parameters was used for modelling. The forcing term was described using a periodic function that uses empirical values for amplitude. The model was able to predict the overall trend and values of the average power and RMS voltage generated across a resistor load with reasonable accuracy. Using the model, the effects of cavity length, incoming velocity and resonant frequency were also studied.

Keywords

Cavity flow oscillations energy harvesting piezoelectricity

1. Introduction

The academic and research interest in energy harvesting has increased tremendously over the last decade– a fact that can be evidenced from the number of publications released that is pertinent to this topic. While 412 research articles were published with energy harvesting as a keyword in the year 2005, the number rose to 7129 in 2021, according to the Scopus database. The research spans exploration of new sources, techniques and methods for energy harvesting as well as simulating and modelling them. Of these, energy harvesting from fluid flow-induced vibrations has attracted a spotlight owing to the abundance and more common occurrence of fluid flow in the form of rivers, wind and oceanic currents.

The main types of motion used for harvesting energy from fluid flow can be classified as flutter, buffeting, galloping and vortex-induced vibrations. In flutter-based energy harvesting, self-excited motions occur when the structure’s damping forces become inadequate to counter the aerodynamic forces and several non-linear models have been used to describe such energy harvesters (Abdelkefi, 2016). In transverse galloping, a bluff body attached to a cantilever beam and exposed to a wind starts oscillating with large amplitudes when the wind speed exceeds a critical value. Several lumped parameter models have been used to describe the energy harvesting characteristics of galloping-based energy harvesting. Zhao et al. (2013) compared three analytical models viz. the lumped parameter single-degree-of-freedom (SDOF) model, single-mode and multi-mode distributed model with Euler-beam representation. Their results showed that all three models were successful in predicting the average power generated by the energy harvester as well as their variation with wind speed and load resistance. The distributed parameter model had a more accurate representation of the aerodynamic force while the SDOF model was able to predict the cut-in wind speed better. They concluded that given the simplicity and ease, the SDOF model faired better in the prediction of the electro-aeroelastic behaviour of the galloping piezoelectric energy harvester. For the forcing term in the SDOF model, they used a polynomial expression of the angle of attack as $C_{F} = \sum_{r = 1, 2 . . .} A_{r} (α)^{r}$ where $C_{F}$ is the aerodynamic force coefficient and $A_{r}$ denotes empirical coefficients obtained using best fit. Sirohi and Mahadik (2011) used piezoelectric sheets to utilise the galloping oscillations created by a triangular prism to an incoming flow of 11.6 mph. They used an electromechanical analytical model for the system, developed using an energy-based variational formulation. The model was validated using experiments carried out in wind tunnel and the effect of wind velocity and load resistance on power generated was studied. The forcing term was modelled by calculating the lift and drag coefficients in wind tunnel tests and assuming a quasi-static angle of attack across the prism. The model, however, showed some discrepancy when compared with the experimental results.

Another category of energy harvesters use vortex-induced vibrations (VIV) wherein the oscillator, typically a piezoelectric beam is placed in the wake region of a body like a cylinder, where periodic vortex shedding takes place. Akaydin et al. (2010) studied the power generated when a PVDF piezoelectric beam was placed in the wake region of a cylinder. The effect of resonance between the vortex shedding frequency and the piezoelectric beam was found to be significant in their studies. They also studied the distribution of voltage amplitudes in an open circuit condition and found that the probability distribution function of the amplitude obeyed a Rayleigh distribution. An SDOF model was proposed for the energy harvesting system, although no model for the forcing term was suggested in the work. Goushcha et al. (2015) studied the prospects of using a turbulent boundary layer for energy harvesting purposes. Along with experimental results, they also proposed a distributed parameter model based on the Euler-Bernoulli beam theory. Their model also incorporated the effects of various turbulent features as well as structural and electrical properties of the beam. Allen and Smits (2001) studied energy harvesting capabilities of a piezoelectric eel placed in the wake region of a bluff body. They used a simplified description of the oscillating system as an Euler-Bernoulli beam and also described a condition for lock-in to occur.

Due to their potential to power sensors using clean renewable energy, energy harvesting from fluid flows has triggered a re-examination of the previously undesirable structural response to fluid forces such as flutter, galloping, vortex-induced vibrations and buffeting. The academic community now views them with a hope to harness these undesirable and at times disastrous motions of the structure and transform them to useable sources of power. A most recent addition to this list is the self-sustained flow oscillations that are generated in cut-outs in aerodynamic surfaces, commonly called cavity flows. Since the exploratory study of Krishnamurty (1955), flow over rectangular cavities have been well-researched and analysed over the following decades. Rossiter (1964) was one of the first researchers to propose an empirical model to predict the frequency of flow oscillations in a cavity. This model is based on the assumption that vortical disturbances triggered from the cavity leading-edge travel downstream and interact with the aft wall. This interaction creates pressure disturbances that travel in the upstream direction to trigger further vortical disturbances at the leading edge – thus completing a feedback cycle. The model is fairly accurate and has been proven to apply to a wide range of flow conditions including supersonic inflows (Bauer and Dix, 1991; Saddington et al., 2016; Thangamani, 2019; Vikramaditya and Kurian, 2009a). However, that is not the case with predicting the amplitudes of the flow oscillations. There is not a widely accepted and applicable analytical model for predicting the amplitude which creates difficulty in describing the forcing term for energy harvesting. Bilanin and Covert (1973) proposed an analytical model wherein the shear layer was assumed as a thin vortex sheet and by using a line pressure force at the leading edge. This model, though matched well at higher supersonic Mach numbers did not show the same agreement at other regimes. Tam and Block (1978) improved the model by incorporating the effects of a finite shear layer and treated the shear layer interactions more rigorously. However, the results had limited success in a narrow regime.

Thangamani and Kok (2022) attempted to harvest energy from the flow oscillations emanating inside a cavity exposed to a low subsonic flow, using a piezoelectric cantilever beam attached perpendicularly to the cavity’s aft wall. The study used experimental and computational means to explore the potential of utilising cavity flows as a source of micro-energy harvesting. This work proposes an SDOF model for predicting the average power generated by the piezoelectric beam by making some assumptions regarding the flow. Unlike some of the models mentioned before, the forcing term for this scenario is challenging since the piezoelectric beam is exposed to a fluctuating incident shear layer. A periodic forcing term based on empirical values have been used and the results obtained has been validated and compared against the experimental values.

2. Experimental details

The key experimental objective was to design, construct and test a small scale cavity model that is capable of generating self-sustaining flow oscillations and utilise this for harvesting micro-wattage power. The experimental results obtained will be used to validate the proposed SDOF model. A test section, made of acrylic, embedded with a cavity of dimensions L × D × W = 150 mm × 50 mm × 50 mm was fabricated. This will yield a length-to-depth (L/D) and length-to-width (L/W) ratio of 3 (see Figure 1). An L/D ratio in the range 2–7 is said to exhibit open flow cavity whereupon the shear layer separating from the leading edge spans across and reaches towards the aft wall without touching the cavity floor (Tracy and Plentovich, 1993). Such open flow cavities exhibit strong oscillations (Ahuja and Mendoza, 1995) and a tuft flow visualisation for this cavity demonstrated the presence of flow oscillations by means of vigorous vibration of the tuft. The test section made was 310 mm long and was attached to the outlet of a converging section of a small scale wind tunnel called AF10, manufactured by TecQuipment^® (Figure 2). The inlet velocity of the test section was measured to be 30 m/s and the boundary layer thickness based on 99% freestream velocity, measured at the leading edge of the cavity was found to be $δ_{0.99} = 1.37$ mm. The displacement thickness, momentum thickness and shape factor were found to be 0.26 mm, 0.13 mm and 2 respectively. The blockage to the flow caused by the boundary layer at the leading edge transverse plane, estimated as $(δ^{*} P_{LE}) / A_{LE}$ , was found to be 1.56%. $P_{LE}$ and $A_{LE}$ denote the area of cross-section and perimeter of the test section at the leading edge transverse plane.

Figure 1.

Test Section with piezoelectric beam located at aft wall. Numbered components show the (1) front wall, (2) cavity floor, (3) aft wall, (4) piezoelectric beam.

Figure 2.

Test section assembly attached to flow bench (flow direction is vertically downwards). Numbered components show the (1) front wall, (2) front wall microphone, (3) cavity, (4) aft wall microphone, (5) rectangular test section with embedded cavity, (6) piezoelectric beam, (7) settling chamber and (8) converging nozzle.

For energy harvesting, a single-layered PZT-5H piezoelectric beam (Piezoelectric Beam abbreviated as PB hereupon), manufactured by Midé was used. The PB weighs 2 g and is suitable for harnessing energy at resonant frequencies up to 660 Hz. Some of the key properties of the PB are given in Table 1.

Table 1.

Properties of the piezoelectric material used in the study.

Property	Symbol	Value
Piezoelectric-charge coefficients	$d_{31}$ $d_{33}$	−320 × $10^{- 12}$ C/N 650 × $10^{- 12}$ C/N
Piezoelectric-voltage coefficients	$g_{31}$ $g_{33}$	−9.5 × $10^{- 3}$ Vm/N 19 × $10^{- 3}$ Vm/N
Coupling coefficients	$k_{31}$ $k_{33}$	0.43 0.75
Density	$ρ_{b}$	7870 kg/ $m^{3}$
Young’s modulus	$Y_{11}$ $Y_{33}$	7 × $10^{10}$ N/ $m^{2}$ 11 × $10^{10}$ N/ $m^{2}$

The PB was installed perpendicularly to the aft wall at a depth of 0.22D from the trailing edge (Figure 3). Results obtained from CFD studies showed this to be a region of very high levels of unsteady pressure and hence this location was chosen (Thangamani and Kok, 2022). The resonant frequency of the PB can be changed by changing the clamping length and/or by adding masses to the tip. The fixtures required for holding the PB was 3D printed. The overall dimensions of the PB are $l_{b} \times w_{b} \times t_{b}$ = 53 mm × 20.8 mm × 0.71 mm.

Figure 3.

Detailed view of piezoelectric beam installed at aft wall.

The frequency of cavity flow oscillations was measured using pre-polarised 130F21 microphones manufactured by PCB. The 130F21 microphone is an Integrated Circuit Piezoelectric (ICP^®) model with a sensitivity of 45 mV/Pa and has a built-in preamplifier. Three microphones of diameter 1/4″ each were flush-mounted on the cavity front wall, floor and aft wall respectively. Their centre locations are (x/L, y/D, z/W) = (0, −0.42, 0), (0.5, −1, 0) and (1, −0.42, 0) respectively. The axis directions are shown in Figure 1 and the origin is kept at the centre of the leading edge.

The tip displacement of the PB in free oscillations was measured using a Keyence manufactured IL-100 high-precision laser sensor that was used along with an IL-1000 amplifier unit. The IL-100 is a CMOS analogue laser sensor that has a displacement measurement range of 55 mm (75 mm–130 mm from the laser head). The system has a repeatability of 4 $μ m$ and linearity of ±0.15% of F.S. To measure the displacement, the laser head of the IL-100 was kept perpendicular to a PB mounted on an anti-vibration platform (Figure 4). The IL-100 outputs 0–5 V analogue signals which were acquired at a rate of 2048 Hz. The IL-1000 also has an LCD which shows the real-time displacement measured by the laser head.

Figure 4.

Keyence IL-100 set up to measure tip displacements.

Data acquisition was done using a National Instruments data acquisition module NI USB-4431 which has a 24-bit resolution and can acquire signals at a sampling rate of up to 102.4 kHz. For transient pressure acquisition, 327,680 samples were acquired at a sampling speed of 32,768 Hz for a measurement time of 10 s. Welch’s method was used to plot power spectral densities in order to identify the various frequencies of interest. For this, the sample was divided to 8192-point segments with a 50% overlap using a Hamming window, yielding a frequency resolution of 4 Hz. To obtain the final power spectral density plots used in this work, the spectrum obtained from 10 different tunnel runs were averaged to obtain even more accuracy. Over the different tunnel runs, no change in dominant frequencies was noticed. The maximum standard uncertainty in amplitude for any of the first three cavity modes, $ε_{0.95}$ , calculated using equation (1) was ±0.71 Pa. The error was calculated for a 95% confidence interval and $σ$ and N are the standard deviation of the tonal amplitude and number of tunnel runs respectively.

ϵ_{0.95} = \pm 1.96 \frac{σ}{\sqrt{N}}

(1)

3. Results and discussion

In this section, the unsteady pressure field of the baseline cavity is discussed first, followed by the cavity with the PB. The details of the energy harvested are also provided here. In the subsequent section, a simplified mathematical model based on single-degree-of-freedom (SDOF) assumption is described and compared with the results obtained in the experiments.

3.1. Baseline cavity

The basic mechanism responsible for self-sustaining oscillations observed in a cavity flow has been well-established in several works (Heller and Bliss, 1975; Rossiter, 1964; Thangamani and Kurian, 2013; Tracy and Plentovich, 1993; Vikramaditya and Kurian, 2009b). Rossiter (1964) was the first to propose a feedback model that was improved by subsequent researchers. According to this fluid-acoustic model, vortical disturbances created at the leading edge convect downstream with a velocity $U_{c}$ and meet the aft wall. The exact nature of the disturbance has been however debated. Heller and Bliss (1975) viewed this as an up-and-down flapping motion of the shear layer bridging the cavity. The uncertainty over this disturbance poses a key challenge in the mathematical modelling of cavity flows. In the context of this work, this leads to difficulty in precisely modelling the external forcing term of the oscillations. The interaction between the vortical disturbance and the aft wall results in the generation of a pressure disturbance that travels in the upstream direction. This pressure disturbance that travels with the speed of sound moves faster within the cavity owing to the lower velocities there. When this curved pressure wavefront interacts with the leading edge, it provokes further disturbances near the separating shear layer. This disturbance again convects in the downstream direction to continue the cycle. With such a physical feedback model, the time period of the cycle can be estimated by using a simple formulation, by accounting for the different fluid-acoustic components. The formulation proposed thus by Rossiter is given by equation (2) as

\frac{f_{n} L}{U_{\infty}} = \frac{n - α}{M_{\infty} + \frac{1}{k}}

(2)

where $f_{n}$ is the frequency of the $n^{th}$ mode of oscillation, k is the ratio of vortex convection velocity ( $U_{c}$ ) to freestream velocity ( $U_{\infty}$ ), $α$ is a term to account for the phase lag between the vortex interaction at the aft wall and the creation of an acoustic disturbance. If $ϕ$ is the phase difference between the impinging vortex and emission of a sound disturbance, then $a = \frac{ϕ}{2 π}$ . The term $\frac{fL}{U_{\infty}}$ on the left-hand side is called as Strouhal number. Equation (2) was improved by Heller and Bliss (1975) for compressible flows where they used a correction factor for the speed of sound within the cavity. In equation (2), although there is a physical meaning for $α$ and k, the two are often treated as empirical factors and found by best fit from experimental data. Various values of $α$ and k have been used by researchers over the years; Rossiter used $α$ and k as 0.25 and 0.57 based on his experimental data. For several studies pertaining to low Mach numbers, the value of $α$ has been found to be negligible (Chatellier et al., 2004; Ma et al., 2009; Verdugo et al., 2012). This is probably due to the incompressible nature of the flow which makes the creation of acoustic disturbance almost instantaneous after the vortex interaction with the aft wall. Taking the values of $α = 0$ and k = 0.63 gave the best fit for the current experimental data. Taking $α = 0$ dictates that the frequencies obtained for the baseline cavity from equation (2), for the different modes, would be harmonics given by $f_{n} = n f_{1}$ where $f_{1}$ is the fundamental frequency. Figure 5 shows the power spectral densities of transient data obtained from microphones flush-mounted with the front wall, cavity floor and aft wall respectively.

Figure 5.

Power spectral densities for the baseline cavity (Thangamani and Kok, 2022).

Three prominent peaks at 116, 244 and 352 Hz can be noticed in the plot. It can be seen that the dominant mode of flow oscillation for the baseline cavity is the third mode (n = 3) followed by the second mode. The first mode is feeble and is spread over a wider band. For brevity, the first, second and third modes of oscillations would be denoted as $f_{1}$ , $f_{2}$ and $f_{3}$ respectively, hereafter. The value of the cavity flow oscillations obtained from the experiment as shown in Figure 5 has been compared with that estimated from the Rossiter model and the agreement between the two was found to be good. The maximum percentage difference between the experimental frequencies and the model occurred for $f_{2}$ (2.8%). Figure 6 shows all the experimental Strouhal numbers for n = 1, 2, 3 plotted against the Rossiter model with a ± 3% tolerance band for each mode.

Figure 6.

Comparison of experimental Strouhal numbers with Rossiter’s model ( $α$ = 0, k = 0.63).

3.2. Cavity with piezoelectric beam

For energy harvesting, the PB was installed perpendicularly to the aft wall at a distance of 0.22 D from the trailing edge. This is a region of high-pressure fluctuation and is expected to apply an external forcing for the PB, thus oscillating it to generate electrical energy. Although the introduction of PB into the cavity does not change the outer dimensions of the cavity, it still may have some impact on the flow dynamics within it due to its intrusiveness and the location near the trailing edge which is vital for the feedback cycle. To study the effect of this, the power spectral density at the aft wall when the PB is fitted is compared with the baseline case. It can be seen from Figure 7 that when the PB is located at the aft wall, the spectrum still has high levels of unsteadiness although a slight reduction in mode frequencies occur. Also, the dominant mode of oscillation shifts from $f_{3}$ to $f_{2}$ . The change in frequencies between the baseline case and the cases with PB are shown in Table 2. The reduction in frequencies that occur is due to the beam interfering with the interaction between the vortex and the aft wall. Previously for the baseline case, it was noted that taking the value of $α$ = 0 gave the best fit for the baseline data. However, with the introduction of the PB, the modified frequencies indicated that a value of $α$ = 0.0892 gave the best fit of data to equation (2). As indicated, $α$ indicates a lag between the vortex interaction at the aft wall and the subsequent generation of an acoustic wave. This lag was found to be negligible for the baseline case but with the placement of a PB beam at the aft wall, there is a phase lag given by $ϕ = 0 . 5605^{c}$ . With the new value of $α$ , the maximum deviation of experimental Strouhal numbers from Rossiter’s model is 1.12 % for $f_{2}$ . Figure 8 shows the experimentally measured Strouhal numbers against the Rossiter model estimation with a $\pm 3 %$ tolerance band.

Figure 7.

Comparison of spectra between baseline cavity and cavity with PB at aft wall (Thangamani and Kok, 2022).

Table 2.

Cavity oscillation frequencies and PB tuned frequency corresponding to the different modes.

Mode	$f_{1}$ (Hz)	$f_{2}$ (Hz)	$f_{3}$ (Hz)
Baseline cavity	116	244	352
Cavity with PB	108	224	348
PB tuned frequency	108	228	356

Figure 8.

Comparison between experimental Strouhal numbers for cavity with PB and Rossiter’s model ( $α$ = 0.0892, k = 0.63).

From the discussion above, it is clear that Rossiter’s model, given by equation (2), is well suited to be used for a theoretical model for estimating the frequencies of cavity flow oscillations. The suitable value of $α$ for a given case can be estimated reliably from tests. To maximise the power yielded from the forced oscillations of the PB, it is optimal to match the frequency of the cavity flow oscillation to that of the PB’s natural frequency. This would imply tuning the PB to match closely and lock-in with $f_{2}$ or $f_{3}$ desirably. The natural frequency of the beam was tuned by changing the length of the beam available for oscillation by adjusting the clamping location.

3.3. Single degree-of-freedom (SDOF) model

To mathematically model the system in order to predict the energy harvested by a PB installed in a given cavity flow, few simplifications and assumptions have been made. The amplitude of vibrations is assumed to be small enough to be described as a linear system. A single degree of freedom is assumed for the vibrations of the PB that is essentially fixed like a cantilever on the downstream end to the aft wall. To estimate the frequency of flow oscillations, equation (2) described by Rossiter is used. The real process occurring behind energy harvesting is a complex fluid-mechano-electrical phenomenon that is computationally expensive and complicated.

For the PB installed as a cantilever exposed to external fluid force, the governing equations for the cantilever beam oscillation and harvester circuit are given by

m_{b} \frac{d^{2} y_{b}}{d t^{2}} + c \frac{d y_{b}}{dt} + k y_{b} + θ v = \frac{1}{2} ρ_{\infty} U_{\infty}^{2} w_{b} l_{b} C_{y} (t)

(3)

C_{pz} \frac{dv}{dt} + \frac{v}{R} - θ \frac{d y_{b}}{dt} = 0

(4)

where $m_{b}$ is the effective mass of the PB, $y_{b}$ is the beam’s tip displacement, c and k are damping coefficient and stiffness of the cantilever beam, $θ$ is the electromechanical coupling coefficient of the PB, $v$ is the instantaneous voltage across a resistor load R and $C_{pz}$ is the piezoelectric capacitance. $C_{y} (t)$ describes the force coefficient in the y-direction acting on the PB. Equation (3) describes the SDOF oscillation of the PB while equation (4) is Kirchoff’s law as applied to the piezoelectric energy harvester circuit. Solving equations (3) and (4) yields the solutions $y_{b} (t)$ and $v (t)$ when the PB is subjected to the cavity flow oscillation forcing. Thus if an approximate form for the forcing term in equation (3) is used, then the average power generated by cavity flow oscillations can be estimated.

For solving the system of equations, we can rewrite the equations by defining a state vector $Y$ as

\begin{matrix} Y [\begin{matrix} Y_{1} \\ Y_{2} \\ Y_{3} \end{matrix}] = [\begin{matrix} y_{b} \\ \frac{d y_{b}}{dt} \\ v \end{matrix}] ∴ \\ Y' = [\begin{matrix} Y_{1'} \\ Y_{2'} \\ Y_{3'} \end{matrix}] = [\begin{matrix} Y_{2} \\ \frac{1}{m_{b}} {F_{y} (t) - c Y_{2} - k Y_{1} - θ Y_{3}} \\ (\frac{θ}{C_{p}}) Y_{2} - \frac{1}{RCp} Y_{3} \end{matrix}] \end{matrix}

(5)

The first-order system of equation represented by equation (5) can be solved numerically using Matlab’s ode45 function and the validity of the model is tested for various cases viz. free oscillations with open circuit condition, forced oscillations with open circuit condition and forced oscillations with a resistor load.

Three different settings of the PB was tested corresponding to the three modes of cavity oscillation frequencies. The PB was tuned by either changing the length of the beam or by adding tip mass. The different tuned frequencies tested are shown in Table 2 and the tuned frequencies lie within 8 Hz of the cavity oscillation mode frequency. This will facilitate lock-in conditions with high amplitude oscillations of the PB resulting in better power harvesting. The clamped length of the PB was set at 37 and 34 mm for locking in with $f_{2}$ and $f_{3}$ respectively. To adjust the PB frequency close to $f_{1}$ , a tip mass of 3.4 g was added to the PB, in addition to the condition at $f_{2}$ . The tuned beam frequency is referred to as $f_{b}$ in this work.

3.4. Case 1: Free oscillations with open circuit condition

In the first case, the PB is allowed to undergo free oscillations without external forcing and the voltage output $v (t)$ is tested against experiments. Under the conditions, $R \to \infty$ and $F_{y} (t) = \frac{1}{2} p_{\infty} U_{\infty}^{2} w_{b} l_{b} C_{y} (t) = 0$ , the system of equations become

m_{b} \frac{d^{2} y_{b}}{d t^{2}} + c \frac{d y_{b}}{dt} + k y_{b} + θ v = 0

(6)

v = \frac{θ}{C_{pz}} y_{b}

(7)

Combining equations (6) and (7) we get

\begin{matrix} m_{b} \frac{d^{2} y_{b}}{d t^{2}} + c \frac{d y_{b}}{dt} + (k + \frac{θ^{2}}{C_{pz}}) y_{b} = 0 \end{matrix}

(8)

It can be seen from equation (8) that the stiffness of the PB in this condition has increased as $K_{t} = k + k_{p}$ where $k_{p} = \frac{θ^{2}}{C_{pz}}$ . $k$ and $k_{p}$ represent the mechanical stiffness and the stiffness due to the piezoelectric effect on the PB. The values of the different constants in equation (8) was obtained using experiments and approximations. The value of $C_{pz}$ is taken as 40 nF, specified by the manufacturer of the PB. The effective mass $m_{b}$ of the oscillating beam is estimated as $m_{b} = \frac{33}{140} (ρ_{b} l_{b} w_{b} t_{b}) + m_{tip}$ , where $ρ_{b}$ , $l_{b}$ , $w_{b}$ and $t_{b}$ denote the density, length, width and thickness of the PB respectively. $m_{tip}$ is the tip mass added to the PB. The value of $θ$ is measured from tests. To do this, the PB was tuned to the required frequency mentioned in Table 2 and then released from a 1 mm tip displacement. The PB thus oscillates freely and the tip displacement ( $y_{b}$ ) was measured using a Keyence IL-100 CMOS laser sensor. The corresponding open-circuit (OC) voltage ( $v$ ) is also measured simultaneously as shown in Figures 9 and 10. From the simultaneous $y_{b} (t)$ and $v (t)$ signals, 20 peaks were identified using the findpeaks function of Matlab and $θ$ for each of the 20 sample calculated using equation (7). The final value of $θ$ was estimated by averaging the 20 values as shown in Figure 11. The standard deviation ( $σ$ ) of the 20 readings were found to be 4.9%, 9.8% and 10.1% of the mean value for modes 1, 2 and 3 respectively. The values of $k_{p}$ and $k$ were estimated from experiments as $k_{p} = \frac{θ^{2}}{C_{pz}}$ and $k = (4 π^{2} m_{b} f_{b}^{2} - \frac{θ^{2}}{C_{pz}})$ . The damping ratio $ζ$ was found using the logarithmic decrement, $δ$ , as $ζ = \sqrt{\frac{δ^{2}}{4 π^{2} + δ^{2}}}$ where $δ = \ln (\frac{A_{n}}{A_{n + 1}})$ where $A_{n}$ and $A_{n + 1}$ are the amplitudes of the $n^{th}$ and $(n + 1)^{th}$ peaks obtained from the tests measuring $y_{b} (t)$ . Since the values of $ζ$ obtained were small, the difference between the damped and natural frequency was found to be negligible. From the value of $ζ$ , the damping coefficient c can be estimated as $c = 2 ζ \sqrt{K_{t} m_{b}}$ . The values of the constants used for the OC case are given in Table 3.

Figure 9.

Measured tip displacement variation for ring out test with PB tuned to 108 Hz.

Figure 10.

Measured voltage variation for ring out test with PB tuned to 108 Hz.

Figure 11.

values obtained for various resonant conditions.

Table 3.

Estimation of constants required for equation (8).

Mode	$f_{1}$	$f_{2}$	$f_{3}$
$m_{b}$ (kg)	$4.4 \times 10^{- 3}$	$1.1 \times 10^{- 3}$	$1.0 \times 10^{- 3}$
$ζ$	0.0101	0.0063	0.0066
$θ$ (N/V)	$0.725 \times 10^{- 3}$	$2.2 \times 10^{- 3}$	$12.9 \times 10^{- 3}$

The open-circuit voltage response of the PB obtained by numerically solving the SDOF equation (8) has been compared with the experimental results in Figures 12 to 17 when PB was tuned to the different cavity flow frequencies. It can be noticed that the SDOF model is fairly reasonable in predicting the temporal response of the oscillating beam. The RMS value of the tip displacement for a period of $Δ t = 25 / f_{b}$ differed by 2.9%, 8.1% and 3.5% between the measurement and predicted data for the three modes respectively.

Figure 12.

Comparison of tip displacement variation between experiments and SDOF model ( $f_{b} \approx f_{1}$ ).

Figure 13.

Comparison of voltage variation between experiments and SDOF model ( $f_{b} \approx f_{1}$ ).

Figure 14.

Comparison of tip displacement variation between experiments and SDOF model ( $f_{b} \approx f_{2}$ ).

Figure 15.

Comparison of voltage variation between experiments and SDOF model ( $f_{b} \approx f_{2}$ ).

Figure 16.

Comparison of tip displacement variation between experiments and SDOF model ( $f_{b} \approx f_{3}$ ).

Figure 17.

Comparison of voltage variation between experiments and SDOF model ( $f_{b} \approx f_{3}$ ).

3.5. Case 2: Forced oscillations with open circuit

In forced oscillations with open circuit, the forcing term $F_{y} (t)$ is non-zero as the PB is now placed inside the cavity flow. The governing equation for the system is now given by equation (9), as

\frac{d^{2} v}{d t^{2}} + (\frac{c}{m_{b}}) \frac{dv}{dt} + (k + \frac{θ^{2}}{C_{pz}}) \frac{1}{m_{b}} v = \frac{θ}{m_{b} C_{pz}} F_{y} (t)

(9)

In equation (9), $F_{y} (t)$ is the force acting on the PB in the y-direction. There is no analytical model that can fairly predict the amplitude of oscillations in a cavity flow, in contrast to the prediction of cavity flow frequencies by equation (2). $F_{y} (t)$ is especially difficult to determine owing to the complex nature of the shear layer oscillation coupled with excitement of other frequencies due to impingement of large scale vortical structures. Assuming a general Fourier form for $v (t)$ and $F_{y} (t)$ as $v (t) = \sum_{i = - N}^{N} V_{i} e^{- j ω_{i} t}$ and $F_{y} (t) = \sum_{i = - N}^{N} F_{i} e^{- j ω_{i} t}$ , where N is an integer, and solving equation (9), we get the amplitude of force term for $i^{th}$ harmonic as

F_{i} = V_{i} m_{b} \frac{C_{pz}}{θ} \sqrt{{[ω_{b}^{2} - ω_{n}^{2}]}^{2} + 4 ζ^{2} ω_{b}^{2} ω_{n}^{2}}

(10)

where $ω_{b}$ and $ω_{n}$ represent the tuned beam frequency and the $n^{th}$ cavity flow frequency respectively. As shown in equation (10), the harvester’s electromechanical coupling coefficient ( $θ$ ) and capacitance ( $C_{pz}$ ) are decisive factors in determining the coefficient of $F_{y}$ . The effective mass term, $m_{b}$ , is dependent on the size of the harvester. If a harvester of different size and material is used, the corresponding values for these terms can be plugged into the equation. In the voltage output for the PB with open circuit, the PB responds almost only to the forcing frequencies close to its tuned frequency $f_{b}$ , as can be observed in the frequency response of the PB in Figure 18. In terms of the current situation, this implies that the force can be simplified and represented using three cosine terms as

F_{y} (t) = \sum_{n = 1}^{3} F_{n} \cos [\frac{2 π U_{\infty} (n - α)}{L (\frac{U_{\infty}}{\sqrt{γ R T_{\infty}}} + \frac{1}{k})} t]

(11)

where F_{n} = V_{n} m_{b} \frac{C_{pz}}{θ} \sqrt{{[ω_{b}^{2} - ω_{n}^{2}]}^{2} + 4 ζ^{2} ω_{b}^{2} ω_{n}^{2}}

(12)

Figure 18.

Frequency response of the PB at $f_{b} \approx f_{2}$ .

However, upon closer inspection of the power spectrum of the peak, it was noted that multiple frequencies close to the $f_{b}$ have high amplitudes. In other words, the peak is spread around a small frequency band centring $f_{b}$ . This can be caused due to small fluctuations inside the cavity that vary $f_{n}$ slightly around the theoretical Rossiter frequency. If there was only a clear singular tone, the amplitude of $V_{n}$ , corresponding to $f_{n}$ , could be estimated easily from the experimental measurement of $v (t)$ as

V_{n} = \frac{2}{t_{m}} \int_{0}^{t_{m}} v (t) e^{- j 2 π f_{n} t} dt

(13)

where $v (t)$ is the voltage signal and $t_{m}$ is the measurement time. However, due to the presence of multiple peaks around $f_{b}$ , this method will underestimate the amplitude of $n^{th}$ mode as the other energetic peaks surrounding $f_{n}$ are ignored in the formulation. The matter is only further complicated by the fact that the amplitudes of $V_{n}$ vary with time. A more representative way of calculating the amplitude of the forcing term, $F_{n}$ , would be to reconstruct $v (t)$ which has time-variant amplitude, $V_{n} (t)$ , with an equivalent signal of constant amplitude, $\bar{V_{n}}$ , but the same energy as $v (t)$ . This approximation retains the total energy incoming on the PB and is suitable for determining the average power generated although not the variation of instantaneous power and voltage. The concept is illustrated in Figures 19 and 20 where an artificially created signal with variable amplitude is shown in green colour. The amplitude of this signal varies between zero to a maximum value in t = 1.5 s. The signal can be found to be non-existent at an instant t = 0.31 s. There is an obvious difficulty in modelling such a signal which has variations in amplitude whose reasons are not well-known. The red colour signal shows an equivalent cosine wave of constant amplitude of 0.6128, which has the same average power and energy content as the green one. The frequency of the two waves is essentially the same as can be noticed from the power spectral density in Figure 20. The variation of the different cavity frequency amplitudes is described in detail in the prior work by Thangamani and Kok (2022). Applying this method, the value of $\bar{V_{n}}$ is determined as

\bar{V_{n}} = \sqrt{\frac{\int_{0}^{t_{m}} v {(t)}^{2} dt}{\int_{0}^{t_{m}} {[\cos (2 π f_{n} t)]}^{2} dt}}

(14)

Figure 19.

Comparison of two periodic signals with same average power – one with variable amplitude and another with constant amplitude.

Figure 20.

Frequency spectral density of the two different signals shown in Figure 19.

Thus the representative and empirical value of the coefficient for the force term corresponding to the $n^{th}$ frequency of cavity flow oscillation can be estimated as

\bar{F_{n}} = \bar{V_{n}} m_{b} \frac{C_{pz}}{θ} \sqrt{{[ω_{b}^{2} - ω_{n}^{2}]}^{2} + 4 ζ^{2} ω_{b}^{2} ω_{n}^{2}}

(15)

The values of $\bar{F_{n}}$ corresponding to the different modes of oscillations were thus obtained by tuning $f_{b}$ to the corresponding modes and determining $\bar{V_{n}}$ from the open circuit condition. $\bar{F_{1}}$ , $\bar{F_{2}}$ and $\bar{F_{3}}$ thus obtained were 0.0065, 0.0025 and 0.0008 N and the corresponding force coefficients, $C_{y} = \frac{F_{y}}{0.5 ρ U_{\infty}^{2} l_{b} w_{b}}$ , were $2.29 \times 10^{- 2}$ , $8.7 \times 10^{- 3}$ and $3.0 \times 10^{- 3}$ respectively.

3.6. Case 3: Forced oscillations with resistor load

In this case, the system of equations is solved in its entirety. Following the discussion so far, the complete system to be solved is given as

\begin{matrix} m_{b} \frac{d^{2} y_{b}}{d t^{2}} + c \frac{d y_{b}}{dt} + k y_{b} + θ v = \\ \frac{1}{2} ρ_{\infty} U_{\infty}^{2} w_{b} l_{b} {0.0229 \cos [\frac{2 π U_{\infty} (1 - α)}{L (\frac{U_{\infty}}{\sqrt{γ R T_{\infty}}} + \frac{1}{k})} t] + \\ 0.0087 \cos [\frac{2 π U_{\infty} (2 - α)}{L (\frac{U_{\infty}}{\sqrt{γ R T_{\infty}}} + \frac{1}{k})} t] + \\ 0.0030 \cos [\frac{2 π U_{\infty} (3 - α)}{L (\frac{U_{\infty}}{\sqrt{γ R T_{\infty}}} + \frac{1}{k})} t]} \end{matrix}

(16)

C_{pz} \frac{dv}{dt} + \frac{v}{R} - θ \frac{d y_{b}}{dt} = 0

(17)

Equations (16) and (17) were solved simultaneously using a Matlab programme and the comparison of the average power across different load resistance values obtained from the SDOF model and the experiments is shown in Figure 21. The RMS voltage ( $V_{RMS}$ ) and average power ( $P_{av}$ ) were calculated as

V_{RMS} = \sqrt{\sum_{i = 1}^{N} \frac{v_{i}^{2}}{N}}

(18)

P_{av} = \frac{V_{RMS}^{2}}{R}

(19)

Figure 21.

Comparison of average power across different load resistance values between the SDOF model and experiments.

From Figure 21, it can be noticed that the overall trend and the orders of magnitude obtained from the SDOF model solution are comparable to the experimental values. The same can be said for the $V_{RMS}$ values shown in Figure 22. The average power generated reaches a maximum value when the mechanical oscillation frequency matches the electrical resonant frequency of the circuit. It is to be noted that in the experiments, it is extremely difficult to tune the PB to the exact value of $f_{n}$ , the cavity flow frequencies. Although the frequency is made as close as possible as denoted in Table 2, a small difference, $| f_{n} - f_{b} | > 0$ , exists. There is also uncertainty in the measured frequencies arising due to the finite frequency resolution used in the measurement and spectra. This could lead to some differences between the results from the SDOF model and the experiments since the resonance condition is not strictly observed in the latter. In the SDOF plots shown in Figures 21 and 22, it is assumed that $f_{b} = f_{n}$ . The results show that the SDOF model offers estimates for energy harvested from cavity flow oscillations to a reasonable accuracy. The effect of the tuned beam frequency, $f_{b}$ , on the power generated is shown in Figure 23 for a range of $f_{b} = f_{2} \pm 8$ Hz. As can be seen, the power generation across the load resistance values differ with the deviation from the forcing frequency, $f_{n}$ . The situation in the experiments for second mode is $f_{b} = f_{2} + 4$ Hz as listed in Table 2. For this situation, the result from the model matches very well with the experimental trend for the $P_{av}$ variation. The slight difference in trend could also be caused due to the limited load resistance values chosen for the experiments. Thirteen resistor values in the range 0–500 $k Ω$ were chosen in the experiments while for the numerical simulation, the values were calculated for every 100 $Ω$ increase in the range of 0–500 $k Ω$ . The variation between the different modes of oscillation occurs due to the varied dominance of the different Rossiter modes temporally. It was reported previously by Thangamani and Kok (2022) that the three modes of oscillations seldom co-exist together but rather switch from one to another. Some modes exist for comparatively longer durations and thus have a higher average value of amplitudes for their corresponding forcing terms.

Figure 22.

Comparison of $V_{RMS}$ across different load resistance values between the SDOF model and experiments.

Figure 23.

Effect of tuned beam frequency $f_{b}$ on the power generated.

3.7. Effect of resonance

The effect of resonance is demonstrated in Figures 24 and 25 for the second mode of oscillation. In Figure 24, the natural frequency of the beam, $f_{b}$ , is plotted against the maximum average power obtained in the load resistance range of 0–500 $k Ω$ . Other modes of oscillation also exhibit similar behaviour. It can be seen that when $f_{b} = f_{2} = 228$ Hz (the forcing frequency), the maximum power generated is peaked due to resonance. The maximum value of $V_{RMS}$ generated by the PB also reaches a maximum when the resonance condition is attained, according to the SDOF model. The RMS voltage generated drops by 50% when the frequency is altered by around 4 Hz off the resonant frequency. The $P_{av}$ has a much flatter peak which is beneficial from an energy-harvesting perspective. Up to 50% of the peak power can be generated in a bandwidth of 8 Hz and up to 16% in a bandwidth of 23 Hz. This has practical importance since the frequencies of flow oscillations is directly proportional to the incoming flow velocity and a 10% tolerance (with reference to a resonant frequency of 228 Hz) of the tuned frequency translate to the same tolerance to the incoming flow velocity value, by the energy harvester.

Figure 24.

Variation of maximum average power with change in $f_{b}$ .

Figure 25.

Variation of $V_{RMS}$ with change in $f_{b}$ .

3.8. Effect of incoming velocity

The effect of the incoming velocity on maximum $P_{av}$ generated and $V_{RMS}$ under resonance condition ( $f_{b} = f_{2}$ ) is shown in Figures 26 and 27. The range of velocities chosen is 10–110 m/s; the uppermost value being close to a Mach number of $M_{\infty}$ = 0.3, the widely accepted limit of flow incompressibility. It can be seen that the maximum value of $P_{av}$ increases almost quadratically with the increase in freestream velocity. This is expected since the amplitude of the forcing terms is proportional to $U_{\infty}^{2}$ .

Figure 26.

Variation of maximum average power with change in $U_{\infty}$ with $f_{b} = f_{2}$ .

Figure 27.

Variation of $V_{RMS}$ with change in $U_{\infty}$ with $f_{b} = f_{2}$ .

Since the velocity of the beam tip $(\frac{d y_{b}}{dt})$ is also obtained as part of the solution, the mechanical power and thus the energy conversion efficiency ( $η$ ) of the beam can be calculated as

η = \frac{P_{av}}{0.5 ρ U_{\infty}^{2} w l_{b} {[\frac{d y_{b}}{dt} (t) C_{y} (t)]}_{RMS}}

(20)

Using equation (20), the efficiency, at resonance conditions, for $f_{b} = f_{2}$ for a range of incoming flow velocities is shown in Figure 28. It can be seen that the energy conversion efficiency of the oscillations drops with increase in velocity. This suggests that energy harvesting from higher flow velocities may not be energy-economic.

Figure 28.

Variation of energy conversion efficiency ( $η_{mech}$ ) of PB across a velocity range.

3.9. Effect of cavity scale

The variation of maximum $P_{av}$ and $V_{RMS}$ are shown in Figures 29 and 30 respectively, again assuming resonance conditions for second mode of oscillation. This shows the effect of increasing the cavity scale tenfold. When the cavity length is doubled from 0.15 m, the experimental value, the average power increases by 65%. It can be noticed the power values level off after a certain cavity size. An increase in the cavity scale decreases the cavity flow frequencies as can be inferred from equation (16). Beyond a certain point, they do not affect the amplitude of the oscillations leading to a diminished consequence on the power.

Figure 29.

Variation of maximum average power with change in cavity size with $f_{b} = f_{2}$ .

Figure 30.

Variation of $V_{RMS}$ with change in cavity size with $f_{b} = f_{2}$ .

The power generated from cavity flow oscillations can be increased by either increasing the velocity or by increasing the scale (cavity dimensions). If the incoming flow velocity is lower, the model’s scale can be increased to yield a higher power. As the wind tunnel used in the current studies restricted the usage of cavities larger than the present case, a higher velocity of 30 m/s was chosen to validate the SDOF model. A nominal wind velocity of 15 m/s should also be suitable for energy harvesting, provided the correct scale of cavity is used.

4. Conclusion

A single-degree-of-freedom model has been proposed to determine the average power harvested and RMS voltage generated by a piezoelectric beam placed in an unsteady cavity flow field that is generating self-sustaining flow oscillations. The model has been validated using experiments from a cavity of length-to-depth ratio of 2 and length 0.15 m, exposed to an incoming velocity of 30 m/s. The power spectral density plots of the transient pressure data from the cavity aft wall showed three peak frequencies corresponding to the first three modes of Rossiter oscillations. The value of the frequencies agreed well with Rossiter’s model for estimation of cavity flow frequencies. For the forcing term in the model, an empirical value was used for the different modes of oscillations. The model has been found to be reasonably accurate in determining the levels of power harvested and their trends with various flow and model parameters.

The results show a variation in average power harvested with load resistance with a maximum value corresponding to the achievement of matching between the electrical resonant frequency and mechanical oscillation frequency. The power generation trends varied slightly with the different modes of oscillations as a result of the varied temporal dominance of the different modes. The effect of resonance between the cavity flow frequency and the piezoelectric beam’s natural frequency was found to be significant. Having said this, it was found that the power generation was reasonably tolerant to the change in the tuned frequency of the oscillator with a 50% power retainment within a frequency bandwidth of 11 Hz for the dominant mode of oscillation. The effect of incoming flow velocity was also found to be significant wherein the harvested power increased quadratically with an increase in incoming flow velocity. However, the energy conversion efficiency of the piezoelectric beam showed an opposite trend with an increase in velocity. The cavity size was found to be not as influential over the power generated as the resonant frequency and flow velocity. While doubling the incoming flow velocity from 30 m/s caused an eight-fold increase in power, doubling the cavity size from 0.15 m increased the power by 1.5 times.

The results of the study indicate that a single-degree-of-freedom model can provide reasonable first estimates for a complex fluid-structure-electromechanical phenomenon like the energy harvesting from cavity flow oscillations. Limitations exist in describing the forcing term due to the absence of a reliable model for predicting amplitudes of pressure oscillations in a cavity flow and this gap has to be currently filled with empirical information. The resonant frequency of the piezoelectric oscillator, the incoming flow velocity and the cavity size are found to be key parameters that determine the power generated from the flow oscillations.

Footnotes

Appendix

Acknowledgements

We wish to gratefully acknowledge and thank the Government of Malaysia for supporting this work.

Declaration of conflicting interests

The author declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This research was supported by the Ministry of Higher Education (MoHE), Malaysia, through the Fundamental Research Grant Scheme FRGS/1/2018/TK10/USMC/03/1.

ORCID iD

Varun Thangamani

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A single-degree-of-freedom mathematical model for energy harvesting from cavity flow oscillations

Abstract

Keywords

1. Introduction

2. Experimental details

3. Results and discussion

3.1. Baseline cavity

3.2. Cavity with piezoelectric beam

3.3. Single degree-of-freedom (SDOF) model

3.4. Case 1: Free oscillations with open circuit condition

3.5. Case 2: Forced oscillations with open circuit

3.6. Case 3: Forced oscillations with resistor load

3.7. Effect of resonance

3.8. Effect of incoming velocity

3.9. Effect of cavity scale

4. Conclusion

Footnotes

Appendix

Acknowledgements

Declaration of conflicting interests

Funding

ORCID iD

References