Fluidic energy harvesting beams in grid turbulence

Abstract

Much of the recent research involving fluidic energy harvesters based on piezoelectricity has focused on excitation through vortex-induced vibration while turbulence-induced excitation has attracted very little attention, and virtually no previous work exists on excitation due to grid-generated turbulence. The present experiments involve placing several piezoelectric cantilever beams of various dimensions and properties in flows where turbulence is generated by passive, active, or semi-passive grids, the latter having a novel design that significantly improves turbulence generation compared to the passive grid and is much less complex than the active grid. We experimentally show for the first time that the average power harvested by a piezoelectric cantilever beam placed in decaying isotropic, homogeneous turbulence depends on mean velocity, velocity and length scales of turbulence as well as the electromechanical properties of the beam. The output power can be modeled as a power law with respect to the distance of the beam from the grid. Furthermore, we show that the rate of decay of this power law closely follows the rate of decay of the turbulent kinetic energy. We also introduce a forcing function used to model approximately the turbulent eddies moving over the cantilever beam and observe that the feedback from the beam motion onto the flow is virtually negligible for most of the cases considered, indicating an effectively one-way interaction for small-velocity fluctuations.

Appendix: Average piezoelectric power decay

To relate the average power of the piezoelectric cantilever beam to the distance from the grid, the coupled electromechanical equations (Elvin and Elvin 2009; Sodano et al. 2004) are utilized:

$$\begin{aligned} {{\mathbf {M}}}{\ddot{\mathbf {r}}}+{\mathbf {D}}{\dot{\mathbf {r}}}+ {\mathbf {Kr}}-{\varvec{\Theta }}v_p= \,& {} {\mathbf {g}} \end{aligned}$$

(12)

$$\begin{aligned} C_p{\dot{v}}_p+\frac{v_p}{R_l}+{\varvec{\Theta }}^T {\dot{\mathbf {r}}}=0 \end{aligned}$$

(13)

where ${{\mathbf {M}}}$, ${\mathbf {D}}$ and ${{\mathbf {K}}}$ are the mass, damping and stiffness matrices, respectively, and ${\varvec{\Theta }}$ is the piezoelectric coupling vector, ${\mathbf {g}}$ is the effective forcing vector, $C_p$ is the capacitance of the piezoelectric layer, $v_p$ is the voltage generated by the piezoelectric layer. The piezoelectric harvester is assumed to be attached to an external resistance $R_l$. The local displacement along the beam $y(X,x/M;t)$ can be expressed as:

$$\begin{aligned} y(X,x/M;t) =\sum _{j=1}^{N} \phi _j(X)r_j(x/M;t) = \varvec{\Phi }^T(X){\mathbf {r}}(x/M;t) \end{aligned}$$

(14)

where ${\varvec{\Phi }}(X)$ is a vector of assumed mode shapes for the beam, ${\mathbf {r}}(x/M;t)$ is the time-dependent component of the displacement vector, and N is the number of modes under consideration.

The $j{{\rm th}}$ entry in the effective forcing vector, $g_j(x/M;t)$, is given as:

$$\begin{aligned} g_j(x/M;t)= \frac{1}{L} \int _0^L {\phi _j(X)f(X,x/M;t)}\, {{\rm d}}X \end{aligned}$$

(15)

where $\phi _j(X)$ is the $j{{\rm th}}$ entry of the $\varvec{\Phi }(X)$ vector and is defined in Erturk and Inman (2008).

Thus, from (12) and (13), the voltage in the frequency domain, $V_p(x/M;\omega )$, is found to be:

$$\begin{aligned} V_p(x/M;\omega )={\mathbf {H}}_v(X;\omega ){\mathbf {G}}(x/M;\omega ) \end{aligned}$$

(16)

where

$$\begin{aligned} {\mathbf {H}}_v(\omega )= & {} \frac{1}{C_p}\left[ \frac{i\omega R_lC_p}{ 1+i\omega R_lC_p}\right] \varvec{\Theta }^T\left[ {\mathbf {K}}- \omega ^2{\mathbf {M}} +i\omega \mathbf {D} \right. \nonumber \\&\left. +\,\frac{1}{C_p}\left[ \frac{i\omega R_lC_p}{1+i\omega R_lC_p} \right] \varvec{\Theta }\varvec{\Theta }^T\right] ^{-1} \end{aligned}$$

(17)

and

$$\begin{aligned} G_j(x/M;\omega )= \frac{1}{L} \int _0^L {\phi _j(X) F(X,x/M;\omega )}\, {{\rm d}}X \end{aligned}$$

(18)

We will limit the analysis to the first mode only, since the beams typically vibrate at or near the first mode. Thus, the expected power ${\mathscr {E}}\left\{ P(x/M)\right\}$ can be expressed as:

$$\begin{aligned} {\mathscr {E}}\left\{ P(x/M)\right\}= & {} \int _{-\infty }^\infty { P(x/M;\omega )}\, {{\rm d}}\omega = \int _{-\infty }^\infty \frac{|V_p(x/M;\omega )|^2}{R_l} {{\rm d}}\omega \nonumber \\= & {} \frac{1}{R_l}\int _{-\infty }^\infty {H_v{\widehat{H_v}} G_1{\widehat{G_1}}}\, {{\rm d}}\omega \end{aligned}$$

(19)

where ${\widehat{H_v}}(\omega )$ and ${\widehat{G_1}}(x/M;\omega )$ are complex conjugates of $H_v(\omega )$ and $G_1(x/M;\omega )$, respectively.

Equation (19) shows that the expected power generated in the beam primarily depends on x/M. The x/M dependence of the expected power is manifested through the term $G_1{\widehat{G_1}}=|G_1|^2$. Parseval’s theorem can be applied to the $|G_1|^2$ term to give:

$$\begin{aligned} \int _{-\infty }^\infty {|g_1(x/M;t)|^2} {{\rm d}}t = \frac{1}{2\pi } \int _{-\infty }^\infty {|G_1(x/M;\omega )|^2} {{\rm d}}\omega \end{aligned}$$

(20)

From (20), the x/M dependence of $|G_1(x/M;\omega )|^2$ is equivalent to that of $|g_1(x/M;t)|^2$. Thus, using the definition of average power, one can conclude that, with respect to x/M:

$$\begin{aligned} {\overline{P}}(x/M)&\propto {\mathscr {E}}\left\{ P(x/M)\right\} \propto |G_1(x/M;\omega )|^2 \nonumber \\&\propto \sigma ^2_{g_1}(x/M) \end{aligned}$$

(21)

where $\sigma ^2_{g_1}(x/M)=\overline{|g_1(x/M)|^2}$ is the variance of $g_1(x/M;t)$.

The effect of the turbulence length scale does not explicitly enter into the result in (21), even though we expect it to be a factor since the beam vibrates because of the presence of turbulence in the flow. The turbulence length scale can be included in (21) by expanding on the relation between the average power and $\sigma ^2_{g_1}(x/M)$:

$$\begin{aligned} {\overline{P}} \propto \sigma ^2_{g_1}(x/M) = \frac{1}{L^2} \int _0^L \int _0^L {\phi _1(X_1)\phi _1(X_2) \overline{f(X_1)f(X_2)}}\, {{\rm d}}X_1 {{\rm d}}X_2 \end{aligned}$$

(22)

The time-averaged term in the integrand of (22) can be expressed in terms of a two-point force (pressure) correlation:

$$\begin{aligned} J(X_1,X_2,x/M)=\overline{f(X_1,x/M)f(X_2,x/M)} \end{aligned}$$

(23)

From Comte-Bellot and Corrsin (1971) and similar to the approach in Goushcha et al. (2015) for the flow in the turbulent boundary layer, the two-point force correlation in isotropic turbulent flow is approximated as an exponential decay, such that:

$$\begin{aligned} J=\gamma (X,x/M){{\rm e}}^{-a(x/M)\left| X_1-X_2\right| } \end{aligned}$$

(24)

where $\gamma (X,x/M)$ and a(x/M) are parameters of the exponential decay. If $X_1=X_2$, equating (23) and (24) leads to:

$$\begin{aligned} \gamma (X,x/M)=\sigma ^2_f(X,x/M) \end{aligned}$$

(25)

Note that in (25), as in Sect. 3.4, the variance of the forcing function $\sigma ^2_f$ is assumed to be weakly dependent on the beam coordinate X. Thus, $\gamma (X,x/M)=\gamma (x/M)$. We posit that a(x/M) is inversely proportional to the integral length-scale $L_I(x/M)$, which characterizes the largest energy-carrying eddies. Thus,

$$\begin{aligned} a(x/M)=\frac{q}{L_I(x/M)} \end{aligned}$$

(26)

where q is a proportionality constant. The reasoning behind the definition in (26) is that we assume that the two-point correlation in (24) is effectively defined by the energy-carrying eddies, i.e., two points that are separated by a distance larger than $L_I$ are virtually uncorrelated. The proportionality constant q in (26) is defined in a manner such that when $|X_1-X_2|=L_I$, the correlation function is a small percentage of the force variance, i.e., $J=0.2\sigma ^2_f$. In this case, $q=-\ln (0.2)=1.61$. Substituting (24)–(26) into (22) and evaluating the integral results in:

$$\begin{aligned} {\overline{P}}(x/M) \propto \xi (\ell ) \sigma ^2_f(x/M) \end{aligned}$$

(27)

where $\xi (\ell )$ is referred to as the local variation parameter, which is a known function of the length ratio $\ell$ that is shown in Fig. 16, and

$$\begin{aligned} \ell =\frac{L}{L_I(x/M)} \end{aligned}$$

(28)

It is important to note that, from Fig. 16, the local variation parameter $\xi (\ell )$ has a maximum value of $\xi _{{\rm max}}=\xi (0)=0.153$. This rather unusual number, which is independent of q, arises from the integrand of (22), where the first-mode shape function $\phi _1(X)$, as defined by Elvin and Elvin (2009), is included.

The variance of the forcing function in (8) is found using Eqs. (1)–(2) to be:

$$\begin{aligned} \sigma ^2_{f}(x/M)=\rho _f^{2} A_f^{2} C_F^{2}{\overline{U}}^{2} {\overline{v^2}}=\frac{2}{3}\rho _f^{2} A_f^{2} C_F^{2} {\overline{U}}^{2} \alpha _k\left( \frac{x}{M}-\frac{x_{0,k}}{M}\right) ^{\beta _{k}} \end{aligned}$$

(29)

From (27) and (29), it is clear that the average power harvested by the piezoelectric cantilever beam displays a power-law behavior in grid turbulence with respect to the distance from the grid. Therefore, we hypothesize that:

$$\begin{aligned} {\overline{P}}(x/M)&= \kappa \left( \rho _f,A_f,C_F,{\overline{U}}, \alpha _k,\ell \right) \left( \frac{x}{M}-\frac{x_{0,k}}{M}\right) ^{\beta _{k}} \nonumber \\&= \alpha _P\left( \frac{x}{M}-\frac{x_{0,P}}{M}\right) ^{\beta _{P}} \end{aligned}$$

(30)

where $\kappa$ is the proportionality constant of the power law. In this theoretical approach, the exponent of the mean TKE power-law expression (2) is identical to the exponent of the average piezoelectric power-law equation (30), provided that $C_F$ is independent of x/M.

Fig. 16

Function $\xi (\ell )$ with respect to $\ell$