Chapter V: Experimental Study of Flextensional Flow-Energy Harvester
5.5 Comparison with Quasi-1D Model
5.5.2 Experimental Comparison to Spanwise Quasi-1D Model
local region. Given the localization, limited density change, and the quasi-static be- havior of structure relative to flow, we expect the incompressible model to be able to replicate flexure 1 experimental response well as compared to those of flexures 2 and 3.
These descriptions from values in table 5.9 are only valid for throat sizes near the equilibrium position of the beam; they are no longer valid once deflection ampli- tudes grow large at unstable flutter cases. The physics that drive the fluid-structure system in high amplitude cases may include a fluctuating Mach number, and signif- icantly alter the flow regime and dynamics discussed here. These are not captured by the linear stability analysis in the subsequent model results.
rates considered, only the flexure mode was excited.
Beginning with flexure setting 1, figure 5.39 shows the spanwise quasi-1D model results for the dimensional critical flow rates and frequency responses over αm, along with the corresponding mean experimental values from table 5.9. The sta- bility boundary trend has a convex shape, with a αm ≈ 3◦representing the lowest critical flow rate over the range of diffuser angles tested. Both quasi-1D critical flow rate of 213 [L/min] and the critical frequency of 190 [Hz] atαm = 7◦appear to replicate the experimental values of 208 [L/min] ans 187 [Hz] remarkably well. Fig- ure 5.40 shows a representative mode 1 shape near the critical point at αm = 7[◦]. The mode shapes, both displacement and spanwise flow segments, are similar for allαm values shown in figure 5.39. The beam shape agrees well qualitatively with the experimental flextensional mode in figure 5.29. This can be seen primarily from the dominant rigid body motion content, evident by the translation of the base. The leakage flow rate mode shows an almost linear trend from channel inlet to outlet, with flow direction switching from into-the-channel, or negative direction, at the inlet to out-of-the-channel, or positive, at the outlet. The inlet spanwise flow direc- tion is opposite that of the inlet axial flow term and of the same magnitude. Hence, the model indicates that as flow enters the inlet axially, it leaves in the spanwise direction at the same time. This trend remains true for all three modeled flexure setting results, and highlights the importance of the leakage flow terms in addition to the axial flow boundary condition.
(a) Critical flow rate versus diffuser angle. (b) Critical frequency versus diffuser angle.
Figure 5.39: Comparison between spanwise quasi-1D critical flow rate (left) and frequency (right) to experimental quantities for flexure setting 1 from table 5.5.
0 0.5 1 x=L
-1 0 1
/=j/j1
Real Part Imaginary Part
(a) Transverse displacement mode 1.
0 0.5 1
x=L -1
0 1
qz=jqzj1
Real Part Imaginary Part
(b) Spanwise leakage flow rate mode 1.
Figure 5.40: Flexure setting 1 representative mode 1 plots of transverse displace- ment (left) and spanwise leakage flow (right) atαm = 7[◦].
Figure 5.41 shows results for flexure setting 2. The early time segment variant was chosen because voltage amplitudes in the data point near the bifurcation in figure 5.34 are ∼ 0.2 [V], well within the early time segment amplitude bound.
Though the stability boundary convexity still remains, the comparison of values between experiments and model are no longer in good agreement. The frequency is overestimated by about 10%, and, if we keep the calibration ofαm = 7◦, we over- predict the critical flow rate by about 25%. A representative mode 1 shape is shown in figure 5.42 for both transverse displacement and spanwise flow. The flexure base displacement is evident though not as pronounced as that of flexure 1, but still reproduces the shape of the experimental result in figure 5.32 qualitatively well.
The spanwise leakage flow is similar to results from flexure 1, with a linear trend and opposing flow directions between inlet and outlet spanwise flows, as well as inlet spanwise and inlet axial flows. Figure 5.43 shows critical property results for flexure 3, with figure 5.44 showing its representative mode 1 shape. These echo the descriptions of flexures 2: the convex shape ofQcras a function ofαm, with a model over-prediction of critical flow rate and frequency at αm = 7 [◦]. Here, however, beam shape only replicates the base motion seen in flexure 3 experimental data in figure 5.35, missing the correct traveling wave form. The leakage flow rate mode shape is similar to those of flexures 1 and 2.
(a) Critical flow rate versus diffuser angle. (b) Critical frequency versus diffuser angle.
Figure 5.41: Comparison between spanwise quasi-1D critical flow rate (left) and frequency (right) to experimental quantities for flexure setting 2 from table 5.5.
0 0.5 1
x=L -1
0 1
/=j/j1
Real Part Imaginary Part
(a) Transverse displacement mode 1.
0 0.5 1
x=L -1
0 1
qz=jqzj1
Real Part Imaginary Part
(b) Spanwise leakage flow rate mode 1.
Figure 5.42: Flexure setting 2 representative mode 1 plots of transverse displace- ment (left) and spanwise leakage flow (right) atαm = 7[◦].
(a) Critical flow rate versus diffuser angle. (b) Critical frequency versus diffuser angle.
Figure 5.43: Comparison between spanwise quasi-1D critical flow rate (left) and frequency (right) to experimental quantities for flexure setting 3 from table 5.5.
0 0.5 1
x=L -1
0 1
/=j/j1
Real Part Imaginary Part
(a) Transverse displacement mode 1.
0 0.5 1
x=L -1
0 1
qz=jqzj1
Real Part Imaginary Part
(b) Spanwise leakage flow rate mode 1.
Figure 5.44: Flexure setting 3 representative mode 1 plots of transverse displace- ment (left) and spanwise leakage flow (right) atαm = 7[◦].
The lack of agreement between experiment and model results for cases of flexures 2 and 3 are expected primarily because compressibility effects cannot be neglected at their respective Mach numbers. This is also the case for flexure 1, although considerably less since its experimental Mach number is less than half the other two cases. The agreement between results however, particularly that of flexure 1, suggests that the model may capture a considerable portion of the relevant physics.
This is in contrast to the quasi-1D model constrained to axial flow, where no aspect of the dynamics could be captured within the parameter space in the experimental geometry and material properties.
C h a p t e r 6
CONCLUSIONS AND OUTLOOK
6.1 Conclusion
This thesis aims to elucidate the fluid-structure instability that drives the flexten- sional based flow energy harvester, primarily focused on the leakage-flow instabil- ity as a hypothesis.
Chapter 2 formulates this in quantitative terms by restricting the coupled fluid- structure equations through assumptions that define leakage-flow. The full equa- tions are subsequently solved through a two-dimensional fluid-structure interaction direct numerical simulation algorithm in chapter 4, and the flutter instability onset and critical values compared to those predicted by the reduced order model derived.
The agreement between FSI DNS and the quasi-1D model is remarkable for small channel gaps in a range of viscous and mass parameters tested. This holds even as the elastic-translating boundary condition is added to the system, along with a linear diffuser though to a lesser degree. Results indicate that the leakage-flow instability mechanism is a strong candidate as the primary source of fluid-structure instability for heavy beams in channels with gaps less than 15% of the beam length. This is valid for systems where the flow is approximately two dimensional.
Chapter 5 experimentally assesses the dynamics of the flextensional based flow en- ergy harvester in air flow. Experiments characterize device mechanical properties first, then appraise the system dynamics in a flowing setting. Critical flow rates and frequencies are measured for three different flexures, with self-sustaining os- cillations reached in tshe flextensional mode in all three tested cases. Furthermore, hysteresis is observed as the flow rate direction is reversed, indicating a bistable region and a subcritical Hopf bifurcation at the neutrally-stable point. Experimen- tally measured values for the flutter onset are then compared with predictions of the spanwise quasi-1D model developed in chapter 2. Results are encouraging: the model is able to predict the critical flow rates and frequency response of the first flexure setting, and is within a factor of 2 of flexure settings 2 and 3. The promising aspect of these predictions is that though many effects are not captured, including fluid compressibility, true frictional pressure loss, and flow separation, the crude ge- ometry of a diffuser at its critical angle is able to estimate within reasonable bounds
the stability properties of the experimental set up. This is in contrast with predic- tions from the quasi-1D model that only accounts for axial leakage-flow: no aspect of the dynamics, including the bifurcation type, was replicated for a wide range of parameters near those in the experimental set up. That distinction in itself indicates the importance of the spanwise flow, and three-dimensionality of the problem.
This work concludes that is it likely leakage-flow instability dominates the behavior and drives the dynamics of the fluid-structure interaction seen in the flextensional flow energy harvester system. More work is necessary to determine effects that are loosely approximated, such as flow separation, or entirely not accounted within the model, as compressibility. Yet the fundamental framework for reduced-order modeling is valuable for predicting the flutter onset and critical values in narrow channels, especially when beams are heavy relative to the fluid. Applications that fall under these parameter ranges will find a relevant tool-set in this work.
6.2 Outlook
Two broad tasks are suggested as a continuation of this thesis. The first is valida- tion of the reduced order models through further experimentation, augmenting the model with more relevant physics as their importance becomes evident. Better flow instrumented flextensional energy harvester experiments are likely a good starting point. Measurements of effective spanwise flow, detailed velocity and pressure pro- files over the beam surfaces, and more accurate assessment of flexure structural properties would help in quantifying or better estimating system unknowns. An- other set of experiments where flexures reach the limit-cycle regime at a Mach num- ber less than 0.1, then with gradually increasing Mach number values, would quan- tify the importance of compressibility in predicting stability boundaries. Lastly, a set of experiments where diffuser angle can be varied would elucidate the effect of flow separation on critical properties, especially when measurements suggested are implemented.
The second task is using the models developed in this work to assess and design new flow energy harvesters. One example is the convexity of the critical flow rate stability boundaries found as a function of diffuser angle in section 5.5. By under- standing the variation of the minimum unstable flow rate inαm, more unstable, and perhaps more powerful, designs may be developed. Building an optimization algo- rithm constrained by the model dynamics in chapter 2 appears a logical next step.
Iteration between model validation and the optimization is an obvious necessity,
hence advancing both tasks simultaneously is seems like a sensible strategy.
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