Chapter V: Experimental Study of Flextensional Flow-Energy Harvester
5.5 Comparison with Quasi-1D Model
5.5.1 Discussion of Modeling Assumptions for Experimental Com-
ters 2 and 4. Though some have been discussed, we will highlight those we believe most relevant in the context of contrasting experimental conditions to model as- sumptions and restrictions in the subsections to follow. We will begin with flow three-dimensionality and turbulence, followed by flow separation, and compress- ibility.
Flow Three-Dimensionality and Turbulence
The flow path in the experiment has a spanwise gap between the channel wall and the flexure of C = (b1 − b)/2 = 1.25 [mm], with spanwise gap-to-length ratio
C
L = 0.031 10−4. This indicates that, at least for the transversely unconfined flag case, the spanwise flow plays a significant role in the stability properties of the system, increasing the critical velocities necessary for the flutter onset in a given flag configuration. As discussed in section 2.1, this is expected to be the case in the experimental flow path as well, which renders both 2D DNS and the “axial only”
quasi-1D model, both discussed and compared in chapter 4, inadequate tools to predict the experimental dynamics. The critical gap Reynolds numberReh 1000 for all flexures tested, ensuring that the flow is fully turbulent as well. This further ascertains that the 2D DNS will not capture the appropriate viscous effects due to its low Reynolds number limitations, and that we must consider a turbulent friction factor correlation within the quasi-1D model (as in equation 2.29). The inadequacy of the axial quasi-1D model is validated when experimental conditions and param- eters are applied, and no aspect for the experimentally observed regime is evident in its results. Specifically, the model predicts an unstable flextensional mode at a critical flow rates < 2 [L/min] and only through a divergence instability rather than flutter. This holds true for all experimental flexure properties and also when parameters are varied through a range of inlet and outlet loss coefficients, diffuser expansion angles, throat sizes, throat positions, denoting that the divergence pre- diction is robust to a wide range of model parameters. These results were essential in the motivation for the development of the quasi-1D model in section 2.4, which considers flow in the the spanwise direction of the beam.
Flow Separation and Simplified Modeled Geometry
A linear plane-asymmetric diffuser consists of a diffuser with a flat lower boundary and an upper surface that expands at a constant angleθ. This diffuser configuration captures the essence of the geometry of our experimental flow path, illustrated in figure 5.2, while the beam is near its equilibrium point. High Reynolds number numerical and experimental studies suggest a θcr ≈ 7◦, independent of Reynolds number for turbulent flows, where flow separation is triggered over the upper dif- fusing wall [99–102]. Asθ =19◦> θcrfor our experimental set up, the flow regime realized is one with a separation bubble that extends the entirety of the beam length for all three flexure settings. Flow separation is not explicitly accounted for within the formulation of the quasi-1D models, as the pressure distribution over x is cal- culated for attached flow: a change in channel geometry would change the force the beam. However, the model allows for anisentropic type phenomena at its inlet and outlet boundary conditions. To account for flow separation within the quasi-1D model, we propose a simplified geometry shown in figure 5.38. The figure shows the model geometry, consisting of a diffuser with angleαm and an abrupt expansion at its outlet. Three distinct features have been simplified: first the contraction sec- tion has been removed under the assumption that losses there are negligible, such thatζin =1in equation 2.45. This assumes that the net force contribution from the converging section is also negligible. Second, we assume that the outlet boundary pressure variation behaves as an abrupt expansion exists at the outlet, whereζout =1 from equation 2.40.
¯h (x, t)
L
x sym
qx0
↵m
✓
¯ a
¯ a
Figure 5.38: Illustration of spanwise quasi-1D geometry for comparison to experi- mental results.
Third is that the separation bubble over the diffusing channel walls effectively serve as a secondary diffuser boundary at an effective expansion angle ofαm. The pres- sure distribution on the beam surface behaves as if the flow had been attached
and expanding at an angle of αm. Because of separation characteristics of plane- asymmetric diffuser, we believeαm ≈ θcr. Though this cannot be shown in experi- mental results, we assess critical flow rates over a range of αm, and discuss model results in the context of a range of effective angles simulated. In addition, in order to remain consistent withh020 1criteria, we upper bound our results toαm ≤ θcr =7 [◦], where h0 ≈ 0.12. The geometrical parameters in the simplified geometry are those of table 5.1, except the throat position L2 = 0. The fluid, structural, and flexure properties are those of table 5.3 and the mean values in table 5.8.
Compressibility
As shown in table 5.9 and described in section 5.4.3, critical flow regimes for flex- ures 2 and 3 are choked, while flexure 1 remains subsonic. Since throat Mach numbers are unity for the former two cases, and a relatively large expansion exists downstream of the throat, it is likely that a combination of shocks and expansions follow. The lack of detailed pressure measurements immediately downstream of the throat make it difficult to ascertain the actual flow regime, as another possibility exists where the flow continues to accelerate into the supersonic range. However, evident by the audible noise from the test section during flexure 2 and 3 runs as similar to that of flexure 1 runs, we do not believe this to be the case, nor the shocks to be particularly strong such that they extend a significant distance downstream of the throat. We believe it reasonable to assume that the flow becomes subsonic relatively quickly after bypassing the channel throat, and that the system expands as suggested in figure 5.38. However, these effects are not negligible and likely account in large part for quantitative discrepancies seen in the results shown in the next section when comparing experimental measured critical values to those pre- dicted by the incompressible quasi-1D model.
In the flexure 1 case, where the flow remains subsonic, compressibility effects may be significant particularly if the structural response time scale is on the order of the sonic flow speed. However, the flutter instability gives rise to a response at f1∼ 200[Hz], while the length based flow acoustic frequency is fa∼ 8.3E3 f1. This means that flow “information” in the form of an acoustic wave can travel mul- tiple times the length of the cantilever, and that the flow has enough time to adapt to changes in the structure’s shape. Hence, we believe that the effect of compress- ibility is likely localized near the throat and that its primary effects is decreasing the density of the fluid by≈10%, consequently changing the mass ratio within that
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.