PUBLISHED CONTENT AND CONTRIBUTIONS

Introduction

Motivation

Recently, decentralized power generation has become a strong candidate to bridge this gap, especially when hydrokinetic technologies become viable. A more immediate application can be found in the direction of distributed sensing: on-site power generation would enable the combination of remote sensing and control to optimize hydrocarbon distribution.

Fluid-Induced Vibration Energy Harvesting

The pressure profile, when a flow with a flow rate Q bypasses a structure moving at a speed V, induces a net force in the direction of the structure's motion [24]. LFI remains largely unexplored in the context of flow energy harvesting, and we will ultimately focus our efforts in this thesis.

Figure 1.1: Leakage-flow instability illustration. The pressure profile as flow with flow rate Q bypasses a structure moving with velocity V induces a net force in the direction of the structure’s motion [24].

Flextensional Flow-Energy Harvester

Figure 1.3 shows the evolution of the flextensional current energy collector by comparing the original, double flextension armature with the latest single flexure design (tested in this chapter). Elements are always compressed and out of the flow path as mentioned earlier.

Figure 1.2 illustrates the two actuator types. In the flextensional actuator , the piezoelectric material is comprised of PZT ceramic stacks that are located inside a flexure and pre-stressed in compression

Stability of Elastic-Member in Confined Flows

However, the mechanism by which the flow drives the translational motion of the cantilever can no longer be discerned among the many instabilities discussed in Section 1.2: the diffuser geometry may operate at transient stall, the flag-type instability may induce pressure differences due to trailing-edge vorticity, and the small size of the throat may causes leakage current instability. 41], the inviscid treatment of flow for small throat-to-length ratios, h, causes difficulties in predicting the instability boundaries, which generally depend on the Reynolds number, especially if it is small.

Leakage Flow Instability Hypothesis

More recent work by Guo and Paidoussis [42] takes an opaque approach to understanding the onset of flutter in a symmetric channel. Finally, the work of Inada and Hayama [31] and Nagakura and Kaneko [33] suggests that quantifying viscous effects and channel shape may also be important in quantifying the flutter boundary.

Thesis Scope and Overview

Therefore, we take the approach of formulating and quantifying the stability effects of a leakage-flow instability type, where channel shape, viscous effects and spanwise flow can be considered and systematically distinguished as significant for the onset of flutter in the flextension flow. energy harvesting system. We first consider the comparison between channel flow geometries, and then extend the results to the elastic-translational boundary condition at the jet leading edge and a diffusing channel.

Summary of Contributions

The geometry of the current energy collector is shown in Figure 2.1, and the dimensional parameters are shown in Table 2.1. This is the case shown by the red circles near the beam tip in Figure 5.28.

Leakage Flow Model

Introduction

We consider the voltage output of the flow rate (characteristic velocity), geometrical parameters and material properties. One of the simplest models of leakage flow instability was investigated by Inada and Hayama [31].

Figure 2.1: Illustration cantilever beam in a converging-diverging channel geometry (right) with simple harmonic boundary condition (left).

Fluid Equations of Motion in Two Dimensions

Lubrication Like Closure for N x and F visc,x
Channel Entrance Length
Pressure Boundary Conditions
Linearization of Pressure

Using nˆ from equation 2.1 and the geometric constraints on the channel gap, along with division by x2− x1, we obtain the ˆx component of the viscous term,. The inertia associated with the motion of the channel walls, especially δ, is captured by Qx and QÛx in Equation 2.27.

Figure 2.2: Two dimensional channel control volume.

Fluid-Structure Coupling

Structure Equations of Motion
Structural and Viscous Damping Models
Structure Boundary Conditions
Discretization Basis Functions
Fluid-Structure Coupled Operator
Non-Dimensional Fluid-Structure Equations

When we apply the elastic translational boundary conditions with basis functions in equation 2.84, we increase the structural constants to include those in the boundary equation 2.74,. First, consider the elastic-translational boundary conditions from Equation 2.70 using the nondimensional structural terms in Equation 2.105, .

Figure 2.5: Illustration hydrodynamic force balance as a function of beam shape.

Fluid Equations of Motion including Spanwise Leakage Flow

Closure Relations for N , F visc , and Evaluated Quantities
Pressure Boundary Condition in z
Linearization
Fluid Structure Coupling and q z1 Discretization

The two-dimensional axial momentum in equation 2.8 is integrated in z and augmented with a non-linear cross term. With the profile ratio specified in the form 2.133, we can define the advection terms as in equation 2.23,. Just as in section 2.2.4, we need to solve the ODE describing the boundary forcing flow rateQ¯x1 in equation 2.147.

The same is done for the boundary value qz1|x=L, with the pressure boundary condition atx = Applied from equation 2.63,.

Figure 2.6: Three-dimensional control volume illustration for the spanwise quasi- quasi-1D leakage flow model.

Linear Stability Analysis

As discussed in section 2.2.1, the results of the quasi-1D model are independent of the inlet velocity profile. This can be seen in the normalized DMD mode, shown in Figure 4.8, in both the real and imaginary parts of the model shape. The set screw and preload system is also illustrated in Figure 5.3 as part of the assembly.

As shown in Figure 5.7, the fundamental mode consists mainly of transverse motion of the bending.

Numerical Methods for Simulations and Data Analysis

Introduction

In this chapter we describe the numerical implementation and validation of the quasi-1D model and the fluid structure direct numerical simulation algorithm. We will also describe the signal processing methods used to obtain relevant quantities from the calculated and measured data sets.

Quasi-1D Model Numerical Implementation

Leakage Flow in Constant Channel Verification

It differs from the model proposed in section 2.3.2, and is only used in this section to compare with existing results. There is also no discussion of other forms of material damping that may dominate the behavior of the system (especially at low flow rates), as discussed by Banks and Inman [59]. We compare two data sets extracted and redrawn from [33] and [45]: the spectrum evolution of the system as qx0 is increased for the parameters in table 3.1, and the neutral stability line, or flutter boundary, characterized by critical dimensional velocityU¯ cr = qx0crh¯ as a function of dimensional throat size h.

Figures 3.2 and 3.3 show a good agreement between the critical variables of the model and our implementation.

Table 3.1: Nagakura and Kaneko [33] constant channel dimensional parameters.

Fluid-Structure Immersed-Boundary Direct Numerical Simulation

Internal Flow Fluid-Structure-Interaction Verification
Immersed-Boundary Elastic-Translating Boundary Condition 69
Dynamic Mode Decomposition
Spectral Proper Orthogonal Decomposition
Hilbert Transform

To begin, consider the definition of the data matrix from Figure 3.8 and in Equation 3.8. The predictions of the quasi-1D model show that the choice of ReL for the inviscid treatment is a strong function of h.ˆ. We implement the circuit in Figure 5.5 and choose a scaled enough that the resonances of the structure satisfy condition 5.9.

Small Vibration Theory of the Clarinet.” In: The Journal of the Acoustical Society of America, p.

Figure 3.4: Illustration of FSI DNS validation geometry by [74].

Two-Dimensional Modeling and Simulation Results

Introduction

In this chapter, we present numerical results for both the quasi-1D analytical model derived in Chapter 2 and the direct numerical simulations of fluid-structure interaction discussed in Chapter 3. The comparison between modeling and simulations allows us to we evaluate the validity of the model assumptions and, in doing so, the physical mechanisms that drive the dynamics of the fluid structure. We start with a constant channel flow geometry and test the model predictions against those of DNS on four relevant non-dimensional parameters: mass ratio, stiffness ratio, throat to length ratio and Reynolds number.

Finally, we define that channel geometry as a spreader and assess model predictions considering also a channel spreader angle for both cases where the elastic member is an elastically mounted translating rigid mean and a cantilever flexible beam.

Cantilever in Constant Channel Flow

Problem Formulation
FSI DNS Discretization and Data Analysis
Grid Convergence and Effective Beam Thickness
Comparison of FSI DNS and Quasi-1D Model Results
Quasi-1D Flutter Boundary Comparison to Inviscid Model . 105

The parameter ranges chosen were loosely based on flow-energy harvester dimensions and proportions (shown in figure 5.2), but were primarily designed to test conditions derived in section 2.2.1 for the validity of the quasi-1D model . The quasi-1D model predicts the flutter limit particularly well for the range M∗ simulated. The quasi-1D model marginally under-predicts the FSI DNS boundary in figure 4.17 (i.e. the quasi-1D model is less stable), with the bias increasing with M∗.

In view of these results and those in section 4.2.4, we use the quasi-1D model to

Figure 4.1: Illustration for fluid-structure constant channel domain and boundary conditions.

Diffusing Channel Flows

Cantilever in Diffusing Channel
Elastically-Mounted Rigid Beam in Diffusing Channel

Representative snapshots of the FSI DNS flow contour of velocity x are shown in Figure 4.34 for the two cases considered in Table 4.5 at α = 6◦ . The results of the critical values are shown in figure 4.35 and the representative shape of the atα= 2◦ mode in figure 4.36. The critical values shown in Figure 4.37, however, show good agreement with FSI DNS up to about α ≤ 2◦ , at which point mode switching occurs in the model prediction.

The mode shapes atα=2◦ are plotted in Figure 4.38 and show qualitatively good agreement similar to the second orthogonal beam mode.

Figure 4.34: Snapshot contours of x velocity for representative stable U ∗ simula- simula-tions in table 4.5 cases at α = 6 ◦

Summary

The first, in Figure 5.29, shows a single representative section of the studied parameter space where the system has reached self-sustained oscillations. The voltage response is shown in Figure 5.31 with its amplitude and frequency closely following the behavior of the primary mode in Figure 5.30. An example is the convexity of the critical flow velocity stability limits found as a function of the diffuser angle in Section 5.5.

Vibration of an elastic rod as an axisymmetric continuous flexible beam induced by axial leakage flow.” In: Journal of Pressure Vessel Technology p.

Experimental Study of Flextensional Flow-Energy Harvester

Introduction

In this chapter, we investigate the design of a current-energy bending energy harvester from an experimental point of view in the context of fluid-structure interaction. Experiments are then carried out to measure structural properties related to bending, including its effective stiffness, mass and damping. The final part of this chapter compares the predicted response of the model proposed in Chapter 2 with experimental results.

We use the output voltage from the piezoelectric arrays as a measuring tool and as an approximation of the dynamics of the system.

Flextensional Flow-Energy Harvester Design

Piezoelectric Stacks
Flexure Dynamics

The amount of torque applied to the adjusting screw biases the piezoelectric stacks, and consequently changes the dynamic properties of the bending. The flexure behaves like a translational spring that transfers motion from the normal direction of the beam surface to compression and expansion of the piezoelectric stacks. An electrical fitting is used to connect the piezoelectric stacks to the data acquisition board on the outside of the test section.

The combination of the flow path, the structure, the piezoelectric elements, and the. electronics consist of the flow-energy harvester design.

Figure 5.1: Current version of flextensional flow-energy harvester with custom de- de-signed flexure.

Flexure Property Measurements

Flexure Static Stiffness Test
Flexure Dynamic Test
Elastic-Translating Boundary Condition Parameters

The loading location is shown as Fa in Figure 5.6 and is chosen because it primarily excites the flexural extension mode described in Section 5.2.2. The subtracted oscillating part is shown in Fig. 5.14b and processed by the Hilbert transform in Fig. 5.15. This deflection produced the cleanest signal of all tested, as can be seen from the linear behavior of the analytical signal amplitude in Figure 5.16a.

A moving average with 300 samples is shown in Fig. 5.24a, where the fit of ζ s] is slightly lower, but within the expected range for RC time-scale variation.

Table 5.5: Table of experimental flexure settings based on qualitative set-screw torque level, with approximate torque values shown.

Flextensional Response to Fluid Flow Experiment

Flow Set up and Test Section
Video Data Processing Algorithm
Flexure Dynamics Results

A detailed experimental procedure can be found in Appendix D. Figure 5.27 shows a picture of the test setup with the data acquisition and control apparatus. Quality assurance (QA) features highlighted in the figure are the origin, marked with a red ×, and the predicted position x of the beam tip, marked with a green ×. The second peak appears at f2 = 341 [Hz], which corresponds to the base mode of the cantilever without clamping.

Since the flow control (needle) valve is upstream of the flowmeter and the test section, it is further.

Figure 5.27 shows a picture of the test set up with the data acquisition and control apparatus.

Comparison with Quasi-1D Model

Discussion of Modeling Assumptions for Experimental Com-
Experimental Comparison to Spanwise Quasi-1D Model

Critical flow rates for collapse of reactor parallel plate fuel assemblies. In: Journal of Engineering for Power pp. Instability of a flexible cantilever plate in viscous channel flow. In:Journal of Fluids and Structures20.7 SPEC. Stability of a flexible cantilever in viscous channel flow. In: Journal of Sound and Vibration pp.

Linear and Nonlinear Analyzes of Skin Flutter Induced by Leakage Flow.” In: Journal of Fluids and Structures pp.

Figure 5.38: Illustration of spanwise quasi-1D geometry for comparison to experi- experi-mental results.

Conclusions and Outlook

Conclusion

The results show that the flow instability mechanism is a strong candidate as the main source of fluid structure instability for heavy beams in channels with gaps less than 15% of the beam length. The experimentally measured values for wave initiation are then compared with the predictions of the quasi-1D spanwise model developed in Chapter 2. This difference in itself shows the importance of the broad, three-dimensional nature of the problem.

This work concludes that it is likely that flow instability dominates the behavior and drives the fluid structure interaction dynamics seen in the bending flow energy harvesting system.

Outlook

Decision Analysis under Uncertainty for Intelligent Well Deployment." In: Journal of Petroleum Science and Engineering pp. Motions, Forces, and State Transitions in Vortex-Induced Vibrations at Low Mass Damping." In: Journal of Fluids and Structures13 (1999), pp. Flutter of an elastic plate in a channel flow: Confinement and finite-size effects." In: Journal of Fluids and Structures pp.

A strongly bound immersed boundary formulation for thin elastic structures. In: Journal of Computational Physics pp.