Chapter II: Leakage Flow Model
2.1 Introduction
In this chapter we develop a model that combines the fluid equations with those from the solid in a relevant but simplified form, to provide insights into the physical mechanisms that drive the coupled fluid-structure system dynamics. The combined fluid-structure equations are simplified by making assumptions and taking param- eter limits that will be verified with numerical simulations in chapter 4, and com- pared with experimental results in chapter 5. Assumptions and limits focus around the leakage-flow definition discussed in chapter 1, which often refers to flow around large center bodies through small passages, but also applies to general flows through narrow passages [24, p. 1221].
The geometry of the flow-energy harvester is illustrated in figure 2.1 and dimen- sional parameters in table 2.1. The elastic cantilever is placed in a symmetric con- verging diverging channel. The flextensional transducer is modeled by connecting the fixed end of the beam to a simple harmonic oscillator. We consider the voltage output of the flow rate (characteristic velocity), geometrical parameters, and ma- terial properties. This gives 23 dimensional quantities spanning four fundamental dimensions: lengthl, massm, timet, and current A. Through the Buckingham-Pie theorem, these yield a total of 19 non-dimensional groups to determine the system dynamics. This large number of a parameters makes a thorough numerical or exper- imental investigation intractable. Hence, the purpose of the theory in this chapter is to provide a simple model that allows us to analytically identify the most important parameters affecting the stability of the system.
One of the simplest models of leakage flow instability was explored by Inada and Hayama [31]. The passage was first constrained to rigid body translation in the streamwise-normal direction and later included rigid body rotation [32]. In both cases the flow rate and pressure profiles only depended on the streamwise direc- tion. A similar analysis extended for a beam in a constant channel by Nagakura and Kaneko [33], maintained the same leakage-flow forcing model but applied a damped Euler-Bernoulli beam equation rather than the rigid-body motion coupling to the elastic body. This analysis was further extended to include geometrical non-
¯ a
¯ a
Figure 2.1: Illustration cantilever beam in a converging-diverging channel geometry (right) with simple harmonic boundary condition (left).
Table 2.1: Table of fluid-structure-electrical dimensional parameters.
Variable Description Dimension
δ beam displacement l
x beam length coordinate l
t time t
V voltage m∗l2∗t−3∗ A−1
p pressure m∗l−1∗t−2
Uc characteristic velocity l∗t−1 k0 flexure stiffness m∗t−2
c0 flexure damping m∗t−1
m0 flexure mass m
Re electric resistance m∗l2∗t−3∗ A−2
L beam length l
hb beam thickness l
b beam width l
ht throat height l
xt throat location l
R radius of curvature l
αi nozzle contraction angle rad αo diffuser expansion angle rad
ρf fluid density m∗l3
µf fluid viscosity m∗l−1∗t−1
ρs beam density m∗l−3
E Young’s modulus m∗l−1∗t−2
ν Poisson’s ratio ND
linearities associated with large displacement of the beam [45]. Fujita and Shintani [48] also explored a similar form of leakage-flow forcing for flexible cylinders, us- ing the Euler-Bernoulli equation for the structure, in a constant cylindrical narrow passage, while adding a flow and pressure component in the azimuthal direction [46, 47]. Comparison to experiments were also made in [48].
Fundamentally, power extraction from the flow is contingent on the energy con- version from fluid kinetic into structural kinetic and potential energy, later to be transduced into electricity through piezoelectric crystals. The characterization of structural energy can be ascertained by the amplitudeδand velocityδÛof the elastic member. We would like to characterize, through the physics, when these quanti- ties will grow or decay as a function of parameters in table 2.1. The purpose is to develop a set of linear equations inδ andδÛand to assess their stability relative to some equilibrium point δ0andδÛ0. We frame our analysis in part inspired by these authors, and apply our framework to the flow-energy harvester.
The analysis within this chapter derives equations of motion from control volume and applies closures consistent with infinitesimal equations through limiting param- eter cases. This is done in order to formulate a physical description that elucidates the appropriate ranges of those parameters in table 2.1 where solutions are applica- ble, which remains an open question through much of the aforementioned literature.
The derivation will consist of the following steps:
1. Develop an expression for pressure of the exerted by the fluid, as a func- tion ofδand its derivatives on the elastic member through the Navier-Stokes equations and conservation of mass;
2. Linearize the result inδand its derivatives;
3. Close the system with the linear elastic description of δ using conservation and constituent relations.
We begin by considering the caseb Lsuch that the flow is approximately two- dimensional. This is further simplified to a quasi-one-dimensional system of equa- tions in the axial direction. We first consider a clamped cantilever, and then add a moving leading-edge boundary condition to model the flextensional device as a simple harmonic oscillator.
Finally, we extend the model to consider the three-dimensional leakage flow for the finite span case. The effect of confinement in the spanwise direction was detailed by Doaré et al. [50, 51] theoretically and with numerical simulations for the unconfined (in the normal direction) flag configuration [50], and in experiments [51]. Their results showed that only at small spanwise gap to length ratios, CL, would the two- dimensional instability boundaries be observed. Furthermore, they showed that even at CL = O(10−4), a large discrepancy between stability boundaries was still seen relative to the two-dimensional model. Though this work is done for standard flag case outside of a channel, the spanwise confinement effect is likely significant in the flow-energy harvester problem, and as discussed in chapter 5, is crucial when comparing model predictions to experiments.
Our work in this chapter is motivated in part by the previous work of Inada and Hayama [31] and Nagakura and Kaneko [33]. We seek, however, tangible paramet- ric bounds on the validity of the model. This required a different approach to the derivation of equations of motion from previous literature: we start with a control volume analysis and look for relations to close openly defined terms. The form of the closure provides the validity bounds we seek. The final equations are only similar to previous work when constrained to specific cases, such as a rigid beam or constant channel. Even in those cases, however, they differ in coefficients derived through our closure relations. Hence, model derivation steps presented are believed to be novel in the context of leakage-flow. In addition, the generalization of equa- tions to arbitrary channel shapes with an flexible member and a moving boundary condition is also believed to be novel, along with its expansion to include flow in the beam spanwise direction.