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Analysis of unsteady flow effects on the Betz limit for flapping foil power generation
- John Young, Fang-Bao Tian, Zhengliang Liu, Joseph C. S. Lai, Nima Nadim, Anthony D. Lucey
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- Journal:
- Journal of Fluid Mechanics / Volume 902 / 10 November 2020
- Published online by Cambridge University Press:
- 14 September 2020, A30
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A control volume based analytical method for calculating the efficiency $\eta$ of flapping foil power generators was developed for single and tandem foil configurations. Ignoring unsteady effects and non-uniform pressures resulted in theoretical limits identical to the Betz ($\eta =16/27$ for a single turbine) and Newman ($\eta =16/25$ for tandem turbines) limits. Inclusion of unsteady flow and non-uniform pressure distributions produced theoretical efficiency maxima in excess of these limits. Simulation of single and tandem foil cases to determine the magnitude of these effects showed that the Betz limit would not be exceeded by a single foil system in practice, but that it is conceivable that a tandem foil system could exceed the Newman limit due to the strong unsteady vortex wake of the upstream turbine entraining additional energy into the path of the downstream turbine and maintaining pressures in the wake below ambient.
The interaction of Blasius boundary-layer flow with a compliant panel: global, local and transient analyses
- Konstantinos Tsigklifis, Anthony D. Lucey
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- Journal:
- Journal of Fluid Mechanics / Volume 827 / 25 September 2017
- Published online by Cambridge University Press:
- 22 August 2017, pp. 155-193
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We study the fluid–structure interaction (FSI) of a compliant panel with developing Blasius boundary-layer flow. The linearised Navier–Stokes equations in velocity–vorticity form are solved using a Helmholtz decomposition coupled with the dynamics of a plate-spring compliant panel couched in finite-difference form. The FSI system is written as an eigenvalue problem and the various flow- and wall-based instabilities are analysed. It is shown that global temporal instability can occur through the interaction of travelling wave flutter (TWF) with a structural mode or as a resonance between Tollmien–Schlichting wave (TSW) instability and discrete structural modes of the compliant panel. The former is independent of compliant panel length and upstream inflow disturbances while the specific behaviour arising from the latter phenomenon is dependent upon the frequency of a disturbance introduced upstream of the compliant panel. The inclusion of axial displacements in the wall model does not lead to any further global instabilities. The dependence of instability-onset Reynolds numbers with structural stiffness and damping for the global modes is quantified. It is also shown that the TWF-based global instability is stabilised as the boundary layer progresses downstream while the TSW-based global instability exhibits discrete resonance-type behaviour as Reynolds number increases. At sufficiently high Reynolds numbers, a globally unstable divergence instability is identified when the wavelength of its wall-based mode is longer than that of the least stable TSW mode. Finally, a non-modal analysis reveals a high level of transient growth when the flow interacts with a compliant panel which has structural properties capable of reducing TSW growth but which is prone to global instability through wall-based modes.
Asymptotic stability and transient growth in pulsatile Poiseuille flow through a compliant channel
- Konstantinos Tsigklifis, Anthony D. Lucey
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- Journal:
- Journal of Fluid Mechanics / Volume 820 / 10 June 2017
- Published online by Cambridge University Press:
- 05 May 2017, pp. 370-399
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The time-asymptotic linear stability of pulsatile flow in a channel with compliant walls is studied together with the evaluation of modal transient growth within the pulsation period of the basic flow as well as non-modal transient growth. Both one (vertical-displacement) and two (vertical and axial) degrees-of-freedom compliant-wall models are implemented. Two approaches are developed to study the dynamics of the coupled fluid–structure system, the first being a Floquet analysis in which disturbances are decomposed into a product of exponential growth and a sum of harmonics, while the second is a time-stepping technique for the evolution of the fundamental solution (monodromy) matrix. A parametric study of stability in the non-dimensional parameter space, principally defined by Reynolds number ($Re$), Womersley number ($Wo$) and amplitude of the applied pressure modulation ($\unicode[STIX]{x1D6EC}$), is then conducted for compliant walls of fixed geometric and material properties. The flow through a rigid channel is shown to be destabilized by pulsation for low $Wo$, stabilized due to Stokes-layer effects at intermediate $Wo$, while the critical $Re$ approaches the steady Poiseuille-flow result at high $Wo$, and that these effects are made more pronounced by increasing $\unicode[STIX]{x1D6EC}$. Wall flexibility is shown to be stabilizing throughout the $Wo$ range but, for the relatively stiff wall used, is more effective at high $Wo$. Axial displacements are shown to have negligible effect on the results based upon only vertical deformation of the compliant wall. The effect of structural damping in the compliant-wall dynamics is destabilizing, thereby suggesting that the dominant inflectional (Rayleigh) instability is of the Class A (negative-energy) type. It is shown that very high levels of modal transient growth can occur at low $Wo$, and this mechanism could therefore be more important than asymptotic amplification in causing transition to turbulent flow for two-dimensional disturbances. Wall flexibility is shown to ameliorate mildly this phenomenon. As $Wo$ is increased, modal transient growth becomes progressively less important and the non-modal mechanism can cause similar levels of transient growth. We also show that oblique waves having non-zero transverse wavenumbers are stable to higher values of critical $Re$ than their two-dimensional counterparts. Finally, we identify an additional instability branch at high $Re$ that corresponds to wall-based travelling-wave flutter. We show that this is stabilized by the inclusion of structural damping, thereby confirming that it is of the Class B (positive-energy) instability type.
A numerical simulation of the interaction of a compliant wall and inviscid flow
- Anthony D. Lucey, Peter W. Carpenter
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- Journal:
- Journal of Fluid Mechanics / Volume 234 / January 1992
- Published online by Cambridge University Press:
- 26 April 2006, pp. 121-146
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A method for numerically simulating the hydroelastic behaviour of a passive compliant wall of finite dimensions is presented. Using unsteady potential flow, the perturbation pressures which arise from wall disturbances of arbitrary form are calculated through a specially developed boundary-element method. These pressures may then be coupled to a suitable solution procedure for the wall mechanics to produce an interactive model for the wall/flow system. The method is used to study the two-dimensional disturbances which may occur on a Kramer-type compliant wall of finite length. Finite-difference methods are used to yield wall solutions driven by the fluid pressure after some perturbation from the equilibrium position. Thus, histories of surface deflection and wall energy are obtained. Such a modelling of the physics of the system requires no presupposition of disturbance form.
A thorough investigation of divergence instability is carried out. Most of the results presented in this paper concern the response of the compliant wall while (and after) a point pressure pulse, carried in the applied flow, travels over the compliant panel. Above a critical flow speed and once sufficient time has passed, the compliant wall is shown to adopt the particular profile of an unstable mode. After this divergence mode has been established, instability is realized as a slowly travelling downstream wave. These features are in agreement with the findings of experimental studies. The role of wall damping is clarified: damping serves only to reduce the growth rate of the instability, leaving its onset flow speed unchanged. The present predictions provide an improvement upon some of the unrealistic aspects of predictions yielded by travelling-wave and standing-wave treatments of divergence instability.
The response of a long compliant panel after a single-point pressure-pulse initiation, applied at its midpoint, is simulated. At flow speeds higher than a critical value, parts of the formerly (at subcritical flow speeds) upstream-travelling wave system change to travel downstream and show amplitude growth. The development of this ‘upstream-incoming’ wave illustrates how divergence instability can occur at locations upstream of the point of initial excitation. Faster flexural waves transmit energy upstream, thereafter these disturbances can evolve into slow downstreamtravelling divergence waves. The spread of the instability to locations both downstream and upstream of the point of initial excitation indicates that divergence is an absolute instability. This behaviour and the effects of wall damping clarified by the present work strongly suggest that divergence is a Class C instability.