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Signature Redacted Author The instability of axial-symmetric gravity-capillary waves generated by a vertically-oscillating sphere by Meng Shen Submitted to the Department of Mechanical Engineering in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Mechanical Engineering at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY February 2019 @ Massachusetts Institute of Technology 2019. All rights reserved. Signature redacted Author ....... .................. 'bepartment of Mechanical Engineering November 15, 2018 Certified by.. ...................... Yuming Liu Senior Research Scientist Thesis Supervisor Signature redacted Accepted by .............. MASSACHUSlS INSTTUTE Nicolas Hadjiconstantinou OF TECHNOLOGY Chairman, Committee on Graduate Students FEB 252019 LIBRARIES ARCHIVES 2 The instability of axial-symmetric gravity-capillary waves generated by a vertically-oscillating sphere by Meng Shen Submitted to the Department of Mechanical Engineering on November 15, 2018, in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Mechanical Engineering Abstract When a floating sphere is forced to oscillate vertically, axial-symmetric outgoing ring waves are generally expected to be produced. Laboratory experiments, however, show that when the oscillation amplitude of the sphere exceeds a threshold value, the axial-symmetric waves abruptly transfigure into asymmetric waves. This problem is related to interfacial instability phenomena widely seen in lab model tests such as sloshing, ship model wakes measurements, etc. Despite its fundamental importance, the mechanism that governs the occurrence of this phenomenon is still unknown. The objective of this thesis is to understand the mechanism of this instability phenomenon using theoretical analysis and direct numerical simulations. We first theoretically show that for an arbitrary three-dimensional body floating in an unbounded free surface, there exists a set of homogeneous solutions at any frequency in the gravity-capillary wave context. The homogeneous solution depends solely on the mean free-surface slope at the waterline of the body and physically represents progressive radial cross-wave. Unlike standing cross-waves, progressive cross-wave loses energy during propagation by overcoming the work done by surface tension at the waterline and through wave radiation to far field. We then theoretically investigate the problem of subharmonic resonant interaction of progressive ring wave with progressive cross-wave. We derive the nonlinear spatial- temporal evolution equation governing the motion of cross-wave by use of the average Lagrangian method. In addition to energy-input terms from the interaction with forced ring wave, the evolution equation contains a damping term associated with energy loss in cross-wave propagation. We show that the presence of the damping term leads to a non-trivial threshold value of oscillation amplitude beyond which the cross-wave becomes unstable and grows with time by taking energy from the ring wave. The theoretical prediction of the characteristic features of generated radial cross-waves agrees well with experimental observations, but the threshold value of oscillation amplitude is about 50% smaller. We finally investigate the instability of finite-amplitude progressive ring waves by direct numerical simulations. The analysis employs the transition matrix (TM) 3 approach and uses a quadratic boundary-element method (QBEM) for computation of the fully-nonlinear wave dynamics. When the nonlinear ring wave effects and viscous effects are accounted for, the predicted threshold value of sphere oscillation amplitude matches the experimental data excellently. In the case of relatively small-amplitude oscillations, the growth rates and shape of the unstable modes from the TM-QBEM computation agree well with the weakly nonlinear theoretical analysis we developed. This further confirms that the fundamental mechanism of the instability is associated with the triad resonance of the progressive ring wave with its subharmonic progressive radial cross-waves. The dependence of threshold value and growth rate of unstable modes on the physical parameters (such as oscillation frequency and amplitude of the body, initial phase of the disturbance) is also investigated and quantified. Thesis Supervisor: Yuming Liu Title: Senior Research Scientist 4 Acknowledgments First of all, I would like to thank my thesis committee members. Professor Yue provided a lot of background and knowledge for the PhD work. His comments are very inspiring and challenging, which really make me thinking. Professor Akylas provided many critical comments for the thesis. Also his lecture notes are very helpful for my theoretical work. Last but not least, great thanks to my advisor, Dr Yuming Liu. We have been working together for long time. I can not work the PhD problem out without his help from all aspects. Special thanks go to all the former and current VFRL members. I learnt a lot of help from the group, and that defines my daily life in MIT. Finally, thanks my family for their constant support for my PhD. I would like to express my great gratitude to them. 5 6 Contents 1 Introduction 21 2 Linear homogenous problem 27 2.1 Mathematic formulation for gravity-capillary wave interaction with a body .......... .................................... 28 2.1.1 Nonlinear initial boundary-value problem ......... ... 29 2.1.2 Linearized frequency-domain problem .. ........ .... 30 2.2 Theoretical proof of the existence of homogeneous solutions ...... 31 2.3 Linear homogeneous cross-wave solution for a vertical circular cylinder 34 2.4 Frequency domain boundary value problem for gravity-capillary wave- body interaction ... ...... ...... ...... ..... .... 36 2.5 Conclusion .......... ............ ........... 43 3 Subharmonic resonant interaction of progressive axil-symmetric wave with homogenous progressive cross-wave 45 3.1 Introduction ................................ 45 3.2 The average Lagrangian method ........... .......... 48 3.3 Linear forced axial-symmetric wave solution .............. 50 3.4 Evolution equation for cross-wave .................... 53 3.5 Results for the vertical circular cylinder case ............. 55 3.6 Results for a half-submerged vertically-oscillating sphere ....... 62 3.7 Conclusion ................................. 64 7 4 Instability of gravity-capillary progressive finite-amplitude ring waves 69 4.1 Introduction ....................... ......... 69 4.2 Mathematical Formulation .... .................... 73 4.2.1 Problem statement . ....................... 73 4.2.2 Instability-analysis method ... ............ ..... 76 4.3 Direct numerical fully-nonlinear simulation method .......... 79 4.3.1 Surface tension boundary condition ...... ......... 81 4.3.2 Far field Orlanski-Sommerfeld boundary condition ....... 82 4.4 Results of direct numerical simulation of instability evolution ..... 83 4.4.1 Validation of QBEM using a vertical swelling-contraction cir- cular cylinder ........................... 83 4.4.2 Validation of base flow and homogenous solution for a vertically- oscillation sphere ......................... 85 4.4.3 Characteristic properties of the instability evolution ...... 86 4.5 Comparison of fully nonlinear simulation with theory and experiment 94 4.5.1 Nonlinear base flow effect on the instability .......... 94 4.5.2 Comparison with theory for a vertically-oscillating half sub- merged sphere ........................... 97 4.5.3 Comparison with experiment data ................ 99 4.6 Dependence of instability on other physical parameters ........ 101 4.6.1 Initial disturbance phase effect .............. .... 101 4.6.2 Initial disturbance frequency effect ............... 101 4.7 Conclusion ................................. 104 5 Conclusion and future work 107 5.1 Conclusions ............. 107 5.2 Future works ............................... 109 5.2.1 The nonlinear interaction involving a interfacial surface and a floating body in high frequency range .............. 110 5.2.2 Viscous effect on nonlinear wave-body interaction .... ... 110 8 A Derivation of the cross-wave solution 113 B The derivation of average Lagrangian 115 B.1 Derivation of the average Lagrangian term Ll ... 115 B.2 Derivation of the average Lagrangian forcing term Lou . 118 B.3 Derivation of the evolution equation using Hamilton's principle. .. 119 C Numerical implement for surface tension term 123 D Base flow asymptotic solution 125 E Estimation of viscous dissipation 127 E.1 Viscous dissipation around the body and free surface ......... 127 E.2 Estimation of damping coefficient in direct numerical simulation ... 128 9 10 List of Figures 2-1 Definition sketch of the wave-body interaction problem. A surface- piercing body floating on the free surface is forced to oscillate, radiating waves propagating away from the body. D(t) represents the instanta- neous fluid domain, SB (t) the body surface, SF (t) the free surface, and S. (t) the fictitious vertical cylindrical closure boundary at far field. The Cartesian coordinate system (x, y, z) and cylindrical coordinate system (r, 0, z) share the same origin located on the mean free surface. 29 2-2 An illustrative sketch of the problem in the proof of the existence of a homogeneous solution. A fluid region E is bounded by two hemispher- ical surfaces, S1 and S2, and a portion of the mean free surface, S. The surface-piercing body with mean body surface SB is encircled by S1. 32 2-3 (a) Real and (b) imaginary parts of cross-wave profiles,
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