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Nonlinear Resonant Interaction of a Surface Wave and Interfacial Waves

Fazeli, Meysam | 2012

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  1. Type of Document: Ph.D. Dissertation
  2. Language: Farsi
  3. Document No: 43025 (09)
  4. University: Sharif University of Technology
  5. Department: Civil Engineering
  6. Advisor(s): Jamali, Mirmosadegh
  7. Abstract:
  8. The results of a theoretical and experimental study of a resonant interaction between a surface wave and two internal waves are presented. It is shown that the motion of a surface wave in a horizontally infinite two-layer fluid can lead to generation of two oblique internal waves. The internal waves have short wave length compared to the surface wave and have nearly opposite propagation directions. The frequencies of the internal waves are approximately half of the frequency of the surface wave.
    In present thesis, Higher-order effects in sub-harmonic resonant interaction of a surface wave with a pair of interfacial waves in a two-layer fluid are studied theoretically and experimentally. Following an initial rapid growth, the interfacial waves approach a steady state of constant amplitude. An explicit solution is presented for transition to the ultimate state of the interaction. It is shown that for interaction in a wave flume it is necessary to include a 2nd pair of the interfacial waves resulting from reflection of the original pair in the analysis. Indeed two types of resonant interaction have been occurred in the done experiments in wave flume with limited width (not open see condition). One type which is called triad resonance has been created between surface wave and two interfacial waves. Other is the quadratic resonance which is produced between four created interfacial waves. All two types satisfy the resonant conditions and it is the curious phenomenon which two type of resonance have been seen simultaneously.
    Weakly nonlinear analysis is the standard tool to study behavior of waves in resonance and we used the perturbation technique to obtain the resonant coefficient in 3rd order the same as previous studies.
    The effects of different parameters on the dynamics of the interaction are investigated. The results indicate that a faster initial growth does not necessarily lead to larger ultimate amplitude. Also, there are two angles at which the interfacial waves continue to grow at the initial growth rate, possibly leading to wave breaking. The results are in qualitative agreement with previous experimental observations.
    There is a need for experimental data to test the theory for large amplitudes. We present laboratory measurements of large interfacial waves forming a 3D standing wave in a two-layer tank when they are generated resonantly by a surface wave. The standard nonlinear theory over-predicts the ultimate amplitude of the standing interfacial wave by a large factor. We show that interfacial mass diffusion due to small-scale instability waves growing on the interfacial wave is responsible for the difference. After the diffusion effect is included, the predictions agree with the measurements. The modified model suggests that the interfacial wave stops growing after the energy transfer from the surface wave to the interface decreases as a result of a frequency mismatch between the waves caused by the interfacial mixing. Based on the current findings, we can speculate on other situations where the standard nonlinear theory may fail.
    Some experiments were motivated by the need for reliable data on the growth rate of the interfacial waves for investigating diffuse layer thickness effects on damping and growth rate. Measurements of the growth and the damping rates are compared with the results of an asymptotic theory. The effect of a diffuse interface on the viscous dissipation and interfacial instability is considered. The theory is well supported by the experiments.
    At the end, we did some experiments base on interaction of a surface wave with interfacial waves at the interface of water and muddy water. Muddy water is a layer of mixed water and kaolin which has a stratified density profile and the natural frequency can be defined for it. The results show in lower natural frequency of muddy water, the lower layer behave look like a layer with average density
  9. Keywords:
  10. Internal Waves ; Nonlinear Interaction ; Perturbation Method ; Frequency Shift ; Muddy Water ; Diffusion Layer

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