應用契比雪夫配置法探討自由傳播波浪下黏性流之穩定性分析

Stability Analysis of Viscous Flow beneath a Free-Propagating Surface Wave Using Chebyshev Collocation Method

胡家宜(Chia-I Hu)

指導教授 ： 蔡武廷

國立臺灣大學/工學院/工程科學及海洋工程學研究所/碩士(2018年)

https://doi.org/10.6342/NTU201800558

摘要

朗謬爾環流(Langmuir Circulation)係因海面波浪與風成流之交互作用形成，然於自由傳播波浪之數值模擬結果中，亦發現類似於朗謬爾環流之渦旋。本研究以Craik-Leibovich equation之線性穩定性分析，判斷Craik-Leibovich type 2 (CL2)不穩定性能否解釋此現象。此穩定性分析以正規模態展開線性擾動方程式，並以契比雪夫配置法(Chebyshev Collocation Method)求解；契比雪夫配置法乃基於契比雪夫多項式、點配置法之擴展，及衍伸出的廣義特徵值之解。研究發現最不穩定模態與於數值模擬及實驗觀察之主要渦旋對間距相近；因此，被觀察到之條痕有很高可能性為CL2不穩定性所造成。

關鍵字

自由傳播波浪 ； 朗謬爾環流 ； Craik-Leibovich equation ； 契比雪夫配置法 ； 不穩定性

並列摘要

Langmuir circulations are formed from the interaction between surface waves and wind driven currents, however in the numerical simulations of free-propagating surface waves, streaks resembling the Langmuir circulation exists. Linear stability analysis of the Craik-Leibovich equation is conducted to determine if the Craik-Leibovich type 2 (CL2) instability can explain the similarities. The analysis is done by representing the perturbations by normal-mode expansion, and solved using Chebyshev collocation method; this method is based on expansions in terms of Chebyshev polynomials, point collocation, and subsequent solution of the generalized eigenvalue problem. The most unstable mode is found to be close to the spacing of predominant vortex pairs observed in numerical simulations and laboratory experiments; hence, it is highly probable that the elongated streaks observed on the surfaces are excited by the CL2 instability.

並列關鍵字

free-propagating wave ； Langmuir circulation ； Craik-Leibovich equation ； Chebyshev collocation method ； instability

參考文獻

Boomkamp, P. A. M., Boersma, B. J., Miesen, R. H. M., & Beijnon, G. V. (1997). A Chebyshev collocation method for solving two-phase flow stability problems. Journal of Computational Physics, 132(2), 191-200.

Craik, A. D. D. (1982). The drift velocity of water waves. Journal of Fluid Mechanics, 116, 187-205.

Craik, A. D., & Leibovich, S. (1976). A rational model for Langmuir circulations. Journal of Fluid Mechanics, 73(3), 401-426.

Langmuir, I. (1938). Surface motion of water induced by wind. Science, 87(2250), 119-123.

Leibovich, S. (1977a). On the evolution of the system of wind drift currents and Langmuir circulations in the ocean. Part 1. Theory and averaged current. Journal of Fluid Mechanics, 79(4), 715-743.