非線性自由液面流場的數值模擬以及其在探討微尺度上層海洋動力之應用

Abstract

本論文建立一完整描述三維非線性自由液面流場運動的數值模式。為於未知的自由液面邊界上滿足非線性邊界條件，本模式利用座標轉換，將非穩態且不規則物理區間的流場控制方程式與邊界條件，映射至規則的計算區間，以進行數值解析。模式的主要數值方法包括：分別以擬頻譜法以及二階有限差分法近似控制方程與邊界條件中的水平與垂直導數；以二階Runge-Kutta法進行非穩態的自由液面位移與流場速度的時間積分；利用改良式割線疊代法求解計算區間中不可分離的壓力Poisson方程式，以滿足不可壓縮流條件。 本論文首先應用所發展的數值模式，探討二維重力–寄生表面張力波之發展與演化。模擬結果顯示，當寄生表面張力波完全發展時，強烈的渦漩自各個表面張力波之波谷剝離，形成高旋度的邊界層，進而大幅提升流場能量的黏滯消散。為進一步顯示本模式解析非線性波浪運動之能力，我們進行三維「短峰型重力波」與「新月型重力波」的計算；其結果完整模擬波浪非線性交互作用之過程，所形成的波型結構與水槽實驗之觀測相同。最後，我們應用本數值模式，模擬正向風壓與風剪應力作用下之三維自由液面邊界層紊流場。模擬結果顯示：寄生表面張力波列自重力波波峯之前波相處產生，並逐漸發展至波背處；同時於波背處，出現平行於沿流方向，高低流速交錯的條痕結構。此流場結構與實驗觀測相同。以上三例有關三維自由液面流場之應用，驗證本論文所發展之模式，可以正確解析自由液面流場不同尺度的物理過程；包括液面重力波、表面張力波、黏滯邊界層流場以及紊流結構。

A numerical model for simulating a three-dimensional flow bounded by a fully-nonlinear, free-surface boundary is developed. The free-surface boundary conditions are satisfied on the free-moving surface exactly without any approximation or linearization. The governing equations and the boundary conditions on the time-dependent, physical domain are transformed onto a regular domain for numerical computations. The major implementations of the developed model include: Pseudo-spectral and finite-difference schemes are employed to approximate the spatial-differential operators in the horizontal and vertical directions, respectively. A second-order Runge-Kutta method is adopted to integrate in time for the velocity field and the free-surface elevation. The solenoidal condition of the velocity field is satisfied by solving the transformed pressure Poisson equation. A modified secant method is used to accelerate the iterative solution of the non-separable Poisson equation. We first apply the numerical model to study the development and evolution of parasitic capillary ripples on a two-dimensional gravity-capillary wave. The simulation results reveal that strong vortices are shed from the troughs of the capillary ripples, convected backward, and form a strong boundary layer. As a result, energy dissipation is enhanced with the generation of parasitic capillaries. The model capabilities for resolving nonlinear interactions among the surface waves are then demonstrated by simulating the three-dimensional evolutions of the short-crested waves and the crescent waves. Finally, the developed model is applied to simulate a wind-driven turbulent boundary layer bounded by a dynamic water surface. The simulation results reveal fine surface structures, including parasitic capillary waves developed from the front face near the crest of the dominant gravity wave and streamwise velocity streaks on the back gravity-wave surface. These surface structures have been widely observed in the laboratory and field experiments. The three simulation examples also indicate the capabilities of the present model in resolving flow processes of various length scales, including gravity and capillary waves, viscous sublayer and coherent eddies of a turbulent boundary layer.