Nonlinear interaction of shear flow with a free surface
Nonlinear interaction of shear flow with a free surface
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剪切流与自由表面的非线性相互作用
DOI：
10.1017/s0022112094003496
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发表时间：
1994
影响因子：
3.7
通讯作者：
G. Triantafyllou
中科院分区：
文献类型：
--
作者：
A. Dimas;G. Triantafyllou
本文研究了近海面二维剪切流不稳定性的非线性演化。该方法是数值的，通过直接模拟的不可压缩的欧拉方程的动态和运动学的边界条件在自由表面。该问题是制定使用边界拟合坐标，并为数值模拟的频谱空间离散化方法，涉及傅立叶模式在流向方向和切比雪夫多项式沿着的深度。使用分裂方案在时间上进行显式积分。流的初始状态被假定为已知的具有平坦自由表面的平行剪切流。然后引入具有剪切流的增长最快的线性不稳定模式的形式的扰动，并且随后数值地跟踪其随后的演变。根据线性理论，具有自由表面的剪切流具有两种线性不稳定模式，对应于色散关系的不同分支：分支I，在低波数下;和分支II，在高波数下，低弗劳德数，和低波数下，高弗劳德数。我们的模拟表明，这两个分支有一个明显不同的非线性演化。分支I：在低弗劳德数时，分支I不稳定波在海洋表面正下方形成强烈的椭圆形漩涡。诱导速度场在水平方向上在自由表面高程的顶部附近呈现非常尖锐的剪切。结果，自由表面波获得陡峭的斜率，而其振幅保持非常小，最终计算机代码崩溃，表明波将破碎。分支二：在低弗劳德数下，分支II不稳定波会发展出弱涡旋，其尺寸远小于其距海面的距离。在海洋表面的诱导速度场在空间中平滑地变化，并且自由表面高程采取传播波的形式。然而，在高弗劳德数下，分支II不稳定波的增长率增加，导致强涡旋的形成。自由表面达到一个大的振幅，并在自由表面上发展强烈的垂直速度剪切。计算机代码最终崩溃，表明波将破裂。即使在无限弗劳德数的极限下，海洋表面的这种行为仍然存在。它的结论是，剪切流不稳定性的自由表面表现获得的传播水波的形式，只有当在海洋表面的诱导速度场沿沿着传播方向平滑地变化。
In this paper the nonlinear evolution of two-dimensional shear-flow instabilities near the ocean surface is studied. The approach is numerical, through direct simulation of the incompressible Euler equations subject to the dynamic and kinematic boundary conditions at the free surface. The problem is formulated using boundary-fitted coordinates, and for the numerical simulation a spectral spatial discretization method is used involving Fourier modes in the streamwise direction and Chebyshev polynomials along the depth. An explicit integration is performed in time using a splitting scheme. The initial state of the flow is assumed to be a known parallel shear flow with a flat free surface. A perturbation having the form of the fastest growing linear instability mode of the shear flow is then introduced, and its subsequent evolution is followed numerically. According to linear theory, a shear flow with a free surface has two linear instability modes, corresponding to different branches of the dispersion relation: Branch I, at low wavenumbers; and Branch II, at high wavenumbers for low Froude numbers, and low wavenumbers for high Froude numbers. Our simulations show that the two branches have a distinctly different nonlinear evolution. Branch I: At low Froude numbers, Branch I instability waves develop strong oval-shaped vortices immediately below the ocean surface. The induced velocity field presents a very sharp shear near the crest of the free-surface elevation in the horizontal direction. As a result, the free-surface wave acquires steep slopes, while its amplitude remains very small, and eventually the computer code crashes suggesting that the wave will break. Branch II: At low Froude numbers, Branch II instability waves develop weak vortices with dimensions considerably smaller than their distance from the ocean surface. The induced velocity field at the ocean surface varies smoothly in space, and the free-surface elevation takes the form of a propagating wave. At high Froude numbers, however, the growing rates of the Branch II instability waves increase, resulting in the formation of strong vortices. The free surface reaches a large amplitude, and strong vertical velocity shear develops at the free surface. The computer code eventually crashes suggesting that the wave will break. This behaviour of the ocean surface persists even in the infinite-Froude-number limit. It is concluded that the free-surface manifestation of shear-flow instabilities acquires the form of a propagating water wave only if the induced velocity field at the ocean surface varies smoothly along the direction of propagation.