Studies of Turbulence as a Nonlinear Science

湍流作为非线性科学的研究

• 批准号: 08404006
• 负责人: KIMURA Yoshifumi
• 金额: $ 12.8万
• 依托单位: Nagoya University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (A)
• 财政年份: 1996
• 资助国家: 日本
• 起止时间: 1996 至 1999
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-08404006/
• 关键词: turbulence; fluid mechanics; vortex motion; diffusion; geophysical flows; Navier-Strokes equations; Hamiltonian dynamics; Group theory; 自己相似性; 特異点; エネルギーカスケード; 渦度; 密度成層; ランダム現象; シンプレクティック積分; ランダムネス

(1) Vortex Formation and Diffusion in Rotating Stratified Turbulence : Rotation and Stratification are two fundamental physical mechanisms in atmospheres and oceans. Studies of vortex formation and diffusion in such flows play a vital role in improving our understanding of geophysical and astrophysical turbulent phenomena. We examine results of direct numerical simulation of the Navier-Stokes equations with the effects of rotation and stably stratification. Both rotation and stratifications tend to make flows two dimensional (or two components), but we verified that they work in different ways in generating vertical structures. As for diffusion, we observed that stratification suppresses the particle migration in the vertical direction and so does rotation, and that the linear growth in time for single particle dispersion persists in the horizontal direction under strong rotation and/or stratification.(2) Vortex Motion on Surfaces with Constant Curvature : Vortex motion on two- dimensi … More onal Riemannian surfaces with constant curvature is formulated. By way of the stereographic projection, the relation and difference between the vortex motion on a sphere and on a hyperbolic plane can be clearly analyzed. The Hamiltonian formalism is presented for the motion of point vortices on a sphere and a hyperbolic plane. As an example of analytic solutions, the motion of a vortex pair (dipole) is considered. It is shown that a dipole draws a geodesic curve as its trajectory on both surfaces.(3) Evolution of decaying two-dimensional turbulence and self similarity : We examine the consequences of self-similarity of the energy spectrum of two-dimensional decaying turbulence, and conclude that traditional closures are consistent with this principle only if the regions of space contributing significantly to energy and enstrophy transfer comprise an ever diminishing region of space as time proceeds from the initial time of Gaussian chaos.(4) Axisymmetrization process for a non-uniform elliptic vortex : The axisymmetrization of a 2D non-uniform elliptic vortex is studied in terms of the growth of palinstrophy, the squared of the vorticity gradient. First, it is pointed out that the equation for the palinstrophy growth, if written in terms of the strain rate tensor, has a similar form to that of enstrophy growth in 3D - the vortex-stretching equation. Then palinstrophy production is analyzed particularly for non-uniform elliptic vortices. It is shown analytically and verified numerically that a non-uniform elliptic vortex in general has quadrupole structure for the palinstrophy production, and that in the positive production regions, vortex filaments are ejected following the gradient enhancement process for vorticity.(5) Pressure distribution for random Gaussian Velocities : Pressure distributions for random Gaussian velocities are studied both analytically and numerically. Arguments based on rotation symmetry allow to clarify the analytical structure of the characteristic function of pressure and to find the power of all its singularities, which in turn, allows to obtain the exact form of the PDF tail, including both exponent and power pre-exponent factors. For the narrow velocity spectrum (velocity restricted to a shell in k-space), the characteristic function is found explicitly, generating pressure cummulants of all orders. Less

(1) 旋转层状湍流中涡旋的形成和扩散：旋转和层化是大气和海洋中的两种基本物理机制。对此类流动中涡流形成和扩散的研究对于提高我们对地球物理和天体物理湍流现象的理解发挥着至关重要的作用。我们研究了具有旋转和稳定分层影响的纳维-斯托克斯方程的直接数值模拟结果。旋转和分层都倾向于使流动二维（或两个分量），但我们验证了它们在生成垂直结构方面以不同的方式工作。至于扩散，我们观察到分层抑制了粒子在垂直方向上的迁移，旋转也抑制了粒子的迁移，并且在强旋转和/或分层下，单粒子分散的时间线性增长在水平方向上持续存在。(2) 具有恒定曲率的表面上的涡旋运动：公式化了具有恒定曲率的二维黎曼表面上的涡旋运动。通过球极投影，可以清楚地分析球面上涡旋运动与双曲平面上涡旋运动的关系和区别。哈密​​顿形式主义是针对球体和双曲平面上的点涡运动而提出的。作为解析解的示例，考虑涡旋对（偶极子）的运动。结果表明，偶极子在两个表面上绘制测地线作为其轨迹。（3）衰减二维湍流的演化和自相似性：我们研究了二维衰减湍流能谱的自相似性的后果，并得出结论，只有当对能量和熵传递有显着贡献的空间区域包含一个永远存在的区域时，传统的闭包才符合这一原理。 从高斯混沌的初始时间开始，空间区域随着时间的推移而减小。(4)非均匀椭圆涡旋的轴对称过程：根据回文增长（涡度梯度的平方）研究二维非均匀椭圆涡旋的轴对称化。首先，指出回文增长方程如果用应变率张量写成，则具有与 3D 中的熵增长相似的形式——涡拉伸方程。然后，特别针对不均匀的椭圆形涡旋，分析回文的产生。分析和数值验证表明，非均匀椭圆涡旋一般具有回文产生的四极结构，并且在正产生区域，涡丝在涡度梯度增强过程中喷射出来。（5）随机高斯速度的压力分布：对随机高斯速度的压力分布进行了分析和数值研究。基于旋转对称性的论证可以阐明压力特征函数的分析结构，并找到其所有奇点的幂，进而获得 PDF 尾部的精确形式，包括指数和幂前因子。对于窄速度谱（限制在 k 空间中的壳的速度），可以明确地找到特征函数，从而生成所有阶次的压力累积量。较少的

项目成果

Y. Kimura and O. Metais: "Vortex Formation and diffusion in rotating stratified turbulence"Development in Geophysical Turbulence. (in press). (2000)

Y. Kimura 和 O. Metais：“旋转分层湍流中的涡流形成和扩散”地球物理湍流的发展。

Yoshifumi Kimura: "Vortex motion on surfaces with constant curvature" Proc.of Royal Soc.of London,Ser.A. (in press).

Yoshifumi Kimura：“具有恒定曲率的表面上的涡运动”Proc.of Royal Soc.of London，Ser.A。

Y. Kimura and J. R. Herring: "Particle dispersion in rotating stratified turbulence"Proc. Fluid Engng Div. Summer Meeting. (in press). (1999)

Y. Kimura 和 J. R. Herring：“旋转分层湍流中的粒子分散”Proc。

Takashi Kumagai: "Short time asymptotic behavior and large deviations for Brownian motion on some affine nested fractals" Publ.RIMS.Kyoto Univ.33-2. 223-240 (1997)

Takashi Kumagai：“某些仿射嵌套分形上布朗运动的短时渐近行为和大偏差”Publ.RIMS.Kyoto Univ.33-2。

Y. Kimura: "Vortex Motion on surfaces with constant curvature"Proc. R. Soc. London Ser A. 455. 245-259 (1999)

Y. Kimura：“具有恒定曲率的表面上的涡运动”Proc。