Analysis of multifractal natures in turbulence based on the dynamics of fine-scale coherent structures

结果表明，二维Boussinesq湍流具有与三维N-S湍流相似的特征。特别是叶栅过程和细尺度相干结构的同时存在，表明湍流应该由某种普遍的机制来维持。我们相信，这些结果将大大有助于对湍流的基本理解。1.发现并定义了由T涡度形成的细尺度相干结构，发现在二维Boussinesq湍流中，由伸展和耗散平衡维持的弦状温度激波是细尺度相干结构，具有比小尺度特征时间更长的寿命和相当长的关联长度。细尺度相干结构的这些特征与三维Navier-Stokes湍流的特征非常相似。此外，还表明这些结构可用肛门…近似与Burgers涡旋层相对应的更多YTIC解。细尺度相干结构在湍流扩散中的作用已经有了一定的认识，虽然细尺度相干结构具有相当长的关联长度，但在Kolmogorov标度意义上，它们本质上对相对湍流扩散有贡献。这意味着，即使在湍流的低阶特性中，细尺度相干结构也起着重要的作用。结果表明有限时间奇异性的存在。我们开发了一种高分辨率的方法，即在需要更高空间分辨率的局部区域堆积新的网格：自适应网格加密方法。在此方案下，我们得到了理想二维Boussinesq系统存在有限时间奇异性的结果，奇异性似乎是由细尺度相干结构的自相似不稳定性触发的。这表明，即使在不可见的极限结构变得不稳定，然后湍流的特征可能不同于目前所接受的。较少

It has shown that two dimensional Boussinesq turbulence has characteristics similar to those of three- dimensional Navier-Stokes turbulence. In particular, the existence of both cascade process and fine-scale coherent structures suggests that turbulence should be maintained by some universal mechanism. We believe that these results will contribute to the fundamental understanding of turbulence considerably. The main results are the following:1. The fine scale coherent structures formed by T vorticity have been found and defined.We found that strings-like temperature shocks maintained by the balance of stretching and dissipation are regarded as fine scale coherent structures in 2D Boussinesq turbulence, which have a life time longer than the characteristic times of small scales and quite long correlation length. These characters of fine-scale coherent structures are quite similar to those in 3D Navier-Stokes turbulence. Moreover it is shown that these structures are approximated by anal … More ytic solutions corresponding to the Burgers' vortex layer.2. The role of fine scale coherent structures in turbulence diffusion has been understood partially.We have shown that although fine scale coherent structures have quite long correlation length, they essentially contribute to the relative turbulent diffusion in a sense pf Kolmogorov scaling. This means that fine scale coherent structures play an important role even in lower order characteristics of turbulence.3. The result suggesting the existence of finite time singularity has been obtained.We developed a high resolution scheme in which new meshes are piled up at local regions where finer spatial resolution is required : adaptive mesh refinement method. With this scheme we have obtained the result suggesting the existence of finite-time singularity in ideal two dimensional Boussinesq system.The singularity seems to be triggered by a self-similar instability of a fine scale coherent structure. This suggests that even in the invisid limit structures become unstable and then characteristics of turbulence might be different from those accepted presently. Less