Collaborative research: Turbulent cascades and dissipation in the 3D Navier-Stokes model

湍流无处不在，在技术应用中既有破坏性又有帮助，例如交通安全和效率，生物医学研究，气候研究或基础设施设计。一方面，抑制湍流阻力是设计和工程节能汽车的关键。了解大型飞机后湍流尾流的成因和动力学是维护机场附近空域安全的关键。另一方面，湍流混合可能是可取的——一个例子是设计更有效的药物输送系统。该项目的主要主题是严格研究由Navier-Stokes方程模拟的三维流体流动中的各种湍流表现。这是从湍流数学理论的角度考虑的，也是作为系统解的可能爆炸（奇点）的物理机制。在任何物理模型中排除奇点形成的可能性都是一个基本问题，当模型适用于诸如三维流体流动等无所不在的物理现象时更是如此。研究生也包括在该项目的工作中。项目分为三个方向：(1)涡旋拉伸和局部各向异性扩散；(2)速度场的各向同性扩散和诱导标度规律；(3)非均匀Navier-Stokes模型中的湍流输运。前两个方向来自研究人员和他们的合作者最近的工作，在三维Navier-Stokes正则性问题的背景下，展示了涡丝形成的数值和分析驱动的几何场景，显示了对数次临界性。在这种情况下，细丝的横向尺度——一种自然的、各向异性的微观尺度——触发了局部、各向异性扩散的机制，防止了奇点的可能形成。研究的第三个方向的动机来自于在流动的物理尺度上建立和加强对Kolmogorov现象学的数学支持。研究人员最近设法采用他们的物理尺度方法来研究强迫湍流。这为研究力的空间分布对湍流叶栅形成的影响以及通过掩蔽反馈/体积惩罚方法研究边界效应开辟了一条途径。研究生也包括在该项目的工作中。

GrujicDMS-1515805DascaliucDMS-1516487 Turbulence is everywhere, both disruptive and helpful in technological applications, such as safety and efficiency of transportation, biomedical research, climate studies, or infrastructure design. On one hand, suppressing turbulent drag is key in designing and engineering energy efficient vehicles. Understanding the genesis and dynamics of a turbulent wake behind a large plane is key in maintaining safety of the airspace in the proximity of an airport. On the other hand, turbulent mixing may be desirable -- an example being designing more efficient drug delivery systems. The main theme of the project is a rigorous study of various manifestations of turbulence in three-dimensional fluid flows modeled by the Navier-Stokes equations. This is considered both from the perspective of the mathematical theory of turbulence, and as a physical mechanism underlying possible blow-ups (singularities) of the solutions of the system. Ruling out the possibility of singularity formation in any physical model is a fundamental question, even more so when the model should be applicable to such an omnipresent class of physical phenomena as are 3D fluid flows. Graduate students are included in the work of the project. The project branches into three directions: (1) vortex stretching and local anisotropic diffusion, (2) isotropic diffusion of the velocity field and induced scaling laws, and (3) turbulent transport in non-homogeneous Navier-Stokes model. The first two directions follow from the recent work of the investigators and their collaborators in presenting a numerically and analytically motivated geometric scenario of vortex filaments formation that exhibits logarithmic sub-criticality in the context of the 3D Navier-Stokes regularity problem. In this scenario, the transversal scale of the filaments -- a natural, anisotropic micro-scale of the flow -- triggers the mechanism of local, anisotropic diffusion, preventing the possible formation of singularities. The motivation for the third direction of the study comes from the realm of building and reinforcing mathematical support for Kolmogorov phenomenology in physical scales of the flow. The investigators have recently managed to adopt their physical scales methodology to the study of forced turbulence. This opens up an avenue for studying the influence of the spatial distribution of the force on formation of turbulent cascades, as well as for the study of boundary effects via the masking feedback/volume penalization approach. Graduate students are included in the work of the project.