The physical foundation of the main mathematical concepts in the nonstandard analysis theory of turbulence are presented and discussed. The basic fact is that there does not exist the absolute zero fluid-volume. Therefore, the corresponding physical object to the absolute point is just the uniform fluid-particle. The fluid-particle, in general, corresponds to the monad. The uniform fluid-particle corresponds to the uniform monad, the nonuniform fluid-particle to the nonuniform monad. There are two kinds of the differentiations, one based on the absolute point, another based on the monad. The former is adopted in the Navier-Stokes equations, the latter in the fundamental equations, the closed forms are the equations (7)-(11) in this article, in the nonstandard analysis theory of turbulence. The continuity of fluid is shown by virtue of the concepts of the fluid-particle and fluid-particle in lower level. The character of the continuity in two cases, the standard and nonstandard analysis, is presented in the paper. And the difference in the discretization between the Navier-Stokes equations and the equations (7)-(11) is pointed out too.
Vortex has been considered as the building block and muscle of turbulence for long time. A new physical quantity called Liutex (previously named Rortex) has been defined as the rigid rotation part of fluid motion. From DNS and experiment, forest of hairpin vortices has been found in flow transition and low Reynolds turbulence, but more and more one leg vortices appear in fully developed turbulence. This letter shows hairpin vortex is unstable. Hairpin vortex will weaken or lose one leg by the shear and vortex interaction. This conclusion is made by Liutex definition and mathematical analysis w…
Foundations of a solved-volatility stochastic turbulence closure: Itô--Hencky kinematics, source-consistent momentum and finite-correlation realisation
Data from experiments and direct simulations of turbulence have historically been used to calibrate simple engineering models such as those based on the Reynolds-averaged Navier--Stokes (RANS) equations. In the past few years, with the availability of large and diverse datasets, researchers have begun to explore methods to systematically inform turbulence models with data, with the goal of quantifying and reducing model uncertainties. This review surveys recent developments in bounding uncertainties in RANS models via physical constraints, in adopting statistical inference to characterize mode…
Emergent Transfer of a Physics Foundation Model from Simulation to Laboratory Turbulence
University of Cambridge（剑桥大学）；CEA, DAM/DIF（法国CEA DAM/DIF）；Flatiron Institute（Flatiron研究所）；New York University（纽约大学）；Princeton University（普林斯顿大学）；Yale University（耶鲁大学）；AIM, Université Paris-Saclay, Université Paris Cité, CEA, CNRS（AIM，巴黎-萨克雷大学，巴黎城市大学，CEA，CNRS）；University of Wisconsin–Madison（威斯康星大学麦迪逊分校）；Polymathic AI（聚合人工智能）
A recent thrust in turbulence closure modeling research is to incorporate machine learning (ML) elements, such as neural networks, for the purpose of enhancing the predictive capability to a broader class of flows. Such a turbulence closure framework entails solving a system of equations comprised of ML functionals coupled with traditional (physics-based - PB) elements. While combining closure elements from fundamentally different ideologies can lead to unprecedented progress, there are many critical challenges that must be overcome. This study examines three such challenges: (i) Physical comp…
Turbulent flow control has numerous applications and building reduced-order models (ROMs) of the flow and the associated feedback control laws is extremely challenging. Despite the complexity of building data-driven ROMs for turbulence, the superior representational capacity of deep neural networks has demonstrated considerable success in learning ROMs. Nevertheless, these strategies are typically devoid of physical foundations and often lack interpretability. Conversely, the Proper Orthogonal Decomposition (POD) based Galerkin projection (GP) approach for ROM has been popular in many problems…
As we all know, the Fourier transform is continuous in the weak sense of tempered distribution; this ensures the weak stability of Fourier pairs. This article investigates a stronger form of stability of the pair of homogeneous profiles $(|x|^{-α}, c_d |ξ|^{d-α})$ on $\mathbb{R}^d$. It encompasses, for example, the case where the homogeneous profiles exist only on a large but finite range. In this case, we provide precise error estimates in terms of the size of the tails outside the homogeneous range. We also prove a series of refined properties of the Fourier transform on related questions in…
The global attractor of the Toner-Tu-Swift-Hohenberg equations of active turbulence and its properties
University of Cambridge（剑桥大学）；Université Côte d’Azur, CNRS, Institut de Physique de Nice（蔚蓝海岸大学、法国国家科学研究中心、尼斯物理研究所）；Imperial College London（伦敦帝国理工学院）；Indian Institute of Science（印度科学学院）
Transition and separation are difficult but important problems in the field of fluid mechanics. Hitherto, separation and transition problems have not been described accurately in mathematical terms, leading to design errors and prediction problems in fluid machine engineering. The nonlinear uncertainty involved in separation and transition makes it difficult to accurately analyze these phenomena using experimental methods. Thus, new ideas and methods are required for the mathematical prediction of fluid separation and transition. In this article, after an axiomatic treatment of fluid mechanics…
In classical viscous fluids, turbulent eddies are known to be responsible for the rapid spreading of embedded particles. But in an inviscid quantum fluid where the turbulence is induced by a chaotic tangle of quantized vortices, dispersion of the particles is achieved via a non-classical mechanism, i.e., their binding to the evolving quantized vortices. However, there is limited existing knowledge on how the vortices diffuse and spread in turbulent quantum fluids. Here we report a systematic numerical study of the apparent diffusion of vortices in a random vortex tangle in superfluid helium-4 …
