A proposal for a calculational program in fluid turbulence is presented. It is proposed that the fluid probability density functional has an attractor for its time-evolution, just as the dynamical system itself has. The evolution of the fluid's statistics can be set up as a space-time path integral. Fluid space-time configuration sampling techniques naturally suggest a useful way, using an arbitrary initial statistics functional, to calculate averages.
This work introduces a formulation of resolvent analysis that uses wavelet transforms rather than Fourier transforms in time. This allows resolvent analysis to be extended to turbulent flows with non-stationary means in addition to statistically-stationary flows. The optimal resolvent modes for this formulation correspond to the potentially time-transient structures that are most amplified by the linearized Navier-Stokes operator. We validate this methodology for turbulent channel flow and show that the wavelet-based and Fourier-based resolvent analyses are equivalent for statistically-station…
Imperial College London（帝国理工学院）；The Alan Turing Institute（艾伦·图灵研究所）；Delft University of Technology（代尔夫特理工大学）
The dynamics of a turbulent flow tend to occupy only a portion of the phase space at a statistically stationary regime. From a dynamical systems point of view, this portion is the attractor. The knowledge of the turbulent attractor is useful for two purposes, at least: (i) We can gain physical insight into turbulence (what is the shape and geometry of the attractor?), and (ii) it provides the minimal number of degrees of freedom to accurately describe the turbulent dynamics. Autoencoders enable the computation of an optimal latent space, which is a low-order representation of the dynamics. If …
Flows within porous media play important roles in many scientific and industrial systems. However, the case wherein such flows become turbulent has not been completely understood, particularly, from a mathematical viewpoint. In this study, the $k-\varepsilon$ model (the variable $k$ denotes the turbulent kinetic energy per unit mass and $\varepsilon$ the dissipation rate for the turbulent kinetic energy), which has been widely used for the usual turbulent flow in a clear fluid, was applied to a turbulent flow through porous media. If a homogeneous, averaged flow and homogeneous isotropic turbu…
Relation between shear parameter and Reynolds number in statistically stationary turbulent shear flows
本文研究了近似均匀的统计定常剪切湍流的剪切参数S^* 与雷诺数Re之间的关系。参数研究与最近概述的关于剪切流中局部各向同性的回归的广义观点一致[Schumacher，Sreenivasan and Yeung，Phys. Fluids，vol.15，84（2003）]。因此，系统地改变两个参数，即恒定剪切速率S和由能量注入速率epsilon_{in}规定的初始湍流波动水平。研究表明，剪切参数的水平为更大的雷诺数，这是由尺寸参数支持。结果发现，偏度的横向导数表现出不同的衰减行为，相对于雷诺数时，模拟运行的序列遵循不同的途径在两个参数的平面。该研究可以为高雷诺数实验中奇阶矩衰减的不同解释提供新的思路。
We present wall-resolved large-eddy simulations (LES) of the incompressible Navier-Stokes equations together with empirical modeling for {turbulent} Taylor-Couette {(TC)} flow where the inner cylinder is rotating with angular velocity $Ω_i$ and the outer cylinder is stationary. A simple empirical model of the turbulent, TC flow is developed consisting of near-wall, log-like turbulent wall layers separated by an annulus of constant angular momentum. The model is closed by a proposed scaling relation concerning the thickness of the wall layer on the inner cylinder. Model results include the Nuss…
We derive precursors of extreme dissipation events in a turbulent channel flow. Using a recently developed method that combines dynamics and statistics for the underlying attractor, we extract a characteristic state that precedes laminarization events that subsequently lead to extreme dissipation episodes. Our approach utilizes coarse statistical information for the turbulent attractor, in the form of second order statistics, to identify high-likelihood regions in the state space. We then search within this high probability manifold for the state that leads to the most finite-time growth of th…
We propose a unified framework to study the turbulent transport problem from the perspective of nonequilibrium statistical mechanics. By combining Krarichnan's turbulence thermalization assumption and Ruelle's recent work on nonequilibrium statistical mechanics settings for fluids, we show that the equation for viscous fluid can be viewed as the non-canonical Hamiltonian system perturbed by different thermostats. This allows an analogy between the viscous fluid and the nonequilibrium heat conduction model where the Fourier modes can be regarded as the ''particles''. With this framework, we ref…
When reporting statistics from simulations of statistically stationary chaotic phenomenon, it is important to verify that the simulations are time-converged. This condition is connected with the statistical error or number of digits with which statistics can be reliably reported. In this work we consider homogeneous and isotropic turbulence as a model problem to investigate statistical convergence over finite simulation times. Specifically, we investigate the time integration requirements that allow meaningful reporting of the statistical error associated with finiteness of the temporal domain…
Topology optimization methods face serious challenges when applied to structural design with fluid-structure interaction (FSI) loads, specially for high Reynolds fluid flow. This paper devises an explicit boundary method that employs separate analysis and optimization grids in FSI systems. A geometry file is created after extracting a smooth contour from a set of binary design variables that defines the structural design. The FSI problem can then be modeled with accurate physics and explicitly defined regions. The Finite Element Method is used to solve the fluid and structural domains. This is…
Turbulent systems exhibit a remarkable multi-scale complexity, in which spatial structures induce scale-dependent statistics with strong departures from Gaussianity. In Fourier space, this is reflected by pronounced phase synchronization. A quantitative relation between real-space structure, statistics, and phase synchronization is currently missing. Here, we address this problem in the framework of a minimal phase-coupling model, which enables a detailed investigation by means of dynamical systems theory and multi-scale high-resolution simulations. We identify the spectral power-law steepness…
University of Cambridge（剑桥大学）；Delft University of Technology（代尔夫特理工大学）；Imperial College London（帝国理工学院）；The Alan Turing Institute（艾伦·图灵研究所）
The spatiotemporal dynamics of turbulent flows is chaotic and difficult to predict. This makes the design of accurate and stable reduced-order models challenging. The overarching objective of this paper is to propose a nonlinear decomposition of the turbulent state for a reduced-order representation of the dynamics. We divide the turbulent flow into a spatial problem and a temporal problem. First, we compute the latent space, which is the manifold onto which the turbulent dynamics live (i.e., it is a numerical approximation of the turbulent attractor). The latent space is found by a series of …
Is there really such a thing as weak turbulence? Here we analyze turbulence of weakly interacting waves using the tools of information theory. It offers a unique perspective for comparing thermal equilibrium and turbulence: the mutual information between modes is shown to be stationary and small in equilibrium but grows linearly with time in weak turbulence. We trace this growth to the concentration of probability on the resonance surfaces, which can go all the way to a singular measure. The surprising conclusion is that no matter how small is the nonlinearity and how close to Gaussian is the …
