In homogeneous isotropic turbulence, slender rods are known to align with the Lagrangian stretching direction. However, how the degree of alignment depends on the aspect ratio of the rod is not understood. Moreover, many flows of practical interest are anisotropic and inhomogeneous. Here we study the alignment of rods with the Lagrangian stretching direction in a channel flow, which is approximately homogeneous and isotropic near the center but inhomogeneous and anisotropic near the walls. Our main question is how the distribution of relative angles between a rod and the Lagrangian stretching direction depends on the aspect ratio of the rod and upon the distance of the rod from the channel wall. We find that the distribution exhibits two regimes: a plateau at small angles that corresponds to random uncorrelated motion, and power-law tails that describe large excursions. The variance of the relative angle is described by the width of the plateau. We find that slender rods near the channel center align better with the Lagrangian stretching direction, compared to those near the channel wall. These observations are explained in terms of simple statistical models based on Jeffery's equation, qualitatively near the channel center and quantitatively near the channel wall. Lastly we discuss the consequences of our results for the distribution of relative angles between the orientations of nearby rods (Zhao et al., Phys. Rev. Fluids, vol. 4, 2019, 054602).
Stretching in continuum mechanics is naturally described using the Cauchy-Green strain tensors. These tensors quantify the Lagrangian stretching experienced by a material element, and provide a powerful way to study processes in turbulent fluid flows that involve stretching such as vortex stretching and alignment of anisotropic particles. Analyzing data from a simulation of isotropic turbulence, we observe preferential alignment between anisotropic particles and vorticity. We show that this alignment arises because both of these quantities independently tend to align with the strongest Lagrang…
Small flexible fibers in a turbulent flow are found to be most of the time as straight as stiff rods. This is due to the cooperative action of flexural rigidity and fluid stretching. However, fibers might bend and buckle when they tumble and experience a strong-enough local compressive shear. Such events are similar to an activation process, where the role of temperature is played by the inverse of Young's modulus. Numerical simulations show that buckling occurs very intermittently in time. This results from unexpected long-range Lagrangian correlations of the turbulent shear.
Viscoelastic flows are pervasive in a host of natural and industrial processes, where the emergence of nonlinear and time-dependent dynamics regulate flow resistance, energy consumption, and particulate dispersal. Polymeric stress induced by the advection and stretching of suspended polymers feeds back on the underlying fluid flow, which ultimately dictates the dynamics, instability, and transport properties of viscoelastic fluids. However, direct experimental quantification of the stress field is challenging, and a fundamental understanding of how Lagrangian flow structure regulates the distr…
拓扑混沌依赖于二维流中障碍物的周期性运动，以形成非平凡的辫子。这种运动产生材料线的指数拉伸，从而产生有效的混合。Boyland等[P. L. Boyland，H. Aref和M. A. Stremler，J.FluidMech.403，277（2000）]研究了在粘性流体中表现出拓扑混沌的杆的特定周期运动。我们表明，这是可能的，以延长他们的工作的情况下，搅拌棒的运动是拓扑平凡的考虑特殊的周期点，我们称之为鬼棒的动力学，因为他们发挥了类似的作用，搅拌棒。鬼棒框架提供了一种新的技术来量化混沌，并深入了解产生混沌和混合的机制。Stokes流的数值模拟支持我们的结果。
We study the relation between flow structure and fluid deformation in steady two-dimensional random flows. Beyond the linear (shear flow) and exponential (chaotic flow) elongation paradigms, we find a broad spectrum of stretching behaviors, ranging from sub- to superlinear, which are dominated by intermittent shear events. We analyze these behaviors from first principles, which uncovers stretching as a result of the non-linear coupling between Lagrangian shear deformation and velocity fluctuations along streamlines. We derive explicit expressions for Lagrangian deformation and demonstrate that…
How pore-scale disorder controls fluid stretching in two-dimensional porous media
University of Oslo（奥斯陆大学）；University of Rennes（雷恩大学）；Univ. Grenoble Alpes, Univ. Savoie Mont Blanc, CNRS, IRD, Univ. Gustave Eiffel, ISTerre（格勒诺布尔阿尔卑斯大学、萨瓦蒙布朗克大学、法国国家科学研究中心、法国发展研究所、居斯塔夫埃菲尔大学、ISTerre研究所）；Norwegian University of Science and Technology（挪威科技大学）
Lagrangian Phase-Lag and Geometric Precedence in Turbulent Vortex Stretching
Mechanical Engineering Deparatment, College of Engineering and Technology（机械工程系，工程学院与技术学院）；Arab Academy for Science, Technology and Maritime Transport（阿拉伯科学、技术与海运学院）
Vortex Stretching in the Navier-Stokes Equations and Information Dissipation in Diffusion Models: A Reformulation from a Partial Differential Equation Viewpoint
Graduate School of Economics, Hitotsubashi University（早稻田大学经济学研究科）
We introduce a new method to measure Lagrangian vorticity and the rotational dynamics of anisotropic particles in a turbulent fluid flow. We use 3D printing technology to fabricate crosses (two perpendicular rods) and jacks (three mutually perpendicular rods). Time-resolved measurements of their orientation and solid-body rotation rate are obtained from stereoscopic video images of their motion in a turbulent flow between oscillating grids with $R_λ$=$91$. The advected particles have a largest dimension of 6 times the Kolmogorov length, making them a good approximation to anisotropic tracer pa…
In this paper we present an extension of the \emph{Recent Fluid Deformation (RFD)} closure introduced by Chevillard and Meneveau (2006) which was developed for modeling the time evolution of Lagrangian fluctuations in incompressible Navier-Stokes turbulence. We apply the RFD closure to study the evolution of magnetic and passive scalar fluctuations. This comparison is especially interesting since the stretching term for the magnetic field and for the gradient of the passive scalar are similar but differ by a sign such that the effect of stretching and compression by the turbulent velocity fiel…
We consider the 3D incompressible Euler equations under the following situation: small-scale vortex blob being stretched by a prescribed large-scale stationary flow. More precisely, we clarify what kind of large-scale stationary flows really stretch small-scale vortex blobs in alignment with the straining direction. The key idea is constructing a Lagrangian coordinate so that the Lie bracket is identically zero (c.f. the Frobenius theorem), and investigate the locality of the pressure term by using it.
In this work, we study the role of viscoelastic instability in the mechanical dispersion of fluid flow through porous media at high Peclet numbers. Using microfluidic experiments and numerical simulations, we show that viscoelastic instability in flow through a hexagonally ordered (staggered) medium strongly enhances dispersion transverse to the mean flow direction with increasing Weissenberg number (Wi). In contrast, preferential flow paths can quench the elastic instability in disordered media, which has two important consequences for transport: First, the lack of chaotic velocity fluctuatio…
We introduce Lagrange2D, a Mathematica package for analysis and characterization of complex fluid flows using Lagrangian transport metrics. Lagrange2D includes built-in functions for integrating ensembles of trajectories subject to time-varying two-dimensional flows, as well as utilities for calculating various quantities of interest, such as finite-time Lyapunov exponents, stretching vector fields, the fractal dimension, and flushing times. The package also includes tools for visualizing transport and pathlines, as well as for generating videos. This package aims to ease rapid characterizatio…
Large eddy simulation allows to incorporate the important driving physics of turbulent flow through forest- or vegetation-like canopies. In this paper we investigate the effects of vortex stretching and coherent structures on the subgrid-scale (SGS) turbulence kinetic energy (TKE). We present three simulations (SGS-d/s/w) of turbulence-canopy interactions. SGS-d assumes a local, dynamic balance of SGS production with SGS dissipation. SGS-s averages the SGS contributions of coherent structures over Lagrangian pathlines. SGS-w assumes that an average cascade of TKE from large- to small-scales oc…
Steady laminar flows through porous media spontaneously generate Lagrangian chaos at pore scale, with qualitative implications for a range of transport, reactive and biological processes. The characterization and understanding of mixing dynamics in these opaque environments is an outstanding challenge. We address this issue by developing a novel technique based upon high-resolution imaging of the scalar signature produced by push-pull flows through porous media samples. Owing to the rapid decorrelation of particle trajectories in chaotic flows, the scalar image measured outside the porous mate…
We study small-scale and high-frequency turbulent fluctuations in three-dimensional flows under Fourier-mode reduction. The Navier-Stokes equations are evolved on a restricted set of modes, obtained as a projection on a fractal or homogeneous Fourier set. We find a strong sensitivity (reduction) of the high-frequency variability of the Lagrangian velocity fluctuations on the degree of mode decimation, similarly to what is already reported for Eulerian statistics. This is quantified by a tendency towards a quasi-Gaussian statistics, i.e., to a reduction of intermittency, at all scales and frequ…
We analyze the dynamics of solute mixing in a vortex flow. The transport of a passive tracer is considered in a Rankine vortex. The action of a shear flow, in general, is to achieve stretching of fluid elements. A vortex flow exhibits stretching and folding of fluid elements in a way which brings adjacent fluid elements closer every turn. A strong stretching along the direction of rotation is accompanied by a concomitant thinning in the radial direction leading to a strong diffusive flux which may cause material from neighbouring regions of the mixing interface to aggregate. Through a Lagrangi…
Non-spherical particles transported by an anisotropic turbulent flow preferentially align with the mean shear and intermittently tumble when the local strain fluctuates. Such an intricate behaviour is here studied for inertialess, rod-shaped particles embedded in a two-dimensional turbulent flow with homogeneous shear. A Lagrangian stochastic model for the rods angular dynamics is introduced and compared to the results of direct numerical simulations. The model consists in superposing a short-correlated random component to the steady large-scale mean shear, and can thereby be integrated analyt…
In this work, we propose a model for the orientation of inertialess spheroidal particles suspended in turbulent flows. This model consists in a stochastic version of the Jeffery equation that can be included in a statistical Lagrangian description of particles suspended in a flow. It is compatible and coherent with turbulence models that are widely used in CFD codes for the simulation of the flow field in practical large-scale applications. In this context, we propose and analyze a numerical scheme based on a splitting scheme algorithm that decouples the orientation dynamics into its main cont…
We develop a stochastic model for the velocity gradients dynamics along a Lagrangian trajectory. Comparing with different attempts proposed in the literature, the present model, at the cost of introducing a free parameter known in turbulence phenomenology as the intermittency coefficient, gives a realistic picture of velocity gradient statistics at any Reynolds number. To achieve this level of accuracy, we use as a first modelling step a regularized self-stretching term in the framework of the Recent Fluid Deformation (RFD) approximation that was shown to give a realistic picture of small scal…
The influence of the underlying flow topology on the shape and size of sub-Kolmogorov droplets dispersed in a turbulent flow is of considerable interest in many industrial and scientific applications. In this work we study the deformation and orientation statistics of sub-Kolmogorov droplets dispersed into a turbulent Taylor-Couette flow. Along with Direct Numerical Simulations (DNS) of the carrier phase and Lagrangian tracking of the dispersed droplets, we solve a phenomenological equation proposed by Maffettone and Minale (\emph{J. Fluid Mech.} 78, 227-241 (1998)) to track the shape evolutio…
