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Browne CA, Datta SS. Harnessing elastic instabilities for enhanced mixing and reaction kinetics in porous media. Proc Natl Acad Sci U S A 2024; 121:e2320962121. [PMID: 38980904 PMCID: PMC11260153 DOI: 10.1073/pnas.2320962121] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/28/2023] [Accepted: 06/07/2024] [Indexed: 07/11/2024] Open
Abstract
Turbulent flows have been used for millennia to mix solutes; a familiar example is stirring cream into coffee. However, many energy, environmental, and industrial processes rely on the mixing of solutes in porous media where confinement suppresses inertial turbulence. As a result, mixing is drastically hindered, requiring fluid to permeate long distances for appreciable mixing and introducing additional steps to drive mixing that can be expensive and environmentally harmful. Here, we demonstrate that this limitation can be overcome just by adding dilute amounts of flexible polymers to the fluid. Flow-driven stretching of the polymers generates an elastic instability, driving turbulent-like chaotic flow fluctuations, despite the pore-scale confinement that prohibits typical inertial turbulence. Using in situ imaging, we show that these fluctuations stretch and fold the fluid within the pores along thin layers ("lamellae") characterized by sharp solute concentration gradients, driving mixing by diffusion in the pores. This process results in a [Formula: see text] reduction in the required mixing length, a [Formula: see text] increase in solute transverse dispersivity, and can be harnessed to increase the rate at which chemical compounds react by [Formula: see text]-enhancements that we rationalize using turbulence-inspired modeling of the underlying transport processes. Our work thereby establishes a simple, robust, versatile, and predictive way to mix solutes in porous media, with potential applications ranging from large-scale chemical production to environmental remediation.
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Affiliation(s)
- Christopher A. Browne
- Department of Chemical and Biological Engineering, Princeton University, Princeton, NJ08544
| | - Sujit S. Datta
- Department of Chemical and Biological Engineering, Princeton University, Princeton, NJ08544
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Musacchio S, Cencini M, Plan ELCVM, Vincenzi D. Enhancement of mixing by rodlike polymers. THE EUROPEAN PHYSICAL JOURNAL. E, SOFT MATTER 2018; 41:84. [PMID: 29987441 DOI: 10.1140/epje/i2018-11692-9] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/04/2018] [Accepted: 06/25/2018] [Indexed: 06/08/2023]
Abstract
We study the mixing of a passive scalar field dispersed in a solution of rodlike polymers in two dimensions, by means of numerical simulations of a rheological model for the polymer solution. The flow is driven by a parallel sinusoidal force (Kolmogorov flow). Although the Reynolds number is lower than the critical value for inertial instabilities, the rotational dynamics of the polymers generates a chaotic flow similar to the so-called elastic-turbulence regime observed in extensible polymer solutions. The temporal decay of the variance of the scalar field and its gradients shows that this chaotic flow strongly enhances mixing.
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Affiliation(s)
| | - Massimo Cencini
- Istituto dei Sistemi Complessi, CNR, via dei Taurini 19, 00185, Roma, Italy
- INFN, Sezione di Roma Tor Vergata, via della Ricerca Scientifica 1, 00133, Roma, Italy
| | - Emmanuel L C Vi M Plan
- Rudolf Peierls Centre for Theoretical Physics, University of Oxford, Oxford OX1, 3PU, UK
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Wendell DM, Pigeonneau F, Gouillart E, Jop P. Intermittent flow in yield-stress fluids slows down chaotic mixing. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2013; 88:023024. [PMID: 24032940 DOI: 10.1103/physreve.88.023024] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/25/2013] [Indexed: 06/02/2023]
Abstract
We present experimental results of chaotic mixing of Newtonian fluids and yield-stress fluids using a rod-stirring protocol with a rotating vessel. We show how the mixing of yield-stress fluids by chaotic advection is reduced compared to the mixing of Newtonian fluids and explain our results, bringing to light the relevant mechanisms: the presence of fluid that only flows intermittently, a phenomenon enhanced by the yield stress, and the importance of the peripheral region. This finding is confirmed via numerical simulations. Anomalously slow mixing is observed when the synchronization of different stirring elements leads to the repetition of slow stretching for the same fluid particles.
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Affiliation(s)
- D M Wendell
- Surface du Verre et Interfaces, UMR 125 CNRS/Saint-Gobain, 39, quai Lucien Lefranc, F-93303 Aubervilliers Cedex, France
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Ait-Chaalal F, Bourqui MS, Bartello P. Fast chemical reaction in two-dimensional Navier-Stokes flow: initial regime. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2012; 85:046306. [PMID: 22680573 DOI: 10.1103/physreve.85.046306] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/06/2011] [Revised: 12/10/2011] [Indexed: 06/01/2023]
Abstract
This paper studies an infinitely fast bimolecular chemical reaction in a two-dimensional biperiodic Navier-Stokes flow. The reactants in stoichiometric quantities are initially segregated by infinite gradients. The focus is placed on the initial stage of the reaction characterized by a well-defined one-dimensional material contact line between the reactants. Particular attention is given to the effect of the diffusion κ of the reactants. This study is an idealized framework for isentropic mixing in the lower stratosphere and is motivated by the need to better understand the effect of resolution on stratospheric chemistry in climate-chemistry models. Adopting a Lagrangian straining theory approach, we relate theoretically the ensemble mean of the length of the contact line, of the gradients along it, and of the modulus of the time derivative of the space-average reactant concentrations (here called the chemical speed) to the joint probability density function of the finite-time Lyapunov exponent λ with two times τ and τ[over ̃]. The time 1/λ measures the stretching time scale of a Lagrangian parcel on a chaotic orbit up to a finite time t, while τ measures it in the recent past before t, and τ[over ̃] in the early part of the trajectory. We show that the chemical speed scales like κ(1/2) and that its time evolution is determined by rare large events in the finite-time Lyapunov exponent distribution. The case of smooth initial gradients is also discussed. The theoretical results are tested with an ensemble of direct numerical simulations (DNSs) using a pseudospectral model.
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Thiffeault JL, Gouillart E, Dauchot O. Moving walls accelerate mixing. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2011; 84:036313. [PMID: 22060498 DOI: 10.1103/physreve.84.036313] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/20/2011] [Indexed: 05/31/2023]
Abstract
Mixing in viscous fluids is challenging, but chaotic advection in principle allows efficient mixing. In the best possible scenario, the decay rate of the concentration profile of a passive scalar should be exponential in time. In practice, several authors have found that the no-slip boundary condition at the walls of a vessel can slow down mixing considerably, turning an exponential decay into a power law. This slowdown affects the whole mixing region, and not just the vicinity of the wall. The reason is that when the chaotic mixing region extends to the wall, a separatrix connects to it. The approach to the wall along that separatrix is polynomial in time and dominates the long-time decay. However, if the walls are moved or rotated, closed orbits appear, separated from the central mixing region by a hyperbolic fixed point with a homoclinic orbit. The long-time approach to the fixed point is exponential, so an overall exponential decay is recovered, albeit with a thin unmixed region near the wall.
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Affiliation(s)
- Jean-Luc Thiffeault
- Department of Mathematics, University of Wisconsin-Madison, Madison, Wisconsin 53706, USA
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Ngan K, Vanneste J. Scalar decay in a three-dimensional chaotic flow. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2011; 83:056306. [PMID: 21728646 DOI: 10.1103/physreve.83.056306] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/01/2011] [Indexed: 05/31/2023]
Abstract
The decay of a passive scalar in a three-dimensional chaotic flow is studied using high-resolution numerical simulations. The (volume-preserving) flow considered is a three-dimensional extension of the randomized alternating sine flow employed extensively in studies of mixing in two dimensions. It is used to show that theoretical predictions for two-dimensional flows with small diffusivity carry over to three dimensions even though the stretching properties differ significantly. The variance decay rate, scalar field structure, and time evolution of statistical moments confirm that there are two distinct regimes of scalar decay: a locally controlled regime, which applies when the domain size is comparable to the characteristic length scale of the velocity field, and a globally controlled regime, which applies when the domain is larger. Asymptotic predictions for the variance decay rate in both regimes show excellent agreement with the numerical results. Consideration of both the forward flow and its time reverse makes it possible to compare the scalar evolution in flows with one or two expanding directions; simulations confirm the theoretical prediction that the decay rate of the scalar is the same in both flows, despite the very different scalar field structures.
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Affiliation(s)
- K Ngan
- Met Office, Exeter EX1 3PB, United Kingdom.
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Sundaram B, Poje AC, Pattanayak AK. Persistent patterns and multifractality in fluid mixing. Phys Rev E 2009; 79:066202. [PMID: 19658578 DOI: 10.1103/physreve.79.066202] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/23/2008] [Revised: 05/12/2009] [Indexed: 11/07/2022]
Abstract
Persistent patterns in periodically driven dynamics have been reported in a wide variety of contexts ranging from table-top and ocean-scale fluid mixing systems to the weak quantum-classical transition in open Hamiltonian systems. We illustrate a common framework for the emergence of these patterns by considering a simple measure of structure maintenance provided by the average radius of the scalar distribution in transform space. Within this framework, scaling laws related to both the formation and persistence of patterns in phase space are presented. Further, preliminary results linking the scaling exponents associated with the persistent patterns to the multifractal nature of the advective phase-space geometry are shown.
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Affiliation(s)
- Bala Sundaram
- Department of Physics, University of Massachusetts, Boston, Massachusetts 02125, USA
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Tsang YK. Predicting the evolution of fast chemical reactions in chaotic flows. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2009; 80:026305. [PMID: 19792247 DOI: 10.1103/physreve.80.026305] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/10/2009] [Indexed: 05/28/2023]
Abstract
We study the fast irreversible bimolecular reaction in a two-dimensional chaotic flow. The reactants are initially segregated and together fill the whole domain. Simulations show that the reactant concentration decays exponentially with rate lambda and then crosses over to the algebraic law of chemical kinetics in the final stage of the reaction. We estimate the crossover time from the reaction rate constant and the flow parameters. The exponential decay phase of the reaction can be described in terms of an equivalent passive scalar problem, allowing us to predict lambda using the theory of passive scalar advection. Depending on the relative length scale between the velocity and the concentration fields, lambda is either related to the distribution of the finite-time Lyapunov exponent of the flow or given in terms of an effective diffusivity which is independent of the small-scale stretching properties of the flow. For the former case, we suggest an optimal choice of flow parameters at which lambda is maximum.
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Affiliation(s)
- Yue-Kin Tsang
- Scripps Institution of Oceanography, University of California-San Diego, La Jolla, California 92093, USA
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Tran CV. Local transfer and spectra of a diffusive field advected by large-scale incompressible flows. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2008; 78:036310. [PMID: 18851146 DOI: 10.1103/physreve.78.036310] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/16/2008] [Indexed: 05/26/2023]
Abstract
This study revisits the problem of advective transfer and spectra of a diffusive scalar field in large-scale incompressible flows in the presence of a (large-scale) source. By "large scale" it is meant that the spectral support of the flows is confined to the wave-number region k<kd , where kd is relatively small compared with the diffusion wave number kkappa. Such flows mediate couplings between neighboring wave numbers within kd of each other only. It is found that the spectral rate of transport (flux) of scalar variance across a high wave number k>kd is bounded from above by UkdkTheta(k,t) , where U denotes the maximum fluid velocity and Theta(k,t) is the spectrum of the scalar variance, defined as its average over the shell (k-kd,k+kd) . For a given flux, say vartheta>0 , across k>kd , this bound requires Theta(k,t)> or =(varthetaUkd)k(-1) . This is consistent with recent numerical studies and with Batchelor's theory that predicts a k(-1) spectrum (with a slightly different proportionality constant) for the viscous-convective range, which could be identified with (kd,kkappa) . Thus, Batchelor's formula for the variance spectrum is recovered by the present method in the form of a critical lower bound. The present result applies to a broad range of large-scale advection problems in space dimensions > or =2 , including some filter models of turbulence, for which the turbulent velocity field is advected by a smoothed version of itself. For this case, Theta(k,t) and vartheta are the kinetic energy spectrum and flux, respectively.
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Affiliation(s)
- Chuong V Tran
- School of Mathematics and Statistics, University of St Andrews, St Andrews KY16 9SS, United Kingdom
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Gouillart E, Dauchot O, Dubrulle B, Roux S, Thiffeault JL. Slow decay of concentration variance due to no-slip walls in chaotic mixing. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2008; 78:026211. [PMID: 18850925 DOI: 10.1103/physreve.78.026211] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/04/2008] [Revised: 05/08/2008] [Indexed: 05/26/2023]
Abstract
Chaotic mixing in a closed vessel is studied experimentally and numerically in different two-dimensional (2D) flow configurations. For a purely hyperbolic phase space, it is well known that concentration fluctuations converge to an eigenmode of the advection-diffusion operator and decay exponentially with time. We illustrate how the unstable manifold of hyperbolic periodic points dominates the resulting persistent pattern. We show for different physical viscous flows that, in the case of a fully chaotic Poincaré section, parabolic periodic points at the walls lead to slower (algebraic) decay. A persistent pattern, the backbone of which is the unstable manifold of parabolic points, can be observed. However, slow stretching at the wall forbids the rapid propagation of stretched filaments throughout the whole domain, and hence delays the formation of an eigenmode until it is no longer experimentally observable. Inspired by the baker's map, we introduce a 1D model with a parabolic point that gives a good account of the slow decay observed in experiments. We derive a universal decay law for such systems parametrized by the rate at which a particle approaches the no-slip wall.
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Affiliation(s)
- E Gouillart
- Surface du Verre et Interfaces, UMR 125 CNRS/Saint-Gobain, 93303 Aubervilliers, France
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Martinand D, Vassilicos JC. Fast chemical reaction and multiple-scale concentration fields in singular vortices. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2007; 75:036315. [PMID: 17500799 DOI: 10.1103/physreve.75.036315] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/09/2006] [Indexed: 05/15/2023]
Abstract
Two species involved in a simple, fast reaction tend to become segregated in patches composed of a single of these reactants. These patches are separated by a boundary where the stoichiometric condition is satisfied and the reaction occurs, fed by diffusion. Stirred by advection, this boundary and the concentration fields within the patches may tend to present multiple-scale characteristics. Based on this segregated state, this paper aims at evaluating the temporal evolutions of the length of the boundary and diffusive flux of reactants across it, when concentrations presenting initial self-similar fluctuations are advected by a singular vortex. First the two sources of singularity, i.e., the self-similar initial conditions and the singular vortex, are considered separately. On the one hand, self-similar initial conditions are imposed to a diffusion-reaction system, for one- and two-dimensional cases. On the other hand, an imposed singular vortex advects initially on/off concentration fields, in combination with diffusion and reaction. This problem is addressed analytically, by characterizing the boundary by a box-counting dimension and the concentration fields by a Hölder exponent, and numerically, by direct numerical simulations of the advection-diffusion-reaction equations. Second, the way the two sources hang together shows that, depending on the self-similar properties of the initial concentration fields, the vortex promotes the chemical activity close to its inner smoothed-out core or close to the outer region where the boundary starts to spiral. For all the considered situations, the length of the boundary and the global reaction speed are found to evolve algebraically with time after a short transient and a good agreement is found between the analytical and numerical scaling laws.
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Affiliation(s)
- D Martinand
- Turbulence and Mixing Group, Department of Aeronautics, Imperial College London, London SW7 2AZ, United Kingdom
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Tsang YK, Antonsen TM, Ott E. Exponential decay of chaotically advected passive scalars in the zero diffusivity limit. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2005; 71:066301. [PMID: 16089861 DOI: 10.1103/physreve.71.066301] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/06/2005] [Indexed: 05/03/2023]
Abstract
The time asymptotic decay of the variance of a passive scalar in a chaotic flow is studied. Two mechanisms for this decay, which involve processes at short and long length scales, respectively, are considered. The validity of the short length scale mechanism, which is based on Lagrangian stretching theory, is discussed. We also investigate the regimes of applicability and observable signatures of the two mechanisms. Supporting evidence is provided by high resolution numerical experiments.
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Affiliation(s)
- Yue-Kin Tsang
- Institute for Research in Electronics and Applied Physics, and Department of Physics, University of Maryland, College Park, Maryland 20742, USA
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Schekochihin AA, Haynes PH, Cowley SC. Diffusion of passive scalar in a finite-scale random flow. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2004; 70:046304. [PMID: 15600516 DOI: 10.1103/physreve.70.046304] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/09/2004] [Indexed: 05/24/2023]
Abstract
We consider a solvable model of the decay of scalar variance in a single-scale random velocity field. We show that if there is a separation between the flow scale k(-1 )(flow ) and the box size k(-1 )(box ) , the decay rate lambda proportional, variant ( k(box) / k(flow) )(2) is determined by the turbulent diffusion of the box-scale mode. Exponential decay at the rate lambda is preceded by a transient powerlike decay (the total scalar variance approximately t(-5/2) if the Corrsin invariant is zero, t(-3/2) otherwise) that lasts a time t approximately 1/lambda . Spectra are sharply peaked at k= k(box) . The box-scale peak acts as a slowly decaying source to a secondary peak at the flow scale. The variance spectrum at scales intermediate between the two peaks ( k(box) <<k<< k(flow) ) is approximately k+a k(2) +... (a>0) . The mixing of the flow-scale modes by the random flow produces, for the case of large Péclet number, a k(-1+delta) spectrum at k>> k(flow) , where delta proportional lambda is a small correction. Our solution thus elucidates the spectral make up of the "strange mode," combining small-scale structure and a decay law set by the largest scales.
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Thiffeault JL. The strange eigenmode in Lagrangian coordinates. CHAOS (WOODBURY, N.Y.) 2004; 14:531-538. [PMID: 15446962 DOI: 10.1063/1.1759431] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/24/2023]
Abstract
For a distribution advected by a simple chaotic map with diffusion, the "strange eigenmode" is investigated from the Lagrangian (material) viewpoint and compared to its Eulerian (spatial) counterpart. The eigenmode embodies the balance between diffusion and exponential stretching by a chaotic flow. It is not strictly an eigenmode in Lagrangian coordinates, because its spectrum is rescaled exponentially rapidly.
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Affiliation(s)
- Jean-Luc Thiffeault
- Department of Mathematics, Imperial College London, London SW7 2AZ, United Kingdom.
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Cerbelli S, Vitacolonna V, Adrover A, Giona M. Eigenvalue–eigenfunction analysis of infinitely fast reactions and micromixing regimes in regular and chaotic bounded flows. Chem Eng Sci 2004. [DOI: 10.1016/j.ces.2004.02.005] [Citation(s) in RCA: 32] [Impact Index Per Article: 1.6] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/26/2022]
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Sukhatme J. Probability density functions of decaying passive scalars in periodic domains: an application of Sinai-Yakhot theory. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2004; 69:056302. [PMID: 15244927 DOI: 10.1103/physreve.69.056302] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/16/2004] [Indexed: 05/24/2023]
Abstract
Employing the formalism introduced by Phys. Rev. Lett. 63, 1962 (1989)], we study the probability density functions (pdf's) of decaying passive scalars in periodic domains under the influence of smooth large scale velocity fields. The particular regime we focus on is one where the normalized scalar pdf's attain a self-similar profile in finite time, i.e., the so-called strange or statistical eigenmode regime. In accordance with the work of Sinai and Yakhot, the central regions of the pdf's are power laws. However, the details of the pdf profiles are dependent on the physical parameters in the problem. Interestingly, for small Peclet numbers the pdf's resemble stretched or pure exponential functions, whereas in the limit of large Peclet numbers, there emerges a universal Gaussian form for the pdf. Numerical simulations are used to verify these predictions.
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Affiliation(s)
- Jai Sukhatme
- National Center for Atmospheric Research, Boulder, Colorado 80305, USA.
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Schekochihin AA, Cowley SC, Maron JL, McWilliams JC. Self-similar turbulent dynamo. PHYSICAL REVIEW LETTERS 2004; 92:064501. [PMID: 14995243 DOI: 10.1103/physrevlett.92.064501] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/24/2003] [Indexed: 05/24/2023]
Abstract
The amplification of magnetic fields in a highly conducting fluid is studied numerically. During growth, the magnetic field is spatially intermittent: it does not uniformly fill the volume, but is concentrated in long thin folded structures. Contrary to a commonly held view, intermittency of the folded field does not increase indefinitely throughout the growth stage if diffusion is present. Instead, as we show, the probability-density function (PDF) of the field-strength becomes self-similar. The normalized moments increase with magnetic Prandtl number in a powerlike fashion. We argue that the self-similarity is to be expected with a finite flow scale and system size. In the nonlinear saturated state, intermittency is reduced and the PDF is exponential. Parallels are noted with self-similar behavior recently observed for passive-scalar mixing and for map dynamos.
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Affiliation(s)
- Alexander A Schekochihin
- Plasma Physics Group, Imperial College, Blackett Laboratory, Prince Consort Road, London SW7 2BW, United Kingdom.
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Dong C. Numerical study of the diapycnal flow through a tidal front with passive tracers. ACTA ACUST UNITED AC 2004. [DOI: 10.1029/2003jc001969] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/10/2022]
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Neufeld Z, Kiss IZ, Zhou C, Kurths J. Synchronization and oscillator death in oscillatory media with stirring. PHYSICAL REVIEW LETTERS 2003; 91:084101. [PMID: 14525240 DOI: 10.1103/physrevlett.91.084101] [Citation(s) in RCA: 10] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/06/2003] [Indexed: 05/24/2023]
Abstract
The effect of stirring in an inhomogeneous oscillatory medium is investigated. We show that the stirring rate can control the macroscopic behavior of the system producing collective oscillations (synchronization) or complete quenching of the oscillations (oscillator death). We interpret the homogenization rate due to mixing as a measure of global coupling and compare the phase diagrams of stirred oscillatory media and of populations of globally coupled oscillators.
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Affiliation(s)
- Zoltán Neufeld
- Center for Nonlinear Studies, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA
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Thiffeault JL, Childress S. Chaotic mixing in a torus map. CHAOS (WOODBURY, N.Y.) 2003; 13:502-507. [PMID: 12777113 DOI: 10.1063/1.1568833] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/24/2023]
Abstract
The advection and diffusion of a passive scalar is investigated for a map of the 2-torus. The map is chaotic, and the limit of almost-uniform stretching is considered. This allows an analytic understanding of the transition from a phase of constant scalar variance (for short times) to exponential decay (for long times). This transition is embodied in a short superexponential phase of decay. The asymptotic state in the exponential phase is an eigenfunction of the advection-diffusion operator, in which most of the scalar variance is concentrated at small scales, even though a large-scale mode sets the decay rate. The duration of the superexponential phase is proportional to the logarithm of the exponential decay rate; if the decay is slow enough then there is no superexponential phase at all.
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Affiliation(s)
- Jean-Luc Thiffeault
- Department of Applied Physics and Applied Mathematics, Columbia University, New York, New York 10027, USA.
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