1
|
Abstract
A theory for turbulent particle pair diffusion in the inertial subrange [Malik NA, PLoS ONE 13(10):e0202940 (2018)] is investigated numerically using a Lagrangian diffusion model, Kinematic Simulations [Kraichnan RH, Phys. Fluids 13:22 (1970); Malik NA, PLoS ONE 12(12):e0189917 (2017)]. All predictions of the theory are observed in flow fields with generalised energy spectra of the type, E(k) ∼ k−p. Most importantly, two non-Richardson regimes are observed: for short inertial subrange of size 102 the simulations yield quasi-local regimes for the pair diffusion coefficient, K(l)∼σl(1+p)/2; and for asymptotically infinite inertial subrange the simulations yield non-local regimes K(l)∼σlγ, with γ intermediate between the purely local scaling γl = (1 + p)/2 and the purely non-local scaling γnl = 2. For intermittent turbulence spectra, E(k) ∼ k−1.72, the simulations yield K∼σl1.556, in agreement with the revised 1926 dataset K∼σl1.564 [Richardson LF, Proc. Roy. Soc. Lond. A 100:709 (1926); Malik NA, PLoS ONE 13(10):e0202940 (2018)]. These results lend support to the physical picture proposed in the new theory that turbulent diffusion in the inertial subrange is governed by both local and non-local diffusion transport processes.
Collapse
Affiliation(s)
- Nadeem A. Malik
- Department of Mathematics and Statistics, King Fahd University of Petroleum and Minerals, Dhahran, Saudi Arabia
- * E-mail: ,
| |
Collapse
|
2
|
Malik NA. Residual sweeping errors in turbulent particle pair diffusion in a Lagrangian diffusion model. PLoS One 2017; 12:e0189917. [PMID: 29253875 PMCID: PMC5734776 DOI: 10.1371/journal.pone.0189917] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/16/2016] [Accepted: 12/05/2017] [Indexed: 11/24/2022] Open
Abstract
Thomson, D. J. & Devenish, B. J. [J. Fluid Mech. 526, 277 (2005)] and others have suggested that sweeping effects make Lagrangian properties in Kinematic Simulations (KS), Fung et al [Fung J. C. H., Hunt J. C. R., Malik N. A. & Perkins R. J. J. Fluid Mech. 236, 281 (1992)], unreliable. However, such a conclusion can only be drawn under the assumption of locality. The major aim here is to quantify the sweeping errors in KS without assuming locality. Through a novel analysis based upon analysing pairs of particle trajectories in a frame of reference moving with the large energy containing scales of motion it is shown that the normalized integrated error eKI in the turbulent pair diffusivity (K) due to the sweeping effect decreases with increasing pair separation (σl), such that eKI→0 as σl/η → ∞; and eKI→∞ as σl/η → 0. η is the Kolmogorov turbulence microscale. There is an intermediate range of separations 1 < σl/η < ∞ in which the error eKI remains negligible. Simulations using KS shows that in the swept frame of reference, this intermediate range is large covering almost the entire inertial subrange simulated, 1 < σl/η < 105, implying that the deviation from locality observed in KS cannot be atributed to sweeping errors. This is important for pair diffusion theory and modeling. PACS numbers: 47.27.E?, 47.27.Gs, 47.27.jv, 47.27.Ak, 47.27.tb, 47.27.eb, 47.11.-j.
Collapse
Affiliation(s)
- Nadeem A. Malik
- Department of Mathematics and Statistics, King Fahd University of Petroleum and Minerals, P.O. Box 5046, Dhahran 31261, Saudi Arabia
- * E-mail: ,
| |
Collapse
|
3
|
Lacorata G, Vulpiani A. Chaotic Lagrangian models for turbulent relative dispersion. Phys Rev E 2017; 95:043106. [PMID: 28505811 DOI: 10.1103/physreve.95.043106] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/20/2016] [Indexed: 11/07/2022]
Abstract
A deterministic multiscale dynamical system is introduced and discussed as a prototype model for relative dispersion in stationary, homogeneous, and isotropic turbulence. Unlike stochastic diffusion models, here trajectory transport and mixing properties are entirely controlled by Lagrangian chaos. The anomalous "sweeping effect," a known drawback common to kinematic simulations, is removed through the use of quasi-Lagrangian coordinates. Lagrangian dispersion statistics of the model are accurately analyzed by computing the finite-scale Lyapunov exponent (FSLE), which is the optimal measure of the scaling properties of dispersion. FSLE scaling exponents provide a severe test to decide whether model simulations are in agreement with theoretical expectations and/or observation. The results of our numerical experiments cover a wide range of "Reynolds numbers" and show that chaotic deterministic flows can be very efficient, and numerically low-cost, models of turbulent trajectories in stationary, homogeneous, and isotropic conditions. The mathematics of the model is relatively simple, and, in a geophysical context, potential applications may regard small-scale parametrization issues in general circulation models, mixed layer, and/or boundary layer turbulence models as well as Lagrangian predictability studies.
Collapse
Affiliation(s)
- Guglielmo Lacorata
- CNR-Istituto di Scienze dell'Atmosfera e del Clima, Via Monteroni, I-73100, Lecce, Italy and Center of Excellence CETEMPS, Università dell'Aquila, Via Vetoio, I-67100, Coppito (AQ), Italy
| | - Angelo Vulpiani
- Dipartimento di Fisica, Universitá "La Sapienza," and CNR-ISC, P. le Aldo Moro 2, I-00185 Rome, Italy and Kavli Institute for Theoretical Physics, Beijing 100190, China
| |
Collapse
|
4
|
Nicolleau FCGA, Farhan M, Nowakowski AF. Effect of the energy-spectrum law on clustering patterns for inertial particles subjected to gravity in kinematic simulation. Phys Rev E 2016; 94:043109. [PMID: 27841627 DOI: 10.1103/physreve.94.043109] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/29/2016] [Indexed: 06/06/2023]
Abstract
We study the clustering of inertial particles using a periodic kinematic simulation. Particle clustering is observed for different pairs of Stokes number and Froude number and different spectral power laws (1.4≤p≤2.1). The main focus is to identify and then quantify the effect of p on the clustering attractor-by attractor we mean the set of points in the physical space where the particles settle when time tends to infinity. It is observed that spectral power laws can have a dramatic effect on the attractor shape. In particular, we observed an attractor type which was not present in previous studies for Kolmogorov spectra (p=5/3).
Collapse
Affiliation(s)
- F C G A Nicolleau
- Sheffield Fluid Mechanics Group, Department of Mechanical Engineering, The University of Sheffield, Sheffield, United Kingdom and Sorbonne Universités, Université Pierre et Marie Curie, Paris 6 and Institut Jean le Rond dAlembert, CNRS UMR 7190, Paris, France
| | - M Farhan
- Department of Mechanical Engineering of the University of Engineering and Technology Lahore, Pakistan and Sheffield Fluid Mechanics Group, Department of Mechanical Engineering, The University of Sheffield, Sheffield, United Kingdom
| | - A F Nowakowski
- Sheffield Fluid Mechanics Group, Department of Mechanical Engineering, The University of Sheffield, Sheffield, United Kingdom
| |
Collapse
|
5
|
Farhan M, Nicolleau FCGA, Nowakowski AF. Effect of gravity on clustering patterns and inertial particle attractors in kinematic simulations. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2015; 91:043021. [PMID: 25974594 DOI: 10.1103/physreve.91.043021] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/28/2014] [Indexed: 06/04/2023]
Abstract
In this paper, we study the clustering of inertial particles using a periodic kinematic simulation. The systematic Lagrangian tracking of particles makes it possible to identify the particles' clustering patterns for different values of particle inertia and drift velocity. The different cases are characterized by different pairs of Stokes number (St) and Froude number (Fr). For the present study, 0≤St≤1 and 0.4≤Fr≤1.4. The main focus is to identify and then quantify the clustering attractor-when it exists-that is the set of points in the physical space where the particles settle when time goes to infinity. Depending on the gravity effect and inertia values, the Lagrangian attractor can have different dimensions, varying from the initial three-dimensional space to two-dimensional layers and one-dimensional attractors that can be shifted from a horizontal to a vertical position.
Collapse
Affiliation(s)
- M Farhan
- Sheffield Fluid Mechanics Group, Department of Mechanical Engineering, The University of Sheffield, Sheffield, United Kingdom
| | - F C G A Nicolleau
- Sheffield Fluid Mechanics Group, Department of Mechanical Engineering, The University of Sheffield, Sheffield, United Kingdom
| | - A F Nowakowski
- Sheffield Fluid Mechanics Group, Department of Mechanical Engineering, The University of Sheffield, Sheffield, United Kingdom
| |
Collapse
|
6
|
Devenish BJ. Geometrical properties of turbulent dispersion. PHYSICAL REVIEW LETTERS 2013; 110:064504. [PMID: 23432254 DOI: 10.1103/physrevlett.110.064504] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/18/2012] [Indexed: 06/01/2023]
Abstract
It is shown that the shape statistics of four correlated particles, known as a tetrad, in a direct numerical simulation of three-dimensional homogeneous isotropic turbulence agree very well with a simple diffusion equation with a separation-dependent eddy diffusivity. The latter is essentially an extension of Richardson's model for particle-pair dispersion to tetrads. It is also shown that the degree of elongation of the tetrad in the inertial subrange of a turbulentlike flow is controlled by the exponent, m, in an eddy diffusivity of the form K(r)[proportionality]r(m) where r is the interparticle separation, becoming more elongated as m increases in the range 0≤m<2.
Collapse
Affiliation(s)
- B J Devenish
- Met Office, FitzRoy Road, Exeter EX1 3PB, United Kingdom
| |
Collapse
|
7
|
Eyink GL, Benveniste D. Suppression of particle dispersion by sweeping effects in synthetic turbulence. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2013; 87:023011. [PMID: 23496614 DOI: 10.1103/physreve.87.023011] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/16/2012] [Indexed: 06/01/2023]
Abstract
Synthetic models of Eulerian turbulence like so-called kinematic simulations (KS) are often used as computational shortcuts for studying Lagrangian properties of turbulence. These models have been criticized by Thomson and Devenish (2005), who argued on physical grounds that sweeping decorrelation effects suppress pair dispersion in such models. We derive analytical results for Eulerian turbulence modeled by Gaussian random fields, in particular for the case with zero mean velocity. Our starting point is an exact integrodifferential equation for the particle pair separation distribution obtained from the Gaussian integration-by-parts identity. When memory times of particle locations are short, a Markovian approximation leads to a Richardson-type diffusion model. We obtain a time-dependent pair diffusivity tensor of the form K(ij)(r,t)=S(ij)(r)τ(r,t), where S(ij)(r) is the structure-function tensor and τ(r,t) is an effective correlation time of velocity increments. Crucially, this is found to be the minimum value of three times: the intrinsic turnover time τ(eddy)(r) at separation r, the overall evolution time t, and the sweeping time r/v(0) with v(0) the rms velocity. We study the diffusion model numerically by a Monte Carlo method. With inertial ranges like the largest achieved in most current KS (about 6 decades long), our model is found to reproduce the t(9/2) power law for pair dispersion predicted by Thomson and Devenish and observed in the KS. However, for much longer ranges, our model exhibits three distinct pair-dispersion laws in the inertial range: a Batchelor t(2) regime, followed by a Kraichnan-model-like t(1) diffusive regime, and then a t(6) regime. Finally, outside the inertial range, there is another t(1) regime with particles undergoing independent Taylor diffusion. These scalings are exactly the same as those predicted by Thomson and Devenish for KS with large mean velocities, which we argue hold also for KS with zero mean velocity. Our results support the basic conclusion of Thomson and Devenish (2005) that sweeping effects make Lagrangian properties of KS fundamentally differ from those of hydrodynamic turbulence for very extended inertial ranges.
Collapse
Affiliation(s)
- Gregory L Eyink
- Department of Applied Mathematics & Statistics, The Johns Hopkins University, Baltimore, Maryland 21218, USA.
| | | |
Collapse
|