Explicit expressions for eddy-diffusivity fields and effective large-scale advection in turbulent transport

2016 ◽  
Vol 795 ◽  
pp. 524-548 ◽  
Author(s):  
S. Boi ◽  
A. Mazzino ◽  
G. Lacorata
Keyword(s):  
Large Scale ◽  
Spectral Gap ◽  
Environmental Flows ◽  
Turbulent Transport ◽  
Eddy Diffusivity ◽  
Small Scale ◽  
Eddy Diffusivities ◽  
Perturbative Methods ◽  
Tracer Dispersion ◽  
Explicit Expressions

Large-scale transport is investigated in terms of new explicit expressions for eddy diffusivities and effective advection obtained from asymptotic perturbative methods. The carrier flow is formed by a large-scale component plus a small-scale contribution mimicking a turbulent flow. The scalar dynamics is observed in its pre-asymptotic regimes (i.e. on scales comparable to those of the large-scale velocity). The resulting eddy diffusivity is thus a tensor field which explicitly depends on the large-scale velocity. Small-scale interactions also cause the emergence of an effective large-scale (compressible) advection field which, as a result of the present study however, turns out to be of negligible importance. Two issues are addressed by means of Lagrangian simulations: quantifying the possible deterioration of the eddy-diffusivity/effective advection description by reducing to zero the spectral gap separating the large-scale velocity component from the small-scale component; comparing the accuracy of our closure against other simple, reasonable, options. Answering these questions is important in view of possible applications of our closure to tracer dispersion in environmental flows.

The Fully Developed Wake of a Circular Cylinder

1949 ◽  
Vol 2 (4) ◽  
pp. 451 ◽  
Author(s):  
AA Townsend
Keyword(s):  
Circular Cylinder ◽  
Diffusion Coefficients ◽  
Large Scale ◽  
Turbulent Transport ◽  
Turbulent Energy ◽  
Small Scale ◽  
Mean Velocity ◽  
Turbulent Fluid ◽  
Air Stream ◽  
Scale Motion

Extending previous work on turbulent diffusion in the wake of a circular-cylinder, a series of measurements have been made of the turbulent transport of mean stream momentum, turbulent energy, and heat in the wake of a cylinder of 0.169 cm. diameter, placed in an air-stream of velocity 1280 cm. sec.-1. It has been possible to extend the measurements to 960 diameters down-stream from the cylinder, and it 1s found that, at distances in excess of 600 diameters, the requirements of dynamical similarity are very nearly satisfied. To account for the observed rates of transport of turbulent energy and heat, it is necessary that only part of this transport be due to bulk convection by the slow large-scale motion of the jets of turbulent fluid emitted by the central, fully turbulent core of the wake, which had been supposed previously to perform most of the transport. The remainder of the transport is carried out by the small-scale diffusive motion of the turbulent eddies within the jets, and may be described by assigning diffusion coefficients to the turbulent fluid. It is found that the diffusion coefficients for momentum and heat are approximately equal, but that for turbulent energy is considerably smaller. On the basis of these hypotheses, it is possible to calculate $he form of the mean velocity distribution in good agreement with experiment, and to give a qualitative explanation of the apparently more rapid diffusion of heat.

scholarly journals Mixed Layer Lateral Eddy Fluxes Mediated by Air–Sea Interaction

2011 ◽  
Vol 41 (1) ◽  
pp. 130-144 ◽  
Author(s):  
Emily Shuckburgh ◽  
Guillaume Maze ◽  
David Ferreira ◽  
John Marshall ◽  
Helen Jones ◽  
...  
Keyword(s):  
Surface Temperature ◽  
Mixed Layer ◽  
Mixed Layer Depth ◽  
Effective Diffusivity ◽  
Eddy Diffusivity ◽  
Heat Fluxes ◽  
Sea Surface ◽  
Temperature Fields ◽  
Small Scale ◽  
Eddy Diffusivities

Abstract The modulation of air–sea heat fluxes by geostrophic eddies due to the stirring of temperature at the sea surface is discussed and quantified. It is argued that the damping of eddy temperature variance by such air–sea fluxes enhances the dissipation of surface temperature fields. Depending on the time scale of damping relative to that of the eddying motions, surface eddy diffusivities can be significantly enhanced over interior values. The issues are explored and quantified in a controlled setting by driving a tracer field, a proxy for sea surface temperature, with surface altimetric observations in the Antarctic Circumpolar Current (ACC) of the Southern Ocean. A new, tracer-based diagnostic of eddy diffusivity is introduced, which is related to the Nakamura effective diffusivity. Using this, the mixed layer lateral eddy diffusivities associated with (i) eddy stirring and small-scale mixing and (ii) surface damping by air–sea interaction is quantified. In the ACC, a diffusivity associated with surface damping of a comparable magnitude to that associated with eddy stirring (∼500 m2 s−1) is found. In frontal regions prevalent in the ACC, an augmentation of surface lateral eddy diffusivities of this magnitude is equivalent to an air–sea flux of 100 W m−2 acting over a mixed layer depth of 100 m, a very significant effect. Finally, the implications for other tracer fields such as salinity, dissolved gases, and chlorophyll are discussed. Different tracers are found to have surface eddy diffusivities that differ significantly in magnitude.

Simple examples with features of renormalization for turbulent transport

1994 ◽  
Vol 346 (1679) ◽  
pp. 205-233 ◽  
Keyword(s):  
Long Range ◽  
Large Scale ◽  
Turbulent Transport ◽  
Gaussian White Noise ◽  
Eddy Diffusivity ◽  
Turbulent Velocity ◽  
Effective Equation ◽  
Long Range Correlations ◽  
Velocity Statistics ◽  
Limiting Case

Two simple exactly solvable models for turbulent transport are introduced and discussed here with complete mathematical rigour. These models illustrate several different facets of super-diffusion and renormalization for turbulent transport. The first model involves time dependent velocity fields with suitable long-range correlations and the complete renormalization theory is developed here in detail. In addition rigorous examples are developed by using variants of this model where the effective equation for the ensemble average at large scales and long times is diffusive despite the fact that each realization exhibits catastrophic large-scale instability. The second model introduced previously by the authors involves transport-diffusion in simple shear layers with turbulent velocity statistics. The theories of renormalized eddy diffusivity and higher-order statistics are surveyed here. An extreme limiting case of the theory involving turbulent velocity statistics with long-range spatial correlations but gaussian white noise in time is discussed in detail. Both the renormalized theory of eddy diffusivity and exact explicit equations for second-order correlations related to the pair distance function are developed in complete detail here in this instructive limiting case.

scholarly journals Oscillating grids turbulence generator for turbulent transport studies

2002 ◽  
Vol 9 (3/4) ◽  
pp. 201-205 ◽  
Author(s):  
A. Eidelman ◽  
T. Elperin ◽  
A. Kapusta ◽  
N. Kleeorin ◽  
A. Krein ◽  
...  
Keyword(s):  
Velocity Field ◽  
Large Scale ◽  
Turbulent Transport ◽  
Observation Time ◽  
Statistical Characteristics ◽  
Small Scale ◽  
Flow Measurements ◽  
Oscillating Grids ◽  
Set Up ◽  
Turbulence Generator

Abstract. An oscillating grids turbulence generator was constructed for studies of two new effects associated with turbulent transport of particles, turbulent thermal diffusion and clustering instability. These effects result in formation of large-scale and small-scale inhomogeneities in the spatial distribution of particles. The advantage of this experimental set-up is the feasibility to study turbulent transport in mixtures with controllable composition and unlimited observation time. For flow measurements we used Particle Image Velocimetry with the adaptive multi-pass algorithm to determine a turbulent velocity field and its statistical characteristics. Instantaneous velocity vector maps, flow streamlines and probability density function of velocity field demonstrate properties of turbulence generated in the device.

Direct and indirect thermal expansion effects in turbulent premixed flames

2011 ◽  
Vol 689 ◽  
pp. 149-182 ◽  
Author(s):  
Vincent Robin ◽  
Arnaud Mura ◽  
Michel Champion
Keyword(s):  
Thermal Expansion ◽  
Velocity Field ◽  
Large Scale ◽  
Turbulent Transport ◽  
Premixed Flames ◽  
Small Scale ◽  
Flame Surface ◽  
Thermal Expansion Effect ◽  
Expansion Effect ◽  
Turbulent Premixed

AbstractThe thermal expansion induced by the exothermic chemical reactions taking place in a turbulent reactive flow affects the velocity field so strongly that the large-scale velocity fluctuations as well as the small-scale velocity gradients can be governed by chemistry rather than by turbulence. Moreover, thermal expansion is well known to be responsible for counter-gradient turbulent diffusion and flame-generated turbulence phenomena. In the present study, by making use of an original splitting procedure applied to the velocity field, we establish the occurrence of two distinct thermal expansion effects in the flamelet regime of turbulent premixed combustion. The first is referred to as the direct thermal expansion effect. It is associated with a local flamelet crossing contribution as previously considered in early analyses of turbulent transport in premixed flames. The second, denoted herein as the indirect thermal expansion effect, is an outcome of the turbulent wrinkling processes that increases the flame surface area. Based on a splitting procedure applied to the velocity field, the respective influences of the two effects are identified and analysed. Furthermore, the theoretical analysis shows that the thermal expansion induced through the local flames can be treated separately in the usual continuity and momentum equations. This description of the turbulent reactive velocity field, leads also to relate all of the usual turbulent quantities to the reactive scalar field. Finally, algebraic closures for the turbulent transport terms of mass and momentum are proposed and successfully validated through comparison with direct numerical simulation data.

scholarly journals Dynamics of fingering convection. Part 1 Small-scale fluxes and large-scale instabilities

2011 ◽  
Vol 677 ◽  
pp. 530-553 ◽  
Author(s):  
A. TRAXLER ◽  
S. STELLMACH ◽  
P. GARAUD ◽  
T. RADKO ◽  
N. BRUMMELL
Keyword(s):  
Large Scale ◽  
Molecular Diffusion ◽  
Turbulent Transport ◽  
Mean Field Theory ◽  
Density Ratio ◽  
Mean Field ◽  
Field Measurements ◽  
Small Scale ◽  
Diapycnal Mixing ◽  
Density Ratios

Double-diffusive instabilities are often invoked to explain enhanced transport in stably stratified fluids. The most-studied natural manifestation of this process, fingering convection, commonly occurs in the ocean's thermocline and typically increases diapycnal mixing by 2 orders of magnitude over molecular diffusion. Fingering convection is also often associated with structures on much larger scales, such as thermohaline intrusions, gravity waves and thermohaline staircases. In this paper, we present an exhaustive study of the phenomenon from small to large scales. We perform the first three-dimensional simulations of the process at realistic values of the heat and salt diffusivities and provide accurate estimates of the induced turbulent transport. Our results are consistent with oceanic field measurements of diapycnal mixing in fingering regions. We then develop a generalized mean-field theory to study the stability of fingering systems to large-scale perturbations using our calculated turbulent fluxes to parameterize small-scale transport. The theory recovers the intrusive instability, the collective instability and the γ-instability as limiting cases. We find that the fastest growing large-scale mode depends sensitively on the ratio of the background gradients of temperature and salinity (the density ratio). While only intrusive modes exist at high density ratios, the collective and γ instabilities dominate the system at the low density ratios where staircases are typically observed. We conclude by discussing our findings in the context of staircase-formation theory.

scholarly journals Momentum and scalar transport at the turbulent/non-turbulent interface of a jet

2009 ◽  
Vol 631 ◽  
pp. 199-230 ◽  
Author(s):  
J. WESTERWEEL ◽  
C. FUKUSHIMA ◽  
J. M. PEDERSEN ◽  
J. C. R. HUNT
Keyword(s):  
Large Scale ◽  
Eddy Viscosity ◽  
Eddy Diffusivity ◽  
Transport Processes ◽  
Scalar Transport ◽  
Small Scale ◽  
Mean Velocity ◽  
Conditional Mean ◽  
Entrainment Process ◽  
The Mean

Conditionally sampled measurements with particle image velocimetry (PIV) of a turbulent round submerged liquid jet in a laboratory have been taken at Re = 2 × 103 between 60 and 100 nozzle diameters from the nozzle in order to investigate the dynamics and transport processes at the continuous and well-defined bounding interface between the turbulent and non-turbulent regions of flow. The jet carries a fluorescent dye measured with planar laser-induced fluorescence (LIF), and the surface discontinuity in the scalar concentration is identified as the fluctuating turbulent jet interface. Thence the mean outward ‘boundary entrainment’ velocity is derived and shown to be a constant fraction (about 0.07) of the the mean jet velocity on the centreline. Profiles of the conditional mean velocity, mean scalar and momentum flux show that at the interface there are clear discontinuities in the mean axial velocity and mean scalar and a tendency towards a singularity in mean vorticity. These actual or asymptotic discontinuities are consistent with the conditional mean momentum and scalar transport equations integrated across the interface. Measurements of the fluxes of turbulent kinetic energy and enstrophy are consistent with computations by Mathew & Basu (Phys. Fluids, vol. 14, 2002, pp. 2065–2072) in showing that for a jet flow (without forcing) the entrainment process is dominated by small-scale eddying at the highly sheared interface (‘nibbling’), with large-scale engulfing making a small (less than 10%) contribution consistent with concentration measurements showing that the interior of the jet is well mixed. (Turbulent jets differ greatly from the free shear layer in this respect.) To explain the difference between velocity and scalar profiles, their conditional mean gradients are defined in terms of a local eddy viscosity and eddy diffusivity and the momentum and scalar fluxes inside the interface. Since the eddy diffusivity is larger than the eddy viscosity, the scalar profile is flatter inside the interface so that the scalar discontinuity is relatively greater than the mean velocity discontinuity. Theoretical arguments, following Hunt, Eames & Westerweel (in Proc. of the IUTAM Symp. on Computational Physics and New Perspectives in Turbulence, ed. Y. Kaneda, vol. 4, 2008, pp. 331–338, Springer), are proposed for how the vortex sheet develops, how the internal structure of the interface layer relates to the inhomogeneous rotational and irrotational motions on each side and why the dominant entrainment process of jets and wakes differs from that of free shear layers.

scholarly journals Effective Isentropic Diffusivity of Tropospheric Transport

2014 ◽  
Vol 71 (9) ◽  
pp. 3499-3520 ◽  
Author(s):  
Gang Chen ◽  
Alan Plumb
Keyword(s):  
Large Scale ◽  
Lower Troposphere ◽  
Effective Diffusivity ◽  
Eddy Diffusivity ◽  
Finite Amplitude ◽  
Equation Model ◽  
Small Scale ◽  
Near Surface ◽  
Diffusion Scheme ◽  
Finite Amplitude Wave Activity

Abstract Tropospheric transport can be described qualitatively by the slow mean diabatic circulation and rapid isentropic mixing, yet a quantitative understanding of the transport circulation is complicated, as nearly half of the isentropic surfaces in the troposphere frequently intersect the ground. A theoretical framework for the effective isentropic diffusivity of tropospheric transport is presented. Compared with previous isentropic analysis of effective diffusivity, a new diagnostic is introduced to quantify the eddy diffusivity of the near-surface isentropic flow. This diagnostic also links the effective eddy diffusivity directly to a diffusive closure of eddy fluxes through a finite-amplitude wave activity equation. The theory is examined in a dry primitive equation model on the sphere. It is found that the upper troposphere is characterized by a diffusivity minimum at the jet’s center with enhanced mixing at the jet’s flanks and that the lower troposphere is dominated by stronger mixing throughout the baroclinic zone. This structure of isentropic diffusivity is generally consistent with the diffusivity obtained from the geostrophic component of the flow. Furthermore, the isentropic diffusivity agrees broadly with the tracer equivalent length obtained from either a spectral diffusion scheme or a semi-Lagrangian advection scheme, indicating that the effective diffusivity of tropospheric transport is largely dictated by large-scale stirring rather than details of the small-scale diffusion acting on the tracers.

scholarly journals Eddy diffusivities of inertial particles under gravity

2012 ◽  
Vol 694 ◽  
pp. 426-463 ◽  
Author(s):  
Marco Martins Afonso ◽  
Andrea Mazzino ◽  
Paolo Muratore-Ginanneschi
Keyword(s):  
Particle Velocity ◽  
Large Scale ◽  
Particle Concentration ◽  
Mass Density ◽  
Concentration Field ◽  
Eddy Diffusivity ◽  
Inertial Particles ◽  
Differential Problem ◽  
Carrier Flow ◽  
Eddy Diffusivities

AbstractThe large-scale/long-time transport of inertial particles of arbitrary mass density under gravity is investigated by means of a formal multiple-scale perturbative expansion in the scale-separation parameter between the carrier flow and the particle concentration field. The resulting large-scale equation for the particle concentration is determined, and is found to be diffusive with a positive definite eddy diffusivity. The calculation of the latter tensor is reduced to the resolution of an auxiliary differential problem, consisting of a coupled set of two differential equations in a $(6+ 1)$-dimensional coordinate system (three space coordinates plus three velocity coordinates plus time). Although expensive, numerical methods can be exploited to obtain the eddy diffusivity, for any desirable non-perturbative limit (e.g. arbitrary Stokes and Froude numbers). The aforementioned large-scale equation is then specialized to deal with two different relevant perturbative limits: (i) vanishing of both Stokes time and sedimenting particle velocity; (ii) vanishing Stokes time and finite sedimenting particle velocity. Both asymptotics lead to a greatly simplified auxiliary differential problem, now involving only space coordinates and thus easily tackled by standard numerical techniques. Explicit, exact expressions for the eddy diffusivities have been calculated, for both asymptotics, for the class of parallel flows, both static and time-dependent. This allows us to investigate analytically the role of gravity and inertia on the diffusion process by varying relevant features of the carrier flow, such as the form of its temporal correlation function. Our results exclude a universal role played by gravity and inertia on the diffusive behaviour: regimes of both enhanced and reduced diffusion may exist, depending on the detailed structure of the carrier flow.

scholarly journals Effective Eddy Diffusivities Inferred from a Point Release Tracer in an Eddy-Resolving Ocean Model

2009 ◽  
Vol 39 (4) ◽  
pp. 894-914 ◽  
Author(s):  
Mei-Man Lee ◽  
A. J. George Nurser ◽  
Andrew C. Coward ◽  
Beverly A. de Cuevas
Keyword(s):  
Flow Regime ◽  
Eddy Diffusivity ◽  
Ocean Model ◽  
Diagnostic Methods ◽  
Equivalent Radius ◽  
Eddy Diffusivities ◽  
Tracer Dispersion ◽  
Tracer Distribution ◽  
Stretching Flow ◽  
Tracer Experiments

Abstract This study uses tracer experiments in a global eddy-resolving ocean model to examine two diagnostic methods for inferring effective eddy isopycnic diffusivity from point release tracers. The first method is based on the growth rate of the area occupied by the tracers (the equivalent variance). During the period when tracer dispersion is dominated by stirring, the equivalent variance is found to increase at a rate between the second power law (for a pure shearing flow regime) and the exponential law (for a pure stretching flow regime). The second method is based on the length of the tracer contours. In the framework of equivalent radius, the two methods of inferring eddy diffusivity can be understood as two different averagings over the tracer patch. Over a shorter period of tracer dispersion the two methods give different eddy diffusivities, and only over a longer time when tracer dispersion approaches the final stage of diffusion do they give a similar value of diffusivity. A new diagnostic quantity called stirring efficiency is introduced to indicate different flow regimes by measuring the efficiency of stirring against mixing. The new diagnostic quantity has the advantage that it can be calculated directly from the gradients of tracer distribution without needing to estimate strain rate or background diffusivity.

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