Far-Field Turbulent Mixing Efficiency and

Large-Scale Outer-Fluid-Interface Dynamics

Jennifer C. Nathman †, Roberto C. Aguirre †, and Haris J. Catrakis ‡

Aeronautics and Fluid Dynamics LaboratoriesUniversity of California, Irvine, CA 92697

Abstract

A new physical point of view useful for examining the mixing efficiency of turbulentflows is the Outer-Fluid-Interface (OFI) approach. Whereas the conventional approachrequires knowledge of all mixed-fluid interfaces, i.e. both the outer and internal interfaces,in the turbulent-flow region of interest, the OFI point of view provides the means tostudy the mixing efficiency based solely on the behavior of the outer interfaces. Inaddition, the OFI approach suggests that the turbulent upper scales, rather than thelower scales, dominate the mixing efficiency. In this work, we consider how the OFIapproach can be used to quantify the contributions of the large interfacial scales vs. smallinterfacial scales, relative to each other. We examine high-resolution measurements inthe far-field behavior of a fully-developed round jet. These measurements captured theentire range of interfacial scales at a Reynolds number of Re ∼ 20, 000 and a Schmidtnumber of Sc ∼ 2, 000, at a downstream distance of 500 jet nozzle exit diameters inthe similarity plane, i.e. in the plane normal to the jet axis. We find, on the basisof these measurements, that the interfacial features from the upper range contribute∼ 90% to the mixing efficiency, as compared to the interfacial features from the lowerscales which contribute the remaining ∼ 10%, at the present flow conditions. This meansthat the large-scale folding, as opposed to the small-scale wrinkling, of the interfacesprovides the dominant contribution. This finding is important because it suggests thatone can examine large-Reynolds-number flows, with emphasis on the upper range ofscales, to identify the dominant behavior of the mixing efficiency. This idea is usefulin applications ranging from far-field aircraft signatures to laser beam propagation andaerooptics in high-speed flight, and is applicable to studies of unforced flows, to determinethe dependence of the mixing efficiency on the Reynolds number, as well as to studies offlow control for characterizing the dominant interfacial contributions to either enhanceor reduce the mixing efficiency.

1. Introduction

The mixing efficiency of turbulent flowsis practically significant in various aero-nautical applications, such as aircraft sig-natures, turbine blade cooling, or laserbeam propagation from high-speed air-

borne platforms. The turbulent mixingefficiency is determined by the behaviorof the mixed-fluid interfaces. Practicaland fundamental aspects of these fluid in-terfaces, such as their dynamics, struc-

†Graduate Student, Member AIAA.‡Asst Professor, Member AIAA. Corresp. Author. Tel: (949) 378-7781. E-mail: catrakis@uci.edu

1

42nd AIAA Aerospace Sciences Meeting and Exhibit5 - 8 January 2004, Reno, Nevada

AIAA 2004-1280

Copyright © 2004 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.

ture, and dependence on Reynolds num-ber, Schmidt number, and other fluid/flowparameters, are subjects of ongoing stud-ies (e.g. [1], [2], [3]). Mixed-fluid interfacescorrespond to isoconcentration or isoscalarsurfaces in the turbulent flow. At largeReynolds numbers, these interfaces are ob-served to exhibit dynamics across a widerange of scales (e.g. [4], [5], [6], [7]). Theknowledge of the dynamics of the inter-faces is crucial for physical descriptions,predictions, and control of the mixing ef-ficiency (e.g. [8]). Although it has beenrecognized for a long time that large-scaleentrainment is important for mixing, onemust recognize that entrainment alone setsan upper bound only on the mixing effi-ciency. In other words, knowledge of thegrowth rate of a turbulent shear flow isnot sufficient to deduce the mixing effi-ciency. The traditional approach to themixing efficiency is to account for all in-terfacial scales, particularly the smallestscales. A new point of view, however, pro-posed by Catrakis et. al. [7] based on theOuter-Fluid-Interface (OFI) behavior indi-cates that a simpler approach to the mix-ing efficiency can be developed where thelarge-scale OFI features are the dominantfeatures. This is because, whereas the sur-face area of the OFI’s is dominated by thesmall scales, the volume enclosed by theOFI’s is actually dominated by the largescales. This is a crucial idea and is espe-cially useful at large Reynolds numbers.

In previous work by Catrakis et. al. (AIAAPaper 2001-2716 [1]), the mixing behaviorin the far field of turbulent jets was investi-gated with high-resolution measurements.In related aerooptics studies (AIAA Paper2002-2269 [9]), and mixing studies ([3]),the interfacial structure was examined atthe small scales in order to investigate boththe extent to which the behavior was scale-independent and the area-volume interfa-

cial properties. To develop physical de-scriptions that capture the dominant con-tributions to the mixing efficiency, how-ever, one must examine the large-scale in-terfacial behavior.

In the present work, we consider the rel-ative contributions of the outer fluid in-terfaces (OFI’s) from the upper range ofscales compared to the lower range ofscales. The objective is to quantify to whatextent the upper-range interfacial struc-ture dominates the mixing efficiency usinghigh-resolution measurements. The upperrange of scales corresponds physically tofolding behavior of the OFI’s, whereas thelower range of scales corresponds to wrin-kling behavior of the OFI’s. The presentapproach enables the way to quantify thelarge-scale interfacial-folding contributionsto the mixing efficiency.

2. Far-field mixing efficiencyand the OFI approach

In the traditional approach to the mixingefficiency, knowledge of the behaviour ofboth the outer interfaces and internal in-terfaces is necessary. In contrast, in theOFI approach pioneered by Catrakis et. al.[7], the mixing efficiency can be describedentirely in terms of the OFI behavior alone.Denoting by αm the mixing efficiency, orfraction of mixed fluid,

αm =∫mixed

p(c) dc , (1)

where the integration is over mixed fluidonly, i.e. excluding concentration valuesthat correspond to pure fluid. While theprobability density function p(c) across therange of concentration values correspond-ing to mixed fluid involves the dynamicsof all mixed-fluid interfaces, the mixing-efficiency integral is actually determinedsolely by the outer interfaces.

2

Figure 1: Left: Whole-field three-dimensional space–time high-resolution (∼ 1, 0003)quantitative images of the Outer-Fluid-Interface (OFI) behavior in a turbulent jet abovethe mixing transition at Re ∼ 20, 000 and Sc ∼ 2, 000 [7]. Right: Transparent visualiza-tion of the OFI dynamics across the entire range of scales along the time axis [7].

This is because

αm =∫mixed

A(c)

Vh(c) dc =

Vouter

V. (2)

where Vouter is the volume of fluid en-closed by the outer interfaces, for constant-density flows, denoting by A(c)/V the in-terfacial area-volume ratio, h(c) the meaninterfacial thickness, and c the concentra-tion.

The crucial point is that it is the volumeenclosed by the outer interfaces, ratherthan the interfacial surface area, that de-termines the mixing efficiency. Realizingthat the mixing efficiency is sensitive tothe volume enclosed by the outer interfacerather than their surface area, it followsthat it is the large-scale dynamics of theouter interface that provide the dominantcontributions to the mixing efficiency.

The dynamics of the outer interfaces,therefore, govern the mixing efficiency.Figure 1 (left) depicts a visualization ofthe three-dimensional space-time outer-

interfacial surface in the far field of incom-pressible turbulent jets. Another visualiza-tion of the OFI behavior is shown in fig-ure 1 (right). These images illustrate thecomplex structure of the OFI’s.

Using the octagonal-tank flow facility atUC Irvine, experiments enable the study ofthe mixing efficiency in jets by direct imag-ing of the outer fluid interfaces. The imagedata, such as the visualization shown infigure 1 (left), were recorded using whole-field three-dimensional space–time imag-ing of the concentration field in the similar-ity plane of each flow. For the jet this planeis normal to the jet axis. The concentra-tion field is thresholded in order to identifythe outer fluid interfaces, i.e. the interfacesbetween pure fluid and mixed fluid. Thecamera resolution is ∼ 1, 0002 in the slic-ing plane and a total of ∼ 1, 000 consec-utive images can be recorded temporally,resulting in ∼ 1, 0003 data volumes of theOFI behavior. These measurements cap-tured three out of the four dimensions atfully-developed turbulent flow conditions.

3

Figure 2: Schematic illustrating the difference in the contributions of large-scale interfa-cial features vs. small-scale interfacial features, to the volume of the fluid region enclosedby the interface and to the surface area of the interface. Whereas the surface area is highlysensitive to the small-scale features, the volume enclosed by the interface is dominatedby the large-scale features.

The minimum Reynolds number for whichturbulent mixing can be considered asfully developed, i.e. above the mixingtransition, is in the range Re ∼ 104.These experiments, therefore, representfully-developed turbulent mixing and canbe expected to be particularly useful fromthe point of view of large-Reynolds-numberturbulent mixing.

A well-known difficulty with direct imaging

of mixing using fluorescence techniques isthat mixing can be overestimated becauseinformation at a pixel that contains smallerscales cannot distinguish mixing variationsbelow that pixel scale.

To alleviate this problem, chemical reac-tions can be done to provide the exactamount of molecular mixing. In the OFIapproach, however, the pure fluid at scalesat or above the pixel scale can be identified

4

by direct imaging , so that the usual diffi-culty of overestimating mixing does not ap-ply to the measurement of the outer fluidinterfaces (OFI’s) at or above the pixelscale.

This is because those pixels which havec = 0 or c = 1 (i.e. correspondingto pure fluid) imply necessarily that onlypure fluid is contained in those pixels. Thiscrucial point provides a new experimentalapproach to the mixing efficiency, at largeReynolds numbers.

Examination of these measurements in thefar field of incompressible turbulent jets in-dicates that the mixing efficiency is

αm ∼ 0.6 , (3)

i.e. 60% of the fluid contained inside thejet outermost boundaries is mixed. TheOFI approach enables a new methodologyto examine the relative contributions ofthe volume of the complex mixed-fluid re-gion contained by the OFI’s as a functionof scale. This can be done by extendingthe box-counting approach, typically usedin this context to examine the interfacialarea, to quantify the volume of the regioncontained by the OFI’s. A new quantityVouter(λ) can be used which quantifies thecumulative dependence of this volume withscale, starting from the large scales, andalso a quantity gV (λ) which quantifies thescale-local volume density, i.e.

Vouter(λ) =∫ L

λgV (λ′) dλ′ . (4)

This facilitates the examination of the con-tributions of the OFI-bounded volume, asa function of scale, to the mixing efficiencyαm particularly in the dominant upperrange of scales. At large Reynolds num-bers, the upper range or energy-containingrange of scales, from the large scale Lto the Liepmann-Taylor scale λLT, is the

dominant range of scales in the OFI ap-proach and is directly measurable with thehigh-resolution imaging. For the presentdata, we can estimate the relative large-scale (upper-range) vs. small-scale (lower-range) mixing-efficiency contributions, de-noted by α

URand α

LRrespectively, i.e.,

αm = αUR

+ αLR

∼ 0.6 , (5)

in terms of the size of each interfacial fea-ture as well as the number (or number den-sity) of the interfacial features as a func-tion of scale λ. From the scale-local area-volume density contributions, which havebeen shown in figure 4e ([10], we can es-timate the relative values of the surfacearea contributions from the large vs. smallscales as well as the ratio of the numberof interfacial features above the thresholdLiepmann-Taylor scale to the number offeatures below that scale. The resultingestimate for the ratio of the upper-rangecontribution to the lower-range contribu-tion is,

αUR

αLR

∼ 10 , (6)

for the present flow conditions. This meansthat the large-scale interfaces yield ∼ 90percent of the mixing efficiency, i.e.,

αUR

αm

∼ 90% , (7)

whereas the small-scale interfaces only ac-count for ∼ 10 percent, i.e.

αLR

αm

∼ 10% . (8)

The large-scale features provide the dom-inant contribution to the volume insidethe outer interfaces. The relative percent-ages will depend on the threshold scaleemployed, which is taken here to be theLiepmann-Taylor scale. This thresholdscale is appropriate because it marks the

5

lower end of the upper scales which are di-rectly influenced by the large-scale vorticalmotions.

3. Conclusions

The Outer-Fluid-Interface (OFI) approachprovides a useful physical point of view forexamining the turbulent mixing efficiency.Whereas the conventional approach re-quires knowledge of both the outer andinternal interfaces, the OFI point of viewprovides the means to study the mixingefficiency based solely on the outer inter-faces. Furthermore, the OFI approach in-dicates that the turbulent upper scales,rather than the lower scales, dominate themixing efficiency. In this work, we consid-ered how the OFI approach can be usedto quantify the contributions of the largeinterfacial scales vs. the small interfacialscales, relative to each other. On the ba-sis of high-resolution measurements in thefar-field of fully-developed round jets, wefound that the interfacial features from theupper range contribute ∼ 90% to the mix-ing efficiency, as compared to the interfa-cial features from the lower scales whichcontribute the remaining ∼ 10%, at thepresent flow conditions. This means thatthe large-scale folding, as opposed to thesmall-scale wrinkling, of the interfaces pro-vides the dominant contribution. Thisis an important finding because it sug-gests that one can examine large-Reynolds-number flows, with emphasis on the up-per range of scales, to characterize thedominant behavior of the mixing efficiency.This idea is applicable to studies of un-forced flows, to determine the dependenceof the mixing efficiency on the Reynoldsnumber, as well as to studies of flow con-trol for characterizing the dominant inter-facial contributions to either enhance orreduce the mixing efficiency. The presentapproach can be expected to be useful in

applications ranging from far-field aircraftsignatures to laser beam propagation andaerooptics in high-speed flight.

Acknowledgements

This work is part of a research programon turbulent fluid interfaces, supported bythe Air Force Office of Scientific Research(Dr. T. Beutner, Program Manager) andby the National Science Foundation (Prof.M. Plesniak, Program Director).

References

[1] H. J. Catrakis, R. D. Thayne, B. A.McDonald, R. C. Aguirre, and J. W.Hearn. Mixing and the three-dimensional structure of fluid inter-faces in turbulent jets. In 31st AIAAFluid Dynamics Conference and Ex-hibit, AIAA 2001-2716, Anaheim,CA, 2001.

[2] P. E. Dimotakis, H. J. Catrakis, andD. C. L. Fourguette. Flow structureand optical beam propagation in high-Reynolds-number gas-phase shear lay-ers and jets. J. Fluid Mech., 433:105–134, 2001.

[3] H. J. Catrakis, R. C. Aguirre, andJ. Ruiz-Plancarte. Area–volume prop-erties of fluid interfaces in turbulence:scale-local self-similarity and cumula-tive scale dependence. J. Fluid Mech.,462:245–254, 2002.

[4] K. R. Sreenivasan. Fractals and mul-tifractals in fluid turbulence. Annu.Rev. Fluid. Mech., 23:539–600, 1991.

[5] N. T. Clemens and M. G. Mungal.Large-scale structure and entrainmentin the supersonic mixing layer. J.Fluid Mech., 284:171–216, 1995.

6

[6] E. Villermaux and C. Innocenti. Onthe geometry of turbulent mixing. J.Fluid Mech., 393:123–147, 1999.

[7] H. J. Catrakis, R. C. Aguirre, J. Ruiz-Plancarte, R. D. Thayne, B. A. Mc-Donald, and J. W. Hearn. Large-scale dynamics in turbulent mixingand the three-dimensional space-timebehaviour of outer fluid interfaces. J.Fluid Mech., 471:381–408, 2002.

[8] M. Gad-el Hak. Flow Control: Pas-sive, Active, and Reactive Flow Man-agement. Cambridge University Press,2000.

[9] H. J. Catrakis and R. C. Aguirre.Inner-scale structure of turbulence-degraded optical wavefronts. In AIAA33rd Plasmadynamics and LasersConference, AIAA 2002-2269, Maui,HI, 2002.

[10] H. J. Catrakis, R. C. Aguirre,J. Ruiz-Plancarte, and R. D. Thayne.Shape complexity of whole-field three-dimensional space-time fluid inter-faces in turbulence. Phys. Fluids,14:3891–3898, 2002.

7