41st Aerospace Sciences Meeting and Exhibit, 6-9 January 2003, Reno, Nevada

Physical Thickness of Turbulent Fluid Interfaces:Structure, Variability, and Applications to Aerooptics

Roberto C. Aguirre,† Jesus Ruiz-Plancarte,† and Haris J. Catrakis ‡

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

Abstract

Basic aspects of the physical thickness of turbulent fluid interfaces, such as density orconcentration interfaces, are considered theoretically and examined on the basis of mea-surements. The variability of the interfacial thickness corresponds to the fluctuations inthe inverse local gradient of the fluid or flow property associated with the interface. Thevariations of the interfacial thickness, throughout the flow or along the interfaces, canbe quantified in terms of probability density functions. Such descriptions are importantpractically, particularly in aerooptics. The variability in the local thickness of refractiveor density interfaces is crucial in establishing the link between the flow behavior and thewavefront-phase variations of optical beams propagating through the flow. The optical-path-length (OPL) integral and wavefront-phase variations can be interpreted physicallyin terms of the behavior of the interfacial thickness. Results are presented on the basisof measurements of interfaces in high-Reynolds-number free-shear flows.

1. Introduction

Fundamental and practical questions inaerooptics (e.g. Jumper & Fitzgerald2001) are intimately tied to the “turbu-lence problem” (e.g. Liepmann 1979) andcan benefit from the development of novelphysical points of view. A useful relatively-recent point of view in both applied andfundamental studies of turbulent flows isto understand the behavior of the variousfluid interfaces generated by the flow dy-namics (e.g., Pope 1988; Dimotakis 1991;Sreenivasan 1991; Villermaux & Innocenti1999; Catrakis, Aguirre, & Ruiz-Plancarte2002). Examples include density inter-

faces, concentration interfaces, vorticity in-terfaces, etc. Such interfaces correspond toregions in the flow where a particular fluidor flow property is constant. The interfacesare typically represented as isosurfaces, i.e.as surfaces on which the fluid/flow prop-erty is constant. However, a physical de-scription of fluid interfaces needs to takeinto account the fact that these interfaceshave a physical thickness. The interfa-cial thickness is associated with the in-verse of the local gradient of the fluid/flowproperty. Regions of high local gradient

†Graduate Student, Member AIAA.‡Assistant Professor, Member AIAA. Corresponding Author. Phone: (949) 824-4028. E-mail:

catrakis@uci.edu

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41st Aerospace Sciences Meeting and Exhibit6-9 January 2003, Reno, Nevada

AIAA 2003-642

Copyright © 2003 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.

correspond to locally-thin interfaces, whilelow-gradient regions are associated withlocally-thick interfaces. In turbulent flows,the interfacial thickness can be expected tobe highly nonuniform spatially as well asin time, as a result of the dynamics of suchflows. Knowledge of the structure and vari-ability of the interfacial thickness is impor-tant both practically and fundamentally.

In various applications involving turbulentflows, such as aerooptics, aeroacoustics,mixing, or combustion, one is interestedin the propagation of optical or acousticwavefronts, molecular diffusion, or chemi-cal reactions across fluid interfaces. Froma practical and fluid-mechanical point ofview, there is a need to establish rela-tions between the flow dynamics and prac-tical quantities such as optical-path-length(OPL) integrals or the mixing efficiency.Such relations will also be useful in stud-ies of turbulence control, for example in thecontext of reducing aerooptical distortions.In aerooptics, in particular, the variabilityin the local thickness of refractive or den-sity interfaces is of central importance inestablishing the link between the flow be-havior and the wavefront-phase variationsof optical beams propagating through theflow. The physical thickness of the inter-faces can be expressed as the local inversegradient of the refractive index or den-sity. For optical beam propagation, it isthe component of the gradient in the localbeam direction that needs to be examined.As will be shown in Section 2, the OPL in-tegrals and wavefront-phase variations canbe interpreted physically in terms of thebehavior of the interfacial thickness.

At high Reynolds numbers, the variationsin the interfacial thickness are related tothe highly-intermittent nature of turbu-lence. As is well known, the intermit-tency becomes increasingly stronger with

increasing Reynolds number (e.g. Coles1962; Sreenivasan 1991) and therefore thenonuniformity in the interfacial thicknessis an important characteristic of high-Reynolds-number turbulent flows. In-termittency, and the interfacial-thicknessvariations, occur throughout the flow. Itis important to understand and quan-tify the nonuniformity of the thicknessboth in terms of its behavior throughoutthe flow as well as on a given interface.The interfacial-thickness variations can bequantified in various ways. To investigatethe distribution of the magnitude of thethickness, in the flow or on an interface,one may examine probability density func-tions of the thickness. To quantify thephysical structure and variability of the in-terfacial thickness, more detailed descrip-tions can be employed such as multifractaldescriptions (e.g. Sreenivasan 1991). Inthe context of aerooptics, it is also impor-tant to understand the effect of compress-ibility on the thickness variations of the re-fractive interfaces.

In Section 2, basic aspects of the physicalthickness of fluid interfaces in turbulenceare considered theoretically and the prac-tical importance of the interfacial thicknessis emphasized in the context of aerooptics.In Section 3, measurements of interfaces inhigh-Reynolds-number free-shear flows areexamined using the ideas from Section 2and results are presented on the variabilityof the interfacial thickness. Implications ofthe present results for developing modelsof the interfacial-thickness variations arestated in the conclusions.

2. Theoretical considerations

Since the pioneering aerooptics study byLiepmann (1952), it has been recognizedthat one of the central goals in aeroopticsresearch is to relate the optical-wavefront

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distortions to the fluid-mechanical behav-ior (e.g. Wissler & Roshko 1992; Jumper& Fitzgerald 2001 and references therein).The optical-wavefront distortions are use-fully quantified by the optical path length(OPL), here denoted as Λ(x, t), for manyaerooptics applications. The most relevantfluid-mechanical quantity in aerooptics isthe refractive-index field,

n(x, t) ≡ c0

c(x, t)≥ 1 , (1)

where c(x, t) is the local speed of lightwhich cannot exceed Einstein’s universalspeed of light c0 in vacuum. In turbu-lent flows, the refractive-index field n(x, t)can be highly nonuniform particularly atlarge Reynolds numbers and at high com-pressibility. How is the OPL behavior re-lated physically to the structure of therefractive-index field? In this section, wepresent a framework to address this ques-tion by emphasizing the role of the re-fractive fluid interfaces and their physicalthickness in determining the variations inthe OPL.

As a starting point, we recall the expres-sion used for the definition of the OPL asan integral of the refractive index alongeach light ray (e.g. Jumper & Fitzgerald2001 and references therein), i.e.,

Λ(x, t) ≡∫ray

n(`, t) d` , (2)

where, as stated above, Λ denotes theOPL, and ` denotes the physical distancealong the propagation path of each lightray. The aerooptical distortions corre-spond to the optical-path difference (OPD)given by ∆Λ(x, t) ≡ Λ(x, t) − Λref(x, t),where Λref(x, t) ≡

∫ray nref(x, t) d` is the

reference OPL that would correspond tothe undistorted wavefronts, with nref de-noting a reference refractive index, e.g.corresponding to freestream conditions.

The optical wavefronts can be representedas isosurfaces of the OPL, i.e.,

Λ(x, t) = const. , (3)

as indicated schematically in figure 1. Aslong as the flow speeds are small relative tothe speed of light, and the propagation dis-tances are small enough for light to propa-gate through the flow before it has evolved,it is sufficient to think of the OPL integralin equation 1 as involving only the spatialstructure of the refractive-index field, ateach instant in time.

Figure 1: Schematic of the refractive-indexfield n(x, y; z, t) in a turbulent shear layervisualized in the streamwise x-y plane withflow from left to right. The refractive fluidinterfaces, i.e. the n = const. surfaces, areevident in the gray-level image at the in-set. The freestream refractive indices aredenoted as n1 and n2, with n2 > n1 > 1.Also shown is a schematic of a planar in-cident optical wavefront, as a Λ = const.surface, and a propagated distorted opticalwavefront, as a Λ + ∆Λ = const. surface,

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where Λ denotes the optical path length.The flow visual thickness is denoted as Lδ

and measures the extent of the outer edgesof the refractive fluid interfaces.

As the optical wavefronts propagatethrough the nonuniform refractive-indexfield n(x, t), the aerooptical interactionsphysically occur across the refractive fluidinterfaces. These are the interfaces onwhich the refractive index n is constant. Itis important to understand the role of theseinterfaces. While these fluid interfaces cor-respond to isosurfaces of the refractive-index field, i.e.,

n(x, t) = const. , (4)

it is crucial to recognize that the refractive-fluid interfaces will have a physical thick-ness whereas the refractive-fluid isosur-faces are geometrical objects with zerothickness. It is the physical thickness of theinterfaces that is very important in aeroop-tics. We can introduce the local interfacialthickness hn, defined per unit n, as the in-verse of the refractive-index gradient mag-nitude, i.e.,

hn(x, t) ≡ 1

|∇n|. (5)

The distance between two neighboring iso-surfaces, corresponding to n and n + dn,will be hn dn. This is shown schemati-cally in figure 2. We can distinguish be-tween the general case, where the gradi-ent magnitude is nonzero, i.e. |∇n| > 0,and the special case of zero gradient mag-nitude, i.e. |∇n| = 0, which will be dis-cussed below. In regions of relatively-largerefractive-index gradients, the isosurfacesare closely spaced and the interfaces are as-sociated with a relatively-small thickness.In regions of weak refractive-index gradi-ents, the isosurfaces will be located fur-ther apart and the interfaces will be rel-atively thicker. The interfacial thickness

can be expected to be highly nonuniform atlarge Reynolds numbers. This is becauseof the strongly-intermittent character offully-developed turbulent flows which be-comes more intermittent with increasingReynolds number for both incompressibleand compressible flows (e.g. Sreenivasan1991 and references therein; Smits & Dus-sauge 1996). As discussed by Jumper &Fitzgerald (2001), refractive-index fluctua-tions can arise in pure fluids, e.g. wheredensity fluctuations are induced by tem-perature variations in low-speed air flows(e.g. Jumper & Hugo 1995) or densityfluctuations in compressible air flows (e.g.Fitzgerald & Jumper 2000), or in mix-tures of dissimilar fluids (e.g. Brown &Roshko 1974; Dimotakis et al. 2001). Inall these different cases, the thickness ofthe refractive interfaces can be defined byequation 4.

n + dn

n hn dn

Figure 2: Schematic representation of twoneighboring fluid interfaces and the varia-tions in the local thickness, in the contextof the refractive-index field n(x, t).

Is the interfacial thickness finite? Is itnonzero? In turbulent flows, the interfa-cial thickness can be expected to be bothnonzero and finite, in general, on physicalgrounds. The thickness must be nonzerowherever the local refractive-index gradi-ent magnitude |∇n(x, t)| is finite, as in-dicated from equation 4. Only an infi-

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nite gradient can lead to a zero interfacialthickness. Physically, it is clear that forflows of real fluids, even at large but finiteReynolds numbers, the finite molecular dif-fusivities of the fluid ensure finite gradi-ents and therefore finite interfacial thick-nesses. A related observation, also indi-cated from equation 4, is that the thick-ness must be finite as long as the gra-dient magnitude is nonzero. If the gra-dient is zero, which would correspond toa region of exactly-uniform refractive in-dex, the thickness would be infinite in thecontext of equation 4 and one then mustinterpret the (infinity-times-zero) producthn dn as the distance given by the extent ofthe uniform-index region in the direction ofthe optical-ray propagation. In summary,we can expect physically that the inter-facial thickness must be finite as long asthe gradient is finite, consistent with equa-tion 4. The thickness is an interfacial prop-erty that has to be taken into account.

Since each optical ray physically propa-gates through refractive interfaces, one canintuitively expect that the local interfa-cial thickness should determine, at least inpart, the local contribution to the OPL.Can this be seen in the OPL integral inequation 1? This can be done by rewritingequation 1, from the point of view of therefractive fluid interfaces, as

Λ(x, t) ≡∫ray

n(`, t) hn,` |dn| , (6)

where the integration is now performedwith respect to the refractive index n,rather than with respect to the spatial dis-tance `, and hn,` is the effective interfacialthickness defined as the component of theinterfacial thickness in the direction of op-tical propagation, given by

hn,` =1

|∇n|`, (7)

with |∇n|` denoting the effective gradientmagnitude, i.e. the magnitude of the com-ponent of the local refractive-index gradi-ent in the ` direction, i.e. in the directionof the optical-ray propagation. The com-ponent of the refractive-index gradient inequation 7 is,

|∇n|` ≡ |dn|d`

, (8)

as required, of course, in order for equa-tions 2 and 6 to be consistent. Since the re-fractive index n could be locally increasingor decreasing as the light rays propagate, itis necessary to express the differential of nas the absolute-valued differential |dn|, inequation 7. For interfaces locally normalto the optical rays, the gradient compo-nent |∇n|` has magnitude identical to themagnitude of |∇n|. Where the interfacesare locally not perpendicular to the opti-cal rays, this component will be of smallermagnitude than |∇n| and the effective in-terfacial thickness will be larger. In otherwords, the effective gradient is always lessthan or equal to the full gradient, i.e.

|∇n|` ≡ |∇n| | cos θ| ≤ |∇n| , (9)

and the effective interfacial thickness is al-ways greater than or equal to the full in-terfacial thickness, i.e.

hn,` =1

|∇n|`≡ hn | sec θ| ≥ hn , (10)

where the angle θ, taken as −π < θ ≤ π,quantifies the interfacial orientation rela-tive to the optical-propagation direction.As long as the refractive-index gradientmagnitude is not zero, i.e. as long as|∇n| > 0, we can define θ as the anglebetween the refractive-index gradient vec-tor and the local optical-ray propagationvector. The refractive-index gradient vec-tor is always normal to the local refrac-tive interface. Combining equations 10 and

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6, we see that the OPL integral of equa-tion 2 can be expressed directly in termsof the interfacial-thickness variations alongthe optical propagation path as,

Λ(x, t) ≡∫ray

n(`, t) hn | sec θ| |dn| ,(11)

where this integral is in terms of the thick-ness hn and relative orientation θ of therefractive interfaces. As mentioned above,in the context of the interfacial thicknessin equation 4, such an integral requiresthat |∇n| 6= 0. In those regions where|∇n| = 0, i.e. in regions of uniform refrac-tive index, the (infinity-time-zero) prod-uct hn dn must be interpreted as the dis-tance, say ∆`, corresponding to the extentof the uniform-index region in the optical-propagation direction, so that the contri-bution to the OPL integral is ∆Λ = n ∆`.We note that, in uniform-index regions,the interfacial orientation θ has no mean-ing and is not needed. In those instanceswhere the fluid interface happens to be lo-cally tangent to the optical-propagation di-rection, i.e. if θ = ±π/2, the term | sec θ|will be infinite but in such cases the refrac-tive index will locally be uniform since theinterface will be aligned with the optical-propagation direction, i.e. |dn| = 0 in suchcases. In those cases, therefore, the contri-bution to the OPL integral will again be∆Λ = n ∆` with ∆` identified as thelength of the interface that is tangentialto the optical-propagation direction. Weshould also note that, in general, one mayalso need to take into account other possi-bilities such as total internal reflection orthe development of caustics.

In summary, the proposed interfacial-thickness approach is based on relating theOPL to the interfacial-thickness variations.Whereas the integral in equation 2 is con-ducted over space, the integrals in equa-tions 6 and 11 are expressed as integrals

over the refractive index and are useful todetermine the manner in which the refrac-tive interfaces physically contribute to theOPL. In addition to the local refractive in-dex n, equation 11 shows that the OPLvariations arise from the variability in theinterfacial thickness hn and the fluctua-tions in the interfacial orientation θ, or thevariations in the effective interfacial thick-ness hn,`. Knowledge of the variability inthe effective interfacial thickness, and itsrelation to the flow dynamics, can be ex-pected therefore to provide physical insightinto the relation between the OPL behav-ior and the interfacial structure.

3. Application of interfacial-thickness approach to turbu-lent high-compressibility shearlayers

In this section, we demonstrate the use ofthe interfacial-thickness approach to tur-bulent highly-compressible fluid interfaces.Figure 3 (top) shows a two-dimensionalspatial slice of the refractive-index fieldin a shear layer between optically-differentgases, with convective Mach number Mc ∼1 and Reynolds number Re ∼ 106 based onthe visual thickness (data from Dimotakis,Catrakis, & Fourguette 2001). The imageshown spans the entire large-scale trans-verse extent of the flow, including smallscales, and indicates that the refractive in-terfaces are highly irregular.

This refractive-index field can be usedto compute the interfacial-thickness fieldwhich is shown in figure 3 (bottom). Thebehavior of the interfacial thickness is im-portant in order to identify the manner inwhich the flow generates the optical dis-tortions. Since the data in figure 3 (top)are two-dimensional spatial data, only twoof the three components of the refractivegradient can be computed in this case.

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Figure 3: Refractive-index field and interfaces (top) and corresponding interfacial-thickness field (bottom) in a compressible shear layer at Mc ∼ 1 and Re ∼ 106, c.f.figure 1. The dark regions in the lower figure correspond to the high-gradient interfaces.

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In figure 3 (bottom), regions of highrefractive-index (in-plane) gradients are in-dicated as dark regions. An importantobservation is that these high-gradient re-gions are spatially isolated, i.e. these re-gions are confined to thin layers in theturbulent-flow region. These high-gradientregions correspond to locally-thin inter-faces. The data indicate that, at high com-pressibility, these high-gradient interfacesare present both in the interior and nearthe outer boundaries of the shear layer.This is in contrast with the behavior atlow compressibility (Truman & Lee 1990;Dimotakis, Catrakis, & Fourguette 2001)where high-gradient regions are confinedonly to the outer parts of the flow. At thehigh-compressibility flow conditions exam-ined in the present work, the high-gradientinterfaces are more highly convoluted andit is evident that they can be located atdifferent transverse positions in the shearlayer.

Four significant observations can be madeon the basis of figure 4 and otherinterfacial-thickness fields at the same flowconditions:

(a) the high-gradient regions occupy arelatively-small part of the turbulentshear-flow region,

(b) the high-gradient regions are sheet-like and can be thought of as high-gradient interfaces,

(c) the high-gradient interfaces exist atvarious transverse locations in theshear-flow region, and

(d) the refractive-index gradient magni-tude exhibits variations along thehigh-gradient interfaces.

It is important to note that these ob-servations refer to the instantaneous spa-

tial structure of the interfacial-thicknessfield, and this is practically very rele-vant for aerooptics since it is the instan-taneous flow structure that needs to beunderstood. The ensemble-averaged be-havior may well be very different and wehope that this will be addressed in futurestudies. A new aerooptics pressure ves-sel, shown in figure 4, has recently beenconstructed in our laboratories in order toexamine these issues further for separatedhigh-compressibility shear layers.

Figure 4: Recently-constructed aeroopticspressure-vessel facility at UC Irvine. Thisfacility will be used to examine separatedhigh-compressibility shear layers and therole of the thickness of refractive fluidinterfaces in large-scale and small-scale

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aerooptical distortions at high Reynoldsnumbers.

The observation that the high-gradient re-gions are spatially isolated, and the inter-pretation of the optical wavefront phase interms of the interfacial thickness variations(c.f. §2), indicate that these isolated re-gions dominate the large-scale aeroopticaldistortions. While the low-gradient regionsare of wide transverse extent, and thereforedo contribute to the optical path lengthintegral, it is the high-gradient (locally-thin) interfaces that can form the basis ofa description that captures the large-scaleaerooptical distortions, as explained below.

The present findings suggest a new wayto model aerooptical distortions in highly-compressible turbulent flows, by utilizingthe fact that the regions of high-gradientinterfaces are spatially isolated. Instead ofthe complete interfacial-thickness field, in-formation on the high-gradient interfacesappears to be sufficient to model the large-scale aerooptical distortions. The loca-tion of the high-gradient interfaces and thevalue of the gradient across (or of the thick-ness of) these interfaces can be expectedto be enough to capture the dominantcontributions that generate the large-scaleaerooptical distortions. In this model, theoptical wavefronts propagate through thehigh-gradient (locally-thin) interfaces andthe gradient value across these interfaces isused to compute the difference in the opti-cal path length, while in the low-gradientregions between the high-gradient inter-faces the wavefronts are propagated ne-glecting the presence of the low gradients,i.e. as if those regions are zero-gradient re-gions. Because the high-gradient regionsare isolated, they span only a small frac-tion of the turbulent-flow field. The pro-posed modeling approach offers a large re-duction in the amount of flow information

needed to capture the large-scale aeroopti-cal distortions at high compressibility.

4. Conclusions

The present work shows that theinterfacial-thickness approach for thestudy of the propagation of optical wave-fronts through turbulent flows can be ex-tended to compressible turbulence. Atlarge Reynolds numbers and high com-pressibility, the interfacial thickness or in-verse of the local refractive-index gradi-ent will be highly variable in space andtime. The interfacial-thickness variationsplay an important role in optical-wavefrontpropagation as can be seen by express-ing the optical path length or wavefrontphase directly in terms of the interfa-cial thickness along the optical propaga-tion path. We have demonstrated thisinterfacial-thickness approach on measure-ments of the refractive-index field in high-compressibility (Mc ∼ 1) high-Reynolds-number (Re ∼ 106) shear layers betweenoptically-different gases. Isolated regionsof high refractive-index gradients are ev-ident both in the interior and near theouter boundaries of the shear layer. Theseisolated regions correspond to locally-thininterfaces which are found to be highlyconvoluted for the present flow conditions.The observation that the high-gradient re-gions are spatially isolated, and the inter-pretation of the optical wavefront phasein terms of the interfacial thickness varia-tions, indicate that these isolated regionsdominate the large-scale aerooptical dis-tortions and this suggests a new approachto model aerooptics at high compressibil-ity and large Reynolds number in terms ofthe high-gradient (locally-thin) interfaces.

Acknowledgements

This work is supported by the Air

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Force Office of Scientific Research throughGrants 01-NA-147 and 01-NA-440 (Dr. T.Beutner, Program Manager) and is partof a research program on turbulent flowsand aerooptics. The authors acknowledgeuseful advice by R. Hugo, M. Jones, E.Jumper, R. Truman.

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