AlAA 89-0370 Optical Propagation Through a Homogenous Turbulent Shear Flow C. Truman, Univ.of New Mexico, Albuquerque, NM; M. Lee, NASA-Ames, Moffett Field, CA

27th Aerospace Sciences Meeting January 9-12, 1989/Reno, Nevada I

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Ai AA-89-0370

Optical Propagation through a

Homogeneous Turbulent Shear Flow

C. Randall Trumant Department of Mechanfcal Engrnccrzng, llnzverszly of New Mexzco, Albuquerque, New Mexzco

Moon J . Lee: C'enier foT Turbulence Research, Stanfoford rJnzverszty, Stanford, Calafornaa

and NASA-Ames Research Center, Moffett Fzeld, Calzfornza

Effects of o rgan ized . t r i rhu len t s t r u c t u r e s on the p r o p a g a t i o n of an o p t i c a l beam in a homogeneous shear flow have been studied. A passive-scalar field in a c o m p u t e d turbulent shear flow is used to represent index-o f - r e f r ac t ion f l u c t u a t i o n s , and phase errors i n d u c e d in a c o h e r e n t o p t i c a l beam b y turbu- lent. f luc . tua t ions are c o m p u t e d . The o r g a n i z e d vo r t i ca l s t r u c t u r e s p r o d u c e a sca l a r dist.ribution with elongated regions of intense f l u c t u a t i o n s which h a v e an inc l ina t ion w i t h r e s p e c t to the mean flow similar to that of the cha rac - t e r i s t i c h a i r p i n eddies. It is found that r.m.s. phase error is minimized b y p r o p a g a t i n g a p p r o x i m a t e l y n o r m a l to the inc l ined vo r t i ca l s t r u c t u r e s . Two- point c o r r e l a t i o n s of vo r t i c i ty and sca la r f l u c t u a t i o n suggest that the regions of int,ense sca l a r f l u c t u a t i o n are p r o d u c e d p r i m a r i l y b y the hairpin edd ies .

1. I n t r o d u c t i o n

The passage of a coherent electromagnetic beam through turbulent flow fields results in degradation of optical quality. Phase distortions by index-of-refraction fluctuations reduce bram intensity in the far field. Prop- agation ofwavr in atmospheric situations has been stud- ied extensively (see Tatarskii 19i1, Chapter 4 ) . The present study, however, is motivated by the need to understand (and predict) losses in optical quality in- duced by thin shear layers through which a beam must pass. These include, for examplr, a mixing layer a t the exit cavit.y of a laser (see Baxter, Truman 8L Masson 1988).

t Senior Member AIAA. $ Member AIAA.

This paper is declared a work of the US. Government and is not subject t,o copyright protection in the United States.

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The physical problem of interest in the present study is a coherent optical beam in the visible range, whose width is large compared to the length scales of turbu- lence in the shear layer through which i t passes. The beam width is assumed to be the same order as the size of the computational flow field, which is about ten times the largest length scale of turbulence. The beam wavelength selected was 4x lo-' m with flow field di- mensions on the order of 0.1 m.

The effect of turbulent fluctuations upon optical quality is commonly modeled assuming isotropic., ho- mogeneous turbulence with a Gaussian distribution (see Tatarskii 1971, $547-49). As noted by Liepmann (1979), however, "the theory of homogeneous [and isotropic] turbulence does not lead t o decisive progress in coping with the shear flow and general mixingproblems." Cur- rent understanding of turbulent shear flow shows the statistical theories, including models ofoptic.al degrada- tion (see Sutton 1969), to be inadequate. Thus success- ful predictions of beam degradation must account for

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instantaneous turbu1enc.e structures of both the large- and small-scale eddies.

This work has been undertaken to gain an under- standing of the relativr importance of the largr- and small-scale turbulent fluc.tuations in optical propaga- tion. A passive-sc.alar field in a homogeneous shear flow computrd by Rogers, Moin k Rrynolds (1986) is usrd to represent an instantaneous index-of-rrfractioii field. It is our primary interest t o explore the effects of or- ganized turbulence structures in turbulent shear flow on the propagation of a c.oherent optical beam. The basis for computing phase error induced by turbulent fluctuations is discussed first.

2. Wave propagation in a turbulent medium

2.1. P a r a b o l i c wave equation

Maxwell's equations govern the brhavior of an de r - tromagnetic beam propagating through a turbulent me- dium. The magnetic permeability is assumed t o be constant while t h e dielectric constant (and thus th? re- frartive index) is assumed to b r space-variant (Good- nian 1985. pp. 363-399), and the effects of depolariza- tion are nFgligiblr (Monin & Yaglom 1Yi5, Chapt r r 9 ) . We neglrct the time-dependence of the rhfractire index, since thr time scale for propagation of hghl through a flow field is much smaller than that of turhulent fluc- tuations.

A scalar rlectric firld E ( x * ) may be considered and the transformation E = UP'"- yields

whcre x* is the distance in the direction of propagation,

of the coherenl optical beam and n(xI;) is the index of refraction. The complex function u ( x * j , which repre- sents the a m p l z t u d c and the p h a s e of the electric field, is then slowly varying i n the propagation direction.

In practice, t.hP timr-averagPd spat,ial variation i n index of refraction is ignored. Such variations may dis- place the beam focus (e.6. beam steering), but induce no loss in far-field intensity (Born k Wolf 1975, pp. 4 F 2 - 4 6 3 ) . Moreover, the effect of these variations can be treated by conventional optical techniques.

Sincp the effects of turbulent struc.turrs arr of pri- mary interest, a fluctuating wave equation is described. Thz instantaneous index of refraction is written as

y72 ~ - a ? ~ / a x ~ i a a / a y i + a 2 / a : $ , K is t h e wavenumber

I n = l + n , ( 2 )

where n', typically of order 0(10-6), is understood to be the fluctuation with respect to t h r spatial distribu- tion of index of refraction.

If backscatter is not important, m e may neglect the second derivative with respect t o x * in ( I ) . Then, after

parabolized Helmholtz equation: neglecting the term quadratic in n', one obtains the u

A pseudospectral method was developed by Clark, n u - man & Masson (1988) t o solve (3) for the phase and amplitude of a coherent optical beam which propa- gates through a turbulent medium with a deterministic index-of-refraction field.

2 .2 . Optical pa th difference

When diffraction of the optical beam is signific.ant, the parahohzed Helmholtz equation (3) must be solvrd. For the present study, however. the selected pirramctcrs ensure that diffraction is negligible, and the restrictions of gpoiuetric optics (or acoustics) hold (Krllt-I 1954). With thc second derivatives representing diffraction nc- glrct,ed, the parabolic equation ( 3 ) reduces to a n or- dinary differential equation wi th a variable cot.ffiricnt n ' ( x * ) . Sinre changes in hram amplitudr arc nrgligi- blr over a short propagation distance, only changcs i n phase need to he considwed.

v I n this case, tlir phasr error can be drtrrminrd tiy

where s is the distance along the propagation path. This integral, known as the o p t i c a l p a t h difference, mul- tiplied by the wavenumber o f the beam yields the phasr error induced by variations in the index of refraction. The use of the optical path differenc.e t o compute phasr errors along arbitrary paths allows the usc of tlir full numerical database, which is heneficial i n analyzing sta- tistical quantities and associated instantaneous turbu- lencr structures.

3. Results and discussion

A flow field in a homogeneous turbulent shear flow compiited on a 128 x 128 x 128 grid by Rogers. Moin k Reynolds (1986) has been analyzed to provide 'data' for the present study. We have chosen a flow field (C128U12 in Rogers et al.'s notation) a t diuiension- less time St = 12 (S is the shear rate), i n which most turbulence statistical correlations are 'fully-devrlopcd.'

u 2

Herr, (z, y, :) drn0t.r the coordinates in the streamwise, transverse and spanwise directions, respectively.

3.1. Passive-scalar field

The instantaneous passive scalar H (0, in Rogers el d ' s notation) subjected to uniform mran velocity and scalar gradients transverse to thc mean flow, d l l l d y and d O / d y , is used to represent thc index-of-rcfraction fluctuations n'; the passive scalar is scaled to an r.m.s. value of As discussed in 52.1, the mean scalar gradient is neglected, since only phase distortions due t o turbulent fluctuations are of interest. The phase er- rors a r c compnted based on the instantaneous index of refraction as determined from the passive scalar distri- bu t ion

In order to examine the spatial distrihution, con- tours of the scalar fluctuations arc shown on a vertical zy-plane in figure 1. I t is clearly shown tha t the scalar field consists of rrgions of intense fluctuation which have a n orientation similar to the slrnctiirw of the vor- t ic i ty firld, i.?. the hairpin vortiws (Rogers Xr Moin 198 i ) . Notice that regions in which scalar fluctuations are IWO or evcn three times as large as the r .n . s . value arr inclincd a t approximately 45' from the gradient di- rrction and are- elongatrd along l,he dirrrtion of inclina- tion. This rrsult, suggests tha t thc large-scale vortical structures strongly influence the scalar distribution. i

3.2. Effect ofpropagation angle on ph,ost erm7

The nonisotropic nature of the scalar distribution immediately lrads to a hypothesis tha t the phase errors depend on the direction of propagation through the tur- bulent field. In order to verify this hypothesis, phase errors have been computcd by using (4 ) for propaga- tion along several dircctions in thr computed flow firld. [For the present problem, solutions from (3) are almost identical to those from (4) . ] The angle of the beam propagation a is measured counter-clockwise from t h e flow direction (z-axis); for example, propagation in the direction of thr mean scalar gradient (y-axis) has an angl? n = 90'. Note tha t propagation at n - 180' and a are identiral.

Figures Z(a-d) show contours of the phase error for propagation between the top and bottom x:-planes of the computational domain a t a = 4 5 O , 90°, 135' and 153.4', respectively. Data in each case has been scaled to account for the different propagation path lengths so tha t contour increments are t h ? same in each fig- ure. The distribution of phase error for propagation at a = 45' (fig. Za) is highly localized and shows the largest phase errors, while the cases at a = 135O, 153.4' +.

(figs. Zc, d ) show distributions somewhat elongated in the flow direction and much less phase errors (see also

The (spatial) 1.m.s. phase error normalized by the value for the vertical propagation ( a = 90') is plotted in figure 3. I n accord with the above qualitative ii1dic.a- tions, th r r.ni.s. phase error attains a maximum for the propagation along the vortical structures (a,,,ax 45' or -135') and a minimum at an angle approximately normal t o them (a1,]i,* 150' or -30'). The max- imum 1.m.s. phase error at amnx is about twic.e the minimum value at a,,,j,<. Consequently, these results confirm tha t the scalar fluctuations are concentrated in regions aligned with the vortical structures and are elongated in tha t direction.

figure 3) .

3.3. Physical modrl jor scalar fivctuntions

The above resolt,s indicate that the distribution of scalar flnrtuations is directly influenced by the vorti- cal structures, namely the numerous hairpin vortices (Rogers Rr Moil1 1987). A simple modpl relating scalar fluctuations to the vortical structures is shown in figure 4 . The flow induced by an 'upright' hairpin has a I('-

gion of n' < 0 betwren and above its legs, and regions of n,' > 0 outside and below its legs. This is a direct consequence of the dominant role of the hairpin vortiws in inducing the scalar fluctuations: 'cold' fluid (n' < 0) is brought up between t h e hairpin legs, while 'hot' fluid (n' > 0) is pulled down outside the lrgs. C'onversel,~. an 'inverted' hairpin has a region of n.' > 0 between and below its legs and regions of n1 < 0 outside and above its legs. The import.ance of this scalar transport by vortical struc.tures t o the scalar flux was discussed by Rogers et a). (1986).

3.4. Two-point correlaiions

The feasibility of the above model for sc.alar fluctu- ations induced by vortical structures can be tested by examining the two-point cross-corrrlations ofscalar and vorticity fluctuations. It is suggested from the concep- tual model for generation of scalar fluctuation shown in figure 4 tha t there should be a strong correlation be- tween vorticity and scalar fluctuation at a separation in the spanwise direction r2 of about half the average spacing A, of the hairpin legs.

In figures 5 and 6 , the spanwise cross-correlations of vorticity and scalar made dimensionless by the resper- tivr r.m.s. values wl and 6 arp shown: I

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where thr overbar denotes the statistical average taken OVPI the threr-dimensional space. I t is evident. that therr exists a fair amount of rorrclation between the scalar and vorticity fluctuations. Because the hairpins are oriented (on the average) a t 45' in the zy-plane, both d z and up correlate well with 8. For positive spanwise separations ( r ; > 0 ) the correlation is positive Q : , , @ ( v ; ) > 0, whrreas for negativr spanwise separa- tions ( 7 : < 0 ) the correlation is negative Q : , @ ( V ~ ) < 0. Notice tha t each leg of the upright as well as inverted hairpins contributes to the correlations in the same manner

T h r two-point spanwise correlations 4?:,e(rz) and Q Z y o [ v z ) in the figures indeed show a pair of distinc- tivc peak and valley antisymmetrically. The distance br twwn the peak and valley corrrsponds t o the average spacing between the hairpin legs A: 2 0.23, estimated froiii the two-point auto-correlation of vorticity. This is rntirelg consistent with the physical picture of scalar- fluctuation genPration sketched in figure 4. The corre- lations romputrd for a fipld ai an earlier time, St = 8 (not shown), ai? nearly identical.

4. Conclusions

The phase distortion induced in a coherent optical beam bg turbr~lent fluctuations in a hotnogeneus shear flow is highly anisotropic, i.e. sensitive to the direc- tion of propagation. This is a result of intense scalar [i.r. it,dex-of-rPfraction) fluctuations in elongated re- gions which ar? inclined to the mean flow similar to the vortical structures (hairpin vortices). A concep- tual nrodel is proposed for scalar fluctuations producrd by t h r hairpin eddies, which are characteristic of ho- mogenrous shear flow. This concept is supportrd by the two-point vorticity-scalar correlations which show a pair of maximum and minimum at spanwise distances which correspond to one-half t h r spacing between the hairpin legs.

The importance oflarge-scale vortical structures and t,he associated scalar distribution to optical dis1,ortion has been established. I t would be of fundaniental in- terest to invrstigate the relative iniportance of sn~al l - and large-scale turbulent structures to phase distortion in propagation through turbulent shear flow. A study of probability density func.tions of th r phase error and two-point correlations of vorticity and scalar gradient would also be of interest. An examination of the ef- fects of inhoinogenity could be c.arried out by using databases front direct numerical simulation of turbu- lent boundary layers and mixing layers.

We would like to thank L. Hesselink, B. S. Masson, M. M. Rogrrs and S. K. Lele for many helpful discus- sions. Support by AFOSR Project 2307Y1, through the Air Force Weapons Laboratory, Albuquerqur, N r w Mexico, is gratefully acknowledged.

R E F E R E N C E S

BAXTER, M. R., T R U M A N , C . R . & MASSON, N. S. 1988 Predicting the optical quality of supersonic shear layers. A I A A Paper 88-2771.

BORN, M . & WOLF, E. 1975 Prznczples of Opiics; Electromagnetic Theory of Propagation, Intcrfrr- ence and DiflTaclion of Li.gh1.. 5th edn. Perganmn: Oxford, England.

CLARK, T. T., TRUMAN. c'. R . d; MASSOK, R . S. 1988 Prediction of optical phasr drgradatinn through a turbulent shear flow. A I A A Paper 8% 3664.

GOODMAN, J . W. 1985 .Stolist?ca/ Op l i c s . bC'ilry Interscienre: New York.

KELLEH, J . B. 1954 Geometrical acoustics. I . Tllc the- ory of wrak shock waves. J. Appl. Phys. 25, 938- 9 4 i .

L I E P M A N N , H . W. 1979 T h r rise and fall of idrrrs i n t,urbulence. AmrT. Scz. 67, 221-228. L.,

M O N l N . A . S . & YAGLOM, A . M . 1975 Sto t i s i zr~r i F l u i d Mrchunics: Mt,chanics of Thrbuitnrt , Vol. 2, MIT I'rrss: ('anibridgc. Mass.

ROGERS, M . M . & M O I N , P . 1987 The strurturr of the vorticity field in homogeneous turbulent flows. J. Fluid Mcch. 176, 33-66,

ROGERS, M. M., MOIN, P. 8r REYNOLDS, n'. C:. 1986 The structure and modeling of the hydrodynamic and passive scalar fields in homogeneous t u r h u l e ~ ~ t shear flow. Dept . Mrch. Fngng. Rep. TF-25, Stanford University: Stanford, California.

SUTTON, G . W. 1969 Effect of turbulent fluctuations in a n optically active fluid mrdiurn. A . I . A . A . J. 7, 1737-1743.

TATARSKII, V . 1. 1971 The efiects of the turbulent a t - mosphere on wave propagalion. NSF Rep. TT 68- 50464, Nat. Tech. Info. Service (NTIS), U.S. Dcpt. of Commerrc. [Translated from the Russian original (1967, C N A V I i A B : Moscow) by Israel Program for Scientific Translations ( IPST) S1,aff.l

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X

F I G V R E 1 . C:ontoors of constant sralar fluctuation 8 on an ve1tic.a) sy-pian? in homogrnrous turbulent shrar flow (Hogrrs cf n l . 1986). ~ , 8 > 0; ----, 0 < 0. The sc.alar field is rlongatrd in the flow direction due to app1ir.d shrar, siniilar to th r vorticity field.

X

FIGURE 2 ( a ) . For caption see next page.

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A

F i ~ i i r t b , 3 Drpendrnce of 1.m.s. phase error on the angle of beam propagation. The r.m.s. values are normalized by thc value for t h r vertical propagation ( n = 90°)

v

Y

X

n’ > 0

n’<O

UPRIGHT HAIRPIN INVERTED HAIRPIN

FIGURE 4. Schematic o f a conceptual picture, showing the scalar fluctuations induced by the ‘upright’ and ‘inverted’ hairpin vortic.es. The arrows with solid line indicate vorticity, and the arrows with broken line describe how the induced flow field transports passive scalar. -

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1 .o l ' l ' l ~ l ' l ' l ' l ' l ' l ~

.8 -

.6 -

,N .4 -

3 .2 - 0

-.2 -

-.4 -

- . 6 - ' I ' I ' I ' I ' ' I ' I ' I ' I '

-

-

- - 1

rn X -

b - - -

-

W

FIGURE 5 . Two-point spanwise correlation Q 2 , 8 ( ~ L ) of wI and 8, showing a pair of peak and valley a t distances half t h r hairpin-leg spacing: I T : / 2 + A r .

1 .o

.8

.6

- .4 N

m L - j= .2

b o -2

-.4

-.6 L . l . l . l . l . l . l . l . l . l . 1 .. -2.5 -2.0 -1.5 -1.0 -.5 0 .5 1.5 1.0 2.0 2.5

r Z

FIGURE 6. Twopo in t spanwise correlation Q: o ( ~ z ) of w y and 0, showing a pair of peak and valley at distances half

the hairpin-leg spacing: IT^ I = : A L . v

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