Shear Layers and Aperture Effects for Aero-Optics

Transcription

1 36th AIAA Plasmadynamics and Lasers onference 6-9 June 5, Toronto, anada AIAA-5-77 Shear Layers and Aerture Effects for Aero-Otics John P Siegenthaler *, Stanislav Gordeyev, and Eric Jumer Hessert Laboratory, University of Notre Dame, Notre Dame, IN, , USA This aer examines the effect of a finite aerture on wavefronts, esecially with regard to aero-otic distortions from a shear layer. When the net deflection of a beam is corrected in real time to bring a beam on target, a common ractice in otic alications even if no other corrections are erformed, this removal of the off-target tilt and the finite aerture of the beam act as a satial filter. This restricts the ti-tilt correction and the wavefront correction to searate frequency ranges, causing the rms magnitude of the remaining distortion to be corrected to vary with aerture size. It has also been found that for a shear layer, aerture size is a key factor in scaling the severity of otical distortions, and it is likely to be a key comonent in scaling other tyes of otically-aberrating flows. 1. Nomenclature a = local seed of sound AO = Adative Otics A = aerture length = amlitude of oscillation DM = Deformable Mirror f = frequency f d = dominant frequency at a location f T/T = maximum T/T frequency filtered g,h = generic functions K GD = Gladstone-Dale constant,.7 m 3 /kg L = length scale M = Mach number OPD = Otical Path Difference OPD rms = root mean square of the OPD over an aerture OPL = Otical Path Length = ressure r u = velocity ratio U /U 1 s = density ratio ρ /ρ 1 T = temerature T/T = Ti-Tilt u = velocity U 1, = sheer layer flow velocities U = convection velocity of the shear layer = deflection angle of a beam in the x direction α x * Graduate Research Assistant, Aerosace and Mechanical Engineering, Notre Dame, Student Member AIAA Assistant Research Professor, Aerosace and Mechanical Engineering, Notre Dame, Member AIAA. Professor, Aerosace and Mechanical Engineering, Notre Dame, Fellow AIAA. oyright 5 by J. Siegenthaler, S. Gordeyev, E. Jumer. Published by the, Inc. with ermission. 1

2 AIAA-5-77 δ δ vis Δt Δx Λ ρ = boundary layer thickness = thickness of the shear layer as estimated by flow visualization. = time ste = satial distance of flow convection in time Δt = streamwise length of large structures in the shear layer = density W. Introduction HEN a laser beam with an otherwise lanar wavefront is rojected through a variable-index-of-refraction turbulent flow, its wavefront becomes aberrated. To the extent that the wavefront is aberrated across the beam s aerture, the beam s ability to create a high-intensity sot at the target in the far field is hamered. 1 An adative-otic (AO) system can be used to measure and then comensate for the aberrations that are to be imrinted on the beam s wavefront by imosing a conjugate wavefront figure on the beam rior to rojecting it through the aberrating medium so that in a erfect case the beam emerges unaberrated. In ractice the conjugate is imosed in at least two stages. First, the aberration must be measured. Because of the linear nature of otics, the aberration that will be imosed by a beam roagating through the aberrating medium in one direction can be determined by measuring the aberration imosed by an otherwise near-lanar wavefront roagated through the medium in the oosite direction. Presumably this would be some sort of return from the target. In the case of a free-sace communication system, the target is cooerative and can emit a diverging beam that will arrive at the entrance uil of the beam director as if it were a inhole source at the target. The wavefront from this far-field return then enters the adative-otic beam train. One realization of the layout of the adative-otic beam train is the Notre Dame adative-otic system designed by Xinetics in cooeration with the Albuquerque Boeing, SVS grou. A schematic of the system is shown in Fig. 1. What can be noted in Fig. 1 is that the first correction element the incoming beam encounters is a ti-tilt (T/T) mirror; an enlargement of the comonents of the T/T mirror is shown in Fig.. As imlemented, this ortion of the system makes use of technology that has long been available for maintaining alignment between multile otical benches and in the resent case is, in fact, used to maintain the alignment of the incoming beam into the remaining elements of the system. The enlargement in Fig. shows one of the critical lacement considerations of the T/T mirror; the T/T mirror is re-imaged on the deformable mirror (DM) which, in effect, adds an ability of the DM to ti and tilt. Figure 1. Schematic of Notre Dame Adative- Otic System cooeratively develoed by Xinetics and Boeing SVS. Note also that the T/T mirror is controlled by a searate, in this case analog, stand-alone rocessor. The beam is focused onto a osition-sensing device, which in this case is a quad cell. Ti-tilt mirrors can be made to have relatively high bandwidth so that one can resume that the remainder of the AO system need not contend with removing overall ti and tilt. In fact, essentially all modern beam-director systems incororate T/T mirrors on their otical benches even when no other AO system comonents are resent. It should be noted that we have also incororated nearstationary correction comonents into our AO system for some exeriments to remove mean aberration. Because of the ubiquitous use of T/T mirrors, when we make time-resolved wavefront measurements to characterize aero-otic environments, we tyically remove both ti and tilt and time-averaged mean aberration leaving only the unsteady comonent over the Figure. Detail of the Ti-Tilt omensation ortion of the system.

3 AIAA-5-77 aerture. It is imortant to note that the aerture itself imoses a satial filter on the overall aberrating character of the medium. Because the aberrations are due to turbulent flow, which by definition is convecting, this satial filter is associated with a concomitant temoral frequency. Thus, low frequency aberrations with coherence length Λ, and a convection velocity U c, will result in low-bandwidth ti and tilt that a T/T mirror will remove. As an aside, this fact allows facilities that might otherwise be resumed to be unaccetable for aero-otic testing to be usable by simly removing ti and tilt from each wavefront frame. 3. Shear Layers Shear layers, also known as mixing layers, occur at the boundary between two arallel flows of different fluids, or even the same fluid moving at different velocities. The transfer of momentum in the vicinity of the boundary causes this boundary region to grow in size as it convects downstream. Kelvin Helmholtz instability roduces riles in the boundary along its san that also grow and contribute to the growth of the layer region. These erturbations eventually roll u into vortical structures that grow, air, and merge as the shear layer grows. 3 When there is a large difference in relative velocity between the two flows, the low ressure well found in the center of these structures, also known as rollers, can be quite ronounced. This dro in ressure is accomanied by a dro in density, with its concomitant change in index Figure 3. The classic shear layer. (Konrad, 1976) of refraction. This makes these structures of interest from an otical standoint. While there are many elements in a comressible flow of this sort that can roduce otical distortions, the low ressure wells associated with these rollers are likely to be the most significant contributor in that regard. Shear layers grow linearly in thickness with distance from their starting oint. There are multile definitions for the thickness of a shear layer, such as momentum thickness and vorticity thickness, and the growth rate of the shear layer is less well understood for comressible flows than for incomressible and weakly-comressible shear layers.. In these exeriments, the rimary method for estimating the layer thickness was by visual observation of the vortical structures in the flow. The literature 5 reorts that this vorticity thickness (δ vis ) can be estimated from the following emirical relation: Λ δ vis dδ dx vis (1 ru )( r s u s), (1) where r u is velocity ratio (U / U 1 ) and s is the density ratio (ρ / ρ 1 ) of the two flows making u the shear layer. For this otical study, the dimension in the layer of rimary interest is not the thickness, but the sacing between rollers. (Λ) This coherence length is related to δ vis by yet another emirical constant. The literature reorts that this ratio of Λ/δ vis, varies from 1.5 (Ref 6) to (Ref 7). In observing a flow assing a oint of measurement, the average length of a reeating structure can be found by the relationshi U Λ = () f d where f d is the frequency or average frequency at which the structure is seen to ass by the oint of measurement. It should be noted that Λ differs from Λ n reorted in Ref 8, where the frequency in Eq. () is relaced with f n. That frequency is biased towards larger structures by the fact that the larger structures roduce a larger-amlitude otical resonse.. Small Beam Aero-Otic Measurements Otical distortions can be thought of in terms of deflection angles for isolated rays of light, or as wavefronts. A wavefront is a sheet of light, with the same wavelength and hase throughout. A lanar wavefront is a flat sheet of this sort, and models what one might exect in a collimated laser beam of some non-zero aerture (A ). A 3

4 AIAA-5-77 wavefront is a more comlete exression of the otical effects of a flow, as a wavefront is comosed of all the rays assing through the flow from that direction or source. While a wavefront contains all the relevant asects of the light and distortions to the light assing through an aberrating flow or medium, there is such a thing as having too much information. Interferometry and hase diversity can measure a wavefront in its entirely, and some other tyes of sensors can be built with imressive satial resolution. However, for many alications, far less information is better. For real-time alications dealing with a time-varying system, there is often a trade-off between how raidly one can samle and rocess data and the amount of data one can samle. The data in this aer was acquired with a Malley robe, 1 which uses two beams that are narrow, relative to the length scale of structures in the flow, and so are deflected as rays rather than distorted as wavefronts. A detailed descrition of the comonents and reconstruction methods for otical wavefronts using a Malley robe can be found in Ref 1. By definition a wavefront is a locus of oints of α constant hase. 11 As shown in Fig., as an initially x lanar wavefront for a collimated laser beam roagates through a region of variable index of refraction, ortions of the wavefront become advanced and retarded from the wavefronts mean osition. If a lane is drawn normal to -α A x the wavefront s mean roagation direction, the beam s hase along that lane will be greater or less than the hase of the wavefront at the mean osition. It is the hase difference over this lane that is referred to as the beam s aberration. According to Huygens rincile the wavefront can be relaced by inhole sources along its surface, each emitting sherical waves and the wavefront Figure. An otical ray as art of a wavefront can be redrawn at some distance by connecting surfaces through a distorting flow field. of constant hase from the inhole sources. 11 This leads to the result that wavefronts roagate normal to themselves and is the basis for geometric otics. As a consequence, a ray, everywhere erendicular to the wavefront can be traced along the wavefronts roagation ath as shown in Fig.. The angle at which the ray emerges from the aberrating medium, being normal to the emerging wavefront will have an off-axis angle, shown as -α x in Fig., which is equal to the wavefront s x-gradient. Hartmann was the first to realize that this fact could be used to measure the figure of wavefronts. 1 He laced an oaque, erforated late in front of the aberrated wavefront with a hotograhic late at a known distance from the erforated late. By exosing the hotograhic late first to an unaberrated beam and then to the aberrated beam he was able to measure the off-axis dislacement of beams emerging from the erforations. Knowing the distance between the lates he could determine the angles and thus the wavefront sloes at each erforation (measurement location) and then through integration determine the wavefront s aberrated figure. Similar measurements can be made by assing small-aerture beams through the aberrating medium. Malley et. al. were the first to recognize that when the aberrating medium is a turbulent flow, the aberrations caused by the convecting flow structures will, convect as well. 9 Thus, a single beam roagated through the flow could be used to measure a continuous time series of wavefront sloes as the wavefront convects by the measurement location. The Malley rincile has been used at Notre Dame to develo a series of wavefrontmeasurement devices. 13,1,15 To the extent that the flow can be treated as slowly varying, the Taylor frozen flow assumtion can be used to comute wavefronts u and downstream of the measurement location, which are reasonably accurate for some distance u and downstream. By roagating two small-aerture, closely-saced, beams through the flow, both the wavefronts sloe and its convection seed can be determined. dopl( dx = α x ( (3) In order to integrate the sloe to roduce a running time series of OPL(, the velocity is required as dopl( dopl( OPL( = dx = U cdt () dx dx

5 AIAA-5-77 where U c = dx/dt. By knowing the distance between beams and cross-correlating the beams to find the delay time between the two beams signals, U c can be determined, 1 so that OPL x ( α x ( τ ) U cdτ = t t (5) From these equations and relationshis, an aroximation of the OPL over a ortion of the flow can be reconstructed from satially coarse data, or even a single oint of measurement. An imortant asect of wavefronts that becomes aarent in this form of reconstruction is the relationshi between length scale of the aberrations and the amlitude of the resulting OPD. If α x were a ure sine wave, then the OPL would be OPL t x ( t = sin(π fτ ) U cdτ = U cos(πf ( t t ) (6) πf By Eqs. () and (6), the amlitude of OPLs associated with deflection data of a given amlitude will be inversely roortionate to the frequency of that deflection and directly roortional to the length of the structures. The OPL roduced by Eq. (5) can be made to any length by setting the uer and lower bounds of the integral. However, the measurement technique is only valid if the frozen flow assumtion is valid, and that is only true for regions close to the oints where the beam asses through the flow. As one extraolates an OPL further ustream or downstream from a location of measurement, the data becomes less valid with increasing distance from that oint. wavefront (μm) 5a wavefront (μm) 5b A x (cm) A x (cm) Figure 5. Examles of the satial filter effect roduced by an aerture and ti-tilt removal. wavefront over the aerture shows a significant degree of net tilt, which, as discussed earlier, would deflect the beam. Removing this tilt to lace the beam on target also reduces the amlitude of the variations in the wavefront in this case by more than 5%. Figure 5b shows the effects of T/T removal when the satial scale of the distortions is smaller than the aerture. In that case, there is very little net tilt and tilt removal has little effect on the magnitude of the aberration. 5. Aerture Effects While this technique of reconstruction and estimation is only accurate over short distances, real-world alications involve beams of a finite diameter, and measurements relating to these alications only need to be accurate over that area. The finite aerture also has an effect on those structures in the flow that will roduce distortions within the boundaries of the beam s wavefront over the aerture that will deflect or tilt the beam as a whole. Figure 5 shows two wavefronts, both aberrated into the form of a sine wave, with the same amlitude, over the same length of aerture. In Fig. 5a, the eriod of the wave is a bit longer than the aerture. A linear fit to this sin(x) linear fit tilt removed Aerture sin(x) linear fit tilt removed Aerture average OPDrms (m) 5.E-5.5E-5.E-5 3.5E-5 3.E-5.5E-5.E-5 1.5E-5 1.E-5 5.E-6.E Λ(cm) Figure 6. Results for simulated α=sin(ω 5 aerture size 5 cm 1 cm x 15 cm Infinite

6 AIAA-5-77 A finite aerture of a hysical beam acts as a satial filter, searating the distortions caused by larger-scale structures from those caused by smaller-scale structures. If A /Λ << 1, then T/T deflection becomes the only significant effect of distortions on that length scale. This would seem to indicate that the most severe wavefront distortions come from distorting structures in the flow that are smaller than the aerture. However, there is a countervailing effect seen in the wavefront reconstruction algorithms and formulas that roduces OPD variations of greater amlitude for aberrations of the largest length scales. This is addressed in Eq. (6) and the associated text. Figure 6 was generated by using a ure sine function of extended coherence length as the beam deflection, α. Each oint is the time averaged rms OPD (OPD rms ) for a fixed eriod of α and a fixed A. Each set of oints shares a common value for A while Λ varies. For an infinite aerture, the average OPD rms grows linearly with Λ, as indicated by Eq. (6). For a finite aerture, this is true only for Λ < A. For Λ > A, OPD rms taers off asymtotically as Λ increases. This filter effect can also be found and exressed analytically. For a wavefront or other inut of the form g(x,, OPD rms, as a function of time over an aerture with T/T removal, will be: OPD rms A ( A = [ g( x, ( A( xb( ) ], + dx (7) where A and B in Eq. (7) are the coefficients of a linear fit to the overall tilt and iston resent in g(x,. It just so haens that finding values for A and B to minimize the function inside the square root in Eq. (7) is the basis for erforming such a linear fit, which exlains why T/T removal tends to reduce the magnitude of aberrations. If g(x, is set to a sine function: x g( x, = sin πf t (8) U and the resulting OPD rms is averaged over time, then that average can be comared to the OPD rms for an infinite aerture with no T/T removal. This ratio can be considered the gain of the satial filter for that length in the distorting structures, Satial filter gain f T/T A /Λ Figure 7. Satial filter normalized frequency resonse OPDrms ( A ) G ( A ) = (9) OPD ( A = ) Figure 7 shows the frequency resonse of this satial filter in terms of a nondimensional frequency, A /Λ. As can be seen in the Fig. 7, this is a high-ass filter, screening out aberrations with a length scale larger than the aerture. This filter gain can be found analytically from Eq. (7) for a sine signal given in Eq (8), and has the form rms A 3 π A Λ G = Λ A + π Λ A 3 A A A A + π cos + 6 sin cos π π π π Λ Λ Λ Λ Λ A π Λ (1) This filtration effect has secial relevance for shear layers. Two of the defining characteristics of shear layers are the resence of coherent structures in the layer, and that the layer and structures grow linearly as they rogress 6

7 AIAA-5-77 downstream. Therefore, one would exect the otical distortions to have a characteristic frequency and length scale at each oint in the flow, and for that length scale to grow linearly with osition downstream in the shear layer. This also has significance for the engineering of T/T removal systems. The bandwidth required for this system corresonds to the frequencies removed by this filter function. Looking at Fig. 7, this corresonds to frequencies lower than A /Λ =.5. Let us call this frequency f T/T. By Eq. () U ft T. 5 A / = (11) The results in Fig. 6 show a similarity in the curves traced for differing values of A, with both the maximum value for OPD rms and the value of Λ at which that maximum is seen having a linear relationshi with A. Both OPD and structure size are in units of length, as is the aerture, 1.E- so it makes sense to non-dimensionalize those values by 9.E-5 A. Fig. 8 shows that this causes all the curves to collase onto one curve. This result is romising, as scaling laws are valuable tools in alying laboratory results to field alications, but flows that roduce otical distortions in the form of a erfect sinewave are hard to come by in the hysical world. On the other hand, suggestions of this scaling, and a rules of scaling aero-otics roblems in general, can be found from basic theory. The receding text defined OPL in terms of the deflection angles often measured and used to reconstruct OPL. The more roer definition of OPL is the integral of the index of refraction along a beam s ath, which for air is a function of density. average OPDrms / A 8.E-5 7.E-5 6.E-5 5.E-5.E-5 3.E-5.E-5 1.E-5.E+ 6 8 Λ/A aerture size 5 cm 1 cm 15 cm infinite Figure 8. Non-dimensionalized results for α=sin(πf ( 1 + K ( )) 1 + n ( x) ds = OPL GD ( x) = ρ ρ ds (1) From here it follows that the magnitude of variation in the OPL, which is OPD, will be roortional to the integral of density variations through the flow field. For isentroic flows, changes in density are roortional to changes in ressure and inversely roortional to changes in temerature. The ressure dro inside a vortical structure, such as the ones seen in shear layers, is aroximately roortional to the square of their characteristic velocity. Thus, one can find an extending chain of equalities and roortionality, Δρ 1 Δ = ρ γ 1 ρ U γ 1 U = γ RT = M (13) where U = (U-U 1 )/ and is the characteristic velocity of the structures, a is the local seed of sound, and M is a convective Mach number. OPD, as noted above, is roortional to an integral of Δρ, which in turn will be roortional to the magnitude of that variance and the length over which that variance is integrated. In Eq. (1), the diameter of the rollers was chosen as the definition of layer thickness for the shear layer. Thus, OPD rms K GD Δρδ vis K GD ρ M δ vis (1) From Eq. (1), δ vis is roortional to x and Λ is roortional to δ vis. OPD rms ρm Λ ρm x (15) 7

8 AIAA-5-77 Thus, the OPD rms tends to grow linearly with Λ, as was the case with the model roblem in Eq. (6) and the infinite aerture case in Fig. 6. If flow conditions are fixed, meaning M and ρ are constants, then Λ is simly roortional to x. Thus, the shear layer has a characteristic structure size, Λ, that is linearly growing with osition, and thus scaling x by A should roortionately scale Λ. If the finite aerture and T/T removal truly act as a filter does over all frequencies, regardless of the mix of those frequencies in the inut, then all the salient oint in the sine function test that make the scaling work should be resent in a hysical shear layer and otical system. Thus, for fixed flow conditions, there is reason to hoe that OPDrms ( x, A) = f A A x (16) It is left now to examine some actual otical data to see if this does aly. 6. Exerimental Setu The exeriments for this study were erformed using the Notre Dame Weakly-omressible Shear Layer (WSL) facility, shown in Fig 8, located in Notre Dame s Hessert Laboratory for Aerosace Research. The WSL facility consists of an inlet nozzle and test section mated with one of Notre Dame s three transonic in-draft, wind-tunnel diffusers. The diffuser section is attached to a large, gated lenum. The lenum is, in turn, High-Seed Flow Figure 8. Notre Dame Weakly-omressible Shear layer facility connected to three Allis halmer 3,31 FM vacuum ums through a sonic throat to revent unsteady effects from roagating ustream from the ums. Deending on the gate-valve arrangements, each of these ums can be used to ower searate diffusers, or they can be used in combination to ower a single diffuser. Low Seed Flow Malley Probe Beams Figure 9. Malley robe ositioning Being an in-draft tunnel, the feeding source is the room total ressure and temerature. The test section is fed from a 1-to-1 inlet nozzle directly from room total ressure on the high-seed side. On the low-seed side, roomtotal-ressure air is first assed through a settling tank with a quiet valve consisting of a bundle of ies that the flow is forced to ass through at high seed. This roduces a loss in total ressure, while keeing its total temerature the same as that of the room air drawn into the high-seed side. In this set of exeriments a Malley robe 1 was used to assess the otical aberrations. In the resent case, the Malley Probe s two laser beams were directed through the test section from below, as shown in Fig. 9. All data was filtered to revent aliasing and to remove low-frequency effects of tunnel vibrations. 7. Physical Results The data resented here is for the shear layer running with Mach-number values of.77 and.6 for the highseed and low-seed flows, resectively. At the ressure and temerature conditions in the test section this translates to velocities of aroximately 6 m/s and 35 m/s, with a convection velocity for the shear layer between them of 18 m/s. By Eq. (1), this indicates a δ vis growth rate of aroximately.5. The redicted growth rate for Λ ranges from.375 to.5, deending on which recommendation for this relationshi is used. 6,7 8

9 AIAA x khz Figure 1. Power density sectrum for deflection data recorded at 13. cm. Unlike the theoretical case exlored in section 5, this hysical shear layer does not roduce otical aberrations in the form of a ure sine function. However, as can be seen in Fig. 1, there is a eak, albeit broad, seen in the ower density sectrum of the beam deflection data, indicating at least some eriodicity in the data. This eak moves to lower frequencies as the measurement location is moved downstream. For this study, the characteristic frequency was chosen as the frequency at which this eak reaches its maximum value, as this corresonds most closely to the methods and definitions associated with δ vis. 7 As mentioned earlier, other otical studies use a weighted average of this sectrum. 8 Figure 11 shows the eak frequency of otical disturbances in the shear layer as a function of osition downstream from the origin of the shear layer. Figure 1 shows the structure length estimated from the frequencies in Fig. 11 and Eq (). The trend does aear linear, which is consistent with the arguments leading vis, which is the maximum value to the scaling relation of Eq. (16). The line in Fig. 1 reresents a value of for Λ/δ for this ratio found in the literature referenced earlier. 7 freqency (khz) osition (cm) Λ (cm) osition (cm) Figure 11. Peak deflection frequencies Figure 1. haracteristic Λ and growth rate. Figure 13 shows the time averaged OPD rms (T/T removed) at a set of measurement locations. Each set of oints in this figure reresents the OPD rms as calculated for a different size of aerture. Smaller aertures filter out longer wavelengths, and so have lower values of OPD rms in the figure. An aerture of 3 cm can effectively be considered an infinite aerture, and for that aerture size, OPD rms grows almost linearly with osition, as it did in.5 16 OPDrms (microns) Aerture size 5 cm 1 cm 15 cm cm 3 cm OPDrms / A Aerture size 5 cm 1 cm 15 cm cm 3 cm 1 3 Position (cm) Figure 13. Aertured OPD 6 8 x/a Figure 1 Non-dimensionalized OPD 9

10 AIAA-5-77 the simlified test shown in Fig. 6. Also note that from Fig. 1, Λ reaches 5 cm in length at a osition somewhere around 11 cm, and in Fig. 13, the curve made u of OPD rms values generated with a 5 cm aerture begins to level off at about that oint. From the results shown in Fig. 6 one might exect the curves to fall off at downstream ositions where the characteristic structure length seen in Fig 1 is larger than the aerture. However, as can be seen in the ower density sectrum for the data in Fig. 1, the data includes high frequency deflections as well as the lower frequencies associated with the ressure wells in the rollers. This may be caused by smaller irregularities and vortices that roll u into the larger structures; however, it is more likely that it is due to the boundary layer forming between the uer surface of the test section and the high seed flow. The satial filter of the aerture does not remove these smaller scale otical distortions. Figure 1 shows that desite these differences, the same ractice of non-dimensionalization shown in Figs. 6 and 8 and Eq. (16) alies to the exerimental data. That the scaling law holds true for the exerimental results indirectly suggests that the filter function shown in 1.5 x 1-7 Fig. 7 and Eq. (1) also holds true, but a more direct T/T sd demonstration can be made. There are difficulties in generating a ower sectral density function for aertured OPDs from Malley robe data, as there are few oints in the 1 1-G aerture to work with. However, it is ossible to generate an aertured OPD for each time ste of the recorded data, record the sloe of the tilt removed, and generate a sd of that tilt. If the T/T removal does indeed filter out frequencies from the.5 OPD, then the dominant frequencies of the tilt should be the ones removed by that filter. Figure 15 shows the T/T PSD based on the data for the PSD in Fig. 1 and a 5 cm aerture. For U = 18 m/s and by Eq. (), a structure 5 cm in length corresonds to a frequency of about 3 khz. Fig 15. also shows the inverse of the filter, (1-G(A /Λ)), scaled by 1-7 khz to fit in the grah vertically and for A /Λ = A f/u = f/3 Hz. Figure 15. T/T ower sectrum and inverse The exact shae does not and could not be exected to match, filter since the signal to be filtered does not have a uniform frequency distribution; however, they are high and low in the roer corresonding laces. This also verifies Eq. (11), as ft/t for this flow would be 7.5 khz, which would indeed fulfill the sectrum requirements of the T/T removal. In fact, a bandwidth of khz on the art of the T/T mirror might suffice. 8. onclusions This aer has examined the effect of removing ti-tilt over an aerture from aberrated wavefronts. As is now common knowledge, the largest source of aberration for a flow over beam directors on airborne otical systems is when the otical signal asses through a searated shear layer. Under the influences of the Kelvin-Helmholtz instability, shear layers are known to be self-similar. This leads to the formation of linearly-growing, coherent structures that generate otical aberrations that are sinuous in both time and sace in the streamwise direction. The results reorted here have shown that tilt removal over an aerture acts as a well-defined satial filter when alied to a sinusoidal train of wavefronts assing through the aerture. Exerimental data obtained for otical roagation through a Mach.77 shear layer have been shown to react in an uncannily-similar fashion to the sinusoidal model wavefronts studied here. The linear growth seen in the average structures in shear layers allowed for the juxtaosition of downstream distance from the shear layer s origin and structure size as related to the filter function. This similarity has revealed some interesting conclusions about the character of aberating structures in shear layers, and in other aberrating flows as well. For examle, The relationshi between the shear layer s growth in OPD rms and the serendiitous choice of a constant amlitude in the sinusoidal model has suggested avenues for simlified models of a shear layer s otical resonse. More imortantly from oerational considerations, this effort has rovided an objective aroach to determining bandwidth requirements for T/T mirrors for airborne otical systems. T/T PSD 1

11 AIAA Acknowledgments These efforts were sonsored by the Air Force Office of Scientific Research, Air Force Material ommand, USAF, under Grant Number F The U.S. government is authorized to reroduce and distribute rerints for governmental uroses notwithstanding any coyright notation thereon. 1. References 1 Jumer, E., and Fitzgerald, E., Recent Advances in Aero-Otics, Progress in Aerosace Sciences, Vol. 37, 1, Tyson, R.K., Princiles of Adative Otics, Academic Press, Inc, San Diego, Ho, -M, and Huerre, P., Perturbed Free Shear Layers, Annual Reviews of Fluid Mechanics, 16, 198, Fitzgerald, E.J., and Jumer E.J., The Otical Distortion Mechanism in a Nearly Incomressible Free Shear Layer, Journal of Fluid Mechanics, Vol. 51,, Paamoschou, D., and Roshko, A., The omressible Turbulent Shear Layer: An Exerimental Study, Journal of fluid Mechanics, Vol. 197, 1988, Brown, G.L., and Roshko, A., On Density Effects and Large Structure in Turbulent Mixing layers, Journal of Fluid Mechanics, Vol 6, No., 197, Dimotakis, P.E., Two-Dimensional Shear Layer Entrainment, AIAA Journal, Vol., No.11, 1986, Nightingale, A.M., Gordeyev, S, Jumer, E.J., et. al., Regularizing Shear Layer for Adative Otics control Alications AIAA Paer 5-77, Toronto, anada, 6-9 June, 5. 9 Malley, M., Sutton, G.W., and Kincheloe, N., Beam-Jitter Measurements for Turbulent Aero-Otical Path Differences, Alied Otics, 31, 199, Gordeyev, S., Jumer, E. J., Ng, T., and ain, A., Aero-Otical haracteristics of omressible, Subsonic Turbulent Boundary Layer. AIAA Paer -366, Orlando, Florida, 3-6 June, Klein M. Otics. Wiley, New York, Malacara D., Otical sho testing, Wiley, New York, Jumer, E. J., and Hugo, R. J., Quantification of Aero-Otical Phase Distortion Using the Small-Aerture Beam Technique. AIAA Journal, 33(11), 1995, Hugo, R., and Jumer, E., Exerimental Measurement of a Time -Varying Otical Path Difference by the Small-Aerture Beam Technique. Alied Otics, Vol. 35(), 1996, Fitzgerald, E.J., and Jumer, E.J., Two-Dimensional Otical Wavefront Measurements Using as Small-Aerture Beam Technique Derivative Instrument, Otical Engineering, Vol. 39, No. 1, Dec.,