wind loads on ground-based telescopes · wind loads on ground-based telescopes douglas g....

Wind loads on ground-based telescopes

Douglas G. MacMynowski, Konstantinos Vogiatzis, George Z. Angeli, Joeleff Fitzsimmons,and Jerry E. Nelson

One of the factors that can influence the performance of large optical telescopes is the vibration of thetelescope structure due to unsteady wind inside the telescope enclosure. Estimating the resulting degra-dation in image quality has been difficult because of the relatively poor understanding of the flow charac-teristics. Significant progress has recently been made, informed by measurements in existing observatories,wind-tunnel tests, and computational fluid dynamic analyses. We combine the information from thesesources to summarize the relevant wind characteristics and enable a model of the dynamic wind loads ona telescope structure within an enclosure. The amplitude, temporal spectrum, and spatial distribution ofwind disturbances are defined as a function of relevant design parameters, providing a significant improve-ment in our understanding of an important design issue. © 2006 Optical Society of America

OCIS codes: 110.6770, 120.7280.

1. Introduction

Design studies are underway for the next generation oflarge ground-based optical telescopes. While the enclo-sure surrounding the telescope provides a significantreduction in wind speeds, the residual wind loads thatresult from large-scale flow structures and turbulenceinside the enclosure may still lead to significant vibra-tion.1,2 Estimates of the wind loads are required earlyin the design process, influencing the design of theenclosure, telescope structure, and control system. Thewind parameterization herein reflects our best cur-rent understanding of the wind environment inside aroughly hemispherical telescope enclosure, informedby three separate sources of data. Full-scale data areavailable from measurements taken at Gemini SouthObservatory.3–5 More extensive data under more con-trolled conditions have been collected in an �1% scale

wind-tunnel experiment.6 Finally, computational fluiddynamic (CFD) analyses7 have been validated againstthese data sources and used primarily to understanddifferences between different sources of data.

Understanding the wind inside a telescope enclo-sure has been of concern for many years. While full-scale tests are ideal, limited data are available bothbecause of the expense and because of the difficulty incontrolling test conditions. The pressure was mea-sured at a few points inside the Multiple Mirror Tele-scope (MMT) dome,8 the torque on the main drivesmeasured for the Very Large Telescope (VLT),9 whilerecent measurements at the Keck Observatory in-clude simultaneous velocity and torque data.10 Incontrast, the Gemini data3 used herein include 32pressure measurements over the primary mirror andmultiple velocity measurements taken under a vari-ety of conditions. Only flow visualization or limiteddata at a few points inside the telescope enclosure isavailable from wind-tunnel tests for Keck11 and theVLT.12 Significantly more data have been collected inrecent tests, leading to both better understanding ofthe flow field and quantitative information that canbe used in modeling and design. Digital particle im-age velocimetry (DPIV) can provide a quantitativespatial map of the flow field in the region around thesecondary mirror,6 while arrays of pressure measure-ments across the primary mirror can be used to un-derstand the spatial variation.13 In addition to thewind-tunnel tests on hemispherical domes that areused to develop the understanding herein, othershave recently been conducted for a different domegeometry.14 Tests have also been conducted for radio

D. G. MacMynowski ([email protected]) is with theCalifornia Institute of Technology, 1200 E. California Boulevard,M/C 104-44 Pasadena; California 91125. K. Vogiatzis is with theNew Initiatives Office, Association of Universities for Researchin Astronomy, NOAO 950 N. Cherry Avenue, Tucson, Arizona85719. G. Z. Angeli is with the Thirty Meter Telescope Project,1200 E. California Boulevard, M/C 102-8 Pasadena, California91125. J. Fitzsimmons is with the Hertzberg Institute for Astro-physics, NRC, 5071 W. Saanich Road, Victoria, British ColumbiaV9E-2E7, Canada. J. E. Nelson is with the University of Califor-nia, Santa Cruz, 1156 High Street, Santa Cruz, California 95064.

Received 14 October 2005; revised 22 May 2006; accepted 1 June2006; posted 6 June 2006 (Doc. ID 65355).

0003-6935/06/307912-12$15.00/0© 2006 Optical Society of America

7912 APPLIED OPTICS � Vol. 45, No. 30 � 20 October 2006

telescopes; however, since these typically do not haveenclosures to shield them from the wind, the flow fieldaround the telescope is quite different. Because of thetime-varying three-dimensional nature of the flow,CFD simulations are computationally intensive andhave only recently been of value for estimating un-steady wind loads. Steady flow simulations were con-ducted for Gemini,15 while more recent steady andunsteady CFD results are incorporated herein.

Wind loads are relevant both over the primary mir-ror16 (M1) and on the secondary mirror (M2) andnearby supporting structure. Although the cross-sectional area of M2 is small compared to that of M1,the structure around the mirror is exposed to higherwind speeds, and thus the resulting forces are also asignificant contributor to image degradation,1 result-ing in the motion of M2 and the deformation of M1through structural coupling. Steady forces can becompensated by active control of the mirror surfaces.The optical consequences due to unsteady forces re-sult primarily from the motion and deformation of M1and M2 and can be computed from an integratedmodel that predicts the actively controlled structuralresponse.2 The performance is less sensitive to themotions of M3 and Nasmyth platforms, and forces onthese surfaces are not estimated. Wind loads on thedome could also be transmitted to the structure, al-though this effect is neglected herein. While air movingwithin the enclosure is bad for telescope vibration,some flow is necessary to ensure thermal equilibriumto avoid turbulence-induced dome seeing.17 Futuretelescope enclosures are likely to have vents that canbe opened to increase air flow across the primary mir-ror, particularly in low external wind conditions.

The wind flow inside the dome involves severalmechanisms. In addition to the mean flow pattern,there is broadband turbulence generated by the flowpassing over and through the dome opening and alsogenerated by the flow entering the dome throughvents. For smooth, air-tight domes, it is possible tohave significant tones associated with shear layer orRossiter modes.18 These modes are attenuated byopen venting and external structure and are not ex-pected to lead to significant forces; the design issuesrequired to ensure that this is true are described inAppendix A.

The relevant characteristics of the turbulence in-clude the spatial distribution within the enclosure,the temporal spectrum, and the spatial correlation.None of the data sources described earlier gives acomplete understanding of the wind loads. Further-more, for design purposes, it is necessary to under-stand how wind loads vary with external wind speedand design parameters such as secondary mirror lo-cation and area or enclosure and aperture size, ratherthan simply applying a specified load pattern ob-tained (e.g., from CFD or experimentally) for a par-ticular design and wind condition. The combination ofexisting data with additional CFD analyses is usedherein to synthesize the required understanding andestimate relevant wind characteristics.

The next section lists key assumptions. Section 3gives a brief overview of the different sources of dataused to develop the parameterization. Section 4 pro-vides details on both the physics and modeling of thebroadband turbulent forces on the telescope.

2. Assumptions

The major assumptions made herein are listed below:

Y Flow patterns and approximate velocity andpressure distributions are assumed to be robust to de-sign details and can therefore be estimated from dis-parate data sources; this assumption is essential tothis undertaking and is validated by the CFD (Fig. 6).The enclosure is assumed to be roughly hemispherical,e.g., a calotte or dome-shutter design. The most criticalscaling parameter is the ratio of aperture to dome di-ameter, and the data used herein fall in a range of28%–40%. Clearly, the further that a dome designdeparts from the designs considered herein, thegreater the error may be.

Y The flow is turbulent (true for any large tele-scope) and the aperture sufficiently high above theground to not have significant velocity gradients overthe opening (�20 m typically sufficient on mountainsites).

Y Wind loads on the dome, tertiary mirror, andinstrument platforms are ignored.

Y The secondary mirror is assumed to be sup-ported by legs similar to radio telescopes. The spidersupport typical for existing optical telescopes wouldsignificantly increase the cross-sectional area exposedto the wind for the upper part of the telescope.

Y The forces on the secondary support structureare represented by an equivalent force on M2. This isa reasonable assumption at frequencies below thefirst resonance of the support structure, which isabove the frequency range of interest. Matching thetorque about the elevation axis is appropriate assum-ing that the dominant effect of forces on M2 is tele-scope rotation.

Y The wind force on the secondary mirror andsupport structure are estimated from velocity mea-surements. This requires an assumption on the dragcoefficient, and an assumption that the presence ofthe structure does not significantly alter the flow pat-terns in the dome. (This would clearly be a poor as-sumption for accurately estimating wind loads on theprimary mirror.)

Y A von Karman broadband turbulence spectrumis assumed, which is consistent with the data. Thespatial correlation characteristics are assumed to beconsistent with frozen turbulence.

Y Turbulence is generated by flow through theaperture and through vents, but the effect of smallerstructures (e.g., tertiary tower, baffles, and the tele-scope structure) is not included. Forces on smalllength scales are therefore not well represented.

Y All forces are assumed to scale with the exter-nal dynamic pressure, q � 1�2�U�

2. For reference,the 50th and 90th percentile external wind speed U�

measured by the Canada–France–Hawaii Telescope

20 October 2006 � Vol. 45, No. 30 � APPLIED OPTICS 7913

(CFHT) on Mauna Kea are 6.5 and 13.5 m�s, res-pectively.

3. Sources of Data

Our understanding of the wind inside a telescopeenclosure is informed by measurements at GeminiSouth,3,4 wind-tunnel experiments focused on theflow near M2 (Ref. 6), the loads on M1,13 and CFD.7The wind-tunnel tests were conducted with simpli-fied dome geometries that result in flow fields thatare not identical to those expected in realistic domes,and CFD is used to understand the differences. Theforces on M1 and M2 are estimated from Gemini dataand M2 wind-tunnel data, while the other sources ofinformation were used to aid understanding.

A. Gemini

Comprehensive wind data were collected during in-tegration of the 8 m Gemini Southern Observatory.Reference 3 gives a full description of the experiment,and earlier analyses of the results appear in Refs. 4,5, and 19. Velocity measurements were made usingultrasonic anemometers above the dome, behind theM2 assembly, and at three points around M1. Thesurface of the (dummy) primary mirror was instru-mented with 32 pressure sensors. The sensors wererecorded at 10 Hz for 300 s, for 40 separate testsvarying the azimuth angle with respect to the wind,telescope elevation angle, and the position of upwindand downwind vent gates. The lower shutter was notraised, and most of the data were collected with ventsopen. This configuration is appropriate for low exter-nal wind speeds to minimize thermal problems but isnot optimal for higher external wind speeds wherewind buffering becomes an issue. Therefore the Gem-ini data are not representative of the minimum windloads that are achievable. The external wind is rela-tively constant in both speed (�10% standard devia-tion) and direction (�9° standard deviation) overthe 5 min test intervals, allowing the internal windspeeds to be meaningfully defined as a ratio of theexternal wind speed for a given orientation.

B. Wind Tunnel Tests

The M2 flow-field wind-tunnel test6 collected datawithin a scaled telescope enclosure to understand theflow field around the region near the dome openingwhere M2 and its supporting structure would be sub-jected to wind loads. The model was representative ofa generic empty telescope dome at approximately 1%of the expected full-scale enclosure size for a 30 mdiameter telescope. The enclosure, shown in Fig. 1(a),is a hemisphere with diameter D � 0.83 m placed ona cylinder of height H � 0.28 m. The interior floor isHf � 0.09 m above the base. The L � 0.32 m squaredome opening was fixed at 30° from the zenith. Avented cross-sectional area up to 25% of the domeopening area could be achieved using 12 vents with adiameter of Dv � 5 cm; data were collected with allof these closed, half open, and all open. The configu-ration corresponds to a full-scale enclosure withD � 90 m, H � 30 m, Hf � 10 m, L � 35 m, andDv � 5.5 m, reduced by a scale factor of �108. Thefull-scale Reynolds number of the flow is sufficientlyhigh ��20 M� so that the flow throughout the enclo-sure is turbulent. Provided that the experimentalReynolds number is sufficiently high to ensure thatthe flow is also turbulent, the flow is largely indepen-dent of the Reynolds number and therefore scale. Thewind-tunnel tests were conducted at 35 m�s, higherthan expected wind speeds, but yielding a sufficientRe �2 M.

DPIV data were collected in a vertical plane nearthe dome opening to obtain the mean and fluctuationvelocity. Data were collected only for azimuth anglesof 0° and 180° with respect to the wind so that themean flow along the centerline remained planar dueto symmetry. Since the DPIV data cannot be recordedat a sufficient rate to give temporal spectra, hot-wireanemometry data was also collected along the tele-scope axis. The most significant error source in theDPIV data is the difference between the measuredparticle velocities and the actual flow velocities inregions of high flow acceleration where the particlesmay not accurately follow the flow.6 Significant errors

Fig. 1. Illustration of enclosure designs. (a) Simplified enclosure used in the M2 flow-field wind-tunnel test. Realistic (nonsmooth) enclosuredesigns based on the Gemini dome geometry were simulated with the CFD with (b) the shutter raised and (c) lowered to its minimum position.


are possible near the dome opening, and thus thepredicted loads on M2 are computed from the errorbound rather than the measured velocity.

C. Computational Fluid Dynamics

Three-dimensional steady and unsteady CFD simula-tions were conducted for a variety of telescope enclo-sure configurations, including wind-tunnel-scale [Fig.1(a)] and Gemini enclosures [Figs. 1(b) and 1(c)] thatallow validation of the computational results. Themore realistic enclosure based on the Gemini geometryincluded vent gates, internal supports, faceted sides,girders, cranes, and other discontinuities. With theexception of wind-tunnel validation, simulations in-cluded a simplified telescope that omitted features lessthan 0.5 m in size. The truss structure under theprimary mirror cannot be resolved and was replacedby a solid surface or omitted, and the structure of thesecondary support was simplified or omitted.

The effects of turbulence were incorporated throughmodeling, and commercial software (STAR-CD) wasused to solve the incompressible isothermal Navier–Stokes equations on a finite volume grid. Two types ofturbulent modeling were used. The renormalizationgroup (RNG) version20 of the k-� approach21 solvesthe Reynolds averaged momentum equations plustwo additional equations for the turbulent kineticenergy and its rate of dissipation. Since k-� methodsare dissipative, a large eddy simulation (LES) Sma-gorinski model22 was used to improve estimates ofbroadband turbulence; this approach explicitly solvesfor the large-scale turbulence motion, while sub-grid-scale motions are approximated by a model. Inaddition to this modeling, the inlet turbulence is in-troduced as a zero-mean perturbation with a stan-dard deviation of �2�3k.

The geometry used in the wind-tunnel experimentwas simulated to assess the ability of the CFD tocapture turbulence for this problem. The computa-tional domain was that of the M2 flow-field wind-tunnel test, with an artificially extended downwindsection to facilitate the application of boundary con-ditions. The basic DPIV and optics equipment pres-ent in the measurements were incorporated. Thespatial resolution of the computational grid variedfrom 0.01 m around and inside the model to 0.002 mnormal to the surface of the model, while the tempo-ral resolution was set at 0.2 ms. The surface of thetunnel and model was given a smooth, no-slip bound-ary. The inlet air velocity was taken to be uniformwith a speed of 35 m�s and very low turbulent inten-sity. The outflow boundary condition set all normalderivatives to zero (convective exit).

For full-scale simulations, the enclosure was placedon a flat square plane nine dome diameters on eachside, with the ceiling of the computational volumeequal to half the domain length. The outflow boundarycondition was convective, while the sides and ceiling ofthe computational domain were given free-slip bound-aries with zero normal velocity. The surface of theenclosure and telescope structure was given a smooth,

no-slip boundary. The spatial resolution varied fromseveral meters in the outer region of the volume, to1 m inside and around the enclosure, 0.5–0.25 mon the telescope surface and 0.2–0.05 m normal tothe structure. The sampling rate was 20 Hz. Thefluid properties used corresponded to air at 4000 m.The horizontal incoming air velocity profile u�z� �U�z�h�1�7 m�s reproduces that of a typical flat plateturbulent boundary layer; the average speed at theopening level was approximately 10 m�s. The incom-ing air turbulent intensity was 2%, and the rate ofturbulent energy dissipation was calculated from theturbulent kinetic energy level by assuming an atmo-spheric eddy length scale of 5 m and Kolmogorovequilibrium between energy production and dissipa-tion.

The spatial distribution along the telescope axis ofmean and rms unsteady velocity measured in thewind tunnel is compared with the computed values inFig. 2 for a case with vents closed. Both the measuredand the maximum velocity from the DPIV error esti-mates are plotted. A comparison of the turbulencepower spectrum with Gemini data is shown in Fig. 3.Since the turbulence amplitudes tend to be underpre-dicted by CFD, Gemini and wind-tunnel data areused to estimate the wind loads. However, based onthese results, the CFD is believed to capture thephysics sufficiently well to understand amplitudechanges with design parameters.

The CFD has also been used to verify that the meanflow pattern near M2 is not significantly affected by

Fig. 2. Comparison between wind-tunnel DPIV data and the CFD.Data are on axis, position normalized by dome radius and the mean(upper) and unsteady rms (lower) normalized by external windspeed. For DPIV data, the actual particle velocity (dashed curve) andworst-case estimate of flow velocity (open circles) are shown.


the presence of M2 and that there are no significant-scale effects on wind loads (comparing wind-tunnelwith full-scale simulations of identical geometry).

4. Broadband Forces

A. Overview

The mean flow pattern in the telescope enclosure isdescribed in Ref. 6. For upwind orientations, someflow enters through the opening, the flow is downalong the back of the enclosure, forward and slightlyupward along the sides, and upward along the front.For downwind orientations, the opening is within therecirculation region of the enclosure wake, and thusthe flow enters the enclosure through the dome open-ing in a direction opposite to that of the free stream.

Steady forces can be compensated by telescope con-trol systems, and it is the unsteady forces that are ofinterest for predicting telescope image degradation.For notation, the velocity u and pressure p can bedecomposed into their mean and fluctuating compo-nents as u � u� � u�, p � p̄ � p�, with

p � 1�2�u2 � 1�2��u�2 � 2u� u� � u�2�, (1)

p� � 1�2�u�2 � 1�2�u�2, (2)

p� � 1�2��2u� u� � u�2 � u�2�, (3)

where u� is the mean velocity and urms � �u�2�1�2 is therms of the fluctuating (unsteady) velocity. The rmsvalue of the fluctuating component of the dynamicpressure can be written in terms of an effective ve-locity:

prms � � p�2�1�2� 1�2�ueff

2, (4)

where, for Gaussian velocity perturbations u�, thedefinition above leads to

ueff2 � ��2u�urms�2 � 2urms

4. (5)

The telescope dome significantly reduces the windspeeds inside the enclosure. The approach we take toestimating wind loads is to first estimate the effectivevelocity near M1 and M2 relative to the externalmean wind speed U�, and then to estimate the sur-face pressure and drag force from

prms � Cp�1�2�ueff2�, (6)

Frms � CDA�1�2�ueff2�. (7)

The expression in Eq. (7) for the force on M2 hasrecently been validated with simultaneous measure-ments of velocity and torque at Keck.10 The dragcoefficient and area depend on particular telescopedesigns and are not discussed herein. Note that thepressure on a surface may be lower than the nearbydynamic pressure away from the surface; estimatesfor the local pressure coefficient Cp are given in Fig. 8.We first discuss the velocity estimates at M1 and M2,and then the temporal spectrum and spatial correla-tion in Subsection 4.D.

B. Wind Speed near M2

Measurements from the wind-tunnel test of the Keckenclosure concluded that the wind speeds near thesecondary mirror would be roughly 25% of the exter-nal wind speed.11 For upwind orientations with theshutter down, the Gemini data indicate 40% of theexternal wind speed at M2 and less than 10% atM1. Wind-tunnel DPIV data suggest that lower windspeeds are possible if the size of the dome opening isminimized; the CFD is used to verify this conclusion.Note that near M2, the flow velocity is nearly orthog-onal to the telescope optical axis, and the velocity inthe axial direction is much smaller than the trans-verse component. Developing estimates of the forceon M2 and the supporting structure is also compli-cated by the desire to predict these forces as a func-tion of the location of the mirror within the dome (i.e.,to estimate the benefit from building a slightly largerdome with greater clearance between the secondarymirror and the shear layer) and also to estimateforces on the support structure.

The mean and perturbation rms velocity measuredat Gemini near M2 are shown as a function of theexternal wind speed in Fig. 4. Each plotted data pointcorresponds to an upwind (0°) test condition and ei-ther a different zenith angle or a different vent con-figuration. Data sets for which any of the velocitysensors were saturated were not included. The meanwind for a 0° orientation is on average 40% of theexternal wind speed, the rms perturbation (turbu-lence component) is on average 23% of the externalwind speed, and the effective turbulent velocity [Eq.(5)] averages 43%.

The wind speeds are much lower in the data fromthe wind-tunnel test, as shown in Fig. 5, from DPIVdata. As noted in Appendix A (see Fig. 13 and alsoRef. 6), the wind-tunnel DPIV data near the domeopening with vents open are dominated by the broad-

Fig. 3. (Color online) Comparison of the CFD velocity spectrumand spectrum from data measured at Gemini: 0° azimuth, 30°zenith angle, closed vents.


band turbulence, rather than the shear layer mode.The DPIV data without vents open contain signifi-cant energy associated with the shear layer mode,rather than solely broadband turbulence, and are notconsidered herein. The normalized effective turbu-lent velocity ueff�U� along the telescope optical axis isplotted for both upwind (0° azimuth) and downwind(180° azimuth) in Fig. 5. The plot is of the upperbound on velocity from the DPIV error analysis. Forsmall variations in the size of the dome compared tothe dome opening it would be reasonable to normal-ize the data by the critical radius Rc

2 � �D�2�2 ��L�2�2 at which the line connecting the edges of theaperture intersects the telescope axis, rather than bythe dome radius; this radius is indicated by the verticaldashed lines. Padin and Davison1 have suggested thatthe internal velocities should also scale with �D�L�2�3.For the wind-tunnel model, L�D � 0.38.

Both the mean and the effective velocity at M2 arebetween 0.15 and 0.2 of the external mean wind

speed, depending on the location of M2 relative to theshear layer. There is a significant drop in wind speedas one moves through the shear layer, while thevelocity over most of the M2 support structure isrelatively constant. Note that analysis of the Keckwind-tunnel data hypothesized a linear decrease inwind speed with radial location; this hypothesis wasbased on extrapolation from limited data taken onlyfor large fractions of the dome radius.

One difference between data sources is the uniformexternal wind speed in the wind-tunnel test, while thereal environment involves some low-frequency varia-tion in wind speed and direction. However, while thismay be a factor (see Fig. 9), it does not explain a sig-nificant fraction of the discrepancy.

The Gemini data were collected with the shutterlowered, so that the open area is much larger than inthe wind-tunnel model, and the shear layer will extenddeeper inside the enclosure. CFD analyses (Fig. 6) ofthe wind-tunnel and Gemini configurations agreequalitatively with the experimental data, confirmingthat the open area is the reason for the higher windspeeds observed at Gemini. It is therefore reasonableto expect that wind loads closer to those estimatedfrom wind-tunnel data are more representative pro-vided that the shutter can be closed. Figure 6 alsoillustrates that while the internal flow speeds aresensitive to the size of the dome opening, they arerelatively insensitive to details of the design. In-cluded in the plot are CFD results for the dome-shutter (square opening, comparable to wind-tunnel)and calotte (round opening) designs. When estimat-ing wind speeds near M2 to be 20%–25% of external,the two designs are equivalent. What matters, fromthe perspective of wind loads on M2, is the location ofthe shear layer that develops across the opening rel-ative to the location of M2.

While the wind-tunnel data were only collected for0° and 180° azimuth angles with respect to the wind,the Gemini data were collected for a variety of azimuthand zenith angles. However, when the dome opening

Fig. 4. Mean (solid line, �) and rms fluctuation (dashed line, Œ)velocity near M2 measured at Gemini for upwind orientations, asa function of external rms wind speed.

Fig. 5. Effective transverse broadband turbulent velocity nearM2 and supporting structure, from wind-tunnel DPIV data as afunction of axial distance from the center of the dome, upstreamorientation (solid curve) and downstream (dashed curve). The ver-tical line is at Rc.

Fig. 6. Comparison of computed upwind axial velocity profiles fordifferent domes. The calotte (�) and Gemini (□) cases include atelescope structure, and the gap in the plot corresponds to thelocation of M2.


was facing downwind, the anemometer mounted abovethe dome was in disturbed air flow, giving a less accu-rate estimate of the external wind speed. Because ofthis, and the simultaneous variation in orientation andventing configuration, it is difficult to reliably estimatethe effect of orientation beyond noting that upwind isthe worst case. In the M2 wind-tunnel data, the tur-bulence is not significantly lower for downwind obser-vations; at some locations, the downwind orientationhas higher effective turbulence due to the entrainmentof the wake flow. The minimum open aperture areasignificantly reduces the wind loads for upwind orien-tations and should be expected to lead to a smallervariation in wind loads between upwind and down-wind orientation.

C. Wind Speed near M1

Estimates of the pressure on the surface of the primarymirror can be derived from the differential pressuremeasurements collected at the Gemini observatory.Converting these into estimates of the distributedforce requires consideration of the spatial correlationdescribed in Subsection 4.D.

Note that pressure variations on a surface can beseparated into those generated by local turbulencepassing over the surface, with a (short) correlationlength related to frequency and the local wind speed,and acoustic perturbations generated by turbulenceelsewhere, with a (long) correlation length related tofrequency and the speed of sound. For turbulence-induced pressure variations, the correlation betweenthe front and back of the primary mirror is small.However over the frequency range of interest, acous-tic pressure variations have wavelengths many timesthe dome diameter and produce essentially zero netforce on the mirror. Absolute pressure measurementsare sensitive to both components. The wind-tunnelmeasurements in Ref. 13 are dominated by the acous-tic component and thus cannot be used to estimatewind forces.

Measurements taken at the Gemini observatorywith the vents closed and the telescope facing upwindshowed, on average, that the effective velocity uM1,effnear M1 was 7% of the external wind speed19 (Fig. 7),and that this was a good predictor of both the rmspressure �Cp � 1� on the primary mirror and theouter-scale frequency. The mean velocity for thesecases is roughly 3%. Measurements taken with thevents open showed, on average, that the mean veloc-ity near M1 was 30% of the external wind speed5 andthe effective velocity was �35%; these are averagesover different orientations and different vent config-urations. The local pressure coefficient relating un-steady surface pressure to local dynamic pressure forthe vented data is �0.25 as shown in Fig. 8. TheGemini data were collected with a flat dummy mirrorin place of M1, which was slightly recessed. The 0.4 mhigher cylinder that surrounded the dummy mirrormay have resulted in slightly reduced pressure fluc-tuations than would be representative of the actualmirror, and thus the actual pressure coefficient maybe higher.

D. Temporal Spectrum and Spatial Correlation

While the turbulence inside the telescope enclosure isclearly not isotropic, observations at Gemini5,19 (e.g.,Figs. 9 and 10) and other telescopes (Keck, VLT,MMT) as well as measurements in wind-tunnel test-ing (see Fig. 12) and computational results suggestthat the broadband turbulence spectrum inside theenclosure can be reasonably approximated with Kol-mogorov scaling, giving a velocity power spectrumwith f�5�3 frequency dependence, while the spectrumof the pressure at a single point has a �7�3 slope.

Incorporating the outer scale with a von Karmanspectrum matches the data; the pressure spectrum isthus proportional to

�VK�f� � � 2��

�2�3��5�6�� 1�f0

�1 � �f�f0�27�6. (8)

Fig. 7. Effective velocity near M1 from Gemini data as a functionof external wind speed. Solid points and line correspond to caseswith vent gates closed. The vent configuration for other cases islabeled o for open, h for half, c for closed; the first letter is theupwind vent, and the second is the downwind vent.

Fig. 8. Comparison between unsteady pressure on M1 and effec-tive velocity from Gemini data as a function of external wind speedfor estimating local pressure coefficient. Solid points and line corre-spond to cases with vent gates closed. Vent labeling is as in Fig. 7.


The frequency f0 is related to the outer scale of tur-bulence by the local convective velocity. The outerscale associated with the turbulence observed nearM2 and the turbulence near M1 when the vents areclosed is roughly equal to the diameter L of the domeopening5,19 for upwind orientations, implying thatthe physical mechanism that creates the turbulenceis due to the flow passing over and through the openaperture, rather than the ingestion of free-streamturbulence from the outside air. The latter effect issmall in the data available from Gemini, althoughvariations in the external wind may result in addi-tional energy at very low frequencies, as shown inFig. 9. This plot is not representative; typical varia-tions in external wind speed are at much lower fre-quency and of much lower energy. For downwindorientations, significant turbulence is generated bythe wake of the dome, and the relevant outer scale isthe dome diameter, rather than the diameter of theopening. The outer scale associated with the turbu-lence over M1 that is caused by flow through openvents is related to the height of the vent gates.5 (See,e.g., the spatially averaged M1 pressure spectrumfrom the Gemini data in Fig. 10, with and withoutopen vents, both at 0° azimuth).

All of the measured data available lead to esti-mates of pressure, but it is the force obtained byintegrating pressure over a finite area that is of rel-evance to predicting telescope performance. This spa-tial integration introduces an additional spectralfactor,

�S�f � � �1 � �f�fs�2�1, (9)

multiplying the von Karman spectrum to give a total�13�3 slope, with the corner frequency fs from thisadditional factor depending on the wind speed andthe spatial scale of the integration. The spatial inte-gration accounts for the frequency dependence of thespatial correlation length assuming frozen turbu-lence. This effect was incorporated in some prior mod-

els with an empirically determined aerodynamicadmittance factor that multiplied the velocity powerspectrum by a factor with frequency dependence f�8�3

to give the correct observed behavior.23–25 The �13�3slope can also be obtained from Kolmogorov scalingarguments.

First, consider the net force on a thin structurealigned with the mean wind velocity U. Frozen tur-bulence implies that the pressure on the surface atfrequency f corresponds to a particular spatial wave-number k � 2�f�U, so that at any instant in time

p�x� � �2p�f �Re�expi�kx � �t��, (10)

where p�f � is the rms pressure at frequency f, ob-tained from the PSD and Re(·) denotes the real part.The total force over a structure of length L and widthw is given by

F � w��L�2

L�2

p�x�dx

� �2p�f �Re�exp�i�wL sinc�kL�2�. (11)

The rms force is obtained by averaging over the time-dependent phase �; this yields a factor of 1��2. Themaxima of the sinc function can be approximated bya first-order low-pass filter with unity gain at zero

Fig. 9. (Color online) M2 velocity spectrum for one case, ventsclosed, compared with a von Karman spectrum. The rms velocity is13% higher than the rms of the fit due to the higher amplitudes atvery low frequencies.

Fig. 10. (Color online) Representative M1 pressure spectra mea-sured at Gemini, compared with von Karman spectrum (dashedcurve); without venting (upper), and with open vents (lower).


frequency and corner frequency kL�2 � 1 or f �U��L�. The overall temporal force spectrum thatresults is thus the product of the original von Karmanpressure spectrum with the corner frequency set bythe turbulence outer scale, and an additional atten-uation factor with a corner frequency that is a func-tion of the spatial length scale of the structure. Bothfactors involve the same convective velocity to scalebetween spatial and temporal frequencies. Thereforeif one is considering structures that are small com-pared to the outer scale of turbulence (e.g., M2 itself),one can ignore the structural spectral factor. Theintegration, however, is important for understandingthe loads on the M2 support structure and the dis-tributed loads on M1.

The equivalent load Feq on M2 due to the loads onthe supporting structure, chosen to match the torqueabout the elevation axis, is given by the integral

LFeq � w�0

L

xp�x�dx. (12)

The relevant velocity scaling between spatial andtemporal frequencies is the projection of the assumedmean velocity along the support structure. Note thatthe wind-tunnel test data (e.g., Fig. 5) show that theunsteady velocities and therefore the unsteady pres-sure are roughly independent of location within theenclosure for radii below the shear layer. While thesecondary mirror may be in a region of higher velocityclose to the shear layer, most of the support structurewill be in the more quiescent interior of the dome.

Integrating and computing the mean-square valueover the phase � of the pressure gives the equivalentmean-square force as

Feq�f �2 � 2L2�1 � cos�kL� � kL sin�kL�� kL�2�2��Lk�4p�f�2w2. (13)

This can be reasonably approximated, as shown inFig. 11, by

Feq�f �2 � �L2�4��1 � �kL�2�2�1p�f�2w2. (14)

The appropriate spectrum of force on M2 to simu-late this torque is therefore the force Fleg�f �2 ��w2L2�4��f � that would be correct at zero frequencyand includes the spectral dependence �(f) of the pres-sure, multiplied by the additional roll-off that ac-counts for the correlation effects upon integration,with corner frequency f � U��L�.

Noting that the pressure at the top of the supportstructure is correlated with the force FM2 on M2 itself,the approach can be extended to show that the netmean-square force has a spectrum that can be ap-proximated by

FM2,net�f�2 � FM2�f �2 � �2 FM2�f �Fleg�f � � Fleg�f �2� �1 � �kL�2�2�1. (15)

Appropriate forces on M1 are more difficult to estimatenot just because the structure is two-dimensional, butbecause we are interested in more than just the totalintegrated force. Rather than defining a frequency-dependent spatial correlation matrix for loads on M1,a computationally simpler approach from Padin16 is tocompute the spectrum corresponding to each term in aZernike expansion of the force distribution on M1. Thisconverges because for the turbulence caused by theflow through the dome opening or vent openings, thewind energy decreases rapidly for shorter lengthscales. Additional short-length-scale turbulence re-sulting from the flow across telescope structural ele-ments is not explicitly considered herein but could bebounded from the maximum pressure observed andadded to the long-length-scale turbulence representedby the Zernike expansion. It is possible to construct abasis so that the correlation between forces along dif-ferent basis functions is minimized. However, the ne-glected correlation between forces applied to differentZernike basis functions Znm is small, and therefore theuse of more familiar basis functions is sufficient.

The approach follows Padin16 with minor modifica-tions. Each temporal frequency is associated with agiven spatial frequency in the wind direction, which wearbitrarily choose as aligned with the x axis. Integra-tion of the sinusoidal pressure projected onto a basisfunction can be computed with the Fourier transformof the windowed basis function, with W�r� � 1�� for

Fig. 11. Exact calculation of frozen-turbulence wind spectrumcompared with a simple fit. For one-dimensional integration (up-per), the structural attenuation is shown. For two-dimensionalintegration, the full spectrum is shown for the n � 0 Zernike basisfunction; the final slope is �13�3.


the normalized radius r � 1, W�r� � 0 otherwise,chosen to give results normalized by the total forceAprms. The Fourier transform of W�r�Znm�r� is

Qnm�k� � ��1��n�m��2�n � 1 Jn�1�2�k��k, (16)

with an additional factor of �2 sin�m� or �2 cos�m�,whose average over all possible wind directions overM1 is unity. The complex wavenumber is k �k exp�i� � �kx

2 � ky2�1�2 exp�i�.

The nondimensionalized wavenumber in thewind direction is kx � fR�U (defined without thefactor of 2� for simplicity). The spatial frequency ky inthe orthogonal direction, however, can take any valueand thus the total mean-square force at frequency fprojected onto a Zernike basis function of radial degreen is obtained by integrating over wavenumber ky:

Fnm2�fR�U� �

2��

�2�3��5�6�n � 1

�2

��

�

�Jn�1�2�k�2

k2k0�1 � �k�k0�27�6dky�Aprms�2.

(17)

The integrand is the amplitude of the pressure sinu-soid at the frequency of interest and particular ky

from the von Karman spectrum, multiplied by theFourier transform of the windowed basis function.This integral can be computed numerically and ap-proximated as before by the product of two terms—the original von Karman pressure spectrum and asecond-order roll-off due to the integration over thestructure with amplitude and corner frequency:

An �n � 1��2, (18)

fn �U�2�R��n � 1�3�2. (19)

The corner frequency increases with increasing ra-dial degree while the zero-frequency amplitude de-creases. This fit is shown with the full integral forn � 0 in Fig. 11; the quality of the fit decreases withincreasing n.

5. Summary

The ability to predict the performance of an observa-tory during its design phase is essential. One factor inthis performance results from the wind-induced vi-bration of the telescope structure, caused by unsteadyflow within the enclosure. Based on recently avail-able data from full-scale measurements, scaled wind-tunnel tests, and new computational fluid dynamicsanalyses, a significant advance in our understandingof the wind loads is possible. The dependency of thewind loads on relevant telescope design parameters(M2 location, aperture size) and external wind speedhas been described.

The predominant unsteady wind mechanism in-side the telescope enclosure is broadband turbulencethat can be approximated with a von Karman spec-trum. The amplitude of the turbulence is high withinthe shear layer that exists over the opening of thedome, but the amplitude drops significantly a shortdistance below this shear layer. An additional sourceof turbulence is due to flow passing through openvents that may be used to control the thermal envi-ronment within the enclosure. A third source of un-steady loads results from tonal shear layer modes,which may couple with the Helmholtz mode of thecavity. While these tones are significant in somewind-tunnel and computational results, they are notbelieved to be a strong contributor to telescope vibra-tion in more realistic geometries owing to a combina-tion of effects. Venting decreases the coupling withthe Helmholtz mode, while surface features on thedome exterior interfere with the shear layer vorticitydevelopment.

The amplitude of the equivalent force on M2 can beobtained from velocity estimates using an assumeddrag coefficient [Eq. (7)]. For a dome with a minimumaperture, the velocity near M2 is less than 20% ofexternal, depending on the location within the dome,while the velocity over the support structure is�0.1U�. This is significantly lower than the windspeeds measured at Gemini without the lower shut-ter raised. Computational fluid dynamic analysescapture the difference in wind speeds with the domeopen area. The equivalent force spectrum on M2 thatcaptures the effect of the forces on the support struc-ture has an additional roll-off relative to the pressurespectrum to account for the spatial decorrelation[Eq. (15)]. The amplitude of pressure forces on M1with vents closed (best case) is roughly prms�1�2��0.07U��2; with vents open, much larger velocities arepossible near M1. The spatial characteristics can berepresented by a decomposition onto Zernike basisfunctions with amplitude and spectrum dependent onradial degree [Eqs. (18) and (19)].

Appendix A: Shear Layer Mode

The shear layer modes observed in wind-tunnel testsof simplified dome geometries,6,13 CFD analyses ofthese same geometries,7,13 and also in airborne ob-servatories26 result from flow passing over an opencavity.18 The differential flow speed across the shearlayer generates vorticity, which rolls up into largevortices. As these encounter the end of the cavityopening, an acoustic reflection propagates upstreamand interacts with the flow at the initial separationpoint. This feedback leads to an organization of theshear layer vorticity into modes with n vortex struc-tures across the length of the cavity opening. Thecharacteristics of these modes include a large oscil-latory vortex structure near the secondary mirrorand supporting structure and a large oscillatoryacoustic pressure throughout the entire enclosure.For downwind orientations, the aperture is within


the dome wake, and there can be no shear layermodes.

A typical spectrum including these modes is shownin Fig. 12. The frequency observed in wind-tunneltests and CFD is roughly fn � 0.66nU��L for integern. The factor of 0.66 corresponds to the shear layerconvection speed across the opening; roughly the av-erage of the speed on either side of the opening, withthe local free-stream velocity at the leading edge ap-proximately 10% higher than U� (from CFD). This isconsistent with other estimates of the frequency fordeep open cavities.27 The acoustic amplitude ob-served in wind-tunnel tests can be as high as 25% ofthe free-stream dynamic pressure 1�2�U�

2. However,the shear layer mode strength is influenced by thedamping of the Helmholtz resonance (see, e.g., Ref.28), which is influenced by the vented area.6 Theamplitude is also influenced by any external struc-ture of the dome, which can affect the boundary layerthickness or the shear layer spanwise vorticity dis-tribution (e.g., by introducing streamwise vorticity29).Finally, the net effect on the mirrors results from thedifferential pressure across them. Since the shearlayer modes affect M1 through long wavelengthacoustic pressure waves, the front and back of theprimary mirror are exposed to similar pressure. In-cluding all of these factors leads to the conclusionthat while the shear layer modes are significant forcertain orientations in wind-tunnel (e.g., Fig. 12) andCFD results, they are not the most important sourceof unsteady wind loads on the telescope structure.

The vented area required to obtain a significantreduction in the amplitude of the shear layer mode isrelatively small (relative to the open area of thedome) as shown in Fig. 13 for hot-wire measurementsof velocity near M2 in the wind-tunnel test. The totalmean-square velocity in the first three shear layermodes was computed, normalized by the total mean-square velocity in the broadband turbulence. Notethat the force is proportional to the square of thevelocity, so a 25% vented area leads to a factor of 25

reduction in force. Note that, in contrast to the loadson M1, the broadband loads on M2 are not signifi-cantly influenced by the open venting. Based on theM1 wind-tunnel test, the reduction in the shear layermode pressure on M1 with venting is similar to thereduction in the forces on M2, although the mecha-nism by which the shear layer mode affects M1 andM2 differs.

Estimates for the reduction in shear layer modeamplitude due to dome effects have been made fromCFD simulations. The smooth dome case in Fig. 1(a)with no vents and the shutter closed to the minimumaperture was compared with simulations with aGemini dome, both with the lower shutter raised, andin its lowest position [Figs. 1(b) and 1(c)]. With theshutter closed to its minimum aperture, the shearlayer tone was reduced by approximately 50%, due tothe nonsmooth external dome surface. Lowering theshutter eliminated the tone completely. For this ge-ometry, the incoming flow at the upstream edge of theopening has not yet been turned parallel to the sur-face, and there is no shear layer mode. Note that thetypical observing condition at both the Gemini andKeck Observatories has the lower shutter completelyopen.

If shear layer modes are present, the resulting pres-sure field across M1 is uniform because of the longacoustic wavelength, this is confirmed by wind-tunneldata. (For example, a 0.25 Hz mode corresponding toa large telescope at high wind speeds would have awavelength longer than 1 km.) The forces on the sec-ondary and supporting structure are nearly in phasewith the forces on M1, but larger pressures are pos-sible due to the drag caused by the large oscillatoryvelocities. The worst-case shear layer mode ampli-tude on M2 could be estimated from wind-tunnelmeasurements of an unvented enclosure.

The authors gratefully acknowledge the support ofthe Thirty Meter Telescope partner institutions. Theyare the Association of Canadian Universities for Re-

Fig. 12. (Color online) Velocity spectrum measured in a wind tun-nel using a hot-wire anemometer for upwind orientation. The fre-quency scale is normalized by the convective time scale across theopening. The dashed curve is a fit with a von Karman spectrum.

Fig. 13. Effect of venting on shear layer amplitude. The inte-grated mean-square velocity for the shear layer modes (solid line)is computed at one location near M2 from wind-tunnel measure-ments with three different levels of vented area. The residualbroadband turbulence amplitude (dashed line) is nearly constant.


search in Astronomy, the Association of Universitiesfor Research in Astronomy (AURA), the California In-stitute of Technology, and the University of California.This work was supported, as well, by the CanadaFoundation for Innovation; the Gordon and BettyMoore Foundation; the National Optical AstronomyObservatory, which is operated by AURA under acooperative agreement with the National ScienceFoundation; the Ontario Ministry of Research andInnovation; and the National Research Council ofCanada.

This work has benefited from the comments andconstructive criticism of Terry Mast (University ofCalifornia, Santa Cruz) and two anonymous review-ers. Various people were instrumental in collectingand interpreting the data used herein. In particular,Myung Cho conducted much of the early analysis ofthe Gemini data, and Tait Pottebaum conducted theM2 flow-field wind-tunnel test.

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