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Study of isotropic structure in atmospheric boundary layer turbulence Webster, Ian Taylor 1972

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A STUDY OF ISOTROPIC STRUCTURE IN ATMOSPHERIC BOUNDARY LAYER TURBULENCE by IAN TAYLOR WEBSTER B'.'Sc, University of British Columbia A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE in the Department of Physics and the Institute of Oceanography We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA June 1972 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make i t freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the Head of my Department or by his representatives. It i s understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department The University of British Columbia Vancouver 8, Canada Date .Art.* ABSTRACT The two purposes of this study were to determine at what turbulent Scales in a high Reynold's number shear flow the transition to isotropy occurs and at what scales Taylor's 'frozen f i e l d ' hypothesis i s applicable. The flow studied was the wind at a height of z = 2 m. above a f l a t land surface. Four hot wire anemometers were mounted in a three dimensional array to collect data on the downwind turbulent velocity fluctuations. Cross spectra were computed from the observed data between three pairs of hot wires having the same spacing in different directions; these were varied between 1.8 m. and 2 cm. Knowing the observed spectrum of downwind velocity fluctuations and assuming the turbulence is isotropic, incompres-sible, and obeys Taylor',s hypothesis, theoretical cross spectra were com-puted. The results of the comparison between the observed and theoretical cross spectra for different spacings revealed that in the flow studied the behaviour 1of the turbulence i s consistent with the assumptions of both isotropy and Taylor's hypothesis for k,z > 20, but for wave numbers l e s s than this range either or both of the assumptions are not valid. However, between k j Z = 4 and k j Z = 20 the turbulence appears to be at least axisymmetric about the downstream direction and for k j Z > 3 that part of Taylor's hypothesis relating observed frequency at a stationary sensor to the downstream wave number component appears to be j u s t i f i e d . TABLE OF CONTENTS Page ABSTRACT X 1 LIST OF TABLES v i LIST OF FIGURES v i i ACKNOWLEDGEMENTS v i i i Chapter 1. INTRODUCTION . 1 2. BACKGROUND 5 S p e c t r a l D e s c r i p t i o n 5 The Experimental Arrangement 7 3. INSTRUMENTATION . . . . . 9. The Array 9 The Hot Wire Anemometers ' 1 1 The Sonic Anemometer 12 The Cup Anemometer 13 Data Recording 13 4. OBSERVATIONS AND SOME RESULTS 14 The S i t e 1 4 The Wind Conditions 14 The Sonic Anemometer Spectra . . . . . . 16 The Hot Wire Anemometer Spectra . I 8 5. THE THEORETICAL CROSS SPECTRA . . . 2 0 The Determination of E (k) 21 i v Page The Computational Procedure 24 The Check on the Computational Procedure 25 6. MEASUREMENT ERRORS 2 7 The E f f e c t of Non L i n e a r i t y of the Hot Wires 27' 97 on Observed Cross Spectra The E f f e c t of Transverse V e l o c i t y Fluctuations on Observed Cross Spectra 28 The E f f e c t of Temperature Fluctuations on Observed Cross Spectra 30 The S t a t i s t i c a l R e l i a b i l i t y of Estimates 30 ' The E f f e c t of Errors i n Mean Wind Speed and D i r e c t i o n on the The o r e t i c a l 32 Cross Spectra The E f f e c t of Spectral D i s t o r t i o n on Theo r e t i c a l Cross Spectra 32 The Summary of Error Estimation . . . . . . . . . . . 33 7. DISCUSSION OF RESULTS 35 The Comparison of Observed and T h e o r e t i c a l Cross Spectra . . . . . 35 The Anisotropic Models 43 8. CONCLUSIONS . . . 46 BIBLIOGRAPHY 4 9 APPENDIX A 50 THE DERIVATION OF THEORETICAL CROSS SPECTRA • 50 APPENDIX B .. 53 THE OBSERVED CROSS SPECTRA . . . . . . . . . . 53 The Th e o r e t i c a l Basis 53 The Computation of Observed Cross Spectra . . . 56 Digitisation 56 V Page FT0E 58 SC0R • 59 R0TATE • 59 SIMPL0T 60 Leakage Correction • 60 PL0TTIN.G 61 LIST OF TABLES Table Page I. The Wind Conditions for the Analyzed Section of each Run 15 •II. The Constants A and B for the Fitted Spectrum, Ak» 22 i LIST OF FIGURES Figure Page 1. The Array, Vector Separations., and Coordinate System . . . 10 2. The Observed Sonic Anemometer Spectra for the Three Runs: 6 = 180 cm., 6 = 50 cm. , 6 = 20 cm .17 "3. The Spectra Observed by the Hot Wire Anemometer at the Origin 19 4. The Observed Hot Wire Spectral Estimates Used for Fitting and the Slopes of the Respective Fits 23 5. The Effect on Normalized Cospectra of Wind Angle Variation, Exponent Variation and Linear and Non Linear Hot Wire Calibrations 29 6. The Normalized, Observed and Theoretical Cospectra Obtained from the Approximately Horizontal Crosswind Separation . . 37 7. The Normalized, Observed and Theoretical Cospectra Obtained from the Vertical Separation 0 0 8. The Normalized, Observed and Theoretical Cospectra Obtained from the Approximately Downwind Separation . .40 9. The Observed and Theoretical Coherences Obtained from the Approximately Downwind Separation • • 10. The Comparison of Normalized Cospectra Computed from Isotropic and from Two Anisotropic Models of the Turbulence 44 11. The Data Analysis Scheme 57 ACKNOWLEDGEMENTS I wish to thank the many staff and students of the Institute of Oceanography, University of British Columbia who have assisted me in this project either with their advice or with their physical labour. I am indebted to Dr. R.W. Burling, my research supervisor, for his many suggestions especially in connection with the preparation of this thesis and to Drs. G.S. Pond and R.W. Stewart for much essential advice. Thanks are due also to Captain D.W. Bastock, Commanding Officer of the Canadian Forces Station at Ladner, for kindly making the site . available for the experiment. Finally, I am grateful to the Department of Mechanical Engineering at U.B.C. for permitting me to use their wind tunnel f a c i l i t i e s and for determining the vibrational characteristics of the probe supports. This work was. done in conjunction with the Air-Sea Interaction Programme of the Institute of Oceanography and was supported by the Defence Research Board and by the National Research Council of Canada. Chapter 1 HJTRODUCTION To date the Navier Stokes eqiiations have not been solved for high Reynold's number shear flow. For high enough Reynold's numbers i t has been postulated that -there exists a subrange of isotropic turbulence scales in which no production and no dissipation of energy take place and in which energy is transferred from scale to smaller scale by i n e r t i a l processes only. In this 'i n e r t i a l subrange' Kolmogoroff (1941) predicted behaviour for structure functions which is equivalent to the one dimensional power spectra of turbulent velocity fluctuations being proportional to 'ic'j J / 3 ' where k x is the downstream wave number. A wave number range in which the spectra have the '-5/3' form has been observed on numerous occasions; for example in atmospheric boundary layer flow by Pond et a l (1963) and in oceanic turbulence by Grant, Stewart and Moilliet (1962). The 'k 5/ 3' spectra of downstream velocity fluctuations observed by Pond et al_ (1963) and many others extend to wave numbers low enough that the turbulence is clearly anisotropic. In a shear flow energy is transferred into the turbulence because of the interaction between the turbulent stresses and the mean velocity shear. Since this process causes energy transfer directly into the velocity component along the mean flow, the turbulence, at the scales involved in the transfer, is necessarily anisotropic; that is i t is not symmetrical in a l l spatial directions. This energy is passed to smaller scales by non-linear velocity interactions and at the same time is redistributed by action of the pressure among the three velocity components. One might expect that as the energy passes down the scales memory of the anisotropic sources at large scales w i l l tend to disappear and that eventually the turbulence w i l l appear to be essentially isotropic at sufficiently small scales. Pond et a l (1963) estimated a lower wave number limit for isotropy in a turbulent shear flow at distance z from a boundary as k j Z > 4.5. This result was based on the determination of the scale of turbulence at which the rate of strain due to the turbulence i t s e l f becomes as big as the rate of strain due to the mean shear in a flow which conforms to the logarithmic law of the wall and in which the local dissipation of turbulent energy equals the local production. A number of experimental attempts have been made to determine at what scales a turbulent shear flow does become isotropic. Using hot wire anemometers (in x-configuration at a single point) Weiler (1966) estimated the ratio of the vertical velocity spectrum to the downwind velocity spectrum and the turbulent shear stress in the boundary layer a.few meters above the sea. In an isotropic i n e r t i a l subrange the ratio of the velocity spectra is predicted-to be 4/3, but the ratio observed was less than 1.1 (except on a few occasions) at scales only slightly larger than the dissipation scales. Also, in an i n e r t i a l subrange the turbulent shear stress must be zero. Although the observed shear stresses were non zero for wave numbers above Pond's limit of isotropic wave numbers the degree of anisotropy associated with this property appeared to be f a i r l y small. Van Atta and Chen (1970) determined a lower wave number limit for isotropj' in the atmospheric boundary layer over the sea. Their lower limit, k j Z = IT, was based on the comparison of observed second order structure functions with those predicted for the i n e r t i a l subrange-The primary objective, of my study was to determine by another means at what scales the transition to isotropy might occur i n a high-Reynold's number shear flow. The measurements were made in a wind at a height of 2 m. over a f l a t land surface. This type of flow is sometimes close to the simplest form of shear flow in which a steady mean wind is parallel to the ground, does not change direction with height and which possesses negligible local heat sources of energy. For the sections of data analyzed the durations of observations of the wind are much shorter than the time scales associated with changing synoptic atmospheric conditions. The degree of isotropy in the turbulence was studied by comparing cross spectra between downwind components at separated points with those expected theoretically. The turbulence data were obtained from a three dimensional array of four hot wire anemometers. If the turbulence is in an incompressible f l u i d , is assumed to be isotropic and i f also Taylor's 'frozen f i e l d ' hypothesis holds i t is possible t c compute theoretical cross spectra from the observed power spectra of downwind velocity fluctuations. A stationary velocity sensor senses some property related to the velocity fluctuations at a given point in space as a function of time. Observed cross spectra are obtained as functions of frequency but the theoretical cross spectra are derived in terms of wave numbers. However, the observed cross spectra can be transformed into a wave number space by assuming that the turbulence is transported in accordance with Taylor's hypothesis, which is that the turbulence transports lik e a 'frozen f i e l d ' and does not change i t s structure or i t s advectipn velocity while advecting through distances comparable with the scales of turbulence studied. Under this hypothesis, observed frequencies, f, are equivalent 2lTf to downstream wave numbers, k , where k = — — . Since Taylor's hypothesis plays such an integral part in the comparison between the observed and theoretical cross spectra the results of the comparison depend strongly on the validity of this hypothesis. To obtain information on the valid i t y of Taylor's hypothesis was a secondary objective of my study. Attempts have been made by various authors to determine theoreti-cally the conditions under which Taylor's hypothesis can be applied in a turbulent shear flow. Lin (1953) theorized that the effect of distor-tion of eddies by the mean shear would be small provided the advection of strain rate in the downstream direction i s very much larger than the turbulent advection of strain rate across the shear. This criterion, applied to the condition of the flow during my measurements and assuming a logarithmic mean velocity profile requires that k 2z » 0.2. Lin also describes conditions under which the rate of distortion in the turbulence due to pressure fluctuations can be considered to be negligible. The criterion he obtains for applicability of Taylor's hypothesis in this case is 5u2/U2 << 1. Typical values of 5uj/U 2 encountered in the present study were about 0.15. A third assumption involved in the derivation of theoretical cross spectra is that the turbulence behaves as i f i t were incompressible. Hinze (1959) estimates the compressibility effect to be negligible i f u 2/c 2 << 1 where u i s the magnitude of the velocity fluctuations and c is the speed of sound under the measuring conditions. For the present study the above ratio was of order 10 ~* making incompressibility a good assumption. Because incompressibility i s such a good approximation the results of the comparison of observed and theoretical cross spectra should depend primarily on the validity of the two assumptions; isotropy and Taylor's hypothesis. Chapter 2 BACKGROUND Spectral Description In order to describe the turbulence a cartesian coordinate system r = (x, y, z) is chosen such that the 'x' axis is in the horizontal direction directed up the mean wind, U, 'z' is the axis perpendicularly upwards, and the 'y' axis is horizontal and transverse to the mean wind. It is desirable to decompose the velocity f i e l d into the fluctuating part, u = (u, v, w), and the mean wind, U = (U, 0, 0). The two quantities, U and u, are defined such that the time averages U = U and u = 0. Consider the product of the velocity components u^ and u^ measured at the two.positions in space, r and r + 6, respectively. Provided u^ and u_. are stationary and homogeneous in the s t a t i s t i c a l sense, we can define a Fourier representation of the product of the velocities averaged over a l l space and a l l time for the n realization of the. flow as: (for example, see Lumley and Panofsky, 1964, p. 25). u.f°(r,t) u j n ) ( r + 6 , t ) r,t ^ ( l O e ^ d k (2.1) where dk is an element of volume in k = (k. , k , k ) space and th is the spectral density tensor for the n realization. We can introduce a cross spectral tensor, Cr^^(k, 6), defined precisely in Appendix A, which has the property that:. r,t f°° u< n )<r,t)u ( n^r+6,t> —oo Cr^\k,6)dk (2.2) Thus C r ^ ( k , 6 ) i s the contribution to the total covariance u^"^ ( j , t ) u j n ^ (r+5 , t) per u n i t wave number volume at k. The smoothed cross spectrum, Cr (lc,6), i s the ensemble average of the C r ^ ^ ( k , 6 ) over an i n f i n i t e number of r e a l i z a t i o n s of the flow (Lumley and Panofsky, 1964). T The t h e o r e t i c a l cross s p e c t r a , Cr„(k,6), are def i n e d to be T averaged i n the manner of (2.3). (The terminology Cr denotes a t h e o r e t i c a l cross spectrum and Cr° an observed cross spectrum.) In the experiment j u s t the one dimensional observed cross s p e c t r a , C r ^ j O C j ^ ) , can be obtained. Furthermore, the Cr° ( k 1 } 6 ) are only estimates of the i d e a l , smoothed, one dimensional cross s p e c t r a because , r , t the average, u_^(r,t)u ( r + 6 , t ) - ' , i d e a l l y over a l l space and a l l time, can only be approximated as an average over f i n i t e time and because the ensemble average can only be oyer a f i n i t e number of r e a l i z a t i o n s of the flow. The t h e o r e t i c a l and the observed one dimensional cross s p e c t r a are estimated i n Appendices A and B r e s p e c t i v e l y . When 6 = 0 and i = j the C r ^ C k ^ S ) are the one dimensional power s p e c t r a , $ ( k j ) . . $^(kj) = C r ^ i ( k 1 , 0 ) (not summed over i ) (2.4) The p h y s i c a l i n t e r p r e t a t i o n of $ ( k x ) , the one dimensional power spectrum of downstream v e l o c i t y f l u c t u a t i o n s , i s that i t i s the c o n t r i b u t i o n to the t o t a l v a r i a n c e u 2 , per u n i t wave number, k j , from a l l wave numbers, k, having the component, -k . . • In t h i s stud)' the cospectrum, C o ( k 1 , 6 ) , and the quadspectrum, Qu(k ,5) are def i n e d as the r e a l and imaginary p a r t s of C r J 1 ( k ,6) which i s the one dimensional cross spectrum of downstream v e l o c i t y fluctuations. Thus: Co(k l 56) = Re(Cr n(k l 56)) (2.5) Qu(k l 96) = Im(Cr n(k 1,6)) (2.6) The cospectrum is a measure of the amount of 'in phase' coherent energy density as a function of k between the velocities at r and r+6, whereas the quadspectrum is a measure of the coherent energy density Coh(k i,6), which is a measure of the normalized coherent energy density between the points, r and r+6. The relative phase, a, of the coherent energy density between the points, r and r + 6 is given by: The Experimental Arrangement The isotropy of the turbulence was studied by estimating the cross spectra between velocity components observed simultaneously•using hot wire anemometers at four points with selected space separations. The ideal set of separations would be to have one sensor situated at an 'origin' of the coordinate system (x, y, z) and the others each at equal distance along the three coordinate axes. The arrays used were designed so the upwind arm was offset in the horizontal plane to avoid which i s 90° out of phase at the same two points. $ (kj), Co(k1,6) and Qu(kj,6) together define the coherence, (2.7) (2.8) 8 wake interference to the sensor at the origin. In practice the coordinate system was always determined by the direction of the mean wind as illustrated in Figure 1 which shows the physical arrangement of the sensor systems. The angle, 0, was some angle between 10° and 20° during each data run. The concept of 'scale'.in turbulence suggests the optimum information on the isotropy at some scale can be obtained by examining cross spectra at separations up tp about the scale distance. The height of the three sensors in the horizontal plane was 2m., so using the criterion (Pond et a l , 1963), the lower limit of wave numbers expected to be isotropic i s somewhat above k s 4.5(2m) 1 z 2.2m 1. 2TT If turbulent scale size be taken as ^— then isotropy is not expected at scales greater than 2m. or so. The largest separation, 1.8 m., used in the measurements was chosen as representative of the possible upper limit of scales in the i n e r t i a l subrange. The closest separation of the sensors was 2 cm. which was the smallest practically measurable spacing under the conditions of the experiment. This separation describes the smallest scales which might be expected to be within the i n e r t i a l subrange near the dissipation end of the spectrum. The four other separations; 50 cm., 20 cm., 10 cm., and 5 cm.; were spaced approximately logarithmically -between the largest separation and the smallest separation. Chapter 3 INSTRUMENTATION The Array In the assembly designed to support the sensors, shown in Figure 1, a l l the members were 1" square aluminum tubing which had a very high resistance to torsion and to bending. The structure was stayed by guy wires fastened to several points. When setting up the array for a run, a l l arms and legs were made vertical or horizontal by adjusting the lengths of the guy wires. The characteristic vibration frequencies of each member of the assembly were different being dominantly near one or two cycles per second. Any vibration excited in an arm of the assembly tended to be damped out after about a cycle. The hot wire anemometers were supported by clamps which could slide along the square tubing to attain any desired sensor separation. The susceptibility of the anemometer probe support to vibration was tested on the Mechanical Engineering Department vibrating machine. The characteristic vibration frequency of each support depended strongly on the precise position of the hot wire probe holder in the supporting clamp. Because vibrations were hard to excite and because the vibration frequencies would be different for each support i t was not expected that vibrations would present a significant problem. Possible effects were looked for but were not detected in any of the data analyzed. The three dimensional sonic anemometer and the cup anemometer also used in the experiment were fastened to the horizontal cross-stream bar of the hot wire anemometer ^<""""1r THE ARRAY position ot hot wire probes 3D sonic anemometer ,1.1 THE VECTOR SEPARATIONS THE COORDINATE SYSTEM g approx. downwind ~" separation g ' approx. crosswind horizontal separation g crosswind vertical ~" separation mean wind-Figure 1: The Array, Vector Separations, and Coordinate System. assembly in such a way that.their sensors were centered at the same 2m. level as the three lower hot wire anemometers. The Hot Wire Anemometers Four hot wire anemometers were used in this experiment to measure the cross spectra of downwind velocity fluctuations. The operating principle of hot wire anemometers depends on the cooling effect of the wind on a very fine piece of heated wire whose resistance depends on temperature. Hot wire anemometers have the two characteristics necessary to the present experiment, small sensor size and fast response. The particular sensors used had a size of about 1.5 mm. and the devices as a whole were adjusted to respond to velocity fluctuations up to at least 10,000 Hz. The hot wire anemometers, which were Disa battery operated constant resistance types*, have a non linear response given by 'King's Law': £ 2 .= A + B/V (3.1) where £ is the output voltage, V i s the instantaneous magnitude of the wind perpendicular to the sensing element and A and B are constants depending on the partictilar anemometer and on the temperature of the a i r . For low turbulence levels output voltage fluctuations, e, are linearly related to the f i r s t order to the downwind velocity fluctuations,'u; the effect of the next order of response i s examined in detail in the section on measurement error on page 27. The calibration constant for * Electronics: Disa type Do55; Probes: Disa type 55A22. the linear response of the hot wire anemometer was determined by matching the hot wire voltage spectra at low frequencies to velocity spectra observed using the sonic anemometer. The sonic anemometer, which responds to scales larger than about lm., is a much better device for measuring absolute velocities because the operating levels of the hot wires are liable to change considerably due to age or to the adherence of specks of dust to the sensing element. Because the hot wire anemometer operation depends on the cooling effect of the wind blowing past the wire, temperature varia-tion in the air i t s e l f w i l l affect the response. The contamination of the output signal by temperature fluctuations can be minimized by operating the hot wire at a sufficiently high temperature above ambient temperature. The magnitude of the temperature effect is estimated in the discussion on measurement error on page 30. The Sonic Anemometer The magnitudes and directions of the mean wind were measured using a three dimensional sonic anemometer mounted in a fixed position on the horizontal cross-stream arm of the array. The sonic anemometer measures wind velocity by comparing the times of f l i g h t of two sound pulses travelling in opposite directions over the same path. The difference in the two times i s proportional to the component of the wind in the direction of the path. The instrument used senses components along such paths oriented so :the three instantaneous velocity components; u, v, w; can be computed. In addition, by measuring the absolute sound velocity, the sonic anemometer xtfill provide the instantaneous density fluc-tuation, p 1 or equivalently the virtual temperature fluctuation, T^ (Lumley & Panofsky, 1964). Besides determining the vector mean wind the sonic 13 was also used to determine the Reynold's stress, uw; the turbulent density flux, p'w ; and the downstream velocity fluctuation spectra for use in calibrating the hot wire anemometers. On the 'Kaijo Denki' sonic anemometer used, the sonic paths were each 20 cm. long so i t s response to velocity fluctuations f a l l s off with scales decreasing from about 1 m. The Cup Anemometer A Kenkusho cup anemometer was used as a check on the mean wind speed measured by the sonic anemometer. Because the cup responds to wind magnitude, and because i t tends to 'overspeed' in turbulent flow i t is expected to register a higher wind speed than the sonic anemo-meter. A comparative study by Tzumi and Barad (1970) indicated that cup mean wind speeds were on the average 10% higher than mean wind speeds in the same flow measured by a sonic anemometer. Data Recording The voltage fluctuations from the various velocity sensors were recorded in FM mode on magnetic tape using an Ampex FR1300 fourteen channel tape recorder. Altogether nine channels of information were recorded simultaneously. Prior to recording, the signals from the cup and the hot wire anemometers were passed through 'gain-offset' amplifiers to ensure satisfactory signal to noise levels in the re-corded signals. Chapter 4 OBSERVATIONS AND SOME RESULTS The Site The site used for the measurements was an abandonned air f i e l d near Ladner, B.C. which is essentially f l a t , horizontal and relatively free of obstructions making i t ideal for the study. The array i t s e l f rested on grass about 15 cm. high. This grassy area extended at least 200 m. in the x^ind approach direction before being crossed by an asphalt runway. Beyond the runway the grass continued further for about 1 km. before i t ended at a dike bordering the shores of Boundary Bay. Because the maximum height of the uppermost sensor was only 4 m. i t could be safely assumed that the runway and the dike had negligible influence on the turbulence seen by any of the sensors. The Wind Conditions A l l the measurements were made on April 28, 1971 which was a cool overcast day. Six runs were made each at a different separation and of approximately one half hour duration; the sta t i s t i c s of the analyzed section of each run are li s t e d in Table I. The wind remained remarkably uniform in magnitude and direction over the whole period of the observations so the data sections analyzed were rather arbi t r a r i l y taken from near the beginning of each run. In Table I, U is the average of the mean wind speeds given by the sonic and by the cup anemometer. The cup mean wind was consistently higher than the sonic mean wind by between 3% to 10%. The angle of the mean wind, TABLE I The Wind C o n d i t i o n s f o r the Analyzed S e c t i o n of each Run ^ Run Time of D u r a t i o n of k, range — — — _ / T I d e n t i f i c a t i o n Run A n a l y z e d of a n a l y s i s . , . 9 /• 2/ 2\ , 2 1 2\ v fi. \ f-n c T \ c J - r \ / s - i (m/sec) (m /sec ) (m /sec ) (by s e p a r a t i o n ) (P.S.T.) S e c t i o n ( s e c ) (m) 6.02x10"3 6 = 180 cm. 13:59 1015 -> 6.60 1 1 . 0° 1.15 0.326 1.91x10 4.09x10"3 6 = 50 cm. 15:16 254 -»• 6.55 2 0 . 1° 0.930 0.299 -0.015 7.73x10 8.19x10"3 6 = 2 0 c m . 12:17 127 -> 6.54 1 3 . 4° 0.888 0.235 -0.027 1.55xl02 1.79xl0"2 6 = 10 cm. 10:36 63.5 -> 6.00 1 2 . 2° 0.858 3.37xl02 2 . 9 8 x l 0- 2 6 = 5 cm. 11:05 31.8 + 7.21 1 2 . 2° 1.01 5.69xl02 6.82xl0~2 6 = 2 cm. 16:50. 15.9 -> 6.29 1 6 . 8° 0.653 1.29xl03 16 6, as defined in Figure 1 is that obtained from the sonic anemometer. The wave number range used for the spectral analysis of the data is given by 'ki range'. The ' k i range' for each run is different because a l l the runs were digitized at different frequencies. Both u 2, the variance of downstream velocity fluctuations, and uw, the Reynold's stress, are calculated from the sonic data. The averages for both are over the duration of the data section analyzed. The density flux, p'w, arises because of variations in density due both to humidity and to temperature. The ratio of the height, z, to the Monin-Obukhov length, Ly, (Lumley and Panofsky, 1964) is a measure of the buoyancy effects in the structure of the turbulence. This ratio was computed for two runs and is shown in Table I; the small values indicate that effects of buoyancy are negligible during the measurements. The Sonic Anemometer Spectra The sonic anemometer spectra, k i * $ ( k i ) , k i * $ ( k i ) and k i * $ ( k i ) , ' U V w computed by the analysis scheme of Appendix B, are plotted for three of the runs on a log-log scale against kiz in Figure 2. At low wave numbers the 'u' spectrum has the most energy and the 'w' spectrum has the least. The '-2/3 slope' line on each of the graphs corresponds -5/3 to the 'k ' form in the spectra predicted for the i n e r t i a l subrange. The 'u' spectrum attains this slope at values of k j Z around 2 but the 'v' and 'w' spectra do not u n t i l k xz is approximately 5 . At s t i l l higher wave numbers the 'v' and 'w' spectra r o l l - o f f due to the effects of using a 20 cm. sound path for the velocity measurements. The difference in the shapes between runs of the 'u' and 'v' spectra at high wave numbers is probably due to aliased electronic noise. None of the sonic spectra should be considered reliable much above k z = 10. O.CH o S=l80cm. A S=50crn. + S = 20crn . -to-] Loqo(k,<^k,)) - 2 . 0 -- 3 . 0 - 3 . 0 - 2 / 3 s l o p e A A 0 + /?o? A + • A+ A t A f A+ O2.0 + -1.0 0 . 0 1.0 L O Q i k , z ) 2 . 0 3 0 Figure 2: The Observed Sonic Anemometer Spectra for the Three Runs: 6 = 180 cm., 6 = 50 cm., 6 = 20 cm. 18 The Hot Wire Anemometer Spectra The hot wire anemometer spectra for a given run are a l l similar in shape to one another and to the 'u' spectra measured by the sonic anemometer in the wave number range where i t gives reliable results. The hot wire anemometers were calibrated by matching the integrals of the low wave.;number voltage measured by the hot wires to the 'u' spectrum measured by the sonic anemometer (see Appendix B). Most of the difference between the details of the spectra of the hot wires for a given run appears to be due to random spatial variation of the turbulence; as the separation of the sensors decreases the details of the spectra become more similar. For the largest separation the upper hot wire at z = 3.8 m. seems to have slightly more energy at low wave numbers than the other three hot wires at z = 2 m. As one would expect, the two spectra from the hot wires separated approximately downstream of one another shox^ed the most agreement in detail at a l l separations. When plotted together on a log-log graph versus k j Z the shapes of the spectra from the different runs f i t t e d one another reasonably well as should be expected i f the wind is s t a t i s t i c a l l y stationary. Figure 3 is a plot of such a composite spectrum obtained from the hot wire situated at the 'origin' of the array. As does the sonic 'u1 spectrum the hot wire spectrum attains i t s i n e r t i a l subrange form for k j Z . > 2. It retains this form until k j Z ~ 1000 in the run analyzed to highest wave number, where i t begins to r o l l - o f f due to dissipation effects. 19 -ao ^20 6.0 10 2D "3D igure 3: The Spectra Observed by the Hot Wire Anemometer at the Origin. Chapter 5 THE THEORETICAL CROSS SPECTRA T The Determination of E (k) The theoretical basis for predictions of cross spectra appropriate to isotropic turbulence obeying Taylor's hypothesis i s outlined in Appendix A. The computational procedure for determining these theoretical cross spectra involved two steps. The f i r s t was the T derivation of E (k), the three-dimensional spectrum as defined in Appen-dix A, appropriate to the observed one-dimensional power spectra of downwind velocity fluctuations and the second was the computation, using T the E (k), of the various cross spectra for the different separations. Providing the turbulence is isotropic and obeys Taylor's T hypothesis then E (k) is related to the one dimensional spectrum of downwind velocity fluctuations by Equation (A.14): ,T, ( k ) - T dk d$ (k) 1 u k dk Using the f^Ckj) measured by the hot wire anemometers a T theoretical E (k) was to be estimated using the above relation. The measured spectra, ^ ( k j ) , are, however, a series of discrete estimates T which can not be used directly in the equation for E (k). To circumvent this d i f f i c u l t y analytic functions were fitted to the observed spectra. T These analytic functions were the $ (k) used in the equation to T T estimate E (k). The analytic functions, E (k), which resulted were then used for the subsequent computation of theoretical cross spectra. . To obtain an analytic form for ^ ( k j ) a least squares f i t t i n g procedure was applied to the geometric means of the spectral estimates from each of the pair of hot wires for which the cross spectrum was to be computed. Each spectral estimate was weighted by the bandwidth over which the estimate was averaged. In a few cases where there were indications of significant noise as, for example, near the high frequency end of the spectrum for the 2 cm. separation where the effects of aliasing of a high frequency pick-up become noticeable, a zero weight was assigned to the estimate. I n i t i a l l y , an attempt was made to f i t a function of polynomial form, or of a polynomial of logarithmic form to the complete hot wire spectrum produced by the computer program SIMPL0T (see Appendix B). In order to get reasonable f i t s over the whole spectrum i t was necessary to go to rather higher order functions than would be easy to handle. By superimposing the spectra from the different separations each of which covers a different wave number range on a log-log plot one can obtain a view of the complete spectrum (see Figure 3). Over a large portion of the wave number range the shape of this composite spectrum appears linear which suggests a simple power law might be a good approximation to the spectrum in these regions. The spectrum at each separation was thus fitted according to (5.1) over this wave number range. ^ ( k i ) = Ak^ (5.1) Here, A and B, the constants determine for each pair of hot wires at each separation, are li s t e d in Table II. Figure 4 illustrates the spectral estimates which were used for this f i t t i n g and the slope of the respective f i t s at each separation for the pair of hot wires having the approximately downwind separation. TABLE II The Constants A and B for the Fitted Spectrum, Ak' Run Approximately Downstream . Separation Approximately Horizontal Transverse Separation Vertical Separation Identification (by separation) A(M.K.S.) B A(M.K.S.) B A(M.K.S. ) B '6 = 180 cm. 0.935 -1.57 0.930 -1.56 0.840 -1.60 6 = 50 cm. 1.06 -1.63 1.03 -1.61 1.00 -1.63 6 = 20 cm. 1.16 -1.62 1.02 -1.60 1.12 -1.62 6 = 10 cm. 1.16 -1.65 1.07 -1.63 1.13 -1.64 6 = 5 cm. 1.16 -1.68 1.09 -1.66 1.14 -1.68 6 = 2 cm. 0.638 -1.56 0.591 -1.55 0.620 -1.56 23 -i.cP ~~ao ID 20 ao Figure 4: The Observed Hot Wire Spectral Estimates Used for Fitting.and the Slopes of the Respective F i t s . 24 The Computational Procedure T From the f i t t e d forms, expressions for E (k) were determined analytically as simple power law functions. On the assumption that an isotropic turbulence corresponds to the observed spectra, the theoretical cospectra, quadspectra, arid coherences were evaluated using the Equations (A.11), (A.12) and (2.7): rCO (CO X T Co(k 1,6) = coskjSj E (k) (k2+k|)cos(k2<52+k36 3)dk2dk3 O TTk Qu(kj,6) = sinkj6, E ^ (k2+k|.)cos(k262+k38 3)dk2dk3 o irk where 6 , 6 , and 6 are cartesian component separations in the coor-dinate system defined by the observed direction of the mean wind. T Because E (k) is assumed to be of simple power law form, the above integrations can be converted to integrations over two other inte-gration variables one of which can be performed analytically whereas the other-can easily be evaluated numerically. The numerical integration was carried out using a Simpson's rule subroutine on the university's computer using a high enough upper wave number limit to ensure proper convergence. Theoretical cross spectra are plotted in Figures 6, 7,8 and 9 with the observed cross spectra; these w i l l be discussed later. T T Although a r e a l i s t i c E (k) like a r e a l i s t i c $ (k) would r o l l -off at low and high wave numbers this was not accounted for in any of the theoretical cross spectra. Because the minimum value of k is just (kj, 0, 0) in any integration the low wave number r o l l - o f f has no influence on the integration as long as kj is above the r o l l - o f f region. 25 On the other hand, the upper limits of integration extended into wave number regions where the high wave number r o l l - o f f might become s i g n i f i -cant. Judging by the slope of the observed spectra at the highest wave T numbers, E (k) might be expected to be truncated considerably by the upper integration limits of k 2 = 1000 m- , and k 3 = 1000 m~ the effect of the r o l l - o f f on the 2 cm. normalized cospectra and coherences was...estimated. Compared to those cross spectra computed using the normal integration limits the cross-stream cospectra were altered by 0.01 or so near kj6 = 10.0 whereas the downstream coherence was increased by 0.04 near kj5 = 10.0. For wave numbers less than k16 = 3.0 the downstream coherence was almost imperceptibly altered. For a l l the larger separations the effect on the cross spectra of the high wave number r o l l - o f f would be much less. The Check On The Computational Procedure The derivations and computational procedures for these calcu-lations were checked by computing cospectra for both a purely downstream separation and a purely cross-stream separation which would result -5/3 from a 'k ' power spectrum. For a purely downstream separation i t can easily be shown theoretically that the normalized cospectrum is given by the particularly simple form: ; T viscous dissipation effects at wave numbers, k, near 1000 m - l Using Co(k 1 ?6) = cos(k 16) (5.2) The result obtained using the numerical procedure agreed with this within ± .0.005. The difference between the spectrum and the cospectrum when integrated with respect to wave number yields the value of the structure function for that particular separation. On dimensional 26 grounds one predicts that in the i n e r t i a l subrange the ratio of the cross stream to downstream structure function should be 1.33 (Lumley and Panofsky, 1964). This ratio was obtained for structure functions estimated from the computed cospectra corresponding to the ideal -5/3 'k ' form. That the downstream cospectrum and ratio of structure functions evaluated numerically from this form agree with the analytical predictions confirms that the expressions, (A.11) and (A.12), were cor-rectly derived and that the numerical procedure i s a correct one. Chapter 6 MEASUREMENT ERRORS Ideally, a sensor would respond linearly to downstream velocity fluctuations; in practice the response of hot wires is non-linear and they also respond somewhat to transverse velocity fluctuations and to temperature fluctuations. The computations of the theoretical cross spectra are subject to error due to uncertainty in the precise vector separation of the sensors and to variance associated with the estimate of a form representative of the observed downwind spectra. In the following each of the above effects is investigated. The Effect of Non-Linearity of the Hot Wires on Observed Cross Spectra In a l l the computation of observed cross spectra the hot wire anemometers were treated as i f they had a voltage response, e, to small downwind fluctuations, u, given by: e = cu (6.1) where c is a calibration constant. However the 'King's Law' response for the hot wires used is (3.1): f2 = A + B /v~ h where V, i s the total horizontal component of wind speed and £ is the h output voltage. Because the velocity fluctuations in the cross stream direction were small compared to the magnitude of the mean wind then V, h is nearly.the instantaneous magnitude of the wind in the downstream 28 direction. . By solving (3.1) for and expanding in a Taylor's series one obtains the second order response for the downstream fluctuations, u: u = C 1 K BV (6.2) C is a constant depending on A, B, the mean voltage, E,, and the mean wind speed, V^. Using the constant B determined by a wind tunnel calibration of the hot wire a section of data was analyzed using both the linear calibration, (6.1), and the more correct non-linear calibration, (6.2). Figure 5c shows a comparison of the two analyses for a normalized down-wind cospectrum for a 50 cm. separation. In the region of the drop-off the corrected (non-linear) calibration is seen to result in normalized cospectral estimates which are as much as 0.05 higher than the corresponding estimates obtained using the uncorrected (linear) c a l i -bration. The alteration to cross-stream cospectra using non-linear calibrations would be expected to be similar. Because the non-linear response affects large amplitude velocity fluctuations more than small amplitude fluctuations and because the large amplitude fluctuations tend to be at the lower wave numbers the cross spectra from the smaller separations, which cover higher wave number ranges, should be affected less by the non-linear response than those from the 50 cm. separation. The Effect of Transverse Velocity Fluctuations on Observed Cross Spectra In the experiment the hot wire anemometers respond to the i n -stantaneous horizontal magnitude of the flow. As a result, the output voltage of the hot wire is somewhat dependent on the transverse horizontal velocity fluctuations, v. Using the 'King's Law' response equation (3.1) one can estimate that for small u and v the variance of the voltage w i l l appear as: 29 a) <Pu(k.) 1.0-Co (!<„§) 0.0 b) lO-i Co(K,§) 0.5 0.0 VERTICAL BARS INDICATE EFFECT OF MEAN WIND ANGLE . VARIATION OF t5* ON COMPUTATION OF. APPROX. DOWNWIND THEORETICAL COSPECTRUM VERTICAL BARS INDICATE EFFECT OF EXPONENT VARIATION OF ±0.1 ON COMPUTATION OF APPROX. DOWNWIND THEORETICAL COSPECTRUM c) Cb(k,a) 0.5-0.0 9 9 d + O + o -1.0 -0.5 0.0 ~0.5o 10 LOG(k,S)6 COMPARISON BETWEEN OBSERVED APPROX. DOWNWIND COSPECTRA UTILIZING LINEAR AND NON-LINEAR CALIBRATIONS ° linear calibration + nonlinear calibration 'Figure 5: The Effect on Normalized Cospectra of Wind Angle Variation, Exponent Variation and Linear and Non Linear Hot Wire Calibrations. 30 e 2 a u 2 + ^ + ^ - (6.3) Both u 2 and v 2 were observed to be around 1 m2 /sec 2 and the mean wind speed, U, to be around 6m /sec, so that both and ^ were about 0.03 u 2. The term —j arises because of the non-linear response of the type already discussed, but ^5- is a measure of the increased variance resulting from the influence of transverse velocity fluctuations. The maximum magnitude of error in the spectra due to the latter effect i s expected to be about the same as the 3% alteration to the variance; the alteration to the cross spectra normalized by the spectra would probably be less than this amount. The Effect of Temperature Fluctuations on Observed Cross Spectra Because the principle of operation of a hot wire anemometer depends on the cooling of a heated wire by the wind i t s output depends somewhat on air temperature fluctuations. This effect can be estimated from formulae given by Bearman (1970). Using a typical observed R.M.S. temperature fluctuation of 0.5°C, the hot wire operating temperature of 600°C as well as typical hot wire calibration constants, typical mean wind speeds and typical hot wire outputs for the experiment the tempera-ture effect is expected to produce equivalent R.M.S. velocity fluctua-tions of about 0.006 m/sec. This is f a i r l y small compared to the observed R.M.S. velocity fluctuations of around 1 m/sec. The Sta t i s t i c a l R e l i a b i l i t y of Estimates Even though the turbulence may be assumed to be stationary, as long as i t s averages are over a f i n i t e number of realizations of the flow, each estimate has a variance associated with i t s s t a t i s t i c a l nature. Jenkins and Watts (1968) derive expressions f o r the variance, C 2, of the coherence estimators, C o h ( k1, 6 ) , under the assumption that the random process considered behaves l i k e white noise: { l - f C o h t k j . f i ) ) 2 } 0 2 ~ Z ( 6.4) Coh V V, the mimber of degrees of freedom of the estimate, i s equal to W i c e the product of the number of estimates i n the wave number band over which i t i s averaged and the number of data records over which i t i s averaged. Because the quadspectrum i s r e l a t i v e l y small, the cospectrum normalized by the spectrum i s s i m i l a r to the coherence (see (2.7)). For a process of the type considered the cross s p e c t r a l estimators are d i s t r i b u t e d approximately normally about the averages that would be obtained from an i n f i n i t e number of r e a l i z a t i o n s of the flow. The 68% confidence . l i m i t s for a normal d i s t r i b u t i o n are given by ± 0"; thus the expected err o r i n the cospectra i s approximately given by ^QQ^- ^ n the region of maximum slope of the coherences and of the normalized cospectra (these features are discussed i n the following sections and are shown i n Figures 6 , 7, 8 and 9) the expected v a r i a t i o n of the estimates i s around ± 0 . 0 5 . O„ , i s smaller both as the coherence goes to 1.0 at Coh low wave numbers and as i t approaches zero at high wave numbers where the bandwidth f or each estimate increases. If the process i s not completely stationary- as was probably the case f o r the turbulence studied, the s t a t i s t i c a l v a r i a t i o n s of the observed cross s p e c t r a l estimates would be expected to be somewhat larger than those predicted. 32 The Effect of Errors in Mean Wind Speed and Direction on the Theoretical  Cross Spectra The important parameters in the computation of theoretical cross spectra are the values of k. 6 and the direction of the mean wind with respect to the vector separations of the four sensors in the array. Inevitably there i s some uncertainty in the exact position of each hot wire sensor on the array which introduces possible errors in both the magnitudes and directions of the separations. Because of errors in estimating the magnitude and direction of the mean wind using the sonic anemometer, the orientation and magnitude of the wind with respect to the array also has some doubt. Whereas the maximum error i n kjS due to both effects i s estimated to be ± 6% at most, the expected error in kjS i s ± 3%. Likewise the maximum and probable error in the direction .of the mean wind are estimated to be ± 5° and ±.3° respectively. The effect of introducing a change in the mean wind angle of ± 5° in the computation of a cospectrum from an approximately downwind separation is shown in Figure 5a. The cross-stream cospectra are negligibly affected by mean wind direction changes of this size. The Effect of Spectral Distortion on Theoretical Cross Spectra Even though a low pass.filter was used prior to di g i t i z i n g the analog signals to help eliminate 'aliasing', some leakage of high wave number energy into the lower wave numbers of analysis did occur. Energy also leaks from low wave numbers to higher wave numbers due to the shape of the transfer function for a f i n i t e data record. Although the latter effect was corrected for prior to the plotting of the normalized cross spectra, the spectral fittings were made using uncorrected estimates. 33 The non-linearity in response of the hot wires discussed also tends to cause low wave number energy to appear at higher wave numbers. The combination of the three sources of distortion on the observed spectra causes errors in both the level, A, and in the exponent, B, of the fit t e d spectra, Akj. Even though the errors in level of each spectrum might be as large as + 15% in the worst case they produce no error in estimates of normalized cross spectra since both numerator and denominator are affected similarly. The exact exponent of the f i t t e d spectrum does, however, alter the theoretical normalized cross spectra. Figure 5b illustrates that the effect of altering the exponent of the fi t t e d spectrum by ± 0.1, the extreme maximum deviation expected, i s to cause shifts in the values of the normalized cross spectra of about ± 0.02. The Summary,of Error Estimation One can summarize the preceding by estimating the maximum expected difference between observed and theoretical normalized cross spectra due to error. The maximum error is evaluated by adding linearly the maximum deviations to the cross spectra expected at each separation arising from mean wind angle uncertainty, from uncertainty in kjS, from non-linearity of the hot wires, from uncertainty in the exponent of the fit t e d spectrum and from the probable s t a t i s t i c a l fluctuations of the observed estimates. As the normalized cross spectra converge to zero at high wave numbers and as they converge to 1.0 at low wave numbers the error in the estimates should approach zero. The maximum error in the normalized cross spectral estimates is expected in the range of wave numbers in which the r o l l - o f f is steepest. In this region a l l observed 34 normalized cospectral and coherence estimates should have s t a t i s t i c a l fluctuations of about ± 0.05. Also in the region of maximum slope the curves representing the theoretical and observed normalized cross spectra may be biased with respect to one another either up or down by ± 0.08 for the cross-stream cospectra and by ± 0.15 for the downstream coherences and cospectra. Chapter 7 DISCUSSION OF RESULTS The Comparison of Observed and Theoretical Cross Spectra Figures 6, 7 and 8 i l l u s t r a t e the comparisons of the observed cospectra with those computed theoretically from the observed spectra assuming isotropy and Taylor's hypothesis. Each point on these plots is normalized: the observed cospectral densities are normalized by the observed spectral densities and the theoretical cospectral densities by the f i t t e d spectrum. The abscissae on the plots.are values of l o g ^ O c ^ ) where 6 is the approximate magnitude of the vector separations for each run. Any power law behavior for the spectrum implies similarity of the normalized cospectra with respect to k j S for different runs i f 6 were the exact magnitude of the separations and i f the angles between the mean wind direction and the vector separations were the same. Because the exact vector separations are used in their computations, the normalized theoretical cross spectra as plotted are only roughly similar from run to run. The representations of theoretical and observed cospectra differ in one important respect; whereas each theoretical estimate of cospectral density is a computation at a discrete value of kj, the observed estimates represent an average over a band of wave numbers near k } (see Appendix B). Only for the f i r s t negative peak of the observed downstream cospectra (see Figure 8) where the curvature of the cospectral curve is relatively large should the effect of band averaging result in noticeable alterations to the observed cospectral estimates. The magnitudes of the observed estimates near this peak are expected to be reduced by no more than 10%. 36 Qualitatively, the theoretical and the observed cospectra are similar to one another for a l l separations:. Both sets of cospectra decrease from the levels of the spectra at low values of k^S and tend to zero at higher values of k^S. At intermediate values of k J6 the observed approximately downstream cospectra oscillate about zero as do their theoretical counterparts. Although normalized cross spectra w i l l converge to 1.0 or to 0.0 at low or high values of k a6 respectively, the shape and position of the cross spectral curve in i t s region of maximum slope is dependent on the structure of the turbulence. Using tolerances outlined in 'The Summary of Error Estimation' (see page 33) i t is evident that there is generally quantitative agreement between the normalized observed and theoretical cross-stream cospectra in this region of maximum slope (see Figures 6 and 7); good agree-ment is evident for the 20 cm., 10 cm. and 5 cm. separations; marginal agreement is evident for the 50 cm. separation and there is definite dis-agreement for the 180 cm. separation. The observed cospectra for 6 = 2 cm. agree well, up to log^O-c^) x 0.5; at higher wave numbers there was evidence of electronic noise due to the presence of the sonic anemometer on the array. The observed cospectral estimates for both cross-stream directions for 6 = 180 cm. seem to be significantly lower than the theoretical estimates for kj6 < 3 or equivalently for k a < 1.7 m-1. Furthermore because low wave number observed cospectral estimates for 6 = 50 cm. and.6 = 20 cm. tend to be somewhat low for both cross-stream directions up to values . of k 26 corresponding to k.1 ~ 2 m-1 (The vertical arrows on Figures 6 and 7 indicate k : = 2 m - 1) i t would seem that the transition from disagreement to agreement of these cospectra occurs at k ~ 2 TO""1, or for k az ~ 4. None of the quadspectra are plotted. The observed cross-stream quadspectra which are expected to be near zero theoretically were scattered 37 1.0-1 8= 180cm 8= 10cm C o l k J ) 0.5-9 • o 0.0 o l o u ¥ + + + t + T ^ " d t d - y ^ i 1.0-1 Co(k„a) *u(k.) 0.5-0.0 8= 50 cm . 9 o + o o + + 8= 5cm -I A-y 6-a-e~€| 1.0n Co(k,.8) cl>(k,) 0.5 H 0.0 -1.0 8- 20 cm — r- 1 -o i, a *-<H -0.5 0.0 LOG(k.S) 8- 2cm P + + 1 ; 1 U-yT0—O-°-' ; >1 1.0 -1.0 -0.5 0.0 1.0 LOGJk/S) • O B S E R V E D ° T H E O R E T I C A L Figure 6: The Normalized, Observed and Theoretical Cospectra Obtained from the Approximately Horizontal Crosswind Separation. (The vertical arrow on each plot indicates kj = 2 m-1) tch CPu(K) 0.5-S= 180cm 8 = 1 0 c m 0.0 1.0-1 C o (!<„$) 4>(i<() 0.5-0.0 8 = 5 0 cm o + o ~l T •y b 6 o 6 * 8 = 5cm n — j -1.0-1 Co(l<^) 0.5 H 0.0 8 = 20cm o + o + ^ , (n--1.0 -0.5 0.0 L0G(k,8) 4 ' U &-t 1.0 8 = 2cm -1.0 -0.5 0.0 LGG{k,8) °ro-£-Tr*i • OBSERVED o THEORETICAL Figure 7: The Normalized, Observed and Theoretical Cospectra Obtained from the Vertical Separation. (The vertical arrow on each plot indicates kj = 2 m~ ) 39 about the theoretical values with what appeared to be random fluctuations of magnitude less than 1" 0.1. Only the approximately downwind quadspectra had appreciable magnitudes, which is consistent with theoretical prediction. The spectrum, cospectrum and quadspectrum together define the coherence (see equation (2.7)). The cross-stream coherences are virtually identical with the cross-stream normalized cospectra which have already been discussed. The approximately downstream normalized cospectra and coherences are shown plotted in Figures 8 and 9 respectively. The maximum expected errors in the region of the drop-off of the observed cospectrum and coherence are uniform shifts of a l l estimates'of ± 0.15 and random fluctuations of estimates of average size ± 0.05. Up to the wave number at which the downstream cospectrum f i r s t crosses the zero axis a l l the observed cospectra agree with the theoretical predictions within experi-mental error. For this range of wave numbers the behavior of the down-stream cospectrum is determined largely by the relative phase of the correlated energy at the two sensors. The observed phase, a , , is r ' obs defined in (2.8): Qu(k r,6)' .CoOc^S), From (A.11) and A.12) the phase of the theoretical cross spectrum is seen to be simply k j O ' j for a given wave number, k x , and a given downwind component of separation, 6 . Because'"from (B.2) k } is inversely propor-tional to the turbulence advection velocity, i t is possible to define an effective advection velocity, U from the observed mean wind, U, J ef f ' from the theoretical phase, k j ^ , and from the observed phase, 0 ^ ^ : l^SjU U . = a , = tan obs - l to-CojkJl 0.5-\ 0.0 8 = 180cm T <=—r 8 = 1 0 c m "i r o ^ 6 ••tr & a i *u(k,) 1-0i Co(k„§) 0.5 H 0.0 ? 0 o + o 8= 50cm + * T a 8 = 5cm s + 0 & 1.0-Co(k„8) 0.5-0.0 8 = 20cm T : i -O-l -1.0 -0.5 0.0 0 L0G(k,S) 1.0 8 = 2cm -r-a r H h r ^ -1.0 -0.5 0.0 o 1.0 LQG(k,8f ? • OBSERVED o THEORETICAL Figure 8: The Normalized, Observed and Theoretical Cospectra Obtained from the Approximately Downwind Separation. . 1.0-C o h e r e n c e 0.5H 8= 180cm 8=10cm 0.0 + * _ j r 1.0-i C o h e r e n c e 0.5-9 ? ? ° o 0 + + o 8= 50 cm o + o 8= 5cm + o 0.0 T r + o • <? i p o 1.0-1 C o h e r e n c e 0.5-8-20cm o + o 0.0 + o + o + o + 6 -1.0 — i — -0.5 0.0 . 0.5 L0G(k,8) 1.0 8= 2cm + . 2 _ ° ° -1.0 <15 O0 0 5 ~ ~ t 0 L0G(k,8) * OBSERVED o THEORETICAL Figure 9 : The Observed and T h e o r e t i c a l Coherences Obtained from the Approximately Downwind Separation. (The v e r t i c a l arrow on each p l o t i n d i c a t e k ; = 10 m ! ) 42 The ratio, U f^/U, was computed for the downwind pair of sensors. The average value of this ratio was 1.22 for wave numbers lower than 1.6 m 1 computed for the 1.8 m separation although for the same range of wave numbers from the 50 cm separation this average was 1.06. The average value of the ratio for a l l but the largest separation was 1.03. In view of the averages of the mean wind speed measured by the sonic and by the cup anemometer being up to 10% different from one another the difference of the average ratio from 1.00 is probably insignificant. Individual values of the ratio were scattered by up to t 0.1 about the average but there was no significant trend in the values at increasing wave numbers. On the other hand, the magnitude of the correlated energy, is estimated by the coherence. The position of the maximum slope in the theoretical downstream coherence curves is determined primarily by the angle between the mean wind and the vector separation of the two velocity sensors; because this angle is smallest for 6 = 50 cm. the coherence for this separation appears to 'hold up' to higher wave numbers than do those for the other separations. In fact, the coherence for a purely downstream separation would be 1.0 at a l l values of kjS. In the region of maximum slope, the observed coherences from both the two largest downstream separations are too low to be explained by possible experimental error. These two coherences suggest that the theoretical and observed coherences do not agree for values of k2S corresponding to k^  less than about 10 m 1. (The vertical arrows on Figure 9 indicate = 10 m However, the observed coherence estimates for 6 = 20 cm., although tending to be low, do agree within experimental error with the theoretical results for k, somewhat less than 10 m-1. At smaller separations there appears to be quantitative agreement at a l l wave numbers except at the three highest 43 ones on the 6 = 2 cm. plot where an electronic pick-up is evident. One can say that the coherence determination is too insensitive to establish at exactly what wave number the observed coherence and theoretical coherence agree, but agreement near k a = 10 m"1 or k az = 20 seems the most l i k e l y . The Anisotropic Models Quantitative agreement occurs betx^een the.observed and theoretical cross spectra at approximately the same wave number for the two cross-^ stream separations but at a higher wave number for the downstream separations.- This behavior suggests that a model of the turbulence which is axisymmetric about the downstream direction might describe the turbu-lence at wave numbers less than the isotropic range. Two such anisotropic models were considered; the f i r s t model has the axisymmetric energy- density E(k) = EOO'cosfJ) whereas tile second also has an axisymmetric energy- density given by- E(k.)_ R E(k.i*C2 *- cos<3>), E(k). is a scalar function of k. and <J> is the angle between the wave numb'er k and the k axis so that the. f i r s t model has- a maximum energy- density- along the k a axis and the second a minimum. The normalized cospectra derived from these two models are plotted on Figure 10 together with the normalized cospectra from the isotropic model. The anisotropic model having the minimum of K(k) along the k2 axis qualitatively describes the observations for low wave numbers; both the observed cospectral estimates and those computed from this model have smaller magnitudes than the corresponding isotropic estimates. It thus seems that some anisotropic model having a d e f i c i t of energy in the k a direction can explain the observations. The equation of continuity requires that V*u=0 or-equivalently that k-u =0 where u, is the vector velocity arising from local integration about k of the energy density 4>(I0 tO-i Co(k„S) I0.5-0.0 1.0-Co(k,.g) 0.5-0.0 MAXIMUM ENERGY ALONG k, AXIS MINIMUM ENERGY ALONG KAXIS APPROX. DOWNWIND SEPARATION -[ 1— + + o o -o-l 6- • o + 0 + o 1—o o I 9 o o • • + APPROX. CROSSWIND o o HORIZONTAL SEPARATION T 'J |i? V O ° 1 o + *u(k.) 1.0-i Co (k J ) 0.5 H 0.0 CROSSWIND VERTICAL SEPARATION •+ o + -1.0 -0.5 — I V" i, y w - o - i 0.0 . 1.0 L0G(k,S) -1.0 -0.5 0.0 6 1.0 L0G(k,S) • ISOTROPIC, o ANISOTROPIC Figure 10: The Comparison of Normalized Cospectra Computed from Isotropic and from Two Anisotropic Models of the Turbulence 45 E(k). Thus, a model having a de f i c i t of energy in the direction would predict an increment of energy in the downstream component of velocity, u. Ideally one could try more sophisticated anisotropic models to duplicate the observed cross spectra. In this experiment the possible error in the observed cross spectrum is too big to warrant the investigation of a reasonably accurate model of the actual turbulence. Cti3.pt er 8 CONCLUSIONS The purpose of this study was to obtain information on the approach to isotropy and on the applicability of Taylor's hypothesis in •a high Reynold's number shear flow. The observed cospectra and coherences within the accuracy of measurement are consistent with the assumptions of isotropy and of Taylor's hypothesis for fejzr > 20. At wave numbers between \t z = 20 and k j Z = 4 the observed coherences for the downstream separation were lower than those predicted from the assumptions whereas the cospectra for both the vertical cross-stream and the horizontal cross-stream directions were within experimental error of the isotropic prediction. Furthermore the sonic 'v' and 'w' spectra are close to the expected i n e r t i a l subrange shape, ' k j 5 / 3 , } for k j Z > 5 and the 'u' spectra measured by the sonic and the hot wire anemometers have this shape for k j Z > 2. It thus appears that within the limits of accuracy of the experiment the turbulence i s at least axisymmetric about the downstream direction for kyz > 4. Coherences for a l l of the separation directions were lower than predicted on the isotropic assumption for k1z > 4. Failure of Taylor's hypothesis due to time evolution of the turbulence as i t passes the pairs of velocity sensors would be expected to lower the coherences. An anisotropic model having an excess of energy spectral density distributed axisymmetrically about the k j axis also gives lower coherences than the isotropic model for the three separation directions. Such a model i s 47 consistent with the behaviour of the observed cross-stream cospectra in which there i s an excess of energy in the 'u' component of velocity. Some anisotropy of this type and a partial failure of Taylor's hypothesis probably account for the observed nature of the cross spectra at low wave numbers. Taylor's hypothesis involves both the assumptions that the time evolution of the turbulence as i t is advected is negligible and that the turbulence i s transported at the mean wind speed. The vali d i t y of the relation: 2'iff i = "IT is directly dependent on the latter assumption. Hie turbulence advection velocity computed from the relative phase of the coherent energy between the downstream pair of sensors does not appear to be significantly different from the wind velocity obtained directly from the anemometers for k, > 1.6 m-1 or for k z > 3. For this range of wave numbers that l i a part of Taylor's hypothesis relating measured frequency to downstream wave number appears to be valid. • In principle the method of comparing theoretical and observed cross spectra could be used to determine the quantitative details of the structure of the turbulence at different wave numbers. From four velocity sensors six different separations are possible for which cospectra can be computed. At each wave number i t is possible to match the six observed cospectra and the spectrum with the cospectra and spectrum computed from a linear combination of seven different isotropic or anisotropic models of the turbulence. In order to determine the con-tribution of each model reasonably accurately the cospectra w i l l have to be determined with less uncertainty than in the present study; the hot 48 wire anemometers should be linearized, the wind angles and separations should be more precise, some account should be taken of the effect of 'V fluctuations on the hot wire response and the s t a t i s t i c a l fluctua-tions of the estimates should be reduced by using longer data sections for example. If one were to use more than four velocity sensors then more separations would be possible and hence the turbulence could be modelled more exactly. BIBLIOGRAPHY Bearman, P.W. (1970) Corrections for the Effect of Ambient Temperature Drift on Hot-wire Measurements in Incompressible Flow. Disa Information 1_1, pp. 25-30. Blackman, R.B. and J.W. Tukey (1958) The Measurement of Power Spectra. Dover Publications, Inc., New York; 190 pps. Grant, H.L., R.W. Stewart and A. Moillet (1962) Turbulence Spectra from a Tidal Channel. J. Fluid Mech. 12, pp. 241-268. Hinze, J.O. (1959) Turbulence. McGraw-Hill Book Co., New York; 586 pps. I.E.E.E. Transactions on Audio and Electroacoustics, Special issue on Fast Fourier Transform and i t s application to d i g i t a l f i l t e r i n g and spectral analysis. June 1967, Vol. AU15, No. 2. Izumi, Y. and M.L. Barad (1970) Wind Speeds as Measured by Cup and Sonic Anemometers and Influenced by Tower Structure. J. Applied Met. £, pp. 851-856. Jenkins, G.M. and D.G. Watts (1968) Spectral Analysis and Its Applications. Holden-Day, San Francisco; 525 pps. Kolmogoroff, A.N. (1941) The Local Structure of Turbulence in Incom-pressible Viscous Fluid For Very Large Reynold's Numbers, in Turbulence. Interscience Publishers, New York, pp. 151-155. Lin, CC. (1953) On Taylor's Hypothesis and the Acceleration Terms in the Navier-Stokes Equations. Quart. J. of Applied Math, 4_, pp. 295-306. Lumley, J.L. and H.A. Panofsky (1964) The Structtire of Atmospheric  Turbulence. Interscience Publishers, New York; 239 pps. Pond, S., R.W. Stewart and R.W. Burling (1963) Turbulence Spectra In the Wind Over Waves. J. Atmos. Sci., 20_, pp. 319-324. Taylor, G.I. (1938) The Spectrum of Turbulence, in Turbulence. Interscience Publishers, New York, pp. 100-114. Van Atta, G.W. and W.Y. Chen (1970) Structure Functions of Turbulence In the Atmospheric Boundary Layer Over the Ocean. J. Fluid Mech., 44, pp. 145-159. Weiler, U.S. and R.W. Burling (1967). Direct Measurement of Stress and Spectra of Turbulence In the Boundary Layer Over the Sea. J. Atmos. Sci.,'24, pp. 653-664. APPENDIX A THE DERIVATION OF THEORETICAL CROSS SPECTRA Cn) The cross spectra Cr„ (k,6) are defined by: Crf n )(k , 6 ) 1 (2ir) 3 u [n ) ( r , t ) u f n ) (r+6+x,t) r,t -ik*x, (A.l) e ~ ~dx We define the theoretical cross spectra Cr^CkjS) as estimates of the cross spectra, Crf ^ ( k , 6 ) , smoothed according to (2.3) that would be obtained i f the turbulence was stationary, isotropic and incompressible: E,t (n) , (n) , l X l -ik«x u' ' ( r , t ) u v '(r+6+sc,t) e ~ - d x . .1, ~ I ~ -v 'v ~ ' 11 J_ , ( 2 7 T ) 3 ( u ! n ) (r,t)u< n ) (r + 6+x,t ) y ' e ' ^ d x _o^. 1 ~ 3 ~ ~ ~ . / n = l ~ (A.2) Likewise using (2.1) we can define an average of the product of velocity components in terms of the smoothed spectral tensor, 3^ (k,<5) . — — r,t (-oo O e ~ ~dk (A. 3) substituting in (A.3) using (A.2) we have: C r i j ( ^ ' ^ ) = C2TFP ik'-(6+x) AT (k')dk' -ik'x' e ~ ~dx ( A . 4 ) Assuming the functions are a l l well behaved one can switch the integration order in (A.4): 51 But: c 0 0 e ^ T ^ ' i ^ d x = (2TT)36(k'-k) (A.6) where 6(k'-k) is the Khrb'necker delta function Hence: Cr^(k,6) = S(k'-k)$T.(k,)ei~'*~dk' ~ ~ 13. ik«6,T . e ~ ~$ (k) (A .7 ) The cross spectra which were observed were the one dimensional cross spectra of downstream velocity fluctuations, Cr^Ck ,<5), in which the total contribution from a l l wave numbers is expressed in terms of the component, k , of wave number. The theoretical equivalent to this, T Cr l x(k ,6), can be computed from the general three dimensional cross T spectrum, Cr^CkjS), by integrating i t over a l l values of k 2 and k 3 > Hence: Cr^Ck, ,6) = GrJjCk.^dkgdk, ,00 J —00 J —00 (A.8) For isotropic incompressible turbulence the tensor, $..(k,6), can be T expressed in terms of a single scalar function of k; E (k): (Hinze, 1959) 3>T.(k) = f - £ ^ (k2<5..-k.k.) (A.9) The integral, (A.8), then becomes: C r i i ^ k i ' ^ 4TT1C2 1-- e ~ ~ dk 2dk 3 (A.10) The theoretical cospectrum, GoCk^S), and the theoretical quadspectrum, 'p T Qu(k 1 56), are the real and imaginary parts respectively of Cr 1 1 ( k 1 , 6 ) . Noting that parts of the integral, (A.10), involving sin(k262+k3<53) are antisymmetric about (k 2,k 3) = (0,0) and that the cosine function i s symmetric about this point, the integration can be done over positive values of k 2 and k 3: Co(k ,'6) = cosk 6, Qu(k 1 56) = sink 1S i •E -(k) 1 2 I k J ) 0 ' TTk2 0 f°° T E X(k) ' 0 ' 1Tk2 0 i. k 2j cos(k 26 2+k 36 3)dk 2dk 3 (A.11) cos(k 26 2+k 36 3)dk 2dk 3 (A.12) If the separation, 6, is zero then the one dimensional cospectrum i s T iust the one dimensional spectrum $ (k ) r U 1 ' E'(k) TTk2 I k 2 J dkgdkj (A.13) T T From (A.13) we can obtain E (k) in terms of $ (k x) T ' E'(k) k _ i _ 2 dk ' l dg(lO k dk • (A. 14) T T In this study the $ (kj) used to compute E (k) are analytic functions fi t t e d to the observed spectral estimates, $ u,(kj). APPENDIX B THE OBSERVED CROSS SPECTRA The Theoretical Basis In this experiment velocity fluctuations were recorded as a function of time at four stationary sensors. The analysis of the data was carried out on velocities sampled at discrete time intervals, At, for a total record duration of NAt, where N is the number of samples in the record. The cross spectra computed from the observed values of uf n^(r,t) and ufn^(r+6,t+T) sampling the n t n realization of the flow can 1 2 3 N only be obtained at the discrete frequencies; f = 0 , —, —, —, : Cr! .(f , 6) = ^ I J ~ M •}j[/2 M uf n )(r,t)uf n )(r+6,t+T) e " 1 2 ^ (B.l) -M/2 1 ~ . . 1 ' ~ ~ The velocity product average can only be taken over the time duration of the record, M. Provided the turbulence is stationary in space and time Cr' . (f,6) is not a function of r. Under Taylor's hypothesis (Taylor, 1938) the turbulence is assumed to be nearly 'frozen' in time as i t is swept past a stationary sensor at a constant rate so the temporal velocity variations seen at the sensor are due only to spatial variations in the velocity along the line of the mean velocity. A cross spectral component having frequency, f, is then equivalent to the one dimensional cross spectral component having downstream wave number, k x, where: (B.2) 54 Thus: Cr!.(k.,6) = I . . . M/2 — y - r M -iV TTT u C n )(r,t)uf n )(r+6,t+T) e l k l U T d T (B.3) In practice the estimates of Cr^ j, (kls6*) were not evaluated according to (B.3). Rather, a Fast Fourier Transform (I.E.E.E. Transactions, Vol. 15, 1967) was used to evaluate the complex Fourier coefficients, A'+iB', for u ^ n ) ( r , t ) and u? n )(r+6,t+T): A ;!(k 1,r)+iB^(k 1,r)= | •M/2 u ! u / ( r , t ) e 1 dt (B.4) -M/2 1 ~ Aj(k1,r+6)+iB^(k1,r+6)= ± •M/2 u! ;(r+6,t)e 1 dt (B.5) -M/2 2 The estimated cross spectra, Cr^. > a r e obtained from combinations of the coefficients in (B.4) and (B.5). Cr!.(k, ,6) = AlAl+BlBl+KA'.Bl+AlB!) (B.6) These estimates are exactly equivalent to those that would be computed by (B.3). Just as (B.l) can be evaluated at discrete frequencies only so (B.3), (B.4), (B.5) and (B.6) are defined at the discrete wave 2TT 4TT TTN numbers; k, = 0, — , — , . . . , — . 1 ' Mir MU MU Because the Cr!.(k,,6) are based on f i n i t e data records of duration, M, they are modified from the cross spectra, Cr^"^ (kj ,6") , computed from i n f i n i t e l y long records. The ensemble average of Gr^(k l 56) is related to the ensemble average of C r f ^ ( k l S 6 ) by: 55 /cr!.(k,,S)> . = 4^n)(K>&h i \ i j 1 ' ~ / n = l , c o \ l j 1 ~ / n = l , c .Cn) , («> Cr^'Ck' 6)2U 2 R M 1 \ <j ^ J T V ^ M - [ ^ ( ( k ^ k p - j j d k ^ ^ (B.7) Although each of the s p e c t r a l e s t i m a t o r s , < ^ r ^ ( k l 9 6 ^ n=]_ oo' ^ s m a d e U P mostly of energy from k j near to k a, the estimator has s m a l l c o n t r i b u t i o n s from a l l v?ave numbers of <Cr f 1^ (k, ,6)) .. . In the observed cross \ i j 1 ~y n=l,°° spe c t r a i t was found that there was a s u f f i c i e n t preponderance of energy at low wave numbers so that according to (B.7), the \ C r ! . ( k ,6)/ , „ & \ ±2 1 ~" n=l,°° would be s i g n i f i c a n t l y modified at h i g h wave numbers. Provided t h a t most of the c o n t r i b u t i o n to the second term on the r i g h t i n (B.7) comes from k j l e s s than k^ and provided that one only considers c o r r e c t i o n s to estimates f o r which k^ . » k^ then t h i s term can be s i m p l i f i e d : « C r f n ) ( k ' 6)2U x3 (k ~k') 2M l - c o s ^ - k ^ f ] dk n=l,°° dk, (B.8) The i n t e g r a l on the r i g h t i n (B.8) can be approximated as a sum over the observed low wave number cross s p e c t r a l estimates, Cr!.(k,,6): ; t<< )*i,5^(i-co.[ck 1-k{^ dk. ^ ^ ^ . ^ ^ - ( ^ ( ( k . - k ; ) ! Ale! (B.9) 56 where Ak! i s the wave number bandwidth over which the estimate Cr!.(k' 6) i s averaged. The quantity under the summation i n (B.9) i s the constant, P_^ ,^ f o r the given run. Thus (B.7) becomes: ^ ( k , . * ) ) n = 1 >„ = (org'*,.«)) „.!,-+ # (B.IO) In p r a c t i c e only the f i n i t e number of data records, L, were analyzed so that the observed cross spectra, Cr?.(k 1,5), which were corrected P. . 1 J i i for the s p e c t r a l leakage term, — - r , were given by: P. . Cr° ( k , S ) = ( c r ' (k , S ) \ - ( B . l l ) i j 1 ~ \ 13 1 ~ / n=l,L k2 1 Because they are based on an average over a f i n i t e number of data records the cross s p e c t r a l estimators, Cr?^.(k 1 }6), w i l l show s t a t i s t i c a l f l u c t u a t i o n s about the smooth cross spectra, (Crf 1?^ (k, ,6) ) ^ r \ i j 1 ~ / n=l, 0 0 The variance of Cr°^(k l 56) i s reduced further i n t h i s study by averaging the estimates over bandwidth as w e l l as over data records. The Computation of Observed Cross Spectra ' The analysis routine which produced the various cross spectra from the raw. data tape i s outlined i n Figure 11. Most of the analysis was done on the University's I.B.M. 360 computer using computer programs developed by students at the I n s t i t u t e of Oceanography. Each step of the analysis procedure i s elaborated i n the following. D i g i t i s a t i o n The process of rewriting the analogue data tape i n d i g i t a l form i s c a l l e d d i g i t i z a t i o n . The d i g i t a l tape produced i s obtained by sampling the analogue voltage at a constant d i g i t i z a t i o n frequency. analog filtered TAPE data A - D FILTER ' data ANALOG TO RECORDER DIGITAL CONVERTER digital tape sine and cosine Fourier coefficients fitting parameters Figure 11: The Data Analysis Scheme. 58 The highest frequency at which energy can appear in the spectral analysis of such a d i g i t a l tape is the Nyquist frequency, di , equal to h the digitization frequency. A l l energy on the analogue tape at frequencies higher than the Nyquist frequency is 'aliased' to some frequency between 0 and in the data on the d i g i t a l tape. (Blackman and Tukey, 1958). To minimize aliasing into the frequencies of interest a low pass f i l t e r is used to f i l t e r out energy above the Nyquist frequency prior to digitization. The f i l t e r s used were a matched set which had a 3db. point at the Nyquist frequency and a 9db. per octave attenuation rate above that. Over the frequency range of interest the phase shifts introduced by the f i l t e r s on each channel were a l l within 1° of one another. The cross spectral phases which depend only on the relative phase of one channel to another are thus only negligibly affected by the use of the f i l t e r s . The attenuation of the spectral estimates below the Nyquist frequency is corrected for in SIMPL0T. The upper frequency of analysis, the Nyquist frequency, is set by the choice of digitization frequency. At each separation the d i g i t i -zation frequency was picked so that the corresponding Nyquist wave number represented scale sizes about h of the size of the separation. By this means only those turbulent scales of the same order of size as the separation were retained for analysis. FT0R FT0R is a computer program designed to evaluate (B.4) and (B.5) from the digitized data. It uses the Fast Fourier Transform to generate complex Fourier coefficients from every consecutive record, each of 1024 sample points from each channel of the d i g i t a l tape. The number 59 of records chosen for analysis was rather a r b i t r a r i l y set at 40 -for a l l of the runs. For each of the 40 records for each channel FT0R computes 512 complex coefficients. SC0R SC0R uses the complex coefficients outputted by FT0R and produces from these smoothed spectral, cospectral and quadspectral estimates. By combining the complex coefficients from one or two channels according to (B.6) i t computes the desired cross spectra for a l l the wave numbers and for a l l the 40 records. It f i r s t averages the estimates over a quarter to a third octave bandwidths for each record. These averages are then them-selves averaged over the 40 different records. The result of this procedure is a cross spectrum which is an estimate of the ensemble average from different realizations of the flow. SC0R has an additional f a c i l i t y for computing cross spectra from the averages of the records. By this means i t is possible to extend the analysis down to lower frequencies but because SC0R does l i t t l e smoothing of these estimates any individual estimate must be considered . fairly.unreliable. In Figures 2 and 3 the estimates for the nine lowest wave numbers for each run are computed by this means. R0TATE The 3D sonic anemometer senses two horizontal components of velocity at 120° to one another. R0TATE transforms the spectra derived from these two components into the spectra of downwind velocity fluc-tuations and into the spectra of cross-stream fluctuations. 60 SIMPL0T SIMPL0T is used primarily to correct the cross spectral estimates for the attenuation introduced by the A-D f i l t e r . Each of the SC0R spectral estimates i s simply multiplied by the inverse of the attenuation factor of the f i l t e r at that frequency. In SIMPL0T too the hot wire anemometers are calibrated. The output of R0TATE gives the integrals under the spectra of the downwind velocity fluctuations measured by the sonic as well as the integrals under the spectra of the uncalibrated hot wire anemometers. The calibration used for a given hot wire was that necessary to make i t s integral over the lower wave numbers where the sonic is expected to give reliable results equal to the sonic integral over the same wave numbers. Because the sonic anemometer and the hot wire anemometers have slightly different shapes to their spectra due to their different operating characteristics and to their sensing the turbulence at d i f -ferent positions in space this calibration w i l l be only roughly accurate. However, except" for Figures 3 and 4 a l l of the hot wire cross spectra are plotted in a normalized form which eliminates the need for any hot wire calibration at a l l . Leakage Correction A l l of the observed cross spectra are corrected using (B.ll) for the leakage of energy from one wave number to another resulting from the analysis of f i n i t e data records. Because the majority of the energy in typical spectra and cospectra occurred in the lower wave numbers of analysis the correction term for them was approximated from their low wave number estimates using (B.9). The magnitude of the 61 correction term which was about the same for both spectra and cospectra ranged between 15% of the spectral level at kjS - 1.0 for the 2 cm. separation down to 1% at k^S =1.0 for the 1.8 m. separation. The magnitudes of the corrections to the quadspectra, which had l i t t l e energy at low wave numbers, were very small. PL0TTING Two different methods were used for plotting the results. The hot wire spectra shown in Figures 3 and 4 and the sonic spectra shown in Figure 2 are plots of the form log^Q (kj^^Ck )) versus log^Ocjz) where z i s 2 m. k , $ ( k , ) is the energy in a wave number band of width f!fi_ l u l k i whereas k j Z is a non dimensional wave number. The theoretical and the observed cospectra plotted in Figures 6, 7, and 8 are a l l normalized by the appropriate spectrum. The abscissa axis in each case i s log ^ C ^ S ) . The cospectra from the different separations should be approximately similar with respect to k,6. 


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