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UBC Theses and Dissertations

Some observed statistical properties of small scale turbulence Wilson, John Ronald 1974

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c I SOME OBSERVED STATISTICAL PROPERTIES OF SMALL SCALE TURBULENCE by JOHN RONALD WILSON M.Sc, Queen's University, 1 9 6 ^ A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in the Department of Physics We accept t h i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA January, 1 9 7 ^ In p r e s e n t i n g t h i s t h e s i s I n p a r t i a l f u l f i l m e n t o f the r e q u i r e m e n t s f o r an advanced degree a t t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a , I agree t h a t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y . I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may be g r a n t e d by the Head o f my Department o r by h i s r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l n o t be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . Department o f Physics The U n i v e r s i t y o f B r i t i s h Columbia Vancouver 8, Canada Date January 10. 197^-. ABSTRACT The purpose of t h i s work was to i n v e s t i g a t e the c h a r a c t e r i s t i c s of the small scales of the t u r b u l e n t v e l o c i t y and temperature f i e l d s i n the atmospheric boundary l a y e r . Of p a r t i c u l a r i n t e r e s t were the d i s t r i b u t i o n s of the s p a t i a l d e r i v a t i v e s , the behaviour of the s t r u c t u r e f u n c t i o n s and the nature of the i n t e r a c t i o n s l e a d i n g to the i n e r t i a l t r a n s f e r of energy between the l a r g e r and smaller s c a l e s of the t u r b u l e n t flow. The observed s t a t i s t i c a l d i s t r i b u t i o n s of the d i f f e r e n t i a t e d downstream and X-wire s i g n a l s d i d not a c c u r a t e l y conform to the t h e o r e t i c a l l y p r e d i c t e d l o g a r i t h m i c normal model although there was good general agreement. The d i s t r i b u t i o n of the sum of the squares of the d i f f e r e n t i a t e d X-wire s i g n a l s was described extremely w e l l by the l o g normal model. The temperature d e r i v a t i v e s followed the l o g normal d i s t r i b u t i o n somewhat b e t t e r than d i d the v e l o c i t y d e r i v a t i v e s , but a poor s i g n a l to noise r a t i o prevented co n c l u s i v e r e s u l t s . The observed s t r u c t u r e f u n c t i o n s conformed to present theory f o r the v e l o c i t y and temperature data. The i n e r t i a l t r a n s f e r of energy i n the downstream component was found to behave as would be i n t u i t i v e l y expected. S p e c i f i c a l l y the net r a t e of a r r i v a l of energy was zero f o r wavenumbers where v i s c o s i t y was unimportant and p o s i t i v e where v i s c o s i t y was important. I n t e r a c t i o n s of s c a l e s i z e s d i f f e r i n g by more than a f a c t o r of ten i n s i z e d i d not c o n t r i b u t e s i g n i f i c a n t l y to the energy t r a n s f e r s . The flow of downstream energy was observed to be from l a r g e r to smaller s c a l e s i z e s i i i f o r h o r i z o n t a l s c a l e s as l a r g e as twenty meters. The mean i n e r t i a l t r a n s f e r of energy a s s o c i a t e d w i t h i n t e r a c t i o n s between downstream components o n l y , was found t o be a p p r o x i m a t e l y e q u a l to the mean r a t e of energy l o s s due to the c o r r e s p o n d i n g d i s s i p a t i o n term. i v TABLE OF CONTENTS page ABSTRACT i i T-ABLE OF CONTENTS i v LIST OF TABLES v i LIST OF FIGURES v i i ACKNOWLEDGEMENTS x i i CHAPTER I i INTRODUCTION 1 Theory of Gurvich and Yaglom 3 Theory of the structure functions 9 The i n e r t i a l transfer of energy 11 CHAPTER I I j DATA COLLECTION 16 Velocity data 16 Temperature data 18 Analog to d i g i t a l conversion 19 CHAPTER I I I : RESULTS AND DISCUSSIONS 22 General remarks 22 Computation of the d i s t r i b u t i o n s 22 E f f e c t of measurement noise on the 25 d i s t r i b u t i o n s A f i l t e r i n g technique for the removal 29 of noise Dis t r i b u t i o n s of the downstream 35 ve l o c i t y data Distributions of the X-wire data 50 Dis t r i b u t i o n s of the temperature data 64 Spectrum of the squares of the log 72 normal quantities Structure functions, skewness and 80 flatness factor The I n e r t i a l t r a n s f e r of energy CHAPTER IV: SUfoKAKY OF CONCLUSIONS LIST CF REFERENCES APPENDIX A: PEN TINE NT CHARACTERISTICS OF RECORDING AND DIGITIZING CIRCUIT: L i n e a r phase s h i f t Thompson f i l t e r Phase s h i f t o f the d i f f e r e n t i a t i n g c i r c u i t Phase s h i f t of the K r o n h i t e f i l t e r APPENDIX B: COMPUTATION OF THE SPSOTHUfc v i LIST OF TABLES Tab l e page I Observed and t h e o r e t i c a l moments f o r v e l o c i t y d a t a 46 I I Observed and. t h e o r e t i c a l moments f o r squared and summed X-wire d a t a 59 I I I C o e f f i c i e n t of excess as a f u n c t i o n o f f i l t e r c u t - o f f f r e q u e n c y f o r d i f f e r e n t i -a t e d S3 temperature s i g n a l 68 IV V a l u e s o f skewness and lv' from the s t r u c t u r e f u n c t i o n s of the downstream v e l o c i t y d a t a 85 7 Skewness .and c o e f f i c i e n t o f exc e s s f o r d i f f e r e n t i a t e d downstream v e l o c i t y d a t a VI Skewness and c o e f f i c i e n t o f e x c e s s f o r 50 second s e c t i o n s of d i f f e r e n t i a t e d v e l o c i t y d a t a 88 V I I S l o p e and s t a n d a r d d e v i a t i o n i n the s l o p e f o r the 2nd and 4 t h o r d e r s t r u c t u r e f u n c t i o n s o f the v e l o c i t y data. 89 V l l LIST OF FIGURES Figure page 1 Block diagram of elect r o n i c s f o r recording the v e l o c i t y data 17 2 Block diagram of electronics for recording the temperature data 19 3 Block diagram of d i g i t i z i n g electronics 20 4 Theoretical log normal d i s t r i b u t i o n 26 5 Theoretical combined normal and log normal d i s t r i b u t i o n s 26 6 D i s t r i b u t i o n of the log of the sum of a normal and a log normal quantity 28 7 Cumulative d i s t r i b u t i o n function of the sum of a normal and a log normal quantity 28 8 Spectrum of the d i f f e r e n t i a t e d temperature, section 2, tape 154 30 9 Mean fo u r i e r amplitude of the d i f f e r e n t i -ated temperature, section 2, tape 154 30 10 D i s t r i b u t i o n of the log of the d i f f e r e n t i -ated temperature, section 2, tape 154 33 11 Cumulative d i s t r i b u t i o n function of the d i f f e r e n t i a t e d temperature, section 2, tape 154 33 12 D i s t r i b u t i o n of the log of the f i l t e r e d d i f f e r e n t i a t e d temperature, section 2, tape 154 3^  13 Cumulative d i s t r i b u t i o n of the f i l t e r e d d i f f e r e n t i a t e d temperature, section 2, tape 154 3^  14 Spectrum of the d i f f e r e n t i a t e d v e l o c i t y , run 2B, tape l4l 36 15 D i s t r i b u t i o n of the log of the d i f f e r e n t i -ated v e l o c i t y , run 2B, tape 141 38 16 Cumulative d i s t r i b u t i o n function for the di f f e r e n t i a t e d v e l o c i t y , run 2B, tape l4l 38 Figure page 17 D i s t r i b u t i o n of the log of the product of 10 random numbers 4l 18 Cumulative d i s t r i b u t i o n function of the product of 10 random numbers 4l 19 D i s t r i b u t i o n of the product of 50 random numbers 42 20 Cumulative d i s t r i b u t i o n function of the product of 50 random numbers 42 21 Contribution to the moments of a l o g normal quantity by the 0 to X portion of the d i s t r i b u t i o n 45 22 D i s t r i b u t i o n of the log of the d i f f e r e n t i -ated v e l o c i t y , run 2, tape 96 48 23 Cumulative d i s t r i b u t i o n function of the di f f e r e n t i a t e d v e l o c i t y , run 2, tape 96 48 24 Spectrum of the d i f f e r e n t i a t e d XI hot wire signal 54 25 Mean fo u r i e r amplitude of the d i f f e r e n t i -ated XI hot wire signal 54 26 D i s t r i b u t i o n of the sum of the squared X-wire signals 56 27 Cumulative d i s t r i b u t i o n of the sum of the squared X-wire signals 56 28 Cumulative d i s t r i b u t i o n of the sum of two normal and two log normal series of random numbers 58 29 D i s t r i b u t i o n of the log of the d i f f e r e n t i -ated XI hot wire 6l 30 Cumulative d i s t r i b u t i o n of the log of the d i f f e r e n t i a t e d XI hot wire 6l 31 D i s t r i b u t i o n of the log of the d i f f e r e n t i -ated X2 hot wire 63 32 Cumulative d i s t r i b u t i o n of the d i f f e r e n t i -ated X2 hot wire 63 33 Spectrum of the d i f f e r e n t i a t e d E3 temperature, tape RCA1 65 34 Mean fo u r i e r amplitude of the d i f f e r e n t i -ated E3 temperature, tape RCA1 65 Cumulative d i s t r i b u t i o n of the d i f f e r e n t i -ated E3 temperature D i s t r i b u t i o n of the f i l t e r e d d i f f e r e n t i a t e d E l temperature Cumulative d i s t r i b u t i o n of the f i l t e r e d d i f f e r e n t i a t e d E l temperature D i s t r i b u t i o n of the f i l t e r e d d i f f e r e n t i a t e d E3 temperature Cumulative d i s t r i b u t i o n of the f i l t e r e d d i f f e r e n t i a t e d E3 temperature Spectrum of the square of the d i f f e r e n t i -ated v e l o c i t y , run 2B, tape Ikl Spectrum of the square of the d i f f e r e n t i -ated v e l o c i t y , run 2, tape 96 Spectrum of the square of the d i f f e r e n t i -ated XI hot wire Spectrum of the square of the d i f f e r e n t i -ated X2 hot wire Spectrum of the sum of the squared X-wire signals Spectrum of squared d i f f e r e n t i a t e d v e l o c i t y signal d i g i t i z e d at 2000 Hz Spectrum of the squared d i f f e r e n t i a t e d E l temperature Spectrum of the squared d i f f e r e n t i a t e d E3 temperature Second order structure function of the v e l o c i t y , run 2B, tape 141 Second order structure function of the v e l o c i t y , run 1, tape 128 Third order structure function of the v e l o c i t y , run 2B, tape l 4 l Third order structure function of the v e l o c i t y , run 1, tape 128 Skewness factor for run 2B, tape l 4 l 1 igure 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 Skewness f a c t o r f o r run 1, taoe 128 Fourth order s t r u c t u r e f u n c t i o n , run 2B, tape 141 Fourth order s t r u c t u r e f u n c t i o n , run 1, tape 128 Fl a t n e s s f a c t o r , run 213, tape 141 Fl a t n e s s f a c t o r , run 1, tape 128 Second order s t r u c t u r e f u n c t i o n , £1 temperature T h i r d order s t r u c t u r e f u n c t i o n , E l temperature Fourth order s t r u c t u r e f u n c t i o n , E l temperature Skewness f a c t o r , E l temperature F l a t n e s s f a c t o r , E l temperature G r a p h i c a l r e p r e s e n t a t i o n f o r an i n e r t i a l i n t e r a c t i o n of type 1 Gr a p h i c a l r e p r e s e n t a t i o n f o r an i n e r t i a l i n t e r a c t i o n of type 2 G r a p h i c a l r e p r e s e n t a t i o n f o r an i n e r t i a l i n t e r a c t i o n of type 3 Values of energy t r a n s f e r due to i n e r t i a l i n t e r a c t i o n s of type 2 f o r the 26 X 26 g r i d Rate of a r r i v a l of energy at frequency f, run 2B, tape l 4 l Confidence i n t e r v a l s f o r the energy flows, run 2B, tape l 4 l Rate of flow of downstream energy from below frequency f to above C o n t r i b u t i o n s to the rate of a r r i v a l of energy f o r i n t e r a c t i o n s of type 1, run 2B, tape l 4 l page 86 90 90 92 92 93 93 96 97 97 99 100 100 103 106 107 109 111 XI Figure page 71 Contributions to the rate of a r r i v a l of energy for interactions of type 1, run 2, tape 96 114 72 Contributions to the rate of a r r i v a l of energy for interactions of type 2, run 2B, tape 141 115 73 Contributions to the rate of a r r i v a l of energy for interactions of type 2, run 2, tape 96 117 74 Contributions to the rate of a r r i v a l of energy for interactions of type 3» run 2B, tape l4l 118 75 Comparison of the rate of a r r i v a l of energy for type 2 and type 3 interactions 121 76 Rates of a r r i v a l of energy i n a wave-number as a function of time 122 77 Response of the d i g i t a l f i l t e r 124 78 Rate of a r r i v a l of energy at frequency f, low pass f i l t e r e d , run 2B, tape l4l 124 79 Rate of flow of downstream energy from below f to above for the low pass f i l t e r e d data 125 80 Ratio of the energy flow across frequency f to the d i s s i p a t i o n at frequencies greater than f. 125 81 Response of Thompson f i l t e r 137 82 Phase s h i f t of Thompson f i l t e r 137 83 Non-ideal phase s h i f t of d i f f e r e n t i a t o r 138 84 Phase s h i f t of Kronhite f i l t e r 138 x i i ACKNOWLEDGEMENTS This work was done i n conjunction with the Air-Sea Interaction program at the Institute of Oceanography of the University of B r i t i s h Columbia. The d i g i t a l computations were done at the U.B.C. computing center which was pa r t l y supported by the National Research Council of Canada. I was on educational leave from the Federal Department of the Environment. I would l i k e to thank Dr. Burling and Dr. Stewart for t h e i r guidance and advice during the course of t h i s work. I would also l i k e to thank Mr. F.E. Jerome, Dr. N. Boston and Dr. J.A. E l l i o t t who allowed me to use some of t h e i r data which had been col l e c t e d for other purposes. I would p a r t i c u l a r l y l i k e to thank Dr. E l l i o t t who helped me to get data d i g i t i z e d when I was i n Ottawa and the data was i n Vancouver. My wife Sandy i s es p e c i a l l y deserving of thanks for her understanding during the preparation of the thesis when I es s e n t i a l l y had two jobs. 1 Chapter I 1.1 Introduction The f a m i l i a r picture of the "cascade" process i n a turbulent f i e l d was f i r s t postulated by Richardson (1922). According to the picture the f i e l d i s comprised of a large number of eddies of various sizes. The energy of the mean motion enters the eddies of the largest scale sizes and by turbulent processes i s transferred to smaller and smaller eddies u n t i l i t f i n a l l y i s dissipated as heat under the influence of v i s c o s i t y at the smallest scale s i z e s . Kolmogoroff (194la) and Oboukhov (1941) were able to develop a quantitative theory based on t h i s q u a l i t a t i v e one. The new theory met with considerable success i n that i t led to the observed "2/3 law" for the second order structure functions and the "-5/3 law" for the power spectrum of the turbulent fluctuations. In developing the theory the assumptions of s e l f s i m i l a r i t y and l o c a l isotropy of the process were made and the idea of a range of scales where the turbulence i s not influenced by the large scale motions or v i s c o s i t y was introduced. In addition i t was assumed that the s t a t i s t i c a l state of the small scale fluctuations depended only on the amount of energy being transferred down the cascade and was also not influenced by the large scale motions. This mean rate of transfer of energy down the cascade was equal to that dissipated and i s defined as V a ^ i e>xc) 2 I t was immediately pointed out by Landau that the turbulent c h a r a c t e r i s t i c s associated with a small volume of space would be determined by the l o c a l value of £ i n the volume and not by the average over a much larger volume. The parameter when averaged over a small volume of space was known to be a random variable e x h i b i t i n g considerable v a r i a b i l i t y i n space and time. Any attempt to form averages of turbulent quantities would be influenced by t h i s v a r i a b i l i t y and the re s u l t would depend on the s t a t i s t i c a l d i s t r i b u t i o n of £ . Consequently Kolmogoroff (1962) and Oboukhov (1962) modified t h e i r theories to take t h i s factor into account. The modifications involved the assumption that the logarithm of £ was a normally d i s t r i b u t e d quantity although no t h e o r e t i c a l evidence of t h i s assumed behaviour was presented at the time. Novikov and Stewart (1964) considered a model of turbulence based purely on the experimentally observed nature of the d i s s i p a t i o n rate. By subdividing space into a series of smaller and smaller volumes and assuming that f o r each stage of subdivision random volumes had either zero d i s s i p a t i o n or some constant d i s s i p a t i o n they were able to predict that the spectrum of the fluctuations of £ would exhibit a power law behaviour with the parameter having a value between 0 and -1. Some measure-ments of quantities which should have behaved l i k e £ were made by Pond (1965) and seemed to bear t h i s out. The l a t e s t contribution to the theory of the s p a t i a l and time v a r i a t i o n of € has been offered by Gurvich and Yaglom (1967). Their paper presents a th e o r e t i c a l treatment leading to 3 the prediction that cert a i n quantities, including the d i s s i -pation rate, should have the logarithmic-normal p r o b a b i l i t y d i s t r i b u t i o n and that the spectrum of the square of these quantities should exhibit a power law behaviour. The bulk of the present work was devoted to an examination of some experimental v e l o c i t y and temperature data i n terms of the log normal model. Some r e s u l t s concerning the structure functions of these data are also presented. The f i n a l portion of the work was devoted to a bispectrum analysis on some turbulent v e l o c i t y data which to the knowledge of the author has not been done before. This permitted computation of the i n e r t i a l transfer of energy due to interactions between downstream wavenumbers. The re s u l t s are discussed i n terms of the present theories of homogeneous, is o t r o p i c turbulence. 1.2 The Theory of Gurvich and Yaglom (for homogeneous turbulence) The theory of Gurvich and Yaglom (196?) i s based on a subdivision of space into smaller and smaller volumes, each located inside the other. The subdivision begins by considering a volume associated with a scale size L of the order of the inhomogeneities i n the mean flow. Inside i s a second volume V 2 and inside V 2 i s a t h i r d volume Vj, etc. Each subdivision i s i d e n t i f i e d with a scale such that the r a t i o of the scales for a volume and the immediately larger volume containing i t i s constant. The subdivision i s continued u n t i l a small enough volume i s reached that the turbulent c h a r a c t e r i s t i c under consideration shows ne g l i g i b l e v a r i a t i o n within the volume. 4 The Kolmogoroff microscale, T^' , discussed below i s defined as • _ ^ ^ £ , ^ where V i s the kinematic v i s c o s i t y and the primes denote l o c a l conditions within the volume. In t u i t i v e notions on the s e l f - s i m i l a r i t y and the random chaotic nature of the turbulent process enabled Gurvich and Yaglom to draw useful conclusions from t h i s picture. I f the length associated with a volume Vj i s of the order of the scale si z e , L, of the macroscopic mean flow, then averaging a non-negative turbulent c h a r a c t e r i s t i c , , over the volume to produce a (p^ , would y i e l d values very nearly equal to the mathematical expectation of the f i e l d . I f however the length scale J l was small with respect to L and large with respect to i\' , the Kolmogoroff microscale for the volume i n question, then 0 i would show considerable f l u c t u a t i o n from volume to volume. The d i s t r i b u t i o n of <Pj would characterize turbulent eddies of sizes & . Furthermore the parameter eC-j = would be a s t a t i s t i c a l quantity, the d i s t r i b u t i o n of which depended only on the r a t i o of the length scales providing these lengths were within the previously accepted l i m i t s . Also i f the Reynolds number were s u f f i c i e n t l y large then there would be a large number of oC j for which the r e s t r i c t i o n s on the length scales were s a t i s f i e d . These would possess i d e n t i c a l d i s t r i -butions and would be mutually independent. The value of cp^ for the n t h volume can be written as C P n = <pa <Vj_ <P± _ . .. <P-~ (Eq. 1) or d?^ <p eu, where 5 <P0 has been assumed equal to the mathematical expectation. The bar denotes s p a t i a l averaging. Therefore 0><n « <P +• + JPQ^ <*•,_ + •f-^j»<^(Eq. 2) The central l i m i t theorem of the p r o b a b i l i t y theory was then invoked to postulate that log <pn w i l l be a normally d i s t r i b u t e d quantity regardless of the d i s t r i b u t i o n s of the log<< A> -This was done subject to the following assumptions. 1. The p r o b a b i l i t y d i s t r i b u t i o n s of a s u f f i c i e n t number of the i<j are dependent only on the associated length scales as hypothesized above and hence are mutually independent. 2. The cp„ can be replaced with the mathematical expectation cp as i t i s averaged over a s u f f i c i e n t l y large volume. 3. The c<- have f i n i t e means and variances. This involves the J assumption that (p* = 0 does not occur. j 4. The value of ' n i s s u f f i c i e n t l y large f o r the above sum to be asymptotically normal. That i s to say, the Reynolds number i s s u f f i c i e n t l y large f o r the existence of many scales of subdivision as described above. I f Jto^ 4?^ i s a normally d i s t r i b u t e d quantity, then i s a logarithmic-normally d i s t r i b u t e d quantity having the p r o b a b i l i t y density function p 0 - i — e x < r ( E q - 3 ) where ^ _ and cr1" - {A°^ 1 )^ ~ C ^oc\ ^ ) The p^*1 moment about zero of the above p r o b a b i l i t y d i s t r i b u t i o n i s given by ~J ^ ( ..t O,. s J . . ( E«-o which y i e l d s on integration CP*9 rr- g (Eq. 5) m and cr defined as above. Gurvich and Yaglom also generalized the model to consider two-dimensional p r o b a b i l i t y density functions at two d i f f e r e n t points i n space. I f <p •=- cpOjt) and <p()u-rtt) and i f the same subdivision scheme i s employed, then for the f i r s t nr\ subdivisions x and y.+r w i l l be i n the same volume. For subdivisions n+1, n+2, etc., the points w i l l be i n d i f f e r e n t volumes. Therefore and (D' -=. CP C{,0^^ O ^ - K t - , o ( '-^o-x I f d V - <p cCtcti. -then Q - CP*. < ^ t c< t-i. and ®' - O V ^ - t - , For the scale of subdivision, n+1, the numerators of tf-vvM and <K'-vx+., are formed from an average over d i f f e r e n t volumes, while the denominators are formed from an average over the same volume. The variables can therefore not 7 be considered mutually independent. The mean value of t h e i r product cannot be replaced with the product of the mean values. For a l l scales of subdivision 3 > ^ t i the oCj can be considered mutually independent. For a l l 3 ; = I Therefore and 00?' » 0?V t * ^ Since the c<-j , J <-vn-i are also mutually independent of <x-*+\ and e ? < t h i s can be written I f r « L , the scale of the mean motion and r ^ ^ ' , the Kolmogoroff microscale, then as previously postulated <<-* + i and <s<^ +, would possess i d e n t i c a l p r o b a b i l i t y d i s t r i b u t i o n s . Hence o^ *., cK.'^ ,., i n t h i s region of r w i l l be a constant independent of r. (Recall that $ j and <X j are positive d e f i n i t e . ) Therefore with a constant factor, 4><p' OC c^C (Eq. 6) Consider equation 2. I f the subdivision had been stopped at some point corresponding to a length scale of the order of the separation r under discussion, what would be the d i s t r i b u t i o n of cp r? Providing the number of mutually independent oij i n the sum was large enough <p>~ would s t i l l possess the log normal d i s t r i b u t i o n but with d i f f e r e n t values of the mean and variance. The mean and variance would be given by -. J?0(j cp" + A, I X , t ) ^ <TX^A. , (Eq. 7) and 0 ; - A o c / t ) + -v, <r, ( E q > 8 ) 8 The A^(x,t) and A(x,t) r e s u l t from the fact that the f i r s t few terms of equation 2 would not have length scales i n the range for the c< j to be i d e n t i c a l l y d i s t r i b u t e d and mutually independent. A l l such terms would be lumped to form a random variable with mean A-^(x,t) and variance A(x tt) which would be independent of the other logo(.. n i s then the number of steps of subdivision remaining to a f i n a l volume of length scale r. i s the mean of these remaining i d e n t i c a l l y d i s t r i b u t e d log^Kj and C,1 i s t h e i r variance. It should be noted that the assumption of s e l f - s i m i l a r i t y of the turbulent process has been made i n describing these d i s t r i b u t i o n s as i d e n t i c a l . Since the r a t i o of the length scales associated with succeeding steps of subdivision were assumed equal, n can be written in terms of some constant length determined by the number of terms lumped into A(x,t) and the f i n a l length scale r . In fact M* , (E<1' 9) where X9 i s the length of the l a s t subdivision included i n the A(x,t) term. By equation 5 and since <s>r i s a log normal quantity Since for a l l r ( f>sv*\r* i ^ ) a ? t - £  which s i m p l i f i e s to , — P i°C|»->> — (Eq. 10) 9 Using p=2 and substituting the r e s u l t s of equations 8 , 9 and 10 into equation 6 the covariance function can be written as G>q>' ~ 8<pv co •= #(x) # c x - i - o C "CPV (Eq. I D I within the previously mentioned constant factor cC 1 The one-dimensional spectrum i s the fo u r i e r cosine transform of t h i s and Gurvich and Yaglom have shown that t h i s spectrum i n the i n e r t i a l subrange varies as ^  ; 0~t s x That i s , the spectrum of the fluctuations i n the log normal quantity should i n the i n e r t i a l subrange exhibit a power law behaviour with the power having a value between 0 and -1 . This i s the same conclusion as was reached by Novikov and Stewart (1964) based on th e i r on-off model of the d i s s i p a t i o n . The assumptions of importance here would seem to be the one that 0"7* i s independent of r and that the corr e l a t i o n e( '^¥l i s also independent of r. 1 . 3 Structure Functions The n ^ order structure function i s defined as ijis-- juiy where/M-and x.' are the values of a component of the turbulent v e l o c i t y at two points separated by a distance r. The o r i g i n a l theory of Kolmogoroff predicted that i n the i n e r t i a l subrange t h i s quantity would be a function of separation r only, and i t s value would be where the are universal constants and £_ i s the di s s i p a t i o n rate. The subsequent modifications to the Kolmogoroff theory to take into account the fluctuations i n the d i s s i p a t i o n rate lead to a new form for t h i s expression. (See Kolmogoroff ( 1 9 6 2 ) ) c r ; < ~ - 3 ) /IS The are no longer universal constants but may depend on the large scale motions. 0~t i s the power law describing the spectrum of the square of the derivative of the turbulent c h a r a c t e r i s t i c . Within the i n e r t i a l subrange 7 T T ^ ^ - - t ^ - T ^ A e (Eq. 12) That i s , the power of r for the second order structure function i s increased, the t h i r d order i s unchanged and the fourth order i s decreased. The normalized t h i r d order structure function (skew-ness) i s given by S i r ) - L*<-. - ju.;) (Eq. 13) (^tA. v - ) Z \ In the i n e r t i a l subrange t h i s quantity would be a constant according to the o r i g i n a l Kolmogoroff theory. According - 0.0& to the modified theory i t should behave approximately as Ir There i s also a d i r e c t r e l a t i o n connecting the skewness i n the i n e r t i a l subrange with the Kolmogoroff constant K'of the " - 5 / 3 region" of the one-dimensional downstream v e l o c i t y spectrum. The r e l a t i o n i s b ( r ) = - - O . i o o ( K ) 11 The normalized fourth order structure function (flatness factor) i s given by Fcr) U L , - A ^ : ) H (Eq- 15) 1.4 The I n e r t i a l Transfer of Energy The Navier-Stokes equation for a Newtonian f l u i d with no body forces can be written i n tensor form as & g r ^ + f r f i + ^ ^ & ^ V i + ^ i l + f ^ + p t e . (Eq. 16) - v v l ( CJ^ 1- ^ - i " ) = ° Here the v e l o c i t y i s assumed to consist of a mean flow L^' and 1 a f l u c t u a t i n g flow , each possessing zero divergence. Also yU.-^ i s equal to zero. The pressure s i m i l a r l y consists of a mean, "fT , plus a fl u c t u a t i n g portion 70 . i s the density and V i s the kinematic v i s c o s i t y . I f t h i s equation i s multiplied by AA~1 and averaged the r e s u l t i s " - i +- y*^L M-K + dg=.-i The term #-i<?uj describes part of the time rate of change of the Reynolds stress. The second term i s an i n e r t i a l transfer term and can be shown to be the only mechanism by which energy can be transferred between turbulent fluctuations of d i f f e r e n t scale sizes. The t h i r d term i s an advection term and i s of no interest here. The fourth term describes the interaction between the turbulence and the mean shear flow. The pressure term can be shown to be related to r e d i s t r i b u t i o n s between components at the This requires the assumption of an incompressible flow. 12 same wavenumber and i s thought of as causing the turbulence to tend towards isotropy. The f i n a l term describes d i s s i p a t i o n of energy due to molecular v i s c o s i t y . Only the terms not involving the mean flow or the pressure are relevant here. For the downstream component these are _ > < . , > c , c ^ - i + S) /U-^ S7*~^<-1 _ -Since the measurements were of the downstream component of v e l o c i t y , only information on the f i r s t , second and f i n a l terms was ava i l a b l e . However the behaviour of the single i n e r t i a l transfer term and i t s r e l a t i o n to the single d i s s i p a t i o n term i s s t i l l important i n understanding turbulence For computation purposes i t has been assumed, as has also been done for the spectrum computations, that the turbulent v e l o c i t y signals and the i r derivatives can be represented over the data i n t e r v a l by a discrete f i n i t e f o u r i e r s e r i e s . That i s c^cx,) „ Co, ( x, t-3 ^ \ /? -=. o Here yi"-Oc,) i s the turbulent s i g n a l , i s the amplitude of the k**1 f o u r i e r constituent, k-j_ an integer, and i s the phase of the constituent r e l a t i v e to X]_ = 0. The length of the record i s X . It should be noted that,experimental data on s p a t i a l derivatives of turbulent parameters are usually obtained from a time d i f f e r e n t i a t i o n of the s i g n a l . Taylor's hypothesis i s then assumed to convert to a s p a t i a l derivative so that X , = " ^ i " ^ -and X- U , T where T i s the duration of the record. 13 For s i m p l i c i t y 2 1 1 jk' w i l l be written as wavenumber k and w i l l be referred to as the k t h constituent. Constants of propor t i o n a l i t y w i l l not be written. Thus, the downstream component Terms involved i n the interactions between downstream wavenumbers can be derived by multiplying equation 16 by and averaging, to get ^ 4-<^^^ , _ ^ ^ f x \ f + ^ 7 ^ (Eq. 17) Since /^-j? <i^:k^o £f-k> the leading term i n the equation becomes This term describes the time rate of change of the energy i n the Jl ^  component; on the assumption of a steady state t h i s term i s zero and i s retained only for the purpose of i d e n t i f i c a t i o n . The term of interest i s the second. It describes the net rate of a r r i v a l of downstream energy at wavenumber Jl due to i n e r t i a l transfer by interactions between downstream v e l o c i t y components. The f i n a l term describes the rate of loss of down-stream energy due to v i s c o s i t y at wavenumber In investigating the i n e r t i a l transfer of energy i t i s therefore necessary to examine the values of the terms i n the double sum — M-A < 2 £ Here = ^-f +- ) and - C-vy +- ©>*0. A^-JI and ^ c ^ a r e f o u r i e r components of the downstream v e l o c i t y . ^ Here A * - ^ - s the f o u r i e r component of ^ t ^ o i . ) • Ik cU<-k a Cfr Cos (h>C S»L, i s the ^ - d e r i v a t i v e of the k^h f o u r i e r component. — x*zt o>^<-k, - Q i C ^ C f c Sun ( £x t&g) Sc--* (-voi f A w ) Cy ( k ^ f c ) This average i s to be computed over the record l e n g t h X. This s i m p l i f i e s to — *- ^ ^ ^-/w C m u l t i p l i e d by the f o l l o w i n g sum of terms Here ^  , m and k are p o s i t i v e i n t e g r a l m u l t i p l e s of the wave-number iLLT The i n t e g r a l of a cosine over the record which i s an i n t e g r a l number of c y c l e s , i s non-zero only when the argument i s zero. Since Jl + m + k y 0 always, the terms i n v o l v i n g cos ( i + m + do not contr i b u t e and the value of - / ^ - i / u ^ ^ h i s one of the f o l l o w i n g : 1. — CjjC^ Cfe C>s C &x - £ - v ~ v t- ) i f +- k * o I t i s thus possible by holding $ constant and summing m and k over a l l t h e i r possible values to examine the mean rate of a r r i v a l of energy at wavenumber J l due to the term dUti i n equation 16. I t i s also worthy of note that any of the above three terms i s , except for m u l t i p l i c a t i o n by -jjr^» within a constant factor of the contribution, by the three f o u r i e r c o e f f i c i e n t s involved, to the t h i r d moment of the v e l o c i t y derivative. The skewness and the bispectrum of the v e l o c i t y derivative are therefore quite closely related to the i n e r t i a l transfer term. 16 Chapter II 2. Data C o l l e c t i o n 2.1 Velocity data Observational data for v e l o c i t y fluctuations were collected i n the atmospheric boundary layer over the ocean using a Disa type 55D05 constant temperature hot wire anemometer. For most of the runs a single wire mounted v e r t i c a l l y to measure only downstream v e l o c i t y fluctuations was used. In one case two Disa anemometers were used i n an X-wire configuration i n an attempt to measure the v e r t i c a l as well as the downstream v e l o c i t y fluctuations. The wires used had diameters of approximately 5 microns and lengths of approximately 1mm, permitting measurements down to scale sizes of les s than 0.5 cm. The probes were mounted at 1.5 to 2.0 meters above the water surface and measurements were made with mean wind speeds varying from 4 to 8 m/sec. The s i t e for the c o l l e c t i o n of the v e l o c i t y data was the Institute of Oceanography f a c i l i t y at Spanish Banks (see E l l i o t 1971). The tape recorder, hot wire electronics and signal conditioning equipment were housed i n a hut constructed on p i l i n g s . The power source was a di e s e l generator and Sorenson AC power regulator combination. The Disa probes were mounted on an aluminum mast located about 5°m to the seaward side and s l i g h t l y upwind of the hut. The mast had a diameter of approximately 15cm and presented l i t t l e obstruction to the wind. The probe holder 1? placed the hot wire sensor about one meter from the mast i n the cross wind d i r e c t i o n . The winds for the measurements discussed here were afternoon sea breezes with a few tens of miles of fetch. The waves were small, of the order of one-half to three-quarters of a meter from crest to trough. The analog signal processing and the recording of the data were e s s e n t i a l l y the same for a l l v e l o c i t y data discussed here. The necessary c i r c u i t s were patched up on an operational amplifier manifold. Figure 1 i s a block diagram of the c i r c u i t used. Disa anemometer off s e t stage gain stage f i l t e r stage V d i f f e r e n -t i a t o r stage to tape recorder s 1 I ^ to tape recorder F i g . 1 The output of the Disa anemometer i s a signal proportional to the wind v e l o c i t y . This was treated as a dc signal representing the e f f e c t of the mean wind v e l o c i t y plus a f l u c t u a t i n g signal representing the turbulent v e l o c i t y 3 fluctuations. The o f f s e t stage e s s e n t i a l l y removed the mean wind. This made i t possible to condition the f l u c t u a t i n g portion of the signal i n the following gain stage to use more e f f e c t i v e l y the dynamic range of the recording system. The f i l t e r stage was a Thompson low pass f i l t e r of order four. This type of f i l t e r has a phase s h i f t which i s l i n e a r with frequency and hence equivalent to a time delay i n The f l u c t u a t i n g portion of the flow was s u f f i c i e n t l y small to make l i n e a r i z a t i o n of the Disa output unnecessary. 18 the signal. The waveform was thus preserved, which was esse n t i a l to the types of analyses to which the data was to be subjected. The l a s t stage was an analog d i f f e r e n t i a t o r and gain stage, the gain once more for the purpose of u t i l i z i n g the f u l l dynamic range of the recording system. The recording system for both the d i f f e r e n t i a t e d and undifferentiated signals was an Ampex FR1300 tape recorder. The recording speed was 7h ips and the recording mode was FM. The c h a r a c t e r i s t i c s of the c i r c u i t above are presented i n more d e t a i l i n appendix A. P a r t i c u l a r attention i s given to undesirable phase s h i f t s . I t s u f f i c e s at t h i s point to say that the phase s h i f t for the v e l o c i t y data was within one or two degrees of l i n e a r i n the band 0 to 2000 Hz. 2.2 Temperature Data The temperature data were collected on August 8, 1969 at Boundary Bay, B.C. The sensors used were platinum resistance thermometers with a wire diameter of 0.25 microns and a length of 0.3mm. The experimental setup featured one wire (referred to as E3) at a height of two meters. Both wires were oriented to measure the downstream temperature fluctuations. The mean wind speed for the run was 4 m/sec. At the Boundary Bay si t e (see E l l i o t 1971) the measure-ments were made over a t i d a l mud f l a t . The mud was not uniformly f l a t . However the depressions were mostly f i l l e d with water pre-senting a rather f l a t boundary to the turbulence. The point at which the measurements were made was approximately 75m upwind from the high tide l i n e . At t h i s point there was a 2m high dike. The land to the north was f l a t farmland. B.C. Hydro power was available and t h i s was used to operate the measuring and recording equipment. Figure 2 below i s a block diagram of the electronic c i r c u i t s . Sensor electronics Kronhite f i l t e r d i f f e r e n -t i a t o r \ to tape recorder s / . — ^ to tape Fig. 2 The Kronhite f i l t e r was used as a low pass f i l t e r with the cut-off frequency set at 10 KHz. The phase s h i f t at 500 Hz was almost 14 degrees but was within •§ degree of being l i n e a r i n the band 0 to 500 Hz. The c h a r a c t e r i s t i c i s shown in Appendix A. The d i f f e r e n t i a t o r was sim i l a r to that described under v e l o c i t y data with phase c h a r a c t e r i s t i c s of a sim i l a r magnitude. It was therefore concluded that the recorded signals were suitable for analyses requiring that the waveform be maintained provided the frequencies were below 500 Hz. The d i f f e r e n t i a t e d and undifferentiated temperatures were recorded as FM signals at 60 ips on the Ampex FR13.00 tape recorder. 2.3 Analog to D i g i t a l Conversion Since i t was intended to do a l l subsequent analyses on a d i g i t a l computer, i t was necessary to convert the data to d i g i t a l form. This was accomplished through the use of a high speed 10 b i t successive approximation analog to d i g i t a l con-20 verter interfaced with a computer and a d i g i t a l tape unit. The d i g i t a l tape unit provided storage of the data i n computer com-patible form. The analog to d i g i t a l converter contained a sample and hold device to eliminate problems of signal changes during the course of a d i g i t i z a t i o n . Figure 3 i s a block diagram of the d i g i t i z a t i o n procedure. FR1300 impedance 2 stage f i n a l Analog to tape matching and > Butterworth -y gain "7> d i g i t a l recorder gain stage f i l t e r stage converter Fi g . 3 The analog data tapes were replayed at the speed at which they were recorded. The f i r s t stage a f t e r the tape recorder was primarily an impedance match between the tape recorder and the f i l t e r . In addition, i t and the f i n a l gain stage permitted signal conditioning to u t i l i z e the f u l l range of the analog to d i g i t a l converter. These stages were e s s e n t i a l l y phase s h i f t l e s s containing only the unavoidable few pico-farads of stray capacitance inherent i n such c i r c u i t s . The f i l t e r served the purpose of preventing high frequency noise o r i g i n a t i n g mainly i n the tape recorder from a l i a s i n g to lower frequencies i n the d i g i t i z e d s i g n a l . I t was of the same design and c h a r a c t e r i s t i c s as the one used i n recording the v e l o c i t y signals, i . e . i t had a l i n e a r with frequency phase s h i f t and did not destroy the waveform. I t should perhaps be pointed out that both i n the recording and d i g i t i z i n g of the data care was taken to insure 21 that the signal was not clipped very often. An attempt was made to choose the gains so that the maximum peak to peak excursions of the signal as observed on an oscilloscope occupied only about two-thirds of the dynamic range of the instruments used. 2 2 Chapter I I I 3• G e n e r a l Remarks on Data P r e s e n t a t i o n P r i o r t o computing the c h a r a c t e r i s t i c s o f any o f the d a t a r e p o r t e d h e r e , a spectrum and a l i n e a r d i s t r i b u t i o n o f each s i g n a l were produced. The spectrum s e r v e d as a check on the q u a l i t y o f the d a t a . The c h a r a c t e r i s t i c s l o o k e d f o r were p r o p e r s p e c t r a l shape, a good s i g n a l t o n o i s e r a t i o and the amount o f the d i s s i p a t i o n range i n c l u d e d i n the a n a l y s i s . Where p e r t i n e n t t o the i n t e r p r e t a t i o n the s p e c t r a are p r e s e n t e d . The l i n e a r d i s t r i b u t i o n s were used as a check on whether optimum use had been made o f the dynamic ranges o f the v a r i o u s i n s t r u m e n t s . I f the g a i n s d u r i n g d i g i t i z i n g were n o t s a t i s f a c t o r y , f o r example, the o p e r a t i o n was r e p e a t e d . 3.1 Computation o f the D i s t r i b u t i o n s o f the S i g n a l s I n the p r e s e n t a t i o n which f o l l o w s , the p r o b a b i l i t y d e n s i t y f u n c t i o n s have u s u a l l y been c o n s t r u c t e d f o r the l o g a r i t h m o f the a b s o l u t e v a l u e o f the d i f f e r e n t i a t e d v e l o c i t y and temperature s i g n a l s r a t h e r than the square o f t h a t q u a n t i t y as was done by G u r v i c h & Yaglom (196?). I f P( y ) i s the p r o b a b i l i t y d e n s i t y f u n c t i o n o f a l o g n o r m a l l y d i s t r i b u t e d q u a n t i t y then ?L^) * 1 £> o ~ x 5 where ".'•'he f o l l o w i n g c o n v e n t i o n s have been adopted f o r the r e m a i n !m diagrams. L0G 1 0 i s w r i t t e n as L O C I . L C : i e i s w r i t t e n as L !\ or NATURAL LOG. I f (X10 n) appears i n an a x i s a n n o t a t i o n the v a l u e o p p o s i t e a t i c i s re a d as y x 1 0 n . 23 I f instead X - <-j *" i s observed then by Cramer (19^-6) » section 14.5i GCDL) di<~ - r \ t j ; c / j where i s the p r o b a b i l i t y density function of X . Therefore Got ; = fttu '33c - fth; Also and where _ T- 7, x ) - H = M e n , Substituting i . £ ( x j = 1 p z-OV^ Hence the positive square root of a log normally dis t r i b u t e d quantity i s also a log normally d i s t r i b u t e d quantity. The values of the mean and standard deviation w i l l be one-half those obtained i f the square root had not been taken. Use of the absolute value rather than the square of the signal allowed a more d i r e c t comparison between the observed and t h e o r e t i c a l values of the moments. In recording or d i g i t i z i n g the turbulent signals logarithmic amplifiers were not used. Thus the values produced by the d i g i t i z e r were i n equal increments of the signal rather than i t s logarithm. To obtain the values at approximately equal increments of the logarithm i t was necessary to combine analog to d i g i t a l converter i n t e r v a l s when c o n s t r u c t i n g the d i s t r i b u t i o n s i n the d i g i t a l computer a n a l y s i s . The method chosen produced about twenty estimates of p r o b a b i l i t y d e n s i t y over the dynamic range of the experiment. The d i s t r i b u t i o n s are presented i n two forms. The f i r s t i s a graph of p r o b a b i l i t y d e n s i t y versus the l o g a r i t h m 5 of the t u r b u l e n t parameter as i n f i g u r e 6 . The observed data i s the histogram. Also shown i s a smooth curve of the normal form which has been obtained from a l e a s t squares f i t to p o i n t s l o c a t e d i n the centers of the histogram bars. The zero of the turbulent parameter was assumed to be the mean value of the d i g i t i z e d s i g n a l . For the d i g i t i z e r i n t e r v a l i n which t h i s mean value f e l l , a p r o b a b i l i t y d e n s i t y could not be c a l c u l a t e d as d i v i s i o n by the l o g a r i t h m of zero would have occurred. Thus there i s information at smaller values of the l o g of the s i g n a l which could not be i n c l u d e d i n the diagram. When c o n s t r u c t i n g the p r o b a b i l i t y d e n s i t y histogram the values corresponding to the l a r g e s t p o s s i b l e p o s i t i v e and negative d i g i t i z e r v alues were ignored. T h i s was done as i t was not p o s s i b l e to know whether the s i g n a l had been equal to or greater than these maximum values. These occurrences and the ones i n the i n t e r v a l c o n t a i n i n g the mean value were however included i n the count of the t o t a l number of samples so that th p r o b a b i l i t y d e n s i t i e s are accurate. The second form of presentation i s shown i n f i g u r e 7. I n f i g u r e 16 the parameter i s a r t i f i c i a l l y c o n s t r u c t e d , 'lowever the methods of d i s p l a y are i d e n t i c a l t o t h o s e f o r the t u r b u l e n t oarameters. 25 In t h i s case the d i s t r i b u t i o n function i s plotted so that a log-normal d i s t r i b u t i o n would appear as a straight l i n e . The l i n e drawn on t h i s diagram i s a least squares f i t to the observed points and was not constructed from the f i t t e d normal curve i n the other f i g u r e . In computing the f i t the f i r s t three points corresponding to small values of the s i g n a l have been ignored. The positions of these points were strongly influenced by the measurement noise. This noise w i l l be discussed further i n the following section. In t h i s second presentation the information which was observed i n the d i g i t i z e r i n t e r v a l which contained the zero of the s i g n a l was used. It yielded the number of occurrences of the log of the s i g n a l between minus i n f i n i t y and the value of the logarithm corresponding to the l e f t side of the f i r s t bar i n figure 6. This cumulative percentage i s the lefthand point i n figure ?. I t should be recognized that f o r a l l the d i s t r i b u t i o n s discussed here, the c h a r a c t e r i s t i c s such as the least squares approximations and the moments were computed using only the values of the signals which were within the dynamic range of the recording-analysis system. The e f f e c t of l o s i n g a few large values i s p o t e n t i a l l y most serious i n computing the higher moments. 3 . 2 The E f f e c t of Measurement Noise on the Logarithmic Normal  D i s t r i b u t i o n Figure 4 shows the p r o b a b i l i t y density function for a logarithmic normal d i s t r i b u t i o n . The mean and standard deviation were chosen as t y p i c a l of those obtained f o r the experimental 26 -2.5 -1.5 -0.5 0.5 1.5 SIGNAL VOLTAGE 2.5 Figure 4. T h e o r e t i c a l l o g a r i t h m i c normal p r o b a b i l i t y d i s t r i b u t i o n , The negative portion of the curve i s a r e f l e c t i o n of the p o s i t i v e nortion. The absissa scale i s l i n e a r . 2.4n 2.0 A 1.6 1.2 0D (X CO o 0.8 oc 0_ 0.4 0.0 • f , ( x ) f 2 ( x ) -l 1 — = — i -2.5 -1.5 -0.5 0.5 1.5 2.5 SIGNAL VOLTAGE Figure 5 . E f f e c t of an independently adding n o r m a l l y distributed noise on a logarithmic normal d i s t r i b u t i o n . f i ( x ) i s the normal d i s t r i b u t i o n . f 2 ^ x ) i s t / h e l o g a r i t h m i c normal d i s t r i b u t i o n . f ( x ) i s the expected r e s u l t of f i and f ? a d d i n g i n d e p e n d e n t l y . 27 v e l o c i t y data to be discussed l a t e r . The curve has been r e f l e c t e d about the v e r t i c a l axis to show more c l e a r l y the experimental s i t u a t i o n . The d i s t r i b u t i o n shows small values of p r o b a b i l i t y i n the neighborhood of zero and the largest values of p r o b a b i l i t y r e l a t i v e l y nearby. independently to the log normally d i s t r i b u t e d quantity would tend to f i l l i n the gap between the peaks. This would r e s u l t i n an excess of small values. To obtain some insight into how large t h i s excess might be, a small numerical exercise was ca r r i e d out. thmic normally d i s t r i b u t e d quantity and f((x) i s the p r o b a b i l i t y d i s t r i b u t i o n of an independent normally d i s t r i b u t e d quantity, then the d i s t r i b u t i o n of t h e i r sum by Cramer ( 1 9 4 6 ) , section 1 5 . 1 2 , i s given by ^ of figure 4 . Normal d i s t r i b u t i o n s with various standard deviations were used as f ^ ( x ) . The above integration was then performed numerically between reasonable f i n i t e l i m i t s to obtain a probabi-l i t y d i s t r i b u t i o n of the sum. I t was found that approximately 80 mv(rms) of noise were required to f i l l i n the gap between the peaks i n figure 4 . (Total dynamic range was 1 0 . 2 4 volts.) The p r o b a b i l i t y d i s t r i b u t i o n s f^(x) , f 2 ^ x ^ f(x) are shown i n figure 5 f o r an f^(x) corresponding to 80 mv rms of noise. The e f f e c t of the noise i s seen to be a smearing both ways of the peaks with a decrease i n t h e i r height'. Figures 6 I t i s clear that even a small amount of noise adding I f fe(x) i s the p r o b a b i l i t y d i s t r i b u t i o n of a l o g a r i The d i s t r i b u t i o n f?(x) was chosen to be exactly that 28 Q.48n F i g u r e 6. P r o b a b i l i t y d i s t r i b u t i o n o f the l o g a r i t h m o f the i n d e p e n d e n t l y added l o g normal and normal q u a n t i t i e s o f f i g u r e 5 ' F i g u r e ?. C u m u l a t i v e d i s t r i b u t i o n f u n c t i o n o f the i n d e p e n d e n t l y added l o g a r i t h m i c normal and normal d i s t r i b u t i o n s o f f i g u r e 5. 29 and 7 show the p r o b a b i l i t y density function and the cumulative d i s t r i b u t i o n function of f(x) plotted i n the manner discussed i n the previous section. As can be seen the f i r s t few points of the d i s t r i b u t i o n have been s i g n i f i c a n t l y influenced by the addition of the random noise. The reason for ignoring the f i r s t three points i n computing the le a s t squares f i t to the d i s t r i b u t i o n function in figure 7 can now be seen. These points could be expected to be far enough from the l i n e to cause an a l t e r a t i o n in the slope. The l i n e would then not provide a good v i s u a l reference for judging the l i n e a r i t y of the more in t e r e s t i n g large value end of the d i s t r i b u t i o n . The other conclusion which can be drawn from an examination of figures k to 7 i s that the measurement noise has n e g l i g i b l e e f f e c t on the large value end of the d i s t r i b u t i o n . 3 . 3 A F i l t e r i n g Technique for Data Contaminated by Noise Several of the signals which were available for processing were contaminated by noise. The method used to recover the d i s t r i b u t i o n s employed the discrete f i n i t e f a s t f o u r i e r transform of Cooley and Tukey. The data selected to demonstrate the technique was a short section of d i f f e r e n t i a t e d temperature data. I t was chosen because i t was extremely noisy and contained both sharp, narrow noise peaks and high frequency d i f f e r e n t i a t e d white noise. Consequently i t provided a severe test of the technique. Figure 8 shows the spectrum of these data plotted as log spectral density versus log frequency. The method used i n 30 -3.0--3.5 >-r— U J Q _ l (X £ - 4 . 5 -C J L U 0 _ o -5.0 _ J -5 .5H -6.0 V 1 1 r i r Q.5 1.0 1.5 2.0 2.5 3.0 LOG OF FREQUENCY IN HZ 3.5 Figure 8 . Spectrum of the d i f f e r e n t i a t e d downstream temperature fluctuations, tape 1 5 ^ section 2 . 7.2-5.6 4.0 1 - 2 . 4 0.8 0.0 0.2 0.4 0.6 0.8 FOURIER COEFFICIENT NUMBER ( X 10 1.0 3 1.2 Figure 9 . Mean fo u r i e r amplitude versus fourier c o e f f i c i e n t number for the d i f f e r e n t i a t e d downstream temperature fluctuations, tape 154 section 2 . The conversion factor from c o e f f i c i e n t number to Hz i s 2 . 9 3 « 31 computing t h i s information i s discussed b r i e f l y i n Appendix B. Figure 9 shows an amplitude spectrum of the same data. The quantities plotted are mean fou r i e r amplitude versus f o u r i e r c o e f f i c i e n t number. The value of t h i s plot was in revealing the presence 6 of very narrow band noise peaks i n the data. In constructing the log-log p l o t several c o e f f i c i e n t s were averaged together masking any sharp narrow peaks. Although such peaks contain l i t t l e energy i t has been shown that they can have a s i g n i f i c a n t e f f e c t on the d i s t r i b u t i o n of the log of the signal i n the region of small values. From inspection of these spectrum diagrams i t can be seen that there i s a large sharp noise peak at 180 Hz and some apparently d i f f e r e n t i a t e d white noise above about 400 Hz. The spectrum also revealed however that i f the d i f f e r e n t i a t e d white noise continued to f a l l towards lower frequencies, then the signal to noise r a t i o should have been good i n the band 0 to 400 Hz. I t seemed probable that the noise could be removed by a d i g i t a l f i l t e r leaving a useful sig n a l . The data was f o u r i e r decomposed i n successive blocks of 2048 points. The complex f o u r i e r c o e f f i c i e n t s corresponding to the frequencies to be f i l t e r e d out were set to zero. The " f i l t e r e d " analog signal was recovered by performing a f o u r i e r synthesis of the remaining c o e f f i c i e n t s . At t h i s point a new d i s t r i b u t i o n was formed by accumulating the number of occurrences of signal values between x and x + A x i n a histogram form. The range of x and the magnitude of A x were chosen to duplicate the __ The bandwidth corresponding to a single f o u r i e r component was 2 . 93 H z . 32 range and fixed increment of the analog to d i g i t a l converter. The magnitude of t h i s f i x e d step determined the resolution of the analysis for the small value portion of the d i s t r i b u t i o n . Decreasing the value of AX af t e r the f i l t e r i n g operation would have increased t h i s resolution. However since a l l d i s t r i -butions to be presented f a i l e d to f i t the theory i n the region of small values t h i s was not done. I t was considered that l i t t l e useful additional information would have accrued. Figures 10 and 11 show the d i s t r i b u t i o n of the d i f f e r e n t i a t e d temperature signal p r i o r to f i l t e r i n g . I t f a i l s to f i t the logarithmic normal d i s t r i b u t i o n throughout the entire range. In fact the u n f i l t e r e d signal had a d i s t r i b u t i o n much closer to normal than log normal, i t s c o e f f i c i e n t of excess being equal to 0 . 8 . Figures 12 and 13 show the d i s t r i b u t i o n for the f i l t e r e d data. In figure 12 the histogram i s the observed d i s t r i b u t i o n corresponding to the f i l t e r i n g described, i . e . , f i l t e r e d to remove the 180 Hz peak and the noise above 400 Hz. The curve l a b e l l e d #1 i s the best l e a s t squares normal curve f i t t e d to that data. The curve l a b e l l e d #2 w i l l be discussed below. The improvement i n the f i t i s s i g n i f i c a n t . The data has also become more "intermittent", the c o e f f i c i e n t of excess becoming 1 9 . 6 . In removing the fo u r i e r c o e f f i c i e n t corresponding to the 180 Hz noise peak the turbulent portion of t h i s r e l a t i v e l y high energy containing c o e f f i c i e n t was also removed. This might be considered as replacing one kind of "noise" with another. To investigate the s e n s i t i v i t y to one or two fo u r i e r Figure 1 0 . D i s t r i b u t i o n of the logarithm of the d i f f e r e n t i a t e d downstream temperature fluctuations for section 2 tape 154 ?he duration of the run was one minute. P ' 99.99-, 99.9 A 99 -, , _ r -6 -4 -2 0 2 NflTURflL LOGRRITHM OF C (DT/DX) i g + u 6 i ^ ' G u r r , u l a t l v e d i s t r i b u t i o n function of the logarithm oi the d i f f e r e n t i a t e d downstream temperature fluctuations for section 2 tape 1 5 4 . The duration of the run was one minute. 0.18 34 NATURAL LOGARITHM OF C (DT/DX) Figure 12. D i s t r i b u t i o n of the logarithm of t h e f i l t e r e d d i f f e r e n t i a t e d downstream temperature f l u c t u a t i o n s f o r s e c t i o n 2 tape 1 5 4 . The duration of the run was one minute. NATURAL LOGARITHM OF C (DT/DX) Figure 1 3 . Cumulative d i s t r i b u t i o n f u n c t i o n of the log a r i t h m of the f i l t e r e d d i f f e r e n t i a t e d downstream temperature f l u c t u a t i o n s f o r s e c t i o n 2 tape 1 5 4 . The duration of the run was one minute. 35 c o e f f i c i e n t s i n the high energy containing portion of the spectrum the analysis was repeated. This time the two adjacent f o u r i e r c o e f f i c i e n t s were set to zero as well. This normal curve obtained was that l a b e l l e d #2 and plotted on figure 12. The r e s u l t demonstrates that the small value end of the d i s t r i -bution can be affected by one or two fourier c o e f f i c i e n t s but not su b s t a n t i a l l y . The largest change produced i n the height of any of the bars i n the histogram by the removal of the extra two c o e f f i c i e n t s was about 5$. I t was concluded that i f the magnitude of the narrow noise peak was greater than the expected magnitude of the turbulent signal at the same frequency i t was probably better to remove the corresponding f o u r i e r c o e f f i c i e n t . 3.4 Downstream Velocit y Data Figure 14 shows a spectrum of one of the d i f f e r e n t i a t e d downstream v e l o c i t y runs reported here. The mean wind speed was 7.8 m/sec and the duration of the run was 10 minutes. The hot wire was mounted at a height of 2 meters above the water. The data shows the expected +1/3 slope of the log spectral density versus log frequency curve below the peak. The 3db point of the f i l t e r (used both i n the recording and d i g i t i z i n g phases) as 2000 Hz. At 1 KHz the signal was attenuated 0.86 i n power. Thus the f i l t e r was just beginning to be f e l t at a point where the signal was down a decade i n power from the peak of the spectrum. Most of the d i s s i p a t i o n range would appear to have been included i n the analysis. The signal to noise r a t i o was sa t i s f a c t o r y as there i s no evidence of white noise at high or low frequencies. 36 Figure 14. Spectrum of the d i f f e r e n t i a t e d downstream v e l o c i t y fluctuations for run 2B tape 141. Hot wire c a l i b r a t i o n and the various gain factors have not been applied to the ordinate. The unit s are therefore e s s e n t i a l l y analog to d i g i t a l converter v o l t s 2 / H z . The l i n e has a slope of 1/3. 37 Figure 15 i s the d i s t r i b u t i o n of the logarithm of the absolute value of the d i f f e r e n t i a t e d downstream velocity-fluctuations constructed as described i n section 3.1. Figure 16 i s the d i s t r i b u t i o n function of t h i s data as described e a r l i e r . As can be seen there i s a s i g n i f i c a n t departure from the t h e o r e t i c a l prediction at both large and small values of amplitude. The skewness of the bar graph i s -0.38 and the 7 c o e f f i c i e n t of excess i s -0.1. This good value of c o e f f i c i e n t of excess i s for t u i t o u s , a r e s u l t of too many small values balancing too few large values i n the even moment. A certain excess of small values was predicted i n the discussion of the measurement noise. I t remained to be determined i f there was s u f f i c i e n t noise to account for the size of the excess. I t was d i f f i c u l t to put a figure on the magnitude of the noise associated with these measurements. Some noise runs with the v e l o c i t y probe capped had been taken and tended to y i e l d rms values of 30 to 40 m i l l i v o l t s at the input to the d i g i t i z e r . However t h i s was not exactly the experimental s i t u a t i o n i n that the currents and voltages i n the instrument and elsewhere would not have corresponded to the mean values at the time of the measure-ment* Also the mechanical tape recorder fluctuations would vary from pass to pass of the analog tape. However i t was f e l t that the electronic and mechanical tape recorder noise was probably by far the largest contribution and that i t was reasonably reproducible. Under t h i s assumption the value of 40 mv rms would be the upper l i m i t of the measurement noise.  7 The c o e f f i c i e n t of excess i s defined as the fourth moment divided by the second moment squared minus 3* 38 0.48 - i 0.4CM N A T U R A L L O G A R I T H M OF C ( D U / Q X ) Figure 1 5 . D i s t r i b u t i o n of the logarithm of the d i f f e r e n t i a t e d downstream v e l o c i t y fluctuations for run 2B tape 141. C i s a c a l i b r a t i o n constant. The duration of the run was 8 minutes. N A T U R A L L O G A R I T H M OF C ( D U / D X ) Figure 1 6 . Cumulative d i s t r i b u t i o n function of the logarithm of the d i f f e r e n t i a t e d downstream v e l o c i t y fluctuations f or run 2B tape l 4 l . C i s a c a l i b r a t i o n constant. The duration of the run was 8 minutes. 39 I t was pointed out i n section 3*1 that i t required 80 mv rms of noise to f i l l i n the gap between the peak and zero of a log normal d i s t r i b u t i o n with the same f i r s t two moments as the one obtained for these v e l o c i t y data. The observational fact was that t h i s f i l l i n g i n did happen. No decrease i n the probabi-l i t y density function occurred i n the neighborhood of zero. I t was therefore concluded that the observed excess of small values was partly due to the f a i l u r e of the downstream v e l o c i t y derivative to conform accurately to the log normal model i n the region of small values. In fact i t would seem d i f f i c u l t for t h i s signal to behave l o g normally i n t h i s region as i t must pass through the value zero i n changing sign. Because of t h i s i t might be expected to spend more time in the v i c i n i t y of zero than would be indicated by the theory. As pointed out i n the e a r l i e r section, the noise did not a f f e c t the large value portion of the d i s t r i b u t i o n . Hence another "explanation" must be sought for the observed deficiency of values i n t h i s region. In pursuit of t h i s "explanation" the fourth assumption made in the development of the logarithmic normal theory i n section 1.2 was considered worthy of examination. This assumption was that the number of scales of subdivision was large which required that the Reynolds number be high. I f the number of scales of subdivision was not large enough then the number of log oc^ would be reduced and the value of t h e i r sum would tend to be l i m i t e d as i n the case of a random walk with a small number of steps. 40 To test t h i s idea groups of random numbers were generated on a computer and multiplied together. For the reasons described i n section 1 . 2 the d i s t r i b u t i o n of the logarithm of t h i s product could be expected to be assympoti-c a l l y normal as the number of m u l t i p l i e r s increased. This exercise could not of course y i e l d information on the range of scales over which the theory holds as the rate of convergence to the normal d i s t r i b u t i o n would depend strongly on the in d i v i d u a l d i s t r i b u t i o n s . The p r o b a b i l i t y d i s t r i b u t i o n from which the random numbers were drawn was defined as Pcx> - o 3 c X c ) P c x ) = l / ( X b O U ) , c x < ^ x ^ x b ) P c x ; = o , ( y > x b ) ?C^and Xh were evaluated by t r i a l and error to y i e l d d i s t r i -butions with the f i r s t two moments corresponding approximately to those of one of the experimental d i f f e r e n t i a t e d v e l o c i t y d i s t r i b u t i o n s . Figure 17 and 18 show respectively the p r o b a b i l i t y density and cumulative d i s t r i b u t i o n functions for the product of 10 such random numbers. There i s a d e f i n i t e deficiency of large values similar to that encountered i n the experimental v e l o c i t y s i g n a l . I t would seem that a not-large-enough Reynolds number might account for the observed r e s u l t s . Figures 19 and 20 show the same functions for a product of 50 random numbers. There i s an obvious improvement i n the f i t i n both diagrams. The e f f e c t of l i m i t i n g the number of m u l t i p l i e r s i s not confined to the large value end although i t i s most obvious 41 0.48-, 0.40 H NATURAL LOG OF SIGNAL Figure 17. D i s t r i b u t i o n of the logarithm of the product of 10 random numbers. NATURAL LOG OF SIGNAL Figure 18. Cumulative d i s t r i b u t i o n function of the logarithm of the product of 10 random numbers. 0.48-1 0.32 -6 -4 -2 0 NATURAL LOG OF S IGNAL Figure 19. D i s t r i b u t i o n of the logarithm of the product of 50 random numbers. 99.99 -1 >- 99.9 o cc UJ 99 2 9 0 1— => Li_ o 50 CD CC 5 10 o £ 1 _ l ZD 3 0.1 0.01 1 1 1 -4 -2 0 NRTURAL LOG OF S IGNAL Figure 20. Cumulative d i s t r i b u t i o n function of the logarithm of the product of 50 random numbers. 43 there. The d i s t r i b u t i o n function of figure 18 shows a curvature over i t s length, there being no portion of i t where a straight l i n e can be drawn through more than a few points. This i s exactly s i m i l a r to the d i f f e r e n t i a t e d downstream v e l o c i t y signals examined. There i s also a s l i g h t excess of small values i n both cases. However t h i s excess i s small compared to the previously discussed one associated with the electronic noise. Neither would i t seem large enough to account for the difference between the excess expected based on the estimated noise i n the measure-ment and that observed. The f a i l u r e of the experimental d i s t r i b u t i o n to f i t the t h e o r e t i c a l model i n the region of large values w i l l have i t s most serious consequences i f the model i s employed i n the prediction of the frequency of occurrence of rare events. In constructing the histograms of observed p r o b a b i l i t y densities the largest and smallest values produced by the analog to d i g i t a l converter were ignored. An occurrence of one of these values usually indicated saturation of the recording-analysis system at some point. Some of these occurrences were undoubtedly due to dropouts on the analog recording tape. This number of times the system was driven to saturation was s t i l l , however, a useful piece of information. It set an upper l i m i t on the number of sig n a l excursions beyond the range of the system. This number could be compared with the expected number assuming log normality. For these v e l o c i t y data the observed maximum number of sig n a l excursions into saturation was 53* The expected number based on the f i t t e d normal curve was 5 5 0 . The 44 observed frequency i s thus less than 10% of the the o r e t i c a l frequency. For the log normal d i s t r i b u t i o n the higher moments are determined mainly by the large values. Since these moments have at times been considered as a good test of the agreement between the theory and experiment, i t i s worthwhile to examine the extent of t h i s dependency. The p^h moment of the logarithmic normal d i s t r i b u t i o n (Eq. 4) i s given by, o o e> which can be written as The f i r s t i n t e g r a l i s the contribution to the moment by the 0 to x portion of the d i s t r i b u t i o n and i s the one of interest here. Since the integral could only be solved a n a l y t i -c a l l y as a d e f i n i t e i n t e g r a l , the problem was treated numerically. The p r o b a b i l i t y density function assumed for f^) was that of the ve l o c i t y data. The contribution to (p? by the portion of the d i s t r i b u t i o n between 0 and x was determined as a function of x. The r e s u l t s are shown in figure 2 1 . The l i m i t of the recording-analysis system occurred at 5 -12 v o l t s . The 4 t h moment i s almost f u l l y determined by the portion of the d i s t r i b u t i o n beyond the range of the measurement. For purposes of the analysis the zero of the signal 45 Figure 21. Contribution to the zeroth to fourth moments of the log normal d i s t r i b u t i o n from the range 0 to X. The upper curve i s the corresponding p r o b a b i l i t y density function. 46 was defined as i t s mean. Therefore the even moments about zero of the one-sided d i s t r i b u t i o n of figure 15 can be d i r e c t l y compared with the even moments of the si g n a l taken about the mean. Table I shows t h i s comparison for the data of figure 1 5 • Column 2 i s the observed even moments of the s i g n a l . Column 3 contains the t h e o r e t i c a l values obtained by integrating numeri-c a l l y over the range of the recording-analysis system. Column 4 contains the t h e o r e t i c a l values. The c o e f f i c i e n t of excess i s defined as the fourth moment divided by the second moment squared minus three. Moment Observed Theoretical Value Theoretical Value Integrated Over Value Range of System (Integrated 0 to ca ) 0.2? 0.41 0 . 56 1.01 2 . 3 8 ^1.3 10 .6 11.0 131.6 Coe f f i c i e n t of Excess Table I. Comparison of observed and t h e o r e t i c a l moments for the d i s t r i b u t i o n of the d i f f e r e n t i a t e d v e l o c i t y data of figure 1$. The discrepancies between the second and fourth columns of the table are due to the f a i l u r e of the data to conform accurately to the log normal model within the range of the recording-analysis system and to the f a i l u r e of the system to measure the larger values. In view of the deficiency of large values found for these data both i n and beyond the range of the analysis i t would seem u n l i k e l y that increasing the dynamic range of the experiment would improve the s i t u a t i o n s i g n i f i c a n t l y . I t i s probably s u f f i c i e n t to say that l i t t l e about the d i s t r i b u t i o n can be inferred from the measured higher moments and the assumption that i t i s log normal. 47 Four other runs of d i f f e r e n t i a t e d v e l o c i t y data were examined i n terms of the logarithmic normal theory. A l l involved the use of the Disa anemometer hut were taken at d i f f e r e n t times in d i f f e r e n t wind conditions (4 to 6 m/sec) and at s l i g h t l y d i f f e r e n t probe heights above the water. A l l the d i s t r i b u t i o n s were q u a l i t a t i v e l y i d e n t i c a l to the one discussed. Each showed an excess of low values si m i l a r to that encountered here and of a similar magnitude. A l l excesses seemed to be s l i g h t l y larger than could be explained by the measurement noise. The deficiency of high values was evident i n each of the four cases as well. The cumulative d i s t r i b u t i o n s showed a curvature over t h e i r entire length. The magnitude of the deficiency of large values was sim i l a r for a l l the runs. Figures 22 and 23 show the p r o b a b i l i t y d i s t r i b u t i o n and cumulative d i s t r i b u t i o n respectively for one of these runs. The mean wind speed was 6 m/sec and the probe was mounted 2 meters above the mean water surface. The other three runs are not presented. At t h i s point i t was f e l t worthwhile to b r i e f l y investigate the nature of the f i r s t v e l o c i t y data discussed. The purpose was to determine whether the f i e l d had been adequately sampled to e s t a b l i s h i t s d i s t r i b u t i o n and i f the f i e l d was reason-ably stationary i n the s t a t i s t i c a l sense. The usual physical "picture" of the turbulence f i e l d used to gain insight into the phenomenon of intermittency i s one of large areas of r e l a t i v e l y l i t t l e a c t i v i t y containing splashes of energy d i s s i p a t i o n . These splashes are observed to occur 48 0.48n 0.4CH NRTURAL LOGARITHM OF C (DU/DX) Figure 2 2 . D i s t r i b u t i o n of the logarithm of the d i f f e r e n t i a t e d downstream v e l o c i t y fluctuations, run 2 tape 9 6 . The duration of the run was 2 minutes. -8 -6 -4 -2 0 2 4 NATURAL LOGARITHM OF C (DU/DX) Figure 2 3 . Cumulative d i s t r i b u t i o n function of the d i f f e r e n t i a t e d downstream v e l o c i t y f l u c t u a t i o n s , run 2 tape 9 6 . The duration of the run was 2 minutes. 49 randomly and c h a o t i c a l l y with no apparent s p a t i a l correlations at l e a s t between widely separated points. The d i f f e r e n t i a t e d v e l o c i t y signals were d i g i t i z e d at a high enough frequency (6000 Hz) to y i e l d an unaliased spectrum. Thus there would have to be a high c o r r e l a t i o n between one value of the f i e l d and the next. However i f the separation was increased i t would be expected that t h i s c o r r e l a t i o n would decrease. At some value of separation measurements could be considered as random samples from a population i n the s t a t i s t i c a l sense. This separation would determine approximately some "number" of independent observations of the phenomenon i n the sample. Assuming Taylor's hypothesis to obtain s p a t i a l separations from time separations t h i s c o r r e l a t i o n was examined. I t was found that i t dropped to about 0 . 1 or le s s for separations of only a few centimeters. It would seem therefore that the sample consisted of a few tens of thousands of independent random observations. How closely the observed d i s t r i b u t i o n would be expected to conform to the true d i s t r i b u t i o n of the f i e l d can now be inferred from the t h e o r e t i c a l data of figure 19 which shows the d i s t r i b u t i o n of the product of 50 random numbers. This d i s t r i b u t i o n had barely passed a chi-squared test for log normality. Since the d i s t r i b u t i o n of the v e l o c i t y data was formed from a larger number of samples, i f i t was t r u l y log normal, the observed histogram would have agreed with the normal curve even better than was the case for the product of 50 random numbers. It i s thus concluded that the sampling was adequate. 50 The question of s t a t i o n a r i t y was investigated through use of the non-parametric runs test described by Hoel ( I 9 6 0 ), Chapter I I . The type of non-stationarity looked for was one which would be associated with fluctuations i n the mean wind speed and would have a time scale of the order of a s i g n i f i c a n t f r a c t i o n of a minute or more. I t was therefore decided to sub-sample the d i g i t i z e d signal at about one sample per second to obtain the data for the t e s t . The observed number of runs obtained was 1.25 standard deviations l e s s than the value expected. While t h i s did not confirm s t a t i o n a r i t y at a high l e v e l of confidence i t was well outside the 5 or 10% c r i t i c a l region of any test hypothesizing that the f i e l d was not stationary. Moving the assumed median, which was located 16 analog to d i g i t a l converter increments from the zero of the record by one increment changed the difference to two standard deviations in d i c a t i n g the test was f a i r l y s e nsitive. I t was concluded therefore that non-stationarity did not contribute s i g n i f i c a n t l y to the disagreement between theory and experiment. 3.5 X-wire Data One run of data which was recorded using two Disa type 55D05 constant temperature hot wire anemometers was also examined in terms of the log normal model. The experimental setup featured the two wires oriented in an X configuration. The X was contained i n a plane defined by the v e r t i c a l and the 51 d i r e c t i o n of the mean wind. I t was designed so that one wire would not operate i n the wake of the other. Each wire formed an angle of 45 degrees with the v e r t i c a l . The wire which sloped downward along the d i r e c t i o n of the mean wind i s henceforth referred to an XI. The wire sloping upward i s referred to as X2. The measurements were made i n the atmospheric boundary layer over the ocean at the Spanish Banks s i t e . The probe was mounted 2 meters above the mean water surface and the mean wind speed was 5«2 m/sec. The run analyzed was of 8 minutes duration. As with the downstream v e l o c i t y data, care was taken during both the recording and d i g i t i z i n g phases to insure that any f i l t e r s used had a l i n e a r with frequency phase s h i f t . I t was also important that pairs of f i l t e r s be phase matched so that the time delay corresponding to the slope of the phase character-i s t i c was the same for each wire. Except for the requirement of phased matched f i l t e r s the recording system was i d e n t i c a l to that described i n section 2.1 for the downstream v e l o c i t y data. The signals were d i g i t i z e d at a frequency of 2000 samples per second per channel. This was s l i g h t l y low for spectrum purposes as there was probably energy present beyond the f o l d i n g frequency of 1 KHz. The spectrum shown (Fig. 24 may be s l i g h t l y a l i a s e d i n the neighborhood of the fo l d i n g frequency but as the signal i s f a l l i n g o f f f a i r l y quickly with frequency the a l i a s i n g should not extend to much lower frequencies. The low sampling frequency would not a f f e c t the observed p r o b a b i l i t y d i s t r i b u t i o n s . 52 During the d i g i t i z i n g process aft e r the occurrence of a clock interrupt the two channels were sampled as quickly as possible. The time between sampling one channel and the other was of the order of 25 usec. Taking into account t h i s and the phase matching of the f i l t e r s , i t was considered the r e s u l t i n g two values were s u f f i c i e n t l y simultaneous observations of the d i f f e r e n t i a t e d XI and X 2 hot wire signals. By assuming the Taylor frozen f i e l d hypothesis these signals were then taken to represent s p a t i a l derivatives of the turbulent f i e l d i n the two d i r e c t i o n s , 45 degrees above and below the horizontal. The theory of Gurvich and Yaglom postulates that the square of any s p a t i a l derivative and the l o c a l d i s s i -pation rate should a l l be logarithmic normal quantities. It was therefore considered of inte r e s t to examine both the i n d i v i d u a l d i s t r i -butions of the above signals and that of a quantity defined as the sum of the squares of the above s p a t i a l derivatives. It was thought that t h i s l a s t quantity might look more l i k e the l o c a l d i s s i p a t i o n rate than a single component. It was also thought that t h i s quantity might conform somewhat better to the log normal model i n the region of small values. The p r o b a b i l i t y that the two signals would be very close to zero simultaneously should be less than for either of them to be close to zero i n d i v i d u a l l y . These comments, and the e a r l i e r one concerning the p r o b a b i l i t y of the downstream v e l o c i t y £ 53 derivative being close to zero, apply to the r e a l turbulent parameters and not to the recorded and d i g i t i z e d values for which these e f f e c t s may have been masked by the presence of noise. The decision to use the X-wire signals rather than the seemingly more l o g i c a l procedure of resolving the signals into horizontal and v e r t i c a l components was due to a d i f f i c u l t y i n e stablishing the c a l i b r a t i o n of the wires. The c a l i b r a t i o n was attempted i n s i t u by mounting the probe inside a sonic anemometer sensor.- Although the run used here exhibited the best agreement, the two measurements could generally not be completely reconciled. I t was f e l t however that the in d i v i d u a l wires were measuring, within a constant factor, v e l o c i t y fluctuations perpendicular to themselves. Therefore, and since the wires were similar and probably had c a l i b r a t i o n factors of the same order, i t seemed worthwhile to do the comparisons described above. A second d i f f i c u l t y with these data evidenced i t s e l f following the routine computation of a spectrum for quality control purposes. The spectrum i s shown in figures 24 and 25. As can be seen the signal contained two sharp noise peaks i n the v i c i n i t y of 5 Hz and 60 Hz. The 5 Hz peak was l i k e l y due to mechanical fluctuations i n the tape recorder speed. The 60 Hz peak i s almost c e r t a i n l y l i n e frequency picked up i n eith e r the recording or d i g i t i z i n g phases. Both noise peaks are quite narrow and would l i k e l y have n e g l i g i b l e e f f e c t on the large value end of the d i s t r i b u t i o n . -3.0--3.5 -4.0 £ -4.5 U J CL. C O L_ D -5.0 -5.5 -H-' -6.0 -| 1 — 1 1 1 1 1 0.5 1.0 1.5 2.0 2.5 3.0 3.5 LOG OF FREQUENCY IN HZ 54 Figure 24. Spectrum of the d i f f e r e n t i a t e d XI hot wire signal. 3.0-, 6.4 £ 4 . 8 H 3.2-0.0-1 r 1 1 1 1 — i 0.0 0.2 0.4 0.6 0.8 1.0 1.2 FOURIER COEFFICIENT NUMBER ( X 10 3 1 Figure 2 5 . Mean fourier amplitude versus fo u r i e r c o e f f i c i e n t number for the XI hot wire signal. Multiplying the fou r i e r c o e f f i c i e n t number by O .98 w i l l y i e l d the frequency corresponding jbo that c o e f f i c i e n t in Hz. 55 However as was pointed out i n the section on the downstream v e l o c i t y data, small amounts of noise can be disastrous to the small value end of the d i s t r i b u t i o n . I t was therefore considered advisable to use the method described i n section 3«3 "to remove th i s noise. The X2 hot wire spectrum evidenced the same ch a r a c t e r i s t i c s as that of the XI wire, i . e . , the 5 and 60 Hz peaks. Consequently both signals were treated to an i d e n t i c a l " f i l t e r i n g " and the d i s t r i b u t i o n s of the r e s u l t i n g XI, X2 and X l ^ + X2^ signals were computed. In obtaining the d i s t r i b u t i o n of X l ^ + X2^, a f t e r the f i l t e r i n g and the squaring and adding had been done a histogram was formed by assuming once again a fixed step i n a simulation of the analog to d i g i t a l converter. The analysis then proceeded i n a manner s i m i l a r to that applied to the other signals already discussed. Figures 26 and 27 show the d i s t r i b u t i o n of the logarithm of the quantity obtained by squaring and adding the f i l t e r e d X-wire signals. Except f o r the excess of small values t h i s data i s 8 very well described by the logarithmic normal d i s t r i b u t i o n . The deficiency of large values does not ex i s t i n t h i s case. There may even have been a s l i g h t excess. The f i t t e d t h e o r e t i c a l curve predicts there should have been observed 81 occurrences of values greater than the range of the analysis. In fact 164 were observed. The e f f e c t of the noise on the small value portion of the d i s t r i b u t i o n was much more d i f f i c u l t to assess for t h i s quantity than for the simple s p a t i a l derivatives. Only part of * G u i m a r d and Stegen (1973) reported a similar r e s u l t for g r i d turbulence. 56 0.24 -i LN SQUARED AND SUMMED X-WIRE5 Figure 26. D i s t r i b u t i o n of the sum of the uncalibrated squared and summed d i f f e r e n t i a t e d X-wire signals. The duration of the run was 8 minutes. 3 o.i 0.01 H 1 1 i 1 r — 1 -8 -6 -4 -2 0 2 4 LN SQUARED AND SUMMED X-NIRES Figure 27. Cumulative d i s t r i b u t i o n function of the uncalibrated squared and summed d i f f e r e n t i a t e d X-wire signals. The duration of the run was 8 minutes. 57 the noise associated with the two channels would be independent. The mechanical noise of the tape recorder and any picked up l i n e frequency or i t s harmonics would be correlated between the two X-wire signals. In addition the turbulent signals themselves would not be independent. Nevertheless an attempt was made to gain some f e e l i n g f o r the e f f e c t of the noise. The dependencies just discussed were ignored. Random normal and logarithmic normal number generators were used on a computer to simulate the two hot wire signals and t h e i r associated noises as closely as possible. The r e s u l t i s the cumulative d i s t r i b u t i o n of figure 28. It does not exhibit a straight l i n e behaviour i n the region of large values as independent l o g normal quantities do not add to produce a log normal quantity. There i s , as would be expected, an excess of occurrences i n the region of small values. Comparing t h i s figure to figure 27 i t seems that the noise once again cannot account for the magnitude of the excess of small values. However the shortcomings of the method of estimating the e f f e c t of the noise must be recognized and the p o s s i b i l i t y of t h i s quantity behaving log normally i n the region of small values cannot be ruled out. The f i t of the sum of the squared X-wire signals was s u f f i c i e n t l y good to encourage the computing of a value of Chi-squared. A second analysis of t h i s data was done using every l 6 t h point of the o r i g i n a l time s e r i e s to insure the independence of adjacent values. The value of Chi-squared obtained assuming the f i t t e d normal curve to be the true p r o b a b i l i t y density function was 3 6 . The value for r e j e c t i n g the hypothesis of 58 3 o . i o.oi -I 1 1 1 1 1 1 -8 - 6 - 4 - 2 0 2 4 NATURAL LOGARITHM OF X Figure 28. Cumulative d i s t r i b u t i o n function of the sum of two log normal and two normal random number series. The f i r s t two moments of each d i s t r i b u t i o n were chosen to approximate the estimated values associated with the X-wire signals. 59 l o g normality at the 95$ confidence l e v e l was 28 for the 17 degrees of freedom. Next the f i r s t two bars of the histogram were ignored under the assumption that they were seriously influenced by the noise. In t h i s case the value of Chi-squared was 9.8 for 15 degrees of freedom. The c r i t i c a l region of the test for 15 degrees of freedom began at a Chi-squared value of 25. Hence there was no evidence to r e j e c t the form of the d i s t r i b u t i o n beyond the second bar of the histogram as being log normal. A similar exercise to the one discussed i n section J.k i n connection with the downstream v e l o c i t y data was performed with the observed and t h e o r e t i c a l moments of the squared and summed X-wire data. Table II. shows the r e s u l t s . Moment Observed Theoretical Value Theoretical Value Integrated Over Value Range of System (Integrated 0 to op) 2 0.059 0.068 0.094 3 0.111 0.117 0.557 4 0.318' 0.321 23.6 C o e f f i c i e n t 86.9 68.5 2543. of Excess Table I I . Comparison of t h e o r e t i c a l and observed moments for the squared and summed d i f f e r e n t i a t e d X-wire data. The discrepancy between the observed value of the second moment and the t h e o r e t i c a l value integrated over the range of the system i s , assuming a log normal d i s t r i b u t i o n , due to the large excess of small values as shown i n figure 27. The t h i r d and fourth moments are l e s s sensitive to these small values, being determined by the large value portion of the d i s t r i b u t i o n . 60 Comparing the t h e o r e t i c a l values integrated over the range of the system to those integrated from 0 to 0 0 however reveals that even the second moment i s determined to a large extent by the portion of the d i s t r i b u t i o n beyond the recording-analysis system. In the case of the fourth moment i t i s more than 98% determined by values beyond the range of the analysis. After obtaining t h i s rather unexpected good agreement between the squared and summed X-wire data and the logarithmic normal theory i t was expected that one or both of the i n d i v i d u a l X-wire signals would show the same good agreement. This expectation turned out not to be r e a l i z e d . The i n d i v i d u a l d i s t r i b u t i o n s are shown i n figures 29 to 3 2 . In figures 29 and 30 the d i f f e r e n t i a t e d XI hot wire i s i n much more serious disagreement with the theory than the downstream v e l o c i t y data. The excess of small values i s t h i s time much too large to be accounted f o r by the electronic and mechanical tape recorder noises. It would seem that since a f a i r l y respectable straight l i n e can be drawn through the upper half of the cumulative d i s t r i b u t i o n , the signal might be composed of a large normal part adding to a logarithmic normal part. However as mentioned above t h i s normal part would have to contain far too much energy too to be explained by c i r c u i t or tape recorder noise. A further test was applied for the presence of a more or less broad band independently adding electronic white noise i n the s i g n a l . It had been found i n the case of the temperature data discussed e a r l i e r i n t h i s section that f i l t e r i n g off high frequency white noise tended to increase the c o e f f i c i e n t of 61 0 . l 4 8 n N A T U R A L L O G OF C ( D X 1 / D X ) Figure 2 9 . D i s t r i b u t i o n of the logarithm of the d i f f e r e n t i a t e d XI hot wire s i g n a l . The duration of the run was 8 minutes. N A T U R A L L O G OF C ( 0 X 1 / D X ) Figure 3 0 . Cumulative d i s t r i b u t i o n function of the logarithm of the d i f f e r e n t i a t e d XI hot wire si g n a l . The duration of the run was 8 minutes. 62 excess and improve the f i t to the theory. I t was also reasoned that i f such a noise was present i t s e f f e c t would be f e l t most severely i n the higher frequencies where the signal to noise r a t i o was poorest. Consequently the XI hot wire signal was f i l t e r e d by removing the f o u r i e r c o e f f i c i e n t s corresponding to the two highest frequency points i n the spectrum plot of figure 24. The c o e f f i c i e n t of excess decreased from 7,k to somewhat less than 7 and the f i t was even poorer. I t was also hard to conceive of the good agreement with the theory obtained for the squared and summed X-wires i f one of them contained that much noise. Based on these obser-vations and a consistency between the U wire and X wire obser-vations to be discussed below, i t i s f e l t that t h i s d i s t r i b u t i o n was the r e a l d i s t r i b u t i o n of that s p a t i a l derivative. The d i s t r i b u t i o n of the X2 hot wire data shown i n figures 31 and 32 i s i n somewhat better agreement with the theory than the downstream data. I f we regard the c o e f f i c i e n t of excess as a measure of the agreement with the t h e o r e t i c a l model and as a measure of the intermittency then the intermittency seems to depend on d i r e c t i o n . The values of 11 to 14 found for the various downstream v e l o c i t y signals examined f a l l n i c e l y between the values of 7.4 and 20.2 observed for the XI and X2 hot wires respectively. Suppose we accept, as evidenced by the e f f e c t of f i l t e r i n g o f f the high frequency d i f f e r e n t i a t e d small scale fluctuations, that the d i s t r i b u t i o n s are to a s i g n i f i c a n t extent determined by these fluctuations. This r e s u l t that the d i s t r i -63 0.48-, NflTURAL LOG OF C ( DX2/DX ) Figure 31• D i s t r i b u t i o n of the logarithm of the d i f f e r e n t i a t e d X2 hot wire sig n a l . The duration of the run was 8 minutes. -8 - 6 - 4 - 2 0 2 4 NATURAL LOG OF C ( DX2/DX ) Figure 3 2 . Cumulative d i s t r i b u t i o n function of the logarithm of the d i f f e r e n t i a t e d X2 hot wire signal. The duration of the run was 8 minutes. 64 bution of the d i f f e r e n t i a t e d turbulent signals depends strongly on d i r e c t i o n would then seem to support other r e s u l t s indicating that the approach to l o c a l isotropy i n the atmospheric boundary layer over the ocean i s somewhat slower than expected. It would also seem that a rather special r e l a t i o n exists between the two s p a t i a l derivatives i n order that the sum of t h e i r squares possesses the logarithmic normal d i s t r i b u t i o n . 3.6 Temperature Data Measurement of the small scale temperature f l u c t u -ations in the atmospheric boundary layer was, at the time t h i s work was i n progress, considerably more d i f f i c u l t than that of the v e l o c i t y fluctuations. The signal to noise r a t i o which could be achieved for the d i f f e r e n t i a t e d temperature was rather i n f e r i o r to the v e l o c i t y case and consequently the signals required much more f i l t e r i n g of both the analog and d i g i t a l v a r i e t i e s before useful d i s t r i b u t i o n s could be recovered. It has already been stated that care was taken with the analog f i l t e r s to insure the waveform was maintained i n a form suitable for computation of the signal d i s t r i b u t i o n s . Figures 33 and 34 show the log-log and l i n e a r spectra of the downstream d i f f e r e n t i a t e d E3 temperature signal (recorded at a height of 2 meters). It i s s i g n i f i c a n t l y "cleaner" than the temperature signal discussed i n section 3«3« However, i t i s characterized by some sharp noise peaks at 60 Hz and three of i t s harmonics and some white noise above a frequency of about 670 Hz. -3.0--3.5 -4. Q H -4.5 <=> -5.0 -5.5 2 T - 6 . 0 i r— -i 1 1 1 1 0.5 1.0 1.5 2.0 2.5 3.0 3.5 LOG OF FREQUENCY IN HZ 65 Figure 3 3 . Spectrum of the d i f f e r e n t i a t e d E3 temperature signal f o r tape RCA 1. The duration of the run was 8 minutes. Figure 3 ^ . Mean fou r i e r amplitude versus fo u r i e r c o e f f i c i e n t number for the d i f f e r e n t i a t e d E3 temperature signal of tape RCA 1. The factor to convert c o e f f i c i e n t number to frequency i n Hz i s 2 . 9 3 . 66 The cumulative d i s t r i b u t i o n function of t h i s data pr i o r to application of any d i g i t a l f i l t e r s i s shown i n figure 35« The data i s not a good f i t , deviating seriously from the th e o r e t i c a l prediction of log normality for a l l values of log c(dT/dx). I f the d i f f e r e n t i a t e d temperature signal does have the log normal d i s t r i b u t i o n then the e f f e c t of the noise i s being f e l t well beyond the mean of the d i s t r i b u t i o n . Also the d i s t r i b u t i o n function does not approach a l i n e a r behaviour even for the largest values of log c(dT/dx). I t would seem that i n spite of the signal to noise r a t i o being much improved over that of the section of temperature data previously discussed the properties of the d i s t r i b u t i o n have been destroyed. The spectra and d i s t r i b u t i o n s of the u n f i l t e r e d E l d i f f e r e n t i a t e d temperature signal, which was recorded simultan-eously at the 4 meter height, were q u a l i t a t i v e l y s i m i l a r to the E3 signal and are not presented. I t was stated in connection with the temperature data discussed in the e a r l i e r section that removing the noise from the signal caused the c o e f f i c i e n t of excess to increase. A consequence of f i l t e r i n g o f f high frequency noise i s that the high frequency turbulent signals are f i l t e r e d o f f with the noise. This corresponds to averaging the parameter over space to some scale size larger than the smallest for which s i g n i f i c a n t v a r i a t i o n s would have been found. In terms of the Yaglom theory as discussed i n section 1.2 t h i s could be interpreted as f a i l i n g to carry the subdivision as far as possible, hence l i m i t i n g the number of «<j i n equation 2. I f the number of ol^ remaining was large enough the f i l t e r e d d i s t r i b u t i o n should s t i l l have Figure 35 . Cumulative d i s t r i b u t i o n function of the d i f f e r e n t i a t e d downstream E3 temperature fluctuations for tape RCA 1. The duration of the run was 8 minutes. 68 conformed to the logarithmic normal model. I f not, we might expect to see a deficiency of large values similar to that observed i n the case of the v e l o c i t y data i n section 3.1 and attributed to a not large enough Reynolds number. a signal a f t e r high frequency noise has been f i l t e r e d o f f i t must be admitted that i t may have been a r e s u l t of the f i l t e r i n g rather than a deficiency i n the f i e l d i t s e l f . signal under discussion was used. The f o u r i e r c o e f f i c i e n t s corresponding to the noise peaks and the high frequency noise were removed as discussed in section 3*3. The cut-off frequency point above which a l l the f o u r i e r c o e f f i c i e n t s were set to 0 was allowed to take on several values. The values are indicated by the v e r t i c a l arrows i n figures 33 l a b e l l e d 1 to 3« For each value of cut-off frequency a d i s t r i b u t i o n was formed and the c o e f f i c i e n t of excess was computed. The r e s u l t s are presented in table I I I . Cut-off Cut-off Co e f f i c i e n t Point Frequency of Excess no f i l t e r i n g 3000 Hz 29.3 Hence i f a deficiency of large values i s observed i n To demonstrate these two e f f e c t s the E3 temperature 1 1266 Hz 39-2 2 710 Hz 43.1 3 448 Hz 36.1 Table III. Behaviour of the c o e f f i c i e n t of excess as a function of f i l t e r cut-off frequency for the d i f f e r e n t i a t e d E3 temperature si g n a l . 69 The c o e f f i c i e n t of excess has behaved as predicted. The value increased as more and more high frequency noise was f i l t e r e d off, reaching a maximum at a point where v i s u a l inspection indicated the signal was becoming greater than the noise. As the turbulent fluctuations began to be f i l t e r e d o f f the e f f e c t of the s p a t i a l averaging caused the frequency of occurrence of large values and hence the c o e f f i c i e n t of excess to decrease. The p r o b a b i l i t y density and cumulative d i s t r i b u t i o n functions corresponding to the f i l t e r i n g for which the maximum value of c o e f f i c i e n t of excess occurred are presented in figures 36 to 39• The f i l t e r i n g has produced a marked improvement in the f i t . There are however s t i l l some s i g n i f i c a n t departures from the t h e o r e t i c a l . In the pr o b a b i l i t y density functions of figures 36 and 38 there i s a noticeable excess of values i n the neighborhood of log c(dT/dx) = 0. This excess i s accompanied by d e f i c i e n c i e s at both higher and lower values of the logarithm. As discussed, i t was expected that the low pass f i l t e r i n g would introduce some deficiency of large values. That t h i s does happen i s demonstrated i n figure 39. The triangular symbols were taken from the cumulative d i s t r i b u t i o n function of the same data for the lowest frequency f i l t e r cut-off l a b e l l e d 3 i n figure 33• The deficiency of large values has increased with the severity of the low pass f i l t e r i n g . It must be admitted that the f i l t e r i n g might account for the deviation at large values. The deficiency of values in the area of log c(dT/dx) = -1.5 i s not explainable by anything done to the data during the 70 0.48-1 Figure 3 6 . D i s t r i b u t i o n of the f i l t e r e d d i f f e r e n t i a t e d downstream E l temperature signal from tape RCA 1 . The duration of the run was 8 minutes. The f i l t e r cut-off frequency was 710 Hz. 9 9 . 9 9 - | 9 9 . 9 9 9 o 50-1 ZD CD OC £ 10 0.01 -6 -4 -2 NATURAL LOGARITHM OF 0 2 C (0T/DX) Figure 3 7 . Cumulative d i s t r i b u t i o n function of the f i l t e r e d d i f f e r e n t i a t e d downstream E l temperature signal from tape RCA 1 . The duration of the run was 8 minutes. The f i l t e r cut-off frequency was 710 Hz. 7 1 0.48-1 0.40H NRTURflL LOGARITHM OF CtDT/DX) Figure 3 8 . D i s t r i b u t i o n of the f i l t e r e d d i f f e r e n t i a t e d downstream E 3 temperature signal from tape RCA 1. The duration of the run was 8 minutes. The f i l t e r cut-off frequency was 7 1 0 Hz. NATURAL LOGARITHM OF C(DT/DX1 Figure 3 9 . Cumulative d i s t r i b u t i o n function of the f i l t e r e d d i f f e r e n t i a t e d downstream E 3 temperature signal from tape RCA 1. The duration of the run was 8 minutes. The f i l t e r cut-off frequency was 7 1 0 Hz. The triangular points are from the cumulative d i s t r i b u t i o n of the data with the lowest frequency f i l t e r cut-off. 72 analysis. In spite of the improvement the chi-squared test denied the p o s s i b i l i t y of the experimental d i s t r i b u t i o n being drawn from a log normal population. This cannot however be interpreted as a re j e c t i o n of the theory due to the presence of the noise and the f i l t e r i n g i t s e l f . A numerical integration s i m i l a r to the one conducted on the v e l o c i t y data revealed that for t h i s data the 4 t h moment was approximately two-thirds determined by occurrences beyond the range of the recording-analysis system. As with a l l the data discussed an excess of small values was observed. In t h i s case i t was rather a smaller excess than with the v e l o c i t y data and i t seems l i k e l y i t could a l l be accounted f o r by the measurement noise. 3 . 7 Spectrum of the Fluctuations i n the Log Normal Quantities As discussed i n section 1 . 2 the extension of the log normal theory to consider two point moments lead to a prediction for the behaviour of the spectrum of the fluctuations i n the square of a s p a t i a l derivative. The prediction was that i n the i n e r t i a l subrange the spectrum would behave as 1^  where O ^  0_7 f_i | The prediction has been tested using the various s p a t i a l derivatives discussed i n the previous sections. Pond et a l . (1963) has suggested that a reasonable c r i t e r i o n f or the existence of l o c a l isotropy and an i n e r t i a l subrange would be that k l » 4 . 5 , wh ere h. i s the radian wave 73 number and 2 i s the height of the probe above the mean sea surface. For the d i f f e r e n t i a t e d v e l o c i t y data presented, the corresponding scale size would appear at a value of log frequency of approximately 0.5> Van Atta and Chen (1970) found that for a probe height of 3 meters the second order structure function began to deviate from the t h e o r e t i c a l l y expected i n e r t i a l subrange behaviour at a scale size of about one-quarter that 9 predicted by kz = 4.5* I t would therefore seem reasonable to expect the spectrum of the square of the fluctuations to follow the anticipated power law for frequencies between about 10 Hz and that corresponding to the peak of the d i s s i p a t i o n spectrum. In the figures which follow the p o s i t i o n of the peak of the d i s s i p a t i o n spectrum i s marked with a v e r t i c a l arrow. The straight l i n e s are regression or l e a s t squares l i n e s and have been f i t t e d to the data to the l e f t of the arrow. In addition a "confidence value" has been computed as I.96 times the standard deviation computed for the slope of the regression l i n e i n the usual manner. The value would represent some sort of 95% confidence l i m i t i n t h i s slope provided each point had the same number of degrees of freedom. This i s not true as the number of degrees of freedom increases with frequency which i s evidenced by the change i n size of the error bars as a function of frequency. The value does however give a f e e l i n g f o r the points of slopes of l i n e s which could reasonably be drawn through the points and i s quoted for that reason. For the v e l o c i t y data i t was found that the l i n e a r behaviour predicted f o r the spectrum of the square was true within 9 Note that kz = 27TA: where "u = 6 m/sec i s the wind speed and 2^  i s approximately 2 meters. 74 the s t a t i s t i c a l accuracy of the determination. The error bars 10 on the spectrum plots describe 95% confidence i n t e r v a l s i n the mean value of the spectral density at each frequency. The assumption i n computing these confidence i n t e r v a l s was that the mean of the d i s t r i b u t i o n was a normally d i s t r i b u t e d quantity independent of the shape of the d i s t r i b u t i o n from which the mean was formed. Figures 40 and 4l show the spectra of the square of the two d i f f e r e n t i a t e d downstream v e l o c i t y signals. The respective values of -0.60 + 0.02 and -0.59 ± 0.05 for slope of the f i t t e d l i n e s represent s l i g h t l y higher negative slopes than the value of -0.5 reported as t y p i c a l by Van Atta and Chen (1970). A further comparison can be made with the values of -0.62 reported by Pond et a l . (1963) and -0.6 reported by Gurvich and Zubkovski (1963). The spectra of the squares of the d i f f e r e n t i a t e d X-wire signals and th e i r sum were also computed. The signals were f i l t e r e d to remove the 60 Hz noise peaks as previously discussed. The mean was then subtracted and the spectrum computation proceeded i n the usual manner. Figures 42 and 43 are the spectra of the in d i v i d u a l squared X-wire signals. Figure 44 i s the spectrum of the sum of these squared signals. The slopes of the l i n e s are a l l generally lower than the downstream case, the values ranging from -0.44 to -0.55« However considering the accuracy of the measurement only the value of -0.44 + 0.05 for the XI wire could be said to be possibly s i g n i f i c a n t l y d i f f e r e n t from the previous values. This _ _ _ _ _ — (+ 2 standard deviation i n the mean) 75 Figure 40. Spectrum of the fluctuations i n the square of the d i f f e r e n t i a t e d downstream v e l o c i t y , run 2B tape 141. The slope of the l i n e i s -0.60 + 0.02. The v e r t i c a l arrow i s at the frequency of the peak of the d i s s i p a t i o n spectrum. Figure 4 l . Spectrum of the fluctuations i n the square of the d i f f e r e n t i a t e d downstream v e l o c i t y , run 2 tape 96. The slope of the l i n e i s -0.59 + 0.05. The v e r t i c a l arrow i s at the frequency of the peak of the d i s s i p a t i o n spectrum. 76 LOG OF FREQUENCY IN HZ Figure 42. Spectrum of the fluctuations i n the square of the d i f f e r e n t i a t e d XI hot wire sig n a l . The slope of the l i n e i s -0.44 + 0 . 0 5 . Figure 43. Spectrum of the fluctuations in the square of the d i f f e r e n t i a t e d X2 hot wire signal. The slope of the l i n e i s - 0 . 5 0 _ 0 . 0 6 . o -5-1.0 1.5 LOG OF 2.0 FREQUENCY 2.5 IN HZ 3.0 3.5 Figure 44. Spectrum of the fluctuations of the sum of the squared XI and X2 hot wire signals. The slope of the l i n e i s -0.55 + 0.05. LOG OF FREQUENCY IN HH Figure 45. Spectrum of the fluctuations i n the squared d i f f e r e n t i a t e d downstream v e l o c i t y s i g n a l . The d i g i t a l sampling frequency was reduced to 2000 Hz for t h i s case. The slope of the l i n e i s -0.57 + 0.02. 78 was also the signal which showed the greatest deviation from log normal behaviour i n a previous section. I t should be r e c a l l e d that the X-wire signals were d i g i t i z e d with a lower sampling frequency than the downstream signals. Consequently there was a small amount of a l i a s i n g i n the X-wire spectra. I t would seem possible that the frequency multiplying e f f e c t of the squaring might have aggravated t h i s problem and t h i s could have contributed to the lower slopes of the regression l i n e s -To v e r i f y that t h i s was u n l i k e l y the data corresponding to the downstream v e l o c i t y signal of figure 40 was subsampled at a frequency of 2000 Hz. The spectrum of i t s square was then computed and i s presented i n figure 4 5 . The new slope of the l i n e i s - 0 . 5 7 + 0 . 0 2 as compared to - 0 . 6 0 . This would indicate the a l i a s i n g was not s u f f i c i e n t l y important to account for the lower slopes provided only frequencies below the peak of the d i s s i p a t i o n spectrum are considered. Figures 46 and 47 show the spectra of the squares of the d i f f e r e n t i a t e d E l and E 3 temperature signals discussed i n the previous section. The signals were f i l t e r e d as were the X-wire signals and had t h e i r means removed. Next they were squared and t h e i r spectra were computed. The regression l i n e slopes of - 0 . 4 4 + 0 . 0 7 and - 0 . 4 7 + 0 . 0 7 are approximately what would be expected and compare well with the X-wire values within the experimental s t a t i s t i c a l accuracy. However the l i n e a r behaviour was not confirmed by t h i s r e s u l t . Considering the signal to noise r a t i o problems t h i s was not surprising. 79 Figure 46. Spectrum of the square of the f i l t e r e d d i f f e r e n t i a t e d downstream E l temperature fluctuations. The slope of the l i n e i s -0 .44 + 0 . 0 7 . >-L O G Q F F R E Q U E N C Y I N H Z Figure 47. Spectrum of the square of the f i l t e r e d d i f f e r e n t i a t e d downstream E3 temperature fluctuations. The slope of the l i n e i s - 0 . 4 7 + 0 . 0 7 . 80 3•8 Structure Functions, Skewness and Flatness Factor The structure functions and the skewness and flatness factors have been computed for two runs of v e l o c i t y data and one of temperature data. The records available f o r the v e l o c i t y data were unfortunately rather short. This was because of technical d i f f i c u l t i e s during the measurements. The undifferentiated v e l o c i t y signal of run 2 tape 96 did not get recorded properly and could not be used. The same signal for run 2B tape l4l was usable for only the f i r s t three minutes before problems developed. Also i t was not f e l t that the c a l i b r a t i o n s available for the various runs were of s u f f i c i e n t accuracy to warrant normalization of the structure functions by the Kolmogoroff microscale or v e l o c i t y scale. The v e l o c i t y data were therefore examined only i n terms of slope. In a l l cases the s p a t i a l separations were derived from time separations by assuming the "frozen f i e l d " hypothesis and do not represent two point measurements. On each figure two v e r t i c a l arrows appear. The one appearing at the smaller separation represents the scale corres-ponding to the peak of the d i s s i p a t i o n spectrum. The one appearing at the larger separation i s defined by kz = 4.5» Here k i s the radian wave number and z i s the probe height above the water. The r e l a t i o n kz » 4.5 has been suggested by Pond et a l . (I963) as a reasonable condition for l o c a l isotropy. 81 In addition the figures showing the observed second and fourth order structure functions have two l i n e s drawn through the observed data points. The s o l i d l i n e i n each case represents the slope of the theory which assumes £ to be a log normal quantity. The dashed l i n e i s the th e o r e t i c a l slope of the structure functions assuming £ i s a constant. The modifications to the slopes of the th e o r e t i c a l l i n e s as discussed i n section 1.3 were calculated from the value of OT where t - was obtained from the slope of the spectrum of the square of the d i f f e r e n t i a t e d downstream v e l o c i t y f l u c t u -ations i n the i n e r t i a l subrange. A value of -0.58 was used for -1 +(r, . This was obtained experimentally from the v e l o c i t y data in the e a r l i e r section. Figures 48 and 49 show the second order structure functions for the two v e l o c i t y runs. In figure 48 the s o l i d l i n e of the modified theory i s a s l i g h t l y better f i t to the data i n the neighborhood of the peak of the d i s s i p a t i o n spectrum and through both s l i g h t l y larger and smaller scale sizes. For scales approaching that corresponding to kz = 4.5 the experimental data has begun to droop below both t h e o r e t i c a l l i n e s . This i s e s s e n t i a l l y the behaviour found by Van Atta and Chen (1970)» p a r t i c u l a r l y f or th e i r lowest probe which was located three meters above mean water l e v e l . Their agreement with the modified theory was also best at the smaller separations. The point at which the structure function begins to droop below the l i n e of the modified theory occurs at a 82 Figure 48. Second order structure function of the downstream v e l o c i t y , run 2B tape 141. The s o l i d l i n e has a slope of 0.71, the dashed a slope of 2/3. The arrows represent the separations corresponding to the peak of the d i s s i p a t i o n spectrum and kz = 4.5. i -i LOG OF S E P A R A T I O N IN CM Figure 49. Second order structure function of the downstream v e l o c i t y , run 1 tape 128. The s o l i d l i n e has a slope of 0.71, the dashed a slope of 2/3. separation of approximately one-tenth that corresponding to kz = 4.5. A corresponding droop was found by Van Atta and Chen (1970) to occur at a separation corresponding to about one-quarter of that defined by kz = 4.5. The difference might be attributable to the lower probe height or d i f f e r i n g wave conditions. The waves during the observations discussed here were s l i g h t l y larger than one-half meter crest to trough. I t was considered beyond the scope of t h i s work to attempt any determination of the e f f e c t , i f any, of the waves on the turbulence. As has been found by others the extent of the conformity of the observed second order structure to the predicted i n e r t i a l subrange behaviour was much l e s s than the extent of the conformity of the spectrum to the predicted -5/3 behaviour. The second order structure function for the other data run i s shown i n figure 49. The behaviour i s q u a l i t a t i v e l y similar to that discussed. I t does begin to deviate from the modified theory at a s l i g h t l y smaller separation however. This could be due to the f a c t that the probe was mounted only 1.5 meters above the surface i n t h i s case. The t h i r d order structure functions for these sections of data are presented in figures 50 and 51* In both cases the f i t to the t h e o r e t i c a l slope of 1 i s f a i r l y good except for droop i n the region of separations of 1 to 2 cm. It was f e l t the droops were not s i g n i f i c a n t as the runs were much too short to determine the odd moment with any accuracy. Figure 5 0 . Third order structure function of the downstream v e l o c i t y , run 2B tape l 4 l . The l i n e has a slope of 1. Figure 5 1 . Third order structure function of the downstream v e l o c i t y , run 1 tape 128. The l i n e has a slope of 1. 85 The skewness factors as defined by equation 13 and for the two runs of v e l o c i t y data are presented i n figures 52 and 53• Considering the behaviour of the t h i r d order structure functions just discussed, i t i s not surprising that there i s l i t t l e or no evidence of the th e o r e t i c a l plateau where the skewness should have behaved as r . This was contrary to the r e s u l t s of Van Atta and Chen (1970) who found a broad plateau The condition that the r e l a t i o n Sc^l --oiooLK') *" of equation 14 holds i s that Y~ l i e s i n the Kolmogoroff i n e r t i a l subrange. Considering the behaviour of the t h i r d order structure function i t appears l i k e l y that i f the r e l a t i o n i s to hold at a l l i t w i l l be at separations of the order of the peak of the d i s s i -pation spectrum and only s l i g h t l y larger. Consequently, i n estimating a value of the skewness factor for purposes of computing K only the points having separations between the peak of the d i s s i p a t i o n spectrum and that where the t h i r d order structure function showed i t s droop were used. This requirement reduced the plateau i n the three v e l o c i t y runs treated to one or two points. Table IV shows the r e s u l t s . Run-Analog Tape Skewness Factor 2-96 -0.25 0.54 2B-141 -0.22 0,59 1-128 -0.23 0.57 Table IV. Values of skewness factor and from the structure functions of the downstream v e l o c i t y data. 86 0 . 6 ° ? 0 . 2 0 . 0 0 a o a a a a a T a t o 0 1 1 1 1 1 I  1  - 1 0 1 2 3 4 L O G O F S E P A R A T I O N IN CM F i g u r e 52. Skewness f a c t o r f o r the downstream v e l o c i t y , r u n 2B tape l 4 l . The arrows r e p r e s e n t the s e p a r a t i o n s c o r r e s p o n d i n g t o the peak o f the d i s s i p a t i o n spectrum, the maximum s e p a r a t i o n c o n s i d e r e d i n e v a l u a t i n g K', and kz = 4.5. 0 . 6 - , E 0-4 U J ? 0 . 2 - T • T o.o-0 1 2 3 L O G O F S E P A R A T I O N IN CM F i g u r e 53. Skewness f a c t o r f o r the downstream v e l o c i t y , r u n 1 tape 128. The arrows r e p r e s e n t the s e p a r a t i o n s c o r r e s p o n d i n g t o the peak o f the d i s s i p a t i o n spectrum and kz = 4.5. 8 7 These values of skewness factor can be compared to a measured value of -0.18 reported by Van Atta and Chen ( 1 9 7 0 ) . Other measured values were -0.39 and -O.36 reported by Stewart as a correction to some values reported by Gurvich (i960). The values of K' ranging from 0.5^  to 0.59 are s l i g h t l y higher than what i s generally thought to be the proper value. Paquin and Pond (1971) reported a value of O.56. Van Atta and Chen (1970), however, reported a value of O.58 determined experimentally from the measured second order structure function. Other higher values have been reported but these have been determined by observing energy spectra, a technique that can produce high values i f contributions to the i n t e g r a l from beyond the i n e r t i a l subrange are non-negligible. That t h i s does happen has been demonstrated by Van Atta and Chen (1970). A further property of the skewness factor i s of i n t e r e s t . There i s a dimensional argument which predicts that as the separation goes to zero the skewness factor w i l l tend to an absolute constant. This constant would be the skewness of the v e l o c i t y derivative. Table V shows the value of skewness for six runs of d i f f e r e n t i a t e d downstream v e l o c i t y data. The c o e f f i c i e n t of excess which i s also tabulated here has been discussed i n the section on the log-normal model. 88 Run-Analog Duration Skewness of Coe f f i c i e n t Mean Probe Tape Derivative of Excess Wind Height 2-96 240 sec -O.69 13.1 6m/sec 2m 4-96 240 sec -0.71 12.3 6m/sec 2m 1-128 120 sec -O.56 9-3 6.5m/sec 1.5m 1-130 90 sec -0.76 17.9 5m/sec 5m 1-131 120 sec -0.71 16.6 4.35m/sec 3.7m 2B-141 515 sec -O.56 10.6 7.8m/sec 2m Table V. Skewness and Co e f f i c i e n t of excess for 6 runs of di f f e r e n t i a t e d downstream v e l o c i t y data. To obtain some f e e l i n g as to whether the values of -O.56 were s i g n i f i c a n t l y d i f f e r e n t from -0.7, run 2B tape 141 was divided into 10 sections of about 50 seconds each. For each section the skewness and c o e f f i c i e n t of excess was computed. The r e s u l t s are presented i n Table VI. Section Skewness Coe f f i c i e n t of Excess 1 -0.63 10.4 2 -0.53 9.5 3 -0.51 9-3 4 -0.55 10.5 5 -0.49 7.4 6 -0.53 9-3 7 -0.59 9.3 8 -0.53 7-0 9 -0.61 10.9 10 -0.42 7.9 Table VI. Skewness and c o e f f i c i e n t of excess for 10 f i f t y second portions of the d i f f e r e n t i a t e d downstream v e l o c i t y run 2B tape l4l. 89 The mean and standard deviation i n the mean for the skewness i s -0.54 + 0.06 and for the c o e f f i c i e n t of excess i s 9 . 2 + 1 . 2 . I t would therefore seem that the differences are in fact s t a t i s t i c a l l y s i g n i f i c a n t . No explanation i s offered. As has "been reported by various people, the fourth order structure functions were smooth monotonically increasing functions of separation. They are presented i n figures 5^ and 55« For* the range of separation from 2 to 25cm the lower slope l i n e of the modified theory i s the better f i t . At t h i s point i t was decided to use a lea s t squares f i t to the second and fourth order structure functions to obtain an objective estimate of the slope. The f i t was confined to the four points having separations between 2 and 25cm. The choice was based solely on the observation that for these separations the data appeared to f i t the modified theory well. In addition to estimating the slope, an expected standard deviation i n the slope was computed by the usual technique for l i n e a r regression. This figure was useful i n estimating the range of slopes of l i n e s which could reasonably be drawn through the data points. The r e s u l t s are presented i n table VII. Tape Structure Function Slope St. Dev. in Slope 141 2nd 0.70 0.007 128 2nd 0.71 0.012 141 4th 1.23 0.013 128 4th 1.22 0.013 Table VII. Slope and expected standard deviation i n the slope computed for the second and fourth order structure functions by l i n e a r regression. The th e o r e t i c a l values are 0.71 and 1.24 respectively. The computation included only points corresponding to separations between 2 and 25cm. L O G O F S E P A R A T I O N IN CM Figure 54. Fourth order structure function of the downstream Ihe°SashedrV3f P* ' T h e S l ° P e ° f t h e S o l i d l i n e i s 1'2^» Figure 55. Fourth order structure function of the downstream v e l o c i t y , run 1 tape 128. The slope of the s o l i d l i n e i s 1.24, the dashed 4/3. 91 It would seem the f i t to the modified theory i n the range of separations 2 to 25cm i s good. It would also seem un l i k e l y that the agreement between the runs and between the data and the theory could be a s t a t i s t i c a l accident. This would suggest that the slope of the second and fourth order structure functions i s a well behaved quantity and i s s t a t i s t i c a l l y s i g n i f i c a n t even over r e l a t i v e l y short periods of time. The flatness factors for the two runs are presented i n figures 56 and S7- As was found by Van Atta and Chen (1970) the flatness factor behaved something l i k e r * over i n t e r -mediate values of separation. However contrary to t h e i r r e s u l t s there was l i t t l e evidence of an r ~ 0 « 2 2 behaviour towards smaller separations. The structure functions of the E l temperature signal which was recorded at the 4 meter height, the skewness factor and the flatness factor were also computed. The duration of t h i s run was 8 minutes. It was noted during a routine spectrum computation for q u a l i t y control purposes that a small noise peak existed at 60 Hz. The data was therefore f i l t e r e d i n the manner discussed previously and the structure functions were computed once more. The r e s u l t s presented correspond to the f i l t e r e d data. There was a change i n the behaviour of the second order structure function after f i l t e r i n g . This change i s shown i n figure 5 8 . The triangular points represent the structure function of the u n f i l t e r e d data, the c i r c u l a r points the structure function a f t e r f i l t e r i n g . The structure function of the f i l t e r e d data i s a reasonable f i t to the l i n e with the 2 / 3 slope. However since the (_> d 1.5 cn 1.0 tn L L J z i — t x LT1 0. 5 • CD O 0 . 0 ' 0 1 2 3 L O G O F S E P A R A T I O N I N CM Figure 5 6 . Flatness factor of the downstream v e l o c i t y , run tape 141. The s o l i d l i n e s have slopes of -0.22 and -0.111. 1.5-1 CI u_ <ji 1.0 tn cn LTJ 0.5 0.0 —1 1 1 \— 0 1 2 3 L O G O F S E P A R A T I O N I N CM Figure 5 7 . Flatness factor of the downstream v e l o c i t y , run tape 128. The s o l i d l i n e s have slopes of -0.22 and -0.111. 93 Figure 5 8 . Second order structure function of the downstream temperature, E l , tape RCA 1. The triangular points represent the structure function p r i o r to f i l t e r i n g o f f the 60 Hz noise. The dashed l i n e has a slope of 0 . 7 2 , the s o l i d 2 / 3 . 0 i L O G O F S E P A R A T I O N I N CM Figure 5 9 . Third order structure function of the f i l t e r e d E l temperature. The l i n e has a slope of 1 . 94 e f f e c t of the f i l t e r i n g was to increase the slope the r e s u l t i s inconclusive. Other noise i n the signal might cause a too low slope. The t h i r d and fourth order structure functions were unchanged by the f i l t e r i n g . The t h i r d order structure function of figure 59 appears somewhat better behaved than i t s v e l o c i t y counterpart. When computing the temperature structure functions i t was thought that some difference might occur which would indicate why the temperature derivative was a better log normal quantity than the v e l o c i t y derivative. This t h i r d order structure function was the only such p o s s i b i l i t y found. It would seem that the temperature structure might not suffer the same physical constraints of the water surface and the waves as the v e l o c i t y f i e l d . The oi-i of the Yaglom theory might be s e l f - s i m i l a r , independent and i d e n t i c a l l y d i s t r i b u t e d to greater scale s i z e s . However the differences i n the behaviour of the t h i r d order structure functions of the v e l o c i t y and temperature would seem i n s u f f i c i e n t to account for the difference. It i s worth noting that the log normal d i s t r i b u t i o n does not require the assumptions of s e l f - s i m i l a r i t y and i d e n t i c a l l y d i s t r i b u t e d <*y ; only independence and f i n i t e means and variances. The theory of both the structure functions and the l i n e a r behaviour of the spectrum of the square of the derivatives does require the notions of a Kolmogoroff i n e r t i a l subrange. It would seem necessary to entertain the p o s s i b i l i t y that the l i n e a r behaviour of the spectra of the squares of the derivatives i s pa r t l y 95 fortuitous i n the manner of the "-5/3" law. This l i n e a r i t y at least i n the v e l o c i t y case extends to much greater scales than would "be expected from the behaviour of the structure functions. Therefore i t does not seem unreasonable that the temperature signal should have an almost log normal d i s t r i b u t i o n even though the structure functions do not indicate a much more extensive i n e r t i a l subrange. The fourth order structure function i s shown i n figure 6 0 . There i s a l i n e a r region over an a p r i o r i reasonable range of separations. The slope i s 0.94. Van Atta (1971, 1973) has suggested a modified theory for the scalar structure functions taking into account the variations i n e and i n the scalar d i s s i p a t i o n x • The slope i s given by 5n (l-n/6)/12 + pn 2/24. Here n i s the order of the structure function and p i s the c o r r e l a t i o n between e and x • I f p i s assumed to be equal to +2/3 the predicted slopes are O.67 and 1 . 0 0 for the 2nd and 4th order cases. This agrees with what was found here within the experimental accuracy. The skewness factor i s shown i n figure 61. I t f a i l s to exhibit a plateau where a lower law behaviour might occur. The flatness factor i s shown i n figure 6?. The low pos i t i v e slope of the fourth order structure function has translated to a large negative slope of -0.42. Using a value of +2/3 for p i n the Van Atta modified theory leads to an expected slope of - 0.34. 96 2n Figure 60. Fourth order structure function of the f i l t e r e d E l temperature. The s o l i d l i n e has a slope of 1.22, the dashed a slope of 0 . 9 4 . Figure 62. Flatness factor of the f i l t e r e d E l temperature sig n a l . The s o l i d l i n e has a slope of -0.42. 9 8 3•9 Experimental Data on the I n e r t i a l Transfer of Energy The t r a n s f e r of energy due to the i n e r t i a l terms i n the equations discussed i n s e c t i o n 1 . 4 was computed f o r two sec t i o n s of downstream v e l o c i t y data. As discussed i n s e c t i o n 1 . 4 the rate of t r a n s f e r of energy between f o u r i e r components having wavenumbers J? , m and k i s given by a sum of three terms which cont r i b u t e only f o r c e r t a i n values of the wavenumbers. S p e c i f i c a l l y the term c o n t r i b u t e s when two wavenumbers sum to equal the t h i r d . The three terms are 1. - C j C ^ C f a Cos (9-jt - P-w, + ) . when X - m + k = 0 2 . - C^C^Cfe Co^(6^ ' W h 6 n * " m " k = ° ( E q * 1 8 ) 3 - +. Cjc^c* C o $ (e* + - ' w h e n ^ + m - k = 0 I t was a l s o pointed out that the r a t e of a r r i v a l of energy at a wavenumber £ could be computed by h o l d i n g Sl constant and summing over a l l values of m and k. In performing the computations t h i s has been done separately f o r each of the above terms. For s i m p l i c i t y the cases have been t r e a t e d as re p r e s e n t i n g three types of i n t e r -a c t i o n s , designated 1 to 3 i n the order i n which they appear above. The type 1 i n t e r a c t i o n can be described as f o l l o w s . Since m = k + k and a l l wavenumbers are p o s i t i v e , m i s always greater than Sl . Al s o k can be greater than or l e s s than Sl but i s always l e s s than m. This i s shown g r a p h i c a l l y i n f i g u r e 6 3 . 99 In the descriptions that follow i t should be r e c a l l e d that the k represents the wavenumber of the fo u r i e r component involving the d i f f e r e n t i a t e d v e l o c i t y signal i n the interaction term. Spectrum of Downstream Velocity wavenumber > Figure 63. Graphical representation of the r e l a t i o n s between the wavenumbers for a type 1 inte r a c t i o n , Jl = m^  - k^. Thus i n the f i r s t case the interaction i s between a v e l o c i t y component of wavenumber Jl , a v e l o c i t y component of wavenumber m > Jl (a smaller scale) and a v e l o c i t y derivative of wavenumber k = m - Jl, either smaller or larger than Jl . The second type of inte r a c t i o n (for Jl - m + k) corresponding to the second term above i s shown graphically i n figure 6 4 . When Jl i s held constant the inte r a c t i o n term 2 represents the rate of a r r i v a l of energy at wavenumber Jl due to interactions of the v e l o c i t y component at wavenumber Jl with a ve l o c i t y component at wavenumber m and a v e l o c i t y derivative component at wavenumber k where k and m are both le s s than J? (both scales are larger than $ ). 100 Spectrum of Downstream Velocity Ki kx. **** JL wavenumber > Figure 64. Graphical representation of the r e l a t i o n s between the wavenumbers for a type 2 i n t e r a c t i o n , £ = + k^ . Spectrum of Downstream Velocity wavenumber > Figure 65. Graphical representation of the r e l a t i o n s between the wavenumbers for a type 3 inte r a c t i o n , Jl = k i - m^ . Figure 65 i s the graphical representation of the t h i r d type of in t e r a c t i o n for which k = J? + m. This interaction has the v e l o c i t y derivative component at the smallest scale s i z e . In terms of the cosine and sine c o e f f i c i e n t s A and B obtained by f o u r i e r transforming a section of d i f f e r e n t i a t e d v e l o c i t y data the term - c - ? C ^ C k Cos -9-^ -reduces to L 101 The other terms reduce to similar products but with the negative sign occurring elsewhere. The negative sign i s i n each case associated with the product of an A and two B's which has the A subscripted with the wavenumber that i s the sum of the other two. Therefore for given values of k, H and m the three terms above w i l l , except for d i v i s i o n by the factor ^m, reduce to the same number. The length of the signal to be analyzed was such that the entire sample could not be held i n computer memory. The runs were therefore broken up into 2048 point sections. For each section the f o u r i e r transform of the data was computed and four running sums of the t r i p l e products of the A*s and B's shown i n equation 1 9 were kept. This allowed computation of the means of the t r i p l e products over the records. The l i m i t a t i o n s of computer memory also did not allow formation of the means of the t r i p l e products for a l l in d i v i d u a l values of the indices. I t was necessary to reduce the number of such means by grouping together f o u r i e r components when forming averages. The grouping scheme was chosen to give estimates of the means of the t r i p l e products at approximately equally spaced values of log frequency or log wavenumber. The r e s u l t was f o r each data run a 26 X 26 array of numbers representing the values of the mean over the record of any of the terms such as """ ^ C ^ C f t Cos - ^ A n - P ^ ) » f o r g i v e n ranges of £ , m and k. 102 Figure 66 shows the array of numbers for one such data run. Only h a l f the array i s shown as i t i s symmetric (m,k). The axes are (m,k) and the elements may be designated S(m,k). The value annotated at a point i s the mean value of the above term for three wavenumbers, two of which are derived from the frequencies on the axes and the t h i r d of which i s derived from the sum of these two frequencies. The rates of a r r i v a l of energy at wavenumber Jl due to the types of interactions discussed were computed separately from the array. Each element was divided by the appropriate value of Pm and then summing was performed over m and k while holding Jl constant. For the inte r a c t i o n of type 1 , wavenumber Jl appears on the horizontal axis i n figure 66, k appears on the v e r t i c a l axis and m i s the sum. For each array element, S(<£, k), the mean value of l/(4jPm) for a l l the fou r i e r components which contributed was formed. This mean value was then multiplied by the value of the element to form a new element Sji^ / % (^+k) . Calculating the rate of a r r i v a l of energy at wavenumber Jl was then accomplished by summing v e r t i c a l l y up the array column for each value of Jl . Computation of the energy flow for the second type of inter a c t i o n was more d i f f i c u l t . The wavenumber which was to be held constant was the sum and did not appear on an axis. I f the axes had been l i n e a r with frequency the path of summing for constant Jl would have proceeded along a diagonal. However since the axes were l i n e a r i n log frequency a l i n e of constant Jl followed a curved path. For purposes of summing over k and m an element S«n)?/(m+k)m of the array was taken to contribute to the energy flow into the wavenumber of the closest l i n e of constant Jl - m+k. 1000 100 10 oyu 0.^ 02 C.U O.^ U 0.^ 11 0^ 12 oy2 0M2 0.08 oyo 0.10 oyo 0.^3 0.09 0.09 oyo oye 0..0B oy 1 0.^7 0.09 0.09 t 0.+13 0.10 0.11 1 0.09 0.07 1 0.07 oyo o^IO oy2 o y 9 o 0. 19 0.21 0.06 0.11 0. II 0.10 0..09 0..06 t t I 0.26 0.22 0.16 0. 13 0 ..11 0..11 0.06 0.07 0 .OS oyo oyu 0.+05 0.06 oys ' ' oya oyu oy>2 o.^ os oyi2 O.jOT O.^ IS 0.07 0.03 0.09 Q.jD7 0.07 0.07 oyi? o.pu oye 0.03 T oya 0.02 0.06 o.oa 0.05 _ o y e o.fx 0.08 0.06 o.p7 a y s oys O.flU 0.08 0.06 I 1 1 \ H oyu a y s oyu oyu oya oy2 oys oy2 0.02 O.JM \ 0..02 o y i T T o.oi o.(oa o y z oya o.oi 0.03 0.02 oya oya oya 0.03 0.03 0.02 oyu oy2 oyu oya oya oy s oys o y i oy2 o.jQi 0.020.01 o.+oo -^.oo o y i o.oo o.oi o.oi o.oo ~r <—r 0.02 0.01 0.00 Q.,02 o y i .01 0.01 ! I o y i o.oi oyz o.oi o y i 0.02 o y i o y i oyo o y i o y i o y i o y i o y i oy2 o y i oyo oy2 o y i o.oi 0.02 o.oi o.oo T r 0.02 o.oi oyo o y i oyo oyo o y i oya oyo o.oi o.oo o.oo 0.38 0.15 0.20 0.17 0.08 0.16 0.11 0.09 0.09 0.011 0.03 0.02 0.03 0.01 0.01 0.01 0.01 0.00 0.00 0.00 0.00 1 I I I 0^09 O.^JO o.^n O.^J o-ps O.+D I n i i 0.U9 0.31 0.07 0.09 0.00 0.07 0.07 0.08 D.,05 O..0tl 0.03 0.02 0.01 0.01 O.OI 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 'I I 1 I 1 -^oi o.oa I 1 T I t o y i oyo ayo o^oi O.^ JO o.^ ao 0.01 0.00 " I I 0.^ 00 0.00 0.00 0.00 oyo oyo O.OT o.oo o.oo O.flO O.OO oyo o.(oo O.,00 Q.,00 oyo oyo o.oo 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 o.^ ao a,as T T 0.^0 O.^ tfi 0.11 0.27 0.42 0.08 0.1U 0.04 0.07 0.01 0.03 0.02 O.OU 0.03 0.03 0.01 0.01 0.00 0.01 0.00 0.00 0  0.00 0.00 0.00 0.1B 0.02 0.12 0..03 0..13 0.05 0.03 0.03 0.02 01 0.00 -0.00 0.00 -I 0.1U 0.12 0.02 0.01 0.00 0.03 0.01 -0.01 0.00 -0.02 -0.02 -0.01 -0.01 -0.01 oo -q.oo -o.oo -o.oo -0..00 -o.oo •0^ .0  -0^ .01 -0 .^00 -0 .^00 -0^ .00 -0^ . 1000 10 100 F R E Q U E N C Y I N H Z ( W A V E N U M B E R M) Figure 66. Values of the term S ^ k = QeOm^A Cos t 9-K) for the 26 X 26 g r i d . The m wavenumber i s horizontal and the k v e r t i c a l . The connecting l i n e s show o the path of the summing for the rate of a r r i v a l of energy at wavenumber Jt = m + k due ^ to the interactions of type 2. Four minutes of data were analyzed from run 2B tape 141. 104 This leads to a certain overlap between adjacent wavenumbers. The value of an array element i s a mean representative of a rectangle of wavenumbers about i t . In a l l cases at le a s t one corner of such a rectangle w i l l be closer to a d i f f e r e n t l i n e of constant Jl than the center. However since the worst that could happen would be that contributions of f o u r i e r components were swapped between adjacent estimates of the rate of a r r i v a l of energy, i t was f e l t t h i s would not seriously a f f e c t the r e s u l t s . The computation for the t h i r d type of interaction was quite s i m i l a r to that for type 1. The wavenumber^ was taken to be on the horizontal axis and the mean summing was v e r t i c a l l y up the columns. The value of l/(4,£rn)was however d i f f e r e n t for these array elements than for the f i r s t type of in t e r a c t i o n . The wavenumber m appeared i n t h i s case on the v e r t i c a l axis instead of as the sum. This resulted i n a symmetrical array SjPm/ J?m. A shortcoming of the method used to compute the energy transfers should be noted. In obtaining the f o u r i e r components the length of record transformed was l i m i t e d to 2048 points. The d i g i t a l sampling frequency was approximately 6000 Hz. Thus the lowest wavenumber fo u r i e r component considered covered only about one-third of a second i n time, or assuming the. frozen f i e l d hypothesis, had a scale size of the order of two meters. Any transfers of energy associated with interactions involving smaller wavenumbers would be ignored by t h i s approach. The r e s u l t s of performing these calculations on a four minute section of the d i f f e r e n t i a t e d downstream v e l o c i t y 105 data are shown i n figure 67 for the three types of interactions and for t h e i r sum. The f i r s t and second types of interactions can be seen on the average to produce a positive rate of a r r i v a l of energy at a l l frequencies. The t h i r d type of int e r a c t i o n leads to a loss of energy from a l l frequencies. The sum fluctuates between negative and positive values at the lower frequencies becoming positive only at scale sizes of the order of the peak of the d i s s i p a t i o n spectrum and smaller. This data run was processed i n such a manner as to make i t possible to compute for each block of 2048 data points the values of energy flow. Thus a 95% confidence i n t e r v a l could be computed based on the assumption that each block was a separate experiment to determine the energy flows. The mean would under t h i s assumption be a normally d i s t r i b u t e d quantity with standard deviation o~/f%. Here cH i s the standard deviation of the values of energy flow at each point and -v> i s the number of values from which the mean was formed. Figure 68 shows the r e s u l t s of t h i s computation. The rates of a r r i v a l of energy for each of the three types of i n t e r -actions are s t a t i s t i c a l l y s i g n i f i c a n t over most of the range of frequencies. The t o t a l rate of a r r i v a l i s however only s i g n i f i -cantly d i f f e r e n t from zero at frequencies where d i s s i p a t i o n has become important. This single term of the downstream equation would seem to be tr a n s f e r r i n g energy as would i n t u i t i v e l y be expected of three dimensional turbulence. Over an a p r i o r i reasonable range 106 LOG OF FREQUENCY IN HZ Figure 6 7 . Rate of a r r i v a l of energy at frequency f due to the three types of interactions and t h e i r sum for four minutes of d i f f e r e n t i a t e d v e l o c i t y data from run 2B tape 141. (type 1 , type 2 - - - - - , type 3 . sum ). The lower s o l i d curve i s f times the dis'sipation spectrum plotted on an a r b i t r a r y l i n e a r scale. 107 CO I o cc oc (X 4.5 2.5 0.5 -1.5 J. J I T " T * M - i ' y ^ 1 1 1 i 1 cc cc cr 0.0 0.5 1.0 1.5 2.0 2.5 LOG OF FREQUENCY IN HZ 3.0 3.5 Figure 68 . 95% confidence i n t e r v a l s of the data presented in figure 74. From the bottom up the traces are f o r inter-actions of types 1 to 3 and the sum. 108 of scale sizes the net rate of a r r i v a l of energy i s not s i g n i f i -cantly d i f f e r e n t from zero as would be expected for an i n e r t i a l subrange of scales. At smaller scale sizes i n the d i s s i p a t i o n range the net rate of a r r i v a l has become po s i t i v e . More energy i s being transferred into a wavenumber than out of i t as energy i s being l o s t to d i s s i p a t i o n . Another computation of interest i s the rate at which energy i s passing from wavenumbers associated with frequencies below a certain value to wavenumbers associated with higher frequencies. In doing t h i s i t was assumed that the net rate of energy a r r i v a l when integrated from f to i n f i n i t y represented the energy crossing frequency f. The r e s u l t s of t h i s computation are presented i n figure 69. This rate of energy flow has also behaved as would be expected f o r three dimensional i s o t r o p i c turbulence. Above the peak of the d i s s i p a t i o n spectrum i t decreases as energy i s l o s t to d i s s i p a t i o n . At frequencies below the peak the rate of energy flow l e v e l s o f f b r i e f l y as v i s c o s i t y becomes unimportant. Towards lower frequencies s t i l l the rate of flow decreases to zero. This decrease to zero represents the f a i l u r e of the model to consider interactions with scale sizes larger than one cycle over the 2048 point sections of data which were fourier transformed. The fact that t h i s i n t e g r a l goes to zero i s also a check on the accuracy of the computations. For each group of three f o u r i e r components which contribute to the transfer of energy a loss i n one component w i l l appear as an exactly equal gain i n the others. Therefore the sum of the gains and losses must be zero. 109 1.2-, CD I O LOG OF FREQUENCY IN HZ Figure 69• Rate of flow of downstream energy from below frequency f to above frequency f as function of l o g f. 110 The trace representing the sum of the energy flows for the three types of interactions i n figure 67 must therefore be negative at the lower frequencies as i t i s positive at higher frequencies and must integrate to give zero. Figure 69 can also be interpreted as giving some information on the length of the interactions. I t would normally be expected that the l e v e l i n g o f f of the rate of energy flow across frequency f i n the i n e r t i a l subrange of scales would continue towards lower frequencies. I f i t begins to decrease i t would seem that the contributions from larger scale interactions, which were ignored, have become important. Assuming t h i s argument to be true i t would seem, from figure 6 9 , that energy transfers by interactions of scale sizes at l e a s t a decade apart i n size are s i g n i f i c a n t . This statement has been q u a l i f i e d by "at l e a s t " due to the uncertainty as to whether i n the diagram the energy flow has i n fact l e v e l l e d o f f , and of course there are contributions to from terms involving X A . , , ^ U , and ^ , , ^ 3 _ CJ>JU. , ^  To gain further insight into the "lengths" of these interactions, the in d i v i d u a l contributions to each of the f i r s t 25 of the 26 estimates of the rate of a r r i v a l of energy at frequency f were plotted for each type of int e r a c t i o n . This information for interactions of type 1 i s shown i n figure 7 0 . Each of the 25 graphs i s numbered i n the upper l e f t hand corner. This number represents the point i n figure 67 which corresponds to that graph. The points are counted from the l e f t . The horizontal axes show the logarithm of frequency. I l l Figure ?0. Contributions to the rate of a r r i v a l of energy at frequency f for interactions of type 1, run 2B tape 141. The absissa i s the frequency corresponding to wavenumber k = m - X . The v e r t i c a l l i n e corresponds to the frequency of wavenumber f. The t i c s on the absissa axis represent 10, 100 and 1000 Hz. The v e r t i c a l scales are a r b i t r a r y . The horizontal dashed l i n e i s zero. 112 The t i c s are respectively 1 0 , 100 and 1000 Hz. The v e r t i c a l axes are not to the same scale for a l l the graphs. The r e l a t i v e v e r t i c a l scales can be implied from figure 67 since the sum of a l l the ordinates for each graph w i l l be equal to the ordinate of the corresponding point for the energy flow of type 1 . The s o l i d v e r t i c a l l i n e shows the frequency for which the rate of a r r i v a l of energy i s being calculated. The horizontal dashed l i n e represents the zero of the ordinate. V/here no h o r i -zontal dashed l i n e appears zero ordinate corresponds to the horizontal axis. As presented i n figure 63 the inte r a c t i o n of type 1 involves a v e l o c i t y of wavenumber Jl i n t e r a c t i n g with a v e l o c i t y of wavenumber m y Jl and with a v e l o c i t y derivative of wavenumber k such that m-k = H. The in d i v i d u a l contributions in the figure under discussion are plotted as a function of the logarithm of the frequency corresponding to the wavenumber k. Thus for the portion of the graph to the right of the s o l i d v e r t i c a l l i n e both int e r a c t i n g wavenumbers are greater than Jl . Except for occassional occurrences the transfer of energy into wavenumber Jl i s positive whether the other in t e r a c t i n g wavenumbers are greater than Jl or one i s l e s s . I f both are greater than Jl t h i s must be a mechanism for the transfer of energy from smaller to larger scale sizes. For the f i r s t 15 graphs there i s more energy a r r i v i n g i n the neighborhood of frequency f from smaller than from larger scale sizes. For the f i r s t few t h i s i s to be expected as the contributions from scales larger than the fou r i e r transform block have been ignored. However by the time 113 the twelfth to f i f t e e n t h graphs have been reached, c o n t r i -butions due to interactions with scale sizes a decade or greater larger have become small. From these graphs (12 to 15)» i t seems that the rate of a r r i v a l of energy from frequencies within a few octaves greater i s systematically larger than from frequencies somewhat smaller. In t h i s range of wavenumbers i t seems t h i s type of i n t e r a c t i o n may be more e f f i c i e n t at tra n s f e r r i n g energy from smaller scales to larger scales. For graphs beyond 15 more energy i s a r r i v i n g i n the neighbourhood of frequency f from larger than from smaller scales. D i s s i p a t i o n would have become important. The energy i n the smaller scales would be dissipated and unavailable for i n e r t i a l transfer back to larger scales. Figure 71 shows the same information for a second data run. This run was of only two minutes duration and the s t a t i s t i c a l scatter i s somewhat larger. The evidence f o r the statement that the e f f i c i e n c y of the transfer of energy by thi s mechanism was greater from smaller to larger than from larger to smaller scale sizes i s not as clear i n t h i s case. The conclusions drawn however do seem to be borne out for most, i f not a l l , of the in d i v i d u a l graphs. Figure 72 shows contributions to the rate of a r r i v a l of energy at frequency f due to interactions of type 2 . Once again the v e r t i c a l l i n e represents the wavenumber Jl f o r which the rate of a r r i v a l has been calculated. The contributions have been plotted as a function of the frequency of wavenumber m. The v e r t i c a l dashed l i n e represents the frequency of wavenumber $/2. When m i s less than Jl/2, k i s greater than Jl/2 and vice versa. 114 -, 1 0 , 15 20 25 Figure 71. Contributions to the rate of a r r i v a l of energy at frequency f for interactions of type 1, run 2 tape 9 6 . The absissa i s the frequency correspolding to wavenumber k = m - $ . The v e r t i c a l l i n e corresponds to the frequency of wavenumber X • The t i c s on the absissa axis represent 10, 100 and 1000 Hz. The v e r t i c a l scales are ar b i t r a r y . The horizontal dashed l i n e i s zero. 115 1 0 , , -, 1 5 2 ° 25 Figure 72. Contributions to the rate of a r r i v a l of energy at frequency f for interactions of type 2, run 2B tape 141, The absissa i s the frequency corresponding to wavenumber m = £ - k. The v e r t i c a l s o l i d l i n e corresponds to the frequency of wave-number J( , the dashed toJp/2. The t i c s on the absissa axis represent 10, 100 and 1000 Hz. The v e r t i c a l scales are arb i t r a r y . The horizontal dashed l i n e i s zero. 116 Except for the f i r s t one or two points of some of the graphs t h i s i n t e r a c t i o n i s a mechanism for the transfer of energy from larger to smaller scale sizes. The transfer must be in t h i s d i r e c t i o n i f the sign i s positive as a gain for wavenumber Jl must be at the net expense of the two smaller wavenumbers. It was expected that the largest contributions would come from the v i c i n i t y of the wavenumber Jl/Z, Figure 67 shows the largest values near the diagonal where the two frequencies are equal. However these values s t i l l require d i v i s i o n by the factor Jim. Since k and m are symmetric about J2/2, as k gets large, or the derivative term moves to the smaller scales, m gets smaller decreasing the d i v i s i o n factor Jl m. The r e s u l t i s apparent large transfers of energy involving interactions of wavenumbers approaching two decades apart i n frequency. The physical interpretation of t h i s would be that although the phase coherence between the larger and smaller scales decreases as the difference i n scale size increases, the transfers remain s i g n i f i c a n t due to the much greater energy content of the larger scales. Figure 73 shows the same information for energy transfers of type 2 for the other data run. The graphs once again show more scatter. However the conclusions reached i n discussion of the previous figure are supported. Figure 7^ shows the ind i v i d u a l contributions to the rate of a r r i v a l of energy by interactions of the t h i r d type. For reasons of c l a r i t y the sign of t h i s term was reversed i n the diagram. Thus positive values indicate energy leaving the 11? Figure 73. Contributions to the rate of a r r i v a l of energy at frequency f for interactions of type 2t run 2 tape 96. The absissa i s the frequency corresponding to the wavenumber m ~ Jt - k. The v e r t i c a l s o l i d l i n e corresponds to the frequency of wave-number Jt , the dashed to Jt/2. The t i c s on the absissa axis represent 10, 100 and 1000 Hz. The v e r t i c a l scales are a r b i t r a r y . The horizontal dashed l i n e i s zero. 118 Figure 74. Contributions to the rate of energy departure from frequency f for interactions of type 3, run 2B tape 141. The absissa i s the frequency corresponding to the wavenumber m = k -The v e r t i c a l s o l i d l i n e corresponds to the frequency of wave-number The t i c s on the absissa axis represent 10, 100 and 1000 Hz. The v e r t i c a l scales are a r b i t r a r y . The horizontal dashed l i n e i s zero. 119 v i c i n i t y of wavenumber J( . The frequency appearing on the horizontal axis i s that corresponding to the v e l o c i t y wavenumber m of figure 6 5 . Thus for values to the l e f t of the v e r t i c a l l i n e the interaction i s between a v e l o c i t y of wavenumber les s than $ and a v e l o c i t y derivative of wavenumber greater than ^ . For values to the r i g h t of the v e r t i c a l l i n e , the v e l o c i t y wavenumber m i s also greater than Jl and a positive sign here indicates a net loss of energy from Jl to smaller scale sizes. For the f i r s t 7 to 8 graphs i n the diagram the transfers seem to occur as would be i n t u i t i v e l y expected with the most energy leaving wavenumber^ for higher wavenumbers. Beyond the eighth graph the s i t u a t i o n appears somewhat d i f f e r e n t . The largest energy transfers occur from interactions involving much smaller wavenumbers with the implication that most of the energy i s being transferred from smaller to larger scales. I f figure 7 2 , which shows the rate of energy flow for interactions of type 2 , i s compared to the present figure, a marked s i m i l a r i t y w i l l be noted. For graphs beyond the tenth, the f i r s t few points corresponding to small wavenumbers are very nearly equal but with the opposite sign. Inspection of the summing scheme i n figure 66 reveals t h i s must be so. For the type 2 interactions the path follows the l i n e s of the diagram. For type 3 the path proceeds v e r t i c a l l y up a column. The values of the product ^m, for the two types of interactions, w i l l also be very nearly equal for the f i r s t few values proceeding up a column. The large transfers and long interactions are thus mathematical 120 rather than physical with the mathematics showing simultaneously about equal and opposite transfers. The quantity of importance i s the difference between these two intera c t i o n s . Also of in t e r e s t i s whether t h i s difference i s s i g n i f i c a n t over large ranges of scale s i z e . Figure 75 shows two t y p i c a l plots from figures 72 and 7^ . The s o l i d curve i s the type 3 i n t e r a c t i o n and the dashed curve the type 2 . The net transfer from these two types of i n t e r a c t i o n has d i f f e r e n t properties than either separately. S i g n i f i c a n t net transfers no longer involve interactions of scale sizes d i f f e r i n g i n size by more than a decade. This i s important to theories which attempt to a f f e c t closure of the energy balance equation by postulating that energy transfers are determined by l o c a l conditions. Figure 76 shows the values to the rate of a r r i v a l of energy i n one wavenumber band for the three types of interactions and t h e i r sum as a function of time. The wavenumber band was that corresponding to the 1 6 t h point on each curve i n figure 6 7 . The points should be counted from the l e f t . The ef f e c t of the simultaneous "mathematical" transfer of energy for the type 2 and 3 interactions can also be seen here. These two transfers are an order of magnitude greater than either the type 1 transfers or the t o t a l and one appears an approximate mirror image of the other. The energy transfers can be seen to be "intermittent" i n time i n the manner of the d i s s i p a t i o n rate. The c o e f f i c i e n t of excess of the traces from bottom to top are respectively 1 0 , 30>, 20 and 9 « 2 . Thus the t o t a l transfer and the type 1 interactions have values of c o e f f i c i e n t of excess quite similar to the downstream v e l o c i t y derivative. 121 13 18 i 1 r -1 1 1—' r Figure 75. Comparison of the rates of a r r i v a l of energy i n wavenumber Sl due to interactions of type 2 (dashed l i n e ) and the rate of departure from Sl due to interactions of type 3 ( s o l i d l i n e ) . The graphs are t y p i c a l ones from figures 72 and 74. 122 -1.2 J l i 6 P r F NI NI.MfiFh IH PHI NT NUMBER 16 3 8 IC ' 4^ HH qH 1 [Ml [ N :>l n iNr: Figure 7 6 . Rates of a r r i v a l of energy i n wavenumber Jl as a function of time for the three types of transfer and t h e i r sum. The data i s that leading to the 16th point i n figure 6 7 . The points are counted from the l e f t . 123 These values of kurtosis were t y p i c a l of those for the other wavenumber bands i n figure 6 7 . No systematic differences were found at higher or lower wavenumbers. An attempt was made to extend the analysis to lower frequencies. The purpose of th i s was to investigate the pos s i -b i l i t y of a net energy transfer by i n e r t i a l means from smaller to larger scales. There i s a hypothesis advanced by Kraichnan (1967) that as the turbulence becomes two dimensional t h i s must happen i n order to conserve both energy and v o r t i c i t y . To e f f e c t the extension to lower frequencies a sharp cutoff, phase s h i f t l e s s d i g i t a l f i l t e r was employed. The response i s shown i n figure 7 7 ' The f i r s t two minutes of the d i f f e r e n t i a t e d v e l o c i t y data from run 2B tape 141 was passed through t h i s f i l t e r . The time series was then subsampled taking every 1 0 t h data point and discarding the intervening points as redundant. The new time series was treated to the same analysis as the o r i g i n a l . However the analysis now yielded information to horizontal scale sizes of 20 meters rather than 2 meters. Figures 78 and 79 show the r e s u l t s . In figure 78 the energy transfers have behaved somewhat s i m i l a r l y to the higher frequency ones. The transfers show generally the same sign i n both cases. However the shapes are quite d i f f e r e n t and there i s a region at a frequency of about 3 Hz where there i s considerable sign switching. This corresponded to a scale size of approximately two meters which was the height of the hot wire probe above the water surface. This analysis of course neglects the effects of i n t e r -124 1.00 -i r 100 200 300 FREQUENCY (HZ) 400 500 Figure 77. Response of the d i g i t a l f i l t e r used to extend the analysis to lower frequencies. ~i r -1.0 -0.5 0.0 0.5 1.0 1.5 LOG OF FREQUENCY IN HZ 2.0 2.5 Figure 78. Rate of a r r i v a l of energy at frequency f due to the three types of interactions and t h e i r sum f o r the low pass f i l t e r e d d i f f e r e n t i a t e d v e l o c i t y data from run 2B tape 141. (type 1 _ , type 2 , type 3 , sum ). LOG OF FREQUENCY IN HZ Figure 79. Rate of flow of downstream energy from below frequency f to above frequency f as a function of l o g f. The data i s from run 2B tape 141 and has been low pass f i l t e r e d . 2 . i H 2.0H 3.5 LOG OF FREQUENCY IN HZ Figure 80. Ratio of the rate of energy flow across frequency f to the rate of energy d i s s i p a t i o n at frequencies greater than f for the downstream component. The data was from the f i r s t and t h i r d two minute sections of run 2B tape 141. 126 a c t i o n s w i t h s c a l e s i z e s l a r g e r t h a n 20 m e t e r s a s w e l l a s w i t h t h e s c a l e s i z e s r e m o v e d by t h e f i l t e r i n g . F i g u r e 79 shows t h e i n t e g r a l o f t h e t o t a l r a t e o f a r r i v a l o f e n e r g y a t wavenumbers c o r r e s p o n d i n g t o f r e q u e n c y f i n t e g r a t e d f r o m f t o oo . The v a l u e o f t h i s d i a g r a m i s somewhat q u e s t i o n a b l e a s t h e K r a i c h n a n a s s u m p t i o n t h a t a l l e n e r g y a r r i v i n g a t a wavenumber comes f r o m l a r g e r wavenumbers i s no l o n g e r v a l i d . The s c a l e s h e r e a r e s u f f i c i e n t l y l a r g e t h a t e n e r g y was p r o b a b l y b e i n g g a i n e d f r o m t h e t u r b u l e n c e i n t e r a c t i n g w i t h t h e s h e a r f l o w . C o n s i d e r a t i o n o f t h e two d i a g r a m s r e v e a l s no e v i d e n c e o f a s y s t e m a t i c f l o w o f e n e r g y f r o m s m a l l e r t o l a r g e r s c a l e s . H owever t h e t u r b u l e n c e c o u l d p r o b a b l y n o t be t h o u g h t o f a s t w o -d i m e n s i o n a l s i n c e t h e p e a k o f t h e v e r t i c a l v e l o c i t y s p e c t r u m o c c u r s a t a p p r o x i m a t e l y t h e l a r g e s t s c a l e s i z e s c o n s i d e r e d h e r e . A n o t h e r q u a n t i t y o f i n t e r e s t i s t h e r a t i o o f t h e r a t e o f e n e r g y f l o w a c r o s s a wavenumber t o t h e r a t e o f v i s c o u s d i s s i -p a t i o n o f e n e r g y a t a l l s m a l l e r s c a l e s . T h i s c o u l d o f c o u r s e o n l y be c o m p u t e d f o r t h e t e r m s i n t h e e q u a t i o n w h i c h c o n t a i n e d o n l y t h e d o w n s t r e a m v e l o c i t y o r i t s d e r i v a t i v e . T h i s i s w r i t t e n ( E q . 20) F ( k ) i s t h e t o t a l r a t e o f a r r i v a l o f e n e r g y i n t h e v i c i n i t y o f wavenumber k, p e r u n i t wavenumber, by i n e r t i a l t r a n s f e r i n t h e d o w n s t r e a m c o m p o n e n t . The i n t e g r a l i n t h e n u m e r a t o r i s , w i t h i n a c o n s t a n t f a c t o r , e q u a l t o t h e f u n c t i o n p l o t t e d i n f i g u r e 6 9 . The d e n o m i n a t o r i s t h e i n t e g r a l u n d e r t h e d i s s i p a t i o n s p e c t r u m . When t h e l o w e r l i m i t o f t h e i n t e g r a t i o n i s 0 t h i s must be e q u a l t o f^-^iV' 127 To do t h i s computation i t was necessary to obtain a c a l i b r a t i o n of the hot wire. As discussed e a r l i e r t h i s was not accomplished at the time of the measurement. The c a l i b r a t i o n was estimated by assuming the flow was i s o t r o p i c so that the d i s s i p a t i o n rate was given by the r e l a t i o n (Hinze, 1959) £ = IT j> ( i ^ - j ( E q * 2 1 ) I f v(t) i s a recorded voltage, l i n e a r l y proportional to the d i f f e r e n t i a t e d downstream v e l o c i t y signal, then g ^ - ' c " v C t ) where c i s the constant of proportionality. I f the "frozen f i e l d " hypothesis i s assumed where Wt i s the downstream mean wind speed. Hence A second equation i s obtained from the r e l a t i o n JO <P(k) » K ' (Eq. 22) where k i s the radian wavenumber, (P(k>) i s the one-dimensional energy spectrum of the downstream v e l o c i t y and K i s the one-dimensional Kolmogoroff constant. I f V(f) i s the spectrum of v(t) i n units of v o l t s -sec, equation 22 can be rewritten as c 128 The "frozen f i e l d " hypothesis has been assumed once more to write the r e l a t i o n connecting the frequency with the radian wavenumber, i . e . K - "z-TT £ The value of V(f) was obtained from a l e a s t squares f i t of a l i n e of slope 1/3 to the appropriate portion of the spectrum of v ( t ) . At a frequency of 1 kz, V(f) had the value 7.53 X 10" 5 v o l t s 2 - s e c . Values of 0.148 cm2 s e c - 1 , 0.5, and 780 cm sec ~ l were used for ~\) , K' and respectively. Solving the above equations for £ and c produced the values £ =165 cm 2/sec3 C = 1.40 X 10^ cm sec" 2 v o l t s - 1 In terms of the recorded signal V(t) and the discussion of the computations of the i n e r t i a l transfer i n section 1.4 ^ ^ . _1_ Au,. * JL_ £."' C K G> y(2-rrKt ) T i s the length of the record which i n t h i s case i s the length of the blocks of data f o u r i e r transformed (0.34 sec). Therefore the integrals of the i n e r t i a l transfer as computed must be multiplied by to convert to the required w Tr 3- U", units of cm2 sec~3. The inte g r a l s under the d i s s i p a t i o n spectrum s i m i l a r l y must be multiplied by S—^~ . Using these factors and the estimated c a l i b r a t i o n constant the r a t i o of equation 20 was computed for the four minutes of data from run 2B tape 141. The r e s u l t i s presented 129 i n figure 8 0 . The author i s not aware of any t h e o r e t i c a l or previous experimental evidence as to how t h i s quantity should behave. The d i v i s i o n of the average d i s s i p a t i o n between the nine viscous terms for the three-dimensional case of i s o t r o p i c turbulence i s understood. For example the value of i s twice as large for the terms for which i / j as for terms for which i = j . No si m i l a r prediction i s available f o r the r e l a t i o n s between terms such as 1 g^if-/ and r1 I n t u i t i v e l y one would expect that for i s o t r o p i c turbulence the i n e r t i a l transfers of energy would not be vastly d i f f e r e n t for these various terms. Hence the r e s u l t of figure 80 (page 125)» that the i n e r t i a l transfer and d i s s i p a t i o n terms are of equal magnitude, i s i n agreement with present i n t u i t i v e notions on the behaviour of i s o t r o p i c turbulence. The sudden increase i n the r a t i o at the highest frequency point could be a r e s u l t of the analog f i l t e r i n g . The Thompson f i l t e r i s down approximately 10% at t h i s point so that the amplitudes here are being decreased 20% (recording and d i g i t i z i n g ) . This would reduce the value of the denominator of the r a t i o more than the numerator as i n general only one of the f o u r i e r amplitudes of the three components contributing to i n e r t i a l transfer would be affected. The drop off i n the r a t i o towards lower frequencies represents once again the f a i l u r e of the analysis to consider energy transfers involving f o u r i e r components of lengths greater than approximately 2 meters i n scale s i z e . 130 Chapter IV Summary of Conclusions The main purpose of t h i s work was to study the s t a t i s t i c a l behaviour of the small scale turbulent fluctuations-of both v e l o c i t y and temperature i n the atmospheric boundary layer to determine i f they were well described by the logarithmic normal theory. Several records of d i f f e r e n t i a t e d v e l o c i t y and temperature were analyzed to determine t h e i r d i s t r i b u t i o n s . I t was found that the d i f f e r e n t i a t e d v e l o c i t y signals were not well described by the log normal d i s t r i b u t i o n s , the frequency of occurrence of small values being too high and of large values being too small. The higher moments of these data and a l l other d i f f e r e n t i a t e d data were shown to be unreliable i n characterizing the d i s t r i b u t i o n and indeed almost impossible to measure accurately because of the li m i t e d dynamic range of the i n s t r u -ments used, in p a r t i c u l a r of the tape recorder. One record of X-wire v e l o c i t y data was analyzed to determine the d i s t r i b u t i o n s of the in d i v i d u a l d i f f e r e n t i a t e d X-wire signals and the sum of the squares of these two signals. I t was found that while neither X-wire signal alone conformed to the log normal model, the d i s t r i b u t i o n of the sum of t h e i r squares was very well described by the log normal model. The agreement of the d i s t r i b u t i o n of t h i s quantity with the model was much better than that of any of three random log normal number generators which were available on the d i g i t a l computer used for t h i s work. 131 Two sections of d i f f e r e n t i a t e d temperature data were analyzed and showed s i g n i f i c a n t l y better agreement with the log normal model than did the d i f f e r e n t i a t e d v e l o c i t y signals. The signal to noise r a t i o , however, was s u f f i c i e n t l y poor that i t was not possible to determine conclusively i f the theory was s a t i s f i e d i n the range of large values. The slopes of the spectrum i n the i n e r t i a l subrange of scales for the squares of the various d i f f e r e n t i a t e d signals were examined and found, as t h e o r e t i c a l l y predicted, to be described within the experimental accuracy by a straight l i n e . The slopes of the straight l i n e s were found to be within the range of values already reported i n the l i t e r a t u r e . The structure functions were computed for two sections of turbulent downstream v e l o c i t y data. In spite of the short duration of the sections, the observed data c l e a r l y supported the modified Kolmogoroff theory which takes into account the l o c a l fluctuations i n the d i s s i p a t i o n rate and assumes these fluctuations to be log normally d i s t r i b u t e d . Some c h a r a c t e r i s t i c s of the i n e r t i a l transfer of energy i n the downstream component of the v e l o c i t y were examined as a function of wavenumber through the use of a bispectrum computation. I t was found that the i n e r t i a l transfer i n the downstream component behaved much as would be expected fo r the t o t a l transfer i n three-dimensional i s o t r o p i c turbulence. The net rate of a r r i v a l of energy at scales f o r which d i s s i p a t i o n had not become important was not s t a t i s t i c a l l y d i f f e r e n t from zero. At scale sizes where d i s s i p a t i o n was important the net rate of a r r i v a l was positive i n d i c a t i n g that l e s s energy was being l o s t 132 to smaller scale sizes than was being gained from larger scales. This was attributed to a loss of energy to viscous forces at these scale s i z e s . Although the analysis was extended to scale sizes where the turbulence might be tending toward two-dimensional behaviour no i n d i c a t i o n of a reverse flow of energy from smaller to larger scale sizes was found. The p o s s i b i l i t y of a reverse flow has been postulated by theories (Kraichnan, 1967) based on conservation of both energy and v o r t i c i t y i n the turbulent flow. It was found that the i n e r t i a l transfer of energy i s almost f u l l y determined by interactions of f o u r i e r components having scale sizes d i f f e r i n g by less than a decade. This i s important for theories which attempt to eff e c t closure of the energy balance equations by postulating the l o c a l i n t e r a c t i o n theory. The f i n a l r e s u l t that the rate of flow of energy from larger to smaller scale sizes was of the same magnitude as the single d i s s i p a t i o n term V (c ^ - i / P x , ) requires further examination. It would be useful to an attempt to construct a theory concerning the d i s t r i b u t i o n of the i n e r t i a l transfer between the various terms of the equation for i s o t r o p i c turbulence. None of the r e s u l t s presented here seriously c o n f l i c t with the present understanding of turbulence i n the atmospheric boundary layer. The experimental evidence of the f a i l u r e of the d i f f e r e n t i a t e d v e l o c i t y data to conform accurately to the log normal model was new for a measurement including the smallest scale fluctuations at the time i t was completed and published (Stewart, Wilson and Burling, 1 9 7 0 ) . The accuracy with which the 133 sum of the squares of the d i f f e r e n t i a t e d X-wire signals i s described by the log normal model i s probably the most i n t r i g u i n g r e s u l t and i s probably not generally known. The calculations concerning the i n e r t i a l transfer of energy have not been done before as far as the author i s aware. In addition to confirming i n t u i t i v e notions of how t h i s transfer of energy occurs i n a turbulent f i e l d the r e s u l t s are important to present theories which postulate l o c a l interactions and to any attempt to develop further theories on the i n e r t i a l transfer of energy. The other r e s u l t s generally add weight to experimental v e r i f i c a t i o n s already i n existence for various theories. 134 LIST OF REFERENCES Cramer, H. (1946) Mathematical Methods of S t a t i s t i c s . Princeton University Press. Guimard, S.I. and G.R. Stegen (1973) Lognormality i n gr i d generated turbulence. B u l l e t i n of the Am. Phy. Soc., November 1 9 7 3 . 1479. Gurvich, A.S. and A.M. Yaglom (1967) Breakdown of eddies and pr o b a b i l i t y d i s t r i b u t i o n s for small-scale turbulence. Phys. Flu i d s Suppl. 10 (part I I ) , S 5 9 . Gurvich, A.S. and S.L. Zubkovski (1963) Izv. Akad. Nauk. S.S.S.R. Ser. Geofis. 1 8 5 6 . Hoel, P.G. ( I 960 ) Elementary S t a t i s t i c s . New York: John Wiley. Kolmogorov, A.N. ( 1 9 4 l a ) The l o c a l structure of turbulence i n an incompressible viscous f l u i d for very large Reynolds number. CR. (Dokl.) Acad. S c i . USSR 3_0, 3 0 1 . Kolmogorov, A.N. (1962) A refinement of previous hypothesis concerning the l o c a l structure of turbulence i n a viscous incompressible f l u i d at high Reynolds number. J. F l u i d Mech. I J , 82. Kraichnan, R.H. (1967) I n e r t i a l ranges i n two-dimensional turbulence. Physics of Fl u i d s , 1 0 , No. 7 , 1417. Novikov. E.A. and R.W. Stewart (1964) The intermittency of turbulence and the spectrum of energy d i s s i p a t i o n f l u c t u a t i o n s . Izv. Akad. Nauk. Ser. Geophys. No .3» 4 0 8 . Oboukov, A.M. (1941a) CR. (Dokl.) Acad. S c i . USSR 32, 19. Oboukov, A.M. (1962) Some s p e c i f i c features of atmospheric turbulence. J. F l u i d Mech. 13_, 7 7 . Paquin, J.E. and S. Pond (1971) The determination of the Kolmogoroff constant f o r v e l o c i t y , temperature and humidity fluctuations from second and t h i r d order structure functions. J. F l u i d Mech. £ 0 ( 2 ) , 257* Pond, S. (1965) Turbulent spectra i n the atmospheric boundary layer over the sea. Unpublished Ph.D. the s i s , I n s t i t u t e of Oceanography and Department of Physics, University of B r i t i s h Columbia (I0UBC Manuscript Report No. 1 9 ) . Pond, S., R.W. Stewart and R.W. Burling (1963) Turbulent spectra i n the wind over waves. J. Atmos. S c i . 2 0 , 3 1 9 « Richardson, L.F. (1922) Weather Prediction by Numerical Process, Cambridge Univ. Press. 135 S t e w a r t , R.W., J.R. W i l s o n and R.W. B u r l i n g (1970) Some s t a t i s t i c a l p r o p e r t i e s of s m a l l - s c a l e t u r b u l e n c e i n an at m o s p h e r i c boundary l a y e r . J . F l u i d Mech. _ 1 , p a r t 1, l 4 l . Van A t t a , G.W. and W.Y. Chen (1970) S t r u c t u r e f u n c t i o n s of t u r b u l e n c e i n the a t m o s p h e r i c boundary l a y e r over the ocean. J . F l u i d Mech. _ , p a r t 1, 145. Van A t t a , C.W. (1971) I n f l u e n c e of f l u c t u a t i o n s i n l o c a l d i s s i p a t i o n r a t e s on t u r b u l e n t s c a l a r c h a r a c t e r i s t i c s i n the i n e r t i a l subrange. P h y s i c s o f F l u i d s , 1_, 1803• Van A t t a , C.W. (1973) E r r a t u m : I n f l u e n c e of f l u c t u a t i o n s i n l o c a l d i s s i p a t i o n r a t e s on t u r b u l e n t s c a l a r c h a r a c t e r i s t i c s i n the i n e r t i a l subrange. P h y s i c s o f F l u i d s , 16, 574. 136 Appendix A Pertinent Characteristics of the Recording and D i g i t i z i n g C i r c u i t s 1. Linear Phase S h i f t Thompson F i l t e r Figure 81 shows the response of the fourth order Thompson f i l t e r used both in the recording and d i g i t i z i n g phases for the downstream v e l o c i t y fluctuations. The response i s the amplitude of the output sine wave for a unit amplitude input sine wave. The measurements were made with an oscilloscope. Figure 82 shows the phase s h i f t of the same f i l t e r . The measurements were made using a L i s s i j o u figure on an oscilloscope. This technique i s inaccurate in the v i c i n i t y of a 90 degree phase s h i f t which accounts for the scatter of the points in that region. As can be seen the f i l t e r had a 3 db point at about 2 KHz and to good accuracy a l i n e a r with frequency phase s h i f t . 2. Phase S h i f t of the D i f f e r e n t i a t i n g C i r c u i t Ideally the phase s h i f t of a d i f f e r e n t i a t i n g c i r c u i t should be 90 degrees at a l l frequencies. P r a c t i c a l l y a small r e s i s t o r must be used i n the input and a small capacitor i n the feedback loop of such an operational amplifier c i r c u i t to prevent small very high frequency noise at the input from saturating the output. This w i l l cause some undesirable phase s h i f t s i n the output. Figure 83 shows the undesirable phase s h i f t for the d i f f e r e n t i a t i n g c i r c u i t used for t h i s work. As can be seen the s h i f t i s 13? -| 1 1 1 1 T 1 1 .0 1 .5 2 . 0 2 . 5 3 . 0 3 . 5 U . O L O G OF F R E Q U E N C Y I N HZ Figure 81. Response of the fourth order Thompson f i l t e r FREQUENCY (HZ) ( X 10 J ) Figure 82. Phase s h i f t of the fourth order Thompson f i l t e r 10 LU CO cx X 0_ 15 H 20 -i 1 1 1 , , 0.0 0.4 0.8 1.2 1.6 2.0 FREQUENCY IN KHZ Figure 8 3 . Non-ideal phase s h i f t o r i g i n a t i n g i n the d i f f e r e n t i a t o r c i r c u i t . Figure 84. Phase s h i f t of the Kronhite f i l t e r used i n recording the temperature data. 139 s i g n i f i c a n t but i s very nearly l i n e a r as the r e l a t i o n £- *- holds well for the range of angles here. 3. Phase S h i f t of the Kronhite F i l t e r A Kronhite analog f i l t e r was used i n the recording c i r c u i t r y for the temperature data reported here. The phase s h i f t of such a f i l t e r i s , over a large range of frequencies, very d i f f e r e n t from l i n e a r . However for the range of frequencies used in forming the d i s t r i b u t i o n s reported the phase c h a r a c t e r i s t i c was approximately l i n e a r with frequency. This response i s shown in figure 84. 140 Appendix B Computational Methods 1. The Spectrum The f a s t fourier transform algorithm developed by Cooley and Tukey (1966) was used to compute a l l spectra reported here. The program computes the complex fo u r i e r c o e f f i c i e n t A(k) for k = 0,1,2, , N-l from the r e l a t i o n The X(j) are the complex data points. The basic assumption of the analysis i s that the r e a l data record can be represented by the l i n e a r superposition of a mean value and a number of o s c i l l a t o r y components equal to one-half the number of discrete data values making up the i n t e r v a l . The highest frequency component w i l l correspond to the fold i n g or Nyquist frequency. A second important assumption i s that the data i s not zero outside the i n t e r v a l being transformed as i s usually assumed, but repeats to i n f i n i t y . The spectral density corresponding to the k"^ f o u r i e r c o e f f i c i e n t A(k) i s given by the square of the modulus of A(k) divided by 2 <_f. £if corresponds to the frequency i n t e r v a l between adjacent A(k) and i s equal to the d i g i t a l sampling frequency divided by the number of data points f o u r i e r transformed. The energy for frequencies occurring i n the data which do not correspond to an i n t e g r a l number of cycles over the data i n t e r v a l appear mostly i n the nearest 141 frequency and according to Blackman & Tukey (1959) w i l l "leak" into that and other nearby c o e f f i c i e n t s according to the r e l a t i o n i /(u>„ - where u ) 0 i s the occurring frequency and i s the frequency of the k^h c o e f f i c i e n t . To obtain the plots of spectral density versus the logarithm of frequency an increasing number of adjacent A(k) were averaged together towards the high frequency end of the spectrum. Thus points were obtained at more or less equal i n t e r v a l s of the logarithm of frequency. The records analyzed a l l consisted of many more points than could be held and f o u r i e r transformed i n a d i g i t a l computer. The runs were therefore broken up into sections (usually 2048 points) and fourier transformed i n d i v i d u a l l y . The mean of t h i s number of estimates of spectral density at each frequency obtained by the breaking up of the record was then plotted as the spectral density. The error bars on the plots represent 95% confidence l i m i t s . They were obtained with the usual assumption that the d i s t r i b u t i o n of an observed mean value i s Gaussian. 

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