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The effects of spatial averaging on airfoil probe measurements of oceanic velocity microstructure Ninnis, Ronald 1984

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THE EFFECTS OF SPATIAL AVERAGING ON AIRFOIL PROBE MEASUREMENTS OF OCEANIC VELOCITY MICROSTRUCTURE by RONALD NINNIS B . S c , U n i v e r s i t y Of B r i t i s h Co lumbia , 1976 M.Sc . ,S imon F r a s e r U n i v e r s i t y , 1972 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF PH.D. i n THE FACULTY OF GRADUATE STUDIES Oceanography We accept t h i s t h e s i s as conforming to the r e q u i r e d s tandard THE UNIVERSITY OF BRITISH COLUMBIA OCTOBER 1984 © RONALD NINNIS, 1984 In p r e s e n t i n g t h i s t h e s i s in p a r t i a l f u l f i l m e n t of the requ i rements fo r an advanced degree at the U n i v e r s i t y of B r i t i s h Co lumbia , I agree that , 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 ence and s tudy . I f u r t h e r agree tha t pe rm i s s i on f o r e x t e n s i v e copy ing of t h i s t h e s i s f o r s c h o l a r l y purposes may be g ran ted by the Head of my Department or by h i s or her r e p r e s e n t a t i v e s . I t i s unders tood tha t copy ing or p u b l i c a t i o n of t h i s t h e s i s fo r f i n a n c i a l ga in s h a l l not be a l l owed wi thout my w r i t t e n p e r m i s s i o n . Department of Oceanography The U n i v e r s i t y of B r i t i s h Columbia 2075 Wesbrook P lace Vancouver , Canada V6T 1W5 Da te : 30 September 1984 i i A b s t r a c t The e f f e c t s of s p a t i a l a ve rag ing on the measurement of v e l o c i t y power s p e c t r a by an a i r f o i l probe have been s t u d i e d . The probe i s a m in i a tu r e a i r f o i l of r e v o l u t i o n wi th a p i e z o -ceramic sensor tha t outputs a v o l t a g e p r o p o r t i o n a l to the t ime-v a r y i n g c ro s s-s t r eam v e l o c i t y component. The s p a t i a l t r a n s f e r f u n c t i o n of the probe was measured in g r i d t u rbu l ence set up in a water t unne l by comparing the power s p e c t r a ob ta ined w i th the a i r f o i l probe to those measured u s i n g h igh s p a t i a l r e s o l u t i o n l a s e r Dopp ler anemometry (LDA). The LDA system was des igned and c o n s t r u c t e d s p e c i f i c a l l y f o r t h i s pu rpose . The LDA was shown to be capab le of measur ing low i n t e n s i t y t u r b u l e n c e , g i v i n g r e s u l t s c o n s i s t e n t w i th s i m i l a r i t y theory and p r e v i o u s cons tan t temperature anemometry measurements. The t r a n s f e r f u n c t i o n e x h i b i t e d unexpected behav ior at low wavenumbers, namely, an i n i t i a l p o s i t i v e s lope and peak away from the o r i g i n . A t h r ee-d imens iona l model i s proposed and used to e x p l a i n the expe r imen ta l t r a n s f e r f u n c t i o n and to d e r i v e a l a r g e - s c a l e t r a n s f e r f u n c t i o n s u i t a b l e fo r c o r r e c t i n g shear probe measurements of ocean i c r a t e s of d i s s i p a t i o n of t u r b u l e n t k i n e t i c ene rgy . A l though i n gene r a l the t r a n s f e r f u n c t i o n depends on the form of the energy spect rum, i t i s shown tha t f o r ocean i c s p e c t r a the t r a n s f e r f u n c t i o n i s independent of s c a l i n g i n the wavenumber domain and hence independent of the r a t e of d i s s i p a t i o n . For the two probe des i gns s t u d i e d , the l a r g e - s c a l e t r a n s f e r f u n c t i o n s have a 1/2-power response at a s c a l e of about 1.5cm and approx imate l y 50% of the t o t a l d i s s i p a t i o n i s r ecove red at the h i g h e s t d i s s i p a t i o n va lue c o n s i d e r e d , 1 0 " 1 c m 2 / s 3 , (which i s l a r g e r than the d i s s i p a t i o n r a t e s encountered in most of the o cean ) . The d i s s i p a t i o n measurements can be c o r r e c t e d to an a ccu racy of 5%. i v Tab le of Contents A b s t r a c t i i L i s t of Tab l e s y L i s t of F i g u r e s y i Acknowledgement v i i Chapter I INTRODUCTION 1 Chapter II SPECTRAL THEORY OF TURBULENCE 4 2.1 The Spectrum Tensor 4 2.2 Measured Spec t r a 5 2.3 The Three-d imens iona l Energy Spectrum 8 2.4 I s o t r o p i c Turbu lence 9 2.5 The S p a t i a l T r a n s f e r F u n c t i o n 12 2.6 The E q u i l i b r i u m Spectrum 13 2.7 The Nasmyth U n i v e r s a l Spectrum 15 Chapter III EXPERIMENTAL PROCEDURE 19 3.1 The A i r f o i l Shear Probe 20 3.2 Shear Probe C a l i b r a t i o n 24 3.3 The Laser Doppler System 25 3.3.1 The O p t i c a l System 25 3.3.2 Frequency S h i f t i n g 32 3 .3 .3 Demodulat ion 33 3.3 .4 Doppler Broadening 37 3.4 The Water Tunne l Arrangement 39 3.5 Measurements 41 3.6 Data A n a l y s i s 41 Chapter IV RESULTS 43 4.1 LDA Measurements 43 4.1.1 Mean Flow 43 4 .1 .2 No ise In LDA Spec t ra 44 4 .1 .3 Power Spec t r a 49 4 .1 .4 G r i d Turbu lence 52 4 .1 .5 C y l i n d e r Wake Spec t ra 56 4.2 Shear Probe Spec t r a 58 4.3 The Measured T r a n s f e r F u n c t i o n s 61 Chapter V MODEL OF SPATIAL AVERAGING .63 5.1 P r e v i ous Work 63 5.2 The Model 64 5.3 A p p l i c a t i o n To The A i r f o i l Probe 65 5.4 C a l c u l a t i o n s Based On Measured G r i d Turbu lence . . . . 7 0 5.5 Model A p p l i e d To Nasmyth Spectrum 78 5.6 E f f e c t On La rge-Sca le D i s s i p a t i o n Measurements . . . . 8 4 V Chapter VI SUMMARY AND CONCLUSIONS 88 6.1 The Measurements 88 6.2 The T r a n s f e r F u n c t i o n 90 BIBLIOGRAPHY 92 APPENDIX A - THE MEASURED PROBE GEOMETRIES 95 APPENDIX B - DETAILS OF CALCULATIONS FOR CHAPTER 5 99 APPENDIX C - DISSIPATION RECOVERY RATES 102 v i L i s t of Tab l e s I. LDA System Components 30 I I . C a l c u l a t e d LDA System O p t i c a l Parameters 31 I I I . G r i d t u rbu l ence a n a l y s i s 55 IV. Po l ynomia l C o - e f f i c i e n t s fo r the La rge-Sca l e T r a n s f e r F u n c t i o n s 81 V I 1 L i s t of F i g u r e s 1. Nasmyth u n i v e r s a l s p e c t r a : F ( k / k s ) , F ^ k / k s ) , F 2 ( k /k s ) 17 2. The t r a n s v e r s e d i s s i p a t i o n spect rum, G 2 ( k /ks ) 18 3. D e t a i l s of the a i r f o i l probe 21 4. The Geomet r i c a l O p t i c s of the LDA System 28 5. F i l t e r e d output from the PMT showing Doppler " b u r s t " . About 50 f r i n g e s are seen . The t r a n s i e n t waveform was d i g i t i z e d at 50 nsec per po in t and then i n t e r p o l a t e d . 29 6. Power s p e c t r a of the f i l t e r e d PMT s i g n a l r e p r e s e n t i n g the v e l o c i t y 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 in a 2.54 cm d iameter p ipe f low 35 7. H igh f requency LDA s p e c t r a showing "wh i t e " ambigu i t y no i s e 45 8. E f f e c t of s u b t r a c t i n g ambigu i ty no i s e from the mixed s p e c t r a 46 9. Ambigu i t y no i se l e v e l , N, as a f u n c t i o n of the mean v e l o c i t y , U 48 10. Norma l i zed LDA c ross-s t r eam s p e c t r a , <j>2 2 (k ,) , f o r mean speeds of U=40,50,60, and 70 cm/s 50 11. LDA t r a n s v e r s e and l o n g i t u d i n a l s p e c t r a measured (above) and the r a t i o of 0 2 2 as d e r i v e d from to the measured 0 2 2 (below) 51 12. Dependence of the v a r i a n c e , u 2 2 , on U 2 , f o r the g r i d LDA s p e c t r a , 0 2 2 ( k , ) 54 13. T r ansve r se g r i d d i s s i p a t i o n s p e c t r a 54 14. C y l i n d e r wake s p e c t r a : compar ison of LDA and hot wire measurements (Uberoi and Freymuth, 1 969) 57 15. Shear probe g r i d s p e c t r a (unsmoothed) at 40cm/s (above) and 70cm/s (below) 59 16. Smoothed shear probe s p e c t r a at 70cm/s, no rma l i zed to u n i t y at peak va lue 60 17. Measured s p a t i a l t r a n s f e r f u n c t i o n s (70 cm/s) no rma l i zed v i i i to u n i t y near the peak 62 18. Component t r a n s f e r f u n c t i o n s (below) c o r r e s p o n d i n g to the s p a t i a l a ve rag ing f u n c t i o n s (above) no rma l i zed to u n i t y at Y=0. Y=k,L f o r A and Y=aR f o r B 67 19. Energy s p e c t r a , E ( k , ) d e r i v e d from the measured one-d imens iona l s p e c t r a , ^ ^ ( k , ) and ^ 2 2 ^ k i ) » u s i ng equa t ion (2 .4 .8 ) 72 20. One-dimens iona l s p e c t r a , ^ ^ ( k , ) and 0 2 2 ^ , ) : measured ( s o l i d ) and r e c o n s t r u c t e d (dash) from E(k) u s i ng i s o t r o p i c r e l a t i o n s 73 21. Model t r a n s f e r f u n c t i o n s v a r y i n g R: R= 0.20 - 0.30cm in s teps of .025cm 74 22. Measured t r a n s f e r f u n c t i o n s (U=70cm/s) and best f i t model t r a n s f e r f u n c t i o n s . Osborn p robe : R=0.21cm; Oakey p robe : R=0.235cm 75 23. Low wavenumber response (LWR); i . e . , X(k,=0) e va l ua t ed as a f u n c t i o n of the probe l a t e r a l s c a l e , R, and no rma l i zed wi th the l o n g i t u d i n a l i n t e g r a l s c a l e , L , . .77 24. L a rge-s ca l e t r a n s f e r f u n c t i o n (Osborn ) , f o r : e = l 0 " 1 , 1 0 - 2 and 10~ 3 c m 2 / s 3 79 25. Comparison of the l a r g e - s c a l e t r a n s f e r f u n c t i o n f o r the two probe des igns and the Garget t t r a n s f e r f u n c t i o n fo r the Osborn probe 80 26. T ransve r se t r a n s f e r f u n c t i o n s fo r power law s p e c t r a . . 83 27. Comparison of the d i s s i p a t i o n recovery r a t e as a f u n c t i o n of the " t r u e " d i s s i p a t i o n , e, a t a temperature of 15°C fo r the two p r o b e s : 86 28. Recovery r a t e s of measured d i s s i p a t i o n pa rame te r i zed w i th temperature i n 5 ° i n t e r v a l s from 0 - 30°C 87 29. The measured probe shapes and the model curve (dash ) , x i = i 0 r 2 , w i th •y0 = 13mm"1 97 ix Acknowledgement I wish to express my g r a t i t u d e to my s u p e r v i s o r , Tom Osborn , f o r sugges t i ng t h i s study and p r o v i d i n g t e c h n i c a l and f i n a n c i a l s u p p o r t . Steve Pond, as c o - s u p e r v i s o r , answered my numerous q u e s t i o n s conce rn ing the theory and measurement of t u r b u l e n t f lows as we l l as a t t e n d i n g to a d m i n i s t r a t i v e d e t a i l . I am t h a n k f u l f o r hav ing had the use of the water tunne l a t the I n s t i t u t e of Ocean S c i e n c e s , P a t r i c i a Bay; i n t h i s r e g a r d , George Chase was most h e l p f u l . I b e n e f i t t e d g r e a t l y from c o n v e r s a t i o n s w i th Roy Douglas and R u s s e l l B r a t kowsk i , who l i s t e n e d p a t i e n t l y to my p rog res s and o f t e n c o n t r i b u t e d v a l u a b l e c r i t i c i s m and a d v i c e . Don Dunbar was most generous w i th h i s t ime and computing e x p e r t i s e . 1 I. INTRODUCTION The a i r f o i l shear probe has been employed to measure ocean v e l o c i t y m i c r o s t r u c t u r e s i n c e 1972 (Osborn, 1974). The probe i s an ax isymmetr i c a i r f o i l which u t i l i z e s aerodynamic l i f t f o r ce to measure one c ross-s t r eam ( t r ansve r se ) component of t u r b u l e n t v e l o c i t y . The probe i s t y p i c a l l y mounted on a f r e e - f a l l v e r t i c a l p r o f i l e r or a towed inst rument v e h i c l e , the drop speed or towing v e l o c i t y p r o v i d i n g a mean a x i a l f l ow . A p i ezoce ramic bimorph beam embedded w i t h i n the a i r f o i l senses the l i f t f o r ce and ou tpu ts a v o l t a g e p r o p o r t i o n a l to the i ns t an taneous c r o s s -stream v e l o c i t y . T h i s vo l t age i s norma l l y d i f f e r e n t i a t e d to improve h igh f requency s i g n a l - t o - n o i s e r a t i o and so the r e s u l t i n g s i g n a l becomes a measure of the f l u c t u a t i n g shea r . The terms " shear p robe " and " a i r f o i l p robe " w i l l be used i n t e r c h a n g e a b l y . I t i s important to o b t a i n es t ima tes of the ra te of d i s s i p a t i o n of t u r b u l e n t k i n e t i c energy per u n i t mass, e, in order to d e s c r i b e the energy f l u x e s of l a r g e - s c a l e mix ing p roces ses and because h i g h Reynolds number t u rbu l ence e x h i b i t s s i m i l a r i t y when no rma l i z ed on the b a s i s of e and the k inemat ic v i s c o s i t y , v. However, d e t e r m i n a t i o n of the d i s s i p a t i o n i m p l i e s the r e s o l u t i o n of the v a r i a n c e of the shear spec t rum. T h i s requi rement i n tu rn e n t a i l s measurement of the h igh wavenumber regime of the v e l o c i t y power s p e c t r a . Now, the f i n i t e s i z e of the a i r f o i l probe i m p l i e s a lower l i m i t on the eddy s i z e tha t can be a c c u r a t e l y measured. A probe wi th d imens ions on the o rder of 1 cm cannot be expected to r e s o l v e v e l o c i t y s t r u c t u r e 2 at a s c a l e , o f 1 mm. Thus the s p a t i a l averaging of the probe w i l l r e s u l t i n a r o l l - o f f of s e n s i t i v i t y at high wavenumbers and any measurement of the shear w i l l r e s u l t i n an underestimate of the true d i s s i p a t i o n . S i nce the shear spectrum s h i f t s t o s m a l l e r s c a l e s as the d i s s i p a t i o n i n c r e a s e s , the e r r o r due to s p a t i a l averaging can be expected to i n c r e a s e with the d i s s i p a t i o n l e v e l . A l s o , the shear spectrum depends on the temperature through the v i s c o s i t y , again s h i f t i n g to higher wavenumbers with i n c r e a s i n g temperature. I t i s c l e a r then that ' a d e t a i l e d understanding of the wavenumber response ( s p a t i a l t r a n s f e r f u n c t i o n ) of the a i r f o i l probe i s necessary to correct^ the measured estimates of the v e l o c i t y s p e c t r a and the r a t e s of t u r b u l e n t k i n e t i c energy d i s s i p a t i o n . I d e a l l y , one would want the r e l a t i o n between the t r u e v a l u e of d i s s i p a t i o n and the measured estimate, as a f u n c t i o n of temperature. There have been s e v e r a l p r e v i o u s attempts at e s t i m a t i n g the e f f e c t of s p a t i a l a veraging. Siddon (1969) compared the response of a probe (designed f o r o p e r a t i o n i n a i r ) with c r o s s e d h o t - w i r e s . Because of the high i n t e n s i t y of the t u r b u l e n c e the behavior of both the probe and the hot wires was p o o r l y understood. I t was concluded only that the probe response began to drop at a s c a l e corresponding to four probe diameters. S e v e r a l attempts were made to f i t the probe measurements of ocean t u r b u l e n c e to the u n i v e r s a l Nasmyth (1970) spectrum. In p a r t i c u l a r , - Gargett ( 1983, p e r s o n a l communication) compared 3 a i r f o i l probe s p e c t r a to l o n g i t u d i n a l s p e c t r a ob ta ined w i th a hot f i l m probe assuming i s o t r o p y and ob ta ined a -3db ( h a l f ) power response at a s c a l e of 1.5cm. Oakey (1982) r epo r t ed a -3db response of the shear probe at a l e n g t h s c a l e of 2±1cm from l a b o r a t o r y compar ison w i th a t h i n f i l m X-probe in water . I t i s the purpose of t h i s t h e s i s to r epo r t measurements of the wavenumber response of the shear probe ob ta ined by compar ison w i th l a s e r Doppler anemometry (LDA) measurements of g r i d t u rbu l ence and to d e r i v e the i m p l i c a t i o n s f o r t u r b u l e n t k i n e t i c energy d i s s i p a t i o n measurements i n the ocean . In order to i n t e r p r e t the l a b o r a t o r y r e s u l t s p r o p e r l y i t was found necessa ry to model the probe response to i s o t r o p i c t u r b u l e n c e t a k i n g i n t o account the t h r e e - d i m e n s i o n a l c h a r a c t e r of both the probe and the v e l o c i t y f i e l d . I t w i l l be shown tha t the wavenumber response depends on the s c a l e of the t u r b u l e n t f low and e x h i b i t s unexpected behav iour at low wavenumbers. The o r g a n i z a t i o n of the t h e s i s i s as f o l l o w s . In Chapter 2 the s p e c t r a l theory of t u rbu l ence i s o u t l i n e d w i th p a r t i c u l a r a t t e n t i o n g i ven to the u n i v e r s a l spectrum of l a r g e - s c a l e mix ing p roces ses found in the ocean . Chapter 3 d e s c r i b e s the expe r imen ta l p rocedure and data a n a l y s i s . The r e s u l t s and some d i s c u s s i o n are p resen ted i n Chapter 4. The t h r ee-d imens iona l model of s p a t i a l a ve r ag ing i s deve loped and a p p l i e d to the l a b o r a t o r y r e s u l t s in Chapter 5. A u n i v e r s a l t r a n s f e r f u n c t i o n f o r l a r g e - s c a l e f lows i s a l s o g i ven i n a form u s e f u l f o r the i n t e r p r e t a t i o n of d i s s i p a t i o n measurements. Chapter 6 i s a summary w i th c o n c l u s i o n s . 4 I I . SPECTRAL THEORY OF TURBULENCE T h i s c h a p t e r o u t l i n e s the theory of s p e c t r a l a n a l y s i s as a p p l i e d to t u r b u l e n c e v e l o c i t y f i e l d s . A complete account of the theory i s necessa r y as a b a s i s f o r the model of s p a t i a l a v e r a g i n g to be p r e s e n t e d in Chapter 5. F u r t h e r d e t a i l can be found in B a t c h e l o r (1953) and H inze (1975) . The s p e c t r a are d e f i n e d in terms of 2-point c o r r e l a t i o n s in a homogeneous, s t a t i o n a r y f l ow . One-d imens iona l and d i s s i p a t i o n s p e c t r a a re d e f i n e d and t h e i r p r o p e r t i e s are summarized f o r the case of i s o t r o p i c t u r b u l e n c e . The e m p i r i c a l Nasmyth u n i v e r s a l spectrum c h a r a c t e r i s t i c of l a r g e - s c a l e p r o c e s s e s found in the ocean i s a l s o i n t r o d u c e d . 2.1 The Spectrum Tensor The s p a t i a l c o - o r d i n a t e system i s chosen so as to move w i th the mean f l ow , U, and i s denoted by x = ( x , , x 2 , x 3 ) w i th x, in the d i r e c t i o n of the mean f l o w . The f l u c t u a t i n g E u l e r i a n v e l o c i t y , measured at x at t i m e , t , i s u ( x , t ) , i = 1 ,2 ,3 . I Time averages a re d e f i n e d by For homogeneous t u r b u l e n c e the two-point v e l o c i t y c o r r e l a t i o n t enso r i s a f u n c t i o n of the s e p a r a t i o n v e c t o r , x, and d e f i n e d by R i : j ( x ) = u±{r) U j ( r + x) (2.1.2) The assumption of homogeneity guarantees that the tensor does not depend on r. The spectrum t enso r * i j (k) i s d e f i n e d as the s p a t i a l t h r ee-d i m e n s i o n a l F o u r i e r t r ans fo rm of the c o r r e l a t i o n t e n s o r . The r e s u l t i n g F o u r i e r t r ans fo rm p a i r i s * i 1 < * > - 77T? >7J R i j ( i , e ' i - - d ^ ( 2 ' 1 ' 3 a ) i J ~~ (2it) 3 - » J J - " —oo -J The r a d i a n wavenumber i s denoted by k = ( k 1 f k 2 , k 3 ) The o f f - d i a g o n a l e lements a re a s s o c i a t e d w i th shear s t r e s s e s . The d i a g o n a l e lements of the spectrum tensor a re r e l a t e d to the d i s t r i b u t i o n of t u r b u l e n t k i n e t i c energy i n wavenumber s p a c e ; i . e . , R (o) = u~2 + u~2 + ^ 2 = j j j s ^ (X)dk ( 2 . 1 . 4 ) 1 1 1 2 3 - ~ Thus the t r a c e of the spectrum t enso r i s tw ice the d e n s i t y of t u r b u l e n t k i n e t i c energy per u n i t mass i n k-space. 2.2 Measured S p e c t r a In p r a c t i c e a component of v e l o c i t y , u 2 ( t )> say , i s measured at a p o i n t past which the t u r b u l e n t s t r u c t u r e i s convec t ed at speed U and i s r e co rded as a t ime s e r i e s . (The d i f f e r e n t i a t e d output r e p r e s e n t s a t ime s e r i e s of the s h e a r , = _ 1 ^ "2 . ) A computer p r o v i d e s an es t ima te of OUj u ot the F o u r i e r t r a n s f o r m , u 2 (co), u , U ) = r u 2 ( t ) e i w t d t ( 2 . 2 . 1 ) T h e power s p e c t r u m a s s o c i a t e d w i t h u 2 ( t ) i s cp2 (to) = |u2<co) | 2 (2.2.2) We a s s u m e t h a t t h e f l o w i s e s s e n t i a l l y " f r o z e n " a s i t i s a d v e c t e d p a s t t h e m e a s u r e m e n t p o s i t i o n ( T a y l o r ' s h y p o t h e s i s , v a l i d f o r |u|=u<<u); t h e n we c a n i d e n t i f y a c o r r e l a t i o n l e n g t h , x 1 f w i t h t h e t i m e l a g , T=X 1/U ; s i m i l a r l y , t h e r a d i a n f r e q u e n c y c a n be r e l a t e d t o t h e d o w n s t r e a m wavenumber s c a l e d w i t h t h e c o n v e c t i o n v e l o c i t y ; u = k,U. I t i s w e l l known t h a t t h e power s p e c t r u m i s t h e F o u r i e r t r a n s f o r m o f t h e t i m e l a g c o r r e l a t i o n , p ( r ) . <MW> = h / ° P ( T ) e " i o ) T dx (2.2.3) — CD • TOT / " P t ^ e - M l dx, <2-2-*> I d e n t i f y i n g t h e s p a t i a l c o r r e l a t i o n a s p ( x , / U ) = R 2 2 ( x , , 0 , 0 ) M") = T H J / E D R 2 2 ( x l ' ° ' o ) e ~ i k l X l d x i (2.2.5) — CD U 22 1 w h e r e 0 2 2 ( k , ) i s t e r m e d t h e o n e - d i m e n s i o n a l t r a n s v e r s e p o w e r s p e c t r u m . C o n s i d e r a t i o n o f t h e o n e - d i m e n s i o n a l F o u r i e r t r a n s f o r m p a i r a n d t h e d e f i n i t i o n o f t h e s p e c t r u m t e n s o r ( 2 . 1 . 3 a ) shows t h a t t h e m e a s u r e d s p e c t r u m i s o b t a i n e d f r o m t h e 7 spectrum tensor by integrating over the transverse plane in wavenumber space for a l l values of k,. R 2 2 ( x i ' 0 ' 0 > = / / J * 2 2 O O e i k i x i d k (2.2.6) = i C D [ / / $ 2 2 ^ ) d k 2 d k 3 ] e i k l X l dk2 (2.2.7) so that, *22 ( k l> = /J$22< k> d k2 d k 3 (2.2.8) The contribution of transverse components of the spectrum tensor to the measured one-dimensional spectra i s an e s s e n t i a l element in the three-dimensional model of s p a t i a l averaging. Two spectra are usually measured: O n ^ i * = /2 $ i i< k > d k 2 d k 3 (2.2.9) * 2 2 0<1> = J7*22<*> d k 2 d k 3 (2.2.10) These are termed the longitudinal (downstream) and transverse (cross-stream) spectra, respectively. They are transforms of auto-covariances with the separation vector aligned p a r a l l e l and perpendicular to the measured v e l o c i t y component. An iso t r o p i c v e l o c i t y f i e l d can be s p e c i f i e d by these spectra (or 8 c o r r e l a t i o n s ) a lone and add ing the c o n s t r a i n t of i n c o m p r e s s i b i l i t y p r o v i d e s an a d d i t i o n a l r e l a t i o n between them. A c ross-s t r eam sensor such as the a i r f o i l probe measures 0 2 2 = 0 3 3 (assuming i s o t r o p y ) wh i le the LDA can measure e i t h e r or 0 2 2 * L D A c a n a l s o measure l i n e a r combina t ions of u , ( t ) and u 2 ( t ) by s u i t a b l e o r i e n t a t i o n of the f r i n g e p a t t e r n . T h i s p rocedure adds a f r a c t i o n of the mean f low to the measurement (used fo r f requency s h i f t i n g as e x p l a i n e d in Chapter 3) and r e s u l t s in c r o s s - s p e c t r a tha t c o n s t i t u t e a measure of the shear s t r e s s . 2.3 The Th ree-d imens iona l Energy Spectrum D i r e c t i o n a l i n f o r m a t i o n can be removed from the spectrum tenso r by i n t e g r a t i n g over the s u r f a c e of a sphere of r a d i u s , k, where k 2 = k i k ^ = ( k , 2 + k 2 2 + k 3 2 ) . The r e s u l t i n g spec t rum, E ( k ) , i n c l u d e s c o n t r i b u t i o n s from the spectrum tensor over a l l d i r e c t i o n s . ( I t i s o f t e n r e f e r r e d to as the " t h r e e - d i m e n s i o n a l s p e c t r u m " ; h e r e , the te rm, "energy spec t rum" , w i l l be used) E(k) = \ //$ i i ( l c ) c la (k ) (2.3.1) where a i s a sphere of r a d i u s k. E (k) g i v e s the c o n t r i b u t i o n to the t u r b u l e n t k i n e t i c energy per u n i t mass due to a l l wavevectors of l eng th k, or on a l eng th s c a l e 1/k, and hence i s the spectrum wi th the most obv ious 9 p h y s i c a l i n t e r p r e t a t i o n , i . e . j " E (k )dk = \ J J J $ i i ( * > d * (2.3.2) 0 — 00 = 2" u ^ U j ^ = k i n e t i c energy/mass 2.4 I s o t r o p i c Turbu lence An i s o t r o p i c f i e l d has no p r e f e r r e d d i r e c t i o n . (For a r i g o r o u s mathemat ica l d e f i n i t i o n of i s o t r o p y see B a t c h e l o r , 1953.) I s o t r o p i c f lows can occur when the re i s no s p e c i a l o r i e n t a t i o n of the s t r a i n i n g f i e l d . He re , , i t w i l l s u f f i c e to demand tha t each component of v e l o c i t y has the same v a r i a n c e (but not the same one-d imens iona l spec t rum) . For example, i s o t r o p y might be expected in the ocean in a s u r f a c e l a y e r tha t has been we l l mixed by the wind s t r e s s . I s o t r o p i c t u rbu l ence a l s o had been expected i n g r i d t u rbu l ence s i n c e the r e g u l a r a r r a y of p r o d u c t i o n shears a t the g r i d s e c t i o n homogenize a few mesh l eng ths downstream, l e a v i n g no shear a s s o c i a t e d wi th the mean f l ow . La t e r s t u d i e s showed tha t g r i d t u r b u l e n c e i s a n i s o t r o p i c to the ex ten t tha t = 1.25 2 T h i s unequal d i s t r i b u t i o n of the energy o c c u r s on l y at sma l l wavenumbers and p e r s i s t s d u r i n g decay . Now, the l a s t r e s u l t can be expec ted in g e n e r a l : a n i s o t r o p y o c c u r s in a l l observed f lows at s c a l e s a s s o c i a t e d w i th 10 p r o d u c t i o n or the mean f l ow . On the o the r hand , the sma l l e r e d d i e s i n t e r a c t w i th the s t r a i n f i e l d of somewhat l a r g e r edd ies which have o r i e n t a t i o n s tha t become p r o g r e s s i v e l y randomized as the f i e l d e v o l v e s . The t ime s c a l e of energy exchange dec reases w i th wavenumber so f o r the s m a l l e r edd ies the a s s o c i a t e d s t r a i n r a t e i s much h ighe r than f o r the mean. As a consequence , the sma l l s c a l e motion becomes decoup led from whatever p r e f e r r e d d i r e c t i o n i s imposed on the mean f l ow . Hence, the v e l o c i t y f i e l d s at h i gh wavenumbers a re expected to be i s o t r o p i c . T h i s " l o c a l i s o t r o p y " i s a u n i v e r s a l p r o p e r t y of h i g h Reyno lds number t u r b u l e n t f l o w s . With the assumpt ion of i s o t r o p y , the spectrum t e n s o r can be g r e a t l y s i m p l i f i e d and exp ressed i n terms of the energy spec t rum, E ( k ) . B a t c h e l o r (1953) has shown tha t In p a r t i c u l a r , 11 4Ttk*4 i 4 i r k H 2 (2.4.3) 11 so t h a t , • 7 /*, ^ J 1 <*2 - **> « (2.4.4) s i m i l a r l y , 0 > 2 2 ( k 1 ) = J / " (*2 + k 2 ) d k ( 2 . A . 5 ) I X 1 The i n t e g r a t i o n i s per formed over the t r a n s v e r s e wavenumber ( k 2 , k 3 ) - p l a n e t a k i n g i n t o account the symmetry about the k , - a x i s . T h i s symmetry a l s o occu r s i n the geometry of the a i r f o i l probe and i s e x p l o i t e d in the c a l c u l a t i o n of the t h e o r e t i c a l t r a n s f e r f u n c t i o n i n Chapter 5 . D i f f e r e n t i a t i o n of the above r e l a t i o n s g i v e s the w e l l known " i s o t r o p i c r e l a t i o n s " : = <*i> - * J aSq [*7 a i q •n<*i>.] ( 2 - 4 - 6 ) Inq- *22<*i> " " T i ^ K i ^ M (2.4.7) 1 These equa t i ons can be used to d e r i v e E ( k 1 ) from the one-d i m e n s i o n a l measured s p e c t r a . Another e x p r e s s i o n fo r E ( k , ) i s a v a i l a b l e i n which on l y f i r s t d e r i v a t i v e s a re employed and which combines both l o n g i t u d i n a l and t r a n s v e r s e s p e c t r a l i n f o r m a t i o n : E ( k x ) = - k j {jjq- + 2 « j » 2 2 ( k 1 ) ] ( 2 . 4 . 8 ) 12 T h i s e q u a t i o n , ob ta ined by summing (2 .4 .2 ) and ( 2 . 4 .3 ) be fo re d i f f e r e n t i a t i o n , was used in t h i s work to c a l c u l a t e E ( k , ) f o r the water t unne l ( g r id ) f low which i s somewhat a n i s o t r o p i c at low wavenumbers, as e x p l a i n e d above . S i n ce the i s o t r o p i c r e l a t i o n s ( 2 .4 .4 ) and ( 2 . 4 .5 ) do not app l y at these s c a l e s , the energy spectrum i s suspec t at low wavenumbers. N e v e r t h e l e s s , i n p u t t i n g both s p e c t r a w i th ( 2 . 4 . 8 ) i s p r o b a b l y a b e t t e r app rox ima t i on than would be o b t a i n e d by u s i n g e i t h e r ^ ^ ( k , ) or 0 2 2 ( k , ) a l o n e . 2.5 The S p a t i a l T r a n s f e r F u n c t i o n The s p a t i a l t r a n s f e r f u n c t i o n ( STF ) , or wavenumber response of the a i r f o i l p robe , c h a r a c t e r i z e s the e f f e c t of s p a t i a l a v e r a g i n g on the s e n s i t i v i t y of the probe as a f u n c t i o n of wavenumber. The' STF i s d e f i n e d to be the r a t i o of the c r o s s -stream spectrum measured by the shear probe t o the " t r u e " spectrum approximated by measurement w i th the h i g h s p a t i a l r e s o l u t i o n LDA. I f 7 2 2 ^ 1 ) i s the spectrum as " s e e n " by the shear probe then we d e f i n e the STF to be (shea£L>robe) <2-5a) A s imp le model of s p a t i a l a v e r ag ing might c o r r e s p o n d to a moving average over a l e n g t h 2L, say . Then , t h i s " t o p h a t " impulse response i s ana logous to the f a m i l i a r prob lem in t ime s e r i e s a n a l y s i s when a s i g n a l i s measured over a f i n i t e t ime 13 i n t e r v a l . The t r a n s f e r f u n c t i o n i n t h i s one-d imens iona l case i s we l l known: sin 2k,L x < * ' ) = o ^ - < 2 - 5 - 2 ) T h i s approach i s shown l a t e r to be o v e r s i m p l i f i e d as i t f a i l s to take i n t o account the t h r e e - d i m e n s i o n a l na ture both of the probe and the t u r b u l e n c e . 2.6 The E q u i l i b r i u m Spectrum The h i g h wavenumber regime of the energy spectrum e x h i b i t i n g l o c a l i s o t r o p y as d i s c u s s e d in 2.2 i s termed the e q u i l i b r i u m range. Here the sma l l e r edd i e s t r a n s f e r energy at a r a t e tha t i s f a s t compared to the e v o l u t i o n (decay) t ime s c a l e of the mean flow ( " l o c a l " e q u i l i b r i u m and i s o t r o p y ) . Then the r a te of d i s s i p a t i o n of t u r b u l e n t k i n e t i c energy , e, a l o n g w i th the k inemat i c v i s c o s i t y , v, shou ld be the o n l y p h y s i c a l parameters govern ing the exchange of energy between n e i g h b o u r i n g wavenumber components and hence shou ld determine the sma l l s c a l e energy spec t rum. D imens iona l a n a l y s i s g i v e s f o r the u n i v e r s a l e q u i l i b r i u m regime (Kolmogorov,1941) E(k) = ( e v 5 ) l ^ F f * - ) (2.611) where, where F (k/ks ) i s a u n i v e r s a l non-d imens iona l s p e c t r a l fo rm. F u r t h e r , i f the e n e r g y - c o n t a i n i n g s c a l e s are w e l l s epa ra t ed in wavenumber space from the d i s s i p a t i o n s c a l e , t he re shou ld e x i s t pa r t of the e q u i l i b r i u m range which i s a l s o independent of 14 v i s c o u s e f f e c t s . In t h i s s o - c a l l e d i n e r t i a l subrange , E (k ) must then have the form E ( k ) = a 0 ( e v 5 ) 1 A ( k / k ) - 5 / 3 s '(2.6.3) = a 0 e 2 / 3 k" 5/ 3 (2.6.4) where, a 0 i s the ( e m p i r i c a l ) Kolmogorov c o n s t a n t . L i k e w i s e , <p-,.{k.) and 0 2 2 ( k i ) c a n be no rma l i z ed to g i v e (the symbol 'k' w i l l be used f o r wavenumber for these normalized spectra) • uOO = ( e v 5 ) l / * F j f J - ) ( 2' 6- 5> s • 2 2(k> = (ev5)l/«» F 2 ( ^ - ) ( 2 . 6 . 6 ) In the i n e r t i a l subrange , • nOO = a l E 2 / 3 k - 5 / 3 f 4>22(k) = a 2 e 2 / 3 k - 5 / 3 § Whereas the e q u i l i b r i u m range can be expec ted i n most t u r b u l e n t f l o w s , the i n e r t i a l subrange can occur on l y at ve ry h i g h Reyno lds numbers. G r a n t , S tewar t , and M o l l i e t (1961) o b t a i n e d a -5/3 s l ope over a range of wavenumbers spann ing more than th ree decades w i th data taken in a t i d a l channe l w i th l a r g e s c a l e s (50 m e t e r s ) , h igh mean speeds (a few mete rs/sec ) and the low v i s c o s i t y ( r e l a t i v e to a i r ) of water . They e s t i m a t e d a, = 9 5T a ° a, = 4 3 1 (2.6.7) (2.6.8) 15 Reynolds numbers on the o rder of 1 0 8 . On the o ther hand, l a b o r a t o r y f lows r a r e l y e x h i b i t an i n e r t i a l subrange . 2.7 The Nasmyth U n i v e r s a l Spectrum Nasmyth (1970), u s i n g three inshore ocean i c s p e c t r a wi th h igh t u rbu l ence l e v e l s and low n o i s e , c o n s t r u c t e d an e m p i r i c a l F ^ k / k s ) , commonly r e f e r r e d to as the Nasmyth u n i v e r s a l spec t rum. Oakey (1982) has p u b l i s h e d 15 data p o i n t s of F , (k/ks ) and c o r r e s p o n d i n g d e r i v e d va lues of F 2 ( k / k s ) and G 2 ( k / k s ) . For t h i s work, a s p l i n e f i t was made to the p u b l i s h e d va lues of F , ( k/ks ) and F 2 ( k /ks ) and the r e s p e c t i v e d e r i v a t i v e s were used in ( 2 .4 .8 ) to c a l c u l a t e the u n i v e r s a l energy spec t rum, F ( k / k s ) . These s p e c t r a are p l o t t e d in F i g . 1. (The s u b s c r i p t s have been dropped from the n o t a t i o n fo r the one-d imens iona l u n i v e r s a l s p e c t r a and a l l the f i g u r e s use c y c l i c wavenumber £ , (cyc les/cm) to s i m p l i f y conve r s i on to c o r r e s p o n d i n g l e n g t h s c a l e s . ) It shou ld be kept in mind tha t on l y the l o n g i t u d i n a l spec t rum, F , ( k / k s ) , was measured by Nasmyth and the t r a n s v e r s e , energy , and d i s s i p a t i o n s p e c t r a were d e r i v e d from these da ta us ing the i s o t r o p i c r e l a t i o n s . The d i s s i p a t i o n spectrum or shear spectrum can be d e f i n e d s t a r t i n g from the e x p r e s s i o n fo r e. e = v S i j S i j (2.7.1) where the f l u c t u a t i n g s t r a i n r a t e (shear ) i s g i v en by _ 1 » uj s i j _ "2" I & X j + "oxjO In i s o t r o p i c t u rbu l ence the s e v e r a l terms in (2 .7 .1 ) can be • 16 summed (H inze , 1975) to g i ve r _ 15 „ (2.7.2). 2 l a x l J The shear probe wi th d i f f e r e n t i a t e d output g i ves the shear spectrum as the d i s t r i b u t i o n of the va r i ance in ( 2 . 7 . 2 ) . The d i s s i p a t i o n spectrum i s d e f i n e d as e = J ° ° D(k) dk (2.7.3) 0 I t can be shown ( B a t che l o r , 1953) that D(k) = 2vk 2 E (k ) (2.7.4) and E = i i v r"k2<|>22(k) d k (2.7.5) 2 0 The l a t t e r e x p r e s s i o n shows how the one-d imens iona l spectrum from the shear probe i s r e l a t e d to the d i s s i p a t i o n , e. The u n i v e r s a l t r ans ve r s e d i s s i p a t i o n spectrum, G 2 ( k / k s ) , i s d e f i n e d by M*H = (h)2 F2(fcl (2.7.6) s s & and shown in F i g . 2. The s p e c t r a l a n a l y s i s of t u rbu l ence g iven in t h i s chapte r w i l l be r e c a l l e d throughout the remainder of the t h e s i s . The r e l a t i o n between the one-d imens iona l spec t r a and the energy spectrum i s c r u c i a l to the development of the ave rag ing mode l . The u n i v e r s a l s p e c t r a w i l l be used to c a l c u l a t e a t r a n s f e r 17 f u n c t i o n f o r l a r g e - s c a l e t u rbu l ence and to determine the f r a c t i o n of the d i s s i p a t i o n v a r i a n c e r e c o v e r a b l e by a shear probe at a g i ven t empera tu re . F i g u r e 1 - Nasmyth u n i v e r s a l s p e c t r a : F ( k / k s ) , • F , ( k / k s ) , F 2 ( k / k s ) 16 F i g u r e 2 - The t r a n s v e r s e d i s s i p a t i o n spec t rum, G 2 ( k / k s ) 19 I I I . EXPERIMENTAL PROCEDURE The expe r imen ta l method cen t e r s around the a p p l i c a t i o n of l a s e r Dopp ler anemometry to measure t u rbu l ence v e l o c i t y w i th h igh s p a t i a l r e s o l u t i o n . P r i o r expe r i ence (Osborn, p e r s o n a l communicat ion) has shown LDA measurement of t u rbu l ence s p e c t r a to be d i f f i c u l t and so a LDA system was des igned and c o n s t r u c t e d to ga in a thorough unders tand ing of the p r i n c i p l e s and l i m i t a t i o n s i n v o l v e d . Measurement of the s p a t i a l t r a n s f e r f u n c t i o n as d e f i n e d by ( 2 .5 .1 ) r e q u i r e s tha t the LDA and shear probe response be measured i n the same t u r b u l e n t f l o w . F u r t h e r , the s i g n a l to n o i s e r a t i o (SNR) must be adequate over s c a l e s of i n t e r e s t , tha t i s , 5cm>X>0.25cm, where X=27r/k=l/k. A f t e r p r e l i m i n a r y i n v e s t i g a t i o n s of f u l l y t u r b u l e n t p i p e f lows and c y l i n d e r wakes, a g r i d f low was chosen because of i t s ex t ens i v e homogenei ty , low i n t e n s i t y , l e n g t h s c a l e o p t i o n s , and approximate i s o t r o p y as d e s c r i b e d i n ( 2 . 4 ) . T h i s chapte r w i l l beg in wi th the theory of the a i r f o i l probe and i t s c a l i b r a t i o n . The LDA theory and inst rument des ign i s then p r e sen t ed w i th a t t e n t i o n g i ven to the o p t i c a l sys tem, f requency s h i f t i n g , demodu la t i on , and the theory of Doppler b roaden ing . The tunne l se t-up , measurement scheme, and data a n a l y s i s p rocedure complete the c h a p t e r . 20 3.1 The A i r f o i l Shear Probe The a i r f o i l shear probe was f i r s t i n t r o d u c e d by S iddon (1971) as a sma l l a i r f o i l s e n s i t i v e to c ross-s t r eam v e l o c i t y f l u c t u a t i o n s . T h i s probe was l a t e r adapted f o r use in water by Osborn and Siddon (1975) . The a l t e r n a t i v e , h o t - f i l m anemometry, has h igh power r equ i r emen t s , i s n o n - l i n e a r , and i t s c a l i b r a t i o n i s a f f e c t e d by v a r i a t i o n s in the ambient t empera tu re . In c o n t r a s t , the shear probe has low power r equ i r emen ts , needs l e s s c o m p l i c a t e d e l e c t r o n i c c i r c u i t r y and has a l i n e a r r e sponse . A complete d e s c r i p t i o n s of the probe i s g i ven in Osborn and Crawford (1977 and 1980). The f o l l o w i n g d i s c u s s i o n w i l l focus on those a spec t s of the probe r e l e v a n t to the water tunne l exper iments and the s p a t i a l a ve rag ing mode l . Two probe des igns were used ; d i f f e r i n g ma in ly in o v e r a l l s i z e . These are r e f e r r e d to as the Osborn probe and the Oakey (1977) p robe , the l a t t e r be ing somewhat l a r g e r than the former w i th maximum r a d i i of 0.24cm and 0.29cm, r e s p e c t i v e l y . The a i r f o i l i s ske tched i n F i g . 3a ; d e t a i l s of the probe d imens ions and shapes are g i ven in Appendix A . The probe c o n s i s t s of th ree p a r t s : a s t a i n l e s s s t e e l rod or " s t i n g " a c t i n g as a support and to ma in ta in the probe in f r o n t of the flow d i s t o r t i o n due to the ins t rument support or v e h i c l e ; a p i ezo-ce ramic bimorph sens ing element tha t ou tputs a v o l t a g e p r o p o r t i o n a l to the t o t a l a p p l i e d bending moment; a c a s t s i l i c o n e rubber nose-p iece tha t p r o t e c t s the sens ing element and conforms to the d e s i r e d c u r v a t u r e . Figure 3 - Schematic of a i r f o i l probe. 22 The behav io r of the shear probe i n an unsteady f low can be unders tood by c o n s i d e r i n g a c ross-s t r eam f l u c t u a t i o n , u 2 ( t ) , superposed on a mean a x i a l f l ow , U, g i v i n g a r e s u l t a n t speed , V ( t ) , and ang le of a t t a c k , a ( t ) as shown in F i g 3b. Then p o t e n t i a l f low theory ( A l l e n and Pe rk ins ,1952 ) g i v e s f o r the c r o s s - f o r c e per un i t l e n g t h , d f / d x , : = 1 n V 2 dA . 0 (3.1.1) where p i s the f l u i d d e n s i t y and A (x , ) i s the c r o s s - s e c t i o n a l a rea of the p robe . The doub le-ang le formula V 2 s i n 2 a = 2Vcosa V s i n a = 2Uu 2 ( t ) a l l ows us to w r i t e d f _ n dA T T„ , ^ (3.1.2) orx"1 " P arx_1 u " 2 ( t ) so that the c r o s s - f o r c e i s a l i n e a r measure of the c ross-s t r eam v e l o c i t y component, u 2 , f o r a cons tan t mean f l ow . I f we d e f i n e 2L as the d i s t a n c e a long the a x i s of the probe from the t i p to where the c u r v a t u r e becomes i n s i g n i f i c a n t (see Appendix A ) , then the f o r ce per u n i t l eng th can be i n t e g r a t e d over t h i s s e n s i t i v e p o r t i o n to g i ve the t o t a l c r o s s - f o r c e , 2L F c = J d x l = P A 0 U U 2 ( 3 - 1 ' 3 ) 0 However, t h i s exp re s s i on fo r the t o t a l c r o s s - f o r c e does not 23 d e s c r i b e how the s e n s i t i v i t y v a r i e s a long x, and w i th the r ad ius r, where r 2 = x 2 2 + x 3 2 . For a p a r a b o l o i d of r e v o l u t i o n ( x , = 7 0 r 2 , fo r 0<x,<2L, r<R, where 7 0 = c o n s t a n t ) , A i s l i n e a r in x, and the s e n s i t i v i t y i s cons tan t in the x , - d i r e c t i o n over a l e n g t h , 2L . df fq- = cons tan t , 0 < x : < 2L (3.1.4) The v a r i a t i o n of the s e n s i t i v i t y in the t r a n s v e r s e p lane i s l i n e a r , i , e . , ojf « r , r < R (3.1.5) T h i s d i s t r i b u t i o n of s e n s i t i v i t y w i l l be a p p l i e d l a t e r in the s p a t i a l a ve rag ing model . I t shou ld be noted tha t there e x i s t s a second c o n t r i b u t i o n to the c r o s s - f o r c e due to v i s c o u s e f f e c t s . T h i s term can be ignored f o r low l e v e l s of t u rbu l ence (<10%) (Osborn and C rawfo rd ,1977 ) . A f i n a l comment shou ld be made conce rn i ng the a p p l i c a t i o n of the (s teady) p o t e n t i a l f low theory to t ime-dependent f l ows . In the a n a l y s i s of A l l e n and P e r k i n s , the c r o s s - f o r c e i s due to the component of the f low in a p lane tha t i s normal to the a x i s of the probe and tha t i s moving w i th the f l ow . The t ime-dependent B e r n o u l l i equa t ion then i n c l u d e s a term due to the t ime ra te of change of the probe r a d i u s , but not due to the t u r b u l e n t v e l o c i t y f l u c t u a t i o n s as long as T a y l o r ' s " f r o z e n 24 f l ow" h y p o t h e s i s i s v a l i d . In o the r words, the t u r b u l e n t f low i s assumed not to be t ime-dependent in a frame of r e f e r e n c e moving w i th the f l ow . I f , however, the eddy s t r u c t u r e e v o l v e s s i g n i f i c a n t l y d u r i n g the t r a n s i t t ime over the l e n g t h of the p r o b e , t h i s t ime dependence would have to be i n c l u d e d in the a n a l y s i s . 3.2 Shear Probe C a l i b r a t i o n The probes were c a l i b r a t e d u s i n g the method d e s c r i b e d by Osborn and Crawford (1980) . The probe was mounted on a s h a f t r o t a t i n g at 6.3 Hz and h e l d at an a n g l e , a , i n a ve ry low t u r b u l e n t i n t e n s i t y n o z z l e f l ow . The dynamic p r e s s u r e was o b t a i n e d w i th p r e s s u r e taps and a d i g i t a l s t r a i n gauge, g i v i n g the n o z z l e v e l o c i t y , V, a c cu r a t e to 2% ( t y p i c a l l y , V=68cm/s). The s i n u s o i d a l output v o l t a g e i s p r e - a m p l i f i e d , averaged over a t l e a s t 1,000 p e r i o d s , and i t s power con ten t de te rmined w i th an HP 3582A spectrum a n a l y s e r . T h i s r . m . s . v o l t a g e , Erms, was c o r r e c t e d f o r the s i g n a l due to the probe d roop ing under i t s own we igh t . The ang le of a t t a c k was v a r i e d over ±15° and the da ta were f i t w i th a c u b i c s p l i n e to de te rmine the c a l i b r a t i o n c o n s t a n t d [ E r m s / P V 2 ] j S ~ dLsin2aJ | a = 0 (3.2.1) The shear probe s e n s i t i v i t y , Ksp [ c m / s / v o l t ] , i s d e f i n e d i n terms of the c a l i b r a t i o n cons t an t and the measurement f low speed , U, by K = ( 2/7 US) (3.2.2) 25 3.3 The Laser Doppler System Lase r Doppler anemometry (or v e l o c i m e t r y ) has been e x t e n s i v e l y rev iewed ( e . g . , D u r s t , M e l l i n g , and White law, 1976). The system c o n s t r u c t e d here i s c o n v e n t i o n a l f o r the most pa r t so on l y a s imple model w i l l be p resen ted wi th a t t e n t i o n focused on the f e a t u r e s r e l e v a n t to the measurement of the s p a t i a l t r a n s f e r f u n c t i o n . There are s e v e r a l e q u i v a l e n t models fo r LDA each of which e x p l a i n s how the t ime-dependent phase of the l i g h t s c a t t e r e d by sma l l p a r t i c l e s can be measured to p rov i de an es t ima te of a component of the p a r t i c l e v e l o c i t y : Lad ing (1972) , V a s i l e n k o (1972) , Wang (1972) . The p a r t i c l e i s assumed sma l l enough to f a i t h f u l l y f o l l ow the f l ow . Hence the LDA can measure a s i n g l e t ime-va r y i ng component of the t u r b u l e n t v e l o c i t y f i e l d as i t i s swept past the s c a t t e r i n g ' v o l u m e . 3.3.1 The O p t i c a l System The o p t i c a l system ( F i g . 4a) i s a c o n v e n t i o n a l dual-beam, f o rwa rd-sca t t e r ar rangement . It^s advantages i n c l u d e ease of a l ignment and h igh s i g n a l - t o - n o i s e r a t i o (SNR). The ( p o l a r i z e d ) l a s e r beam i s focused by LI onto the r a d i a l d i f f r a c t i o n g r a t i n g (RDG) which se rves as a beam s p l i t t e r . The d i f f r a c t i o n p lane i s tangent to the RDG where the l a s e r beam i s i n c i d e n t . By t r a n s v e r s e adjustment of the RDG p o s i t i o n , the measurement p lane can be r o t a t e d to an a r b i t r a r y a n g l e , /3. T h i s ad justment pe rmi t s measurement of o b l i q u e v e l o c i t y . components c o n s i s t i n g of l i n e a r combina t ions of U+u, ( t ) and u 2 ( t ) . ( F i g s . 4b and 4c) 26 The g r a t i n g i s des igned to keep 90% of the d i f f r a c t e d power i n the 0th and the two 1st order modes. A l l but the l a t t e r are b l o c k e d , r e s u l t i n g in two coherent beams which are c o l l i m a t e d by L2 and then focused by L 3 . Each beam i s focused onto i t s e l f and at the same time the two beams are brought to conve rgence . A Gauss ian l i g h t beam w i l l focus to i t s d i f f r a c t i o n l i m i t d iameter be fo re d i v e r g i n g a g a i n . There r e s u l t s a r e l a t i v e l y l ong beam wa is t where the l i g h t waves are approx imate l y p l a n a r . The r eg ion where the two beam wa i s t s i n t e r s e c t i s termed the probe volume. I t i s easy to show that these two p lane waves, of wavelength X, i n t e r s e c t i n g at an a n g l e , 26, w i l l form a f r i n g e p a t t e r n of p a r a l l e l p l anes o r i e n t e d p e r p e n d i c u l a r to the p lane c o n t a i n i n g the beams and p a r a l l e l to the o p t i c a x i s . The spac ing of the p l anes i s d=X/ (2s in0 ) . The i n t e n s i t y of the f r i n g e p a t t e r n w i l l be modulated by a t r i p l y Gauss ian s p a t i a l d i s t r i b u t i o n , forming a sma l l (approx. 0.1mm in the x, d i r e c t i o n and 1.0 mm a long the o p t i c a x i s ) e l l i p s o i d a l probe volume, norma l l y d e f i n e d by the s u r f a c e over which the i n t e n s i t y of the f r i n g e p a t t e r n i s 1/e 2 of i t s maximum. T h i s s u r f a c e and the probe d imens ions are ske tched in F i g . 4d . A p a r t i c l e moving w i th a speed , U , normal to the f r i n g e p l a n e s , w i l l s c a t t e r the l i g h t w i th an i n t e n s i t y modulated by the f r i n g e p a t t e r n . The s c a t t e r e d l i g h t w i l l s c i n t i l l a t e w i th a f requency depending on l y on the v e l o c i t y and f r i n g e s p a c i n g . T h i s Doppler f requency i s j u s t 27 f = 2sinQ „ _ K U (3.3.1) rD X u l ~ * u l The scattered l i g h t i s col l e c t e d by L4 and focused onto a 0.1mm pinhole, passed through a narrow band o p t i c a l f i l t e r and detected by a photo-multiplier tube (PMT). The output of the PMT i s a voltage that is frequency modulated with an instantaneous frequency proportional to the measured velocity component. The system i s l i n e a r , calibrated only in terms of the beam convergence angle and wavelength, non-intrusive, and has high s p a t i a l resolution (small probe volume). The accuracy of the LDA i s limited by the measurement of the convergence angle, 6, as the arctangent of the r a t i o of the beam separation to the focal length. This results in a 2% uncertainty in the LDA conversion factor, K, and hence in the measurement of U. Table I l i s t s the components and Table II contains the design parameters for the system constructed, including the d e f i n i t i o n s and values of the probe volume dimensions. F i g . 5 shows the output of the PMT as a large p a r t i c l e crosses the probe volume. This "Doppler burst" i s due to a large p a r t i c l e (diameter>1n) and shows agreement with the approximate number of fringes expected based on cal c u l a t i o n s using the system o p t i c a l parameters (Table I I ) . 28 F i g u r e 4 - The G e o m e t r i c a l O p t i c s of the LDA System 29 i r H 1 1 ^ + K- 1 1 r time (15.4 ysec/div) F i g u r e 5 - F i l t e r e d ouput from the PMT showing Dopp le r " b u r s t " . About 50 f r i n g e s are seen . The t r a n s i e n t waveform was d i g i t i z e d at 50 nsec per p o i n t and then i n t e r p o l a t e d . 30 Table I - LDA System Components L a s e r He-Ne Spe c t r a P h y s i c s Model 120 power: 7mW wavelength: 632.8nm beam diameter ( 1 / e a ) : 0.79mm R a d i a l D i f f r a c t i o n G r a t i n g TPP Type H no. of l i n e p a i r s : 16,384 r o t a t i o n speed: 50Hz beam d i v e r g e n c e a n g l e (1st mode): 5.98 deg % power i n each beam: 25-30% Lenses M e l l e s - G r i o t d iameter: 95mm f o c a l l e n g t h : 305mm AR-coated F i l t e r narrow band, He-Ne pass PMT EMI 9798B 31 Table II - Calculated LDA System Optical Parameters Laser beam diameter: d = 0.79mm Beam waist diameter: d 0 •= 4Xf/iTd = 0.31mm Fringe spacing: X* = 1/K = X/2sin0 = 3.1mm Beam Convergence angle: 9 =5.9 deg Probe Volume Dimensions (at 1/e2 = 10%) ai = d0/4cos6 = 0.080mm c?2 = do/4 = 0.076mm 03 = do/4sin6 = 0.75mm Approximate number of fringes: 52 32 3.3.2 Frequency S h i f t i n g The mean Doppler f r equency must be s h i f t e d away from the no i se which occurs near the o r i g i n at f r e q u e n c i e s a s s o c i a t e d w i th p a r t i c l e t r a n s i t t i m e s . A l s o , f requency s h i f t i n g i s used to d i s t i n g u i s h f low r e v e r s a l s and to vary the rms-to-mean Doppler f requency r a t i o f o r optimum demodu la t i on . Measurement of the zero-mean c ro s s-s t r e am v e l o c i t y component r e q u i r e s f requency s h i f t i n g which was to have been ob t a i ned by r o t a t i n g the RDG at a cons tan t f r equency . When the RDG i s r o t a t i n g , the movement of the g r a t i n g l i n e s Doppler s h i f t s the d i f f r a c t e d beams r e s u l t i n g in a cons t an t f requency d i f f e r e n t i a l between the two beams and a t ime-va ry ing i n t e r f e r e n c e p a t t e r n e q u i v a l e n t to a v i r t u a l mean f l ow , i . e . the d e s i r e d f requency s h i f t . However, i t was found tha t the no i se due to i m p e r f e c t i o n s in the RDG, e s p e c i a l l y at i t s 50Hz r o t a t i o n f r equency , a l ong wi th the need fo r subsequent e l e c t r o n i c d o w n s h i f t i n g fo r optimum demodu la t i on , f o r c ed the adop t i on of an a l t e r n a t i v e p rocedu re . As ment ioned in ( 3 . 4 ) , t r a n s v e r s e p o s i t i o n i n g of the RDG pe rmi t s measurement of " o b l i q u e " v e l o c i t y components ( F i g . 4 b , c ) U m ( t ) = [U + U l ( t ) ] s i n p + u 2 ( t ) c o s p (3.5.1) The mixed s p e c t r a computed from t h i s l i n e a r combina t ion can be w r i t t e n as 33 •m*kl) = 4>n( ki) s i n 2P + * 2 2 ^  1 ^ G O S 2 P + 4>12sin2(2P) (3.5.2) I f the re i s no mean shear (and hence no Reynolds s t r e s s ) the c r o s s term i s ze ro and the c ross-s t r eam power spec t rum, . 0 2 2 ^ 1 ) t i s known g i ven tha t the downstream spect rum, tf>ii(k,), has been measured s e p a r a t e l y (with 0 = 9 0 ° ) , so tha t . , v s _ t>m<kl> - O u P c ^ s i n Z p (3.5.3) Q l o o V K , ) -1 1 1 c o s 2 B Any. non-zero c ross-spec t rum would be an odd f u n c t i o n of 0 and so i t i s easy to check f o r mean shear by comparing 0 m ( k i ) f o r +/3 and - / 3 . (The d i f f e r e n c e of these mixed s p e c t r a p r o v i d e s a measure of the Reynolds s t r e s s . ) The f requency s h i f t i n g i s thus a ch i e ved (without r o t a t i n g the RDG) by hav ing a f r a c t i o n of the mean f low , Usin/3, mixed i n t o the c ross-s t r eam measurement. A va lue f o r /3 of 19° was found conven ien t r e s u l t i n g in a f requency s h i f t of 48kHz at U=50cm/s. The " o b l i q u e " or "mixed s p e c t r a " , 0 m ( k , ) , a re ma in l y the c ross-s t r eam s p e c t r a w i th a sma l l c o r r e c t i o n at l a r g e - s c a l e s where the l o n g i t u d i n a l power dominates . 3 .3 .3 Demodulat ion The PMT s i g n a l depends on the nature of the s c a t t e r i n g p r o c e s s . The PMT v o l t a g e can on ly r i s e above the no i se l e v e l when a p a r t i c l e i s p resen t in the probe volume. With an o s c i l l o s c o p e mon i t o r i ng the PMT v o l t a g e , a h igh SNR s i g n a l was 34 observed between 1/2 to 1/3 of the t ime sugges t ing an i n t e r m i t t e n t s i g n a l w i th an e f f e c t i v e sampl ing r a t e (assuming a p u l s e l eng th equa l to the probe volume t r a n s i t t ime) of 2-3 kHz. The s i g n a l s t r e n g t h and sampl ing ra te had been improved by a r t i f i c i a l l y seed ing the f low wi th l a t ex p a r t i c l e s ob ta ined from d i l u t e d house p a i n t . A l though the above d e s c r i p t i o n of the PMT s i g n a l i m p l i e s a d i s c r e t e LDA v e l o c i t y s i g n a l , i t was found tha t the q u a s i -con t inuous o p e r a t i o n of the f requency t r a c k e r ( d e s c r i b e d below) n e v e r t h e l e s s r e s u l t e d in an e f f e c t i v e l y con t i nuous demodulated output v o l t a g e . I t i s p o s s i b l e tha t the c o n c e n t r a t i o n and the s i z e d i s t r i b u t i o n of the s c a t t e r i n g p a r t i c l e s were such tha t the re were many p a r t i c l e s in the probe volume at any t ime , w i th on l y the l a r g e s t p a s s i n g through the cen te r appear ing on the o s c i l l o s c o p e as Doppler " b u r s t s " w i th h igh SNR. The f requency of the p h o t o - m u l t i p l i e r s i g n a l has a mean va lue determined by the mean f low ra te (p lus any a p r i o r i s h i f t i n g ) and i s modulated about t h i s mean by t u r b u l e n t f l u c t u a t i o n s and n o i s e . I f the s i g n a l i s f ed d i r e c t l y to a spectrum a n a l y s e r , one o b t a i n 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 (PDF) of the f requency and hence of the v e l o c i t y . The width of t h i s spectrum i s a measure of the rms tu rbu l ence i n t e n s i t y p l u s broaden ing due to n o i s e . The PDF p r o v i d e s the v a r i a n c e but not i t s s p e c t r a l d e c o m p o s i t i o n . For example, F i g . 6 shows th ree PDF 's ob ta ined as 35 CM O CM F i g u r e 6 - Power s p e c t r a of the f i l t e r e d PMT s i g n a l e p r e s e n t i n g the v e l o c i t y 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 in 2.54 cm d iamete r p i pe f low 36 the power spectrum of the f i l t e r e d PMT output fo r 2.54cm p ipe f low w i th a f i n e mesh p l a c e d at the p ipe e n t r a n c e . The mean speeds are determined from the cen te r Doppler f requency and the LDA c o n v e r s i o n f a c t o r , K=3.2 kHz/cm/s (cf Tab le I I) and the widths of the " l i n e s " a re determined by broaden ing due to t u r b u l e n t f l u c t u a t i o n s , ambigu i t y no i s e ( to be d i s c u s s e d in s e c t i o n 3 . 3 . 4 ) , and any v a r i a t i o n s i n the mean f l ow . Hence, the widths i n c r e a s e at l e a s t as f a s t as U. The r e a l t ime i n f o r m a t i o n was ob ta ined w i th a f requency t r a c k e r (DISA 55N20) which employs nega t i ve feedback to f o l l ow the input f r equency . A phase- lock l oop ma in ta ins the output of a v o l t a g e - c o n t r o l l e d o s c i l l a t o r (VCO) in a d e f i n i t e phase r e l a t i o n w i th the input s i g n a l . When t h i s " l o c k " c o n d i t i o n h o l d s , the VCO c o n t r o l v o l t a g e f o l l o w s the changes in the input f requency and so p r o v i d e s a demodula t ion of the f i l t e r e d FM s i g n a l from the PMT. Be fore demodu la t i on , the PMT s i g n a l i s band-pass f i l t e r e d to remove low f requency v a r i a t i o n and to min imize broadband h igh f requency no i s e power such as shot no i s e produced as a r e s u l t of ( Po i sson d i s t r i b u t e d ) random e l e c t r o n emiss ion at the PMT photo-ca thode . T h i s ( s e l e c t e d ) pass band i s wider at h ighe r f r e q u e n c i e s r e s u l t i n g i n h i ghe r no i s e l e v e l s in the demodulated s i g n a l . The t r a c k e r i n c o r p o r a t e s a " l o c k - i n d e t e c t o r " to a v o i d t r a c k i n g s p u r i o u s s i g n a l s . For example, i f a " s p i k e " in the PMT s i g n a l causes the t r a c k e r to l o se l ock ( l o s s of c o r r e l a t i o n between the input and the VCO output s i g n a l s ) , the t r a c k e r ho lds 37 the l a s t VCO c o n t r o l v o l t a g e u n t i l l ock i s r e g a i n e d . In most c a s e s , l o ck i s q u i c k l y r e - a c q u i r e d , o therw ise the VCO w i l l s t a r t a s y s t ema t i c sea rch f o r a v a l i d s i g n a l . I t has been suggested tha t t h i s mode of o p e r a t i o n c o u l d reduce the e f f e c t of phase f l u c t u a t i o n s ; on the o the r hand, w i th a h igh drop-out r a te a sample-and-hold spectrum can appear i n the demodulated spect rum. As w i l l be seen , there was no ev idence of t h i s " d r o p - o u t " no i se (which would r o l l o f f as 1/f 2 at h i g h f r e q u e n c i e s ) . 3 .3 .4 Dopp le r Broadening LDA genera tes broad-band no i se power which can 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 power s p e c t r a at h i gh wavenumbers. T h i s no i s e a r i s e s from the s t o c h a s t i c nature of the s c a t t e r i n g p rocess w i t h i n the probe volume. The random a r r i v a l of the p a r t i c l e s and the f i n i t e d u r a t i o n of the s i g n a l from each p a r t i c l e r e s u l t s in s e v e r a l c o n t r i b u t i o n s (other than t u rbu l ence ) to the b roaden ing of the PDF and the appearance of broad-band f requency (almost "wh i t e " ) no i s e in t u rbu l ence s p e c t r a . C o l l e c t i v e l y , these n o i s e sources are termed " amb igu i t y n o i s e " which i s rev iewed by Buchave, et a l (1979) . They c o n s i d e r s e v e r a l c o n t r i b u t i o n s (each assumed to be Gauss ian ) to the v a r i a n c e of a s i g n a l f o r a mean f low wi th t u r b u l e n c e . 1) T r a n s i t t ime b r o a d e n i n g : (3.7.1) 38 2) G r a d i e n t b r o a d e n i n g : (3.7.2) 3) T u r b u l e n c e b r o a d e n i n g : ( A f rp) = T 5 i T S T ' v (3.7.3) The f i r s t t e r m i s o f t h e o r d e r o f t h e i n v e r s e o f t h e t r a n s i t t i m e f o r a p a r t i c l e c r o s s i n g t h e p r o b e v o l u m e , 1/T, w h e r e T=4a,/U, a n d i s what one w o u l d e x p e c t f o r t h e s p e c t r u m o f a t r u n c a t e d s i n e w a v e . I n t h e c a s e o f many p a r t i c l e s i n t h e p r o b e v o l u m e s i m u l t a n e o u s l y , t h e s i g n a l i s c o n t i n u o u s b u t s i n c e i t c o n s i s t s o f t h e s u p e r p o s i t i o n o f r a n d o m l y p h a s e d s i g n a l s , t h e r e r e s u l t m i n i m a i n t h e e n v e l o p e , s e p a r a t e d on t h e a v e r a g e by t h e c o r r e l a t i o n t i m e f o r t h e p a r t i c l e e n s e m b l e . So t h e c h a r a c t e r i s t i c t i m e i s t h a t r e q u i r e d t o r e f r e s h t h e p o p u l a t i o n o f p a r t i c l e s , i . e . a s b e f o r e , t h e t r a n s i t t i m e . H e n c e , t h e b r o a d e n i n g f o r t h e d i s c r e t e a n d t h e c o n t i n u o u s mode i s a b o u t t h e s ame . The g r a d i e n t b r o a d e n i n g a r i s e s when t h e s c a l e o f t h e mean s h e a r i s on t h e o r d e r o f t h e p r o b e d i m e n s i o n a l o n g t h e o p t i c a x i s , 4a 3. T h i s e f f e c t may be i m p o r t a n t i n a b o u n d a r y l a y e r b u t c a n be i g n o r e d f o r t h e f l o w s s t u d i e d h e r e . The l a s t t e r m a r i s e s f r o m t u r b u l e n t v e l o c i t y f l u c t u a t i o n s w i t h i n t h e p r o b e v o l u m e and t h i s v a r i a n c e s h o u l d be p r o p o r t i o n a l t o U 3 . The e s t i m a t e o f e was o b t a i n e d by u s i n g B a t c h e l o r ' s d e c a y l aw f o r g r i d t u r b u l e n c e a s d i s c u s s e d i n C h a p t e r 4. T h e s e c o n t r i b u t i o n s sum t o t h e t o t a l D o p p l e r b r o a d e n i n g 39 ( A f ) = ( A f F ) 2 + ( A f G ) 2 + ( Af rp) 2 <3-7-4> o r , s i n c e t h e m e a n s h e a r i s a s s u m e d t o b e n e g l i g i b l e , we c a n i g n o r e t h e g r a d i e n t b r o a d e n i n g t e r m a n d w r i t e ( 3 . 7 . 5 ) ( A f ) 2 = ( A f F ) 2 + ( A f T ) 2 G e o r g e a n d L u m l e y ( 1 9 7 3 ) h a v e s h o w n t h a t t h e p o w e r s p e c t r u m o f D o p p l e r a m b i g u i t y n o i s e i s n e a r l y c o n s t a n t ( " w h i t e " ) f o r t u r b u l e n c e f r e q u e n c i e s a n d t h e p o w e r d e n s i t y i s p r o p o r t i o n a l t o A f a t l o w f r e q u e n c i e s . N = 0.73Af ( 3 . 7 . 6 ) S o a t u r b u l e n c e v e l o c i t y p o w e r s p e c t r u m w o u l d b e a d d e d t o a p l a t e a u o f n e a r l y c o n s t a n t p o w e r . T h i s p r o p e r t y o f t h e n o i s e s p e c t r u m a l l o w s o n e t o o b t a i n a l o w e r b o u n d ( t h e p o w e r a t f r e q u e n c i e s a b o v e t h e v i s c o u s c u t - o f f r e g i o n ) w h i c h c a n b e s u b t r a c t e d f r o m t h e m e a s u r e d s p e c t r a t o i m p r o v e t h e s i g n a l - t o -n o i s e r a t i o a t h i g h f r e q u e n c i e s . 3 . 4 T h e W a t e r T u n n e l A r r a n g e m e n t T h e t e s t s e c t i o n o f t h e w a t e r t u n n e l ( l o c a t e d a t t h e I n s t i t u t e o f O c e a n S c i e n c e s , P a t r i c i a B a y , B . C . ) i s 2 m e t e r s i n l e n g t h w i t h a 3 0 c m x 3 0 c m c r o s s s e c t i o n . T h e w a l l s a r e c o n s t r u c t e d w i t h 1" p l e x i g l a s s p e r m i t t i n g o p t i c a l a c c e s s t o t h e f l o w . T h e g r i d o r c y l i n d e r was m o u n t e d a t t h e e n t r a n c e t o t h e t e s t s e c t i o n a n d m e a s u r e m e n t s t a k e n a b o u t 4 0 c m d o w m s t r e a m . 40 The tunne l f a c i l i t y i s equipped w i th a P i t o t tube to moni tor the mean f low (most u s e f u l as a check on the LDA c a l i b r a t i o n ) . A s t r a i n gauge senses the dynamic p r e s s u r e and a s i g n a l w i th a f requency p r o p o r t i o n a l to the dynamic p r e s s u r e i s counted to g i ve a 2 second ave rage . A computer g i v e s updated readouts of the mean speed and p r o v i d e s nega t i v e feedback to s t a b i l i z e the mean f l ow . In t h i s way, the tunne l f low speed can be determined w i th an accu racy of b e t t e r than 1% at a l l but very low speeds . A b i - p l a n a r g r i d w i th a mesh l eng th of 1.27 cm and rod d iameter of 0.3175 cm was c o n s t r u c t e d and measurements taken 30 mesh l eng ths downstream in the water t unne l f a c i l i t y . The LDA probe volume was he l d f i x e d wh i l e the p o s i t i o n of the shear probe was a d j u s t a b l e in the streamwise d i r e c t i o n . I t was found tha t f low d i s t o r t i o n upstream of the probe p r e c l uded s imu l taneous measurements at the same p o s i t i o n . I n s t e a d , the shear probes were moved back from the LDA p o s i t i o n wh i le measurements w i th the LDA con f i rmed the r e p e a t a b i l i t y of the f low as i t was i n t e r r u p t e d and r e s t a r t e d . The probe was mounted on a b rass c y l i n d e r which c o n t a i n e d a pre-amp (-3db at 0.3 Hz) f o r i s o l a t i o n pu rposes . (The s i g n a l was not d i f f e r e n t i a t e d s i n c e the v e l o c i t y r a the r than the shear s p e c t r a were r e q u i r e d . ) The pre-amp assembly was suppor ted from a r a i l at the top of the tunne l so i t s streamwise p o s i t i o n c o u l d be a d j u s t e d . Measurements were a l s o made in the wake c e n t e r , 40cm downstream from a 1/4" c y l i n d e r , to p rov ide data which c o u l d be 41 compared w i th the p u b l i s h e d r e s u l t s of Ubero i and Freymuth (1969) . 3.5 Measurements The measurements were s e l e c t e d in order to p rov i de data which would c o n f i r m the v a l i d i t y of the LDA techn ique as we l l as to compare the shear probe response w i th the LDA to c a l c u l a t e the s p a t i a l t r a n s f e r f u n c t i o n . A l l the g r i d s p e c t r a were taken at a p o s i t i o n 37.5cm downstream of the g r i d . The l o n g i t u d i n a l and c ross-s t r eam LDA s p e c t r a were measured r epea t ed l y throughout the exper iment to ensure tha t the flow was not chang ing and to look f o r ev idence of mean shea r . The tunne l speeds were 4 0 , 5 0 , 6 0 , and 70 cm/s. There were two probe des igns and each probe response was measured at two tunne l speeds , 40cm/s and 70cm/s. The c y l i n d e r wake s p e c t r a were measured 40 cm downstream c l o s e to the cen te r l i n e at convec t i on speeds of 25 and 50 cm/s. 3.6 Data A n a l y s i s Ampl i tude s p e c t r a (256 F o u r i e r c o e f f i c i e n t s ) were c a l c u l a t e d from each probe s i g n a l w i th a HP3582A spectrum ana l y se r which was c o n t r o l l e d by an HP-85 computer w i th p l o t t e r . The spectrum ana l y se r t r ans fo rmed 256 s e q u e n t i a l t ime s e r i e s and ensemble averaged the 256 ampl i tude e s t ima tes at each f requency (so tha t each c o - e f f i c i e n t was ch i - squa red d i s t r i b u t e d w i th 512 degrees of f reedom) . The ensemble-averaged spectrum was s t o r ed on c a s s e t t e . These da ta were c o p i e d v i a an LSI-11 computer to tape and mounted onto the main-frame computer at UBC. The LDA s p e c t r a were taken over th ree s p e c t r a l r anges : 0-25Hz ( fo r long 42 term ave rag ing at lower f r e q u e n c i e s ) , 0-250Hz (the t u rbu l ence r eg ime ) , and 0-1kHz ( fo r h igh f requency no i se a n a l y s i s ) . The shear probe s p e c t r a were each taken over 0-lOOHz. A l l the s p e c t r a were smoothed w i th 3 passes of a 7 p o i n t equa l weight moving ave rage . The low f requency (25Hz) LDA s p e c t r a were s p l i c e d to the co r r e spond ing 250Hz s p e c t r a at 20Hz and the r e s u l t was smoothed aga in w i th 3 passes of a 3-point moving ave rage . 43 IV. RESULTS LDA s p e c t r a have not been r epo r t ed e x t e n s i v e l y i n the l i t e r a t u r e and most tu rbu lence measurements have been of the mean flow and v a r i a n c e , q u a n t i t i e s norma l l y c a l c u l a t e d from ensemble averages of Dopp le r f requency P D F ' s . Consequen t l y , in t h i s chapter the LDA s p e c t r a w i l l be examined in d e t a i l to e s t a b l i s h con f i dence in t h e i r v a l i d i t y and s u i t a b i l i t y as the " t r u e " h i g h r e s o l u t i o n es t ima te of the c ross-s t r eam spec t rum. The shear probe s p e c t r a w i l l be examined w i th p a r t i c u l a r a t t e n t i o n g i ven to the q u e s t i o n of c a l i b r a t i o n v a r i a b i l i t y . F i n a l l y , the measured t r a n s f e r f u n c t i o n s f o r two probe types w i l l be shown and d i s c u s s e d . 4.1 LDA Measurements 4.1.1 Mean Flow The mean f low was averaged by the t r a c k e r f o r a minimum of 20 seconds . The s e n s i t i v i t y was c a l c u l a t e d from the system o p t i c a l parameters (Table I I) and the t r a c k e r c o n v e r s i o n f a c t o r . I t was found that the LDA measurements f o l l o w e d the tunn'el P i t o t - c o u n t e r w i t h i n 1 cm/s (maximum e r r o r was 2%). The f r i n g e t i l t a n g l e , B, was measured to w i t h i n 1° and the c a l c u l a t e d mean f low component, Usin/3, employed in f requency s h i f t i n g , agreed w i th the measurement of the co r r e spond ing Dopp le r s h i f t w i t h i n 5% which i s c o n s i s t e n t w i th the 1° accuracy of B a t 1 9 ° . In any c a s e , t h i s mean f low , Usin/3, i s used on ly f o r f requency s h i f t i n g and hence i s not a c r i t i c a l f a c t o r . A l s o , B en te r s the c a l c u l a t i o n of the c ross-s t r eam spectrum in ( 3 .5 .3 ) on ly as a sma l l c o r r e c t i o n ( l e s s than 1% e r r o r fo r 1° e r r o r i n B). 44 A t r end i n the mean f low (an o s c i l l a t i o n w i th an ampl i tude of 1 cm/s and p e r i o d of 5 seconds) was observed and may account f o r some of the unexpected energy fo r the f i r s t 2 Hz i n the power s p e c t r a . For t h i s r ea son , the s p e c t r a were c o n s i d e r e d v a l i d on l y above 2 Hz . I t shou ld be noted tha t the mean f low en t e r s the s p a t i a l power s p e c t r a as U fo r LDA and 1/U fo r the shear p robe . The shear probe s p e c t r a a l s o depend on the c a l i b r a t i o n n o z z l e speed, V, as V. 4.1 .2 No i se In LDA Spec t ra The power s p e c t r a w i l l be contaminated by ambigu i t y no i se as d i s c u s s e d in s e c t i o n 3 . 3 . 4 . T h i s n o i s e shou ld appear as an almost f l a t p l a t e a u fo r t u rbu l ence f r e q u e n c i e s . The power s p e c t r a s i g n a l - t o - n o i s e r a t i o (SNR) w i th r espec t to the no i se l e v e l was 170 (200) a t 70cm/s (40cm/s) f o r l a r g e - s c a l e s (5 cm) and dec reased to 10 (7) at a s c a l e of 0.5 cm. Once the no i s e l e v e l has been measured f o r g i ven f low c o n d i t i o n s , i t can be s u b t r a c t e d from the power spectrum to improve the sma l l s c a l e SNR. The s p e c t r a taken out to 1 kHz at 40, 50, 60, and 70 cm/s are shown in F i g . 7. The no i s e i s e s s e n t i a l l y f l a t pas t 400Hz and the va lue at 500Hz was assumed to be due to no i s e a l o n e ; N=\//(500 Hz) r ep re sen t s an upper bound f o r the es t ima te of the low f requency ambigu i ty no i s e l e v e l . T h i s 500 Hz va lue was s u b t r a c t e d from two LDA power s p e c t r a , i//(f) at 40 and 70 cm/s, and the e f f e c t i s shown in F i g . 8. 45 T h e i m p r o v e m e n t i n t h e SNR o b t a i n e d i n t h i s manner c a n be e s t i m a t e d by s u p p o s i n g t h a t t h e n o i s e l e v e l , N , i s known t o a a c c u r a c y o f ejq% a n d we demand t h e c o r r e c t e d s p e c t r u m t o be a c c u r a t e t o e %. S i n c e s u b t r a c t i o n i s l i n e a r , t h e e r r o r i n t h e c o r r e c t e d s p e c t r u m i s e N N a n d we c a n a l l o w \j/= (e^ / e ) N . H e r e , we a s s u m e t h a t t h e n o i s e ( m e a s u r e d s e p a r a t e l y ) i s a c c u r a t e t o e N =5% a n d demand a c o r r e c t e d s p e c t r a l u n c e r t a i n t y o f e^=lO%. T h i s p r o c e d u r e p e r m i t s c o r r e c t i o n t o 3db b e l o w t h e n o i s e l e v e l . ( T h i s a n a l y s i s d e p e n d s on h i g h r e l a t i v e a c c u r a c y w i t h i n t h e s p e c t r u m . ) T h e -3db l e v e l h a s b e e n i n c l u d e d i n F i g . 8. 0.0-, -2.0 T 600 Frequency (Hz) 1000 F i g u r e 7 - H i g h f r e q u e n c y LDA s p e c t r a s h o w i n g " w h i t e " a m b i g u i t y n o i s e 46 F i g u r e 8 - E f f e c t of s u b t r a c t i n g amb igu i t y n o i s e from the mixed s p e c t r a . 47 I t i s c l e a r tha t the LDA amb igu i t y n o i s e i s not a problem f o r the s c a l e s of i n t e r e s t (5 cm to 0.3 cm). The n o i s e l e v e l s p r e d i c t e d by the a n a l y s i s of George and Lumley (1973) were c a l c u l a t e d from equa t i ons (3 .7 .1-4 ) u s i ng the v a l u e s of e g i ven l a t e r i n s e c t i o n 4 . 1 . 4 . The t u r b u l e n c e b roaden ing term tu rned out to be l a r g e r than the t r a n s i t t ime b roaden ing and the dependence of N on U can be w r i t t e n here as ' • N = 0.73 A f where, ( A f ) 2 = ( A f p ) 2 + ( A f ^ 2 A f p = (22.1 cm" 1) U rui = Sin L U J s (4.1.1) ( A f T ) 2 = (14.1 c m - 3 s - 2 ) U3 The measured v a l ues a re compared w i th theory i n F i g 9 where they have been f i t t e d to the s c a l e d t h e o r e t i c a l c u r v e . The measured v a l u e s are app rox ima te l y tw ice tha t p r e d i c t e d from o p t i c a l c o n s i d e r a t i o n s a l o n e . T h i s r e s u l t i s l i k e l y due to shot n o i s e as sugges ted by the s i m u l a t i o n s t u d i e s of L ad i ng and Edwards (1974) . They found tha t add ing shot n o i s e to t h e i r s i m u l a t e d Dopp le r s i g n a l r e s u l t e d i n a wide-band demodulated n o i s e power which behaved i n a manner e q u i v a l e n t to a r e d u c t i o n i n the probe volume d i m e n s i o n s . C l e a r l y , the n o i s e i s a lmost whi te and can be s u b t r a c t e d to improve the SNR,of the power s p e c t r a . However, i n the p resence of shot n o i s e , the amb igu i t y n o i s e l e v e l cannot be s a f e l y c a l c u l a t e d from the o p t i c a l parameters a l o n e . 4'8 •a-Mean v e l o c i t y , U (cm/s) F i g u r e 9 Amigu i t y n o i s e l e v e l , N, as a f u n c t i o n of the mean v e l o c i t y , U 49 4 .1 .3 Power Spec t ra The mixed c r o s s - s p e c t r a , 0 ( k , ) , were compared f o r m c l o c k w i s e and coun te r-c l o ckw i se r o t a t i o n s about the ( o p t i c ) x 3 - a x i s , and were found to be the same under c o n d i t i o n s of ze ro mean s h e a r . These s p e c t r a were checked at v a r i o u s s tages throughout the exper iment to ensure both an absence of mean shear (symmetry) and the (necessary ) r e p e a t a b i l i t y of the f low c o n d i t i o n s . The LDA c ross-s t r eam power s p e c t r a , 022 ( ^ 1 ) a re shown in F i g . 10 fo r 11=40,50,60,70 cm/s. They have been c a l c u l a t e d from the mixed s p e c t r a , 0 m ( k , ) , u s ing the co r r e spond ing downstream s p e c t r a , 0 i i ( k 1 ) , in equa t ion 3 . 5 . 3 , and no rma l i zed w i th U 2 . Good s i m i l a r i t y i s seen . The downstream and c ros s-s t r eam s p e c t r a are compared in F i g . 11 where the approach to i s o t r o p y at h i g h wavenumbers i s d i s p l a y e d as the r a t i o of the downstream spectrum c a l c u l a t e d by the i s o t r o p i c r e l a t i o n <t» 2 2 (k 1 ) = \ ^ ( k ^ - \ k t o u d ^ ) to the measured va lue fo r U=70 cm/s; a va lue of u n i t y i m p l i e s the v a l i d i t y of the i s o t r o p i c r e l a t i o n . As e x p e c t e d , the a n i s o t r o p y i s more ev iden t at the l a r g e r s c a l e s . The a n i s o t r o p y was u , 2 / u 2 2 = 1 . 1 9 ( 1 . 3 6 ) f o r U=70(40)cm/s; c f s e c t i o n 2 . 4 ) . 50 F i g u r e 10 - Normalized LDA c r o s s - s t r e a m s p e c t r a , 4>2 2 ( k 1) > f o r mean speeds of U=40,50,60, and 70 cm/s. rt o T J 0) CD U > 3 •r-l (0 cfl <D 01 TD E o f — I I • n ( k i ) U«70cm/s -1.0 ~ I — 0.0 l o g ^ k (cycles/cm) —1 1.0 k i (cycles/cm) F i g u r e 11 - LDA t r a n s v e r s e and l o n g i t u d i n a l s p e c t r a measured (above) and the r a t i o of 0 2 2 as d e r i v e d from to measured 0 2 2 (below) 52 4 . 1 . 4 G r i d Turbu lence G r i d t u r b u l e n c e has been e x t e n s i v e l y s t u d i e d (Townsend, 1976) because of i t s homogeneity and approximate i s o t r o p y . In t h i s s e c t i o n the a n a l y s i s of the v a r i a n c e and d i s s i p a t i o n i s compared w i th theory in order to ga in c o n f i d e n c e i n the LDA measurements. The Ba t che l o r decay law f o r such f lows can be w r i t t e n (Stewart and Townsend, 1951) as U-2 x, - x Q ^=r- - * „ ° (4.1.2) u 2 M 1 Stewart and Townsend (1951) r e p o r t e d X O /M=10 w i th (M/d=5) wh i le B r i t t e r e t a l (1979) ob ta ined x 0 / M = 5 . 5 wi th M / d = 4 , where M i s the mesh l e n g t h and d i s the rod d i amete r . Comparison of the works c i t e d above showed tha t the cons tan t of p r o p o r t i o n a l i t y depends on the exper imenta l ar rangement , e s p e c i a l l y f o r x , / M<50 . A decay c o n s t a n t , D 2 f o r the cross-stream- s p e c t r a was computed by f i t t i n g the i n t e g r a t e d downstream s p e c t r a to ( 4 . 1 . 2 ) u s i ng x0/M=io.O. The r e s u l t was D 2 =102. Equa t ion ( 4 . 1 . 2 ) fo r the c ross-s t r eam component can be d i f f e r e n t i a t e d to g i ve an e x p r e s s i o n f o r the r a te of d i s s i p a t i o n , e, assuming i s o t r o p y , e = - 3 d _ ( U 2 , = _ 3 „ d . ^ (4.1.3) - 3 r M .2 TJ3 (4.1.4) ~ TTJJ Ix - x 0 i M~ 53 The d i s s i p a t i o n s p e c t r a were c a l c u l a t e d u s i n g ' e = - I | v j"k2<J>22(k) dk 0 f o r the measured s p e c t r a ( c o r r e c t e d f o r ambigu i ty no i se ) and are shown i n F i g . 13. The i n t e g r a t e d d i s s i p a t i o n va lues (Table I I I ) compare we l l w i th v a l u e s ob t a i ned u s i n g (4 .1 .4 ) w i th D 2 =102, e s p e c i a l l y a t low speeds , sugges t ing tha t most of the d i s s i p a t i o n spectrum was r e c o v e r e d . I f the d i s s i p a t i o n s p e c t r a were extended us i ng the u n i v e r s a l spectrum in the manner of Pond (1965) i t i s c l e a r tha t the v a l u e s of the d i s s i p a t i o n would agree more c l o s e l y wi th ( 4 . 1 . 4 ) . I t shou ld be kept in mind tha t equa t i on (4 .1 .4 ) i s d e r i v e d on the b a s i s of i s o t r o p y , and f o r t h i s reason must be c o n s i d e r e d as an approx imat ion f o r the t rue d i s s i p a t i o n . The measured and t h e o r e t i c a l va lues of the va r i ance and d i s s i p a t i o n are shown in Tab l e I I I . The LDA measurements of the g r i d s p e c t r a d i s p l a y the expected s e l f - s i m i l a r i t y and a n i s o t r o p y . The success in measur ing the d i s s i p a t i o n p r o v i d e s c o n f i d e n c e i n the s p e c t r a at the sma l l s c a l e s where the shear probe " r o l l - o f f " due to s p a t i a l a ve rag ing o c c u r s . 54 1 2 u l (cm/s) 2 F i g u r e 12 - Dependence of the v a r i a n c e , u 2 2 , on U 2 , f o r the , s g r i d LDA s p e c t r a , 0 2 2 ( ^ 1 ) • F i g u r e 13 - T ransve r se g r i d d i s s i p a t i o n s p e c t r a . 55 Tab le III - G r i d t u r b u l e n c e a n a l y s i s U (cm/s) u (cm/s) u/U (%) c (CID^/B^) Theory* Measured 40 0.87 2.17 1.94 1.83 50 1.14 2.28 3.80 3.58 60 1.35 2.24 6.56 4.84 70 1.58 2.25 10.4 6.59 * Theory: equation (4.1.4) with D=102 and x0/M=!0.0 U2 = u2 2 56 4 . 1 . 5 C y l i n d e r Wake Spec t r a The l o n g i t u d i n a l s p e c t r a , ^ ^ ( k , ) , were measured 25 d i ame te r s downstream of a 1/4" rod on the wake c e n t e r l i n e . Ube ro i and Freymuth (1969) have s t u d i e d c y l i n d e r wake s p e c t r a i n a wind t unne l w i th hot w i re anemometry. They found an e m p i r i c a l decay law: 1— = 0.1 = cons t an t (A. 1.5) " c U C = U(5)1/2 (4.1.6) where D i s the rod d iameter and x i s the d i s t a n c e downstream from the r o d . The LDA da ta show good agreement w i th t h i s r e s u l t . The cons tan t i n (4 .1 .5 ) was found to be 0.11 at the low speed and 0.12 f o r the h i ghe r mean v e l o c i t y . The s p e c t r a , no rma l i z ed w i th the l a r g e - s c a l e pa r ame te r s , 1 and U , (4.1.7) 1 = ( x D)W2 are compared w i th the hot wire measurements of Ubero i and Freymuth i n F i g . 14. The u n c e r t a i n t y in the s p e c t r a l l e v e l s was e s t i m a t e d from the s c a t t e r of the hot-wi re d a t a . There i s r e a sonab l e agreement out to a l e n g t h s c a l e of about 5mm. The g r i d s p e c t r a had a b e t t e r SNR at s m a l l e r s c a l e s , and f o r t h i s reason were used to measure the s p a t i a l t r a n s f e r f u n c t i o n . t-" tr c o n rt fD * ->-•• i O ro i 3 ro OJ cn c n ro 3 ro 3 ri-01 3 ro •-» «: QJ ro G cn D"0 ro ro >-i o O rr 0) o> • • 3 Qj O o TJ 3 ro 3 c r t 3* 3 0) 3 «l » o a . « N •r4 o SB -1.0 -2.0 -3.0 -4.0 -5.0 -6.0 •7.0 -8.0 AD O A 0 ~ 0 Rp = UD/O Hot. wire A • R.- 2160 Re- 4320 O R - 1080 * e r o Reynolds e number limit LDA Re- 1600 R « 3200 e A-10 cm 1 - ( xD) 1 - 5.04 c a -5/3 slope 1.0 0.0 1.0 2.0 logjgd^l) (k i n rad / c tn) 3.0 58 T h i s compar ison wi th p r e v i o u s c y l i n d e r s p e c t r a a long wi th the behav io r of the g r i d s p e c t r a suppor t to the v a l i d i t y of the LDA measurements. 4.2 Shear Probe Spec t ra A composi te of the unsmoothed probe s p e c t r a , 7 2 2 ^ 1 ) i s d i s p l a y e d in F i g . 15 fo r 40cm/s and 70 cm/s. A l though the r e s u l t s seem to i n d i c a t e u n r e p e a t a b i 1 i t y in the f l ow , I w i l l argue tha t the v a r i a b i l i t y i s due to the u n c e r t a i n t y in the probe c a l i b r a t i o n . In F i g . 16 the s p e c t r a have been smoothed and no rma l i zed w i th r espec t to the peak v a l u e s . (The vagueness of "peak v a l u e s " f o r the 40 cm/s s p e c t r a a l l owed them to c o l l a p s e l e s s d e f i n i t i v e l y than the 70 cm/s s p e c t r a . ) The two probe s i z e s are appa ren t , w i th the s l i g h t l y l a r g e r Oakey probes r o l l i n g o f f e a r l i e r than the sma l l e r Osborn p r o b e s , as e x p e c t e d . The no rma l i zed s p e c t r a at 70 cm/s were used to c a l c u l a t e the t r a n s f e r f u n c t i o n s . The LDA s p e c t r a were moni tored throughout the exper iment and showed r e p e a t a b i l i t y , so i t i s r easonab le to argue tha t the v a r i a b i l i t y i n the probe peak s p e c t r a l v a l ues can be accounted f o r by the u n c e r t a i n t y i n v o l v e d in the c a l i b r a t i o n p r o c e d u r e . The e r r o r in the c a l i b r a t i o n j e t speed was put at 2%. T h i s e r r o r c o n v e r t s to 8% u n c e r t a i n t y i n the power s p e c t r a , wi thout c o n s i d e r i n g any e r r o r i n the tunne l f low speed . The s tandard d e v i a t i o n from the mean peak va lue was 10% f o r the h igh speed (70 cm/s) s p e c t r a (and 15% fo r the low speed s p e c t r a , but where the "peak" i s p o o r l y d e f i n e d ) . FREQUENCY HZ F i g u r e 15 Shear probe g r i d s p e c t r a (unsmoothed) a t 40cm/s (above) and 70cm/s (be low) . 60 F i g u r e 16 - Smoothed shear probe s p e c t r a at 70cm/s, no rma l i z ed to u n i t y a t peak v a l u e . 61 S ince the LDA and repeated probe measurements show much l e s s v a r i a b i l i t y , we can conc lude that the c a l i b r a t i o n i s the dominant source of e r r o r . An un fo r tuna te i m p l i c a t i o n of t h i s u n c e r t a i n t y in the c a l i b r a t i o n of the probes i s the u n c e r t a i n t y in the t r a n s f e r f u n c t i o n at low wavenumbers. 4.3 The Measured T r a n s f e r F u n c t i o n s The 70 cm/s probe s p e c t r a were d i v i d e d by the c o r r e s p o n d i n g LDA s p e c t r a and no rma l i zed to a peak va lue of u n i t y , i . e . , the s p a t i a l t r a n s f e r f u n c t i o n d e f i n e d by (2 .5 .1 ) was c a l c u l a t e d and n o r m a l i z e d . The mean peak va lue was 0 .90 . The r e s u l t s are shown in F i g . 17. The ha l f-power (-3db) wavenumbers co r respond to l e n g t h s c a l e s of 1.6 cm and 2.0 cm fo r the Osborn and Oakey probes r e s p e c t i v e l y . A response f u n c t i o n such as s i n 2 ( x ) / x 2 or a s i n g l e - p o l e f i l t e r c o u l d be f i t t e d to t h i s hal f-power p o i n t and the r e s u l t termed the s p a t i a l t r a n s f e r f u n c t i o n of the shear p robe . However, i t i s obv ious from F i g . 17 tha t the shape of the measured t r a n s f e r f u n c t i o n at sma l l wavenumbers cannot be d e s c r i b e d by such a s imple r o l l - o f f f u n c t i o n . In p a r t i c u l a r , the i n i t i a l p o s i t i v e s lope and the subsequent peak away from the o r i g i n suggest tha t a model more s o p h i s t i c a t e d than a one-d imens iona l moving average i s demanded. T h i s i s the sub j e c t of the next chap te r where the q u e s t i o n of how to i n t e r p r e t the t r a n s f e r f u n c t i o n f o r l a r g e - s c a l e ocean i c measurements w i l l a l s o be s t u d i e d . 62 F i g u r e 17 - Measured s p a t i a l t r a n s f e r f u n c t i o n s (70 cm/s) no rma l i zed to u n i t y near the peak . 63 V. MODEL OF SPATIAL AVERAGING In an e f f o r t to unders tand the shape of the t r a n s f e r f u n c t i o n ob ta ined in Chapter 4 (namely, the i n i t i a l p o s i t i v e s l ope and consequent peak away from the o r i g i n ) , a model i s deve loped to e x p l a i n how th ree-d imens iona l s p a t i a l a ve rag ing a f f e c t s the measurement of i s o t r o p i c t u r b u l e n c e . Due to the i n t e r n a l mechan ica l comp lex i t y of the probe and the known a n i s o t r o p y of the g r i d t u r b u l e n c e , the model i s not expected to p rov i de good q u a n t i t a t i v e agreement at the lowest wavenumbers; n e v e r t h e l e s s , important q u a l i t a t i v e f e a t u r e s of the t r a n s f e r f u n c t i o n w i l l be e x p l a i n e d . An i n t e r p r e t a t i o n of the expe r imen ta l t r a n s f e r f u n c t i o n a p p l i e d to l a r g e - s c a l e t u rbu l ence s p e c t r a i s g i v e n . 5.1 P r e v i ous Work Skramstad (1937) gave an e x p r e s s i o n fo r the c o r r e c t i o n f a c t o r ( f o r the va r i ance ) to take i n t o account the s p a t i a l a ve rag ing i n v o l v e d in hot-wire anemometry. Working in terms of the l a t e r a l c o r r e l a t i o n f u n c t i o n , g ( r ) , g ( r ) = ^ -r R , , ( x , , o ( o ) u^ ^ the f a c t o r c o u l d be e va lua t ed f o r cases where the wire was e i t h e r much longer or sho r t e r than the i n t e g r a l l e n g t h s c a l e . For the case when the wire l e n g t h and the i n t e g r a l s c a l e were of the same o r d e r , the c o r r e c t i o n depends on the shape of c o r r e l a t i o n f u n c t i o n . C a l c u l a t i o n s were made assuming e x p o n e n t i a l and e r r o r f u n c t i o n forms fo r g ( r ) (H inze , 1975,p120) . The important r e s u l t tha t the response f u n c t i o n depends on 64 the form of g ( r ) (o r , e q u i v a l e n t l y , E ( k ) ) , i s p e r t i n e n t to the q u e s t i o n of how to i n t e r p r e t the g r i d t u rbu l ence response f u n c t i o n f o r l a r g e - s c a l e ocean t u r b u l e n c e . The shear probe i s an order of magnitude l a r g e r than the hot-wi re and the s p a t i a l a ve rag ing can be thought of as t a k i n g p l a ce over the (ax isymmetr i c ) a i r f o i l s u r f a c e , thus a t t e n u a t i n g the spectrum tensor at h igh wavenumbers both a long the d i r e c t i o n of mean f low and in the t r a n s v e r s e p l a n e . C l e a r l y , the a n a l y s i s must i n c o r p o r a t e the th ree-d imens iona l c h a r a c t e r of the s p a t i a l a v e r a g i n g . A gene ra l mathemat ica l framework s u i t a b l e f o r a n a l y s i n g the e f f e c t of s p a t i a l a ve rag ing on the s p e c t r a d i r e c t l y i s p r o v i d e d by the work of Ubero i and Kovasznay (1953) . E s s e n t i a l l y , t h i s f o r m u l a t i o n i s a t h r ee-d imens iona l s p a t i a l g e n e r a l i z a t i o n of the method used to c a l c u l a t e f i l t e r t r a n s f e r f u n c t i o n s in t ime s e r i e s a n a l y s i s . S i m i l a r methods were used by George and Lumley (1973) f o r the LDA probe volume and Wyngaard (1968) f o r c r o s s e d hot w i r e s . 5.2 The Model For a l i n e a r probe r e sponse , the t u rbu l ence v e l o c i t y f i e l d component, u^, can be r e l a t e d to the "measured" component, w^, by a l i n e a r mapping of the form w ± (x ) = / f ( s ) U ; L ( s + x )dV(s ) (5.2.1) where dv (s ) i s a volume element at s. I t has been asssumed tha t the f i e l d i s homogeneous and that the re i s no c r o s s - f e e d i n g between components (so tha t f ( x ) i s a s c a l a r r a the r than a 65 second order t e n s o r ) . Two-point v e l o c i t y c o r r e l a t i o n s can be d e f i n e d as i n ( 2 .1 .2 ) f o r the measured f i e l d , W £ . The c o n v o l u t i o n theorem a p p l i e d to the a u t o - c o r r e l a t i o n s p r o v i d e s a gene ra l r e l a t i o n between the " t r u e " spectrum t e n s o r , $. j j (k ) , and the measured t e n s o r , I\^ (k ) : and F(k_) i s the ( th ree-d imens iona l ) F o u r i e r t r ans fo rm of f ( x ) . . A g a i n , by analogy wi th c i r c u i t a n a l y s i s , S(k) i s the ( th ree-d imens iona l ) t r a n s f e r f u n c t i o n co r r e spond ing to the f i l t e r r esponse , f ( x ) . 5.3 A p p l i c a t i o n To The A i r f o i l Probe In order to app l y the above f o r m u l a t i o n i t w i l l be necessa ry to d e s c r i b e the way the shear probe averages in 3-space. Equa t ions (3 .1 .4 ) and (3 .1 .5 ) w i th 70=13mm~1 approximate c l o s e l y the shape of both probes (see Appendix A) and w i l l be used to d e f i n e a no rma l i zed s e n s i t i v i t y as a f u n c t i o n of x, r ± j(k) = s ( k ) $ i j ( k ) (5.2.2) where S(k) = | F ( k ) | 2 (5.2.3) o t h e r w i s e 0 < x, < 2L (5.3.1) 66 and s i m i l a r l y i n the t r a n s v e r s e p lane we have f n ^ r , r < R b ( r ) = < ( 5 . 3 . 2 ) [ o, o t h e r w i s e For the shear p robe , the a v e r a g i n g t akes p l a c e over the probe s u r f a c e . The volume a v e r a g i n g comes about as the " f r o z e n " t u r b u l e n c e f i e l d i s swept over the l e n g t h of the p r o b e . The a v e r a g i n g i s un i fo rm in the a x i a l d i r e c t i o n , wh i l e i n the t r a n s v e r s e p l ane the a v e r a g i n g i s we ighted i n p r o p o r t i o n to the r a d i u s . Thus , we can w r i t e the k e r n e l i n ( 5 . 2 . 1 ) as f ( x ) = a ( x 2 ) b ( r ) ( 5 .3 .3 ) In t h i s case the t r a n s f e r f u n c t i o n f a c t o r s i n t o a l o n g i t u d i n a l component, A ( k , ) , and a t e rm, Bio), t ha t d e s c r i b e s the a t t e n u a t i o n of the spec t rum t enso r i n the t r a n s v e r s e wavenumber k 2 , k 3 - p l a n e . That i s , S (k ) = A ( k j ) B(o) ( 5 . 3 . 4 ) where the l a t e r a l wavenumber modulus i s d e f i n e d as c 2 = X 2 + k 2 = k 2 - k 2 ( 5 . 3 . 5 ) 2 3 1 The component t r a n s f e r f u n c t i o n s n o r m a l i z e d w i th t h e i r v a l u e at k, = 0 and a=0 have been c a l c u l a t e d (see Appendix B.I f o r d e t a i l s ) and a re shown i n F i g . 1 8 . 67 F i g u r e 18 - Component t r a n s f e r f u n c t i o n s (below) c o r r e s p o n d i n g to the s p a t i a l a ve r ag ing f u n c t i o n s (above) n o r m a l i z e d to u n i t y at Y=0. Y=k,L f o r A and Y=oR f o r B. 6B They can be w r i t t e n a s : A(k.) = |F r . a ( x . n | 2 1 ( 5 . 3 . 6 ) " ( ( k ^ ) 2 ' B ( a ) = \F2[b(o,6)]| = | — 3 — f ( o R ) 2 J 0 ( a R ) d ( a R ) } ( o R ) 3 0 ( 5 . 3 . 7 ) = { i j / Y x 2 J Q ( x ) d x } 2 , Y = aR Y 0 where Fm denotes the m-dimensional F o u r i e r t r a n s f o r m and 6 i s the p o l a r ang le about the x , - a x i s . The a x i a l symmetry of S(k) pe rm i t s i n t e g r a t i o n of ( 5 .2 .2 ) over the p l a n e , k ^ c o n s t a n t , to o b t a i n the one-d imens iona l spect rum as " s e e n " by the p robe , 7 2 2 ( k , ) , in terms of E (k ) (Appendix B . 2 ) : 69 The model t r a n s f e r f u n c t i o n i s then d e f i n e d as i n (2 .5 .1 ) by / " B(o) (k2 + k2) dk X ( k x ) = A ^ ) *1 . * ( 5 ' 3 - 9 > f EikJ. ( k 2 + k 2 ) d k J k 1 k 3 l I f the re were on l y a x i a l s p a t i a l a v e r a g i n g , A ( k , ) , ( i . e . no l a t e r a l e x t e n s i o n so tha t B (o )=1) , then the t r a n s f e r f u n c t i o n would be s imp ly the response f o r a l o n g i t u d i n a l s p a t i a l average over a l e n g t h 2L . However, the i n c l u s i o n of B(o) i n ( 5 . 3 . 9 ) , i m p l i e s a reduced response even at low wavenumbers. In f a c t , ( no t i ng tha t k 2 = a 2 + k , 2 so t ha t k=o f o r k,=0) the probe response in the l i m i t of z e ro wavenumber becomes J ° °B (k ) dk l i m X ( k ^ = X(o) = 0 . . . < 1 (5.3.10) k,-Ki f dk 0 K s i n c e E(k) > 0 f o r k > 0 and B(k) < 1 The one-d imens iona l response i s l e s s than u n i t y at z e ro wavenumber, and the magnitude of t h i s e f f e c t w i l l depend on the s c a l e ( r e l a t i v e to R) of the e n e r g y - c o n t a i n i n g e d d i e s . . I t i s a l s o p o s s i b l e t o have a peak in the t r a n s f e r f u n c t i o n away from the o r i g i n depend ing on the d e t a i l e d shape of E ( k ) . The one-70 d imens iona l spectrum at a g i ven va lue of k, i s ob ta ined by i n t e g r a t i o n over a l l t r ans ve r s e wavenumbers; a t t e n u a t i o n of the h i g h t r a n s v e r s e wavenumbers reduces the one-d imens iona l response f o r a l l v a l ues of k,. 5.4 C a l c u l a t i o n s Based On Measured G r i d Turbu lence An approx imat ion to E(k) f o r the g r i d t u rbu l ence can be o b t a i n e d through (2 .4 .8 ) s i n ce both 0 , , ( k , ) and 0 2 2 ( k i ) were measured. ( D e r i v i n g E(k) from the measured da ta would be d i f f i c u l t i f the second d e r i v a t i v e s i n v o l v e d in (2 .4 .6 ) were r e q u i r e d . ) The energy s p e c t r a c a l c u l a t e d from the d e r i v a t i v e s of the measured <p^(k^) and # 2 2 ( ^ 1 ) are shown in F i g 19 f o r 40 cm/s and 70 cm/s. The i n i t i a l nega t i ve v a l ues of E(k) f o r U=70cm/s a re due to the a p p l i c a t i o n of i s o t r o p i c r e l a t i o n s to a n i s o t r o p i c one-d imens iona l s p e c t r a ; measured power s p e c t r a must be p o s i t i v e , of c o u r s e . The " s h o u l d e r " which appears in these d e r i v e d energy s p e c t r a i s t y p i c a l f o r g r i d t u rbu l ence f o r x,/M<40, a c c o r d i n g to Stewart and Townsend (1951) who show a b i -modal spectrum fo r measurements c l o s e to the g r i d . The one-d imens iona l s p e c t r a were r e c o n s t r u c t e d us ing ( 2 .4 .4 ) and (2 .4 .5 ) and are compared w i th the o r i g i n a l s in F i g . 20. The d isagreement at low wavenumbers i s due the l a r g e - s c a l e a n i s o t r o p y of the g r i d t u rbu l ence ( the r e c o n s t r u c t i o n employed the i s o t r o p i c r e l a t i o n s ) and the c o i n c i d e n c e at h ighe r wavenumbers i n d i c a t e s the approach to " l o c a l " i s o t r o p y as p r e d i c t e d by the Kolmogorov t h e o r y . Numer ica l i n t e g r a t i o n of (5 .3 .9 ) u s i ng the above E(k) g i v e s the model t r a n s f e r f u n c t i o n s shown in F i g . 21 fo r a cons tan t 71 probe shape f a c t o r , 7 0 , and v a r i a b l e s i z e parameter , R. The model when a p p l i e d to the g r i d da ta does indeed show a t r a n s f e r f u n c t i o n hav ing a zero wavenumber response l e s s than u n i t y . A l s o , the re i s a sma l l i n i t i a l p o s i t i v e s l o p e , the magnitude of wh ich , u n f o r t u n a t e l y , i s very s e n s i t i v e to the exact form of the d e r i v e d E(k) over the f i r s t few F o u r i e r . c o e f f i c i e n t s . The measured t r a n s f e r f u n c t i o n s fo r the two probe des igns are compared w i th the model in F i g . 22. The measured t r a n s f e r f u n c t i o n peak va lues were s c a l e d from the mean va lue of 0.90 to 0.85 which was the peak va lue of the model responses tha t showed good agreement at the 1/2-power p o i n t . T h i s p rocedure was c o n s i d e r e d reasonab le inasmuch as the probe s p e c t r a are c o n s i d e r e d a ccu ra t e to w i t h i n 10%. Then the r a d i u s R was v a r i e d to o b t a i n the best f i t w i t h i n the wavenumber range where the maximum s lope occu r r ed i n the t r a n s f e r f u n c t i o n s . The d isagreement at low wavenumbers i s not unexpected s i n c e the flow i s a n i s o t r o p i c there and n e i t h e r shear d i s t o r t i o n nor the i n t e r n a l s t r u c t u r e of the probe have been c o n s i d e r e d . The best f i t r a d i i are R=0.21cm and 0.235cm (2L=0.57cm and 0.72cm) f o r the Osborn and Oakey probes r e s p e c t i v e l y . These r e s u l t s compare w e l l w i th the s e n s i t i v e l eng ths determined from the probe geomet r i es i n Appendix A. I t seems reasonab le to f i t the model to the t u n n e l t r a n s f e r f u n c t i o n and use the best f i t va lue of R to c a l c u l a t e the t r a n s f e r f u n c t i o n fo r l a r g e - s c a l e f lows us ing the a p p r o p r i a t e E ( k ) ; i . e . , F ( k / k s ) . T h i s w i l l be done i n s e c t i o n 5 .5 . 7 2 B i ' 1 1 • 1 . , — , , 0 5 ? «> 1.5 2 . 0 J . 5 k, (cycles/cm) F i g u r e 19 - Energy s p e c t r a , E ( k , ) d e r i v e d from the measured one-d imens iona l s p e c t r a , ^ ^ ( k , ) and 0 2 2 u s i n g e q u a t i o n ( 2 . 4 . 8 ) . 73 10 . 01 -] 1 1 1 1 , 1 0 0.5 A 1.0 1.5 k i (cycles/cm) 3.0 i k i (cycles/cm) F i g u r e 20 - One-d imens iona l s p e c t r a , ^ ^ ( k , ) and 0 2 2 ( k i ) : measured ( s o l i d ) and r e c o n s t r u c t e d (dash) from E ( k ) u s i n g i s o t r o p i c r e l a t i o n s . 74 k i (cycles/cm) F i g u r e 21 - Model t r a n s f e r f u n c t i o n s v a r y i n g R: R= 0.20 - 0.30cm i n ste p s of .025cm. 75 F i g u r e 22 - Measured t r a n s f e r f u n c t i o n s (U=70cm/s) and bes t f i t model t r a n s f e r f u n c t i o n s . Osborn p r o b e : R=0.2lcm; Oakey p r o b e : R=0.235cm. 76 The low wavenumber response (LWR), X (k ,=0 ) , can be c a l c u l a t e d from ( 5 . 3 . 1 0 ) . In F i g . 23 the LWR i s p l o t t e d as a f u n c t i o n of R/L,, where R i s the probe r a d i u s and L, i s the l o n g i t u d i n a l l eng th s c a l e d e f i n e d by •L l = - t r - J°° R n ( r , o , o ) d r (5.4.1) u 2 o l T h i s s c a l e was es t imated by e v a l u a t i n g the one-d imens iona l LDA spect rum, 0 i i ( k , ) , at the o r i g i n ; L = $ ( 0) (5.4.2) u 2 1 F i g . 23 shows that the LWR w i l l be s i g n i f i c a n t l y l e s s than u n i t y when the i n t e g r a l l e n g t h s c a l e i s of the same order as the t r ans ve r s e s p a t i a l a ve rag ing s c a l e . F o r t u n a t e l y , the i n t e g r a l s c a l e s of most g e o p h y s i c a l f lows are much l a r g e r than the s c a l e of the p robe . However, the g r i d t u rbu l ence i s a sma l l s c a l e f l ow , L ,=1.0 cm, and so the low wavenumber response of the a i r f o i l probe i s l e s s than the c a l i b r a t i o n v a l u e . In g e n e r a l , the LWR w i l l be l e s s than u n i t y whenever t r a n s v e r s e s p a t i a l a ve rag ing o c c u r s ; c o n v e r s e l y , the LWR w i l l a lways be u n i t y i f on l y l o n g i t u d i n a l s p a t i a l a ve rag ing i s i n v o l v e d . 77 F i g u r e 23 - Low wavenumber response (LWR); i . e . , X t k ^ O ) e v a l u a t e d as a f u n c t i o n of the probe l a t e r a l s c a l e , R, and n o r m a l i z e d w i th the l o n g i t u d i n a l i n t e g r a l s c a l e , L , . 78 5.5 Model A p p l i e d To Nasmyth Spectrum The bes t f i t R was used to c a l c u l a t e the t r a n s f e r f u n c t i o n fo r the Nasmyth energy spect rum, F ( k / k s ) , f o r s e v e r a l v a lues of the d i s s i p a t i o n r a t e , e. The r e s u l t s are shown in F i g . 24. T h i s l a r g e - s c a l e t r a n s f e r f u n c t i o n i s seen to be e s s e n t i a l l y independent of the r a te of d i s s i p a t i o n ; i . e . , there i s a unique t r a n s f e r f u n c t i o n f o r l a r g e - s c a l e f lows which i s independent of the d i s s i p a t i o n s c a l e s . The l a r g e - s c a l e t r a n s f e r f u n c t i o n s , F i g . 24, can be a c c u r a t e l y r ep resen ted by a f i t t o a f o u r t h degree po l ynomia l when t r u n c a t e d at the the f i r s t z e r o , k=k 0 . The domain has been no rma l i zed w i th k 0 so tha t the t r a n s f e r f u n c t i o n i s d e f i n e d over the i n t e r v a l 0<k/k o <1. The c o e f f i c i e n t s are g iven i n Tab le IV. The l a r g e - s c a l e t r a n s f e r f u n c t i o n s f o r the two probes are compared i n F i g . 25. A l s o shown i s the t r a n s f e r f u n c t i o n ob ta ined by Garge t t ( pe r sona l communicat ion) by f i t t i n g measured (Osborn probe) ocean s p e c t r a to the Nasmyth u n i v e r s a l cu r ve . The agreement i s very good. The l a r g e - s c a l e t r a n s f e r f u n c t i o n i s u n i t y at the o r i g i n . For the Nasmyth s p e c t r a , most of the energy i s c o n t a i n e d at s c a l e s much l a r g e r than the s c a l e of the probe so tha t the r a t i o of i n t e g r a l s in equa t ion (5 .3 .10 ) i s nea r l y u n i t y . P rev ious workers on l y c o n s i d e r e d the h igh wavenumber range by assuming a form such as tha t due to C o r r s i n (1964) or Pao (1965) f o r the energy spec t rum, E ( k ) , thus o v e r l o o k i n g the reduced LWR due to t r a n s v e r s e a v e r a g i n g . 7 9 F i g u r e 24 - L a r g e - s c a l e t r a n s f e r f u n c t i o n (Osborn ) , f o r : e = 1 0 - \ 1 0 " 2 and 1 0 " 3 c m 2 / s 3 . 80 o k, ( c y c l e s / c m ) F i g u r e 25 - Compar ison of the l a r g e - s c a l e t r a n s f e r f u n c t i o n f o r the two probe d e s i g n s and the Ga rge t t t r a n s f e r f u n c t i o n fo r the Osborn probe 81 Table IV - Polynomial C o - e f f i c e n t s f o r the L a r g e - S c a l e T r a n s f e r F u n c t i o n s The l a r g e - s c a l e t r a n s f e r f u n c t i o n has been t r u n c a t e d a t the A f i r s t z e r o c r o s s i n g , k c, and f i t t e d t o a q u a r t i c p o l y n o m i a l A A over the domain, 0<k/k o<1. The t r a n s f e r f u n c t i o n i s very A A w e l l approximated as T ( k / k e ) , where T(k/k 0) = Z a n < k / k <>) n n=0 A _ | A _ | Osborn probe: 1^=1.7cm Oakey probe :.ke = 1 .4cm n n an 0 1 .000 0 1 .000 1 -0.164 1 -0.165 2 -4.537 2 -4.763 3 5.503 3 5.900 4 -1.804 4 -1.986 82 In f a c t , by approx imat ing the t r a n s v e r s e a t t e n u a t i o n by a " b r i c k - W a l l " low pass f i l t e r at a c r i t i c a l t r a n s v e r s e wavenumber, a =1/R, say , (so tha t k 2 = a 2 + k ! 2 ) , ( 5 .3 .9 ) can be i n t e g r a t e d e x a c t l y fo r any power law spectrum E (k )=k n , where n<0 and r e a l . (Appendix B .3 ) . The u n i v e r s a l spectrum has an approx imate l y p i ecew ise power law form, w i th most of the energy in the (n=-5/3) i n e r t i a l subrange. For t r a n s v e r s e ave rag ing o n l y , A ( k 1 ) = 1 , and we have that X d c j ) = 1 - y n ' 2 ( „ - 2 + § ) ( 5 - 4 > 3 ) w h e r e y = 1 + ( k ^ ) - 2 In the low wavenumber l i m i t , X(0) = 1.0 and so f o r t h i s c l a s s of s p e c t r a there i s no q u e s t i o n of reduced response at low wavenumbers. (Only the h igh wavenumber regime i s d e s c r i b e d . ) The power law t r a n s f e r f u n c t i o n i s p l o t t e d in F i g . 26 fo r n = - 1 , - 2 , - 4 , - 6 . The r o l l - o f f occurs at sma l l e r s c a l e s f o r l a rge n s i n c e , i n t h i s c a s e , a g rea te r f r a c t i o n of the energy occurs at s c a l e s much longer than the s c a l e of the p robe . Equa t i on (5 .4 .3 ) c o u l d e q u a l l y w e l l have been w r i t t e n fo r E ( k / k s ) = ( k / k s ) n , showing tha t the t r a n s f e r f u n c t i o n f o r a power law energy spectrum w i l l be independent of any s c a l e change (such as a v a r i a t i o n in the d i s s i p a t i o n ) . 83 F i g u r e 26 - T r a n s v e r s e t r a n s f e r f u n c t i o n s f o r power law s p e c t r a 84 5.6 E f f e c t On La rge-Sca l e D i s s i p a t i o n Measurements S i n c e , f o r a g i ven p r o b e , the l a r g e - s c a l e t r a n s f e r f u n c t i o n i s independent of e or v, i t i s a s imp le task to c a l c u l a t e the f r a c t i o n of d i s s i p a t i o n r e cove red from the Nasmyth s p e c t r a for a se t of temperatures ( v i s c o s i t i e s ) r e p r e s e n t i n g the range o c c u r r i n g i n ocean . For v i s c o s i t i e s c o r r e s p o n d i n g to T= 5 , 1 0 , 1 5 , 2 0 , 2 5 , 3 0 ° C , ( d e r i v ed a c c o r d i n g to Miyake and Koizumi ( 1948 ) ) , the measured d i s s i p a t i o n , - "4 * f * 2 Y 2 2 < * > d k ( 5 ' 6 - 1 ) * 0 = 15 v j " i c 2 X ( k ) « 2 2 ( k ) dk (5.6.2) 2 0 was c a l c u l a t e d from the u n i v e r s a l d i s s i p a t i o n spec t rum, G ( k / k s ) , u s i n g (2 .7 .3 ) and ( 2 . 7 . 8 ) ; i . e . , E M = 1 1 ( e v ) 3 M j " x ( k ) G 2 ( k f -) dk (5.6.3) 2 o s F i g . 27 compares the f r a c t i o n of the d i s s i p a t i o n - r e cove red by the two probe d e s i g n s at 15°C as a f u n c t i o n of the t rue d i s s i p a t i o n . About one h a l f of the d i s s i p a t i o n i s r e c o v e r e d at e=0.1 c m 2 s - 3 . F i g . 28 shows how the d i s s i p a t i o n r e c o v e r y r a t e s depend on temperature f o r each p r o b e . These da ta a r e p r e sen ted f o r conven ience in Appendix C . The f i t f o r R was e s t ima ted to be a c cu r a t e w i t h i n 10%. At the l a r g e s t v a l ues of d i s s i p a t i o n , t h i s c o r r e s p o n d s to an u n c e r t a i n t y i n the c o r r e c t i o n f a c t o r (he re , 50%) of about 10% as de te rmined by examin ing the r e cove r y r a t e s f o r the two p r o b e s . 85 Consequen t l y , the s p a t i a l t r a n s f e r f u n c t i o n can be used to c o r r e c t measurements of l a r g e - s c a l e d i s s i p a t i o n so as to reduce the e r r o r due to s p a t i a l a ve rag ing to l e s s than 5% over the complete range of va lues found in oceanograph ic m i c r o s t r u c t u r e measurements. 86 F i g u r e 27 - Comparison of the d i s s i p a t i o n r e cove r y r a t e as a f u n c t i o n of the " t r u e " d i s s i p a t i o n , e, at a t empera tu re of 15°C f o r the two p r o b e s . 87 F i g u r e 28 - Recovery r a t e s of measured d i s s i p a t i o n p a r a m e t e r i z e d with temperature i n 5 deg i n t e r v a l s from 0 -deg C. 88 V I . SUMMARY AND CONCLUSIONS The c o n c l u d i n g remarks w i l l be d i v i d e d i n t o two p a r t s . The measurements, e s p e c i a l l y the LDA t e c h n i q u e , w i l l be rev iewed f i r s t and then the measured and model t r a n s f e r f u n c t i o n s w i l l be d i s c u s s e d . 6.1 The Measurements The LDA p r o v i d e d a s imple and accu ra t e measure of the mean f low as demonstrated by compar ison w i th the tunne l P i t o t counter sys tem. In f a c t , i t p rov i des a wider range of measurement by i n c l u d i n g the very low va lues not measured r e l i a b l y w i th the tunne l sys tem. The accuracy of the LDA was l i m i t e d by the accu racy w i th which the beam convergence ang le c o u l d be de te rm ined . T h i s was es t ima ted as 2% a l t h o u g h , i f the t r a n s m i t t i n g o p t i c s geometry i s indeed symmetr ic , the convergence ang le shou ld be equa l to the d i ve rgence ang le on d i f f r a c t i o n at the g r a t i n g . The LDA system was capab le of measur ing the r e a l t ime s p e c t r a w i th good s i g n a l - t o - n o i s e r a t i o w i t h i n the s c a l e "window" of 5.0 - 0.25 cm r e q u i r e d to determine the shear probe t r a n s f e r f u n c t i o n . However, t h i s c a p a b i l i t y demanded low t u rbu l ence i n t e n s i t y and c a r e f u l s e e d i n g . In f a c t , the f low must be " t a i l o r e d " to the l i m i t a t i o n s of the LDA in order to o b t a i n u s e f u l s p e c t r a . The amb igu i t y or phase f l u c t u a t i o n no i se o c c u r r i n g in the LDA s p e c t r a proved to be e s s e n t i a l l y white at f r e q u e n c i e s of i n t e r e s t and c o u l d t h e r e f o r e be s a f e l y s u b t r a c t e d to improve the SNR at h igh wavenumbers. The theory of George and Lumley (1973) 89 underes t imated t h i s no i se l e v e l by a f a c t o r of 2. The theory assumed the re were many p a r t i c l e s in the probe volume and ignored the ( expe r imen ta l l y observed) presence of shot n o i s e . N e v e r t h e l e s s , the f l a t n o i s e spectrum at t u rbu l ence f r e q u e n c i e s and i t s dependence on the mean f low conformed to the t h e o r y . The r e l a t i o n between the low f requency no i s e power and the t o t a l ambigu i t y bandwidth, ( 4 . 1 . 1 ) , had not been con f i rmed d i r e c t l y u s i ng the o p t i c a l pa ramete rs . In the c l a s s i c r e f e r e n c e in t h i s r e g a r d , Berman and Dunning (1971) , ob t a i ned t h e i r probe volume d imens ions from PDF l i n e w id ths , and the apparent probe volume d imens ions were s i g n i f i c a n t l y sma l l e r than those p r e d i c t e d from the o p t i c a l pa ramete rs . In t h i s sense , the no i se was too h i g h ; aga in p o s s i b l y due to shot n o i s e . A good (but c o s t l y ) way to dea l w i th amb igu i t y no i s e i s to employ dua l probe volumes (dua l o p t i c s and d e t e c t o r s ) o f f s e t s l i g h t l y from each other and to measure the c ross-spec t rum to average out the no i se as demonstrated by Veth (1983) . The o b l i q u e f r i n g e p a t t e r n o r i e n t a t i o n method used to ach ieve f requency s h i f t i n g was s u c c e s s f u l and avo ided the n o i s e inheren t in r o t a t i n g the r a d i a l d i f f r a c t i o n g r a t i n g or the c o s t of dua l Bragg c e l l s h i f t i n g which i s a v a i l a b l e w i th commerc ia l LDA systems. However, t h i s t echn ique i s l i m i t e d to s t a t i s t i c a l l y s t a t i o n a r y f l o w s . 90 6.2 The Transfer Function The experimental transfer function did not exhibit the expected low wavenumber behavior. The measured transfer function had an i n i t i a l p o s i t i v e slope and peaked away from the o r i g i n with a value less than unity. A model i s proposed for s p a t i a l averaging in three dimensions that takes into account the attenuation of the transverse wavenumber contributions to the one-dimensional power spectrum. The model equation was integrated using the energy spectrum derived from the gr i d flow data using the "isotropic r e l a t i o n s " and a probe geometry having a shape similar to the actual probes allowing a free parameter which spe c i f i e d the o v e r a l l size in terms of the maximum radius or (equivalently) length. The resulting model transfer function exhibited a low wavenumber response that was q u a l i t a t i v e l y similar to that of the measured transfer function. The transverse s p a t i a l averaging modified the low wavenumber part of the transfer function but had l i t t l e e f f e c t for the range of wavenumbers over which s i g n i f i c a n t attenuation occurs. The "size" parameter of the model was varied so as to obtain the best f i t over th i s range of wavenumbers and these values for the two probes were used to calculate the transfer functions appropriate for large-scale turbulence as characterized by the Nasmyth universal spectrum. For large-scale flows the transfer function decreased monotonically from unity at the o r i g i n and was shown to be nearly independent of the wavenumber scaling. Hence, the 91 t r a n s f e r f u n c t i o n i s e s s e n t i a l l y i n d e p e n d e n t o f t h e r a t e o f d i s s i p a t i o n f o r l a r g e - s c a l e s p e c t r a . T h i s p e r m i t s t h e a p p l i c a t i o n o f t h e m o d e l t o t h e N a s m y t h s p e c t r a a n d r e s u l t s i n l a r g e - s c a l e t r a n s f e r f u n c t i o n s f o r e a c h p r o b e w h i c h c a n be u s e d t o c a l c u l a t e t h e r e c o v e r y r a t e o f t h e m e a s u r e d d i s s i p a t i o n a s a f u n c t i o n o f t e m p e r a t u r e f o r e a c h p r o b e . T h i s c o r r e c t i o n f a c t o r i s t a b u l a t e d a n d c a n be u s e d t o o b t a i n d i s s i p a t i o n r a t e t o w i t h i n 5% o f t h e r a t e t h a t c o u l d be e x p e c t e d w i t h no s p a t i a l a v e r a g i n g . My l a r g e - s c a l e L D A - m o d e l t r a n s f e r f u n c t i o n c o m p a r e s v e r y w e l l w i t h t h e r e s u l t s o b t a i n e d by G a r g e t t ( p e r s o n a l c o m m u n i c a t i o n ) . The r e d u c e d r e s p o n s e a t l a r g e s c a l e s r e s u l t s f r o m s p a t i a l a v e r a g i n g i n t h e t r a n s v e r s e p l a n e i n a f l o w w h e r e t h e i n t e g r a l l e n g t h s c a l e i s o f t h e same o r d e r a s t h e l a t e r a l s c a l e o f t h e p r o b e . H o w e v e r , t h e t r a n s v e r s e a v e r a g i n g d i d n o t s u c c e s s f u l l y a c c o u n t f o r t h e s h a p e o f t h e l a r g e - s c a l e r e s p o n s e ; t h e m e a s u r e d p o s i t i o n o f t h e p ea k was a t a h i g h e r wavenumber t h a n p r e d i c t e d by t h e m o d e l . The d e t a i l e d r e s p o n s e o f t h e s h e a r p r o b e l i k e l y c a n n o t be e x p l a i n e d w i t h o u t i n c l u d i n g e f f e c t s due t o t h e d i s t o r t i o n o f t h e i n c i d e n t t u r b u l e n t f i e l d by t h e p r o b e i t s e l f . The d e t e r m i n a t i o n o f t h e t r a n s f e r f u n c t i o n w o u l d h a v e b een more s t r a i g h t f o r w a r d i f t h e L D A - s h e a r p r o b e m e a s u r e m e n t s h a d b e e n done i n a l a r g e - s c a l e f l o w . H o w e v e r , t h e l i m i t a t i o n s o f t h e LDA p r e c l u d e d t h i s a p p r o a c h s i n c e e i t h e r t h e h i g h i n t e n s i t y o f t h e t u r b u l e n c e w o u l d h a v e r e s u l t e d i n p o o r LDA s i g n a l q u a l i t y ( i n c r e a s e d d r o p - o u t ) o r t h e LDA SNR w o u l d h a v e b een t o o s m a l l a t t h e s c a l e s o f i n t e r e s t . 92 BIBLIOGRAPHY A l l e n , H . J . a n d E . W . P e r k i n s , 1 9 5 2 . A s t u d y o f e f f e c t s o f f l o w o v e r s l e n d e r i n c l i n e d b o d i e s o f r e v o l u t i o n . N a t i o n a l A d v i s o r y C o u n c i l f o r A e r o n a u t i c s R e p o r t N o . 1 0 4 8 . B a t c h e l o r , G . K . , 1 9 5 3 . The T h e o r y o f Homogeneous T u r b u l e n c e . C a m b r i d g e U n i v e r s i t y P r e s s , 1 9 3 p p . B e r m a n , J . W . a n d D u n n i n g , N . S . , 1 9 7 1 . T u r b u l e n c e m e a s u r e m e n t s u s i n g t h e l a s e r - D o p p l e r v e l o c i m e t e r . P r o c . 2nd B i e n n i a l S y m p o s i u m on T u r b u l e n c e i n L i q u i d s , R o l l a , 1 7 5 . B r i t t e r , R . E . , J . C . R . Hunt a n d J . C . M u m f o r d , 1 9 7 9 . T h e d i s t o r t i o n o f t u r b u l e n c e by a c i r c u l a r c y l i n d e r . J . o f F l u i d M e c h . , v o l 9 2 , pp 2 6 9 - 3 0 1 . B u c h a v e , P . , W . K . G e o r g e a n d J . L . L u m l e y , 1 9 7 9 . T h e m e a s u r e m e n t o f t u r b u l e n c e w i t h t h e l a s e r D o p p l e r a n e m o m e t e r . Ann R e v . o f F l u i d M e c h . j j_ , 4 4 3 - 5 0 3 . C o r r s i n S . , 1 9 6 4 . F u r t h e r g e n e r a l i z a t i o n s o f O n s a g e r ' s c a s c a d e m o d e l f o r t u r b u l e n t s p e c t r a . P h y s i c s o f F l u i d s 7 , 1156 . D u r s t , F . , A . M e l l i n g a n d J . H . W h i t e l a w , 1 9 7 6 . P r i n c i p l e s a n d • P r a c t i c e o f L a s e r - D o p p l e r A n e m o m e t r y , A c a d e m i c P r e s s , 4 0 5 p p . G e o r g e W . K . a n d J . L . L u m l e y , 1 9 7 3 . T h e l a s e r - D o p p l e r v e l o c i m e t e r a n d i t s a p p l i c a t i o n t o t h e m e a s u r e m e n t o f t u r b u l e n c e , J . F l u i d M e c h . 6_0 , 3 2 1 - 6 2 . G r a n t , H . L . , R.W. S t e w a r t a n d A . M o l l i e t . 1 9 6 2 . T u r b u l e n c e s p e c t r a f r o m a t i d a l c h a n n e l . J . F l u i d M e c h . J_2 , 2 4 1 . H i n z e , J . , 1 9 7 5 . T u r b u l e n c e , M c G r a w - H i l l , I n c . , 7 9 0 p p . K o l m o g o r o v , A . N . , 1 9 4 1 . C o m p t . r e n d . a c a d . s c i . , U . R . S . S . , 3_1 , 5 3 8 . L a d i n g , L . a n d R . V . E d w a r d s , 1 9 7 6 . The e f f e c t o f m e a s u r e m e n t v o l u m e on l a s e r D o p p l e r a n e m o m e t e r m e a s u r e m e n t s a s m e a s u r e d on s i m u l a t e d s i g n a l s . P r o c . L D A - S y m p o s i u m , C o p e n h a g e n 1 9 7 5 . L a d i n g , L . , 1 9 7 2 . A F o u r i e r o p t i c a l m o d e l f o r t h e l a s e r - D o p p l e r v e l o c i m e t e r . O p t o - e l e c t r o n i c s , 4 , 3 8 5 - 3 8 9 . M i y a k e , Y . a n d M. K o i z u m i , 1 9 4 8 . The m e a s u r e m e n t o f t h e v i s c o s i t y c o - e f f i c i e n t o f s e a w a t e r . J . M a r . R e s . 7 , 6 3 -6 6 . N a s m y t h , P . , 1 9 7 0 . O c e a n t u r b u l e n c e . P h . D . t h e s i s , I n s t i t u t e 93 of Oceanography, U n i v e r s i t y of B r i t i s h Co lumbia , 69pp. Oakey, N . S . , 1977. An ins t rument to measure ocean ic tu rbu lence and m i c r o s t r u c t u r e . Bedford I n s t i t u t e of Oceanography, Rep. Se r . B1-R-77-3, 52pp Oakey, N . S . , 1982. De te rm ina t i on of the r a te of d i s s i p a t i o n of t u r b u l e n t k i n e t i c energy from s imul taneous temperature and v e l o c i t y shear m i c r o s t r u c t u r e measurements. J . Phys. Oceanography, 12. Osborn , T .R . and T . E . S i ddon , 1975. Oceanic shear measurements us ing the ocean ic shear p robe . P r o c . 3rd B i e n n i a l Symposium on Turbu lence in L i q u i d s , G.K. P a t t e r son and J . L . Zaken, E d s . , U n i v e r s i t y of M i s s o u r i -R o l l a . Osborn , T .R . and W.R. C rawfo rd , 1977. Tu rbu l en t v e l o c i t y measurements wi th an a i r f o i l shear p robe . Manusc r ip t Report 3J_ , U n i v e r s i t y of B r i t i s h Co lumbia , I n s t i t u t e of Oceanography, Vancouver , B.C. Osborn , T .R . and W.R. C rawfo rd , 1980. An a i r f o i l probe fo r measur ing t u r b u l e n t v e l o c i t y f l u c t u a t i o n s in water . Ch . 19 of A i r -Sea I n t e r a c t i o n : Inst ruments and Methods, ed . by F .Dobson, L. Hasse and R. D a v i s , Plenum, New York . Osborn , T . R . , 1974. V e r t i c a l p r o f i l i n g of v e l o c i t y m i c r o s t r u c t u r e . J . Phys. Oceanography, 4 , 109-115. Pao, Y . H . , 1965. S t r u c t u r e of t u r b u l e n t v e l o c i t y and s c a l a r f i e l d s at l a rge wavenumbers. Phys. F l u i d s 8 , 1063-75. Pond, S . , 1965. Tu rbu l en t s p e c t r a in the a tmospher ic boundary l a y e r over the s ea . Ph .D. t h e s i s . I n s t i t u t e of Oceanography, U n i v e r s i t y of B r i t i s h Co lumbia , Vancouver , B . C . , 68pp. S i ddon , T . E . , 1969. On the response of p ressu re measur ing i n s t r u m e n t a t i o n in unsteady f l ow . U n i v e r s i t y of T o r o n t o , I n s t i t u t e f o r Aerospace S t u d i e s , Report No. 136 S i ddon , T . E . , 1971. A m i n i a t u r e t u rbu l ence gauge u t i l i z i n g aerodynamic l i f t . Rev. S c i . I ns t ruments , 42 ,653-656. Skramstad, H .K . , H.L . D ryden , G.B. Schubauer and W.C. Mock. NACA Report No. 581, 22. S tewar t , R.W. and A . A . Townsend, 1951. S i m i l a r i t y and s e l f -p r e s e r v a t i o n in i s o t r o p i c t u r b u l e n c e . P h i l . T r a n s . Roy. Soc . L o n d . , v o l 243, pp 359-86. Townsend, A . A . , 1976. The S t r u c t u r e of Tu rbu len t Shear F low, 2nd edn , Cambridge U n i v e r s i t y P r e s s . 94 U b e r o i , M.S. and L . S . G . Kovasznay, 1953. On mapping and measurement of random f i e l d s . Q. A p p l . Math . j_0 ,375-93. U b e r o i , M.S. and P. Freymuth, 1969. Spec t r a of t u rbu l ence in wakes behind c i r c u l a r c y l i n d e r s . P h y s i c s of F l u i d s , \2 , 1 359. V a s i l e n k o , Y . G . , 1972. Laser v e l o c i t y meters - a comparat ive s t udy . O p t i c s and Laser Techno logy , December, 270-272. V e t h , C , 1983. Turbu lence measurements i n the s t r a t i f i e d c e n t r a l Nor th Sea w i th a l a s e r - D o p p l e r v e l o c i m e t e r system. Nor th Sea Dynamics , eds J . Sundermann and W. Lenz , S p r i n g e r - V e r l a g , 693p. Wang, C . P., 1972. A u n i f i e d a n a l y s i s on l a s e r Doppler v e l o c i m e t e r s . J . S c i . Instruments (J. P h y s i c s E ) , 5 , 763-766. Wyngaard, J . C , 1968. Measurement of s m a l l - s c a l e tu rbu l ence s t r u c t u r e w i th hot w i r e s . J . S c i . Ins t ruments (J. Phy s i c s E ) , S e r i e s 2, J_ , 1105-1108 95 APPENDIX A - THE MEASURED PROBE GEOMETRIES The shape and s i z e of the two probes used in t h i s study were o b t a i n e d by measur ing the s i l o u e t t e s of the probes imaged by an overhead p r o j e c t o r . The measured v a l u e s were f i t w i th a cub i c s p l i n e r o u t i n e and are shown p l o t t e d in F i g . 29. A l s o shown f o r compar ison i s the curve used in the model as an approx ima t ion f o r the shapes of both probes and d e f i n e d by x , = 7 0 r 2 w i th 7 o =l3 .0mm~ 1 . The model f i t t i n g procedure co r responds to v a r y i n g the maximum probe r a d i u s , R (and hence , 2 L = 7 0 R 2 ) . T h i s i s to say , the o v e r a l l s i z e (but not the shape) was v a r i e d . I t can be seen from F i g . 29 tha t the model curve i s a good approx imat ion to the shape of both probes over the s e n s i t i v e (curved) p o r t i o n s . The r a t e of change of c r o s s - s e c t i o n a l a rea a long the l eng th of each probe was computed and t h i s r e l a t i v e s e n s i t i v i t y , i s p l o t t e d in F i g . 30 w i th a ske t ch showing the l o c a t i o n and l eng th of the c o r r e s p o n d i n g sens ing e lement . In a d d i t i o n , the l eng ths of cons tan t s e n s i t i v i t y from the model best f i t are i n c l u d e d . The Oakey probe has a b l un t t i p and shows a co r r e spond ing peak in s e n s i t i v i t y . The Osborn probe i s f i n e r than a pa rabo l a and has a lower s e n s i t i v i t y near the t i p . Of c o u r s e , the model s e n s i t i v i t y i s a c o n s t a n t . In both des igns the a i r f o i l shape r e s u l t s in a marked decrease in s e n s i t i v i t y we l l be fo re the fu l c rum of the sens ing e lement . S ince i t was not c l e a r to me how the c r o s s - f o r c e per u n i t l eng th c o n t r i b u t e s to the l o a d i n g of the beam, e s p e c i a l l y 96 near the t i p where there i s a gap ahead of the sens ing e lement , the model was s i m p l i f i e d by assuming a cons tan t l e ve rage fo r the whole l oad d i s t r i b u t i o n . T h i s assumpt ion appears more r easonab le as the d i s t a n c e between the s e n s i t i v e p o r t i o n and the fu l c rum i s i n c r e a s e d . The e f f e c t i v e l eng th from the model f i t compares we l l w i th the l e n g t h of the s e n s i t i v e r eg ion in the two p robes . I t would appear that a long sens ing element w i th a shor t a i r f o i l i s a good des ign s t r a t e g y to maximize the s e n s i t i v i t y and min imize the s p a t i a l a v e r a g i n g . uQ C f-t ro NJ KD I - -3 0) ro cn : r 3 —' ro - 0) in X C - i-l ii ro ~* Q J o >1 TJ K) r-l - o cr € ro ri" W> TJ o ro il tn to 0) 3 3 3 a - r f • V ro 3 o Qi ro m-1 tn • — i -a ro or A Oakey probe X Osborn probe model shape vo 8. 10. 12. Distance From Probe Tip (mm) 14 o c < ro 98 Oakey probe Distance from Probe Tip (mm) F i g u r e 30 - The r e l a t i v e s e n s i t i v i t i e s ( in a r b i t r a r y u n i t s ) c a l c u l a t e d from the probe shapes and compared to the b e s t - f i t v a l u e s f o r the s e n s i t i v e l e n g t h s , 2L . The s ens i ng e lements a re a l s o shown. 9 9 APPENDIX B - DETAILS OF CALCULATIONS FOR CHAPTER 5 B . l C a l c u l a t i o n s o f t h e component t r a n s f e r f u n c t i o n s , A ( k 2 ) and B(a), g i v e n i n e q u a t i o n s ( 5 . 3 . 6 ) and ( 5 . 3 . 7 ) . S ( k ) = A ( k , ) B ( a ) , where a 2 = k 2 + k 2 - 1 2 3 w i t h r 2 = x 2 + x 2 2 3 and a * r = a r c o s cj> L o n g i t u d i n a l A ( k x ) = |/ r 1 Ca(x 1) ] | 2 , n o r m a l i z e d t o u n i t y a t k x = 0, oo • 2L now, flla{xl)l = / " a f x ^ e ^ ^ l dx1 = / e l k l x l d x x < 5 _ i k , L = 1 f k 7 i - Binder) and f jtXx^] kj=0 = 2L s i n ( k . L ) 2 s o ' A < V = [-T7L ] T r a n s v e r s e f 2 [ b ( r ) ] = J J m Q r e i ( k 2 x 2 + k 3 x 3 ) d x 2 d x ; — CO = m Q / R r 2 J 2 \ i a r c o s ^ ^ d r o o oR 2 - r c m 0 J r 2 J Q ( o - r ) d r 2 ^ J F F R ( O R ) 2 J o ( a r ) d ( a r ) a 3 o 100 2nm_R3 and F2lh(r)la=0 = —f— s o t h a t t h e t r a n s v e r s e a v e r a g i n g f u n c t i o n i s B ( c ) = | F 2 [ b ( r ) ] | 2 { — ^ - r / a R ( o r ) 2 J ( a r ) 2 d ( a r ) } ( a R ) 3 o ° [ 3 - / Y x 2 j ( x ) d x ] 2 , Y = aR Y 3 o ° B.2 C a l c u l a t i o n o f e q u a t i o n ( 5 . 3 . 8 ) , g i v i n g t h e m e a s u r e d o n e - d i m e n s i o n a l t r a n s f e r f u n c t i o n i n terms o f t h e e n e r g y s p e c t r u m E ( k ) and t h e component t r a n s f e r f u n c t i o n , A ( k ) and B ( a ) . Y 2 2 ( k , ) = /" B ( a ) ZlKl ( k2 + k 2 ) dk (5.3.8) z z 1 4 k : k 3 1 From t h e g e n e r a l r e l a t i o n between t h e " t r u e " s p e c t r u m t e n s o r and t h e s p a t i a l l y a v e r a g e d t e n s o r , we h a v e f o r t h e t r a n s v e r s e component: T 2 2 ( k ) = S ( k ) $ 2 2 ( k ) The measured o n e - d i m e n s i o n a l power s p e c t r u m i s Y 2 2 ( k 1 ) = / / r 2 2 ( k ) d k 2 d k 3 = // S ( k ) $ 2 2 ( k ) d k 2 d k 3 — 00 "~ CO = A ( k , ) // B ( a ) (k2 - k2) dk, dk, = A ( k l ) // B ( a ) (* 2 + k 2 ) d * 9 d k * 2 - i 4nk 3 1 2 3 where, y 2 2 = Y33 = \ ^ Y 2 2 + Y 3 3)» n a s b e e n u s e d . 101 N o t i n g t h a t a2 = k 2 - k 2 , acta = kdk f o r k, = c o n s t a n t , 1 1 and d k 2 d k 3 = adcdrj) we c a n w r i t e t h a t Y 2 2 ( k i > = ^1T^- Ja>/2llB(a) £ f £ * ( k 2 + k 2 ) k dk d* 2 2 1 8-rt J o o kk 1 . ^ i U l l J- B ( C ) ^  ( k 2 + k 2 ) dk 4 k=k x k 3 1 B.3 The i n t e g r a t i o n o f power law s p e c t r a , e q u a t i o n ( 5 . 5 . 1 ) . Assume t h a t E ( k ) i s o f t h e f o r m E ( k ) = k n , n < 0, and B(a) i s r e p r e s e n t e d by {1 , a <a 0 , a > c - _ 1 B(a) = ^  where a c = ^ and k 2 = a2 + k 2 . Then, f o r t r a n s v e r s e a v e r a g i n g a l o n e ( t h a t i s A ( k x ) = 1) we h a v e t h a t X ( k x ) = 1 -f" (k2 + k2) dk / " ( * 2 + * 2 ) ^ k3 1 .k c . n ( ) 1 - [ n - 2 + n ( ^ ) " 2 ] (5.5.1) 2 ( n - l ) L 1 vn / 2 = 1 - — [n - 2 + S ] 2 ( n - l ) L y J 102 APPENDIX C - DISSIPATION RECOVERY RATES DISSIPATION RECOVERY RATE: OSBORN PROBE R= 0.21cm TEMP- 0.0 DEG C VISCOSITY* 0.0175 CGS TRUE DISSIPATION MEASURED DISSIPATION FRACTION RECOVERED 0.10E-05 0.10E-05 1 .00 0.19E-05 0.19E-05 0.99 0.35E-05 0.35E-05 0.99 0.64E-05 0.63E-05 0.99 0.12E-04 0.12E-04 0.99 0.21E-04 0.21E-04 0.98 0.39E-04 0.38E-04 0.98 0.72E-04 0.70E-04 0.97 0.13E-03 0.13E-03 0.97 0.24E-03 0.23E-03 0.96 0.44E-03 0.42E-03 0.95 0.81E-03 0.76E-03 0.93 0.15E-02 0.14E-02 0.92 0.27E-02 0.24E-02 0.90 0.50E-02 0.44E-02 0.87 0.92E-02 0.77E-02 0.84 0.17E-01 0.14E-01 0.81 0.31E-01 0.24E-01 0.77 0.57E-01 0.41E-01 0.72 0.10E+00 0.70E-01 0.67 TEMP= 5.0 DEG C VISCOSITY* 0.0148 CGS TRUE DISSIPATION MEASURED DISSIPATION FRACTION RECOVERED 0.10E-05 0.10E-05 0.99 0.19E-05 0.19E-05 0.99 0.35E-05 0.34E-05 0.99 0.64E-05 0.63E-05 0.99 0.12E-04 *. 0.11E-04 0.98 0.21E-04 0.21E-04 0.98 0.39E-04 0.38E-04 0.97 0.72E-04 0.70E-04 0.97 0.13E-03 0.13E-03 0.96 0.24E-03 0.23E-03 0.95 0.44E-03 0.41E-03 0.94 0.81E-03 0.75E-03 0.92 0.15E-02 0.13E-02 0.90 0.27E-02 0.24E-02 0.88 0.50E-02 0.42E-02 0.85 0.92E-02 0.75E-02 0 .81 0.17E-01 0.13E-01 0.78 0.31E-01 0.23E-01 0.73 0.57E-01 0.39E-01 0.68 0.10E+00 0.65E-01 0.63 103 TEMP* 10,0 DEG C VISCOSITY* 0.0128 CGS TRUE DISSIPATION MEASURED DISSIPATION FRACTION RECOVERED 0.10E-05 0.10E-05 0.99 0.19E-05 0.19E-05 0.99 0.35E-05 0.34E-05 0.99 0.64E-05 0.63E-05 0.98 0.12E-04 0.11E-04 0.98 0.21E-04 0.21E-04 0.97 0.39E-04 0.38E-04 0.97 0.72E-04 0.69E-04 0.96 0.13E-03 0.13E-03 0.95 0.24E-03 0.23E-03 0.94 0.44E-03 0.41E-03 0.92 0.81E-03 0.74E-03 0.91 0.15E-02 0.13E-02 0.88 . 0.27E-02 0.23E-02 0.86 0.50E-02 0.41E-02 0.82 0.92E-02 0.72E-02 0.79 0.17E-01 0.13E-01 0.74 0.31E-01 0.21E-01 0.70 0.57E-01 0.36E-01 0.64 0.10E+00 0.61E-01 0.59 TEMP= 15.0 DEG C VISCOSITY= 0.0112 CGS TRUE DISSIPATION MEASURED DISSIPATION FRACTION RECOVERED 0.10E-05 0.10E-05 0.99 0.19E-05 0.19E-05 0.99 0.35E-05 0.34E-05 0.99 0.64E-05 0.63E-05 0.98 0.12E-04 0.11E-04 0.98 0.21E-04 0.21E-04 0.97 0.39E-04 0.38E-04 0.96 0.72E-04 0.69E-04 0.95 0.13E-03 *. 0.12E-03 0.94 0.24E-03 0.22E-03 0.93 0.44E-03 0.40E-03 0.91 0.81E-03 0.72E-03 0.89 0.15E-02 0.13E-02 0.87 0.27E-02 0.23E-02 0.84 0.50E-02 0.40E-02 0.80 0.92E-02 0.70E-02 0.76 0.17E-01 0.12E-01 0.71 0.31E-01 0.20E-01 0.66 0.57E-01 0.34E-01 0.61 0.10E+00 0.57E-01 0.55 104 TEMP* 20,0 DEG C VISCOSITY* 0,0098 CGS TRUE DISSIPATION MEASURED DISSIPATION FRACTION RECOVERED 0.10E-05 0.10E-05 0.99 0.19E-05 0.19E-05 0.99 0.35E-05 0.34E-05 0.98 0.64E-05 0.62E-05 0.98 0.12E-04 0.11E-04 0.97 0.21E-04 0.21E-04 0.97 0.39E-04 0.38E-04 0.96 0.72E-04 0.68E-04 0.95 0.13E-03 0.12E-03 0.93 0.24E-03 0.22E-03 0.92 0.44E-03 0.40E-03 0.90 0.81E-03 0.71E-03 0.88 0.15E-02 0.13E-02 0.85 0.27E-02 0.22E-02 0.81 0.50E-02 0.39E-02 0.78 0.92E-02 0.67E-02 0.73 0.17E-01 0.11E-01 0.68 0.31E-01 0.19E-01 0.63 0.57E-01 0.32E-01 0.57 0.10E+00 0.53E-01 0.51 TEMP* 25.0 DEG C VISCOSITY* 0.0088 CGS TRUE DISSIPATION MEASURED DISSIPATION FRACTION RECOVERED 0.10E-05 0.10E-05 0.99 0. 19E-05 0.19E-05 0.98 0.35E-05 0.34E-05 0.98 0.64E-05 0.62E-05 0.98 0.12E-04 0.11E-04 0.97 0.21E-04 0.21E-04 0.96 0.39E-04 0.37E-04 0.95 0.72E-04 0.68E-04 0.94 0.13E-03 0.12E-03 0.93 0.24E-03 « 0.22E-03 0.91 0.44E-03 0.39E-03 0.89 0.81E-03 0.70E-03 0.86 0.15E-02 0.12E-02 0.83 0.27E-02 0.22E-02 0.79 0.50E-02 0.38E-02 0.75 0.92E-02 0.65E-02 0.70 0.17E-01 0.11E-01 0.65 0.31E-01 0.18E-01 0.60 0.57E-01 0.30E-01 0.54 0.10E+00 0.50E-01 0.48 105 TEMP= 30,0 DEG C VISCOSITY" 0,0078 CGS TRUE DISSIPATION MEASURED DISSIPATION FRACTION RECOVERED 0.10E-05 0.10E-05 0.99 0.19E-05 0.19E-05 0.98 0.35E-05 0.34E-05 0.98 0.64E-05 0.62E-05 0.12E-04 0.11E-04 0.97 0.21E-04 0.21E-04 0.96 0.39E-04 0.37E-04 0.72E-04 0.67E-04 0.93 0.13E-03 0.12E-03 0.92 0.24E-03 0.22E-03 0.44E-03 0.39E-03 0.87 0.81E-03 0.69E-03 0.84 0.15E-02 0.12E-02 0.27E-02 0.21E-02 0.77 0.50E-02 0.36E-02 0.72 0.92E-02 0.62E-02 0.17E-01 0.10E-01 0.62 0.31E-01 0.17E-01 0.56 0.57E-01 ... . 0.29E-01 0.10E+00 0.46E-01 0.45 * 106 DISSIPATION RECOVERY RATE: OAKEY PROBE R=0,235cm TEMP* 0.0 DEG C VISCOSITY* 0.0175 CGS TRUE DISSIPATION MEASURED DISSIPATION FRACTION RECOVERED 0.10E-05 0.10E-05 0.99 0.19E-05 0.19E-05 0,99 0.35E-05 0.34E-05 0.99 0.64E-05 0.63E-05 0.99 0.12E-04 0.11E-04 0.98 0.21E-04 0.21E-04 0.98 0.39E-04 0.38E-04 0.97 0.72E-04 0.69E-04 0.96 0.13E-03 0.13E-03 0.95 0.24E-03 0.23E-03 0.94 0.44E-03 0.41E-03 0.93 0.81E-03 0.74E-03 0.91 0.15E-02 0.13E-02 0.89 0.27E-02 0.23E-02 0.86 0.50E-02 0.41E-02 0.83 0.92E-02 0.73E-02 0.79 0.17E-01 0.13E-01 0.75 0.31E-01 0.22E-01 0.70 0.57E-01 0.37E-01 0.65 0.10E+00 0.62E-01 0.59 'TEMP* 5.0 DEG C VISCOSITY* 0.0148 CGS TRUE DISSIPATION MEASURED DISSIPATION FRACTION RECOVERED 0.10E-05 0.10E-05 0.99 0.19E-05 0.19E-05 0.99 0.35E-05 0.34E-05 0.99 0.64E-05 0.63E-05 0.98 0.12E-04 0.11E-04 0.98 0.21E-04 0.21E-04 0.97 0.39E-04 0.38E-04 0.96 0.72E-04 0.69E-04 0.95 0.13E-03 0.12E-03 0.94 0.24E-03 0.22E-03 0.93 0.44E-03 0.40E-03 0.91 0.81E-03 0.72E-03 0.89 0.15E-02 0.13E-02 0.86 0.27E-02 0.23E-02 0.83 0.50E-02 0.40E-02 0.80 0.92E-02 0.70E-02 0.76 0.17E-01 0.12E-01 0.71 0.31E-01 0.20E-01 0.66 0.57E-01 0.34E-01 0.60 0.10E+00 0.56E-01 0.54 107 TEMP* 10,0 DEG C VISCOSITY* 0,0128 CGS TRUE DISSIPATION MEASURED DISSIPATION FRACTION RECOVERED 0.10E-05 0.10E-05 0.99 0.19E-05 0.19E-05 0.99 0.35E-05 0.34E-05 0.98 0.64E-05 0.62E-05 0.98 0.12E-04 0.11E-04 0.97 0.21E-04 ' 0.21E-04 0.97 0.39E-04 0.38E-04 0.96 0.72E-04 0.68E-04 0.95 0.13E-03 0.12E-03 0.93 0.24E-03 0.22E-03 0.92 0.44E-03 0.40E-03 0.90 0.B1E-03 0.71E-03 0.87 0.15E-02 0.13E-02 0.84 0.27E-02 0.22E-02 0.81 0.50E-02 0.39E-02 0.77 0.92E-02 0.67E-02 0.72 0.17E-01 0.11E-01 0.67 0.31E-01 0.19E-01 0.62 0.57E-01 0.32E-01 0.56 0.10E+00 0.52E-01 0.50 TEMP= 15.0 DEG C VISCOSITY* 0.0112 CGS TRUE DISSIPATION MEASURED DISSIPATION FRACTION RECOVERED 0.10E-05 0.10E-05 0.99 0.19E-05 0.19E-05 0.98 0.35E-05 0.34E-05 0.98 0.64E-05 0.62E-05 0.97 0.12E-04 0.11E-04 0.97 0.21E-04 0.21E-04 0.96 0.39E-04 0.37E-04 0.95 0.72E-04 0.68E-04 0.94 0.13E-03 , 0.12E-03 0.92 0.24E-03 0.22E-03 0.90 0.44E-03 0.39E-03 0.88 0.81E-03 0.69E-03 0.85 0.15E-02 0.12E-02 0.82 0.27E-02 0.21E-02 0.78 0.50E-02 0.37E-02 0.74 0.92E-02 0.64E-02 0.69 0.17E-01 0.11E-01 0.64 0.31E-01 0.18E-01 0.58 0.57E-01 0.30E-01 0.52 0.10E+00 0.48E-01 0.46 108 TEMP* 20.0 DEG C VISCOSITY* 0.0098 CGS TRUE DISSIPATION MEASURED DISSIPATION FRACTION RECOVERED 0.10E-05 0.10E-05 0.99 0.19E-05 0.19E-05 0.98 0.35E-05 0.34E-05 0.98 0.64E-05 0.62E-05 0.97 0.12E-04 0.11E-04 0.96 0.21E-04 0.20E-04 0.95 0.39E-04 0.37E-04 0.94 0.72E-04 0.67E-04 0.93 0.13E-03 0.12E-03 0.91 0,24E-03 0.22E-03 0.89 0.44E-03 0.38E-03 0.86 0.81E-03 0.68E-03 0.83 0.15E-02 0.12E-02 0.80 0.27E-02 0.21E-02 0.76 0.50E-02 0.36E-02 0.71 0.92E-02 0.60E-02 0.66 0.17E-01 0.10E-01 0.60 0.31E-01 0.17E-01 0.54 0.57E-01 0.27E-01 0.48 0.10E+00 0.44E-01 0.42 TEMP= 25.0 DEG C VISCOSITY* 0.0088 CGS TRUE DISSIPATION MEASURED DISSIPATION FRACTION RECOVERED 0.10E-05 0.10E-05 0.98 0.19E-05 0.19E-05 0.98 0.35E-05 0.34E-05 0.97 0.64E-05 0.62E-05 0.97 0.12E-04 0.11E-04 0.96 0.21E-04 0.20E-04 0.95 0.39E-04 0.37E-04 0.94 0.72E-04 0.66E-04 0.92 0.13E-03 0.12E-03 0.90 0.24E-03 0.21E-03 0.88 0.44E-03 0.38E-03 0.85 0.81E-03 0.66E-03 0.81 0.15E-02 0.12E-02 0.78 0.27E-02 0.20E-02 0.73 0.50E-02 0.34E-02 0.68 0.92E-02 0.58E-02 0.63 0.17E-01 0.96E-02 0.57 0.31E-01 0.16E-01 0.51 0.57E-01 0.25E-01 0.45 0.10E+00 0.41E-01 0.39 109 TEMP*....30, 0 PEG C yiSCOSITY* . 0 , 0 0 7 8 CGS TRUE DISSIPATION MEASURED DISSIPATION FRACTION RECOVERED 0.10E-05 ... 0.10E-Q5 0.98 0.19E-05 0.19E-05 6.98 0.35E-05 0.34E-05 0.97 0.64E-05 0.61E-05 0.96 0.12E-04 0.11E-04 6.95 0.21E-04 0.20E-04 0.94 0,39E-04 0.36E-04 0.93 0.72E-04 0.65E-04 6.91 0.13E-03 0.12E-03 0.89 0.24E-03 0.21E-03 0.86 6.44E-03 0.37E-03 0.83 0.81E-03 0.64E-03 0.79 0.15E-02 0.11E-02 0.75 0.27E-02 0.19E-02 0.70 0.50E-02 0.33E-02 0.65 0.92E-02 0.55E-02 0.59 0.17E-01 0.90E-02 6.53 0.31E-01 0.15E-01 0.47 0.57E-01 0.23E-01 0.42 0.10E+00 0.37E-01 0.36 

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