UBC Theses and Dissertations

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UBC Theses and Dissertations

Heated anemometry and thermometry in water Lueck, Rolf Gero 1979

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HEATED ANEMOMETRY AND THERMOMETRY IN WATER by ROLF GERO LUECK B . A . S c . , U n i v e r s i t y o f B r i t i s h C o l u m b i a , 1973 A T H E S I S SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n t h e DEPARTMENT OF PHYSICS a n d t h e INSTITUTE OF OCEANOGRAPHY We a c c e p t t h i s t h e s i s a s c o n f o r m i n g t o t h e r e q u i r e d s t a n d a r d THE UNIVERSITY OF B R I T I S H COLUMBIA 1979 @ R o l f G e r o L u e c k , 1979 I n p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f t h e r e q u i r e m e n t s f o r a n a d v a n c e d d e g r e e a t t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g r e e t h a t t h e L i b r a r y s h a l l m a k e i t f r e e l y a v a i l a b l e f o r r e f e r e n c e a n d s t u d y . I f u r t h e r a g r e e t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s m a y b e g r a n t e d b y t h e H e a d o f my D e p a r t m e n t o r b y h i s r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l n o t b e a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . D e p a r t m e n t o f T h e U n i v e r s i t y o f B r i t i s h C o l u m b i a 2 0 7 5 W e s b r o o k P l a c e V a n c o u v e r , C a n a d a V 6 T 1W5 D E - 6 B P 7 5 - 5 1 1 E i i A b s t r a c t The c h a r a c t e r i s t i c s o f s e v e r a l t y p e s o f heated s e n s o r s used f o r measuring o c e a n i c t u r b u l e n c e have been examined. The a u t h o r measured t h e s t e a d y heat f l u x from g l a s s c o a t e d e l l i p s o i d a l microbead t h e r m i s t o r s , computed n u m e r i c a l l y t h e unsteady h e a t . f l u x from an i d e a l c o n s t a n t t e m p e r a t u r e f l a t p l a t e , and a n a l y z e d t h e s t e a d y f l o w c a l i b r a t i o n d a t a o f a p a r a l e n e - c c o a t e d p l a t e t h e r m i s t o r as w e l l as t h e s t e a d y and unsteady c a l i b r a t i o n d a t a of two c o n i c a l c o n s t a n t t e m p e r a t u r e hot f i l m anemometers. In o r d e r t o u n d e r s t a n d t h e b e h a v i o u r o f probes a l o n e , t h e r m a l models of t h e s e s e n s o r s have been d e v e l o p e d . The models i n c o r p o r a t e a N u s s e l t number governed s t e a d y heat f l u x from t h e w e t t e d s u r f a c e and t h e the r m a l e f f e c t s o f a s u r f a c e c o a t i n g and a s u p p o r t i v e s u b s t r a t e . D e r i v e d f u n c t i o n a l r e l a t i o n s h i p s between t h e s t e a d y heat f l u x and t h e f l o w r a t e a g r e e f a v o u r a b l y w i t h t h e a v a i l a b l e c a l i b r a t i o n d a t a . The q u a s i - s t e a d y s e n s i t i v i t y o f t h e s e probes when used as anemometers o r thermometers as w e l l as t h e i r s i g n a l c o n t a m i n a t i o n by t e m p e r a t u r e o r v e l o c i t y a r e c a l c u l a t e d u s i n g t h e f u n c t i o n a l heat f l u x r e l a t i o n s h i p s . The s u b s t r a t e and t h e c o a t i n g reduce t h e s e n s i t i v i t y t o t e m p e r a t u r e and t o speed as w e l l as t h e r a t i o o f s p e e d - t o - t e m p e r a t u r e s e n s i t i v i t y . The r e s p o n s e of s e n s o r s i s not governed by t h e N u s s e l t number when t h e boundary l a y e r i s unsteady. The unsteady r e s p o n s e of a s e n s o r t o v e l o c i t y o s c i l l a t i o n s i s governed by i t s unsteady v i s c o u s boundary l a y e r and may i n c r e a s e w i t h i n c r e a s i n g f r e q u e n c y o v e r some f r e q u e n c y r a n g e s . The r e s p o n s e bandwidth i s w i d e r f o r v e l o c i t y t h a n f o r tempera-t u r e . The r a t i o o f unsteady t e m p e r a t u r e - t o - v e l o c i t y s e n s i t i v i t y i s h i g h e s t a t z e r o f r e q u e n c y . Frequency r e s p o n s e c a l i b r a t i o n s methods i i i must r e a l i s t i c a l l y s i m u l a t e t h e s e n s o r ' s u n s t e a d y v i s c o u s b o u n d a r y I a y e r . i V T a b l e o f C o n t e n t s A b s t r a c t i i L i s t o f T a b l e s v L i s t o f F i g u r e s v i Acknowledgements x I . I n t r o d u c t i o n I 2. . Background 2 3. S t e a d y Heat T r a n s f e r 7 3. I I d e a l P r o b e s 7 3.2 R e a l P r o b e s II 4. U n s t e a d y Heat T r a n s f e r 18 4. I I d e a l P r o b e s . . . f 18 4.2 Real P r o b e s 22 5. C o n c l u s i o n s 25 5. I S t e a d y Heat T r a n s f e r 25 5.2 U n s t e a d y Heat T r a n s f e r 26 B i b I i o g r a p h y 28 A p p e n d i x A: The Unsteady F l u x o f Heat and Momentum f r o m a F l a t P l a t e 31 A p p e n d i x B: The C a l i b r a t i o n o f a Hot F i l m T u r b u l e n c e P r o b e . . . 78 A p p e n d i x C: The C h a r a c t e r i s t i c s o f I n t e r n a l l y Heated T h e r m i s t o r s I 17 V L i s t o f T a b l e s I The k i n e m a t i c v i s c o s i t y and P r a n d t l number o f pure w a t e r a f t e r B a t c h e l o r (1976) 9 A l The u n s t e a d y - t o - s t e a d y s t r e s s r a t i o T J / E T , compared t o Cheng and E l l i o t t ' s (1957) f o u r t h o r d e r f o r m u l a and L i g h t h i l l ' s (1954) f i r s t o r d e r formuI a . . . . 48 A l l A c o m p a r i s o n o f t h e d e l e t e d t e r m 0 a g a i n s t t h e r e t a i n e d t e r m s o f t h e unsteady momentum e q u a t i o n (9a) a t t h e f r e q u e n c y Q=I.O f o r p o s i t i o n s n=0.5, 2.0, and 5.0. The v a l u e s f o r g " ' a r e a p p r o x i m a t i o n s . 4 9 Bl Thermal p r o p e r t i e s o f t h e hot f i l m p r obes d e r i v e d f r o m f i t t i n g t h e t h e r m a l model t o t h e s t e a d y c a I i b r a t i o n d a t a 94 C i The p a r a m e t e r v a l u e s a s s o c i a t e d w i t h t h e t h e r m i s t o r s examined. The d i m e n s i o n a l d a t a f o r Lange's t h e r m i s t o r s a r e nominal v a l u e s . The c o n d u c t i v i t y o f p a r a l e n e - c (X| f o r Lange's t h e r m i s t o r s ) i s a f t e r A l p a u g h and Morrow (1 9 7 4 ) . The v i s c o s i t y ( v ) , t h e d i f f u s i v i t y ( K ) , and t h e P r a n d t l number ( P r ) a r e a f t e r B a t c h I o r (1970, p. 5 9 7 ) . The c o n d u c t i v i t y o f g l a s s (A| f o r t h e a u t h o r s ' t h e r m i s t o r s ) i s a f t e r Lueck, e t a I . (1977) 143 V i L i s t of Figures I. Diagrammatic summary of oceanic turbulence sensors . . . . 3 Appendix A I. A sketch of the i d e a l i z e d p l a t e showing the c o o r d i n a t e system employed f o r the analyses. The s u r f a c e temperature i s f i x e d a t 0, a constant 39 2. The real and imaginary components of the boundary layer p e r t u r b a t i o n v e l o c i t y p r o f i l e s a t values of the frequency parameters fi=0.0, 0.25, 1.0 and 5.29. The imaginary component of g' i s zero f o r Q=0 44 3. A comparison between the computed p e r t u r b a t i o n v e l o c i t y p r o f i l e and L i g h t h i l l ' s p r o f i l e at the " c r i t i c a l " frequency ft=0.588 45 4(a). A comparison of the unsteady-to-steady l o c a l s u r f a c e s t r e s s r a t i o ( s o l i d curves) a g a i n s t L i g h t h i l l ' s r e s u l t (dashed curves) and Cheng and E l l i o t t ' s r e s u l t s ( s o l i d dots f o r the magnitude and s o l i d t r i a n g l e s f o r the phase) 47 4(b). The unsteady-to-steady t o t a l s t r e s s r a t i o f o r a p l a t e of length L 51 5. The p r o f i l e of the f u n c t i o n t h a t generates the unsteady anemometric e f f e c t , f o r Prandtl numbers I and 8 at fi=0 (dashed l i n e ) and fj= I ( s o l i d L i n e ) . The phase i s the same f o r Pr=l and Pr=8 58 6(a). The l o c a l p e r t u r b a t i o n heat f l u x response t o speed modulations, normalized by the steady heat f l u x ; l o g a r i t h m i c p l o t 60 6(b). The l o c a l p e r t u r b a t i o n heat f l u x response t o speed modulations, normalized by the steady heat f l u x ; l i n e a r p l o t d e t a i l i n g the heat f l u x maxima 61 6 ( c ) . The l o c a l p e r t u r b a t i o n heat f l u x response t o speed modulations, normalized by the steady heat f l u x ; a frequency scaled v e r s i o n of (a) 62 7(a). The p e r t u r b a t i o n - t o - s t e a d y t o t a l heat f l u x r a t i o f o r speed modulations; unsealed frequency a x i s 64 v i i 7 ( b ) . The p e r t u r b a t i o n - t o - s t e a d y t o t a l h eat f l u x r a t i o for. speed m o d u l a t i o n s ; s c a l e d f r e q u e n c y a x i s 65 8(a). The l o c a l h e a t f l u x r e s p o n s e t o a t e m p e r a t u r e m o d u l a t i o n o f t h e ambient f l u i d , n o r m a l i z e d by i t s q u a s i - s t e a d y v a l u e ; magnitude and phase 67 8(b). The l o c a l heat f l u x r e s p o n s e t o a t e m p e r a t u r e m o d u l a t i o n o f t h e ambient f l u i d , n o r m a l i z e d by i t s q u a s i - s t e a d y v a l u e ; magnitude o n l y p l o t t e d a g a i n s t a s c a l e d f r e q u e n c y a x i s . . .' 68 9(a). The t o t a l heat f l u x r e s p o n s e t o a t e m p e r a t u r e m o d u l a t i o n o f t h e ambient f l u i d , n o r m a l i z e d by i t s q u a s i - s t e a d y v a l u e ; u n s e a l e d f r e q u e n c y a x i s 70 9(b). The t o t a l h eat f l u x r e s p o n s e t o a t e m p e r a t u r e m o d u l a t i o n o f t h e a m b i e n t f l u i d , n o r m a l i z e d by i t s q u a s i - s t e a d y v a l u e ; s c a l e d f r e q u e n c y a x i s 71 A p p e n d i x B I. A s k e t c h o f p r o b e s V—31 and V-38, t h e i r r e l e v a n t d i m e n s i o n s and t h e c o o r d i n a t e system employed i n t h e ana I y s e s 82 2(a). The p r o b e s ' t h e r m a l c o n d u c t a n c e Q/AT v e r s u s R ? f o r V-31 83 e 2(b). The p r o b e s ' t h e r m a l c o n d u c t a n c e Q/AT v e r s u s R ? f o r V-38. 84 e 3. The u n s t e a d y p r o b e v o l t a g e per a m p l i t u d e of speed m o d u l a t i o n i n a r b i t r a r y u n i t s a g a i n s t t h e f r e q u e n c y o f t h e speed m o d u l a t i o n i n H e r t z . The l i n e s a r e c a s u a l f a i r i n g t o t h e d a t a 87 4. The e l e c t r i c a l a n a l o g u e t o t h e s t e a d y t h e r m a l model o f t h e p r o b e s 89 5(a). The p r o b e s ' s t e a d y t h e r m a l r e s i s t a n c e AT/Q v e r s u s R The s o l i d l i n e s r e p r e s e n t t h e l e a s t s q u a r e f i t o f t h e t h e r m a l model t o t h e d a t a . The q u a l i t y o f t h e f i t i s i n d i c a t e d n ear t h e t o p o f t h e f i g u r e ; f o r V-31 92 5(b). The p r o b e s ' s t e a d y t h e r m a l r e s i s t a n c e AT/Q v e r s u s R The s o l i d l i n e s r e p r e s e n t t h e l e a s t s q u a r e f i t o f t h e t h e r m a l model t o t h e d a t a . The q u a l i t y o f t h e f i t i s i n d i c a t e d near t h e t o p o f t h e f i g u r e ; f o r V-38 93 v i i i 6. The p r o b e s ' u n s t e a d y t h e r m a l c o n d u c t a n c e s c a l e d by ,U 0 5 and t h e f r a c t i o n a l speed a m p l i t u d e e v e r s u s t h e f r e q u e n c y p a r ameter P. = a)L/3U 0. The s o l i d l i n e i s t w o - t h i r d s o f t h e u n s t e a d y -t o - s t e a d y s u r f a c e s t r e s s r a t i o a f t e r Lueck (1979). The dashed l i n e i n d i c a t e s t h e q u a l i t y o f t h e f i t i f t h e f r e q u e n c y had been s c a l e d by oiL'/3U . L' i s shown i n F i g u r e 1 110 App e n d i x C 1. ( a ) A c r o s s - s e c t i o n o f t h e model s p h e r i c a l t h e r m i s t o r and (b) a c r o s s - s e c t i o n o f t h e model p l a t e t h e r m i s t o r 122 2. Heat f l u x v e r s u s t e m p e r a t u r e f o r v a r i o u s speeds f o r Lange's 0.051 x 0.051 x 0.0025 cm p a r a l e n e - c c o a t e d f l a k e t h e r m i s t o r 127 i 3. Thermal r e s i s t a n c e (T-9)/Q v e r s u s U - 2 f o r Lange's 0.051 x 0.051 x 0.0025 cm p a r a l e n e - c c o a t e d f l a k e t h e r m i s t o r . -. 128 4. Heat f l u x v e r s u s t e m p e r a t u r e f o r v a r i o u s speeds f o r Lange's p a r a l e n e - c c o a t e d m i c r o - b e a d t h e r m i s t o r 130 5. Thermal r e s i s t a n c e (T-8)/Q v e r s u s t h e i n v e r s e N u s s e l t number o f Lange's p a r a l e n e - c c o a t e d m i c r o - b e a d t h e r m i s t o r . The N u s s e l t number i s a f t e r e q u a t i o n 3 and t h e p a r a m e t e r s i n t a b l e C l 131 6. Back l i g h t e d m i c r o g r a p h of t h e a u t h o r s ' g l a s s c o a t e d m i c r o b e a d s and a s k e t c h o f t h e t h e r m i s t o r s m o u n t i n g f o r k s 133 7 ( a ) . Heat f l u x v e r s u s " t e m p e r a t u r e f o r v a r i o u s f l o w speeds f o r t h e r m i s t o r number 9 134 7 ( b ) . Heat f l u x v e r s u s t e m p e r a t u r e f o r v a r i o u s f l o w speeds f o r t h e r m i s t o r number 10 135 8 ( a ) . Thermal r e s i s t a n c e (T-6)/Q v e r s u s t h e i n v e r s e N u s s e l t number f o r t h e r m i s t o r number 9. The N u s s e l t number i s a f t e r e q u a t i o n 3 and t h e pa r a m e t e r s i n t a b l e C l 136 i X 8(b). Thermal r e s i s t a n c e (T-9)/Q versus the inverse Nusselt number f o r t h e r m i s t o r number 10. The Nusselt number i s a f t e r equation 3 and the parameters in t a b l e Cl 137 9. Speed s e n s i t i v i t y per degree of overheat f o r the authors' g l a s s coated t h e r m i s t o r number 10 and Lange's paralene-c coated f l a k e t h e r m i s t o r . The curves are based on the observed thermal r e s i s t a n c e s given by equations 8 and 9 145 0. R e l a t i v e speed-to-temperature s e n s i t i v i t y per degree of overheat f o r the authors' g l a s s coated t h e r m i s t o r number 10 and Lange's paralene-c coated f l a k e t h e r m i s t o r . The curves are based on the observed thermal r e s i s t a n c e s given by equations 8 and 9 146 X AcknowIedqements P r o f e s s o r T.R. Osborn has p r o v i d e d encouragement, g u i d a n c e and s u p p o r t t h r o u g h o u t my g r a d u a t e s t u d i e s . I would a l s o l i k e t o e x p r e s s a p p r e c i a t i o n t o t h e f o l l o w i n g : - Dr. A. G a r g e t t , Dr. P. Nasmyth and Mr. G. Chase o f t h e Ocean M i x i n g Group a t t h e I n s t i t u t e o f Ocean S c i e n c e s , P a t r i c i a Bay and Mr. D. Evans, f o r m e r l y o f t h e Defense R e s e a r c h E s t a b l i s h m e n t , P a c i f i c , f o r p r o v i d i n g t h e hot f i l m c a l i b r a t i o n d a t a and needed i n f o r m a t i o n . - Dr. E. Lange o f t h e S c r i p p s I n s t i t u t e o f Oceanography f o r p r o v i d i n g h i s t h e r m i s t o r c a l i b r a t i o n d a t a . - P r o f e s s o r s P.H. L e B l o n d and L. Mysak f o r t h e i r comments and d i s c u s s i o n s . - Ms. P. L u s t f o r her d i l i g e n t t y p i n g o f t h i s m a n u s c r i p t . The N a t i o n a l R e s e a r c h C o u n c i l o f Canada has s u p p o r t e d me p e r s o n a l l y f o r f o u r y e a r s o f my G r a d u a t e S t u d i e s a t U.B.C. I. 1ntroduction This t h e s i s discusses some s a l i e n t features of heated anemometry and thermometry in water with emphasis on oceanic applications,. The discussion separates the hydrodynamic c h a r a c t e r i s t i c s of probes from t h e i r associated e l e c t r o n i c s . Many pertinent features of a probe's behaviour can now be deduced unobscured by the c h a r a c t e r i s t i c s of the probe's e l e c t r o n i c s . The three appendices in t h i s t h e s i s are manuscripts that have been submitted to refereed journals and deal with three d i f f e r e n t . b u t related aspects of the subject of thermometry and heated anemometry. This t h e s i s i n t e r - r e l a t e s and summarizes the contents of the three manuscripts. - 2 -2. Background The d i r e c t o b s e r v a t i o n o f t u r b u l e n c e i s i m p o r t a n t t o oceano-graphy. O b s e r v a t i o n s o f t h e m i x i n g o f momentum, s a l i n i t y and t e m p e r a t u r e between w a t e r masses c o u l d c l a r i f y t h e d e t a i l s of f r o n t a l dynamics ( G a r g e t t , 1978) and t h e r o l e of t u r b u l e n c e i n g e n e r a l c i r c u l a t i o n s . The measurement o f t u r b u l e n c e on s c a l e s s m a l l enough t o r e s o l v e t h e d i s s i p a t i o n s p e c t r u m i s p r e s e n t l y t h e o n l y p r a c t i c a l way t o e s t i m a t e t h e v e r t i c a l R e y n o l d s s h e a r s t r e s s i n open ocean s h e a r f l o w s ( C r a w f o r d and O s b o r n , 1978). The c o r r e l a t i o n o f o r t h o g o n a l v e l o c i t y f l u c t u a t i o n s i n r i v e r e s t u a r i e s c o u l d y i e l d t h e i n t e r n a l s t r e s s , an i m p o r t a n t but so f a r unmeasured t e r m i n t h e b a l a n c e o f f o r c e s . An. u n d e r s t a n d i n g o f t h e h y d r o d y n a m i c s of t h e employed s e n s o r s i s c r u c i a l t o t h e s u c c e s s f u l i n t e r p r e t a t i o n o f t u r b u l e n c e d a t a . I n -v a r i a b l y a s e n s o r must be mounted on a p l a t f o r m such as a towed body ( G a r g e t t , 1978; G r a n t , S t e w a r t and M o i l l i e t , 1962), a s e l f p r o p e l l e d v e h i c l e ( G r a n t , M o i l l i e t and V o g e l , 1968), a f r e e f a l l i n g package ( O s b o r n , 1978), a t e t h e r e d f a l l i n g package (Mamorino and C a l d w e l l , 1978) o r a p l a t f o r m a t t a c h e d t o t h e bottom o r t h e s h o r e . The m o t i o n s o f t h e p l a t f o r m a r e a l s o i m p o r t a n t f o r d a t a i n t e r p r e t a t i o n b u t a r e beyond t h e scope o f t h i s t h e s i s . A d i a g r a m m a t i c summary o f t u r b u l e n c e s e n s o r s t h a t have been used f o r o c e a n i c measurements i s shown i n F i g u r e I. A summary o f s a l i n i t y s e n s o r s has been g i v e n by Gregg and P e d e r s o n ( 1 9 7 8 ) . C y l i n d r i c a l f i l m and w i r e s e n s o r s , a b s e n t from F i g u r e I, have so f a r o n l y seen use i n l a b o r a t o r i e s ( T a n - a t i c h a t , Nagib and P l u i s t e r , 1973 and G u s t , 1976). - 3 -Figure I. Diagrammatic summary of oceanic turbulence sensors. (I) Osborn, 1978. Mamorino and C a l d w e l l , 1978. Gargett, 1978. Gregg and Pederson, 1978. Lange, personal communication. Belyaev, Gezentsvey, Monin, Ozmidov and Paka, 1975. John Woods, personal communication. Crawford and Osborn, 1978. (2) (3) (4) (5) (6) (7) (8) The following c h a r a c t e r i s t i c s of thermometers and anemometers used in water must be examined to determine t h e i r s u i t a b i l i t y : ( 1 ) environmental a d a p t a b i l i t y (2) e l e c t r i c a l i n s u l a t i o n from the f l u i d (3) s p a t i a l r e s o l u t i o n (4) frequency response (5) s e n s i t i v i t y to temperature and v e l o c i t y Environmental a d a p t a b i l i t y includes the requirements of mechanical strength, r i d i g i d y and longevity as well as the a b i l i t y to shed contaminants l i k e plankton and d i r t . Choosing a sensor that meets the requirements of environmental a d a p t a b i l i t y is an a r t as much as i t is a science. For example the Ocean Mixing Group of the I n s t i t u t e of Ocean Sciences at P a t r i c i a Bay, Canada, has found that t h e i r cone shaped hot f i l m sensors remain r e l a t i v e l y free from contaminants and can be success-f u l l y back flushed, in s i t u , to remove larger plankton such as j e l l y - f i s h . The sensing element of heated anemometers and thermometers is a conductor with a temperature c o e f f i c i e n t of e l e c t r i c a l resistance. The sensing element can be a metal-oxide core or a th i n metal f i l m . Suitable e l e c t r o n i c s measure the temperature and the heating of the sensing element. Spurious signals are introduced by conduction through sea water i f the sensing element is not e l e c t r i c a l l y insulated. The s p a t i a l r e s o l u t i o n of a probe is primarily governed by i t s s i z e . The turbulence cannot be f u l l y resolved if i t s s p a t i a l scale is comparable to or smaller than the probe. A deta i l e d discussion of the spa t i a l r e s o l u t i o n of probes is outside the scope of t h i s t h e s i s . A spa t i a l structure of wavelength X advected past a probe at speed U has, - 5 -f o r t h e p r o b e , an a p p a r e n t f r e q u e n c y U/X. If U/X exceeds t h e f r e q u e n c y r e s o l u t i o n o f t h e probe t h e n t h e r e i s an a p p a r e n t s p a t i a l r e s p o n s e I i m i t a t i o n c a u s e d by f r e q u e n c y r e s p o n s e I i m i t a t i o n s . Frequency r e s p o n s e i s d i s c u s s e d i n C h a p t e r 4. D e t e r m i n i n g s e n s i t i v i t y t o t e m p e r a t u r e o r v e l o c i t y i s a fundamental p r o b l e m o f thermometry and h e a t e d anemometry. Temperature s e n s o r s and heated anemometers a r e s i m u l t a n e o u s l y s e n s i t i v e t o both t e m p e r a t u r e and v e l o c i t y . Thermometers have a s m a l l amount of i n t e r n a l h e a t i n g because o f t h e e l e c t r i c c u r r e n t p a s s i n g t h r o u g h them and c o n s e q u e n t l y have some s e n s i t i v i t y t o s p e e d . The l e v e l o f i n t e r n a l h e a t i n g t h a t c a n be t o l e r a t e d i s not c l e a r . G r e g g , Meagher, P e d e r s o n and Aagaard (.1978) w r i t e : "A t h i r d , more o b v i o u s , a p p r o a c h [to increase the sensitivity of thermistors] would be t o i n c r e a s e t h e c u r r e n t t h r o u g h t h e t h e r m i s t o r s ; however, by i n c r e a s i n g t h e power d i s s i p a t i o n , t h i s i n c r e a s e s t h e v e n t i l a t i o n n o i s e [sensitivity to speed]. As n e i t h e r t h e v e l o c i t y s p e c t r a nor t h e c o e f f i c i e n t s f o r c o m p u t i n g t h e c o o l i n g have been w e l l known, we have f o l l o w e d t h e c o n s e r v a -t i v e p o s i t i o n o f Cox, e t a I (1969) i n m a i n t a i n -i n g a low power d i s s i p a t i o n , i . e . 4 t o 13 uw [10~6 watts]. With f u r t h e r i n f o r m a t i o n i t may be p o s s i b l e s a f e l y t o i n c r e a s e t h e power." The c o n t e n t s of t h e s q u a r e b r a c k e t s have been added f o r c l a r i t y . The h e a t f l u x from g r o s s l y heated s e n s o r s r e s p o n d s t o v a r i a t i o n s of t h e ambient f l u i d ' s speed as w e l l as i t s t e m p e r a t u r e . I n c r e a s i n g t h e s e n s o r ' s t e m p e r a t u r e i n c r e a s e s t h e r a t i o o f s p e e d - t o - t e m p e r a t u r e s e n s i t i v i t y . E v e n t u a l l y t h e f o r m a t i o n o f b u b b l e s and a s h o r t e n i n g o f t h e o p e r a t i n g l i f e l i m i t t h e a l l o w a b l e o v e r h e a t . G a r g e t t (1978) comments on t h e problems o f l i m i t e d o v e r h e a t : - "6 -" . . . a second p r o b l e m a r i s e s because t h e ho t f i l m i s n o t , i n f a c t , v e r y hot (measure-ments were t a k e n w i t h an o v e r h e a t o f 20 C, a v a I u e c h o s e n • a s a compromise between v e l o c i t y s e n s i t i v i t y and f i l m c o r r o s i o n , both of which i n c r e a s e w i t h i n c r e a s i n g o v e r h e a t ) , and as such i t i s s e n s i t i v e t o t e m p e r a t u r e as w e l l as v e l o c i t y . A s i n g l e t e s t o f t h e hot f i l m probe i n t h e plume t a n k used t o c a l i b r a t e c o l d f i l m t e m p e r a t u r e p r o b e s [ F a b u l a , 1968] s u g g e s t s t h a t t h i s t e m p e r a t u r e s e n s i t i v i t y may be h i g h e r t h a n would be e x p e c t e d a t t h e h i g h - f r e q u e n c y end of t h e v e l o c i t y d i s s i p a t i o n r a n g e , h i g h enough t o be a s i g n i f i c a n t problem i n most r e g i o n s o f t h e main t h e r m o c l i n e , where m i c r o s c a l e t e m p e r a t u r e f l u c t u a t i o n s a l m o s t a l w a y s accompany d i s s i p a t i v e v e l o c i t y e v e n t s . U n t i l t h i s p r o b l e m r e c e i v e s f u r t h e r a t t e n t i o n , t h e v e l o c i t y d a t a from o u r tows i n t h e main t h e r m o c l i n e remain s u s p e c t . " - 7 -3. Steady Heat Transfer Much work has been published on the steady heat t r a n s f e r from ideal bodies, that i s , probes with a constant temperature wetted surface. Real probes have a coating that insulates the sensing element and some have a substrate that conducts heat away from, the sensing element. However, the concepts of heat t r a n s f e r from ideal probes can be f r u i t f u l l y applied to real probes used f o r oceanic work to deduce t h e i r steady and quasi-steady c h a r a c t e r i s t i c s . 3.1 IdeaI Probes The rate of heat t r a n s f e r from a constant temperature wetted surface (an ideal probe) to i t s fJ u i d environment depends on (Hinze, (1) the flow v e l o c i t y (2) the temperature d i f f e r e n c e between the surface and the ambient f l u i d (over-heat A8) and (3) the physical properties of the f l u i d . The rate of heat t r a n s f e r from a given constant temperature wetted surface is c haracterized by a dimens ion I ess parameter c a l l e d the Nusselt number (Nu). The dependence of the-Nusselt number on other dimension I ess groups of importance may be expressed by (Hinze, 1975) 1975) Nu = f {R e' C A9 P , Z, 0} ( I ) where R = e v UL Pr = V / K Gr = qL3SA9 v 2 A6= temperature d i f f e r e n c e between the wetted surface and the ambient f l u i d (overheat), - 8 -L = a relevant dimension of the probe, K = thermal d i f f u s i v i t y of the f l u i d , v = kinematic v i s c o s i t y of the f l u i d , p = density of the f l u i d , C = s p e c i f i c heat of the f l u i d , S = c o e f f i c i e n t of thermal expansion, g = g r a v i t a t i o n a l a c c e l e r a t i o n , U = speed of the f l u i d , 1 T = absolute temperature of the f l u i d , Z = a parameter group representing the e f f e c t of.geometry, tj) = angle between the flow d i r e c t i o n and the axis of the probe ( i f i t has one). The term U2/C_A9 represents the r e l a t i v e importance of flow induced heating (stagnation remperature, U2/C ). which equals ^ 1/4000 C for t y p i c a l f l u i d speeds of 100 cm/sec. The stagnation temperature is possibly important for thermometry CA9 <V IO - 3 °C) but not for anemometry where A9 ^ I0°C. C o l l i n s and Williams (.1959) showed that, f or wires in a i r , buoyancy e f f e c t s (thermal convection) may be neglected whenever G < R 3 r e If the above c o n d i t i o n also holds for water and other shapes then buoyancy e f f e c t s are small whenever L)3 o 500 — > I in units of cm/sec and C A9 a c o n d i t i o n that i s always well s a t i s f i e d in oceanic work. The term A9/T is important i f the overheat is s u f f i c i e n t to change e i t h e r the Reynolds number or the Prandtl number. For water both the Prandtl - 9 -number and the kinematic v i s c o s i t y depend strongly on temperature (table I), consequently A0/T may be an important dimensionless group. However, the author is unaware of any attempt to r e l a t e the Nusselt number to A6/T in water. For anemometry in water the Nusselt number can be expressed as Aft Nu = f {R g, Pr, , Z, <}>} (2) and fur t h e r discussion is limited to Nu - f {-R Pr, Z} (3) when Re>>I and I $ Pr $ 10. Kinematic v i s c o s i t y T C cm 2/sec Prandtl Number 0 1.787 x IO - 2 13.44 5 1.514 1 0 1 .304 9.45 15 1 . 138 8. 13 20 1 .004 7.07 25 0.894 30 0.802 5.49 35 0.725 40 0.659 4.34 50 0. 554 60 0.474 3.01 70 0.414 80 0.366 2.25 90 0.327 100 0.295 1 .78 TABLE I. The kinematic v i s c o s i t y and Prandtl number of pure water a f t e r Batchelor (1976). The Nusselt number for a sphere and a c y l i n d e r is known from observations. The Nusselt number has been derived a n a l y t i c a l l y and e m p i r i c a l l y for wedges, including a f l a t plate, and for cones. The - 10 -Nusselt number of a constant temperature sphere is (Kramers, 1946) n i s n ^ i 1 Nu = 2.0 + 1.3 Pr 1 3 + 0.66 Pr J R 2- (4) e where the Reynolds number is based on the sphere's diameter. The rate of heat t r a n s f e r to the f l u i d is 0 = i r D X fA0Nu (5) where X ^ is the conductivity of the f l u i d at the sphere's surface and D is the diameter of the sphere. The Nusselt number for two-dimensional flow over a wedge (Fa Ikner-Skan flow) has been tabulated by White (1974) for a wide range of Prandtl numbers and wedge angles. A f l a t plate is 0 34 I l a wedge of zero angle and has a Nusselt number of 0.664 Pr ' Rg The Nusselt number of a two-dimensional stagnation area is .715 Pr' e .37 R g 2. The rate of heat t r a n s f e r from both faces of a wedge is 0 = bXfA9Nu (6) where b is the width of the wedge across the flow. There is a transformation due to Mangier (1948), v a l i d for both incompressible and compressible laminar flow, which converts the axi-symmetric boundary layer equations of a cone into the equations of an equivalent two-dimensional wedge flow. The heat flux from any a x i -symmetric body can, in p r i n c i p l e , be obtained from the heat f l u x of an equivalent two-dimensional wedge. Hinze (1975) gives an extensive discussion on the Nusselt number of a c y l i n d e r . A l l findings are not completely in agreement but a r e l a t i o n s h i p , claimed to be s a t i s f a c t o r y f o r many gases and l i q u i d s , is (Kramers, 1946) ,, n _ 0.20 . D 1/3 D 1/2 Nu = 0.42 Pr + 0.57 Pr R g (7) - I I -The heat f l u x from a c y l i n d e r of length b is 0 = TrbXfA9Nu. The quasi-steady s e n s i t i v i t y of the heat f l u x to speed f o r an ideal probe is 30 R e 30 , A6 , , , ^  , 3Nu _ 3U = LTiR = ( U } A f ( lb ) W R e e b e where the fac t o r TTb is used for c y l i n d e r s , TTD for spheres and the factor b f o r wedges and cones. The quasi-steady s e n s i t i v i t y to temperature o s c i l l a t i o n s of the ambient f l u i d i s 30 1 7 0 ' - A^ ( Trb ) Nu 3(A6) " f b The r a t i o of the quasi-steady speed-to-temperature s e n s i t i v i t y is 30/3U ( A6 . ^ e 3Nu 3Q/3(A6) U Nu 3R g i This r a t i o is A6/2U i f Nu <= R g 2 which is the case f o r wedges and cones but is less than A9/2U for cy l i n d e r s and spheres. For very large Reynolds numbers the r a t i o of the quasi-steady speed-to-temperature s e n s i t i v i t y f o r c y l i n d e r s and spheres appraoches A8/2U asymptotically. Thus the maximum value of the r a t i o of quasi-steady speed-to-temperature s e n s i t i v i t y of heated sensors never exceeds A8/2U. 3.2 Real Probes In p r i n c i p l e , i f the Nusselt number of a probe is known and if i t s temperature and i t s heat f l u x can be measured then the f l u i d ' s speed can be uniquely determined. Two d i f f i c u l t i e s with using the Nusselt number of an ideal probe for a real probe are (aside from geometric imperfections): - 12 -(1) A l l probes are e l e c t r i c a l l y and hence thermally insulated from the f l u i d . The wetted surface is cooler than the e l e c t r i c a l l y heated element and may not be at a constant or uniform temperature even i f the heated element is at a constant and uniform temperature, (2) Cones, wedges and plates may have a substrate that is not completely enveloped by the heated element. If so, there is a heat loss into the substrate. S i m i l a r l y c y l i n d e r s and spheres may have a heat loss into t h e i r supports. In p r i n c i p l e , the heat f l u x from any real probe can be calculated by solv i n g the coupled temperature equations of the boundary layer and the s o l i d probe but, in p r a c t i c e , t h i s would e n t a i l an inordinate amount of e f f o r t . The heat f l u x from any probe can also be determined e m p i r i c a l l y but t h i s by i t s e l f would shed l i t t l e l i g h t on the para-meters governing the heat f l u x unless there is insight to help interpret the r e s u l t s . In p r a c t i c e , only the average temperature of the sensing element and the heat f l u x from i t are known. The temperature of the wetted surface is not d i r e c t l y assessable. Thermal models of real probes that incorporate the concept of a Nusselt number governed heat f l u x from t h e i r wetted surfaces and that also include the thermal e f f e c t s of the coating and the substrate have been developed (appendices B and C). The temperature of the wetted surface is v a r i a b l e but is assumed to be, uniform. With these models a t h e o r e t i c a l functional r e l a t i o n s h i p between the heat flux and the flow can be derived without any knowledge of the probe's e l e c t r o n i c s . These r e l a t i o n s h i p s take the form - 13 -T r = h(R *) (8) 0 e and agree favourably with the a v a i l a b l e c a l i b r a t i o n data of hot f i l m probes (appendix B) and heated thermistors (appendix C).' Regardless of geometry, a constant temperature surface has a heat flux that is e i t h e r a proportional or a linear function of U 2. Tan-Atichat, et al (1973) found i t impossible to f i t the heat flux i versus U 2 c a l i b r a t i o n data of t h e i r 0.015 cm diameter, glass coated, c y l i n d r i c a l , hot f i l m probe to a l i n e a r r e l a t i o n s h i p . They attempted the more general l i n e a r i z a t i o n S- = A + BU n (9) A0 and claimed a best f i t when A = 0 and n = 0.23. Equation 9 is often f i t t e d to c a l i b r a t i o n data from hot wires in a i r but A is never zero and the exponent n appears to be r e s t r i c t e d to 0.45 to 0.52. There is no physical model to j u s t i f y equation 9 when n does not equal The steady c a l i b r a t i o n data of two coni c a l hot f i l m probes (appendix B) i also show that the heat f l u x i s not a proportional function of U 2, i as expected from equation' 6, or even a li n e a r function of U 2. S i m i l a r l y , the c a l i b r a t i o n data of a paralene-c coated plate-shaped thermistor, a paralene-c coated e l l i p s o i d a l microbead thermistor and two g l a s s ' coated microbead thermistors (appendix C) show that the heat flux is not a li n e a r function of U 2. The heat f l u x r e l a t i o n s h i p does not take the form ~ = h(R *) = [A + B R 2 ] _ 1 (J e e (equation 9 with n=4) because of the thermal e f f e c t s of the coating and - 14 -the substrate. The heat flux r e l a t i o n s h i p s are ^ ~ A + — i Q R 2 e for a coated plate shaped thermistor, Q A Nu for a coated e l l i p s o i d a l microbead thermistor and A6 = c + __B 9 " ' .A + R * e for a coated conical f i l m sensor on a substrate. The parameters, A, B and C depend on the probe's geometry, s i z e , coating thickness, the thermal conductivity of the f l u i d , the coating and the substrate, and the f l u i d ' s Prandtl number. The use of a t h i r d parameter for the conical probe model r e f l e c t s the presence of the substrate. With a knowledge of the function h the quasi-steady s e n s i t i v i t y to speed 90/aU, the s e n s i t i v i t y to temperature 8Q/9(A8) and the r a t i o of speed-to-temperature s e n s i t i v i t y are r e a d i l y determined. The coating and the substrate each reduce the quasi-steady s e n s i t i v i t y to speed and to temperature o s c i l l a t i o n s of the ambient f l u i d . The substrate and the coating a l s o reduce the r a t i o of speed-to-temperature s e n s i t i v i t y below i t s maximum value of A0/2U. The e f f e c t of a f i n i t e coating thickness can be very s i g n i f i c a n t in water. At 15 cm/sec a t y p i c a l glass coating of 0.0028 cm on a 0.017 cm diameter microbead thermistor reduces the speed s e n s i t i v i t y by a f a c t o r of 10 and the r a t i o of speed-to-temperature s e n s i t i v i t y by a factor of 3.2 from i t s value in the absence of a coating. This e f f e c t increases with increasing speed. A t y p i c a l glass coating of I or 2 x 10 - l + cm over a t h i n f i l m probe reduces the temperature of - 15 -the wetted surface by approximately 20%. The glass substrate of a hot f i l m probe can c a r r y about 20% of the t o t a l heat f l u x from the f i l m . The e f f e c t of the coating and the substrate depends strongly on the r e l a t i v e c o n d u c t i v i t y of the f l u i d . There is a strong coating e f f e c t but a moderate substrate e f f e c t on probes in water whereas in a i r the coating e f f e c t is not very important but the substrate e f f e c t is i mportant. A knowledge of a probe's s e n s i t i v i t y to both speed and temperature is c r u c i a l f or oceanic work because temperature a c t i v i t y often accompanies v e l o c i t y a c t i v i t y . Some measurements of the i n t e n s i t y of temperature and v e l o c i t y turbulence are now a v a i l a b l e . System noise and the probe's working l i f e are two other important and s e n s i t i v i t y r e l a t e d considerations. An a p r i o r i evaluation of a sensor's perfor-mance is p o s s i b l e . The expected signal can be used to judge a sensor's s i g n a l - t o - n o i s e r a t i o , i t s signal contamination and i t s performance r e l a t i v e to other sensors. Because the temperature-heat f l u x r e l a t i o n -'ship is s e n s i t i v e to geometry the merits of various coating materials, coating thicknesses, and probe shapes and s i z e s can be determined from i the appropriate function h ( R g 2 ) . Oceanic microscale temperature gradients can, at some times and in some locations, be small enough to p r o h i b i t t h e i r r e s o l u t i o n with the best present techniques (Gregg, et a l , 1978 using thermistors). An increase of the s i g n a l - t o - n o i s e r a t i o is therefore important. Increasing a thermistor's current increases i t s s i g n a l - t o - n o i s e r a t i o but a l s o increases i t s s e n s i t i v i t y to v e l o c i t y f l u c t u a t i o n s . Because the heat f l u x r e l a t i o n s h i p - 16 -is independent of the level of s e l f heating i t applies to thermometers as well as anemometers. The l i m i t to which the current through a thermistor used for thermometry can be increased is governed by i t s r e l a t i v e velocity-to-temperature s e n s i t i v i t y , the r e l a t i v e v e l o c i t y -to-temperature a c t i v i t y in the f l u i d being probed and the level of contamination of the temperature signal that one is w i l l i n g to accept. Circumstances may vary greatly. By using the observed values f o r the rates of temperature and k i n e t i c energy d i s s i p a t i o n in the ocean i t is shown in appendix C that temperature measurements made with glass coated microbead thermistors may be contaminated by v e l o c i t y in the mixed layer and in other regions of high v e l o c i t y but low temperature a c t i v i t y when the over-heat A 8 is only ^ICT 2 °C. Heated anemometers, such as grossly heated th i n f i l m probes and thermistors, are also s e n s i t i v e to tempera-ture. Several people have suggested warmed thermistors for high frequency thermometry. However, because even low levels of internal heating may cause v e l o c i t y contamination i t is not l i k e l y that even moderately heated probes are p r a c t i c a l f o r thermometry in the ocean. As a f i n a l comment to t h i s s e c t i o n , c a l i b r a t i o n data is often presented in terms of a bridge voltage and an over-heat r a t i o . This makes c a l i b r a t i o n comparisons d i f f i c u l t . To determine the heat f l u x versus flow r e l a t i o n s h i p one then needs some knowledge of the probe's e l e c t r o n i c s and the temperature-resistance c a l i b r a t i o n of the probe. Often the needed information is not published. This is acceptable i f the only purpose of p u b l i c a t i o n is t o demonstrate that a given operational system works but is not acceptable i f the purpose is to - 17 -demonstrate the fundamental c h a r a c t e r i s t i c s of a probe. Understanding of probe c h a r a c t e r i s t i c s might proceed more s w i f t l y i f c a l i b r a t i o n data were published in terms of relevant parameters such as over-heat A8, heat f l u x , Reynolds numbers, Prandtl numbers, and sensor dimensions instead of bridge voltages and over-heat r a t i o s . - 18 -4. Unsteady Heat Transfer The Nusselt number is a steady heat t r a n s f e r concept. The response of sensors is not governed by the Nusselt number when the viscous boundary layer is unsteady. An understanding of the unsteady c h a r a c t e r i s t i c s of ideal probes can be used advantageously to under-stand the unsteady behaviour of real probes. 4. I Idea I Probes : The behaviour of the unsteady viscous boundary layer over a f l a t plate has been examined by the author (appendix A), Cheng and E l l i o t t (1957), H i l l and Stenning (I960), L i g h t h i l l (1954), Ostrach (1955), Rott and Rosenzweig (I960) and others. Only Ostrach (1955) and the author have extended t h e i r examination to the unsteady heat flux from a plate. Ostrach's r e s u l t s are limited to very low frequencies whereas the author's r e s u l t s apply to a l l frequencies. L i g h t h i l l (1954) examined the unsteady viscous boundary layer and the heat f l u x over the frontal surface of a c y l i n d e r (two-dimensional stagnation flow). The author (appendix B) has examined the unsteady boundary layer and the heat f l u x over a cone of half angle 15° under axi-symmetric flow. The boundary layer over a f l a t plate of length L departs from i t s quasi-steady behaviour when the frequency parameter uL/U>0.3 where OJ is the angular frequency in rad i ans/second. Any c r i t e r i o n for unsteady behaviour is (necessarily) a r b i t r a r y . The frequency parameter chosen is that f o r which the unsteady surface shear st r e s s on any part of the surface exceeds the quasi-steady stress by more than 20%. The boundary layer over a cone of half angle less than approximately 20° - 19 -becomes unsteady when <JOL/U>0.9 and becomes unsteady over the front of a c y l i n d e r of diameter D when u)D/U>5. An examination of the unsteady f i r s t order perturbation equations of an ideal constant temperature sensor, l i k e the plate discussed in appendix A, is i n s t r u c t i v e . The equations are 3 U | 3u j . 3u 3u a 2u! icou. + u - — + v - — + u, -~- + v. ~ ~ = lull + v - •. 2 CO) I o 3 x o 3 y I 3 x I 3 y o 3y^ . U j = V | = 0 at y = 0 and x > 0 u. = U • . at y -»- °= 3T. 3T. 3 2T. 3T 3T . - r . I . I I 0 0 / i i \ I 0)T. t U ~ t V K . o ~ - U i 7 V I (II) I o 3x o 3y 3y^ I 3x I 3y Tj = 0 at y = 0, °° and x > 0 and iw6 + u — + v —— - K r — r = 0 (12) o 3x o 3y 3y^ 6 = 0 a t y = 0 , x > 0 6 = I at y = oo from equations 5(.al, 16 and 17 in appendix A, where (u,v) = Cu , v ) + ee'a)"*"(u, ,v, ) o o I I T = T + ee T, + ve 6 \o ' I U = U (I + e e i w + ) o is the free stream speed, T(x,°0 = ye'u>"'" is the free stream temperature and the constants y and e are small compared to unity. The boundary layer v e l o c i t y is represented by the steady f i e l d ( U Q , V ) and a small perturba-t i o n ee' < 1 )^(u, ,v. ) due to a small f l u c t u a t i o n of the main stream ell e'W"^. I I o - 20 -The steady temperature f i e l d due to steady streaming and the constant surface temperature is represented by T , ee l a 3^T| is the temperature f i e l d due to the unsteady v e l o c i t y perturbation, and yee'"^ represents the temperature f i e l d due to a small independent perturbation of the main stream temperature ye'10"*". Because of the non-homogeneous term iujUo in equation 10, which represents the unsteady pressure gradient in the f l u i d , the unsteady boundary layer v e l o c i t y ( U | , V | ) near the plate increases in magnitude and advances in phase from i t s quasi-steady value. A comparison of equations II and 12 makes i t quite c l e a r that the anemometric response (TjI must be quite d i f f e r e n t from the tempera-ture response (6). Equation 10 is non-homogeneous but has homogeneous boundary conditions. The heating source that generates the.perturbation temperature f i e l d T^ i s not at the boundaries but i s i n t r i n s i c a l l y d i s t r i b u t e d throughout the e n t i r e boundary layer and r e s u l t s from the advection of the steady boundary layer temperature gradient OT /3x, o 3T_/3y) by the unsteady boundary layer v e l o c i t y ( . U j , V | ) . This heating source U|*vT is a function of Prandtl number, frequency and p o s i t i o n . Proper frequency response c a l i b r a t i o n requires a f a i t h f u l simulation of the f o r c i n g term CUj'VT ). which can be provided by v i b r a t i n g the probe l o n g i t u d i n a l l y in a steady flow (.Lighthill, 1954). C a l i b r a t i o n s in a steady flow made by i n t e r n a l l y modulating the probe's temperature by, say, an unsteady e l e c t r i c current (Freymuth, 1977, 1978; Ling, 1955; Tan-atichat, 1973; Gust, 1976 and others) can infer only the quasi-steady response because the unsteady boundary layer is absent ( d e t a i l s are discussed in appendix A). - 21 -The temperature response equation (II) is homogeneous and driven by a non-homogeneity at the boundary y = °°. This heating source is outside of the boundary layer and can only penetrate, the boundary layer by steady advection and by d i f f u s i o n . Independent of the level of heating, the anemometric response bandwidth is wider than the temperature response bandwidth because (I) the heating source U|"vT f o r the anemometric response increases in magnitude whereas the heating source f o r the temperature response is constant and (2) the heating source for the anemometric response is i n t r i n s i c to the boundary layer but the heating source for the temperature response is external to the boundary layer. The -3 dB response bandwidth for a f l a t plate in the Prandtl number range 0.7 < Pr < 15 is U 3.7 Cl + 2/Pr) (I + / P r ) 2 3 dB for the anemometric response and u_L U '= 0.56 Pr • 3 8 3 dB for the temperature response. Wedges and cones should have a wider bandwidth because t h e i r boundary layers are thinner. L i g h t h i l l (1954) found that the -3 dB bandwidth of the heat f l u x from the front of a c y l i n d e r in response to speed o s c i l l a t i o n s is approximately O J D / U = 5 at a Prandtl of 0.7 ( a i r ) . At a Prandtl number of 0.7 the bandwidth of a c y l i n d e r is 1.7 times larger than the bandwidth of a plate ( i f D=L) - 22 -The above bandwidths are independent of the level of steady heating but apply s t r i c t l y only to constant temperature surfaces. For v a r i a b l e temperature probes the thermal i n e r t i a of the probe may also be impor-tant and further l i m i t the response bandwidth. For e i t h e r case, constant temperature or v a r i a b l e temperature probes, the anemometric bandwidth must be larger than the temperature response bandwidth because of the boundary layer's behaviour. However, the d i f f e r e n c e between the two responses wiI I be less pronounced for v a r i a b l e temperature probes and the amount of t h i s d i f f e r e n c e w i l l depend on the r e l a t i v e importance of the probe's i n e r t i a . 4.2 Real Probes The high frequency heat f l u x from the two conical probes examined in appendix B d i f f e r s s i g n i f i c a n t l y from i t s quasi-steady value. The heated f i l m on these probes spans the e n t i r e circumference at a position "downstream" from the apex. The f i l m is narrow compared to i t s distance from the apex. These probes have a frequency response which at f i r s t seems s u r p r i s i n g ; the response r i s e s with increasing frequency. Because of the film ' s aspect r a t i o the unsteady heat f l u x is proportional to the viscous surface s t r e s s over the f i l m which increases in magnitude and phase advance with increasing.vaIue of the frequency parameter = u)L/3U. Increasing the r e l a t i v e width of the f i l m by ei t h e r moving the f i l m towards the apex or by widening the f i l m would tend to diminish the r i s e of the frequency response. If the e n t i r e cone of 15° half-angle were covered by the f i l m then the frequency response should be s i m i l a r to the response of a f l a t plate. - 23 -S i g n i f i c a n t departure from quasi-steady behaviour of the heat flux from a wedge shaped sensor in a i r at frequencies where the boundary layer must s t i l l be quasi-steady have been reported by Bel I house and Schultz (1967). This e f f e c t is caused by the substrate. The heat flux into the substrate must, somewhere, pass through the probe's boundary layer to enter the ambient f l u i d . If the substrate's conductivity is large compared to the f l u i d ' s c o n d u c t i v i t y , the substrate's heat flux is s e n s i t i v e to the flow because of the boundary layer in i t s path. The substrate's unsteady heat f l u x in response to speed o s c i l l a t i o n s can decrease with increasing frequency because of the substrate's f i n i t e thermal d i f f u s i v i t y . If the substrate's con d u c t i v i t y is not large compared to the f l u i d ' s conductivity then the heat f l u x through the substrate is nearly independent of the flow and does not respond to speed modulations. The heat f l u x through the substrate in response to o s c i l l a t i o n s of the ambient f l u i d ' s temperature on the other hand, must decrease with increasing frequency regardless of the value of the s u b s t r a t e - t o - f I u i d c o n d u c t i v i t y r a t i o . Low frequency attenuation of the heat f l u x in response to speed o s c i l l a t i o n s is evidently small in water but large in a i r . The unsteady c a l i b r a t i o n data of the two conical hot f i l m probes discussed in appendix B show very l i t t l e low frequency attenuation. Frequency response c a l i b r a t i o n s in water by Nowell (1974) do not show any low frequency attenuation. Because the response bandwidth is wider for speed than for temperature and because the substrate may a f f e c t a low frequency attenuation of the temperature response but not the speed response (in water), the r a t i o of temperature-tb-veIocity s e n s i t i v i t y is highest - 24 -at zero frequency. An upper bound for the temperature contamination of a measured v e l o c i t y spectrum can be c a l c u l a t e d with the temperature spectrum and the quasi-steady s e n s i t i v i t y r a t i o . Conversely, a lower bound for the v e l o c i t y contamination of the temperature spectrum can be c a l c u l a t e d with the quasi-steady value of the s e n s i t i v i t y r a t i o . The high frequency response bandwidth of t h i n f i l m anemometers is given approximately by I O L / U ^ I and depends somewhat on the Prandtl number. If one assumes that a probe traverses through a "frozen" turbulence f i e l d at a speed U then the largest wave number o that can be resolved because of frequency response l i m i t a t i o n s is 0 * U which implies that at ^ I. Thus the scale s i z e corresponding to frequency response l i m i t a t i o n s is a l s o approximately the scale s i z e corresponding to s p a t i a l response l i m i t a t i o n s . - 25 -5. Cone I us ions I This t h e s i s has discussed the steady and unsteady heat transfer from various types of sensors. The s a l i e n t conclusions are as follows. 5.I Steady Heat Transfer (i ) The Nusselt number is a steady heat t r a n s f e r concept and is limited to the frequency range where the boundary layer is quasi-steady A functional r e l a t i o n s h i p for the steady heat t r a n s f e r from probes with an i n s u l a t i n g c o a t i n g and a substrate can be derived by incorporating the concept of a Nusselt number governed heat f l u x from the wetted -surface. ( i i ) The functional r e l a t i o n s h i p f o r the steady heat f l u x has been e x p l i c i t l y derived f o r . f l a k e thermistors, microbead thermistors and conical hot f i l m probes. The functional r e l a t i o n s h i p is AT/0 = h(R 2 ) where h(R 2 ) ± (A + BR 2 ) - 1 because of the thermal e f f e c t s of e e e the co a t i n g and the substrate. ( i i i ) The function h ( R g 2)'can be used to c a l c u l a t e the quasi-steady s e n s i t i v i t y of thermometers and anemometers as well as the quasi-steady contamination of thermometers by v e l o c i t y and anemometers by temperature. (iv) The coating and the substrate each reduce the quasi-steady s e n s i t i v i t y to speed and temperature o s c i l l a t i o n s of the ambient f l u i d as well as the r a t i o of speed-to-temperature s e n s i t i v i t y . The maximum value of the r a t i o of speed-to-temperature s e n s i t i v i t y under forced convective heat t r a n s f e r is A 9 / 2 U . i (v) With some knowledge of h ( R ^ 2 ) an a p r i o r i evaluation of a - 26 -sensor's performance is possible. The expected signal can be used to judge a sensor's signal-to-noise r a t i o , i t s level of contamination, and x i t s merits r e l a t i v e to other sensors. Because the function h(R 2) is e s e n s i t i v e to geometry the r e l a t i v e value of various coating materials, coating thicknesses and probe configurations can be assessed. The i function h ( R g 2 ) is independent of the probe's e l e c t r o n i c s . Inter-comparison of published c a l i b r a t i o n data would be easier if i t were presented in terms of relevant n o n - e l e c t r i c a l parameters such as over-heat A8, Reynolds numbers, and Prandtl numbers rather than bridge voltages and over-heat r a t i o s . 5.2 Unsteady Heat Transfer ( i ) The viscous boundary layer over a probe departs from i t s quasi-steady behaviour when (1) coL/U > 0.3 for a f l a t p l a t e , (2) ( J J L / U > 0.9 for cone of half-angle less than approximately 20°, and (3) (oL/U > 5 for a c y l i n d e r where L is the stream-wise dimension of the probe or the diameter of the cy I i nder. ( i i ) The heating source that generates the unsteady anemometric e f f e c t i s , to a f i r s t order perturbation, the advection of the steady boundary layer temperature gradient by the unsteady boundary layer v e l o c i t y , namely U j-VT . This heating source is i n t r i n s i c a l l y d i s t r i b u t e d through-out the boundary layer ana is a function of frequency, Prandtl number and p o s i t i o n . - 27 -( i i i ) F r e q u e n c y r e s p o n s e c a I i b r a t i o n s must r e a l i s t i c a l l y s i m u l a t e t h e g e n e r a t i o n t e r m U | ' v T 0 when t h e boundary l a y e r i s u n s t e a d y ; probe v i b r a t i o n does but i n t e r n a l h e a t i n g does n o t . ( i v ) The r e s p o n s e o f t h e h e a t . f l u x t o o s c i l l a t i o n o f t h e ambient f l u i d ' s t e m p e r a t u r e i s g e n e r a t e d by t h e u n s t e a d y t e m p e r a t u r e boundary c o n d i t i o n a t t h e e x t e r i o r o f t h e boundary l a y e r . C o n s e q u e n t l y t h e r e s p o n s e b a n d w i d t h i s l a r g e r f o r speed o s c i l l a t i o n s t h a n f o r t e m p e r a t u r e o s c i l l a t i o n s . B o t h bandwidths depend s t r o n g l y on t h e P r a n d t l number. Because both b a n d w i d t h s a r e independent of t h e l e v e l o f i n t e r n a l h e a t i n g t h e above c o n c l u s i o n a l s o h o l d s f o r n e a r l y unheated t e m p e r a t u r e s e n s o r s . (v) The h e a t f l u x i n t o t h e s u b s t r a t e i s s e n s i t i v e t o t h e boundary l a y e r i f t h e r a t i o o f s u b s t r a t e - t o - f I u i d c o n d u c t i v i t y k_/k^. i s l a r g e . If k g / k f i s n o t l a r g e t h e n t h e heat f l u x i n t o t h e s u b s t r a t e i s n e a r l y i n d e p e n d e n t o f t h e f l o w . If k_/k^. >> I t h e n t h e u n s t e a d y h e a t f l u x i n t o t h e s u b s t r a t e i n r e s p o n s e t o speed o s c i l l a t i o n s d e c r e a s e s w i t h i n c r e a s i n g f r e q u e n c y . R e g a r d l e s s o f t h e v a l u e of k_/k^. t h e heat f l u x i n t o t h e s u b s t r a t e i n r e s p o n s e t o t e m p e r a t u r e o s c i l l a t i o n s o f t h e a m b i e n t f l u i d d e c r e a s e s w i t h i n c r e a s i n g f r e q u e n c y . ( v i ) The r a t i o o f t e m p e r a t u r e - t o - v e l o c i t y s e n s i t i v i t y i s h i g h e s t a t z e r o f r e q u e n c y . ( v i i ) An upper bound f o r t h e t e m p e r a t u r e c o n t a m i n a t i o n o f a measured v e l o c i t y s p e c t r u m c a n be c a l c u l a t e d w i t h t h e t e m p e r a t u r e s p e c t r u m and t h e q u a s i - s t e a d y s e n s i t i v i t y r a t i o . C o n v e r s e l y , a lower bound f o r t h e v e l o c i t y c o n t a m i n a t i o n of t h e t e m p e r a t u r e s p e c t r u m can be c a l c u l a t e d w i t h t h e q u a s i - s t e a d y v a l u e of t h e s e n s i t i v i t y r a t i o . - 28 -Bi bI i o q r a p h v B a t c h e l o r , G.K. 1970. An I n t r o d u c t i o n t o F l u i d M e c h a n i c s . Cambridge U n i v e r s i t y P r e s s , 615 pages. B e l l h o u s e , B . J . and D.L. S c h u l t z . 1967. The d e t e r m i n a t i o n o f f l u c t u a -t i n g v e l o c i t y i n a i r w i t h h eated t h i n f i l m gauges. J o u r n a l o f F l u i d M e c h a n i c s , 29: 289-295. B e l y a e v , V.S., A.N. G e z e n t s v e y , A.S. M o n i n , R.V. Ozmidov and V.T. Paka. ' 1975. S p e c t r a l c h a r a c t e r i s t i c s o f smaI I - s e a I e f I u c t u a t i o n s o f t h e h y d r o p h y s i c a l f i e l d s i n t h e upper l a y e r o f t h e ocean. J o u r n a l o f P h y s i c a l Oceanography, 5_: 492-498. B o s t o n , N.E.J. 1970. An i n v e s t i g a t i o n o f h i g h wave number t e m p e r a t u r e and v e l o c i t y s p e c t r a i n a i r . Ph.D. T h e s i s , U n i v e r s i t y o f B r i t i s h C o l u m b i a . Cheng, S i n - I and D. E l l i o t t . 1957. The u n s t e a d y l a m i n a r boundary l a y e r on a f l a t p l a t e . T r a n s a c t i o n s o f t h e A merican S o c i e t y o f Mechan i ca I - Eng i n e e r s , 79_: 725-733. C o l l i n s , D.C. and M.J. W i l l i a m s . 1959. Two d i m e n s i o n a l c o n v e c t i o n from heated w i r e s a t low R e y n o l d s number. J o u r n a l of F l u i d M e c h a n i c s , 6K 357-384. C r a w f o r d , W.R. and T.R. Osborn. 1979. M i c r o s t r u c t u r e Measurements i n t h e A t l a n t i c E q u a t o r i a l U n d e r c u r r e n t d u r i n g GATE. S u b m i t t e d t o Deep-Sea R e s e a r c h . F r e y m u t h , P. 1977. F requency r e s p o n s e and e l e c t r o n i c t e s t i n g f o r c o n s t a n t - t e m p e r a t u r e hot w i r e anemometers. J o u r n a l o f P h y s i c s E; S c i e n t i f i c I n s t r u m e n t s , J_0: 705-710. - 29 -Freymuth, P. 1978. Theory of frequency optimization for hot f i l m anemometers. Journal of Physics E; S c i e n t i f i c Instruments, J J _ : 177-179. Gargett, A.E. 1978. Microstructure and f i n e structure in an upper ocean frontal regime. Journal of Geophysical Research, 83, CIO: 5123-5134. Grant, H.L., R.W. Stewart and A. M o i l l i e t . 1962. Turbulence spectra from a t i d a l channel. Journal of F l u i d Mechanics, J_2: 241-268. Grant, H.L., A. M o i l l i e t and V.M. Vogel. 1968. Some observations of the occurrence of turbulence in and above the thermocline. Journal of F l u i d Mechanics, 34: 443-448. Gregg, M.C., T. Meagher, A. Pederson and E. Aagaard. 1978. Low noise temperature microstructure measurements with thermistors. Deep-Sea Research, __5: 843-856. Gregg, M.C. and A.M. Pederson. 1978. High r e s o l u t i o n s a l i n i t y measurements, NATO Advanced Study In s t i t u t e on Instrumentation for Air-Sea-Interaction, in press. Gust, G. 1976. Observations on turbulent-drag reduction in a d i l u t e suspension of clay in sea-water. Journal of F l u i d Mechanics, 75: 29-47. H i l l , P.G. and A.H. Stenning. I960. Laminar boundary layers in o s c i l l a t o r y flow. Transactions of the ASME Series D, pp. 593-608. Hinze, J.O. 1975. TurbuIence. Second e d i t i o n , McGraw-Hill, 790 pages. Kramers, H. 1946. Heat t r a n s f e r from spheres to flowing media. Physica, J_2: 61-80. - 30 -L i g h t h i l l , M.J. 1954. The response of laminar skin f r i c t i o n and heat t r a n s f e r to f l u c t u a t i o n s in the stream v e l o c i t y . Proceedings of the Royal Society of London, Series A, Vol. 224, pp. 1-23. Ling, S-E. 1955. Measurement of flow c h a r a c t e r i s t i c s by the h o t - f i l m technique. Ph.D. Thesis, State University of Iowa. Marmorino, G.O. and D.R. Caldwell. 1978. Horizontal v a r i a t i o n of v e r t i c a l temperature.gradients measured by thermocouple arrays. Deep-Sea Research, 25;: 221-230. Mangier, W. 1948. Zusammenhang zwischen ebenen und r o t a t i o n s -symmetrischen Grenzschichten in kampressibI en F l u s s i g -k i r t e n , Z e i t s c h r i f t fur angewandte Mathematic und Mechanik, 28, 95. Nowell, Arthur R.M. 1974. Some response c h a r a c t e r i s t i c s of parabolic hot f i l m s in water. J. of Hydronautics 8_: 4, pp. 169-171. Osborn, T.R. 1978. Measurement of energy d i s s i p a t i o n adjacent to an isl a n d . Journal of Geophysical Research, 83, C6, 2939-2957. Rott, Nicholas and Martin L. Rosenzweig. I960. On the response of the laminar boundary layer to small f l u c t u a t i o n s of the . free-stream v e l o c i t y . J. of Aeronautical and Space S c i . 27_: 741-747. Tan-atichat, J . , H.M. Nagib and J.W. P l u i s t e r . 1973. On the i n t e r -p retation of the output of hot-film anemometers and a scheme of dynamic compensation for water temperature v a r i a t i o n s . Proceedings of the Symposium on Turbulence Measurements in Liquids, Department of Chemical Engineering, U n i v e r s i t y of Mi ssouri-RoIla. White, Frank. 1974. Viscous F l u i d Flow. McGraw-Hill, 725 pages. - 31 -Appendix A The Unsteady Flux of Heat and Momentum from a F l a t P l a t e by R o l f G. Lueck I n s t i t u t e of Oceanography The U n i v e r s i t y of B r i t i s h Columbia Vancouver, B.C., Canada V6T IW5 March 1979 - 32 -Abstract The unsteady viscous stress on a f l a t plate computed numerically from a s i m p l i f i e d unsteady momentum equation that is not r e s t r i c t e d by frequency compares favourably with the frequency r e s t r i c t e d c a l c u l a t i o n s a v a i l a b l e . The unsteady heat flux from the same plate maintained at a constant temperature was also computed. The heat f l u x responds d i f f e r e n t l y to o s c i l l a t i o n s of the ambient f l u i d ' s speed than i t does to o s c i l l a t i o n s of the ambient f l u i d ' s temperature. The response to speed o s c i l l a t i o n s has a maximum value at low frequencies and a larger bandwidth than the response to temperature o s c i l l a t i o n s . An examination of the unsteady temperature equations shows that an o s c i l l a t i o n of the plate's surface temperature does not generate a heat flux that is comparable to the heat fl u x in response to e i t h e r speed o s c i l l a t i o n s or to temperature o s c i l l a t i o n s of the ambient f l u i d when the boundary layer is unsteady. - 33 -I . I ntroduct ion A s i g n i f i c a n t motive f o r understanding the unsteady heat f l u x from bodies exposed t o streaming comes from the common use of heated sensors f o r high frequency turbulence flow measurements. Spectra of v e l o c i t y and temperature may be s i g n i f i c a n t l y biased ( f i l t e r e d ) i f the sensor's un-steady response i s d i f f e r e n t from i t s quasi-steady response, in which case, some data c o r r e c t i o n i s r e q u i r e d . Wire-shaped sensors have t r a d i t i o n -al I y been used in a i r . In water, however, i t i s more common t o use t h i n f i l m sensors t h a t are wedge-shaped (NowelI, 1974), cone-shaped (Gargett, I 976, Grant et a I . , 1962, I 968), and cyI i n d r i c a I . The dimensions of f i l m type sensors d i f f e r by several f a c t o r s of 10 from wire sensors; wires have t y p i c a l diameters of I O - 6 meters, c y l i n -d r i c a l f i l m sensors 10 - 1 + meters, and wedge and cone shaped sensors have f i l m s about I O - 3 meters long. As shown l a t e r , a frequency parameter t h a t c h a r a c t e r i z e s the unsteady boundary layer i s W L / U q where OJ i s the angular frequency of o s c i l l a t i o n ( i n radians per second), L i s a r e l e v a n t dimension (diameter or length in the flow d i r e c t i o n ) , and U q i s the ambient f l u i d ' s steady speed. If W L / U _ << | the flow i s e s s e n t i a l l y quasi-steady, i t s behaviour at any i n s t a n t i s t h a t a p p r o p r i a t e t o steady c o n d i t i o n s at the instantaneous value of the f l u i d ' s speed and temperature. If U L / U O z I the flow i s unsteady and may behave d i f f e r e n t l y from the quasi-steady flow. It i s r e a d i l y seen t h a t wire type sensors operate in the quasi-steady frequency range f o r a l l p r a c t i c a l purposes. However, the boundary layer on c y l i n d r i c a l f i l m s i s unsteady when to £ \0k U Q ( U q in meters/second) and i s unsteady on wedge and cone shaped sensors when ID 2 I 0 3 U q . Thus unsteady e f f e c t s are w i t h i n the frequency domain of t y p i c a l measurements made with - 34 -wedge and cone shaped f i l m turbulence sensors. Unsteady boundary layer e f f e c t s have been reported by NowelI (1974) and Evans (1963). In both cases the response t o external speed o s c i l l a t i o n s was measured by v i b r a t i n g the probe in the d i r e c t i o n of the steady stream. NowelI showed that the unsteady response of wedge shaped sensors diminished with increasing frequency when W / U Q 2 I0~ 3metre 1. The response was down by 3 dB at 120 cyles/sec f o r U =0.31 m/sec and 500 cycles/sec for U =1.12 m/sec. o ' o For cone shaped sensors with a narrow f i l m at a distance L from the rear of the apex, Evans reported that the response r i s e s when <*>L/U_ i 0.9 The frequency response of heated sensors is often inferred from internal heating methods; the probe is exposed to a steady streaming, i t s temperature i s o s c i l l a t e d by an e l e c t r i c current and the r e s u l t i n g heat fl u x is measured. Whereas one wants to know the response to o s c i l l a t i o n s from outside of the boundary layer one i s c a l i b r a t i n g the response to an internal o s c i l l a t i o n . The c a l i b r a t i o n s made by internal heating are related to the response to external speed and temperature o s c i l l a t i o n s by i m p l i c i t l y assuming that the response (.to external o s c i l l a t i o n s ) is quasi-steady (Freymuth, 1978). When the boundary layer is quasi-steady i t does not matter whether the o s c i l l a t i o n s are external or internal because the i n e r t i a 'and the thermal i n e r t i a of the boundary layer are n e g l i g i b l e . However, when the boundary layer is unsteady the heat f l u x in response to external o s c i l l a t i o n s is d i f f e r e n t from the heat f l u x in response to internal temperature o s c i l l a t i o n s ; at unsteady frequencies ( G J L / U _> I ) the former o decreases with increasing frequency while the l a t t e r increases with increas-ing frequency (as shown l a t e r ) . At high frequencies the f l u x in response to internal o s c i l l a t i o n s depends only on the f l u i d ' s thermal properties and O J , whereas the fl u x in response to external o s c i l l a t i o n s always depends on the - 35 -f l u i d ' s speed and Prandtl number. The point here is that i f the response of a sensor is not affected by the i n e r t i a or the thermal i n e r t i a of the boundary layer then internal heating can give useful information. If the sensor is operated in a. constant temperature mode then internal heating can t e l l about the e l e c t r o n i c s . If the sensor is operated in a constant current mode then internal heating can measure the thermal i n e r t i a of the probe and the l i m i t a t i o n of the e l e c t r o n i c s . But if the response is limited by boundary layer processes then internal heating cannot give any i n f o r -mation about the e f f e c t of the boundary layer because (I.) the unsteady v e l o c i t y in the boundary layer is not present during c a l i b r a t i o n and because (2) the heat f l u x from the probe responds d i f f e r e n t l y to external o s c i l l a t i o n than to internal o s c i l l a t i o n . The probe's constant temperature e l e c t r o n i c s cannot control the behaviour of the boundary layer. Hence, frequency response c a l i b r a t i o n s of c y l i n d e r , wedge and cone shaped sensors made by internal heating may be erroneous in the frequency range where the boundary layer i s unsteady. In p r a c t i c e the response may be complicated by substrate e f f e c t s (Bellhouse and Schultz, 1967), coating e f f e c t s and possibly l i m i t a t i o n s of the e l e c t r o n i c s . However, an examination of the inherent hydrodynamic l i m i t a t i o n s of an ideal sensor l i k e a f l a t plate has p r a c t i c a l implications because these l i m i t a t i o n s must also apply to real sensors and other heated surfaces exposed to an unsteady streaming. A considerable amount of work has been published on the plate's non-steady v e l o c i t y boundary layer (Cheng and E l l i o t t 1.957, Hi.ll and Stenning - 36 -I960, R o t t and R e s e n z w e i g I960). However, l i t t l e o f t h i s work has been e x t e n d e d t o t h e u n s t e a d y r e s p o n s e o f t h e h e a t f l u x t o speed o r t o t e m p e r a t u r e o s c i l l a t i o n s i n t h e ambient s t r e a m . Lig ' h t h i I I (.1954). i n v e s t i g a t e d t h e u n s t e a d y f l o w o v e r a f l a t p l a t e f o r t h e c a s e o f s m a l l o s c i l l a t i o n s i n t h e m a g n i t u d e , but not t h e d i r e c t i o n , o f t h e ambient f l u i d . By u s i n g t h e Ka"rma*n-Poh I hausen method, L i g h t h i l l d e r i v e d an a p p r o x i m a t e a n a l y t i c s o l u t i o n f o r t h e v e l o c i t y f i e l d f o r low f r e q u e n c i e s . The u n s t e a d y h e a t f l u x , however, c o u l d not be d e r i v e d by t h e same method, m a i n l y because o f t h e p l a t e ' s z e r o s t e a d y p r e s s u r e g r a d i e n t . O s t r a c h ( 1 9 5 5 ) , u s i n g a n u m e r i c a l method v a l i d f o r low f r e q u e n c i e s (coL/U o << I ) , found t h a t t h e heat f l u x i n c r e a s e s s l i g h t l y i n magnitude and has a phase advance o v e r i t s q u a s i - s t e a d y v a l u e . The a u t h o r i s unaware o f any work on t h e r e s p o n s e of heated s e n s o r s t o t e m p e r a t u r e o s c i l l a t i o n s o f t h e ambient f l u i d . In t h e f o l l o w i n g , a t t e n t i o n i s r e s t r i c t e d t o t h e c a s e where t h e magnitude but n o t t h e d i r e c t i o n o f t h e a m b i e n t f l u i d o s c i l l a t e s a b o ut a s t e a d y mean, and where t h e a m p l i t u d e of o s c i l l a t i o n i s s m a l l enough t o a v o i d harmonic d i s t o r t i o n s . The R e y n o l d s number range i s r e s t r i c t e d t o where t h e boundary l a y e r a p p r o x i m a t i o n s a r e v a l i d and where t h e boundary l a y e r i s l a m i n a r . I t i s assumed t h a t t h e t e m p e r a t u r e f i e l d does not i n f l u e n c e t h e v e l o c i t y f i e l d . The s u r f a c e t e m p e r a t u r e o f t h e p l a t e i s t a k e n as c o n s t a n t a t 0 and t h e a m p l i t u d e o f o s c i l l a t i o n o f t h e ambient f l u i d ' s t e m p e r a t u r e i s assumed t o be s m a l l compared t o 9. The a n a l y s i s has a p p l i c a t i o n s f o r c o n s t a n t t e m p e r a t u r e s e n s o r s and o t h e r s u r f a c e s whose t e m p e r a t u r e r e m a i n s c o n s t a n t o r p r a c t i c a l l y c o n s t a n t when exposed t o u n s t e a d y s t r e a m i n g . - 37 -The viscous boundary layer is examined in section 2. Section 2a deals with the steady boundary layer v e l o c i t y and 2b deals with unsteady flow in the boundary layer. The heat flux is c a l c u l a t e d in section 3. The unsteady f l u x in response to speed o s c i l l a t i o n s in ambient f l u i d is c a l c u l a t e d in 3a and the response to ambient f l u i d temperature o s c i l l a t i o n s is examined in 3b. - 38 -2. The Viscous Boundary Layer The boundary layer equations for two dimensional incompressible flow with an ambient f l u i d speed U(t) p a r a l l e l to the plate are ( L i g h t h i l l , 1954) 3u , u3u , v3u , I 3P 3 2u ' + ~ r — + — — i - — -— = v -—*r = v TTT?" (la) 3t 3x 3y ' p 3x v 3y with boundary conditions u = v = 0 @ y = 0 for x > 0 and u U(t) as y -*• <*> where p i s the f l u i d density and v the kinematic v i s c o s i t y (Figure 1). We now le t the q u a n t i t i e s U, u, and v perform small o s c i l l a t i o n s about a steady mean and write U(t) = U ( l + e e l u + ) (2a) o u = U Q(x,y) + e U | (x,y,to)e'U)"'" (2b) v = v (x,y) + ev|(x,y,_)e' w^ (2c) where e << I is the r e l a t i v e amplitude of the speed o s c i l l a t i o n s and is therefore a s u i t a b l e perturbation parameter. 2a. Steady Flow The well known Blasius transformation given in S c h l i c h t i n g (1955, p. 103), reduces the steady momentum equation (la) to ff + 2f = 0 f(0) = f'(0) = 0 f' (») = I , (31 4 y.v Figure I. A sketch of the idealized plate showing the coordinate system employed for the analyses. The surface temperature is fixed at G, a constant. 40 -'U where n = y/ — 7 vx u = U f ' ( n ) , 0 o and v o ( x ' V ) = ^ / i f Cr,f' - f ) (4) o and primes denote d i f f e r e n t i a t i o n with respect to n. 2b. Non-Steady Flow e << I By s u b s t i t u t i n g equations (2a,b,c) into Cla,b,c) and by only r e t a i n i n g terms up to order e one gets ( L i g h t h i l l , 1954) 3 u j 3u j 3U Q 9U q 9 2U| ia>u. + u - — + v - — + u, - — + v. - — = iuU + v , o" (-5a) I o 3x . o3y I 3x I 3y o 3y / 3u 3v. —-L + — ± = o (5b) 3x 3y where U| = V | = 0 @ y = 0 and x > 0 and u, = U as v + ». I o L i g h t h i l l (1954) found an approximate s o l u t i o n to equation (5a) for small values of the frequency parameter fi = uix/U . For very large values of the parameter ux/U o L i g h t h i l l reduced equation (5a) to a "shear wave" equation a 2u! 'o' * 3y 2 ico(u, - U_) = v ~T~2 ( 6 ) which has the s o l u t i o n u. = U { I - exp(-yvM)}, 7p » I . (7) I o ' v U o The s u b s t i t u t i o n of U|(x,y,oj) = U og (n,fi) (8a) U i— , v,(x,y,_> = {ng - g - 2fi |£} (8b) o ) - 41 -wh i c h s a t i s f i e s c o n t i n u i t y ( 5 b ) , i n t o (5a) y i e l d s i t i I i t t I t i g + y f g - i ^ g + - f g - 0 = - in (9a) where , , 0 = n ( f g - f g) (9b) i g(0,Q) = g (0,Q) = 0 g(n, 0 ) = - j (f + n f ' ) . R o t t and R o s e n s w e i g (I960) have p a r t i a l l y s o l v e d e q u a t i o n (9b) n u m e r i c a l l y by w r i t i n g g as a power s e r i e s i n Q, t h a t i s 00 g(n,n) = £ ( i f t ) n h (n) f o r Q < I , n=0 and c a l c u l a t e d h Q and h j . However, a power s e r i e s e x p a n s i o n o f t h e above t y p e i s i m p r a c t i c a l f o r Q ^ I. Cheng and E l l i o t t (1957) have s o l v e d t h e momentum e q u a t i o n ( e q u a t i o n ( 5 a ) ) t o h i g h e r o r d e r s i n Q and have t a b u l a t e d s u f f i c i e n t c o e f f i c i e n t s t o e v a l u a t e t h e d i m e n s i o n I e s s s u r f a c e s t r e s s g (0,f2) t o t h e f o u r t h o r d e r i n U. E q u a t i o n (9a) becomes an o r d i n a r y d i f f e r e n t i a l e q u a t i o n where 9, i s o n l y a p a r a m e t e r i f t h e term 0 i n e q u a t i o n (9a) c a n be i g n o r e d . C l e a r l y as Q •+ 0 0 -»- 0. F o r v e r y h i g h f r e q u e n c i e s (ft>>l) we have from e q u a t i o n ( 7 ) , ( 8 a ) , and (9b) t h a t I T i l T i t I I 0 = — { n f g - n f g + f g ) . (10) For n < 2 t h e f u n c t i o n f i s t o a good a p p r o x i m a t i o n (SchI i c h t i n g , 1954, p. 107) l ' ' f = j n 2 f (0) w h i c h makes t h e t e r m 0 f o r n<2 0 ^ 2 f (0) ( n 2 g - ng + g> - 42 -whereas the advective term is I II i t | i i I i t J { f g + f 9 ) % 7 f ( 0 ) n 2 g + g } . t Because the magnitude of g is never much larger than unity the term 0 is comparable to the advective terms for n<2 at high frequencies. For n<4 an adequate approximation for f is ( S c h l i c h t i n g 1954, p. 107) f = n which makes the term 0 approximately 0 % ^ n g " and equal to the advective term. However, for high frequencies the ad-vective term is small r e l a t i v e to the pressure gradient i n , the i n e r t i a t 111 term -ifig and the d i f f u s i o n term g . The r e s u l t i n g equation, v a l i d for fi>> I , is g ' " = ifiCg'-l) which is equivalent to the "shear wave" equation 6. For low frequencies, that is fi<<l, a f i r s t order expression for g is given by L i g h t h i l l (1954) and can be written as g(ri,fi) = g(.n,0) + ifihCn), where h is a real function of order unity. Consequently. 0 = ifi { f ' h 1 - f ' h } (I la) whereas the advective term is ^•{[fg"(n,0> + f " g ( n , 0 ) ] + i f i ( f h " + f ' h ) } . (Mb) The term 0 is small compared to the advective term because fi<<I and because the negative sign in (Ila) causes a p a r t i a l c a n c e l l i n g except for large n where h ! may be s l i g h t l y negative. In summary then, 0 is i d e n t i c a l zero at fi=0, i s small compared to the advective terms for small fi, and is comparable to the advective terms for large fi but small compared to the - 43 -dominant terms which are d i f f u s i o n , i n e r t i a and pressure. i f the v e l o c i t y f i e l d does not have any s i n g u l a r i t i e s with respect to £2, such as resonances and notches, then one can reasonably expect that the magnitude of 0 t r a n s i t s smoothly, i f not monotonicaI Iy, from zero at £2=0 to a value comparable to the advective term at very large values of £2. Thus the deletion of the term 0 from equation (9) cannot lead to serious errors in computing g, i t s incl u s i o n on the other hand would lead to formidable computational d i f f i c u l t i e s near £2=1. (In comparison the advective terms are e a s i l y handled by numerical routines.) The s i m p l i f i e d unsteady momentum equation g ' " + j f g " - i£2g' + \ f ' g = i£2 g(0,£2) = g'(0,£2) = 0, g'(°°,£2) = I (12) and the steady momentum equation (3) have been solved numerically to an accuracy of I part in I0 5 or better, by a c o l l o c a t i o n routine c u r r e n t l y being phased into operation at the Computing Science Department of The University of B r i t i s h Columbia (Ascher, C h r i s t i a n s e n , and R u s s e l l , 1977). The frequencies £2 examined range from 0 to 8 2 in steps of (O.I) 2. The computed values for f, f' , and f'' agree with the values tabulated in Sc h l i c h t i n g to within one unit in the least s i g n i f i c a n t d i g i t . An estimate of 0 at £2=1 based on the solu t i o n to equation (12) w i l l be given l a t e r . The unsteady v e l o c i t y p r o f i l e g' is plotted in Figure 2 for £2=0, 0.25, I.00 and 5.29. Figure 3 compares L i g h t h i l l ' s low frequency s o l u t i o n at his c r i t i c a l frequency of £2=0.588, which in the notation used here is I - (I - O . I 7 l n ) 2 { ( l - 0.17 In - 0.0882n2) - i 0.5l5n(l - 0.229n)} against the computed v e l o c i t y p r o f i l e at the same frequency. The frequency - 44 -1.21 0.1 0.2 0.3 0.4 0.5 0.6 0.7 / O o " Figure 2. The real and imaginary components of the boundary layer perturbation v e l o c i t y p r o f i l e s at values of the frequency parameters £2=0.0, 0.25, 1.0 and. 5.29. The imaginary component of g' is zero f or £2=0. Figure 3. A comparison between the computed perturbation v e l o c i t y p r o f i l e and L i g h t h i l l ' s p r o f i l e at the " c r i t i c a l " frequency ft=0.588. - 46 -£2=0.588 is "where L i g h t h i l l joined his low frequency so l u t i o n to the shear-wave solu t i o n (7). The real part of g' compares quite favourably with L i g h t h i l l ' s s o l u t i o n , the discrepancy between the two curves being approxi-mately 10$. The imaginary component of g', however, d i f f e r s s i g n i f i c a n t l y from L i g h t h i l l ' s . Using the computed data for g at ft=l.0 and ft=l.24 i t is possible to estimate the neglected term 0 and compare i t to the other terms in the momentum equation, t a b l e 2. The magnitude of 0 is less than \% of the pressure gradient ift at n=0.5 and n=5 and approximately 5% at n=2. The local steady st r e s s on a plate is (SchIichting) AT T (x) — y U / — f " CO). o o v x S i m i l a r l y , the local unsteady stress is n r T l ( x , n ) = yUoe\| 7J^ g " <0,Sl) The stress r a t i o g' ' (0,ft)/f' ' (0)e is plotted in Figure 4(a), and included for comparison is L i g h t h i l l ' s low frequency formula | (I t 1.7 in) and the fourth order formula of Cheng and E l l i o t t | {(I + 0.94294ft2 - 0.54105^) + i ( I . 70353ft - 0.73828ft3)} The magnitude of L i g h t h i l l ' s f i r s t order stress formula is smaller than the computed str e s s f o r a l l frequencies below ft=0.9 and the phase is co n s i s t e n t l y larger. On the other hand the agreement between Cheng and E l l i o t t ' s fourth order formula and the computed stress is remarkable, the maximum d i f f e r e n c e amounting to less than 1.6$' in magnitude and 1.4$ in phase for ft<0.64. For ft>0.7 the agreement is quite poor; as expected t h e i r s e r i e s expression diverges. Table I summarizes the unsteady I 8 r UJ Q 3 H < 3 o cnl 0 . 0 1 0 . 0 2 0 . 0 5 0 . 1 F i g u r e 4 ( a ) . A c o m p a r i s o n o f t h e u n s t e a d y - t o - s t e a d y l o c a l s u r f a c e s t r e s s r a t i o ( s o l i d c u r v e s ) a g a i n s t L i g h t h i l l ' s r e s u l t (dashed c u r v e s ) and Cheng and E l l i o t t ' s r e s u l t s ( s o l i d d o t s f o r t h e magnitude and s o l i d t r i a n g l e s f o r t h e p h a s e ) . 4-- 48 -Author Chen & E l l i o t L i g h t h i l l fl Magnitude Phase Magni tude Phase Magni tude Phase 0 . 0 1 . 5 0 0 0 . 0 0 1 . 5 0 0 0 . 0 0 1 . 5 0 0 0 . 0 0 0 . 0 1 1 . 5 0 0 1 . 0 2 1 . 5 0 0 0 . 9 8 1 . 5 0 0 0 . 9 7 0.0k 1 . 5 0 8 4 . 0 6 1 . 5 0 6 3 - 9 8 1 . 5 0 3 3 - 8 9 0 . 0 9 1 . 5 3 7 8 . 9 0 1 . 5 2 9 8.62 1 . 5 1 7 8 . 7 0 0 . 1 6 1 . 6 0 7 14.85 1 . 5 8 8 14.75 1 . 5 5 4 1 5 - 2 2 0 . 2 5 1 . 7 3 0 2 0 . 8 9 1 . 7 0 3 21.41 1 . 6 3 0 2 3 . 0 0 0 . 3 6 1 . 9 0 0 26.24 1 . 8 8 2 2 7 . 4 7 1 . 7 5 9 3 1 . 4 7 0 . 4 9 2 . 1 0 7 3 0 . 6 9 2 . 1 1 5 32.04 1 . 9 5 2 3 9 . 7 9 0.64 2 . 3 ^ 5 34.27 2 . 3 6 3 3 4 . 6 9 2 . 2 1 7 ( 4 7.41) 0.81 2 . 6 0 9 3 7 . 0 6 2 . 5 5 2 3 5 . 4 7 1 . 0 0 2 . 8 9 3 39.16 ( 2 . 5 5 3 ) ( 3 4 . 5 5 ) 1 . 2 1 3.190 40.69 1 . 4 4 3 . 4 9 6 41.78 1 . 9 6 4.116 4 3 . 1 1 3.24 5 . 3 5 4 4 4 . 1 9 9 . 0 0 9 . 0 1 0 44.84 6 * 1 . 0 0 24.09 4 4 . 9 9 Table A l The unsteady-to-steady stress r a t i o IX/ZTQ, compared to Cheng 6 E l l i o t ' s ( 1 9 5 7 ) fourth order formula and L i g h t h i l l ' s (1954) f i r s t order formula. n=0. 5 n=2.0 n=5.0 Real Imaginary Real Imaginary Real Imaginary g ' " -0.237 -0.62 -0.30 +0.069 +0.034 -0.022 hfg'' + i + V ' g J -0.29 +0.013 +0.232 -0.014 -0.016 +0.043 - i n g ' +0.193 • -0.362 +0.074 -1.043 -0.019 -1.021 i n — + 1.0 — + 1.0 — + 1.0 0 +0.010 +0.006 -0.023 -0.056 -0.024 -0.003 Table A l l A comparison of the d e l e t e d term 0 a g a i n s t the ret a i n e d terms of the unsteady momentum equation (9a) at the frequency 0=1.0 f o r p o s i t i o n s 0=0.5, 2.0, and 5.0 . The values f o r g ' 1 ' are approximat ions. - 50 -dimensionless surface stress r a t i o s . For very large frequencies the non-dimensional surface stress approaches g' 1(0 , n ) = /nr in agreement with the "shear wave" solution given in equation (7). The t o t a l steady drag per unit width on a plate of length L wetted on both sides i s ( S c h l i c h t i n g , 1955, p. 108) ATT D = 4 yU / - 2 - f " (0) . 0 0 V S i m i l a r l y the t o t a l unsteady drag is L AT D. = 2eyU / / — q " (0 , n)dx 1 o Q vx = 4 EyU Z - 2 - -4" / g"(0,n)d/fT o v _ 2 0 where E, z = W L / U q is the non-dimensional plate frequency. The r a t i o of these two drags, namely D l I ^ r~ e U o f " ( 0 ) ? 2 0 is plotted in f i g u r e 4(b), and e x h i b i t s c h a r a c t e r i s t i c s s i m i l a r to the local s t r e s s r a t i o , namely an asymptotic trend to J~\\\ at large frequencies. - 52 -3. The Thermal Boundary Layer The heat f l u x from a plate, maintained at a constant temperature 0, can be modulated by a change of the ambient f l u i d ' s speed and by a change of the ambient f l u i d ' s temperature. For both cases i t is convenient to measure temperature with reference to the ambient f l u i d ' s steady tempera-ture. The equations for the d i s t r i b u t i o n of the temperature T i n a boundary layer without f r i c t i o n a l heating are ( L i g h t h i l l , 1954) 3 I + U _ I + V ! I _ K_£L 9t 3x 8y 3 y 2 (.13) T = 0 @ y = O x > O T = T (t) = ve as y -*• » where K is the thermal d i f f u s i v i t y of the f l u i d which is taken to be uniform. Because equation (13) is I inear in T, we can l e t T be the sum of three component terms. That i s , let T = T_(x,y) + ET,(x,y,_) e l a ) + + y6(x,y,u)) e ' w + (14) where T_ is the temperature d i s t r i b u t i o n due to steady streaming over the constant temperature plate, eT| is the perturbation temperature f i e l d due to the speed modulation eU Q at frequency u), and y9 i s the temperature f i e l d due to an independent i n f i n i t e s i m a l modulation of amplitude y of the ambient temperature at the same frequency. The s u b s t i t u t i o n of equation (14) and the v e l o c i t y expressions (2b) and (2c) into equation (13) y i e l d s , to f i r s t order in e, the following three sets of equations. * The reader should be careful to d i s t i n g u i s h 0 from 0 in t h i s section. - 53 -u 3T o o 3x + v ST c o 3y 32T ay' T = e @ y = O x > o 0 T = 0 as y o ' (15) 3T, 3T, 3 2T, iuiT, + u o 3x + v o 3y - K = -u, 3y 3T c 3x - v, 3T c 3y and T = 0 % y = 0 T. = 0 as y -*• <* • Q . 36. . 36 iu)6 + u — + v K o 3x o 3y (16) 3 29 3y = 0 (17) 0 = 0 % y = 0 6 = I as y °° A comparison of equations (16) and (17) makes i t quite c l e a r that the anemometric response (Tj) must be very d i f f e r e n t from the temperature response (9). Equation (16) is non-homogeneous but has homogeneous boundary conditions. The unsteady heating source that generates the temperature f i e l d Tj is not at the boundaries but is i n t r i n s i c a l l y d i s t r i b u t e d throughout the e n t i r e boundary layer and r e s u l t s from the advection of the steady boundary layer temperature gradient (3T q/3X , 3T Q/3y) by the unsteady boundary layer v e l o c i t y ( U | , V | ) . This heating source U j . V T , as shown l a t e r , is a function of Prandtl number, frequency and p o s i t i o n . The temperature response (9) equation (17), on the other hand, is a homogeneous equation driven by a non-homogeneity at the boundary y = c o. The heating source is outside of the boundary layer and can only penetrate the boundary layer through advection by the steady v e l o c i t y f i e l d (u and v ) and by d i f f u s i o n , o o ' - 5 4 -Proper frequency response c a l i b r a t i o n requires a f a i t h f u l simulation of the generating term, U|*VT o when determining the anemo-metric response and a simulation of the boundary condition 9=1 at y-*» when determining the temperature response. The equations f o r an internal temperature o s c i l l a t i o n of exp(iojt) are 9T. iojT. + u r — I o 9x 9T. + v o 9y - K 9 2T. I 9 y 2 ( 1 8 ) T . ( X , 0 , O J ) = 1 i T. (X , ° ° , O J ) = 0 . i For quasi-steady frequencies ( O J X / U _ < < I ) the thermal i n e r t i a of the boundary layer iojT. is small compared to the other terms and i t is c l e a r that T. = 1 -9 and, using equation ( 1 5 ) , that T Using equation ( 1 6 ) one has f o r O J X / U Q < < I 9T o 9U and hence there is a simple r e l a t i o n s h i p between Tj , 8 , and T. when d)x/U o<<l. When O J X / U Q > I there does not e x i s t any simple r e l a t i o n s h i p between Tj , 8 and T., because ( I ) the generating term U j'VT becomes unsteady but is not present in the equation for T. and (2) the thermal i n e r t i a of the boundary layer becomes important. For very high frequencies ( O J X / U >>l) equation ( 1 8 ) reduces to o io>T. - K — i 9y a 2 T d i _ _ T ( 0 , O J ) = I 2 ~ u > T(»,_) = 0 because only terms involving OJ and the d e r i v a t i v e s of highest order need - 55 -to be retained. The so l u t i o n is T. (y,u>) = exp {-y } The e f f e c t of the internal temperature o s c i l l a t i o n s on.the f l u i d is r e s t r i c t e d to a layer whose thickness decreases with increasing i frequency. The surface heat f l u x is k(iu>/K)2 which increases with increasing frequency and does not depend on the f l u i d ' s speed or Prandtl number. Solutions f or Tj and 6 (.section 3 a & bl show that the heat flux in response to speed or temperature o s c i l l a t i o n s in the main stream decreases with increasing frequency when aix/Uo>0.3 and always depends on the speed and Prandtl number of the main stream. In summary, the response as measured by an internal heating method can only be rela t e d to the response to external o s c i l l a t i o n s when u>x/U << I . When aix/U £ I o o internal heating methods do not c o r r e c t l y measure the response to e i t h e r speed or temperature o s c i l l a t i o n s in the main stream. Using the Blasius transformation, equation (15) can be written as ( S c h l i c h t i n g , 1954, p. 264) T " + PrfT ' = 0 o / o T = e n = 0 o T = 0 n -»• °° o which has the so l u t i o n (19) T o ( n , P r ) = f [f"(z)]P rdz where Pr = v/K - 56 -and F(Pr) • = / [ f " ( z ) J P r d z 0 By making use of the Blasius transformation and equations (8a), (8b) and (20), the perturbation temperature equation (16) becomes j - T | . . + i n i . . I a T | . | M ^ l g t 2 a ^ } + f , f l » T -3ft (20) T|(0,ft) = 0 T ( (°°,ft) = 0 T i ^ 0 ) = U o i U a = - i ? t f , , ^ ) J P r 0 and s i m i l a r l y the temperature response equation becomes 0(0,ft) = o 0(«,ft) = i 9T | z=n p 9(n,0) = I - -r~- = \=- / [f»'(z)] r rdz (21) 30 F Q 3a. The Anemometric Response A perturbation temperature f i e l d s olution (T|) for very high frequencies that is v a l i d where the steady temperature T , is a l i n e a r function of n was given by L i g h t h i l l (1954) and is T I ( T 1 ' Q ) = ^ 9 [ ? ' p r ) ) F P r ^ ( 1 - P D 2 + P r [ n ( l - P D t ^ e " ^ /Tft 2Pr -n/iftPr , e s /Tft - 57 -Thus for high frequencies the l a s t term in equation 20 and has the same magnitude as the advective term f T j ' but i s small compared to the dominant terms d i f f u s i o n , thermal i n e r t i a and heating The 'reasons given f o r dropping the term 0 from the momentum equation apply equally to the term f'„3T./3fi in equation (20). The s i m p l i f i e d perturbation temperature equation is T M + ! fPrT ' - ifiPrT = 6 P r [ f " C n ) ] P r { 2 __. } i j 2 1 I 2FfPr) l y s T = 0 @ n = 0,». The term g + 2fi ag/90. can be evaluated from the solution of the non-steady momentum equation. Equation 122) has been solved numerically by a f i n i t e d i f f e r e n c e formulation with 881 points for n in the range 0 to 8.8. The tri - d i a g o n a l matrix representing the f i n i t e d i f f e r e n c e term of C22) was solved by Gaussian e l i m i n a t i o n . The generating term which represents the heat source f o r the anemometric e f f e c t is calculated from the solutions to the momentum equations (.4 and 12) and is plotted in Figure 5 for Prandtl numbers I and 8 at frequencies fi= 0 and I . The forc i n g increases in magnitude and phase advance with increasing frequency; i t has vanishing boundary values and an amplitude maximum with a value and location that depends strongly on the Prandtl number. When O J X / U ^ I the unsteady anemometer response can only be measured by a method that 1 ' V V X Figure 5. The p r o f i l e of the function that generates the unsteady anemometric e f f e c t , f o r Prandtl numbers I and 8 at n=0 (dashed line) and Q=I ( s o l i d l i n e ) . The phase is the same for Pr=l and Pr=8. - 59 -simulates t h i s generating term. The steady local heat f l u x from the plate is A T -k / — T '(0,Pr) vx o where k is the f l u i d ' s c o n d u c t i v i t y . S i m i l a r l y the local perturbation heat f l u x , that produces the non-steady anemometric e f f e c t , is L T ' ( n,Pr> The perturbation-to-steady local heat f l u x r a t i o T,'(fi,Pr)| eT '(Pr) <n=0 which is e f f e c t i v e l y the local unsteady Nusselt number is plotted in Figure 6(a). The heat f l u x magnitude r i s e s above the quasi-steady value of 0.5 for a l l Prandtl numbers from 0.7 to 15. The highest heat f l u x peak is attained at a Prandtl number of two, for larger Prandtl numbers the peak height decreases and i t s location s h i f t s to lower frequencies. A more d e t a i l e d p l o t is in Figure 6(b). A power s e r i e s expression f o r the local perturbation heat flux would have to be of t h i r d or higher order to d u p l i c a t e the low frequency curves shown in Figure 6(b'). For high frequencies the heat flux is proportional to Uft) 1 . The phase i s negative f o r a l l frequencies in s p i t e of the low frequency amplitude r i s e The r a t i o of the perturbation-to-steady local heat f l u x has been plotted in Figure 6(c) against a frequency axis scaled by (I + /Pr")2 I + 2/PF Except for some discrepancies associated with the various low frequency C 2 3 ) CQ C ~I 0 CT* 3 —I O rr - i 0 3 QJ — — o — n N OJ o — Q. C T 0 -< -1 0 CT QJ C/> -+ •-+ — 0 O Q) ZJ Q. ~< ZT 0 =T OJ 0 -+ QJ — X c X -I -• 0 cn — T3 O O CQ ZJ Q) cn -I 0 — cn n -o 0 " O 0 — Q. o -+ 3 • o Q. c 0) + 2 . 0 r - 5 8 . 0 1 -n 3 0 . 0 2 2 . 0 1 4 . 0 + 6 . 0 - 2 . 0 - 1 0 . 0 - 1 8 . 0 - 5 8 . 0 - 6 6 . 0 - 7 4 . 0 - 8 2 . 0 - 9 0 . 0 (0 ui U l cc o Ui o X 3 - 2 6 . 0 § - 3 4 . 0 5 - 4 2 . 0 o oc - 5 0 . 0 0. < O o _ l x ( V I o o o cn - 61 -o G O o ZD x 3 3 a n ± i N 9 v i / \ i x n i d i v 3 H N o i ± v a u n ± u 3 d I V D C T I Figure 6(b). The local perturbation heat f l u x response to speed modulations, normalized by the steady heat flux; linear plot d e t a i l i n g the heat fIux maxima. - 62 -S33M93Q NI 3SVHd Xflld ±V3H NOI±V9HrUM3d ""IVDOT x Z O 6 ro 9 Q (M T}-CVJ — T o t o + o CM i § S 3 Q 5 N i i i o ro I O CM I O d m i O S Q tO tO I O I Q Q CM O 00 Co I I • I ' I I I ' I ' I I I I I I I ' I » I ' | I | I O cJ t o CM 00 d d o d o i 00 d CM x ° 3 3 o I I I I • I • I • I q CM + q CM I 0 o o o to' d * c o 1 i i i I I I 0 o CM' t o CM CM 1 I I ' I ' I • I t I . I • I • I o d r o l O t o I o oo r o l o CM I o t o I o d in i o o cd m i BP NI Xnid 1V3H N0llV8HfUH3d 1 V 0 0 1 * Z Figure 6(c). The local perturbation heat f l u x response to speed modulations, normalized by the steady heat f l u x ; a frequency scaled version of (a). - 63 -peaks, both the magnitude and phase of the heat flux for the various Prandtl numbers s a t i s f a c t o r i l y c o l l a p s e into one curve. The 3 dB band-width of the local perturbation heat flux is Q = 1.45 { ' + 2 / p T } 3 d B (I + /Pr") 2 In p r a c t i c e one senses only the t o t a l heat f l u x . The t o t a l heat f l u x , per unit width, from a plate wetted on both sides is (Sch l i c h t i n g 1955, pp. 268-269) 4k /U L v o = 4k R 2 | [ f " ( 0 ) ] P r n e F where R = U L/v. S i m i l a r l y the t o t a l perturbation heat flux is e o 2ek r AT / — T ' (0,ft,Pr)dx vx I i i 2 k R 2 -V T '(0,O,Pr)d v ^ " e 0 where E. - wL/U . The r a t i o of the perturbation-to-steady t o t a l heat f l u x o 2e[f " ( 0 ) ] P r ? 2 Tj '(0,ft,Pr)d/ft 0 is plotted in Figure 7(a). Asymptotically, the t o t a l heat f l u x is propor-t i o n a l to (i£) 2 , in contr a s t to the local heat flux r a t i o which is asymptotically proportional to (ifi) '. The peaks are less pronounced than the local heat f l u x peaks. The Prandtl number s c a l i n g of equation (23) s a t i s f a c t o r i l y c o l l a p s e s the curves for various Prandtl numbers into - 64 -S33U93C1 Nl Xmd J.V3H N0 l lV9dnid3d "IViOl x z 8P NI xmd ±V3H Nouveaniasd ivioi x z Figure 7(a). The perturbation-to-steady t o t a l heat modulations; unsealed frequency axis. flux r a t i o f o r speed - 65 -S 3 3 H 9 3 0 NI 3 S V H d X f l l d 1 V 3 H N O I l V S a f U H S d "IV101 X 2 8P NI X n i J 1 V 3 H N O I ± V 8 d r U H 3 d 1 V 1 0 1 X z Figure 7(b). The perturbation-to-steady t o t a l heat flux r a t i o for speed modulations; scaled frequency axis. - 66 -one curve (Figure 7(b)). The 3 dB bandwidth of the to t a l perturbation heat f l u x , the anemometric response bandwidth, is K _ coL_ = 3 7 { I + 2/Pr j 3 d B ~ °o 3 dB " ( 1 + / F ) 2 3b. The Temperature Response Constant temperature sensors can in p r i n c i p l e be used to measure temperature; however, in p r a c t i c e t h i s i s not usually the case. The e f f e c t of temperature o s c i l l a t i o n s of the ambient f l u i d is in pr a c t i c e a contamination of the anemometric response. The t r a n s f e r function of the temperature response i s therefore useful for estimating the contamination of a measured v e l o c i t y spectrum. The s i m p l i f i e d version of the temperature response equation (21), name Iy 9 " + J r fPr6' - iftPr0 = 0 9 = 0 @ n = o' 9 = 1 n -*• °° (24) has been solved by the same routine used to solve the momentum equation (12) The numerical grid extended to n=10 and the derived temperature f i e l d s a t i s f i e s equation (24) to within I part in I0 5 or better. The local heat f l u x response to v a r i a t i o n s of the ambient f l u i d temperature, normalized by i t s quasi-steady value i s F [ f ' ' ( 0 ) ] P r e ' ( o , n , P r ) and is plotted in Figure 8(a), for the Prandtl number range 0.7 to 15. In contrast to the anemometer response, the local heat f l u x response to temperature decreases r a p i d l y with increasing frequency. The phase s h i f t - 67 -gure 8(a). The local heat flux response to a temperature modulation of the ambient f l u i d , normalized by i t s quasi-steady value magnitude and phase. - 68 -+ l.0r — i — i — i — i — i i i i i i i • i i - 0 . 8 - 0 . 4 0.0 0.4 --- , . i /wx D 0.2751 0 Figure 8(b). The local heat f l u x response to a temperature modulation of the ambient f l u i d , normalized by i t s quasi-steady value; magnitude only plotted against a scaled frequency ax i s. - 69 -and the rate of decrease of magnitude (in decibels per decade) do not tend towards an asymptotic l i m i t . The curve for the magnitude of the local heat f l u x f o r various Prandtl numbers nearly c o l l a p s e s into one curve when the frequency axis is scaled by Pr^'^^^, Figure 8(b). The phase, however, does not have a Prandtl number exponent s c a l i n g . The phase s h i f t is large, even at very low frequencies. Consequently there are regions on the plate where the heat f l u x d i r e c t i o n is opposite to that at the leading edge, where the phase s h i f t is zero.. The t o t a l heat f l u x response to v a r i a t i o n s of the ambient f I u i d ' s temperature, normalized by i t s quasi-steady value, is i e 9' (0,n,Pr)d^fT [ f " ( 0 ) ] P r K 2 0 and is plotted in Figure 9(a). The character of the curves changes around 5=1. In t h i s frequency region the local heat f l u x phase delay near the t r a i l i n g edge of the plate exceeds 270°. This local " i n phase" f l u x moderates the rate of f l u x decrease with increasing frequency. For frequencies above £2=1 the r e l a t i v e local heat flux magnitude i s , however, quite small and consequently the t o t a l heat f l u x is proportional to ( \ V~^ for £>> I . A s a t i s f a c t o r y frequency s c a l i n g , f o r both the magnitude and 0 38 phase, is given by Pr , Figure 9(b). There i s , however, some spreading 0 38 of the curves near £Pr ' =| where the character of the individual curves depends strongly on the Prandtl number. The 3 dB bandwidths for the response to temperature o s c i l l a t i o n s in the main stream is •sixg Aousnbajj. pe|90sun Jen|eA Ape8j.s-1 senb s+i Aq p 9 Z | | 9 i i u o u * p i n | 4. + U 8 i q u j e 9144. 4.0 uoj+einpoiu 9jnj rejadwe-j r 9 04. esuodsej xn | 4. 4.9914 194.04. 9U_L • ( 9 ) 6 9 j n 6 1 j TOTAL TEMPERATURE RESPONSE HEAT FLUX IN dB TOTAL TEMPERATURE RESPONSE HEAT FLUX PHASE IN DEGREES - OZ. -• S J X B A o u e n b e j j pe | QDS :en|BA Apee+s - j S 9 n b s+ i Aq pez i | BUJJOU ' p m i j . u^eiqaiB 9114. j.o uo j+Binpoiu 3jn4.BJ8duj8+ e 04. e s u o d s e j xn 14. 4-Beq | B4.04. au,i (q)6 e j n 6 jj TOTAL TEMPERATURE RESPONSE HEAT FLUX IN dB o OJ CO o ro to b i l l ro ro ^ • 1 ± : 1 • 1 - 1 ~g 0 1 OJ — to .g ui UJ = to 0 o o 0 o b 0 b o b 1 1 I 1 I • • — I N O l b 1 b TOTAL TEMPERATURE RESPONSE HEAT FLUX PHASE IN DEGREES - \L -- 7 2 -r , ML , n cc o -0.38 ? 3 dB = IT = °'56 P r o 3 d B This makes the radian frequency temperature band-width 0 . 5 6 U Q / L at Pr-and 0 . 2 5 U Q / L at Pr= 8 . In contrast, the corresponding values for the anemometric frequency response are 2 . 8 U Q / L and 1 .7 U Q / L , larger by a factor of 5 and 6 . 8 r e s p e c t i v e l y . - 73 -4. Cone I us ions The unsteady viscous stress computed numerically from a s i m p l i f i e d unsteady momentum equation that is not r e s t r i c t e d by frequency compares favourably with the frequency r e s t r i c t e d r e s u l t s of Cheng and E l l i o t t (1957) and LighthiI I (1954). When the boundary layer is unsteady (ux/U^>> I) the equation governing the heat f l u x from the plate in response to speed o s c i l l a t i o n s in the main streams i s s i g n i f i c a n t l y d i f f e r e n t from the equation governing the response to temperature o s c i l l a t i o n s in the main stream. Both equations are also d i f f e r e n t from the equation governing the response to o s c i l l a t i o n s of the plate's temperature. The unsteady heating source that generates the anemometric response is i n t r i n s i c a l l y d i s t r i b u t e d throughout the boundary layer and has a p r o f i l e that depends strongly on the Prandtl number and the frequency of the speed o s c i l l a t i o n . The heating source that generates the temperature response, on the other hand, is constant for a l l frequencies and is at the boundary y=°° f o r external temperature o s c i l l a t i o n s and at y=0 f o r internal temperature o s c i l l a t i o n s . The anemometric heat f l u x response increases in magnitude with increasing frequency (from i t s quasi-steady value of 0.5) to a maximum value. The heat f l u x then decreases with increasing frequency to an asymptotic value proportional to ( i f t ) - ' f o r the local heat f l u x and (i?) 2 for the t o t a l heat f l u x from the plate. The location and the value of the heat f l u x maxima depend on the Prandtl number. The largest maximum occurs at a Prandtl number of 2. The phase of the heat flux is always negative. - 74 -The local heat f l u x in response to external temperature o s c i l l a t i o n s decreases with increasing frequency and has neither an asymptotic logarithmic rate of decrease nor an asymptotic phase s h i f t at high frequencies. The t o t a l heat f l u x in response to temperature o s c i l l a t i o n s decreases ra p i d l y from i t s quasi-steady value with i n -0 38 creasing frequency up to the frequency 5=1.4 Pr ' and then becomes 0 38 —— proportional to (£Pr ' ) 2 for larger frequencies. The -3 dB bandwidth is 3.7(l+2/Pr)/(l+/Pr) 2 for the t o t a l —0 38 heat f l u x in response to speed o s c i l l a t i o n s and is 0.56Pr * for the t o t a l heat f l u x in response to temperature'osci I I at ions in the main stream. Frequency response c a l i b r a t i o n s made by internal heating methods cannot c o r r e c t l y measure the response to speed or temperature o s c i l l a t i o n s in the main stream, for constant temperature anemometers or thermometers, when the boundary layer is not quasi-steady. - 75 -Bi bI ioqraphy Ascher, U., J . Christiansen and R.D. R u s s e l l . 1977. A c o l l o c a t i o n solver fo r mixed order systems of boundary value problems. Technical Report 77-13, Department of Computer Science, The University of B r i t i s h Columbia. Bellhouse, B.J. and D.L. Schultz. 1967. The determination of f l u c t u a t i n g v e l o c i t y in a i r with heated t h i n f i l m gauges, Journal of F l u i d Mechanics, 29, 2, pp. 289-295. Cheng, Sin-I and D. E l l i o t t . 1957. The unsteady laminar boundary layer on a f l a t plate. Transactions of the American Society of Mechanical Engineers, 79: 725-733. Evans, D. 1963. An instrument for the measurement of ocean turbulence, Technical Memorandum 63-8, P a c i f i c Naval Laboratory, Defense Research Board, Canada. Freymuth, Peter. 1978. A comparative study of the signal-to-noise r a t i o of hot-film and hot-wire anemometers. J. Physics E: S c i . Instrum. I N 915-18. Gargett, A.E. 1976. An investigation of the occurrence of oceanic turbulence with respect to f i n e s t r u c t u r e . J. of Phys. Oceanog. 6:2, 139-156. Grant, H.L., R.W. Stewart and A. Moi I Met. 1962. Turbulence spectra from a t i d a l channel. J . o f Flu i d Mech. J_2: 241-268. Grant, H. L. , A. MoiI I iet and W.M. Vogel. 1968. Some observations of the occurrence of turbulence in and above the thermocline. J . of Fl u i d Mech. 34:443-448. H i l l , P.G. and A.H. Stenning. I960. Laminar boundary layers in o s c i l l a t o r y flow. Transactions of the ASME Series D, pp. 593-608. - 76 -L i g h t h i l l , M.J. 1954. The response of laminar skin f r i c t i o n and heat t r a n s f e r to f l u c t u a t i o n s in the stream v e l o c i t y . Proceedings of the Royal Society of London, Series 4, 224: 1-23. Nowell, Arthur R.M. 1974. Some response c h a r a c t e r i s t i c s of parabolic hot films in water. J. of Hydronautics 8_: 4, pp. 169-171. Ostrach, Simon. 1955. Compressible laminar boundary layer and heat t r a n s f e r for unsteady motions of a f l a t plate. National Advisory Committee for Aeronautics, Technical Note 3569. Rott, Nicholas and Martin L. Rosenzweig. I960. On the response of the laminar boundary layer to small f l u c t u a t i o n s of the free-stream v e l o c i t y . J. of Aeronautical and Space S c i . 27: 741-747. S c h l i c h t i n g , M. 1955. Boundary Layer Theory, Pergamon Press, 535 pages. - 77 -AcknowIedqements I would l i k e to express appreciation to the following: Dr. U. Ascher and Mr. Chris Hermansen of the Computing Science Department of the University of B r i t i s h Columbia f or providing the numerical routines for solving the equations and for other co-operation that they have shown; Professors P.H. LeBlond and L.A. Mysak of the Instit u t e of Oceanography at U.B.C. for t h e i r reviews and c r i t i c i s m s ; and Professor T.R. Osborn for his encouragement and support throughout t h i s work. This work has been supported by the O f f i c e of Naval Research, Contract Number 00014-083-207. - 78 -Appendix B The CaI i b r a t i o n of a Hot Film Turbulence Probe by Rolf G. Lueck Ins t i t u t e of Oceanography University of B r i t i s h Columbia Vancouver, B.C., V6T IW5 Canada - 79 -Abstract The steady flow c a l i b r a t i o n data of two platinum t h i n f i l m turbulence probes show that t h e i r thermal conductance Q/AT is not pro-i portional to U O 2 as expected for a constant temperature conical surface. A thermal model of the probes that agrees favourably with the c a l i b r a t i o n data indicates that t h i s non-proportionality is due to the thermal e f f e c t of the glass coating over the f i l m and glass substrate below i t . The unsteady heat f l u x from the f i l m in response to v a r i a t i o n s in the ambfent f l u i d ' s speed increases with increasing frequency above 100 Hertz. It is shown that the unsteady heat f l u x is proportional to the unsteady viscous surface stress over the f i l m which increases in magnitude with increasing value of the frequency parameter U J L / 3 U O . The r a t i o of the temperature-to-velocity s e n s i t i v i t y is highest at zero frequency. - 80 -I . Introduction The steady and unsteady c a l i b r a t i o n of two (similar) conical platinum t h i n f i l m turbulence sensors used for oceanic measurements are discussed in t h i s paper. Comparable probes have been used by Grant, Stewart and M o i l l i e t (1962) to derive the Kolmogorov spectrum from d i r e c t turbulence measurements in a t i d a l channel, and by Grant, M o i l l i e t and Vogel (1968) to measure turbulence in and above the thermocline. More recently, the same probes discussed here have been mounted on a towed body for open ocean measurements (Nasmyth, 1970) and have been mounted on a small submersible for coastal and estuarine studies (Gargett, personal communication). The most outstanding feature of the conical hot f i l m probe is i t s high frequency response. The frequency response is c a l i b r a t e d from 10 to 1000 Hertz by v i b r a t i n g the probe in the d i r e c t i o n of the steady flow in a low turbulence water tunnel. A l l of the c a l i b r a t i o n data presented here have been taken by the Ocean Mixing Group of the Institute of Ocean Science at P a t r i c i a Bay, B r i t i s h Columbia, Canada. In c a l i b r a -t i o n , the modulation in v e l o c i t y , as seen by the probe, a r i s e s from the o s c i l l a t i o n of the probe p a r a l l e l to a steady stream. However, during f i e l d measurements the modulation in v e l o c i t y stems from the o s c i l l a t i o n of the ambient f l u i d speed parallel- to the steady motion of the probe. This flow d i f f e r e n c e is not important; the behaviour of the boundary layer is the same in both cases when the f l u i d is incompressible ( L i g h t h i l l , 1954). Hence, c a l i b r a t i o n by probe v i b r a t i o n c o r r e c t l y simulates the probe's frequency response to ambient f l u i d speed o s c i l l a t i o n s . - 81 -It is more common to measure the frequency response of heated sensors by e l e c t r o n i c a l l y modulating i t s internal temperature in the presence of steady streaming (.Freymuth, 1977, 1978). However, t h i s method does not measure a probe's frequency response when the boundary layer is not quasi-steady (.Lueck, 19791. The unsteady c a l i b r a t i o n data presented indicates that the boundary layer is not quasi-steady at frequencies above approximately |QO Hertz. A reproduction of the probes is in Figure I. The basis of the probe is a blunt glass cone having a half-angle of 15 degrees. The sensing element is a narrow t h i n platinum f i l m which is deposited around the e n t i r e circumference at a position 'downstream' from the cone's apex. E l e c t r i c a l contact with the f i l m is made on opposite sides of the cone via two platinum sponges in the glass substrate. That portion of the f i l m that overlays the sponges (hashed area in Figure I) is not considered as part of the fi l m ' s sensing surface. A t h i n quartz layer of unknown thickness covers the e n t i r e probe to provide e l e c t r i c a l i n s u l a t i o n from the f l u i d being probed. The probes were made under special contract at the Un i v e r s i t y of B r i t i s h Columbia. The probes are operated in a constant, temperature mode by an AC servo-amplifier centered at 12.5 kHz. A d e t a i l e d account of the probe's e l e c t r o n i c s and i t s operation has been given by Evans (.1962),. The steady flow c a l i b r a t i o n curves for the two probes (V-31 and V-38) are given in Figure 2. Plotted is the probe's thermal conductance Q/AT against the square root of the Reynolds number where 0 is the e l e c t r i c power d i s s i p a t e d in the f i l m and AT is the temperature d i f f e r e n c e - 8 2 -V-31 d/L" 0.047 Film Surface Area s 0.92(2x)Ldsin4> « 1.5 x 10- 3 cm 2 V - 3 8 Hypothetical Apex x • 0 L • 1.38 mm L! - 1.17 mm d «0.056 mm Film Surface Area « 0.94 (2r)Ldsina> A s k e t c h o f p r o b e s V - 3 1 a n d V - 3 8 , t h e i r r e l e v a n t d i m e n s i o n s a n d t h e c o o r d i n a t e s y s t e m e m p l o y e d i n t h e a n a l y s e s . - 83 -0.016-^ 0.0I2H W l 1 O o 0.008H ° 5 0.004 V-31 AT o 1973 April 6B 20.0 °C A . 1973 April 5 20.0 °C + 1973 April 6A 18.5 °C x 1973 April 4 18.5 °C 20 40 DJ/2 . (H|ty / 2 60 2(a) The probes' thermal conductance Q/AT versus R 2 tor V-31 e - 84 -probes' thermal conductance Q/AT versus R * for - 85 -between the f i l m and the ambient water. The Reynolds number is defined U L as — — where U q is the steady ambient f l u i d speed r e l a t i v e to the probe, L the distance from the hypothetical apex to the s t a r t of the f i l m (see Figure I) and v is the kinematic v i s c o s i t y of the ambient f l u i d . The heat flux from an ideal constant temperature axi-symmetric surface i s proportional to the square root of the Reynolds number (White, 1974, p. 282) i f the usual boundary layer approximations are v a l i d . Bellhouse and Schultz (1967) reported a I inear re I ationship between the heat f l u x and the square root of the ambient f l u i d ' s speed f o r a two-dimensional sensor in a i r . However, the data in Figure 2 show curvature at large Reynolds numbers and a systematic deviation away from a s t r a i g h t l i n e for Reynolds numbers below 500. This deviation at lower Reynolds numbers cannot be a t t r i b u t e d to a fundamental change in the Nusselt number - Reynolds number r e l a t i o n s h i p of the flow. Although changes in the Nusselt number - Reynolds number r e l a t i o n s h i p have been reported (Hinze, 1975), they appear to be r e s t r i c t e d to Reynolds numbers below 0.5. In a d d i t i o n , Kramers (1946) reported for spheres a li n e a r r e l a t i o n -ship between the Nusselt number and the square root of the Reynolds -number to Reynolds numbers as low as 50. Tan-atichat, Nagib and P l u i s t e r (1973) had s i m i l a r d i f f i c u l t i e s with f i t t i n g the heat f l u x t o the square root of the speed, for a coated c y l i n d r i c a l sensor in water. They found that p l o t t i n g the heat f l u x against U m where m = 0.225 gave a moderately successful f i t to the data; however, no physical explanation was given for t h e i r choice of m. An explanation for the observed steady flow response is presented in Section 2. - 86 -A plot of the unsteady response of probe V-38 is shown in Figure 3. Plotted is the probe's unsteady voltage-to-speed amplitude r a t i o in a r b i t r a r y units against the frequency, in Hertz, of the speed o s c i l l a t i o n . The lines j o i n i n g the data points are a casual f a i r i n g of the data. The unsteady response, at a given steady speed, decreases s l i g h t l y with increasing frequency between 10 and 50 Hertz and then increases sharply with increasing frequency up to 1000 Hz. The character of the frequency response depends strongly on the ambient f l u i d ' s speed. A more s i g n i f i c a n t response decrease at low frequencies has been reported by Bellhouse and Schultz (1967) for a two-dimensional probe in a i r . These authors a t t r i b u t e d the response decrease to an attenuation of heat waves through the probe's substrate and predicted the e f f e c t to be much smaller in water than in a i r . A small d i f f e r e n c e between the quasi-steady response of a hot wire anemometer c a l c u l a t e d by d i f f e r e n t i a t i n g the steady response and the low frequency response observed by v i b r a t i n g the probe has also been reported by Morrison, Perry and Samuel (.1972). Their c a l i b r a t i o n s , however, were r e s t r i c t e d to one frequency. An a n a l y s i s of the bridge c i r c u i t r y by Evans (1963) indicates that the increase in response with increasing frequency above 50 Hz cannot be a t t r i b u t e d to the probe's e l e c t r o n i c s and must therefore represent a real hydrodynamic e f f e c t . Evans (1963) showed that the heat f l u x at high frequencies may be connected to the unsteady surface shear. The nature of the probe's unsteady response is examined in Section 3. - 87 -CO r o o in o ^ CM o> CM £ O CM if) CD u _o • * O X o o o m ro NouvinaoiAi a33ds J O 3an±ndwv U3d 3DV110A 390188 A0V31SNH The unsteady probe vol tage per amplitude of speed modulation in a rb i t r a r y un i t s aga ins t the frequency of the speed modulation in Her tz . The l i nes are casual f a i r i n g s to the data. - 88 -2. Steady Response / The steady thermal network connecting the constant temperature f i l m to the ambient f l u i d w i l l be modelled by 1) a resistance due to the glass coating over the f i l m which is in s e r i e s w i th 2) the thermal resistance of the f l u i d boundary layer over the f i l m and by 3) a shunt resistance through the glass substrate which i s p a r a l l e l to the coating and the boundary layer resistance. The e l e c t r i c a l analogue of the proposed thermal network is shown in Figure.4 where the subscripts f, c and s refer to the f l u i d , the coating and the substrate r e s p e c t i v e l y . Because the heat f l u x through the substrate Q s eventually enters the ambient f l u i d , i t is to be expected that t h i s heat flux a l s o encounters some thermal resistance from the boundary layer denoted by ' in Figure 4. The s l i g h t decrease in the frequency response between 10 and 50 Hertz is evidence for a small boundary layer e f f e c t in the substrate heat path. However, the bulk of the thermal resistance of the substrate must come from the region near the f i l m where the f l u x lines are r e l a t i v e l y concentrated. The thermal resistance R^' is determined by the surface area through which the substrate's heat f l u x Q s passes upon entering the f l u i d , by the thermal condu c t i v i t y of the f l u i d and by the e f f i c i e n c y of forced convective heat t r a n s f e r in the boundary layer. Because t h i s surface area is comparable to or greater than the surface area of the f i l m R^'sR . Moreover, the c o n d u c t i v i t y of water at 20°C i>5.9xl0 - 3 watts/ °C-cm) is comparable to the c o n d u c t i v i t y of glass (M.OxlO - 2 watts/°C-cm) whereas the conductivity of a i r is much smaller (only ^2.5x1 0 - l + watts/°C-cm). Subsequent analyses show that for the probes in question R + < < ;R e and hence Hot Film Shunt - 90 -R f ' < < R s i n w a t e r . In a i r , however, i t i s p o s s i b l e t h a t R.p'M^. In t h e r e m a i n i n g work R^' i s i g n o r e d r e l a t i v e t o R g, t h e s u b s t r a t e ' s r e s i s t a n c e , The probe's c o a t i n g r e s i s t a n c e i s a p p r o x i m a t e l y R c =TS <"> c where D . i s t h e c o a t i n g t h i c k n e s s , X c i s i t s t h e r m a l c o n d u c t i v i t y and S i s t h e s u r f a c e a r e a o f t h e f i l m . The r e s i s t a n c e o f t h e boundary l a y e r c a n be w r i t t e n as R, = a e i f i t i s dominated by f o r c e d c o n v e c t i o n , where R g i s t h e R e y n o l d s number and a , which w i l l be d e r i v e d l a t e r , i s a f l o w i n d e p e n d e n t c o n s t a n t t h a t depends on t h e p r o p e r t i e s o f t h e f l u i d and t h e p r o b e ' s geometry. The a p p r o p r i a t e n e s s of e q u a t i o n 2 has been d e m o n s t r a t e d by W h i t e CI974, p. 282), The e q u i v a l e n t impedance o f t h e c i r c u i t i n F i g u r e 4 w i t h R f'=0 i s Z = £ L = C + - A _ (3) A + SR~ e R R where C = c s R + R ' (4a) c s ctR 2 B = 2 ' ( 4 b ) (R + R ) c s A = R + R ' (4c) s c AT i s t h e t e m p e r a t u r e d i f f e r e n c e between t h e f i l m and t h e ambient f l u i d and Q i s t h e h e a t f l u x from t h e f i l m . A t an i n f i n i t e l y l a r g e R e y n o l d s - 91 -number equation 3 gives a thermal resistance equal to the p a r a l l e l equivalent of the coating and substrate resistance and at very small Reynolds numbers a resistance equal to the substrate resistance. Therma convection, which must be present when R <I, is nor considered. e~ The a v a i l a b l e steady flow c a l i b r a t i o n data for 1 probes V-31 and V-38 has been f i t t e d to equation 3 by a least square error routine to determine the constants A, B and C. A l l measurements were made at an ambient water temperature of 20°C. The data are plotted in Figure 5. The s o l i d l i n e is the thermal resistance Z=AT/Q determined by the least square routine and the symbols represent the observed resistances. The q u a l i t y of the f i t in terms of the percent d i f f e r e n c e between the f i t and the observed values is plotted near the top of the graphs. The f i t t e d impedance of the probes are AT _ „ . 3220_ A e f o r V-31 and ~- = 25.A + = ^ °C/watt (5a) Q 7.77 t v/R-^ = 1 2 . 5 + 4 4 1 0 °C/watt (5b) V 12.3 t /R~ e f o r V-38.. Table I summarizes the experimentally determined values of R g, R c and a. The data of zero flow were not used to c a l c u l a t e the least square f i t . Some care was taken to attempt to minimize thermal convection at zero flow speeds. AlI of the data for V-31 agree to better than 4% and a l l except three of the lower speed points agree to better than 2% with the f i t t e d curve. A l l of the data for V-38 agrees to better than 0.8$ with the f i t t e d curve. The two data sets for V-38 were taken one 5(a) The probes' steady thermal resistance AT/Q versus R 2 . The s o l i d e lines represent the least square f i t of the thermal model to the data. The q u a l i t y of the f i t is indicated near the top of the f i gure; for V-31. - 93 -40Ch 0.8-1 0 A o o CD © CD A A A -0.4-r r o cr r r V - 3 8 © 1971 Dec. 21 20.82 °C A 1972 Jan. 18 20.02 °C £ 1 . | 2 5 + _ ^ 4 j p _ Q 12.3+VR e 5(b). The probes' steady thermal resistance AT/Q versus R g 2. The s o l i d lines represent the least square f i t of the thermal model to the data. The q u a l i t y of the f i t is indicated near the top of the f i g u r e ; for V-38. - 94 -V - 3 1 A B C S 7.77 R = 440 C/watt 3220 C/watt 25.4 °C/watt (0.92)2TrLdsin<f> = I . 50 x IO" 3 cm 2 R = 27 C/watt a = 3630 C/watt A = 12.3 B = 4410 °C/watt C = 12.5 °C/watt S = (0.94)27rLdsin<)) V-38 R s R c a 1.20 x IO" 3 cm 2 370 °C/watt 13 °C/watt = 4720 °C/watt Table Bl Thermal p r o p e r t i e s of the hot f i l m probes derived from f i t t i n g the thermal model to the steady c a l i b r a t i o n data. - 95 -month apart and show a systematic d i f f e r e n c e of approximately 0.8%. It appears that the d i f f i c u l t y with f i t t i n g the probe's heat f l u x to a i proportionate r e l a t i o n s h i p with U* i s due t o the thermal e f f e c t of the coating and the substrate. The coating thickness over the probes, using equation (I) and the surface areas l i s t e d in Table I, is 4.1 x I0~k cm f o r V-31 and 1.6 x 10 ^ cm for V-38. The coating thickness on these probes is other-wise unknown but two commercial manufacturers of s i m i l a r probes coat t h e i r s to a nominal thickness of 2 x 10 - 1 + cm. The substrate resistance should be given by D' R = ~ 7 f (6) s A S s where D' is a c h a r a c t e r i s t i c depth of thermal penetration and should be comparable to the width of the f i l m . Using equation (6) gives D' = 67 x 10 _ 1 + cm for V-31 which is almost exactly equal to the width of the f i l m and D' = 46 x 10 _ 1 + cm f o r V-38 which is 20$ less than the film's width. 2(a). Boundary Layer Resistance The hydrodynamic equations f o r the probe's boundary layer, using the coordinates shown in Figure I, are (White, 1974, p. 340) | - (Ru ) + R | - v = 0 (7) 9x o 9y o 9u 9u 9 2u dV 0 , 0 0 , ., 0 u - — + v T — = v — — + V - — (8) o 9x o 9y . 2 o dx 7 9y u (x,y) = v (x,y) = 0 , u (x,°°) = V (x) o ' • o ' o . o - 96 where V q i s t h e s t e a d y p o t e n t i a l f l o w a t t h e probe's s u r f a c e , U q and V q a r e t h e s t e a d y boundary l a y e r v e l o c i t i e s and R = R ( x ) i s t h e r a d i u s o f t h e pro b e . The p o t e n t i a l f l o w o v e r an i d e a l a x i s y m m e t r i c cone i s ( W h i t e , 1974, p. 341) V Cx) o Ex 19) where E i s a c o n s t a n t and n=0.035 f o r a cone o f h a l f - a n g l e 15°. E q u a t i o n 8 r e d u c e s t o t h e o r d i n a r y d i f f e r e n t i a l e q u a t i o n ( W h i t e , 1974, p. 242) t i t + f f '» . 2n ( l - f 2 ) 0 where 3+n f ' ( 0 ) = f ( 0 ) = 0 f ' ( » ) = I u 0 < x , y ) . = V q ( X ) f ' ( n ) = E x n f ' ( n ) (10) , o ( x , y ) = - 2 r- n- I r n+3 , , n-1 , t • , 3 ^ Ex { — f + — n f } ( I I ) (12) n = y and p r i m e d e n o t e s d i f f e r e n t i a t i o n w i t h r e s p e c t t o r\. E q u a t i o n 10 i s i d e n t i c a l t o t h e e q u a t i o n f o r t w o - d i m e n s i o n a l f l o w o v e r a wedge o f h a l f -a n g l e 2.1 d e g r e e s . S o l u t i o n s t o e q u a t i o n (10) f o r v a r i o u s v a l u e s of n can be found i n White (1974, p. 276) and S c h l i c h t i n g (1955, p. 129). A t t h e b l u n t t i p o f t h e cone and a t t h e r e a r o f t h e c o n i c a l s e c t i o n t h e p o t e n t i a l f l o w a t t h e s u r f a c e c a n n o t be r e p r e s e n t e d by e q u a t i o n 9. Over most o f t h e c o n i c a l s e c t i o n o f t h e probe e q u a t i o n 9 s h o u l d be a c c u r a t e , and because n=0.035«l t h e p o t e n t i a l f l o w V q s h o u l d not d i f f e r s i g n i f i c a n t -l y f r o m t h e f r e e - s t r e a m speed U . However, a d e t a i l e d a n a l y s i s o f t h e p o t e n t i a l f l o w o v e r p r o b e s V—31 and V - 3 8 has not been made. - 97 -The e q u a t i o n g o v e r n i n g t h e boundary l a y e r t e m p e r a t u r e T Q ( x , y ) , measured w i t h r e s p e c t t o t h e ambient f l u i d ' s s t e a d y t e m p e r a t u r e , i s ( L i g h t h i l l , 1954) u -r— + v T K = 0 (14) o 3x o 9y 2 9y where 6 = A T q - T ( x , y ) , A T q i s t h e t e m p e r a t u r e d i f f e r e n c e between t h e pr o b e ' s w e t t e d s u r f a c e and t h e amb i e n t f l u i d , and K i s t h e t h e r m a l d i f f u s i v i t y o f t h e f l u i d . Assuming n e g l i g i b l e h e a t i n g upstream o f t h e f i l m and a u n i f o r m t e m p e r a t u r e A T q a t t h e we t t e d s u r f a c e o v e r t h e f i l m , t h e boundary c o n d i t i o n s on e q u a t i o n 14 a r e = 0 a t y = 0 L s x s L t d = A T f o r a I I x<L o = A T as y->-°° f o r a l l x. o ' (15) Because t h e momentum boundary l a y e r s t a r t s a t t h e t i p o f t h e cone whereas t h e t h e r m a l boundary l a y e r s t a r t s a t t h e f i l m (x = L) t h e s i m i l a r i t y v a r i a b l e r\ i s not a p p r o p r i a t e f o r t h e t e m p e r a t u r e e q u a t i o n ( 1 4 ) . F o l l o w i n g Kays (1966) l e t t h e v e l o c i t y p r o f i l e be a p p r o x i m a t e d by t h e c u b i c p a r a b o l a u ( x , y ) . 3 o where 6 i s an a r t i f i c i a l v i s c o u s boundary l a y e r t h i c k n e s s . The c h o i c e 6 = 3 .04 / — r (17) (3+n) E X makes t h e s u r f a c e v e l o c i t y g r a d i e n t c a l c u l a t e d from e q u a t i o n (16) a g r e e w i t h t h e v a l u e t a b u l a t e d i n Whit e (1974, p. 2 7 0 ) . Assuming a s i m i l a r p r o f i l e f o r t e m p e r a t u r e we can w r i t e - 98 -AT 1 ( ) 2 5 j (18) where 6 y is the a r t i f i c i a l thermal boundary layer thickness. The thermal-to-viscous boundary layer thickness r a t i o r = ^ / s is (Kays, 1966, p. 218) r = JT Pr 1/3 (19) where ! < ^  ) - 3/4 (20) and Pr = V / K is the Prandtl number. The steady heat f l u x from the probe's surface can be calculated using equations 17, 18 and 19, and is , AT 3 o X fPr 1/3 2 3.04. y=0 /3+n j 2 X [ l - ^ c ] 1 / 3 x 'U x 0 (21 ) where Xf is the thermal c o n d u c t i v i t y of the f l u i d . The v a r i a t i o n of the heat flux over the wetted surface is dominated by the fac t o r 1/3 ( - ( k ) c ) x because the f i l m is r e l a t i v e l y narrow (d/L << I). Accordingly, x can be set equal to L where i t occurs outside of the above fa c t o r and t h i s f a c t o r can be set equal to (cB / L ) l / / 3 where B = x - L << L. "The t o t a l heat f l u x into the f l u i d above the f i l m is then obtained by integrating equation 21 over the surface area of the f i l m ; L+d L+d O o ( x ) d x = ( 0.'94 ) 2 7 l L s i n * 0 (x)dx (22) o L 3.04 2 ' v3+n 9 , 0.92 2 T r L s i r i * ,3+n/'\ 4 / / J , d , T , " ( 0.94 ) ^ nA ( — > (r> A.AT R / P r V 3 d 2 / 3 — ) L f o e 1/3 - 99 -The thermal resistance c o e f f i c i e n t , equation 2, is }o_ 9 .0.92. 2TTLSinj> ,3+n. l / 2 4 , l / 3 , d 2 / 3 _ ^ 3  D i 4 ^0.94; 3.04 1 2 ; C3+n ; V A f r ' (23a) a" 1 AT R o e which f o r both probes equals (to within \%) I.53 Xf L Op Pr . (23b) Using the probes' dimensions (figure I), X^ = 6.3 x I0~ 3 watts/ °C-cm (appropriate for water at 40°C), and a Prandtl number of 7.1 (water at 20°C) in equation 23 predicts values f o r a of 2880°C/watt for V—31 and 3320°C/watt for V-38. The experimentally derived values for a are 3630 and 4720 °C/watt. The boundary layer heat f l u x infered from f i t t i n g the thermal model to the c a l i b r a t i o n data i s smaller than the c a l c u l a t e d heat f l u x by 2\% for V-31 and 30$ for V-38. Some of t h i s discrepancy i s due to a v a r i a b l e Prandtl number in the thermal bounday layer. At 40°C the Prandtl number is 4.3, for instance. In addition the o r i g i n of the momentum boundary layer has been assigned to the hypothetical apex (x=0) rather than the real (blunted) apex. Placing the o r i g i n of the momentum boundary layer at the real apex would accentuate, the discrepancy whereas decreasing the Prandtl number to account f o r the warming of the thermal boundary layer would diminish the discrepancy. Some implications of the proposed thermal model can now be examined. Oceanic measurements with V-31 and V-38 are t y p i c a l l y taken at 130 cm/sec, This corresponds to a Reynolds number of approximately 1800. A t y p i c a l value for the boundary layer resistance at that speed is - 100 -Rx = -2- = 85°C/watt. e Taking R c = 27 and R g = 440°C/watt (Table I) one c a l c u l a t e s the temperature drop across the coating, AT c, from R AT = AT Q x D ~ 0.24AT. c R, + R f c Twenty-four percent of the film-to-ambient f l u i d temperature drop occurs across the coating. The boundary layer resistance c o e f f i c i e n t , a, is considerably larger in a i r than in water. Placing the appropriate values of f l u i d c o n d u c t i v i t y , X^, and Prandtl number into equation 23 gives a . = I .7 x |0 5 °C/watt a i r A Reynolds number based on a more representative a i r flow speed of 10 meters per second gives R f * 4500 °C/watt. a i r According to the model the temperature drop across the coating in a i r then is A T c = A T ^ S o O = 6 X l 0 " 3 A T -Only 0.6 percent of the film-to-ambient f l u i d temperature drop occurs across the coating when the same probes are operated in a i r . The coating's thickness and the coating's thermal conductivity are obviously much more important to a probe's heat f l u x response in water than in a i r . - 101 -A f r a c t i o n of the heat flux from the f i l m goes through the substrate and consequently contributes nothing, i f R |<<R^, to the probe's speed s e n s i t i v i t y . Let Q G denote the substrate f l u x and 0 Q +he boundary layer f l u x (Figure 4). The f l u x r a t i o is 0 R, + R i s _ f c 0 R o s For water at U q = 130 cm/sec, Q S/Q Q is approximately 0.25; i . e . 20$ of the t o t a l flux from the f i l m goes into the substrate. In contrast, QS/QQ should be approximately 14 in a i r , i n d i c a t i n g that more than 90$ of the t o t a l f l u x is into the substrate. However, t h i s part of the model is n6t completely app l i c a b l e in a i r because R| may be comparable to R . Nonetheless, the same probes operated in a i r s would lose a s i g n i f i c a n t amount of speed s e n s i t i v i t y because of conduction into the substrate. A t h i n f i l m turbulence probe s u i t a b l e for use in water may be unsuitable for use in a i r . The quasi-steady s e n s i t i v i t y to both speed and temperature f l u c t u a t i o n s in the ambient f l u i d are now r e a d i l y attained. The quasi-steady s e n s i t i v i t y to speed is 90 = 9Q_ ^ e = 9£_ 9U 3R 9U U 9R o e o o e and to temperature is 9Q = _Q 9(AT) " AT -The r a t i o of the quasi-steady speed-to-+emperature s e n s i t i v i t y is - 102 -3Q/3U R o _ _e 3Q , AT . 3Q/3(AT) Q 3R 1 U e o B R 2 A T e , AT 1 1 71 I (A+R 2)(AC+B+CR 2) o The ideal conical probe has no coating resistance and an i n f i n i t e substrate resistance, i .e. A = C = 0. Thus the optimum r e l a t i v e s e n s i t i v i t y is AT/2U . The real probe (V-31) has at U =130 cm/sec o o a r e l a t i v e s e n s i t i v i t y of 0.61 o which is 6\% of the optimum s e n s i t i v i t y at 130 cm/sec. Using AT = 20°C gives 3Q/3U 3Q/9(AT) = ° - 0 4 7 ° C P e r c m / s e c which means that the probe cannot d i s t i n g u i s h a quasi-steady f l u c t u a t i o n of 0.047°C from a f l u c t u a t i o n of I cm/sec at 130 cm/sec. - 103 -3. Unsteady Response 3(a). Viscous Boundary Layer If the ambient f l u i d ' s speed, r e l a t i v e to the probe, has an i n f i n i t e s i m a l sinusoidal o s c i l l a t i o n about i t s steady mean, then one can write the potential flow at the probe's surface as V(x,t) = V (x)(l+ee"° +) = Ex"(I+ee' W +) = U (l+ee' W +) (24) o o where co i s the angular frequency in radians/sec and where E , small compared to unity, is the normalized speed amplitude of the probe shaker used f o r response c a l i b r a t i o n s . The boundary layer v e l o c i t y w i l l c o n sist of a steady mean v e l o c i t y plus a sinusoidal perturbation, i.e. u(x,y,t) = u Q(x,y) + ee"° + u (x,y,w) v(x,y,t) = v (x,y) + ee i < i ) + V ^ X ^ O J ) (25) The appropriate perturbation momentum equation, to f i r s t order in e is ( L i g h t h i I I , 1954) 3u 3u, • 3u 3u icou + u - — L + v - — + u - — + v . - r — 1 o3x o3y l3x I3y dV 3 2u = iwv + 2V -T-9- + v h (26) o odx . 2 3y with boundary conditions U ^ X J O J O J ) = v ( X , O , O J ) = 0 u, ( X , « > , O J ) = V (x) i o If we l e t u1(x,y,u>) = V Q (x)g'(n,fi) = Ex ng'(n,ft) (27) - 104 -where ^~^T~ > then c o n t i n u i t y o |- (Ru. ) t R — = 0 9x 1 9y requires that v ^ x ^ i o ) = ~Jy^ Ex { — g t ( — ) n g + (|-n) Q (28) Substituting the above expressions f o r u^, V j , and P. into the perturba-t i o n momentum equation (26).and using the expressions for u • and V Q given in equations (II) and (12) y i e l d s g , " + f g " " ^  ^ n f ' t 3iQ} g ' + f " g = = z 2 _ { 2 n + 3 I n ) . 2 C J ^ n . ) n | _ I f , , g _ f , g , } ( 2 9 ) with boundary conditions g(o,PJ = g'(o,fi) = 0 g ' < « , n ) = I g(n,0) = ±(f + n f ' ) . The presence of n in the momentum equation (29) re s u l t s from the steady pressure gradient along the cone's surface. Its e f f e c t r e l a t i v e to a zero pressure gradient flow, is an increase in the f l u i d ' s i n e r t i a (the t h i r d term on the left-hand side in (29)) and an increase in unsteady pressure gradient (the f i r s t term on the right-hand s i d e ) . The steady pressure gradient'a Iong a cone of half-angle 15° is evidently quite small because n = 0.035. Thus n can be ignored r e l a t i v e to 3 and, because 0 < f' < I, 2n can also be ignored r e l a t i v e to 3ift when - 105 -ft >> 2n/3 = 0.023. Within the above c o n s t r a i n t s , the perturbation momentum equation (29) s i m p l i f i e s to g M ' + f g " - 2iftg' + f " g = -2 ift + § ft (f'»g-f 'g') (30) g(0,ft) = g'(0,ft) = 0 g'(»,«) = I g(n,0) = ±(f+nf'). Except f o r the d e f i n i t i o n of the var i a b l e s n and ft equation (30) is ident i c a l to the corresponding perturbation momentum equation for two-dimensional flow over a f l a t plate (Lueck, 1979). The s o l u t i o n f o r the unsteady flow over a plate also a p p l i e s to the unsteady flow over a cone of half-angle 15°, within the l i m i t s c i t e d . 3(b). Thermal Boundary Layer Let . w + T(x,y,t) = T (x,y) + ee T^(x,y,ui) where T q is the steady temperature f i e l d given e a r l i e r and Tj is the perturbation temperature f i e l d due to main stream flow osciI I at Ions. The governing equation f o r T^, to f i r s t order in z, is ( L i g h t h i l l , 1954) 3T, 9T 9 2T 3T 9T IOJT + u — — + v - — - K — = -u, - — - v, - — (32) 1 o 9x o 9y 2 1 9x 1 9y dy (a) (b) (c) (d) T = 0 at y = 0 and ». ( - 106 -The a r t i f i c i a l concept of a f i n i t e boundary layer thickness w i l l be used to solve T because TI cannot be a simi l a r i t y v a r i a b l e f o r T , as 1 1 was the case for T . o Let Tj (x,y,o>) = Tj (5,ft) where, using equations (13), (17) and (19) ^ 6 r6 3.04r T and 0 < £ < I • The steady viscous boundary layer thickness, 6, is e s s e n t i a l l y constant over the f i l m because the aspect r a t i o d/L<<l. Hence, x can be set equal to L except where i t appears in r. The d i f f e r e n t i a l operators in the temperature equation (32) then become /3U 9y r6 9£ 3.04r/ 2v L dE, { 3 a ) p,2 , „2 . 3U 2 3 _ I 9 I o 9 ,,,, , 2 " 2 2 " 2 2vT 2 ( 3 4 b ) 9y ( r 6 r 9? (3.04rr Z U L 9? 9_ = £ 9_ 8 X ~ " 4Lr 3Pr ^ ( 3 4 c ) - 107 -Using the above operators along with equations ( I I ) , (12), (18), (19), (27) and (28) transforms the terms in the perturbation temperature equation (32) to U (a) r 2 - x 3iftT (35a) U , 2 p . 9T (b) - T - ^ X {(l+2r 3Pr)gf ' - % 7 7 } — z — (35b) 3 > 0 4 4 r P r 3 5 U 9 ?T (c) - r ^ x 5-r-2 \ (35c) 2(3.04) r Pr 9£ U 3(AT )(l-£ 2) 2 p <d) - L2" x f ^ — {<l +2r P r ) 5 g - ( 6 g + 4 ^ ) . (35d) 3 For the two probes considered, r is always less than 0.17. Thus 2r Pr is small compared to unity. Throughout the domain 0 < £ < I the v a r i a b l e n never exceeds 0.51; consequently, the steady v e l o c i t y p r o f i l e f' is e s s e n t i a l l y proportional to T\ CSctTI rent i ng, T955, p. 107). A l i n e a r approximation for the unsteady v e l o c i t y p r o f i l e g' is reasonable up to frequencies ft not much greater than unity (Lueck, 1979). Thus for r) < { and ft < I the boundary layer v e l o c i t y p r o f i l e s over the f i l m can be approximated by f = ± nf " (0) = i ( 3 . 0 4 ) 2 r V f " (0) f ' = nf " (0) = 3.04 r-£f " (0) g = J- ng"(0,ft) = { ( 3 . 0 4 ) 2 r 2 ? 2 g " (0,ft) g' = ng'(0,ft) = 3.04 rE, g"(0,ft) S u b s t i t u t i n g the above into the perturbation temperature equation (35 and 32) y i e l d s f i n a I ly - 108 -; 2 9 1 (3.04)V »» 8 T 1 2 2 — " * f (0) — 1 - 2(3.04) r PriftT = 35 6 95 1 _ (3.04) 3 2,, 2 3 5 (l-£)(AT >g" (0',n> +~0(r JRr) (36) Tj = 0 at 5 = 0, I 2 2 The term'2(3.04) r PriQT 1 which represents the thermal i n e r t i a of the boundary layer, is evidently small compared to the other terms in equation 36 f o r the frequency range considered here ft<I• Dropping the t h i r d term in equation 36 reduces i t to the ordinary d i f f e r e n t i a l equat ion d T l + (3.04) 3 r2 »• d T l _ (3.04) 3 2 2. II — 2 + 6 ? dF~ 4 — ? ? ( A T o 9 ' d5 (37) T = 0 at 5 = 0, I . Equation 37 does not have to be solved e x p l i c i t l y . The perturbation temperature, T j , is everywhere proportional to (AT )g''(0,fi). Hence the t o t a l heat f l u x from the f i l m is Q = QQ + q ( f i ) e l ( i ) T = Q q + e e ' W + F ( A T Q ) g " (0,ft) (38) where F is a frequency independent constant. In the l i m i t JH0, the perturbation heat flux must equal the quasi-steady heat f l u x , i .e. 3Q = U - r = ± 0 = F(AT )g"(0,0) e o 3U o o o The constant F is r e a d i l y determined by using II 3 II g (0,0) = |- f (0). - 109 -The t o t a l heat f l u x into the f l u i d above the f i l m then is Q = 0 [I + - | e i a > + g " ( 0 ' n ) ] (39) 3 f " ( 0 ) or using (23b) A 2/3 1/3 1/2 E a'' (o o) 0 = I .53A,L(f> Pr R AT [I + f e l a ) T 3 — ] (40) f L e 0 3 f , , ( 0 ) The unsteady response of the two t h i n f i l m probes is plotted in figur e 6 f o r the steady speeds 102, 129, 154 and 182 cm/sec. Plotted i is the unsteady thermal conductance q/AT Q divided by U 2 and the f r a c t i o n a l amplitude £ against the frequency parameter O J L / 3 U o . The unsteady conductance has been scaled by an a r b i t r a r y f a c t o r , y, to bring the low frequency values close to 0 dB. (The c a l i b r a t i o n f o r the v e l o c i t y monitor on the shaker is unavailable.) The s o l i d l i n e in figur e 6 is 2/3 g''(0,Q) / f ' ( 0 ) which according to equation 40 is 1 proportional to q/(eAT U z ) . The dashed l i n e has been drawn to show the 0 0 q u a l i t y of the f i t i f the frequency parameter wL'/3U had been used o instead of a)L/3U (figure I). The data f o r both V-31 and V-38 e x h i b i t 0 a r i s e with increasing frequency c h a r a c t e r i s t i c of g''(0,ft) but appear to be s h i f t e d by approximately 40$ to the r i g h t along the frequency axis (less i f scaled by o)L'/3U ).. This s h i f t along the frequency axis may r e f l e c t some inadequacies with modelling the flow over a blunted cone by the flow over an ideal "needle sharp" cone. The s c a t t e r among the data points for Q>I.5 is noticeable. This scat t e r may be due to some spurious motions of the probe support at high frequencies (Evans, personal communication); ft>I.5 corresponds to real frequencies of 500 to 1000 Hertz. Thermal i n e r t i a should become important f o r fi>l"and — 110 — 7 r CO XI o < 5h 4 h 0.01 U 0 cm/sec A T 0 C C • 102 18.0 A 130 17.5 O 154 17. 1 X 182 16.7 AT • 23.1 °C L • 0.144 cm L' • 0.120 cm V - 3 1 tu L 3U 0 The p r o b e s ' u n s t e a d y t h e r m a l c o n d u c t a n c e s c a l e d by U O 2 and t h e f r a c t i o n a l speed a m p l i t u d e e v e r s u s t h e f r e q u e n c y p a r a m e t e r ft = C J L / 3 U 0 . The s o l i d l i n e i s t w o - t h i r d s o f t h e u n s t e a d y - t o -s t e a d y s u r f a c e s t r e s s r a t i o a f t e r Lueck ( 1 9 7 9 ) . The dashed l i n e i n d i c a t e s t h e q u a l i t y o f t h e f i t i f t h e f r e q u e n c y had been s c a l e d by a ) L ' / 3 U 0 . L ! i s shown i n f i g u r e - I. - I l l -cause a " r o l l - o f f " in the frequency response, but only the data for U Q = 1 0 2 cm/sec shows a d e f i n i t e trend to r o l l - o f f . Although the match between the data and the t h e o r e t i c a l l y predicted response i s not complete i t does appear that the unsteady heat f l u x from the f i l m is strongly influenced by the unsteady viscous boundary layer and that for probes V - 3 1 and V - 3 8 i t is proportional to the unsteady surface shear g ' ' ( 0 , f i ) . The pe c u l i a r r i s e of the response with increasing frequency is due to the probes' aspect r a t i o d/L<<I. Most commercial probes have aspect r a t i o s c l o s e r to unity and therefore should show much less i f any r i s e , at high frequencies due to the thermal i n e r t i a of the boundary layer. Observations by Howell ( 1 9 7 4 ) of commercial constant temperature hot f i l m probes show a decrease in response with increasing frequency above approximately 1 0 0 Hertz. The s e n s i t i v i t y of these probes to temperature o s c i l l a t i o n s in the main-stream causes a contamination of the measured v e l o c i t y signal which can be very severe . (Gargett, 1 9 7 8 ) . The response of these probes to temperature o s c i l l a t i o n s is d i f f i c u l t to c a l c u l a t e . However, at least three aspects of the response to i n f i n i t e s i m a l external temperature o s c i l l a t i o n s can be deduced. F i r s t l y , the heat flux into the boundary layer above the f i l m cannot increase with increasing frequency because there is no mechanism to accentuate the response to temperature o s c i l l a t i o n s in the main stream. As well, c a l c u l a t i o n s for an ideal f l a t p l ate sensor by Lueck ( 1 9 7 9 ) show that the bandwidth of the response to temperature o s c i l l a t i o n s is smaller than the bandwidth for the response to speed o s c i l l a t i o n s . Secondly, the heat flux into - 1 1 2 -the substrate induced by external temperature o s c i l l a t i o n s w i l l decrease with increasing frequency because of the substrate's f i n i t e d i f f u s i v i t y . The e f f e c t i v e depth of penetration into the substrate is approximately the width of the f i l m d = 6 x IO - 3 cm, hence the heat f l u x decreases when u)d 2/K s ;> I, i.e. above approximately 30 Hertz. T h i r d l y , the glass upstream from the f i l m acts as a passive heat sink and reduces somewhat the temperature o s c i l l a t i o n s of the f l u i d c l o s e t o the surface before t h i s f l u i d reaches the area above the f i l m . Thus the r a t i o of temperature-t o - v e l o c i t y s e n s i t i v i t y is highest at zero frequency and decreases with increasing frequency. An upper bound to the contamination of a measured v e l o c i t y spectrum by temperature f l u c t u a t i o n s can be c a l c u l a t e d from the r a t i o of quasi-steady temperature-to-velocity s e n s i t i v i t y and an independent estimate of the temperature spectrum. — 113 — 4. Cone I us ions The steady c a l i b r a t i o n data presented show that the probes' thermal x conductance Q/AT is not proportional to U 2 as expected for a constant temperature conical surface. A thermal model of the probes indicates that i Q/AT is neither proportional nor l i n e a r l y related to U 2 because of the thermal e f f e c t s of the coating over the f i l m and the substrate below i t . The model pr e d i c t s that a glass coating of only M x 10-1* cm can s i g n i f i c a n t l y reduce the temperature d i f f e r e n c e between the probe's wetted surface and the ambient f l u i d and that the heat f l u x into the substrate is moderately important for probes operated in water. For probes operated in a i r , on the other hand, the model predicts the opposite c h a r a c t e r i s t i c s : a r e l a t i v e l y small e f f e c t due to the coating and a very large heat f l u x into the substrate. The coating and the substrate reduce the quasi-steady speed-to-temperature s e n s i t i v i t y of the probes by about 40$ from i t s optimum value of AT/2U . o The unsteady heat flux from the hot f i l m probes in response to v a r i a -t i o n s of the ambient f l u i d ' s speed is strongly influenced by the unsteady viscous boundary layer. For the probes considered, the unsteady heat flux is proportional to the unsteady viscous surface s t r e s s over the f i l m . The unsteady heat f l u x from the probes examined increases in magnitude, over i t s quasi-steady value, with increasing frequency. The r a t i o of temperature-to-velocity s e n s i t i v i t y is highest.at zero frequency and decreases with increasing frequency. An upDer bound to the contamina-t i o n of a measured v e l o c i t y spectrum by temperature f l u c t u a t i o n s in the main stream can be c a l c u l a t e d from t h e r a t i o o f t h e quasi-steady temoerature-to-v e l o c i t y s e n s i t i v i t y and an independent estimate of the tempeature spectrum. - 1 1 4 -Bi bI ioqraphy Bellhouse, B.J. and D.L. Schultz. 1967. The determination of f l u c t u a t i n g v e l o c i t y in a i r with heated t h i n f i l m gauges. Journal of F l u i d Mechanics, 29: 289-295. Evans, D.J. 1963. An instrument for the measurement of ocean turbulence. Technical Memorandum 63-8, P a c i f i c Naval Lab-oratory, Defense Research Board, Canada. Freymuth, P. 1977. Frequency response and e l e c t r o n i c t e s t i n g f o r constant-temperature hot wire anemometers. Journal of Physics E; S c i e n t i f i c Instruments, 10: 705-710. Freymuth, P. 1978. Theory of frequency optimization for hot f i l m anemometers. Journal of Physics E; S c i e n t i f i c Instruments, II: 177-179. Gargett, A.E. 1978. Microstructure and f i n e s t r u c t u r e in an upper ocean fr o n t a l regime. Journal of Geophysical Research, 83(CI0): 5123-5134. Grant, H.L., R.W. Stewart and-A. M o i l l i e t . 1962. Turbulence spectra from a t i d a l channel. Journal of F l u i d Mechanics, J_2: 241-288. Grant, H.L., A. M o i l l i e t and V.M. Vogel. 1968. Some observations of the occurrence of turbulence in and above the thermocline. Journal of F l u i d Mechanics, 34: 443-448. Hinze, J.O. 1975. TurbuIence, second e d i t i o n . McGraw-Hill, 790 pp. Kays, W.M. 1966. Convective Heat and Mass Transfer. McGraw-Hill, 387 pp. - 1 1 5 -Kramers, H. 1946. Heat t r a n s f e r from spheres to flowing media. Physics, 12: 61-80. L i g h t h i l l , M.J. 1954. The response of laminar skin f r i c t i o n and heat t r a n s f e r to f l u c t u a t i o n s in the stream v e l o c i t y . Proceedings of the Royal Society of London, Series A, v o l . 224, pp. 1-23. Lueck, R.G. 1979. The unsteady flux of heat and momentum from a f l a t p l ate. Submitted to Journal of F l u i d Mechanics. Morrison, G.L., A.E. Perry and A.E. Samuel. 1972. Dynamic C a l i b r a t i o n of Inclined and Crossed Hot Wires. Journal of F l u i d Mechanics, 52: 405-474. Nasmyth, P.W. 1970. Oceanic Turbulence. Ph.D. Thesis, I n s t i t u t e of Oceanography, Un i v e r s i t y of B r i t i s h Columbia. NowelI, Arthur R.M. 1974. Some response c h a r a c t e r i s t i c s of parabolic hot f i l m s in water. Journal of Hydronautics 8_:4, pp. 169-171. S c h l i c h t i n g , H. 1955. Boundary Layer Theory. Pergamon Press, 535 pp. Tan-atichat, J . , H.M. Nagib and J.W. P l u i s t e r . 1973. On the interpre-t a t i o n of the output of hot-fiIm anemometers and a scheme of dynamic compensation for water temperature v a r i a t i o n . Symposium on Turbulence in Liquids, University of Missouri, R o l l a . White, F.M. 1974. Viscous F l u i d Flow. McGraw-Hill, 725 pp. - 1 1 6 -Acknow1edqements This paper could not have been produced without the co-operation of the Ocean Mixing Group at the I n s t i t u t e of Ocean Sciences, P a t r i c i a Bay, p a r t i c u l a r l y Dr. A. Gargett, Dr. P. Nasmyth and Mr. G. Chase. Mr. D. Evans, formerly with the Defense Research Establishment, P a c i f i c , has provided needed information on the probes' e l e c t r o n i c s . Prof. T.R. Osborn deserves appreciation for his encouragements and support. This work has been supported by the O f f i c e of Naval Research, Contract Number - 1 1 7 -Appendix C The Characteristics of Internally-Heated Thermistors by Rolf G. Lueck and Thomas R. Osborn Institute of Oceanography University of British Columbia Vancouver, B.C. V6T 1W5 Canada January 1979 - 1 1 8 -Abstract A model for the heat f l u x from coated spherical and plate-shaped thermistors to t h e i r f l u i d environment has been developed and agrees favourably with the available data. The heat f l u x model i s used to derive the thermistor's quasi-steady s e n s i t i v i t y to speed and to temperature. The coating reduces s i g n i f i c a n t l y both the s e n s i t i v i t y to speed and to temperature as well as the r a t i o of speed-to-temperature s e n s t i v i t y . The available values f o r the rates of temperature and k i n e t i c energy dissipation i n the ocean indicate that (1) v e l o c i t y -measurements by heated thermistors may be contaminated by temperature i n the main thermocline and (2) temperature measurements by a nearly unheated thermistor may be contaminated by v e l o c i t y i n the mixed layer. - 1 1 9 -1. Introduction Thermistors are now commonly used to measure temperature micro-structure i n the ocean. Their large sensitivity to temperature, due to a high fractional temperature coefficient of resistance (approxi-mately 0.04/°C) and a large resistance (up to 25 x 106 ohms), make thermistors attractive for oceanographic use. There are, however, several problems with using thermistors. Three of these problems are (1) the thermistor's signal-to-noise ratio which has been examined by Gregg, Meagher, Pederson, and Aagaard (1978), (2) i t s frequency (and spatial) response which has been examined by Lueck, Hertzman, and Osborn (1977), and (3) the sensitivity to velocity due to sel f -heating which has been raised by Gregg, et a l . (1978). Inevitably, a small but f i n i t e e l e c t r i c a l current must be passed through the thermistor to measure i t s e l e c t r i c a l resistance and hence its temperature. The e l e c t r i c power dissipated by the ohmic-heating raises the temperature of the thermistor above the ambient fluid's temperature and thereby subjects the thermistor to a velocity-dependent forced convective cooling. At a given resistance, the sensitivity to temperature is propor-tional to the current passing through the thermistor. Hence, the signal-to-noise ratio is readily increased by increasing the thermistor current. The limit to which the current can be increased is governed by the thermistor's relative velocity-to-temperature sensitivity, the relative velocity-to-temperature ac t i v i t y in the f l u i d being probed, and the level of velocity contamination in the temperature signal that one is willing to accept. - 120 -A deliberate large self-heating is also interesting because i t turns the thermistor into an anemometer with some level of temperature contamination. The use of thermistors as anemometers has so far been limited. Caldwell (1968) reported on an indirectly heated thermistor to measure bottom currents. Lange (personal communication) has mounted both 0.015 cm micro-bead and 0.05 cm flake thermistors on a f r e e - f a l l vehicle for microstructure measurements. The thermistors were heated by a constant current source to approximately 50°C above the ambient temperature. Their relatively large temperature coefficient and large resistance compared to thin metal film sensors also make thermistors attractive for measuring velocity microstructure. In this paper we propose, in section 2, a thermal model for the heat flux from thermistors. Section 3 compares the model with the available data. The quasi-steady sensi-t i v i t y of the thermistor to speed and temperature in the ambient f l u i d is derived in section 4 both for constant temperature operation, 4a, and constant current operation, 4b. The implications for thermometry and anemometry in the ocean are discussed in section 5. r - I 2 T -2. Steady Response Following Lueck, et a l . (1977) we w i l l use a sphere to model microbead thermistors and a f l a t plate for flake thermistors. Both models are shown in Figure 1. The models consist of an inner core of metal-oxide, a uniform coating of thickness A and a boundary layer of unspecified thickness. Because the ohmic heating in the metal-oxide is distributed over the volume of the core we w i l l assign a spatially uniform temperature T to the metal-oxide core. The temper-ature of the wetted surface is 9^  and the temperature in the ambient f l u i d , at the outside limit of the boundary layer, is 8. 2a. Sphere The heat flux, Q, from a sphere with a uniform surface temperature 6 is Q = 2TT(R + A)A 2 (9S - 6)Nu (1) where Nu is the Nusselt number, A2 the fluid's thermal conductivity, and 2(R + A) is the diameter of the sphere. In the coating the temperature distribution, i f spherically symmetric, is governed by 1 _9_ r2 8r = 0 which is satisfied by T (r) = T - (T R + A 1 - * r R < r < R + A - 122 -Metal - oxide (a) ft. ft e L ft 1 T JL A A U U (b) Figure 1. ( a) A c r o s s - s e c t i o n of the model spher ica l the rmis to r , and (b) a c r o s s - s e c t i o n of the model p la te thermis to r . - 123 -Under steady conditions the heat flux through the coating must equal the heat flux, Q, into the f l u i d . Hence, we use = 0 . \ 47T R2 9 - I ^ i 9 r R where Aj is the thermal conductivity of the coating, to eliminate the surface temperature, 9 g, and derive A / R . : (2) Q 4TT(R + A)Aj T 21T(R + A)A2Nu Equation 2 relates the heat flux from the model thermistor to the temperature difference (T - 9), to the properties of the coating and to the properties of the flow. If the ratio (T - 9)/0 is regarded as a thermal resistance, in analogy to Ohm's Law, where T - 8 is the thermal-"voltage" and Q the thermal "current", then the f i r s t term on the right side in equation 2 represents the thermal resistance of the coating and the last represents the thermal resistance of the boundary layer. The thermistor's inner core thus "sees" two thermal resistances between i t and the ambient f l u i d , namely the coating and the boundary layer. The velocity dependence of the heat flux is implicit in the Nusselt number. The Nusselt number for spheres of approximately uniform surface temperature was empirically determined by Kramers (1946) as Nu = 2.0 + 1.3 P r 0 ' 1 5 + 0.66 P r ° - 3 1 Re 1 / 2 (3) - 124 -where R e = 2 ( R _ + ^ ) U Pr = 1 K Re is the Reynolds number, Pr the Prandtl number, U the ambient fluid's speed, V the ambient fluid's kinematic viscosity, and < i t s thermal di f f u s i v i t y . The effect of the coating on the thermistor's heat flux is more readily determined by rearranging equation 2 to read 2TT(R + A)A 2 (T - 9)N U ( 4 ) For an uncoated sphere A = 0 and hence the heat flux relates to the fluid's speed in a fashion after King's Law because of the Nusselt-Reynolds number relationship shown in equation 3. A f i n i t e coating thickness, however, reduces the heat flux's dependence on speed through the term - • ft f r The term DF depends on the ratio of the fluid-to-coating conductivity, the relative coating thickness, A/2R, and the Nusselt number of the flow. It can also be regarded as the ratio of the coating-to-boundary layer resistance. Evidently, DF is an important design parameter because i f DF << 1, there is no coating effect, and i f DF >> 1, the heat flux is independent of the flow. In addition, equation 4 shows that the ratio - 125 -(T - 9)/Q is a constant at a given Nusselt number and that the heat flux does not follow King's Law i f DF is not small compared to unity. The term DF can be quite large for commonly available thermistors. For example, 0.015 cm appears to be the lower limit for the diameter of the metal-oxide core (M. Sapoff, personal communication) and at a speed of 10 cm/sec the Nusselt number i s approximately 10 in water. Accordingly, a 0.015 cm diameter microbead coated with paralene-C (Xi = 0.0010 watts/°C cm) to a thickness of A = 0.0018 cm has a DF value of 7. The same thermistor under the same flow conditions but coated with 0.0025 cm of glass (Xj = 0.0105 watts/°C cm) has a DF value of 0.9. 2b. Plate The total heat flux from both wetted surfaces of a plate held at a uniform temperature 6 g i s CSchlichting, 1955, p. 269) Q = 2bX_(9 - 9)Nu 2 s where Nu = 0.664 Pr 7 Re ' Re = — v b i s the plate's width and L i t s length in the flow direction. The heat flux through the coating i s 2bLX Q = — r - i (T - 9 ) (6) A S The surface temperature i s readily eliminated to yield the plate's thermal resistance as - 126 -3. Observations 3.a Plate Lange (personal communication) measured the temperature of a flake thermistor for various heating currents. The flake thermistor had the nominal dimensions of 0.051 by 0.051 by 0.0025 cm and was coated with paralene-C to a nominal thickness of 0.0018 cm. The measurements were made by towing the thermistor, with i t s face par a l l e l to the flow, down the length of a 30 by 2.4 by 0.24 metre tow tank at speeds of 10, 20, 30, 40, 50 and 60 cm/sec. The accuracy of the speed measure-ments is 15% or better. We used Lange's data to calculate the heat flux and the thermistor's (metal-oxide's) temperature for the various heating currents and flow speeds. The results are shown in Figure 2. The slopes of the temperature vs heat flux plots for a given speed equal the thermal resistance (T - 8)/Q (because T = 0 when Q = 0). The -1/2 thermistor's thermal resistance is plotted against U in Figure 3. The observed relationship, by a least square f i t , is = 0.437 ± 0.001 + ° - 5 3 8 * ° - 0 0 5 ^ (8) Using the appropriate parameter values l i s t e d in Table 1 yields, according to equation 7, T - 8 .„ o c 0.59 °C = 0.35 + —r-pr Q U m W The observed and the predicted thermal resistances agree favorably considering that the flake's dimensions are known only by the nominal values specified by the manufacturer. - 127 -l l I I I I I I I IQ o o o o o o o o o o MUJ 0 x m d 1V3H Figure 2. Heat f l u x versus temperature for var ious speeds for Lange's 0.051 x 0.051 x 0.0025 cm paralene-c coated f l ake thermis to r . - 128 -0.15 0.20 0.25 0.30 U J /2 (cm/sec)-'^ i Figure 3. Thermal res is tance (T-8)/Q versus I T 5 fo r Lange's 0.051 x 0.051 x 0.0025 cm para lene-c coated f l ake thermi s to r . - 129 -3b. Sphere Using the equipment specified above, Lange (personal communication) also tested an e l l i p s o i d a l microbead thermistor with a nominal semi-minor axis diameter of 0.013 cm coated with paralene-C to a thickness of 0.0018 cm. The temperature vs heat flux curves for the various tow speeds are shown in Figure 4.. We have calculated the microbead's Nusselt number using equation 3 and the parameter values in Table 1. The microbead's thermal resistance is plotted against the inverse of i t s Nusselt number (in Figure 5). The least square f i t t e d line in Figure 5 is T - 6 n c n 2.51 °C — - — = '0.59 + — — — - . Q Nu mw In comparison, the relationship predicted by equation 2 is • 1 ^ 6 = 2 > 8 5 + 3^5 °C Q Nu mw The agreement between the predicted and the observed coating resistance (2.85 vs 0.59 °C/mw) is poor and the agreement between the predicted and the observed Nusselt number coefficient (3.35 vs 2.51) is only f a i r . However, in the absence of more precise knowledge about the thermistor's diameter and i t s coating thickness i t is not clear i f the thermal model is in error or i f the parameters used are erroneous. For further tests we ordered a set of 20 microbead thermistors from one production batch. The thermistors' semi-minor axis are, nominally, 0.015 cm. Half (10) of the thermistors were coated with glass to a nominal thickness of 0.0025M- cm. A l l 20 microbeads were F i g u r e 4. H e a t f l u x v e r s u s t e m p e r a t u r e f o r v a r i o u s s p e e d s for L a n g e ' s p a r a l e n e - c c o a t e d m i c r o - b e a d t h e r m i s t o r . - 131 M W 0 0-1 T h e r m a l r e s i s t a n c e ( T-9 ) / Q v e r s u s t h e i n v e r s e N u s s e l t n u m b e r o f L a n g e ' s p a r a l e n e - c c o a t e d m i c r o - b e a d t h e r m i s t o r . T h e N u s s e l t n u m b e r i s a f t e r e q u a t i o n 3 a n d t h e p a r a m e t e r s i n t a b l e 1. - 132 -photographed through a microscope to determine their outside dimensions. The difference between the dimensions of the coated and the uncoated microbeads was used to derive a s t a t i s t i c a l estimate for the average coating thickness. The derived coating thickness is 0.0028 +_ 0.007 cm which is only about 10% larger than the nominal value. A back lighted microphotograph of the two coated thermistors tested, namely #9 and #10, is shown in Figure 6. The tank used to test the thermistors is an annulus of width 4 cm, depth 4 cm and outer radius 14 cm. The annulus was placed on a rotating table. The speed of the f l u i d past the stationary thermistor was determined from measurements of the rotation rate and the radial position of the thermistor under the assumption of solid body rotation in the f l u i d . The walls of the annulus were roughened to reduce f l u i d motions relative to the rotating table. Samples of the temperature vs heat flux relationship for the thermistors are shown in Figure 7. The thermal resistance is plotted against the inverse of the Nusselt number in Figure 8. It should be noted that each point on these plots represents 10 observations of the thermistor's temperature and heat flux. The Nusselt number used is based on equation 3 and the parameters in Table 1. One data point in Figure 8(b) is at variance with the straight line f i t t e d through the other four points. This data point comes from the f i r s t flow speed tested. The point's deviation cannot be explained as a "blunder" in either the speed measurement (33.6 cm/sec) or the heat flux vs temperature regression. The consistency of the zero heat flux intercept (Fig. 9b) for various speeds also rules out a sporadic change in the e l e c t r i c a l resistance temperature relation for the - 133 -Figure 6. Back l ighted micrograph of the authors ' g lass coated microbeads and a sketch of the thermis tors mounting f o r k s . - 134 -- 135 -(MW) 0 X f l l J 1V3H Figure 7(b) Heat f l u x versus temperature for various flow speeds for thermistor number 10. - 136 -Figure 8(a) Thermal resistance (T-e)/Q versus the inverse Nusselt number f o r thermistor number 9. The Nusselt number is a f t e r equation 3 and the parameters in ta b l e I. Do 0 0-1 3DNV1SIS3U n V l A l d 3 H l - 138 -thermistor. The point's variance is most l i k e l y due to a surface contamination, such as l i n t , which was too small to be noticed during the measurements and dislodged prior to the readings at other speeds. Breakage of the probe precludes a re-examination of the thermistor's heat flux at that speed. Because the other thermistors tested do not have such a deviant data point we have not used i t to f i t the line shown in Figure 8(b). We have, however, retained the data point because we do not have an explicit reason for removing i t . The least square f i t t e d thermal resistances are L^l = 0.527 10.001 t 2 ' ° V ° - 0 1 (9) Q Nu mw for microbead #10 and 1 ^ 1 = 0.422 ±0.001 t X - 9 2 * °-° 2 °-± Q Nu mw for microbead #9. In comparison, the predicted thermal resistance, using equation 2, is T - 6 2.4 °C — - — = 0.21 + - — — Q Nu mw for both thermistors. The observed Nusselt number coefficients for the two thermistors are in reasonable agreement with the predicted values. The observed thermal resistance of the coating i s , however, 2 to 2.5 times larger than predicted and consequently warrants some consideration. The model for the coating's temperature is based upon spherical symmetry. The fact that the thermistors are el l i p s o i d a l rather than spherical cannot alone account for the observed discrepancy. However, i t is known that the heat flux from bluff bodies is preferentially - 139 -biased toward their front surface where the boundary layer is thinnest. Thus the surface through which a large fraction of the heat passes is much smaller than the total surface area of the sphere. Perhaps the front hemisphere, or even less, represents the area that transmits the heat from the core to the ambient f l u i d . An examination of equation 2 then indicates that the term representing the coating's resistance A  4TTR(R + A H i should really be larger to account for this effective surface area reduction. Until the heat flow process is better understood, multi-plying the above coating resistance by a factor of about 2 or 2.5 might maintain the usefulness of equation 2. The Nusselt-Prandtl-Reynolds number relationship for spheres, on the other hand, is an empirical observation independent of the local heat flux distribution over the sphere's surface. The d i f f i c u l t y with using Kramers' (1946) Nusselt number formulation (equation 3) is that i t applies only to spheres with a nearly uniform surface temperature. Because the heat flux from a sphere is not uniform over i t s surface, the presence of the coating leads to a non-uniform temperature over the wetted surface. Even for simple geometries such as a plate, the effect of a non-uniformity of the surface temperature on the heat flux is d i f f i c u l t to predict. Calculations by White (1974) suggest that the coating should make the heat flux more uniform over the wetted surface. The fact that the observed Nusselt number coefficients differ from the predicted coefficients by only 13 and 20% indicates that Kramers' relationship has some valid i t y for the thermistors tested. - 140 -There i s , however, a more general problem with predicting the response of thermistors. There is evidence for non-uniformity between nominally identical thermistors. This is particularly evident in the coating thickness which may vary considerably over the thermistor's surface and between thermistors, as noted by Gregg et a l . (1978) and by us.i Although we were not able to observe directly the coating thicknesses of the particular thermistors reported here, the difference between the observed response of thermistors #9 and #10, whose outside dimensions differed by less than 2%, suggest that their coatings may be quite different in thickness and in thickness distribution. Because of this non-uniformity among nominally identical thermistors i t is not clear that improvements of the thermal model, although intellectually interesting, can lead to corresponding improvements in response predictability. - 141 -4-. Quasi-Steady Response The response is considered to be quasi-steady when the heat flux at any instant is that appropriate to steady conditions at the instantaneous value of the fluid's speed and temperature; i.e. the response in the limit of zero frequency. 4a. Constant Temperature T (i) Sphere The variation of equation 2 with respect to speed and temperature gives T - 8 dQ ae Q Q - Q 0.33 P r ° - 3 1 R e 1 / 2 dU_ 27T(R + A)A,Nu2 U (10) The sensitivity of the heat flux to temperature is dQ de -Q dU=0 T -(11) which is a constant at a given speed and readily determined from the temperature vs heat flux relationship. For thermistor #10, for example, the sensitivity to temperature is 1.39 mw/°C at 15.4 cm/sec. The sensitivity of the heat flux to speed is dQ dU d8=0 T - 9 .31 1/2 0.33 Pr Re ' 2rr(R + A)X 2Nu 2 T - e (12) which at 15.4 cm/sec, using only the dimensional information in Table 1. equals 0.0111 mw/cm/sec for every centigrade degree of overheat (T - 8). Choosing, say, T - 8 = 50°C results in a sensitivity of 0.553 mw/cm/sec. The relative speed-to-temperature sensitivity then is - 142 -(dQ/dU)d9=0 (dQ/d9)dU=0 T - e 0.33 P r - 3 1 R e 1 / 2 2TT(R + A)A.,Nu2 T - e (13) and equals 0.40°C/cm/sec for thermistor #10 under the conditions specified, namely U = 15.4 cm/sec and T - 9 = 50°C. The implications are that a quasi-steady fluctuation of 1 cm/sec is indistinguishable from a fluctuation of 0.4 °C. Alternate expressions for the sensi-t i v i t i e s that c l a r i f y the effect of the coating are dQ dU dQ=0 2 (R + A)X2[0.33 P r , 3 1 R p 1 / 2 ] T - Q (1 + DF) 2 U (14) and d9 dU?0 (dQ/dU)dQ=0 (dQ/d9)dU=0 2TT(R + A)AgNu 1 + DF 0.33 P r - 3 1 R e 1 / 2 T - 9 (1 + DF)Nu U (15) (16) where DF represents the coating-to-boundary layer resistance ratio. For thermistor #10 the observed DF value at 15.4 cm/sec is. 2.2, which implies that the speed sensitivity has been reduced by a factor of (1 + 2.2) 2 = 10 from i t s value in the absence of a coating. The temperature and the relative speed-to-temperature sensitivity, using equations 15 and 16, is reduced by a factor of (1 + 2.2) ~ 3.2 by the coating. Clearly the coating requires careful consideration when one chooses a thermistor for heated anemometry and heated thermometry. We have used the observed thermal resistance of thermistor number 10 (eq. 9) to predict the speed sensitivity of that thermistor for various flow speeds. The speed sensitivity per degree of overheat i s , using equation 9 and the parameters in Table 1, - 143 -Lange Authors Plate Microbead Microbeads #9 #10 R (cm) 0.0064 0.0087 0.0087 L (cm) 0.051 -* -b (cm) 0.051 A (cm) 0.0018 0.0018 0.0028 0.0028 K (watts/cm°C) 0.0010 0.0010 0.010 0.010 (watts/cm°C) 0.0059 0.0059 0.0059 0.0059 V (cm2/sec) • 0.0123 0.0120 0.-0097 0.010 K (cm2/sec) 0.00139 0.00139 0.0014 0.0014 Pr 8.9 8.63 7.1 6.8 Table CI The parameter value associated with the thermistors examined. The dimensional data for Lange's thermistors are nominal values. The conductivity of paralene-C (Aj for Lange's thermistors) is after Alpaugh and Morrow (1974). The viscosity (v), the d i f f u s i v i t y ( K ) , and the Prandtl number (Pr) are after Batchelor (1970, p. 597). The conductivity of glass ("X for the authors' thermistors) is after Lueck, et a l . (1977). - 144 -1 dQ T - 9 dU 0.230yT? 1 " 77 d9=0 (1 + 0.235^ ) 2 2 U in units of °C per cm/sec per degree centigrade, which is plotted in Figure 9. Both the speed sensitivity and the ratio of speed-to-temperature sensitivity decrease rapidly with increasing flow speed. Consequently the usefulness of heated thermistors for anemometry may be restricted to low flow speeds. ( i i ) Elate The variation of equation 7 gives the sensitivity of the plate's heat flux as dQ dU dQ dQ = 0-664 bX P r 1 / 3 R e 1 / 2 T - 9 . d 9 = 0 " T T T I ^ V . u A j Li Q _ -2bX20.664 P r 1 / 3 R e 1 / 2 ~ " " X, A „.. , - ( 1 8 ) d u = o T " 9 ( l + T 2 r Nu) A j L and (dQ/dU)d0=Q _ 1 T - 6 , . (dQ/d9)dU=0 ~ X2 A 2U U J x" - L~ Using the observed thermal resistance shown in equation (8) we calculate the relative sensitivity as (dQ/dU)d9=0 _ 1.00 . T - 8 (dQ/d9)dU=0 1 + 0.825U1'2 2U which, for comparison with the glass-coated microbead #10 equals 0.38 °C/cm/sec at T - 6 = 50 °C and U = 15.4 cm/sec. The term DF 1/2 equals 0.825 U for the plate tested by Lange. - 145 -Figure 9. Speed s e n s i t i v i t y per degree of overheat fo r the authors ' g lass coated thermistor number 10 and Lanqe's para lene-c coated f l ake thermis tor . The curves are based on the observed thermaI res is tances given by equat ions 8 and 9. a y Qo n p e-i oas/uio/Miu 0 p — 1V3HU3A0 33U93Q 83d A1IAI1ISN3S Q33dS o -+• AMBIENT FLUID SPEED U cm/sec - 147 -The relative speed-to-temperature sensitivity of Lange's flake thermistor i s plotted in Figure 10. The speed sensitivity per degree of overheat i s given by dQ dU d9=0 1.86 _1_ (1 + 0.812/U) 2U which is plotted in Figure 9. The assymptotic value of the speed sensitivity of a heated sensor -1/2 -3/2 at high speeds is proportional to U without a coating and U with a coating. The assymptotic value of the ratio of the speed-to-— 3/2 temperature sensitivity is U - 1 without a coating and U with a coating. 4b. Constant Current I (i) Sphere In the constant current mode the temperature of the thermistor's core is variable and i s , in practice, the sensed quantity. The variation of the heat flux dQ is related to the variation of the metal-oxide's temperature by dQ = IdE = I 2dR T = QadT (20) where I is the constant current passing through the thermistor, dE is the fluctuation of the thermistor's voltage, dRT the fluctuation in i t s el e c t r i c a l resistance, dT the fluctuation in i t s temperature, and a i t s fractional temperature coefficient of e l e c t r i c a l resistance defined by 1 d R T a = - — = -0.04/oc .. - 148 -The variation of equation 2 is dT _ d9 _ T - 9 dQ Q Q Q Q 0.33 P r ' 3 1 R e 1 / 2 dU 27r(R + A)A 7Nu 2 U which, when combined with equation 20, yields for the sphere, n _ n(T aMdT d e - 0-33 P r ' 3 1 R e 1 / 2 dU Li a u Q ~ Q ~ 2TT(R + A)A2Nu2 U~ (21) The constant current temperature sensitivity is dT d9 dU=0 1 - a(T - 9) (22) According to (22) the temperature sensitivity is unity at zero overheat which is desired i f the thermistor is to be used as a temperature sensor with infinitesimal internal heating. The temperature sensi-t i v i t y decreases with increasing internal heating (noting that a < 0) and vanishes at very high levels of internal heating. It is very interesting to observe that sensors with positive temperature coeffi-cients (a > 0) increase their temperature sensitivity with increasing internal heating in the constant current mode. The condition T - 6 = a corresponds to an i n s t a b i l i t y for positive temperature coefficient sensors. In contrast, the temperature sensitivity in the constant temperature mode is independent of the overheat T - 9. The speed sensitivity of the spherical thermistor's heat flux is - I dT dU d9=0 Q T - 9 0.33 P r - 3 1 R e 1 / 2 [1 - cc(T - 9)]2TT(R + AU 2Nu 2 (23) - 149 -At zero overheat (T = 9) the thermistor has no speed sensitivity. The speed sensitivity increases with increasing overheat to the limit dT dU _ I 0-33 P r , 3 1 R e 1 / 2 Q 1 a 2TT(R + A)-A„Nu2 T - 9 U d9 = o 2 limit Q •*• 0 0 which is independent of the overheat because Q/(T - 9) is a constant at a given Nusselt number. The ratio of the quasi-steady speed-to-temperature sensitivity is (dT/dU)d9=0 (dT/d9)dU=0 T - 9 0.33 P r ' 3 1 R e 1 / 2 T -2TT(R + A)A 2Nu 2 U which is identical to the relative sensitivity in the constant temperature heating mode. The advantages, i f any, of operating the thermistor in the constant temperature mode over operating i t in the constant current mode are reflected in the unsteady response which has, so far, not been measured. - 150 -5. Discussion Some knowledge of the r e l a t i v e a c t i v i t y of temperature and v e l o c i t y i n the f l u i d being probed (such as the ocean) i s necessary to determine the s u i t a b i l i t y of heated thermistors f o r turbulence measurements and to assess the v e l o c i t y contamination of (.nearly) unheated thermistors used for thermo-metry. We w i l l use as a basis for discussion some of the available observa-tions of the k i n e t i c energy d i s s i p a t i o n rate e and the temperature d i s s i p a t i o n rate x where, assuming isotropy 15v 9w' 3z 2 (.24a) , x = 6< 89 ' 3z 2 (24b) 3w'/3z i s the v e r t i c a l gradient of the v e r t i c a l component of the v e l o c i t y fluctuations and 3 0'/3z i s the v e r t i c a l gradient of the temperature f l u c t u a -t i o n s . Gregg (1976) reported temperature dissipation rates of 4.3 x 10~ 9 to 1.1 x 1 0 - 1 0 °C2/sec averaged over several tens of metres i n the mixed layer and values of 3.6 x 1 0 - 8 to 2.1 x 10~ 9 °C2/sec averaged over several tens of metres i n the thermocline of the central North. P a c i f i c . Gregg (1977) also observed temperature d i s s i p a t i o n values of 2.2 x 1Q~ 1 2 to 6 x 1 0 - 1 1 °C2/sec for depths below 1000 metres i n several widely spaced locations i n the P a c i f i c . Gargett (1978) measured d i s s i p a t i o n rates as large as 9.8 x I O - 7 °C2/sec i n an upper ocean f r o n t a l regime. The rate of k i n e t i c energy d i s s i p a t i o n near the Azores has been measured by Osborn (1978). Averaged over 5 metres of depth the values were t y p i c a l l y 10 _ l + cm 2/sec 3 but ranged as high as 10~ 2 cm 2/sec 3. Crawford and Osborn (1978) reported averaged k i n e t i c energy di s s i p a t i o n rates of up to 7 x 10~ 3, 4 x 10" 5 and 2 x 1 0 - 3 cm 2/sec 3, for the regions above, i n and below the core of the A t l a n t i c Equatorial Undercurrent. - 151 -If we denote the ratio of a probe's speed-to-temperature sensi-t i v i t y by 8, then the velocity contamination of a temperature d i s s i -pation rate measurement i s , using 24a and b, X +. = r i ~ B 2 e • (25) Acont 5Pr Correspondingly, the temperature contamination of a kinetic energy dissipation rate measurement is 5Pr 1 . . £cont = I " " F X ( 2 6 ) Using Gargett's (1978) value of 9.8 x 10-7 °C2/sec as an upper limit for "X in the thermocline, and using the observed relative sensi-t i v i t y of 0.29 °C/cm/sec for thermistor #10 (Fig. 10), at T - 9 = 50°C, U = 15.4 cm, and Pr = 8' gives a contamination of e = 2.3 x IO - 4 cm 2/sec 3 cont which is comparable to the values of e reported by Osborn (1978) in the seasonal thermocline adjacent to the island of Santa Maria in the Azores. In the mixed layer, using Gregg's (1976) upper value of 4.3 x IO"9 °C 2/sec, e ^ = 1.0 x 10~6 cm 2/sec 3 cont which may be small compared to the kinetic energy dissipation in the mixed layer. At large depth the contamination i s , using X = 6 x 10" 1 1 °C2/sec, 1.4 XJQ-8 cm 2/sec 3. However, there are no values of e av a i l -able for depths of ^ 1000 metres. - 152 -If thermistor .#10 were to be used for temperature measurements then the level of velocity contamination in the measured temperature dissipation rate i s from equation 25 X = 1.7 x 10~6 (T - 9) 2e Acont where we have used, from Figure 10, O 5.8 x 10-3 "c/cm/sec-l ^ at U = 15.4 cm/sec. Taking, for example, an upper value of IO - 2 cm 2/sec 3 for e in the mixed layer, and x = 1 0 - 1 0 °C2/sec as the lowest dissipation rate to be resolved implies a 1% contamination at an overheat (T - 6) of only 7.7 x 10"3 °C. Using Figure 7b we calculate that this temperature r i s e results from only Q = 1.39 2 | x 7.7 x 10 - 3 °C = 1.1 x 10 - 5 watts of internal heating in the thermistor. Gregg et a l . (19.78) reports internal heating levels of 4 to 13 x 10~6 watts for his thermistors which, however, have a different mounting and are 2.5 times larger than the microbeads discussed here. Several people have suggested warmed thermistors for high frequency thermometry, however, because even very small amounts of self heating may lead to velocity contamination i t appears that grossly and moderately heated thermistors cannot be very useful for thermometry in the ocean. We have used, in the above discussion, temperature dissipation rates that were averaged over several tens of metres or more and kinetic energy dissipation rates that were averaged over 5 metres or more. On a more local scale, however, the dissipation rates are - 153 -larger than c i t e d , and consequently the amount of signal contamina-t i o n may be more severe than the values given above because tempera-ture and v e l o c i t y a c t i v i t y are not necessarily correlated. We have also ignored the effect of the heated thermistor's frequency response. Lueck (1979) has shown that the frequency response bandwidth of a heated plate i s larger for speed than for temperature. However, i n the ocean.the wavenumber range of the temperature gradient spectrum i s t y p i c a l l y larger than the range of the veloc i t y gradient spectrum. - 154 -6. Conclusions A model for the heat flux from internally heated spherical and plate-shaped thermistors to their f l u i d environment has been derived. The model incorporates the effects of a thermistor's coating and the effects of forced convection in i t s boundary layer through the use of an established Nusselt number. The model agrees favourably with the heat flux from a paralene-C coated flake thermistor as observed by Lange. Agreement between the model and our observations for the heat flux from two glass coated microbead thermistors is also favourable except that the effect of the coating is approximately two times larger than predicted. Preferential heat conduction through the frontal surface of the thermistor could account for this difference. The model agrees well with the observations i f the coating's surface area is reduced to an effective surface area. The heat flux from heated sensors, in general, depends on both the temperature and the speed of the ambient f l u i d . The simultaneous sensitivity of thermistors to speed and temperature has been ex p l i c i t l y derived. The coating reduces both the thermistor's sensitivity to speed and to temperature from i t s value in the absence of a coating. The reduction in sensitivity is greater for speed than for temperature, and hence, the relative speed-to-temperature sensitivity is also reduced by the coating. Some implication of the results for anemometry and thermometry in the ocean have been discussed. The level of contamination of a .measurement, due to the thermistor's simultaneous sensitivity to speed and temperature, depends on the amount of internal heating and on the - 155 -r e l a t i v e i n t e n s i t y of v e l o c i t y and temperature turbulence i n the f l u i d being probed. Consequently the contamination of a measurement depends strongly on one's p a r t i c u l a r circumstances. Using observed values for the rate of temperature and k i n e t i c energy d i s s i p a t i o n i n the ocean we conclude that i t i s possible that (1) v e l o c i t y measurements made by heated thermistors may be contaminated by temperature i n the main thermocline, that '(2) temperature measurements made by nearly unheated thermistors may be contaminated by v e l o c i t y i n the mixed layer and i n other regions of high v e l o c i t y but low temperature a c t i v i t y , and that (3) moderately or grossly heated thermistors are not use f u l f o r thermometry. - 156 -Bibliography Alpaugh, W.A. and D.R. Morrow, 1974, The thermal properties of monochloro-para-xylylene. Thermochimica Acta, 9_, 171-204. Batchelor, G.K., 1970, An Introduction to Fluid Mechanics, Cambridge University Press, paperback, 615 pp. Crawford, W.R. and T.R. Osborn, 1979, Microstructure measurements in the Atlantic Equatorial Undercurrent during GATE, submitted to Deep-Sea Research. Gargett, A.E., 1978, Microstructure and finestructure in an upper ocean frontal regime, Journal of Geophysical Research, 83(C10), 5123-5134. Gregg, M.C, 1976, Finestructure and microstructure observations during the passage of a mild storm, Journal of Physical Oceanography, 6, 528-555. Gregg, M.C, 1977, Variations in the intensity of small-scale mixing in the main thermocline, Journal of Physical Oceanography, 7, 436-454. Gregg, M.C., T. Meagher, A. Pederson and E. Aagaard, 1978, Low noise temperature microstructure measurements with thermistors, Deep-Sea Research, 25_, 843-856. Kramers, H., 1946, Heat transfer from spheres to flowing media, Physica, 12_, 61-80. Lueck, R.G., 0. Hertzman, and T.R. Osborn, 1977, The spectral response of thermistors, Deep-Sea Research, _24, 951-970. Lueck, R.G., 1979, The non-steady flux of heat and momentum from a f l a t plate, submitted to Journal of Fluid Mechanics. - 157 -Osborn, T.R., 1978, Measurement of energy dissipation adjacent to an island, Journal of Geophysical Research, 8£(C6), 2939-2957. Schlichting, H.-, 1955, Boundary Layer Theory, Pergamon Press, 535 pp. White, F. , 1974, Viscous Fluid Flow, McGraw-Hill, 725 pages. 

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