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Two-layer exchange flow through a contraction with frictional effects 1990

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TWO-LAYER EXCHANGE FLOW THROUGH A CONTRACTION WITH FRICTIONAL EFFECTS By EMILY ANNE CHEUNG B.Sc, The University of B r i t i s h Columbia, 1988 • A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTERS OF APPLIED SCIENCE i n THE FACULTY OF GRADUATE STUDIES ( C i v i l Engineering) We accept t h i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA September 1990 ® Emily Anne Cheung, 1990 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of C \ M \ c btA<Z\ t*i&ae\ N1<̂ The University of British Columbia Vancouver, Canada Date QcJc V\ , y ° ) 9 o DE-6 (2/88) A b s t r a c t T h e g r a v i t a t i o n a l e x c h a n g e o f two f l u i d s o f d i f f e r e n t d e n s i t y t h r o u g h a c o n v e r g e n t - d i v e r g e n t c o n t r a c t i o n i s c o n s i d e r e d . T w o - l a y e r e x c h a n g e f l o w t h e o r y i s e x t e n d e d t o i n c l u d e f r i c t i o n a l e f f e c t s w i t h a n e m p h a s i s o n t h e i n t e r f a c i a l f r i c t i o n . T h e m a g n i t u d e o f t h e i n t e r f a c i a l f r i c t i o n i s f o u n d t o b e g r e a t e r t h a n p r e v i o u s l y s u g g e s t e d a n d may b e v i t a l t o t h e a n a l y s i s o f e x c h a n g e f l o w s . E x p e r i m e n t s m o d e l l i n g g r a v i t a t i o n a l e x c h a n g e f l o w t h r o u g h a c o n v e r g e n t - d i v e r g e n t c o n t r a c t i o n w e r e c o n d u c t e d i n t h e h y d r a u l i c s l a b o r a t o r y 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 t o t e s t t h e h y d r a u l i c s o l u t i o n s t h a t h a v e b e e n d e v e l o p e d o n t w o - l a y e r e x c h a n g e f l o w . A c o m p a r i s o n o f t h e t h e o r e t i c a l s o l u t i o n s a n d e x p e r i m e n t a l r e s u l t s i s made . E x p e r i m e n t s c o n d u c t e d p r o v i d e d a t a f o r e v a l u a t i n g t h e t h e o r e t i c a l f i n d i n g s a n d h e l p i n l o c a t i n g t h e h y d r a u l i c c o n t r o l s o f t h e e x p e r i m e n t a l o n g w i t h q u a n t i f y i n g t h e m a g n i t u d e o f i n t e r f a c i a l f r i c t i o n c o e f f i c i e n t . A c o m p a r i s o n i s made b e t w e e n n u m e r o u s v a l u e s o b t a i n e d f o r t h e i n t e r f a c i a l f r i c t i o n c o e f f i c i e n t s b y p r e v i o u s i n v e s t i g a t o r s a n d t h e e x p e r i m e n t a l r e s u l t s o f t h e p r e s e n t s t u d y . F l o w v i s u a l i z a t i o n i s u s e d t o s t u d y t h e K e l v i n - H e l m h o l t z a n d H o l m b o e i n s t a b i l i t i e s t h a t f o r m a t t h e i n t e r f a c e o f t h e two l a y e r s . i i i 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 L i s t o f S y m b o l s v i i i A c k n o w l e d g e m e n t s . ' x i 1. INTRODUCTION 1 1 .1 O b j e c t i v e s 3 2 . L I T E R A T U R E REVIEW 5 2 . 1 H y d r a u l i c s o f T w o - l a y e r F l o w 5 2 . 2 E x c h a n g e F l o w 6 2 . 3 S t a b i l i t y a n d M i x i n g 7 2 . 4 F r i c t i o n C o e f f i c i e n t s 8 3 . REVIEW OF H Y D R A U L I C THEORY 10 3 . 1 B a s i c A s s u m p t i o n s 10 3 . 2 E q u a t i o n s o f M o t i o n 10 3 .3 R e v i e w o f F r o u d e Numbers 11 3 . 4 T h e S t a b i l i t y F r o u d e Number 15 3 . 5 I n t e r f a c i a l I n s t a b i l i t i e s 16 3 . 5 . 1 K e l v i n - H e l m h o l t z I n s t a b i l i t i e s 16 3 . 5 . 2 H o l m b o e I n s t a b i l i t i e s 17 4 . T H E O R E T I C A L DEVELOPMENT 18 4 . 1 E n e r g y a n d S h e a r S t r e s s 18 4 . 2 E x c h a n g e F l o w T h r o u g h a C o n t r a c t i o n 20 i v 5 . EXPERIMENTS 25 5.1 Experimental Apparatus 25 5.2 Experimental Procedure 2 6 5.3 V e l o c i t y Measurements 27 6. NUMERICAL ANALYSIS 28 6.1 The Numerical Model 28 6.2 Evaluation of F r i c t i o n C o e f f i c i e n t s 30 6.3 Control Point Location 31 6.4 Interface P r o f i l e 32 7. DISCUSSION ON THE INTERFACIAL FRICTION COEFFICIENT . . 33 8. RESULTS AND DISCUSSION 39 8.1 Hydraulic Solutions . . . . . 39 8.2 Flow v i s u a l i z a t i o n 41 8.3 I n t e r f a c i a l F r i c t i o n C o e f f i c i e n t 44 8.4 Control Point Location 4 6 9. CONCLUSION 49 10. RECOMMENDATIONS 51 Bibliography 52 Appendix A. Derivation of Equation for Interface Slope . 56 Appendix B. Example of Numerical Spreadsheet 57 Appendix C. Derivation of k and a Relation 60 V L i s t of Tables I . Summary of I n t e r f a c i a l F r i c t i o n C o e f f i c i e n t s 37 I I . Calculated S t a b i l i t y Parameters 43 I I I . Experimental Data and F r i c t i o n C o e f f i c i e n t s 45 v i L i s t of Figures 1. V a r i a t i o n of the i n t e r n a l , composite and s t a b i l i t y . . 61 Froude numbers throughout a contracted channel f o r i n v i s c i d flows 2. D e f i n i t i o n sketch of assumed l i n e a r approximations . . 62 of both the v e l o c i t y and density p r o f i l e s 3. D e f i n i t i o n sketch of the shear stresses acting on . . 63 a volume of f l u i d for both the upper and lower layers 4. Experimental set-up 64 5. Photographs of the experiment i n progress 65 6. Photograph of the experimental apparatus 66 7. Long exposure photograph showing movement of v e l o c i t y 66 beads 8. V e l o c i t y p r o f i l e at the throat for Experiment #18 . . 67 9. V e l o c i t y p r o f i l e at the throat for Experiment #19 . . 68 10. V e l o c i t y p r o f i l e at the throat for Experiment #22 . . 69 11. Long term v a r i a t i o n of the average v e l o c i t y at the . . 70 throat for Experiment #18 12. Comparison of various i n t e r f a c i a l f r i c t i o n c o e f f i c i e n t s 71 plot t e d as a function of the Reynolds number 13. V a r i a t i o n of the i n t e r n a l , composite and s t a b i l i t y . . 72 Froude numbers throughout the contracted channel 14. Comparison plot of the experimental and t h e o r e t i c a l . 73 interface height along the channel 15. a) Comparison of f r i c t i o n l e s s and f r i c t i o n a l interface 74 p r o f i l e s and b) Vari a t i o n of So, Sf, and 1-G2 approaching the control 16. Long term v a r i a t i o n of the interface height at the . . 75 throat for Experiment #18 17. Sequence of photographs showing Kelvin-Helmholtz . . . 76 i n s t a b i l i t i e s from Experiment #19 18. Sequence of photographs from Experiment #24 77 v i i L i s t of Figures (continued) 19. Sequence of photographs from Experiment #25 78 20. Sequence of photographs with Kelvin-Helmholtz and . . 79 Holmboe i n s t a b i l i t i e s from Experiment #21 21. Sequence of photographs showing Holmboe i n s t a b i l i t i e s 80 22. V a r i a t i o n of dimensionless maximum interface thickness 81 5 ,max = 5max/y' with t h e s t a b i l i t y Froude number 23. Photograph of the interface thickness 82 24. Photograph of wavelengths of the Kelvin-Helmholtz mode 82 25. S t a b i l i t y diagram (Lawrence et a l , 1990) 83 26. V a r i a t i o n of calculated k and a with 13=0, 0.5 and 1 . 84 27. Theoretical v a r i a t i o n of k and a with )3=0, 0.5 and 1 . 85 28. Comparison of t h e o r e t i c a l and calculated curves of . . 86 k and a for 0=0 29. V a r i a t i o n of: a)the composite Froude number through . 87 a contraction b)interface height along a section of a contracted channel and c)the Froude numbers for each layer shown on a Froude number plane VX11 L i s t of Symbols Aj cross sectional area of layer j b(x) width of flow b Q width of flow at throat d r e l a t i v e displacement of v e l o c i t y and density p r o f i l e interfaces E mechanical energy per unit volume f o r single layer flow Ej mechanical energy per unit volume f o r layer j (Eq. 4.1 and 4.2) EI i n t e r n a l energy for two-layer flow (Eq. 4.3) f Darcy c o e f f i c i e n t F A 2 s t a b i l i t y Froude number (Eq. 3.11) Fr or F Froude number (Eq. 3.4) F E 2 external Froude number F x 2 i n t e r n a l Froude number (Eq. 3.10) fj_ i n t e r f a c i a l f r i c t i o n c o e f f i c i e n t f w wall f r i c t i o n c o e f f i c i e n t f s surface f r i c t i o n c o e f f i c i e n t h density interface thickness h s(x) height of bottom va r i a t i o n s i n topography g g r a v i t a t i o n a l acceleration g'=eg modified g r a v i t a t i o n a l acceleration G 2 composite Froude number (Eq. 3.6) J bulk Richardson number n number of layers P wetted perimeter i x L i s t of Symbols (continued) q volumetric flow rate per unit width Q volumetric flow rate r= p 1/p 2 density r a t i o Re Reynolds number Rh hydraulic radius Ri gradient Richardson number S surface area Sf f r i c t i o n slope (Eq. 4.18) S Q topographical slope (Eq. 4.21) t time u c r i t i c a l v e l o c i t y (Eq. 6.2) U j horizontal component of v e l o c i t y for layer j x horizontal d i r e c t i o n y t o t a l depth of flow y j depth of flow for layer j y 2 1 nondimensional interface height (Eq. 6.1) X L i s t of Symbols (continued) Greek a r a t i o of i n t e r f a c i a l f r i c t i o n to wall f r i c t i o n a=kh, §8.2 i n s t a b i l i t y wave number /3 r a t i o of surface f r i c i t o n to wall f r i c t i o n 6" mixing layer thickness c S ' m a x =<S/y dimensionless maximum in t e r f a c e thickness e = ( p 2 - p 1 ) / p 2 r e l a t i v e density difference X wavelength 7r mathematical constant, 3.14 pj density of layer j i|r f r i c t i o n parameter (Kaneko) T shear stress x i Acknowledgements Many thanks to Dr. Gregory A. Lawrence who has given me endless hours of support and encouragement. His advice and suggestions on the work were invaluable to the completion of t h i s project. Without h i s support and patience, t h i s work would not have been possible. I would also l i k e to thank my husband, B i l l , f o r a l l h i s moral support during the two years of my masters program and h i s comments on numerous dr a f t s of my th e s i s . This work i s dedicated to my son, Benjamin, who f o r the f i r s t seven months of his l i f e shared h i s mother with t h i s t hesis project. 1 1. INTRODUCTION Density s t r a t i f i c a t i o n s due to temperature and/or s a l i n i t y v a r i a t i o n s occur naturally around the world i n bodies of water and the atmosphere. The exchange of two f l u i d s of d i f f e r e n t density often occurs when two bodies of water are connected by a s t r a i t or channel. In the past fo r t y years, the i n t e r e s t i n exchange flows i n many f i e l d s of study including oceanography, the atmospheric sciences, hydraulics and environmental engineering has grown considerably. A prime example that has attracted perhaps the greatest i n t e r e s t i s the S t r a i t of Gi b r a l t a r where the more sa l i n e Mediterranean Sea exchanges water with the A t l a n t i c Ocean v i a the S t r a i t . This i s due to the high evaporation rate that exceeds the p r e c i p i t a t i o n i n the Mediterranean. Other S t r a i t s i n which important exchange flow occur are discussed i n Defant (1961). They include the S t r a i t of Bab e l Mandeb which connects the Indian Ocean with the highly s a l i n e Red Sea, and the S t r a i t of Hormuz which connects the Arabian Sea and the Persian Gulf. Excess run-off flowing into the ocean through a connecting S t r a i t i n more humid regions also cause exchange flows. For example, the Black Sea i s joined by the Bosporus to the Sea of Marmara which i n turn i s connected by the Dardanelles with the Mediterranean. As well, Cabot S t r a i t l i e s between the Gulf of St. Lawrence and the A t l a n t i c Ocean. Other waterways that experience flows that may be 2 s t r a t i f i e d due to density differences include fjords, lakes, reservoirs, locks and estuaries. However i n some of these situ a t i o n s the flow may not necessarily be exchanging as i n the case of arrested s a l t wedges i n fjords and estuaries where the lower layer of f l u i d i s stagnant for long periods of time. In the S t r a i t of Georgia and Knight Inlet along the coast of B r i t i s h Columbia, s t r a t i f i e d flows are found during times of large r i v e r discharge (Thomson, 1981) . I n t e r f a c i a l mixing between the layers of s t r a t i f i e d flows has created concern i n many areas where p o l l u t i o n or s a l i n i t y threaten the qua l i t y of water. For example, an exchange flow occurs i n the ship canal connecting Lake Ontario with the heavily polluted Hamilton Harbour. An understanding of t h i s exchange flow i s c r u c i a l to the evaluation of p o t e n t i a l measures to improve water qua l i t y i n Hamilton Harbour, see Hamblin and Lawrence (1990). The environmental e f f e c t s caused by the construction of a bridge and tunnel system for the Great Belt Link i n Denmark that i s proposed to span the Great Belt, a main artery for exchange flow to the B a l t i c Sea, are to be minimized (Hansen and Moeller 199 0). The government of Denmark has decreed that the exchange flow s h a l l remain the same a f t e r the construction of the l i n k ; therefore, the added resistance of the tunnel and bridge piers must be compensated such that exchange to the B a l t i c remains unchanged. The shear produced at the interface of s t r a t i f i e d flows has 3 also attracted the in t e r e s t of numerous authors (e.g. Koop and Browand 1979 and Thorpe 1987). The generation of turbulence and mixing i n s t r a t i f i e d flows i s a function of the s t a b i l i t y of the sheared density interface. Understanding the processes involved i n the shear and mixing i s pertinent to the advance of the study of exchange flows. The shear between the s t r a t i f i e d layers generates i n s t a b i l i t i e s known as Kelvin-Helmholtz and Holmboe i n s t a b i l i t i e s . The magnitude of t h i s shear and the s i z e of the Kelvin-Helmholtz billows can be predicted from i n t e r n a l hydraulic theory; however, good agreement between observations and predictions has not as yet been obtained. An i n t e r f a c i a l f r i c t i o n c o e f f i c i e n t i s used to quantify the shear that i s produced at the interface of two flows. Although there seems to be no consensus on a method of quantifying t h i s c o e f f i c i e n t , various authors have examined t h i s problem and further research i s included i n the present study. 1.1 Objectives There are three primary objectives of t h i s study: 1. To t e s t experimentally the e x i s t i n g hydraulic solutions for exchange flow through a contraction. 2. To obtain good flow v i s u a l i z a t i o n of the exchange flow i n experimental conditions. 3. To extend the theory of exchange flows to include f r i c t i o n a l e f f e c t s and locate the i n t e r n a l hydraulic controls. 4 A review of the hi s t o r y of exchange flows i s presented i n §2 including two-layer flow, mixing and f r i c t i o n c o e f f i c i e n t s . The i n t e r n a l hydraulic theory i s reviewed i n §3 covering assumptions, equations of motion, Froude numbers and a discussion of i n t e r f a c i a l i n s t a b i l i t i e s . The t h e o r e t i c a l development of exchange flow i s covered i n §4 with a p p l i c a t i o n to exchange flow through a convergent-divergent contraction presenting formulation with f r i c t i o n a l considerations. Details of the experimental work are provided i n §5 followed by a description of the solution technique and the evaluation of the f r i c t i o n c o e f f i c i e n t s i n § 6 . A discussion on the i n t e r f a c i a l f r i c t i o n follows i n §7 with the r e s u l t s and conclusions given i n § 8 and § 9 . F i n a l l y , further recommendations f o r additional research are discussed i n § 1 0 . 5 2. LITERATURE REVIEW 2.1 Hydraulics of Two-layer Flow Among the f i r s t t h e o r e t i c a l analyses of two-layer flow were those of S c h i j f and Schonfeld (1953) and Stommel and Farmer (1953). They examined numerous examples including s a l t wedges, s a l t i n t r u s i o n i n estuaries and lock exchange flow. The analysis of S c h i j f and Schonfeld (1953) includes f r i c t i o n a l e f f e c t s i n the formulation of two-layer flow equations, but assumes n e g l i g i b l e surface shear and side wall shear stresses. They have also expanded the theory to lock exchange flow, s a l t wedges and s a l t i n t rusion i n an estuary, but focus p r i m a r i l y on turbulence and other d i f f u s i o n mechanisms. Subsequently, Wood (1968, 1970) also examined lock exchange flow and the i d e n t i f i c a t i o n of points of in t e r n a l hydraulic control. Wood (1968) i d e n t i f i e s two points of control for s t r a t i f i e d flows through a contraction: one at the point of minimum width and the other c a l l e d the point of v i r t u a l c o n t r o l . Both control points can be located again by s a t i s f y i n g the condition of the composite Froude number, G 2=l. Wood (1970) also looks at lock exchange flow but does not deal with f r i c t i o n . Lawrence (1985) presents additional work on two-layer flow with an emphasis on flow over an obstacle. A c l a s s i f i c a t i o n scheme was developed to i d e n t i f y regimes of steady two-layer flow and the location of int e r n a l hydraulic controls. Denton (1987) looks at locating and i d e n t i f y i n g hydraulic 6 controls and analyzes single layer flow including f r i c t i o n a l e f f e c t s and two-layer flow assuming n e g l i g i b l e f r i c t i o n by int e r n a l energy methods. Recently, Denton (1990) extended the work to u n i d i r e c t i o n a l three-layer flow over a hump and through a contraction and c l a s s i f i e s d i f f e r e n t flow regimes, each with a d i f f e r e n t set of locations for i n t e r n a l hydraulic cont r o l . 2.2 Exchange Flow Exchange flow through a long s t r a i t was examined by Assaf and Hecht (1974) whereby the flow i s bounded by controls at either end of the s t r a i t . Their analysis models an enclosed basin by balancing f r i c t i o n , s a l t and mass through the s t r a i t . Results of the model are compared with observations made of the S t r a i t of Gibr a l t a r , the Bosphorus and the Bab e l Mandeb S t r a i t . Papers by Armi (1986) , Armi and Farmer (1986) and Farmer and Armi (1986) have presented i n t e r n a l hydraulic theory on exchange flows. In p a r t i c u l a r , g r a v i t a t i o n a l exchange flow of two f l u i d s of s l i g h t l y d i f f e r e n t density through a contraction has been examined by Armi and Farmer (1986). Armi and Farmer (198 6) examine maximal two-layer exchange flow through a contraction with barotropic flow, that i s q ^ s ^ , where q i s the volumetric flow rate, and also acknowledge the presence of the two control points, the narrowest section (throat) and the v i r t u a l control, separated by a region of s u b c r i t i c a l flow. In the absence of any barotropic component or any f r i c t i o n a l e f f e c t s , these two control points coalesce. 7 Farmer and Armi (1986) have presented d e t a i l e d exchange flow analysis for both flow over a s i l l and the combination of a s i l l and a contraction with application of the hydraulics of layered flows to f i e l d measurements of the S t r a i t of G i b r a l t a r . A considerable amount of f i e l d work and a p p l i c a t i o n of exchange flow theory was done by Armi and Farmer (1988) i n the S t r a i t of G i b r a l t a r and recently applied to a model of the S t r a i t . 2.3 S t a b i l i t y and Mixing At the interface of two s t a t i c a l l y stable f l u i d s i s a horizontal shear which may generate i n s t a b i l i t i e s that cause mixing between the two layers. This process has been examined by many authors including Taylor (1931), Goldstein (1931), Turner (1973), Sherman, Imberger & Corcos (1978) and Thorpe (1987) . Koop and Browand (1979) studied some of the c h a r a c t e r i s t i c s of turbulence i n s t r a t i f i e d f l u i d s with emphasis on conditions which may approximate those found i n oceans. They also suggest an upper bound of turbulent mixing and conclude that the Richardson number, Reynolds number and Schmidt number a l l become important i n the analysis of turbulence. Lawrence (1985) presents work on two-layer flow with an emphasis on flow over an obstacle. He examines in t e r n a l hydraulic controls i n the flow as i t approaches an obstacle and also investigates the dynamics of mixing of the two layers. The extent to which mixing occurs was examined by Lawrence (1989) and he poses the question, "Can mixing i n exchange flows be 8 predicted using i n t e r n a l hydraulics?". Lawrence states that an upper bound on the extent of mixing can be predicted from the i n t e r n a l hydraulics. A d e t a i l e d examination of the s t a b i l i t y Froude number, F A 2, was done recently by Lawrence (1990a) and he notes that i t i s of great s i g n i f i c a n c e i n the prediction of mixing i n two-layer flows. Lawrence, Browand & Redekopp (1990) discuss the s t a b i l i t y of a sheared interface s t a t i n g that i t i s fundamental to the generation of mixing i n s t r a t i f i e d flows and i s dependent on the v e l o c i t y and density differences of the flowing layers of f l u i d . Their t h e o r e t i c a l and experimental r e s u l t s are presented covering a more general study of i n t e r f a c i a l i n s t a b i l i t i e s and presented are s t a b i l i t y diagrams used i n the p r e d i c t i o n of wavelengths of both the Kelvin-Helmholtz and Holmboe modes of i n s t a b i l i t y . The shear at the interface leads to the formation of Kelvin-Helmholtz and Holmboe i n s t a b i l i t i e s which are also discussed by Thorpe (1987). 2 .4 F r i c t i o n C o e f f i c i e n t s The dynamics of f r i c t i o n a l exchange flows involves an examination of the shear stresses on the walls and interface of s t r a t i f i e d flows which are mainly dependent on two c o e f f i c i e n t s of f r i c t i o n namely the wall f r i c t i o n c o e f f i c i e n t , f w , and the i n t e r f a c i a l f r i c t i o n c o e f f i c i e n t , f j . Although evaluation of the wall f r i c t i o n c o e f f i c i e n t has been well documented (Henderson 1966), there has proven to be great d i f f i c u l t i e s i n evaluating 9 the i n t e r f a c i a l f r i c t i o n c o e f f i c i e n t . Often an order of magnitude estimate of, 10"3, i s used to quantify the c o e f f i c i e n t . Although methods of defining the i n t e r f a c i a l f r i c t i o n c o e f f i c i e n t were presented as early as 1953 by S c h i j f and Schdnfeld, there has been no conclusion as to which method i s most appropriate or i f any of the presented methods are adequate for c a l c u l a t i n g the c o e f f i c i e n t . A det a i l e d review and discussion of various authors' methods of determining the i n t e r f a c i a l f r i c t i o n c o e f f i c i e n t are presented i n §7 a f t e r some of the pertinent theory has been discussed. 10 3. REVIEW OF HYDRAULIC THEORY 3.1 Basic Assumptions The focus of t h i s analysis of layered flows i s based on the assumptions of steady two-dimensional, non-rotating, i n v i s c i d flow. The assumption i s also made that there i s n e g l i g i b l e free surface d e f l e c t i o n . There i s assumed to be no external forcing, f o r example, t i d a l l y driven flows, such that the exchange flow i s due to the density differences i n the two layers. The analysis focuses primarily on two-layer flows; however, extension to any number of layers has been formulated by others including Benton (1954), Baines (1988), and Denton (1990). I t i s useful here to begin by understanding the hydraulics of single layer flow before expanding to two-layer flow. Armi (1986) and Lawrence (1989) discuss the basic hydraulic theory involved i n two-layer flow; however, a review of some of the pertinent theory follows i n the next section. 3.2 Equations of Motion Assuming steady flow, the motion of layered flows i s governed by (3.1) 11 and the continuity equation 1 dQj =0 (3.2) b dx where j i s the number of layers (j=l for single layer flow), Pj i s the density of layer j , b i s the width of the flow, Qj i s the volumetric flow rate and Ej i s the mechanical energy per unit volume defined as where p i s the pressure, assumed hydrostatic, g i s the acceleration due to gravity, y j and Uj are the depth and the v e l o c i t y of layer j and n i s the number of layers considered. Although Equations 3.1 and 3.2 are one-dimensional, they can be applied to flow through a contraction provided the contraction i s gentle. 3.3 Review of Froude Numbers Hydraulic flow i s t r a d i t i o n a l l y c l a s s i f i e d by the nondimensional Froude number that i s best introduced as a r a t i o of convective v e l o c i t y to phase speed. In single layer flow, a flow i s c l a s s i f i e d as s u b c r i t i c a l where the Froude number of the flow i s less than 1; s u p e r c r i t i c a l where the Froude number i s greater than 1; and the control point of the flow i s located where the Froude number has a value of 1. The control point i s ( j = l.n) (3.3) 12 a t r a n s i t i o n point and provides a s t a r t i n g point required to study flows. Henderson (1966) gives a general d e f i n i t i o n of a hydraulic control as a point at which there i s a known depth-discharge re l a t i o n s h i p . An example of a control can be seen at a s l u i c e gate at which upstream flow i s s u b c r i t i c a l and downstream flow i s s u p e r c r i t i c a l ; here, the s l u i c e gate acts as a control device (Henderson, 1966). The Froude number i s evaluated by Fr2 = -Hi (3.4) gy This theory can e a s i l y be expanded to two-layer flow where the densimetric Froude number for each layer simply becomes Fl) = -4- (3.5) g'yj where g 1 i s the modified acceleration due to gravity; that i s , g'= eg and 8 i s defined by e = (p2~P]_)/P2' However, i n two-layer flow i t i s important to recognize a composite Froude number denoted G 2 and defined by Equation 3.6. G2 = Frl + Fr% - eFr?Fri (3.6) In t h i s case the s i n g u l a r i t y condition or control occurs where the composite Froude number, G 2=l analogous to Fr 2 = l i n single layer flow. T y p i c a l l y the Boussinesq approximation i s made that 13 e<l (of order 10°) such that Gl i s approximated by G2 = Frl + Frl (3.7) The l o c a t i o n of the control point i n a convergent-divergent channel for i n v i s c i d flows i s h i s t o r i c a l l y the narrowest point or throat of the channel (Armi and Farmer, 1986). V a r i a t i o n of the Froude numbers through a contraction i s best diagrammed on the Froude number plane as shown by Armi and Farmer (1986). The flow i s c r i t i c a l at the throat (G2=l) and on e i t h e r side of the control point or throat i s s u p e r c r i t i c a l flow (G2>1). In conjunction with G , three additional Froude numbers, F x 2, F E 2, and F A 2; the i n t e r n a l , external and s t a b i l i t y Froude numbers respectively, become important i n two-layer flow. Recall that the Froude number i s defined as the r a t i o of convective v e l o c i t y to the phase speed. This i s also applicable to the i n t e r n a l and external Froude numbers. The c e l e r i t y or c h a r a c t e r i s t i c v e l o c i t y of long waves both on the surface (external) and along the interface of two-layer flow (internal) i s a sum of a convective v e l o c i t y and a phase speed. Note that although the composite Froude number may determine the c r i t i c a l i t y of two-layer flow, i t cannot be defined as the single layer Froude number as i t i s not a r a t i o of convective v e l o c i t y to phase speed for both i n t e r n a l and external waves (Lawrence 1990a). Perhaps the most useful of these three Froude numbers i s 14 the s t a b i l i t y Froude number. I t can be regarded as the inverse of a bulk Richardson number and i s used i n quantifying the mixing layer thickness, the s t a b i l i t y of the flow and i t s s u s c e p t i b i l i t y to i n s t a b i l i t i e s at the interface. A fundamental re l a t i o n s h i p (Lawrence 1985, 1990a) exi s t s between these four Froude numbers given by: However, for exchange flow through a contraction, F E Z ~ 0 from the assumption of n e g l i g i b l e free surface d e f l e c t i o n . Therefore the r e l a t i o n s h i p between the Froude numbers, Equation 3.8 i s reduced to the following. V a r i a t i o n of these three Froude numbers along the contraction for i n v i s c i d flows i s better understood by the i l l u s t r a t i o n shown i n Figure 1. Note that the location of the control i s the point at which G 2=l. For Boussinesq two-layer flows the external Froude number i s the same as the single layer Froude number (3.4) . The in t e r n a l Froude number (Lawrence 1985, 1990a) i s defined ( 1 - G 2 ) = (1-Fl) (1-F2E) ( 1 - F | ) (3.8) ( 1 - G 2 ) = ( 1 - F | ) ( I - F J ) (3.9) uiV2 + u2Vi (3.10) where y = y i _ + y 2 • The sign i f i c a n c e of the s t a b i l i t y Froude number i s discussed further i n the following section. 15 3.4 The S t a b i l i t y Froude Number The i n t e r f a c i a l long wave s t a b i l i t y Froude number i s a representation of the strength of the v e l o c i t y shear across the interface of the two f l u i d layers. This s t a b i l i t y Froude number i s defined as: Fl = -AH! (3.11) g'y where A u = u^-u 2 and y i s the t o t a l depth of flow. I f f r i c t i o n a l e f f e c t s are ignored, Lawrence (1990b) has shown that F A 2=1 throughout the channel i f the flow rate r a t i o , q r=l. However, taking into account the f r i c t i o n a l e f f e c t s reduces F A 2 proportionally as the flow rates i n each layer are a c t u a l l y less than the t h e o r e t i c a l ideal values. Lawrence (1990a) notes from Equation 3.10, Long's s t a b i l i t y c r i t e r i o n for long int e r n a l waves, F A 2 < 1, must be s a t i s f i e d i n order for the int e r n a l Froude number to have r e a l values. Long's s t a b i l i t y c r i t e r i o n applies only to long i n t e r n a l waves, since the assumption of a hydrostatic pressure d i s t r i b u t i o n precludes the existence of short waves. To quote Long (1956), *If we abandon the hydrostatic assumption momentarily, we f i n d that s u f f i c i e n t l y short i n f i n i t e s i m a l waves are unstable for any shear. 1 Thorpe (1987) notes that the interface i s unstable to i n s t a b i l i t i e s even for F A 2<1 including the Kelvin-Helmholtz and Holmboe i n s t a b i l i t i e s . Note that the higher F A 2 the larger the i n s t a b i l i t i e s . 16 3 . 5 I n t e r f a c i a l I n s t a b i l i t i e s 3 . 5 . 1 K e l v i n - H e l m h o l t z I n s t a b i l i t i e s T h e K e l v i n - H e l m h o l t z i n s t a b i l i t i e s a t t h e i n t e r f a c e o f t h e two f l u i d s c a u s e c o n s i d e r a b l e m i x i n g t o o c c u r . T u r n e r (1973) s t a t e s , "When a s u f f i c i e n t l y l a r g e s h e a r i s a p p l i e d a c r o s s a d e n s i t y i n t e r f a c e a n d i s s u c h t h a t t h e g r a d i e n t R i c h a r d s o n number f a l l s b e l o w a c r i t i c a l v a l u e o f a b o u t 0 . 2 5 , K e l v i n - H e l m h o l t z w a v e s w i l l g r o w a n d o v e r t u r n t o p r o d u c e p a t c h e s o f t u r b u l e n t m i x i n g . " Where t h e g r a d i e n t R i c h a r d s o n n u m b e r i s d e f i n e d , R i = N 2 / ( 3 u / d z ) 2 , a n d t h e b u o y a n c y f r e q u e n c y , N 2 = ( g / p ) ( d p / d z ) . T h i s m e c h a n i s m o f m i x i n g c a u s e s t h e p r o d u c t i o n o f i n t e r f a c i a l l a y e r s i n s t r a t i f i e d f l u i d s . W i l k i n s o n a n d Wood (1983) d e s c r i b e t h e K e l v i n - H e l m h o l t z i n s t a b i l i t y a s o n e w h i c h c o n v e r t s k i n e t i c e n e r g y o f l a r g e - s c a l e s h e a r f l o w s t o s m a l l e r d i s s i p a t i v e s c a l e s . T h e s e t h r e e - d i m e n s i o n a l d i s t u r b a n c e s e f f e c t i v e l y m i x m o s t o f t h e f l u i d t h a t i s e n t r a i n e d b y t h e K e l v i n - H e l m h o l t z b i l l o w s . A s a r e s u l t , maximum i n t e r f a c e t h i c k n e s s , <S m a x , c a n b e p r e d i c t e d . L a w r e n c e (1990b) p r e s e n t s a d i a g r a m s h o w i n g t h e r e l a t i o n s h i p o f <5m a x a n d F A 2 f o r t h e c a s e o f F A 2 = 0 ( 1 ) w h i c h i s t h e s i t u a t i o n c o n s i d e r e d h e r e . F u r t h e r c o n s i d e r a t i o n o f t h i s d i a g r a m i s made i n § 8 . 1 . V e l o c i t y a n d d e n s i t y p r o f i l e s a r e m o d e l l e d u s i n g t h e p i e c e w i s e l i n e a r a p p r o x i m a t i o n s shown i n F i g u r e 2 . Two l a y e r s a r e shown o f d i f f e r e n t d e n s i t i e s w i t h a d e n s i t y i n t e r f a c e t h i c k n e s s o f h . T h e s h e a r a t t h e i n t e r f a c e o f t h e two f l o w s 17 leads to the formation of Kelvin-Helmholtz and Holmboe type i n s t a b i l i t i e s . 3 . 5 . 2 Holmboe I n s t a b i l i t i e s The Holmboe i n s t a b i l i t y was f i r s t studied by Holmboe (1962) and more recently by Browand and Winant (1973), Koop and Browand (1979), Smyth, Klaassen and P e l t i e r (1988) and Lawrence, Browand & Redekopp (1990). Although the occurrence of Kelvin-Helmholtz i n s t a b i l i t i e s have been well documented, there i s l i t t l e documented evidence of Holmboe i n s t a b i l i t i e s . The Holmboe mode of i n s t a b i l i t y i s known to dominate only i n the case where the v e l o c i t y and density interfaces are not displaced v e r t i c a l l y with respect to each other. Linear s t a b i l i t y theory predicts the formation of the Holmboe i n s t a b i l i t y when the gradient Richardson number exceeds a c r i t i c a l value of approximately 0.2 5 (For further discussion see Smyth et a l , 1988 and 1989) . These i n s t a b i l i t i e s are depicted by a series of sharply cusped crests which protrude into each layer of f l u i d . Portions of the top of the cusps are occasionally torn away and become mixed with the layers of f l u i d flowing by. More f l u i d i s drawn up by these cusps as the Richardson number approaches zero ultimately forming Kelvin-Helmholtz billows. 18 4. THEORETICAL DEVELOPMENT 4.1 Energy and Shear Stress Consider the Mechanical Energy per unit volume f o r each layer separately. E1 = PiffV + -jPiUi (4.1) E2 = pxgy + (p2-p1)gy2 + \ p 2 u % ( 4 - 2 ) Subtracting Equation 4.1 from 4.2 and d i v i d i n g by the unit weight of the lower layer, p2<?'/ gives us an equation f o r the int e r n a l energy head. For two-layer flow, the i n t e r n a l head i s obtained from EI = y2 + -—(u2 - ul) (4.3) si m i l a r to that of the t o t a l energy or Bernoulli equation for single layer flow given by Equation 4.4. E = y + -ii - ' (4.4) D i f f e r e n t i a t i n g Equations 4.1 and 4.2, the r e s u l t i s two simultaneous equations which when solved produce an equation for the slope of the interface. Denton shows these equations neglecting f r i c t i o n a l e f f e c t s . S c h i j f and Schonfeld include the 19 bottom and i n t e r f a c i a l shear stresses and assume n e g l i g i b l e side wall and surface shear stresses. They also state that t h e i r formulas are approximate, based on the assumption that e<l. In t h i s section the theory i s expanded to evaluate f r i c t i o n a l e f f e c t s caused by the surface, the side walls, the interface and the bottom surface. Energy losses due to f r i c t i o n are denoted by the shear stresses at the four surfaces respectively and can be represented by the following equations. Surface xs Walls xw Upper layer Interface xT1 Lower layer Interface xXi Bottom xb For Boussinesq flows, p" i s estimated as p-̂  for the upper layer and p 2 for the lower layer. The convention has been adopted that the p o s i t i v e d i r e c t i o n of flow i s that of the upper layer such that | U ] _ | = U]_ and |u2|= -u 2. The shear stresses are defined diagrammatically i n Figure 3 by a sketch of an element of f l u i d from each layer. Introduced i n the shear stress equations are the f r i c t i o n c o e f f i c i e n t s where f s i s the surface f r i c t i o n c o e f f i c i e n t , f w i s the wall f r i c t i o n c o e f f i c i e n t and f1 i s the i n t e r f a c i a l f r i c t i o n c o e f f i c i e n t . Note that throughout the - r - s P i u i K I -fwPjUi\Uj\ - f x " p ( A u ) 2 f x p ( A u ) 2 ( 4 . 5 ) ( 4 . 6 ) (4.8) analysis f w=f/8; so that conventionally f, the Darcy c o e f f i c i e n t , i s equivalent to 8 f w . Note that f1 does not acquire the same value as f w nor f s since the Darcy c o e f f i c i e n t would be d i f f e r e n t for the interface, the surface and the walls. Such formulations for the i n t e r f a c i a l shear and bottom shear are s i m i l a r to those presented by S c h i j f and Schonfeld (1953). We assume that the energy losses are due to these shear stresses on a l l the walls, the surface and at the interface of the two flows such that d E J = E t J ( 4 > 1 0 ) dx Aj dx where S i s the surface area and Aj i s the cross s e c t i o n a l area. The l e f t side of Equation 4.9 i s evaluated by d i f f e r e n t i a t i n g (4.1) and (4.2) for each layer respectively. The r i g h t side of (4.10) for each layer are evaluated by \XfS = - f w P i " i 4 - ^ i P i A u 2 - ^ - f . P ^ i 2 — (4.11) A1dx w 1 b 1 1 y1 s 1 yx \XfS = fw?2ul\ * fiP2^2— + ^ P 2 " l — (4-12) A2dx w * b 1 z y2 y2 with the shear stresses defined e a r l i e r i n Equations 4.5 - 4.9. 4.2 Exchange Flow Through a Contraction The use of the one-dimensional equations (3.1 and 3.2) to multi-layer flows accounting for the shear stresses acting on 21 the i n t e r f a c i a l and boundary surfaces can be reduced to Equation 4 .13 . C | X = D _|£ + S (4.13) ox ox For single layer flow u g u -g o V v = D = . o c f = 8 -y u y. and for two-layer flow 0 sr 9 " l 0' 0 U2 9 0 Vi 0 " i 0 V = D = 0 Oi 0 y 2 0 U2. 0 °2. f = b-1 2 1 f „ u 2 2 ( - | + — ) + f j A u 2 - 0 0 Generally h s(x) are the bottom va r i a t i o n s i n topography, but are not considered i n the present study. Note the f r i c t i o n a l components appear only i n matrix S where f w , f j and f s are the wall, i n t e r f a c i a l and surface f r i c t i o n c o e f f i c i e n t s . Since Equation 4.13 i s quasi-linear, the dependent variables v x can be expressed as functions of the independent variables f x . Thus the following four equations are derived. 2 2 1 l - F | ( l ^ / y J 1 db 1 ul dx 1-G2 b dx 1-G2. dx 1 du2 \ i-^ 2K y v y 2 ) 1 db I-FI i dhg U2 dx l-G2 b dx . 1-G2 . 2̂ dx 1 dyx _ G 2 - F | ( l + ^ / y i ) " l db ' Fl ' 1 dhs y i dx 1-G2 b dx 1-G2. 1̂ dx i dy2 G*-Fi(l+**/y2) 1 db 1 - F i 1 dhs y 2 dx 1-G2 b dx 1-G2 y 2 dx yi\ ASf 1-G2 1 y 2 A S f 1-G 2 1 A S , 1-G2 A S , 1-G2 (4.14) (4.15) (4.16) (4.17) Details of the derivation of Equation 4.17 as an example are given i n Appendix A. The f r i c t i o n slope, As f, which includes the wall, i n t e r f a c i a l , bottom, and surface f r i c t i o n , i s given by A S f = ASfw + ASfI + ASfb +ASfs that i s AS, 2f, -[Fly, * Fly2] + fxFl + f j r l + fsFl (4.18) where A s f = S f 2 ~ S f l and Sf! and S f 2 are the f r i c t i o n slopes for the upper and lower layers respectively. Note that the density 23 r a t i o , r=p^/p2 which i s also equivalent to 1-e. The Froude number, Fr, i s now denoted simply as F. I t i s useful here to examine the problem by looking at the i n t e r n a l energy head. The flow within the channel depends on the i n t e r n a l resistance equation from Henderson (1966). = - ASf (4.19) dx where S f can be termed the energy slope or f r i c t i o n slope. Substituting the i n t e r n a l energy head, Equation 4.3, into Equation 4.19, r e s u l t s i n an equation for the slope of the interface. dy^ S0 - A S , { A 2 Q ) dx l-G2 where S Q here i s defined as a topographical slope a t t r i b u t a b l e in the present study to the v a r i a t i o n of width i n the channel and any v a r i a t i o n i n depth which i s considered n e g l i g i b l e here. Note that the form of the Equation 4.20 i s s i m i l a r to that derived i n Equation 4.17; therefore, expressions for both the topographical slope and the f r i c t i o n slope are determined with the f r i c t i o n slope given by Equation 4.18 and the topographical slope by Equation 4.21. SQ - (Fly2-rF?yi) A _g (4.21) 24 The singular points of Equation 4.20 occur again where the composite Froude number, G 2=l. Therefore the control points are found from the numerator of Equation 4.20 where Sa = A5 f (4.22) Two points of control are then i d e n t i f i e d ; one for and one for y 2 which occur at equal distances on opposite sides of the throat. For each experiment the control points were i d e n t i f i e d using the above formulation. 25 5. EXPERIMENTS 5.1 Experimental Apparatus A two-layer g r a v i t a t i o n a l exchange of s a l t and fresh water through a convergent-divergent contraction i s modelled i n a 3.7 x 1.1 x 0.3 m tank. The simplest configuration that incorporates both f r i c t i o n a l e f f e c t s and var i a b l e topography was constructed as a f i r s t step towards modelling natural configurations. Two reservoirs of approximately 500 l i t r e s are joined v i a a contraction which may be altered i n both width and curvature. Both the elevation and plan view of the apparatus are shown i n Figure 4. Each reservoir i s independently f i l l e d with f l u i d from the same source by i n s t a l l i n g a b a r r i e r i n the throat of the contraction. The density of the r i g h t r e s e r v o i r i s increased by di s s o l v i n g a known quantity of s a l t as well as fluorescein dye to d i f f e r e n t i a t e between the two flowing layers of f l u i d . I n i t i a l measurements are taken of the t o t a l water depth as well as the channel widths at the throat, channel end points and midway points between the throat and channel ends. These measurements are taken to ensure a width p r o f i l e s i m i l a r to the assumed p r o f i l e (discussed further i n §6) since the channel i s variable i n width along the entire channel length. The modified acceleration due to gravity, g', i s calculated knowing the s a l t content by weight and reservoir volume. Temperature measurements are also taken to a t t a i n a value f o r v i s c o s i t y 26 which i s required to calculate Reynold's numbers. The experiment i s conducted i n a darkened room with two s l i d e projectors mounted on the c e i l i n g above the contraction of the channel. Each projector contains a s l i d e which has mounted on i t two razor blades aligned with about only a 1 mm gap to allow for a t h i n sheet of l i g h t (approx. 3 or 4 mm at the water surface) to illuminate a two dimensional view of the flow i n the channel. Dissolving fluorescein dye i n the more dense layer produces a lower fluorescing "green" layer flowing leftward and a top c l e a r layer or "black" layer flowing r i g h t as shown by Figure 5. A photograph of the experimental apparatus i n shown in Figure 6. 5.2 Experimental Procedure A f t e r preparation of the two reservoirs and allowing a b r i e f moment for the reservoirs to s e t t l e , the b a r r i e r at the throat i s removed. Due to the density gradient, the f l u i d i n the r i g h t r e s e r v o i r i s forced to flow under the l e s s dense f l u i d s e t t i n g up an exchange flow. I t may take up to a h a l f of a minute before the experiment becomes quasi-steady and the duration i s t y p i c a l l y approximately 10 minutes enabling s u f f i c i e n t time for measurements and photography to be done. Polystyrene beads are used as ne u t r a l l y buoyant p a r t i c l e s which are seen when they pass through the t h i n sheet of l i g h t projected through the layers of f l u i d . O r i g i n a l l y these beads are s l i g h t l y more dense than water ( s p e c i f i c gravity of 1.04) 27 and therefore sink when submersed. However a technique was developed to heat and expand the polystyrene beads so as to make them neu t r a l l y buoyant and hence suitable for experimental purposes. 5.3 V e l o c i t y Measurements In order to obtain v e l o c i t y measurements, long exposures of 3 or 4 seconds are taken i n which the polystyrene beads show up c l e a r l y as streaks on the photographs. An example photograph, Figure 7, shows the streaks from the beads i n both the upper and lower layers. Although t h i s proves worthy for obtaining i n d i v i d u a l v e l o c i t i e s at c e r t a i n depth locations, i t i s inadequate for producing v e l o c i t y p r o f i l e s throughout the depth of the flow because of i n s u f f i c i e n t beads passing through the single frame i n a p a r t i c u l a r instance. To obtain f u l l v e l o c i t y p r o f i l e s throughout the depth, a video camera i s use to record the experiment i n progress. Using an image processor, several images are captured i n sequence from the video tape over a known time span and v e l o c i t i e s are determined from the r e l a t i v e movement of i n d i v i d u a l beads throughout the depth of flow. Three examples of the v e l o c i t y p r o f i l e s obtained are shown i n Figures 8,9, and 10. To v a l i d a t e the assumption of steady flow, a long term analysis of the v e l o c i t i e s was done (Figure 11) which shows the startup, a period of quasi-steady flow for approximately 10 minutes and the uncontrolled l a t e r portions of the experiment. 28 6. NUMERICAL ANALYSIS 6.1 The Numerical Model In order to evaluate the location of the control points and the i n t e r f a c i a l f r i c t i o n c o e f f i c i e n t , a numerical spreadsheet model was developed using the hydraulic theory discussed i n §3 along with §4. The channel p r o f i l e i s assumed to take on the shape represented by b'=exp(x') 2 with b 1 and x' being nondimensional channel width and length respectively defined by b'=b/bo and x'=x/2L, L i s the distance from the throat to the channel ends and bo i s the width at the narrowest point of the channel c a l l e d the throat. Therefore, the nondimensional channel width can be determined as a function of nondimensional channel length. Choosing an increment i n t e r v a l for the channel length, b 1 i s then determined at each of the increment points along the channel. The height of the interface at each of these points i s then evaluated by x<0 (6.1) x>0 where y 2' i s the nondimensional interface height. This equation i s adapted from Lawrence (1990b) assuming that the flow r a t i o , q r=l where q r = q^/q2 and q i s the flow per unit width. The t h e o r e t i c a l v e l o c i t y at the throat, u, i s calculated from Equations 3.5 and 3.7, assuming that u 1=-u 2 and y^=y2 at the throat so that the v e l o c i t y at the throat, u, for c r i t i c a l flow i s evaluated from Equation 6.2. However, t h i s r e s u l t s i n a t h e o r e t i c a l v e l o c i t y which ignores f r i c t i o n a l e f f e c t s . To compensate, t h i s v e l o c i t y i s reduced to allow for f r i c t i o n by using v e l o c i t i e s measured from photographs and analysed from the video tapes of each of the experiments. From the experimental data, a reduction c o e f f i c i e n t , k, i s determined to factor down the v e l o c i t y to correspond to the physical values measured such that the t h e o r e t i c a l v e l o c i t y i s equal to ku. P r o f i l e s taken from numerous experiments at d i f f e r e n t locations were plotted and a regression curve f i t to the data to evaluate the experimental v e l o c i t i e s and determine a value of the v e l o c i t y reduction c o e f f i c i e n t . A value of k = 0.74±0.05, equivalent to 74% of the t h e o r e t i c a l v e l o c i t y was determined to be appropriate. Both the v e l o c i t i e s across the channel as well as the v a r i a t i o n throughout the depth of the channel are considered i n determination of the v e l o c i t y c o e f f i c i e n t . The flow rate i s calculated using the v e l o c i t y , depth, and width at the throat. Subsequent v e l o c i t i e s downstream of the throat are then evaluated using the continuity equation. Froude 30 numbers for both layers are computed from Equation 3.5. Now a l l of the required data i s available for c a l c u l a t i n g the composite Froude number which determines the locations of the control points. To substantiate the assumption that the flow r a t i o of the two-layers, q r = q^/q2, i s indeed unity as prescribed f o r no external forcing, q r i s also evaluated. The s t a b i l i t y Froude number (Equation 3.11) evaluated at a l l locations i n the channel proved to be constant for each experiment. 6.2 Evaluation of F r i c t i o n C o e f f i c i e n t s Due to discrepancies i n the l i t e r a t u r e as to the r e l a t i v e importance of d i f f e r e n t f r i c t i o n a l terms, i t i s necessary to use the experimental data to look at t h i s problem. A l l of the f r i c t i o n terms from Equation 4.18 are evaluated to examine the r e l a t i v e magnitude of each. Since a l l terms are within one order of magnitude, each of the terms are considered s i g n i f i c a n t enough to be included i n the analysis. Values for f w are calculated from the H. Blasius* s o l u t i o n for a f l a t plate boundary layer theory (Schlichting 1979) given by f 2.656 f = (6.3) yfRex r e c a l l i n g that f w = f/8 and Re x i s the Reynolds number, Re = ux/v, using a length parameter, x, of the length of the contracted channel. Note that t h i s equation i s v a l i d for Re x < 5xl0 5 and since the Reynolds number, Re x, i s used, then the wall f r i c t i o n c o e f f i c i e n t , f w , i s a function of x. I t i s also important to r e a l i z e that the flow i s not f u l l y developed thereby the boundary layers do not extend throughout each layer. This i s good for examining the i n t e r f a c i a l stress since there w i l l be no interference from the wall stresses and the i n t e r f a c i a l stress can be determined more accurately. A constant, /3, i s introduced to r e l a t e the surface f r i c t i o n c o e f f i c i e n t to f w such that f s = /3fw. Since a value of /3 i s not determined experimentally, the numerical analysis i s done allowing for complete v a r i a b i l i t y of /3, 0</3<l. Once f w i s determined from the Blasius equation, the only remaining variable i s the i n t e r f a c i a l f r i c t i o n c o e f f i c i e n t , fj_. 6.3 Control Point Location Experimental data which include g', b 0, y Q, and u are used i n conjunction with the numerical spreadsheet model to determine the location of the control points on e i t h e r side of the throat • • • 9 • 9 • which s a t i s f y the s i n g u l a r i t y condition, G -1. F i r s t Ĝ  i s evaluated everywhere. Then each term of Equation 4.17 i s evaluated along the length of the channel to i d e n t i f y the l o c a t i o n at which Equation 4.17 i s s a t i s f i e d . In order to evaluate both the f r i c t i o n and topographical slopes, a l l three f r i c t i o n c o e f f i c i e n t s are needed. For each of the experiments 32 conducted the i n t e r f a c i a l f r i c t i o n c o e f f i c i e n t can now determined by adjusting i t s value u n t i l the lo c a t i o n where S Q=S f corresponds to the location at which G 2=l. An example of the numerical spreadsheet i s given i n Appendix B. 6.4 Interface P r o f i l e The interface p r o f i l e assumed for the analysis (Equation 6.1) was derived from f r i c t i o n l e s s theory (Lawrence 1990b). An analysis i s done using the experimental data to attempt to reevaluate the p r o f i l e and the i n t e r f a c i a l f r i c t i o n c o e f f i c i e n t . S t a r t i n g with Equation 4.20, a slope of the inter f a c e at the throat i s estimated from the experimental data. At the throat, the topographical slope, So=0 and height of the interface, y 2 l = %• Using the value of fj_ determined from the o r i g i n a l model as a f i r s t estimate, the slope of the interface can be reevaluated at the next increment of x and subsequently along the length of the channel. The i n i t i a l slope at the throat and fj are adjusted u n t i l S 0=S f at the point where G 2=l. 33 7. DISCUSSION ON THE INTERFACIAL FRICTION COEFFICIENT Although previous work suggests that the magnitude of the i n t e r f a c i a l f r i c t i o n c o e f f i c i e n t i s small r e l a t i v e to the wall f r i c t i o n c o e f f i c i e n t , t h i s i s not necessarily the case as shown by the numerical work of which the r e s u l t s are discussed i n §8.3. In fact, from the numerical and experimental work the i n t e r f a c i a l f r i c t i o n i s of the same order of magnitude as the wall f r i c t i o n from the Blasius Equation. Bertelsen and Warren (1977) also state that the i n t e r f a c i a l shear stress has proved to be of greater importance than expected i n the movement of the lower layer. Dermisses and Partheniades (1984) summarized p r i o r prominent investigations of Keulegan (1949), Ippen and Harleman (1951), Abraham and Eysink (1971), and Lofquist (1960) and found that a wide discrepancy among graphical and a n a l y t i c a l equations for f j e x i s t . A f t e r applying some of these equations to the same problem, Dermisses and Partheniades found that f j may i n fact d i f f e r by orders of magnitude. There are also d i f f e r i n g opinions as to the appropriate dimensionless parameters to use i n c o r r e l a t i n g fj_. For example, Keulegan introduced as c r i t e r i o n , the Keulegan Number which i s a function of the Reynolds and Densimetric Froude numbers. Other authors have also taken t h i s approach including Macagno and Rouse (1962) as well as Shi-Igai (1965). However, Abraham and Eysink, Ippen and Harleman, and Lofquist a l l came to the conclusion that f j i s a 34 function of the Reynolds Number only, with Lofquist's and Ippen and Harleman's r e s u l t s being extremely s i m i l a r with f x = 139/Re (Lofquist) and f T = 140/Re (Ippen and Harleman). Grubert (1990) follows the work of Keulegan and Lofquist. Using data from the South Pass of the M i s s i s s i p p i River (arrested s a l t wedge situation) he concludes that f/8 = 0.012 R"1/4. Since f here i s the Darcy c o e f f i c i e n t then the i n t e r f a c i a l f r i c t i o n c o e f f i c i e n t f x = 0.012 R"1/4 Dermisses and Partheniades conducted experiments i n a rectangular duct and came to the conclusion that f j can best be correlated with the Reynolds number and a regular nondensimetric Froude number together as RF 2 as well as with the r e l a t i v e density difference, Ap/p. They present a family of curves based on these parameters which are i n close agreement with both t h e i r laboratory data and f i e l d data from the M i s s i s s i p p i estuary. E x p l i c i t l y expressed i s t h e i r conclusion that neither the densimetric Froude number nor the Reynolds numbers can be used as single c o r r e l a t i o n parameters. Eidnes (1986) presents a method for determining the i n t e r f a c i a l f r i c t i o n c o e f f i c i e n t based on a Richardson number for pressure driven shear flow for a top stationary and a bottom flowing layer of f l u i d and suggests that f j = 2.63*10~ 3/ Ri where Ri i s a gradient Richardson number, Ri = g*y/Au 2. Note that t h i s Richardson number i s indeed the inverse of the s t a b i l i t y Froude number defined e a r l i e r (3.11) and that f x i s proportional to F A Z which i s proportional to the height of the i n s t a b i l i t i e s which can be regarded as roughness. Eidnes states that the corresponding values for a bottom current are stated to be 1.6 times higher and s t i p u l a t e s that t h i s equation i s v a l i d only for Ri<10. Bertelsen and Warren (1977) suggest a value of f j = 0.001 or approximately h a l f of the bed stress c o e f f i c i e n t , f w . This i s a r e s u l t of c a l i b r a t i n g data taken from the Danish Belts, applied to t h e i r computer simulation of two-layer flows. Di S i l v i o (1975) used a constant value for the Darcy c o e f f i c i e n t of 0.05 which i s equivalent to a f j value of 0.0062. C a l i b r a t i o n of t h i s c o e f f i c i e n t came from data of the Adige River i n N.E. I t a l y . Macagno (19 62) presents a more rigorous derivation for rectangular pipe flow of what he terms the resistance c o e f f i c i e n t based on density and v e l o c i t y p r o f i l e s and the geometry of the system. He derives an equation f o r t h i s s i t u a t i o n based on the hydraulic radius, Rh, the width, b, and the displacement thickness, 6", and the wetted perimeter, P. An attempt to correlate the resistance c o e f f i c i e n t with the Froude and Reynolds numbers was made; however, although a d e f i n i t e c o r r e l a t i o n was apparent, no d i s t i n c t i v e quantitative conclusion was drawn. Numerous Japanese authors have examined the f r i c t i o n c o e f f i c i e n t i n great d e t a i l . S t i l l many seem to agree with or 36 approve of e a r l i e r works done by Kaneko i n the early 1950's. Kaneko correlates the i n t e r f a c i a l f r i c t i o n c o e f f i c i e n t with the Froude and Reynolds numbers as many others have subsequently done and introduces the parameter 7, where 7 = ReFr 2 and the i n t e r f a c i a l f r i c t i o n c o e f f i c i e n t , fI=0.2Y"i!!. Georgiev (1990) states that f j should depend on the type of flow (bottom density current, arrested s a l t wedge or exchange flow), the Reynolds number and the s t a b i l i t y c h a r a c t e r i s t i c s quantified by a densimetric Froude number he defines by Fr'. Georgiev defines the Reynolds number, Re = u2Rh/v and the densimetric Froude number, Fr' = u 2 2/(g'Rh) where Rh i s the hydraulic radius with wetted perimeter that includes not only the walls and surface but also the interface. He also attempted to p l o t a r e l a t i o n of the i n t e r f a c i a l f r i c t i o n c o e f f i c i e n t with Re and shows curves of constant Fr' that can be examined as curves of constant roughness. Recall that f j i s proportional to F A 2 which i s proportional to the i n s t a b i l i t y height which can be regarded as a roughness. Values of the i n t e r f a c i a l f r i c t i o n c o e f f i c i e n t from h i s data ranged from 0.0006 to 0.008. Using the r e l a t i o n plotted by Georgiev would suggest a value of f j i n the present study of approximately 0.004. Although many authors have used f i e l d data to a r r i v e at a value for the i n t e r f a c i a l f r i c t i o n c o e f f i c i e n t and some also present formulas based on such data, i t i s s t i l l inconclusive as to how to cal c u l a t e such a c o e f f i c i e n t . The present objective 37 i s not to t r y to i d e n t i f y which of the methods are suitable for determining the i n t e r f a c i a l c o e f f i c i e n t , but rather to t e s t whether the r e s u l t s of the present study give values that are comparable to other methods previously published and perhaps present a numerical approach to c a l c u l a t i n g the i n t e r f a c i a l f r i c t i o n c o e f f i c i e n t . Therefore a comprehensive review was done using several of the above mentioned methods which are summarized i n Table I. Table I: Summary of I n t e r f a c i a l F r i c t i o n C o e f f i c i e n t s Author Formulation Conditions Ippen f l - 140/Re Laminar underflows Grubert f l = 0.012 Re"5* S t r a t i f i e d estuaries and fjords Eidnes f l = 4.21(10" 3)/Ri Bottom flowing layer Ri < 10 Macagno f l = 4g'/u 2 (Rh-b<S/4P) Pipe flow Experimental Kaneko f l = 0.2 ill"*4 S a l t wedge Experimental A graph showing these various c o e f f i c i e n t s i s plotted as a function of Reynold's Number (Figure 12). Shown are values calculated from the 5 equations given i n Table I using data from several of the experiments conducted, constant values suggested by Di S i l v i o and Georgiev, along with the experimental values 38 determined from the numerical model. I t i s i n t e r e s t i n g to note i n Table I that Grubert's equation i s based on f i e l d data and the others equations on experimental data. The r e s u l t s of using h i s equation give the largest values for the i n t e r f a c i a l f r i c t i o n c o e f f i c i e n t . Perhaps t h i s i s due to the use of f i e l d data which would imply larger Reynolds numbers i n the f i e l d as opposed to the laboratory and t h i s would be r e f l e c t e d i n h i s c o r r e l a t i o n . 39 8. RESULTS AND DISCUSSION 8.1 Hydraulic Solutions The s t a b i l i t y Froude number i s v i t a l to the analysis of exchange flows and with the in c l u s i o n of f r i c t i o n a l e f f e c t s i t i s found that F A 2<1. Since t h i s i s the case, Long's s t a b i l i t y c r i t e r i o n i s s a t i s f i e d f or long i n t e r n a l waves and confidence can be placed i n the in t e r n a l hydraulic theory. Using experimental data from Experiment #18 reveals the r e l a t i o n of the three pertinent Froude numbers: the composite Froude number, the in t e r n a l Froude number and the s t a b i l i t y Froude number (see Figure 13). The Froude numbers are calculated using Equations 3.7, 3.10, and 3.11 respectively and plotted against the nondimensional length of the contracted channel, x'. Notice that F A 2 i s constant throughout the channel length and the control point i s located where G 2=l which coincides with Fj_ 2=l. These values are calculated assuming a v e l o c i t y reduction or flow r a t i o of experimental values to f r i c t i o n l e s s theory (Equation 6.2), of k = 0.74. A comparison p l o t of Equation 6.1, the calculated t h e o r e t i c a l height of the interface, and the height of the interface measured from various experiments i s given i n Figure 14. Although good agreement can be seen near the channel ends, there i s some deviation from the t h e o r e t i c a l values j u s t on either side of the throat. Given the d i f f i c u l t y i n determining the p o s i t i o n of the interface due to movement and the presence of i n s t a b i l i t i e s , these values match the f r i c t i o n l e s s p r o f i l e remarkably well. Reevaluation of the interface p r o f i l e using Equation 4.20 to include f r i c t i o n a l e f f e c t s reveals l i t t l e d ifference i n the shape of the p r o f i l e and does not appear to improve the match of the experimental data either. A p l o t of both the f r i c t i o n l e s s and reevaluated f r i c t i o n a l p r o f i l e s i s shown i n Figure 15a along with the v a r i a t i o n of So, Sf, and 1-G2 i n Figure 15b. With f r i c t i o n a l e f f e c t s included i n reevaluation of the p r o f i l e , the lo c a t i o n of the control points were pushed farther along the channel close to the channel ends. Entrance and e x i t e f f e c t s may explain some of the discrepancy between the model and the experimental data. Refinement of the experimental p r o f i l e through more exact data c o l l e c t i o n may be needed before a precise experimental p r o f i l e can be assumed. The long term v a r i a t i o n of the interface height at the throat taken from photographs and video recordings i s shown i n Figure 16. Although t h e o r e t i c a l l y the interface height at the throat, y 2 1 , i s assumed to be %, there appears to be an o s c i l l a t i o n about the mid-depth. I t i s not c l e a r as to the cause of these deviations; however, i t i s proposed that they may be caused by the apparatus i t s e l f by a period of "rebound" or c i r c u l a t i o n which i s a function of the s i z e of the end reservoirs. 41 8.2 Flow v i s u a l i z a t i o n The use of s l i d e projectors to a t t a i n a two-dimensional image of the two-layer flow experiment was extremely successful. Photographs of the experiment resulted i n good v i s u a l i z a t i o n of both the Kelvin-Helmholtz and Holmboe i n s t a b i l i t i e s at the interface. Several series of photographs are shown i n Figures 17 through 21. Growth and development of Kelvin-Helmholtz billows can be seen i n Figures 17, 18, and 19. In Figure 17 (Experiment #19), the throat i s located on the l e f t edge of the r u l e r v i s i b l e near the center of the photographs. The series of photographs i n Figure 18 taken from Experiment #24 show a t r a n s i t i o n from a f a i r l y smooth interface to the development of much turbulence at the interface. The photographs are taken of the l e f t channel with the throat located at the edge of the r u l e r on the r i g h t side on the photographs. Again the growth of Kelvin-Helmholtz billows i s shown i n Figure 19 (Experiment #25) showing a series of photographs taken of the channel to the l e f t of the throat. A series of both Kelvin-Helmholtz and Holmboe i n s t a b i l i t i e s can be seen i n Figure 2 0 photographed centered at the throat of the channel from Experiment #21. The development of Holmboe i n s t a b i l i t i e s are shown i n Figure 21 beginning with a single cusp and the l a s t photograph showing 4 cusps. D i s t i n c t wavelengths of the Holmboe mode are taken from these photographs. A f i n a l interface thickness was obtained from the photographs and video recordings of the experiments. Figure 22, o r i g i n a l l y presented by Lawrence (1990b), i s shown with several data points from the present study added. As best as can be determined, S m a x'=0.15±0.05, but varies for each experiment. V a r i a t i o n of the dimensionless maximum interface thickness o*'max with the s t a b i l i t y Froude number are shown i n Figure 22 with l i n e s i n d i c a t i n g values of the bulk Richardson number, J = g'S/Au2. A photograph of the interface thickness i s shown i n Figure 2 3 taken from Experiment #23 with the throat at the r i g h t side of the photograph. The interface thickness can also c l e a r l y be seen i n the long exposure photograph (Figure 7). Wavelengths of both the Holmboe and Kelvin-Helmholtz modes of i n s t a b i l i t y are measured from photographs of the experiments. A photograph showing wavelengths of Kelvin-Helmholtz mode i s given i n Figure 24. These wavelengths and calculated values of the s t a b i l i t y parameters are given i n Table I I . 43 Table I I . Calculated S t a b i l i t y Parameters Exp # h J A. a (cm) (cm) 18 3.20 0.20 18.0 1.12 19 3.70 0.23 17.0 1.37 20 3.50* 0.22 18.2 1.21 21 3.50* 0.22 16.4 1.34 22 3.53 0.22 21.1 1.05 23 3.53 0.22 29.3 0.76 24 3 .80* 0.23 29.4 0.81 25 3.80* 0.23 33.1 0.72 * Estimated from Video Recordings or Photographs J=g'h / ( A u ) 2 a=2n/\ A=average wavelength for experiment Koop and Browand (1979) determined a value of J=0.32 from t h e i r experiments. Note that the value of J i n the present study varies from 0.20 to 0.23 with Reynold's numbers being larger than (6 to 8 times) those of Koop and Browand*s experiments. However, Koop and Browand (1979) also state that the maximum Richardson number decreases to as l i t t l e as 0.15 with increasing i n i t i a l Richardson numbers. Figure 25 i s a s t a b i l i t y diagram plotted by Lawrence, Browand & Redekopp (1990) where a here i s the i n s t a b i l i t y wave number, a = kh, k = 2TT/X, and A i s the wavelength. Additional explanation of the s t a b i l i t y diagram i s given by Lawrence et a l (1990). Also shown on the diagram are the wavelengths measured 44 from the experiments. Note that a l l of the data points l i e within the region predicting the i n s t a b i l i t i e s . The p o s s i b i l i t y of p a i r i n g of the billows might explain the higher values of X and therefore lower values of a which i s consistent with the points p l o t t e d i n Figure 25. Although photographs were used to i n i t i a l l y obtain v e l o c i t y p r o f i l e s , using a video recording of the experiment proved to be the more complete and accurate method of obtaining v e l o c i t y p r o f i l e s . An example long exposure photograph showing the streaks l e f t by the beads used for v e l o c i t i e s was shown i n Figure 7. 8.3 I n t e r f a c i a l F r i c t i o n C o e f f i c i e n t From the experimental and numerical work conducted i t i s shown that f o r the s i t u a t i o n of exchange flow, the i n t e r f a c i a l f r i c t i o n c o e f f i c i e n t i s of the same order of magnitude and several times larger than the wall f r i c t i o n . On average, the i n t e r f a c i a l f r i c t i o n , fI=0.0088 for /J=1.0 and as high as 0.0096 for /3=0. The v a r i a t i o n of a, where a=fj/f w, with the v e l o c i t y c o e f f i c i e n t , k, determined numerically from the data of Experiment #22 i s shown i n Figure 26 allowing for three surface conditions of (3: 0, 0.5 and 1. The s i g n i f i c a n c e of the surface f r i c t i o n i s small as can be seen by the r e l a t i v e difference of the three curves. Values of both the i n t e r f a c i a l f r i c t i o n and wall f r i c t i o n c o e f f i c i e n t s along with additional experimental data for a l l experiments are given i n Table I I I . 45 Table Hf: Experimental Data and F r i c t i o n C o e f f i c i e n t s v e l • u Exp # bo cm y cm g' cm/s 2 Eg.6.3 cm/s exp cm/s Rh cm Re (Rh) Re (x) f fw f/8 f l 1 10. 2 16. 8 2 .48 3. 23 2.39 3 .17 2316 18246 0. 020 0 .0025 0 .0047 2 10. 0 26. 1 1.85 3. 47 2.57 3 .61 2834 19602 0. 019 0 .0024 0 .0082 3 5. 5 27. 8 1.50 3. 23 2.39 2 .30 1676 18246 0. 020 0 .0025 0 .0093 4 5. 5 28. 6 1.51 3. 29 2.43 2 .31 1715 18585 0. 019 0 . 0024 0 .0094 5 10 . 6 28 . 2 1.26 2 . 98 2.21 3 .85 2594 16834 0. 020 0 .0026 0 .0088 6 10. 6 26. 0 0.99 2 . 70 2.00 3 .77 2297 15252 0. 022 0 .0027 0 .0093 7 10. 5 29 . 5 0 . 99 2. 70 2.00 3 .87 2362 15252 0. 022 0 .0027 0 .0093 8 10. 5 29. 5 0.86 2 . 52 1.86 3 .87 2205 14235 0. 022 0 .0028 0 .0096 9 10. 3 29. 5 0.94 2 . 64 1.95 3 .82 2277 14913 0. 022 0 .0027 0 .0094 10 10. 1 29 . 3 1.10 2 . 85 2.11 3 .76 2418 16099 0. 021 0 .0026 0 .0094 11 10. 1 29. 6 1.53 3. 37 2.49 3 .77 2867 19037 0. 019 0 .0024 0 .0096 13 10. 4 29. 5 0.94 2. 64 1.95 3 .84 2293 14913 0. 022 0 .0027 0 .0095 14 10. 3 29. 7 0.94 2. 64 1.95 3 .82 2281 14913 0. 022 0 .0027 0 .0095 15 10. 3 29. 7 0.94 2. 64 1.95 3 .82 2281 14913 0. 022 0 .0027 0 .0095 16 10. 5 29 . 6 1.05 2 . 78 2.06 3 .88 2434 15704 0. 021 0 .0026 0 .0097 18 10. 4 29 . 6 1.02 2. 74 2.03 3 .85 2382 15478 0. 021 0 .0027 0 . 0097 V i s c o s i t y = 1.31 (10- 2) cm/s 2 (10 C) 19 10.7 29 . 6 1 .46 3 .29 2 .43 3 .93 2676 17025 0. 020 0 .0025 0 .0101 20 10.6 29 .5 1 .29 3 .09 2.29 3 .90 2494 15990 0. 021 0 .0026 0 .0100 21 10.6 29 .5 0 .98 2 .69 1.99 3 .90 2171 13920 0. 023 0 .0028 0 . 0098 V i s c o s i t y = 1 .43 (10--2) cm/s2 (7 C) 22 10.4 28 .8 1 .77 3 .57 2.64 3 .82 4037 18474 0. 020 0 .0024 0 .0099 23 10.4 29 .4 1 .77 3 .57 2.64 3 .84 4059 18474 0. 020 0 .0024 0 .0102 24 10 . 5 29 . 6 1 . 72 3 .57 2 . 64 3 .88 4095 18474 0 . 020 0 .0024 0 .0103 25 10 . 5 29 . 6 1 .72 3 .57 2.64 3 .88 4095 18474 0. 020 0 .0024 0 . 0103 V i s c o s i t y = 1 . 00 (10--2) cm/s2 (21 C) E x p e r i m e n t a l v e l o c i t i e s are 74% of t h e o r e t i c a l The f u l l w e tted p e r i m e t e r has been i n c l u d e d f = 2.656/sqrt(Rex) W a l l f r i c t i o n f a c t o r (Eq. 6.3) fw = f/8 S t a n d a r d i z e d w a l l f r i c t i o n f a c t o r b e t a = 0 Re(Rh) = 4Rh u / v i s c o s i t y Reynolds number based on h y d r a u l i c r a d i u s Re(x) = u L / v i s c o s i t y Reynolds number based on c h a n n e l l e n g t h Note that the value of f j for experiment #1 i s lower than that of the other experiments. This i s due to the shallow depth of t h i s experiment which r e s t r i c t s growth of the billows and hence lowers the e f f e c t i v e roughness and r e s u l t s i n a lower f j . A t h e o r e t i c a l r e l a t i o n between k and a was derived using conditions at the throat. Although t h i s i s may not be appropriate f o r the whole channel, i t provides a good f i r s t estimate of the value of the i n t e r f a c i a l f r i c t i o n factor. Figure 27 show the t h e o r e t i c a l curves calculated. Note again the small difference i n the three curves for the e n t i r e range of f3 i n d i c a t i n g that the surface f r i c t i o n has l i t t l e impact on the determination of the i n t e r f a c i a l f r i c t i o n c o e f f i c i e n t . Details of the t h e o r e t i c a l derivation are given i n Appendix C. A comparison p l o t of the t h e o r e t i c a l and experimental r e l a t i o n between k and a i s shown i n Figure 28 for the case, /3=0. Although some previous authors have elected to ignore or assume very small values for the i n t e r f a c i a l f r i c t i o n i n the theory of layered flows, i t i s shown by the numerical and experimental work that i t i s pertinent to the theory and should be included i n any analysis. 8.4 Control Point Location With estimates of the i n t e r f a c i a l , wall, and surface f r i c t i o n c o e f f i c i e n t s , the hydraulic equations derived can be used to help locate the control points of a two-layer exchange flow. A f t e r determining the location of the controls f o r each experiment from the numerical model using a f r i c t i o n l e s s i nterface p r o f i l e as discussed i n §6, i t was found that the controls were located between x'=±0.40 and x'=±0.45. However, a f t e r reevaluation of the interface to include f r i c t i o n the controls were located within the v i c i n i t y of the ends of the contraction (x'=±0.55). Depending on the v e l o c i t y reduction c o e f f i c i e n t , k, that was used, the location of the controls varies within t h i s region. A f t e r examining the v e l o c i t y p r o f i l e s both across the width of the channel and throughout the depth of the channel, a value of k=0.74±0.05 was found to be appropriate. For k=0.74, the controls were located at x'=±0.51 (f^O.0104, Exp. #18) and for k=0.70, x'=±0.68 (f I=0.0130). With the consideration of f r i c t i o n a l e f f e c t s , the control points of an exchange flow do not occur at the narrowest section as suggested by i n v i s c i d theory but l i e on e i t h e r side of the throat and perhaps at the ends of the contraction. S t i l l the controls must s a t i s f y the s i n g u l a r i t y condition G =1 and are separated by a region of s u b c r i t i c a l flow. The r e l a t i o n of the control points and the composite Froude number i s diagrammed i n Figure 29 calculated from Experiment #5. I l l u s t r a t e d are the two control points on either side of the throat separated by a region of s u b c r i t i c a l flow (G2<1) . At the two controls, the composite Froude number takes on a value of unity. Beyond the control points, the flow becomes s u p e r c r i t i c a l (Figure 29a). A t h e o r e t i c a l interface p r o f i l e along the channel i s shown i n Figure 29b. The Froude number plane showing the l o c a t i o n of the 48 control points (b c) and the throat (b Q) r e l a t i v e to the Froude numbers for each layer i s given i n Figure 29c. 49 9. CONCLUSION The i n t e r f a c i a l f r i c t i o n c o e f f i c i e n t i s of great importance to the analysis of layered flows e s p e c i a l l y i n the lo c a t i n g of the hydraulic controls. The addition of f r i c t i o n a l considerations moves the t h e o r e t i c a l l o c a t i o n of the control points away from the throat of the contraction. For the experiments conducted the control points were calculated to be located i n the v i c i n i t y of the ends of the contraction. A value of 0.74±0.05 for the v e l o c i t y reduction c o e f f i c i e n t was determined from the data analyzed and resulted i n an average i n t e r f a c i a l f r i c t i o n factor of 0.0096 (j0=O) and 0.0088 ((3=1.0) using a f r i c t i o n l e s s interface p r o f i l e . These values increase with the use of the f r i c t i o n a l p r o f i l e by approximately 7%. The a b i l i t y to obtain good flow v i s u a l i z a t i o n both through photographs and video recordings has d e f i n i t e l y proven worthy i n order to obtain more accurate data for t e s t i n g the hydraulic theory. V i s u a l i z a t i o n of the interface provides data for better understanding as well as additional observation of the phenomenon of both the Kelvin-Helmholtz and Holmboe i n s t a b i l i t i e s . The t h e o r e t i c a l p r o f i l e (Equation 6.1) i s shown to be adequate i n estimating the height of the interface along the channel. Measurements of the maximum interface thickness are found to agree with the v a r i a t i o n of the thickness and the s t a b i l i t y Froude number presented by Lawrence (1989) . As well, the measurements made of the i n s t a b i l i t y wavelengths found i n the experiments l i e within the predicted region of i n t e r f a c i a l i n s t a b i l i t y occurrence given by Lawrence et a l (1990). 51 10. RECOMMENDATIONS The use of a larger f a c i l i t y with a more f l e x i b l e apparatus may help to obtain additional data p a r t i c u l a r l y f o r higher Reynold's numbers to further examine the hydraulic theory. As well, additional f i e l d data i s needed to t e s t the theory i n large scale s i t u a t i o n s . A more extensive review of ex i s t i n g data on the i n t e r f a c i a l f r i c t i o n c o e f f i c i e n t i s s t i l l needed. To a t t a i n and assemble together a l l e x i s t i n g data to t r y and achieve some sort of co r r e l a t i o n i n the re s u l t s may be necessary. An emphasis may include f i e l d data and categorization of d i f f e r e n t types of flow and the r e s u l t i n g i n t e r f a c i a l f r i c t i o n c o e f f i c i e n t s . I t i s d i f f i c u l t to obtain an accurate p r o f i l e of the interface along the entir e channel due to i t s length. A more precise method of obtaining t h i s p r o f i l e i s needed to r e f i n e the analysis i f increased accuracy i s required. I t i s suggested that v e r t i c a l p r o f i l e s of the density be taken using conductivity probes at multiple location along the channel. Va r i a t i o n of f w along the channel, as a constant value was assumed may also help to increase the accuracy of r e s u l t s . The analysis may be approached by assuming the controls of the experiment occur at the ends of the contraction and calcul a t e i n toward the throat. This requires the slope of the interface at the control points and hence the use of L'Hopital's ru l e to d i f f e r e n t i a t e Equation 4.20 which i s beyond the scope of the present study. 52 B i b l i o g r a p h y A r m i , L . (1986) . "The h y d r a u l i c s o f two f l o w i n g l a y e r s w i t h d i f f e r e n t d e n s i t i e s " , J . F l u i d M e c h . . Vol. 163, p p . 2 7 - 5 8 . A r m i , L . a n d F a r m e r , D . M . ( 1 9 8 6 ) . " M a x i m a l t w o - l a y e r e x c h a n g e t h r o u g h a c o n t r a c t i o n w i t h b a r o t r o p i c n e t f l o w " . J . F l u i d M e c h . . Vol. 164, p p . 2 7 - 5 1 . A r m i , L . a n d F a r m e r , D . M . (1988) . 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" T w o - l a y e r m o d e l l i n g o f t h e D a n i s h b e l t s : c o l l e c t i o n a n d p r o c e s s i n g o f d a t a a n d c a l i b r a t i o n o f t h e m o d e l s " , I A H R . p p . 3 7 9 - 3 8 6 . C h r i s t o d o u l o u , G . C . ( 1 9 8 6 ) . " I n t e r f a c i a l m i x i n g i n s t r a t i f i e d f l o w s " , J . H y d . R e s . , Vol. 24, N o . 2 . , p p . 7 7 - 8 9 . D e n t o n , R . A . ( 1 9 8 6 ) . " L o c a t i n g a n d i d e n t i f y i n g h y d r a u l i c c o n t r o l s f o r l a y e r e d f l o w t h r o u g h a n o b s t r u c t i o n " , J . H y d . R e s . , Vol. 25, N o . 3 , p p . 2 8 1 - 2 9 9 . D e n t o n . R . A . ( 1 9 9 0 ) . " C l a s s i f i c a t i o n o f u n i d i r e c t i n a l t h r e e - l a y e r f l o w o v e r a hump", J . H y d . R e s . . Vol. 28, N o . 2 , p p . 2 1 5 - 2 3 3 . D e r m i s s e s , V . a n d P a r t h e n i a d e s , E . ( 1 9 8 4 ) . 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"A lock exchange flow". J . F l u i d Mech., Vol. 42, pp. 671-687. A p p e n d i x A D e r i v a t i o n o f t h e E q u a t i o n o f t h e S l o p e o f t h e I n t e r f a c e F r o m E q u a t i o n 4 . 1 3 C^L = + S ox dx ( 4 . 1 3 ) r e a r r a n g i n g | Z = c-iD-lf + C^S ox ox a g a i n w h e r e v ( x ) , D , f ( x ) , a n d S a r e g i v e n b y -g 0 -9 0 Yi D = 0 Oi 0 °2. f = he Jb"1 v b 1 Yi Yi b y2 x y 2 S i n c e we a r e p r i m a r i l y i n t e r e s t e d i n t h e h e i g h t o f t h e i n t e r f a c e , o n l y t h e l a s t c o m p o n e n t o f m a t r i x v w i l l b e shown h e r e . T h e f i r s t t e r m o n t h e r i g h t h a n d s i d e i s e v a l u a t e d C ~XD df dx F i - l 1 - G 2 dft. dx y2 1-G2 y2 db b dx a n d t h e s e c o n d t e r m : 2f\. 1-G2 yflFf^flEiYf^ y±y2 N o t e t h a t C'-'-S i s d e f i n e d a s A S f g i v e n i n E q u a t i o n 4 . 1 8 . A d d i n g t o g e t h e r t h e p r e c e e d i n g two e q u a t i o n s g i v e s dy2 dx G 2-F 1 2(l+ry 1/y 2) 1-G' y2 db b dx 1-F£ 1-G2 dhc dx 1-G' a n d m o v i n g y 2 t o t h e l e f t h a n d s i d e r e s u l t s i n E q u a t i o n 4 . 1 7 . APPENDIX B EXPERIMENTAL CALCULATIONS FOR EXPERIMENT # 18 Experimental Data = 2.03 cm/sec Uth = 2.74 cm/sec r = 0.9989 1 - 10.4 cm k = 0.74 epsi = 0.0010 29.6 cm 1.02 cm/s2 fw = 0.0027 ) - 30.0 cm2/sec fi = 0.0097 alpha = 3.6 = 312 cm3/sec xinc = 0.01 X' b' yl' y2' yi y2 b ul u2 Flsq F2sq Gsq Fdsq So fw fi Sf So-Sf terms terms 0.00 1 0.50 0.50 14.80 14.80 10.4 2.03 2.03 0.27 0.27 0.545 0.54 0.0000 0.0007 0.0211 0.0219 0.0219 0.01 1.000 0.50 0.50 14.95 14.65 10.40 2.01 2.05 0.26 0.28 0.545 0.54 0.0000 0.0007 0.0211 0.0219 0.0219 0.02 1.000 0.51 0.49 15.10 14.50 10.40 1.99 2.07 0.26 0.29 0.546 0.54 0.0001 0.0008 0.0211 0.0220 0.0220 0.03 1.000 0.51 0.49 15.24 14.36 10.40 1.97 2.09 0.25 0.30 0.547 0.54 0.0001 0.0008 0.0211 0.0220 0.0219 0.04 1.001 0.52 0.48 15.39 14.21 10.41 1.95 2.11 0.24 0.31 0.548 0.54 0.0003 0.0009 0.0212 0.0221 0.0219 0.05 1.002 0.52 0.48 15.54 14.06 10.42 1.93 2.13 0.23 0.32 0.550 0.54 0.0004 0.0009 0.0212 0.0222 0.0218 0.06 1.003 0.53 0.47 15.69 13.91 10.43 1.91 2.15 0.23 0.33 0.553 0.54 0.0006 0.0010 0.0212 0.0223 0.0217 0.07 1.004 0.53 0.47 15.83 13.77 10.45 1.89 2.17 0.22 0.34 0.555 0.54 0.0008 0.0011 0.0212 0.0223 0.0216 0.08 1.006 0.54 0.46 15.98 13.62 10.46 1.87 2.19 0.21 0.35 0.559 0.54 0.0010 0.0011 0.0213 0.0224 0.0214 0.09 1.008 0.54 0.46 16.13 13.47 10.48 1.85 2.21 0.21 0.36 0.562 0.54 0.0013 0.0012 0.0213 0.0225 0.0213 0.10 1.010 0.55 0.45 16.28 13.32 10.50 1.83 2.23 0.20 0.37 0.567 0.54 0.0016 0.0012 0.0213 0.0226 0.0211 0.11 1.012 0.55 0.45 16.42 13.18 10.52 1.81 2.25 0.19 0.38 0.571 0.54 0.0019 0.0013 0.0214 0.0227 0.0208 0.12 1.014 0.56 0.44 16.57 13.03 10.55 1.79 2.27 0.19 0.39 0.576 0.54 0.0023 0.0013 0.0214 0.0228 0.0206 0.13 1.017 0.56 0.44 16.72 12.88 10.57 1.77 2.29 0.18 0.40 0.582 0.54 0.0027 0.0014 0.0215 0.0229 0.0203 0.14 1.019 0.57 0.43 16.86 12.74 10.60 1.75 2.31 0.18 0.41 0.588 0.54 0.0031 0.0015 0.0215 0.0231 0.0200 0.15 1.022 0.57 0.43 17.01 12.59 10.63 0.16 1.025 0.58 0.42 17.15 12.45 10.66 0.17 1.029 0.58 0.42 17.30 12.30 10.70 0.18 1.032 0.59 0.41 17.44 12.16 10.74 0.19 1.036 0.59 0.41 17.59 12.01 10.78 0.20 1.040 0.60 0.40 17.73 11.87 10.82 0.21 1.045 0.60 0.40 17.87 11.73 10.86 0.22 1.049 0.61 0.39 18.02 11.58 10.91 0.23 1.054 0.61 0.39 18.16 11.44 10.96 0.24 1.059 0.62 0.38 18.30 11.30 11.01 0.25 1.064 0.62 0.38 18.44 11.16 11.07 0.26 1.069 0.63 0.37 18.58 11.02 11.12 0.27 1.075 0.63 0.37 18.72 10.88 11.18 0.28 1.081 0.64 0.36 18.86 10.74 11.24 0.29 1.087 0.64 0.36 19.00 10.60 11.31 0.30 1.094 0.65 0.35 19.14 10.46 11.37 0.31 1.100 0.65 0.35 19.28 10.32 11.44 0.32 1.107 0.66 0.34 19.42 10.18 11.52 0.33 1.115 0.66 0.34 19.55 10.05 11.59 0.34 1.122 0.67 0.33 19.69 9.91 11.67 0.35 1.130 0.67 0.33 19.83 9.77 11.75 0.36 1.138 0.67 0.33 19.96 9.64 11.83 0.37 1.146 0.68 0.32 20.09 9.51 11.92 0.38 1.155 0.68 0.32 20.23 9.37 12.01 0.39 1.164 0.69 0.31 20.36 9.24 12.10 0.40 1.173 0.69 0.31 20.49 9.11 12.20 0.41 1.183 0.70 0.30 20.62 8.98 12.30 0.42 1.192 0.70 0.30 20.75 8.85 12.40 0.43 1.203 0.71 0.29 20.88 8.72 12.51 0.44 1.213 0.71 0.29 21.01 8.59 12.62 0.45 1.224 0.71 0.29 21.14 8.46 12.73 0.46 1.235 0.72 0.28 21.26 8.34 12.85 0.47 1.247 0.72 0.28 21.39 8.21 12.97 1.73 1.71 1.69 1.67 1.65 1.63 1.61 1.59 1.57 1.55 1.53 1.51 1.49 1.47 1.45 1.43 1.41 1.40 1.38 1.36 1.34 1.32 1.30 1.28 1.27 1.25 1.23 1.21 1.19 1.18 1.16 1.14 1.12 2.33 2.35 2.37 2.39 2.41 2.43 2.45 2.47 2.49 2.51 2.53 2.55 2.57 2.58 2.60 2.62 2.64 2.66 2.68 2.70 2.72 2.73 2.75 2.77 2.79 2.81 2.83 2.84 2.86 2.88 2.90 2.91 2.93 0.17 0.17 0.16 0.16 0.15 0.15 0.14 0.14 0.13 0.13 0.12 0.12 0.12 0.11 0.11 0.11 0.10 0.10 0.09 0.09 0.09 0.09 0.08 0.08 0.08 0.07 0.07 0.07 0.07 0.06 0.06 0.06 0.06 0.42 0.594 0.43 0.601 0.45 0.609 0.46 0.616 0.47 0.625 0.49 0.634 0.50 0.643 0.52 0.653 0.53 0.663 0.55 0.674 0.56 0.685 0.58 0.697 0.59 0.709 0.61 0.722 0.63 0.736 0.64 0.750 0.66 0.764 0.68 0.780 0.70 0.795 0.72 0.812 0.74 0.829 0.76 0.846 0.78 0.864 0.80 0.883 0.83 0.903 0.85 0.923 0.87 0.943 0.90 0.965 0.92 0.987 0.95 1.010 0.97 1.034 1.00 1.058 1.03 1.083 0.0036 0.0041 0.0046 0.0052 0.0058 0.0064 0.0070 0.54 0.0077 0.54 0.0084 0.0092 0.0099 0.0107 0.0115 0.0124 0.0133 0.54 0.0142 0.54 0.0151 0.0161 0.0171 0.0181 0.54 0.0192 0.54 0.0202 0.54 0.0213 0.54 0.0225 0.54 0.0236 0.0248 0.0260 0.0272 0.0285 0.54 0.0298 0.54 0.0311 0.54 0.0324 0.54 0.0337 0.54 0.54 0.54 0.54 0.54 0.54 0.54 0.54 0.54 0.54 0.54 0.54 0.54 0.54 0.54 0.54 0.54 0.54 0.54 0.54 0.0015 0.0016 0.0016 0.0017 0.0018 0.0018 0.0019 0.0020 0.0020 0.0021 0.0021 0.0022 0.0023 0.0023 0.0024 0.0025 0.0025 0.0026 0.0027 0.0028 0.0028 0.0029 0.0030 0.0031 0.0031 0.0032 0.0033 0.0034 0.0034 0.0035 0.0036 0.0037 0.0038 0.0216 0.0217 0.0217 0.0218 0.0219 0.0220 0.0221 0.0222 0.0223 0.0224 0.0225-;, 0.0226 0.0227 0.0229 0.0230 0.0231 0.0233 0.0234 0.0236 0.0237 0.0239 0.0241 0.0242 0.0244 0.0246 0.0248 0.0250 0.0252 0.0254 0.0257 0.0259 0.0261 0.0264 0.0232 0.0233 0.0234 0.0236 0.0237 0.0239 0.0240 0.0242 0.0243 0.0245 , 0.0247 0.0249 0.0251 0.0252 0.0254 0.0257 0.0259 0.0261 0.0263 0.0265 0.0268 0.0270 0.0273 0.0275 0.0278 0.0281 0.0283 0.0286 0.0289 0.0292 0.0295 0.0299 0.0302 0.0196 0.0193 0.0189 0.0184 0.0180 0.0175 0.0170 0.0165 0.0160 0.0154 0.0148 0.0142 0.0136 0.0129 0.0122 0.0115 0.0108 0.0100 0.0093 0.0085 0.0077 0.0068 0.0060 0.0051 0.0042 0.0033 0.0024 0.0014 0.0005 -0.0005 -0.0015 -0.0025 -0.0035 APPENDIX B continued: Refinement of Main Spreadsheet Program to compute the value of the velocity coefficient, k, from the Shear equations where G2= 1 @ So=Sf Initial conditions for Experiment # 25 gprime = 1.72 cm/s2 bo = 10.5 cm Utheor = 3.57 cm/sec H = 29.6 cm viscos = 0.01 cm/s2 L = 110 cm Reynold = 4701 fw = 0.0024 alpha=fi/fw xprime=x/2L alpha = 4.29 beta=fs/fw Sf=Sfw+Sfb+Sfi beta = 0 y={l+sqrt[(l-k~2)/(l+3k~2)]}/2 fi = 0.0103 fs = 0.0023 k y (y2) b' x/2L Flsq F2sq Gsq Sfw Sfb Sfi Sf So So-Sf k x' 0.730 0.71 1.22 0.45 0.938 0.062 1.0 0.0012 0.0001 0.03 0.0286 0.0298 -0.0011884 0.730 0.445 0.732 0.71 1.22 0.44 0.937 0.062 1.0 0.0012 0.0002 0.03 0.0287 0.0296 -0.0009478 0.732 0.443 0.734 0.71 1.21 0.44 0.936 0.063 1.0 0.0013 0.0002 0.03 0.0288 0.0295 -0.0007056 0.734 0.440 0.736 0.71 1.21 0.44 0.935 0.064 1.0 0.0013 0.0002 0.03 0.0289 0.0293 -0.0004617 0.736 0.438 0.738 0.71 1.21 0.44 0.934 0.065 1.0 0.0013 0.0002 0.03 0.0290 0.0292 -0.0002162 0.738 0.436 0.740 0.71 1.21 0.43 0.933 0.066 1.0 0.0013 0.0002 0.03 0.0291 0.0291 0.00003079 0.740 0.433 0.742 0.71 1.20 0.43 0.933 0.067 1.0 0.0013 0.0002 0.03 0.0292 0.0289 0.00027948 0.742 0.431 0.744 0.70 1.20 0.43 0.932 0.068 1.0 0.0013 0.0002 0.03 0.0293 0.0288 0.00052976 0.744 0.429 0.746 0.70 1.20 0.43 0.931 0.069 1.0 0.0013 0.0002 0.03 0.0294 0.0286 0.00078164 0.746 0.426 60 A p p e n d i x C D e r i v a t i o n o f k a n d a r e l a t i o n Sa - bSf S t a r t i n g w i t h E q u a t i o n 4 . 2 0 dy, dx l-G2 a t t h e t h r o a t , S Q = 0, t h e r e f o r e , ( 4 . 2 0 ) dy2 dx A S , F r o m t h e i n t e r f a c e p r o f i l e we c a n a p p o x i m a t e d y 2 / d x b y i t s s l o p e a t t h e t h r o a t , y / 4 L , d e t e r m i n e d f r o m t h e f r i c t i o n l e s s i n t e r f a c e p r o f i l e ( E q u a t i o n 6 . 1 ) s u c h t h a t y AL A S , 1-G' A t t h e t h r o a t , we know t h a t G2 = k2 Fl =Fl = *? 1 2 2 y V - - V 2 = f y2 y 2 yiy2 y2/4 = 4 s u c h t h a t 4L k2 y_ + k2 y_ 2 2 2 2 + fw— = AfTk2 = f \ —  2 i s 2 ( 1 ~ k 2 ) T L = " I T J c 2 y + i f » + f s ) J T + 4 f * k 2 R e a r r a n g i n g , we f i n a l l y g e t a r e l a t i o n f o r f^ a n d k . f = d - ^ 2 ) y _ f (±+_y_\ _ IJL 1 16k2L " 8 4 i ; 8 N o t e t h e r e l a t i o n s h i p p l o t t e d i n F i g u r e 27 i s k a n d a , w h e r e a = f i / f w 61 Figure 1. Variation of the internal, composite and stability Froude numbers throughout a contracted channel for inviscid flows Figure 2. Definition sketch of assumed linear approximations of both the velocity and density profiles fc3 Figure 3. Definition sketch of the shear stresses acting on a volume of fluid for both the upper and lower layers FIXED BARRIER ADJUSTABLE SECTION REMOVABLE BARRIER P L A N Figure 4. Experimental set-up (all dimensions in millimeters) Figure 5. Pho tog raphs of the experiment in p rog ress Figure 6. Photograph of the experimental apparatus Figure 7 Long exposure photograph showing movement of velocity beads t7 Q_ 0) Q ro c g c C D E Xi c o 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 Experiment #18 g' = 1.02 cm/s 2 _ + - " + o o 0 • l 1 1 o o o oo + + + A AA • ++ + _ . • - i . * o A O 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 -1 -0.8 -0.6 -0.4 -0.2 0.2 0.4 0.6 0.8 Nondimensional Velocity, U/Uo Time of Velocity Measurement • 1:48 + 2:28 o 3:53 5:43 Figure 8. Velocity Profile at the throat for Experiment #18 <«6 Q. Q) Q 15 c g "w c cu E T3 C o z 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 Experiment #19 ~ g' = 1.46 cm/s 2 A X ++ $ • A A • "V A Q B X o 4A A ' X A <> + " A + + I I I I I I I I I I I I I I I I I I -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 Nondimensional Velocity, U/Uo 0.8 Time of Velocity Measurement • 1:25 + 2:05 o 4:05 A 5:30 x 7:30 Figure 9. Velocity profile at the throat for Experiment #19 b<5 Experiment #22 g' = 1.77 cm/s 2 PC £3 i § m m B H H , — i 1 1 i i i 1 1 " -1 -0.8 -0.6 -0.4 -0.2 ~ i 1 1 1 1 1 1 1 1 r 0 0.2 0.4 0.6 0.8 Nondimensional Velocity, U/Uo Figure 10. Velocity profile at the throat for Experiment #22 Initial Startup Experiment #18 a Top layer + Bottom layer Quasi-Steady Flow (Controlled) + a • m + + + Unsteady Flow (Uncontrolled) 0 240 480 Time, seconds 720 960 Figure 11. Long term variation of the average velocity at the throat for Experiment #18 0.016 0.014 - 0.012 H 0.010 0.008 - 0.006 - 0.004 0.002 A A A A • • V v X *f X $ V «> <*><£> O o A A o o« i r "i i i r 1000 1400 1800 • Cheung o Ippen x Macagno Di Silvio 2200 2600 + Eidnes A Grubert v Kaneko Georgiev 3000 3400 Re Figure 12. Comparison of various interfacial friction coefficients plotted as a function of Reynolds number 72 -0.5 -0.3 -0.1 0.1 0.3 0.5 Nondimensional Channel Length, x' = x/2L Figure 13. Variation of the internal, composite and stability Froude numbers throughout the contracted channel 73 Figure 14. Comparison plot of the experimental and theoretical interface height along the channel 1 0.9 0.8 0.7 0.6 H 0.5 0.4 0.3 H 0.2 0.1 H k = 0.74 f : = 0.0104 frictional frictionless theory i r location of controls l r- ~i r -0.6 -0.4 -0.2 0 0.2 0.4 Nondimensional Channel Length Figure 15a. Comparison of frictionless and friction interface profiles 0.05 0.04 0.03 H k = 0.74 fj = 0.0104 Sf 1-G : 0.6 control 0.02 0.01 S o 0.2 0.4 Right channel Figure 15b. Variation of Sf, So and 1-G 2 approaching the control 0.6 75 1 I o 9 Experiment #18 g' = 1.02 cm/s 0.8 0.7 0.6 0.5 I a a - - - — a y B g b H B i ™ " U H P H - i g l 0.4 J h H 0.3 0.2 0.1 0 u 1 1 1 r 0 240 480 720 960 Time, seconds Figure 16. Long term variation of the interface height at the throat for Experiment #18 Figure 17 N E C . 1 4 / 8 9 2 2 : 8 8 : 4 2 N O . 1 9 1 9 Sequence of photographs showing Kelvin-Helmholtz instabilities from Experiment #19 Figure 18. Sequence of photographs from Experiment #24 78 Figure 19. Sequence of photographs from Experiment #25 Figure 20. Sequence of photographs with Kelvin-Helmhoftz and Holmboe instabilities from Experiment #21 Figure 21. Sequence of photographs showing Holmboe instabilities 8A Figure 22. Variation of dimensionless maximum interface thickness with the stability Froude number • Koop. and Browand (1979) n Lawrence (1985) A Lawrence, Browand & Redekopp (1990) • Lawrence, Guez & Browand (unpublished) 0 CVi&unq  6 3 1.0 2 .0 A E x p . #18 o E x p . #22 a E x p . #19 » E x p . #23 * E x p . #2 0 A E x p . #24 * E x p . #21 • E x p . #25 Figure 25. Stability Diagram (Lawrence et al. 1990) 84 Figure 26. Variation of calculated k and alpha with beta = 0, 0.5, and 1.0 6fT Figure 27. Theoretical variation of k and alpha with beta = 0, 0.5, 1.0 8b Figure 28. Comparison of theoretical and calculated curves of k and alpha for beta = 0 a ) t c b o t c G 2 >1 G 2 <1 G 2 <1 G 2 >1 II C V J O Figure 29. Variation of: a)the composite Froude number through a contraction, b)interface height along a section of a contracted channel, c)the Froude numbers for each layer shown on a Froude number plane

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