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Model of frictional two-layer exchange flow Zaremba, Lillian 2001

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MODEL OFFRICTIONAL TWO-LAYER EXCHANGE  FLOW  by Lillian Zaremba B.Sc.(Eng), University of Guelph, 1998  A T H E S I S S U B M I T T E D IN P A R T I A L F U L F I L L M E N T O F T H E REQUIREMENTS FOR T H E DEGREE O F M A S T E R OF APPLIED SCIENCE  in T H E F A C U L T Y O F G R A D U A T E STUDIES (CIVIL E N G I N E E R I N G )  We accept this thesis as conforming to the required standard  T H E UNIVERSITY O F BRITISH C O L U M B I A  December 2000 © Lillian Zaremba, 2000  In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t of the requirements f o r an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia, I agree t h a t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and study. I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g of t h i s t h e s i s f o r s c h o l a r l y purposes may be g r a n t e d by the head of my department or by h i s or her r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g or p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n .  The U n i v e r s i t y of B r i t i s h Columbia Vancouver, Canada Date  Abstract  An unsteady model is developed for two-layer exchange through a channel with friction on the bottom, sidewalls, surface and interface. Steady or time-varying barotropic forcing can be specified. The unsteady model is first used to solve for the steady solution with zero barotropic forcing starting from initial conditions of the lock exchange problem. The effects of friction on steady exchange are investigated for four channel configurations: a contraction with constant depth and with an offset sill, and a constant-width channel with constant depth and with a sill near one end. Exchange flow decreases substantially with increasing friction. The interface position and locations of internal hydraulic control are affected by varying friction. Solutions are asymmetrical when surface friction is absent. Internal hydraulic jumps form when friction is increased. Flow becomes hydraulically uncontrolled for high friction in all channel geometries considered. The model predictions are compared to experiments in a constant-width channel with constant depth and with a sill. The model is also applied to the Burlington Ship Canal which connects Hamilton Harbour to Lake Ontario. The exchange in the Burlington Ship Canal is modeled with zero and net steady barotropic components. Field observations from boat-mounted instruments show barotropic components and unsteadiness in flows. The magnitude of the observed barotropic variations is not great enough to influence exchange so that friction is the dominant factor governing exchange in the Burlington Ship Canal. The unsteady model is finally used with a periodic barotropic forcing in the contraction geometry. Exchange increases with forcing period and magnitude for the frictionless case. The model results are inconclusive for the effect of increasing friction with the  ii  periodic barotropic forcing. The numerical methods of the model do not' allow it to be generally applied to other channel geometries with time-varying barotropic forcing.  iii  Table of Contents  Abstract  ii  List of Tables  vi  List of Figures  vii  List of Symbols  ix  Acknowledgments  xi  1  2  3  Introduction  1  1.1  Objectives  2  1.2  Outline  3  Literature Review  4  2.1  Internal hydraulics  4  2.2  Time-dependence  6  2.3  Friction  7  Theory  9  3.1  Equations of motion  9  3.2  Model formulation  3.3  Steady hydraulics of two-layer  12 flow  13  3.3.1  Hydraulic controls  13  3.3.2  Energy  14 iv  4  Steady unforced solutions  17  4.1  Methods  17  4.2  Contraction  18  4.2.1  21  4.3  Contraction and offset sill  4.4  Constant-width channel  4.5  22 .  24  4.4.1  Comparison with experiment  26  4.4.2  Comparison with theory  28  Constant-width channel with sill 4.5.1  5  Varying friction ratios  29  Comparison with experiment and theory  .  31  Application to the Burlington Ship Canal  54  5.1  Field study  55  5.2  Modeling  55  5.2.1  Unforced steady solutions  58  5.2.2  Steady barotropic forcing  59  6  Periodic barotropic forcing  68  7  Discussion  72  7.1  Effect of friction on unforced steady exchange  72  7.1.1  Friction factors  74  7.1.2  Limitations  75  7.2 8  Burlington Ship Canal  76  Conclusions and Recommendations  Bibliography  80 83  v  List of Tables  5.1  Observed and modeled flows for Burlington Ship Canal  61  7.1  Natural sea straits  73  7.2  Effect of friction on hydraulic control for four geometries  73  vi  L i s t of Figures  3.1  Flow configuration for model  16  4.1  Geometry for contraction  33  4.2  Evolution of interface for inviscid lock exchange in contraction  34  4.3  Steady solution for contraction with a = 0.02  35  4.4  Steady solution for contraction with a = 0.1  36  4.5  Steady solution for contraction with a = 0.5  37  4.6  Effect of friction on exchange, for contraction  38  4.7  Effect of friction on controls with zero surface friction, for contraction . .  39  4.8  Change in location of controls with friction, for contraction  40  4.9  Effect of varying friction ratios on exchange, for contraction  . . . . . . . .  41  4.10 Geometry for contraction with offset sill  42  4.11 Effect of varying friction on steady solution, for contraction with offset sill  43  4.12 Effect of friction on exchange, for contraction with offset sill  44  4.13 Geometry for constant width channel  45  4.14 Effect of varying friction on steady solution, for constant-width channel .  46  4.15 Effect of friction on exchange, for constant-width channel  47  4.16 Comparison of model and experimental results, for constant width channel  48  4.17 Comparison of model and analytical solutions, for constant width channel  49  4.18 Geometry for constant width channel with sill  50  4.19 Effect of varying friction on steady solution, for constant-width channel with sill  51  vii  4.20 Effect of friction on exchange, for constant-width channel with sill . . . .  52  4.21 Comparison with experiment, for constant-width channel with sill . . . .  53  5.1  Map of Hamilton Harbour and Burlington Ship Canal  62  5.2  Hyperbolic tangent fit for A D C P velocity data  63  5.3  Geometry for Burlington Ship Canal  64  5.4  Comparison of unforced steady solution and field data in Burlington Ship Canal  5.5  65  Comparison of model with steady barotropic forcing and field data in Burlington Ship Canal  5.6  66  Comparison of model with field data in Burlington Ship Canal for high barotropic component (Drift C)  67  6.1  Effect of periodic barotropic forcing on exchange in contraction  71  7.1  Reduction in exchange due to friction, for four channel geometries . . . .  78  7.2  Summary of changing control locations with friction, for four channel geometries  79  viii  List of Symbols  b  width  B  width scale  Ei  energy per unit volume of layer i  Ei  internal energy head of two-layer flow  fb  friction factor for channel bottom  //  friction factor for interface  f  s  friction factor for surface  f  w  friction factor for sidewalls  Fi  layer Froude number for layer %  Fr  Froude number for single-layer flow  g  acceleration due to gravity  g' = eg  reduced gravitational acceleration  G  composite Froude number  h  total depth of flow  hi  thickness (depth) of layer i  h  height of sill  H  depth scale  i = 1,2  index of upper, lower layer  L  length scale  s  Q  non-dimensional magnitude of barotropic forcing  Q(t)  net barotropic volumetric flow rate  b  ix  ri  friction ratio for interface  r  friction ratio for surface  1*11)  friction ratio for sidewalls  Sf  friction slope  So  topographic slope  t  time  Ai  timestep of numerical model  T  characteristic period of barotropic forcing  u  velocity of single-layer flow  Ui  velocity of layer i  A u = U2 — U i  shear  X  horizontal distance along channel  Aa;  grid size of numerical model  s  a = £={P2-  f L/H b  Pl)/P2  frictional parameter relative density difference  7 = TyW/i  time-dependent parameter  V  numerical viscosity  ^max = A x / 2 A i 2  Pi  m a x i m u m stable numerical viscosity density of layer i shear stress on bottom due to friction  T/  shear stress on interface due to friction  T  shear stress on surface due to friction  7~w  shear stress on sidewalls due to friction  s  Acknowledgments  I thank my advisor Greg Lawrence for his guidance and encouragement throughout this project. I appreciate the insights and patience of Roger Pieters in delving into problems ranging from theory to numerical methods to Matlab. Helpful comments on the final draft were provided by Noboru Yonemitsu. The completion of this thesis was made easier by the help of fellow students in the Departments of Oceanography and Civil Engineering: Debby Ianson, Ramzi Mirshak, T i m Fisher and Ted Tedford. Thanks to former students Sue Greco, L i Gu and David Zhu for supplying field and experimental data and to Karl Helfrich at Woods Hole Oceanographic Institute for generously sharing his model code and answering questions about his paper. A special thank you goes to family and friends near and far for their support over the years of my Masters. Financial assistance from the Natural Sciences and Engineering Research Council is gratefully acknowledged.  xi  Chapter 1  Introduction  Two-layer exchange flows often occur when a constriction separates two bodies of water with different densities. The density difference may arise due to differences in temperature, salinity or sediment concentration. Understanding exchange flow is important when addressing water quality issues in semi-enclosed bodies such as harbours, bays, fjords and inlets. One example which has attracted considerable research attention is the exchange of more saline Mediterranean water with Atlantic water through the strait of Gibraltar (e.g. Farmer and Armi / Armi and Farmer, 1988; Armi and Farmer, 1985). Important exchange flows occur in other straits including the Bosphorus and Dardanelles Straits which connect the Aegean and Black Seas via the Marmara Sea (Oguz et al., 1990; Oguz and Sur, 1989) and the Bab-el-Mandeb which connects the Indian Ocean to the Red Sea (Assaf and Hecht, 1974). Understanding exchange flow can be important in engineering problems. For example, the design of a bridge linking Denmark and Sweden required that exchange flow between the Baltic and North Seas through the Great Belt not be reduced (Ottesen-Hansen and Moeller, 1990). Another exchange of environmental interest is that of heavily polluted water from Hamilton Harbour with Lake Ontario water through the Burlington Ship Canal (Hamblin and Lawrence, 1990). Many important features of exchange flows can be described by steady hydraulic theory of two-layer inviscid (frictionless) flows (e.g. Armi and Farmer, 1986; Farmer and Armi, 1986). Maximal exchange occurs when the flow through a strait is isolated from reservoir conditions by supercritical exit regions. Solution of the fully non-linear  1  Chapter!.  Introduction  2  hydraulic equations can be achieved by specifying the location of internal hydraulic controls (Gu and Lawrence, 2000). In the absence of friction these are located at topographic features such as contractions or expansions in width and sills (Armi and Farmer, 1986). The locations of controls are difficult to predict with the introduction of friction and/or complicated geometries. In the case of a dynamically short channel, inertial forces dominate and frictional forces can be neglected (Anati et al, 1977). In dynamically marginal or long channels friction significantly reduces the magnitude of exchange flows (Hamblin and Lawrence, 1990; Gu and Lawrence, 2000). Many natural channels can be classified as long. The application of steady hydraulics has been reasonably successful in oceanographic contexts.  However, exchange flows are often subjected to unsteady barotropic  forcing. Exchange through channels connected to the ocean is affected by astronomical tides. Flow in channels connected to lakes can be influenced by tides, intermittent seiching and meteorological phenomena. Steady hydraulic theory is applicable when the forcing period is short compared to the time for an internal wave to travel the strait length. The quasi-steady limit is applicable when the period is long enough that each point in the tidal cycle can be analyzed as steady (e.g. Armi and Farmer, 1986). For intermediate forcing periods, the usual concept of hydraulic control no longer applies and time-dependence is important (Helfrich, 1995). There have been few studies of timedependent exchange flows.  1.1  Objectives  This thesis aims to extend hydraulic theory to include both frictional and time-dependent effects. The objectives are to:  Chapter 1.  Introduction  3  • Develop a time-dependent model which includes bottom, sidewall, surface and interfacial friction. • Apply the model to various channel geometries to establish the solution for steady two-layer exchange. • Investigate the impact of friction on exchange rate and hydraulic control of the steady solutions. • Verify model results against theoretical predictions and experimental and field observations. • Examine the effect of a periodic barotropic forcing on exchange with varying friction.  1.2  Outline  In Chapter 2, the literature on exchange flows is reviewed. The formulation of the model and steady hydraulic theory are presented in Chapter 3. The steady solutions for different geometries subjected to varying friction are investigated in Chapter 4. The results are also compared to analytical solutions and laboratory experiments. In Chapter 5 the model is applied to the Burlington Ship Canal with zero and net steady barotropic forcing. The results are compared to field data from the canal. The effect of periodic barotropic forcing on exchange is briefly examined in Chapter 6. Discussion is given in Chapter 7 and conclusions and recommendations are outlined in Chapter 8.  Chapter 2  Literature Review  2.1  Internal hydraulics  Many flows of atmospheric and oceanographic interest can be modeled as homogeneous, inviscid layers with little variation in vertical velocities in each layer. Hydrostatic pressure and uniform horizontal velocity can be assumed. This leads to the development of the hydraulic or shallow water equations for single layer flow (e.g. Henderson, 1966). This theory was extended to two-layer flow by Stommel and Farmer (1953) in their seminal work on exchange flow through a narrow channel between a semi-enclosed estuary and the ocean. They found that the exchange rate and the amount of mixing between fresh and salt water in the basin was limited by the critical condition at the mouth of the estuary. In this example, two-layer stratification was maintained in the estuary. Wood (1970) later studied the lock-exchange problem of flow through a contraction initiated by opening a gate separating fluids of different densities. Layered flow over an obstacle was studied experimentally and analytically by Long (1954, 1970, 1974) and numerically by Houghton and Isaacson (1970).  These studies  considered the blocking of two-layer flow over a mountain ridge, starting from conditions of uniform flow upstream and downstream. Later Baines (1984) examined stratified flow over topography in experiments with a towed obstacle and developed predictions from a two-layer hydrostatic model. These analyses vary fundamentally from the current study and other studies discussed below in that the reservoir conditions (interface depths and  4  5  Chapter 2. Literature Review  velocities) were imposed. The solutions over the obstacle were matched to the reservoir depths and velocities by jumps and/or rarefactions.  In the present study and others  described below, the interface position and velocities are determined. Maximal two-way exchange, where flow is supercritical at each end of the channel, is not affected by reservoir conditions but is governed only by channel geometry and fluid densities. In their study of exchange through a contraction, Armi and Farmer (1986) identified the contribution of Stommel and Farmer (1953) as a special limiting case of submaximal exchange, not generally applicable to many flows. Armi and Farmer (1986) used a more general theoretical approach that encompassed both lock exchange and Stommel and Farmer's (1953) analysis and was extended to include steady barotropic flows. The existence of a "virtual" control was first identified by Wood (1968). Flow which accelerated from a stagnant reservoir through a contraction was governed by a hydraulic control at the narrows and a second "virtual" control upstream. When a net barotropic flow is applied to an inviscid contraction, maximal exchange also requires two "virtual" controls. Subcritical flow between the controls in the channel is isolated from reservoir conditions by regions of supercritical flow (Armi and Farmer, 1986).  In the absence  of barotropic flow, the two controls coincide at the narrows and flow is supercritical everywhere else. If barotropic forcing is strong enough, the hydraulic control can be overcome and unidirectional flow occurs. For single-layer flows, contractions and sills control the flow in a similar manner. The control of two-layer flow over a sill differs from that of two-layer flow through a contraction (Armi, 1986; Farmer and Armi, 1986). In a contraction, the change in width affects both layers equally so the control at the narrows is symmetrical and characterized by layers of equal depth. A sill extends only into the lower layer so that the control acts on the total flow indirectly through its effect on the bottom layer (Farmer and Armi, 1986).  Chapter 2.  Literature  6  Review  This results in asymmetrical control and a maximal exchange less than that through a contraction with constant depth. For the combination of a contraction and sill, exchange is maximal when a control is present at each. Submaximal exchange occurs when one of the controls is lost due to reservoir conditions (Armi and Farmer, 1986).  2.2  Time-dependence  There have been few studies of time-dependent forcing on two-layer exchange flows. Armi and Farmer (1986) and Farmer and Armi (1986) extended their analysis of exchange flow through a contraction with and without a sill to include barotropic forcing. Flows such as those induced by meteorological events or tides were simulated by quasi-steady flows where the internal hydraulic adjustments are rapid compared with the forcing. The solution for a periodic flow was achieved by treating each point in the cycle as steady and integrating over a tidal cycle to determine the exchange. As the barotropic forcing increased, the exchange averaged over a cycle increased.  Different types of flow were  observed throughout the cycle, including maximal exchange, submaximal exchange, the formation of fronts, single-layer flow, and reverse flow. The quasi-steady approximation no longer applies if either the time for internal waves to propagate through the strait is of the same order as the barotropic timescale or if the temporal accelerations (du/dt) erations (udu/dx)  are of the same order as the convective accel-  (Helfrich, 1995). Helfrich (1995) determined the parametric region of  validity of the steady unforced and quasi-steady theories. The parameter 7 =  T\/g'H/L,  where T is the period of the barotropic forcing, g' is reduced gravity, and H and L are strait depth and length scales, is a measure of the length of the strait relative to the distance an internal signal will travel during one forcing period. The steady unforced  Chapter 2. Literature  Review  7  solution corresponds to 7 = 0 while the quasi-steady limit is reached as 7 —> 0 0 . For intermediate forcing periods, the usual concept of hydraulic control no longer applies and time-dependence is important. Helfrich (1995) determined how average exchange over a tidal cycle is affected by the forcing period and magnitude and by the strait geometry (three variations of a convergent-divergent contraction with constant depth and one with an offset sill were considered).  2.3  Friction  The effect of friction on two-layer flow was first considered by Schijf and Schonfeld (1953) in their classic study of a salt wedge. Anati et al. (1977) examined the relative importance of frictional and inertial forces in constant width channels. dynamic length of the channel by the parameter f L/H, b  where  They classified the  is bottom drag coeffi-  cient, L is channel length, and H is channel depth. In a short channel fbL/H  <C 1 and  bottom friction can be neglected compared to the convective acceleration udu/dx. long channel f L/H b  In a  » 1 and friction is important and inertial forces can be neglected  compared to bottom friction. In channels of marginal length fbL/H  ~ 1 and both terms  are important. Exchange rate was reduced significantly by friction in theory (Gu and Lawrence, 2000) and in observations (Hamblin and Lawrence, 1990). The position of the interface and the location of controls were also affected by friction (Pratt, 1976; Bormans and Garrett, 1989). The analysis of Anati et al. (1977) ignored interfacial friction. G u and Lawrence (2000) analysed the relative importance of interfacial friction on exchange and found its effect significant, especially for channels of marginal length. A model of flow through the Dardanelles was more sensitive to interfacial friction than bottom friction (Oguz and Sur, 1989). Increasing interfacial friction resulted in the removal of one of the  Chapter 2.  Literature  Review  8  hydraulic controls so that flow became submaximal. Interfacial friction was also found to be important in determining the shape and intrusion of an arrested salt wedge (Grubert, 1990). Bottom and wall friction can be determined theoretically (White, 1991) or empirically (Henderson, 1966). Determining interfacial friction directly is difficult as it requires the measurement of Reynolds stresses as well as velocity profiles (Zhu, 1996). Interfacial friction is more commonly estimated from the balance of forces (Dermissis and Partheniades, 1984). This method has been used in saline wedges where the interfacial friction is estimated from observed wedge shapes (Arita and Jirka, 1987).  Chapter 3  Theory  3.1  Equations of motion  We consider a system of two layers of homogeneous fluid separated by a sharp interface with variables as defined in Figure 3.1. We include variations in depth and width to modify the approach of Schijf and Schonfeld (1953). The dimensional equations of continuity and momentum in the two layers, neglecting vertical accelerations, can be written  dhi dt dh 2  dt du\ dt  d d  du  2  d dx  d  dt  x  +  (3.2)  + dx  — - + — ^+g(h —- H  (3.1)  +— dx  + h + h) 2  (l-e)  g  +  s  - 9'S  fl  h + h) 2  s  (3-3)  = 0  g'S  f2  = 0  (3.4)  dx  where t and x are time and space co-ordinates, i = 1,2 indicates upper and lower layer respectively, b is channel width and h is sill height which are given functions of x, and s  hi and Ui are layer thickness and velocity respectively. between the layers is e = (p — p\)/p2 2  The relative density difference  where pi are layer densities. The reduced gravity  is g' — eg where g is gravitational acceleration. The friction slope for the upper layer is given by  9  Chapter 3.  Theory  10  n  f s ui\ui\  S  f  fi  ui\ui\  i - - j Y h T -  f  w  ^ r  Au\Au\  ^ - V h r  +  ( 3  -  5 )  where A u = u — ui is the shear. The friction factors are defined by the shear stresses 2  on the surface, sidewalls and interface, respectively: /, fw  //  =  -2r  =  - 2 T  =  s  /(pxUxM),  (3.6)  |MI|),  (3.7)  / ( P I U I  W  2r,/(p Au|Au|)  (3.8)  1  The friction slope for the lower layer is c, b  f  _  fbU \u \ 2  2  t  2 g'h  *~  J w 2  f Au\Au\  u \u \ 2  2  T  g'b  2  g'h  ^  2  where the friction factors for the bottom, sidewalls and interface, respectively, are defined by: fb  =  ~2r  b  /(p u \u \),  (3.10)  /„,  =  -2T  w  / (p u \u \),  (3.11)  //  =  -2r /(p Au|Au|)  2  2  2  /  2  2  2  (3.12)  2  We consider cases where the density difference between layers is sufficiently small that the Boussinesq approximation applies and pi ~ p so that the friction factors f 2  w  and / /  are the same for each layer. Assuming the free surface deflections are small, we can take the equation for continuity in the upper layer alone (3.1). We also subtract the equation for momentum in the upper layer from that in the lower layer (3.4-3.3). This study differs from that of Schijf and Schonfeld (1953) as, like Helfrich (1995), we consider the application of barotropic forcing to the system, with characteristic period T. Non-dimensionalizing t by T , x by  11  Chapter 3. Theory  length scale L, hi and h by depth scale H, b by width scale B, and  and A u by  \/g'H,  we obtain the following two equations:  where the friction slope Sj defined in terms of the non-dimensional variables is S = -a j ^ T ^ M f  + n (^J^j  Au|Au| + ^ " I M  + y  ( 2\u2\ - u i M ) j u  (3.15)  This non-dimensionalization has introduced two parameters:  = ^ ^ H ,  7  a  = H~ fb  ( 3  '  1 6 )  The ratio of the convective inertia to unsteady inertia is given by 7. It was first defined in Helfrich's (1995) study of exchange flow subject to a periodic barotropic forcing. It is the ratio of the barotropic forcing period to the time for an internal wave to travel the length of the canal. The ratio of the friction forces to the inertia forces is given by a. This parameter was used to classify channel length by Anati et al. (1977). In a short strait (a -C 1) inertia dominates and frictional forces can be neglected, while in a long strait ( a > l )  friction dominates and inertial forces are negligible in comparison.  Additional friction ratios have been introduced in equation 3.15:  //  ri = ~r, Jb  r = s  Is T  h  ,  fw H  r  w  = ——  Jb -£>  .  .  (3.17)  These parameters indicate the relative importance of interfacial, surface and sidewall friction compared to the bottom friction.  Chapter 3. Theory  3.2  12  Model formulation  In order to solve the two equations of motion (3.13 and 3.14), we need to reduce the number of unknowns to two. Following the approach of Helfrich (1995), the barotropic transport Q(t) is specified. Recognizing that the net barotropic transport is the sum of the layer transports, i.e. Q(t) = Qi + Q = b (uihi + u h ), we use the substitutions 2  Q(t)M l  2  Aub(h-hi)  Q(t) + Aubhi  bh  =  2  '  U 2  ,  bh  =  ( 3  0  -  1  0  .  1 8 )  to reduce the two equations 3.13 and 3.14 to two unknowns, the depth of the upper layer hi and the shear Au. Note that exchange flow and unidirectional flow are possible and as a result layer flows Qi and velocities barotropic forcing, Q\ = Q  can be positive or negative. For flow without  and the flow for both layers will be denoted Qi. For the  2  inviscid case the layer flow Qi  will be denoted Q  inv  for convenience.  inv  This allows us write the following non-dimensional equations for use in the numerical model:  a / b dx \  i  iat, •y dt  i_ _Au  a  7  ox \  a  dt  f  ^  bh  f  e  Q  l  M  A 6V_ M  +  A  \  a  h  +  ^  _  h  =  0  J  l  *  f  _  2  A  _  s  0  ( 3  .  2 0 )  j  These equations are essentially the same as those used in Helfrich's (1995) frictionless model with the addition of the friction slope Sf which is given by equation 3.15. Note that determining the friction slope Sf from equation 3.15 in terms of the two variables hi and Au requires the calculation of layer velocities from equation 3.18 and the calculation of lower layer depth from h = h - hi. The solution of equations 3.19 and 3.20 is governed 2  by an additional parameter, Qb, the dimensionless magnitude of the barotropic forcing Q(t).  Chapter 3.  3.3  Theory  13  Steady hydraulics of two-layer flow  When barotropic forcing Q(t) is zero or steady, the model reaches a steady solution. That solution can be analysed based on steady two-layer hydraulics.  3.3.1  Hydraulic controls  Single-layer flow is classified by the Froude number Fr = u/(gh), where u is the dimensional flow velocity, g is gravitational acceleration, and h is the dimensional depth of flow. The Froude number is the ratio of the convective velocity to the phase speed of an infinitesimal long gravity wave. Flow is subcritical when Fr < 1, critical when Fr = 1 and supercritical when Fr > 1. Critical flow occurs at control points such as contractions and changes in bottom elevation. Two-layer flow is classified by the composite Froude number G  2  = F  2 1  +  F -eF F 2  2  2  1  2 2  (3.21)  where the densimetric layer Froude numbers are (3.22) where Ui and hi are non-dimensionalized as in § 3 . 1 . In the present study, the density difference between layers is small (e <C 1) and the last term in equation 3.21 is ignored. Two-layer flow is internally critical when the composite Froude number is unity. Although the composite Froude number is no longer a ratio of convective and phase velocities, it correctly determines criticality of flow (Armi, 1986). Long waves can propagate along the interface as well as along the free surface. Interfacial waves cannot propagate out of regions of supercritical flow (G  2  > 1), thus if flow in a channel is bounded by two  controls with supercritical flow beyond, flow is isolated from reservoir conditions. This is known as maximal exchange (Armi and Farmer, 1986).  Chapter 3.  14  Theory  Often the Froude number is much greater in one layer than the other. In this case the layer with the higher Froude number will be called the active layer. Flow dynamics are governed primarily by the active layer (Lawrence, 1985). The stability of two-layer flow is determined by the stability Froude number F  = (Au)  2 A  (3.23)  2  where A u is the non-dimensional shear as in §3.1. Flow is stable for F  2 A  FA  2  < 1. When  > 1, internal phase speeds are imaginary and internal hydraulics may no longer apply  (Long, 1954).  3.3.2  Energy  We examine the steady solution assuming hydrostatic flow of two homogeneous layers. The internal energy (or Bernoulli constant) for each layer is  E = g(h 1  Pl  + h) + ^  (3.24)  s  2  E  2  = g(h Pl  + h ) + (ft a  )g(h  Pl  2  + h) + ^  (3.25)  s  where variables are dimensional. This study is concerned with flows where the layer Froude numbers are of order one and the density difference between layers is small, i.e. e < l .  We eliminate the large  term p\ g (h + h ) by defining the internal energy of the two-layer flow as s  =  E^-Ei  p2 g' Non-dimensionalizing E , E{, and hi by H and u^ by y^g'H, the internal energy for r  two-layer flow becomes Er = 1 - h, + ^ J f l !  (3.27)  Chapter 3.  15  Theory  where variables are non-dimensional as above in § 3 . 1 .  The internal energy describes  the behaviour of the interface in two-layer flows similar to the total energy or Bernoulli equation for single layer flow. For the inviscid case, internal energy is conserved along a channel in the absence of hydraulic jumps. When friction is included, shear stresses at the bottom, sidewalls, interface and surface result in energy losses. The change in internal energy due to friction is  similar to the approach for single layer flows (e.g. Henderson, 1966). This is equivalent to the steady solution to equation 3.14. Substituting the equation for internal energy head (3.27) into the above equation (3.28), we obtain an expression for the slope of the interface  dh\ dx  S  0  (3.29)  1-G  2  where the topographical slope due to changes in depth and width is  where variables have been non-dimensionalized as in § 3.1. The flow is subject to hydraulic controls where the composite Froude number is unity, G  2  topographic slope must equal the friction slope, i.e. S = Sf. 0  — 1.  At these points the  Chapter 3.  Theory  16  Figure 3.1: Flow configuration for model. Side view of channel with sill. Width (b) may vary along channel. Convention is source of less dense water at left (pi < p?)- During exchange upper layer flows left to right and lower layer flows right to left.  Chapter 4  Steady unforced solutions  The steady case can be solved analytically by setting the unsteady terms equal to zero in equations 3.13 and 3.14 (dhi/dt  = 0, d(Au)/dt  = 0). This solution is achieved only  if the locations of hydraulic control are known. Armi and Farmer (1986) and Farmer and Armi (1986) have shown these solutions for the inviscid case with simple geometries (contraction, sill, and combination of contraction and sill). With the addition of friction or with a complicated geometry, it is difficult to predict the location of controls. To avoid the need to prescribe control locations in advance, the present unsteady model can be run from arbitrary initial conditions to the steady solution. The control locations are then determined from the solution.  4.1  Methods  Instead of solving for the steady case directly (7 = 0, dh\/dt  — 0, d(/\u)/dt  = 0), the  unsteady model was used to attain the steady solution by running until steady state was reached. The value of 7 in equations 3.19 and 3.20 affects only the timestep; it was arbitrarily set at 7 = 1. For the steady unforced case Q(t) = 0 while for the steady barotropic case Q(t) is a constant.  Following Helfrich (1995), equations 3.19 and 3.20  are solved numerically using the two-step Lax-Wendroff method as outlined in Press et al. (1986). The method is second-order accurate in time and space. It is able to model the development and propagation of shocks which occur in such exchange flows. Sommerfeld radiation boundary conditions (Orlanski, 1976) are applied at each end of the strait so 17  Chapter 4.  18  Steady unforced solutions  that information propagates out of the domain. In some cases, the inclusion of numerical dissipation was required to control the growth of high wavenumbers in regions of unstable shear. term u(Au)  As in Helfrich (1995), the  was added to the right hand side of equation 3.20.  xx  noted, the applied numerical viscosity was v = 0.02 v  where the maximum stable  max  value is determined by the Courant condition, v  = Ax /2At. 2  max  Unless otherwise  Although other more  sophisticated methods exist, the present method was adequate for controlling instabilities without significantly changing the solutions. The model was typically run from initial conditions of the lock exchange problem, mimicking the removal of a vertical barrier in a channel. The choice of initial conditions did not affect the final solution of the steady case. When other arbitrary initial conditions were specified, the model eventually reached the same steady solution as for the lock exchange. The solution was considered steady when the variation in time of the interface position (hi  t+At  — hi) became sufficiently small.  The model was run with a number of strait geometries, discussed in § 4 . 2 - § 4 . 5 below.  4.2  Contraction  A convergent-divergent channel was considered, using the same expression for width as Helfrich (1995):  (4.1) The depth is constant throughout the channel (see Figure 4.1). This strait geometry is representative of contractions; Helfrich (1995) showed that results change quantitatively but not qualitatively when the contraction geometry is modified.  Chapter 4. Steady unforced solutions  19  The model was run from initial conditions of the lock exchange problem.  The  exchange flow is established by gravity currents which travel out from the centre of the strait (see Figure 4.2). Once they reach the ends of channel, the gravity heads are swept out of the model domain by the open boundary conditions. After the unsteady period of flow establishment, the solution reaches steady state. The inviscid solution of Helfrich (1995) for the steady unforced case was reproduced. During the initial unsteady portion, the gravity currents travel with a velocity of approximately \^Jg'H  as expected. Once the steady solution is established, the com-  posite Froude number is unity at the narrows, with supercritical flow everywhere else in the strait. This control at the narrows is actually two "virtual" controls which coincide in the absence of barotropic forcing (Armi and Farmer, 1986). The stability Froude number is equal to one throughout the strait, indicating marginal stability. The layer velocities are U{ — ± 0 . 5 at the narrows. The layer flows are Qi = 0.25 throughout the strait, as expected for maximal exchange. Internal energy is conserved along the channel (dEi/dx  = 0).  The effect of friction on the flow through the contraction was investigated by increasing the parameter a.  For the initial analysis, the surface and interfacial friction  ratios were set equal to one (77 = l,r  s  = 1). The wall friction factor was set equal to  bottom friction (/„, = f ) but since most natural channels are wide (H/B b  friction ratio was set at r  w  <C 1), the wall  = 0.1. Representative cases from the range of a values are  illustrated in Figures 4.3-4.5. With increasing friction the two controls at the centre move outward (Figure 4.3). The controls are located equal distances from the centre; the solution is symmetrical since friction is applied equally to the bottom and surface. The flow is subcritical in the central region of the canal between the controls and supercritical beyond. Although flow is reduced compared to the inviscid solution, this flow is still the maximal exchange for  Chapter 4.  20  Steady unforced solutions  the specified friction parameters, asflowin the channel is isolated from outside conditions by supercritical regions (Farmer and Armi / Armi and Farmer, 1988). Internal energy is no longer conserved along the channel as there are losses due to friction. The change in internal energy along the canal dEj/dx balances the friction slope Sf (equation 3.28). The controls occur where the friction slope Sf equals the topographic slope due to changes in width S (equation 3.29). 0  When friction is increased beyond a = 0.055, hydraulic jumps are present near the ends of the strait (Figure 4.4). The jumps appear to be nearly vertical in the model solutions, as a minimal amount of numerical viscosity was applied to the model. In reality, internal hydraulic jumps would be subject to extensive mixing between layers. Horizontal stretching of the jumps as they appear when mixing occurs can be achieved by increasing the viscosity. With the jumps present, the flow is subcritical beyond the jumps as well as in the centre between the two virtual controls, with supercritical regions between the virtual controls and the jumps. Since exchange is defined by theflowthrough the narrows, it is still maximal as the flow in the centre is isolated by the supercritical flow. As a increases, the jumps move toward the centre of the strait. At the same time, the virtual controls move away from the centre. When a = 0.23 the hydraulic jumps and virtual controls meet. For higher a, the flow is subcritical throughout the channel (Figure 4.5). This flow is no longer hydraulically controlled by the contraction. For all positive a, the flow is stable along the channel ( i * A < 1). The stability 2  Froude number is highest at the narrows (x = 0). As friction increases, the maximum value of F A  2  decreases (Figure 4.6). The layer flow rate of the steady solution similarly  decreases from its inviscid value of Q  inv  = 0.25 with increasing friction (Figure 4.6). The  r  reduction is significant even for relatively short straits - for a = 1 the layerflowis reduced by 57 % from the inviscid prediction.  Chapter 4.  Steady unforced solutions  21  The specification of zero net flow forces the flow in the two layers to be equal (Q = Q ) . With friction applied symmetrically to the surface and bottom, the two layers x  2  are of equal thickness (hi = h = 0.5) and equal velocity (ui = u ) at the narrows, so that 2  2  Au = 2u{. The results in the composite Froude number equalling the stability Froude number at the narrows since now G = yj2ui /hi 2  width at the narrows is b = Qi = Au/4.  1  = 2UJ and F A = Au.  so that layer flow Qi  =  Mj/i;.  Since hi — 0.5 and Au = 2ui,  The flowrate normalized by the inviscid value of. Qi  nv  = 0.25 is thus equal  to the composite and stability Froude numbers at the narrows (Qi/Qi  nv  4.2.1  In addition, the  = G = FA).  Varying friction ratios  The effect of the friction ratios on exchange flow was investigated. In real straits, friction factors would not be equal on all surfaces.  Estimates of interfacial friction factors are  typically within one order of magnitude of bottom friction factors (0.1 < 77 < 1) (Gu, 2000).  Reducing the interfacial friction results in solutions which are similar to those  presented in Figures  4.3-4.5  and are still symmetric. When interfacial friction ratio rj  is decreased, the first appearance of internal hydraulic jumps occurs at higher values of 01. (Figure 4.8). The transition to uncontrolled flow occurs at higher a as well. In the absence of wind or ice cover, surface friction is zero.  When r — 0, the s  solution is no longer symmetric along the canal. A representative series with varying a which shows the change in location of controls with r = 0 is given in Figure 4.7. The s  virtual controls move out from the centre with increasing friction. A hydraulic jump forms at the left hand end of the canal when friction is increased above a = 0.05. The flow is still supercritical to the right of the virtual controls. A second hydraulic jump, in the right end of the canal, forms when a — 0.1. The left-hand control is drowned for a > 0.3; flow is now subcritical except for the region between the right-hand virtual control and the right-hand hydraulic jump. Flow is uncontrolled for a > 0.4, with subcritical flow  Chapter 4.  22  Steady unforced solutions  throughout. Reducing interfacial friction in the absence of surface friction caused the appearance of hydraulic jumps and the transition to uncontrolled flow to occur at higher values of a (Figure 4.8). The effect of changing friction ratios on exchange is shown in Figure 4.9. The impact is significant for a > 0.1. For example, when a — 1 and r = 1, halving the interfacial s  friction from 77 = 1 to 77 = 0.5 causes exchange to increase by 26%. Further decreasing the interfacial friction by an order of magnitude from ?7 = 1 to 77 = 0.1 results in an increase in flow of 69 % when a = 1 and r = 1. s  Changing the surface friction ratio has a greater impact when the interfacial friction ratio is small. The effect is most significant for a > 1. For example, when a = 1 and 77 = 1, removing surface friction (i.e. changing r = 1 to r = 0) causes exchange to s  s  increase by only 1 %. When a = 1 and 77 = 0.1 the same change results in an increase in exchange of 15 %.  4.3  Contraction and offset sill  In most straits, variations in bottom topography such as sills are present. A combination of sill and contraction was considered. The same strait width and depth of Helfrich's (1995) example are used: (4.2) and h(x) = 1 — - cosh  2  Bx  (4.3)  where 6 = 3.75 and <f> = 0.637 for x < 1 and <f> = 1.273 for x > 1. The sill crest is at x = 0 and the narrows is at x = 1 (Figure 4.10). The steady unforced solution for the frictionless case (Figure 4.11) matches the solution for the composite and stability Froude numbers and the interface position of  Chapter 4. Steady unforced solutions  23  Helfrich (1995). Flow is critical at the sill crest and at the narrows, with subcritical flow between and supercritical flow beyond. The stability Froude number is above unity where the lower layer descends to the left of the sill crest, indicating unstable shear. Layer flow rates are Qi = 0.137. The effect of friction on the solution was examined. The parameter a was varied while holding the friction ratios constant at T\, = 1, 77 = 1, r  w  tions for a range of values of a are shown in Figure 4.11.  = 0.1. The steady solu-  Friction has a minor effect  on the position of the interface and controls for a < 0.1. The exit layers thicken slightly at the ends of the canal and the controls migrate outwards slightly with increasing a. The region of unstable shear to the left of the sill crest seen in Figure 4.11 persists until a > 0.07. When a > 0.1, an internal hydraulic jump forms to the left of the sill crest with corresponding increase in lower layer thickness.  Another hydraulic jump forms to the  right of the narrows. Flow is now subcritical in both exit regions. The subcritical flow between the sill and narrows is isolated from the exit regions by supercritical flow, so that exchange is still maximal for the frictional conditions. With increasing a, the control at the narrows moves outward while the right-hand jump moves inward, until the control is drowned for a > 0.4.  Flow is now subcritical  from the control on the lee of the sill to the right-hand end of the channel. A region of supercritical flow remains to the left of the sill crest, with a jump to subcritical flow further to the left. When one control is present, flow is defined as submaximal (Farmer and Armi, 1986). Further increase in friction above a = 3 results in the drowning of the left-hand control as well, so that flow is subcritical throughout the strait. Increasing a causes a decrease in layer flows (Figure 4.12). The decrease relative to the inviscid case (Qi/Qi ) nv  is somewhat less than that for the contraction without a  sill. For example, when a = 1, for the contraction with offset sill the flow is reduced  Chapter 4.  24  Steady unforced solutions  from the inviscid value by 45%, while for the contraction with constant depth it was reduced by 57%. For the contraction with offset sill, both the width and the depth vary from those for the contraction, so the behaviour of the exchange would not be expected to be the same. The difference is partially due to the submaximal exchange conditions which exist in the channel with the sill compared to uncontrolled (subcritical) flow in the channel without sill. The sill forces the left hand control to remain for much higher a. For the contraction with constant depth, flow became subcritical throughout the channel for a > 0.23, while for the contraction with offset sill flow became completely subcritical throughout for a > 3 when the sill was present.  4.4  Constant-width channel  Constructed or natural canals can take the form of a long channel of relatively constant width opening into reservoirs via sudden expansions.  Instantaneous expansions were  difficult to model numerically, so a rapid expansion was used at the ends of the channel. The non-dimensional width of the expansions is given by  b (x < 0, x > 1) = 1 + 6.1 ( l - - (*-V-) ) 100  2  e  (4.4)  where ip = 0 for x < 0 and ij) = 1 for x > 1 (see Figure 4.13). The portion of the channel between x = 0 and x = 1 is of constant width 6 = 1. Modifying the expansion geometry slightly caused no observable impact on exchange. Again the model was run from initial conditions of lock exchange. The steady solution for the inviscid case gives a flat interface located at half-depth within the constantwidth section (Figure 4.14).  The layer flows are ± 0 . 2 5 as expected.  The composite  Froude number is critical throughout the straight section and supercritical beyond. The stability Froude number is one throughout the channel.  Chapter 4.  25  Steady unforced solutions  Friction was applied to the model with r = 1,77 = 1, and r s  w  = 0.1. As friction (a)  increases, the slope of the interface in the channel increases (Figure 4.14). The interface tilts to compensate for energy losses due to friction. Internal hydraulic controls occur where the friction slope (5/, not shown) balances the topographic slope (S , not shown) due to changes in width. The topographic slope is 0  zero throughout the channel and in the reservoirs beyond. In the case of an instantaneous expansion, the topographic slope varies only at the ends of the channel, forcing controls there.  Flow is critical at the ends of the channel, subcritical within the channel and  supercritical beyond (Figure 4.14). The minimum value of G  2  occurs at the channel  midpoint. When a > 1 flow becomes subcritical throughout the model domain. In every case, the flow is stable along the channel ( F A < 1). The stability Froude 2  number is highest at the ends of the channel and lowest in the centre. As friction increases, the minimum value of F A  2  decreases. As in the contraction, the stability Froude number  equals the composite Froude number at the centre of the channel, and both equal the flowrate normalized by the inviscid value of 0.25 ( F A =  G  =  Qi/Qinv)-  Gu and Lawrence (2000) developed an analytical solution for exchange rate and interface position by direct integration of the fully non-linear hydraulic equation for a wide channel, i.e. with bottom and interfacial friction only. The unique solution for given frictional parameters was achieved by specifying internal hydraulic controls at each end of the channel. For comparison, the present model was used to solve for exchange rate with increasing friction and changing interfacial friction ratio in the absence of sidewall and surface friction. The layer flow rates decrease with increasing friction (Figure 4.15). Even for short channels the effect is significant. For example, when a — I, the reduction in flowrate from the inviscid value ranges from 38% to 65% for 77 = 0.1 to 77 = 1, respectively.  Thus reducing the interfacial friction ratio has a substantial impact on  exchange, especially for marginal straits of 0.1 < a < 10 (Figure 4.15). These results  Chapter 4.  Steady unforced  solutions  26  agree with the findings of Gu and Lawrence (2000).  4.4.1  Comparison with experiment  Gu (2000) investigated steady maximal frictional two-layer exchange flows and conducted laboratory experiments of exchange flow in a straight channel. A tank 370 cm long and 106 cm wide was divided into two reservoirs connected by a channel 200 cm long, 15.2 cm wide and 30 cm deep, with zero bottom slope. The reservoirs were filled with water and allowed to come to room temperature. A removable barrier was installed in the channel. Salt was added to the reservoirs to create the density difference required to drive the exchange flow. After the water became quiescent, the barrier was removed, allowing exchange between the reservoirs to begin. Initially exchange flow was uncontrolled. Gradually hydraulic controls established at each end of the channel. Only the data from the subsequent period of steady maximal exchange were processed as Gu (2000) was interested only in the steady regime. Simultaneous measurements included capture of the density interface by still and video cameras, density profiling by a conductivity probe on an automated traversing mechanism, and mean velocity measurements by particle image velocimetry. The present study considers one experiment (E5) which was repeated sixteen times. The total flow depth was H = 28 cm and the reduced gravity was g' = 1.14 cm/s . 2  The interface position can be determined by three methods: from the position of maximum velocity shear, from the zero velocity line, or from the position of maximum density gradient. In ideal two-layer exchange flows with no barotropic forcing, the three are the same. The positions of maximum velocity shear and zero velocity coincided in the experiment. The density interface was slightly higher in the channel than the velocity interface. The velocity interface was measured only at the midpoint of the channel while the density interface was measured along the channel length.  The shift between the  Chapter 4.  Steady unforced  solutions  27  density and velocity interfaces at the midpoint of the channel was less than 2 % of the total channel depth for a l l experimental runs ( G u , 2000). T h e exchange rate was obtained by integrating the mid-channel velocity profiles to the zero velocity point w i t h respect to depth. G u (2000) determined average bottom and wall friction factors from experimental data using the integral momentum (Thwaites) method. Refer also to W h i t e (1991) and Zhu (1996) for the theoretical formulation and assumptions of applying the Thwaites method to exchange flows. In contrast to bottom and wall friction, an accurate method for estimating interfacial friction coefficient has not been established ( G u , 2000). G u (2000) determined the interfacial friction factor / / from experimental data based on the principle of conservation of energy. In the experiment, the effective friction factors were calculated to be f = 0.0104 and / / = 0.0039 giving an interfacial friction ratio b  of rj = 0.375. T h e experimental channel was dynamically short, w i t h a = 0.074. T h e measured layer flow rate of 30.9 c m / s (Qi = 0.195) is a 22 % reduction from the inviscid 2  prediction (Qi = 0.25). T h i s implies that frictional effects may be important even for channels w i t h « « 1 and thus the applicability of inviscid theory may be very l i m i t e d . In the present study, the numerical model was r u n i n a constant-width channel w i t h expansions given by equation 4.4 to represent the experimental channel. T h e model geometry specifies sharply curving exits as an approximation of the instantaneous expansions from the channel to the reservoirs i n the laboratory. T h e model parameters were set to a = 0.074 and 77 = 0.375 as determined from the experiment. W a l l friction was set equal to bottom friction (r  w  = H/B = 1.8) and surface friction set equal to zero  (r = 0). T h e model was r u n from lock exchange i n i t i a l conditions to m i m i c the res  moval of the barrier from the laboratory channel. A small amount of artificial viscosity (v = 0.01 Vmax) was applied i n order to model the heads of the gravity currents.  Here  "max is the m a x i m u m stable value determined by the Courant condition ( § 4 . 1 ) . It was  Chapter 4.  28  Steady unforced solutions  discovered that this viscosity affected the solution slightly so the model was subsequently run with zero viscosity (v = 0), starting from the previous v = 0.01 v  max  lock exchange  solution until a new steady solution was reached. The steady-state non-dimensional layer flow rate of the model, Qi = 0.19, is close to the experimental value of Qi = 0.195.  The model solutions for interface position,  composite Froude number, and internal energy agree closely with experimental observations (Figure 4.16). The offset between the density and velocity interfaces introduces uncertainty into the experimental results. Error is also introduced by fluctuations in the interface and measurement of the flowrate during the experiment. Both the predicted and measured interfaces are slightly higher than mid-depth at the centre of the canal due to the absence of surface friction. The model predicts internal hydraulic controls at the ends of the canal, as expected for a sudden expansion in width. Hydraulic controls were observed at the ends of the canal during the experiment. The predicted composite Froude number G  2  is lower than measured throughout the canal. The difference may result from  the modeled layer velocities being less than experimental values. The internal energy Ei changes inside the channel are due to the friction slope Sf since the topographic slope is zero (S = 0) except at the expansions into the reservoirs. 0  4.4.2  Comparison with theory  The analytical solution of Gu and Lawrence (2000) is a wide channel approximation which assumes no wall friction. To compare the analytical solution with the experiment in the narrow laboratory channel, Gu (2000) developed effective friction factors for the bottom and interface to incorporate wall friction in the analytical solution. The effective bottom friction was equal to the narrow channel value (a = 0.074). The effective interfacial e  friction factor is increased to compensate for the wide channel approximation (r  Ie  = 0.84,  compared to 77 = 0.375). The present model was run with these effective values and  Chapter 4.  Steady unforced  29  solutions  r = 0, r = 0. Results were the same as those for the model run above with 77 = 0.375 w  s  and r = H/B = 1.8. This verifies the method used by Gu (2000) in calculating the w  effective factors. This also verifies that the unsteady model of the present study run to steady state gives comparable results to a direct solution of the steady equations for the given parameters. The model results are compared with the analytical solution of Gu (2000) in Figure 4.17. The modeled interface shows less curvature near the channel ends than the analytical solution. This difference is amplified in the composite Froude number, which is lower than the analytical solution near the channel ends. The hydraulic controls of the model solution occur within one grid space of the channel ends while the analytical solution specifies controls exactly at the ends. 4.5  Constant-width channel with sill  Sills are often present in natural channels. The model was applied to a constant-width channel with a sill near one end. The sill has gradually varying topography defined by (4.5) where the sill crest is located at x = 0, L is the sill half-length and H is the sill crest s  s  height (Figure 4.18). Here the channel length scale L is the distance from the sill crest to the right-hand exit and the channel depth scale H is the constant depth away from the sill. The left end of the channel is located just beyond the sill at x = —0.3. The sill is a prominent feature in the channel, with L /L = 0.25 and H /H = 0.30. These values s  s  were chosen to represent a laboratory experiment of Zhu and Lawrence (2000) described in §4.5.1. The expansions are specified by b (x < -0.3, x > 1) = 1 + 9.8 (l -  -i°°(*-V0 ) 2  e  (4.6)  Chapter 4.  30  Steady unforced solutions  where ip = —0.3 for x < —0.3 and tp = 1 for x > 1. The portion of the channel between x = —0.3 and x = 1 is of constant width 6 = 1. The model was run with friction ratios of 77 = 1, r = 1, r s  w  = 0.1.  Due to the  presence of unstable flow (F& > 1) on the lee of the sill, it was necessary to include viscosity in the model. The minimum viscosity for the model to succeed starting from lock exchange initial conditions was v = 0 . 1 5 f  mai  where v  max  is determined by the Courant  condition (§4.1). The case with zero friction is no longer inviscid as the viscosity term v (Au)  xx  simulates interfacial friction. The internal energy is not constant along the  channel as it would be in the truly inviscid case. For the inviscid case the topographic slope alone determines the location of controls. Internal hydraulic controls are predicted at the sill crest and at the right hand end of the channel at which points the topographic slope equals zero. The topographic slope also equals zero in the section of constant width and depth, with critical flow predicted along the entirety of this section. Examining equation 3.29, we expect supercritical flow to the left of the sill crest and to the right of the channel, and subcritical flow along the right hand portion of the sill. The viscosity term in the model creates a friction slope which influences the control locations for the inviscid case. A control remains at the sill crest; however the flow is no longer critical along the flat, constant-width portion of the channel. Instead of a control at the right hand end of the channel, the control marking the transition from subcritical to supercritical is midway along the channel (Figure 4.19). With the inclusion of friction, both the friction slope and the topographic slope determine the location of controls. The controls at both the sill and the right hand exit move outward with increasing friction. For a > 0.02 hydraulic jumps form in the reservoir regions of the model and move inward with increasing friction. A third hydraulic jump forms on the lee of the sill for a > 0.4. The flow is still maximal as the subcritical region in the channel is isolated from the reservoirs by supercritical flow. The controls at both  Chapter 4. Steady unforced solutions  31  ends of the channel are drowned when OL > 1 so that flow is uncontrolled and subcritical throughout the channel. T h e flow was unstable (F  A  OL <  > 1) to the left of the sill control for  0.1. T h e modeled inviscid maximal exchange flow is Qi = 0.13 which corresponds to  the analysis of Zhu and Lawrence (2000).  It is recognized that the artificial viscosity  term affects the flowrate slightly. T h e same amount of viscosity was applied for each a and the flowrates for the frictional solutions are normalized by the frictionless case with the artificial viscosity so that the effect is relative. Flowrate decreased with increasing friction (Figure 4.20). For a dynamically marginal strait (a = 1) with selected friction ratios rj — 1, r  s  4.5.1  = 1, r  = 0.1, flowrate was reduced by 57% from the frictionless value.  w  Comparison with experiment and theory  Zhu and Lawrence (2000) extended internal hydraulic theory to include the effects of friction and streamline curvature over a sill.  Theoretical solutions were obtained by  determining control locations where the frictional and topographic slopes were equal. Laboratory experiments were conducted to verify theoretical solutions.  A channel of  constant width 10 cm was placed in a tank 370 cm long, 106 cm wide, and 30 cm deep. A sill with half-length L  s  — 25 cm and crest height H  s  — 8 cm was used. T h e exits of  the channel were located 31 cm to the left and 103 cm to right of the sill crest which was located at x = 0 (Figure 4.18). A partition was placed in the middle of the channel and salt dissolved in the right hand reservoir to provide the driving buoyancy force. Here an experiment with reduced gravity g'=1.56 c m / s  2  will be examined (Zhu and Lawrence,  2000). Dye and particles were added to the water to permit recording of interface position and velocity field by video cameras. Exchange flow was initiated by removing the barrier. After an initial unsteady period, maximal exchange flow was established with controls at the crest and the right hand exit.  Chapter 4.  32  Steady unforced solutions  The frictionless solutions of Zhu and Lawrence (2000) and of the present model (§ 4.5) are similar to each other but are quite different from the experimental observations (Figure 4.21). The solution of the present model deviates from the analytical solution due to the control located midway along the canal as explained in §4.5. The match with the experiment is greatly improved by the inclusion of friction. An average friction factor for the channel bottom and walls was estimated by Zhu and Lawrence (2000) using the Thwaites method (fb — f  w  — 0.019). The interfacial friction  factor was then estimated from the measured friction slope (// = 0.016). Considerable error is introduced in the estimation of / / because of errors in determining the interface position and the bottom-wall friction factor fb = fw  The present model was run using  these values, which give a = 0.07 and 77 = 0.84, and with r  w  = H/B = 2.8 and r = 0. s  When friction is included in the model, the predicted flowrate is reduced to Qi = 0.11, a decrease of 15% from the inviscid value of Qi  nv  = 0.13.  The model prediction falls  in the range of exchange rates Qi = 0.108 to Qi = 0.119 estimated from experimental measurements (Zhu and Lawrence, 2000). The modeled interface is slightly steeper than the analytical solution of Zhu and Lawrence (2000).  The internal hydraulic controls  remain near the sill crest and the right-hand exit of the channel. With friction, flow was no longer unstable to the left of the sill control. Zhu and Lawrence (2000) achieved even better agreement with the experiment by considering nonhydrostatic effects. Non-hydrostatic (curvature) effects are not considered in the present model.  Chapter 4.  Steady unforced  solutions  33  Figure 4.1: Geometry for convergent-divergent contraction: Plan view of width. Width varies according to equation 4.1. Strait length scale is twice the distance from the narrows to the point where b = IB. Width at boundaries is b = 4.7J5. Depth is constant throughout.  Chapter 4.  Steady unforced solutions  34  Figure 4.2: Evolution of interface position with time for inviscid lock exchange in contraction. Consecutive lines are Ay = 0.5 apart. The channel bottom and surface are located at 0 and 1, respectively (non-dimensional height). Dashed lines show velocity of \yJg'H for heads of gravity currents.  Chapter 4.  <D O CO t  0  Steady unforced  35  solutions  0.75 0.5 0.25 -  Figure 4.3: Steady solution for contraction with a = 0.02. Friction ratios are ti = l , r = l,r = 0.1. Top: Interface position along canal. Middle: Composite Froude number, G (—) and stability Froude number, F ( ) along canal. Controls occur where G = 1. Flow is stable [F < 1). A t narrows G = F . Bottom: Rate of change in internal energy along the canal equals friction slope, dEj/dx = S/ (—). At controls, friction slope equals topographic slope, S , ( ). s  w  2  2  A  2  2  2  A  2  A  0  36  Chapter 4. Steady unforced solutions  -1.5  -0.5  -1  0  1  0.5  1.5  X  Figure  4.4:  n = l,r  s  Steady  solution for contraction  = l,rw = 0.1.  Froude number,  G  with  a = 0.02.  Top: Interface p o s i t i o n a l o n g c a n a l .  (—) a n d s t a b i l i t y F r o u d e n u m b e r ,  2  t r o l s o c c u r w h e r e G = 1. F l o w is s t a b l e ( F 2  2 A  F& ( 2  Friction  ratios are  Middle:  Composite  ) along canal.  < 1). A t narrows G = F . 2  2  A  Con-  Bottom:  R a t e o f c h a n g e i n i n t e r n a l energy a l o n g t h e c a n a l equals f r i c t i o n s l o p e , dE[/dx — Sf (—). A t c o n t r o l s , f r i c t i o n slope equals t o p o g r a p h i c s l o p e , S , ( 0  ).  Chapter  4.  Steady unforced  solutions  37  0 1|  1  1  1  r  1  o  II 0.5 -  QI  I  -1.5  -1  ~~  i  l_  -0.5  I ^  I  0  ~  0.5  I  1  _J  1.5  X  Figure 4.5: Steady solution for contraction w i t h a = 0.02. Friction ratios are n = 1, r = 1, r = 0.1. Top: Interface position along canal. M i d d l e : Composite Froude number, G (—) and stability Froude number, F& ( ) along canal. F l o w is hydraulically uncontrolled (G < 1) and stable ( F A < 1). B o t t o m : Rate of change in internal energy along the canal equals friction slope, dEjjdx = Sf (—). A t controls, friction slope equals topographic slope, S , ( ). s  w  2  2  2  2  0  Chapter 4. Steady unforced solutions  38  Figure 4.6: Effect of friction on exchange, for contraction. Friction ratios are rj = l,r = l,r = 0.1. Layer flow Q i normalized by inviscid layer flow Q = 0.25 equals the stability and composite Froude numbers (F&, G) at the narrows. The maximum value of stability Froude number occurs at the narrows for all a. s  w  inv  Chapter 4.  39  Steady unforced solutions  CM  01 o  •1.5  -1  -0.5  0  0.5  1.5  CM  Figure 4.7: Effect of friction on controls with zero surface friction, for contraction. Friction ratios are 77 = l,r = 0,r = 0.1. T o p : Interface position along canal. M i d d l e : Composite Froude number. B o t t o m : Stability Froude number. Selected values are: a = 0.06 (—), a = 0.1 (- - ) , a = 0.3 ( ),<* = ! ( • • •)• s  w  Chapter  4.  Steady unforced  40  solutions  Figure 4.8: Change in location of controls with changing friction and friction ratios. With increasing friction (a), hydraulic jumps move in from ends of canal and virtual controls move out from centre (x = 0) until flow is no longer hydraulically controlled. When surface friction equals bottom friction (r — 1) controls are equidistant from narrows (—). When surface friction is removed (r = 0) the solution is asymmetric ( ). Interfacial friction decreases from bottom line to top line (77 = 1,77 = 0.5,77 = 0.2,77 = 0.1). Wall friction ratio is r = 0.1 for all cases. s  s  w  Chapter 4.  41  Steady unforced solutions  Figure 4.9: Effect of varying friction ratios on exchange, for contraction. Friction ratio for wall is held constant at r = 0.1. Exchange is measured by layer flow Qi normalized by the inviscid value Qi = 0.25. Interfacial friction decreases from left to right (77 = 1,77 = 0.5,77 = 0.2,77 = 0.1), with surface friction equal to bottom friction (r = 1, —) and with zero surface friction (r = 0, ). w  nv  s  s  Chapter 4.  Steady unforced  solutions  42  Figure 4.10: Geometry for contraction with offset sill. Top: Plan view. Width varies according to equation 4.2. Bottom: Side view. Depth is defined by equation 4.3. Strait length scale L is the distance from the sill crest to the narrows.  Chapter 4. Steady unforced solutions  43  Figure 4.11: Effect of varying friction on steady solution, for contraction with offset sill. T o p : Interface position along canal. M i d d l e : Composite Froude number. B o t t o m : Stability Froude number. Selected values are a = 0 (—), a = 0.2 ( ), a = 1 (—•—), a = 10 (•••)•  Chapter 4.  44  Steady unforced solutions  Figure 4.12: Effect of friction on exchange, for contraction with offset sill. Layer flow Qi is normalized by inviscid layer flow Qi = 0.137. Friction ratios are 77 = 1, r = 1, r = 0.1. nv  s  w  Chapter 4.  -0.5  45  Steady unforced solutions  0  0.5  1  1.5  X  Figure 4.13: Geometry for constant width channel. Plan view. Depth is constant.  Chapter 4.  0  0.2  1 0.75 LL  46  Steady unforced solutions  " / /  0.4  0.6  0.8  1  i  i  i  1  \  0.5 0.25  r. •  /'  /  \  "  \  0  0  0.2  0.4  0.6  0.8  1  Figure 4.14: Effect of varying friction on steady solution, for constant-width channel. Top: Interface position along canal. Middle: Composite Froude number. Bottom: Stability Froude number. Selected values are a = 0 (—), a = 0.1 ( ), OJ = 1 (—•—), a = 10 (• • •). Friction ratios are 77 = 1, r = 1, r = 0.1. s  w  Chapter 4.  47  Steady unforced solutions  Figure 4.15: Effect of friction on exchange, for constant-width channel. Layer flow rate Qi normalized by inviscid value Q i = 0.25 decreases with increasing friction a and interfacial friction ratio 77. Surface friction is zero (r = 0) and wide channel is assumed [r — 0). The composite and stability Froude numbers at the midpoint of the channel equal the normalized flowrate (G = = Qi/Qmv)n v  s  w  Chapter 4. Steady unforced solutions  48  Figure 4.16: Comparison of model and experimental results, for constant width channel. Experimental measurements (•) and error bars for run E5 (Gu, 2000). Model prediction with a — 0.074,77 = 0.375,r = l,r = 0 (— solid line). Top: Interface position. Middle: Composite Froude number. Bottom: Internal energy. w  s  Chapter 4.  49  Steady unforced solutions  Figure 4.17: Comparison of model and analytical solutions, for constant width channel. Model prediction of current study (—) and analytical solution of Gu (2000) ( ) for wide-channel approximation with a = 0.074, rj = 0.84, r = 0,r — 0. T o p : Interface position. M i d d l e : Composite Froude number. B o t t o m : Internal energy. e  e  w  s  Chapter 4.  Steady unforced solutions  50  Figure 4.18: Geometry for constant width channel with sill. Top: Plan view. Expansions in width are defined by equation 4.6. Bottom: Side view. Depth is defined by equation 4.5. Strait length scale L is the distance from the sill crest to the narrows.  Chapter 4. Steady unforced  solutions  51  Figure 4.19: Effect of varying friction on steady solution, for constant-width channel with sill. Friction ratios are 77 = 1, r = 1, r = 0.1. Channel ends are at x = —0.3 and x = 1. T o p : Interface position along canal. M i d d l e : Composite Froude number. B o t t o m : Stability Froude number. Selected values are a — 0 (—), a = 0.1 ( ), a = 1 (— • —), s  a =10 (•••)•  w  Chapter 4.  52  Steady unforced solutions  Figure 4.20: Effect of friction on exchange, for constant-width channel with sill. Layer flow Qi is normalized by inviscid layer flow Q i = 0.13. Friction ratios are 77 = l,r = l,r = 0.1. n  s  w  v  Chapter 4.  Steady unforced solutions  53  Figure 4.21: Comparison with experiment, for constant-width channel with sill. Interface position from experimental measurements (o) (Zhu and Lawrence, 2000); analytical solution of Zhu and Lawrence (2000) for inviscid (• • •) and frictional (— • —) cases; solution of present model for inviscid ( ) and frictional (—) cases.  Chapter 5  Application to the Burlington Ship Canal  An exchange flow of environmental interest is that between Hamilton Harbour and Lake Ontario through the Burlington Ship Canal. The harbour is heavily polluted and has been recognized as an area of concern by the International Joint Commission (Barica, 1989). A 500 k m watershed of urban, industrial and agricultural lands drains into the 2  harbour (126 • 10 m /yr). Four sewage treatment plants serving a population of 500,000 6  3  discharge effluent into the harbour (95 • 10 m /yr). The harbour is an enclosed body of 6  3  water whose sole outlet is the Burlington Ship Canal which connects it to the western end of Lake Ontario (see map, Figure 5.1). The canal is 830 m long with a rectangular cross-section of constant width of 89 m and average depth of 10.6 m. In summer, the temperature-dominated density difference between water in Hamilton Harbour and Lake Ontario drives two-layer exchange flow through the canal; superimposed on this is an oscillating barotropic component.  Warmer Hamilton Harbour water flows out of the  canal above Lake Ontario water flowing into the harbour. During periods of exchange flow, large volumes of water are exchanged; the exchange flow is one order of magnitude greater than all other inputs combined. Understanding this phenomenon is necessary for determining the water quality of both Hamilton Harbour and the western end of Lake Ontario.  54  Chapter 5.  5.1  Application  to the Burlington  55  Ship Canal  F i e l d study  An extensive field study was conducted in the summer of 1996 to investigate the exchange flow dynamics in Burlington Ship Canal. On July 25, 1996, velocity and conductivitytemperature-depth (CTD) profiles were collected from boat-mounted instruments during five drifts along the canal. Here data from four drifts (A, B, C, E) will be presented, not including one drift in the wake of a dry bulk carrier (D). Velocity profiles were created from data from the Acoustic Doppler Current Profiler (ADCP). Density profiles were calculated from the conductivity and temperature data. Following Greco (1998), data were processed and a hyperbolic tangent function was fitted to each profile (see Figure 5.2). Layer velocities, interface location and interface thickness were determined from the fitted hyperbolic tangent functions,  (5.1) where u is velocity, z is depth, and a, b, c, d are fit parameters. The upper layer velocity is u\ = a — b, the lower layer velocity is u = a 4- b, and the shear is Au = u — ui — 2b. The 2  2  barotropic component of the velocity is a. The interface is typically taken at the position of maximum shear, hi — c (where position is measured from the surface). The interface can alternately be determined from the position of zero velocity, hi = c + d t a n h (—a/b). - 1  5.2  Modeling  There have been various previous attempts to quantify exchange through the Burlington Ship Canal. Dick and Marsalek (1973) numerically integrated equation 3.29 with zero topographic slope (S = 0) and with bottom and interfacial friction only. 0  Instead of  assuming controls, the solution proceeded outward from a measurement at the midpoint of the canal. The results were compared with ten observed estimates of interface position  Chapter 5.  Application  to the Burlington  56  Ship Canal  from field measurements at either end of the canal. Hamblin and Lawrence (1990) used the same equation assuming internal hydraulic controls at either end of the canal with a linear interface connecting them. The calculated flow rates were compared with flow measurements from Spigel (1989).  Greco (1998) assumed a linear interface between  controls at each end of the canal and neglected friction. The solution was compared to observed interface position determined from hyperbolic tangent fits of the drift data as described above ( § 5 ) . Gu (2000) used his analytical solution for a wide channel with controls at each end (§4.4.1) with an "effective" bottom friction factor to account for irregularity in geometry, three-dimensional effects, and wall friction. Flow rates agreed reasonably with measurements by Dick and Marsalek (1973), Spigel (1989) and Greco (1998). No comparison with interface position was made. The present model was applied to the Burlington Ship Canal. The bottom material is fine sand with some variations. The walls are vertical steel sheet piles (Dick and Marsalek, 1973). Bottom and wall friction coefficients of fb = f  w  = 0.0026 and an inter-  facial friction coefficient of / / = 0.001 were chosen for the model (Hamblin and Lawrence,' 1990). The low friction coefficients are a result of the high Reynolds number and relatively smooth material. The resulting model parameters are a = 0.2, r/ = 0.38 and r  w  = 0.0003. Surface friction was set to zero (r = 0). s  The model requires the specification of depth and width along the domain. The model geometry chosen is not meant to represent every detail of topography around the canal but to provide a schematic of the canal with which to begin exploring exchange flow and barotropic effects.  The chosen domain extended beyond the canal so that  effects near the ends of the canal could be modeled without being influenced by the model boundaries. Depth soundings from 1994 gave bathymetry within the canal and for roughly 500 m outside either end. These were compared to the water depths recorded by the boat-mounted A D C P during the 1996 field study. The cross-section averaged depth  Chapter 5.  Application  to the Burlington  Ship Canal  57  from the 1994 soundings, adjusted by the water level offset from the 1996 data, was used for model bathymetry (Figure 5.3). The model assumes that depth does not vary in the vertical, so that the width of the upper and lower layers is the same. This holds within the Burlington Ship Canal where width is constant.  However outside either end of the canal there are variations in un-  derwater topography. The choices for model width beyond the canal were based on the behaviour of the active layer, i.e. the thinner layer which has a higher layer Froude number. Outside the canal at the harbour end, the lower layer is active. The Burlington Ship Canal opens into Hamilton Harbour beyond docks to the north. As the model cannot deal with an instantaneous expansion, a smoothed transition is used. To the south, mounds of dredgings as high as 7 m above the canal bottom constrain the inflow. This is represented as a gradual expansion. The model solution for the unforced case showed a control outside the canal at the Hamilton Harbour end above the sill at x = —0.14 (-116 m). At the Lake Ontario end, mean lake depth is 5 m, with a deeper dredged channel leading to the ship canal. This has little effect on exchange as the upper layer is active. Warm Hamilton Harbour water forms a thin (active) surface layer which discharges into Lake Ontario. Jet-like behaviour has been observed where the surface layer enters the lake without undergoing lateral spreading for some distance. However, as the discharge does not always behave like a jet, the model was run for the unforced case assuming various rates of lateral spreading in Lake Ontario. The runs showed sensitivity to the varying geometry prescribed at the Lake Ontario end. For no or small expansion, no control formed at the lake end (submaximal exchange).  For expansion rates above a  certain value (db/dx > 0.05) a control formed at the lake exit of the canal. This is the maximal exchange case where subcritical flow in the canal is isolated from the basins by supercritical regions at both ends. The control at the lake exit remained for additional  Chapter  5.  Application  to the Burlington  58  Ship Canal  expansions in width; exchange rate and interface position within the canal were not affected. The only change was an increase in the interface level beyond the end of the canal in Lake Ontario with increasing expansion rate. A n expansion in width of 1:1 with distance (db/dx = 1) was chosen as it resulted in the best match between the interface level and the (sparse) field data beyond the lake exit.  5.2.1  U n f o r c e d steady solutions  The model results for unforced, steady maximal exchange are compared to the data from the boat drifts A , B, and E in Figure 5.4.  (Drift C was not included in this analysis  due to a large observed net barotropic flow.) Note that the field data vary in time along the canal since the boat drifts each lasted around half an hour. The predicted interface matches roughly with the observed interface position near the ends of the canal. Within the canal the predicted interface was below the observed position for Drifts A and B. The observed interface for Drift E was characterized by what appear to be interfacial waves, which are not reproduced in the modeled interface (which is to be expected). The values of the composite Froude number along the canal predicted by the model are generally lower than those calculated from field data. The value of G is often at or near 1 in each of the drifts, but no distinct points of control could be identified from the field data, either at the ends of the canal or elsewhere within the canal. The model predicts a small peak in G near x = 0.2. The corresponding dip in the predicted interface at that position is not clearly seen in the field observations of interface position. The predicted layer flow rates are approximately equal to field values.  Friction  has reduced the layer flows to Qi = 0.18, a reduction of 30% from the inviscid value of Qinv — 0.25. According to the analysis of a constant-width and constant-depth channel ( § 4 . 4 ) , for the frictional parameters a = 0.2,77 = 0.385, r — 0,r s  w  = 0, a 26% reduction  would be expected (see Figure 4.15). While the model specifies no barotropic component  Chapter 5.  Application  to the Burlington  Ship Canal  59  so that layer flow rates are equal, the field data exhibit a net barotropic flow component (Table 5.1).  5.2.2  Steady barotropic forcing  Since net barotropic flows were observed for all drifts, the modeling was repeated with steady barotropic forcing to examine its effect on exchange and to attempt to improve the agreement with field observations.  For each drift a steady barotropic component  Q(t) = Q equal to the canal-averaged mean of the observed net flow was specified in b  the model equations 3.19 and 3.20. The agreement between observed and modeled layer flows improved (Table 5.1). The net flow rates for the field and model are equal since the model input Q was specified from field values. b  Drift A showed a negative net barotropic flow. Modeling with a steady negative barotropic forcing resulted in a control and hydraulic jump appearing at x ~ 0.2 (Figure 5.5). The fit between the predicted and observed interfaces improved to the right of this control. Drifts B and E exhibited small positive net barotropic flows. The model solution for interface position and composite Froude number did not change significantly with the introduction of steady barotropic forcing for Drifts B and E (Figure 5.5). The model predicts that the control at the right-hand end (Hamilton Harbour) is on the verge of being drowned due to the barotropic forcing for Drifts B and E . Drift C had a significant net barotropic component. During a portion of the drift, barotropic flow was high enough that the lower layer was arrested (Figure 5.6). Drift C was modeled with a steady barotropic forcing equal to the canal-averaged mean of the measured net flow. The modeled interface position is significantly lower than observed within the canal although the ends match well (Figure 5.6). In the model run, the imposition of net left-to-right barotropic flow removes the control at the Hamilton Harbour exit. The control at the Lake Ontario end remains in the same position. Supercritical  Chapter 5.  Application  to the Burlington  Ship Canal  60  flow observed in the canal is not present in the model results (Figure 5.6). This corresponds to the discrepancy in interface position; the observed upper layer is thinner and has a higher velocity which results in a higher Froude number than modeled. The interface can be determined from the A D C P data by two methods - from the position of zero velocity or from the position of maximum shear (see Figure 5.2). In the field, a shear layer or mixed layer is present so that the two positions are offset when the velocity exhibits a barotropic component. The model assumes perfectly two-layered flow so that the two positions are coincident. The interface and composite Froude number were recalculated from the hyperbolic tangent fits of the field data using the position of zero velocity. For Drifts A , B and E , determining the interface positions from the two methods resulted in only slight differences (Greco, 1998). For Drift C the difference was more noticeable. The interface at the zero-velocity position is much closer to the model prediction (Figure 5.6). Using this interface position to calculate the composite Froude number results in reduced values of G and an improved match with model predictions. Unsteadiness observed in all drifts limits the applicability of the steady analysis. In Figures 5.4-5.6, the measured flow rates vary along the canal. This indicates variation with time since each boat drift lasted over half an hour. The results for the steady model cannot be expected to match with the field observations as the data from each drift are influenced by changing barotropic flows.  Chapter 5.  Application  to the Burlington  61  Ship Canal  Table 5.1: Comparison of observed flows in Burlington Ship Canal and predicted flows for unforced solution and for steady barotropic forcing. Values are flow per unit width in m /s (non-dimensionalize by ^/g'H = m/s (10.6 m) = 4.9 m/s . Field values have been averaged along canal. 2  3  2  3  2  Upper Layer  Lower Layer  Net  Unforced Model  0.86  -0.86  0  A - Field A - Model  0.63 0.62  -1.11 -1.10  -0.48 -0.48  B - Field B - Model  0.98 1.04  -0.62 -0.68  0.36 0.36  C - Field C - Model  1.53 1.61  -0.27 -0.26  1.35 1.35  E - Field E - Model  0.98 1.00  -0.69 -0.71  0.29 0.29  Drift  Chapter 5.  Application  to the Burlington  62  Ship Canal  Creek  Figure 5.1: Map of Hamilton Harbour, its tributaries, and Burlington Ship Canal. Note that representation of the harbour end of the canal is inaccurate.  63  Chapter 5. Application to the Burlington Ship Canal  Position of zero velocity  o  0  o  o  a -20  -10  0 10 Velocity (cm/s)  20  30  Figure 5.2: Hyperbolic tangent fit (—) and data (o) for velocity profile from one ADCP ping during Drift C (17:35:49.40 GMT). Exchange flow is occurring with upper layer flowing from harbour to lake and bottom layerflowingfrom lake to harbour. Layers of approximately uniform velocity are separated by a mixed shear layer. A net barotropic component results in a vertical shift between the position of zero velocity and the position of maximum shear.  64  Chapter 5. Application to the Burlington Ship Canal  -415 01  1  0  415  830  1245  1  1  1  r  415  830  1245  24h  -415  0 x(m)  Figure 5.3: Geometry for Burlington Ship Canal. Ends of canal are at x = 0 m and x = 830 m. Top: Plan view. Beyond the ends of the canal, width is prescribed for the active layers: lower layer in Hamilton Harbour and upper layer in Lake Ontario. Bottom: Side view. Cross-channel average of bathymetry soundings were used for depth. (Values are non-dimensionalized for model by width scale B = 89 m, depth scale H — 10.6 m and length scale L = 830 m).  Chapter 5. Application to the Burlington Ship Canal  Drift A  0  0.5 X  65  Drift B  1  0 X  0.5  Drift E  1 X  0  0.5  1  Figure 5.4: Comparison of the unforced steady solution and three sets of field data in Burlington Ship Canal. Field data (•) are from hyperbolic tangent fits of A D C P data. Model results (—) are for unforced steady solution. Ends of canal are x = 0 (harbour) and x = 1 (lake) T o p : Interface. M i d d l e : Composite Froude number. B o t t o m : Layer and net flow rates. Positive values are for upper layer flowing from harbour to lake (Qi); negative values are for lower layer flowing from lake to harbour (Qz). for drifts ( x ) ; model specifies zero net flow.  Net flow is shown  Chapter 5. Application to the Burlington Ship Canal  Drift A  CD  66  Drift B  Drift E  1  0.25  * *. h °  0  X  CM  1  •'J.;.">^ :  o  O -0.25 -0.5  0.5  0.5  0.5  X  X  X  Figure 5.5: Comparison of steady solution with barotropic forcing and three sets of field data in Burlington Ship Canal. Field data (•) are from hyperbolic tangent fits of A D C P data. Model results (—) are for steady barotropic forcing. Ends of canal are x = 0 (harbour) and x = 1 (lake) Top: Interface. M i d d l e : Composite Froude number. B o t t o m : Layer and net flow rates. Positive values are for upper layer flowing from harbour to lake ( Q i ) ; negative values are for lower layer flowing from lake to harbour (0,2)- Net flow is shown for drifts (x) and model ( ).  Chapter 5. Application to the Burlington Ship Canal  Drift C  67  Drift C  <D 1  0.5  O 0.25  y--.v.v  O  cr  0  -0.25  0.5  0.5  X  X  Figure 5.6: Comparison of steady solution with field data in Burlington Ship Canal for high barotropic component (Drift C). See caption of Figure 5.5 for details. Field values (•) of interface and Froude number are determined from hyperbolic tangent fits by two methods: left-hand panels, from position of maximum shear; right-hand panels, from position of zero velocity. Flows in both cases are integrated to zero velocity position.  Chapter 6  Periodic barotropic forcing  A periodic barotropic forcing Q(t) =Q sm(2irt)  (6.1)  b  was applied in model equations 3.19 and 3.20, where Q is the dimensionless magnitude of b  the barotropic forcing and time t is non-dimensionalized by the period of the barotropic forcing T. The parameters 7 and Q now govern the response in addition to the frictional b  parameters a, ri, r , and r . s  w  Problems were encountered in running the model code with this time-varying barotropic forcing. Dispersive ripples overwhelmed the solution and caused it to fail. Dispersion is a feature of the advection scheme chosen (Pietrzak, 1995).  Subsequent to  code development, the code used by Helfrich (1995) was acquired for comparison. A smoothing step was included in the code provided by Helfrich. Only with the inclusion of the smoothing step could the unsteady inviscid results of Helfrich (1995) be replicated for the contraction, with either Helfrich's code or the present code which was developed independently. Unfortunately this smoothing was not detailed in Helfrich (1995). The smoothing eliminates dispersive ripples by averaging the two variables hi and Au along the canal at each time step. The following summarizes the effect of barotropic forcing as described by Helfrich (1995) and describes the attempt to include friction and extend the model to geometries other than those used by Helfrich (1995). The model was run for the inviscid case starting from initial conditions corresponding to the steady unforced solution. A periodic response developed within two to three 68  Chapter 6.  Periodic barotropic  69  forcing  periods. If different initial conditions were specified, the model response evolved to the same periodic solution. The effect of the barotropic parameters 7 and Q on the exchange b  rate is measured by the ratio of the period-averaged layer flow to the steady unforced layer flow ((Qi) ve/Qinv)a  The exchange rate increased with increasing 7 and Q  b  ure 6.1). Little deviation from the steady limit of Qi  nv  (Fig-  = 0.25 was observed for 7 < 0.5  for all Q and for Q < 0.5 for all 7. The quasi-steady limit (Armi and Farmer, 1986) b  b  was approached when 7 > 30. In all cases the flow is hydraulically controlled by the narrows only at certain times in the cycle. For low forcing Q , the response is nearly sinusoidal so that integrating the b  layer flow over one period gives no increase above the steady layer flow. The interface moves back and forth with the barotropic forcing with little change in shape from the steady unforced solution.  The layers never reverse direction.  For higher barotropic  forcing layer flows are reversed during a portion of the cycle. The asymmetric response to increased Q results in increased layer transport integrated over the cycle. Each layer b  is expelled from the narrows during a portion of the cycle. Bores form when the layer is released by the reversal of the barotropic forcing. Without the viscosity term  u(Au) , xx  the bores steepened and caused the model to fail. The minimum viscosity for the model to succeed was applied in each case as well as the smoothing step. Increasing friction was applied with the sinusoidal barotropic forcing in the contraction. The inviscid steady unforced solution was used as the initial condition in order to allow controls to form away from the narrows without being previously specified. The solutions did not all become periodic and the exchange rate did not follow any discernable patterns with changing barotropic parameters 7 and Q and frictional parameter a. b  Results are briefly described below but no figures are presented as the meaningfulness of the results is uncertain.  Chapter 6.  Periodic  barotropic  forcing  70  For low friction, when the steady case showed only two virtual controls near the narrows (a < 0.06), the response was similar to that for the inviscid case. For intermediate values of 7, the response ceased to be periodic after a number of forcing periods. When friction increased to the range where the steady solutions showed internal hydraulic jumps (0.06 < a < 0.23), the model succeeded only for values of 7 close to the steady and quasi-steady limits. For low 7, internal jumps formed after many periods, in the same location where they had formed for the steady unforced case. For intermediate values of 7 in the time-dependent range, the response was not periodic even after many periods. For high 7, the jumps formed for low forcing Qb and were obliterated for high forcing Q . b  For a further increase in friction to the range which resulted in uncontrolled (subcritical) flow for the steady solutions (a > 0.23), flow in both layers followed the barotropic forcing so that unidirectional plug flow rather than exchange flow was occurring. For small 7 near the steady limit, the interface was similar to the steady unforced solution, oscillating back and forth. For intermediate and high 7, the interface was nearly flat and located near middepth, but the response did not become periodic so that the interface migrated up and down with time. The layer flows averaged over one period were less than the inviscid unforced solution for all combinations of 7 and Qb and even became negative. (However the solutions were not always periodic when the period-averaged flow was determined so these results may not be accurate). The model with periodic barotropic forcing could not be applied successfully to other geometries. For the contraction with offset sill, the model failed numerically when friction was included, due to zero layer depth occurring on the sill. When the barotropic forcing was applied to the constant-width channel, the smoothing step was required as for the other geometries. This created over- and under-shoots in the interface at either end of the channel which then propagated through the strait, obscuring the solution.  Chapter 6. Periodic barotropic forcing  I  I  71  I  I  I  I  y=32  2  // 1.75 > c  // // /  d" ^sj 1.5  cr  ////  / /  1  //  6  "  8  4  1.25  1  0  "  i  i  i  0.25  0.5  0.75  I  1 Q  ~  " 7=0.5  i  I  1.25  1.5  b  Figure 6.1: Effect of periodic barotropic forcing on exchange in contraction. Sinusoidal barotropic forcing was applied for the inviscid case. Layer flow Qi averaged over one period and normalized by inviscid layer flow Qinv = 0.25 increases with magnitude (Q0) and period (7) of barotropic forcing. Thisfigurereproduces the work of Helfrich (1995).  Chapter 7  Discussion  7.1  Effect of friction on unforced steady exchange  The impact of friction on steady exchange rate is similar for all geometries studied (Figure 7.1). The layer flow rate decreases from the inviscid value with increasing friction. The exchange is roughly halved for a strait of marginal length (a = 1). Even for short channels (a -C 1), flow is reduced substantially from the inviscid prediction. In the experiment of Gu (2000), flow rate was reduced from the inviscid prediction by 22 % in the short laboratory channel (a = 0.074). This verifies that frictional effects are important even in short straits. The applicability of inviscid theory in predicting flow in natural channels is thus limited. The channel dimensions of several natural straits are listed in Table 7.1. Bottom friction is typically of order 10~ < f 3  < 10~ (Gu, 2000) which give 2  b  the estimates of a in Table 7.1. According to the present study, the frictional exchange would be much lower than inviscid predictions in all of these natural channels. Internal hydraulic control of exchange flow occurs at topographic features such as a sudden expansion in width, the narrowest point of a contraction, and the crest of a sill. Friction changes the location of controls. Internal hydraulic jumps and uncontrolled (subcritical) flow were observed in the channel geometries considered in the present study. These are summarized in a schematic of the changes in hydraulic controls with increasing friction (Figure 7.2). Traditionally, studies of two-layer exchange flow have assumed that flow is governed  72  Chapter 7.  73  Discussion  Table 7.1: Natural sea straits. Dimensions are from Assaf and Hecht (1974), Maderich and Efroimson (1990) and Helfrich (1995). Strait Gibraltar Bosphorus Dardanelles Bab-el-Mandeb Messina Oslofjord Lombok Tiran  L (km)  H (m)  a  50 30 60 130 10 10 40 3  280 40 70 185 80 15 350 270  0.2 - 2 0.8 - 8 0.9 - 9 0.7- 7 0.1 - 1 0.7- 7 0.1 - 1 0.01 - 0.1  by controls at topographic features. In the present study, uncontrolled flow occurs for all geometries when a is high. The transition from maximal exchange to uncontrolled flow was characterized by the presence of internal jumps within the channel for all geometries except the constant-width channel. Table 7.2 shows the values of a at which the transition to uncontrolled flow occurs. The values are for the case with friction applied equally to bottom, surface and interface (r = 1,77 = 1), with wall friction ratio r s  w  — 0.1.  The  values of a at which the transition occurs are particular to the specific geometries used. The results would change somewhat for varying geometric parameters.  Table 7.2: Effect of friction on hydraulic control for four geometries: values of a which mark transitions to uncontrolled flow. Geometry Contraction Offset sill and narrows Constant-width channel Constant-width channel with sill  a for uncontrolled flow 0.23 3 1 2  Chapter 7.  74  Discussion  The transition to uncontrolled flow occurs at higher a when a sill is present for both the contraction and the constant-width channel. The control on the sill is not affected by friction as easily as controls at changes in width. The impact of friction on exchange rate is somewhat reduced by the presence of the sill for this reason (see Figure 7.1). Flow was subcritical between "virtual" controls or topographic features as expected for each geometry. Flow was also subcritical beyond the internal hydraulic jumps when they were present and along the entire channel when flow was uncontrolled. For inviscid theory, subcritical flow occurs when reservoir conditions drown the internal hydraulic controls. In the present frictional model, reservoir conditions are not specified. Rather, boundary conditions are open so that information propagates outward. The subcritical flow here is not governed by downstream reservoir conditions. Flow becomes subcritical as the friction forces along the surfaces of the canal cause energy losses.  The active  layer velocity decreases with corresponding thickening of the layer so that the composite Froude number is reduced below one (G  2  < 1). The location of internal hydraulic jumps  is determined by both the friction and topographic slopes.  7.1.1  F r i c t i o n factors  The friction ratios contribute to the friction slope (equation 3.15) with varying degrees of importance. Most natural channels are wide so that the aspect ratio and thus wall friction ratio are small {H/B C l , r „ € l ) .  Surface friction acts on the upper layer only.  Changing surface friction has an impact on the criticality of the layered flow primarily where the upper layer is active (i.e. where it is thinner). The interfacial friction acts on both layers. It is proportional to the square of the shear (Au) while the other friction 2  terms are proportional to the square of layer velocities u . 2  geometry (§4.2).  For the inviscid case, (Aw) = Au 2  2  Consider the contraction  at the narrows; the difference is  much more pronounced farther away from the narrows where the passive layer has a very  Chapter 7. Discussion  small velocity  75  and the shear A u is greater. For this reason changes in interfacial friction  have a larger impact on the flow than changes in the surface friction (see Figure 4.9). When modeling exchange flow, determining the interfacial friction ratio is especially important since it has a strong effect.  This is compounded by the fact that it is very  difficult to estimate the interfacial friction factor (Zhu, 1996). The use of average friction parameters over the length of the canal is satisfactory for modeling exchange. This is confirmed by the good agreement between model and experimental results in § 4.4.1 and § 4.5.1. In reality the friction factors vary with Reynolds number and boundary layers. The present model could be modified so that the friction factors vary along the canal as a function of Reynolds number, if a more sophisticated analysis were desired.  7.1.2  Limitations  Flow in laboratory or natural channels may violate some of the basic assumptions of twolayer hydraulic theory. When the stability Froude number is above unity ( i A > 1), the 7  long wave speed is imaginary and hydraulic theory is violated. The model assumes perfectly two-layered flow where two homogeneous layers are separated by a sharp interface of zero thickness. In reality, mixing between the two layers occurs due to shear instabilities at the interface such as Kelvin-Helmholtz and Holmboe instabilities (Zhu, 1996). This creates an interfacial layer of thickness greater than zero. As well, the density and shear interfaces are often offset (Lawrence et al., 1991). Interfacial waves observed by Gu (2000) in the constant-width channel (§4.4.1) affected the laboratory measurements, especially near the ends of the channel as indicated by the error bars in Figure 4.16. As a result, discrepancies between observed and predicted values are expected. The hydraulic theory of this study assumed a hydrostatic pressure distribution. Non-hydrostatic effects can be important in stratified flows over obstacles (Zhu, 1996).  Chapter 7. Discussion  76  Zhu and Lawrence (2000) extended hydraulic theory to include the effects of streamline curvature over a sill. The match between analytical solutions and laboratory observations was greatly improved by the inclusion of non-hydrostatic effects compared to the consideration of friction alone (§4.5.1).  7.2  B u r l i n g t o n Ship C a n a l  The steady solution of the model was compared to the field observations in the Burlington Ship Canal. The observed barotropic flow changed during each drift which indicates unsteadiness and limits the comparison with the model results. The presence of barotropic flows was investigated in an analysis of continuous measurements made by moored instrumentation (Tedford, 1999). A bottom-mounted A D C P at the harbour end of the canal collected velocity profiles from July 4 to August 15, 1996.  The barotropic cur-  rent in the canal is approximated by the mean velocity observed at the moored A D C P . The barotropic current oscillated around zero. For flow from the harbour to the lake, the mean velocity was 0.08 m/s and the maximum 0.89 m/s.  For flows towards the  harbour, the mean velocity was -0.07 m/s and the maximum -0.63 m/s.  A spectrum  of the barotropic velocity revealed several significant peaks including the semi-diurnal lunar tidal oscillation and the first, second, third, fourth, sixth, eighth and ninth modes of oscillation of Lake Ontario. The modes of Lake Ontario are due to standing waves initiated by wind and/or barometric pressure changes. The periods ranged from 1 hour for the ninth mode of Lake Ontario to 12.5 hours for the lunar tide. This range of periods gives 1 < 7 < 25, which places the Burlington Ship Canal in the time-dependent range identified by Helfrich (1995). The mean magnitude of the barotropic current is approximately 0.08 m/s which corresponds to a non-dimensional value of Qb = 0.17. In  Chapter 7.  Discussion  77  Helfrich's (1995) study of sinusoidal barotropic forcing in an inviscid contraction, exchange increased above the steady value only for barotropic forcing above Qb > 0.5 (see Figure 6.1). In addition, the signal in the Burlington Ship Canal is a composite of many different periods which would likely have a lesser influence than a single periodic forcing. This means that the current model is applicable to the Burlington Ship Canal without the addition of unsteady barotropic forcing.  Chapter 7. Discussion  0  78  1  i -2  10  -1  10  1  1  1  1  0  10 a  10  2  10  Figure 7.1: Reduction in steady exchange rate due to friction, for four channel geometries: contraction (—), contraction with offset sill ( ), constant-width channel (— • —), constant-width channel with sill (• • •). Layer flows Qi are normalized by inviscid layer flow Q for each geometry. Friction ratios are ri = l,r = l,r = 0.1. inv  s  w  Chapter 7.  Discussion  79  Figure 7.2: Summary of changing control locations with friction, for four channel geometries. For each geometry, plan view of width (—) and side view of sill (shaded) are shown, with schematic of flow below. Friction (a) increases from top to bottom in each schematic. Control locations (o) and internal hydraulic jumps (A) separate regions of supercritical (—) and subcritical ( ) flow. Locations of topographic features are shown (:).  Chapter 8 Conclusions and Recommendations  An unsteady, one-dimensional model of frictional two-layer exchange flow through a strait was developed. Friction can be applied on the bottom, sidewalls, surface and interface of the channel. Steady or time-varying barotropic forcing can be applied to the flow. The frictional parameters are  LL Jb -rj !  — ll  — t  &  n  fl  p  — ll }  S  P  r  Jb  _ fw H i  W  p  r  T}  Jb &  Jb  frictional to inertial forces andThe 77, rbarotropic and r for the interface, surface and sidewalls, respectively. parameters are s  l  T  (8.2)  where 7 is a measure of the forcing period and Q is a measure of the forcing amplitude. 0  To solve the model equations, the channel geometry must be specified in addition to these parameters. Width and depth can vary along the channel. The model was solved numerically for the unforced steady solution (Q — 0) by runb  ning from initial conditions of the lock exchange problem until steady state was reached. The unsteady model is useful for finding the steady solution when it is difficult to predict control locations in frictional channels with complex geometry. The model was applied to four channel configurations: a contraction with constant depth and with an offset sill, and a constant-width channel with abrupt expansions, with constant depth and with a  80  Chapter 8.  Conclusions  and  81  Recommendations  sill near one end. The effect of friction on the interface position, layer flows, and composite and stability Froude numbers were investigated. The following conclusions can be drawn from the study of steady unforced exchange: • Friction reduces exchange substantially from the inviscid prediction, even in dynamically short and marginal channels. • When surface friction is absent, the control locations and the interface position are asymmetrical along the channel. • Internal hydraulic jumps may form in a channel due to friction. • Friction can remove controls that are traditionally assumed at topographic features such as the ends of channels or sills, so that flow is hydraulically uncontrolled. Model results compared well with laboratory studies of flow in constant-width channels with and without a sill. Even with the small values of a in the channels, the inclusion of friction greatly improved the match between the model results and experimental observations compared to the inviscid solutions.  The model also agreed with analytical  solutions of steady flow in these channels. This validates the use of the unsteady model to achieve the steady solution. The model was compared to field observations in the Burlington Ship Canal during four boat drifts on July 25, 1996. The model was solved with zero barotropic forcing and the results compared with the field data for Drifts A , B, and E which showed small net barotropic flows. With frictional parameters of a = 0.2 and 77 = 0.38, modeled exchange was reduced by 30 % from the inviscid prediction. The field data did not reveal distinct points of hydraulic control in the canal while the model predicted controls near both ends of the canal. When a strong steady barotropic forcing was applied for Drift C, the model predicted only one control. The strong barotropic component in Drift C resulted in a  Chapter 8.  Conclusions  and  82  Recommendations  shift between the position of zero velocity and the position of maximum shear in the field data. The model results agreed better with the interface and composite Froude number calculated from field observations for the position of zero velocity. Unsteadiness observed in all drifts limits the applicability of the steady solutions of the model.  While the  significant periods of observed barotropic currents place the canal in the time-dependent range of the parameter 7, the magnitude of the barotropic forcing Qb is not strong enough to influence the exchange significantly. Friction dominates exchange in Burlington Ship Canal rather than barotropic effects. Predictions of exchange between Hamilton Harbour and Lake Ontario should consider friction. When a periodic barotropic forcing is applied in a contraction with little or no friction, exchange increases with increasing barotropic forcing magnitude Qb and period T above threshold values of Qb and 7.  The model developed by Helfrich (1995) for  the inviscid case succeeded only in specialized geometries and could not be generalized to other geometries. Results with friction in the convergent-divergent contraction were inconclusive. The increase in exchange may be reduced by friction. The present numerical model is not generally applicable to various strait geometries for both the inviscid and frictional cases. A better advection scheme is needed to eliminate dispersion (see for example Pietrzak, 1995). It is recommended that the numerical methods be improved so that the model can include friction and be applied to different geometries. The relative importance of friction and barotropic effects on exchange can then be investigated.  Bibliography  Anati, D., G . Assaf, and R. Thompson, 1977: Laboratory models of sea straits. J. Fluid Mech. 81:341-351. Arita, M . and G . Jirka, 1987: Two-layer model of saline wedge, I: Entrainment and interfacial friction, II: Prediction of mean properties. J. Hydraul. Eng. 113:12291263. Armi, L . 1986: The hydraulics of two flowing layers with different densities. J. Fluid Mech. 163:27-58. Armi, L . and D. Farmer, 1985: The internal hydraulics of the Strait of Gibraltar and associated sill and narrows. Oceanologica Acta 8:37-46. Armi, L . and D. Farmer, 1986: Maximal two-layer exchange flow through a contraction with barotropic net flow. J. Fluid Mech. 164:27-51. Assaf, G . and A . Hecht, 1974: Sea straits: a dynamical model. Deep Sea Res. 21:947958. Baines, P. 1984: A unified description of two-layer flow over topography. J. Fluid  Mech.  146:127-167. Barica, J. 1989:  Unique limnological phenomena affecting water quality of Hamilton  Harbour, Lake Ontario. J. Great Lakes Res. 15:519-530. Bormans, M . and C. Garrett, 1989: The effects of non-rectangular cross section, friction, and barotropic fluctuations. J. Phys. Oceanogr. 19:1543-1557. Dermissis, V . and E . Partheniades, 1984: Interfacial resistance in stratified flow. J. Hydraul. Res. 28:215-233. Dick, T . and J . Marsalek, 1973: Exchange flow between Lake Ontario and Hamilton Harbour. Scientific Series No. 36. Environment Canada Inland Waters Directorate. Farmer, D. and L . Armi, 1986:  Maximal two-layer exchange over a sill and through  the combination of a sill and contraction with barotropic flow. J. Fluid 164:53-76. 83  Mech.  Bibliography  84  Farmer, D . M . and L . Armi / Armi, L . and D . M . Farmer, 1988: The Flow of Atlantic Water Through the Strait of Gibraltar / The Flow of Mediterranean Water through the Strait of Gibraltar. Progress in Oceanography 21:1-105. Greco, S. 1998: Two-Layer Exchange Flow Through thesis, University of British Columbia.  the Burlington Ship Canal. Master's  Grubert, J. 1990: Interfacial mixing in estuaries and fjords. J. Hydraul. Eng. 116:176195.  Frictional Exchange Flow Through a Wide Channel with Application to the Burlington Ship Canal. PhD thesis (draft), University of British Columbia.  Gu, L . 2000:  Gu, L . and G . Lawrence, 2000: Frictional Exchange Flow: A n Analytical Approach. In Lawrence, G . , R. Pieters, and N. Yonemitsu, editors, Stratified Flows, Fifth International Symposium on Stratified Flows, Volume I, pages 543-548. Hamblin, P. and G . Lawrence, 1990: Exchange flows between Hamilton Harbour and Lake Ontario. In Proc. 1st Biennial Environmental Specialty Conference, C S C E , pages 140-148. Helfrich, K . R. 1995: Time-Dependent Two-Layer Hydraulic Exchange Flows. J. Phys. Oceanogr. 25:359-373. Henderson, F . M . 1966:  Open Channel Flow. MacMillan.  Houghton, D. and E . Isaacson, 1970: Mountain winds. Stud, in Num. Analysis 2:21-52. Lawrence, G . 1985:  The Hydraulics and Mixing of Two-Layer Flow over an Obstacle.  PhD thesis, University of California, Berkeley. Lawrence, G . , F . Browand, and L . Redekopp, 1991: The stability of a sheared density interface.  Physics of Fluids 3:2360-2370.  Long, R. R. 1954: Some aspects of the flow of stratified fluids II: Experiments with two fluids.  Tellus 6:97-115.  Long, R. R. 1970: Blocking effects in flow over obstacles.  Tellus 22:471-480.  Long, R. R. 1974: Some experimental observations of upstream disturbances in a twofluid system.  Tellus 26:313-317.  85  Bibliography  Maderich, V . S. and V . O. Efroimson, 1990: Theory for Water Exchange across a Strait. Oceanology 30:415-420. Orlanski, I. 1976: A Simple Boundary Condition for Unbounded Hyperbolic Flows. J. Comput. Phys. 21:251-269. Ottesen-Hansen, N. and J. Moeller, 1990: Zero blocking solution for the Great Belt Link. In Pratt, L . , editor, The Physical Oceanography of Sea Straits, pages 153170. Kluwer Academic. Oguz, T . , E . Ozsoy, M . Latif, H. Sur, and U. Unliiata, 1986: Modeling of hydraulically controlled exchange flow in the Bosphorus Strait. J. Phys. Oceanogr. 16:19701980. Oguz, T . and H . Sur, 1989: A two-layer model of water exchange through the Dardanelles Strait. Oceanol. Acta 12:23-31. Pietrzak, J . D. 1995: A Comparison of Advection Schemes for Ocean Modelling. Scientific Report 95-8. Danish Meteorological Institute. Pratt, L . 1976: Hydraulic control of sill flow with bottom friction. J. Phys. 21:251-269. Press, W . , B. Flannery, S. Teukolsky, and W. Vetterling, 1986: Cambridge University Press.  Numerical  Oceanog.  Recipes.  Schijf, J. and J. Schonfeld, 1953: Theoretical Considerations on the Motion of Salt and Fresh Water. In Proc. of the Minn. Int. Hydraulics Conv., Joint meeting IAHR and Hyd. Div., A S C E , pages 321-333. Spigel, R. 1989:  Some aspects of the physical limnology of Hamilton  Harbour.  Environ.  Can. NWRI Contribution No. 89-08. Stommel, H . and G . Farmer, 1953: Control of salinity in an estuary by a transition. J. Mar. Res. 12:13-20. Tedford, E . 1999:  Exchange Flow Through the Burlington  Ship Canal.  Master's thesis,  University of British Columbia. White, F . M . 1991:  Viscous fluid flow. McGraw-Hill.  Wood, I. R. 1968: Selective withdrawal from a stably stratified fluid. J. Fluid 32:209-223.  Mech.  Bibliography  86  Wood, I. R. 1970: A lock exchange flow. J. Fluid Mech. 42:671-687. Zhu, D. Z. and G. Lawrence, 2000: Hydraulics of Exchange Flows. J. Hydraul. Eng. 126:921-928. Zhu, Z. 1996: Exchange Flow Through a Channel with an Underwater Sill. PhD thesis,  University of British Columbia.  

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