The unsteady model is first used to solve for the steady solution with zero barotropic forcing .... 5.1 Map of Hamilton Harbour and Burlington Ship Canal. 62.
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
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 = 0.637 for x < 1 and = 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.
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
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