Mean flow structure in meanders of the Squamish River, British Columbia
EDWARD
J. HICKIN
Department of Geography, Simon Fraser University, Burnaby, B.C., Canada V5A IS6
Received May 16, 1978
Revision accepted July 6, 1978
The primary velocity field and pattern of secondary flow are described for nine consecutive
bends of the Squamish River in southwest British Columbia.
The velocity field largely can be explained in terms of variation in channel form, advective
acceleration responses, and water transfers by secondary flow.
The pattern of secondary flow accords with the general model of spiral flow in meanders.
Divergences from this ideal pattern can be explained by bend-flow interaction induced by the
variable planform geometry of the channel.
The strength of secondary circulation increases rapidly as the ratio of the radius of bend
curvature to channel width (r,/w) declines from 4.0 to the data minimum of 1.41. There is no
discontinuity phenomenon in the flow structure over the measured range of r,/w; the Bagnold
sevaration-collavse model does not apply
- . - to the Squamish River.
As r,/w declines to values less than 3.0, the maximum velocity filament shifts from the concave
to the convex bank zone. The resulting high shear stresses over the point bar and declining shear
stresses at the concave bank markedly reduce the channel migration rate.
Separation zones developed at the concave bank of tightly curved bends provide the mechanism for completely halting (and indeed reversing) the process of channel migration.
On decrit le champ d e vitesse primaire et laconfiguration de I'ecoulement secondaire pour neuf
meandres de la riviere Squamish dans le sud-ouest de la Colombie-Britannique.
Le champ d e vitesse peut s'expliquer surtout en termes d e la variation dans la forme du chenal,
des reponses advectives i I'acceleration et des transferts d'eau par ecoulement secondaire.
L a configuration de I'ecoulement secondaire est compatible avec le modele general d'ecoulement en spirale dans les meandres. On peut expliquer les divergences a ce patron ideal par
l'interaction meandre-ecoulement causee par la variation d e la geometrie en plan du chenal.
L'amplitude de la circulation secondaire s'accroit rapidement quand le rapport entre le rayon
d e courbure du meandre et la largeur du chenal (r,/w) diminue de 4.0 jusqu'a 1.41, la valeur
minimum mesuree. I1 n'y a pas de phenomene de discontinuite dans la structure de I'ecoulement
dans I'intervalle de r,/w mesure; le modele de separation-effondrement de Bagnold ne s'applique
pas a la riviere Squamish.
Lorsque r,/w diminue jusqu'a des valeurs plus faibles que 3.0, le fil d e vitesse maximum se
deplace de la rive concave i la rive convexe. Les contraintes de cisaillement elevees sur le
bourrelet arque et la diminution des contraintes de cisaillement sur la rive concave reduisent d e
faqon marquee le taux de migration du chenal.
Les zones de separation qui se developpent du c8te concave des meandres tres courbes
fournissent un mecanisme capable d'arrtter (ou mtme de renverser) le processus de migration du
chenal.
[Traduit par le journal]
Can. J . Earth Sci., 15, 1833-1849 (1978)
Introduction
In several previous papers concerned with the
lateral stability of river channels, I have offered
speculative comment on the hydraulic control of
lateral migration (Hickin 1974, 1977). Field
studies on the Beatton River in northeastern British
Columbia (Hickin and Nanson 1975; Nanson 1977)
indicate that there, the migration rate N at the axis
of a channel bend varies with channel curvature
(bend radiuslchannel width, rm/w) in the manner
shown in Fig. 1. Although the increasing rate of
migration as r,lw declines is an expected response
to the increasing inertial force applied to the concave bank, the reason for the rapid decline in N for
r,lw < 3.0 is not obvious.
My earlier tentative explanation of this phenomenon (Hickin 1974) relied on Bagnold's (1960)
idea that, during bend development, a separation
zone forms at the convex bank and subsequently
collapses at rm/w = 2.0, disrupting the entire flow
pattern. Observations on two tightly curved bends
on the Beatton River, however, confirmed neither
the presence nor collapse of flow separation at the
convex bank (Hickin 1977).
A search of the literature reveals a surprising
paucity of information on flow structures in river
bends. Although numerous experimental studies
have addressed problems relating to flow processes
in open channel bends (e.g., Mockmore 1943;
Rozovskii 1961; Ippen and Drinker 1962; Francis
1834
CAN. J. EARTH SCI. VOL. IS, 1978
Unfortunately, no reliable quantitative data exist
for sediment-load characteristics. Nevertheless,
studies of the sediments of the Squamish River
delta (see Hoos and Vold (1975) for a review), together with observations made during the present
survey, indicate that the test reach is essentially a
gravel-bed channel through which large amounts of
sand and silt move in suspension. Point bars consist
of broad gravel aprons of about 5-50 mm diameter
gravel capped with a thin (about 30 cm) veneer of
0
sand (mean grain diameter: 0.25 mm). These
10
20
:I0
40
50
60
70
boundary materials are considerably coarser than
Radius of channel curvatureichannd width
those of rivers examined in most other studies conFIG.1 . The relation between migration rate and channel curvature ratio for 10 bends o f the Beatton River, British Columbia cerned with flow structure in channel bends.
(after Hickin and Nanson 1975).
Spectral analysis of bedforms (Rood 1978) in the
test reach reveal that the pool and riffle sequence is
and Asfari 1971) far fewer have dealt with natural the major source of bed deformation; higher frechannels. Furthermore, the field studies are gener- quency bedforms such as dunes and waves, alally confined to detailed examination of a small though present, are relatively small. There are also
isolated river bend, disregarding bend-flow inter- minor perturbations associated with a few ranaction (e.g., Apmann 1'972; Hey and Thorne 1975; domly placed boulders and logs.
Bridge and Jarvis 1976).
Two important characteristics recommend the
Studies of this type in larger rivers, where the test reach for the present type of study. First, besubtleties of flow structure are amplified to more cause it is a proglacial river, the discharge was
easily measurable scales, are virtually nonexistent relatively constant throughout the survey; flow(work on the Wabash River by Jackson (1975a ,b, structure variation due to discharge fluctuation is
1976) provides the only recent exception of which I thus minimized. Second, the test reach consists of a
am aware). The reason for the lack of studies of this series of nine channel bends ranging from several of
latter type is probably the fact that current meters very tight curvature (rm/w < 2.0) through an interwith the facility to measure accurately both veloc- mediate range to several open bends (rm/w > 8.0);
ity and flow direction have only been commercially thus there is an opportunity to examine the
available in the last few years.
influence on flow structure of a wide range of bend
Thus the aims of this paper are to describe the curvature.
mean flow characteristics through a continuous
Field Survey Procedures
series of river meanders and to identify likely hydraulic controls on lateral channel migration.
Twenty-nine cross sections were located at
A 5 km reach of the Squamish River was selected about 180 m intervals along the test reach. In genfor study (see Fig. 2) and all flow measurements eral these corresponded with bend axes, bend
were obtained in the period June-August 1977. The inflection points, and with a series of intermediate
river, located about 45 km north of Vancouver, locations (see Fig. 5). At each cross section a cable
British Columbia, drains about 3600 km2 of the was stretched across the river to serve as an anchor
Coast Mountains.
line for the survey boat. The position of the boat on
The long-term (20 years) mean-monthly flow of the river cross section was determined with the use
the Squamish River, recorded at Brackendale, is of an optical range finder and a theodolite. Cross245 m31s, with the maximum usually occurring in sectional form and longitudinal bed profiles were
June (770 m31s) and the minimum usually in March determined from a combination of line soundings
(70 m3/s). Runoff during the summer months is and echo-depth profiles recorded by a Raytheon
supplied largely by glacial meltwater and the DE 719B fathometer (accurate to +0.05% 2
Squamish River maintains a high (425 m31s) and 0.025 m of the indicated depth). The longitudinal
regular discharge during this period.
water-surface profile was constructed by averaging
During the present survey, the following average left and right bank water-surface elevations, meahydraulic conditions characterized the 5 km test sured with a theodolite, along the test reach.
reach: discharge, 402 m'ls; flow velocity, 1.14 mls;
Velocity and current-direction measurements
water-surface slope, 0.0016; water-surface width, were obtained at horizontal intervals of approxi106 m; and mean flow depth, 3.54 m.
mately w/10 m and at vertical intervals of 0.5 or
1835
HICKIN
BRITISH COLUMBIA
OOeoOoO
The study reach
Squamish River watershed
CAN. J. EARTH SCI. VOL. 15, 1978
\
\
-- -----
Channel axis
Bend axis
0
Inflection point
X
Bend apex
FIG.3. Definition of channel geometry parameters.
1.0 m for flow depths less or greater than 5.0 m,
Methods of Analysis
Several derived parameters are discussed in the
respectively. Thus for each cross section an average of about 90 point measurements were obtained. analvsis to follow: channel curvature. downstream
These were recorded by a cable-suspended model and cross components of velocity, indices of secDNC-3 (N.B.A. Controls Ltd.) direct-reading cur- ondary circulation strength and direction, mean
rent meter. Electrical pulses from this meter are shear stress at the boundarv. and flow resistance.
The channel curvature index is the ratio of the
converted to an analogue signal, which is then
smoothed with a 3 s averaging constant for display radius of channel curvature (r,) to channel width
at the surface meter; accuracy is &2%. Flow direc- (w) as defined in Fig. 3. Channel width is defined as
tion is determined by a compass with an optical water-surface width at the inflection points of river
encoding disc and is sampled approximately once bends; here width measurement is not complicated
every second. The refading is continually updated by the presence of separated flow, which often
and transmitted as an analogue output from the occurs elsewhere in the channel bends.
current meter to the surface display unit; accuracy
Flow velocity has been resolved into two comis +5". Velocity and direction measurements were ponents: that in the mean downstream flow direcrecorded for the present study as averages for a 30 s tion (Vz) and that for the orthogonal cross compoperiod. An analysis of the continuous output indi- nent (Vu). The mean downstream flow direction
cates that the velocity means for 20 consecutive was determined by averaging all the direction mea30s periods of a typical record range from a surements for each cross section. Although considminimum of 1.38m/s to a maximum of 1.56 mls, or eration of secondary circulation in the present con&6% about the stable 10min mean velocity. Thus text ideally should be based on channel bank orienthe mean velocity based on the 30 s period averages tation, this reference direction and mean flow direction are sensibly identical in most cases. Furtherout most of the eddy fluctuations in the flow.
d
,
HICKIN
more, the use of mean flow direction overcomes the
difficult problem of determining bank orientation at
the axis of channel bends and at sites with irregular
bank alignment.
The strength of secondary circulation at each
cross section is represented by two indices. The
first is the dimensionless mean cross velocity, M ,
defined as
n
M
=
C l vyilvxilln
i=1
where n is the number of equally spaced point measurements (adapted from Yen 1967, 1972). The
second index, M * , designed to reflect the sense of
circulation, is defined as
N
M*
=
C (Vysii= 1
Vy,,)lNvx
Here VIJsand
are, respectively, the cross velocities at the water surface and at the channel
boundary for a given vertical velocity profile, and
N is the number of vertical profiles in the cross
section. The cross velocities were taken as those
indicated by the regression line through the Vu
profile at each vertical. Thus M* is a sLlmmati0n of
local circulation standardized on the mean flow
velocity for each cross Section. Positive values of
M* indicate clockwise flow circulation with respect
to the downstream primary current direction and
negative values indicate the reverse or counterclockwise sense of circulation.
IVIean shear stress at the channel boundary (Yz)
was estimated as the mean depth-slope product
( ~ g y s ) This
.
expedient is necessary because it is
not possible to specify local boundary shear stress
from the vertical velocity distribution in any meaningful way because the logarithmic velocity law of
the wall does not seem to apply to much of the flow.
In fact, in some cases there is evidence that the first
25 cm above the bed displays wake flow. Nevertheless, in general the average velocity gradient in this
25 cm boundary zone is likely to be proportional to
the boundary stress and the velocity at 25 cm above
the bed is probably a surrogate n~easureof shear
stress.
Resistance to flow, expressed in terms of the
D'Arcy Weisbach resistance coefficient, ff, has
been derived from the mean shear-stress values (ff
= 8y,/VZ2).
Field Observations
Flow Velocity
Vertical distributions of downstream velocity in
the Squamish River, examples of which are shown
in Fig. 4, rarely conform to the normal logarithmic
1837
law. They are characterized by very low velocity
gradients throughout much of the flow and most
display markedly irregular profiles. The low velocity gradients result from the intense mixing brought
about by the dense pattern of eddies in the flow.
Eddies generated in the highly sheared flows very
close to the boundary are rapidly advected to all
parts of the flow and produce a dense pattern of
boils at the water surface.
Local reductions in flow velocity, such as those
appearing about 2.5 m from the water surface in
profiles 12 and 14 in Fig. 4, are common and appear
to be persistent. It seems likely that this type of
mean velocity anomaly is produced when the current meter is located asymmetrically within a vertical vortex attached to a fixed boundary element
such as a boulder or bar. Alternatively it could be
similarly produced by intermittent eddy generation
of sufficiently high frequency that it appears as a
persistent structure in the mean velocity pattern.
Profile 15 in Fig. 4 displays a velocity reduction
for the first 3 m above the bed. This type of negative
velocity gradient is not uncommon along the test
reach and is the result of advective and very
strong vortex-flow interaction.
A three-dimensional picture of the channel form
and the maximum velocity field is shown in Fig. 5.
~ l t h the
~ isovel
~ ~ h
pattern is somewhat irregular
the persistent eddy effect, there is a
because
general tendency for flow velocity to decrease from
the water surface to the boundary. Velocities close
to most of the boundary (0.25 m above the bed),
however, generally remain relatively high (0.80l.Oomls).
The velocity distribution at station 1 (not shown
in Fig. 5) is symmetrical about the channel axis. By
the time the flow reaches station 2, however, the
velocity distribution has become distinctly asymmetrical with the highest velocities occurring near
the convex bank. The asymmetry persists through
station 3 after which the flow divides around a
central bar in the channel.
Similar but less well-developed velocity asymmetry of this type is present in other channel bends
in the test reach (see stations 7,15, and 25 in Fig. 5).
It is a velocity pattern that also has been observed
elsewhere in both flume and field studies of flow
through open channel bends (see Mockmore 1943;
Einstein and Harder 1954; Ippen and Drinker 1962;
Hooke 1975; Bridge and Jarvis 1976). Nevertheless, the generally accepted model of velocity distribution in channel bends indicates that, at the
entrance to channel bends, the filament of maximum downstream velocity is near the convex bank
and is moved across to the concave bank by the
developing secondary circulation induced by radial
1838
CAN. J. EARTH SCI. VOL. 15, 1978
Map scale
1840
CAN. J. EARTH SCI. VOL. IS, 1978
1
-
...
- 3
+
..
Observed veloc~ty
distr~bution
Veloc~tydistribution
based on contlnultv
Velocity distribution
based on contlnutty and
secondary c~rculat~on
CONCAVE
BANK
\
I
\
I
\
I
.--1
10
20
30
50
40
I
I
60
70
80
90
:L\O
Channel width, metres
FIG.6. Observed and computed lateral velocity distributions at two stations on the Squamish River.
acceleration (see Einstein and Harder 1954;
Rozovskii 1961; Ippen and Drinker 1962; Fox and
Ball 1968; Francis and Asfari 1971; Jackson
1975a,b). Although this flow transition is usually
interpreted as progressive suppression of the velocity distribution expected in an irrotational vortex,
Bridge and Jarvis (1976)reject this ideal-fluid model
as inappropriate for natural channel bends. They
argue that, if relatively high velocities occur near
the convex banks of river bends, they are most
likely a remnant of the velocity field in the adjacent
upstream bend.
The rigid assumptions of the irrotational-vortex
model are certainly violated for the flow at station
2. Indeed, in this channel bend there is a very large
separation zone at the concave bank and secondary
circulation in the main channel is here more intense
than in any other section of the survey reach. But it
is also the case, as I noted earlier, that the velocity
distribution in the entrance to this bend (station 1 ) is
sensibly symmetrical. Furthermore, conditions at
the entrance to bends on which stations 7, 15, and
25 are located are similarly free of any obvious
'memory effects' evident in the isovel patterns
(note the symmetry of the velocity fields at stations
5, 13, and 24).
It seems more likely that the velocity distributions in the Squamish River bends are simply the
result of advective acceleration that has been
modified by the pattern of secondary circulation.
As flow moves from the bend entrance around the
convex bank it encounters the shoaling zone of the
point bar. If there is no net loss of water by secondary flow, then the mean velocity must increase
locally to maintain flow continuity. Conversely, the
mean velocity of flow at the concave bank must
decrease as it moves into the pool zone of the bend.
In Fig. 6 examples of this effect are shown for
stations 2 and 15; curves 1 represent the observed
mean velocity pattern across the channel width and
curves 2 are those predicted from a stream-line
analysis. At the bend entrance (stations 1 and 13)
the channel cross section was divided into five cells
defined by the channel banks and four verticals
located across the channel width. Five cells were
similarly defined for the cross sections at the axes
of the two bends (stations 2 and 15). For the purpose of the analysis it was assumed that the left
'
I
bank cell at the entrance (cell 1E) and the left bank
cell at the channel axis (cell 1A) were the end sections ofa single continuous flow filament 1. Cells 2E
and 2A through cells 5E and 5A were assumed to
similarly represent the end sections of separate
flow filaments 2-5. In other words, the river was
viewed as five sepal-ate flow filaments along which
flow continuity must be satisfied. Mean velocity for
cell 1A was calculated by dividing the discharge
through cell 1E by the cross-sectional area of cell
lA, and so on. In this phase of the analysis, all
factors other than continuity requirements were
ignored.
Although the predicted velocity distribution (see
curve 2 in Fig. 6) is similar to the observed distribution with respect to skewness direction, it is markedly exaggerated. It should not be surprising,
however, that the predicted velocity distributions
are considerably more skewed than their observed
counterparts because in reality there is a net transfer of fluid from the convex to concave banks by
secondary flow.
Although the pattern of secondary flow between
each pair of stations is not known in detail, the net
transfer of fluid among the various flow filaments
can be approximated by a linear model based on
conditions at the entrance and axis of each bend
(see Fig. 8). For example, in cell 1E it is assumed
that there is no net fluid transfer through the left
side of the cell formed by the channel bank. The net
transfer of fluid from the right side of cell 1E is the
product of the mean transverse velocity for the
bounding vertical, VulEp,and the flow depth, YIER.
A similar two-dimensional transverse discharge
can
- be calculated for the corresponding cell 1A as
YIAR.The mean two-dimensional transverse
V
dw-%arge from the entire right side of filament 1 can
be taken as the average of the two cell discharges. If
it is assumed that all flow filaments move at the
mean velocity for the whole reach in question, v,,
then the total volume of fluid lost or gained from a
unit length of filament 1 as it moves from the bend
entrance a distance L1 to the bend axis is ( L , / ~ , )
(VUlERYlER
+ VulARYlAR)/2.
This volume of fluid is
also the corresponding amount lost or gained by the
left side of filament 2. A similar calculation for the
right side of filament 2 allows the determination of
the net transfer of water to that filament, and so on.
Mean velocities through the cells at the bend axis,
calculated on the basis of continuity and net fluid
transfers, yield the horizontal velocity distributions
described by the curves 3 in Fig. 6. These curves
clearly conform much more closely to the observed
distributions than do curves 2.
The discussion above is not a statement of a
formal model. It is merely intended to demonstrate
that it is neither necessary to have velocity asymmetry at the entrance to bends nor is it necessary to
conserve angular momzntum in order to develop
bend flows in which the maximum downstream
velocity filament is skewed towards the convex
bank (see Yen (1967) for a comparison of lateral
velocity distributions in trapezoidal and selfformed channels).
An important implication of this discussion is
that the location of fully developed bend flows (see
Jackson 1975a,b) in natural channels, unlike that in
experimental studies of trapezoidal channels (e.g.,
Mockmore 1943), will depend to a large degree on
the rate of change in channel form through the bend
(see also Hooke 1975).
Shear Stress and Flow Resistance
Downstream variation in mean values of boundary shear stress and the D'Arcy Weisbach resistance coefficient are shown in Fig. 7; station numbers correspond with the axes of the eight bends in
the reach. There is clearly a correlation between
channel planform and both shear stress and flow
resistance: channel bends are associated with relatively high shear stress and flow resistance and the
intermediate inflection points have considerably
lower values of these parameters. Although flow
depth increases (and relative roughness declines) in
the bends, energy dissipation through secondary
circulation and increased shear rate (internal distortion resistance; see Leopold et al. (1960)) results
in a net increase in the resistance to flow. These
relationships are better defined for the first four
bends of relatively tight curvature than they are for
the remaining more open bends in which internal
distortion resistance is probably negligible.
Shear stress and flow resistance at station 11 are
anomalously high because the channel there is 30%
contracted by an alluvial fan deposited into the
channel by a tributary stream. Comparison of flow
conditions at the bend axis and at immediately adjacent sections (for the bracketing bend stations 7
and 15) indicates that the ratios of shear stress and
flow resistance at the bend axis to the corresponding values at the adjacent sections are, respectively, 1.55 and 1.38. Application of these ratios to
the average values of shear stress and flow resistance at stations 10 and 12, uncontracted sections
immediately adjacent to station 11, yields more
realistic values of y, = 39.0 N/mZandff= 0.25.
Secondary Circulation
The pattern of flow circulation in the survey
reach is indicated by the selected cross sections and
cross velocity profiles shown in Fig. 8 and by the
downstream variation in circulation strength and
direction shown in Fig. 7.
lo-
- 05
0"-
I
0
05 -
z
*
2
1
1
1
1
1500
1
11
,
,
I
2000
,
Distance, rn
,
15
I
2500
1
,
18
I
3000
I
n
1
3500
I
I
23
4000
I
I
I
25
FIG.7. Downstream variation in flow resistance, shear stress, and secondary circulation along the study reach.
I
1000
1
7
500
I
I
45ar
I
27
I
5000
'I
I
29 Station
*
--
#
Zone of amiclockwise
1844
CAN. J. EARTH SCI. VOL. 15, 1978
At station 1 there exist two cells of circulation,
one near the left bank (with respect to primary flow
direction) representing the last remnant of the spiral flow developed in the upstream bend, and a
larger and stronger cell at the right bank representing the initial develop.ment of a reverse spiral in the
downstream bend. The right-bank cell at station 1 is
rapidly enlarged to fill the complete channel section
at station 2; clockwi:se circulation is very strong
here with bed velocities of more than 0.5 mls and
surface velocities as high as 1.O m/s. Circulation
strength at station 2 is in fact higher than at any
other station in the survey reach (see Fig. 7).
The intense pattern of circulation at station 2
rapidly decays downstream so that by station 3 the
cell is only a little stronger than it was at station 1; a
short distance beyond station 3 the flow circulation
is completely disrupted where the flow divides
around a mid-channel bar.
Downstream from 1.hisbar, in the relatively wide
and shallow channel at stations 4 and 5, there appear to be at least three cells of circulation related
to the confused patteirn of converging flow. At the
bed near the left bank at station 5 there is a local
counterclockwise circ:ulation pattern probably representing the incipient spiral-flow filament, which
enlarges to almost completely fill the channel at
stations 7 and 8. This spiral decays through station
9 and is completely displaced by a developing
clockwise spiral in the next downstream channel
bend. This new filament of spiral flow is best developed at station 11 (the bend axis) and in turn
decays through stations 12 and 13.
A weak counterclockwise spiral begins to develop at the left-bank approach to the short and
rather open bend at s,tation 14. It reaches a maximum, but nevertheless relatively low, intensity at
the bend axis (station 15) and decays through station 16 and is completely dissipated by station 17.
The rapidly strengthening clockwise spiral flow associated with the next bend appears to be present in
incipient form as far upstream as station 16. It is
well developed by station 18, occupying almost
the complete channel. The clockwise spiral is intensified by flow convergence (maximum downstream velocities here reach 2.0 m/s) to a maximum
at station 19. Normally this clockwise spiral would
decay and be replaced by reverse circulation in the
next downstream bend. The bend at station 21,
however, has an open curvature and the intense
clockwise circulation at the entrance persists
through the bend; the weakening spiral is subsequently reinforced as it comes into the influence
of radial acceleration induced by the downstream
bend at station 23.
The secondary circulation beyond station 23 appears to resume the more normal pattern of inphase alternation of bends and flow helices.
There is a tendency for the intensity of spiral flow
to decline along the study reach (see Fig. 7) reflecting the decline in the channel bend curvature with
distance downstream.
The pattern of secondary circulation in the
Squamish River accords well with the general
model of spiral flow in meanders (see Rozovskii
1961; Yen 1967; Wilson 1973). Divergences from
this ideal pattern of circulation with its single cell at
bend axes and overlapping double vortices at
inflection points can be explained by the variable
planform geometry. Local shear stress variation
around bars in wide shallow sections of channel can
produce complex multi-celled circulation, which
can dominate the general pattern (stations 3-5);
long relatively straight bend inflections may result
in complete vortex decay and no spiral overlap;
intense spiral flow generated in tightly curved
bends can dominate the normal reverse circulation
in more open bends immediately downstream (stations 19-25). This latter persistence effect, well
illustrated in Fig. 7, demonstrates the importance
of bend-flow interaction to an understanding of the
behaviour of individual channel bends.
Discussion of Flow Conditions in Relation
to Channel Curvature and Migration Rates
Changes in flow resistance, shear stress, bed velocities, and the strength of secondary circulation,
as the degree of channel curvature increases, are
indicated in Fig. 9.
There appears to be very little change in flow
resistance and shear stress as the degree of channel
curvature increases in the range 4.0 < r,/w < 9.0.
As rm/w declines below 4.0, both parameters display marked increases to maxima at the data limit of
rm/w = 1.41. At this curvature ratio the rates of
increase in resistance and in shear stress also appear to reach maxima. These trends, consistent
with observations made elsewhere (see Leopold et
al. 1960; Onishi et al. 1972), are the result of the
rapidly increasing internal distortion resistance as
the bend curvature ratio declines below 4.0.
It is also clear from Fig. 9 that, in the tighter
bends, there must be a considerable increase in the
rate of energy expenditure associated with the
rapid strengthening of the secondary flow system.
Flow circulation is weak and changes little as the
curvature ratio declines from 9.0 to 4.0. For the
range 1.40 < rm/w < 4.0, however, the strength of
secondary flow displays an exponential increase.
The abrupt reduction in the migration rate along
,
1
Station number
2
7
1
I
11
23 15
I
I
I
21
19
I
I
27
1 25
t
ff
0.70.60 5-
I
\
\
\
8'\,
0.4-
0 3-
02010-
0
\
'
\
ax.,
..-.
'
-.
.-.
--
r,. ~ / r n ~
80
9--
.-----------,-----J----------*
8
ff
60
50
......
-....-----a-
40
30
0
@
a
--__
--o----- ', -------- ----------
'---
A = convex bankzone
B = concave bank zone
20
4
FIG.9. Variation in flow resistance, shear stress, boundary velocities, and secondary circulation strength with the degree
channel curvature in the study reach.
1846
CAN. J. EARTH SCI. VOL. 1.5, 1978
the axial direction of a bend that has developed a
curvature ratio of 2.0 (see Fig. 1) does not appear to
be the result of changes in mean shear stress, flow
resistance, nor in the intensity of secondary circulation. Mean shear stress and flow resistance do
increase as bend curvature tightens, but the change
occurs gradually over a wide range of rm/w . If the
rate of channel migration were primarily limited by
sediment supply from secondary circulation, then
maximum migration irates should correspond with
bends in which rm/w is a minimum (i.e., <2.0); in
fact, the reverse correspondence is true.
It is also clearly the case that flow resistance
variation of the type suggested by Bagnold (1960)
does not apply to 'the Squamish River bends.
Furthermore, the general flow structure remains
intact through the complete range of bend curvatures; there is no flow disruption associated with
unstable separation zones.
Channel bend migration generally can only continue if there is erosiion at the concave bank and
concomitant deposition on the point bar of the convex bank. Because bank slumping is probably the
most important single: process of concave bank retreat, flow character at the base of this bank is
probably very important in conditioning it for erosion.
Velocities near the boundary of the convex and
concave banks vary with the curvature ratio as
depicted in Fig. 4. Concave-bank velocities are not
strongly related to rm/walthough there does appear
to be a general increase over the range 3.0 < rm/w
< 9.0, and a subsequent decline in velocities for
rmlw < 3.0. The choice of this generalized trend line
in part reflects the fact: that the value of Voat station
19 is known to be unusually high. Here the channel
has been contracted by the infringement of a tributary fan of gravel at the concave bank. Furthermore, the magnitudes of Voat stations 15 and 23 are
very likely depressed because the length of curved
channel is relatively short. That is, secondary flow
does not have sufficient time (or space) to develop
the velocity asymmetry that would normally be
associated with a bend of this curvature ratio.
The relationship between velocities over the
point bar surface at the convex bank and the curvature ratio is more clearly defined. As bend curvature increases, velocities decline to a minimum at
about rmlw = 3.0 and subsequently increase to their
greatest values for the tight bends at stations 2 and
7.
These two comp1e:mentary velocity trends on
either side of the channel boundary are a direct
expression of the variation in the isovel patterns
noted earlier in this paper. Much of the scatter in
these data not associated with channel curvature is
the result of bend-flow interaction.
Because these bank velocities were all measured
at a constant distance (0.25 m) above the boundary,
it also follows that, if the logarithmic velocity law is
applicable (in an average sense) and if the von
Karman constant is indeed constant, then the bank
velocities squared are proportional to the local
boundary shear stress. Thus the bank velocity
trends in Fig. 9 suggest a still more pronounced
pattern of shear-stress variation as the curvature
ratio declines. Shear stress at the convex bank,
where rm/w < 2.0 (stations 2 and 7), is almost twice
as great as that in the range 5.0 < rm/w < 9.0 and
more than 30 times the minimum at rm/w = 3.0.
Similarly, shear stress at the concave bank where
rm/w < 2.0 is only about one quarter of that for the
range 5.0 < rm/w < 9.0 and less than 7% of the
maximum shear stress at rm/w = 3.0.
The observed increase in velocity at the convex
bank and the decrease at the concave bank (and the
implied changes in shear stress) in bends with a
curvature ratio less than 3.0 accord with findings in
similar conditions elsewhere (see Hooke 1975;
Jackson 19756; Hickin 1977). It seems likely that
these velocity adjustments provide the direct
mechanism for controlling lateral migration rates
and yielding the commonly observed limiting
meander geometry in which rm/w = 2.0.
The present findings differ, however, from those
of Hooke (1975) and of most other experimental
studies in one very important respect: large-scale
flow separation in the Squamish River study reach
is not a feature of the flow structure at the convex
banks of channel bends. It is, however, a feature of
the flow structure at the concave banks of the
tightly curved channel bends (stations 2 and 7).
Concave- rather than convex-bank flow separation
was also found to be a persistent structure in tight
bends of the Beatton River (Hickin 1977).
The large separation zone between stations 1 and
2 is a depositional environment in which a
concave-bank bench (see Woodyer 1975) is forming. Similarly, at station 7 much of the concave
boundary consists of a well-developed concavebank bench. Both of these channel bends are presently undergoing negative migration rates in the
sense that deposition is presently occurring at the
concave banks at a faster rate than at the convex
banks.
The Squamish River flow structure data confirm
the suggestion made elsewhere (Hickin 1977) that
concave-bank flow separation is an important
mechanism of migration control in very tightly
curved channels. It tends not to be observed in
1
,
1847
HICKIN
(A - A')
+
- (B - B
lnlt~atlonstage ( rm/w
> 4 0)
Growth stage ( r m / w 30-40)
(C- c')
Termlnat~onstage ( r m/w
+-&
AXISof the hlgh veloclty fllament
< 3 0)
Zone of maxlmurn veloc~ty
Separat~onzone
FIG.10. A model of changing flow structure in a developing channel bend.
experimental work (with some exceptions, e.g.,
Mockmore 1943; Hooke 1975) because constantwidth channels suppress its development (see
Hickin (1977) for a more detailed discussion).
The present study suggests the ideal model of
flow structure in channel bends depicted in Fig. 10.
Three stages in a continuum of bend development
are represented, corresponding to the initiation,
growth, and termination phases of migration shown
in Fig. 1. The model applies to a developing channel
bend of regular planform in which the flow structure is completely free of bend-flow interaction
effects. Until further data are forthcoming, I must
assume that it will best apply to gravel-bed meandering channels like those of the Squamish and
Beatton Rivers.
In the initiation stage, in which r,/w > 4.0, there
is only a slight asymmetry of the channel cross
section and of the velocity distribution, towards the
concave bank. Secondary circulation is weak and
flow-depth adjustments from the inflection points
to the bend axis are minimal; rates of channel migration are very small.
As the curvature of the bend tightens into the
growth stage, in which r,/w = 3.0 to 4.0, the increased inertial forces are expressed as an asymmetrical velocity distribution, and high shear and
erosion rates at the concave bank. The general
asymmetry of the channel geometry and of the velocity distribution is strengthened by the rapidly
intensifying helical flow. It is likely that in this stage
that bed scour at the concave bank markedly reduces the width-depth ratio of the channel. Velocities near the convex bank are relatively low,
promoting active deposition on the point bar.
The termination stage, in which r,lw < 3.0, is
characterized by further deepening of the channel
near the concave bank, and by movement of the
maximum velocity filament of the flow into the
convex-bank zone. Although helical flow is greatly
strengthened in this stage, it tends to be localized
near the bend axis and does not substantially alter
1848
CAN. J. EARTH SCI. VOL. 15, 1978
the sense of flow asy~mmetryimposed by advecAlthough I believe that the mean flow structure in
tive acceleration. Velocities and shear stresses at the Squamish River is probably representative of
the boundary increase at the convex bank and de- that in most rivers, there is clearly a need to
cline at the opposite bank. In consequence, examine other rivers in order to verify this assumpconcave-bank erosion is reduced while convex- tion.
Acknowledgments
bank erosion increases; channel migration virtually
ceases.
I gratefully acknowledge that this study forms
In tightly curved bends (r,/w < 2.0) further ero- part of a project that is funded by the National
sion of the concave bank leads to channel widening Research Council of Canada.
(high velocities prevent deposition on the point bar)
and a rapidly declining radius of curvature along the APMANN,R. P. 1972. Flow processes in open channel bends.
ASCE Journal of the Hydraulics Division, 98, pp. 795-810.
concave bank near the bend axis. These conditions BAGNOLD,
R. A. 1960. Some aspects of river meanders. United
promote concave-bank; flow separation and a comStates, Geological Survey, Professional Paper 282-E, 10 p.
plete cessation (or in some cases, reversal) of chan- BRIDGE,J. S., and JARVIS,J. 1976. Flow and sedimentary processes in the meandering river South Esk, Glen Clova, Scotnel migration.
Summary ,and Conclusions
The principal conclusions drawn from this study
may be summarized as follows:
1. The velocity field through the bends of the
Squamish River displays patterns that can largely
be explained in terms of variation in channel form,
advective acceleration responses, and water
transfers by secondarly circulation. It is therefore
not necessary to assurne free vortex flow, nor is it
necessary to assume inherited flow asymmetry at
bend entrances, in order to account for relatively
high velocities over the point bars of tightly curved
bends.
2. The pattern of secondary flow in the Squamish
River accords with the general model of spiral flow
in meanders. Divergences from this ideal pattern of
circulation with its single helix at bend axes and
overlapping double vortices at inflection points can
be explained by the variable planform geometry of
the channel.
3. The strength of secondary circulation increases rapidly as r,/w declines from 4.0 to the data
minimum of 1.41. There is no discontinuity phenomenon in the flow structure as r,/w declines; the
Bagnold separation-collapse model is inappropriate to conditions in the Squamish River.
4. As r,/w declines to values less than 3.0, the
maximum velocity filament shifts from the concave to the convex bank. The resulting high shear
stresses over the point bar prevent further deposition and declining shear stresses at the concave
bank reduce the rate of erosion there; channel migration becomes negligible.
5. Separation zones;, developed at the concave
bank of very tightly curved bends of the Squamish
River, are depositional environments during nearbankfull discharge and provide the mechanism for
completely halting (and indeed reversing) the process of channel migration.
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