Measurements of the rate of adj.ustment of the secondary flow in a curved open channel with varying discharge.
R. Booij and Joh.G.S. Pennekamp
Report no. 15-84
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Laboratory of Fluid MechanicsDepartment of Civil EngineeringI
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Measurements of the rate of adjustment
of the secondary flow in a curved open
channel with varying dischárge.
R. Booij and Joh.G.S.Pennekamp
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Report no. 15-84Laboratory of Fluid Mechanics
Department of Civil Engineering
Delft University of Technology
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2 Contents List of figures~
~~~~cim~nt~~_á~t=~~_~nQ_~CQç~áÉiQg_Qf_tn~_m~~§~c~m~QtÉ
3.1 Flume and discharge
3.2 The measurements
3.3 Processing of the measurements
~
~~~g~imgnt~~_~g§~~t§
4.1 Secondary flow profiles
4.2 Adjustment of the secondary flow to a time-varying main flow
4.3 Relaxation length References Notation Figures page 2 4 6 9 9 10 11 14 14 15 16 17 18 19 20
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3b!§t_Q!_fig!:!r:~§
1. Definition sketch 2. 3. 4. 5. 6.Distribution functions of the secondary flow velocity for various chézy coefflcients
The Laboratory of Fluid Mechanics flume The Laser Doppier Velocimeter
Smoothing of the data
a. Example of velocity measurements
b. The ensemble average of three consecutive periods c. Smoothed velocity measurements
Flow diagram of datp processing 7. Depth averaging methods
8. Calculation of the velocity profile of the secondary flow a. Main and secondary flow velocities
b. Smoothed water level mea.surements c. Velocity profiles of the ma-in flow
d. Velocity profiles of the secondary flow
e. Normalized velocity profiles of the main flow
f. Normalized velocity profiles of the secondary flow
9. Comparison of secondary flow profiles with measurements of de Vriend in another curved flume
10. Second secondary flow cell
11. Time dependence of secondary flow velocity profiles Overview of the whole field
Profiles along the axis
Profiles along the outer wall
Normalized secondary flow intensities averaged over the cental part of the flume
Table of normalized secondary flow intensities throughout the field (measured).
a. b. c. 12 a. 12 b.
...
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Prediction of bottom changes in rivers and tidal channels
require a thorough knowledge of the secondary flow, as this flow
gives rise to bottom slopes transverse to the main flow
direction. In the model used in this research, the cOfflPutation
of the secondary flow field is based upon an, independently
computed, depth averaged main flow field.
In the computation of the secondary flow it can of ten be
assumed fully developed, i.e. adjusted to the local variables:
depth, main flow velocity, etc. In case of variation of bottom
topography or of main flow properties in the flow direction, the
secondary flow will not be adjusted to these local variables, but
will retain information about the situation considered parcels
of water have met earl ier during their flow. An analogous
misadjustment will occur in case of time dependent main flow
properties •
.A theoretical investigation about the degree of adjustment
of secondary flow was reported by Booij and Kalkwijk (1982).
They describe the rate of adjustment by a relaxation length in
case of variation in the flow direction and by a relaxation time
in case of variation in time. Both are defined by the reduction
of the misadjustment to a factor l/e of the original
misadjustment with respect to the local variables. For
the value of the Ch~zy coefficient of 50 m1/2/s, the
relaxation length L of the secondary flow was found to be ab out
15 times the water depth hand the relaxation time T was found to
be about the time needed by the dep th averaged flow to cover a
distance equal to this relaxation length.
Comparison with the adjustment of the secondary flow as
measured in curved flumes showed a satisfactory agreement with
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the theoretically obtained values (de Vriend, 1981). Noexperimental affirmation of the theoretical relaxation time in
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case of fluctuating main flow was, however, present. To obtain
experimental verification, measurements of secondary flow were
executed in a curved flume in the Laboratory of Fluid Mechanics
of the Delft University of Technology with a sinusoidally varying
discharge. Because of the limited possibilities in this
investigation for the variation of discharge, this varying
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5
water from flowing backwards. The magnitude and the period of the
varying discharge were chosen in such a way as to give a fair
reproduction of the flow in a tidal channel. The measurements
confirmed the theoretical result that the misadjustment of the
secondary flow in tidal channels caused by the tidal variation
of the main flow is negligible.
The value of the relaxation length, derived theoretically,
was again confirmed. The misadjustment of the secondary flow in
tidal channels caused by variations in the flow direction can
occasionally be important. The bed shear stresses adjust more
rapidly to the local values and hence here the assumption of full
adjustment is more readily justified. Consequently local
variables will generally suffice for the computation of the bed
laad part of the sediment transport. For the suspended load the
entire secondary flow profile can be of importance. Whereas
·tidal variation appears to have a negligible influence on the
degree of development of the secondary flow, variation in the
flow direction may influence the degree of development and as a
consequence the suspended load.
All measurements considered above were executed in flumes
and concerned the secondary flow due to the curvature of the main
flow. Another component of the secondary flow, the secondary
flow due to the Coriolis acceleration, is important only in
channels with a large radius of curvature (Booij and Kalkwijk,
1982)~ Experimental confirmation of the theoretical results,
namely about the same relaxation lengths and times as for the
curvature component, can only be verified by extensive
measurements in t1dal channels or large rivers.
A more comprehensive report about the measurements of the
secondary flow in the curved flume with varying discharge is
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6~~
IbêQ~g~i~2!_8g§~!~§
The complex flow in river and channel bends can be
considered as a superposition of a more or less parallel main
flow and a secondary circulation. To describe this flow pattern,
following Kalkwijk and de Vriend (1980), the orthogonal
curvilinear coordinate system (s,n,z) of fig. lis used. It is
formed by the streamlines (s) and the normal lines (n) of the
depth averaged horizontal velocity field. The main flow is
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defined by the horizontal velocity component in the streamwise ors-direction, us' and the secondary flow is defined by the
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horizontal component in the normal or n-direction, un0 Because
of this definition the depth averaged secondary velocity
vanishes
Ün
=
o
(1)where the overbar means dep th averaging.
Assuming a quasi-hydrostatic pressure distribution and an
eddy viscosity A, only depending on the main flow, yields a
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momentum equation for the secondary flow component for thecurvature of the form (Booij and Kalkwijk, 1982)
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....
... u":'-u"" S s R (2) "tn + ph=
0with t is the time,
p
the mass density, "tn the normal componentof the bed shear stress and
R
the radius of curvature of the main.1
flow. To get rid of the bed shear stress "tn in equation (2) anassumption about the behaviour of the secondary flow velocity
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near the bottom is required.
When the first three terms in equation (2) can be neglected,
the solution of the resulting equation
'"'
..,
u""-u"'-s sR
"tn
+ ph
=
0 (3 )is the fully developed secondary flow. This fully developed flow
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7curved flow that is both,steady and uniform in the s-direction
the secondary flow would be fully developed. When a logarithmic
main flow velocity profile and a parabolic eddy viscosity
distribution are assumed, then the expression for the fully
developed secondary flow reads (Booij and Kalkwijk, 1982)
(4)
Here ~ is van Karman's constant and f is a complicated function
of the vertical coordinate and the friction parameter ~, which is
a measure
0+
the steepness0+
the logarithmic velocity profile.
(see fig. 2). ~ depends on the Chezy coefficient C.
(5)
9 is the acceleration due to gravity.
Real curved open channel flows vary in the streamwise
direction. In particular the changes of the radius of curvature
of the main flow near the entrance and the exit of a bend can
give rise to important changes in the secondary flow. Th~
neglect of the s-derivatives in equation (2) is only justified
at places where these changes are very graduale Otherwise the
more e~tensive equation
(6)
-en
+ ph
=
0has to be used. Equation (6) describes the adjustment of the
s~condary flow to the local variables. It does not allow exactly
similar secondary flow profiles. The departure of the secondary
flow velecity profiles from similarity with the fully developed
profile is, however, quite smalle Only the secondary flow near
the bottom and with it the bed shear stress adjust much faster.
The misadjustment of the similar secondary flow velocity
profile to the fully developed profile fellows quite exactly a
negative e-power. The relaxation length L, defined by the
reduction of the misadJustment to a factor 1/e of its original
value, is about 15 h for a Ch~zy coefficient of abeut 50 m1/2/s
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8concluded.that the relaxation length is roughly proportional to
C, or
L ~ 15h~ 50Jlms
=
0. .,.)'/m
"'1' S hC (7)Measured values of the relaxation le~gth all stem from
experiments in flumes, as measurements in rivers and tidal
channels canno~ be executed with the required precision. The
measured values agree within the possible precision with the
theoretical value.
In time dependent flow Booij and Kalkwijk (1982) find a
rela)~ation time
s hC
O.3./m
-Us
(8)
No experimental evidence was available about the relaxation time.
As this relaxation time could play a part in the velocity field
in a tidal channel, it seemed worthwhile to try to measure the
relaxation time in a curved flume with a varying discharge.
The theoretically derived bed shear stresses in the
n-direction relaxate, much more quickly to the values that
agree with the fully developed secondary flow (Booij and
Kal kwi jk, 1982). Measurements of these bed shear stresses are
however not easily executed. The Coriolis component of the
secondary flow is about proportional to the depth of the flow.
However,generally the curvature component will dominate in deep
rivers or channels, except in cases with a large radius of
curvature like tidal channels. The theoretically derived
relaxation of the Coriolis component hardly deviates from the
relaxation of the curvature component. For experimental
verification extensive measurements in a tidal channel or a very
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9~
~
S~Qg~i~gD~~l_§g~=~Q_~Dg_E~Q~g§§iDg_Qf_~bg_~g~§~~g~gD~§
~~!
__
El~~g_~Dg_gi§~b~~gg
The measurements of the development of the secondary flow in
a time-varying main flow were executed in a curved flume in the
Laboratory of Fluid Mechanics of the Delft University of
Technology (see fig. 3). To fit in within the framework of this
research, the measured varying flow should be a fair model of the
flow in a tidal channel.
The available curved flume, however, does not satisfy all
the requirements in this respect. The depth is about 5 cm and
the width 50 cm, providing a depth to width ratio of about 1/10,
whereas in tidal channels this ratio is generally much lower,
è.g. 1/40. A more severe deviation is the rectangular form of
the cross-section of the flume. The bot tom and the straight
sidewalls of the flume were made of glass and curved sidewalls
'of perspex. The transparancy of siäewalls and bottom and the
rectangular cross-section made possible Laser Doppier Velocity
measurements in both horizontal and vertical directions. The
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rectangular form and the large depth to width ratio do also notagree with the conditions required in the theoretical model
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(Kalkwijk and de Vriend, 1980).
At the inflow of the flume a sinusoidally varying discharge
with an amplitude of 2.3 x 10-3 m3/s. was imposed. The control
system requfred an unconditionally positive discharge. To this
end th~ varying discharged was superimposed on a constant
discharge of 6.8 x 10-3 m3/s. In this way the following two
requirements were met:
-The Froude numbers should not be too large in order to
exclude severe water surface gradients effects.
The velocities have to be quite large to make reliable
velocity measurements possible.
This is of special importance for the measurements of secondary
flow because of the small velocity compared to the main flow
velocity.
The tidal period Tp uBed in the experiments was 9 minutes.
This value of Tp was chosen in order to keep the Strouhal number
(=h/uTp)' which gives an indication of the relative importance of
the time-derivative term in the momentum equation, roughly equal
to the Strouhal number encountered in a tidal channel. In this
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10reproduced the best.
~
~~_Inê_mê~§~Lêmênt§
For the determination of the degree of development of the secondary flow, the following variables h~ve to be known (see eq. 4): the depth aVèraged main flow velocity, the water depth, the radius of curvature of the main flow and a measure of the
secondary flow velocity, e.g. the secondary flow intensity, In' defined by
To this end measurements were executed in several cross-sections (see f i9 . 3). In all cross-sections measurements were done in the two verticals at 5 cm from the inner and at 5 cm from the outer wall and in the vertical in the middle of the flume. In some cross-sections measurements were done in two additional verticals at 15 cm from the inner and the outer wall (see fig.
3). The water level at each vertical and two horizontal velocity components in various points in the vertical were determined.
In order to obtain two horizontal velocity components, the laserbeams crossed the flume in the vertical di~ection. Because af the limited space underneath the flume, the optics th at cr~ate the measuring volume, had to be placed above the flume. The
disadvantage of this set-up was the varying distance of the measuring volume to the flume bottom during the measurements caused by the refraction of the laserbeams by the varying water surface (see fig. 4).
The two v~locity components were both measured at an angle of 450 to the flume direction. From these two components the horizontal velocity components in the direction af the flume and normal to the flume were obtained. The difference between the main flow direction and the flume direction, which earl ier
experiments in this flume proved to be very small, is neglected. This means also that the radius of curvature of the main flow corresponds with the radius af curvature of the flume.
The velocities in each point were measured during three connected 'tidal . periods. The sampling frequency of the velocity measurements was 10 Hz. Registration af the velocity
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measurements was in digi~al form with a data acquisition system, which made processing of the measurements by computer possible.I
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To determine the phase of each measurement, the discharge through the inflow pipe was measured and registered by the data
acquisition system simultaneously.
as zero phase.
Minimum discharge was chosen
~
~
~_
_ECQç~§§ing_Qf_tn~_m~~§~c~m~nt§
The rough data contained much turbulent fluctuation and noise (see fig. 5a). Because of the tidal variation the turbulent fluctuation and the noise could not be effectively suppressed by common averaging methods. More advanced methods were used to obtain useful instantaneous values (see fig. 6).
When the measurements over the 3 tidal periods were
'comp Iete,
then the following scheme was used:
the three values at corresponding phases in the three tidal periods were averaged, which means a weak ensemble averaging.
(see fig. 5b)
the resulting values over one period underwent a fast Fourier transform.
components above the 4th harmonic were neglected. These
componènts contained many disturbances, mostly standing surface waves.
the smoothed data were recovered by an inverse fast Fourier transform of the remaining 5 components (including the Oth harmonic).
The resulting smoothèd data provide useful instantaneous values (see
f ig. 5c).
If the measured data lacked appropriate information in parts of the periods which makes the usage the fast Fourier transformation difficult another smoothing procedure was followed. This was the case in measurements near the bottom and the surface, where the measuring volume was outside the water in a part of the tidal period because of the varying water depth.
In these cases the scheme followed was:
A harmonie analysis to ·the first harmonies (including the
Oth harmonie) was made over the complete series of measurements. From these 5 components the smoothed series was recomputed.
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12Af ter such a smoothing procedure the main flow velocity and
the secondary flow velocity were obtained from the 2 velocity
components measured, uI and u2. To reduce the amount of
calculations, these calculations are limited to timesteps of
1/100 period.
ul and u2 were measured at an angle of 450 with the flume
direction, 50
us -- ul+u2~and
ul-u2
un
=
±--n-
(10)This value of the secondary flow is, however, quite
unreliable, because of the relatively inaccurate
determination of the angle of the Laser Doppier Velocimeter
with respect to the direction of the flume. A small error in
this angle has an important effect on the measured secondary
flow • The correct angle was found by rotating the orientation of
.the s,n coordinate system with respect to the laser doppier
velocimeter until the depth averaged secondary flow velocity
vanished.
( 11>
Using this angle the correct un' us' the depth averaged main
flow velocity
Ü
s and the secondary flow intensity In (seeequation 9) were calculated.
Because of the small number of points in a vertical in which
the velocities were measured, special precautions were required
to yield trustworthy depth averaged values. In the following
schemes knowledge about the expected vertical distributions is
used (see fig. 7).
In the depth averaging of thé main flow velocity Us was
integrated over the depth by means of the simple procedure
shown in fig. 7. This procedure corresponds with the usage
of the trapezoidal rule of integration when zero is assumed
for the velocity at the bottom and the value of het velocity
of the highest point for the velocity at the water surface.
This methad yields a slightly underestimated Gs'
In the depth averaging of the secondary flow velocity the
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13points of measurement. Above the highest point the
difference between this point and the next measuring point beneath it was extrapolated linearly. When the lowest point was more than h/4 above the bottom, the same procedure was used beneath this point, otherwise the value of the ~elocity of the lowest measuring point was assumed at the bottom. Wh en the lowest point was at less than h/20 the logarithmi~ behaviour of the secondary flow near the bottom was
corrected for by multiplying the height of this interval with 0.8.
In the calculation of the secondary flow intensity the same procedure was used as for the depth averaged secondary flow, but for the depth inteval where un changes sign.
The depths at which the measurements were executed, had to be determined at each time, taking account of the instantaneous place of the varying surface level.·
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~~
s~E~riID~n~~!-[~§~!~§
~~l__
§g~QnQ§[~_E!Q~_E[Qfi!~§
The processing of the measurements af ter their collection had one drawback. M~asurements had sometimes to be rejected af ter processing, although the original signals from the Laser Doppier
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Velocimeter did not give rise to suspicion. Most of ten this happened in points near the bottom or near the surface.
the velocity measuements were spoiled in this regions by
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reflections of the laserbeams. The vertical at the center of theI
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flume at 450 is an example where the consequences are severe.
O~ly two measuring points in this vertical remained instead of the usual 4 or 5. The vertical at the inner side of the bend at
900 was even rejected completely. Here the cause was a bad
separation between the two velocity measurements, a disadvantage of the separation by polarization of the beams in the Laser Doppier Velocimeter used.
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From the smoothed velocity measurements (see fig. 5c) the velocities of the main f10t.>J and the secandary flow were
calculated as discussed in 3.3 (see fig.Ba). To plot velocity profiles, the varying place of the measuring volume in the
vertical had to be calculated from the water level measurements (see
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fig. Bb). The fitting of the time base of the water level measurements was made possible by the simultaneous measurementsI
of the discharge in the inflow pipe (see 3.2).The resulting velocity profiles for the main flow and the secondary flow are
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plotted in figs. 8c and 8d). Only 10 velocity profiles covering
·
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the tidal period are given and the measured values are connected by straight lines to enhance the distinctions.
To compare the velocity profiles, the vertical coordinate and the velocity components are normalized in the following way
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...*
=
z...
n
u
*
=
Us ( 12) Us s u*=
unhu
s n II
II
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.15In the example of the normalized secondary flow velocity profiles, showed, the error in the measurements, when the measuring volume approaches the water surface, as discussed above, is obvious. The sharp decline of the secondary velocity is contrary to the theoretical profile and to the visual inspection of the movement of small floating objects at the water surface.
The secondary flow velocity profiles measured at 1350,where the secondary flow is about fully adjusted to the curvature of the bend, compare weIl to secondary velocity profiles measured by de Vriend (1980) (see fig. 9). These latter measurements were
executed in a former flume in the Laboratory of Fluid Mechanics of the Delft University of Technology. The depth to width ratio had about the same value, but the depth to curvature ratio of the flume was much sharper compared to the width. The profile at the centre of the flume resembles closely the profile measured by de .Vriend and the theoretical profile •. The secondary flow velocity
profile near the inner sidewall shows a small influence of this wall. The profile near the outer sidewall shows the influence of a second contra-rotating secondary flow cell near the surface
(see fig. 10). This second secondary flow cell is caused by an instability of the flow along a concave wall (de Vriend, 1980). The strengths of the secondary flow cells in the two flumes do not correspond (see 4.2). As aresuit the secondary flow
velocity profiles near the outer sidewalls differ.
1~~__B~jy§tmgnt_Qf_thg_§g~QQ~~c~_f!Q~_tQ_~_timg=~~c~ing_m~iQ_f!Q~
The expected-relaxation time in a time dependent flow isgiven in expression (8). Using for the flow considered h ~ 0.05 m, Us ~ 0.25 mis and C ~ 60 m1/2/s (averaged values for the
tidal period, this expression yields a relaxation time of ab out 4 seconds or about 1'l.of the tidal period. This expected small relaxation time will have hardly any influence on the adjustment of the secondary flow.
The normalized secondary flow veloeities, measured in this investigation, have indeed nearly fully similar profiles at different times, so the time dependence of the flow has no
measurable influence on the .adjustment of the secondary flow (see fig. l1a,b). This confirms the theoretical result of a very short relaxation time. In the computation of secondary flow in a tidal
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channel a instantaneous adjustment can consequently be assumed.The normalized profiles near the outer wall show a small
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dependence on the depth of the flow (see fig. l1c). This can beI
explained by the influence of the second secondary flow cello The
importance of this second cel! increases with the Dean number,
defined by
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DE (13)I
where Rax is the radius of curvature of the flume axis. The Deannumber depends on the water depth. Because ~ is usually considered
pr~portional to ush, the Dean number does not necessarily depend
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directlyon the discharge. In a tidal channel na second cell is
expected because of the less steep sidewalls.
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~~~_
_
Bg!~~~~iQ~_!g~g~b
This investigation was not aimed at a determination of the
adjustment in case of spatial variation. Information on this
adjustment is more easily obtained from measurements in a steady
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flow. In steady flow much less effort is needed for each
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measurement, allowing hence measurements in more cross-sections.
A rough guess of the relaxation length can, hdwever, be obtained
from the few secondary flow intensities, measured.
In the table of fig. 12 the measured intensities, normalized
by the values, theoretically calculated for fully developed
secondary flow, are reproduced af ter averaging over the tidal
periode Not all measured intensities are equally trustworthy.
·
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E.g. the measurements of the secondary flow in the upper half of
the central vertical at 1350 failed, which introduces an
uncertainty in the value of the intensity. In fig. 12 the
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normalized intensities are averaged over the central part of thef Iume. In the first part of the bend a relaxation length of
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ab6ut 17 h can be concluded to, when for h the averaged value
over the tidal period is used. This value is indeed nearly equal
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to the theoretical value calculated from equation (7), which
gives for C ~ 60 m1/2/s and L ~ 18 h. The relaxation length in
the straight part af ter the bend could not be determined with
sufficient accuracy.
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17§~
ÇQQ~l~§iQQ§Measurements of the adJustment of the secondary flow to the local and instantaneous variables were executed in a curved flume with a varying discharge in the laboratory of fluid mechanics, of the Delft technical university.
The small theoretical value of the relaxation time was confirmed. Although the flow in the flume was not in every aspect a good model of the flow in a tidal channel, it may be ioncluded that, for computations of the secondary flow in a tidal channel, a instantaneous-adjustment with respect to the tidal variation of the flow can be assumed.
This investigation was mainly aimed at the determination of the adJustment in time and less to the adJustment in the flow direction. A rough determination of the relaxation length, however, confirmed the theoretical value.
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18
Booij, R
.
and Kalkwijk, J.P.Th., 1982, Secondary flow in
estuaries due to the curvature of the main flow and to the
rotation of the earth and its development, Delft Univ. of
Techn., Dept. of Civil Engrg., Lab. of Fluid Mech., report
9-82.
K
alkwijk, J.P.Th., and Vriend, H.J. De, 1980, Computation of the
flow in shallow river bends, J. Hydr. Res., 18, no
.
4, p
327-342.
Vriend, H.J. De, 1981, Steady flow in shallow channel bends,
Thesis, Delft Univ. of Techn.; Comm. on Hydraulics, Delft
Univ. of Techn., report no. 81-3.
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A CDE
f 9 hIn
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n RI
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19eddy viscosity
,Chezy coefficient
Dean number
distribution function of the secondary flow velocity
acceleration due to gravity
depth of flow
secondary flow intensity
relaxation length of the secondary flow
normal coordinate
Radius of curvature of the main flow
Radius of curvature of the flum axis
longitudinal coordinate
time
relaxation time of the secondary flow
period of the flow
measured horizontal velocity components
horizontal main flow and secondary flow velocity
components
vertical coordinate
friction parameter
Von Karman's constant
mass density
normal component of bed shear stress
an asterix denotes a normalized value
an overbar denotes depth averaging.
I
1
1
1
1
1
I
·
1
I
I
I
1
1
I
·
1
Fig. 1.
Definition sketch.
I
I
I
1
I
I
.0; .---0.0
Zlhl
-0.2 -0.8 C=iO -tO -0.8 -0.6 -0.4 -0.2 0.0 -1.0 0.2 0.4 0.6· 0.8 1.0 fI
I
o.
+
~
B 1800• 1.64m R200 doorsnede A-B O.SOmr
OO_2m.i-.
R002 r0- 1 1 I I I I I I E I I,
I I I/)I
I I I I -0 I ~ I I ---I 1I
lo.oSnj,
0.10 m ol. 0.10 m ol, 0.10 mJo
0.10 mJ.o.osd
S 4
INNER BEND
3 2 1
OUTER BEND
PATH OF THE BEAMS WITH VARYING WATERLEVEL
I
1
I
·
1
1
1
1
I
1
1
I
·
1
1
1
1
I
·
1
1
1
I
I
LASER EAM-DISPLA ER RAGG-CELL LENS PHOTO-DETECTOR SIGNAL-PROCESSING&
OATA-ACOUISITIONFig. 4. The Laser-Doppler Velocimeter.
- - - _
.
_
- - - -
-
- -
-
-
-ORIG
.
51GNRLS;
FIRST
PERIOD
-0 111 0
-
.
(0 N E_ Cf) I o .-4-
Wo =>0 _Ja) a: :> o o::;;0'.00 't~.OC
9b.oo
1~5.00 lao.oo 225.00 2170.00 3115.00 360.00ilhs.oo
ij~o.oo
il95.00 540.002
),(10' (discharge)
,.._o 111-
E ,." f\J 315.00 360.00 ij1
),(10' (velocity)
0.00-
111È'
,." f\JI"
10.00 . 00 .360.00 Ij),(10' {ve l oc
ity)
4Fig. Sa.
Smoothing of the data.
- - - _
..
_
- - -
-
-
-
-
.-
- -
.
-
-ORIG.
5IGNRLS;
1 PERIOD
COMPOSED
OF 3
---
11'10 _0 ('t) •eN
('t) -I o .-4 '-" uJo ::::>0 _Ja::Î a:: > oor
.. r i I I I I I i i I I J :2'0.00 '15.00 90.00 135.00 180.00 225.00 210.00 315.00.360.00 il05.00 1j50.00 '195.00 5'!0.00MEASUREMENTS
CHRNNEL
2
)(10·
(discharge)
-
11'1-
e 0.00,.,
o ... 1 11'1 'E-'-" couJ-=>ö
_J. a:: >,.,
N
ltr~r'
Iö
'I" ~51Y 10.00 . .. __ .. _ ..~_ ..~ 30
15.00101
360.00 L!05.00 '150.00 il95.00 SilO. 00)(
(veloc
i
ty)
Fig. 5b
.
Smoothing of the data
.
- - - -
.
_
,
_
_ _
-FFT
F I'L TER EOS
I GNALS.
-0
~o
('I') •eN
('I') -I 0--
wo
::)0 _Ja;)ex:
>
0 0.
::2"0.00 ij5.00 90.00 135.00I 180.00 225.00 270.00 315.00 360.00
&jos.oo
Y50.00 i!95.00 SiW.OOMERSUREMENT 5 CHANNEL
2
~ 10
1(di scharge)
('t')
""
0.
...
111 <,e
... Wee ::)-_Jóex:
>
('t')-=r=
180.00 2~S.00 250.00 3~S.003bo:9
0ilfs.oo
I I I I I IYS.OO 90.00 135.00 i!SO.OO i!9S.00 SYO.OO
('t')
MER5 URE MENT 5 CHANNEL
1
)( 101
(ve oei ty
-
.
0 . I...
111 <,e
·
-co w-::).
_J'Tex:
>
('t') 1'\1 H?~~I'i.
0la.
00 135.00 ·lBO.OO· 225.00 315.00 360.00 YOS.OO 1150.00 Y95.00 5YO.00MERSURËMENTS
o
)(
101 (velocity)
Fi g. Sc.
Smoothing of the data.
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1
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.
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1
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I
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I
,
I
Data of measuremen~s of consecutive
discharge oscilations
Rotation te s and n direction and
diminishing data
,
by representing
a series
by 100samples
Ensemble average over
oscilations
\
Fast Fourier transiorm
Inverse fast Fourier transform
of the first five components
-IHarmonic analysis up
te the 4th harmonic
\lnverse harmon1C
analysis
I
I
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1
·
,
1
I
I
I
I
I
\Computation
of
l
Us(Z).AZ~
L
Un(Z).AZ
Rotation of the orientation of
s,n coordinate
system
so
I
Un(Z).~Z.At
=0
Final computation of
2
Us(z).AZ,
L
un(Z).AZ and
2
IUn (z ,
J • AZI
I
I
-I
I
I
II
I
1
I
II
-I
I
I
I
I
-I
I
I
I
I
/
,
I I / .I"
,
t----~., _--I,
...
---FFT
FILTERED
ENSEMBLE
PERIOD/2700
SPL.
5
HRR;
...0 VlO ...
.
M N !=;._, M I 0 ... ...wO
::JO ....Ja) er :> 0 0·
:::2'0.00 22.50 i!5.00 67.50 90.00 112.50 135.00 157.50 160.00 202.50 225.00 21!7.50 270.00MERSUREMENTS
CHRNNEL
2
)(10' (discharge)
0 :r
·
0 -VI <, E--
Wl'\l ::J"" ....Jó er :> :::2' t\I·
.00.00 22.50 &!5.00 67.50 90.00 112.50 135.00 157.50iao.oo
202.50 225.00 2Y7.50 270.00MERSUREMENTS
CHRNNEL
1
)(10· (us velocity)
co """"0 VI. ê'0 ... -; 0
-
:tg
·
0w
::J ....J a:co :>~ 0 10'.00 22.50 LI5.00 6'7.50 90.00 112.50 135.00 157.50 IrO.OO 202.50 225.00 2~7.50 270. 00Fig. 8a .
Calculation
MERSUREMENTS
of the velocity profile of the secondary flow.
CHANNEL
0
)(10'
un velocity)
Main and secondary flow velocities.
---FFT FILTERED
ENSEMBLE
PERIOD/2700
SPL.
5 HAR;
CD IJ'
·
o E,
o
-0 x;...·
o w :J _Ja:
:;:..'":l'~ ~ _ ~ I'IS.oo
i.:::-,
i en onI nn ff-"tI ~n ,~r-i r.,,,,,, .__I __ 1___ ,__ I I I 61.0,1" 90.00 112.50 -135.00 151.50 180.00 202.50MERSUREMENTS
CHRNNEL
2
MlO'
(vertical
2)
225.00 2in.50 210.00CD U"\
·
o E ,o
-0 x;.."l·
o w :J _Ja:
:;:..'" :l'1~ ~ i· ~ i I I 1 22.50 a..l nn ~." en nn rin , • ~ rn '~r __ I I I I I i 225.00 2~ 1.50 210.00 -0 Vl.o -....
ME-'" -M-I o r-t...
wO :JO _Jcè eI: :;:.. g~---:l'0-.00 22.50 "5.00 67.50 90.00. 112.50 135.00 J57.50 lBO.OO 202.50 225.00-
MERSUAEMENTS
CHRNNEL
0
MlO' (discharge)
Fig.
-
Sb.
Calculation
of the velocity
profile
of the secondary
flow
.
Smoothed water
level
measurements.
o o
. ~---~----~---+---~---~----~---,
o _ 0.00 -30.00 -20.00 -10.00 0.00 10.00 20.00 30.00 110.00MAIN FL~W VEL~CITY
(CM/S)
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I
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I
I
I
I
o o·
CD o ::r·
U) o CD·
::r o N.
~ o CD.
-(I') ~ U .-±o
1-0 0..' W(\') Cl o ::r·
N o CD·
...
o N·
...
o CD • o QJ ~ 5 u X 11 I'd+
6 l+-s, ::::J+
3 V) s, X 7 QJ +> I'd 3 IJ. 2 0'1 Y 8 r::...
~ ~ ~ I'd > I!l 0n
I
I
IVELOCITI PROFILES OF R135-2
Fig. Sc. Calculation
of the velocity
profile
.
of the secondary flow.
Velocity profiles
of the main flow.
I
I
c c •I
co ~ 5I
c Q.I X 11 ::JO U+
6 • n:I In l+-s,+
3 :::lI
s,VIX 7
C Q.I Cl)..,
•
n:I • 2 -:!' 3:I
0"1 Y 8 c:: .,..t'
i ~
I
°
N n:I> (!J 0 • ::JOI
0 co • _('I)I
(J~ :J:oI
t-0....0 LLJ('I) ClI
c ::JO·
NI
0 Cl)·
....
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I
I
I
I
2.60 1.95 1.30 0.65 -0.00 -0.65 -1.30SEC~NDARY
FLOW
VELOCITY
(CH/S)
-1.95 - .60
VELOCITY
PROFILES
OF R135-2
Fig. 8d.
Calculation of the velocity profiles of the secondary flow.
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I
I
o o •....
o 01 • o·
o o r-• o :J: I-0...0 lIJm 0' o o lilen •
(f')O lIJ_.
Z El _0en=:
Zo lIJ ~-
Cl o ('l.
o o....
.
o~----~---~---~
-
---,---.~---,
-0.50
-0.25
0.00
0.25
0.50
0.75
1.00
HAIN FLOW VELOCITY
SCALED
WITH
DEPTH AVERAGED
MAIN FLOW VELOCITY
o
o
.
o
VEL~CITI PR~FILES ~F R135-2
Fig. 8e.
Calculation of the velocity profile of the secondary flow.
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I
I
I
I
I
o 11)en·
en
o LIJ ...J Z Do 1-4~en.
zo
LIJ ~ 1-4o
o 0') • o::r:
t-Q..o LlJeDo ·
o o o·
....
o 01 • o o CD • o o,...
•
o o N • o o...
·
og
• o2.60
1.95
1.30
0.65
-0.00
-0.65
-1.30
-1.95
SEC.VELOCITY SCALED WITH LOCAL
Q
(10MM-2/CHl
- .60
VEL~CITY
~R~FILES
~F R135-2
Fig. Sf.
Calculation of the velocity profile of the secondary flow.
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I
Fig. 9.
I
I RtJ5-5 (INNER BiND).
I,
I,
,
I.
I R135-J (AXIS).
,,
,
.
,
I ~,
,
•.,' --- DE VRIEND (1980) ~ -- PRESENT DATA R13S-t (OUTER BEND)Comparison of secondary flow profiles with measurements of
de Vriend in another curved LFM-fl~me.
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.
1
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,
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·
1
I
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I
Inner bend
Outer bend
I
I
o -
°
2.00
m
I
Clg
z ..:
UJco
a::
UJz
z
o oI
oI
I
s
é2',60 o o O~OO -2,80 é2",60 RO02-5 o o o In Ó'I
I
g
Ó' 2",60 , 0",00 -2,80 O2',60 ROOO-ij 0',00 -2,80 R023-ijI
o o oo0°
23°
o o...
o In Ó' o o1.5°
o In, o o o . 0' . 0,00 -2,80 2,60 ROOO-5 o o 0",00 -2,80 02',80 0',00 -2,80 R023-5 ROij5-5I
, O~OO -2,80 02~60 R002-3...
I
o In Ó' o In, o o oI
I
I
I
o o Ó 2',80 o o oo,...
o In, o o In Ó'I
o o é 2',80 o o, 0",00 -2,80 O2',60 R002-2 , 0",00 -2,80 O2",80 ROOO-2 O~00 -2,80 R023-2 o o oo...
o o -'.80 é2~80I
I
Cl 0 InZ
éw
al o In.
oI
0::W 0~ t- O2',80::::>
o
0",00 R002-1Vertical axis: O;mens;onless water depth.
I
o o o In Ó o o...
oo...
" o In, o o o.
0",00 -2.80 0 2~80 ROijS-3~orizontal axis: Secondary flow velocity, scaled with the local discharge
.
(lO-2/cm)
I
o o o o o Ino
o In Ó' o o O~OO -2.80 é'2~80 ROOO-3 o o 0",00 -2,80 Ó 2'.80 R023-3 o o o 11).
o o o o o.
...
o In.
o o o é 2",80 o o,...
90°
135°
g
fg óg
.
O2",80 0".00 -2.80 R135-5 o o...
o In.
o o o é 2'.60 0'.00 -2.80 R135-ij o 0 o . 0 "...
o In·
o o o.
0".00 -2.80 O2'.60 R135-3 o oc.co
-2.60 O2'.80 R180-3 o o . 0' . 0.00 -2.80 2.80 R090-3 o In Ó'g
é 2'.80 0".00 A090-1 o o.,
...
o In·
os
·
o 2~60 0".00 -2.80 R135-2 o o "...
o In·
o -...10 o o Ó' 2',80 0".00 R135-1 o o -".80 é2'.801
80°
o o oo...
o In é o o . Ö . 0.00 -2.80 2.60 R180-5 o o "...
o In Ó' o180
+1.64
m
0".00 -2.80 R200-5 o o...
o In Ó o o o 11)·
o o o Ó 2.'80 o o "-
oo·
...
o In é o 11)·
o o o·
- ....80 0 2'.80 - ....80 O~OO R180-1Fig. lla. Time dependence of secondary flow
velocity orofiles
.
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I
•
I
I
I
I
I
11"
!"
I
•
"
VELaCJTY PftaFJLES aF ft002-SI
I
I
II
I"
I
I cl I! olI
I
I
11elI
I olI
VELGCITY PftGFILES ft02S-3 I 1.1 cl !I•
I•
VELGCITY PftGFILES aF ROOO-S
I I ol lil cl I D
VELGCITY PftGFILES aF ROqS-S
I
Fig. llb
1•Time dependenee of secondary flow velocity profiles.
•
I tiI
I clI
I!•I
I
I
11..
I
!!..
I
VELeCITY PRerILES er R090-3I
I
11I
~ •I
I cl I! olI
I
I
I
Jl..
!! olI
I
VELeCITY PRerILES er RIBO-3Ir cl I! .: !! cl 11 á a.lo VELeCITY PRerILES er R135-3 IJ .: Ir cl I! á !! cl 11 á VELeCITY PRerILES er R200-3
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I
I
I
YELaCJTY PRarJLES aF R002-1I
•
I
I
I
I
•
..
I
I
I
!I..
•
YEL8CJTY PRaFJLES R023-1•
YELaCJTY PRarJLES aF ROOO-l
I
..
I•
..
!I"
I" ••10SEC.'UNI. Y\:UCJTY 'CIILED1 10 •• -
NIl"
• L tiL• a-1
1&.- ~
i)
VELaCJTY PRaFJLES aF RO~5-1
I
Fig. llc
1•
Time dependenee of secondary flow velocity profiles.
Profiles along the outer wall.
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•
1
•
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I
VELDCITY PRDrILES Dr R080-1I
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I
I
VELDCITY PRDrILES Dr Rl10-1 VELDCITY PRDrILES Dr R135-1 VELDCITY PRDrILES Dr R200-1Fig.
11e
2•Time dependenee of
'
seeondary flow veloeity profiles.
---5,
*
E
u
M'
'0 ~*
-N "0 -*c3
.Jo':::'or-*
I-IN
11 ~c...
2
I •1
0.00 0° 1.61 23°R023
3.22 45°R045
6.44 90°R090
9,66 135°R135
12.88 180°R1BO
1~2[m]
ROOO
Fig. 12a. Normalized secondary flow intensities averaged over the central part
of the flume
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I
---
-
--
-
-
-
---_
.
_---
-
---, VERTICAL
,
1 2 3 4 5, C~:OSS