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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 Engineering

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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. 15-84

Laboratory 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

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~~~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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b!§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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!D~~Q~~~~iQn

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). No

experimental 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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~~

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 or

s-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 the

curvature of the form (Booij and Kalkwijk, 1982)

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....

... u":'-u"" S s R (2) "tn + ph

=

0

with t is the time,

p

the mass density, "tn the normal component

of the bed shear stress and

R

the radius of curvature of the main

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flow. To get rid of the bed shear stress "tn in equation (2) an

assumption 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 s

R

"tn

+ ph

=

0 (3 )

is the fully developed secondary flow. This fully developed flow

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7

curved 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 steepness

0+

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

=

0

has 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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concluded.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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~

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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 not

agree 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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10

reproduced 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.

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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

~

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_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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12

Af 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.

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Using this angle the correct un' us' the depth averaged main

flow velocity

Ü

s and the secondary flow intensity In (see

equation 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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13

points 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 the

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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 measurements

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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

·

1

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*

=

un

hu

s n I

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.15

In 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 is

given 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 be

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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)

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where Rax is the radius of curvature of the flume axis. The Dean

number 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.

·

1

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 the

f 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.

(20)

I

I

I

I

A C

DE

f 9 h

In

L

I

I

I

1

n R

I

I

I

1

1

I

*

I

·

I

I

I

I

I

19

eddy 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.

(21)

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; .

(22)

---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 f

(23)

I

I

o

.

+

~

B 1800• 1.64m R200 doorsnede A-B O.SOm

r

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 1

I

lo.oSnj,

0.10 m ol. 0.10 m ol, 0.10 m

Jo

0.10 m

J.o.osd

S 4

INNER BEND

3 2 1

OUTER BEND

(24)

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-ACOUISITION

Fig. 4. The Laser-Doppler Velocimeter.

(25)

- - - _

.

_

- - - -

-

- -

-

-

-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 3115.00 360.00

ilhs.oo

ij~o.oo

il95.00 540.00

2

),(10' (discharge)

,.._o 111

-

E ,." f\J 315.00 360.00 ij

1

),(10' (velocity)

0.00

-

111

È'

,." f\J

I"

10.00 . 00 .360.00 Ij

),(10' {ve l oc

ity)

4

Fig. Sa.

Smoothing of the data.

(26)

- - - _

.

.

_

- - -

-

-

-

-

.

-

- -

.

-

-ORIG.

5IGNRLS;

1 PERIOD

COMPOSED

OF 3

---

11'10 _0 ('t) •

eN

('t) -I o .-4 '-" uJo ::::>0 _Ja::Î a:: > o

or

.. 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'!0.00

MEASUREMENTS

CHRNNEL

2

)(10·

(discharge)

-

11'1

-

e 0.00

,.,

o ... 1 11'1

'E-'-" co

uJ-=>ö

_J. a:: >

,.,

N

ltr~r'

I

ö

'I" ~51Y 10.00 . .. __ .. _ ..~_ ..~ 3

0

15.00

101

360.00 L!05.00 '150.00 il95.00 SilO. 00

)(

(veloc

i

ty)

Fig. 5b

.

Smoothing of the data

.

(27)

- - - -

.

_

,

_

_ _

-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.OO

MERSUREMENT 5 CHANNEL

2

~ 10

1

(di scharge)

('t')

""

0

.

...

111 <,

e

... Wee ::)-_Jó

ex:

>

('t')

-=r=

180.00 2~S.00 250.00 3~S.00

3bo:9

0

ilfs.oo

I I I I I I

YS.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'T

ex:

>

('t') 1'\1 H?~~I'i

.

0

la.

00 135.00 ·lBO.OO· 225.00 315.00 360.00 YOS.OO 1150.00 Y95.00 5YO.00

MERSURËMENTS

o

)(

101 (velocity)

Fi g. Sc.

Smoothing of the data.

(28)

I

1

I

.

I

I

1

I

I

I

I

,

I

Data of measuremen~s of consecutive

discharge oscilations

Rotation te s and n direction and

diminishing data

,

by representing

a series

by 100

samples

Ensemble average over

oscilations

\

Fast Fourier transiorm

Inverse fast Fourier transform

of the first five components

-I

Harmonic analysis up

te the 4th harmonic

\

lnverse harmon1C

analysis

I

I

I

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

IU

n (z ,

J • AZ

(29)

I

I

I

-I

I

I

I

I

I

1

I

I

I

-I

I

I

I

I

-I

I

I

I

I

/

,

I I / .I

"

,

t----~.,

_--I

,

...

(30)

---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!7.50 270.00

MERSUREMENTS

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 90.00 112.50 135.00 157.50

iao.oo

202.50 225.00 2Y7.50 270.00

MERSUREMENTS

CHRNNEL

1

)(10· (us velocity)

co """"0 VI. ê'0 ... -; 0

-

:tg

·

0

w

::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. 00

Fig. 8a .

Calculation

MERSUREMENTS

of the velocity profile of the secondary flow.

CHANNEL

0

)(10'

un velocity)

Main and secondary flow velocities.

(31)

---FFT FILTERED

ENSEMBLE

PERIOD/2700

SPL.

5 HAR;

CD IJ'

·

o E

,

o

-0 x;...

·

o w :J _J

a:

:;:..'":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 -135.00 151.50 180.00 202.50

MERSUREMENTS

CHRNNEL

2

MlO'

(vertical

2)

225.00 2in.50 210.00

CD U"\

·

o E ,

o

-0 x;.."l

·

o w :J _J

a:

:;:..'" :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.

(32)

o o

. ~---~----~---+---~---~----~---,

o _ 0.00 -30.00 -20.00 -10.00 0.00 10.00 20.00 30.00 110.00

MAIN FL~W VEL~CITY

(CM/S)

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

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 0

n

I

I

I

VELOCITI PROFILES OF R135-2

Fig. Sc. Calculation

of the velocity

profile

.

of the secondary flow.

Velocity profiles

of the main flow.

(33)

I

I

c c •

I

co ~ 5

I

c Q.I X 11 ::JO U

+

6 • n:I In l+-s,

+

3 :::l

I

s,VI

X 7

C Q.I Cl)

..,

n:I 2 -:!' 3:

I

0"1 Y 8 c:: .,..

t'

i ~

I

°

N n:I> (!J 0 • ::JO

I

0 co • _('I)

I

(J~ :J:o

I

t-0....0 LLJ('I) Cl

I

c ::JO

·

N

I

0 Cl)

·

....

I

I

I

I

I

2.60 1.95 1.30 0.65 -0.00 -0.65 -1.30

SEC~NDARY

FLOW

VELOCITY

(CH/S)

-1.95 - .60

VELOCITY

PROFILES

OF R135-2

Fig. 8d.

Calculation of the velocity profiles of the secondary flow.

(34)

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

o o •

....

o 01 • o

·

o o r-• o :J: I-0...0 lIJm 0' o o lil

en •

(f')O lIJ

_.

Z El _0

en=:

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.

(35)

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

o 11)

en·

en

o LIJ ...J Z Do 1-4~

en.

zo

LIJ ~ 1-4

o

o 0') • o

::r:

t-Q..o LlJeD

o ·

o o o

·

....

o 01 • o o CD • o o

,...

o o N • o o

...

·

o

g

• o

2.60

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.

(36)

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

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.

(37)

I

I

1

I

.

1

I

I

I

I

I

I

I

I

,

I

I

·

1

I

I

I

I

I

Inner bend

Outer bend

(38)

I

I

o -

°

2.00

m

I

Cl

g

z ..:

UJ

co

a::

UJ

z

z

o o

I

o

I

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-ij

I

o o oo

23°

o o

...

o In Ó' o o

1.5°

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 ROij5-5

I

, O~OO -2,80 02~60 R002-3

...

I

o In Ó' o In, o o o

I

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 é2~80

I

I

Cl 0 In

Z

é

w

al o In

.

o

I

0::W 0~ t- O2',80

::::>

o

0",00 R002-1

Vertical 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 In

o

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 ó

g

.

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 o

c.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

·

o

s

·

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'.80

1

80°

o o oo

...

o In é o o . Ö . 0.00 -2.80 2.60 R180-5 o o "

...

o In Ó' o

180

+

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 R180-1

Fig. lla. Time dependence of secondary flow

velocity orofiles

.

(39)

I

I

I

I

I

I

I

11

"

!

"

I

"

VELaCJTY PftaFJLES aF ft002-S

I

I

I

I

I

I

"

I

I cl I! ol

I

I

I

11el

I

I ol

I

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.

(40)

I ti

I

I cl

I

I!•

I

I

I

11

..

I

!!

..

I

VELeCITY PRerILES er R090-3

I

I

11

I

~ •

I

I cl I! ol

I

I

I

I

Jl

..

!! ol

I

I

VELeCITY PRerILES er RIBO-3

Ir 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

(41)

I

I

I

I

I

I

I

I

I

I

I

I

YELaCJTY PRarJLES aF R002-1

I

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 tiLa

-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.

(42)

I

I

I

1

I

I

I

I

I

I

I

I

VELDCITY PRDrILES Dr R080-1

I

I

I

I

I

I

I

I

I

I

VELDCITY PRDrILES Dr Rl10-1 VELDCITY PRDrILES Dr R135-1 VELDCITY PRDrILES Dr R200-1

Fig.

11e

2•

Time dependenee of

'

seeondary flow veloeity profiles.

(43)

---5,

*

E

u

M'

'0 ~

*

-N "0

-*c

3

.J

o':::'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

(44)

I

I

I

I

I

I

---

-

--

-

-

-

---_

.

_---

-

---, VERTICAL

,

1 2 3 4 5

, C~:OSS

(outer)

(ax i s)

(innet-)

I

'_§~ÇIIQ~

__'

---,---,---,---I

I

R002 X ROOO 0.59 0.41 0.37 R023 1.94 3.92 4.04 4.33 2.68 R045 1.56 X 5.00 X 2.56 R085 1.47 X 4.86 X X R135 1.73 4.97 4.36 5.05 4.05 R180 1.91 X 4.79 X 3.87 R200 0.85 1.91 1.06 0.73 0.41

I

I

I

I

I

I

I

"

"

--- --- --- ---

---

---<*

10-3 lcm)

I

Fig. 12b. Table of normblized secondary flow intensities

throughout the field

(measured).

I

I

I

I

(45)

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

'

"

:

.

I

I

. '" , ,

'.

'

,'I, . ,', """ , ' -c,,' " .' [ .' .

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