Flow measurements in a curved rectangular channel

H.J. de Vriend

Internal report no. 9-79

Laboratory of Fluid mechanics Department of Civil Engineering

H.J. de Vriend

Internal report ~-79

Laboratory of Fluid Hechanics Department of Civil Engineering Delft University of Technology Delft, The Netherlands

List of tables List of figures List of symbols Summary 1. Introduction ' 1 2. Experimental apparatus ' 2 2.1. The flum.e _ 2

2.2. Combined current-velocity/direction meter •...••...•..•.•..••.... 2 2.3. Water level measurement.s •••••.•••••.•...•..••••.•••.•...••...•.•••.. 3

3. Experimental procedure •...•....••..•.••..•.••..•..•.•..•..••••.•.•• 4

3.1. Configuration of the measuring grid ..••••.•••••••••••••.••.•.•.•... 4 3.2. Observation period .••••••..••.•••.•.•..••••••••••••••••••.•••.•••.• 4 3.3. Calibration of the current meter .••••...•••••'•••••••..•..•.•••.••.. 6

3.4. ,Measuring procedure ' '... 7

3.5. Elaboration of the measured data .•••.•.•••••••••.••••.••'... 9

4. Results and discussion .

..,4.1.Vertical distribution of the _{main veloci.ty .}

4.2. Depth-averaged main velocity •••••••••••••••••••••• e-••••••••••••••••

4.3. Vertical distribution of the secondary flow ••••••..••.•.••...•....•

4.4. Secondary flow intensity ••.••••...••.••••••••••.••.•...•...•...

4.5. Water surface configuration •••••.•.••...•.•••••••.•••••.•.•••••••••

4.6. Comparison with earlier measurements under the sawe conditions •..•.

5. Conclusions .

Acknowledgements

References

Tables

I Determination of the observation period 11 Measured velocity data

111 Measured water levels

LIST OE FIGURES

1. Flume geometry

2. Combined current-velocity/direction meter 3. Configuration of the measuring grid

4. Adaptation time for the water level measurements

5. Vertical distribution of the main velocity (cross-sections)

6. Vertical distrihtition of the main velocity .(longitûdinal sectLons.) 7. Depth-averaged main velocity

a.

Vertical distribution of the horizontal secondary velocity component (cross-sections)

9. Vertical distribution of the horizontal secondary velocity component (longitudinal sections)

10. Secondary flow intensity (cross-sections)

11. Secondary flow intensity (longitudinal sections)

12. Transverse configuration of the water surface

13. Longitudinal configuration of the water surface 14. Transverse fall of the water surface

15. Velocity at 0.4 h as an approximation of the depth-averaged velocity

16. T~nsversedistribution of the velocity at 0.4 h compared with the results of the earlier measurements

B C DO f . mal.n f sec g h -z, s I lil I s n N r=R/R c R R c T V v m :m'vR,vcp v m v _m

O

.

4

v m

O

.

4

v

R

vR m vR s v s Vtot Vtot channel width Chezy's factor

calibration constants of the current meter characteristic depth of flow

time-integral over the observation period of the output signal from the direct ion meter

value of D if the vane is directed along the channel axis vertical distribution function of the main flow

vertical distribution function of the secondary flow acceleration due to gravity

local depth of flow secondary flow intensity

integral of the normalized main velocity distribution over the bottom interval

integral of the normalized secondary velocity distribution over the bottom interv~l

number of grid points in a vertical

number of pulses counted by the velocity meter during the observation period

normalized radial coordinate

radial coordinate in a cylindrical coordinate system radius of curvature of the channel axis

duration of the observation period overall mean velocity

horizontal component of the main velocity depth-averaged velocities

cross-sectional mean value of v m main velocity at z

=

0.4 h

cross-sectional mean value of v m

O.4

radial velocity component

part of vR due to the main flow part of vR due to the secondary flow

horizontal component of the secondary flow total horizontal velocity

magnitude of the depth-averaged velocity vector (depth-averaged main velocity)

z z s z s z

so

K bottom level

vertical coordinate of the k-th grid point from the bottom

water surface elevation

cross-sectional mean value of z

s

water level in the channel axis in the last cross-section of the measuring grid

flow angle (Cl

=

0 in the direction of the channel axis) direction of the depth-averaged velocity vector

(main flow direction) Von Karman's constant

turbulent flow in a rather sharply curved U-shaped flume with a shallow rectangular cross-section, under alroost the same conditions as earlier, less extensive measurements in the same flume. The experiment was carried out as a part of a research project aiming at a mathematical model of the flow in shallow river bends. Extensive

measurements of the water level and the magnitude and direction of the horizontal velocity vector in a three-dimensional measuring grid are described and the results are analysed in tenns of main and secondary flow.

I. Introduction

The development of a mathematical model of steady flow in river bends with a fixed uneven bed forms one of the research projects of the Laboratory of Fluid Mechanics of the Delft University of Technology, as a part of the river bend investigation project of the joint hydraulic

*

research programme T.O.W. ).

One of the intermediate phases of this research project is the development of a mathematical model of steady turbulent flow in curved open channels of shallow rectangular wet cross-section. Experimental information for testing this model was drawn from the literature and from experiments conducted in a rather sharply ,curved flume in the Laboratory of Fluid Mechanics**).

During the model tests the need for more extensive measurements in this flume came forward (DE VRIEND, 1979), because of the following reasons: • the old experiments were conducted in a flume with a very smooth

outer wall in the bend, which may have influenced the results; • the water surface configuration measured during these experiments

differs considerably from the model predictions, whereas the agreement is fair for all other experiments used for the test;

• the prediction of the secondary flow appears to be one of the weak points of the model: its "intensity tends to be underestimated and its decay beyond the bend is described rather poorly; the picture of these shortcomings, however, is not quite uniform for all experiments considered, so that it would be most interesting to have secondary "flow data from the present flume, which is about the sharpliest curved one for which the model works.

In order to provide for this need" one of the old experiments 1n this flume was repeated, but now with an outer wallof the same roughness as the bottom and the inner wall and with a measuring programme concerning the horizontal components of the main and the secondary flow and the water surface configuration.'

M)

'Toegepast Onderzoek Waterstaat', in which Rijkswaterstaat, the Delft Hydraulics Laboratory and the Delft University of Technology participate.

MM)

The results of these experiments, concerning the main flow and the water surface configuration, are still unpublished, but they were kindly put at the author's disposal by the Group of River Engineering.

2. Experimental apparatus

2.1. !he flume

The outlines of the flume 1n.which the experiments were conducted are given in figure 1. It concerns a 1.70 m wide fluroehaving a

U-shaped ground plan, with a horizontal bottom and vertical sidewalls. The radius of curvature of the flume-axis in the bend is 4.25 mand the upstream and downstream straight reaches have an effective length (i.e. the length available for measurements) of about 6 m.

The upstream straight reach is preceded by an inflow section consisting of a stilling basin, a screen of vertical timber laths, alm long pack of corrugated plastic plates mounted in such a way that they form a number of small streamwise tubes in which the larger eddies are damped out, and a 3 m long reach with an artificially roughened bottom

(gravel of gradually decreasing size, fixed in a mortar bed) for the transverse distribution of the flow. Small surface waves are damped by a 1.2 m long foam-plastic plate floating on the surface at the end of the inflow section.

The downstream straight reach is followed by a 5 m long settling bas in to be used during alluvial bed e~periments;. the bottom of this bas in lies 0.3 m below the channel bottom. The depth of flow in the flume can be regulated by a tailgate immediately downstream of the settling basin.

The horizontal bottom of the floor has a concrete top layer and the sidewalls are formed by plastered brick walis, except for the outer wall in the bende The glass panels of which this outer wall consists are covered with thin iron plates roughened by sticking fine-grained sand (grain size 0.006 m) to them.

The flume has a 'half-open' water supply system, i.e. the water is pumped at a constant flow ra te from one of the reservoirs into the fluroe and is discharged into this reservoir again af ter passing through the flume.

The flow ra te is measured by an orifi.cenieter_~inthe supply-pipe.

2.2. Combined current-velocity/direction meter

The magnitude and the direction of the horizontal velocity vector were measured using a combined current-velocityfdirection meter. This device,

developed by the Delft Hydraulics Laboratory (DHL, 1975) and shown in figure 2, is a combination of a micropropeller (diameter 0.011 m) for the determination of the magnitude of the velocity, and a bifurcated vane (height 0.020 m; total length 0.050 m) to indicate the flow

direction.

The propeller and the vane are mounted in a vertically adjustable frame consisting of a bow with two sensors between the branches of the vane and a long vertical shaft. The distance between the centre of the propeller and the lowest póint of the frame is about 0.025 mand the distance from the axis 0f the vertical shaft to the outermost point of the bow is about 0.06 rn.

The vertical shaft passes through a housing, with a fixed vertical

position, that contains the electronics of the device and a small servo-motor guided by the vane-sensors in such a way, that the orientation of

the frame is continuously adjusted to the instantaneous flow direction. Thus the flow direction can be determined from the orientation of the shaft.

For further details of this combined current-velocity/direction meter it is referred to the technical description by the Delft Hydraulics Laboratory

(DHL, 1975).

The veloeities and flow directions measured by this device show fluctuations with time due to turbulence. As the instrument has a certain response time, however, the actual turbulent fluctuations are represented only in part. Therefore the meter was used only for determining the time-rneanvalues of the measured quantities. In practice this implied that the output signals of the meter were averaged over a set period (see section 3.2)"by an electronic integrator; the resulting potentials were translated into

velocities and flow directions through aalibration factors (see section 3.5)

2.3. Water level measurements

The level of the water surface was measured using a single static tube mounted on a bar across the flume in such a way that it could easily be displaced from vertical to vertical and from cross-section to cross-section without pulling the tube out of the water. The water level was transferred from the tube to a measuring glass outside the flume through a plastic hose. The water level in the measuring glass, which was mounted on the

3. Experimental procedure

J.I. Configuration of the measuring grid

Figure 3 shows the configuration of the measuring grid used for the experiments. It consists of 21 cross-sections containing II verticals with 9 grid points each. The cross-sections are situated at 3 mand

m ahead of the bend entrance, at every Iso'in the bend and at every m beyond the bend exit. The verticals are taken at every 0.17 m from the left wall and as closely to the sidewalls as practically possible: 0.10 m. In each vertical the lowest grid point lies at 0.025 m above the mean bottom level in the relevant cross-section and the other points are

situated at every 0.020 m above this point; if the highest point would then lie too closely to the water surface, it is taken at a level as close to the surface as possible, i.e. such that the micropropeller is just immersed.

The number of cross-sections in the upstream straight reach is limited, as the effects of curvature are not likely to extend far upstream. Besides, the vertical distribution of the main velocity in the inflow reg ion is affected by the artificial bottom roughness utiliied to ,obtain a uniform inflow distribution and by the plate on the surface used as a wave damper. Hence.some distance is needed for the main velocity distribution to adapt

"

to the smooth bottom in the actual channel (see Appendix A). Therefore the first three meters beyond the inflaw section are not incorporated in the measuring 8rid.

The minimum distance of the outermost verticals to the sidewalls is determined by the requirement that the vane of the flow direction meter should be able to turn around without touching the walls. The position of the lowest grid point in a vertical follows from the dimensions of the current meter and the deviations of the bottam from the horizontal plane (about 5.10-3 m in a cross-section).

It should be noted that this 2079-point measuring grid is rather fine for the gradually varying main flow, but certainly not for the secondary flow, which shows much steeper variations.

3.2. Observation period

The instantaneous veloeities and flow directions in turbulent flow are fluctuating about a certain mean value. The current meter used..heze is

quite suited for measuring these mean values, but its own inertia makes it an unreliable tool for gathering information on the turbulent fluctuations. Therefore the present experiments concern turbulence-averaged quantities only.

The turbulence-averaged velocities and flow directions are obtained by averaging the instantaneous signals from the current meter over a certain period, the duration of which has to be determined from

preliminary rneasurements. In the present ca~e, the following procedure was chosen: a long series (90) of velocity and flow direction samples with an averaging period of 10 s is taken in a fixed point in the bend; smaller series of samples with an averaging period of 20, 30, 60

and 100 s are generated by combining subsequent 10 s observations; of each of the series of samples the mean value and the standard

deviation are determined. The results of this preliminary investigation are shown in table I. As was to be expected, the standard deviation of the turbulence-averaged velocities and flow directions decreases as the averaging period grows longer. Even for a period as short as 10 s the standard deviation in the turbulence-averaged velocity is less than 1%,

which is acceptable for the present purpose. The standard deviation in the average readings of the flow angle should not be compared with the mean value 7663, but rather with the reading that corresponds with the maximum flow angle to be expected. Adopting a commonly used expression for the secondary flow (ROZOVSKII, 1961; DE VRIEND, 1976), this angle can be approximated by

d tan ~ ::::6

-max R

c

, in which d and R denote a characteristic depth of flow and the c

radius of curvature of the channel axis, respectively.

For the present experiments this means that qmax = ISo which corresponds with about 500 units in table 1. Then the standard deviation in the average flow angle is about 1% of a,. if the averaging period is 30s.

max

Therefore the observation period for the velocity and the flow direction measurements was taken 30 s.

the static tube from one vertical to another. The following procedure was chosen for determining this adaptation time: the static tube 1.S

placed in an arbitrary point of the flow and af ter about an hour of

-adaptation, water is removed from the measuring glass until the water level has been lowered by about 0.02 m, which is much more than expected water level differences between two adjacent verticals;

subsequently, the water level in the glass is measured every 15s (later on every 30 s) until now it has reached its original position. The results of this test, represented in figure 4, show an adaptation time of 5

minutes to be sufficient.

3.3. Calibration of the current meter

r,

The relation between the reading of the velocity meter and the measured v~locity, which is almost linear in the domain of interest, was known from earlier calibrations. As a check, the meter was recalibrated in a towing tank .at the Laboratory of Fluid Mechanics. The differences between the two calibration curves turned out to be negligible. The relation between the reading of the direction meter and the measured flow angle was not

given. Besides, it was known from earlier experiments that the vertical transport system of the current meter may give rise to important systematic errors in the measured flow angles (DE VRIEND AND KOCH, 1977; DE VRIEND, 1978a) especially if the vertical shaft is moved by hand. Therefore the direction meter waa calibrated every day and a series of preliminary measurements was carried out to determine the backlash in the vertical transport system. For the dayly calibration the tailgate was raised and the flume was filled with water. Af ter the filling a very low discharge (a few liters per second) was maintained in order to compensate the leaking along the tailgate.

Under these conditions the current meter, which was mounted on a carriage (see figure 5) was towed along the straight .inflow reach of the flume,

several times in upstream direction and several times in downstream direction. The mean reading of the tipstream towing was taken as the zero reference

upstream and the downstream towings was assumed to correspond

with a rotation of 1800•As became evident from these dayly calibrations,

the zero reference angle showed considerable variations from day to day (a few degrees), but the scalefactor for the reading varied only a few percents.

The preliminary measurements were made both in a fixed point of the actual flow and when towing upstream under the same conditions as

the dayly calibration. In either case the meter was given an arbitrary initial rotation and after some adaptation time needed for the vane to return to its equilibrium position, the flow angle was measured. This was done several times for.positi.veand for negative initial rotations.and the difference between the mean "positive" and "negative" readings (about 30 in either case) was taken as a measure for the backlash

in the vertical transportation system.

3.4. Measuring procedure

In order to facilitate the positioning of the current meter, it was placed on the carriage shown in figure 5, providing the possibility of automatical displacement from cross-section to cross-section and from vertical to

vertical. The displacement in a vertical, however, was not automatized: the vertical shaft was moved by an electromotor (thus avoiding the effect of the backLash in the vertical transport system; see section 3.3), which had to be operated by hand. As this vertical positioning could only be done accurately close to the sidewalls, taking the measurements vertical by

vertical would not have been quite efficient here. Therefore the measurements were taken level by level, i.e. the current meter was set in a vertical position and the velocities and flow directions were measured at this level in all verticals of the relevent cross-section before proceeding to the next vertical position.

As the measured data are e'laboratedvertical by vertical (see section ), this 'level by level' procedure is more vulne~able than the 'vertical by vertical' one, but the measurements in one cross-section took only about half a day, so that this was no real drawback. For each measuring point, the output signals from the current meter, representing the magnitude and the direction of the measured velocity vector, was integrated over the observation period of 30 s by a pulse counter and an electronic integrator, respectively, and the resulting data were stored on paper tape in order to be

elaborated in due time. Although the current meter itself provides the facility of measuring magnitude and direction of the velocity simultaneously, this additional instrumentation could deal with only one signal at a time. Therefore the direction and the magnitude of the velocity in a grid point were measured separately during two subsequent periods of 30 s.

The accuracy of the depth of flow is not quite important for the accuracy of the elaborated velocity data (DE VRIEND, 1978a), so that no accurate water level measurements are needed Ln relation to the velocity data,

i.e. when taking velocity measurements in a cross-section a global

indication of the depth of flow there is sufficient. Therefore the depht of flow in such a cross-section was measured near the two sidewalls using a thin flat ruler and the two values were averaged to yield the depth to

.be used in the elaboration of the measured velocity data.

The water level measurements for the determination of the water surface configuration, which should have a much higher accuracy, were carried out independently of the flow measurements. In order to reduce the difference in water level between subsequent verticals, the static tube was

displaced from one vertical to the adjacent one in the same cross-section, or, af ter a cross-section had been finished, to the same vertical in an adjacent cross-section. Af ter positioning the tube, an adaptation time of about 5 minutes was taken before reading the point gauge.

The water level measurements were carried out cross-section by cross-section, in a rather arbitrary order of succession. Between the measurements in two subsequent cross-sections several days may have passed and the experiment may have been stopped and restarted several times. As the water level differences of interest are very small, this implies that the measurements

taken in this way provide only reliable information on the transverse configuration of the water surface. Information on the longitudinal

configuration was obtained from an additional series of water level measurements in three longitudinal sections, viz. near the inner wall, in the axis and near the outer wall. The longitudinal profiles obtained from either series of measurements are shown in figures 13 and 14. Although there is a slight systematic discrepancy in the second half of the bend, there is hardly

any difference in the spread of the data, i.e. the water surface configuration was well-reproducible.

3.5. Elaboration of the measured data

The data gathered from the experiment represent the voltages issued by the current meter, after integration over the observation period. They can easily be translated into velocities and flow angles through siraple linear relations to be determined by calibration (see section 3.3). Eor a good understanding of the phenomena and for the testing of the mathematical model, however, the main and secondary flow are more convenient quantities.

Therefore the measured data were elaborated to main and secondary velocity components, making use of the following definition (see also DE VRIEND,

1973c and 1979): the horizontal component of the main velocity is the component of the total velocity in the direction of the depth-averaged

flow, i.e. the main flow takes place in vertical plane through the streamlines of the depth-averaged flow field; the secondary flow is perpendicular to

these strearolines, i.e. it takes place in vertical plane through the normal lines of the depth-averaged flow field.

For the elaboration a channel-fitted coordinate system ~s adapted,

consisting of two cartesian'systems (x,y,z) for the upstream and downstrearo straight reaches and a cylindrical system (<jl,R,z)for the bend. The x- and <jl-axesare directed downstream along the channel axis and the y- and R-axes are perpendicular to the channel axis and directed from the left to the right wall. The z-axis is vertical in all systems. Yor convenience, the elaboration is described for the cylindrical system only.

The elaboration proceeds as follows:

• the output of the measuring system is translated into the magnitude v and the direction a(a=U in the direction of the channel axis) of

tot

the measured horizontal velocity vector,

• this vector is decomposed into a tangential component v<jland a radial component vR'

• the depth-averaged values v<jl and vR are determined,

• the magnitude ~ tand the direction~ H~ of the depth-averaged velocitYt

to

i.e. the depth-averaged main velocity and the main flow direction, ~an now be calculated,

• the local horizontal velocity vector is decomposed into a main component vm and a secondary component vs'

the intensity of the secondary flow i , defines as 0.5 times the

. s

depth-averaged value of Ivsl~ can now be determined.

A more extensive description of this elaboration procedure ~s given

in Appendix B.•

Although thiselaboration procedure and the ones used for earlier experiments

in the present project are much alike (DE VRIEND AND KOCH, 1977 and 1973),

there are a few important differences. The earlier experiments were carried out in a large flume of mild curvature (depth to width ratio d/B

=

0.042, curvature ratio d/R

=

0.005), in which the vertical distributions of the

c

main and the secondary velocity were fairly self-similar in almost the entire flume. This provided the possibility to determine the depth-averaged

qnantities ~ t'

ä

and

i

by fitting theoretical velocity curves to the

to s

measured data. The values obtained in this way were considerably more accurate than those found by simple numerical integration over the depth of flow.

The bend in the present flume is much sharper (d/B

=

0.12, d/R

=

0.047) c

and the main and especially the secondary velocity profiles are much less self-similar. Consequently, the curve-fitting procedure yields unreliable results and a numerical integration procedure has to be applied. The main source of the errors in this procedure li~s in the approximation of the velocity distributions below the lowest measuring point (DE VRIEND, 1978a). In order to reduce these errors, the velocity components are assumed to have their theoretical form~~between the bottom and the lowest point (cf. the wall function technique applied in many numerical flow models; LAUNDER ~D SPALDING, 1974; DE VRIEND, 1979).

;K) In terms of the :st.re.am.function of the secondary flow (see DE VRIEND, 1979), this means that

i

corresponds with the maximum of the stream function in

s the vertical considered.

**) In the present case the main veloCity profile ~s assurued tG be

logarithI:li:c"near the bottom and the secondary velocity profile is assumed t.obe gi:ven by the expression derived by DE VRIEND (1976).

4. Results and discussion

A complete survey of the measured data is given ~n tab les 11

(veloeities and flow directions) and 111 (water surface levels). In the following paragraphs these results will be discussed in terms of main and secondary flow.

4.1. Vertical distribution of Ilhemain velocity

The measured vertical distributionsof:the main velocity are shown in figure 5 (for various cross-sections) and in figure 6 (for various

longitudinal sections). Obviously, the main velocity distribution is not as fairly self-similar as in flumes of mild curvature (DE VRIEND AND KOCH, 1977; cf. also YEN, 1965), but still there are some systematic features. First there is the deformation of the main velocity profile in and beyond

the bend. After entering the bend, the -veloerty"maximum shifts·downwards. rather strongly near the inner wall, less strongly, but still considerably, further outwards in the bend. This dèformation persists in the first few meters beyond the bend and then gradually decays, the first near the "outer" wall. These observations are consistent with many other experiroents on

curved channel flow (see, for instanee, SHUKRY (1949), YEN (1965), FOX AND BALL (1968), FRANCIS AND ASFARI (1971), l1ECKEL (1977)). As was

indicated briefly by RUZOVSKII (1961) and shown more extensively by DE VRIEND (197Bb), this phenomenon can be explained at least qualitatively from the transverse riedistribution of main flow momentum caused by the secondary flow. On the other hand, the velocity maximum in the upstream straight reach does not occur at the water surface, either: especially near the sidewalls the velocitiy at the surface is considerably smaller than somewhat lower in the vertical (see figure 5). This is also ancoften observed phenomenon (CHOW, 1959), which is mostly attributed to the secondary flow caused by the transverse

anisotropy of tutlbulence in non-circular channel flow (ROUSE, 1961;

GESSNER AND JONES, 1965; GERARD, 1974). Accordingly, the velocity reduction at the surface in the central part of a straight rectangular channel should become smaller as the channel becomes shallower (see also: TRACY, 1965). This is in conflict, however, with observations in a very shallow

channel (DE VRIEND AND KOCH, 1977), where velocity reductions at the surface were found troughout the flume. So far, no satisfactory explanation of this contradiction could be given.

4.2. Depth-averaged main velocity

The depth-averaged main velocity distribution (see figure 7) shows, 1n a rathér pronounced way, the typical features of curved channel flow (see a1so ROZOVSKII (1961) and YEN (1965». In the first part of

the bend the velocity maximum shifts from the centre towards the inner wall, main1y as a consequence of the redistribution of the pressure. Af ter about 450 the influence of the secondary flow becomes perceptible (DE VRIEND, 1978c and 1979): alocal velocity reduction develops near the inner wall and gradually extends further outwards. Consequently, the transverse distribution of the main velocity becomes more and more skewed outwards in the second part of the bend.

It should be noted that near the outer wall alocal velocity reduction occurs, as well. This reduction, caused by the reverse secondary

circulation in the outer bend (see section 4.3), hardly extends further inwards.

The pressure redistribution near the bend exit leads to further decelerations in the inner bend and accelerations in the outer bends, yielding a rather strongly skewed velocity distribution in the downstream straight reach, with the maximum near the "outer" wall.

The skewness of this distribution is hardly evened out 1n the first 6 m beyond the bend, partly as a consequence of the outward skewihg tendency due to the remainder of the secondary flow (see section 4.3), partly

because this skewness is evened out mainly by the bottom resistance, 1.e. at a length scale of C2/g times the depth of flow (DE VRIEND, 1976)*. Finally it should be noted that the influence of the friction at the

sidewalls, if perceptible at all, is very small in all points of the measuring grid, even in the ones closest to the walls.

4.3. Vertical distribution of the secondary flow

The vertical distribution of the horizontal component of the secondary flow is given in figure 8 (for various cross-sections in and beyond the bend) and ~ (for various longitudinal sections).

*)In rectilinear shallow channel flow the decay of the skewness of the depth-averaged velocity can be describe~ by the vorticity transport equation

v dW = _

za,

w~ (w = aVm).

üne af the mast striking features of the secondary flow field is the rather persistent reverse circulation near the water surface in the outer bend. This phenomenon was also observed during other experiments

(YEN, (1965-),GÖTZ (1975), RAO (1975), CHOUDHARY AND NARASIMHAN (1977), DE VRIEND AND KOCH (1977» and it has been shawn bath experimentally and numerically to exist in laminar flow as well (CHENG, LIN AND OU (1976), DE VRIEND (197~b». The explanation for it is essentially the same as

the one given by DE VRIEND (1978b) for fully developed laminar flow in

ft

curved rectangular channels ).

Although the influence of the deformation of the main velocity profile is perceptible.in the vertical distributions of the horizontal secondary velocity component, the secondary velocity profiles away from the outer wall are more self-siillilarthan the main velocity profiles. In the downstream

straight reach the point of zero secondary velocity shifts upwards, which ~s likely to be caused by a difference in rate of decay between the ~e~?cities in .the lower anl1l.·the uppae p ar.tof che vertical.

4.4. Secondary flow intensity

The transverse and the longitudinal·distributions of the secondary flow intensity are given in figures 10 and 11, respectively.

The influence of the transverse redistribution of the main velocity (cf. section 4.2 and figure 7) is identifiable ~n the inner half of the bend and the first few meters beyond it (see figure 10). Irithe outer half of the bend and the downstream straight reaeh, however, the picture is disturbed by the reverse secondary circulation, the influence of which on the secondary flow intensity extends almost to the channel axis.

The longitudinal distribution of

i

shows that nowhere in the bend the s

fully developed curved flow stage is reached. In the first part of the bend, up to about 75°, the intensity of the secondary flow increases rather sharply and then it starts decaying gradually, without a sharp discontinuity in the rate of decay at the bend exit.

The decay in the second part of the bend must be attributed mainly to the flattening of tihemain velocity profile: if the main velocity at the surface decreases, the principal souree term in the equation for the secondary flow

ft) As this explanation is rather extensive, reference is made to

intensity, viz. the term arising from the centripe~al acceleration, decreases and hence the secondary flow intensity itself (DE VRIEND,

1979) .

The gradual decay beyond the bend is in contrast with most of the

present mathematical modeis, in which the secondary flow has no streamwise inertia at all (i.e. it is determined by the local flow conditions; see DE VRIEND, 1976 an~ many others), or a much smaller inertia than

corresponds with the present data (DE VRIEND, 1979). In either case the growth and the decay of the secondary flow are much faster than observed here. The same discrepancy between measured and pre~icted secondary flow intensity was encountered hy DE VRIEND (1979) when simulating an experiment reported by YEN (1965).

As a consequence of the streamwise inertia, the secondary flow intensity does not reach its "equ_ilihrium" value in the bend. A strongly simplified

prediction method for the secondary flow, neglecting the streamwise inertia and the main velocity reduction at the surface, yields for a

logarithmic main velocity profile and C

=

60 mils (DE VRIEND, 1976 and 1977)

v h m

is

'"

1.67

a-s

(4.1)

, 1IRs denoting the local curvature of the main flow. This means that for

the present experiments the predicted value of

i

is about 0.05 mis, s

whereas the observed values nowhere exceed 0.04 mis and are mostly smaller than 0.03 mis.

4.5. Water surface configuration

The configuration of the water surface, represented in figures 12 through 14, shows the familiar transverse slope due to the centrifugal force acting on the water mass. This slope starts to develop just before the bend entrance, remains almost constant in the bend and vanishes within a short distance

about the bend exit. The magnitude of the transverse faU (about 0.014 m)

agrees weil with the theoretical value (see for instance, YEN AND YEN

(1971), DE VRIEND (1976» z_s: outer

- z

s. ~nner R outer f -2 v ~ dR '" gR 0.013 m (4.2) R. ~nner

The shape of the transverse configuration of the water surf ace (figure 12) reflects the redistribution of the main velocity along the bend: in the first part, where the velocity maximum lies close to the inner wall (see figure 7), the surface has a somewhat concAve shape and in the last part, where the main velocity distribution is almost uniform, the surface follows.almost a straight line. This effect is also illustrated by figure 14: in the first part of the bend the water level in the axis lies the closest to the level at the outer wall, whereas in the last part it lies almost halfway the levels at the inner and the outer wall.

Though the longitudinal water surface configuration is influenced by backwater effects (figure 13), it looks as though there is an extra fall over the bende If so, it must be attributed to an increase of

the bed shear stress in the bend as a consequence of the vertical deform-ation of the main velocity distribution (see section 4.1).

4.6. Comparison with earlier measurments under the same conditions As was stated in the introduction (section 1.1), the present experiment is an extensive repetition of an earlier one, carried out in the same

I

flume and under almost the same conditions. Only the roughness of the outer sidewall in the bend is different: at present this waU, consisting

'ofvery smooth glas s+pane Ls , has been made about as rough as 'the bottom (see section 2.1).

The measuring p~ogramme of the original experiment was limited to water level.measurements and tot~l velocity measurements at the level z

=

0.4 h. As was mentioned before, the results of the water level measurements

deviated considerably from the theory and from the results of similar experiments (see section 1.1 and also DE VRIEND, 1979). The present water

level data, however, agree weU with theltheory and with experimental results obtained by others (see section 4.5). In the following it will be shown that the flow is hardly influenced by the sidewall roughness, so that the old water level data are likely to be erroneous.

The velocity measurements were restricted to the level z

=

0.4 h, starting

from the assumption that, on the analogy of unifmrm straight channel flow

depth-averaged velocity. Regarding the deformation of the main velocity verticals in and beyond the bend, however, this assumption seems

to be rather crude here. Therefore it will be verified using the present main velocity data (the contribution of the secondary velocity component

to the total horizontal velocity is neglected).

Figure 15a shows that, especially near the inner wall, the velocity at 0.4 h differs considerably (up to 15%) from the depth-averaged velocity and that the differences gradually increase as the flow proceeds through the bend. So the magnitude of the local depth-averaged velocity is approximated not too weil by the velocity at 0.4 h. When considering the transverse distribution of the velocity, however, the results are far better, as becomes evident from figure 15b. Apart from a rather small reg ion near

the inner wall, the transverse distribution of the velocity at 0.4 h

agrees with the depth-averaged velocity distribution: though the differences are systematic, they are mostly smaller than a few per cents and the

typical features of the horizontal redistribution of the velocity are quite weil represented.

Finally, the influence.of the smooth outer wall during the old experiment will be investigated by comparing the transverse distributions of the old and the present velocity at 0.4 h. According to figure 16, the differences are quite small and there is certainly no tendency of the old velocity near the outer wall in the bend to be systematically higher than the present velocity. Hence it is concluded that the smooth outer wall in the old experiment has hardly influenced the flow.

5.

Conclusions

The conclusions to be drawn from the experiments and the analysis of the results can be summarized as follows:

displacing the current meter in a vertical may give rise to

unacceptable errors in the zero reference angle if this is done

by hand;

• the rather sharp spatial variations of the veloeities in the present flume neccesitate a rather fine measuring grid;

• the water levels are so well-reproducible, that the transverse and the longitudinal configurations of the water surface cán be derived from the same series of water level measurements;

• the vertical distribution of the main velocity shows considerable deformations, both in transverse and in longitudianl direction; hence a similarity hypothesis for the main flow (cf. DE VRIEND, 1979)·

seems not quite appropriate here;

• in all verticals the main velmcity distribution shows the typical

features of curved channel flow: it is skewed inwards in the first part of the bend, then the velocity maximum gradually shifts ~utwards, starting from the inner wall, and near the bend exit the distribution becomes

rather strongly skewed outwards; this skewness damps out very slowly; • even in the wall-nearest verticals the influence of the friction at the

sidewalls is hardly perceptible in the distribution of the depth-averaged main velocity;

• in the bend and in the downstream straight reach the expected clockwise secondary circulation is found in the greater part of the cross-section; near the outer wall, however, areverse secondary circulation occurs close to the surface;

• the vertical distribution of the horizontal secondary velocity~.component away from the outer wall is more sélf-similar than the vertical

distribution of the main velocity;

• the reverse secondary circulation near the outer wall has a.considerable influence on the secondary flow intensity in the outer half of the bend; • the streamwise inertia of the secondary flow is much larger than expected

the secondary flow intensity does not reach its equilibrium va1ue in the present bend; its maximum occurs at about 750 and

in the second part of the bend it gradua11y decays;

in contrast with the configuration measured during earlier experiments in the same f1ume and under a1most the same conditions, the measured water surface configuration agrees we11 with the theoretica1 predictions;

the velocity at the level z

=

0.4 h gives a rather poor indication of the magnitude of the 10ca1 depth-averaged velocity; the transverse distribution of this velocity, however, agrees we11 with the transverse distribution of the depth-averaged velocity in the greater part of the cross-section;

• during the earlier experiments under the same conditions the smooth surface of the outer wa11 has hard1y inf1uenced the flow.

Acknow1edgements

The author wishes to thank Messrs D.C. Pest and A.M. den Toom, both of the Laboratory of F1uid Mechanics, for the way they have carried out the measurements.

Referenees

1. CHENG, K.C., LIN, R.-C. AND OU, J.-W. (1976), Fully deve Loped laminar flow in curved rectangu1ar channels, Trans. ASME, Jnl. of F1uids Engng., 98, series I, no . 1, p , 41.

2. CHUUDHARY, U.K. AND NARASHIMHAN, S. (1977), Flow in 1800open channe1

rigid boundary bends, Proc. ASCE, Jn1. Hydr. Div., 1.03,no. HY6, p. 651.

3. CRUW, V.~T. (1959), Open channe1 hydrau1ics, McGraw-Hi11, New York.

4. DHL (1975), Combined current-ve10city/direction meter, Delft Hydrau1ics Laboratory, Technica1 description, March 1975.

5. FO~, J.A. AND BALL, D.J. (1968), The ana1ysis of secondary flow in bends in open channe1s, Proc. Inst. Civi1 Engrs, London, 39, p. 467.

6. FRAl~CIS, J.R.D. AND ASFARI, A.F. (1971), Velocity distributions in wide, curved open channel flow, Jnl. Hydr. Research, ~, no. 1, p. 73.

7. GESSNER, F.B. AND JONES, J.B. (1965), On some aspects of fully-developed turbulent flow in rectangular channels, J. Fluid Mech., 23, part 4, p. 689.

8. GOTZ, W. (1975), Sekundärströmungen in aufeinander folgenden

Gerinnekrümmungen, Universität Fredericiana Karlsruhe, Mitt. Theodor-Rehbock-Flussbaulaboratorium, Heft 163.

9. LAUNDER, B.E. AND SPALDING, D.B. (1974), The numerical calculation of turbulent flows, Computer Methods in Applied Hechanics and Engineering,

1,

p. 269.

10. MECKEL, H. (1977), Transport des sediments et topographie du lit sedimentaire dû au courant spiral dans les courbes consecutives

d'un canal, Proc. 17th Congress of the IAHR, Baden-Baden, August 1977, paper A33.

11. RAO, K.V. (1975), Secondary flow in a curved channel as revealed by a laser Doppier anemometer, Proc. LDA-Symposium, Copenhagen, p. 710.

12. ROUSE, H. (1961), Fluid Mechanics for hydraulic engineers, McGraw-Hill, New York.

13. ROZOVSKII, I.L. (1961), Flow of water in bends of open channels,

Israel Program for Scientific Translation, Jerusalem (original publication in Russian: 1957)

14. SHUKRY, A. (1949), Flow around bends in an open flume, Proc. ASCE, ~, p , 713 (also: Trans. ASCE ,

.!.!2_,

1950, p , 751).

15. TRACY, H.J. (1965), Turbulent flow in a three-dimensional channel, Proc. ASCE, Jnl. Hydr. Div.,

2l,

no. HY6, p. 9.

16. VANONI, V.A. (1946), Transportation of suspended sediments by water, Trans. ASCE, ~, p. 67.

17. VRIEND, H.J. DE (1976), A mathematical model of steady flow in

curved shallow channels, Delft Univ:. of Techn., Dept. of Civil Engng., Communications on Hydraulics, Report 76-1.

I~. VRIEND, H.J. DE (1977), A mathematical model of steady flow in curved shallow channels, Jnl. Hydr. Res.,

21,

no. I, p. 37.

19. VRIEND, H.J. DE (1978a), Accuracy of measurements in a curved open channel, Delft Hydraulics Laboratory/Delft University of Technology, TOW-report R657-VII/MI415-III.

20. VRIEND, H.J. DE (1978b), Fully developed laminar flow in curved ducts, Delft Univ. of Techn., Dept. of Civil Engng., Laboratory of Fluid

l1eehanics, Internal report 2-78.

21. VRIEND, H.J. DE (1978c), Developing laminar flow in curved rectangular channels, Delft Univ. of Techn., Dept. of Civil Engng., Laboratory of Fluid Meahanics, Internal report 6-78.

22. VRIEND, H.J. DE (1979), Steady turbulent flow in curved rectangular channels, Delft Univ. of Techn., Dept. of Civil Engng., Laboratory of Fluid l1echanics, Internal report 5-79.

23~ VRIEND, H.J. DE AND KOCH, F.G. (1977), Flow of water in a curved open channel with a fixed plane bed, Delft Hydraulics Laboratory/Delft University of ïechnology, TOW-report R657-V/MI415-I.

24. VRIEND, H.J. DE ANU KOCH, F.G. (1978), Flow of water in a curved open channel with a fixed uneven:bed, Delft Hydraulics Laboratory/ Delft University of Technology, TOW-report R657-VI/MI415-II.

25. YEN, B.C. (1965), Characteristics of subcritical flow in a meandering channel, Report of the Institute of Hydraulic Research, University of Iowa.

26. YEN, B.C. and YEN, C.L. (1971),Water surface configuration in channel bends, Proc. ASCE, Jnl. Hydr. Div.,

2l,

no. HY2, p. 303.

(s) samples Mean ,·~L -dey. Mean St ....rdev ,

la

90 6393 ·60 7663 :.11 20 45 6393 41 7664

₈

30 30 6393 37 7664 5 60 15 6393 .,25 7663 4 100 9 6393 .20 7663 3

Table I. Determination of the observation period (all data reduced to the

la

s-period)

0.025 0.463 -2.1 0.025 0.431 -2.8 0.025 0.476 -3.1 0.025 0.435 -3.9 0.025 0.476 -3.1 0.025 0.465 -3.9 0.045 0.485 -2.9 0.045 0.446 -2.9 0.045 0.504 -3.8 0.045 0.458 -4.1 0.045 0.503 -2.6 0.045 0.496 -3.6 0.065 0.499 -4.1 0.065 0.473 -3.1 0.065 0.507 -3.1 0.065 0.484 -3.6 0.065 0.526 -3.3 0.Ob5 0.515 -3.0 0.085 0.525 -3.5 0.085 0.491 -2.9 0.085 0.532 -2.9 0.085 0.499 -3.1 0.085 0.550 -3.0 0.085 0.533 -3.0 0.105 0.534. -3.5 0.105 0.522 -2.5 0.105 0.554 -2.9 0.105 0.523 -2.9 0.105 0.568 -3.2 0.105 0.554 -3.2 0.125 0.54b -3.2 0.125 0.531 -2.4 0.125 0.551 -2.4 0.125 0.539 -3.4 0.125 0.590 -3.1 0.125 0.516 -2.9 0.145 0.553 -3.3 0.145 0.568 -2.5 0.145 0.558 -2.5 0.145 0.558 -3.6 0.145 0.603 -3.1 0.145 0.588 -2.8 0.165 0.542 -2.9 0.165 0.555 -2.1 0.165 0.544 -2.2 0.16:; 0.564 -3.6 0.165 0.581 -3.0 0.165 0.590 -2.9 0.185 0.525 -1.3_ 0.185 0.'539 -1.1 O.l§;; 0.506 -1.5 ~.185 0.527 -2.6 0.185 0.560 -2.5 0.185 0.554 -2.1

VERTICAL 1 VERTlCA' 8 VERTICAL 9 vERTlCAL 10 VERTICAL 11

Z VTOT ALfA Z VTOT ALfA Z VTOT ALfA Z VTOT ALFA Z VTOT ALfA

(14) (MIS) WEGI (MI (04/S) IOEGI (M) IM/S) WEG) IMI IH/S) WEG) (104) (MIS) IOEG)

0.025 0.452 -3.8 0.025 0.425 -2.6 0.025 0.468 -3.0 0.025 0.457 -3.7 0.025 0.488 -3.7 0.045 0.487 -4.1 0.045 0.452 -3.2 0.045 0.503 .-3.3 0.045 0.480 -3.2 0.045 0.511 -2.7 0.065 0.518 -3.7 0.Ob5 0.466 -3.1" 0.065 0.523 -3.2 0.065 0.499 -3.3 0.065 0.528 -2.5 0.085 0.527 -3.6 0.085 0.488 -3.1 0.085 0.54b -3.6 0.085 0.524 -2.1 0.085 0.540 -2.1 0.105 0.559 -3.3 0.105 0.511 -3.0 0.105 0,562 -3.1 0.105 0.542 -2.9 0.105 0.554 -2.0 0.125 0.576 -3.0 0.125 0.527 -3.1 0.125 0.581 -3.7 0.125 0.565 -2.5 0.125 0.555 -1.6 0.145 0.590 -3.0 0.145 0.539 -2.9 0.145 0.583 -3.7 0.145 0.577 -2.9 0.145 0.553 -1.8 0.165 0.600 -2.9 0.165 0.542 -2.8 0.165 0.568 -3.6 0.165 0.517 -3.1 0.165 0.526 -2.3 0.185 0.558 -2.3 0.185 0.514 -1.7 0.185 0.529 .-2.1 0.185 0.545 -~.7 0.185 0.493 -2.2

CROSS-SECTION 05 (ESTIHATEO OEPTH Of fLOw 0.210 MI

VERTICAL 1 VERTICA 2 VERTICAL 3 .VERTICAL 4 VERTJCAL 5 VERTICAL 6

Z VTOT ALfA Z VTOT ALfA Z VTOT ALfA Z VTOT ALfA 1 VTOT ALfA Z VTOT ALfA

(H) (HlS) WEGI (1'4) (415) WEG) (M) (MIS) IOEGI (M) (1"'5) IOEG) 041 (HlS) ( nEG) (04) (MIS) InfGI

0.025 0.418 1.8 0.025 0.430 0.9 0.025 0.449 1.0 0.025 0.438 0.8 0.025 0.463 1.6 0.025 0.449 0.3-0.045 0.448 1.7 0.045 0.448 1.7 0.045 0.470 2.6 0.045 0.410 1.0 0.045 0.486 0.8 0.045 0.471 0.8 .0.065 0.473 1.8 0.065 0.466 1.0 0.065 0.493 1.7 0.065 0.481 0.1 0.065 0.501 0.7 0.Ob5 0.499 0.5 0.085 0.490 1.3 0.085 0.489 1.9 0.085 0.508 2.3 0~085 0.503 1.7 0.085 0.510 2.0 0.085 0.516 1.9 0.105 0.514 1.1 0.105 0.521 2.0 0.105 0.539 2.5 0.105 0.518 1.2 0.105 0.537 1.2 Q.105 0.564 1.7 0.125 0.515 2.9 .0.125 0.520 3.4 0.125 0.529 3.0 0.125 0.534 1.ti 0.125 0.559 1.5 0.125 0.539 1.5 0.145 0.530 2.7 0.145 0.533 3.3 0.145 0.552 3.5 0.145 0.528 2.5 0.145 0.575 2.7 0.1105 0.560 2.9 0.165 0.498 3.4 0.lb5 0.533 2.8 0.165 0.535 2.4 0.165 0.538 1.6 0.165 0.560 1.8 0.165 0.549 4.0 0.185 0.453 3.9 0.185 0.510 3.9 0.185 Q~519

~.e

0.185 0.522 4.6 0.185 0.553 2.6 0.185 0.553 2.9

V RTlCAL 1 V RTICA 8 VERTICA 9 IIERTJCAL 10 VERTICAL 11

Z VTOT ALfA Z VTOT ALfA Z vTOT ALfA Z VTOT ALfA Z 'ITOT ALfA

(H) (HlS) WEG) (MI ( ..IS) WEG) (M) _{(M/SI WEG)} (M) (MIS) (OEG) (MI (M/SI WEG)

0.025 0.445 0.6 0.025 0.427 1.2 0.025 0.450 2.8 0.025 0.436 1.1 0.025 0.426 1.8 0.045 0.478 0.7 0.045 0.449 0.1 0.045 0.485 1.1 0.045 0.454 2.2 0.045 0.466 2.5 0.065 0.497 0.7 0.065 0.471 0.6 0.065 0.494 0.9 0.065 0.474 2.5 0.065 0.480 2.7 0.085 0.506 1.5 0.085 0.483 2.1 0.085 0.510 1.2 0.085 0.489 2.3 0.085 0.493 3.0 0.105 0.521 1.0 0.105 0.496 1.0 0.105 0.523 0.9 0.105 0.509 2.4 0.105 0.501 3.4 0.125 0.533 1.6 0.125 0.523 1.9 0.125 0.546 1.6 0.125 0.531 2.9 0.125 0.514 3.7 0.145 0.564 1.6 0.145 0.534 1.9 0.145 0.552 1.6 0.145 0.542 2.9 0.145 0.532 3.6 0.165 0.554 1.7 0.165 0.529 1.9 0.165 0.540 1.7 0.165 0.520 2.5 0.165 0.527 3.0 0.185 0.563 2.9 0.185 0.508 4.6 0.185 0.529 2.1 0.185 0.493 1.5 0.1!~ .0~480 .5.6

0.025 0.519 0.9 0.025 0.513 -1.6 0.025 0.520 -1.9 0.025 0.520 -3.ä 0.025 0.512 -2.3 0.025 0.494 -3.-6 0.045 0.539 0.2 0.045 0.542 -1.6 0.045 0.539 -1.5 0.045 0.542 -2.9 0.045 0.531 -2.8 0.045 0.530 -3.3 0.065 0.558 0.2 0.065 0.557 -0.7 0.065 0.557 -1.3 0.065 0.550 -2.5 0.065 0.552 -2.2 0.065 0.536 -2.3 0.085 0.562 0.0 0.085 0.561 -0.4 0.085 0.564 -0.8 0.085 0.564 -1.9 0.085 0.560 -1.8 0.085 0.550 -·1.7 0.105 0.563 0.4 0.105 0.564 0.0 0.105 0.565 -0.6 0.105 0.568 -1.4 0.105 0.571 -1.5 0.105 0.558 -1.7 0.125 0.569 0.5 0.125 0.573 0.2 0.125 0.569 -0.5 0.125 0.573 -1.6 0.125 0.579 -1.1 0.125 0.570 -1.2 0.145 0.579 -0.5 0.145 0.592 -0.8 0.145 0.587 -2.0 0.145 0.595 -Z.7 0.145 0.604 -2.8 0.145 0.596 -2.4 0.165 0.558 0.1 0.165 0.584 -0.1 0.16S 0.586 -1.5 0.165 0.595 -2.3 0.165 0.602 -1.9 0.165 0.590 -1.9 0.185 0.498 1.8 0.185 0.543 0.9 0.185 0.552 -1.5 0.185 0.559 -1.4 0.185 0.578 -1.5 0.185 0.571 -l.Z

VERTICAL 7 VEATICAL 8 VERTICAL 9 VEf<TlCAL10 VEATICAL 11

Z VTOT ALfA Z VTOT ALfA Z VTOT ALfA Z VlOT ALfA Z VTOT ALfA

CM) (MIS' CDEG' IM' (MIS) WEG' CM) CMIS' WEG' CM' CMIS' (DEG' CM' CMIS, (DEG'

0.025 0.482 -3.2 0.025 0.448 -3.4 0.025 0.477 -1.2 0.025 0.4!~ -1.3 0.025 0.417 -1.8 0.045 0.509 -2.8 0.045 0.473 -2.8 0.045 0.506 -1.3 0.045 0.451 -O.Z 0.045 0.455 -0.2 0.065 0.533 -2.1 0.065 0.490 -2.1 0.065 0.521 -0.8 0.065 0.472 0.5 0.065 0.474 1.0 0.085 0.545 -1.6 0.085 0.501 -1.4 0.085 0.536 -0.5 0.085 0.495 1.2 0.085 0.496 1.6 0.105 0.544 -1.6 0.105 0.512 -1.1 0.105 0.528 -0.6 0.105 0.502 1.0 0.105 0.497 2.1 0.125 0.553 -1.0 0.125 0.508 -0.9 0.125 0.540 -0.4 0.125 0.511 1.4 0.125 0.498 2.1 0.145 0.581 -Z.l 0.145 0.538 -Z.O 0.145 0.564 -1.9 0.145 0.528 -0.2 0.145 0.494 0.5 0.165 0.577 -1.7 0.165 0.542 -1.6 0.165 0.555 -1.8 0.165 0.495 -0.8 0.165 0.470 -0.2 0.185 0.549 -1.2 0.185 0.517 -1.0 0.185 0.528 -1.4 0.185 0.455 -1.9 0.185 0.414 -1.9 '

CROSS-5ECTION 07 CESTIMAfED OEPTH Of fLOW 0.202 M)

VERTICAL 1 VERTICAL 2 VERTICAL 3 VERTICAL 4 VEATICAL 5 V RTICAL 6

Z VTOT ALfA Z VTOT ALfA Z VTOT ALfA Z VTOT ALfA Z VTOT ALfA Z \/TOT ALfA

CM' (MIS' WEG' CM' CH/S) COEG) (M) _{(MIS' IOEG' CM'} (MIS) WEG) CM) CMIS' CDEG)

_"U

CM/S) CDEG)

0.025 0.552 -3.3 0.025 0.561 -4.6 ~.U2510.:'42

-3.

Ijl 0.025 IU.:::»4o

-:».3

_{0.025 IU.~Ol} -5}'5

~:~~5

·u.~,:,~ -5.Ijl 0.045 0.585 -1.5 0.045 0.589 -2.4 0.045 0.571 -1.B 0.045 0.570 -2.6 0.045 0.536 -3.0 0.517 -3.1 0.065 0.596 -0.5 0.065 0.598 -1.2 0.065 0.590 -1.0 0.065 0.589 -1.5 0.065 0.571 -1.7 0.065 0.536 -Z.1 0.085 0.604· 0.0 0.085 0.605 -0.0 0.085 0.596 -0.1 0.085 0.595 -0.7 0.085 0.569 -0.4 0.085 0.55t. -0.3 0.105 0.618 0.3 0.105 0.615 0.3 0.105 0.608 0.5 0.105 0.596 0.2 0.105 0.587 0.6 0.105 0.567 0.8 0.125 0.614' 0.6 0.125 0.615 0.6 0.125 0.611 0.7 0.125 0.600 0.7 0.125 0.616 1.1 0.125 0.577 1.4 0.145 0.610 1.0 0.145 0.615 1.1 0.145 0.601 1.3 0.145 0.603 1.2 0.145 0.597 1.5 0.145 0.588 2.0 0.165 0.597 1.8 0.165 0.642 1.9 0.165 0.619 1.9 0.165 0.616 1.7 0.165 0.593 2.1 0.165 0.574 2.5 0.185 0.541 2.4 0.185 0.573 3.1 0.185 0.574 2.4 0.185 0.565 2.5 0.185 0.581 2.9 0.185 0.560 3.6 ~

V RTlCA 7 V RTICAL 8 VERTICAL 9 VERTICAL 10 VERTICA 11

z

VTOT ALfA Z VTOT ALfA Z VrOT ALrA Z vrOT A~~A Z VTOT ALFA

CM) CM/SI IOEG) (1'4) CH/S) COEG) CM) CMIS' (DEG) CH) CMiS) COEG) CM) CM/S) (OEG)

0.025 0.457 -5.7 0.025 0.449 -6.4 0.025 0.414 -5.2 0.025 0.373 -6.0 0.025 0.376 -4-.8 0.045 0.495 -3.4- 0.045 0.473 -3.6 0.045 0.454 -3.2 0.045 0.404 -Z.5 0.045 0.412 -1.0 0.065 0.510 -1.6 0.065 0.483 -2.1 0.065 0.473 -1.0 .0.065 0.433 -0.2 0.065 0.1t40 0.6 0.085 0.524· -0.0 0.085 0.490 -0.6 0.085 0.500 .0.4·0.085 0.451 2.1 0.085 0.1t63 2.4 0.105 0.548 0.6 0.105 0.509 0.3 0.105 0.520 1.0 0.105 0.475 2.7 0.105 0.466 3.1 0.125 0.556 1.3 0.125 0.519 1.2 0.125 0.520 1.6 0.125 0.487 3.6 0.125 0.4·73 3.4 0.145 0.563 1.9 0.145 0.529 2.2 0.145 0.535 1.9 0.145 0.482 2.9 0.11t50.469 2.9 0.165 0.552 2.3 0.165 0.517 2.6 0.165 0.519 2.3 0.165 0.465 1.8 0.165 0.435 _1.11 0.185 0.535 3.4 0.185 0.498 3.6 0.185 0.513 2.0 0.185 0.428 -0.7 0.185 0.384· -2.4 Table 11 (continued)

0.025 0.564 -8.5 0.025 0.564 -8.7 0.025 D.542 -8.7 0.025 0.527 -10.0 0.025 0.511 -10.8 0.025 0.482 -10.7 0.045 0.•582 "'4.7 0.045 0.590 -5.2 0.045 0.575 -S.2 0.045 0.556 -6.3 0.045 0.539 -7~1 0.046 0.522 -1.4 0.065 0.590 -2.2 0.065 0.598 -2.9 0.065 0.588 -2.3 0.065 0.581 -3.5 0.065 0.565 -4.0 0.065 0.533 -3.9 0.085 0.602 -0.4 0.085 0.603 -0.9 0.Otl50.600 -0.8 0.085 0.589 -1.3 0.085 0.577 -1.1 0.085 0.548 -1.7 0.105 0.598 1.5 0.105 0.601 1.3 0.105 0.605 1.5 0.105 0.588 1.1 0.105 0.581 1.4 0.105 0.566 1.2 0.125 0.605 1.1 0.125 0.618 2.2 0.125 0.614 2.3 0.125 0.595 2.3 0.125 0.588 2.8 0.125 0.575 3.1 0.145 0.604 2.0 0.145 0.608 2.4 0.145 0.613 3.0 0.145 0.598 2.8 0.145 0.608 3.5 0.145 0.583 3.7 0.165 0.581 2.0 0.165 0.598 3.1 0.165 0.600 3.2 0.165 0.598 3.7 0.165 0.599 4.5 0.165 0.576 4.7 0.185 0.548 3.7 0.185 0.558 3.7 0.185 0.576 4.8 0.185 0.558 4.9 0.185 0.567 5.4 0.185 0.549 6.4 --

--VERTICAL 7 VERTICAL 8 .vlRTICAL 9 VERTlC.AL 10 VERTICA 11

Z VTOT ALFA Z VTOT ALfA Z VTOT ALfA Z VTOT ALFA Z VTOT ALfA

CM' (MIS) WEG' CM' (MIS) WEG) (14) (MIS) WEG) (14) CM/S) WEG) (14) CM/S) COEG)

0.02510•45(1 -9.3 0.025 0.438 -8.4 0~025 0.421 -8.6 0.025 0.402 -5.0 0.025 0.411 -2.6 0.045 0.485 -6.1 0.045 0.471 -5.1 0.045 0.454 -4.3 0.045 0.440 -0.3 0.045 0.441 0.7 0.065 0.504· -2.2 0.065 0.490 -2.4 0.065 0.466 -1.0 0.065 0.460 1.8 0.065 0.462 3.1 0.085 0.526 -1.5 0.085 0.506 -0.1 0.085 0.497 0.6 0.085 0.477 3.6 0.085 0.473 4.6 0.105 0.547 1.7 0.105 0.510 2.0 0.105 0.507 2.3 0.105 0.491 5.2 0.105 0.480 4.9 0.125 0.554· 3.3 0.125 0.518 3.3 0.125 0.510 3.3 0.125 0.4.86 5.3 0.125 0.477 5.0 0.145 0.565 4.3 0'.145 0.518 4.2 0.145 0.516 3.6 0.145 0.471 3.2 0.145 0.463 2.0 0.165 0.559 5.1 0.165 0.528 5.3 0.165 0.50~ 3.4 0.l65 0.437 0.3 0.165 0.444 -0.1 0.185 0.536 6.8 0.185 0.502 6.6 0.185 0.486 4.8 0.185 0.386 -4.3 0.185 c!.38~ -3.5

CROSS-SECTION 09 (ESTIMATEO OEPTH Of fLO. 0.200 14)

V RTICAL 1 VERTICAL 2 VERTICA 3 VERTlCAL 4 V RTl CA 5 V RTICAL 6

I VTOT ALFA Z VTOT ALFA Z VTOT ALFA Z VTOT ALFA Z VTOT ALFA Z VTOT ALFA

CM) (MIS) COEG) CM) (MIS' WEG) (14) CM/S) WEG' (14' (MIS' (OEG' (14' (MIS' (OEG' (lol' (MIS) (QEGJ

U.Oil!:::I

I~:~;;

'

-'1."

U.UZ:' U.55&

-~.,

u.u~:,

~:~~:

-Y.lt u.u~:,!U.:'l' -lU.lt

~:~!~

IU.lt"l'"-'11....

t~~:

U ...ötl -11.1

0.045 -4.4 0.045 0.576 -4.8 0.045 -5.1 0.045 0.546 -5.8 0.532 -6.6 0.520 _-6.81 0.065 0.573 -1.8 0.065 0.595' -1.5 0.065 0.586 -1.4 0.065 0.566 -2.3 0.065 0.551 -2.0 0.065 0.527 -3.5. 0.085 0.566 0.4 0.085 0.591 1.3 0.085 0.597 1.;l 0.085 0.582 0.4 0.085 0.566 0.8 0.085 0.549 0.6I 0.105 0.541 1.7 0.105 0.600 2.4 0.105 0.606 2.9 0.105 0.579 2.7 0.105 0.580 3.0 0.105 0.550 2.4 0.125 0.533 1.5 0.125 0.612 3.3 0.125 0.627 3.5 0.125 0.604 3.8 0.125 0.603 4.0 0.125 0.582 4·.0 0.145 0.524· 1.8 0.145 0.598 3.6 0.145 0.617 4.3 0.145 0'.603 4.4 0.145 0.606 4.6 0.145 0.574 5.6 0.165 0.499 1.8 0.165 0.590 3.9 0.165 0.600 4.8 0.165 0.592 5.6 0.165 0.590 6.1 0.165 0.574 7.1 0.115 0.456 3.5 0.185 0~543 5.3 0~185 0.566 6.5 0.185 0.566 5.8 0.185 0.556 6.9 0.186 0.553 8.4 .

VERTICAL 7 VERTICA 8 VERTICAL 9 VERTJCAL 10 V RTl CAL 11

(~,

~~~~,

ALI'A Z VTOT ALFA Z VTOT ALFA Z VTOT ALFA Z nOT ALFA

IOEG' (14' (MIS) WEG) (M) (.14/5,1 (DEG) (14' (MIS) WEG' (M' (MIS' (OE~)

0.025 0.464- -9.3 0.025 0.458 -8.2 0.025 0.459 -5.9 0.025 0.431 -0.4· 0.025 0.422 -0.0 0.045 0.496 -7.0 0.045 0.488 .,.6.1 0.045 0.484 -2.9 0.045 0.459 2.6 0.045 0.459 3.1 '0.065'0.508 -1.7 0.065 0.501 -0.8 0.065 0.491 0.8 0.065 0.481 5.3 0.065 0.469 5.7 0.oa5 0.523 1.5 0.085 0.513 1.9 0.085 0.511 2.8 0.085 0.488 6.7 0.085 0.476 6.2 0.105 0.540 2.7 0.105 0.518 3.6 0.105 0.504 4.0 0.105 0.484 6.4 0 •.105 0.470 5.9 0.125 0.559 4.2 0.125 0.543 5.6 0.125 0.521 .4.8 0.125 0.466 5.2 0.125 0.473 5.3 0.145 0.567 6.0 0.145 0.54'9 7.3 0.145 0.518 5.1 0.145 0.460 3.1 0.145 0.466 2.1 0.165 0.561 6.5 0.165 0.532 8.7 0.165 0.501 5.9 0,165 0.444 -0.6 0.165 0.453 -0.9 0.185 0.532 9.3 0.185.0.518 10.3 0.185 0.477 7.1 0.185 0.4en -3.9 0.185 0.417 '-4.2 -- _.- -- ---Table 11 (continued)

0.025 0.552 -10.2 0.025 0.544 -10.6 0.025 0.533 -11.6 0.025 0.522 -12.6 0.025 0.509 -12.9 0.025 0.492

-rrrr

0.045 0.577 -5.8 0.045 0.578 -5.6 0.045 0.56b -6.2 0.045 0.552 -7.4 0.045 0.535 -7.9 0.04~ 0.518 -7.5 0.065 0.573 -2.7 0.065 0.588 -1.8 0.065 0.585 -2.8 0.065 0.571 -3.8 0.065 0.553 -4.1 0.065 0.543 -4.2 0.085 0.557 -0.3 0.085 0.594 0.4 0.085 0.59b 0.5 0.085 0.517 -0.2 0.085 0.510 -1.0 0.085 0.550 -0.6 0.105 0.532 1.5 0.105 0.567 1.6 0.105 0.618 2.4 0.105 0.604 2.2 0.105 0.584- 2.1 0.105 0.563 1.9 0.125 0.494· 3.4 0.125 0.548 2.1 0.125 0.622 3.8 0.125 0.61ij 4.1 0.125 0.600 4.1 0.125 0.574 3.8 0.145 0.453 4.9 0.145 0.524 1.9· 0.145 0.615 5.2 0.145 0.611 5.3 0.145 0.601 5.7 0.145 0.585 5~6 0.165 0.405 6.5 0.165 0.485 2.5 0.lb5 O.59~ 6.0 0.165 0.609 7.0 0.165 0.593 7.6 0.165 0.584 1.7 0.185 0.336 6.6 0.185 0.428 3.5 0.185 0.573 1.2 0.185 0.580 8.2 0.185 0.558 8.1 0.185 0.551 8.6

VERTICAL 7 VERTICAL 8 VERTICAL 9 VERTJCAL 10 VERTICAL 11

Z VTOT ALFA Z VTOT ALFA Z VTOT ALFA Z VTOT ALFA Z VTOT ALFA

(M) (MIS) WEG) (M) (M/SI IOEGI (MI (MIS) WEG) (M) (M/SI COEG) (M). (MIS) _IOEGI

0.025 0.489 -12.7 0.025 0.493 -11.8 0.025 0.483 -6.6 0.025 0.424 -2.9 0.025 0.429 ";'-Y.3 0.045 0.514· -1.7 0.045 0.514 -7.6 0.045 0.517 -3.5 0.045 0.465 1.7 0.045 0.467 1.3 0.065 0.527 -4.7 0.065 0.519 -4.0 0.065 0.528 -2.1 0.065 0.499 3.3 0.065 0.488 3.2 0.085 0.539 -0.9 0.085 0.524 -1.5 0.085 0.525 -0.1 0.085 0.500 3.8 0.085 0.493 4.0 0.105 0.556 1.0 0.105 0.541 0.1 0.105 0.536 0.7 0.105 0.499 3.2 0.105 0.497 3.1 0.125 0.564 3.3 0.125 0.555 2.7 0.125 0.531 1.3 0.125 0.493 1.3 0.125 0.491 1.7 0.145 0.57l 5.5 0.145 0.551 4.7 0.145 0.537 2.5 0.145 0.482 -0.1 0.145 0.489 -0.3 0.165 0.567 7.0 0.165 0.547 6.8 0.165 0.526 3.3 0.165 0.475 -3.9 0.165 0.473 -3.4 0.185 0.539 ·9.8 0.185 0.527 8.8 0.185 0.494 4.3 0.185 0.438 -5.8 0.185 0.429 -6.3

CROSS-SECTION 11 (ESTIMATEO OEPTH OF f~UW 0.195 M)

VERTlCAL VERTICAL 2 VERTICAL 3 VERTICAL 4 VERTICAL 5 VERTICAI 6

Z VTOT' ALFA Z VTOT ALFA Z VTOT ALFA Z VTOT ALFA Z VTOT ALFA Z VTOT ALFA

CM) (MIS) COEG) (14) (MIS) WEG) (14) (HlS) WEG) (lol) CHISI (OEGI (lol) (MIS) (OEG) (MI (MIS) (OEGI

0.025 0.~4'11-'11.:>0.025 0.551 -10.3 0.025 0.553 -11.9 0.025 u.!)35-12.9 0.025 10.527 -13.3 0.02~ O.:,~e.-l;:::~:' 0.045 0.580 -5.0 0.045 0.580 ;"5.3 0.045 0.583 -6.2 0.045 0.561 -7.8 0.045 0.554 -7.7 0.045 0.54·7 -8.l 0.065 0.582 -1.8 0.065 0.585 -1.2 0.065 0.595 -2.0 0.065 0.571 -3.1 0.065 0.513 -3.9 0.065 0.556 -4.3 0.085 0.514 0.6 0.085 0.556 1.2 0.085 0.604 2.0 0.085 0.592 0~8 0.085 0.579 0.2 0.085 0.568 -0.4. 0.105 0.547 l.3 0.105 0.520 2.4 0.105 0.601 4.2 0.105 0.602 3.5 0.105 0.586 3.2 0.105 0.575 2.3 0.125 0.532 4.5 0.ll5 0.481 3.3 0.ll5 0.599 5.3 0.125 0.607 5.8 0.125 0.595 5.5 0.125 0.580 4.9 c..1450.484 5.9 0.145 0.457 4.8 0.145 0.585 6.3 0.145 0.610 7.2 0.145 0.600 7.2 0.145 0.586 7.2 0.165 0.446 8.3 0.165 0.425 6.5 0.165 0.565 6.7 0.165 0.595 8.7 0.165 0.593 9.1 0.165 0.575 ·8.8 0.185 0.350 10.7 0.185 0.372 8.5 0.185 0.526 8.3 0.185 0.570 9.9 0.185 0.561 10.8 0.185 0.544 10.3

--VERTICAL 7 VERTICAL 8 VERTICAL 9 VERTICAL 10 .VERTICAL 11

Z

~~~~,

ALI'A Z vrOT IALFA Z vror IALI'A z VIUI IA~I'A Z Ivror IALI'A

UU WEG) (lol) (MIS) WEGI (MI (MIS) COEG) (MI (14/51 COEGI CH) CH/S) (OEG)

0.025 0.:,16-12.5 0.025 0.516 -10.2 0.025 0.492 -6.2 0.025 0.438 -2.6 0.025 1°·443 -J.o 0.045 0.540 -7.5 0.045 0.538 -6.3 0.045 0.534 ..,2.60.045 0.480 1.0 0.045 0.466 0.8 0.065 0.549 -4.1 0.065 0.551 -2.6 0.065 0.543 0.1 0.065 0.50j 2.7 0.065 0.491 3.3 0.085 0.553 ·-0.9 0.085 0.55Z -1.1 0.085 0.545 0.5 0.085 0.506 4.1 0.085 0.498 4.7 0.105 0.564 1.5 0.105 0.549 1.5 0.105 0.540 1.6 ·0.1050.508 4.3 0.105 0.495 4.5 0.12$ 0.561 4.4 0.125 0.553 3.2 0.lZ5 0.540 1.8 0.125 0.499 Z.7 0.125 0.497 2.6 0.145 0.569 6.0 0.145 0.559 4.6 0.145 0.536 2.0 0.145 0.497 0.7 0.145 0.493 0.6 0.165 0.564 8.3 0.165 0.549 7.1 0.165 O.SZO 2.5 0.165 0.471 -1.9 0.165 0.416 -1.4 0.185 0.532 9.9 0.185 0.520 8.3 0.185 0.481 3.5 0.185 0.431 -5.2 0.185 0.436 -4.4 Table 11 (continued)

0.025 0.548 '-9.4 0.025 0.51t7-10.0 0.025 0.51t6-11.6 0.025 0.543 -13.1t 0.025 0.51t0-12.8 0.025 0.537 -12.7 0.045 0.566 -5.6 0.01t50.566 -1t.9 0.01t50.570 -6.2 0.01t50.561t -8.0 0.01t50.560 -8.6 0.046 0.558 -'1.6 0.065 0.57lt -l.3 0.065 0.568 -1.2 0.065 0.586 -1.6 0.065 0.519 -3.1 0.065 0~577 -3.6 0.065 0.573 -3.lt· 0.085 0.576 1.5 0.085 0.548 2.1 0.085 0.591t 3.l 0.085 0.596 1.3 0.085 0.593 1.0 0.085 0.580 0.7 '0.105 0.564 1.8 0.105 0.516 2.0 0.105 0.587 1t-.20.105 0.603 3.6 0.105 0.596 2.9 0.105 0.584 2.2 0.125 0.5lt6 3.4 0.125 0.502 3.1 0.125 0.568 4.9 0.125 0.608 5.4 0.125 0.609 5.3 0.125 0.590 lt-.6 -0.145 0.506 5.0 0.145 0.486 4.1 0.145 0.537 5.2 0.U5 0.596 6.4 0.145 0.597 6.8 0.146 0.590 6.5 0.165 0.452 7.1 0.165 0.446 6.i 0.165 0.487 4.6 0.165 0.512 1.5 0.165 0.587 8.6 0.166 0.578 8.2 0.185 0.374 10.7 0.185 0.392 9.4 0.185 0.437 6.1 0.185 0.528· 9.8 0.185 0.542 10.3 0.185 0.542 10.1

V RTl CA 1 V RTl CAI 8 VERlICAL 9 VERHCAL ·10 VERTICAI 11

l VTOT ALfA l VTOT ALFA Z VTOT ALfA Z VTOT ALFA Z VlOT ALfA

CM) CM/S) (DEG) _CM' CM/S) eDEG) CM' CM/S' eDEG) (!OU _{C"'/S) WEG)} (M) CM/S) COEG)

0.025 0.528 -9.7 0.025 0.522 -7.8 0.025 0.485 -2.2 0.025 0.447 -0.6 0.025 0.462 -0.7 0.045.0.556 -7.0 0.046 0.550 -4.7 0.01t50.525 0.1 0.045 0.481 3.0 0.045 0.481 2.5 0.065 0.563 -1.7 0.065 0.557 -1.1 0.065 0.51t8 2.5 0.065 0.509 4.9 0.065 0.499 5.3 0.085 0.570 0.8 0.085 0.564 1.4 0.085 0.554 1t.0 0.085 0.519 6.7 0.085 0.505 6.3 0.105 0.579 3.5 0.105 0.573 3.5 0.105 0.552 4.3 0.105 0.523 6.4 0.105 0.507 5.7 0.125 0.581 5.4 0.125 0.573 5.1 0.125 0.553 4.1 0.125 0.515 5.0 0.125 0.511 4.8 0.145 0.573 5.7 0.145 0.565 5.3 0.145 0.538 4.2 0.145 0.514 3.2 o.11t50.506 3.0 0.165 0.564- 8.2 0.165 0.552 7.4 0.165 0.528 It.o 0.165 0.487 0.4 0.165 0.481 0.3 0.1"850.526 8.1 0.185 0.521 7.3 0.185 0.501 3.6 0.185 0.41t4 -2.3 0.185 0.439 -2.5 ----

-CROSS-SECTION 13 CESTIHATED DEPTH OF FLOW 0.195 M)

V RTlCAL 1 VERTICAL 2 VERTICAL 3 VERTICAL 4- VERTICA 5 V RTICA 6

C~) VTOTCM/S' ALFAWEG) CM)Z VTOTIMIS, ALFAWEG) CH)Z VlOT114/5) ALFA(DEG) CM)Z VlOTCH/S) ALFACOEG' CM)Z

"'Of

CM/S) ALfACOEG) CM)Z IVTOICM/S' ALFA(DEG·)

0.025 0.544 -9.7 0.025 0.539 -9.9 0.025 0.554 -11.1 0.025 0.551 -13.4 0.025 0.51t7.;r2.8 0.Ol5 0.540 -11.9 0.045 0.570 -5.9 0.045 0.556 -5.0 0.045 0.571 -b.O 0.045 0.514 -7.7 0.045 0.568 -8.0 0.045 0.562 -1.8 0.06l 0.565 -2.7 0.065 0.549 -2.4 0.065 0.578 -1.7 0.065 0.586 -J.4 0.065 0.579 -J.8 0.065 0.570 -4·.2 0.08 0.565 -0.5 0.085 0.537 -0.2 0.085 0.566 1.3 0.085 0.589 0.6 0.085 0.586 -0.2 0.085 0.577 -1.0 0.105 0.551 0.9 0.105 0.521 1.2 0.105 0.555 3.5 0.105 0.587 2.9 0.105 0.591 2.0 0.105 0.578 1.6 0.125 0.524· 2.8 0.125 0.505 2.2 0.125 0.517 3.5 0.125 0.580. 1t.5 0.125 0.588 4.2 0.125 0.581 3.9 0.145 0.492 4.6' 0.145 0.1t80 3.4 0.145 0.490 3.6 0.145 0.566 5.7 0.145 0.584· 6.1 0.145 0.585 5.4 0.165 0.436 6.7 0.165 0.447 5.0 0.165 0.458 3.5 0.165 0.536 6.0 0.165 0.568 7.1t 0.165 0.568 6.9 0.185 0.363 9.2 0.185 0.383 7.6 0.185 0.422 1t.5 0.165 0.495 7.3 0.185 0.519 8.6 0.185 0.528 8.3

.,,~

... V.<I~~ - .... v u.,,~.., V.;J~V

-...".~

v.vc;;;;, v ...gg -;J. , u.v~~ V."DV

-~

.

.:: "'I!.UC:J u.

0.045 0.560 -6.9. 0.045 0.546 -5.6 0.045 0.5Z6 -2.7 0.045 0.494 0.1 0.045 0.492 ..0·.2 0.065.0.565 -3.5 0.065 0.557 -2.8 0.065 0.51t8 -0.6 0.065 0.513 2.2 0.065 0.507 2.1 0.085 0.569 -1.4 0.085 0.563 -0.8 0.085 0.550 0.8 0.085 0.520 3.5 0.085 0.511 3.0 0.105 0.568 1.2 0.105 0.564 0.6 0.105 0.555 1.4 0.105 0.521 3.5 0.105 0.511 3.2 0.125 0.573 3.3 0.125 0.561 1·.9 0.125 0.549 1.6 0.125 0.517 2.6 0.125 0.517 2.6 .,0.1·450.569 ••6 0.145 0.565 3.1 0.145 0.544 1.4 0.145 0.510 -0.0 0.11t50.508 0.2 0.165 0.562 6.1 0.165 0.554 4.5 0.165 0.530 f.4 0.165 0.492 -2.5 0.165 0.491 -3.0 0.115 0.526 . 8.4 0.185 0.517 6.1 0.185 0.1t89 1.9 0.185 0.449 -5.5 0.185 0.441t·-6.1 Table II (continued)

0.025 0.547 -8.4 0.025 0.548 -8.8 0.025 0.5;-8 -9.5 0.025 0.557 -10.~ 0.025 1°·555 -12.0 0.025 1°·552 -10.9 0.045 0.571 -5.1 0.045 0.566 -4.8 0.045 0.582 -5.2 0.045 0.580 -6.7 0.045 0.581 -6.8 0.046 0.577 -6.7 0.065 0.583 -2.7 0.065 0.565 -2.0 0.065 0.575 -1.5 0.065 0.590 -2.2 0.065 0.594 -3.0 0.065 0.585 -3.5 0.085 0.576 -0.7 0.085 0.557 0.2 0.'HI50.570 1.5 0.085 0.593 1.1 0.085 0.599 0.4 0.085 0.587 0.1 0.105 0.567 1.0 0.105 0.545 1.5 0.105 0.5'45 2.7 0.105 0.586 3.1 0.105 0.598 2.5 0.105 0.592 2.2 0.125 0.540 2.7 0.125 0.531 2.3 0.125 0.529 3.4 0.125 0.573 4.3 0.125 0.589 4.1 0.125 0.597 3.4 0.145 0.510 3.6 0.145 0.502 3.0 0.145 0.502 3.0 0.145 0.551 4.4 0.145 0.583 5.4 0.145 0.589 5.0 0.165 0.459 6.7 0.165 0.463 4.1 O.lb~ 0.473 3.2 0.165 0.527 4.6 0.165 0.556 5.9 0.165 0.569 6.1 0.185 0.374 8.3 0.185 0.401 7.0 0.185 0.416 _'+.q 0.185 0.475 5.6 0.185 0.511 6.9 0.185 0.528 7.5

VERTlCAL 7 VERlICAL 8 VEIHICAL 9 VERTICAL 10 VER1ICAL 11

Z VTOT ALfA Z VlOT ALFA 1 '1101 ALfA Z VlOT ALFA Z VTOT ALfA

(MI (101/51 WEG) (Hl (H/SI WEG) (MI (lol/SIWEG) (M) (H/SI WEGI (MI (lol/SI(OEGI

0.025 0.542 -10.3 0.025 0.531 -8.2 0.025 0.495 -5.2 0.025 0.471 -3.6 0.025 0.487 -3.2 , ,0.045 0.571 -5.8 0.045 0.558 -4.7 0.045 0.539 -2.4 0.045 0.512 0.1 0.045 0.508 -0.2 I 0.065 0.579 -2.7 0.065 0.568 -1.9 Ó.065 0.554 0.1 0.065 0.523 2.7 0.065 0.520 2.5 0.085 0.582 -0.5 0.085 0.576 -0.2 0.01:150.563 1.5 0.085 0.534 3.9 0.085 0.524 3.8 , 0.105 0.582 1.7 0.105 0.576 1.3 0.105 0.562 2.3 0.105 0.534 4.1 0.105 0.531 3.6 0.125 0.586 3.0 0.125 0.517 2.2 0.125 0.557 2.0 0.125 0.523 3.0 0.125 0.526 2.9 0.145 0.590 4.5 0.145'0.578 3.4 0.145 0.556 1.6 0.145 0.523 0.6 0.145 0.527 0.5 I 0.165 0.569 6.2 0.165 0.564 4.2 0.165 0.538 0.4 0.165 0.506 -2.5 0.165 0.506 -2.0 0.185 0.535 6.7 0.185 0.535 6.1 0.185 0.501 0.5 0.185 0.464 -5.5 0.185 0.459 -5.2 ---

---CROSS-SECTION 15 (ESTIMA1EO DEP1H Of fLOw 0.190 MI

VERTICAL 1 VERTICAL 2 vERTICAL 3 VER1ICAL 4 VEiHICAL 5 VERTICAL 6

Z .VlOT ALFA Z VlOT ALfA Z VTOT ALfA Z VTOT ALfA Z vTOl ALfA Z V10l ALfA

(lol) (M/SI WEGI (MI (M/S) (uEGI (MI (M/SI'WEG) (MI (lol/SI(OEGI (lol) (MIS) WEG' (MI (M/S) WEG)

0.025.0.551 -8.4 0.025 0.545 -8.1 0.025 0.548 -9.4 0.025 0.551 -10.7 0.025 0.554 -11.1 0.025 0.547 -10.9 0.045 0.572 -4.7 0.045 0.560 -4.4 0.04~ 0.576 -4.7 0.045 0.581 -5.2 0.045 0.583 -6.2 0.045 0.574 -6.1 0.065 0.576 -2.2 0.065 0.567 -1.2 0.065 0.572 -0.6 .p.065 0.587 -1.6 0'.0650.594 -1.8 0.065 0.588 -2.6 0.085 0.572 0.3 0.085 0.551:1 0.8 0.01:150.562 2.0 0.085 0.583 1.7 0.085 0.597 1.2 0.085 0.592 0.9 0.105 0.555 1.7 0.105 0.543 1.8 0.105 0.548 3.0 0.105 0.517 3.6 0.105 0.590 3.3 0.105 0.599 3.1 0.125 0.530 3.4 0.125 0.523 2.6 0.125 0.535 3.6 0.125 0.556 4.5 0.125 0.584 5.0 0.125 0.593 4.1 0.145 0.500 4.9 0.145 0.514 3.6 0.145 0.517 3.6 0.145 0.537 4.7 0.145 0.573 5.8 0.145 0.591 5.9 0.165 0.456 6.9 0.165 0.469 5.6 0.165 0.488 4.0 0.165 0.516 4.4 0.165 0.553 6.3 0.165 0.573 7.3 0.185 0.374 8.3 0.185 0.406 7.3 0.185 0.441 5.3 0.185 0.469 5.2 0.185 0.513 7.8 0.185 0.526 7.5

VERTICAL 7 VERTICAL 8 VERTICAL 9 VERTICAL 10 vERTICAL 11

Z VlOT ALFA Z VlOT ALfA 1 VlOT ALfA Z VlOT ALFA Z VTOT ALfA

(MI (MIS) COEG) (lol) (M/SI WEG) (MI (lol/SIWEG) (lol) (MIS) COEG) (lol) 01/5 ) (OEG)

0.025 0.546 -9.0 0.025 0.525 -5.7 0.025 0.501 -3.6 0.025 0.483 ';2.0 0.025 0.495 -1.2 0.045 0.571 -6.1 0.045 0.561 -2.6 0.045 0.545 0.3 0.045 0.520 2.4 0.045 0.519 2.0 0.065 0.580 -0.9 0.065 0.570 0.6 0.065 0.560 2.4 0.065 0.530 4.5 0.065 0.521 4.0 0.085 0.584 0.7 0.085 0.578 2.0 0.085 0.570 3.1:i 0.085 0.541 5.6 0.085 0.533 5.0 0.105 0.593 2.9 0.105 0.582 .3.1 0.105 0.573 4.2 0.105 0.542 5.5 0.105 0.534 5.6 0.125 0.591 4.2 0.125 0.588 4.8 0.125 0.569 3.'J 0.125 0.539 4.8 0.125 0.531 4.0 0.145 0.597 5.6 0.145 0.589 4.5 0.145 0.568 2.9 0.145 0.533 2.1' 0.145 0.535 1.8 0.165 0.583 7.1 0.165 0.579 5.9 0.165 0.553 2.4 0.165 0.524 -0.3 0.165 0.512 0.2 0.185 0.543 7.9 0.185 0.537 5.8 0.185 0.507 2.3 0.185 0.477 -3.6 0.185 0.458 ·4.0 Table 11 (continued)