Improved simulation of main flow and secondary flow in a curved open channel

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simulation of main flow and secondary flow in a curved open channel

J,oh.G.S.Pennekamp and R. Booij

Report no. 1 - 84

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Laboratory of Fluid Mechanics Department of Civil Engineering

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Delft University of Technology

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Improved simulatien ef main flow and secondary flow in a curved open channel

Joh.G.S. Pennekamp and R. Booij

Report no. 1 - 84

Laboratory of Fluid Mechanics Department of Civil Engineering Delft University of Technology

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Summary

For the computation of depth averaged flows in tidal

channels and rivers a recently developed fully.implicit finite

difference method of the ADI-type proved vastly superior to the

partly explicit variantsf commonly used. The fully implicit

variant allowed a relatively large time step in combination with

arealistic lateral diffusion coefficient.

The fully implicit method, used, requires a square grid for

the time being. The irregular numerical representation of the

sidewalls of a curved channel or flume gives rise to disturbances

in the computed flow field. In this report a correction scheme

to keep the disturbances small is considered. In this scheme a

correction is realized by a modification of the depth near the

osidewalls. The correction scheme is actually a combination of

twe different corrections. The first correction is aimed at

forcing the velocity at corner points of the computational grid

in the direction of the physical boundary. The second correction

compensates for the local narrowing and widening of the flow by

the irregular numerical representation of the ~idewalls.

The influences of both corrections appear to be small when

applied separately, but the combination of the two corrections

has, however, important consequences. The application of the

combined corrections on the simulation of steady flow in a curved

flume with a rectangular cross-section annihilates nearly the

important disturbances caused by the use of a rectangular grid.

For a curved flume with the geometrical proportions of a river, in

which case the disturbances proved much less important, a less

important amelioration is obtained. For flow in tidal channels

even smaller disturbances and less improvement are ~xpected. The

effect of the corrections is very sensitive to the exact waterdepth

at the sidewalls. In time dependent flows it is therefore

difficult to apply the right corrections. Further improvement

\~iII have to await the possibility te use a curvilinear grid in

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This research is aimed at the computation of secondary flow in tidal channels based on the depth averaged velocity field. The improved computation of the depth averaged velocity field

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brings about an improved computation of the secondary flow. Knowledge of the seccndary flow in a tidal channel is essential for predictions about the morphology of the bottom.

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Contents

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·Summary ₂ Contents 4 List of figures ₅ 1. Introóuction ₆

~

.

...::.

.

Mathematica! description ₈

8 9

2.1 Depth averaged flow

2.2 Secondary flow

·3. The computation of depth avpraged flow

3.1 The flow configurations

3.2 Reproduction of the velocity field

11. 11 12

4. Correction schemes for the computation

of depth averaged flow

4.1 Velocity direction correction

4.2 Flow area correction

14

14 17

5. Improved reproduction of the depth averaged flow 19

6. Reproduction of secondary flow 22

7. Conclusions

References 24

Notation 25

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5 Li st of f igUt-es 1. Definition sketch.

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~_"_{_}_'

.

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4a=. ..J.

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6.

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7.

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8. 9.

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10.

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11.

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12.

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_13.

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14.

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15.

Geometry of the DHL-flume with the plane bed.

Geemetry of the DHL-flume with the uneven bed.

Computational grid with !J.

=

0.40 m.

Depfh averaged velocity distributions in several

cross-sections (plane bed configuration).

Depth averaged velocity distributions in several

cross-sections (uneven bed configuration).

Obstruction of the flow at the euter side of the bend at

Widening 6f the flow at the inner side of the bend at 25°.

cross-sections (plane bed configuratien), combined corrections.

Depth averaged velocity distributions in several cr oss-sections' (plane bed configuration) , flow area correction.

Depth averaged velocity distributions in ~everal

cross-sections (plane bed configuration) , velocity direction

correction.

Depth averaged velocity distributions in ~everal

cross-sections (uneven bed configuration), combined cerrections.

cross-sections (uneven bed configuration) , determined by the bottom

friction.

Secondary flow intensity di'stributions in several

cross-sections (plane bed configuration). No correction applied.

Secondary flow intensity distributions in several

cross-sections (plane bed configuration). Combined corrections.

16. Secondary flow intensity distributions in several

cross-I

sections (uneven bed configuration). No correctien applied.

17. Secondary flow intensity distributions in several

cross-I

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6 1. Introduction

A thorough knowledge of the secondary flow in tidal

channels with alluvial bottoms is required for the predictions of

their morphology~ because this secondary flow gives rise to

hottom slopes transverse to the main flow. This research, which

is financially supported by.the directorate of the Deltadienst of

Rijkswaterstaat, concerns the determination of the secondary flow

in tidal channels of estuaries like the Eastern Scheldt, based on

a known depth averaged velocity field. The depth averaged

veloeities must be computed with a high accuracy in order to make

possible areasonabie determination of the secondary flow.

For the computation of the depth averaged veloeities.

generally an implicit finite difference method of the ADI-type is

used. In such a method the depth averaged equations of motion and

the depth averaged continuity equatien, together cal led the

'shallow water equations, are solved by means of an Alternating

Direction Implicit computatian using a staggered spatial grid.

Although the velocity and waterlevel parameters are treated

implicitly, in general the cenvective and diffusion terms are,

however, treated explicitly in the difference equations. In this

partly explicit representation a large diffusi~n coefficient is

required in order to suppress a possible instability lest an

unecanomically small time step has ta be used eVreugdenhil and

Wijbenga, 1982}. Such a large diffusion caefficient, compared to

the physical eddy viscasity, severely hampers the representation

of the velocity distributions in the considered steady or

quasi-steady flow (Pennekamp and Booij, 1983 and Booij, 1983).

Recently the Dienst Informatieverwerking of Rijkswaterstaat

developed a fully implicit finite difference method of the

ADI-type. This method is usually referred to as Miniwaqua. In this

fully implicit methad no diffusion coefficient is r~quired'for

stability, so arealistic diffusion ~oefficient can be introduced

(Stelling, 1983). To investigate the reproduction of the depth

averaged flow in circumstances comparable to bends in tidal

channels, using this fully implicit method, computations were

executed for steady flow in a curved flume of the Delft

Hydraulics Laboratory (Booij and Pennekamp, 1983). The

computations concerned two different battom topographies, for

bath of which extensive masurements were available: a rectangular

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cr-oss-sectian <de Vr-iend and Koc h, 1977) and an uneven bott.om

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topogr:aphy as found in river b.ends (de Vriend arrdKoc h , 1978). The reproduction of the depth averaged velocity field was satisfactory in the uneven bed case. The influence of the

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secondary flow on the main flow cannot be reproduced by Miniwaqua. This influence is not very large because of the gentie curvature of the flume. The influence of the secondary flow is very small in the case of the rectangular cross-section. Here disturbances connected with the numerical representation, which also appear in the uneven bed configuration,

but are not very important there~ are however very strong,

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because of the large sidewalls in this plane bed configuration.

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The reproduction of the secondary flow is reasonable. Only in regions in which the depth averaged vel~city field is strongly ·influence~ by the sidewall disturbances, a defective reproduction

occurs.

In this report a correctien procedure te suppress the disturbances ~onnected with the numeri cal representation is investigated.

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8 2. Mathematical description

2.1 Depth averaged flow

The computation of the depth averaged flow is based on

shallow water equations of the form (Booij and Pennekamp, 1983).

a

u

a

u

au

aÇ

a

t + u

ax

+ v

a

y + g

ax

/2 2' r g

uVu +v

_ ~

_ (')...

+ -

-c-

2- h ph H V (1)

a

v

+ u

a

v +

v

a

v +

as

at

a

x

a

y

g

ay

J 2 2'

1L vVu

+v

_

'rwy

+

n

u

è2 h ph +

o

(2)

a

ç

a

(hu)

- + ___.;._,-+

a

· t

ox

a(hv)

a

y

=

0 (3)

In these equations the following notation is used (see also

definition sketch, fig. 1):

x ,y horizontal coordinated, z is the vertical coordinate;

t time;

u,v depth-averaged velocity-component in x-,y-direction;

waterlevel above reference level;

acceleration due to gravity;

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h watet-depth;

p ma ss density;

components of surface shear stress;

L ,'r

wx wy

n

Cor-iolis parameter-: 2w sin <P ,wher-e<P is the geographic

C

latitude and

w

is the angular velocity of the rotation

of the earth;

Chézy coefficient;

diffusion coefficien~

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Shallow water equatio~s ~an be obtained by integrating the

Reynolds' equations for turbulent, flow over the depth, assuming a

hydrostatic pressure distribution along each'vertical. Same

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additional assumptions about the shear stresses are reflected in

the form of the shallow water equations used (eqs. 1,2 and 3).

Jhe bottom shear stresses are assumed to act opposite to the

directien of the mean velocity vector and to vary with the mean

velocity squared. The effective stresses in vertical planes are

replaced by diffusion terms, with an isotropic diffusion

coefficient,E, which is constant in time and throughout the flow.

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A goed choice for E is an average value of the lateral eddy

viscosity (Booij and Pennekamp, 1983)

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The overbar in expression (4) indicates averaging over the

flow field.

The shallow water equations are solved nu~erically with

Miniwaqua.

2.2 Secondarv flow

The flow pattern in river and channel bends is quite

comple}~. A main flow can be defined by the horizontal velocity

component, us(z) in the direction,s, of the depth averaged

velocity. In addition to this main flow a secondary flow,

defined by the horizontal velocity component, un(z), in the

normal direction, n, can be important. The main flow velocity

can be described properly by its depth averaged value, us' but

the depth averaged value of the secondary flow velocity is

zero. The secondary flow can be described by its intensity, i.e.

half the averaged absolute value (de Vriend, 1981),

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n = 2~

J

Iun(z) I dz

depth

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In tidal channels two contributions to the secondary flow can be disti nguished, curvature of the mai n f1ow and the Corrol is

acceleration. In the curved flume considered in this report the

enly important souree of secondary flow is the curvature of the

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flows the secondary flow can be assumed fully developed

everywhere. The intensity of the fully developed secondary flow

caused by a curvature of the main flow with a radius of curvature

R is (Booij and Pennekamp, 1983) I

=

n u h c

12--2 R K (6)

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where K is von Karman's constant and c is a function of the Chézy

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coefficient only. The value of c derived in most theoretical e~aminations is slightly lower than the value suggested by

·measurements in flumes (de Vriend~ 1981). For C

=

50 mIls the

theoretical value is about 0.25 but the value obtained from

measurements is about ~ij3 times as large.

The computation of the intensity of the secondary flow in

.this report is based on equation 6. The radius of curvature of

the main flow, R, is calculated using

au

_n

=

R

uas

s

(7)

where un is the depth averaged velocity component normal to the

direction of the flow at the point in which R is calculated.

The intensity of the secondary flow as given by equation (6) does

only depend on the depth averaged flow. An incor-rect

reproduction of the depth averaged flow is reflected in a

corresponding incorrect reproduétion of the secondary flow.

Disturbances of the depth averaged flow near the sidewalls of the

flume can deteriorate the computation of the secondary flow as

the radius of curvature of the main flow depends on a gradient of

the main flow velocity (see eq. 7). lts correct computation is

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hence severely impaired by a scatter of the depth a~eraged"

velocities in neighbouring grid poiMts, caused by the irregular

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3. The computation of depth averaged flow

3.1 The flow confiqurations

To investigate the accuracy of the computations of the depth

averaged flow by Miniwa~ua a comparison with measurements was

executed CBooij and Pennekamp, 1983). No measurements of flow in

tidal channels, with enough precision and detail to make an

investig~tion of this accuracy feasable, are known. Dnly flows

in Laboratory flumes are investigated thoroughly enough.

The calculations are executed for a flume in the Delft

Hydraulics Laboratory,called the"DHL-flume in this report. In

this large flume, with a rather gentie bend CB/Rf

=

0.12, with B

the width and Rf the radius of curvature of the channel axis) of

almost 900, extensive measurements were e~ecuted for two

different bed configurations. In the first series of experiments

the bed of the flume was plane and the cross-section rectangular

(see fig. 2) (de Vriend and Kocri, 1977). In the other series of

experiments the flume was provided with a fixed uneven bottom of

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more or less the same shape as in a natural river bend (see fig.

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3)(de Vriend and Koch, 1978). The flume with the uneven bed is

then also a fair model of a bend in a tidal channel (Pennekamp

and Booij, 1983). The cross-sections in which·the measurèments

were executed are indicated in fig. 2 and fig. 3. The

measurements were limited to steady flow. Measurements of time

dependent flow in a curved flume in the Laboratory of Fluid

Mechanics of the Delft University of Technology are being

elaborated.

The flow is mainly controlled by the bottem friction. The

distributions of the depth averaged velocity in the

cross-I

~ections reflect therefore mainly the depth distributions.

Deviations from the measurements are somewhat easier to analyse

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in the plane bed configuratien than they are in the uneven bed

configuration., Besides, the effects of the irregular numerical

representation of the sidewalls by the square grid, used, are

much larger in the plane bed configuration. The plane bed

configuration is properly speaking too strong a test for the

re~roduction of flow in a tidal channel. It gives however an

indication of the result~ to be expected from the numeri cal

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reproduction of the flow, measured at the Delft University of

Technology. The cross-section of the flume in which these

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masurements were executed is also rectangular.

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The distance between neighbouring grid points in the

considered computation was ~= 0.40 m (see fig. 4) ~ and the time

·step used was ~t= 1.5 s. These values were chosen because of

available memory space, accuracy, stability an~ efficiency.

The computations were e~écuted with a free-slip boundary

condition

au

s

an

_wal1

=

0

(8)

.The direction of n is perpendicular tcithe wall. The wall shear

.stre~ses, when using the physically more attractive no-slip

boundary condition~ are much too large because of the relatively

large grid spacing (Booij and Pennekamp, 1983).

3.2 Reproduction of the velocity field

The reproduction by Miniwaqua of the depth averaged velocity

fields of the DHL-flume for both bottom configurations was

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satisfactory (Booij and Pennekamp, 1983). The comparison of the

cornputations and the measurements are given i~ fig. 5 and fig. 6.

Fig. 5 shows the results for the rectangular cross-section and

fig. 6 for the uneven bed configuration. In both figures the

reproduction of the shifting of the maximum velocity to the inner

side of the bend at the beginning of the bend can be appreciated.

This effect shows the flow in this region to behave like a

potential flow. The overall distribution of the depth averaged

velocities are satisfactory. The importance of the

bottom friction is evident and is reproduced in the computations.

This is an important improvement compared to the finite

difference schemes of the ADI-type hitherto, in which hori~ontal

diffusion of momentum appeared to be tOD important.

The influence of secondary flow on the main flow is not

reproduced by Miniwaqua. This influence is very small in the

plane bed configuration, but somewhat stronger in the uneven bed

configuration. Consequently, the gradual shifting of the main

flow to the outer side of the bend is slightly too small in the

computation, as only the shift caused by the bottom topography is

accounted for. This effect is to be expected in the computation

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of flow in tidal channels tOD, -but it will be somewhat smaller

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th~re, because the flow is relatively shallower.

The most important failure in the reproduction of the depth

averaged flow is connected with the irregu.lar numerical representation of the stdewalls. Two different kinds of

disturbances can be distinguished (Booij and Pennekamp, 1983).

One disturbance is a scatter in the values 6f the depth averaged

velocity in neighbouring gid points, caused by the irregular

boundary. This effect is especially obvious at the outer side in

the first half of the bend and at the inner side in the secend

half, where the irregularities i~ the"representation of the

sidewalls appear to act as obstructions to the flow (see fig. 7).

A more important disturbance connected with the numeri cal

representatien of the sidewalls is found at the inner side in the

first part of the bend and at the outer side in the second part of the bend. There the flow does not follow the local widenings

of the flow~ presented by the irregularities in the representation

of the sidewalls (see fig. 8).

In Chapter 4 correction schemes ta suppress

_.

these

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4. Corrpction schemes fer the computatio~ of deptM avpraoed flow

4.1 Velnritv directien correction

Correction schemes' to improve the computation of the depth

averaged velocity near the sidewall boundaries in the curved part

of the f!ume are rel~ted to the exact numeri cal representation of

the flow there. In order to discuss pessible correctien schemes

a short description of the staggered grid~.used, can be helpfull.

In Miniwaqua the surface level~ the b6ttom level and the two

c omporierrts o+ the vel oc ity at-e def ined at; "dif ferent pI aces. The

locations in the horizontal plan~ of the various places, makinç

up a staggered grid, are shown below.

place where the surface level is

+

defined.

place where the bottOffilevel,b,

o

_{is defined}_.

+

place where the x-component of

the velocity is defined.

Cirection

X -

d

i

reet ion

place where the v-component of

the velocity is defined.

Four different neighbouring elements make up a computational

molecule in.which these different elements bear the same indices

although their locations in the horizontal plane are not the

same. 0 0 0

+

0

I

0 0

v

" "

b.

"

I

J

I

J

+

+ç

_j

_molecule

_i

_,

_j

"

I

0

I

0 0

j

t

+

-?

1

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Whenever an impermeabie ~oundary occurs in the space- ""

staggered grid~ the existence of.this boundary is simulated in

the ~umerical model by set~ing the velocity" of the nearest

velocity element perman~ntly to zero, where the ohysical bo~ncary

crosses a line between two neighbouring surface level points!

V=-O

~I

.:

U=-O

t

+

)

+

qu

0 0

iqv

0

+

u=-o

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As é consequence in flow regions where the sidewall is not in a

coordinate dir~ction some surface level points are surrounded by

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only two non-zero velocity elements. In a stationary flow the

continuity 'of the flow around such surface level points requires

that the discharges through these two velocity elements are

"equal, q~

=

quo When the local bottomlevel is horizontal, this

means that the veloeities in the two velocity elements have to be

·equal, u

=

v. The velo~ity at the considered surface level

points is the mean velocity of the f6ur surrounding velecity

elements. Consequently, the velocity at the surface level points

will always be oriented at an angle of 45 degrees with respect

to the grid orientation.

This means an important difference between the flow in a

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fl~m2! river or ti~al channel and its numerical reproductien by

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Mi~iwaqua or another ADI simulating system. In the prototype a

velocity direction near the sidewall parallel to this wall is

expected and measured. In the numeri cal simulation the velocity

direction in the considered points does not depend on the

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fOI sloping bottoms other

angles

angle 1..1th the direct.ion. For a horîzontal local·

beottom level

This ill-fitting of the veloc~ty direction. near the

·beundary is passet on t~ a more extensive region by the computation.

The impcrtance of this boundary directio~ effect depends on

the relative flow depth at the boundaries. Fer a flume with a

~ectangular cross-section a large disturbance is found. In the

DHL-flume with an uneven bed the effect is smaller because of the

s·maller-depth ·~,tthe side~·J2.s112.!l'j the corlsequE:IT. 1\,. '1esser

impor~ance of the flow direction· at the walIs.

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1

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even less important effect can te expected.

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Tc correct this iIl-fitting velocity direction~ local

bottem slopes car te assumed. In this way the velocity directien

can be made to correspond with the sidewall directien. These

1oC.?,lbottom sl opes at-eirrtroduc ed by a change of the bottom

eleVë.tions arourid tr:e cc,ns.idered sw-fa.ce level point. The

equality qu

=

qv still applies but the flow direction is changed.

This can be understood by the

in an element can be cemputed

following reasoning. The vel ccity

by the division of the local discharge

by the area of the ~ Iow s.ection •..Th e area of the

flow section in the numeri cal simulation is the width of the grid

spacing multiplied by the mean differences between surface level and

battom level in two neighbouring grid points.

The hattom levels of the

computational molecules of which

the surface level point is

just cutside the sidewall

0

I

v

=

0 Ob

A

u

=

O

+

,.

+

qu

Ob

jQV

Ob

B

c

.b ourid ar y , ar e changed to

give the good velocity direction!

without violating continuity in the

+

considered corner surface level

point. Tc impose an angle a with

respect to the grid direction in

correspondence with the angle of

th~ sidewall! the bed levels have to s2tisfy

v ₍₉₎

I

tal1 a u

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wh~~~ bA~bBlbC are the bottom levels in PDlrts

A,E

and C

re~pectiYely. Substitutin~ cy

=

qu' the values of 0A and bB hBve

ta s.e:tisfy

tan Cl

=

r - l(b +b )

., 2 A C

(10)

The bott.om le'/el b,_,-,~_. in point C ma._.v not te altered because this

bottam level directly influences other velocitie~.

the '·/a1LI.eof one of the remaining bottom levels bA

t-!

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determines the other.

Very unrealistic battom levels have ta be imposed in regions

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with small an~les between sidewall and gGid direction. Other

regions where very unrealistic values car.develop are regions

where the angle is about 45°. In the~.e reç ions the choice of one

bottom level determines a whole chain of bottom

4.2 F

l

n

w

area correctinn

The physical boundaries are translated into the numerical

configuratien by imposing the value of the v~locity in the

nearest velocity elements at zerb permanently. The exa.ct

position of the physical boundary within the grid spacing

is lost.

+

Espec~211y.wh2n the physical boundary has 2 small angle with

crientation this coarse repre~entation of the.

r=..-, r,

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boundary impairs the flow simulation. The relation tletween the

local discharge qv and the veloçity v in the numerical

representat ion depends on Iy on the d~pth h. In the prot o.type the

'exact placE of the physical boundary with r~spect to the

coordinates (and so to the grid) is important,.hawever, for the

area of the flow section.

To correct for this deviation in the area of the flow

section in the numerical simulation, the battom level of the

computa.tional molecule of which the surfa~e level point is just

outside the flow is to be changed in order to obtain the correct

area of the flow section of the molecule at the sidewall.

Bath the corrections are dependent on the waterlevel. This

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=~.

Improved rporoduction of thp depth averaged flow

Application of the c~rre~tion schemes modifies the bottom

configuration near the sid~walls. This modified bottom

configuration is used in the computation of the depth averaged

flow by Miniwaqua. The correction on the bottom configuration

depends on the water depth at the sidewalls, hence the correction

should be determined at every time step, especially in time

dependent flow. This procedure would consume much computation

time. In this investigation the correction~ were obtained from

the water depths computed in the 'comp~tation without a correction

for the numerical representation of the sidewalls.

Computations based on the combination of both correction

schemes as weIl as computations based on each correction scheme

separately were executed. The resulting depth averaged

velocities are compared to those resulting from the uncorrected

computation and to the depth averaged values of the measured

veloeities. The improvement of the reproduction of the depth

I

averaged flow by the various correction schemes can be appreciated

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in this way. Most comparisons are carried out for the

rectangular cross-section, where the effect of the use of the

correction schemes is largest, because of the serious disturbances

in the uncorrected flow in connection with the high vertical

sidewalls.

The depth averaged velocities computed for an originally

plane bottom configuration, modified by the combination of the

two correction schemes are plotted in fig. 9. (The veloeities at

all the grid points within a strip with a width of 1.5 m around each

cross-section are used to compose the plot, in order to provide a

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better velocity distribution by the inclusion of more grid points.

The use of this relatively broad strip also gives an estimate of

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the scatter in the velocity distribution.)

The improvement obtained by the application of this

correction is obvious. From the large disturbances at the inner

side in the first part of the bend and at the outer side in the

second part (see fig. 5) only traces remain. This means that

af~er this correction the flow is capable to follow the flume

wall at the local widenings in the irregular numerical

representation 9f the sidewalls. The cross-section at 13.8

degrees shows a small remainder of the originally very important

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The correctien of the bettom levels of the grid points near the

wal I in these cross-sections is e:{tremely 1arge becau,se of the

small angle of the flow 'with respect to the grid direction.

Hence exactly at these cross-sections a deviation of the measured

value will show up easily.

It is remarkable that both corrections, flow area correction

and velocity direction correction, separately barely improve the

depth averaged velocity field (see figs. 10 and 11), whereas the

combined correction is sa effective. The ~orrect reproduction

appears to be quite sensitive to the exact correction applied.

The scatter in the values of the depth averaged velocity~

,mainly caused by the ~ocal obstructions of the flow in the

irregular numerical representation of the sidewalls, is also

diminished. The combined correction gives the best results for

this kind of disturbance tOD.

The corrections have less effect on the reproduction of the depth

averaged flow in the uneven bed configuration (see fig. 12). The

reproduction by the uncorrected computation was already quite

good, because of the low sidewalls in this configuration. The

same low sidewalls make an important improvemént difficult.

Comparison of the results of the uncorrected computation (see

fig. 6) and of the corrected computation show an improvement with

respect to both kinds of disturbances that is small but

relatively not unimportant. Only at 13.8 degrees and at

82.5 degrees considerable disturbances remain because of the

small angles with respect to the grid directions.

The influence of the secondary flow on the main flow is

appreciable in this configuration. This influence is not

accounted for in Miniwaqua, so an important deviation of the

computed depth averaged velocities in the last cros~-seciohs is

to be expected. The velocity field obtained by the computation

appears to be mainly determined by the bottom friction as can be

concluded from fig. 13. In fig. 13 the depth averaged velocities

calculated from the surface slope and alocal Chézy coefficient

are plotted. The local Chézy coefficient is assumed to be only

dependent on the water depth. These velocities that are determined

by the bottom friction and the computed velocities based on the

combined corrections are more or less s~milar except for the

I

first part of the bend, where ~he flow behaves like a potential

(22)

I

.

I

21

flow,.and the region ar ound 84.5 degrees, y.Jhet-ethe flow is pushed

slightly to the imner side of the bend by the remaining

disturbance at the outer sidewall.

It may be concluded that the improvement of the reproduction

pf the depth averaged flow is impressive for the plane bed

configuration. The uncorrected reproduction of the uneven bed

flmaJ vJë\S already better- and the impt-ovement is smaller. A still

less deviant reproduction can be expected far- tidal channels, but

the effect of the corrections will probably be much smaller toD.

In all cases the combined cor-rections for velocity dir-ection and

(23)

"

I

22

6. Reproduction of spcondary flow

The reproduction of the secondary flow in this research is

based on the computed d~pth averaged flow. Deviations in the

.repr-oduct t on of the depth averaged flow ~'Jill,hence, have direct

consequences for the reproduction of ~he secon~ary flow. The

reproduction of the secondary flow in the plane bed configuration

was severely hampered by the disturbances of the depth averaged

flow computed without the bottom level corrections at the

sidewalls (see fig. 14). Especially the SQatter by local

obstructi6ns has a devastating effect. The use of the depth

averaged velocity field computed with the corrections applied'

increases greatly the part of the flume where reasonable values

of the secondary flow appeal'"(see fig. 15).

The reproductien of the secendary flew in the uneven bed

cbnfiguration was already satisfactory when the uncorrected

cemputation was used (see fig. 16). Ä factor- of about (1/3

between the computed and the measur-ed values, connected ~ith th~

uncer-tainty of c in equatien 6, was discussed in chapter- 2. The

reproduction using the corrected depth averaged velocity field is

slightly better (see fig. 17) because of the more correct depth

averaged field, and especially the smaller scatter. The scatter

at 82.5 degrees is caused by the large variation of the radius of

curvature of the main flow over a short distance at the end of

the flume, because of the bad reproduction of the flow there,

(24)

I

7. Conclusions

The reproduction of the depth averaged flow in a large

curved flume of the Delft ~ydraulic Laboratory with a fully

implicit finite difference method of the ADI-type, Miniwaqua, is

satisfactory on the whole but certain defects remain (Booij and

Pennekamp, 1983). These defects can, in sofar as they are

I

connected with the numerical representatien of the sidewalls in

the bend, considerably be reduced by means of a correction

I

scheme. The correction scheme combines two kinds of corrections

which have only a slight effect separately. Both corrections

modify the bottom level at the sidewalls to correct the velocity

direction in corner points of the computational grid and to

correct the area of the flow section at the sidewalls.

The reproduction of the depth averaged flow using this

correction scheme is importantly improved. For a rectangular

cross-secti6n, in which configuratien the uncorrected computation

showed quite large disturbances caused by the representation of

I

the sidewalls, only traces of the original disturbances remain

when the correction scheme is used. The effect of the correction

I

scheme is less in the case of a flume with the geometrical

proportions of a river, for which flow the disturbances obtained

by the uncorrected computation are small, however, becausé of the

smaller sidewalls. In tidal channels smaller disturbances and

less improvement is to be expected. Further improvement requires

a more natural choice of the grid configuration of the sidewalls

and the bed. The use of a curvilinear grid is not vet possible

I

in Miniwaqua, but it is foreseen in the near future.

The reproduction of the secondary flow, which is based on

I

.

I

the computed depth avera~ed flow, is consequently also much

improved when the correction scheme for the computation of the

depth averaged flow is used. The part of the flume, where

reasenabie values of the secondary flow are computed, increases

(25)

""

I

7.

I

8.

I

9.

I

24 F:eferences 1. Booij, R., 1983, di~cussion to: Vreugdenhil~ C.S. and

Wijbenga, J.H.A.! Computation of +low patterns in

2.

rivers, J. Hydr. Div. ASCE, to be pu~lished.

800ij, 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.

800ij, R. and Pennekamp, Joh.G.S., 1983, Simulation of

main flow and secondary fl00 in a curved open channel~

6.

Delft Univ. of Techn., Dept. of Civil Engrg.! Lab. of

Fluid Mech., report 10-83.

Pennekamp, Joh.G.S. and Booij, R., 1983, Simulation of flow

in rivers and tidal channels with an implicit finite

difference method of the ADI-type, Delft Univ.·of

Techn., Dept. of Civil Ençrg., Lab. of Fluid Mech.,

report no. 3-83.

Stelling, G.S., 1983, Thesis, Delft Univ. of Techn •.

Vreugdenhil, C.B. and Wijbenga, J.H.A., L982, Computation

of flm'llpatterns in rivers, ASCE-proc., 108, no.

HY 11.

Vriend, H.J. de, 1981, Steady flow in shallow channel bends,

Thesis, Delft Univ. of Techn.; Comm. on Hydraulics,

Delft Univ. of Techn. i report no. 81-3.

Vriend, H.J. de and Koch, F.G., 1977, Flow of water in a

curved open channel with a fixed plane bed, TOW, Report

on experimental and theoretical investigations, R657-V,

M 1415 part I.

Vriend, H.J. de and Koch, F.G., 1978, Flow of water in a

curved open channel with a fixed uneven bed, TOW,

Report on experimental and theoretical investigations,

(26)

I

Notation

I

c

I

.

C 9 h

I

_{i ,}_j

I

s

I

t

I

u Un un(z) Us us(z) v

I

y z

I

_Cl

I

e:

I

Kp

I

T ,T wx wy 4> w

I

bottom level bottom level ~n grid point P width of the flume

coefficient in the secondary flow intensity

Chézy coefficient

acceleration due to gravity

depth of flm'l

subscripts indicating a com~utational molecule

secondary flow intensity

local flow coordinate perpendicular to the direction

~f the depth averaged flow

discharges through velocity elements

radius of curvature of the main flow

ràdius of curvature of the channel axis

local flow coordinate in the direction of the depth

averaged flow

time

depth averaged velocity in x-direction

depth averaged velocity in n-direct~on

secondary flow velocity at level z

depth averaged velocity

main flow velocity at level z

depth averaged velocity in y-direction

horizontal coordinate

vertical coordinate

local angie bet ween flow direction and grid direction

distance bet ween grid points

numeri cal time increment

diffusion coefficient

water level with respect to a horizontal reference level

Von Karman's constant

mass density

components of the surface shear stress

1ati t_ude

angular rotation of the earth

(27)

1 I

-1

1 I

z

I

h

!

y

(28)

I

_.

-.>: \ \

I

\ \ \ \ \ 55.00

I

_E

I

u_c R:; SO.Om 0

I

0>

.s

I

QJ 00 -0 cu .tl

f

- 39

I

-41

--r-...:

_~ ~

f-F':

I-I

-40 -42

I

11 t

j , -43 _-23_._0m-U_._5m ₀ 0 13.8027.50 4:13055.00 68.80 82.50

---3>~longitudinal distance along channel axis

-11.5m

I

~3m

r ....

12x0.45m 123 4 5 6 7 8 9 10 11 12 -23.0 m CROSS - SECTION

(29)

1

1 I

·

1

1 I

I

1

1 I

I

1

1 I

CONTOUR KEY 1 - 0.2000 2 - 0.2500 3 - 0.3000 4 -0.3500 5 -0.4000 6 -0.4500 -7 - 0.5000 8 -0.5200

1 I

V

-l

1/

:1

1\

.>

i-.

1/\

1\

'\

r--

I-! 1A5m trom inner wall channel axis 1.55m trom outer wall -23.0 m -11.sm 0° 13.8° 27.5° 41.3° 55.0° 68.8° 82.5°

---'»longitudinal distance along channel axis

.-. - 25

.,.

0 )( E - 30 ~ c 0 _{- 35}

....

0 > ClI _{- 40} ~ "0 -11.5m ClI - 45 .0

1

- 50 I.(') -55 0 1 2 3 4 5 6 ~

distonce trom outer wall (m)

-23.0 m

(30)

-I

1 I

I

1

10

20

30

40

50

60

70

80 90 100 110 120

111I1I111J 111 IjI I "11 11IjII"111 I 1II111I1I II1I I I II'I I I1I I I I1' I I I111111' I I , 111 11II , 11111I ,1' I,,11 I"I'i1111"1 1"1 111111I

-=

i180

170 =

i1

70

1

1 I

· .

130 =

I

1

90 =

90

1 I

I

1 ~

11111111111111 I111 , 111111111 , 111tIllIl' 1111 1,1, 1,111 11" 1,, ,, 1, 11, 11111111"" , 111 , 1" 11,",,11" 1, 11,, 1,, 11 , 11, ,!11' 1, ,, , 1I,,,1, , , 11I

1

10

20

30

40

50

60

70

80 90 100 110 120

=

--

80 =

-

-70

=

-=

--

60 --=

-=

₅₀

--

40 =

-30

-

=

-20

--

---=

-10

-

-1

-80

=

1

60 =

50 =

1

1 I

INITIRL

GEOMETRY

(VERT.)

NMRX=

193 (HOR.)

MMRX=

120

Fig. 4. Computational grid with ~

=

0.40

m.

I

(31)

I

++++ 02.50EG_{++ +}₊ _...++ .,.++ +

.

,

...+ ++ ++++ ₊₊₊ o o +....'" o o o o o o o o o

..

+ o 68.8 OEG + + ... ... ... ..t+ .. ·...+ ... ++ + o _o ++ + +.,.+ ... .t-+... 1-' f-+ + ... o .. + + 0 o o o o o o o o + + + ++ 55.0 OEG .,.+ -I-+ + + -:- ...+ o +.~'" '''++++.} o '" .,; Ol ...

..

.

,. o +

..

... + o

....

++ o + ...ot- o El o

"

+ .. 0 + + +

_..

..

o Cl è en + _., +... ₊ + _+._, +

..

ijI.3 OEG ++ +... +++ "+ + o o o o o o o El o

..

_o

..

+ Cl Cl

..

....

..

+ 27.5 DEy ... + .. + + ... + + +

...

+

..

+ + Cl '" '" .... _o _o o o o e o

..

_o o o o _o ... Cl o Ol> Ol> + + + 13.6 OEG ++++ ++t+ +++.,. +.t- ..+ ++++

..

Cl <> Xc -V> o

..

o + o o o o

..

o O' o o o o + 0.0 OEG + f> o + + + + .. + o +0

..

o + ., o + + _o o o o Cl Zo D_,O_. erf; e +0 +

..

+ + o + + + ... + + o o o _o _o e o <> o o

'"

r o o '" " o

..

+0 -I- -I-o

..

+ ... + + +

..

+ o _o _o o o o o o CJ ,,; <> '" N e +

HEASUAEO VALUE tOE VRIlNO ANO ~OCH)

COHPUTEO VRLUE tMINIWROURJ

o

°or.-o.,.o--....,.or.-::,j-::O---lr.O-::O:---:1I-:,~:-:O:----:2T.-:O-:O---::2r.::5::0---=,)r.O:-:O:---3::r.-:~-:·O:---~T.-:(I-:O---~r.-5-"O---5r.-O-O<-,O---sr.---G'. (10 t-IIDTH lMI Fig. 5 Dcpth averaged velocity distributions in several cross-sections (plane bed configuration). Cl Il' è ++ 0.0 FOR -23.0 M o If) <, :>: 0-""

oW

..J a: u If) o

..

o ~ 0.0 FOA °02.50EG + o 0.0 rOR 68.8 OEG o 0.0 FOR 55.0 OEG o 0.0 FOR ijl.3 OEG o + 0.0 FOA 27.5 OEG +

..

+ 0.0 FOA 13.80EG o

..

0.0 FOR 0.0 Of.G o 0.0 FOA -11.0 H

(32)

I

+ Cl Cl C) en +... o ...0 o ...

..

+ o + +

..

+ ..

..

o +

..

....

+ o o + o

..

() +....,. o +

..

.

,

.....·H· o o o +

..

o o o ...

..

+

..

+ o '" c.i + + .. Ó 0.0 rOR -23.0 M C) ;:> ". Cl)

..

o o _._,...._._._._.. o o + o I./J <, r 0

-.,

c.iw ..J er U lf) + + o ~ 0.0 rOR <>62.5 OEG o <> '" e-o o Cl '" '"

.,

I./J

-

xc cr~ '" Win :r ::;) ..J LLo CJ w"; r... Cl Zo DO ..J' cr '"

...

..

_e o ... o

..

_._.._._.. + o + o + 0.0 FOA 66.6 OEG Cl o <> '" o

'"

en ·t++.. 68.6 OEG

..

₊ ·f + 0.0 FOR 55.0 DEG o 0.0 FOR ijl.3 DEG + o

..

o o o ·f o + + + o +

..

o 0.0 FOA 27.5 DEG

..

+

..

0.. 0.0 FOR IJ.6 DEG o + 0.0 rOR 0.0 OEG o ... ₀_'_.0 _FOA -11.0 H o + ·f o o +~ '+ o '"

èor.-ûo---o-r.'-jO---'I.-û~0----~lT,-5-0----~2r.~ÛO~----~2r,~~·0~--~3~.-(-10----~JT,~5~0----~~r,~ü~0---~~r"~jO~----r.,,.r(-'0---5'.-5-·0---,6,00

r-IIDTf1 (MI

I

·

1

·f o ++ + o +.. 0 +

..

0+ cf 0 + + .. 0++ ..0 + +_o

_..

o ++

..

₊ o + o +

..

_o

_..

o + o +'f o + o + +0 + o o +

~ ~ MEASURED VALUE IOEVRIlND AND KOCHI

+ COMPUTED VRLUE IHINIWAQUAJ

Fig. 6 Depth averaged velocity distributions in several cross-sections

(uneven bed configuration).

++ ijl.J OEG o _..₊₊ + 27.50EG

..

+ o

..

.

,

o

..

+

...

o + o

..

.."..t 13.60EG ++++++++++++ o 0 ..0....... + ., +

..

..·to·~ 0+ 0+ 0.0 DEG é + o + o

..

I) + -11.0H 0+ +0 o .. o. + + o + -23.0 M 0" +0 o .. o .. + o ...

(33)

1 I

'

1

1 I

I

.

I

.

1 I

I

1 I

I

/ / / I

-:I / /

I

1 /1/

//111/

/ I I I I /

/ / / I 1 I /

!

I I 1 / / /

·I///I//!

_I

---~

. . 0 .

(34)

I

.

:

.

..

f-j--;-/- ;--/--

!--I--~--~--I

////I/If

:

1//1111.

1

·

· //1///1

I

.

· //111/.

:

/ I

I / ..

1 /

I

~

I

· L

_

I

(35)

I

· I

I

02." O[G o ..;0+ o ++_o++ ..H·++ o .~+.~+o ++'.. +++ ++-H ++++ o o + o o o o

..

• ..

+ + o '" o o 0.0 fOR -23.0 H 60.60EG + IJ ++++++_o'''0 o _o o (/) <, :>:" o~

.

,

oW

.J cr U sn + o

N>-0::

u o .J o_._{_>}W o .+ • 55.0 OEG C> o cO '" o o +..;.+ + ·t ++ o .;.o +Y·t+ 0"-t ...+ ..;. + o + + o ~ O.0 FOA 062.5 DEG o + + .p... ... ..+0++.H-.;o+ ... +0... ...·iOt '1 ... + +o+ + ... 0+... _"'_·_t _-_t_.... o o + ₆₆0.0_.6 F_OOA_E_G C> o o '" _~1_.₃ _DEG C> o '" CD + IJ H.o 'H .... H-o

.

..

o +_o

..

.. '1 o .. 0+ o

..

0.0 FOA 55.0 DEG C> o en ....

..

+ o ++...'''++++

..

....

'"

... o 0 0 .. '4+ '0

•

+ + o o + 0.0 FOA ~ I.3 OEG o + + + 0.0 FOR 27.5 OEG 27.50EG o o '" .... .. + o o+ .. + .. o o .0 ₊ ₊ ₊ o o

..

0.0FOR 13.6 DEG + + o

..

o +0 ₊ + .. o + ₀_._{0 FOR} 0.0 OEG

..

+ IJ .. _o+ o o o o o tv

~ MEA5UAED VALUE 10EVRllNO ANO ~OCH)

+ •COHPUTEO VALUE IHINIWAOUAI o· 0.0 FOA -11.0 H Cl Cl

°Or:o-O---OT:-~-0---1~:-O~O----~I'r.5~O----~2r:O~O~--~2T:~5~O----~3~:~O~O-·----~j'~.5~O~--~4rl.~OO~----~T:~5~O---5~:~O~O~--~5r:~50~--~6j,OO NIOTI-I (MI

I

13.8 DEG ++ ++++++++++++++++ o + + + ... .,. + ...0" ..-t-t ri'" ... o + IJ o o o + _o

..

0.0 aEG + cl> o + + o + + _o o C) Zo DO .J' cr '",.. IJ + + + + 0+ -11.o!o0 H IJ o .. + o o o o

'"

.;

..,

-23.0t1 o' 0

..

o o o ". "

..

+ + +

..

o o _o <> Cl .,;

Fig. 9 Depth averaged veiocity distributions in several cross-sections

(36)

I

02,50EG ++++ ++++ ++- .. o}-++ +++.+ <> '" à El 0.0 rOR -11.0 M ++++ ++++ ++++ ++++ _o ." à +

'"

<, :.: 0 -'"_oW ...J 0: U

'"

o "'>-ö~ u o ...J OW' -> à IJ o _o o

..

o "! 0,0 FOR '"62,5 OEG o o 0 o o o

..

o

•

_{0,0 FOR} 68,60EG 0,0 FOR 55,0 OEG I) + 0,0 FOR 111.3 OEG I) +

• •

0,0 rnn 21,5 DEG I) + _{13.6 DEG}0.0 FOR o I)

•

0.0 fOF! 0.0 OEG o 60,80EG + ++i-+ o + +++-: ... o 0

.

..

I) <J 0.0 FOA <J -23.0 M

oOr.-00---0-'-. '-jO---'L-O-O---l"T,-~-:' O---ZT.-:0-=-0---::2r.'::-jO::---)::>,-:O:-:O:---J=",-:~-:O---:~T,-=-O-::O::'S::-O----:~r,--::S(:-r.'0::---," r,'-:;0,---.,,6.(10

I'lIDTH (MI

,

I

.. ++ + o ++-}o+ • + of. .. 0

•

f) o _o o + o

•

.. +

..

55,0OEG o o à Ol Cl + MEASUREO VALUE IDE VRllNO ANO ~OCHI

COM"UT[Q VALUE II11NIHAQUAI

++ +

..

.

,

"

₊

..

+. +

..

+ 1I1.30EG + ₊₊ + H+ +++ o _o o '" ". en + IJ o o + e e I) 21.5 OEG.

•

+ ,... + +

_..

_.

_... o + ...+ + ... o + ...

..

... + o e o o o ... o El _El I) I)

..

o

'"

ui co ~. 13.8 OEG ₊₊₊₊ ... + +:++·t o + o + + ++++++++++++++++ o e

..

.

,

o +

..

_o o I) o o e 0.0 OEG + • o

..

+ +

..

o +0 + El + + + I) + + _o El _o o I) -11.0 H 0+ l!l +0 .' _o +

• ..

+ + + + + I) + El El _e I) El c.> c.> c.> '" +0 + • +

_•

+

•

+ • + + _o o e I) _o _El o _o o CJ .; c.>

'"

c.> '" .;

Fig. 10 Depth averaged velocity distributions in several cross-sections

(37)

I

<> '" '" en ++++ 0++2.5+... DEG++++ v·..·H. + + + + + + C) 11' Ó .,.

.

,

0.0 rOR -23.0 M <> o è en <> '" '" en <> '" In

...

<> o '"

...

<> o (t) '" '-" Zo co ..J' a~

'"

<> '" -,.++-" + Il o _o .,. .o_, c.i o o o () o _o

..

Cf) <, ::E 0

-.,

oW ..J Cl: U Cf) +

..

., o o

..

o + .. 0 + .,. +"+" .:-60.0 DEG +++- ... ,. ++ + ot..,. ...+ + o + o~ 0.0 FOR °02.50EG + ...,. ·t ++ .:.... +

..

o o () o + o +. 0.0FOR 68.6DEG 0.0rOR 55.0DEG + + ., 0.0FOR ~1.3 DEG e + 0.0 FOA 27.50EG + + + _{13.8 OEG}0.0 rOA o o o

"

+ _{0.0 rOR} 0.0 OEG o o

"

0:0 rOA -11.0 M '"

'"

è

or.-o-o----~Or.-~-O----~lr,t-,O~----I~.~~-'O~--~2T.~O~O----~2r.~5~O----~Jr.O~O~----J~.~~~'O~----~~.~U~O---~T.~5~O---5r.-OO---5r.~-'(-,---6.UO I'nOTfi (MI

I

+ + o + + ₅₅_.₀_DEG +..+ +++-t .... +..-t- ++ +++ ++ +++ +-t+ 000 0 0 0 o ++ + .... ~++

..

...

++ o ++_o+ _{+ +} ₊ o ++ + + " +' + e» + + + ~1.3 DEG +1-2- +.... + +

+,

+

,

"-tot "

.

'"

"

o o o o

..

.. 0+ + + o o o o o .

,

.~. + .y + + .. + 27.5 DEC ..

..

.,

..

+ + +

..

+ +

"

o o o _o o o o o o

..

_o + 0 t + 13.00EG ....+ + ...t ++ .... ·t .,....jo·t.. ... + + + _o _o ₀ o o + o o o o o

"

+

..

o

.

,

+ 0.0 OE(, + e o o

.

,

+ + o o + ..+ o +0 + _o o o + + + o + ,,+ -11.0t!> M o

"

..

+ " + + + o + +

"

o o

_"

+ + ..

"

.,Il ₊ + + + _o

..

o Il o o o I) o ~MERSURED VRLUE (DEVRllND RND ~OCHI + COMPUTEO VRLUE (MINIHROURJ

Fig. 11 Dcpth averaged velocity distributions in several cross-sections

(38)

I

· I

I

o ., o en <> <> ".

.,

<> <> <0 CD IJ) ... xo cr~ ". UJ'" :>: ::> ...J u..0 <.> UJC;; r ... ,_ Cl Zo 0<> ...J' cr~ <> <> C

'"

o <> ". ... o +

"

.. O'

"

+ ++ + 02.5 DEC o _o o

..

o

....

o o o +e, + o '" è o 0.0 FOR -23.0 M + + .. + IJ

.. ..

o

..

+

.,

o

..

a> .. ..+ 0...

....

.

0+ + o

..

++ _o +....

'"

<, :>: 0-'" oi.JJ ..J er U

'"

o "' >-à~ U o ..J oUJ .~> è o "! 0.0 FOR 082.5 DEC 0.0 FOR 66.8 DEG + 0.0 FOR 55.0 DEG +<0 -#0 0.0 FOR 4 I. 3 DEG +

..

o + +

..

+ + .. IJ

..

IJ

..

o o

..

60.80EG +_o + ...0 +<+ tO + o + ol'

..

o 0.0 FOR 27.5 OEG +

..

_..

0 .. 0.0 FOR 13.8 OEG o

..

0.0 FOR 0.0 DEG o + _0.0 _FOR -11.0 M + o +0 +

....

0+.+ + +0 + + o + +

I

o <>

°Or.-o-o---or.,-jo---'1-.o-o---I'.-~-O---~T.-O-0----~2r.~~~O----~3r.O~O~----J~.~~~·O~----~'.~O~O----~~T.~5~O---=5r.~O~O---=Sr.S~O~----S~.UO

NIDTfi lMI

I

+ o o o +.. e ++++.,. + + 0 ..+ 55.0 OEG +.. o ++..

..

₊

_.

o + + 41.3 OEG o ++> ++ ++. o o 27.5 DEG

"

..

~.

..

o ..._o + e + + o + + ++0 + .. +ct 13.6 OEG ++++++++++4 . .-o 0 '~+'''+ .. e " + .,0++ + 0+ 0.0 DEG 0+ l!l o + + o + o ... -11.0 M

•

" + '0 + e + + o + -23.0 M

•

+0 0+ o + " +

..

o + (!) +

HERSURED VALUE IOE VR IlNO ANO ,(OCHI

~ COMPUTED VRLUE IMINIHAOUAI

o + e + o + o + o +

"

+ e + o +

Fig. 12 Depth averaged velocity distributions inseveral cross-sections

(39)

I

,

I

,

I

o '" cO en o o o en o o o o N r-. o '" CO IC ~ Zo 00 _J' cr~ o o d '" o o ...; 0000 02.5 DEG Ili5Ó+ o o 00 ot

.,

..

o '" o o 0.0 FOA -23.0 M o IJ) <, :r 0-'"

oW

_J er u IJ) o 'J >-o~ U o _J oW -> o o '! 0.0 FOA 062.5 DEG ..+ o o 0.0 FOA 66.6DEG o

..

0.0 FOA 55.0 OEG 0.0 FOA ~1.3 OEG o o

..

o .. "0

..

0.0 FOA 27.5 OEG o +

..

+ 0 o o

..

o ..+ o .. o + 0.0 FOA 13.6 OEG o

..

0.0 FOA 0.0 OEG o

..

0.0 FOA -11.0 M + + .. + o o o + o o

I

o <>

°or.-Oo---oT.-~-O---,I.-O-O---,lr.~-·O---2r.-OO----~2T.-5~0---3~.~O~O----~Jr.~~·O~--~~r.~O~O---~T.~5~O----~5r.O~O~--~5r.~50~--~6.00

tolIOTH (Hl

I

_,., H> 41.3 DEG ~~

...

.

o + + 0' 00 + o _o 0 _o

..

o o o o o +

..

o₊ 27.5 OEG

..

o + + o + o ₊

..

o

..

o .. o o

..

Cl> t o ₊ 0000 0000 ++++ 00 13.60EG ++~G>t!lÓ· ..+++ .... 0°0, -t+ ...+ 0°0 o o t o

..

o 0000 o ....+ + + o

..

+..+ + o o o 0 0000 ++ +.+ + +0 .. 0 + 0000 + + 00 + 00 0.0 OEG ~ ~ Ijl

•

o

•

-11.0 M tJ '" + o + o -23.0 M

• ..

•

o

•

+ ~ L value determined by the bottom friction

+ computed value (Miniwaqua)

o

Fig. 13 Depth averagcd velocity distributions in scveral cross-scctions

(40)

I

_..

..

I

_..

..

_....

_..

..

_..

.. ..

..

_..

..

_..

..

_..

..

_..

I

..

...

s

oe: co

• ..

..

++

....

I

_..

..

...

•

+ ++ + +.+. + ++ +

..

....

..

+ + + +

I

..

_•

..

o o

..

+++'

..

I

..

55.00

..

... ++ +++ ... + ..

..

+ +

..

+ +

..

+

..

+

..

+ +

I

<>

..

+ ₊ <>

....

..

+

I

<>o oe:

'"

..

+..

..

+

..

....

_..

..

₊ + ..+

..

"

..

"

..

....

..

+ + + -, ..+

..

++ VI

....

xo a:~ wf;'; ::I: ::l ...J "-0_o w"; :1:,.. l-C) Zo 00 ...J' a:::: w I-....0 Vlo .;; ++

...

..

_..

..

_..

..

_~

_..

+ ... + ...

..

+ +

..

+ + + +

..

13.So ...++++++++++ +++ +

..

_{+ +}+ + + + + + +

..

_..

..

_..

..

+ .. +

I

..

.. + ..

I

s

ó

_...

I

o o ,,; N

..

+

..

+

..

+

..

+ + + +

a

..;

-I

-11.5 m + .. + + + + + + + + + +

..

-23.0 m

I

o

..

C>

óOi~.O-O---OT:-SO----~I:rO-O---IT:-SO---2~:-O-0---2r:s-O---3T:-o-o----~3:rS-O---qT:-OO---q~~-S-O---Sr~O-O---ST~-S-O----·~6:00

HIDTH (Hl

+

..

+

I

+

==

computed value

I

Fig. 14 S(ecplondanearbed coy flow intnfiguraentsion)ity dis. tributiNo corroenctios in sn appliedeveral.cross-sections

I

I o

..

.... + 0.0 for 68.S(

..

0.0 for 55.0 ++ 0.0 for 41.3( + 0.0 for 27.5( 0.0 for 13.S

..

0.0 for 00

..

0.0 for -11.5 0.0 for -23.C

..