MODEL FOR FLOW AND SEDIMENT TRANSPORT IN t … U.S. Army Engineer Waterways Experiment Station ......

AD-RI38 035 A NUMERICAL MODEL FOR FLOW AND SEDIMENT TRANSPORT IN tRRLLUYIAL-RIVER BENDS(U) IOWA INST OF HYDRAULIC RESEARCHIOWA CITY T NAKATO ET AL. DEC 83 IIHR-271

UNCLASSIFIED DACW39-80C 8129 F/'G 8/8 N

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MICROOPY ESOLTIONTESTCHAR

A NUMERICAL MODEL FOR FLOW ANDSEDIMENT TRANSPORT IN

o ALLUVIAL-RIVEk BENDS

by

Tatsuald Nakato, John F. Kennedy, and John L. Vadnal

Sponsored by

U.S. Army Engineer Waterways Experiment StationVicksburg, Mississippi

2 IIHR Report No. 271 DTICIowa Institute of Hydraulic Research .',(-.

The University of Iowa, "

Iowa City, Iowa 52242 F'E8 " (34

December 1983Final Report A

This documerni has boon cppr,'vve"f_- publi': release and sole: its

I d-tribution is un-miied.

84 02 16 092

. . -. . , .. .

ROWT-ACUMENTTION3. Rocipievirs Accession No.-- PAGE I H eprt #271_

4. Title and Sulmiftl L Report Dta

December, 1983A NUMERICAL MODEL FOR FLOW AND SEDIMENT TRANSPORT IN_- AlU VIA' -RIVER BRENDS7. Authos) B. Perfonning OrganIztlon Rapt. No.

No. 2719. Performing Organization Name and Addre 10. ProJect/Tesk/Work Unit No.

Iowa Institute of Hydraulic Research ,o aot(G) No.The University of IowaIowa City, Iowa 52242 (cDACW39-80-C-O129

12. Sponsoring Organization Name and Address 13. Type of Report & Period Covered

U.S. Army Engineer Waterways Experiment Station Final Report

~14.

I. Supplementary Notes

1L Abstract (Limit 200 words)

- Closed-form mathematical models are formulated for the rate of development andstrength of secondary current, and for bed topography in meandering, erodible-bed streamsof arbitrary plan form. The formulations utilize conservation of flux-of-moment-ofmomentum as the additional relation required to complete the formulation for the radialbed-shear stress. The secondary-flow-velocity calculation was uncoupled from the calculatioof the radial mass-shift velocity and streamwise velocity distributions, which'wereobtained from the depth-integrated equations expressing conservation of mass and streamwisemomentum. Numerical simulations were made for one laboratory flow and two SacramentoRiver flows with constant centerline curvature. Good agreement was obtained betweenmeasured and computed distrinutions of flow depth and depth-averaged streamwise velocity.

17. Document Analysis a. Descriptors

b. Identifies/Opn-Ended Terms

Rivers; Secondary Flow; Numerical Analysis; Sediment Transport

c. COSATI Field/Group

SO. Availability Statemen' 19. Security Class CThis Report) 21. No. of Pages

Release Unlimited Unclassfied___ ___146_20. Security Class (This Page) 22. Price

%I(See ANSI-Z39.18) See Instructions on Reverse OPTIONAL FORM 272 (4-77)

(Formerly NTIS-35)if D.P.,...,.of Commerce". " ,- ", -'' . , '' ." . ," . , -"-" -" , , -" ."- .","-" . . ," . -' ." " , . -' . , ' ., . .. ' . .V. " .

A NUMERICAL MODEL FOR FLOW ANDSEDIMENT TRANSPORT INALLUVIAL-RIVER BENDS

by

Tatsuaki Nakato, John F. Kennedy, and John L. Vadnal

Sponsored by

U.S. Army Engineer Waterways Experiment StationVicksburg, Mississippi

Li

IIHR Report No. 271

Iowa Institute of Hydraulic ResearchThe University of IowaIowa City, Iowa 52242

December 1983

Final Report*6o.

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I-w.F. .; -U U %-

PREFACE

The investigation reported herein was conducted for and sponsored

by the U.S. Army Engineer Waterways Experiment Station (WES, Contract

No. DACW39-80-C-0129). The numerical aspects of the investigation

served as the basis for the Ph.D. thesis of Mr. John Vadnal. The

companion, experimental investigation has been reported by Odgaard and

Kennedy 3. The numerical model developed in this phase of the

investigation analyzes and predicts flow and sediment-transport

distributions in alluvial-channel bends. The authors wish to

acknowledge their gratitude to Mr. Steve Maynord of WES for his

- continuing encouragement and assistance during the course of this study.

m4

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11

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CONTENTS

Page

PART 1: INTRODUCTION. ..... . . . .... . . .......... .. . 4

Anac lyt I c l St ra e ... . . ..................... ......

PART 11: ANALYTICAL MODEL...... ... . ............. . ... .. . . .. . 9* Secondary Flow in Rectangular Channels with* ~~~~~Nonuni formCuvtr............... 9

Bed Tpgah....................1Equations for Fluid Motion.................. .~oe*99**14Sediment-Discharge Relation. ........ se.............15

* PART [II: NUMERICAL MODEL..................................... .17Nume ri calStat... .............. 1Numerical Solution for U o.......... .o .20Numerical Solution for Vo.oo.........................22Boundary Cniin..~...... . .. ..... 2

* Computer ... r.... . ooe...... ......................... 23SensitivityAnlss..... ... ... . ..... 2

PART IV: RESULTS OF NUMERICAL SIMULATIONS.....................27

Sacramento Rvr..................2Idealized Single-Bend Model with Gradually

Varying Radius of Curvature.......................29Idealized Two-Bend Model with Gradually

Decreasing Radius of Curvature...................30

PART V: SUMMARY AND CONCLUSIONS..............................32

APPENDIX A: FLOW IN ALLUVIAL-RIVER CURVES.o...................A1

APPENDIX B: LISTING OF COMPUTER PROGRAM PR-SEG6 ANDINPUT-OUTPUT SAMPLES...... .................... . . .Bi

APPENDIX C: NOTATION............................ ..............C1

2

CONVERSION FACTORS, US. CUSTOMARY TO METRIC(SI).m. UNITS OF MEASUREMENT

U.S. customary units of measurement used in this report can beconverted to metric (SI) units as follows:

Multiply To Obtain

.inches 0.0254 metersfeet 0.3048 metersfeet per second 0.3048 meters per secondcubic feet per second 0.02831685 cubic meters per secondmiles (U.S. statute) 1.6093 kilometerspounds (mass) 0.4535924 kilogramspounds (force) per squareinch 6894.757 pascalstons (short) per footper day 2.9763 tons per meter per day

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A NUMERICAL MODEL FOR FLOW AND SEDIMENT TRANSPORT INALLUVIAL-RIVER BENDS

PART I: INTRODUCTION

Background

1. Two of the most striking and intriguing features of natural

alluvial streams are their tendency to meander, and the downstream

migration of the meanders. In addition to being a fascinating natural

phenomenon and posing some of the most nettling problems in the whole of

river mechanics, river meandering, and in particular the bank erosion

attendant to the growth and migration of the meander loops, has become a

major international problem. According to the final report on work

conducted under the Streambank Erosion Control Evaluation and

Demonstration Act of 19741 (Section 32, Public Law 32-251, submitted in

December 1981), approximately 142,000 bank-miles of streams and

waterways are in need of erosion protection. The cost to arrest or

control this erosion by means of conventional bank-protection methods

currently available is estimated to be in excess of $1 billion

annually. For the Upper-Mississippi River basin alone, the cost was

estimated to be in excess of $21 million annually. These figures exceed

the benefits derived by a large margin, thereby rendering many of the

bank-erosion-control projects uneconomical on a cost/benefit basis. As

a result, most bank-erosion losses continue unabated. Attempts to halt

.the erosion are often limited to piecemeal protection along isolated

bank reaches, at public or private facilities on streambanks, or at

highway crossings. However, as such facilities increase in value and as

the consequences of failure become greater, the threshold level of

acceptable risk becomes smaller. At the same time, traditional channel-

stabilization measures have become extremely expensive and are not

acceptable to environmentalists in many instances.

2. Nowhere has the problem come to sharper focus than along the

Sacramento River, California. The Sacramento River Valley contains the

4

.

. . . . . . . . . . . . . . . . . . . . . . . . .. . . .

. . . . . . . . . ..'. . . -.-.- .'... '.-....'." .. 4 .-,. ' J .-. " -.. '......-.. .. '.', ... .. " . ." .".. - ' . . ' .". ."

* " ."""""° "" "" .4-.-"" *" " """"."'.- .. ".""

- ".- ."... . . , " ".. . . . . .-. .". . . , " " " "..

" i -d . >'.-X-<,i ; i~i' - : -- .- ; i .< C Z, i -. 6 , *. *.. .- .

" Nation's finest (and most rapidly disappearing) agricultural land.

According to Brice 2, river-bank erosion along the unprotected stretches

of the approximately 200-mile-long reach of the Middle and Lower5"Sacramento River is producing an average annual loss of nearly two acres

of farmland per mile. Even when evaluated against current inflated land

values, traditional means of bank protection (for example rock riprap)

are so expensive that they cannot be justified economically. The

problem is compounded by some of the material eroded from the banks

being transported to the dredged navigation channels of the Lower

Sacramento River system and San Francisco Bay. Bank protection along

the upper reaches by traditional means can be justified economically

only if it can be demonstrated that the reduced erosion will result in

less dredging for navigation-channel maintenance. Thus, the problem

" poses two general questions: (1) Will it be possible to develop

-; alternative bank-protection measures that are effective, environmentally

acceptable, and economically justifiable when evaluated against land

values alone?; and (2) Will reduced bank erosion upstream be reflected

in reduced downstream dredging (and how much, and when), or is the

'* material eroded from the banks being deposited at other locations (e.g.,

on point bars) along the river?

3. It is against this backdrop that the Institute of HydraulicResearch at The University of Iowa entered into a contract with the Army

Corps of Engineers, Sacramento District, in 1980 to conduct an

investigation directed at developing improved, "unconventional" bank-

protection methods for application along the Sacramento River. It was

realized that the investigation should also include laboratory testing

of the techniques or the devices proposed, and development of an

analytical model, likely a computer-based one, for routing of flow and

sediment through channel bends. Funds for conduct of the laboratory

investigation and development of the routing model were not available

from the Sacramento District, but were provided by the Waterways

Experiment Station.

%.

d5

.. .. . . - . ... , ., .4, -. . , , , - , " "., ' ' ' . '. ' '. "".. .- ' '" ' " i ' 2 " " . " " i . " . " " " . ' '. i "i " '

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.. " ," .'-- "v - .--. . , .- ' ' .*.. " .- , -. . , '. ' . • ,, ' . .'. .. '- ,. - .: . ' -. .-. : ... , .'-

- 4. A repcrt describing the new bank-protection method developed4.

o' under the contract with the Sacramento District (Contract No. DACW05-80-

C-0083) and the laboratory testing supported jointly by the Sacramento

District and the Waterways Experiment Station (Contract No. DACW39-80-C-

0129) was submitted in May 19823. The present report is concerned

solely with the second phase of the WES-sponsored project: development

of a numerical model for analysis and prediction of flow and sediment-

transport characteristics in the bends of meandering alluvial channels.

Analytical Strategy

5. The point of departure for development of the numerical model

.. is the analytical work reported by Falcon and Kennedy 4. The manuscript

of this paper is included herewith as Appendix A, and is to be

considered an integral part of this report. An understanding of the

analysis presented in PART II requires considerable familiarity with the

Falcon-Kennedy analysis described in Appendix A.

6. The principal stumbling blocks encountered in the analysis of

river-bend flow are the interdependency of the bed topography, flow

distribution, and sediment-discharge distribution. The interaction of.;.the vertically nonuniform distribution of streamwise velocity and

channel curvature produces a secondary flow which spirals about the

channel-section axis and moves the higher velocity, near-surface fluid

toward the outside of the bend and the near-bed fluid toward the

inside. The radially inward bed shear-stress transports bed sediment

* .radially inward until the bed becomes inclined such that the radial-

• -plane shear stress is balanced by the component of the moving bed

layer's weight radially outward along the bed. The resulting warping of

the channel bed, which produces larger depths near the outer bank, as

shown in figure 1, leads to a redistribution of the streamwise flow, and

- produces much larger velocities, I)oundary shear stress, and unit

discharge near the outer bank. This, in turn, affects the lateral

distribution of unit sediment discharge. It is emphasized that the

6

,.

.........................................

................................................... S S .** S*S...-

. . - S .* ................ .,' .-... •.<'

secondary flow itself has a relatively minor impact on the distribution

of flow and sediment transport in a channel bend. It is the bed warping

produced by the secondary flow that is primarily responsible for the

redistributions outlined above.

7. In the case of fully developed flow in a uniform curved

channel (i.e., one of whose channel axis is circular in plan view), the

torque generated about the channel axis by the interaction of the

velocity profile and channel curvature is balanced almost exclusively by

* boundary shear stress. In the case of channels with nonuniform plan-

form curvature, as occurs in meandering streams, it is necessary in the

I' calculation of the secondary-flow strength to include the nonuniformity

of the flux of moment-of-momentum (or, more simply, the torsionalinertia of the flowing fluid) in the torque balance. The inclusion ofthe inertial effects in this case leads to a phase shift between local

channel curvature and local secondary-current strength and transverse

bed slope.

8. A hallmark of the numerical flow and sediment routing model

developed in PART II is a partial uncoupling of the secondary flow from

• the distributions of primary flow and sediment transport. The principal

steps in the development of the analytical framework for the numerical

model are summarized with annotation as follows:

,

i. The strength of the secondary flow is computed for any

section along a channel of nonuniform curvature. The

integral-form analysis of conservation of moment-of-momentum

(or torsion) is performed for a channel of rectangular cross

section with depth equal to that of the same flow in a

straight rectangular channel of equal slope.

ii. It is assumed that the bed topography can be adequately

represented by an inclined straight line passing through the

mid-width point of the equivalent rectangular channel

7A.'.

S.-......-...:.:;....... .........;....;,,.:..........:. ......... :... :..--..

." . , , ,,',w~~~~~~~~~~~w .. ._ ..... -.. ,,.,.. ,....-,...... ,..... ...... .. , ... .. ... ,,* , . . . . - . .* . * A il . . p * * * * * ,

utilized in step i, above. It is further argued that thetransverse slope of the channel bed varies linearly with the

strength of the secondary current. The force equilibrium of

the bed layer is analyzed to relate the mean bed slope to

the secondary-flow strength.

iii. The depth-integrated differential equations of continuityand of conservation of streamwise momentum then are employed

to calculate the velocity field. It is assumed that the

secondary-flow velocity has linear vertical distribution at

any point across the channel, and the magnitu' of the

secondary-flow velocity is obtained from th analysis

- presented in Appendix A. A radial, mass-shift % )city is

also included to account for the movement of f 1 icross

the channel as the channel curvature and trans .se bed

slope change and even reverse. The mass-shift velocity is

assumed to be uniformly distributed over the depth.

iv. The lateral distribution of unit sediment discharge across

the channel at any section is computed on the basis of a

power law using the local flow properties computed in step

lii.

9. The analysis is limited to steady flow in channels with

constant width and centerline streamwise slope. Utilization of the

model requires an estimate of the stream's friction factor using some

other method, such as that of Alam and Lovera 5. The model does not

allow for transverse variations of local friction factor, due to the

lateral variations of local depth and velocity, nor does it compute

sediment discharge by size fraction. However, the computer program is

structured to accommodate these features.

.x -

' '.,,.,-., -,,,.- 'v ,,,.-,+,..,'.- "-.",-,.",,. ,,'.- .-. -. ..... - ....''.. ..-.. .. ."...-. .".. .-..-. ..-..-. ..... . . . .. - . .

-. . 'T

+ . " - s ". 4I.. . . - S +. •.+'. . • . . .. -*

PART I: ANALYTICAL MODEL

Secondary Flow in Rectangular Channels with Nonuniform Curvature

10. As indicated above, the secondary-flow calculation will be

made for a rectangular channel that is "equivalent" to the warped

sections. Because the analysis is of the integral form, and considers

only the streamwise and lateral fluxes, over the whole charnel section,

of the quantities of interest, neglect of the lateral variations of the

primary and secondary currents that occur in a warped channel is not

judged to be a major limitation. Support for this conclusion is

provided by the generally good agreement between measured transverse bed

slopes and those calculated on the basis of the radial bed shear stress

in the "equivalent" rectangular channel (see figure 4A).

11. The control volume to be utilized in the moment-of-momentum

analysis is depicted in figure 2, and the coordinate system is defined

in figure 3. The control volume can be envisioned as the central region

(see figure 1) of the flow as it existed before the bed became warped by

the secondary current. The primary-flow velocity profile will be

described by the power law,

v . n+1 d) /nV -n d

where, in addition to the quantities defined* in figures 1 and 2, V =

depth-averaged flow velocity and n = reciprocal of the power-law

exponent. The secondary-flow velocity distribution will be approximated

by a linear profile,

S--(2)L d

*For convenience, symbols and unusual abbreviations are listed anddefined in the Notation (Appendix C).

9

J.%.

In general, n is greater than about 4, and figure 2A indicates that (2)

is an adequate representation of the profiles for an integral analysis,

in which small deviations between the actual and formulated profileshave relatively small effects. The equation expressing conservation of

moment-of-momentum (torque) about the centroid of the control volume

shown in figure 2 is

d r 2 dd d_ 7(y dr dy rd -Wp [ f uv (y dy d

o r i 2d 2 2)or - (ro - ri = 0 (3)

in which Tor = radial component of the bed shear stress. The radial

shear force exerted on the bed results principally from the velocityprofile just above the bed being skewed by the secondary-flow

velocity. The secondary velocity itself is relatively small compared to

the primary velocity, and alone produces a minor increase in the total

shear stress. It appears reasonable to assume, therefore, that it is

the skewing of the primary-flow bed shear stress that produces theradial component of the bed shear, and that the latter is proportional

I,.'..to the skewing of the velocity profile. This will be expressed as

or= _ U (4)

' - -TOS V

where 8 = proportionality factor to be determined from measured rates of

streamwise development of secondary flow in curved channels, and V =

mean streamwise flow velocity. The quantity Tos (hereinafter denoted

simply as T ) will be related to the local mean velocity by means of the

Darcy-Weisbach equation,

f -2 (5)

10

AA

".07W IF, 7777 o

. where f = Darcy-Weisbach friction factor. Substitution of (1), (2),

(4), and (5) into (3) and carrying out the indicated integrations yields

dc dU dcrc g.U= Rc g2V (6)

in which

i -R (r + ro) (7)

91g = (3n+l)(2n+l) of (8)- 2n +n+1

and(3n+l)(2n+l)(n+l)

(9)92 n(n+2)(2n 2+n+l)

Equation 6 is a linear, ordinary differential equation which has for its

solution

S-S s d s-sU(s) = U(so) + g2V exp [-g1 (- d )] f R(sC exp [gl ( d T)]d10)

0 s )

where the change of variable ds = Rc d has been made. Note that the

subscript c is used hereinafter to refer to centerline values. The

quantity U(so) is the secondary-current strength at s = so. In a field

application, the centerline curvature, 1/Rc(S), would be determined from

a map or survey and, in the case of complex channel lineament, the

quadrature appearing in (10) likely would have to be evaluatednumerically. This poses no problem inasmuch as the governing equation

themselves must be treated numerically.

Bed Topography

12. To determine the bed topography, and therefrom the streamwise

and transverse distributions of flow depth for utilization in the

'-'--11

........... .

"°, . -'

;' --.< - - s * ' ' . - , , .' ' - v -. w .. .'

'.- . .- - .-.-. - , _ , .. . , - -. . ..- _,- .. .. .. -.

numerical solution of the equations of continuity and motion of the

fluid, two assumptions will be made, as follows:

i. The transverse bed profile is a straight line at every channel

section. Figures 4A and 6A, and also the results presented by

Zimmermann and Kennedy 6 , demonstrate that the deviations of

both measured and more accurately computed transverse profiles

from a straight line are rlatively small.

ii. The transverse bed slope varies linearly with U. This is

consistent with (4) and the bed-layer equilibrium analysis

presented in Appendix A, where it is shown that the local bed

slope varies linearly with the local stress (14, App. A). In

terms of the mean transverse bed slope, ST(= sin 0 in (14,

App. A)), this equation reads

or = Yb(-p) Ap g ST (11)

where p = bed-layer porosity; Ap = pS-p, in which

Ps = particle density and p = fluid density; and

g = gravitational constant. Substitution of (4), (5), and

a-.. (10), (15, App. A), (16, App. A), and (17, App. A), into (11)

yieldsa-.':"" ST aT T ____

, f c (12)L /Vg .AR D5

where e = Shields parameter defined by (16, App. A) and a =C

proportionality constant between the bed-layer thickness, Yb

(see figure 1), and shear velocity, u., used by Karim 7.

Equation (12) can be simplified to the following expression:

12p.

a; a-..' ...................-.- '"-"'-*"- "-" "- .'- ." -""-- -.."

."-.". ..... .-. . .

•~s " " ' •° , . -. . *. * . \ ° % *'*" , .. '* . .* -. . "..-• • . . .. -. ' .

r7|' i L 'VV . *L 4, . E , ' . ', ,.' o.' ' ° ' '

U

ST (12A)

where

=T Vc (I (2B),P D50p 5

Note that n and f are related by Nunner's relation (17, App.

A), which is

n = I/i (13)

13. It will be assumed further that the depth changes across any

section due to curvature-induced inclination of the water surface are

very small compared to those due to bed warping. (Note, however, that

the effect of radial water-surface inclination is retained initially in

the equations of motion developed in the next section, but is then shown

to be negligible in the streamwise momentum equation for most meandering

river situations.) The depth at any point across the channel is then

given by

d(r,s) = dc i STr (14)

where the sign, t, is adopted according as Rc (see figure 3) is positive

or negative. The bed elevation at any point in the channel then is

given by

h(r,s) = hc(Oso) Sc(so) * STr (15)

in which, again, the sign is chosen to be the same as that of Rc-

,-'.

":_'. 13

.N.-... N-N.-.-" "- '.'' ' ' '' . - . . . . . -. . . . ,•.. ... , . , . -. . • *-N. . . . ., , .-

"'- '",,", ,',-- " '' ' '""r,,, ,"""- -N " " ... . "" " '-- ""N.'*m , " ,," N m . • .•

Equations for Fluid Motion

4' 14. The steady-flow, depth-integrated conservation equations for

mass and for radial and streamwise momenta, expressed in radial

coordinates, will be used for calculation of the velocity field. The

continuity equation is

a- [-h)h) =0a V(H-h)] + [r 1(H-h)] : 0 (16)

-

in which IJ = shift velocity (see figure 1) which accounts for thetransverse mass shift that occurs in channels with nonuniform curvature

(e.g., along meandering channels as the thalweg moves from the vicinity

of one bank to the other).

15. The radial-momentum equation is

H v2 a) 12- [f pr-dr dy] rd* + a 1 pg(H-h)2rd*]dr

h21 (H-h)2 dr d* - pg(H-h) dr rd, - + T rd dr =2 -r or

a 2- p(u+)v dr dy]d + r [ p(u+U) rdf dy]dr (17)

Substitution of (1) and (2) into (17) yields for the depth-integrated

radial -momentum equationS.._

H-1 r" v2 - - g(H-h) -n~n2) r 8 r

-[V(H-h)('f + U I + [r(H-h)(- + 2)] (18)II - 2n ) ] + I T18)

16. The corresponding equation expressing conservation of

streamwise momentum is

7.0 pg(H-h) drds- pg(H-h) rd dr i + T rdd ah

14

. -

S'i"" """ " " -"- " -. . -S . ". • ". - . - -~ ,',i.S . '-. , " * , ". " . -'"" "

pv dr dyjd* + pv(u+')rd dy]dr (19)

which, after introduction of the velocity distributions adopted for this

analysis, (1), (2), and the uniformly distributed transverse shift

velocity, yields the following depth-integrated streamwise-momentum

equation:

aH f 2-g(H-h) Ts - V

[rV(H-h)(U + U-+)] + [V2(H-h)] (20)

The numerical treatment of these equations is described in PART III.

Sediment-Discharge Relation

17. The local sediment discharge will be calcualted on the basis

-, of the local streamwise velocity using a power-law relation,

qt = a Vb (21)

in which = total sediment discharge per unit width; and the

coefficient a and exponent b are to be determined on the basis of a

sediment-discharge predictor or data on the channel under consideration

for its particular flow regime, bed-material size, etc., etc. The

- numerical program is structured such that other sediment-discharge

relations can be incorporated into it. In particular, it is envisioned

that the future development might utilize a formulation such as that

ecently developed by Karim 7, which uses an iterative procedure to

calculate sediment discharge and friction factor as interdependent

variables. This would permit incorporation of laterally nonuniformfriction factor into the program, and calculation of the sediment

discharge of each bed-material size fraction. However, time and funds

% 15

........- ..-.... . . ...... ...-. ............. . .,... ............. ,.................. ....... ...

did not permit undertaking of this rather major effort in the present

study.

18. It is recognized that the nonlinearity of (21) can lead to

calculated streamwise variations in the sediment discharge along a

nonuniformly curved channel with warped bed. A correction procedure is

incorporated into the numerical model which compensates for this

artifact in the following way:

i. The sediment discharges computed from (21) for radial

computational increments across each computation section are

summed to obtain the computed total sediment discharge for the

secti on.

ii. The sediment discharge in each radial computational increment

is corrected by a factor equal to the ratio of the sediment

discharge into the bend divided by the computed total sediment

discharge across the section.

This insures that sediment continuity is preserved along the channel

bend.

"- 16

-p.45,',

5m' ,' .' '.,. ','- - . - ., ..'. . ' ...., . - . - ... ... . ,... .- . .. . - . • . ,. . .- . . ,- , - . . .

-p '..., ' '' ''.' ' '' °'' ' .'' . - . "" , ." " """"" " ." " .. """ . " . ." " • " . ' ' ' . ." . ." - .' . '_, .5 """ " """""""" ' , " Y ," ' -"" " . ''" " , ' . " . "' "" " " -"" '

""

'* N , .N ;., 'N - " , ' °- -"-" "- "-.-:-," -/ -, . ;". -. "--. :::- -.--..... .-

FJ- - -rw- *- .-. 7b7 -:

PART III: NUMERICAL MODEL

Numerical Strategy

19. The three governing equations, (16), (18), and (20), contain

three unknowns: the depth-integrated streamwise velocity, V(r,s); the

shift velocity, U9(r,s); and water-surface elevation, H(r,s). The

secondary-flow velocity U(s), was calculated from the torsion-balance

analysis and is given by (10), and the bed elevation, h(r,s), was

obtained from the computed average radial bed slope and expressed by

(15). Numerical solution of these three strongly coupled equations

proved to be quite difficult, but was greatly simplified by introducing

the following restriction. Note that in (16) and (20), H appears only

in the combination (H-h), which is the local depth given by (14), except

in the first term of (20). The streamwise water-surface slope comprises

two parts: one due to the friction slope, which is of order

II (_)f g(f -- (22)

and a second resulting from the centrifugally-induced superelevation of

the water surface and of order

A = O(V W

in which L = characteristic length of the curve (say, the half-

wavelength of a meander). The second of these can be neglected compared

to the first if

4. . Wdc

c' 16 TT << f (24)

which is satisfied by most natural, sand-bed, meandering streams flowing

in the ripple- or dune-bed regimes. If (24) is satisfied, -L in (20)asmay be replaced by

17

-... . . .. .. . . . . ... . . . . . . .Z :.- .a< . L - .... .. . .: . .. . -.

~...-;............ ........-. -..- ,,.wo ...... b. * *" II.....= "

aH R cWS Sc~ (25)

C +

which states that the water-surface elevation is constant across all

sections. Substitution of (25) into (20) yields an equation which,

together with (16) forms a pair of simultaneous equations for the two

- velocities of interest, U and V.

20. It is convenient for numerical analysis to simplify (16) and

(20) as much as possible. The radial coordinate, r, is first replaced

by Rc + r in (16) and the expressions for d(r,s) and ST(s), (14) and

(12), are introduced, which yields

FlU + F2V + dcF4 U' + )-0 (26)

d + S T(s)rF1 SC T (27)= R~c rF-r2 g ST( ) (28)

2, = 9293 RdT c sand

F = 1 + r S (29)' 4 = d i+c ST ( s )

c

The flow-continuity equation (26) is normalized using the followingvariables:

SV Rc dcU i V R c' = ; d c' = W-

V V

r , s (0r = and s = (30)

The normalized expression of (26) becomes, after dropping the prime

superscripts,

F 1 U+ F2V + dcF 4 (-r + 1s) = 0 (26A)

Similarly, substitution of (14) into (20) and nondimensionalization of

the equation yields

18

%'°" '.'"." "." "." -*- .- .• .• ." " . .. - •.-. . . . - • . -'- .. •.• . -, *.- ,- "

• " *-" "" '" " " -" -' " " "'- "'" " " . .. . " ,. . i .* ,

F U V + d F (V)+ U d F 1V-

I C +r +T* I 4 2r

(n+1)L~ 2Sf F24

N"SF= r (32)

4.

by~~ th us of (2A(3ih2h)rsl

rn+ gr & +

by the 4 use of(6Awihth esl

Fr F

dc-Weisbac eqain,+.6

S F c (33A)

in. whh d nuei a given eg byloe (14)ve fo and V

proceded s folo1s

i. Te lcldphaeae elct a prxmtd b h

DarcyWeisbch eqation

V . 28;2 (33)

in whc d is gie by (14 an Sb

19.. . . . . . . . . .. . .4

Lj'. ." "".-v ""*- *-**- " "° *" - -"" " - - - " "" "" ' . "

S =S c T (34)c

which expresses continuity of energy slope across the channel.

ii. The velocity V given by (33) was substituted into (16) which

* .- was then integrated numerically to obtain the first estimate

f or TT.

iii. The value of U was substituted into (31A) which was integrated

to obtain the next estimate for V.

iv. The V computed in step iii was substituted into (16) and a new

estimate of 'U obtained.

v, The iteration procedure between (16) and (31A) was continued

until satisfactory convergence, as measured by the differences

.- between successive values of U and V, was obtained.

-. Further details are presented below.

Numerical Solutions for UJ

23. In order to solve (26A) and (31A) numerically for the two

unknown variables, 9 and V, a backward finite-difference scheme was

employed. Figure 4 shows the coordinate-grid layout that was utilized.

The indicies i and j represent the streamwise and radial positions,

respectively. Note that the origin of the transverse coordinate was

taken at the channel centerline. In discretizing both radial and

streamwise derivatives of an arbitrary variable F, the following

dIN, backward finite-difference scheme was utilized:

aF F i j Fi 1 (35)ar rj -

(35)

20

• .o °• .- ,.. ,,." ,-.%,-,. ~~~~~~~~~...... .. -. . - ..... . . ... ...-.. ,... . ... ... ,. .° . ,... -. ,. ,- -, -,

and

aF-- F

"siF Fi'j F i'l' (36)as "-si - Sil j

24. An approximate solution for U can be obtained from the flow-

continuity equation, (16), by introducing the Darcy-Wei sbach

relationship and (34):

V2 8gdS c Rc:- 2 F (37)

Substitution of (37) into (16), use of (14) and (12A), and subsequent

discretization yields the following explicit expression for U .:

_ d(Rc+r) i -_ 2g c exp[gl dc - I

."Ui : Uj_1 d(Rc+r)lj - 3929 3 f d(Rc+r)lj (38)

where

•2 (R cST-dc)T1T = 2R (RcST - )Fc dcT

+ (5RcST'dc)T 2 - (dc + 3RCST)T 38R sT

T IT (40)

T2 = 2t -H (41)t R RcST J

t -A7,

T3 I CT (42)

I tan- [___ ]: RcST < 0'RcST /-RcST

and

21C, ..

*,*, ,./ -. - - -- " - CC . . . . . . . . . . . ... -.> ...--.,_-.. - - -.,.-.-,. .. ..- -...,-; ..- .- , ,. . .- -.,, . ,- .."- C. ."""""""" *"-' " " " "-."""*' .. . . . .. . . . . . . . . . .." " " "" " " .' ""' " " -"""''" "" '""' .""

t c + T (43)Rc+r

Note that the boundary condition 11 = 0 was imposed at the inside

- (convex) bank. Equation 38 gives the approximate solution for U which

can he substituted into the streamwise-momentum equation, (31A), to

*; solve for V.

25. Once V is computed at Section I = i, the flow-continuity"." equation can be again utilized to solve for iT. in the iterative

1,'

process without utilizing the Darcy-Weisbach relationship. The final

• .discretized form of (16) in terms of V is

-- d(Rc+r) _1 (Vi,jdi,j - Vi_l,jdi_l,j )•U: . = u-cj_ .. vd,~ i_ si1,j I- d(R c+r) l 2(si 1,j -1,j

[(R c+d)2j - (Rc+d) 21 j_( ]

d(Rc+r)i j (44)

Numerical Solution for V

26. Disc:etization of the streamwise-momentum equation, (31A),

yields the following quadratic equation for Vij:

AV 2 + BV1 + C = 0 (45)

" where

F 2 f dc F4 1_ 1A - 2 + q + si 1 Si~lj + (46)

n~~n+2)' - si-1 n1 +2

cF4 - F 1B- rj - r_ 1 [i,j 2n+1] + +- Uij (47)

' and

-dcF 4 U.C r - [ j + 11 Vi j-1

22

........................................

.. . . . . .

.* *.b.

* .. .. * *

- -.-.. ; w- . - -"-

dcF4 1 1 2 1 R cSi j - Sithat i V 11 72 c 4 r*:*

It should be noted that the total water discharge calculated with the

computed transverse distribution of V did not equal the imposed total

discharge due to discretization errors. Therefore, an adjustment was

made to V at each cross section by multiplying V 11 by the ratio of theimposed water discharge to the computed water discharge. This ratio was

typically of the order of 1.0005.

Boundary Conditions

27. The streamwise velocity, V, was specified at the inlet

2 section (s so) and along the inside bank of the computational reach by

*the Darcy-Weisbach relationship, (37). Along the inside bank, the shift

*velocity, U, was set equal to zero.

Computer Program

28. The program PR-SEG6 consists of a main program, four

subroutines, and seven sub-subroutines. Listinqs of the main program,

the subroutines, a sample input file, and a sample output file are

included in Appendix B. Note that the sample input and output shown in

Appendix B are for the idealized two-bend model which is discussed in

PART IV.

29. The main program first reads the common input variables from

the input file called SEGDAT: V, dc, W, Sc, P, PSI and NSEG (number of

channel segments in the reach that requires new input parameters). The

program, then reads the following parameters at each new channel

segment: a, b, M (number of radial positions), N (number of streamwise

positions), Rc, so, Sl (centerline streamwise coordinate of the

23

- * - * . . . . . . . * - *

, . .. • . ... ,, .. .... . .. . ... 'w- . - ,' .- . -..--. ... -"' "-"-" ", "."-"," ' " " " """""" ' "• " "• " ' ' - ' ' " " " '" "- -"% ""' ' " t *'w "

' ' ""

' %"

-.- 7 7, -7- -- -- -. -

downstream end of the segment) , 0, , and NSTEPS (number of

cross sections into which the channel segment is divided). The program

computes the boundary values of U and V across the inlet section using

(10) and (37), and the transverse distribution of U is subsequently

computed from (38). The program then advances to the next downstream" section, and computes U and V according to the iterative scheme

described in paragraphs 22, 24, 25, and 26.

30. Subroutine PQN was used to determine the total water

discharge for a given cross section with computed transverse

distributions of streamwise velocity and depth. Subroutine PG

determines the g-parameters defined by (8), (9), and (12B). Subroutine

EVAL evaluates the shift velocity, 9, given by (38).

31. The main program writes the following outputs in the output

file, called OUTT, for each cross section: ST, Uc, number of iterations

required to compute satisfactory convergence of U'and V, and transverse

distributions of-U, V, d, U + U, and qt,

, "

Sensitivity Analysis

32. The effects of the specified error tolerance for U, the grid

size, the transverse derivative of U, and parameters o and a on U and V

were tested using the basic hydraulic and sediment parameters that were

utilized in the Oakdale flume experiments conducted at the Iowa

Institute of Hydraulic Research, The University of Iowa, by Odgaard and

Kennedy3. The basic parameters were: = 1.56 ft/s, dc = 0.505 ft, W =

8.0 ft, Rc = 43 ft (see figure 5), Sc = 0.00104, p = 0.4, ps/p =

2.65, v 1.21 x 10- 5 ft2/s, D50 = 0.3 mm, and 0c = 0.032.

33. The relative errors for 'Uand V computed at each cross

section were defined by

j=L2 (M-1) L

E and E J=2 (49),.U V v

24

°. .. . .. . .. . .. .. . .. ...

°... ... ~;:-

I7K

where AG and AV are changes in UJ and V between adjacent radial

positions, respectively. Because U was typically at least two orders of

magntiude smaller than V, the error tolerance for U was selected to be

• ]an order of magnitude larger than that for V. In the sensitivity tests,

.. parameters a and B were set equal to 1.0 and 3.35, respectively, and a- grid size of 6 in. was used. For EV of 0.1%, two tests were run

for EUf values of 2% and 0.4%. There were no discernible differences

• .between two sets of values computed. An additional run with EU equal to

0.2% demonstrated that this criterion was not able to be satisfied with

single-precision computations. Note that EV was of the order of 10- 5

between successive iterations for V. It was concluded that satisfactory

results could be obtained with the error tolerances of E-- and EV equal

to approximately 2% and 0.1%, respectively.

34. Sensitivity tests were run for different square-grid sizes.

* The grid size was reduced step by step until no significant changes in

estimates of U and V resulted. As shown in figures 6 and 7, grid sizes

of 4 in. and 6 in. yielded quite similar transverse distributions

*of U and V; in fact, the two sets were almost identical at the

downstream end of the channel bend. From the sensitivity analysis, it

was concluded that the grid size should be approximately equal to the

mean flow depth. Note that the mean flow depth of the Oakdale flume was

about 6 in. (see figure 5).

35. In obtaining the simplified streamwise momentum equation,

(31), the secondary-flow velocity, U, was treated as a function of only

s because of the assumption of constant transverse bed slope, as given

by (12). However, in computing V by means of (45), the computer program

utilized a radially-varied secondary-flow velocity distribution derived

by Falcon and Kennedy4

. R.. 4f - C (50)U = V d R (50c c c c

*G 25

-. . ...... . . . . . . . . . .. . . . . . . . . . . . . .. . . . . . ... . .

was utilized. When (50) is substituted into (20), the discretized

streamwise-momentum equation, (45), yields coefficients A, B, and C that

are slightly different from those given by (46), (47), and (48). In

order to ascertain the validity of the computational scheme, a special

test run was made with the term, aU/ar, retained in the streamwise

momentum equation. The computed distributions of Uand V are compared

in figures 8 and 9 with those obtained using (45), which was developed

without the term, 3U/ar. As can be seen in these figures, the effects

of the term, aU/ar, on overall estimates of U and V are minor.

36. The parameters a and a control the transverse bed slope, ST,

and the development rate of the secondary-flow velocity, respectively,

as can be seen from (12), and (8) and (10). Figures 10 and 11 show the

effects of a on the transverse distributions of U and V. As can be seen

in these figures, the smaller a resulted in larger ST. and consequently

in much larger shift velocities along the initial entrance reach of the

bend. The smaller a also resulted in much smaller streamwise velocities

along the inside bank, because the larger ST decreased the flow depth

there. Similar effects of a on "U and V are seen in figures 12 and 13.

The smaller a resulted in a slower development rate of the secondary-

flow velocity, and reduced the rate of the development of the transverse

nonuniformity in V.

26

.,-.. .. . . .°.-. . ....... . . .- . -. * .

,,.,.o ,.,., .,., .,% . .. ,% . .... ... , , . . , ,,,. ., .. .., .-. .. . . ,. .. . , .. .. .• •. ,.' ...

PART IV: RESULTS OF NUMERICAL SIMULATIONS

Oakdale Flume

37. The Oakdale flume shown in figure 5 is a 1:48-scale, highlyidealized, undistorted model of the Sacramento River bend lying between

R.M. 188 and 189, approximately. Experimental data on the streaiwise

distribution of the equilibrium transverse bed slope and transverseIdistributions of the depth-averaged streamwise velocity reported by

N Odgaard and Kennedy 3 were compared with the computer-simulated

results. The basic hydraulic and sediment parameters described in

paragraph 32 were utilized in the simulation. Additional parameters

specified were: a = 1.00, B = 3.35, grid size = 6 in., n = 4.24, E-G

, 2%, and EV = 0.1%.

38. Figures 14 and 15 demonstrate generally good agreementsbetween the measured and computed transverse distributions of the flow

depth and the depth-averaged streamwise velocity, respectively, at =

200. The results for * = 1140 are shown in figures 16 and 17, in which

the observed streamwise velocities are seen to be somewhat larger than

the computed values in the outside portion of the channel. The larger

measured velocities near the outside bank are believed to be

attributable to the very low roughness of the exposed plywood bank ofthe trapezoidal flume section. Note that the friction factor was kept

constant in the whole flow field in the numerical model. Figure 18depicts extremely good agreement between the measured and computed

transverse bed profiles for * = 1460.

* Sacramento River

39. The Sacramento River bend between R.M. 188 and 189 shown in

figure 19 was simulated for two water discharges (Q = 9,000 cfs and25,800 cfs). Basic field data were collected in the reach in 191) ind1

* 27

* K... . . .9 . . . . % . . . . I... . °o . ° .

_ , , % ". -, -, . . . . . . .. .~ • . - -, . - . • . . . - • . . . .f -.. .

1980 by the U.S. Geological Survey (USGS) (Odgaard and Kennedy 3 ). As

can be seen in table 1, both the hydraulic and sediment parameters

varied widely along the bend. Therefore, average values of the various

quantities listed in table 1 were utilized for the numerical

simulations.

40. The transverse distributions of the measured and computed

flow depth and streamwise velocity for Q = 9,000 cfs are shown for * =

800 dnd 1260 in figures 20 and 21, and figures 22 and 23,

respectively. Note that at * = 800, velocities and depths were measured

at only four verticals across the channel, while they were measured at

ten verticals at * = 1260. Despite the fact that averaged input data

were adopted for the simulation, the numerical model reproduced the

field distributions surprisingly well for the low river discharge of

9,000 cfs. The distributions obtained for the higher discharge of

25,800 cfs are shown in figures 24 through 27. The agreements between

the measured and predicted values are seen to be not as good as those

for Q = 9,000 cfs; however, it is believed that during high flows the

channel bed had not attained an equilibrium configuration. For example,

the measured transverse bed slopes shown in figures 22 for Q = 9,000 cfs

and figure 26 for Q = 25,800 cfs are entirely different. The field

transverse bed slope was, paradoxically, much smaller during the high

flow, resulting in the decreasing streamwise velocity toward the outside

bank, as seen in figure 27. This type of abnormal transient phenomenon

likely is a consequence of the rapidly changing flow conditions, and

cannot be simulated by a steady-state numerical model. It should be

" noted that each Sacramento River simulation required approximately 0.7

second CPU time per 100-grid points using the PRIME-750 computer at The

University of Iowa.

28

*' .". -- C", "* - "," -" "-.." . " .* .- ,---...'."'' , - - - , •" -- " -"-"--*- " " " *-" ." - ' -"

, . C *, - . *' " ", *' *. " ". .', .'. ,.-C. .. ....-.... C. .'. .'._C . .... '.'. . .*.•- ." " .'

Idealized Single-Bend Model with Gradually

Varying Radius of Curvature

41. The numerical results presented in paragraphs 37 through 40

for the Oakdale flume and the Sacramento River were obtained using

constant centerline curvature. In order to demonstrate the ability ofthe computer program to handle nonuniform curvature, two simulationswere made for single bends with gradually varying centerline

curvature. These numerical simulations were made also to illustrate the

behavior of flow in idealized, nonuniform river bends.

42. The first simulation was for a four-segment channel bend withstepped decreases in curvature in the downstream direction, as depictedin figure 28. The centerline radius of the first segment was 2,000 ft,

and this value was increased by 2.5% for each of the subsequent three

segments, resulting in a total channel length of 7,400 ft. It was found

that a 5% increase in Rc produced such large transverse-bed-slope

changes, which appear as sloped steps in the bed elevation, that theprogram would not run. Therefore, in cases in which Rc increases along

a bend, the curve should be subdivided into sufficiently short

subreaches that the increments in Rc are less than about 2.5%, although,

as discussed in the next example, the model can accommodate larger

changes in the case of decreasing Rc. The basic hydraulic and sedimentparameters used were identical to those for the Sacramento River at high

. flow, listed in table 1. A grid size of 14.5 ft was used, and theparameters a and a were set at 0.86 and 7.13, respectively. The

computed longitudinal and transverse distributions of the normalized

shift velocity are shown in figures 29 and 30, respectively, In figure

29, the shift velocities computed for sections 65, 193, 321, and 449 are

connected by straight lines. The shift velocity developed rapidly inthe first segment, with its maximum values occurring near r/W equal to

-0.25, and diminished gradually after section 193. At section 385, theshift velocity along r/W equal to -0.25 became negative, and remained sountil section 469. This flow redistribution directed radially inward

29

do ,

.........................................................................

""" "- ' '.- -*- - '- " . . ." . ." . . ..".". ..". . .*"""'" ,.• ' o"-'• " " .. " -... -. """. ."", h'% ""%"".,"," .'* , * "

was a consequence of the increased Rc. Figures 31 and 32 depict the

longitudinal and transverse distributions of the depth-averaged

. streamwise velocity, respectively. Along the inside bank, the

streamwise velocity decreased initially; however, it increased farther

downstream as the larger radii of curvature produced less steep

transverse bed slopes. The values of ST at sections 1, 65, 193, 321,

449, and 513 were 0, 0.058, 0.063, 0.062, and 0.060, respectively.

43. The second idealized case simulated was a single bend with

radius of curvature that decreased 10% between curve subreaches. The

* numerical results are not presented herein, because the qualitative

characteristics are very similar to those for the two-bend curve with

decreasing radius of curvature presented in the following section.

Idealized Two-Bend Model with Gradually

Decreasing Radius of Curvature

44. An idealized two-bend model, shown in figure 33, was

tested. The two-bend reach consisted of four segments with equal

centerline length of 67.5 ft. The centerline radius of curvature of the

first segment was 43.0 ft, and was reduced by 10% for each subsequent

subreach. The sign of Rc was reversed after the second subreach. The

simulation was made on the basis of the principal parameters used in the

Oakdale flume simulation. These parameters are described in paragraph

37, except that a = 1.42 and a = 3.28 were used in the present

simulation. Figures 34 and 35 show the longitudinal and transverse

distributions of the normalized shift velocity, U/V, respectively. The

Nshift velocity increased rapidly in the first segment and decreased

gradually toward the end of the first bend. Once the flow entered the

second bend, a mass shift took place toward the right bank due to the

change in sign of the channel curvature. Note that in figure 34, the

computed data points at sections 69, 205, 341, 477, and 545 are

connected by straight lines. As shown in figure 35, the maximum value

30

* 4 ... ... . * '_

% / -% ' . " •m ' % . /-f, . . . . ,7 . - " . ° " ." . . . • . . - .. . "* .~ " . - .- *. , . - .

L ".° . -4

° D.4

D ". -. °

r of the shift velocity across the cross section was closer to the convexside of the bend. Figures 36 show the transverse distributions of the

9 depth-averaged streamwise velocity computed at sections 1, 205, 341, and

, 545. The transverse location of the maximum V gradually shifted

9 radially outward in the first segment, and reached the outside bank atsection 71. The maximum V remained along the left bank until section273, after which the flow became concentrated near the right bank. The

maximum V reached the right bank at section 417 in the second bend.Because the streamwise velocity at the left bank at section 273 was muchlarger than that at section 1, a larger streamwise distance was required

to attain redistribution of the flow in the second bend.

45. Figure 37 shows the transverse distributions of the unittotal-load discharge, qt, computed at various cross sections. Thesediment-transport coefficients a and b in (21) were taken to be 0.108and 4.0, respectively. Note that the units of V and qt are ft/s andtons/ft/day, respectively. These coefficients yielded a mean total-loadconcentration of 300 mg/l (or about 5 tons/day) for the Oakdale flume.The distribution curves shown in this figure are seen to be generally

congruent with those transverse distributions of the streamwise velocityshown in figure 36, because of the sediment-transport relation adopted

being a power function of V.

' I.P131

, . .* .o

.o. . . . - .

PART V: SUMMARY AND CONCLUSIONS

Conclusions

46. The principal features of the numerical model developed

herein for calculation of flow and sediment-transport distributions in

alluvial-river bends may be summarized as follows:

i. The secondary-flow strength and the bed topography are

.- uncoupled from the calculation of distributions of lateral

shift velocity and streamwise velocity. This is accomplished

by, first, calculating the secondary-flow strength on the

basis of conservation of flux of moment-of-momentum, and,second, determining the bed topography on the basis of radial

force equilibrium of the moving bed layer.

ii. The distributions of lateral shift velocity and depth-averaged

streamwise velocity are calculated, for the warped channel

determined as described in step i above, from the depth-

integrated equations expressing conservation of mass and

momentum. It was concluded that for flows which satisfy (24),

it is not necessary to include the third conservation

equation, that for radial-direction momentum, or to iterate

among three equations to obtain a solution. The numerical

scheme utilizes the backward finite-difference method, and

evaluates transverse and streamwise distributions of the

radial mass-shift velocity and the depth-averaged streamwise

velocity.

47. Numerical simulations utilizing the model developed were made

for one laboratory flow, two Sacramento River flows, and three different

idealized channel bends. The principal conclusions obtained from the

simulations are as follows:

..

i. Generally satisfactory agreement between computed and measured

results was obtained by utilizing error tolerances

of E1. and Ev of 2% and 0.2%, respectively. In the absence of

better information, it is recommended that a = 1.00 and a -

3.50 be utilized. In instances where actual field data are

available on the rate of development and equilibrium values of

.P ST, a and 8 should be adjusted on the basis of the data.

ii. The most cost-effective square-grid size is approximately

equal to the mean flow depth.

iii. The computer program is capable of simulating flow in

multiple-bend channels with stepwise-varying radius of

curvature. On the basis of the numerical simulations, it was

found that the maximum permissible stepwise change of

centerline curvature for which the program will run is about

2.5% in the case of increasing Rc, and about 10% for

decreasing Rc.

48. Further development and improvement of the model should

include the following:

i. More complete and modern sediment-discharge and friction-

factor models should be incorporated into the model. In

particular, it is recommended that Karim's 7 model be

incorporated into the program to permit calculation of lateral

and streamwise variations of friction factor based on local

flow depth, velocity, and sediment discharge. Karin's model

is unique in that it formally takes into account the

interdependence between sediment discharge and frictionfactor, an interdependency which appears to be very important

in channel-bend flows.

'.

33

!N..

* *.*.*.-.

ii. A further refinement of the flow calculation would involve

incorporation of the radial-momentum equation, (18). This

would permit application of the model to bends with relatively

short radius of curvature. However, the numerical model would

become much more complex, and would require significantly more

computer time. The model developed herein is believed to be

-* . adequate in its flow-calculation aspects for most natural

alluvial-channel bends.

iii. An effort should be made to incorporate features into the

model to permit prediction of the occurrence and

characteristics of point bars and their effects on the flow

field. It is believed that this likely will require

incorporation of the radial-momentum equation and a more

*, refined sediment-discharge predictor, as described above.

iv. As is generally the case in river-flow analysis, there is a

pressing need for detailed, diagnostic-quality data on the

distributions of velocity and sediment discharge from both

natural and laboratory streams.

v. After some experience is gained with the model, the computer

program should be reviewed, made more compact and concise

where possible, and a user's manual for the program should be

prepared.

.

... 3

, °°.° .d°.-o" ..r-...- . .,° .. . .* . . •. .r , . . oo . " -. ". •. . . . . . . . . . .. -.

-.i W. W. -. T7 W

p

REFERENCES

1. U.S. Army Corps of Engineers, Final Report on the StreambankErosion Control Evaluation and Demonstration Act (Section 32,Public Law 93-251), U.S. Government Printing Office, Dec., 1981.

2. Brice, B.J., "Lateral Migration of the Middle Sacramento River,California," U.S. Geological Survey, Water-Resource Investigations,77-43, U.S. Department of the Interior, July 1977.

3. Odgaard, A.J., and Kennedy, J.F., "Analysis of Sacramento RiverBend Flows, and Development of a New Method for Bank Protection",IIHR Report, No. 241, Iowa Institute of Hydraulic Research, TheUniversity of Iowa, Iowa City, Iowa, May 1982.

4. Falcon, M. and Kennedy, J.F., "Flow in Alluvial-River Curves,"accepted for publication in Journal of Fluid Mechanics, Jan., 1983.

5. Vanoni, V.A., (ed.), Sedimentation Engineering, American Society ofCivil Engineers, Manuals of Engineering Practice, No. 54, 1975.

6. Zimmermann, C. and Kennedy, J.F., "Transverse Bed Slopes in CurvedAlluvial Stream," Journal of the Hydraulic Division, ASCE, Vol.104, No. HY1, Jan., 1978.

7. Karim, M.F., "Computer-Based Predictors for Sediment Discharge andFriction Factor of Alluvial Streams," Ph.D. thesis submitted to theDepartment of Mechanics and Hydraulics, The University of Iowa,December 1981 (Also available as IIHR Report No. 242, IowaInstitute of Hydraulic Research, The University of Iowa.)

35

- . . - . . . ..!

. . . . . . . . . . . . . . ..,

Table 1

Hydraulic and Sediment Parameters Used in Simulating the Sacramento River

Parameter Low Flow High Flow

Measured Average Measured AverageRange Value Range Value

Used Used

*Q(cfs) 7,800-9,900 9,000 24,000-28,400 25,800

A (ft 2 3,190-4,340 3,960 6,370-7,600 6,950

V(ft/s) 2.12-2.60 2.28 3.15-4.00 3.72

d c(ft) 5.6-15.2 10.28 8.6-23.1 15.0

R (ft) 1,800-3,920 2,540 1,800-3,920 2,430c

W (ft) 263-570 385 275-778 463

n 6.3-10.5 8.2 5.8-12.5 8.6

D so(Mm) 0.7-6.3 1.0 0.7-10.8 1.3

ST0.01-0.15 0.053 0.018-0.145 0.065

cx0.374 0.711

2.39 3.82

0 0.045 0.050

Grid Size (ft) 9.6 9.6

F 7)0.1 0.1

11 (% 1.0 1.0

*Maximum equilibrium transverse bed slope

*~-07

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'ft..APPENDIX A: FLOW IN ALLUVIAL-RIVER CURVES

by

Falcon, M., and Kennedy, J.F.

(A~cepted for publication in theJournal of Fluid Mechanics, Jan. 1983.)

46

FLOW IN ALLUVIAL-RIVER CURVES

by

Marco Falcon AscanioI

and

John F. Kennedy2

I. INTRODUCTION

Even casual observers of Earth's geological features soon notice that

natural alluvial streams are seldom straight along reaches of more than a few

channel widths. Hydraulic engineers and other students of fluvial processes

long have recognized meandering to be not only an intriguing geometrical and

kinematical feature of rivers, but also one that has major effects on their

sediment-transport and roughness characteristics. Fluid mechanicians

appreciate further that the internal structure of flow in meandering rivers is

as fascinating as their migrating, serpentine channel lineament. Especially

engaging is the interaction between the vertical profile of the primary flow

and the centrifugal forces resulting from the flow's curvature. The principal

result is the well known spiraling or secondary flow in planes normal to the

channel axis. Because the bed-surface sediment of a stream actively

transporting its bed material is in a quasi-fluidized state, even the

relatively small radial component of the bed shear stress and small near-bed

1Professor and Director, Instituto de Mecanica de Fluidos, Facultad deIngeniereia, Universidad Central, Caracas, Venezuela. Formerly, Benedict

. Fellow, The University of Iowa, Iowa City, USA.

2Carver Professor and Director, Institute of Hydraulic Research, TheUniversity of Iowa, Iowa City, USA.

4, Al

* o2

velocities produced by the secondary flow transport sediment toward the inner

(convex) banks until the bed becomes inclined such that the gravity and shear

forces exerted radially along the bed on the moving bed-load particles are in

balance. The resulting greater depth near the outer banks increases the

primary-flow velocities there, which in turn intensifies the erosive attack on

the banks and also undermines them. Both of these effects aggravate bank

erosion and thereby promote further growth of the meanders.

Although the seemingly disproportionate effects of channel meandering on

river flow have been appreciated for several decades, attempts to develop a

mathematical model for the secondary flow and its interactions with the

primary flow and sediment motion have met with only limited success. The

principal stumbling block encountered arises from the radial shear-stress

force exerted on an elemental control volume at any elevation (the vertical

distribution of which is the principal determinant of the radial-plane

velocity profile) being the small difference between two much larger

" quantities: the centrifugal body force and the radial pressure force

resulting from superelevation of the water surface. It is important to bear

in mind that even though the integrals of these two forces over the depth are

very nearly equal, locally they are grossly out of balance. The radial

gradient of pressure resulting from the transverse inclination of the free

surface is very nearly constant over the depth, while the centrifugal force

varies from zero at bed level to a maximum near the free surface. In fact, it

is precisely the difference between the distributions of these two nearly

equal forces that is responsible for the secondary flow. Moreover, the

secondary flow (or, viewed differently, the vertical gradient of the primary

___ velocity) causes the radial water-surface slope to be greater than it would be

A2

i+ . . . . . - ,4 -* . + . . .. ... . . . .

I - S .. _ _ + 42+ ,, . . . , - . - ,

3

for a flow with vertically uniform primary velocity (which would not produce a

secondary current). This is because the secondary flow produces an inward

radial shear force on the bed; the corresponding radially outward force on the

flow must be balanced by part of the radial pressure-gradient force. Thus,

just in -determining the distributions of the three principal radial forces--

pressure, shear, and centrifugal--exerted on the flow, one is faced with the

problem of having two of them--shear and pressure--unknown, even if the

velocity distribution of the primary flow and hence also the centrifugal-force

distribution are known. Clearly to proceed with the calculation of these

forces, another relation, in addition to the equation expressing the balance

of radial forces, is needed. Further physical considerations or assumptions

must be introduced to calculate the distribution of the radial velocity.

In the analytical model developed herein for vertical distributions of

radial shear stress and velocity, and radial distributions of depth and

streamwise velocity, an expression for the conservation of moment-of-momentum

is the additional relation utilized to close the formulation of the radial

forces. This aspect of the analysis is similar, for example, to use of

equations expressing balances of forces and moments to calculate the

supporting forces on a loaded, simply supported beam. One of the two unknown

forces does not appear in the formulation of moments about one of the

supports, and therefore the other can be calculated directly. A roughly

parallel approach is followed in the present analysis. Formulation of the

flux of moment-of-momentum about the longitudinal axis at the bed surface

yields an expression for the radial pressure gradient. The radial momentum

equation then is used to obtain the vertical distribution of radial shear

stress. The transverse bed profile is determined from consideration of the

A3

* .* . . .- * . . .* .-.i .. ; L"" . *

"." '-.-'-""-" "-'--" " "- " "-" - ". " ." ' . "** E - . ' " " '

4

force balance for the moving bed-load particles. The radial-velocity profile

is calculated by introducing into the radial momentum equation a linear

primary-shear-stress distribution and the eddy-viscosity distribution obtained

from the power-law distribution utilized for the primary velocity. Finally,

the radial distribution of local depth corresponding to the bed profile is

used in the calculation of the transverse distribution of depth-averaged

streamwise velocity. The analysis is limited to a channel of constant

centerline radius, which is a good approximation to extended reaches of bows

of many strongly meandering natural channels. Extension of the analytical

model to weakly meandering channels is developed in Falcon's thesis (1979).

Some of the background literature on this problem is cited in connection

with development of the present model. For a more complete review, reference

is made to the surveys by Callander (1968, 1978), to Falcon's (1979) thesis,

and to Odgaard's (1981) paper on river-bend topography.

.II ANALYSIS

General. The idealized channel treated here has infinite length,

constant width, an erodible sediment bed, and banks with a common center of

curvature. The central, longitudinal channel axis at the level of the bed has

constant mean slope Sc , and describes a helix in space which traces a circle

of radius rc when projected onto a horizontal plane. The flow is conveniently

described in cylindrical coordinates: the vertical z axis passes through the

curvature center of the channel and is positive in the direction opposite to

gravity; in planes perpendicular to the z axis, locations are specified by

radial distance from the z axis, r, and polar angle, 6, as shown in figure 1.

In order for the radial slopes of the bed and water surface to be constant

along the channel, the local streamwise slope, S(r), of both must be

A4

.:-.'.--''. ,'. --''.-.-'- .'- ... . .... . .. . . .'-"-. . ... , .. i. . . . . "-' f."- ''°", .""" , % '" ° "" , ,"* - " ~ .'.. " " " " .> "." ' ' ' " .. "- - *. ". . .""." . -.. -

,.* -. rrcI5S c ' -" (c

The flow is treated as uniform in the sense that its properties are

invariant along any helix with constant radius r and slope S. The analysis

will be restricted further to a central region of the channel, delineated in

figure 1, throughout which the vertical velocity is much smaller than the

characteristic velocities in the r and e directions. The channel slope Sc

will be limited to small values, so that forces and velocities parallel to theunderlying bed-surface helices may be taken to be equal to those along the o

coordinate. Finally, the restriction d/r << 1 will be imposed, for reasons

that become apparent in the next section.

Vertical distribution of radial shear stress. Calculation of the radial

shear stress at any elevation requires, first, that the radial water-surface

slope and associated pressure gradient be determined, for use in calculation

of the vertical distribution of radial shear stress from the radial momentum

equation. The radial water-surface slope will be calculated from a

simplified, by means of an ordering analysis, formulation of the conservation

of flux of moment-of-momentum for the control volume shown in figure 1, which

extends over the whole flow depth and has base dimensions Ar and AS = r Ae.

- For this control volume equation of moment-of-momentum about an axis r =

-4 constant at the bed surface is

1 1 2 1 1ap v 1 a 2

d f - n dn d f p - Yn dn + f T dn + --- [d f puvn dT]or r z da0 rz das 0 (2)

+ [rd f pu2 n dn] - f puw dn = 0

A5

%"

.•4 * * * - . . . * . . .

a' " . '.'. - ,. .. -. 'q,- * . *- .. . .* . ". . . .. . ..-.. . . . . ., .. % . 4" , ., -. "%- • . .- "%"p .

~~. . .. . .......... _.._-...... . . . . . •

6

* where, in addition to the quantities defined in figure 1, p = pressure;

z-hn ----; p = fluid density; u(r,n), v(r,n), and w(r,n) = velocities in r,

* o, and z directions, respectively; and T (r,n) = T (r,n) = shear stressrz zr

i acting on surfaces normal to r and z axes, respectively. The fourth term in

(2) is zero for uniform flow. The remaining terms will be ordered by taking v

= O(V), where V(r) = depth-averaged flow velocity; u = 0(u(r,1) Um); andm

the z-direction velocity w = 0(wmax W ). Relative to the second term, the

fifth and sixth terms are of order r( and o(-VV respectively. Yenand V repcieyVe

V"-" (1965) concluded from his measurements of flow in curved channels that

V = O(). This result is suggested also by the analysis of Zimmermann and

Kennedy (1978, Eq. 7), which shows the ratio of radial to streamwise

dcomponents of bed shear stress to be 0(-). If z = 0(d), the continuity

equation,

- au + av + aw = 0 (3)r 3r as 3z

inhchri - z e o nr te uniform flow being considered, requires

concluded then that the fifth and sixth terms of (2) are both 0(- r relative to

the centrifugal-force (second) term, and can be dropped.

The shear-stress (third) term of (2) may be ordered by utilizing the

equality of shear stresses and the Boussinesq shear-stress relation for

turbulent flow, and treating the eddy viscosity, £o, as constant over the

•-'- depth:

A6

.. . . . . . . . . . . .

T 1 1 T 1 -oUm (4)

Trz dn b Tzr do = p - J- dn = P (4)

The eddy viscosity may be expressed as a product of the shear velocity, u*,

and depth; therefore,

f T dn O(apu*U) (5)

0

where a E0 /u*d (a 0.079, according to Hinze (1975)). From (5) it follows

that

1

0 rz (r u (

1 v2 d V ) = __(

d f p '-n dn0 r

in which f = Darcy-Weisbach friction factor. Both a and V71 are (10-1), and

therefore the third term of (2) is two orders of magnitude smaller than the

second and may be disregarded. Because p = 0(pV 2), the first and second

K terms are of the same order. Incorporation of the simplifications resulting

from the foregoing ordering analysis reduces (2) to

1 1 2I i v2

f n d p d (7)ar r f- d0 0

.U WIn the central region, where W << 1 and V << 1, it is reasonable to

assume that the vertical distribution of p is hydrostatic:

2r pg 3H pgH' (8)

A7

.. ,. . .. . . , . . .

8

where g = gravitational acceleration and H is defined in figure 1. It has

been demonstrated by Yen (1965) that the deviation of the pressure from the

hydrostatic distribution is 0(-) or smaller in even moderately curved open-r

channel flow. The primary-flow velocity, v(r,n), will be expressed by the

power law,

v n+1 1/n"-'- q= n (9 )n

where I/n = exponent, which is related to the Darcy-Weisbach friction factor

by

18 =(10)n K

where K Karman's constant. The background of this relation is reviewed by

Zimmermann and Kennedy (1978). Karim (1981) examined (10) critically, veri-

fied it with laboratory data, and formulated the dependence of K on sediment

concentration; this refinement will not be included in the present analysis.

- Substitution of (8) and (9) into (7) yields

H'(r) - n+1 V 2

n rg

By means of an ordering analysis similar to the one developed above and

. guided in some measure by his experimental results, Yen (1965) simplified the

radial momentum equation for curved open-channel flow to

2 T zr

gH' - v 0 (12)r pd Dn

A8

. _ • - - D . .. . °- . ... .8 o .. .i. . .. ,.'\..

°(4

It is noteworthy that T makes a first-order contribution to the radial-zr

momentum relation, but the corresponding vertical shear stress, T rz' makes

only a higher-order contribution to the moment-of-momentum equation if the

moment axis is taken at bed level to avoid inclusion of the bed shear

stress. This is, in fact, the motivation for utilizing the moment relation:

it avoids specification of T rz at the bed in the determination of H'.

Substitution of (9) into (12), integration of the resulting expression from

arbitrary n to n = 1, and application of the boundary

condition T (1) 0 yields

rz'" 2+n

Tzr(r,n) = pgd [H*(n-1) - r(gn1 n n 1)] (13)

Traverse bed profile and depth distribution. Equilibrium of the trans-

verse bed profile, h(r,e) in figure 1, and of the depth, d(r), are attained

when the radial-plane forces acting on the moving, bed-load particles sum to

zero. Bed-load movement will be treated as occurring in a layer of thickness

Yb, as shown in figure 1. The shear forces exerted on this agitated, somewhat

dilated, moving layer are in reality diffuse, "seepage" forces caused by the

, flow within the layer's intersticies, and any net force, however small, in the

radial direction will produce transverse motion of the bed-load particles.

* Therefore, radial equilibrium will be reached when the local bed inclination,

8(r), is such that

. Tor = zr(0 ) = Y Ap g sin 8 (14)

A9

where p = porosity of the bed-layer; Ap p ;-p, and ps = density of bed parti-

cles. In the development of his detailed, computer-simulation model of sedi-

ment transport in streams, Karim (1981) concluded from inferential evidence

* that

' = D (15)

where D50 = median bed-material size, u*(r) = local shear

velocity= V/f/8; and u*c = critical shear velocity for incipient particle

motion. In terms of the Shields parameter, 6, u*c may be expressed

UAc -gpD 5 0 (16)

which defines 0 Substitution of (13), (11), (15) and (16) into (14), and

incorporation of the simplification of (10) to Nunner's (1956) relation (Hinze

1975),

n I/f (17)

which corresponds to = 0.354, yields

'"d r /8 I + VfST sin 6= FD 8 (18)

T r

"--1 + 2Vo

wher FD o • 0

b"A10

S- - - K. . , " ' -. . . : .

. K - -. . .. .. . - - '

11

The traverse bed profile may be calculated by neglecting the effect of H'

on d(r); then

dd (Ig)

sin 0 = dd (19)

The velocity V to be used in (18) is obtained by incorporating (1) into the

local expression of the Darcy-Weisbach friction factor,

8 O /8 sSpgd(r) / rc g6d(r)

pf pf c r f

where To = longitudinal shear stress acting on the bed; and 6(r) = bed-shear-

stress reduction factor defined by

T oe (r) = 6Spgd(r) (21)

which takes into account the transport of primary-flow momentum out of the

central region to the vicinity of the outer bank, where it is balanced by bank

shear. An analysis of 6 is developed below. Substitution of (19), (20), andm"' dd

(1) into (18) and integration of the resulting expression for dd yields

1~~ r [L _ _ S g6. + V c (22)

c c p 50

where the subscript c denotes the centerline values used in setting the inte-

gration constant. Elimination of 6Sc from (22) by means of (20) and replacing

V by its section averaged value for the whole flow, V,to facilitate

verification, leads to

All

"." ". % " " - .

12

C 1+

where FD V 62 DD V gP 50

Utilization of the mean velocity for the whole cross section in the

calculation of d(-) from (23), or in calculating an average value of ST from

(18), by replacing FD with FD neglects the effects of the variation of V

across the channel, but nevertheless yields satisfactory results, as is

demonstrated in Section I1.

Equations 20 and 23 give the radial distribution of mean velocity for

uniform flow. In practical applications, it generally suffices to take

6 = 1.

Vertical distribution of transverse velocity. Calculation of the radial-

plane velocity will incorporate the following assumptions:

1. The primary-flow shear stress, Tze ' is linearly distributedaT ro

and T-r makes a negligible contribution to the streamwise force balance:a..r

T (r,n) = r (1-n) 6pgdS(l-n) (24)Ze 00

. 2. The eddy viscosity is isotropic and is given by

(rn d ze (25)P av

i, - an

3. Because of the isotropy of c, the radial velocity and shear stress

are related by

A12

4.. Kl% -1

13

T (r,n) __ u (26)zr

Substitution of (9) and (24) into (25) yields

S2 n-iS6gdSrc n2 n (27)

cc nT--. E - V n+1l (l-n) n n(7

which, when substituted along with (13) into (26), leads to

3 1-n1-nu 1 r n+ln n V n , n[ , 'n (28)-.SS c c n2 0 rg n(n+2) 1--n (28)

For steady, uniform flow, u must satisfy

1f u(n) dn 0 (29)0

There is no assurance that (28) will satisfy this requirement if H' given by

(11) is utilized, because of errors inherent in the Boussinesq eddy-viscosity

model and other assumptions that have been made in the derivation of the

relation for u. Therefore, the integral of (29) will be evaluated for arbi-

trary H', denoted by H'u and expressed as

2H' (r) _=T n+1 V2 (0

u n rg

where T = Hu/H' and H' is given by (11). Substituion of (28) and (30) into

(29), utilization of the expansion

A13

4°-.

14

S _ n < 1 (31)I-n j =0

and term-by-term integration of the resulting series yields

3 1 +j

" V2 n [T (n n (32)ur n j=0 - +1+j

nn

Equation 29 is satisfied if T is given by

T(n) (n+1) Er 1 31i_] (33)2 L3 1 1(

n(n+2) j=O (- + 2 + j)( + +j) (+ +j)(- + j)

Incorporation of (17), (20), and (33) into (32) gives

u r 1(n+l)4 [ 1 1 n

Vn + ) n + + + j)

3 +1+j 1 +j(- [ n - ]} - G(n,n) (34)" "' * ( +1+j) (- +j)

Equation 34 is portrayed in figure 2 for four values of n which span the range

from very rough (n = 2.5; f = 0.16) to relatively smooth (n = 10, f = 0.01)

channels. It is noteworthy that for all but very low values of n, the

velocity profiles are nearly linear except near the bed, with u = 0 at about

mid-depth.

A remark on H' and H',. In the foregoing analysis, two expressions were

derived for the transverse slope of the water surface: (11), and (30) and

(33). Corresponding to each of these is a different value of T zr(0), the

radial component of the bed shear stress. Their ratio T = H'u/H' given by

4" A14

U; " " " " " "" -- - "-"" "-" -- " -" " " " ": :: : " ' -"' -"" ' -".I il -i a I I ! i - i : I

: -" : ., ' ' . - " - ' " " " '' ' ' - " ' ' - ,'

-,'," +-,-. , $ - ' . . .- -• . .+ J . -+ -. . -" - .. -'+- . ..+.i _ , °

-. '

15

(33) has a nearly constant value of 0.9 for 2 < n < 8. In view of the near

equality of H'u and H', why was it necessary to utilize different values in

the formulations of Tzr and of u(n) ? The problem is one of sensitivity, asV.r

will now be demonstrated. Equation 13 gives

- . V2 n1 2 ]

T or(r) = Tzr(r,O) = pgd 2 n+ - H'] (35)

which together with (11) for H' shows that T is the difference between two

small quantities multiplied by a large one, (pgd). Therefore, small errors in

the expression for the transverse water slope produce large errors in T or* For

example, the ratio of Tor given by (35) with H' replaced by H'u obtained from

* (30) and (33), to Tr yielded by (35) and (11) varies widely with n, from

about 0.6 for n = 2 to nearly 0.2 for n = 8. Because the radial bed slope and

thus also the bed profile depend directly on T or ' as is shown by (14), it is

important in their derivation to have an accurate estimate of T or --one whose

calculation avoids use of such artifices as the Boussinesq relation, and

instead directly utilizes a mechanics principle such as conservation of moment

of momentum. The effect of the radial water-surface slope on u(n) calculated

from the Boussinesq relation may be examined by substituting (26) into (12)

and treating c as constant, say o , which results in

2 I2 2S! H (36)

an2 0 co

,, 2Equation 36 shows that a , and hence u(n) , also is very sensitive to the

ansmall difference between two nearly equal quantities multiplied by a large

one. Accordingly, only a very small adjustment in the radial water-surface

A15

16

slope is required to compensate for the effects on u of the assumptions made

in its calculation, and thereby to permit u to satisfy the continuity

requirement, (29). But this small adjustment in H'--amounting to only about

10 percent, as just noted--has a major effect on Tor , as (35) demonstrates.O-or

Secondary-flow effects on T and v. In a laterally nonuniform curved

flow, the secondary current produces a net radial transport of streamwise

(0-direction) momentum out of the central region, which in turn

reduces Tze and in so doing modifies v(r,n)(or n). It was anticipation of this

effect which prompted incorporation of the factor 6 (r) into the shear-stress

expressions, (20) and (21). Calculation of 6 and the secondary-flow effect

on v(n) proceeds from the 0 -direction momentum equation with Tr neglected,

a v + v + v +uv H + z," ¥+v - gl 1 + ZO (37)

-.s r a s P z (3

Multiplication of (3) by v, addition of the result to (37), integration of the

new relation from n to 1, and imposition of the boundary condition

T ze (r,1,0) = 0 gives

11 1 2a.. ZB=d{ (uv)d 2 av2

par d -r f uv dn - - dn- (38)n n n

_I [(wv) - wv] + g S(l-n)}.

_ allwhere S - - The z velocity, w, is evaluated from the continuity equation,

(3), the relation for u, (32), and the identities,

ahd ad.d dd 3H-c m- (z-h)d -d nd _d (1-n) - - r

an arr ar a rr3 dr ar r"a.."r d2 d2 d

d..d

i.A16

17

*and, similarly,

. nS (40)

* The result is

d dd dV 1 d§ nw(r,n) = - r + Tr + d(- dr - 6 d f u d (41)

V 0+DH dd[Tr - d (1-n)] u Sv.

Substitution of (9), (34), and (41) for v, u, and w along with

J2

a 2 _ S av2

as (42)

into (38) yields

T1 1d dd d d dV d 1

= (1-n) - 646 A n(n+l){[ r + 2 - + 4 - ] f G(n,n) n d.pgSd r r d- 6 dr

1

dd d d dV d d6 n+ + + d3r- T -] n f G(n,n) dn} (43)Edrr 6 dr 0

. where G is defined by (34).

The first terms on the right-hand side of (43) is the linear shear-stress

distribution, and the second term expresses the shear-stress reduction due to

. the transverse gradient of lateral flux of streamwise momentum. In the calcu-

latlon of T it is assumed that the 3-6 term is negligible in comparison toar

the other derivative terms in brackets. Substitution of (18) for dd and ofdr

A17° .A.

K 18

(20) for V then permits calculation of the shear stress at any location. The

bed-shear-stress reduction factor, 6, introduced in (20) and (21), is obtained

by letting n = 0 in (43). To gain some idea of its magnitude, a representa-

tive constant value of 6 for the whole central region, say 6, will be deter-

mined. For this calculation it is appropriate to replace V and d by their

section-averaged values, V and d, after carrying out the substitutions and

taking the derivatives in (43); and to take r = rc. The result is

To 192 n (n+l) ST f G(n,n)n l/n dn1- (44)pgSd =[r 192 c 0

where ST is obtained from (18) by replacing d and V by d and V. The integral

of (44) was evaluated numerically, with the result shown in figure 3. Values

of 6 for some field and laboratory flows are presented in the next section.

The effect of the secondary flow on the primary-flow velocity

distribution may be estimated by substituting (17) into (20) and replacing

6 by 6. If V is considered to be constant, n is increased, as 1/16 , which

corresponds to the velocity becoming blunter. This is the observed effect of

secondary flow oF v(n) (Falcon 1978).

III. VERIFICATION

Data utilized in the verifications reported here are summarized in table

S1. Falcon (1978) presents additional comparisons of measured and computed

quantities.

Bed topography. Zimmermann (1974) and Zimmermann and Kennedy (1978)

reported the results of experiments conducted in three, concentric, 60-cm,'j

wide, circular-plan-form flumes with a central angle that approached 27. Two

A18

i ..... ..... " -" "- ,-" "-'''" " ".....',.,."....,""..-...-..-,.-..,.............."..."'.-..".".'.".".."..."."-'".""

19

different sediments, with median diameters of 0.21 mm and 0.55 mm, were

used. Longitudinal bed profiles were measured at 11 different radii, and the

transverse bed profiles obtained from them for numerous cross-sections in the

reaches of fully developed flow were plotted and averaged. The transverse

profiles were found to be slightly convex upward, as illustrated in figure

4. Mean transverse bed slopes, averaged across numerous sections for each

run, were then computed. Figure 5 shows the transverse slopes, ST 5 so deter-

mined plotted in the format of (18) based on cross-section-averaged

properties. Excellent agreement between measured and computed values is".'., V8e

obtained if-- = 1.3. If the limiting value (for fully turbulent boundary

layers) of the Shields parameter, e = 0.06, is adopted, the resulting porosity

4% is p = 0.47, a not unreasonable value for the agitated, dilated, moving bed-

load particles. The computed profile for Zimmermann's (1974) Run No. RII-13

shown in figure 4 was obtained from (23), using these values of e and p. The

centerline depth, dc = 9.66 cm utilized in computing the profile was obtained

by equating the reported mean depth, 10.1 cm, to the mean depth calculated

by integration of d given by (23) across the channel width:

(__ 2 3/2 3/24, d 2r (2 )(r - r.

d - r)

where = - 1 + - and ri and ro = radii of the inner and outer

banks, respectively. Calculation of the relation between d and dc in this way

- is consistent with the measurement procedure that was used. Falcon (1978)

describes calculation of d in a way that is consistent with conservation of

bed-material volume in a curved channel. The friction factor utilized, f =

0.165 for this flow, is the value for the bed section obtained from the side-

A19. .

i'.[...''. .''1 -" - "" '" "" "" '"'-'-" - * "'-" " ' " . - -* • . "* -. . ". ". - - . - " -- "

20

wall-correction procedure (Vanoni 1976, p. 152). The measured and computed

profiles are in excellent agreement. The bed-shear-stress reduction factor

obtdined from (44) for this flow is 6 = 0.43, which shows that in this

relatively narrow channel the secondary current produced a major reduction in

the bed shear.

Figure 6 compares measured and computed transverse bed profiles for a

Missouri River section (Falcon 1978). In computing the profile, d was taken

to be equal to dc, because of the difficulty of determining rc for a wide

natural stream, and of the insensitivity of dc to rc (see (45)). Included in

figure 6 is the mean bed profile given by (18), which also is seen to give

quite good results. Equation 44 gives 6 = 0.97 for this relatively wide,

shallow flow, demonstrating the bed shear stress at any r in the central

region of this natural channel is nearly equal the local value of pgdS. Other

comparisons of measured and computed bed profiles yielded conformities as good

as those demonstrated in figures 4 and 6. In evaluating data from natural

streams, in which the flow is seldom steady for appreciable periods, there is

always uncertainty about the equilibrium of the bed topography. Moreover, the

bed-material size often varies widely across a section, often by a factor of

five or more. In view of these difficulties, it is suggested that in the

calculation of bed topography by means of (18) ind (23), averaged (across

channel sections, and along subreaches that are sufficiently short that rc is

practically constant) values of d and V be used, and that the median diameter

" of the material that can be moved by the flow be utilized for D50.

Furthermore, for most natural-stream situations, the refinement given by (23)

is probably not justified; a straight-line profile with slope ST given by (18

and passing through d = d at r rc is generally as accurate a- the

A20IU

i i i i i . .i -| .i " ' ' :

LHi~sb8 015 AI NUEk HL MODEL FOR FLOW AIND SEDLIMENT TRANSPORT IN 212,RALLUVIAL RIVER BENDS(IJ) IOWA INST OF HYDRAULIC RESEARCH

IOWA CITY T NAKATO ET AL- DEC 83 IIHR-271

UNCLASSIFIED DACW39-SN C 8129 F/G 8/8 N

J&.

116 111

1.18

NAIO A BUEU O SA DRS-16 -

%4%

21

data warrant, and perhaps within the reproducibility of bed topography of

natural streams with their vagaries of discharge and bed-sediment

characteristics.

Velocity distributions. It is very difficult to obtain reliable data on

u(n,r) in erodible-bed channels, because of the small values of the

secondary-flow velocities, and of the problems posed by the moving sediment

" and the continuous bed changes attendant to migration of bed forms.

Therefore, the two measured profiles obtained by Kikkawa, Ikeda, and Kitagawa

(1976) from uniform flow in a circular-plan-form, rigid channel were utilized

* in the verification of (34), with the results shown in figure 7. Kikkawa et

al (1976) developed an analytic model for u(n) , which can be seen in their

paper also to yield generally satisfactory results except near the bed, where

: it does not satisfy the no-slip condition. Comparisons presented by Falcon

(1979) of (34) with the rigid, sinuous-channel data on transverse-velocities

reported by Yen (1965) also demonstrate very satisfactory agreement.

5 The transverse distributions of V, the depth-averaged streamwise

velocity, in erodible-bed channels are somewhat easier to measure than the

radial-velocity distributions. Velocity data obtained by Onishi (1972) at the

.. apex cross sections in two of his rigid-bank, erodible-bed, meandering-channel

flows were used to validate the distribution of V obtained from (20) and

(23). The average friction factors, T, used in the computations were obtained

*from the reported mean values of velocity, depth and slope for the flows, and

the Darcy-Weisbach relation in the form

= 2 (46)

A21

... * . ... .- .... . -.,,.,..- , ,' ,,-... * ** ,- ** ,-.., - * -., ,-. , ,, ,, .,1,,,*- ,> '.'-',',',,.€ ,., .,S,

22

Note that computation of T from (46), which assumes 6 1, and of the

corresponding n from (17), computation of T from (44) for this n, and use of

i and 7 in (20) to determine V(r) is slightly inconsistent. Falcon recommends

iteration between (20) with V = V , (44), and (17) to obtain consistent values

of n, f, and 6 . However, the convergence is quite rapid and the effect on

the computed V(r) or u(n) is not great. The computed and measured distribu-

tions of V(r) shown in figure 8 agree quite well except near the banks, where

V must tend to zero.

IV. CONCLUDING REMARKS

It must be borne in mind that the model developed here is strictly valid

only for uniform, curved-channel flows. However, the available experimental

data on flows in strongly curved channels (Zimmermann 1974, Zimmermann and

Kennedy 1978, Odgaard and Kennedy 1982) indicate that they are characterized

by relatively small phase shifts or lag distances between local secondary-flow

properties or bed topography and local channel curvature. Therefore,

application of the present model utilizing local channel and flowCo

characteristics in nonuniform flows, as was done in the foregoing comparison

with Onishi's (1972) data, will generally yield satisfactory results.However, in the case of flow in weakly meandering sinuous channels, as

investigated by Gottlieb (1976) and Falcon (1979), the phase shift between

local channel curvature and secondary-flow strength approaches W/2.

The analytical model developed here is valid only for the central regions

of curved-channel flows, which generally extend to about one local depth from

the bank. In the near-bank regions, the flow becomes strongly three-

dimensional and heavily influenced by local bank characteristics (erodibility,

A22

..:. .. -. : .. . .. , : .. -.. .... . ... , -.C*. -. . .. . , . .. .. . . . . .. . . . ..C..-. ....°• .°• C••• .°• o . . . o " . . o ' , °

%' , . -, . .. . , . q . .. , .-- , . . , . . . ... , . -, - - .. * , .,, . ., - , -. . .

23

slope, roughness, etc). Analysis of the flow and sediment transport in these

regions is correspondingly more difficult than for the central region, and

likely must await availability of better experimental data for its guidance.

ACKNOWLEDGEMENTS

The analysis presented here was largely developed in the course of the

Ph.D. thesis research of the first author. Further work was done after his

departure from The University of Iowa under the sponsorship of the U.S. Army

Corps of Engineers Waterways Experiment Station (Contract No. DACW39-80-C-

0129). The analysis was completed and the manuscript prepared under support

provided by the National Science Foundation (Grant No. CEE-8023003).

REFERENCES

CALLANDER, R.A., "Instability and River Meanders," Ph.D. Thesis, Univ. ofAuckland, New Zealand, 1968.

CALLANDER, R.A., "River Meandering," Ann. Rev. Fluid Mech., Vol. 10, pp. 129-158, 1978.

GOTTLIEB, L., "Three-Dimensional Flow Pattern and Bed Topography in MeanderingChannels," Series Paper 11, Institute of Hydrodynamics and Hydraulic

- Engineering, Tech. Univ. of Denmark, Lyngby, February 1976.

HINZE, J.O., Turbulence, Second Edition, McGraw-Hill Book Company, New York,1975.

KARIM, M.F., "Computer-Based Predictors for Sediment Discharge and FrictionFactor of Alluvial Streams," Ph.D. Thesis, Dept. of Mech. and Hyd.,Univ. of Iowa, 1981.

FALCON, M.A., "Analysis of Flow in Alluvial Channel Bends," Ph.D. Thesis,Dept. of Mech. and Hydr., Univ. of Iowa, 1979.

KIKKAWA, H., IKEDA, S., and KITAGAWA, A., "Flow and Bed Topography in CurvedOpen Channels," Proc. ASCE, Jour. Hyd. Div., HY9, Vol. 102, pp.1327-42 September 1976.

NUNNER, W., Warmeubergang und druckabfall in rauhen Rohren", VDI-Forschung-scheft 455, B, Band 2, 1956.

S'Ak'°

*5

'> """'""' '"'"' "" '"° " "" " " "" ". .Zg2L:

24

ODGAARD, A.J., "Transverse Bed Slope in Alluvial Channel Bends", Proc. ASCE,Jour. Hyd. Div., HY12, pp. 1677-94, December 1981.

ODGAARD, A.J. and KENNEDY, J.F., "Analysis of Sacramento River Bend Flows, andDevelopment of a New Method for Bank Protection", IIHR Rtport No.241, Iowa Institute of Hydraulic Research, Univ. of Iowa, May 1982.

ONISHI, Y., "Effect of Meandering on Sediment Discharges and Friction Fatorsof Alluvial Streams", Ph.D. Thesis, Dept. of Mech. and Hydr., Univ.of Iowa, 1972.

VANONI, V.A. (ed)., Sedimentation Engineering, Am. Soc. of Civ. Engrs., ManualNo. 54, ASCE, New York, 1975.

YEN, B.C., "Characteristics of Subcritical Flow in a Meandering Channel",Ph.D. Thesis, Dept. of Mech. and Hydr., Univ. of Iowa, Iowa City,Iowa, 1965. (Also available as unnumbered report from the IowaInstitute of Hydraulic Research).

ZIMMERMANN, C., "Sohlausbildung, Reibungsfaktoren, und Sediment-Transport inGleichformig Gekrummten und Geraden Gerinnen," Ph.D. thesis, Inst. ofHydromechanics, Univ. of Karlsruhe, Karlsruhe, Germany, 1974.

ZIMMERMANN, C., and KENNEDY, J.F., "Transverse Bed Slopes in Curved AlluvialStreams," Proc. ASCE, Jour. Hyd. Div., Vol. 104, HY1, pp. 33-48,January 1978.

FIGURE CAPTIONS

1. Definition sketch for flow in an alluvial-channel bend.

2. Distribution of radial-plane, secondary-flow velocity given by (34).-'., 1

' 1 n3. Result of the numerical evaluation of f G(n,n)nn dn where G is

given by (34). 0

4. Transverse bed profiles measured in Zimmermann's (1974) Run RII-13,and computed from (23).

5. Comparison of (18) and Zimmermann and Kennedy's (1978) measured"A-. transverse bed slopes.

6. Transverse bed profile measured in Missouri River (Falcon 1978) and,those computed from (18) (--.-) and (23) (-)

7. Secondary-flow velocity profiles measured by Kikkawa et al (1976) andcomputed from (34).

8. Comparison of Onishi's (1972) measured transverse distributions ofdepth-averaged velocity and those computed from (20) and (23).

.- A2, ."A2 4

4 4 4 . - 4 - 4

60 m

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A28-

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. A29

,"'-".- .:.,.'' .''','",..' .. '': . c''.. ',,. -, '','-' ,-.,-' •, s - " '-' .", ',." -. ',.'.-..'... '.-,-' '. ' -.,' . .' .-- '.--'. " ,. '_ ' - -' .. ..-. ''._ ... .'' .., "- .- . " . . "-"- . . . ",h, ,, . "-"

10

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F0 1 + 2V/COMPARISON Of PEASURED AND CC.4PuTED TRANSV4ASE BED SLOPES IN ALLUVIAL CHA.'JtEs.

,,.(DATA: hKMENI.A.4 AND KENEDY 1973)

Figure 5. Comparison of (18) and Zimimermnann and Kennedy's (1978)measured transverse bed slopes.

A30

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> (b)

---

849 869 889 909 929 949 969Figure 8. Comparison of Onishi's (1972) measured transverse

distributions of depth-averaged velocity and thosecomputed from (20) and (23).

Follow 114QWA33 lwd, O

*2e.. available- 'S.

APPENDIX B: LISTING OF COMPUTER PROGRAM PR-SEG6

AND INPUT-OUTPUT SAMPLES

MAIN PROGRAM PR-SEG6

C ************** MAIN PROGRAM PH-SEG6 *******************7 THIS IS A GENERAL PROGRAM FOR A RIVER RF.CH COMPOSE. *

.:IF SEVERAL SFGM.NTED BENDS WITH POSITIVE OR NEGATIVE *'---:r RADII OF CURVATURF:, THIS PROGRAM SIMULTANEOUSLY *

S. OLVS THE CONTINUITY AND MOMENTUM EQUATIONS BYC ITERATING BETWEEN THEM* EACH NEW SEGMENT INPUT DATA *

MUST BE READ AT THE FIRST SECTION OF EACH SFGMENT. *

4 * 4 * * * * * * * * * * * * * * * * * * * * 4 * * * * *

C "UBROUTINES:C FALL : DETFRMINES THE F-PARAMETERS, WHICH DEPEND ON THE*C FOLLOWING: •C F1 = ( T, RC, R, HC F2 = ( ST, RC9 R, 01, G2, 639 H ) *C F3 = ( ST9 RC, V, R, G1, G2, G3, H, F, VBAR ) *C F4 = ( ST, R, H ) *

C GALL : DETERMINES THE G-PARAMETERS, WHICH DEPEND ON: *

C G = ( RHOS* BETA, ALPHA )C G1 = ( F, BETAr G2 F )C G3 CV9AR, PH0,9 r"09 F, BETA, ALPHA, THETAC,

C POR G) *C INTGiL : EVALUATES THE INTEGRAL: •C SUM = I'JT( R * SQRT( (A * R + B) / (R * ) ) ) *

FPOM R(J-1) TO R(J) *C CALQ : DETFRMINES THE DISCHARGE OVER THE CROSS *

C SECTION GIVEN: *C C M, V, HL, R --- GETS VA, VO, AT, (T )r 4 * * * * * * * * * * * * * * * * * * * * * *

-" C VARIABLES: •C LII :TRANSVERSE SHIFT VELOCITY (FT/SEC) *C UBIMI = VALUES FOR UBAR AT THE PREVIOUS SECTION. USED *f WH[% CHANGING ORIENTATION OR WHEN USED AS AN *

C APPROXIMATION TO THE CURRENT SECTION'S VALUES *

C (KOPTI=I) *C TUP = TEMPORARY STORAGF OF UBAR WHEN CHANGING *C DRIENTATION •C UBNI = UPDATED TRANSVERSE SHIFT VELOCITY (FT/SEC) *C VI = STRCAMWISE VELOCITY (FT/SEC). V1;4 1= PRFVIOUS VALUE OF VI AT ( I - 1 ) *C T _MV = TEMPORARY STORAGE OF V WHEN CHANGIG ORIENTATICN*

OR VALUES OF ETA-MODIFIED V *

C VNI = UPOATED STREAMWISF VELOCITY (FT/SEC) *

C U = LOCAL SECONDARY FLOW VELOCITY--MOOIFIEC (FT/SEC)*r HLIV = LOCAL FLOW DEPTH AT SECTION I (FT) •C NONrIMFNSIONALIZEO BY THE VERTICAL LENCTH---H *

HLIH = LOCAL =LOW DEPTH AT SECTION I (FT) *c NONCIMENSIONALIZED PY THE HORIZONTAL LFNGT---i *

•.4 " 1LIMlV = LOCAL FLOW DEPTH AT DREVIOUS SECTION (1-1) FT *

C THLV = TEM PORARY STORAGE OF HLV WHEN CHAN;,ING *

C" r ORIENTATION *THLP = " " HLH " *

% Bl

........ .... .'. . ... . . .-.......- . .. . /.*. .-. .- - ..'."°. "* ....'*.. -. '. ..-,-.". . .I, •."" -i: -" "",". '" """ "',-'. -' *"-" ''" "'-"."' ".-.'.>,'- .'.'"".','-.';',, Z" " - - '.:'"--* ", /'"'',w*" -'

p- 7

C ST = LOCAL STPEAMWISE POSITION AT SECTION I (FT) *

. SIMI = LOCAL STREAMWISE POSITION IT PREVIOUS SECTION -C (I - 1) (FT)C TEMPS = TEMPORARY S STORAGE FOR REVERSING ORIENTATIONC UC = CENTERLINE SECONDARY FLOW VELOCITY (FT/SEC) aC U0 = INITIAL CONDITION FOR THE SECONDARY FLnW *

C INTEGRATION

C STI = CUMULATIVE TPANSVEPSE BED SLOPE *C R = RADIAL COORDINATE FROM THE CENTERLINE (FT) *C TEMPR = TEMPORARY R STORAGE FOR REVERSING ORIENTATION *

C VA = LOCAL FLOW AREA (SQ FT) •C VQ = LOCAL FLOW DISCHARGE (CFS) *C V = STORAGE ARRAY FOR OUTPUT *C VB = nt It *

C U = " " "

C HL 7*

C ILOC = ARRAY DENOTING THE SECTION NUMPER WHERE EACH NEW*C SEGMENT BEGINS *C IST = ARRAY DENOTING THE SECTION NUMBER ;HERE THF *C VELOCITIES AND DEPTHS ARE KEPT UNTIL FINAL a

C TABULATION MAXIMUM OF 6 VALUES STORED-TNLET, a

C 4 OTHERS* OUTLETC ISFG = VARIABLE FOP THE ILOC ARRAYC NIST = VARIABLE FOR THE IST ARRAY *C ISTO = LAST VALUE OF NIST IN IST-ARRAY : DIMENSIO' *C OF IST *C IKEEP = PARAMETER TO TELL WHETHER TO STORE FOR TABULATIONC (IKEEP=I) OR PASS ON TO NFXT SECTION (IKEEP-O) *

C * * * * * * * a * a * * * * * * * * k * a * A * a * "*

C H = MEAN FLOW DEPTH (FT) *C HV = H NONOIMENSIONALIZED BY H---THUS UNITY *C HH = H W *C RC = RADIUS OF CURVATURE (FT) *C RC IS POSITIVE IF THE ORIGIN IS LOCATED ON THE *

C RIGHT BANK SIDE a

C PC IS NEGATIVE IF THE ORIGIN IS LOCATEr ON THE *

C LEFT BANK SIDE a

C Rn = PREVIOUS SECTION'S RC TO CHECK ORIFNTATTON a

C SI1 S2 = STARTING* ENDING CENTFRLINE COORDINATECI OF EACI a

C BEND SEGMENTC SC = CUMULATIVE CENTERLINE STREAMWISE CORDTNATf *C 'CO = PREVIOUS SECTION'S CENTER LINE COORDINATE *C OS = CENTER LINE DISTANCE BETWEEN SECTIONS *C F = DARCY-WCISBACH FRICTION FACTOR aC NOTE THAT F IS USUALLY DETERMINED BY B

,- C F = 8.O # G R * S / C V*a ) *C THUS, SPECIFYING 3 OF C F, P. S, OR V ) *

, C DETERMINES THE FOURTH PARAMETER *C VRAR = MEAN STREAMWISF FLOW VELOCITY (FT/SEC) a

C REMAINS IN ORIGINAL UNITS PY THE NAME VACT a

C CL = CENTERLINE WATER-SURFACE SLOPE aC W = RIVER %IDTH (FT)

*" B2

, .~.5 . . .- . . . . .-. . , . • . • -.

-7 a.-.6

C REM4AINS INTHE ORIGINAL UNITS DY THE NAME WACTc N =NUMBER OF STREAMWISF POSITIONSr NUM3ER OF RADIAL POSITIONS---TAKEN 000 FOR*r CENTERLINE

a.C NSLG =NUMBER OF CONSECUTIVE SEGMENTED BEN'DSr NSTFPS = NUMHER OF SECTIONS THAT INTERVAL C S1, S2 )IS

a: C TO 3E DIVIDED

C ALPHA =CONSTANIT USED IN BED-LAYER RELATIONSHIPr BETA =CONSTANT USEn IN SHEAR-STRESS RELATIONSHIP*c PO0R =PED-LAYFR POROSITYc SG = SprCIFIC GRAVITY OF SEDIMENTC5-C = MEDIAN BED-MATERIAL PARTICLE SIZE (MM)C RMU =DYNAMIC VISCOSITYc 7HETAC =SHIELDS PARAMETERC. ********** * * * * * * * * * * * * * * * * *

C DIMENSION VT'(17), VTMi(17), VNI(17)DIMENSION UPI(17), UflIM1(17), UBNI(17), UI(17)nIMENSION SI(17), SIMiC17), HLIV(17), HLIMiV(17)OIMrNSION HLIH(17), HLIMlH(17)DIMENSION R(17)9 VA(i?), VU(17)DIMENSION TUB(17)9 TEMV(17)9 THLV(17), THU-i(17)DIMENSION TEMPR(17), TEMPS(17)OIMNSION ILOC(5), IST(6)DIMENSION VC6,17)o UB(6917), U(6tl7), HL(6917)DIMENSION UDPU(6,17)t UTRAN(6917), ANGLE(6917)

*DIMENSION rAOSS(6,17)

c PRIFOS I/0 COMMENTc

OPEN ( 5,F ILE z SFGDAT)OPcN C69F ILEr 'OUTT9)

* REAP C5,10) VbARHWSCL ,POR9SGRMUNSECWR I TE(6,95)

5 FORU0AT(//,2flXSMATHEMATICAL MODEL FOR THE PREDICTION OF1 9/,24X,'THE VELOCITY FIELD IN RIVER FLOW99//)WRITE (6,6)

6 FORMAT(lX,78(l**))*13 FORMAT(6F1C.5,E15*5974)

* *NSF!Pl NSEG + 1READ(5912) ( ILOC(1)I),1,NS1GDI

12 FORMAT(711J)* . WRITE(6914)

14 FORMAT(l//15X,@VALlJFS FOP Vi3AP9 H, U, SCL, POR9 SC, RFU,

WRITr(6,15) V8AP9 H, U, SCL, POR, S6, RPU, NSEGI FOQ iAT(5X,3F1D.3,El3.5,2F7.3,Eli .3,14,1)

WRI TE(6,16)16 FORMAT(14X,'OSECTION NUMRrRS UHIQE NEW SEGMENTS BEGIN

WRITE(6,12) CILOC(I)gI1,NSEGPI

B3

-3 u. -

WRITE(696)

C rEFINITIONS OF VARIOUS TERMS USEnC

C MCL = RADIAL CFNTERLINE POSITION .* C IMI = DESIGNATED RADIAL POSITION TO BE PRINTED *

C IM2 = a ofi f

C IM3 =, ,.. C IIA4 =

: C NOTE 5 POSITIONS PRINTED IN ALL:

vC I = NEGATIVEF RIGHT -.5ANW,

C 2 = QUARTER POINT *C 3 = CENTER LINE a

C 4 = 3-QUARTER PCINTC 5 =-POSITIVE LEFT BANK •C MP = INCREMENT FOR RADIAL LOCATION OUTPUTf NP = INCREMENT FOR DOWNSTREAM SECTION OUTPUT *C T = DOWNSTREA4 SECTION INDEXC TOUT = D/S PRINTING OPTION PARAMETER FOR RESULTS aC EVERY IOUT-TH SECTION IS PRINTED *C NOUT = TEST PARAMETER/VARIABLE FOR IOUT-TH SECTION *C EPSV = EPSILON FOR ERROR IN V BETWEEN ITERATIONS aC EPSU =" IJBAR BETWEEN ITERATIONS *C KOUNT = ITERATION COUNTER OF UBARC KMAX = MAXIMUM NUMBER OF ITERATIONS OFSIRED FOR UBAR *C CALCULATION a

KTIM = FREQUENCY OF OUTPUT DURING ITERATION PROCErnuRF FCR*- C UBAR

r, KO = PRINTING PARAMETER FCP KTIM-TH ITERATION )UT'dJT *

C KCPTI = OPTION PAKAMETER FOR INITIAL APPROXIMATICN FOR UfiAPC= 1 FOR USING PREVIOUS SECTION'S VALUESC = " FOR USING DISCRETIZED CONTINUITY EQ.)UATION *C = 3 FOR ANALYTIC SOLUTION OF THE CONTINUITY EQULATICN,C KCPT2 = OPTION PARAMETER FOR SOLVING QUADRATIC FORMULAC FOR VC = 1 FOR ORIGINAL DISCRETIZED MOMENTUM EQUATIONC = FOR MODIFIED DISCRETIED MOMENTUM EQUATION WHICH.C INCLUDES THE CONTINUITY EQUATION *

,. C KOPT 3 -OPTION PARAMETER FOR RADIAL INTEGRATION P'IRECTION *C = 1 FOR DIRECTION FROM THE POSITIVE LEFT PfANK ACROSSC TO THE NEGATIVE RIGHT BANK *

C = P FOR DIRECTION FROM THE NEGATIVE RIGHT BANK ACROSSC TO THE POSITIVF LEFT BANK *C * * * * * * * * * * * * * * * * * * * * * * * * * 6 * * * 0 *

C YORF DEFINITION'S, C PI : PI !!! BOSTON CREME, APPLE, PUMPKIN, LTC*"" C G = GRAVITATIONAL CONSTANT *

r AAA = COEFFICI92T IN SEDIMENT RELATIONC BBP = FXPONENT IN SEDIMENT RELATIONC SEDI' FT RELATION --- POWER LAW OF THE FCRM aC QS AAA VBAR *BBf•

B4

•. * .. .C.... . ,. *- .- , . . *°. , . . . . . ....-,.... , . .. ... . , ., • ' -° ." IF • " • " • # • " . , * e •. ° ~** . * . .. .* .

°

. - -. " ' .. .. .*. , -.. . °.. - *. • ° . o*; * • ° .. ". - .. - . .- - * " . .•

• .. " .- • . . " " . •

"'"..• .,a ","r"- #' .;.'." . . ., ..,'.." . " .' ._., ,'"." &, , "" ._ ''..'*'.-' .., ..." , ... .''''.- *..-...."

IT-I

-C FTMM = r ONVERSION NUMBER OF MM IN ONE FOOT *C KUVM = MAX NUPlB[R OF ITFRATIONS FOR U-V COMPATIILITY *

C ISTn = SIZE OF IST-ARRAY --- PRINT 19 ISTD --- LAST VALLE*

r WATER DISCHARGE •(7 011, = SEDIMENT fISCHARGE ,C Q-FACT = SEDIMENT DISCHARGE---ACTUAL *C V STAR = SHEAR VELOCITY *

FRCUDF = FROUDE NUMBER *

c R?-J = POWER LAW EXPONENT *C rELR = RADIAL STEP SIZE *

C BL, PQ = POSITIVE LEFT AND NEGATIVE RIGHT BANKS, *C RESPECTIVELY

C CPIGP? = DIMENSIONLESS GRAVITATIONAL TFqMS BY DEPTH H ANDC WIDTH W *C RESTAR = BOUNDARY REYNOLDS NUMRER *r VSTARC = CRITICAL SHEAR VELOCITY

C FRO = DENSIMETPIC FROUDE NUMBFR *

.- VARIABLFS TO DEFINE EVERY RUN

PI 3.1415926-14G 32.174FTMM 30498AAA 0.108

H~BO 400N = 545m =17MCL =IMi = 5IM = MCLIM3 = 13IM'4 = MlOUT = 2MP =NP = IEPSV = 0,011rp'll = C.01KMAX 2r)

KTI M 10KOPTI = 3

..... TIKOPT2 = 2-O-T3 2KUVM = 20SIST = 6IST 1) =

IS T 2) = 619IST(3) = 205IST( 4) = 141IT(5) = 477IST(ISTD) N

B5

row.

WR ITEC6, 20) N, VM, MCL,9 IM 1 1 M29 1M39 I M49 IOUT 9 MF, NP23 FORMAT(5X99DOWNI TREAM STEPS ='01590 RADIAL ST'FPS = 6915,

1 9 CENTER AT M = ,15,/95X,'PAr)IAL POSITION'S5 e TOR D AT'S J =1 09419,/,SX99RESULTS OUTPUT LVFRY '.I,plS ' SECTIONS',/95X9D/S AND RADIAL OUTPUT FR'EQUENCY IS ',

s 215,'O STFPS0,/)WRITE(6,22) EPSV, EPSU, I(MAX, KTIM, KOPT19 KOPT?, KOPT2

22 FORMAT (5X,'RELArIVE ERROR CRITERIA FrR V Aflr UPAR ARE '

I 2FlO.5q/o5X9'MAX1MUM ITERATIONS *9159' PR~INTED EACH 09159'vI 'ITERATIONS9, /,5X,'PPOGRAM OPTIONhS FOR U %AR9 OENU

$FOR?' 9DIRECTION ARE '9 3159/)WRITE(6924) NN, KUVM, IST0, (IST(KhvK=191STP)

24 FORMAT(5X,'INITIAL NUMPER OF SUB3INTERVALS FOR SIMPrON1 RULE + 99159 /,SXt*'MAX NUMBER OF IJ-V ITE:RATION", I,, = 09S 15,/95X9 *DIMENSION OR NUMBER OF SECTIONS TO PE T.APULATEDS IS ='t159/,5X9 'AND ARE AT SZCTIONS: 09(/5W97110))WRITE(6925) AAAv BRR

25 FOR VAT(/95X 9 SE'lIMENT POWER LAW OF THE FORM Q A ~S C V )**Hot 19X90 WITH A, B = 9F24/

WRI TE(696)rC COMPUTE QUANTITIES TO BE USED IN THE PROGRA'

0 = VBAR * H * WOSACT =AAA * BR*RVSTAR =SQRT( G * H * !.CL)FROUOE =VBAR / SQRT( G HF R.O *G * H *SCL / CVHAR**2)RN 1.0 /SGRT( F )RN2 RN *10 )**2 /(RN QN + ),n*20DELR W / CM - 1)

BL=W ~'2.0BR =-W/ 2.0WRI TE(6965) GFRN,r)ELR.ELgeRVSTARFROUOERN2,OSA CT

65 FORMAT (5Xo9DISCHARGE = 99FI3.49' DARCY-WEISBACP F =0I E14*69/,5X, 'POWER-LAW N = 9E16.899 RADIAL STEP =9S E12949/95X9 OLFFT 9 RIGHT RANK AT R 09E24/=XI *SHEAR VELOCITY 09E14.699 FROUDE NO: *qFl49,/95X#S$ 'N-TERM GIVEN UY 99E14*69/95X99QS= oleo

.e 1'ENG TONS/DAY99/)WRITE(696)

C .***.*********,*.*********.*.*.

C PETERMINT THE NONPIMENSIONAL TERMSVACT =VBAR

*WACT =UGPI = G H /VHAR**2

G P 7 =/ . V!IAR -2HH =H wBL = 1% wBR =HR WDELR = DELR /wVB k= 1.0

B6

. . - . * - . .n- .. . * , . -. , . 4.. . h, .: ,.*; . .ir, %,+

*.p.

dC

''4%

4,e

HV 1.0W 1.0WRITE(6 9 70)

70 FORMAT(//27Y9*NONOIMENSIONAL OUANTITIES : ')WRITE(6972) GP1 GP29 HH, BL BR, DELR

72 FOR"'AT(5XOTWO GRAVITY TERMS FOR -VERT-, -HOR- ARE ',S 2Fi4.b,/q5X, *DEPTH = 09E14.69 LEFT AND RIGHT BANKS AT 9,2EI4969/,5X, ORADIAL STEP ',E14.6)

, WRITE(6,6)' R * ****T**********E69**************************************

- .. i PEFINE THE INITIAL ORIENTATION OF THE RADIAL POSITIONSIF( KOPT3 *EQ. 2 ) GO TO 75

C 4ADTAL POSITIONS DEFINED FROM THE POSITIVE LEFT BANKc TO THE NEGATIVE RIGHT BANKC ************** KOPT3 = 1 OPTION ************&*******

DO 74 J = If MR(J) 93L - ( J - I ) DELR

74 CONTINUEGO TC 7q

, 75 CONTINUEC RADIAL POSITIONS ARF DEFINED FROM THE NEGATIVEC RIGHT BANK TO THE POSITIVE LEFT BANKC **********.** KOPT3 2 OPTION ************************

R(J) =R * J - I ) ' DELR7A CONTINUE79 CONTINUE

ISE G = 1i I = 0

NIST = 1r ** .**.'. NEW SECTION *******************************

Ra CONTINUEp I = I + I

IF( IST(NIST) .E. I ) GO TO 95C ClO NOT WANT TO STORE THIS SECTIONOS DEPTHS AND VELOCITIES

IKFEP = 0GO TO 100

C r, CONTINUE

C 6ANT TO STOR_ THIS SECT!CNOS DEPTHS AND VELOCITIES FOR LATERC TAB'JLATION

IKFEF = 1

IK = NISTNIST = NIST * I

100 CON TINUEIF( I *F0. ) 'n TO 22nIJK = ILOCCISEG) I IIF( I *CQ. IJK ) GO TO 120IF( I .NE. ) GO TO 229

120 CONTINUr

B7

, ,,.-. . . - . . .'.,'."" '" w2/-"" .,..' '. -"-";"."" -'-"" '"- - " " _ .4. P" .. "' " . d -"" ." ." ".--" " " " . "" " . " •

. ~ J W~ .V . - '. w ., ~ - ~

C *****EITHER OnEI SECTION DEEP INTO A NEW SrGMENT *ac OR AT THE INLET SECTION

ISFG ISEG + 1REAr5C5,125) RC, S19 S2. ALPHA, BETA, THFTAC, [ThOp IiSTEF'"

WR I TE (6 , 6)WRITFC6, 127) IRCSlS2,ALPHARETATHETACefl5ON'STEPS.'

127 FOR'PAT(//v17X9NEW SEGM4ENT-IMPORTANT PARAMtTERS GIVEN AS1 ,/, 5X,'FOR SECTION I =',15*0 RC 9,El4.6/5XSF,'MENT1 LOCATION B3ETWEEN '.2E15o69/95X99AL0HAv BETA =02~~S ' THETAC, D50 =', 2F8.q,/,5X,9NUM3ER OF SECTIONS BETWEE%Isi R S", is 09I59//)

C COMPUTE OTHER VARIABLESRESTAR =VSTAR - D5n, / ( RMIJ *FTNM )

* VSTARC =SQRTC (SG - 1.0) * G D 59 THETAC / rTPM )RATIO =VSTAR / VSTARCFRD VACT /SO~RT( (SG - 1.0) * G D0 / FTMF'DS S 2 -Si ) / NSTEPS

C COMPUTE NONDIMENSIONAL QUANTITIESRC =RC /WACTSl 51 / WACTS2 =52 /WACTDS = S /WACT05C0 050 / ( FTMM * H)CALL PG(FVBAR,050,THETACtPO RALPHABETASGGPIG.2,C5N)

C NO T: 01 = GiC N,9 F, BETA)*C G72 = 2( N )

C G3 =G3( RETA, ALPHA, 0OR, F, THETAC, VDAR,4 C G9 050, DRHOS

WRI TE(6, 130)130 FORMAT//95X,'COMPU7ED VARIABLES FOR THE NEW SEGM4ENT

I GIVEN 9Yt/)WRITE(69132) RES TAR9 VSTARC9 RATIO9 FRO, 61, G2 9 G3

132 FOR'4AT(5X,'RESTAR, VSTARC, RATIO = ,3El5e~,,I,'X,'PF.NST-* IMETTC FROUDE = 9Ei4*69/95X,'Gl, G?, G.S 093E51579/)

W RI T E (6,6 )WRY TE(6, 135)

135 FORMAT(5X9*NONnIMENSIONALIZED QUANTITIES GIVEN PY 9//)WRITE(69137) RC, Si, S?, 050, DS

137 FORUAT(5X,9RC, S19 S2 = 93E14*699 053 *,EI's.6,/95X,*I 'P!4TERVAL BET4FEN SECTIONS DS 9 ',F5.79/)

WRY TE(6,6)C TEST FOR INLETC TEST FOR PEGtIIREC NEW RAIAL ORIFNTATION

IF( I .Eklo I ) GO TO 145ORI-N =RC / Ri

* 1F( ORIEN eGE. (i.Q ) Gn TO 220138 CONTINUE

IF( KnRT3 *Erl. I )CO TO 13QKOPT3 1

B8

p GO TO 140139 CONTINUE

KOPT3 =2140 CONTINUE

00 14 J = ItMTFOPR(J) =R(J)TEMPS(J) =SIMICJ)TU3(hJ) =URIM1(J)TE'iV(J) VIM1(J)THLV(J) HLIMlV(J)THLH(J) HLIMIH(J)

142 CONTINUJEDO 144 J = I. MJJ =( Mi + 1 )- JR(J) =TEMPR(JJ)SIMI(J) =TFMPS(JJ)VIMI(J) = TEMV(JJ)LIBIMI(J) =TUB(JJ)HLIM1V(J) rTHLV(JJ)HLIP1H(J) =THLH(JJ)

144 CON TINUEGO TO 220

145 CONTINUEC *a*.******** INLET SECTION *******a*C SET ARBITRARY CONDITIONS --- CAN IMPOSE ANYTHING

RDO RC% SC =CO

UO =000160 CONTINUE

C ESTIMATE INITIAL DOJNSTREAM V-VELOCITIES AT INLETC' ~SECTION USING LARCY-WEISBACH APPROXIMATION --- NOTE Tl-AT

r VALIIES AT BANKS ARE NONZERO00 170 J It M,'HLIV(J) HVH-LIH(J) HHSI(J) 000UI(-j) = 0flVNI(J) SORT(8.O*GP1*HLIV(J)*SCL*RC/(F*(RC + R(J))))

170 CONTINUE:WRITFr6, 174)

174 FOROAT(/,5X,9INLFT SECTION V =I/WRI TF(69176) (VNJICJ),J=1 ,MMP)

176 FOR ?AT(/,(3X95E15.7))WRI TE(6, 177)

177 FORMAT(M/r CALCULATE THE FLOW DISCHARGE AND THEN' ADJUST TI-F V-c VELOCITIES TO SATISFY THE MEAN FLOW CONTINUITY

CALL PQN(MqR ,VNIHLIV9VA9VOAT9tQT)FTA =1.0ETA = / OTWRTTF:6,17P) Q9 QT ETA

B9

178 FOcPAT(//,5lX,9(n OT, ETA 9,3r16 sag//)

DO 180 1 = 19 mVNICJ) = VNI(J) F TA

*180 CNIUCALL PQN(MR9VNI9HLIVqVAVQ*AT'lT)

* * WRITE(69182)

182 FORPAT(5X99'ETA-MODIFIED V-VELOCITIrES WITH UNITI DISCHARGES")

DO 190 J = 19 Mg MP

185 FORvAT(5Xv'J =9913,' V = 1,716.890 VA =',El6o89$ Vtr* ='9I E16*8)

150 CONTINUEWRTE6g~l OT

191 FO'V'IAT(5X99NEW VALUE OF OT WITH MODIFIED V tC16*8)C CCMPUTE SEDIMENT DISCHARGE

QS =3.0% D00 193 J =1,t M

VVV VNI(J) *VACT

RATA VA(J) /VA(MCL)Q= QS + RATA VVV**B3RB

193 CONTINUEQS OS * AAA /CM - I)OS11 = S / QSACT

WRITE(69194) GS. QSND194 FORVAT(/,5X,'QS ttE14*6,# ENG, TONS/DAY QS/tlSACT 09

SE15969/)Ct * * * * * * * * * * * * * * * * * * * * * * * * *

C FIN4D INITIAL VALUES FOR UBAR AT INLET SECTICN~C FROM THE ANALYTICAL SOLUTION OF THE SIMPLIFIEDC CONTINUITY EQUATION USING A ZERO-VALUE 6OUNCARYC CONTINUITY AT THE OUTSIDF bANKC USE CONTINUITY EQUATION WITH DARCY-WEISBACH FORc 0(V*HL)/DS AT INLET TO APPROXIMATE IJBAR

* TEMP = 0.0UDNI(1) =0.0IF(KCPT3 .110. 1) TJMl (-?.D.I3L/RC) *SORT(lor+PLIRC)

IF(KOPT3 *EQ. 2) TJMI (-2*QeBP/RC) *S~qT(1.J4RR/RC)Cl =8.0 * GP2 *sCL /(F HCl =-190 * ETA *62 *G3 *SORT( Cl * *r*

DO 200 J =2, MtRAnR RC + R(J)UAL C-2.0 + RUj) / PC I*S;)RT( 1.0 + R(J) / PCUBNI(J) =C T!7MP * Cl (JC AL - TJM1 I AflRTEMP = U3NI (J) 0 RADRTJ4*1 JAL

200 CON TTPriUFWRITE(69210) 19 (U!)NI(J) ,J=19M*MP)

210 FOR"AT(/,* 1 =991399 VALUES FOR UBAR = ,/,(4X,5[E1%7))WRITE (6, 12)

212 FOR'0AT(//,5X,9E-ND OF INLFT S;ECTICN'/)

.6

WR I TE(696)GO TO 763

22U CONTINUFC **•**•••'*••*•*• NEW SECTION *****************o****C SAME OR NEW SEGMENTC FCR THE NEW SECTION, DEFINE THE ANGULAR COORDINATEc qAND THE STREAMWISE POSITIONS ACROSS THE TRANSVERSE,C SI. THEN, DETERMINE THE CENTERLINE SECONDARY FLOWc VELOCITY UCLI, AND THE TRANSVERSE BED SLOPE, STI,C USING THE CENTERLINE STREAMWISE DISTANCE." FINALLY* CALCULATE THE LOCAL FLOW DEPTH AT EACH OFC THE RADIAL POSITIONS.

221 CON T INUESC, = SCSC = SC + nSUUI = L I fXP( Gi * ( SC - SCO ) I HH )UU2 = G2 • VEAR / EXP( G1 SC / HH )UU.5 = HH ( ( EXP(GI•SC/HH) - EXP(GI*SCO/HH) )/( GI * RC )UC = UU1 * UU2 * UU3

STI = UC • G3 / VBAR

UO = UC

DO 225 J It M1HLIH(J) = HH + R(J) * STI

HLIV(J) = HLIH(J) I HH" TEST FOR NON-POSITIVE DEPTH. IF SO, MUST MCDIFYC 'ROGRAM.C FXIT IF ENCOUNTERED.

IF(HLIV(J) oLE. 0.0) WRITE(6,224) 19 J, HLIV(J)"24 FORMAT(//,5x,'NON-POSITIVE FLOW DEPTH OCCURS AT SECTIOK

$ I = ',159 /,5X,'RADIAL POSITION J = 9915p5X,'FLOWI DEPTH= ',E2•4,/95X, 'FXIT FROM PROGRAM TO MODIFY')IFCHLIV(J) ,LE, 0.0) GO TO 950

229 CO'JTINUEC PETERMINF INITIAL VALUES FOR SECONDARY FLOW USING

" C FALCON'S RELATION AND THE PREVIOUS SECTION'S V-VALUES0 00 226 J = 19 MFAC = HLIV(J)•VIMl(J)*RC I C HH*VIMI(MCL)*(RC * R(J)) )

UI(J) : UC * FAC226 CONTINUE

DANCLE = SC - .CC ) / RCDO 23s J 1 1, '4SI(J) C RC * R(J) ) * DANGLE + SIM1(J)

231 CONTINUEr NCTF: FOR RC < Ct THE TWO NECATIVE QUANTITIFS WILLC STILL YIELD A POSITIVE STREAMWISE COORDINATE, So

IF(KOPT1 ,Nr, I) GO TO 235

C* ft * * * KnPT1 1 CPTIOj * * * * * •

C FOR A ROUGH APPROXIMATION, ASSUME THAT THE CURRENTr VALUES OF UPAR ARF THE SAMF AS THE VALUES CALCULATEr

AtT THE POLVIOUS SECTION.

.-..

wRu,,".-'."".' -" ."" " " "- - ". ". " ". . -. "

., . . . .- - - . -. .. .' . , . . ' ' . " - " . " - - . . " . . . . -, ., . - ' . " " .

110 232 J 2, MUBItJ) =UpIml(d)

232 CONTINUEGO TO 250l

235 CON,,)TI NU FIF(KOPTI OJE. 2) GO TO 241

C* * * * * * * * * KOPTI 2 OPTION* *********

C AS A ANJOTHER ROUIJ-H APPROXIMATION, DISCRFTIZE THFc CCNT INUITY rQUAT ION TG APPROXIMATE THE VALUVS F F0 U;'ARC IN THE FIRST ITCRATION.

TEMP 1.

81 =2.0 * CP2 * SCL /F832 =-1.0 Gi * SI(MCL) / P1HRl1 -,5.0 G2 * G5 * ;f)PT( R1 ) XPC P2n0 ?40 J =,DELR = R(J) -~Jl

* TJ = HLIH(J) *(RC + R(d))FACJ R(J) /((RC + R(J) )*SfQRT(TJ /RC))UAL 01l * DELR *FACJ

U8I(J) =UAL + TEMP TJTEMP =URI(J) *TJ

240 CONTINUEGO TO 250

241 CONTINUE

* * * . * * K(OPT1 3 OPTION *********.

c ANALYTICAL SOLUTION OF THE CONTINUITY EFIiATION Tr F.-C USE) TO SOLV-K FOR UPAR AS AN INITIAL APPROXTMAT ION 'HENc TAPTING A NEW SE:CTIONe

*I = -3.9 * G2 * G3 *SQRT( 2.1 * GP? SCL /FZl = 21 * EXP( -GI* SI(MCL) / HH)TE*NT =0.900 ?4F, J =29 MCALL EVAL(STIHH9RCvR(J) ,R(J-l),RIA)ROTJ =HLIH(J) *( RC + R(J) )UB1CJ) =( 21 R IA + TENT) /BOTJTENT =UPI(J) *POTJ

246 CONITINUF250 CONTINUE

C * * * ** * * * * * * * * * * * * * * * * * * * * *

255 CONTINUEC FIND STREAMWISE VELOCITY V 9Y SOLVING MOIMENTUM FOUATION

C THAT HAS PEEN DISCRFTIZFP INTO A QUAr)RATIC FORM USING AC ' 'IMPLE 9ACKWARl DIFFERENCING

C ~~~,A0Y Tt OIECIN cOLUTION nV 7?ICQFTI/f") MFUM L'TO

c 'OF V ,hUT F IRNT9 NEEO A PIJNDARY CONO! TION FOi, V *US

r flARCY-WFIBACH FOUAT ION AT RANK.IF(KCOPT3 .O., 1) RJ =IF(KCPT3 *Er, -) RJ dVI(l) SQRT(8.0*GPI*HLIV(1)*SCL*RC / (F*URC RjV NI1(1) VI (1)

B12

.................... .....

*TE*'V(l) VI(l)K0IJI4T = nETA =1.0KlJVT=KOT =IKTT =0

2! 12 CON~ T INIJE

316~ CON T INUEKlJV

2 20 C(PITINUEC**44******** DISCRETIZEn TMO!,FNITUM EQUATION FOR V

----------------GFNTEP HERL FOR NEW ITFRATION Ar SAM1E SECTION- -- -- -[PRy 0.0DO !,00 J = , ) s M

r PETERMINF F-FUNCTIONS AT SPECIFIC POSITION (4 9R)CALL Pr(FHHiRCVI3APGl,(;2,'O3-,R(J) STIUI(J),FI ,F2,F3,F'.)

STFPR R(J) -R(J-1)

CALCULATE COFFFICIENTS FOR GJADPATIC FUNxCTION IN U:C A.A * U**2 + H El * V *CC 0.1i*

TP=HH * F4. S TEPPTS = HH * F4~ / ELS

IF(KOPT2 fENe. 1) GO TO 34t0r KOPT2 1I

C COEFFICIENTS FOR PEGULAP DISCRETIZED S.TREAMJISE MOMENTUP EQ.C CEPF'!O ON: V(I-1,J),VNEW IJ-l),U(1,J)IJBAR(1,J)U31R(IJ-1)

AA =RNP F? *TS + F / 890P R ( F 1 T R) U (F I(J) + kf(J I ( 2. 2 ~ * )CC = P1 HV *SCL *F4. * RC / (RC *R(J))CC zCC + RP12 *TS *VIMI(J)**?

C TF< * CUf3I(J-1) *UI(J) /C2.01* RN *1.0 )CC -1.0 '(CC *C4. VNI(J-1))Go Tfl 360

_j 4.3 COIJTIIJJ* * * * * * * * * * K!PT2 2* * * * *

COEFFICIFNTS FOR MODIFIED DISCRFETI7En STREAPWISE MCI/ENTUMc FOUiATION WHICH INCLUDES THF CONTINUITY E.lUATION

*-:OTF : NO RAtrIAL PIFFER~t..TIATIOrq IN UBAR H'E:ENDON: V(I-1,J), VNFW(IJ-1)9 U(IgJ), UPAR(IJ)

A- A= F2 + ( RN2 - 1.0 )+ TS C RN2 - 0.5 )4F/8.0PR =TR * ( JE31(J) + UI(J) /(2.0 * RN + 1.0oRr R P + IT(J) *Fl R . N + 1.0CCI rPl * HV *Sr.L F4. PC /(PC *RCJ))CC"' Ti;*( UH3I(J) + UI(J) / I2.*QtN + 1.0 )*)VNT(J-1)CC = TS *CRN2 - 0.5 )*Vlwl(J)**?CC =-1.0 * C l + CC2 CC3)

B13

CHECK -B- 49n AA *CCIF(,-HECK *LT. g:l) CO TO 400

I () %J -:3F + SORT(C7HECK) )/C2.0 AA)*C SUiM THE DIFFERENCES 8ETWEE'J THE OLD A~n NEW VAL(UF.----C C JOTE THAT THE BOUNDARY VALUE IS NOT INCL(Il.' TNr.C THE ERROIR S114CE IT IS FIXED

FRRV =ERRV + A5( VNIC.J) - VI(J))I 1IF( VNI(J) *GE. Co'j ) r0 TO 5f00K C NEGATIVr STREAMWISE VELOCITY *~~~*.*

WRITC7(6,393) I, J9 VNI(J)

!qO FORtFAT(/95XINEGATIVE STREAMWISF VELOCITY',/95X,$ SECT!ON I =99 T5,' RADIAL STEP J = 9'gIS, V 9-l)7/GO TO 415

400 CONTI!JUEWRITE(6,419) CHU*CK

410 FOR'lAT(//,5X,'l~lFGATIVE RADICAL IN U-O)UAORATIC---CHErK to

415 CONTINUEC EXIT FROM PROGRAM FOR A NEGATIVE RADICAL

GO TO 950590 COPITINUE

C FIND NEW U-VALUES HASED ON V JUST CALCULATED *** DO ,05 J = 1.1 M* FAC = H-LV(J)*VNI(J)*RC / (HV+VNI( tCL)*(RC *R(.J)))

UI(J) =UC a, FAC505 CON~TINUE

CRV FRRV / M-1) )/VIARKUVT =KUVT + IIF( ERRV2 9LE. EPSV )GO TO 513

* ~~C *******************************C RESULT OF V CHANGINO TOO MUCH --- TRY NEW U-VALUFS U..;C CALCULATION OF NFW V-VALUES UNTIL CHANGES ARF MINOP

4 KIJV =KUV + 1*C REPLACE OLD VALUES WITH NEW VALUES AND ' OLVr FOR V

C AGAIN BUT NOW WITH THE. N-,WLY UPDATED V-VALUV!S. NOTEC TIFAT U-BnUNDARY REMAINS UINCHANGED.

DO '938 J = 2, 1-VI(J) =VNI(J)VNI(,,) =0.0

C P~ CO0N T INU EI'FCKUV oLTo KUVv') GO TO 32r

* WRITE(6,509) It KUV, EFRV~0 FOR 'AT(/ 95X9*MAY IMUF NUMOER OF I TF; ATION", 9 OR L-V I XCFU; E:D

I 9,Xv /t5X,'SECTTON I = 915,' KUV = $I"** FR4V1 E b5 6v/ )

GO TO 990Otl1l CflNTT"JUE

C **0******.V - U COMPATIBILITY

CALL PN(MRVNI ,HLTVVAV'.;,AT,.-IT)PATC =All3S( 0 O T)/KQT KOT + I

B14

................. ... .... ..- ...

ETA' ETA

-. DO ':35 J 2. m

TE"1V(J) VNI(J) *ETA

CALL P0(MR ,TFMVHLJVVAW,9ATsQT)C ~ ~ FIND NEW UBAR FROM ACTUAL (V-H) VALUES**h

.

" T ,lf ri I .

KOUIPT KOUNT + IIF(rOPT3 o[(9 1) TI = ( RC + .L )**2'F(KOT 3 91Q9. P) TI ( RC + RR )**2

c USE CONTINUITY EUATION WITH ACTUALLY CALCULATEc VALUES FOR D(V*HL)/DS AT THS SAME SECTION

C TO GET THE SECOND APPROXIMATION OF UBAR,C AGAIN WITH UAR EQUALS ZERO AT THE V3ANKL

.D r 0 0 J 2,

CAPIX =-0.5 *(TEMVCJ) * HLIH(J) - VIM1CJ) * h-LIMlH(J))CAP, v= CAPX ( ) - SIMICJ))T2 (ORC + R(d) ) 2DOlT HLIH(J) RC ( RC P() )U SCNT(J) ICAPX .UT2 - Ti ) + TFMP / CPOTJ

TCAP = UV(J) * L)TJ

ERPL = RIJ + A", UINI(J) - LI(J) )tr TI O T INIJFC"/ * * * * * * * * * * * 4 * * * * * * * * * * * * * * * * * * *

C CONVERGENCE CRITEPIA FOR ERRORS: ITERATE UNTIL THE AVERAGE. CIFFERENCE PETWEEN 2 CONSECUTIVE ITERATIONS FOR ALL THEC RADIAL STEPS ACROSS THE TRANSVERSE, DIVIDED BY THE MEANC ')OWNSTRFAM FLOW VELOCITY, IS SMALLER THAN SOME PRESCRIBEr)C SMALL VALUE.

EPRU2 ERKU / ( ( I - ) * ABS( UBNI(MCL) ) )KTT = KTT + I

IF(wRRU2 LF.o FPSU) GO TO 750IF(KCUNT ,GE KMAX) GO TC 770

C t"EACHED AS A RESULT OF LARGE ERROR WITHOUT EXCEEDINCTHE mAXI m UN ALLOwABLE NUMBER OF ITERATIONS.

c %0 CONVERGENCE YET, SOC17 ITERATE THE WHOLF PROCESC AGAIN FOR A NEW V AND UBAP.

KP = KOUNT / KTIMKv = KP * KTIMIF(KP ,NEo KOUNT) GO TO 741WRIT (,97%O) I, KOUNT, ETA, ETA.I, FRRV2% ERRL

7i FR0'AT(/9Y, pOF0 SECTION I = ',15,' KOUNT = 0,T,/,5X9i ' rU4RNT AND PREVIOUS ETA AFE '.2F15e6./v9X,

4 ' LATIVE F.Rr;. IN V AND IIEAR ARF ',2E15.9, )

B15

. \ ;.. ya g% - K£..:>: K* .. - B15- *-'-.*

C WRITE(69705)

705 FORM'AT(bXqOPREVIOUS VALUES F00 V GIVEN 9Y 09/)C 01TTE(6,710) (VI(J)qJrltm)

710 FORM~AT(5Xq6El2*4)WRITE C6,712 )

712 FOP MAT (5X,'OUN-ET A-MOflIFIED MOMENTUM VALUES FORI V GIVEN 9Y OP/)WRITE(69710) (VNI(J)9J=1,")

C WRITE(6,714)'7 14 FORmAT(5X9 9 OLD VALUES FOR UPAR GIVEN BY O9,1)

C WRITE(6971l) (URI(J)*J=19M)WRITE(6, 716)

716 FORMAT(5X,'NEW VALUES FOR LIBAR GIVEN BY go/)

* .C WRTTE(6,718)

718 FORMAT(5Xv*LATFST VALUFS FOR U ARE 99/)C WRITE(6,710) (UICJh9J=19M)

748 CONTINUIEC RESET NEW VALUES BEFORE QEPEATINCG THE ITERATION

DO 749i J =2, MVI(J) VNI(J)UE3I C) Un-NICJ)VNI (J) 0.0Ut3NICJ) 0.0

7499 CONTINUEGO TO 312

* 750 CONTINUE

C REACHED AS A RFSULT OF A SATISFACTORY SMALL r.RRiR

r TN UBAR OR V THIS IS WANT WE WANT.C ETERMINE FINAL ETA-MODIFIED V-VALUE

00 752 J = 2,9VNI(J) =TEMV(J)

*752 CONTINUEC D*.****.* 7FTERMINEF SEDIMENT DISCHARGE ****

OS =0.0DO '153 J =19 MVVV VNI(J) *VACT

RATA VA(J) /VACMCL)QS = QS *RATA *VVV**F9P

753 CON~.TINUEQS =AAA OS (,/Cm - 1)QSN OS / SACT

NO1.1T III /'IO'JTNOUT NOUT * IO'UTIFCNCIJT *NE-4 ITI) GO TO 763WRITE(.9755') Iq ST!, UC* ERRV2, FRRUL'? KOUNT, HATO

755 FOR4AT(I,5X,'FOR SECTIONj I =',149' ST =9F46I *UCL 09EI4.6 /95X9 9SATISFAC TORY TTEPATION9,

B16

i CtX9"ERRV2 9 'El4*F,' FQPIJ2 =9 46/'X

't ':U'4"FR OF ITERATIONS XOUINT = tP4/!XI OR7LATIVE DIFFEREiNCE OF OT TO G I" ',r15o79/)WRTT70 757) SC, RC, QT, ETA, (QS9 uSNO

7!77 FcWm AT(5)XoPCFNTEFkLINE POSITION S = 9914&6*9 WITH Qr14.96, 9 /5 X 9 0N EW rT r; 'E 16 o A WITH ETA =tf197/'XI 'O0M.ENT DISCHARGE = VE14.599 LiS/QSACT 99F14,59/)WRITE(4,976l) KTT, KOT, KUVT

769 FORMAT(5X99 NiLJ'KER OF ITERATIONS FOR UE19 ,p UV ARE 9,316)

WRITTE-(6971b) UlIJ9lMM)

WPI TE(69761)761 FORP'ATI5X,'[TA-MOOTFTED MOMEiNTUM VALUP FOR V rIVE.' BY')

WRI TEC69710) CVNI (J),J:1,MMP)r RTTL(6,718)

c WPITE(69713) CVl(JJ1,F9MP)WRITE (6, 762 )

765 CON TINUEC < TOPE VALUEcS FOR V9 U9AR, ANfl U FOR LATEP TAFULATION

I F I( TKFEP *%Eo 1 ) Cin TO 7b7IF( KOPT3 oEQ. 1) GO TO 765DO 764 J 1g 9V ( I K9J) VNI(J)UHij( TK9d) LHINT (J)U(,J) UI(J)HL(IK9J) HLIV(J)00Q '(IKpJ)=AAA*(VNI(J)*VACT)**-4EUf3PU (IK ,J )UT d) UPNI Cd)US.)RT=UBPUCIKJ)**2,VN!(J)**2UTR AN( IKJ)=SQRT (USGRT)

IF(UCRII.OFQ.) GO TO 1AN7mJU=URPUC !KqJ)/VNI Cd)ANGLECIK ,J)=j7.?q578*ATAN(ANGUU)GO TC 764

1 ANGLE(TR9J) =90.0764 CONTINUE

.d.. GO TO 767165 CO!TINt!E

DO 766 J =J,

VO(lKtJJ) VNE!I(J)

* U(T"(,JJ) = UI(~J)fL(IKoJJ) = HLTV(J)

U~tPI.(IKJJ)=UI (J)),,IrNI(J)* USO1 T=1JB3PU(IKg,Jj)**2#.I (J)t**?

UTL.ANIKqJJ)=S0RTC1JSORT)

B17

-~~~ - - -. .-. .-.. . . . . . . . -- -.

IF(LJCPII*Ekl).O.O) GO TO 3AN6U0=UBPU( IK9JJ)/VNI(J)ANcLE( I' JJ) =57.2?9578*ATAN(ANGI)U)CO TO 75b

3 ANGLE( IK9JJ)=9O0766 CON~TINUIE767 CONTINUE

DO 76A J 19 mHLI'OIV(J) =HLIV(J)

HLI'ilH(J) HLIH(J)SIMl(J) = ST(J)

UBIM1(J) UE3NI(J)768 CONTINUE

DO 769 J 19VI(j) 00VNI (J) 0. 0

*TEMV(J) = 9

UBI(J) = *

*UBI (J) 0.0HLIV(J) 0.0HLIH(J) =0.0SI(j) = 00

76q CAJ T INtJEETA = 1.90IF( I *GE* 'Y GO TO 8,10

* C ******e* TRIAL '(TOPSC IF( I *GE9 35 ) GO TO 990

GO TO 80710 CONTINUF

C REACHED AS A RESULT OF EXCEEDING THE ALLOWAI:LF rJ)McifP* C OF ITERATIONS

WRITE(69780) It KOUNT9 ST1, SC, ERRV29 !RRU2700 FORMAT(/,5X,9OQ SECTION I = 0,15,' KOUNT 09'1590 cJI

19E15-69 /,5X,'CENTERLINE POSITION = 0S E14o6,f,!iX96RELATIVE ERRORS IN V AND U13AR AkE 992rl5e()WRITE(69716)

WRI TEC69712)

WRI TE(6,719)WRITE(6,71V ) (UT (J),J=1,M)GO TO 950

P 03 COiTINUE

c-------------- PRINT FINJAL RESULTS ORTATNEP-------------------c t!.SULTFS ARE TAfWLATrEO FROM~ THE NE6ATIVE RIGHT PANK TC THL- FPtSIcTIVE 'IGHT tANK- --KPT!,

B18

....................... a. * * . a

IF( KOPT3 .10. 2 )GC TO SOPDO ?'02 J = , '4

P02 CONTINUE0O F0)4 J =19 M4

R(J) = TEM4PR(JJ)P04 CONT7INIEeO8 CONTINUE

C PESULTS FOR TRANSVERSE SHIFT VELOCITY UPARWRITE(6, RiO)

F10) FORVAT(//932X99VALUES FOR UEIAR',//)WRITE (6, 820)

820 FORMAT(3X,'R99,3X99 I ',/)

EMFORtOAT(4X96 112)no 15O J =, I I

8493 FO~vAT(F7*3 ,bEl2o4)E50O CONTINUE

WR!TE(69852)852 FOR'4AT(//,o3X,'o I *,32X9'R')

F54 FORP'AT(6X95Fl2Ll)DO P58 1 =It ISTO

8 FO'RMAT(I5,bX95EI294)\f. Eb CONTINUE

r RESULTS FOR SECONDARY FLOW VELOCITY UWRI TE(6,86")

P60 FOPf'AT(//,3?X,9VALUES FCR U'/)

-~~~~ ~WRITE(69833) (S()1,SO

WRITE(69840) R(J), (U(19J),1=1,ISTD)FPC CONTINUE

C RESULTS FOR STREAMWISE VELOCITY VWPTTE(6,98qO)

F 50 FOR PAT( //932X9 OVALJES FOR V*,//)* WRT(69820)

WRITE(6,833') (TST(II1,I9STO)DO 910 J = ~1

* WRITE(69840) RCJ), (V(IJ),11,*ISTC)5 0U CONTINUE

WQITE(69852)

DO 'nG8 I1 1, I'ST1)WRITE(6,8) I ,VI,91 ,VI,91IV(IIM2V(IL'43),V(1IM4

498 C 0'T INU F

B19

C RESULTS FOR LOCAL FLOW DEPTHWRITE (6,91 0)

510 FORMAT(//%32X9'VALUES FOR HL'./I)I-. WRITE(69823)

DO q2?0 j= 19 M~WRITF(6,84U) R(J), (HL(IJhiI=1.ISTD)

920 CONTINUEWRITE(6, 921)

S 21 FOP MAT (//925X9,$VALUES FRU13AR * U9/WRITE (6, 82,I)

K WRITE(6,83!) (IST(I ),IllISTf))r1o 930 J=I, M

530 CONTINUEWRITE(6qq3l)

931 FOO'4ATd//925X99VALUES FOlR SORT((UBAR*U)tr*?*V**?)09//)WRITFC69820)

* WRITE(6983N%) (!STU ,9I=19ISTD)DO 94 0 J=19 MWRITE(69840) R(J), (UTiRAN(IJ)g 1=191STI))

S40 CONTINUEWRITE(69941)

941 FORMAT(/29X,'VALUES FOR VELOCITY VECTOR ANGLES',9MWRI TE(b.,820)WRITEU6,83C) (ST(I ),I=191FTn)DO q45 J~ls M

545 CONTINUFWRI TE(6,V46)

S46 FORAAT(//,2VX99VALU[S FOR UNIT SED141ENT DIS;CHAPCGES,*//)WRIT!7(6982l)

0094 J=19 M

* WRITE(6,84l) RCJ), (QQSS(I.J)v 1=19ISTD)947 CONJTINUE950 CONTINUE

Cc PRHAOS 1/0 COMMFNTC

CLO SE (b)CLO! F(:)STO)PEN)

SUIROIJTINE PQN(M9RVHLqVA9VQATQT)C THIS SU4ROUTINE DFTERMINFS THE FLOW DISCHAREF FnR A*r rC]VIN RECTAPNriULAR CROSS SECTION OF C'nNSTANTC 110TT)M rLOPI. WITH KNOCWN VELOCITIES AND ()LPTH-

C VARIAHLrAS:

B20

r I ' j! ,EA SF INCREF NTS OR VERTICALS IN TH. CkOSS SECTIONC ; = LOCAL RArlIAL POSITION ARRAY (FT) *

r = LOCAL FLOW VELOCITY ARRAY (FT/SEC) *

r HL = LCCAL FLnW OEPTH ARRAY (FT) *

C VA = UNIT FLOW AREA ARRAY (SU FT)C VC. = UNIT FLOW DISCHAkGE (CFS) ,c AT = TOTAL FLOW AREA (CUnIr FT)

C UT = TOTAL FLOW DISCHARGE (CFS) *

C * * * * * * * * * * * * * * * * * * * * * * * * * * * *

DIMENSION R(17)9 V(17)9 HL(17)9 VA(17)9 VQ(I?)

AT 0.0T = 0.0

Do 1 I 9lMVA( I = Vo10

Vt I )= 0.010 CONTINUE

DO ?03 I i,94

IF(J *Ego M) GO TO 100IF(HL(J+I) *LT. 0,0) GO TO 120IF(J ,,T. 1) GO TO 50W2 - ',5 * ABS( R(J*I) - RIJ)

-'" H HL(J) * HL(J*I) ) / ?.C

VA(J) = w2 * ( HL(J) + H2 ) / 2.0

VO(J) = VA(,J) - V(J)GO TO 110

50 CONTINUEWI = W2

HI = H?.W2 = 5 * ABS( R(J+I) - R(J) )

H? = ( HL(J) + HL(J.1) ) I 2,rA1 = Wi * ( HI + HL(J) ) / 2.0

A2 = W2 * ( HL(J) + H2 ) / P&PVA(J) = Al + A?

VQ(J) = VA(J) V(J)GO TO 110

ICG CONTINUEWi = W2Hi = H?

VA(J) = 1 ( Hi + HL(J) ) I 2.0VO(J) = VA(J) * V(J)

110 CONTINUEAT = AT + VA(J)-T = OT + VQ(J)

GO TC 2no

120 CONTINUEWi W2Hl H2)W2 = 0.5 , ABS( P(J+l) - R(J) )H2 = HL(J) + HL(J*l) ) 1 2.0

* FAC = i I ( HI - HL(J+I)

WO = FAC * ADS( P(J+I) - C R(J) * 4(d-I) ) I 2.0 )

B21

-%~~~~~~~~~~~~. . . •.. . .. . . . •. . . . . .. . ... . . . . .I .° .- . , o o. . •.

• : ; .,..." .,....-............-....-.............,... , ............ .,.-. '.',' . .,.. . -, '>,'%, ' ',.,,

IF( H2 oLT. 0o. ) GO TO 15rlDELq = ABS( WO - C WI 4 W2 ) )VA(J+) = 5 * H2 * DELWVQ( J+l) = VA(J+l) * V(J)Al = WI * ( HI + HL(J) ) 1 2.0A2 = w2 * ( HL(J) * H2 ) / 2.0VA(J) = Al + A?VA (J) = VA(J) * V(J)AT = AT + VA(J + VA(Jl)

" QT = OT * VQ(J) * VG(JI)GO TC 220

153 CONTINUEVA(J) = 0.5 * WO * HIVQ(J) = VA(J) * V(J)AT = AT + VA(J)"T = OT + VQ(J)GO TO 220

230 CONTINUE220 CONTINUE

RFTURNE ND

C * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *SU.POUTINE PG(FVBARtD.,rTHETACPOktABRHOSG, GI , 2,G. )

C THIS SUPROUTINE DETERFINES DIMENSIONLESS PARAMETERS •C GI G2 AND G3 FOR THE DEFINED AND INPUT QUANTITIESC * 4 * * * * * * * * * * * * * * * * * * * * * * * * * * * * *

C DEFINED: *C RFCS = SPECIFIC GRAVITY OF THE SEDIMENT PARTICLES *C B = PARAMETER IN THE VELOCITY-SHEAR RATIO RELATIONC A = PARAMETFR IN THE SHEAR-BED LAYER RATIO RCLATIO, *C INPUT VARIA13LES: *c F = DARCY-WEISBACH FRICTION FACTORC VBAR = MEAN STREAMWISE FLOW VELOCITY (FT/ EC) *C D50 = MEDIAN BED MATERIAL PARTICLE SIZE (MM) *C THETAC= SHIELDS CRITICAL SHEAR PARAMETERC PCR = BED LAYER POROSITY *C G = GRAVITATIONAL CONSTANT *

CALL PGI(FoBoGl)CALL PG2(FG2)CALL PG3 (VHAR 9RHOS9050,F qBATHETACPORGG3)RETURN

"4 END

SURROUTINE PGI(FBtGI)

RN 1.9 / .ORT(F)T (3.0O RfJ + I.O)*(2.r,*RN 1 l,0)/(C2.O*RN*&? * RN I.f')Gi = T * H - F / R.0RETURN

C * * * * * * * * * * * * * * * * * * - * B * *

S° B22

• " "-""i "% " " " "°% ' '

" " " '

"' % '•

" . - *" -w" : " ' "" " "" "" "" """ ''" . " " "" "' "

".- StJ''(,UTINF PG2(FG "')

RN 1.0 / SURT(F)1 -(3.0 * RN * IC) f (2.0 *RN lo)) • (RN 1.0)

T6 (?.o 9 RN**? * RN * 1.0) • (RNI + 2.0) * RN62 Ti T2RET U PEN)

C * * aa * w * * * 0 & * * * * * * * * * * * * *

SUBPCIJTINE PG3(VL3ARRHOSDC90F,.,At THETACPORGC-,G3)Ti = F * THETACT2 = S.0 * G * 950 * (RHOS - 1*0)T3 = StRT(T1 T?)IT4 * V'4AR / (A • (1.0 - POR))G3 = T4 T3

R FT U R N

SU~i OUTINE PF(F ,HRCVBAP ,GI'G2G3,R,STUFl ,F2,F3,F4)'C THIS SUBROUTINE DETERMINES ALL FOUR OF THE *

C r;IMrNSI0NLEqS FUNCTIONS Fl* F2, F3, AND F4 FOR THE *

r INPUT QUANTITIES GIVEN *

.4C k * * * * * * * * * * * * * * * * * * A * 0 * * *

. INPUT VARIALrS:

C F = nARCY-WrISBACH FRICTION FACTOR •CH = MEAN DEPTH OF FLOW OVER CROSS SFCTICN (FT)

C RC = RADIUS OF CURVATURE OF CHANNEL CENTERLINE (FT)C VHAR = MEAN STREAMWISE FLOW VELOCITY (FT/SEC) *

C Gor,?,G.3 = DIMENSIONLESS PARAMETERSC R = PADIAL POSITION FROM CHANNEL CENTERLINE (FT) *

C ST = TRANSVERSE BED SLOPE •C J = MAXIMUM SECONDARY FLO4 VELOCITY (FT/SEC) •C F1oF2,F3,F4 = DIMENSIONLESS FUNCTIONS TO BE EVALUATLDC * * * * * * * * * * * * * * * * * * * * * k * * * * *

CALL PF1(STRCRoHF1)" CALL PF2(S TqRC, RGlvG2,G3,HF2)

CALL oF3 (STRCURG1,G2,G3,HFVBARF")CALL PF4 (ST qRHF4)R FT URNFNf

SUnROUTINE PFl(STPCqR,H9FI)

Fl = ST * ( H + R * ST ) / ( RC + P )RETURN

ENOC * a * * * * * * a * * * a*

SU'4OUTINE PF2( STRCR9 GlqG?9G3,HF2)TI = G2 * G. m / RC

T2 R * T / HF2 RC * TI - T2 ) / ( PC * R )R ET ,J riF FB2

• . B2 3

V . .. . • •

-• " • - , • . . _', W. '.' , # ,./, ., . ;wt ,• - . , - . - . - , . . . . • .' . . "

SUPIOUTINE PF3( OT RC109R,9G) 9G2,G3 9HF VSAPqF5)Ti1 = F * LI / (S*O VB3AR)T2 = G 2 aH )/(RC + R)T22 r-1 aRC a 'T /(G3 *(PC aR)

RN =1.0 /SQRT(F)T3 =( R *ST *H )/HT 3 3 G3 7 R U H * VEFAR)

F3 =T4 -Ti

RETURNE NO

C a** * * **a * * * * * * * * * a *

S U3 OUT IN E P F 4 ST,9R 9 H 9F 4T R * ST / H

F4 1.0 *TRET URN~END

C * * * * * * * * * * * * * * * * * * . * a * * *

SU9ROUTINE EVAL (AP ,CRJsRJM1SUM)C THIS SUPROUTINE EVALUATES THE ANALYTICAL SOLUITIfN OF

THE C CONTINUITY EQUATION AS THE PROGRAM ?IEGIrNS A NEWC SFCTION. THE GENERAL FORM OF THE INTEGRAL ISC G-IVEN BY:C I =IN~TEGRAL( X *SORT( C*(A*X + '3)/(X # C) )*

C WHERE:aC X =RADIAL COORDINATE, R(J)aC A =TRANSVERSE BED SLOPE* STc C3 MEAN FL3W DEPTH, HC C =RADIUS OF CURVATURE. RCaC T =TRANSFORMED COORDINATE, DEFINED AS:*C =SORT( C *(A * X aP) / (X *C) )aC =SORT( RC ( R(J) aST + H )/ ( R(J) + PC)) *

C RJ9RJ'4l =UPPER AND LOWER INTEGRATION LIMITS.p RESPECTIVELYC SUM =FINAL EVALUATION OF THE INTEGRALa

T2 = SQRT( C * ( A * RJ aR~ Rj + C))Ti= SORT( C * ( A * RJm1 + R RJMI C))SUM Of.0TT =A * CIF(TT 9%'E9 0.0) GO TO 70WRITE(69.6U) A

60 FORMAT(/ 95X9 ZFRO TRANSVFRSF RED SLOPE ST = s2041STOP

-'70 CONTINUEIF(TT *6T. 0.0) GO TO InURIA =ATA'4 T2 / SORT(-TT) )-ATAN( Ti / ',QPT(-TT)RIA r'IA / SORT( -TT)GO TC 15'G

100 CONTINUERIA2 APS( (T2? - SRT( TT ))/CT2 SORT( TT)))

a, B24

I .-- a. . .' ,,* o.." ." . -. o.'- , . .' • , - . . - .. ,.-.i .. ... o. ,.,, .,, ... jw.... . ..-.

- ..°.- - . . . .

4/ .°y,° . .. c .* ..-.o

774 . .. .

RIA) ABSE( Ti SGRT( TT Ti SIPT( TTRIA (ALO0,(RJAP) - ALOG(RIAI) 2 .U SO~RT( TT)

F AC2 T?)*2 - TTFAC~C I Tl *2 - TTSI = T2 /FAC2**2 )-(Tj / FACI**2)Si -I = ( TI - B ) 4.0S2 = T2 /FAC2 Ti FACt

S 2- = 5.3 * TT B )/C8.0 *TT

= 1. w( 3.0 TT + P )*RIA /C8.0 *TTSUM 2.f C TT F Si + S2 *S3 ) C**2R -T ORN

-a B2 5

INPUT FILE: SEGDAT

1.56 0.505 8.0 .00104 0.40 2.65 i.IE.-S 4I t7 273 409 545

43.0 0.0 67.5 i.416 3.276 0.032 0.30 3638 7 67.5 35.0 1.416 3.276 0.032 0.30 36-34.83 135. 0 202. L 1.416 3.276 0.032 0.30 136-1.34, 202.5 270.0 1.416 3.276 0.032 0.30 136

B26

,¢~

.3k

OUTPUT FILE: OUTT

MATHEMATICAL MODEL FOR THE PREDICTION OFIHE VELOCITY FIELD IN RIVER FLOW

VALUES FOR VAR, ., W, SCL, POR, SG; RMU, NSEGi. SG E j 3.00 0.10400E-02 3,400 2.650 0.110E-04 4

sFTIN NUMBERS WiE NEW SEGMENTS LEGI ARE. 27 27 409 545

DOWNSTREAM STEPS 54S RADIAL STEPS 47 CENTEr ATM= 9RADIAL POSITIONS STORED ATJ= S 9 2 17

,. RESULTS OUTPUT EVERY 2 SrpTONS

, D/S AND RADIAL OUTPUT FReQuENCY I i STEPS

RELATIVE ERROR CRITERIA FOP . AND UBAR ARE O.Oo100 0.010oMAXIMUM ITERATIONS 20 PRINTED EACH 10 ITERATIONSPROGRAM OOTIP.NP FOR UBAR, MOMENTUM FORM, DIRECTION ARE 3 n

INITIAL NUMDCR OF SUBINTERVALS FOR SIMPSON RULE 0MAX NUMBER OF U-V ITERATIONS IS 20DIMENSION CR NUMBER OF SECTIONS TO BE TABULATED IS =

AND ARE AT SECTIONS.I 69 205 341 477 545

SEDIMENT PWER !AW OF THE FOR QS A t V )1BWiTY. A, P = 10B0 4 0001

DISCHARGE 6.3024 DARCY-WEBACH r = 2.355483E-0iPOWER-LAW N 0.42429190E 01 RADIAL STEP = 0.S000E+00LEFT & RIGHT [ANK AT R 0.40.OE+01 -0.4000E+01EAR V C"",' O.1299iEO0t rROUDE 40 0.387I14E+00

N-TERM "VE ' O .i0377",QS 0.639621E+00 ENG TONSDAY

a ...

NfINDIMENSIONAL QUANTITIES. TWO GRAVITY TERMS FOR -VERT--, HOR- ARE 0.667648E+0i 0 OS766E+03

DEPTH 0.631250E-Oi LEFT AND RIGHT BANKS AT 0.500000E+00 -0.SOOOOOE+0ORADIAL STEP 0.625000E-01

Ni W SEGMENT --- IMPOPTANT PARAMETERS GIVEN ASFOR SECTION I = i RC 0.430000E+02

B27

4*a %,"

76 j, .1

SEGMENT LOCATION BETWEEN O.000000E+00 0.675000E+02ALPHA, BETA = 1.4160 3.2760 THETAC, D59 0.0320 0 3000NUMBER OF SECTIONS BETWEEN S! & S21IS 136

COMPUTED VARIABLES IFOR THE NEW SEGMENT GIVEN BY

RESTAR, VSTARC, RATIO : 0.116313JE+02 0.409E-0 0.317991E+0iDENSIMETRIC FROUDE = 0.682463E+OiG!, G21 GZi 0.718179E-01 0.6249260E+00 0.3922577E+00

NONDINENSICHALIZED QUANTITIES G'IVEN BY

RC, 5, '02 = 0.537500.E+i 0.OOOO00 0.O4Z75O0E10 DSJO 3.94901E-02INTERVAL BE-TW.EEN SECTITAONS DS 0.6204044E-01

INLT SECTION IV

0 - 100 E L1 0.0406[ 0. J 03682E+Oi 0. i030402E 41 0.1024100E+010.1171201 0.l0IiOSE+01 0,1005865901i 0.1800000901 0.9942364E+000.0871E40 0.98390i6E+00 0.1775252E+00 0 9721i3949+00 0.9663416E.00

.§7, ET 0.1900000090 3.i0010936E+31 0.99890757E+00

ETA-MODIFIED V-VELOCITIES WIT: VNIT DISCHARGES3 = i V = 0.10488832E901 VA =0.3if'SOCO0E-0i VQ : 0.32777600E-Ot

3. 2 V 0422237E+0 1 VA = 0.625000OOOC[-014 VQ =0.65138981E-013 3V 0.10356894E+0! VA = 0.62500000901i Q = 0.64730585E-013: 4V 0.10292764E+01L VA = 1.62500000F 04 VQ =0.64329773E-01

J =5 V = 0.102129009E+01 VA = 0.62500000E-0i V9 = 0.63936308E-013 6 V = 0.i0167997E+01 VA = 0.62500008E-01 VQ =0.63549981E-013 = 71 V = 0.1017291E+Oi VA =0.62500000E-01 VQ =0.63070567E-013:= 8 V = 0.100476629+01 VA =0.62500000E-Oi VQ : 0.62797889E-0i3 9 V = 0.99090757E+00 VA =0.62 500000E-0i VQ : 0 62431723E-Di3 : 10 V = l.9935012E+0 VA =0.62500000E-0i VQ = 0.6281882E-0i3:= i v: 0.987149i13E+00 VA:= 0.62500000E11i VQ : 0.607119E-0i

t 2 V = 0.98192763E900 VA = 0.6250000 VQL 0.6i370477E-013 = 1 V = 0,771645724900 VA : 0.62500000901i VQ : 0.61028M7E-0iJ3=14 V: = .97107732E+00 VA = 0.62500000r-fll VQ = 0 60692333E-01

B2 8

J is 45V = C.9657026E400 VA =0 6250008CEL0i V9 .0.6036i579E-01J 16 V = 0.96CS7092E+0 'VA =0.62500000E-Di VQ =0.60036M8E-01I =1' V = 0 95')4557OE*00 VA =0.31250000E--0i VQ =0.298 57993F-0iNEW VALUE or QT WITH MODIFIED Y 0.10Q00000Ct0i

Q= 0.642432ENC ENG TONS/DAY QS/QSACT 0 100440E+Oi

1 VALUES FO.O UBAR0 ODDOOOOE0ro 0.3637iEli0 .6641i 531-gi 0.9lii969E-0i 0.!i0S05iE+l00 12407311~0 0,0431290E+00 O.I3922L'8E+Cvl 0 397235E+0 0 0.0360422E+000 i!0343S1+Gl 0.ii687471+00 0.!O16711E+00 1 8302411E-0! 0 L61f4374E-Ot0.3508C 0 1 tiI7L79~0E-02

END OF INLET SECTION

roR SECTION 1 3 ST 0 5277541-02 UCL L.134S43E-giSATTOFACTC Y :ATEATION ERR~' 0 22643"'E-03 ERRUZ 0,41,7332E-O2NUMBER OF IT7RAT:0N' XcOUNT 2RELATIVE DIFFERENC.E OF QT TO 9 1 S 0. 46111"L-03

C~T lNC r TSION 3 .:2400S1E 0 WITH RE' 0.S37SQELi1NEWEST Q = .9?990546E+00 WITIh ETA = .100P34tiE+O1SEDIMENT DIECCHARlE 0.'J42."F.+00 9G2/VZACrT 0.i0342E+0i

1-DE Ow ITERMTION5 FOR UP~, 1, UV. ARE 2 3NEW VALUE3 !XI UBPAP iUE~j P

L.000CEF0C 0.2017E-h1 C&7-I 0.4863F-01 0 S7qJE-11j 0.6392E-Oi0.676EC 0. L77-01-01 761-0i~C 0.6i47C .0 0 559CE-01 C.4648E-010.3606E-I' 0.238L.E-01A A.i0iIE0i -I.5390E-02 -0I.2222E-01

[TA -MOIE7 lOMENTUl-, VALUE -~ "IVEN D

lo iO3ELI iO17E+01 LI. ivE+6' 0.9946E-LIC I.'788E40I 9.9930E+006.9773E+00 3.9716E+00 0.?661[400 0.9606E+00 0.V952E+00

FOR SECTION 5 ST = O.9'66028E-0c' UCL = 0.2513?3E-OiSATISFACTORYLITERAT7ON rRRV2 = .t916Q7E-07j ERRU2 = 0.489446E-02NUMBER OF ITERATIONS KOUNT = 2RELATIVE DIFFERENCE OF QT TO Q IS 0.4338r0%E 0'

CENTEILINr POSTION S = 0.244.62E+00 WITH PC = 0.537SDDE+I)lNEWEST 0 0.99798736E+00 WITH ETA 0 0080434F+01SEDIMENT DIGC ~C L.427Ei QS/QSACT LI 1004iE.0!

N'MERO ?rAOIFOR Ufl: Q, L" ARE i 3

B2 9

V NEWI VALUES FOR UBAR GIVEN BY

0.00800[00 0.IGOZE-Ct 0.333iE-Bi 4.4443E.-O 0.51249E-01 0.5772E-010.b83SE-4i 0.6057E-Oi 1.5857E-011 I.S453E-Oi 0.4862E-Oi 0.4097E-Oi0.3074-01 O2106E-Oi iC.98331-02 -0.42i3cr-02 -8.iOSE-Ci

ETA-4IODTFIED MOMENTUM VALUES FOR V GIVEN DYOi098E4O1 0.1044E+Oi 0.0839[+01 010O33111 O.iO27E40i 0.iC2CE+Ci

-~~ 8 IAi4[*8i 0.~8E8 .101ZE4i 0.996014D tt.MlBE49 0.9941E4.000 ,9733E+00 0.1 '725E+0l C.7668EDOO tL6IiEl38 3 .9S5E+03

~-B30

VAES FOR UBAR

169 20S 34 1 477 S451- 500 8.OOOOE'90 10000OE+01 1 0000'00o 0 fli82E-04' 9 1787E-03 0.6042E-04-0 438 0 36W4-01 C.M44IE-02 9.K037 -O 8245-03 -0 iMBE-03 -9.6320E-S4A 757 0 664iE-C-1 0 2746C-011 0.416SE-07i -0 .262iE--2 -0.4552E-03 -0 1663E-130 0 0. ?i IC- 1 0.3708BE-02 0I~965E-03 -. 0 E-92I -0.7049E-03 -1 .2498E-C!

-~25 i9E~9 0.435?E-011 0.729"E-03 -0 SE-02 -0.9135E-03 -C.3150E-03018 0.1N482,9 0 47S4E-O 02 ..9 -~93 -f 67T)'. E -V 082E-02 -0.362SE-13

-0.15 0 l4~9 0.934E9 iE L2- ?741 91 1 11091-C, -0 3931E-0300 0 13?92i99 0 4919E 0 , 229 -08S6ZE 92L -Q.12?4K-02 -0.4164E-130000 0.139710 0.476~0E-02n 0. MIE-13 -0 4i2-2 -0. 035E-12 -6. 493E-030.163 60 i601CM 0.4449E-011 0.91?0E-03 -0.94141.-B0 -0i329E-02l -0.3830E-13i0 .125 0 .12831 t. 0.4012E-02 0 .8338-03 -0 -9406E-02 -0.12%iE-02 -0.3467E-03V.138 0.111160E.10 3.3464E-02 0.77SHE-07 -9 907ir-o, -0.11601-02 -C.2963E-03I ZS 0 1017[' G" .2G171-02 1 :,oo Q: -0 8775E-92 -0 995SE-C3 -0 '"(44E-03

0.31 L 03 2 !I - L921 I.~4- L. 7 L.92- 82 -9.1653E-53e937S 0.6i04EL-n 0 26~9C-02 9 .230SE-07 -0 S,2S57E 02 -0 5311-03 -P. 9562E-04

9 43 035G~~I C. C?79 -" E 0 -. 275'6E r 2 -0 273E- 03 -0. 35iSE-04t .500 0 76721 02 *9.SrS3E-0] -0.1,2IE-0; O.9rJ9PH 0 COQ0E+00 C 0000ED00

-0 50003 -0.2500 0000 2~~ 0.50001 9 00O9 ~ ~ ~ ~ i ' 01i~0 0,7672E-AU2

4 2 fl.0 ' 0 94759E-02 0 4?O-2 021~9 -0.5353E-033 0099!9 72-C7 I T72ff .07 6 C. C9- 3 -0 112,,E-03

1 .O2-4-.~2 -9431E0 -0.244E-03 0.00DOE-00

VAL E s 1C-9 U

69 NI .i47" P

!YOO 9 OOC99 9 ~60E0 0~98L9 -6.14SEr'0 -01i940[400 -0.1979E+10C.4703 0AIW!C! C .'?3?OE-Ci 39 'I77-0i -0.145E.9 0 .18E0 I U~ 3E-'10.37S 0.0000C400 0.7768E-Ot C.CiSKE01 -O.1430EM0 .--0.AMEMO -OiB4SE'O0-0.313 0.OOOOE+00 C.M41E-01 0.3?9E-Bi -0.1407E-100 -0.17SSEA'0 -0.75EWO-0.2L 00OE909 879ZE-Oi 0.9269E-0i -8.i383ED0 -0 1688E+00 -0.1704E+8

-0 M8 0100E040 0 914GE-01 0.9804E-al -0, 153E49 -0.1601E400 -0.163E4D9-15 J' f.001CEA0O 0.940"E-01 9.1933E+00 -0.191M0 -0.548EL+00 -9.iS56E+00 .0 6 'l 0.000KNO0 0.9807E-011 0.0i04ED0 -0.i202E09 -0.i476EP00 -0.9A80E+001.000 0.0000r-40 0 l014E+00 0 i13SE+0 -0.242[009 -0.1401EL0 -0.1402,6+00

9.6 90 9'O .94E09 9134~0-0.1!97NO -0.i324EM9 -9.1322E+09 12 0.0 9 C 0 .49 074N 9 9. i 33E 00 - .i 5 E- 9 .1 4 +0 -.24 E 0

B31I

41

~~~~~i L~ .~EO ~i3+ -019i4Efil -0 1164E+00 -t~Ai59EBO1.2C 0.3000[400 ODI EMO fl.4327ELOO -6.10SIEIC ABIM~Er -0.16075E+000.3117 0.0010EWQ CA5E4O B A3 E+00 -C ?j- -0 ??5E-Gi -1 99OE-010 §.I 11.00E-flO *.118SE-40 *a.14OE+8C 'w9O"ECl l 9E'4Q-0

VALUES FOR V

1 69 1jr 341 477 S4S1.91~ ViWTEki 0.869iE+iO 1.84 3LIO Q.?641E Bva.1.885E+0i QtiIOQi1*.430 C. ID01O B .73i'LEMB I.757+og O.MSD~EOC 0iO7CEOII 1.1099E+01

0.3S o0ME+O,0 0.9643EDO 9.873QE+00 0.9937E+00 Q0iB7CEMO OiO79E+0t1.31 0.1-027E40i 0.?8CiE+00 0.jtQ7E+Go 0,004rEf14 OAW.1060Q C.iC67E+Di

rC ~C1C:*e BB8E+O 0 93?6SE+0O P. ifii-4H 0iO0CE+C! l.J,34E+Qi0.O.0!~2:0.994SE2060 0.STAD40O 0 C.1,17E,.0 . 1~039rE-01 '1.104KE+i

v i-: Vigli041 0.9%E+00 O.Th9-EiO 0l. 10,21E 0 i 0. 26EII OlC25E+Di?. P ~. 10QIEi 0i 0004F+01 i/L7~0 0 .10231+i0I1E0t0iOEI

Z 00 0.99?9EM( OBIDO7E+oa 0.191101M 0 10I24N~I D.?971E400 1.9924E+00* o.o 0.992E4~ .i~GEBI .!19~0I0.1024L ., 0.9807[400 0 974tE+gO

0.i2 ~ ~ ~ ) 0.82B .~3+ i 12EB B L .02EC 0.76I7E400 13.9545EMB113C8 9.807D10 01MIE+Bi 0.I1032rEfti 0.1019E~0t O.?420E400 0.9334E+00C

50 .9765E400 0AM1EM O. 0i43E+Oi 0.102i O .9207E400 0.9107E+00C 3~ o.7u:C0 10i1+O 0 0'3*0i0.100E 0 .8961C+00 0.8862E+00

S377 0 9Lr:300 0,1i?~: 1062+01 0.97~TEll0~ .8688[+00 0.8600E+00'Y42~~~ 0%0E: B2E'4 fi7E+01 02814P 0,0384C,'04 1.9323E+BC,

0 0610 -0.930"LAL 0 '7 0.'251 E' 0 -fL:0 E+

7 0 S-4701 0 .93 7 6E +CF0 0.9r?7BE00 Gr. I14? 7 10i 1 .iBSOE+li4 0.164111101 0.iBilE'0: O.iC24E+0i 0.i0i2Edli 0.8330E-+PO

CA '11CI C.Hr050E+01 C.9924P+00 0.7410171100 0.60446E+00

VPI.UcES FOR HL

1 69 L0 '41 477 545-0.90 6001!01'0 0.685111+00 0.L647ElD01B~3C61401 V1479SEli 3 1436E+Ci

01. 0 10CfC C, , 02 45E0 ) .l71l)E n0 0.1378[ N1 0 .13101 0 i381EM0- .77 0.1 1-O1 0.7638O "i4 0' C0 0 G. 121'.' n 1726C401 '. I1327E*0

B32

~.*.~ *.*~.-. .. -~*.:~-5.

*% %-A .

-0.~30 0. tOO81+01 0.803"E+00 C.7797E+00 012~441+01 0 1272E+Di 0.i272E+Bi

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0 ioC0rs0 Q.tii7E+Ci 4 !176E+01 0.8071[tol 0.7824E+00 0.7822E+00O 312 CAiCO01O 0.i97E+Di 0 i'1201+0i 0.7SGCf4'30 0.72SOE400 0.71178E+00

37S D~I0DON: O.1236E+Ci V.A264E+Di 0.710./1,00 0.6736E+00C 0.67341+00431 3 tcoo3,O! 01i276EL~ 0 :0ODN'0. .0.66240 3.i2 0 u.1?E

MCC0 0 1000Cr0i 0.131SE+01 O.i7#:52E+Ci 0.6i4,rjC .54+0 3S4E

VAlUES FUR (UBAR +U

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n

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VLUES FOR UNIT SDIMEN T DI~c!:9qcr2

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-B3

-a APPENDIX C: NOTATION

NOTATION

P a Constant used in sediment-transport formula

A Constant in quadratic equation

b Constant used in sediment-transport formula

B Constant in quadratic equation

C Constant in quadritic equation

* d Local flow depth

dc Centerline flow depth

D50 Median bed-material size

' f Darcy-Weisbach friction factor

F Arbitrary variable

- F1 ,... Dimensionless functions

Fr Froude number

g Gravitational constant

g1 o.o. Dimensionless functions

h Local bed elevation

hc Centerline bed elevation

H Local water-surface elevation

i,.. Streamwise grid locations

I Streamwise index for section numbers

j,°,. Transverse grid locations

J Transverse index for section numbers

M Number of transverse grid points

n Exponential parameter used in power-law velocity distribution

Cl

-.- . -- .. .'- -. - . - . .. .. ...-. .- - - - - - - --~..---

N Number of streamwise grid points

• - p Bed-material porosity

qt Total-load discharge per unit width

ot QTotal-load discharge

Ot Total water discharge

r Transverse coordinate

r i Radius of curvature at inside bank

ro Radius of curvature at outside bank

rj.... Radial grid locations

Rc Centerline radius of curvature

s Streamwise coordinate

SO Origin of streamwise coordinate

-si,... Streamwise grid locations along channel centerline

Sc Streamwise water-surface slope along channel centerline

ST Transverse bed slope

. t Integral function

TTI.... Integral functions

u Local secondary-flow velocity

* uLocal shear velocity

U Local secondary-flow velocity at water surface

Uc Secondary-flow velocity at channel centerline

"U Mass-shift velocity

v Local streamwise velocity

V Depth-averaged streamwise velocity

Vc Depth-averaged velocity at channel centerline

..

C2

. .. . . . . . . .

.--. V Area-averaged streamwise velocity

.- ,-, W Channel width

y Vertical coordinate

Yb Bed-layer thickness

a Proportionality constant for bed-layer thickness

* Proportionality constant for bed-shear stresses

.Bc Critical shear-stress parameter

p Fluid mass density

Ps Sediment mass density

T Streamwise bed-shear stress =T0 os

T or Transverse bed-shear stress

*Streamwise angular coordinate

C3

" -..., ~ *

MODEL FOR FLOW AND SEDIMENT TRANSPORT IN t … U.S. Army Engineer Waterways Experiment Station ......

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Transcript of MODEL FOR FLOW AND SEDIMENT TRANSPORT IN t … U.S. Army Engineer Waterways Experiment Station ......