Measurement of Mixing 

Characteristics of the 

Missouri River Between 

Sioux City, Iowa, and 

Plattsmouth, Nebraska

GEOLOGICAL SURVEY WATER-SUPPLY PAPER 1899-G
Measurement of Mixing 

Characteristics of the 

Missouri River Between 

Sioux City, Iowa, and 

Plattsmouth, Nebraska

By NOBUHIRO YOTSUKURA, HUGO B. FISCHER, and WILLIAM W. SAYRE 

CONTRIBUTIONS TO THE HYDROLOGY OF THE UNITED STATES

GEOLOGICAL SURVEY WATER-SUPPLY PAPER 1899-G

Measurement of longitudinal dispersion, 

transverse mixing, channel geometry, 

and transverse velocity distribution

UNITED STATES GOVERNMENT PRINTING OFFICE, WASHINGTON : 1970
UNITED STATES DEPARTMENT OF THE INTERIOR 

WALTER J. HICKEL, Secretary

GEOLOGICAL SURVEY 

William T. Pecora, Director

For sale by the Superintendent of Documents, U.S. Government Printing Office 

Washington, D.G. 20402 - Price 25 cents (paper cover)
CONTENTS

Page

Symbols..__-_--__-___-___-__-_----_____-_________________________ iv

Abstract- ________________________________________________________ Gl

Introduction, _____________________________________________________ 1

Acknowledgments-____---_-_--__--__--____-__-___-____________ 3

Longitudinal dispersion.___________________________________________ 3

Description of the test reach.___________________________________ 3

Test procedure.__-___--__-_____--__-_-_-__--__________________ 5

Data-.____-___-_--___--_-_-__-__--_--____________________ 6

Computation of the longitudinal dispersion coefficient-_____________ 8

Routing procedure-__-___-_-___--______-__-__--_-_________ 8

Method of moments.______________________________________ 10

Transverse mixing.._______________________________________________ 11

Description of the test reach.____________________________________ 11

Test procedure.__-_-___-_____-_-___---_-_-_-__-__-_______-_-__ 13

Data---_--_---_---_-___-_-_--__-___-__-_--____________-_-. 18

Computation of the transverse mixing coefficient-_________________ 20

Method of moments.______________________________________ 20

Computer simulation method.--__-____-_-_-_______-________ 21

Summary and conclusions._________________________________________ 27

References-____________--___-___-_-----_----_---_--__-__-___-____ 29

ILLUSTRATIONS

Page

FIGURE 1. Map of study reach, Missouri River between Sioux City,

Iowa, and Plattsmouth, Nebr_______________________ G4

2. Graph showing longitudinal dispersion on the Missouri 

River comparison of the November 1967 data with the 

October 1966 data________-__-__--__-_-________ 7

3. Graph showing measured mean time of passage and variance as functions of distance.______-___-___-_-____-_ 12

4. Sketch showing reach of the Missouri River for the

transverse mixing test-_____________________________ 14

5. Graph showing observed transverse profiles of velocity 

and depth at cross sections not affected by bridge 

abutments._______________________________________ 15

6. Photograph showing mixing of dye immediately downstream from injection site at Blair Highway Bridge____ 16

7. Sketch illustrating method for locating lateral sampling

positions.__________-____________--_____--__-__--_ 17

8-10. Graphs:

8. Variance of transverse tracer distribution, observed and calculated__-______-_-_-_---_--- 21

9. Comparison of observed and calculated transverse 

distribution of dye concentration at nine cross 

sections. ______________-__-_-___-__-_----- 25

10. Influence of E2 on transverse distribution of dye

concentration._ __________-___-__----_-_--_ 26
TABLES

Page

TABLE 1. Distribution of cross-sectional average dye concentration 

with time, longitudinal dispersion test on the Missouri 

River, November 1967_____________________________ G6

2. Hydraulic data, longitudinal dispersion test on the Missouri River, November 1967________________________ 8

3. Time-of-travel data, longitudinal dispersion test on the

Missouri River, November 1967___--____--__________ 9

4. Dispersion coefficients by the routing procedure, longitudinal dispersion test on the Missouri River, November 

1967________-_________--_-____-______-___--_____- 10

5. Dispersion coefficients by the method of moments, longitudinal dispersion test on the Missouri River, November 

1967___-____-_____-_--__-_____-______--____--___- 13

6. Distribution of dye concentration in transverse direction, 

transverse mixing test on the Missouri River, November 17, 1967______--_____---_________-____-----___ 18

SYMBOLS

A Cross-sectional flow area

a s Dividing area between adjacent stream tubes

C Cross-sectional mean tracer concentration

c Local tracer concentration

D Longitudinal dispersion coefficient

d Mean flow depth

Ez Transverse mixing coefficient

g Acceleration of gravity

k Constant of transverse mixing coefficient

1 Characteristic length

M Mass flux of tracer per unit time

q Discharge in a stream tube

Q Discharge

RR Recovery ratio

s Mean energy slope

T Lagrangian time scale

t Time measured from the instant of injection

t' Dimensionless time scale

t Mean time of passage for tracer cloud

~t' Dimensionless mean time of passage for tracer cloud

U Mean tracer velocity (mean flow velocity)

U* Shear velocity

u Local velocity

V0 Volume

W Surface width

x Longitudinal distance measured from injection site

Ax Increment of longitudinal distance

2 Transverse distance from a boundary

Az Transverse distance between centroids of adjacent stream tubes

cr2 t Variance of longitudinal tracer distribution

cr2 2 Variance of transverse tracer distribution
CONTRIBUTIONS TO THE HYDROLOGY OF THE 

UNITED STATES

MIXING CHARACTERISTICS OF THE 

MISSOURI RIVER BETWEEN SIOUX CITY, 

IOWA, AND PLATTSMOUTH, NEBRASKA

By NOBUHIRO YOTSUKURA, HucoB. FISCHER, and WIILLIAM W. SAYRE

ABSTRACT

Measurements of longitudinal dispersion, transverse mixing, channel geometry, 

and transverse velocity distribution were made in the Missouri River at a flow 

of about 33,000 cubic feet per second. The results show that the longitudinal 

dispersion coefficient for the 141-mile reach from Sioux City, Iowa, to Plattsmouth, Nebr., is about 16,000 square feet per second (approximately 5,600 

U*d, where U* is the shear velocity and d is mean depth). The transverse mixing 

coefficient, Ez, for a 6-mile reach immediately downstream from Blair, Nebr., 

is about 1.3 square feet per second (approximately 0.6 U*d). The value of the 

longitudinal dispersion coefficient is one of the largest ever measured, rnd the 

value of the ratio Ez/U*d is approximately three times as large as that frequently 

reported for small straight channels.

INTRODUCTION

Interest in dispersion processes in open channels has accelerated 

during recent years because of mounting concern over water pollution. 

A knowledge of the dispersion capacities of streams is necessary in 

order to control pollution, and important improvements in tracer 

technology and theories for predicting dispersion have made possible 

for the first time significant headway in observing and understanding 

dispersion phenomena in large natural streams.

The application of tracer technology to large streams became 

feasible following the introduction of radioactive and fluorometric 

tracing techniques, both of which are capable of measuring extremely 

small concentrations of tracer materials. Godfrey and Frederick 

(1963) have reported on the use of radioactive-tracer techniques in 

natural streams. The application of fluorescent-dye methods in large 

streams has been described by Wilson and Forrest (1965), Thackston 

and Krenkel (1966, 1967), and Bowie and Petri (1968).

Gl
G2 CONTRIBUTIONS TO THE HYDROLOGY OF THE UNITED STATES

An important step in the development of a method for predicting 

longitudinal dispersion in natural streams has been the adaptation by 

Fischer (1966b, 1967b) of Taylor's (1954) theory of convective dispersion in axisymmetric pipe flow. According to Fischer's formulation, 

transverse mixing combined with the variation of velocity with respect 

to lateral position in the channel is the dominant mechanism contributing to longitudinal dispersion in most natural streams. One important 

aspect of Fischer's result is that, other factors being equal, the longitudinal dispersion coefficient is proportional to the square of the channel 

width. This accounts for the extremely large values of the longitudinal 

dispersion coefficient that have been observed in large natural streams. 

For example, Yotsukura (1967) has reported values on the order of 

10,000 sq ft per sec (square feet per second) for the Missouri River. 

For open channels, earlier applications of Taylor's theory, which 

assume that vertical mixing combined with the variation of velocity 

with respect to depth is the dominant mechanism, typically predict 

longitudinal dispersion coefficients for natural streams that are two or 

more orders of magnitude too small.

Prediction of the longitudinal dispersion coefficient by Fischer's 

method requires that the geometry, the transverse velocity distribution, and the value of the transverse mixing coefficient be known for a 

typical cross section. The necessary velocity and cross-sectional 

geometry information can be readily obtained by conventional streamgaging procedure. However, the value of the transverse rrixing coefficient for large natural streams is not so easily determined.

The transverse mixing coefficient, in addition to being one of the 

important factors controlling the rate of longitudinal dispersion, is 

important in its own right because it controls the rate at which crosschannel mixing occurs. This is often an important consideration in the 

design of pollutant-outfall and water-intake systems. As of 1968, no 

theoretical b.asis for predicting the value of the transverse mixing 

coefficient had yet been formulated. On dimensional grounds and by 

analogy with the form derived from the logarithmic velocity profile for 

the vertical mixing coefficient, the transverse mixing coefficient may 

be assumed to have the form

Ez=kU*d (1)

where d is depth of flow, U* is shear velocity, and k is a numerical 

constant, which in straight uniform channels has a value of approximately 0.2. Elder (1959), for flows on the order of 1 centimeter deep 

in a laboratory flume, obtained fc=0.23; Sayre and Chang (1968), in 

flows up to 1 foot deep in an 8-foot-wide flume, obtained fc = 0.17; and 

Fischer (1967a), in a flow 2 feet deep in a 60-foot-wide sand-bed canal,
MIXING CHARACTERISTICS OF THE MISSOURI RIVER G3

found &=0.24. Equation 1, with k = Q.2, is probably a valid representation of the transverse mixing coefficient in straight uniform channels 

where turbulent mass transfer is the main contributing mechanism. 

In natural streams, additional mechanisms such as secondary currents 

induced by bends may significantly increase the rate of transverse 

mixing. For example, Glover (1964) has reported &=0.72 for the 

Columbia River near Richland, Wash. Therefore, it is of considerable 

practical importance to obtain additional experimental data on transverse mixing in large natural streams.

Ideally, the objectives of the experimental phase of the investigation reported in this paper would have been to obtain in a large natural 

stream sufficient data to define simultaneously: (1) the longitudinal 

dispersion coefficient, (2) the transverse mixing coefficient, and (3) 

the typical cross-sectional gometry and transverse velocity distribution. This information, if obtained for a long reach, would be sufficient 

for a complete evaluation of Fischer's theory. Lack of time and funds, 

however, restricted the transverse mixing phase of the experiment to 

one relatively short reach, and the cross-sectional geometry and 

velocity observations to just a few cross sections.

ACKNOWLEDGMENTS

The authors would like to express their sincere gratitude to Charles 

L. Hipp, Chief, Engineering Division, and to other members of the 

U.S. Army Corps of Engineers, Omaha District, for the generous 

cooperation and support which they rendered this investigation.

The tests were made possible by the voluntary participation of the 

following members of the research staff of the Water Resources 

Division: F. A. Kilpatrick, E. V. Richardson, R. S. McQuivey, and 

W. E. Gaskill. J. K. Okoye, a graduate student at the California 

Institute of Technology, also participated. Assistance in equipment 

and manpower was supplied by the Council Bluffs Subdistrict Office, 

Iowa District of the Water Resources Division, and also by the 

Nebraska and Missouri Districts. The water agencies of the municipalities of Omaha and Council Bluffs conducted the dye sampling 

at the water intakes. All these contributions are gratefully 

acknowledged.

LONGITUDINAL DISPERSION 

DESCRIPTION OF THE TEST REACH

The longitudinal dispersion test was conducted in a 141-mile reach 

of the Missouri River between Sioux City, Iowa, and Plattsmouth, 

Nebr. (fig. 1). Discharge in this section is controlled by a system of 

upstream reservoirs, including the Gavins Point Dam near Yankton, 

South Dakota, releases from which are coordinated with downstream
G4 CONTRIBUTIONS TO THE HYDROLOGY OF THE UNITED STATES

42°

41°

Combination Bridge injection site for longitudinal dispersion test

5 10 15 20 MILES 

I I I I

FIGURE 1. Study reach, Missouri River between Sioux City, Iowa, 

and Plattsmouth, Nebr.

tributary inflows to maintain a minimum flow of 30,000 to 35,000 

cfs (cubic feet per second) during the navigation season. Within 

the study reach the only significant tributary is the P'atte River, 

which joins the Missouri a few miles upstream from Plattsmouth. 

The channel is maintained at a width of 500 to 800 feet by a system 

of dikes and jetties constructed to stabilize it. The depth of the 

thalweg is generally less than 25 feet.

An extensive survey conducted by the U.S. Army Corps of Engineers, Omaha District (1967) has disclosed the following character-
MIXING CHARACTERISTICS OF THE MISSOURI RIVER G5

istics of sediments and flow regimes of the reach: The channel bottom 

consists predominantly of sand, 90 percent of which is coarser than 

0.15 mm in size. The total suspended sediment load shows seasonal 

variation and, in the late fall, reaches a maximum of between 700 and 

1,000 ppm (parts per million). During the late fall, as the temperature 

drops about 20° to 30°F, the channel bed changes from a dune-bed 

to a plane-bed configuration. Consequently the water surface falls 

1 to 2 feet, with a correspondent increase in average velocity, even 

though the discharge remains steady. The channel degradation has 

been known to be minimal. The change in resistance due to change 

in bed form is reflected in the Manning n value which ranges from 

0.020 for a dune bed to about 0.015 for a plane bed.

TEST PROCEDURE

A total of 600 pounds of rhodamine WT 20-percent solutior was 

injected downstream from the Combination Bridge at Sioux City, at 

about 1700 hours, November 13, 1967, in a line source extending across 

the middle half of the channel. The mode of injection was similar to 

that for the 1966 time-of-travel test reported by Bowie and Petri 

(1968). Rhodamine BA dye was employed in the 1966 test. The amount 

of pure dye used in both tests was about the same (120 pounds).

Samples for dye-concentration analysis were obtained at four 

downstream locations: Decatur Highway Bridge, Blair Highway 

Bridge, Ak-sar-ben Bridge in Omaha, and Plattsmouth Highway 

Bridge. The distance from the injection point to the respective bridges 

is 40.8, 83.5, 116.0, and 141.3 river miles. The samples were collected 

from near the surface by lowering 25-cc glass bottles, attached to the 

end of a weighted handline, into the water. At Decatur Highway 

Bridge, five sampling stations were maintained across the channel, 

whereas at the other bridges enough transverse mixing had occurred 

for three stations to suffice. The sampling stations were located so as to 

represent increments of cross-sectional area carrying approximately 

equal discharges. The time interval between samples was varied from 

15 minutes to 2 hours depending on the predicted rate of change of 

concentration.

All test samples were taken to the Hydraulic Laboratory at Colorado 

State University and analyzed for dye concentration with a Turner 

Model 111 fluorometer. Before readings were taken, sample temperatures were brought to 25.1±0.1°C by means of a conrtanttemperature water bath to eliminate the need for temperature 

corrections. The fluorometer was calibrated by obtaining readings for 

standard solutions of known dye concentrations. The standards were 

prepared by successive dulitions of rhodamine WT 20-percent solution
G6 CONTRIBUTIONS TO THE HYDROLOGY OF THE UNITED STATES

in distilled water. The calibration was linear except for concentrations 

less than 0.1 ppb (part per billion).

Hydraulic data were obtained at a number of cross sections during 

the dispersion experiment. Discharge measurements by the currentmeter method were obtained from bridges at Sioux City, Decatur, and 

Omaha. Discharge measurements were made at four additional cross 

sections by the current-meter method from a boat anchored to a tag 

line. Data from these measurements were used to obtain the geometric 

characteristics of the cross section and the lateral velocity distribution 

as well as the discharge.

DATA

The longitudinal dispersion data consist of cross-sectional average 

dye concentration as a function of time at each sampling cross section. 

An average concentration was obtained as an arithmetic average of all 

concentrations obtained at the cross section during erch sampling 

period. The sampling time was taken as that at the middle of the 

sampling period, which was normally less than 8 minutes. The data 

are presented in table 1. The concentrations in table 1 are net values

TABLE 1. Distribution of cross-sectional average dye concentration with time, 

longitudinal dispersion test on the Missouri River, November 1967

Time after Dye coninjection, centrain hours tion, in 

and parts per 

minutes billion

Decatur Bridge 

RR =0.882

10:30.......

10:45... .

11:00.......

11:25.. .....

11:36..... 

11:49.. __...

11:58.. .....

16:16. _._.__

12:35-.... .

12:47._. .._.

13:03_._-___

13:17-

13:33-_. .__.

13:47.

14:03.. ___.-

14:18-

14:33. .._ .

14:47-----.-

15:02-..-...

15:17-. _._._

15:32- .-. . 

15:48-.__._.

16:03-.-..-

16:23.......

16:43. .

17:03- 

17:22--- 

17:42.... ...

18:04.

18:37--- -

19:02.

19:35-_. _.-.

20:02.

0 

.25 

.80 

1.89 

2.32 

2.84 

3.21 

3.55 

3.92 

4.05 

4.03 

3.88 

3.64 

3.36 

3.10 

2.70 

2.38 

2.07 

1.64 

1.39 

1.12 

.86 

.72 

.49 

.37 

.28 

.19 

.14 

.11 

.09 

.08 

.06 

.06

Time after Dye coninjection, centrain hours tion. in 

and parts per 

minutes billion

Blair Bridge 

RR =0.780

20:59-....._

22:10.......

22:30.-... .

22:47-....-

23:01-___._.

24:12. 

24:34- _____ _

24:55-_.-__

25:27-___._.

25:55.____._

26:53__..._.

27:46-

29:02.

29:35. . .

30:02. ,____.

30:32. .._..

31:02.

32:03. .

33:01.......

34:04-..-.-.

35:06-_.._..

36:03. . .

38:03. .._ .

40:02

43:05. . . .

0.01 

.02

99

.35 

.60 

.88 

1.09 

1.38 

1.88 

2.23 

2.45 

2.50 

2.52 

2.39 

2.10 

1.78 

1.20 

1.01 

.74 

.61 

.42 

.30 

.21 

.14 

.09 

.09 

.06 

.07 

.05 

.03 

.02 

.02

Time after Dye coninjection, centrain hours tion, in 

and parts per 

minutes billion

Ak-sar-ben Bridge 

RR =0.775

28:33-_. -._.

30:03- _..._.

30:33. . . .

31:03-.--.

31:32. . __

32:02-.._._.

32:33. - -.

33:02-___ 

34:02- _.-.._

34:35-. _____

35:05-. .....

36:05. ---_.

37:03-, _.__.

38:00--.....

39:01.......

42:01._ 

43:02-_.....

45:00. .....

47:10.. _____

0 

.04 

.11

.29 

.53 

.88 

1.16 

1.52 

1.79 

2.01 

2.09 

1.84 

1.73 

1.28 

.88 

.54 

.32 

.20 

.13 

.09 

.07 

.05 

.03

Time after Dye coninjection, centrain hours tion, in 

and parts per 

minutes billion

Plattsmouth Bridge 

RR= 0.775

34:02----__

35-02. 

36-02-. ..-. .

37:01- --__-

38 02. ...._.

39.00-. .... .

40.00-. ..

41:02. 

42.02--- 

4300- 

44:30- ..

46-11------

47:58--- 

5003- 

52.05- 

55-12----.

59.06-----..

63?0-___ 

66-58- -----

0 

.03 

.15 

.49 

1.00 

1.37 

1.64 

1.57 

1.39 

1.07 

.62 

.29 

.15 

.08 

.06 

.03 

.02 

.01 

.01
MIXING CHARACTERISTICS OF THE MISSOURI RIVER G7

(background subtracted) and have not been compensated for dye loss 

that occurred in the stream. The data are also presented graphically 

in figure 2. Corresponding data from the 1966 test are included in 

figure 2 for comparison. Two striking features are evident in this 

comparison: the differences in the time displacements of the curves 

and the areas under them. Since the description of these aspects is not 

the major topic of this report, it is sufficient to say that the shortened 

travel time in the present test was caused by the reduction of bottom 

resistance due to a shift from dune-bed to plane-bed. The 1966 test 

was done under the condition of a dune bed. The difference in the 

areas under the curves indicates the difference in the observed dye 

recoveries and is attributed to the difference in the adsorption 

characteristics between WT dye (present test) and BA dye (1966 test).

10 15 20 25 30 35 40 

TIME, IN HOURS AFTER INJECTION

EXPLANATION

50

Cross-sectional average concentration versus time 

curve resulting from the injection of 120 Ib.of 

rhodamine WT dye (600 Ib. of 20-percent solution) 

at Sioux City at 1700 hr., Nov. 13, 1967. Discharge of 33,300 cfs at Omaha, Nov. 13, 1967.

Cross-sectional average concentration versus time 

curve resulting from the injection of 125 Ib. rhodamine BA dye (313 Ib. of 40-percent solution) injected at Sioux City at 1000 hr., Oct. 4, 1966. 

Discharge of 31,600 cfs at Omaha, Oct. 3, 1966.

FIGURE 2. Longitudinal dispersion on the Missouri River comparison of the 

November 1967 data with the October 1966 data.
G8 CONTRIBUTIONS TO THE HYDROLOGY OF THE UNITED STATES

Values of the recovery ratio, RR, for the present test ar3 also shown 

in table 1. It is the latio of the amount of dye actually recovered at 

the cross section to the amount that was injected initially and is 

computed by a formula

«TJo

Gdt

where C is the cross-sectional average concentration observed at 

time t, Q is the discharge, and C0 and V0 are respectively the concentration and the volume of the injected solution. It wrs concluded 

from the high percentage of recovery that the effect of dye loss on 

the analysis of dispersion characteristics is negligible in the present 

test.

The hydraulic data are presented in table 2. The average depth, 

d, was computed by dividing the area, A, by the surfac^ width, W, 

while the shear velocity, U*, was computed as the square root of the 

product gds, where g is the acceleration of gravity and s i^ the energy 

slope which was assumed to be a constant, 0.0002, for the entire 

reach.

TABLE 2. Hydraulic data, longitudinal dispersion test on the Missouri River,

November 1967

Date 

Location of cross section November 

Combination Bridge, Sioux City.__.

Near the mouth of the Little Sioux

1,000 ft upstream from Blair HighRiver mile 645, 17,500 ft downstream

Near Plattsmouth Highway Bridge.

13

15

13

14

14

15

13

16

15

Cross- Average Average 

Dis- Channel sectional velocity Average slear 

charge width area A YT (fovt depth d velocity

Q (cfs) W (feet)

33, 500

34, 100

31, 200

33,700

33, 300

33,500

34, 000

750

755

610

600

740

577

577

585

(sq ft) per sec)

8,220

8,780

7,650

6

OQO

5,870

6,340

6,340

5,634

4.08

3.88

A fift

5.10

5.74

5.25

5.28

6.04

(feet) U*

11.0

11.6

12.5

10.7

10.0

7.9

11.0

11.0

9.6

(ft per 

sec)

0.27

.28

OQ

.26

.25

.23

.27

.27

.25

COMPUTATION OF THE LONGITUDINAL DISPERSION COEFFICIENT

ROUTING PROCEDURE

Although several other methods for computing the longitudinal 

dispersion coefficient are available, the routing procedure presented 

by Fischer (1968) appears to be the least sensitive to either human 

judgment or data scatter and produces a coefficient which matches 

the data as closely as possible. Fischer (1967b) has divided dispersion
MIXING CHARACTERISTICS OF THE MISSOURI RIVER G9

into two time periods according to a dimensionless system in which 

dimensionless time, t' ', is given by

«'=! (2)

In equation 2, t is a time measured from the instant of injection, and 

T is a time scale given by the relation

I2

in which I is a characteristic length (distance between the maximum 

velocity thread and the more distant bank) and Ez is the transverse 

mixing coefficient. As shown in a later section, the measured value 

of the transverse mixing coefficient for the study reach of the Missouri 

River is approximately EZ =O.Q U*d. Fischer (1967b) states that if 

tf, the dimensionless mean time of passage of the tracer cloud past 

the measuring station, is less than 6, the cloud is in the "convective 

period," in which the dispersion theory is not applicable. If possible, 

the routing procedure should be applied using data obtained at 

measuring stations sufficiently far downstream that the dimensionless 

mean time of passage past each station is greater than 6.

Table 3 shows the distance downstream from injection, mear time 

of passage of the tracer cloud, and dimensionless mean time of passage 

for each of the measurement stations of the present experiment. The 

reach from Blair to Plattsmouth is entirely within the "difusive 

period" (£'i>6) and is sufficiently long to permit an accurate calculation of the dispersion coefficient. The reach beginning at Decatur 

is partly within the "convective period," so it yields a somewhat 

erroneous coefficient; the remainder of the reaches are probably too 

short to permit accurate calculations.

Table 4 shows the result of applying the routing procedure ir turn 

from each measured section to each subsequent measured section.

TABLE 3. Time-of-travel data, longitudinal dispersion test on 

the Missouri River, November 1967

[? is calculated on the basis of the average values Z=450 ft, d=9.68 ft, Z7*=0.250 

ft per sec, and T=167 min]

Measurement station

Blair.

Omaha

Plattsmouth.................

Distance 

from injection 

x, in feet

216,000

441 000

61? 000

796.000

Mean time 

_of passage 

t, in minutes

800

2,080

2.520

Dimensionless 

mean time of 

passage ?'

4.8

0.5

12.5

15.0
G10 CONTRIBUTIONS TO THE HYDROLOGY OF THE UNITED STATES

TABLE 4. Dispersion coefficients by the routing procedure, 

longitudinal dispersion test on the Missouri River, November 1967

Dispersion coefficients, in sq'iare 

Initial Station feet per second, at final stations

Blair Omaha Plattsrrouth

Reported data:

Data with reduced tails: l

Blair_, __________________________

..._. 7,420 8,600

-- -.......... 9,160

----- 6,540 8,660

_---___-_-__.____ 10,400

14,300

16,100

26, 600

11,900

16, 700

23,400

1 Reductions; 0.05 ppb at Decatur and Blair; 0.03 ppb at Omaha; 0.02 ppb at 

Plattsmouth.

For the first set of values, the data are used exactly as rep orted in the 

previous tables; for the second, a portion of the trailing end of the 

tracer distribution has been subtracted out by the procedure outlined and for the reasons given by Fischer (1968). As expected, the 

results of the routing procedure are insensitive to whether or not the 

tails are included, as is seen by comparing the two parts of the table. 

The value of the computed dispersion coefficient increases steadily 

for points successively farther downstream in the measurement 

reach. This may be in part because the velocity of flow at Omaha and 

Plattsmouth is slightly higher than at Decatur and Blair, or it may 

be because of the low values of dimensionless time at tH upstream 

stations. For this reach of the river and flow condition, the value 

obtained by routing from Blair to Plattsmouth,

Z> = 16,000 sq ft per sec,

is probably the best approximation to the true dispersion coefficient 

obtainable from the present data.

METHOD OF MOMENTS

For comparison, longitudinal dispersion coefficients as calculated by 

the method of moments (Fischer, 1966a; Sayre and Chang, 1968) 

are also presented here. In this method, the longitudinal dispersion 

coefficient can be defined as

»- 

In equation 4, U is the mean rate of longitudinal displacement of the 

dye cloud which can be expressed as

U=^ (5) 

dt
MIXING CHARACTERISTICS OF THE MISSOURI RIVER Gil

where tis the mean time of passage of the dye cloud from the injection 

station to a point at distance x downstream from the injection station,

f^tCdt

7-Jo - (6) 7T

Jo

Cdt

Another term in equation 4, o-/, is the variance of the concentration 

versus time curve at x,

/»0

t2Cdt

 [~t] 2. (7) « .

Cdt

Jo

The main difficulty in evaluating D by the method of moments is 

the large contribution to the variance by the tails of the concentration 

distributions. Even a small amount of tracer that is temporarily 

trapped in slow-moving flow near the banks and subsequently released 

to the main channel, where it shows up as a tail on the concentration 

distribution, can greatly inflate the value oi D. In anticipation of this 

problem the curves were truncated at times when concentrations on 

the recession limbs of the curves reached 3 percent and 1 percent of 

the peak concentration. The truncations were performed before 

calculating 7 and a ts by equations 6 and 7. The results are shown in 

figure 3 and table 5. If the variances corresponding to the 1- and 3- 

percent truncation levels are considered as maximum and minimum 

estimates respectively, a reasonable estimate of an average value of D 

for the entire reach is 15,000 sq ft per sec as shown in figure 3. The 

values of Z> given in table 5, although somewhat larger than the values 

for the corresponding subreaches given in table 4, exhibit the same 

general trend and variation within the reach. On the whole, the values 

of D calculated by the method of moments compare reasonably well 

with those determined by the routing procedure, evidently because 

the channel of the Missouri River in the study reach is sufficiently 

well maintained that there are few dead zones near the banks crpable 

of significantly affecting the longitudinal dispersion process.

TRANSVERSE MIXING 

DESCRIPTION OF THE TEST REACH

The transverse mixing test was performed in a reach immediately 

downstream from Blair Highway Bridge, about 85 miles downstream 

from Sioux City and 56 miles upstream from Plattsmouth. It was
G12 CONTRIBUTIONS TO THE HYDROLOGY OF THE UNITED STATES

40

30

20

10

I I T

Mean time of passage

1 I

(J=-«!*-=3.64mph 

= 5.34fps

12

10

cc 

I 6

 

I I I T

Variance

x Truncation at 1 percent of peak 

° Truncation at 3 percent of peak

 T

=-^- dfff = 1.93 sq mi per hr

= 15,000 sq ft per sec

I I I I I

20 40 60 80 100 120 140

x, IN MILES

FIGURE 3. Measured mean time of passage and variance as 

functions of distance.
MIXING CHARACTERISTICS OF THE MISSOURI RIVER G13

TABLE 5. Dispersion coefficients by the method of moments, 

longitudinal dispersion test on the Missouri River, November 

1967

Dispersion coefficients, in square feet 

per second, at final stations

Truncation at 1 percent of peak concentration:

Truncation at 3 percent of peak concentration:

Blair.. .................................

Blair Omaha

16, 300 16, 300

........... 16,300

9, 730 11, 200

........... 13,500

Plattsmouth

20,600

24, 000

33, 600

14, 800

18, 300

24, 900

selected for the test because of fairly gentle meandering for a 33,000- 

foot long stretch, availability of a bridge for installing the injection 

equipment, and not much river traffic. Channel geometry and hydraulic characteristics are as described in the section on the longitudinal dispersion test. Looking downstream from Blair Highway 

Bridge, the river meanders gently to the left for about 3.5 miles with 

the thalweg running near the right bank (see fig. 4), after which the 

direction of the meander reverses, and the thalweg crosses to the left 

side. The channel width ranges from 500 to 700 feet, and the average 

depth is about 9 feet. The variation in water depth and mean velocity 

across the channel is illustrated in figure 5; one cross section is at 

1,000 feet upstream from Blair Highway Bridge, and another is at 

River Mile 645 or 17,500 feet downstream from the same bridge.

TEST PROCEDURE

A mixture consisting of 150 pounds of 40-percent rhodamine BA 

solution, 20 gallons of tapwater, and 10 gallons of methyl cellusolve 

was prepared in a 50-gallon mariotte vessel installed at Blair Highway 

Bridge on the morning of November 17, 1967. The vessel, a con^tanthead tank designed for maintaining a uniform rate of discharge, 

was set 200 feet from the pier on the west bank, so that the river 

discharge was about equally divided on either side. A 50-foot-long 

garden hose was used to carry the mixture, coming out of a 0.1065- 

inch orifice, down to a level about 5 feet above the water surface. 

The mixture dripped freely from the end of the hose into the water. 

The injection started at 1040 hours and continued until 1420 hours 

emptying most of the 48-gallon mixture in the vessel. The overall
G14 CONTRIBUTIONS TO THE HYDROLOGY OF THE UNITED STATES

CHICAGO AND 

NORTHWESTERN

4 

 U.S 30

Injection site on 

I \ Blair Highway Bridge

EXPLANATION

Cross section for 

dye sampling

Cross section 

for velocity 

measurement

643 River rrrleage

0 :/4 Vi 34 1 MILE I___i I___i___i

FIGURE 4. Reach of the Missouri River for the transverse mixing test.

rate of injection was 13.05 ml per sec, and the dye concentration in 

the mixture was 1.51X108 ppb or 15.1 percent. The pattern of mixing 

immediately downstream from the injection site is shown in figure 6. 

The boat ramp on the right bank is about 1,200 feet downstream 

from the injection site.

Samples for determining the transverse distribution of dye concentration were collected at 10 cross sections at distances between 

1,700 and 33,000 feet downstream from the injection site. The sampling 

started at the 1,700-foot cross section at 1100 hours and continued 

until 1509 hours when the sampling at the 33,000-foot cross section
MIXING CHARACTERISTICS OF THE MISSOURI RIVER G15

was completed. At each cross section, a sampling boat traversed the 

channel in a zigzag path along the cross-sectional line marked by 

three range poles as shown in figure 7. Samples were collected by 

dipping glass bottles into the water at reasonably uniform intervals 

so that a minimum of 20 samples was obtained during each traverse. 

Referring again to the sketch in figure 7, since the length of the base 

line AB and the angle /3 had been measured previously, the transverse 

sampling positions could be determined by reading the angle a with 

a transit at the instant each sample was obtained. A sample was taken 

immediately upon receiving a signal given by a control party at B 

when the sampler was in line with the three range poles. Since two 

transits were available, considerable time was saved by having an 

advance party set up a transit at the next section downstream. A set

I i

1000 feet upstream from Blair Bridge

100 200 300 400 500 600 

DISTANCE FROM LEFT BANK, IN FEET

700

FIGURE 5. Observed transverse profiles of velocity and depth at cross 

sections not affected by bridge abutments.
G16 CONTRIBUTIONS TO THE HYDROLOGY OF THE UNITED STATES

FIGURE 6. Mixing of dye immediately downstream from injection site at Blair

Highway Bridge.

of walkie-talkie radios was used for communication between the 

sampling, transit, and control parties.

The samples were analyzed for dye concentration with a Turner 

Model 111 fluorometer in the same manner as in the longitudinal
MIXING CHARACTERISTICS OF THE MISSOURI RIVER G17

pole

FIGURE 7. Method for locating lateral sampling positions.

dispersion experiment. Neither the calibration nor the sample readings were as satisfactory for the BA dye as for the WT dye used in the 

longitudinal dispersion test. Because of absorption of BA dye on the 

walls of sampling bottles and perhaps also because of chemical reactions between the BA dye and impurities in the river water, the calibration shifted with time. The resulting calibration error was probably 

on the order of ± 5 percent. The calibration errors together with the 

sampling scheme, which allowed only one traverse at each cross 

section, probably caused many of the observed instataneous point 

concentrations to differ from the time-averaged point concentrations 

by considerably more than 5 percent.

Velocity measurements by current meter were obtained prior to 

the test at two cross sections, one 1,000 feet up stream from Blair 

Highway Bridge on November 14 and the other 17,500 feet downstream from the same bridge on November 15 (see fig. 5). Several 

available staff gages were read before and after the test to determine 

the water-surface slope for this reach.
G18 CONTRIBUTIONS TO THE HYDROLOGY OF THE UNITED STATES

DATA

The transverse mixing data presented in table 6 cons'st of dye 

concentrations and lateral sampling positions at each cro^s section. 

Each concentration determined from the surface sampling is assumed 

to be uniform over the depth and also to be representative, at the 

particular location, of the time-averaged concentration resulting 

from the continuous uniform-rate injection of the solute. Values of 

the recovery ratio, RR, are also shown for each cross section. It is 

computed by the formula

(*

Q\Jo

cdz

RR=

where W is the width of the cross section, and c0 and q$ are respectively 

the concentration and volumetric discharge rate of the injected dye 

solution.

TABLE 6.- -Distribution of dye concentration in transverse direction, transverse 

mixing test on the Missouri River, November 17, 1967

Distance Dye confrom left centraStation bank, in tion, in 

feet parts per 

billion

Distance Dye confrom left centraStation bank, in tion, in 

feet parts per 

billion

Distance Dye confront left centraStation bank, in tion, in 

feet parts per 

billion

Cross section 1 

X=l,692 ft 

JF=665ft

1

2

3

4

5

6

7

8

9

649.1

635.6

628.5

621.2

609.5

587.9

568.2

569.6

544.3

0.00

.13

.00

.05

.00

.11

.13

.03

.00

10

11

12

13

14

15

16

17

18

528.7

498.1

478.1

452.8

420.6

397.8

365.4

336.4

315.5

.00

.00

.00

.00

.79

51.8

11.4

1.2

.08

19

20

21

22

23

24

25

26

286.4

263.8

208.9

168.6

146.7

109.4

68.4

30.3

.03

.00

.00

.00

.02

.00

.00

.00

Cross section 2 

X==3,692 ft 

IF=735 ft

1

2

3

4

5

6

7

8

706.6

687.3

682.9

668.4

647.8

622.4

608.4

587.1

0.00

.00

.00

.02

.00

.00

.00

.00

9

10

11

12

13

14

15

16

580.7

562.2

552.7

547.1

533.6

493.7

470.4

452.2

.00

.00

1.55

.15

4.3

9.4

9.6

5.8

17

18

20

22

23

24

25

449.5

409.4

355.1

288.1

235.1

157.5

79.3

3.09

2.67

.13

.00

.00

.00

.00

Cross section 3 

X=5,730 ft 

W=689 ft

RR=0.5S>

1

2

3

4

5

6

7

8

9

655.9

645.1

642.1

609.9

593.0

577.4

561.6

536.1

512.3

0.03

.00

.00

.00

.00

.00

.00

.00

.00

10

11

12

13

14

15

16

17

498.8

480.7

445.0

409.8

400.7

388.3

373.2

343.1

.00

.81

6.37

6.56

7.00

6.12

5.10

2.70

18

19

20

21

22

23

24

25

325.8

290.9

268.1

237.3

207.2

161.9

109.9

44.9

1.41

.37

1.20

.44

.00

.00

.00

.00
MIXING CHARACTERISTICS OF THE MISSOURI RIVER G19

TABLE 6. Distribution of dye concentration in transverse direction, transverse mixing 

test on the Missouri River, November 17, 1967 Continued

Distance Dye confrom left centraStation bank, in tion, in 

feet parts per 

billion

Distance Dye confrom left centraStation bank, in tion, in 

feet parts per 

billion

Distance Dye confrom left centraStation bank, in tion, in 

feet parts per 

billion

Cross section 4 

^"=8,730 ft 

W=644 ft 

##=0.49

1

3

4

5

6

7

8

9

620.4

579.9

576.7

556.3

537.3

511.2

480.4

474.3

0.00

.02

.00

.00

.02

.02

2.20

1.10

10

11

12

13

14

15

16

17

438.9

418.7

418.2

411.8

392.9

379.7

356.2

335.4

5.25

5.78

5.68

5.59

4.27

2.65

3.05

3.19

18

19

20

21

22

23

24

308.2

275.3

225.1

200.6

157.3

112.3

91.7

.73

.13

.00

.05

.03

.03

.00

Cross section 5 

JT=11,850 ft

##=0.65

1

2

3

4

5

6

7

8

9

605.4

591.9

580.8

564.7

549.1

533.2

518.9

497.6

481.5

0.00

.00

.00

.00

.09

.00

.13

.44

.40

10

11

12

13

14

15

16

17

18

446.4

411.7

389.8

382.1

369.4

342.1

320.3

327.8

330.2

1.43

1.87

3.98

3.29

3.82

3.58

4.12

4.02

5.10

19

20

21

22

23

24

25

311.0

300.3

261.7

222. 2

178.7

136.0

64.1

4.21

4.31

2.80

1.10

.54

.23

.11

Cross section 6 

JT= 15,490 ft 

TF=733 ft 

##=0.56

1

3

4

5

6

7

8

9

731.7

702.2

669.8

663.7

637.0

610.2

586.1

559.6

0.05

.15

.23

.15

.33

.75

.69

.77

10

11

12

13

14

15

16

17

523.8

485.8

438.1

445.4

451.6

405.2

406.6

364.1

2.39

3.63

4.27

4.16

3.00

3.34

3.44

2.49

18

19

20

21

22

23

309.5

264.3

218.5

162.1

121.3

48.2

1.10

.37

.27

.35

.25

.17

Cross section 7 

X=18,720 ft 

W=657 ft 

##=0.66

1

2

3

4

5

6

7

8

627.0

607.2

590.6

568.2

549.3

516.0

560.1

524.2

0.33

.23

.29

.23 . 29

.87

.50

.58

9

10

11

12

13

14

15

16

494.5

471.2

449. 0

419! 6

449.0

439.0

441.8

387.0

.83

1.22

2.05

1.64

1.89

2.36

1.51

2.76

17

18

19

20

21

22

23

24

334.1

283.9

258.7

226.5

191.9

146.0

105.7

76.2

3.19

3.82

1.85

1.10

.38

.37

.50

Cross section 8 

JT=22,410 ft 

W=657 ft 

##=0.64

1

2

3

4

5

6

7

8

9

583.0

570.9

566.2

545.0

534.4

522.6

500.7

490.1

475.1

0.43

.27

.25

.28

.25

.31

.34

.38

.44

10

11

12

13

14

15

16

17

18

474.7

442.2

424.6

398.4

366.5

338.5

307.8

291.4

272.3

.50

.54

.91

.72

2.10

1.55

2.24

2.68

2.51

19

20

21

22

23

24

25

225.2

195.7

188.9

146.1

98.6

60.2

35.8

3.10

2.26

3.25

3.20

.81

.58

.44
G20 CONTRIBUTIONS TO THE HYDROLOGY OF THE UNITED STATES

TABLE 6. Distribution of dye concentration in transverse direction, transverse 

mixing test on the Missouri River, November 17, 1967 Continued

Distance Dye confrom left centraStation bank, in tion, in 

feet parts per 

billion

Distance Dye confrom left centraStation bank, in tion, in 

feet parts per 

billion

D ist ance D y e co nfron- left centraStation ban!', in tion, in 

feet parts per 

billion

Cross section 9 

JT=27,690 ft 

W=605 ft

I

2

3

4

5

6

7

8

9

586.6

576.0

561.9

555.1

540.3

529.9

514.2

496.4

483.7

0.27

.48

.19

.27

.23

.25

.23

.33

.23

10

11

12

13

14

15

16

17

18

467.8

453.1

440.0

414.3

371.4

364.8

337.3

308.3

255.6

.31

.31

.34

.62

.93

.94

1.62

1.80

2.47

19

20

21

22

23

24

25

26

27

222.1

189.1

190.0

171.7

168.8

141.4

112.8

72.9

22.7

2.83

2.39

2.63

2.74

2.35

2.05

1.51

1.08

.88

Cross section 10 

JT=32,970 ft 

W=509ft 

RR=0.5&

8

9

10

11

12

13

14

15

16

490.6

470.4

442.5

427.3

411.1

399.5

381.8

367.1

362.8

0.27

.27

.31

.27

.33

.40

.43

.44

17

18

19

20

21

22

23

24

25

354.6

339.6

320.3

295.6

263.1

252.7

205.8

180.7

172.0

.44

.46

.48

1.12

1.58

1.82

2.01

2.13

2.26

26

27

28

29

30

31

32

172,9

163,6

135 6

93 0

74 1

60,9

23 5

2.03

2.08

2.13

2.14

1.47

1.64

1.35

COMPUTATION OF THE TRANSVERSE MIXING COEFFICIENT

METHOD OF MOMENTS

For a flow in an infinitely wide, open channel, flowing at a uniform 

depth and uniform velocity throughout its width, Sayre and Chang 

(1968) have shown that the transverse mixing coefficient may be 

calculated by the relationship

(8)

in which Ez is the transverse mixing coefficient, a-2z is the variance of 

the transverse distribution of the tracer, x is distance downstream 

from the point of injection of the tracer, and JJ is the mean flow 

velocity. In a nonuniform stream, such as the Missouri, equation 8, 

which is derived assuming an equal convective velocity at every point, 

is not strictly applicable because of the changing pattern of convective 

velocity. Nevertheless, equation 8 can be applied to the present data 

to provide a useful first approximation to the correct mixing coefficient, at least so long as the tracer distribution is not significantly 

affected by the presence of the boundaries.
MIXING CHARACTERISTICS OF THE MISSOURI RIVER G21

Figure 8 shows a plot of variance of the tracer cloud at each of the 

first seven measuring stations versus distance downstream from the 

injection point. Downstream from station 7 significant quantities of 

tracer had reached the boundaries. The variances were calculated 

directly from the transverse concentration distributions, without discharge weighting. Figure 8 shows a reasonably linear increase of the 

tracer variance for measuring sections 1 through 5; the slope of the 

straight line of best fit yields a transverse mixing coefficient EZ =1.5G 

sq ft per sec. On the basis of mean shear velocity U* = 0.24 ft per sec 

and a mean depth c?=9.1 ft, this yields a dimensionless transverse 

mixing coefficient El/U*d=0.71.

COMPUTER SIMULATION METHOD

The mixing coefficient calculated by equation 8 is only an approximation; because the transverse mixing experiment was conducted 

over a meandering reach of a river where the velocity is nonuniform 

across the width and along the length. A more valid method of calculating a mixing coefficient must take into account the pattern of convective velocities. In view of the complex pattern of velocity distribution, 

however, such calculation is only possible through a step-by-step 

simulation of convective diffusion process by a numerical method. 

This leads to a trial-and-error method of evaluating the mixing

14,000

Distribution calculated by computer 

using E, =0.6 u*d

2000 4000 6000 8000 10,000 12,000 14,000 16,000 18.0CO 20,000 

DISTANCE DOWNSTREAM FROM INJECTION, IN FEET

FIGURE 8. Variance of transverse tracer distribution, observed and calculated.
G22 CONTRIBUTIONS TO THE HYDROLOGY OF THE UNITED STATES

coefficient in which the coefficient that yields the calculated concentration distribution closest to the observed data is considered as the 

correct value. To establish accurately the pattern of convective 

velocities and to deduce the effect of diffusive mixing, one would need 

point velocity measurements at every point within the stream. 

Although this is not possible, it would be useful for future experiments 

to have at least a complete transverse velocity profile at each measuring 

station. In the present experiment only two velocity profiles were 

measured within the reach of the transverse mixing experiment, one 

1,000 feet upstream from the injection point and the other halfway 

between cross sections six and seven at mile 645. The^e velocity 

profiles are shown in figure 5. The first velocity measurement is in 

the upstream section of a long curve to the left, and the second is 

somewhat downstream from the end of the same curve, where the 

flow is beginning to shift from the right to the left bank.

The computer program which is to be described requires that the 

cross-sectional geometry and velocity distribution be knowT n at the 

beginning of every distance step. On the basis of the two measured 

sections, the following velocity distributions were assumed: For the 

reach from the injection point to measuring station 5 (at the end of 

the bend to the left), the velocity distribution measured upstream 

from the injection point was assumed to apply to all sections. At mile 

645 the measured velocity distribution was used, except that the 70- 

foot width of shallow water near the left bank was omitted as being, 

from visual observations, not representative of the reach. At a point 

halfway from section 8 to section 9, the velocity distribution was 

assumed to be the inverse of the distribution at mile 645; that is, the 

same velocities and depths were used as at mile 645, except that distances measured from the right bank were assumed to b° measured 

from the left bank. At station 10 (in the middle of a bend to the 

right), the velocity distribution was assumed to be the mirror image 

of the velocity distribution upstream from the injection point, except 

that at this point the width of the stream is only 510 feet, so that all 

transverse distances were reduced by the factor 510/600 and all depths 

were increased by the inverse of the same factor. For cross sections 

between the above sections, depths, widths, and velocities were interplated linearly from the nearest assumed distributions.

The computer program divides the stream longitudinally into distance steps, Ax, 200 feet long, and transversely into 20 stream tubes of 

equal discharge. The velocity within each stream tube and the transverse location of its end point are found from the measured or assumed distribution of velocity within the cross section. Diffusion is
MIXING CHARACTERISTICS OF THE MISSOURI RIVER G23

assumed to be occurring between the stream tubes at a rate given by 

the Fickian equation

M=Ezas ^ (9)

in which M is the mass transport through the dividing surface between 

stream tubes per unit time, as is the dividing area between the stream 

tubes, and z is the transverse distance. A steady-state equation for the 

conservation of mass within each stream tube for each distance step is

q.ct+i, i=qct, j+Mj-i, } Mj, j+l (10)

in which q is the discharge per stream tube (constant for all stream 

tubes), Ct,j is the concentration in the jth stream tube at the start of 

the -ith distance step, and Mj, j+i is the rate of mass transfer between 

streams j' andj + 1. The mass transfer between stream tubes is assumed 

to be entirely diffusive (that is, any effects of secondary currents are 

absorbed in the diffusive term), and is calculated by the formula

Mj, J+l=E4u+l*x ^F^1 (11)

in which djtj+i is the depth at the interface between stream tubes, j 

andj'+l; and Azjij+i is the distance between the centroids of adjacent 

stream tubes. The derivation of this formula assumes that for computing mass transfer the concentration in each of the stream tubes is 

constant throughout the distance step. The boundary condition of no 

transport through either bank is satisfied by allowing stream tubes 1 

and 20 to exchange only with stream tubes 2 and 19, respectively.

The variation in stream-tube velocity is not shown explicitly in 

equation 10, but the functional dependence can be seen by substituting 

equation 11 into equation 10 and dividing the result by g. Dropping 

the stream-tube subscripts, taking q=uAzd, where u is the local 

velocity, and, for simplicity, replacing the finite difference form of the 

transverse concentration gradient by the differential form yield

u 02*

This shows that for a constant mixing coefficient and a constant distance step, the change of concentration within a stream tube caused 

by transverse mixing is inversely proportional to the local velocity, «. 

To complete one distance step, the computer carries out the computation given in equation 10 for each of the 20 stream tubes lining the
G24 CONTRIBUTIONS TO THE HYDROLOGY OF THE UNITED STATES

concentration values at the beginning of the distance step, replaces 

each concentration value by its new value at the end of the distance 

step, and prints the 20 concentrations. Between distance steps the 

positions of the midpoints of each of the stream tubes and the depths 

of the stream tubes may be changed, according to the pattern of convective velocities assumed at the outset. The program is designed to 

read velocity-distribution data normally recorded in the Geological 

Survey stream-gaging operations. From these data the coniDuter constructs a set of stream tubes of equal discharge and interpolates streamtube velocities and dimensions between measuring stations.

Before the program was used in the present experiment, it was 

verified by assuming a flow having a width of 500 feet, r, depth of 

10 feet, a uniform transverse velocity distribution, a mean velocity 

of 5 ft per sec, and a shear velocity of 0.25 ft per sec. The distance 

step was taken to be 500 feet and the mixing coefficient 0.23U*d, or 

(0.575 sq ft per sec). The variance of the tracer cloud produced by 

the computer program increased exactly linearly with distance, the 

rate of increase giving a mixing coefficient of 0.56 sq ft per sec by 

equation 8. The difference between the given mixing coefficient and 

the computer result is 2.6 percent, well within the level of other 

experimental errors.

The program was next applied to predict the transverse mixing 

pattern in the experimental reach by introducing the velocity distributions previously assumed. The initial tracer distribution was taken 

to be that measured at measuring section 1, because upstream from 

section 1 the tracer probably was not fully mixed over the depth. 

The program then generated tracer distributions at all downstream 

sections. The computer program is written to accept up to 10 values 

of k in the mixing coefficient, kU*d, and to generate tracer distributions simultaneously for each k value; for this experiment k values 

of 0.4, 0.5, 0.6, 0.7, and 0.8 were tried. Figure 9 shows the observed 

tracer distributions and those which were generated by the program 

using the value fc=0.6. This value produced what appeared to be 

the best match between observed and computer-predicted concentration distributions; much of the error probably results from incorrect 

assumption of the velocity distribution. However, the comparison is 

reasonably good at all sections and shows that the equation

#z = 0.6 U*d (12)

gives an adequate prediction of the mixing pattern for the reach in 

question.

The sensitivity of the mixing pattern to changes in the value of k 

can be seen in figure 10, which shows the transverse concentration
MIXING CHARACTERISTICS OF THE MISSOURI RIVER G25

0 100 200 300 400 500 600 700 

TRANSVERSE DISTANCE FROM RIGHT BANK, IN FEET

800

FIGURE 9. Comparison of observed and calculated transverse 

distribution of dye concentration at nine cross sections
G26 CONTRIBUTIONS TO THE HYDROLOGY OF THE UNITED STATES

distribution at sections 5 and 9 for k equal to 0.4 and 0.8. The wide 

range of k produces only a modest change in the distribution; at 

section 5, 0.4 appears to give the better match, but at section 9, 0.8 

appears to be the better choice. Thus the value 0.6 can be regarded 

only as an adequate approximation, probably accurate to within db 20 

percent. Unfortunately there appears to be no more accurate method 

of measuring the value of the coefficient. The variances of the tracer

10 n i r

EXPLANATION 

Observed concentration

Concentration calculated 

by computer

with £~ = Q.4U*d

CO 7

04

Section 5

with f, =0.8U*d

100 200 300 400 500 

TRANSVERSE DISTANCE FROM RIGHT BANK, IN FEET

600

100 200 300 400 500 

TRANSVERSE DISTANCE FROM RIGHT BANK, IN FEET

600

FIGURE 10. Influence of E, on transverse distribution of dye concentration.
MIXING CHARACTERISTICS OF THE MISSOURI RIVER G27

cloud predicted using the value k=O.Q have been plotted in figure 8; 

the excellent agreement with the variance calculated from the observed 

data lends support to the value fc = 0.6.

SUMMARY AND CONCLUSIONS

In November 1967 a longitudinal dispersion experiment WPS conducted in a reach of the Missouri River by introducing a concentrated 

slug of 600 pounds of Rhodamine WT dye at Sioux City, Iowa. Observations of dye concentration as a function of tune were made at 

four downstream locations, ranging from 41 to 141 miles below Sioux 

City, over a period of 3 days. Longitudinal dispersion coeffcients 

were calculated from these data, using both the routing and changeof-moment methods. During the experiment the channel geometry 

and transverse distribution of velocity were measured at seven cross 

sections distributed along the reach. Channel widths varied from 

about 500 to 750 feet, average depths from about 8 to 12 feet, average 

velocities from about 3.9 to 6.0 ft per sec, and discharges from about 

31,000 to 34,000 cfs.

On the day following the completion of the longitudinal dispersion 

experiment, a transverse mixing experiment was performed in a 

6-mile reach immediately downstream from Blair, Nebr., son:ewhat 

more than halfway down the longitudinal dispersion study reach. In 

this experiment, a continuous stream of Rhodamine BA dye was 

injected near the center of the river at Blair, and sets of samples 

defining the transverse distribution of the dye plume were ottained 

at 10 cross sections ranging from 1,700 to 33,000 feet downstream. 

Values of the transverse mixing coefficient were calculated by both 

the method of moments and a computer simulation method which 

takes into account the longitudinal variations of channel geometry 

and the transverse distribution of velocity.

Analysis of the experimental data leads to the following conclusions:

1. The best estimate of the longitudinal dispersion coefficient for the 

entire study reach at the time of the experiment is

D~ 16,000 sq ft per sec.

Estimates for individual subreaches range from 10,000 to about 

30,000 sq ft per sec. Estimates based on the routing and moment 

methods are in reasonably good agreement. However, it is 

concluded that in general the routing method gives more consistent results.

2. The average rate of movement of the dye cloud through the 141- 

mile study reach was 5.34 ft per sec. This compares well with 

average velocities ranging from about 3.9 to 6.0 ft p0,r sec
G28 CONTRIBUTIONS TO THE HYDROLOGY OF THE UNITED STATES

measured by current meter at various cross sections through the 

reach.

3. The best estimates of the transverse mixing coefficient are, by the 

moment method,

#2=0.7 U*d,

and, by the computer simulation method,

#2 =0.6 U*d.

The data suggest that Ez varied, perhaps by as much as 100 

percent within the 6-mile study reach. It is concluded that the 

ratio Ez/U*d is apt to be significantly larger and more variable 

in large meandering channels than in small straight channels. 

This underlines the need for further study of transverse mixing 

in large natural streams.

4. Bhodamine WT is far superior to Rhodamine BA as a water 

tracer, both with respect to dye loss in the channel and aclsorbtion of dye on glassware used for storing experimental and 

calibration samples. Whereas the longitudinal dispersion experiment with Rhodamine WT gave a dye loss of less than 25 

percent for the 141-mile study reach, the transverse mixing 

experiment with Rhodamine BA gave a dye loss on the order of 

40 percent for a 6-mile reach. The relative amounts due to dye 

loss in the river and to adsorption on glassware are not known, 

but both are probably important.
MIXING CHARACTERISTICS OF THE MISSOURI RIVER G29

REFERENCES

Bowie, J. E., and Petri, L. R., 1968, Travel of solutes in lower Missouri River:

U.S. Geol. Survey Hydrol. Inv. Atlas HA-332. 

Elder, J. W., 1959, The dispersion of marked fluid in turbulent shear flew: Jour.

Fluid Mechanics, v. 5, no. 4, p. 544-560. 

Fischer, H. B., 1966a, A note on the one dimensional dispersion model: Air

and Water Pollution, an Internat. Jour., v. 10, p. 443-452. 

Fischer, H. B., 1966b, Longitudinal dispersion in laboratory and natural streams:

California Inst. Tech., W. M. Keck Lab. Hydraulic and Water Resources,

Rept. KH-R-12, 250 p. 

Fischer, H. B., 1967a, Transverse mixing in a sand-bed channel: U.S. Geol.

Survey Prof. Paper 575-D, p. D277-D282. 

Fischer, H. B., 1967b, The mechanics of dispersion in natural streams: Jour.

Hydraulics Div., Am. Soc. Civil Engineers, v. 93, no. HY6, p. 187-216. 

Fischer, H. B., 1968, Dispersion predictions in natural streams: Jour. Sanitary

Engineering Div., Am. Soc. Civil Engineers, v. 94, no. SA5, p. 927-943. 

Glover, R. E., 1964, Dispersion of dissolved or suspended materials in flowing

streams: U.S. Geol. Survey Prof. Paper 433-B, 32 p. 

Godfrey, R. G., and Frederick, B. J., 1963, Dispersion in natural streams: U.S.

Geol. Survey open-file report, 75 p.

Sayre, W. W., and Chang, F. M., 1968, A laboratory investigation of openchannel dispersion processes for dissolved, suspended, and floating dispersants, U.S. Geological Survey Prof. Paper 433-E, 71 p. 

Taylor, G. E., 1954, The dispersion of matter in turbulent flow through a pipe:

Royal Soc. London Proc., v. 223A, p. 446-468. 

Thackston, E. L., and Krenkel, P. A., 1966, Longitudinal mixing and reaeration

in natural streams: Vanderbilt Univ. Sanitary and Water Resources Eng.,

Tech. Rept. 7,212 p. 

Thackston, E. L., and Krenkel, P. A., 1967, Longitudinal mixing in natural

streams: Jour. Sanitary Engineering Div., Am. Soc. Civil Engineers, v. 93,

no. SA5, p. 67-190. 

U.S. Army Corps of Engineers, Omaha District, 1967, Missouri River Channel

regime studies: M.R.D. (Missouri River Division) Sediment Series 13, 13 p. 

Wilson, J. F., and Forrest, W. E., 1965, Potomac River time-of-travel measurements: Lamont Geol. Observatory Symposium on Diffusion in Oceans and

Fresh Waters, Palisades, N.Y., 1964, Proc., p. 1-18. 

Yotsukura, Nobuhiro, 1967, General discussion in subject D microturbulent

diffusion and dispersion: Internat. Assoc. Hydraulic Research Cone;., 12th,

Fort Collins, Colo., 1967, Proc., v. 5, p. 542-551.

US. GOVERNMENT PRINTING OFFICE: 1970 O 37F-385