Flow Characteristics

of Elkhorn River

Near Waterloo, Nebraska

GEOLOGICAL SURVEY WATER-SUPPLY PAPER 1498-B
AUG 61l^

Flow Characteristics

of Elkhorn River

Near Waterloo, Nebraska

By E. W. BECKMAN and L. W. FURNESS

STUDIES OF FLOW IN ALLUVIAL CHANNELS

GEOLOGICAL SURVEY WATER-SUPPLY PAPER 1498-B

Analysis of depth-discharge 

relationships with changing 

bed configuration

UNITED STATES GOVERNMENT PRINTING OFFICE, WASHINGTON : 1962
UNITED STATES DEPARTMENT OF THE INTERIOR 

STEWART L. UDALL, Secretary

GEOLOGICAL SURVEY 

Thomas B. Nolan, Director

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

Washington 25, D.G.
CONTENTS

	Pa«<f 

Abstract__ __________________________________________________ B-1

Introduction-_____________________________________________________ 1

Description of reach.______________________________________________ 1

Observation program_______________________________________________ 4

Stability of channel--___---_-_-_-_________-_-_--__--__-_-___--__-__ 6-

Width_____-_-------------___-_-_____-_--_--_--_--_-_____-_-__ 6

Average bed elevations_________________________________________ 12

Point bed elevations-_________ _________________________________ 12

Bed configuration._________________________________________________ 12

Depth-discharge relationship._______________________________________ 17

Gage height-discharge relationship_-__-__--___-_____-_______________. 20

Roughness coefficient______________________________________________ 20

Einstein's parameter._____----_______,-_-_________-___________- 27

Simons' equation______________________________________________ 31

Liu's equation________________________________________________ 31

Conclusions_ _ _________-_-_--_-----__--_-__--___-_---_-_---_-____ 31

References cited________-_____-___-_-_-_-_-_____-_-__--------_-____ 34

ILLUSTKATIONS

Page 

FIGURE 1. Map showing site of investigation._______________________ B-2

2. Size distribution of bed material, June 1956_______________ 3

3. Size distribution of bed material, June 1958.______________ 4

4. Size distribution of bed material, June 1959_______________ 6

5. Variation in width of stream at upper cableway-___________ 7

6. Variation in width of stream at lower cableway____________ 8

7. Average channel bed elevations at upper and lower cableway sections.________________________________________ 13

8. Elevation of channel bed at upper cableway section at time

of measurements.____________________________________ 14

9. Elevation of channel bed at lower cableway section at time of

measurements- ______________-__-________-_____-_----- 15

10. Comparison of Elkhorn River bed configuration with Albertson's relationship_ _ _________________________________ 16

11. Comparison of Elkhorn River bed configuration with Garde's

relationship_ _ ______________________________________ 18

12. Relation between depth and discharge at lower cableway

section. _____________________________________________ 19

13. Relation between depth and discharge at upper cableway section_______________________________________________ 21

HI
IV CONTENTS

Page

FIGURE 14. Relation between gage height and discharge at upper cableway section_________________________________________ B-22

15. Relation between gage height and discharge at upper cableway section during flood of May 1951-. _----_--_--.---__ 23

16. Relation between gage height and depth of water at upper

cableway section during flood of May 1951-_____________ 24

17. Relation between gage height and mean velocity at upper

cableway section during flood of May 1951._____________ 25

18. Relation between gage height and slope of water surface during flood of May 1951_____-___..__-__.__-__-_____ 26

19. Relation between Chezy's C and mean velocity for measurements made at both cableways.____-__-__-__-__-_____ 28

20. Comparison of data from Elkhorn River with data from other

rivers supporting Einstein's curve of relationship..------- 29

21. Relation between Chezy's C and Einstein's X', Elkhorn

River data.__________-__-_-_-___--_--__----__----__- 30

22. Comparison of relationship between Chezy's C and Simons'

C, Elkhorn River data-_-__-_--------------__-_-____ 32

23. Comparison between observed velocity and velocity computed

by Liu's formula_.__--_-__-_----_-_-__--------------- 33

TABLES

Page 

TABLE 1. Particle-size analysis of two samples of bed material from

upper cableway section______________________________ B-5

2. Summary of observed data____--_-----_-_--_---_-------- 9

3. Particle-size analysis of 10 samples of suspended sediment.__ 11

Symbol

A 

B 

C

ca

C,

D 

d or

F

LETTERS, SYMBOLS, AND UNITS

Area of cross section of stream

Top width of cross section of stream

Chezy coefficient of roughness

Empirical discharge coefficient proposed by Liu and

Hwang 

Concentration of sediment in suspension, in parts per

million

Mean depth of flow in cross section 

Median fall diameter of streambed material 

Grain size of bed material at which 65 percent are finer;

used as K, by Einstein and Barbarossa 

Grain size of bed material at which 35 percent are finer;

used as DM by Einstein and Barbarossa 

Observed fall in a reach 

Froude number =V/-\/gD 

Gravity acceleration=32.2 

A constant

Units 

ft* 

ft

fi^/sec 

ft»~ 

*/sec

ft

ft

ft

ft 

ft

ft/sec2
CONTENTS V

Symbol Units 

n Manning coefficient of roughness ft* 

P Wetted perimeter of stream cross section ft 

0 Total discharge of water-sediment mixture ft3 /sec 

q Unit discharge: the average discharge of water-sediment 

mixture in a unit width of the stream cross section= 

Q/B ft3 /sec 

<r Standard deviation of bed material 

R and Rm Hydraulic radius=A/P; also used as average hydraulic

radius when combining several sections in a reach ft 

R' Hydraulic radius pertaining to grain roughness ft 

R" Hydraulic radius pertaining to channel irregularities

where R=R' + R" ft 

S Slope of energy gradient

T Temperature of water °F 

V Average velocity in a cross section, also used as average

mean velocity when combining sections ft/sec 

V* Shear velocity=^JgRS or as -^gDS by Simons, Richardson, and Albertson ___ ft/sec 

V'f Velocity pertaining to grain roughness=-JgR'S___ ft/sec 

V"f Velocity pertaining to channel irregularities=iJgR"S ft/sec 

w Fall velocity of sediment particles usually of the mediangrain size, d ft/sec

X' A parameter describing sediment transport=1.68 -57% K £>

X An exponent for R which varies with the median-grain 

size (d) of the bed material in the equation V= CaRxSu

y An exponent for S which varies with the median-grain 

size (d) of the bed material

v Kinematic viscosity of water ft2 /sec

P Unit mass density of water Ib sec2

ft*

p. Unit mass density of sediment in air Ib sec2

ft*

7 Specific weight of water lb/ft3

 y, Specific weight of sediment lb/ft3

STUDIES OF FLOW IN ALLUVIAL CHANNELS

FLOW CHARACTERISTICS OF ELKHORN RIVER NEAR 

WATERLOO, NEBRASKA

By E. W. BECKMAN and L. W. FURNESS

ABSTRACT

The characteristics of flow in a reach of alluvial channels on the Elkhorn River 

were observed. The variations in channel width, bed elevations, bed configurations, and roughness coefficients are described and related to certain 

hydraulic parameters previously defined in laboratory and field studies of flow 

in alluvial channels.

INTRODUCTION

A better understanding of the relationship of discharge to depth 

and slope in alluvial channels was sought through observations on a 

reach of channel of the Elkhorn River in Nebraska. Measurement of 

bed elevation, velocity, and slope were obtained for 45 different 

flow events during the period 1951-56. Several determinations of 

size of bed material and suspended-sediment concentration were also 

made.

The hydraulic relationships in alluvial channels are complex because 

the configuration and elevation of the channel bed are unstable. 

It is recognized that general principles cannot be established from the 

history of events in a single reach in which slope and size of bed material are virtually constant. Laboratory studies during recent years 

have defined the regimes of bed configuration and the primary variables which determine flow depth. However, the results of laboratory 

studies cannot yet be applied to natural streams because the relationships of model to prototype flow of water-sediment mixtures is imperfectly understood. Studies of the mechanics of flow in natural 

channels should contribute towards a better understanding of these 

relationships.

The hydraulic characteristics of the Elkhorn reach are described 

in this report. Certain analyses of the data are presented in order 

to make comparisons with the results of laboratory and field studies 

previously published in the literature.

DESCRIPTION OF REACH

A reach 1,670 feet long on the Elkhorn River at King Lake Camp 

about 2 miles northwest of Waterloo, Nebr., was selected for investiB-l
B-2 STUDIES OP PLOW IN ALLUVIAL CHANNELS

EXPLANATION

< > 

Gages

< > 

Gages and cableways

1000 2000 FEET

FIGXTBE 1. Map showing site of investigation.
FLOW CHARACTERISTICS, ELKHORN RIVER, WATERLOO, NEBR. B~3

gation (fig. 1). The channel alinement is relatively straight, and, 

except for unusually high floods, the flow is confined to the channel. 

Vegetation tends to stabilize the relatively steep banks. The channel 

widens from about 250 feet at the upper end of the reach to about 285 

feet at the lower end of the reach; the depth of flow decreases in a 

corresponding manner. The depth of flow ranges from 2 to 7 feet 

with a corresponding range in discharge from 1,000 to 13,000 cfs 

(cubic feet per second). The energy gradient through the reach is on 

the order of 4 X 10~4 foot per foot.

The stream channel is in alluvial sand. Samples of the bed material 

were taken in June 1956 and 1958 at points along the upper cableway 

section (site 2, fig. 1). The results of the analysis of these samples 

by the visual accumulation tube or sieve methods are shown in table 1 

and figures 2 and 3. The size of the material ranges from 0.062 to

PERCENT PARTICLES OF FINER THAN INDICATED DIAMETER 

«to 

3 P 9}°5°!*tO f\>» 0<»1 -4 to.-<0 - 

ooooooo r\> oicaooioaa o - 01

X ^

/

/

V

/

isual accumulatio 

tube analysis

1

/

/

'

j

~7~/

1

'

Co

/

^

rc

n

/

1- C

lcr=

mpo 

at s 

200 

cab

site 

tat 

,2 

ew

/

f

/

/

).23 m 

.49

/

m

^

/

/

Sf 

Sieve an<alysis

o 

> of 7 samples taken -

ions 80, HO, 140, 170,. 

30 and 260, at upper, 

ay section

0.02 O.I 

DIAME TE R,

l.O 3.0

IN MILLIMETERS

FIGUBE 2. Size distribution of bed material, June 1956.

606&58r «1 2
B-4 STUDIES OF FLOW IN ALLUVIAL CHANNELS

99.8

99.5 

99.0 

98

95 

90

80 

70 

60 

50 

40 

30

20

10 

5

2

1.0 

0.5

0.2 

n. i /

r

i

\

/

isual ace 

tube a

 r

/

/

/

:umul 

nalys

/

A/

ation

is

/

/

~{

/

f

/

X

X

s

rf = 0.24 mr 

a= 1.81

^

n

Sieve a

^

nalysi

^

o 

Composite of 32 s 

at 5- ft intervals

t 

c

o 190, inclusive 

ableway sectio

s

-?

amp 

,stc 

!, at

n

/

/

/*

>les take 

tions 2 

upper

/

5

0.06 O.I

OIA METER, IN

i.a 

MILLIMETERS

9.0

FIGURE 3. Size distribution of bed material, June 1958.

8 mm; the median size is about 0.24 mm and the standard deviation 

is about 1.65. The median size of bed material was largest in the 

middle part of the channel.

Forty tons of bed material was taken from the reach in June 1959 

for use in laboratory studies. The result of the analysis of 39 samples 

of this material is shown in figure 4. The median size is 0.28 mm and 

the standard deviation is 1.68.

Reference sizes ^50=0.24 mm, ^35=0.20 mm, ^65=0.28 mm have 

been used in the analysis in this report. The subscript for d shows the 

percentage of particles finer than the diameter indicated.

OBSERVATION PROGRAM

A summary of the data obtained during the period 1951-56 is given 

in table 2. Information on velocity, depth, width, and suspended sediment was obtained at the cableway site at the upper end of the reach; 

simultaneous measurements of depth and width were obtained at the 

cableway site at the lower end of the reach (site 11, fig. 1). Water-
TABLE 1. Particle-size analysis of two samples of bed material from 

[Methods of analysis: S, sieve; V, visual accumulation tub

Date

1956 

June 27 ...

June 27..-- _

June 27.-.- _

June 27_-___

June 27-.-._ _

June 27 ....

June 27-.. _

June 27 _ _

1958 

June 20-..- ...

June 20... ._

June 20---.-

June20-- _

Number of 

sampling 

points

1 

1 

1 

1 

1 

1 

1

7

7 

18 

7 

32

Percent of bed material finer than indicated size, in millimeters

0.062

1 

0 

0 

0

1 

1 

1 

1

0 

0

1

0

0.125

11

4 

1 

1 

25 

2 

7 

7

3 

1 

27

7

0.250

78 

40 

34 

38 

82 

58 

89 

60

68 

30 

99 

53

0.500

100 

96 

84 

89 

98 

93 

100 

94

94

74 

100 

84

1.000

100 

95 

99 

100 

98

99

99 

90

94

2.000

98 

100

99

100

99 

95

97

4.000

100

100

99 

99

99

8.000

100 

100

100

Meth

anal

V

sv 

sv 

sv 

sv 

sv

V

sv

sv 

sv

V

sv
B-6 STUDIES OF FLOW IN ALLUVIAL CHANNELS

0.04

99.8 

99.5 

99.0 

98

95 

90

80 

70 

60 

50 

40 

30 

20

10 

5

2

l.O 

0.5

0.2 

O.I

0

<*

/

/

yx

*

/ /i -?

/

/

/

s

^

s

^

^50= 0.28 mm 

o- = l.68

Co m 

4C 

at 

tu

U

30 

)

r< 

be

si 

fo 

;a

>-

tt 

n 

c

31

^

^

^-o

; of 39 samples from 

s of bed material taken, 

h. Visual accumulation

lalysis

 

O.I 1.0 6.O

DIAMETER, IN MILLIMETERS 

FIGURE 4. Size distribution of bed material, June 1959.

surface elevations were read on a series of staff gages spaced along the 

stream channel at locations shown in figure 1. The fluctuations in the 

water surface due to surges did not exceed 0.10 foot. Each set of data 

was adjusted to a common time.

The results of the analysis of particle size for 10 samples of suspended sediment are given in table 3.

STABILITY OF CHANNEL 

WIDTH

The width of the channel of the upper and lower cableway sections 

at time measurements were made is shown in figures 5 and 6. The 

banks of the channel are relatively stable, and variations in width for 

a given stage are not pronounced.
FLOW CHARACTERISTICS, ELKHORN RIVER, WATERLOO, NEBR. B~7

240 250 260 270 

WIDTH OF SECTION, IN FEET

280

FIGURE 5. Variation in width of stream at upper cableway. Figures correspond to measurement numbers in table 2.
B-8 STUDIES OF FLOW IN ALLUVIAL CHANNELS

-\

270 280 290 300 

WIDTH OF SECTION, IN FEET

FIGUEE 6. Variation in width of stream at lower cableway. Figures correspond to measurement numbers In table 2.
TABLE 2. Summary of observed data

[Q, Discharge, in cubic feet per second; T, water temperature (°F); B, surface width, in feet; D, mean depth of sect

radius, in feet; A, area of cross section, in square feet; C» concentration of suspended sediment, in parts per mi

slope of water between sections; n, Manning coefficient of roughness; C, Chezy coefficient of roughness]

Measurement 

No.

567 ___

568 

569. _ . 

570. ....

571 

572 .. 

573 .

574 __ .

576,.... 

577.....

578 

580 

581 

582 

583 _.

587 

589 

591 .....

599.....

628.....

630 

632..... 

633... __

634. . 

See

Date

1961 

May 18.......

May 21.. __ .

May21,__ .

May22.......

May 22.......

May 23... _ May 24. ....

May 26... 

May 31. ......

June 1 .........

Jnn e 1

June 28 ........ 

July 5- 

July 19- 

Sept. 11 

1962 

Mar. 26.. 

Apr. 1 

Apr. 1. ........ 

Apr. 2 .

Apr. 2... ...... 

botnotes at end

Q

2,680 

6,630 

6,800 

7,660 

7,560

7,700 

7,750 

6,780 

5,790 

4,860

12,600 

6,830 

6,160

6,440 

4,370

2,760 

8,560 

4,380 

3,430

2,830 

8,960 

8,850 

7,900

7,860 

of table

Time

4:42 p.m ......

2:54 p.m. 

10:22 a.m .....

9:53 a.m. .....

12:48 p.m...,. 

12:17 p.m. 

1:22 p.m. 

11:23 a.m ..... 

2:59 p.m ......

2:07 p.m. .. ... 

2:53 p.m

3:34 p.m ...... 

9:54 a.m ......

11:20 a.m.- .. 

p. 11.

T

168 

166 

i 66 

169 

69

169 

66 

68

70 

70 

70 

63 

63

73

68 

176 

73

135 

44 

46 

44 

46

Upper cableway section

Stage

1.685 

3.19 

3.225 

3.655 

3.69

3.72 

3.89 

3.68 

3.21 

2.63

5.39 

3.465 

3.25 

2.725 

1.96

1.82 

3.92 

3.08 

1.81

1.06 

3.84 

3.68 

3.52 

3.52

B

248 

261 

265 

269 

271

272 

273 

268 

262 

262

283 

266 

265 

262 

258

249 

268 

262 

262

243 

266 

264 

263 

262

D

3.76 

4.97 

4.92 

6.65 

5.75

5.92 

6.58 

6.55 

5.97 

5.03

7.16 

6.20 

5.97 

5.13 

4.52

4.30 

6.44 

5.07 

4.01

3.07 

6.19 

6.08 

5.65 

5.73

V

2.87 

5.11 

5.22 

4.28 

4.86

4.78 

4.32 

3.86 

3.70 

3.69

6.22 

4.14 

3.89 

4.79 

3.75

2.58 

4.96 

3.30 

3.26

3.79 

5.44 

5.51 

5.32 

5.23

B

3.74 

4.95 

4.80 

6.57 

5.71

5.87 

6.51 

6.46

7.10 

6.15 

5.92 

5.09 

4.48

4.27 

6.37 

5.02 

4.00

3.06 

6.11 

6.02 

5.58 

5.66

A

934 

1,298 

1,303 

1,788 

1,557

1,610 

1,796 

1,756 

1,565

2, 025 

1,648 

1,583 

1,345 

1,165

1,071 

1,726 

1,329 

1,051

747 

1,647 

1,606 

1,485 

1,501

C,

23,430 

6,260 

2 6, 010 

5,700 

2 5, 010

L

Stage

0.985 

2.465 

2.465 

2.87 

2.905

2.90 

2.99 

2.815

0 CC

4.75 

2.615 

2,415 

2.22 

1.37

1.115 

3.315 

2.445 

1.21

.405 

3.205 

3.02 

2.79

2.785

B

28

28

28

28

28

29

28

28

29

28

28

28

28

28

28

29

28

28

28

28

28

28
TABLE 2. Summary of observed data C

Measurement 

No.

635 

636. 

637 

638.--- 

640 

642 

644 

646 

648 

650 

651 

652 

654 

655 

656 

657 

658 

659 

660 

661 

662 

663 

664 

665 

666 

667 

668 

669 

670 

671 

672 

673., 

Date

1S6S

Apr. 2 __ .... 

Apr. 3... ___ 

Apr. 4 ___ 

Apr. 9 ._ 

Apr. 18 - 

Apr. 23.. __ ..

May 15... 

May 28--. -

July 10. 

July 24 ... -

Aug. 7.-. 

Aug. 12--. 

Aug. 21-- 

Aug. 27--- 

Sept. 2_- __ -

Sept. 9. ....... 

Sept. 15 . 

Sept. 22 ___ .

196$

Sept. 30 ____ 

Oct. 7 

Oct. 13. 

Oct. 23 .

Q

7,540 

6,520 

5,610 

4,020 

2,860

4,100 

2,150

1,700 

1,500 

1,540

5,740 

2,080 

1,260 

978 

879

1,820 

1,890 

1,190 

749 

673

967 

799 

2,840 

1,000 

1,150

650 

622 

685

517 

488 

630 

616

Time

3:52 p.m ......

10:58a.m __ 

10:26 a.m. . 

10:10 a.m ..... 

11:09 a.m __

11:14 a.m. 

12:36 p.m . 

11:17 a.m. 

11:49 a.m .... . 

3:30 p.m ......

11:00 a.m ... 

12:20 p.m. . 

12:02 p.m. 

3:29 p.m ......

12:59 p.m. 

10:42 a.m. 

11:55 a.m. __ 

12:30 p.m. .

12:25 p.m. 

11:52 a.m ..... 

3:37 p.m ......

11:15 a.m. 

2:56 p.m -

10:32 a.m. 

12:40 p.m.- 

2:22 p.m ...... 

2:07 D.IO- ....

T

47 

46 

44 

44 

52

53 

68 

68 

66 

64

68 

76 

86 

77 

78

79 

79 

80 

81 

78

74 

75 

74 

80 

67

74 

75 

66

72 

52 

55 

54

Upper cableway section

Stage

3.49 

3.11 

2.53 

1.92 

1.22

2.075 

1.485 

1.03 

1.01 

1.11

2.56 

1.16 

.52 

.13

.22

1.33

1.06 

.54 

-.13

.29 

.01 

2.45 

.35 

.625

-.17 

-.13 

.00

-0.28 

-.31 

-.19 

-.02

B

264 

262 

264 

264 

268

262 

256 

253 

264 

254

261 

248 

248 

246 

248

258 

252 

237

224

224

236 

235 

266 

251 

258

250 

249 

252

248 

252 

252 

249

D

5.70 

5.46 

4.65 

3.99 

2.90

3.92 

3.02 

2.71 

2.84 

2.84

4.44 

3.42 

2.41 

2.17 

2.21

3.44 

3.10

2.42 

1.88 

1.72

2.05 

1.73 

3.75 

2.03 

2.28

1.67 

1.59 

1.66

1.35 

1.17 

1.31 

1.41

V

5.01 

4.55 

4.57 

3,82 

3.68

3.99

2.78 

2.48 

2.08 

2.14

4.95 

2.45 

2.11 

1.83 

1.60

2.05 

2.42 

2.07 

1.78 

1.74

2.00 

1.96 

2.85 

1.96 

1.95

1.56 

1.57 

1.75

1.54 

1.65 

1.91 

1,76

R

5.63 

5.36 

4.60 

3.96 

2.89

3.90 

3.01 

2.70 

2.82 

2.82

4.42 

3.38

2.20

3.43 

3.09

2.05

3.73

2.27

1,40

A

1,504 

1,431

1,227 

1,053

777

1,026

774 

686 

721 

721

1,159 

848 

597 

533 

549

888 

781 

574 

420 

386

483 

407 

998 

510 

589

418 

397 

392

335 

295 

329 

350

C.

5,300 

2 4, 140 

3,780 

4,460 

2 1, 520

2,570

n,ooo

513 

2600 

2698

813 

1,210 

398 

349 

604

2 3, 410 

3,400 

1,440 

319 

213

978 

311 

3,790

578

830

Stage

2.71 

2.29 

1.81 

1.27 

.62

1.45 

.82 

.31 

.27 

.40

1.96 

.475

-.26

.715 

.58

-.43

1.755

-.15

-6.89

B

28

28

28

28

28

28

28

28

28

28

28

28

28

28

29

28

29

28

27
703.- 

704 

709 

710 

876- 

1953 

May 11- 

May 11.. 

1956

8,180

8,160

11,800

13,600

4:05 p.m... _

6:18 p.m ......

11:22 p.m. .

164

264

176

71

3.59

3.64

4.93

5.64

3.63

9178

278

9178

283

264

4.90

4.82

6.38

7.28

4.38

6.00

6.10

6.65

6.61

6.00

4.88

4.79

7.20

4 S4

1,363

1,339

1,774

2,059

1,160 2 91 nno

4.99

2 Q1

2

O

i Estimated water temperature, based on air temperature observed at Omaha Airport. J Sediment-size data 

TABLE 3. Particle-size analyses of 10 samples of susp

[Methods of analysis: P, pipet; S, sieve; W, in distilled water; C, chemically disperse

Measurement 

No.

628 .-

632 

634.. _ - 

636- 

640 ....

644. ......

648 .

650 .

657 

876. 

Date

1952 

Mar. 26 -

Apr. 1 

Apr. 2. - 

Apr. 3. .. 

Apr. 18 . 

May 1 ---------

May 15 .

May 21 

June 30. ... .

1956 

June 27.. 

Time of sample

11:30 a.m. 

5:30 p.m... __ .

12:45 p.m. 

12:50 p.m. _ . 

11:20 a.m. ___

12:45 p.m. ....... 

1:15 p.m _ ---.

11:45 a.m. .

5:10 p.m 

1:20 a.m.. 

Water 

temperature 

(°F)

135 

46 

46 

46 

52

68 

66 

64 

79

71

Concentration 

(ppm)

3,430 

6,010 

5,010 

4,140 

1,520

1,000 

600 

698 

3,410

21,000

Percent of suspended sediment finer th

0.002

18 

22 

19 

17 

18

19 

23 

18 

50

38

0.004

18 

25 

24 

20 

18

19 

23 

25 

63

48

0.008

24 

29 

27 

24 

21

22 

26 

27 

71

60

0.016

28 

35 

32 

29 

24

27 

31 

33

77

1 72

0.031

49 

48 

45 

42 

36

42 

46 

42 

85

82

0

Pipet

61 

68 

68 

63 

52

60 

60 

61 

93

90

Estimated from air temperature observations at Omaha Airport.
B-12 STUDIES OF FLOW IN ALLUVIAL CHANNELS 

AVERAGE BED ELEVATIONS

The plot, figure 7, of the average bed elevation at time of each 

measurement at the upper and lower cableway sections shows shortterm variations of 0.6 foot from the mean and a long-term variation 

of about 1 foot during the period 1951-56. The short-term variations 

appear to occur randomly because there is no consistent correlation 

with discharge of other variable. These differences may be associated 

with movement of dunes or sand bars through the reach.

POINT BED ELEVATIONS

Although the average bed elevation at a cross section is relatively 

stable, scouring and filling is continuously occurring at any point in 

the cross section. A change of 5.5 feet at one point in the upper cross 

section occurred during a period of 4 hours. Point bed elevations 

observed for the upper and lower sections during the period May 21- 

24, 1951, are shown in figures 8 and 9. The discharge during this 

period varied from 6,630 to 7,750 cfs. The scour and fill pattern 

is much more pronounced at the upper section than at the lower 

section.

The maximum discharge of major floods is frequently determined 

by the slope-area method. The computations are based on field surveys of channel characteristics and high-water marks and on the Manning flow equation. The average elevation of the bed at the time of 

the peak is estimated after the flood is over by prodding the bed of the 

stream with a steel rod until a firm bottom is reached. The method 

assumes that the maximum scour at all points in the cross section 

occurred at the time of the peak discharge. It is evident from figures 

8 and 9 that the prodding method would give incorrect results on the 

Elkhorn reach. The average bed elevation remains virtually constant 

as the scour hole moves back and forth across the channel.

BED CONFIGURATION

On the basis of laboratory investigation, Simons (Simons and others, 1961) has described the regimes of bed configuration of sand channel 

streams as ripples, dunes, plane bed, standing waves, and antidunes. 

This sequence of bed configuration occurs with increasing discharge. 

As the dunes wash out and the sand is rearranged to form a plane bed, 

there is a marked decrease in resistance to flow. After this rearrangement of sand the resistance to flow remains in effect the same for higher 

regimes of flow.

Albertson (Albertson and others, 1957) defined the occurrence of 

different forms of bed roughness from laboratory data as a function of 

T^/co and ud/v; this relationship is shown in figure 10. Accordingly,
FLOW CHARACTERISTICS, ELKHORN RIVER, WATERLOO, NEBR. B-13

133d Ml ' 39V9 M 0 1 3 9 NOI1VA313
B-14 STUDIES OF FLOW IN ALLUVIAL CHANNELS

en oj co CD oj CD co 

 CM co co N co

fO to to to ro to to

in m 10 m in m m

i i i i i i i

OJ OJ ro <fr

OJ OJ OJ OJ OJ OJ OJ i i i i i i i

m m in m in m m

E E E E E E E

Q. Q. O Q. Q. O O

oo m « o 10 to

to to OJ in ro <fr in

ro co O m o CD

oo CD O M to

co co

m in m in m if>

co ^

133d 'NOI1 VA3 T3
FLOW CHARACTERISTICS, ELKHORN RIVER, WATERLOO, NEBR. B~15
B-16 STUDIES OF FLOW IN ALLUVIAL CHANNELS
FLOW CHARACTERISTICS, ELKHORN RIVER, WATERLOO, NEBR. B~17

these parameters were computed from the Elkhorn data and are plotted on figure 10. Their location indicates that most of the observed 

flows were in the plane-bed and antidune regimes. Although no direct 

observations of bed configuraton were obtained on the Elkhorn River, 

it is apparent from analysis of the depth-discharge relation and from 

the absence of waves on the water surface that most of the flows were 

in the dune regime and the remainder in the plane-bed regime. The 

laboratory relationship shown in figure 10 does not satisfactorily 

explain the regimes of bed configuration experienced on the Elkhorn.

Garde * defined the regimes of bed roughness in terms of RS/l.Q5d 

and V/T/gD. His relationship, which is shown in figure 11, predicts 

that all observed flows on the Elkhorn River would be in the rippledune regime; actually some of the flows were considered to be in the 

plane-bed regime.

There is no satisfactory method at present of predicting the occurrence of the various forms of bed roughness in alluvial channels.

DEPTH-DISCHARGE RELATIONSHIP

The effect of changes in bed configuration on the relation of mean 

depth to unit discharge at the lower cableway section is shown in 

figure 12. Two distinct relationships are apparent: the curve for 

ripples and dunes on the bed and the curve for a plane bed. The 

dunes begin to wash out at a unit discharge of about 10 cfs, and are 

completely washed out at a unit discharge of about 28 cfs. In this 

region of transition between dune and plane-bed regimes, part of 

the bed is plane and part is covered with dunes owing to the nonuniform distribution of depth and velocity in the reach. The depthdischarge relationship is very unstable in this region because changes 

in bed configuration can be triggered by changes in temperature, 

velocity, and depth of the water or by variation in the washload. 

Under certain conditions the discharge may double with no increase 

in mean depth.

Dawdy (1961) studied the relation of mean depth to unit discharge 

on 25 different streams. He found that equation 1 described the 

relationship for the plane bed through antidune regimes for wide 

channels.

(1)

This implies a constant value of Chezy's G in these regimes if the 

energy slope remains constant at a station. As shown in figure 12, 

equation 1 appears to fit the limited range of data for the plane-bed 

condition at the lower cableway section.

i Garde, E. J., 1959, Total sediment transport in alluvial channels: Doctor of philosophy dissertation, 

Colorado State University.
B-18 STUDIES OF FLOW IN ALLUVIAL CHANNELS

in

CO

0.02

[ value of RS/\ 

for c'oarse material

o.i 

FIGURE 11. Comparison of Elkhorn River bed configuration with Garde's 

relationship. Figures correspond to measurement number in table 2.
1 u

9

7

6

H

U p

U.

Z 4

I

t»

LJ 

O

Z

< 

1,1 2

2

1

X

6:

-*>/

^m \>,S

555" \ /

<^

<£r

/*673

^r $***, t>&£

 666

 6i

^648

"¥

^658

646

^

 65

 5

X

44_ 

Z

37s 

4

 5

 « 

<

/ 

S K

^59^ 6

64 6

$7 

fisin <

xf

S28 9

2 3 4 5 6 7 8 9 10 

DISCHARGE, IN CUBIC FEET PER SECOND, 

FIGURE 12, Relation between deDth and discharge at. inwar ra.hlRwa-0- aopttnn irtfm^o^ />/
B-20 STUDIES OF FLOW IN ALLUVIAL CHANNELS

The mean depth-unit discharge relation for the upper cableway 

section (fig. 13) does not show a consistent pattern. Several measurements are obviously in the plane-bed regime, but the scatter of 

the data obscures any relationship that may exist.

GAGE HEIGHT-DISCHARGE RELATIONSHIP

Sixty discharge measurements made at the upper cable are shown 

plotted against the corresponding gage height in figure 14. A rating 

curve has been drawn as an average of the points. Thirty-seven of 

the measurements are within 10 percent of the curve, 45 are within 

15 percent, and 52 are within 20 percent. The extreme measurements 

are 628, 640, and 664, which are +62, +45, and 36 percent, 

respectively, from the curve. This scatter is typical on stage-discharge 

plots of data secured at gaging stations on alluvial streams.

A series of 10 measurements were made during a flood period in 

May 1951. Discharge, mean depth, mean velocity, and slope are 

shown plotted against gage height in figures 15 to 18. The mean 

velocity is much lower, and depths are greater on the falling side of 

the hydrograph than on the rising side; slope, however, shows no 

consistent pattern. These compensating trends reduce the width of 

the stage-discharge loop. Data on suspended sediment were not 

collected during this flood.

ROUGHNESS COEFFICIENT

The roughness coefficients n and C used in the Manning and Chezy 

equations as computed for the Elkhorn reach are shown in table 2. 

The computations were based on the properties of the sections at the 

ends of the reach, and water-surface elevations observed on staff 

gages along the reach. Although the reach is fairly uniform, the 

area of the two sections was never exactly the same. If the area of 

the upper section was larger, the energy slope was computed by subtracting the change in velocity head from the fall of the water surface 

and dividing this quantity by the length of the reach; if the upper 

section was smaller, only 50 percent of the difference in velocity 

heads was added to the fall. The geometric mean value of AR* for 

the two sections was used in computing the Manning coefficient, and 

the average value of A and R for the two sections was used in computing the Chezy coefficient. The value of n varied from 0.0143 to 

0.030 and the value of C from 61 to 143.

The variation in roughness coefficient is caused primarily by the 

variation in configuration of the channel bed. At low discharges the 

entire bed is covered by dunes; as the discharge increases, the dunes 

are washed out in the part of the channel where the velocities are 

highest; and finally all the dunes are washed out and a plane-bed
10

9

7

6

r 

U 5 UJ s

u.

2 4

I

t»

UJ

Q

Z

^

_ 

2

1

656

*

/

£$£.

64*.

X

^«^_

657

P

658^

*Zr

S +646

s

.S

t*O7 

$ 

/ 599 

e'ef"'

^652

*6 '?4 .628

*640

1 2 3 4 5 6 7 8 9 10 

DISCHARGE, IN CUBIC FEET PER SECOND

FIGURE 13. Relation between depth and discharge at upper cabloway section. Figures co
2000 4000 6000 8000 

DISCHARGE, IN CUBIC FEET PER 

FIGURE 14. Relation between gage height and discharge at upper cableway section. Figu
H 

U 

LJ

5? 2 

LJ

I

LJ 

O

0

576^

1000 2000 3000 4000 5000 6000 

DISCHARGE, IN CUBIC FEET PE

FIGUBE 15. Relation between gage height and discharge at upper cableway section during flood of May 195
B-24 STUDIES OF FLOW IN ALLUVIAL CHANNELS

133d Nl 1 1H9I3H
FLOW CHARACTERISTICS, ELKHORN RIVER, WATERLOO, NEBR. B~25

tO (M 

133d Nl '1H9I3H 39V9

o 

o

LU 

C/>

QL 

O.

U 

O

< 

LU

o

W r~,

b-S

3
2.20 2.60

WATER SURFACE SLOPE, IN FEET P

FIGURE 18. Relation between gage height and slope of water surface during flood of May 1951. F
FLOW CHARACTERISTICS, ELKHORN RIVER, WATERLOO, NEBR. B~27

condition exists throughout the reach. It is apparent that the roughness coefficient is a function of some parameter which in turn determines the bed configuration. The effect of mean velocity on the bed 

configuration and on the roughness coefficient is indicated by the 

S-shape of the curve in figure 19. The dunes begin to wash out at a 

velocity of 2.5 feet per second and are probably completely washed 

out at a velocity of 5 or 6 feet per second.

EINSTEIN'S PARAMETER

Einstein (Einstein and Barbarossa, 1952) suggested that the total 

shear force RS could be divided into two parts, R'S and R"S, when 

R=R'-\-R". The force R'S is dissipated by sand-grain resistance 

according to Nikuradse's definition, and R"S is dissipated by the 

dunes. He correlated resistance of the dunes V/V"f with a parameter 

X' previously used to describe sediment transport.

The data for the Elkhorn reach are compared in figure 20 with the 

relation which Einstein developed on basis of data for eight streams. 

Although there is considerable scatter on the plot, the general trend 

is evident.

The values of Chezy's C for the Elkhorn reach are shown plotted 

against X' in figure 21. There is a tendency for C to remain constant 

in the dune and plane-bed regimes and thus the curve tends to flatten 

out at the extremities. The variation of C in the transition from 

dunes to plane bed appears to be definitely related to Xf, but additional parameters must be used to define completely the roughness 

coefficient. Other investigators have noted the effect of temperature 

on the rate of sediment transport and bed configuration. Dunes are 

less likely to persist or form as the temperature of the water decreases. 

The effect of temperature is apparent in figure 21; the temperature of 

the water for measurement 628, 638, 640, and 642 was about 20°F 

lower than for measurements 574 and 581.

The amount of material in suspension of bed origin is not considered 

to be an independent variable in determining the roughness coefficient, 

but washload must be considered. Washload is composed of grains 

which are smaller in size than those found in appreciable quantities 

in the bed material. For example, only about 1 percent of the 

grains of bed material in the Elkhorn reach are smaller than 0.06 mm, 

but 63 percent of the grains in one sample of suspended sediment 

was smaller than 0.004 mm (table 3).
B-28 STUDIES OF FLOW IN ALLUVIAL CHANNELS

I 2 3 456789 10 

MEAN VELOCITY, IN FEET PER SECOND

FIGUEE 19. Relation-between Chezy's C and mean velocity for measurements made at both cableways. 

Figures correspond to measurement numbers in table 2.
FLOW CHARACTERISTICS, ELKHORN RIVER, WATERLOO, NEBR. B~29
300

40

0.2

P fi'S

FIGUBB 21. Relation between Chezy's C and Einstein's X', Elkhorn River data. Figures c
FLOW CHARACTERISTICS, ELKHORN RIVER, WATERLOO, NEBR. B~31

Simons (Simons and others, 1961) found that washload has no 

appreciable effect on the roughness coefficient if the concentration 

was less than 20,000 ppm (parts per million). The maximum concentration of sizes less than 0.06 mm observed on the Elkhorn was 

only 3,000 ppm, and thus the scatter on figure 21 cannot be attributed 

to this variable.

SIMONS' EQUATION

The equation which Simons (Simons and others, 1961) developed for 

Chezy's C from laboratory tests of one sand size is 

L

(3)

A comparison of C computed from this equation with Vi/RS for 

the Elkhorn data is shown in figure 22. It is apparent that equation 

3 cannot be used to predict the value of the roughness coefficient for 

the Elkhorn reach.

LIU'S EQUATION

Liu and Hwang (1959) determined a general equation for sandbed 

streams on the basis of all available laboratory data as

V=CaR*S«. (4)

The values of Ca, x, and y are related to the median size of bed material. A comparison of V computed from equation 4 with the observed velocity for measurements on the Elkhorn reach which were 

definitely in the dune or plane-bed regime is shown in figure 23. The 

velocities predicted from Liu's equation are about 40 percent too high 

in the dune regime and 40 percent too low in the plane-bed regime.

CONCLUSIONS

The configuration of the bed of the Elkhorn River changes from 

dunes to plane bed with increasing discharge. This causes a discontinuity in the depth-discharge relationship and a large variation 

in the roughness coefficient. The change from dunes to plane bed 

does not occur simultaneously at all points in the reach and thus a 

multiple roughness condition exists in the transition region.

Scour and fill occur simultaneously at different points in the cross 

section, but the average bed elevation remains relatively constant. 

The true cross section at the time of the peak discharge cannot be 

determined by prodding the channel bed after the flood event.

The roughness coefficient for the Elkhorn reach cannot be predicted 

from the relationships published by Simons, Liu, or Einstein. The
150

140

50

C = JJ X 0.90 X 10

FIGURE 22, Comparison of relationship between Chezy's C and Simons' C, Elkhorn River data. Fi
VELOCITY OBSERVED 

SECOND PER FEET IN 

'soLLSiaajLOvavHO .NaoirsrTa: 'aaAia 'oonaaj/vAi £8-9
B-34 STUDIES OF FLOW IN ALLUVIAL CHANNELS

configuration of the channel bed does not agree with the classification 

proposed by Albertson or Garde.

REFERENCES CITED

Albertson, M. L., Simons, D. B., and Richardson, E. V., 1957, Discussion of,

Mechanics of ripple formation, by Hsin-Kuan Liu: Jour. Hydraulics Div.,

Am. Soc. Civil Engineers, v. 84, No. HY-1, Pt. 1. 

Dawdy, D. R., 1961, Depth-discharge relations of alluvial streams discontinuous

rating curves: U.S. Geol. Survey Water-Supply Paper 1498-C. 

Einstein, H. A., and Barbarossa, N. L., 1952, River channel roughness: Am.

Soc. Civil Engineers Trans., v. 117, p. 1121-1132. 

Liu, Hsin-Kuan, and Hwang, Shoi-Yan, 1959, A discharge formula for flow in

straight alluvial channels: Am. Soc. Civil Engineers Jour. Hydraulics Div.,

v. 85, No. HY-11. 

Simons, D. B., Richardson, E. V., and Albertson, M. L., 1961, Flume studies

using median sand (0.45 mm): U.S. Geol. Survey Water-Supply Paper

1498-A.

o