Discharge Characteristics

of Embankment-Shaped Weirs

GEOLOGICAL SURVEY WATER-SUPPLY PAPER 1617-A

Prepared in cooperation with the 

Georgia Institute of Technology
Discharge Characteristics

of Embankment-Shaped Weirs

By CARL E. KINDSVATER

STUDIES OF FLOW OF WATER OVER WEIRS AND DAMS

GEOLOGICAL SURVEY WATER-SUPPLY PAPER 1617-A

Prepared in cooperation with the 

Georgia Institute of Technology

UNITED STATES GOVERNMENT PRINTING OFFICE, WASHINGTON : 1964
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, D.C., 20402
CONTENTS

Page

Abstract_ _______________________________________________________ Al

Introduction. _____________________________________________________ 1

Purpose._ ____________________________________________________ 1

History of the investigation--__________________________________ 2

Acknowledgments _____________________________________________ 3

Review of the literature _______________________________________ 4

Description and analysis of the problem._____________________________ 4

Nature of the problem__ ______________________________________ 4

Flow patterns.-_______________________________________________ 5

Nomenclature.__ ______________________________________________ 8

Dimensional analysis.-___-__-___________-______---_______-__--- 9

Free-flow discharge equations ___________________________________ 14

Effect of boundary resistance ___________________________________ 15

Submerged flow_________-___________-________-__-_____-____-_- 18

Laboratory investigation ___________________________________________ 19

Experimental equipment_______________________________________ 19

General arrangement________________________________.______ 19

Flume____________________________________________________ 21

Embankment models.-_____________________________________ 21

Discharge measurements___________________________________ 22

Piezometric-head and profile measurements ___________________ 24

Velocity measurements.-------------------.---------------- 28

Scope of embankment design variations__________________________ 32

Significance of roughness variations-_____________________________ 33

Influence of embankment form and roughness. ______._._---_--_- --._---_ 34

Objective._ ___________________________________________________ 34

Analysis of results_______________________--________-__-_------_ 35

Coefficient of discharge for free flow________-_________-__--- 35

Coefficient of discharge for submerged flow.__________-____--- 39

Incipient submergence_____-________-____________-_-_------ 45

Free-flow transition range_____-__-___--___________----_^-_- 51

Depth at crown line for free flow_ __________________________ 54

Summary and evaluation.______________________________________ 58

Influence of the boundary layer on free-flow characteristics------------ 62

Objective.. _-__________________________________________ 62

Definition of 5 e and X--___________________-_-__--_____-__------ 63

Boundary layers in accelerated motion___________________________ 64

Boundary-layer measurements_______________-__-___------------ 66

Purpose and scope._____________-__-__________-_-___--__-- 66

Summary of results-_________ ___________________-----__-- 67

Computation of C based on boundary-layer measurements. _________ 74

Modified discharge equation. _________._________-______-_---__-- 76

Influence of the sidewall boundary layer._____________-__-_----_-- 77

General solution for -<,________________________________-_----_-- 79

Computation of C for the prototype __-_______-____-_--_-__------ 86

Conclusions._ _____________________________________________________ 87

References cited---_-______-_________________________-__-_--------- 88

Experimental data____________________-________-_______-__---_--_-- 89

ni
IV CONTENTS

ILLUSTRATIONS

Page 

FIGURE 1. Typical water-surface profiles in 1:6 scale model___________ A7

2. Three significant flow patterns in l:6-scale model._________ 8

3. Changing flow pattern with rising tail water _ _______________ 10

4. Changing flow pattern with falling tailwater______________ 12

5. Principal variables needed to describe flow over an embankment. _______________________________________________ 13

6. Partially established boundary-layer flow__________________ 16

7. Arrangement of experimental equipment __________________ 20

8. View of l:9-scale model embankment in test flume_________ 21

9. Basic embankment design______________________-----_-_- 23

10. Special design variations (models K, L, and M) ____________ 24

11. Model AC with birdshot roughness--___---------------_-- 25

12. Weighing-tank system for discharge measurements________ 26

13. Point gage and pitot-static tube._________________________ 27

14. Manometer for the pitot-static tube- _________--__----_-_- 29

15. Stagnation tube for boundary-layer velocity measurements-_ 30

16. Manometer for the stagnation tube_____________________ 31

17. Coefficient of discharge for free flow; basic design, model A__ 35

18. Coefficient of discharge for free flow; smooth surfaced models. 37

19. Coefficient of discharge for free flow, rough-surfaced models.- 38

20. Coefficient of discharge for submerged flow; influence of h/P,

with t/h as the submergence ratio___-__--_-----_-------- 40

21. Coefficient of discharge for submerged flow; influence of h/P,

with t/Hi as the submergence ratio_------_----_---_---_- 41

22. Coefficient of discharge for submerged flow; basic design,

model A-_________________-____-__-_-_--._----------- 42

23. Coefficient of discharge for submerged flow; influence of pavement cross slope, Sv ---------------------------------- 43

24. Coefficient of discharge for submerged flow; influence of

shoulder slope, St ----------------------------------- 44

25. Coefficient of discharge for submerged flow; influence of

special design variations------------------------------- 45

26. Coefficient of discharge for submerged flow; influence of

screen-wire roughness on all surfaces____-_________-__--- 46

27. Coefficient of discharge for submerged flow; influence of birdshot roughness on shoulders and embankment slopes--___- 46

28. Coefficient of discharge for submerged flow; influence of birdshot roughness on all surfaces_______________-___---_--- 47

29. Incipient submergence and free-flow transition range; influence of h/P, with t/h as the submergence ratio--------- 48

30. Incipient submergence and free-flow transition range; influence

of h/P, with tjHi as the submergence ratio-_____-____---- 49

31. Incipient submergence and free-flow transition range; basic

design, model A_______________-______-__-_----------- 50

32. Incipient submergence and free-flow transition range; influence

of pavement cross slope, Sv ---------------------------- 50

33. Incipient submergence and free-flow transition range; influence

of shoulder slope, Ss------------------------------------ 51

34. Incipient submergence and free-flow transition range; influence

of special design variations.____________-_---__------- 52
CONTENTS V

Page 

FIGTJBE 35. Incipient submergence and free-flow transition range; influence

of screen-wire roughness on all surfaces-_________________ A53

36. Incipient submergence and free-flow transition range; influence

of birdshot roughness on shoulder and embankment slopes.- 54

37. Incipient submergence and free-flow transition range; influence

of birdshot roughness on all surfaces___-________________ 55

38. Depth at crown line for free flow; basic design, model A_ _____ 55

39. Depth at crown line for free flow; smooth-surfaced models.__ 56

40. Depth at crown line for free flow; rough-surfaced models----- 57

41. Summary of the coefficient of discharge for free flow________ 59

42. Summary of the coefficient of discharge for submerged flow__ 60

43. Summary of incipient submergence and free-flow transition

range--_-________-_______________-___________________ 61

44. Velocity distribution in boundary layer on upstream side of

roadway; basic design, typical discharge _________________ 68

45. Velocity distribution in boundary layer on upstream side of

roadway; rounded shoulder, typical discharge__-_-______- 69

46. Boundary-layer velocity distribution near crown line (station

1.65), models.A-l and AA-1_____________________- 70

47. Boundary-layer velocity distribution near crown line (station

1.65), models K-l and KA_______________________ 71

.48. Boundary layer near crown line (station 1.65)______________ 72

49. Boundary layer on upstream side of roadway for a typical

discharge.___________________________________________ 73

50. Values of (5 a X) from boundary-layer velocity measurements. 75

51. Values of C computed from equation 20 and figure 50_______ 76

52. Values of 5 Q from boundary-layer velocity measurements---- 78

53. Values of C computed from equation 47 and figure 52_______ 78

54. Velocity outside the boundary layer, model of basic design. _ 81

55. Velocity outside the boundary layer, model with rounded

upstream shoulder.___________________________________ 82

56. Values of 5 Q computed from equation 52 compared with

experimental values from figure 49__-_________-_-------- 85

57. Computed prototype-values and measured model values of C,

smooth-surfaced embankments.-,----------------------- 86

TABLES

Page 

TABLE 1. Summary of designs tested, l:9-scale models.------------- A33

2. Scope of boundary-layer tests made by Davidian___________ 67

3. Summary of data for discharge characteristics--____-_-___-_ 89

4. Summary of data for boundary-layer velocity distribution,--. 106
STUDIES OF FLOW OF WATER OVER DAMS AND WEIRS

DISCHARGE CHARACTERISTICS OF EMBANKMENT-SHAPED WEIRS

By CARL.E. KINDSVATER*

ABSTRACT

, An embankment-shaped weir is an embankment overtopped by flood waters. 

Among the engineering problems frequently resulting from this occurrence is 

the need to compute the peak discharge from postflood iield observations. The 

research described in this, report was concerned with the theoretical and experimental-bases for the computation procedure.

The research had two main objectives. One was to determine the relationship 

. between embankment form and roughness and -.some of the more important

-discharge- characteristics. - The second was to define, theoretically and experi-

-mentally, the.relationship between free-flow discharge and the boundary layer 

on the-roadway. The first-objective was accomplished with the experimental 

determination of coefficients of discharge and other significant flow characteristics for a variety of boundary and flow conditions. The second objective was

-accomplished with the development and experimental verification of a discharge 

equation which involved the boundary-layer displacement thickness. This phase 

of the research included "a general investigation of boundary-layer-growth on 

the roadway.

 It-is-concluded that both free- and submerged-flow "^discharge-are virtually 

independent of the influence of embankment shape-and relative height. The 

influence of boundary resistance is appreciable only for smaller heads. The 

most practical solution for discharge is one which is based on the simple weir 

equation and experimentally determined coefficients. A completely analytical 

equation of discharge is impractical.

- The-report contains the results of 936 experiments on the discharge characteristics of 17 different models, plus 106 boundary-layer velocity traverses 

on 4 different-models. The data are summarized in-both graphical and tabular 

form.

INTRODUCTION

PURPOSE

The subject of this report is the highway embankment, which becomes a weir when it is overtopped by flood waters. Several problems 

which result from this occurrence are of interest to hydraulic engineers as well .as highway engineers. One of these is the problem of 

destructive erosion. Another is the - backwater which results from 

the obstruction of the flood channel. Still another is the problem- of 

determining the magnitude of the flood discharge. The last-named

-problem is the one which especially-concerns the Geological Survey.

- *Regents Professor of Civil Engineering, Georgia Institute of Technology, Atlanta, Ga.; consultant to 

the U.S. Geological Survey.

Al
A2 FLOW OF WATER OVER DAMS AND WEIRS

The Geological Survey's interest in the hydraulics of embankments 

is related to one of the so-called indirect methods of discharge determination. Indirect methods involve the use of computation procedures in lieu of discharge or velocity-area measurements. One such 

method is based on the computation of the discharge capacity of 

spillways and weirs. The highway embankment belongs in this category because it is 'a form of broad-crested weir.

The common ingredients in all indirect methods of discharge determination are postflood field measurements and observations. Thus, 

the accuracy of the discharge determination depends on the accuracy 

and sufficiency of the field data as well as the computation procedure. 

As it pertains to the work of the Geological Survey, this investigation 

was concerned primarily with the principles involved in the computation procedure. The primary objective was an understanding of 

the mechanics of the flow over some typical highway embankments. 

A natural consequence of the investigation was the development of 

equations and coefficients which can be used to compute the discharge 

for a wide range of embankment forms and flow conditions.

HISTORY OF THE INVESTIGATION

The first of a series of related research projects on the hydraulics 

of embankments was begun at the Georgia Institute of Technology 

in 1947. On that occasion the Institute cooperated with the Georgia 

State Highway Department in a brief, exploratory study of a 1:6- 

scale model of a typical two-lane highway embankment. Laboratory 

tests were performed by R. L. Chapman, >a State Highway Department engineer. From the results of the investigation, preliminary 

conclusions were drawn regarding the nature and scope of the problem.

Tests on a rebuilt version of the 1:6-scale model were made in 1948 

by C. J. Chi and H. R. Henry, graduate students in the School of 

Civil Engineering. A major contribution of this investigation was 

a comprehensive record of water-surface profiles, velocity measurements, and photographs required to describe the external flow 

characteristics.

In 1949, tests on a 1:12-scale model were made by H. Y. Lu, another 

graduate student. The purpose of his investigation was to explore the 

scale effect related to the formation of the boundary layer on the 

surface of the model. The kind of data recorded was similar to that 

obtained from the previous tests on a 1: 6-scale models. The investigation was terminated before completion.

After a lapse of several years, embankment research was reactivated 

in 1954 by Sigurdsson.1 In addition to summarizing and reanalyzing

1 Sigurdsson, Gunnar, 1956, Discharge characteristics of an embankment shaped weir: Georgia Inst. 

Technology, Master's degree thesis, 83 p., 38 figs.
DISCHARGE OVER EMBANKMENT-SHAPED WEIRS A3

the results of the previous investigations, Sigurdsson made additional 

tests on a 1:9-scale model. The purpose of his tests was to explore more 

critically the boundary-layer influence on free-flow discharge 

characteristics.

Subsequently, from 1956 to 1958, two more laboratory studies of the 

1:9-scale model were made. One of these, by Davidian,2 was especially 

concerned with the relationship between the free-flow coefficient of 

discharge and the boundary layer on the roadway. The other, by 

Prawel,3 , was principally concerned with the influence of changes in 

boundary form on significant flow characteristics. Finally, in 1959, 

some tests on the influence of surface roughness and some verification 

tests were made by W. W. Emmett.

For most of the work described above, one embankment form was 

used as the basis of design for all the models. In one of the recent 

investigations, the basic form was systematically varied in order to 

determine the separate influence of some of the more critical geometric 

characteristics. For most of the tests, the model surfaces were essentially smooth. For some tests, however, the model surfaces were 

artificially roughened. The test procedure, scope and kind of recorded 

data, and experimental techniques varied with the objectives of the 

individual investigators.

ACKNOWLEDGMENTS

The laboratory investigations were performed in the Hydraulics 

Laboratory, School of Civil Engineering, Georgia Institute of Technology, Atlanta, Ga. The experimental data which are used in this 

report were obtained primarily from the investigations during 1948 

and 1956-59. All of the data accumulated since 1947, whether used 

herein or not, contributed'to the experience and understanding upon 

which the conclusions are based.

The principal investigators were Chapman, Chi, Henry, Lu, Sigurdsson, Prawel, Davidian, and Emmett. Chapman's work on the 

initial^ exploratory study was contributed by the Georgia State Highway Department. Chi, Henry, and Lu were graduate research assistants employed by the Institute. Sigurdsson's work was supported by 

the American Society of Civil Engineers through its J. Waldo Smith 

Hydraulic Fellowship. Prawel's research was partly supported by 

the Geological Survey. Emmett's work was supported by the Geological Survey and the Institute. For the purpose of preparing this report, 

the author was employed as a consultant by the Geological Survey.

8 Davidian, Jacob, 1959, Influence of the boundary layer on embankment-shaped weirs: Georgia Inst. 

Technology, Master's degree thesis, 97 p., 38 figs.

3 Prawel, S. P., Jr., 1959, Discharge characteristics of an embankment-shaped weir: Georgia Inst. Technology, Master's degree thesis, 58 p., 38 figs.

690-195 O - 64 - 2
A4 FLOW OF WATER OVER DAMS AND WEIRS

Homer Bates, Laboratory Technician in the School of Civil Engineering, constructed special instruments and assisted in the construction of the models. A complete photographic record of the 1948 tests 

was made by the author with the assistance of Chi. Most of the data 

used in the report were computed from original laboratory data by 

Emmett, who also assisted in the preparation of illustrations. John 

Shen, Geological Survey, and E. R. Holley, Jr., graduate student, 

assisted with the computations. All the work was done under the 

direction of the author.

Three Master's degree theses (Sigurdsson, 1956; Davidian, 1959; 

Prawel, 1959) have been based on parts of the investigation. A 

summary of certain results was contained in a previous publication 

(Kindsvater, 1957).

BEVIEW OF THE LITEBATUBE

A paper by Yarnell and Nagler (1930) was the only research publication on embankment hydraulics available to the investigators. The 

research described in that paper was concerned primarily with railroad 

embankments. The basic embankment was flattopped, with a short, 

flat berm on both the upstream and downstream sides. The embankment slopes were covered with grouted gravel, and the top surface 

contained timber ties embedded in srravel ballast. The test data cono

sisted of information needed to compute discharge coefficients for 

both free and submerged flow. Water-surface levels were measured 

with staff gages, and the discharge was measured with a full-width 

thin-plate weir. The results of a few tests made without rails on the 

ties have been used widely to describe the flow over highway 

embankments.

Because the highway embankment is classified as a form of broadcrested weir, the literature in this field is pertinent. A compilation 

and analysis of the most important information on broad-crested weirs 

is contained in a report by Tracy (1957). Most of the published work 

on the subject is concerned with the experimental determination of 

the discharge coefficient for smooth, level weirs. Very little is known 

about the influence of the boundary layer on the crest, separation at 

the upstream edge of the crest, boundary roughness, or variations in 

boundary form.

DESCRIPTION AND ANALYSIS OF THE PROBLEM 

NATUBE OF THE PBOBLEM

Theoretical analyses of the embankment-shaped weir are complicated by a combination of effects related to boundary resistance and 

boundary form. Experimental studies are complicated by the occurrence of several significantly different flow patterns. Both analysis
DISCHARGE OVER EMBANKMENT-SHAPED WEIRS A5

and experiment are complicated by characteristic difficulties associated 

with free-surface phenomena.

Typical flow patterns ior an embankment-shaped weir involve 

boundary-layer growth under conditions of acceleration and separation, nonhydrostatic pressure distribution due to curvilinearity, and 

an unstable transition between a boundary flow and a separation flow 

on the downstream slope. It is evident that a general analytical solution for discharge is impossible. Nevertheless,.existing theory can be 

used to .explain some, of the different flow regimes. Experiment, 

guided by dimensional reasoning, must be depended upon to fix the 

limits of the flow regimes and evaluate the coefficients needed for practical solutions.

FLOW PATTERNS

An important contribution of the 1948 tests was a classification of 

the different flow patterns which characterize the flow of water over 

highway embankments. Doubtless the most significant classification 

is that which distinguishes between free and submerged flow. For 

the low-tailwater condition known as free flow, critical-flow control 

occurs on the roadway and the discharge is determined by the upstream head. At higher tailwater levels, when the depth of flow over 

the roadway is everywhere greater than the critical depth, the discharge is controlled by the capacity of the tailwater channel as well as 

the head. Under conditions of tailwater control, the flow is said to be 

submerged. With a rising tailwater level, the change from free flow 

to submerged flow occurs rather abruptly. The flow pattern antecedent to the change is described as incipient submergence.

Free flow is subclassified into plunging flow and surface flow. 

Plunging flow occurs when the jet plunges under the tailwater surface, 

producing a submerged hydraulic jump on the downstream slope. 

Surface flow occurs when the jet separates from the roadway surface 

at the downstream shoulder and "rides" over the tailwater surface. 

Whereas free flow can be either a plunging or a surface flow, submerged flow is always a surface flow.

The free-flow transition range is the range of tailwater levels within 

which a given discharge can produce either a plunging flow or a surface flow, depending on the antecedent conditions. Thus, if the tailwater is initially low and the flow plunging, this pattern persists as the 

tailwater level rises until it reaches the upper limit of the transition 

range, whereupon the plunging flow changes abruptly to a surface 

flow. However, if the tailwater is initially high and the flow is a surface flow, this pattern persists as the tailwater drops until it reaches 

the lower limit of the transition range, whereupon the flow pattern 

changes abruptly to plunging flow. The stability or persistence of
A6 FLOW OF WATER OVER DAMS AND WEIES

the flow patterns within the transition range is related to the inertia 

of the large, horizontal-axis rollers which occur on the downstream 

side of the embankment.

The tailwater-level limits of the transition range were recorded 

for all the models investigated. In addition to their general significance in the description of the characteristic flow pattern, the transition-range data are also significant in determining the safety of the 

structure against destructive erosion. This conclusion is based on the 

observation that surface flows are doubtless less erosive than plunging 

flows. Although the safety of flooded highway embankments was not 

a primary objective of the investigation, this information is believed 

to provide an important contribution to erosion prevention and 

control.

Figure 1 shows water-surface profiles for typical tests on the 1:6- 

scale model. Attention is called to the fact that the profiles are plotted 

to a distorted scale. Thus, vertical dimensions are shown 2.5 times 

larger than equivalent horizontal dimensions. For 3 different discharges (3.01, 14.7, and 25.0 cfs per ft in the prototype), the profiles 

illustrate the major flow-pattern classifications described above. Only 

1 of the 2 possible flow patterns is shown at each of the free-flow 

transition limits. Profile a is a typical example of submerged flow, 

whereas e is an example of free, plunging flow at a tailwater level considerably below the lower limit of the transition range. Characteristic 

standing-wave patterns mark incipient submergence and the upper 

limit of the transition range. The theoretical value of the critical 

depth and its intercept on each of the free-flow profiles is shown on the 

figure. It is apparent that the depth at the crown line is very nearly 

equal to the theoretical critical depth for all discharges.

Figure 2 shows photographs of the three intermediate classifications 

of flow illustrated on figure 1. The discharge shown in figure 2 is 

20.4 cfs per ft in the prototype.

Figures 3 and 4 show closeup photographs of the flow in the vicinity 

of the downstream shoulder for the conditions illustrated in figure Iff. 

The discharge is constant at 14.T cfs per ft (prototype). The photographs shown in figure 3 represent a rising-tailwater sequence, beginning with a free, plunging flow. Figure 4 shows a falling-tailwater 

sequence, beginning with submerged flow. In all of the photogaphs, 

the flow is from left to right. The streaks shown in some of the 

photographs were made with an injection of potassium permanganate 

solution.

Attention is called to the similarity between figures 3B and 4Z>, 3^7 

and 4^7, and 3Z> and 4ff. It is observed that the flow patterns which
DISCHARGE OVER EMBANKMENT-SHAPED WEIRS A7

-2

-4

x a Submerged flow

Db Incipient submergence

A c Upper limit of transition 

range (surface flow)

+d Lower limit of transition 

range (plunging flow) 

Headwater level

Oe Free flow (plunging)

Note: Data taken from tests on 

l:6-scale model with P=10.5 

feet. All discharges and dimensions shown as in prototype

Tailwater level

I

es.

Headwater level Tailwater level

e-SL

Headwater level Tailwater level

-40 -30 -20 -10 10 20 30 40

Distance, in feet 

FIOUEE 1. Typical water-surface profiles in l:6-scale model.
A8 FLOW OF WATER OVER DAMS AND WEIRS

A. Free, surface flow at the upper limit of the transition range

B. Surface flow, incipient submergence

C. Free, plunging flow at the lower limit of the transition range

FIGURE 2. Three significant flow patterns in the l:6-scale model. The discharge is equivalent to 20.4 cfs

per ft in the prototype.

appear to be almost identical in surface and streak-line configuration 

actually correspond to a difference in tailwater elevation of approximately 1.5 ft in the prototype. This difference is the difference in 

tailwater levels at the upper and lower limits of the free-flow transition range for this discharge.

NOMENCLATURE

The principal variables needed to describe the flow of water over a 

highway embankment are shown in figure 5. The selection of the 

variables is based on the following assumptions : The cross section is 

symmetrical about the vertical centerline; the embankment surface, 

shoulder surface, and each half of the pavement surface are planes; 

the crown line is horizontal, straight, and perpendicular to the flow;
DISCHARGE OVER .EMBANKMENT-SHAPED WEIRS A9

the channel bottoms upstream and downstream from the embankment 

are -horizontal, plane, smooth surfaces, and both are at the same elevation. Under these conditions, the variables needed to describe the 

embankment are the total height of the embankment (P) , the width 

of the shoulder (L8 ) , the width of the pavement (Lp) , the total width 

of the roadway, including the pavement and two shoulders (Z), the 

embankment slope ($<>), the shoulder slope ($«), and the pavement 

cross slope (Sp ) .

The variables needed to describe the one-dimensional flow characteristics are the discharge per foot of length of embankment (q) , the 

average velocity in a channel section upstream from the embankment 

(Fi), the piezometric head of the upstream water surface measured 

with respect to the level of the crown line (A), the average total 

energy head referred to the crown level (#1), the depth of flow at 

the crown line (2/0), and the piezometric head of the downstream 

water surface referred to the crown level (t).

In addition to the variables: shown on figure 5, quantities needed to 

describe the flow over an embankment include the absolute roughness 

of the boundary surfaces (A?), the specific weight of the fluid (y), 

the density of the fluid (p) , and the dynamic viscosity of the fluid (//,) .

DIMENSIONAL ANALYSIS

Because some of the quantities defined above are not independent, 

they can be excluded from a general expression of the discharge function. For example, Z, F1} and H± can be excluded because their meanings can be described in terms of two or more of the other variables 

listed. An expression which includes the -minimum number of independent variables required to describe the one-dimensional discharge 

characteristics is

/(P, Ls, Lp , Se , Ss, Sp, q, h, t, k, y, p, /x)=0. (1) 

From- this array, 10 -independent ratios can be formed, as follows : 

Ls ~ k h t q

The next to the last ratio in equation 2 is equivalent to a coefficient 

of discharge, <7i, and the last ratio is the Reynolds number, R. Select- 

.ing the coefficient of discharge as the dependent ratio,

Cl=f (pM7 S" Ss' Sp' h' Tv I' R) (3)

In the experimental investigation, LS/LP and 'Se were constants. 

Consequently, for the purposes of this report, both ratios can be 

omitted from the functional relation for the coefficient of discharge.
A10

A. Free, plunging flow

B. Free, plunging flow at the upper limit of the transition range

C. Upper limit of the transition range, changing from plunging to surface flow
DISCHARGE OVER EMBANKMENT-SHAPED WEIRS All

D. Free, surface flow at the upper limit of the transition range

E. Free, surface flow between the upper limit and incipient submergence

F. Surface flow, incipient submergence

690-195 O - 64 - 3
A12 FLOW OF WATER OVER DAMS AND WEIRS

A. Submerged, surface flow

B. Free, surface flow at the lower limit of the transition range

C. Lower limit of the transition range, changing from surface flow to plunging flow

D. Free, plunging flow at the lower limit of the transition range

FIGURE 4. Changing flow pattern with falling tailwater. The discharge shown here in the l:6-scale model 

is equivalent to 14.7 cfs per ft in the prototype.
DISCHARGE OVER EMBANKMENT-SHAPED WEIRS A13
A14 FLOW OF WATER OVER DAMS AND WEIRS

Furthermore, for convenience, L can be substituted for Lp in the 

h/Lp ratio. Thus, the scope of the investigation is represented by 

the expression

(4)

FREE-FLOW DISCHARGE EQUATIONS

For free flow, discharge control occurs at a critical-flow section on 

the roadway. This means that a unique relationship exists between 

the discharge and the head on the embankment. Consequently, a 

one-dimensional discharge equation can be derived from a simple 

energy analysis.

The discharge equation for a so-called ideal fluid is based on the assumption of potential flow (no boundary resistance, no energy losses) , 

with uniform velocity distribution and hydrostatic pressure distribution in the approach channel and on the roadway. From the onedimensional energy and continuity equations for the reach between a 

section at the crown line and a section in the channel immediately 

upstream from the embankment, the free discharge of an ideal fluid 

is

_y0 ), (5)

in which y0 is the depth of flow at the crown line. From the additional assumption that y0 is equal to the critical depth for the discharge <?,, and from the fact that the critical depth is theoretically 

equal to two-thirds the total head,

Because the conditions assumed in the derivation of equation 6 are 

never actually satisfied, the discharge equation for an ideal flow must 

be adjusted to the flow of a real fluid. Thus, in the following equation,

J f0Hi*i (7)

q is the true discharge of a real fluid, and C3 is an experimentally determined coefficient which accounts for the inaccuracies inherent in 

equation 6.

It is customary in American engineering practice to use a coefficient 

O which combines the coefficient of discharge and the constants, in-
DISCHARGE OVER EMBANKMENT-SHAPED WEIRS A15

eluding <7, in equation 7. Thus, the equation of discharge in its simplest form is

q=CHj, (8)

in which C has the dimensions of V<7- The obvious practical advantages of the simple equation have been weighed against the fact that 

C is a dimensional qauntity. The decision to use equation 8 as the 

basic equation of discharge for this investigation is a concession to 

the convenience of the practicing American engineer. Those who insist that the coefficient of discharge must be nondimensional will be 

compelled to divide <7 by a constant involving -\/g.

The independent variables which govern C are the same as those 

which govern Cz in the general functional expression for the coefficient of discharge. Therefore, from equation 4,

For free flow, t/h is significant only as it defines the condition of 

incipient submergence.

From equation 6, a simple equation for the discharge of an ideal 

fluid is

««=f -^ (32.2) #^=3.09 Hj, (10)

and the coefficient 3.09 can be described as the ideal value of C in 

equation 8.

EFFECT OF BOUNDARY RESISTANCE

Two equations for free-flow discharge have been described. One, 

illustrated by equation 6, is sometimes called a theoretical or ideal 

equation, because certain real-fluid characteristics were ignored in its 

derivation. The second, illustrated by equation 7, is a simple modification of equation 6 which depends on the experimental evaluation of 

a coefficient of discharge for real-fluid flows. To the second can be 

added the result of the dimensional analysis, which provides a means 

of correlating the variables which govern the coefficient of discharge. 

Still lacking is a completely analytical equation of discharge for real 

fluids.

An analytical equation must contain terms capable of describing 

the influence of boundary characteristics and real-fluid characteristics 

which were ignored in the development of equation 6. Of these, the 

most important for free-flow discharge conditions are the effects of
A16 FLOW OF WATER OVER DAMS AND WEIRS

boundary resistance. These effects are incorporated symbolically in 

an equation which begins, as did equation 6, with the one-dimensional 

equations of energy and continuity.

FIGURE 6. Partially established boundary-layer flow.

In figure 6 the flow at a typical section on the roadway is represented as a partially established boundary-layer flow. In the figure, 

V is the average velocity in the section; U is the uniform velocity in 

the potential flow outside the boundary layer; A, is the difference between U2 /(2g) and the true average velocity head, aV2 /(2g) ; HI is 

the total head at the upstream measuring section (referred to the local 

level of the boundary); H is the true total head at the roadway section ; and HL is the head loss between the upstream section and the 

roadway section. Because the flow is uniform (the velocity vectors 

are everywhere straight and parallel), y in figure 6 is both the depth 

of flow and the true piezometric head referred to the boundary. The 

boundary-layer thickness, 8, is represented as the nominal distance
DISCHARGE OVER EMBANKMENT-SHAPED WEIRS A17

from the boundary to the point where the velocity, u, is equal to U. 

Also shown on figure 6 is 8a, the boundary-layer displacement thickness, which is defined in the equation

q=yU-lf7=(y-lJU, (11)

in which q is the true discharge, yV is the potential discharge, and 

SqU is the discharge decrement due to the boundary layer.

From figure 6 and the preceding definitions, a one-dimensional 

energy equation is

772 

H=H1 -HL=y+l^-\, (12)

and from the simultaneous solution of equations 11 and 12,

(13)

If it is assumed that the roadway section is located at the crown line, 

then y="y0 (see fig. 5) and, without special identification, 8q and A, are 

the crown-line values of the boundary-layer quantities. It follows 

that the counterpart of equation 5 is

(14)

in which HL is now defined as the head loss between the measuring 

section and the crown line.

When the measuring section Js a considerable distance from the 

roadway, HL must be evaluated in order to determine the net head on 

the embankment. Actually, 'J£L can only be estimated, for it depends 

on the resistance characteristics of the upstream channel as well as 

the embankment. On the other hand, if the measuring section is located close to the embankment, as it was for the model tests described 

herein, HL is very small. At this point it is convenient to assume that 

HL is negligible. Consequently, from equation 14,

(15)

In the derivation of equation 6 from equation 5 it was assumed 

that the depth at the crown line is equal to the critical depth. With 

this assumption and the assumption that the critical depth is again 

equal, to two-thirds the total head, y0 can be eliminated from equation 15, whence

(16)
A18 FLOW OF WATER OVER DAMS AND WEIRS

Simplifying,

(17)

Expanding, and discarding terms which are negligible when Sg and 

A are small relative to H^

g=3.09 r#if-| ^(5? -X)l> (18) 

or,

g=3.09 Hj |~1-| (^^)T (19) 

Therefore, from equations 8 and 19,

(20)

From equation 10, the quantity 3.09 Hj2 in equation 19 is equal 

to <?i, the discharge of an ideal fluid. Thus, the quantity in brackets 

in the last two equations is a measure of the total influence of boundary 

resistance.

The quantity in brackets in equations 19 and 20 involves the relative magnitude of the quantities SQ and A. It is significant to recall 

that Sq is a measure of the influence of boundary resistance on the 

continuity equation, and A is a measure of the influence of boundary 

resistance on the energy equation. The usefulness of equations 19 

and 20 depends on the determination of a means of evaluating 8q and 

A.

It should be emphasized that the preceding development involves 

several assumptions which must be verified or delimited by experiment. For example, the equations are misleading when (for large 

values of h/L) form effects are dominant in comparison with the 

effects of boundary resistance. Similarly, it is obviously wrong to 

neglect HL under some circumstances. The determination of the real 

significance of such qualifications is an important objective of the 

experimental investigation.

SUBMERGED FLOW

Equations of discharge for free flow have been derived on the basis 

of a simple energy analysis. The analysis was made possible because 

critical-flow control occurs on the roadway when the flow is free. 

When the flow is submerged, however, the discharge capacity of the
DISCHARGE OVER EMBANKMENT- SHAPED WEIRS A19

downstream channel is a primary control. Thus, for submerged flow, 

the discharge depends on the tailwater level 'as well as the head.

Complications introduced by the influence of the tailwater make 

it impractical to derive an independent equation for submerged flow. 

The most expedient alternative is an empirical solution based on experiment and the free-flow discharge equation.

In the general functional relationship for the coefficient of discharge (eq 4), a single ratio (t/Ji) distinguishes submerged flow from 

free flow. This observation leads to the conclusion that the effect of 

submergence can be expressed in terms of Ctree and t/h, as in

^subm=/ I Cfree> T I' \"l)

Frequently it leads to another conclusion, namely,

(22)

which is difficult to justify physically as well as mathematically. Actually, of course, the coefficient of discharge for submerged flow 

should be expected to be independently related to all the ratios listed 

in equation 4. In other words, there is no basis for believing that all 

ratios must have the same significance in both free and submerged 

flow. Thus, empirical solutions based on equation 22 are adequate 

as well as convenient only to the extent that Ctree is constant over 

the full range of conditions considered. When Ctree is not constant, 

application of equation 22 results in a vague definition of the conditions which govern the value of Ctr^ to be used as a reference parameter for C^ubmFor purposes of reducing and analyzing the experimental data, the 

simple equation of discharge, with C defined as in equations 8 and 9, 

is used for both free and submerged flow.

LABORATORY INVESTIGATION

EXPERIMENTAL EQUIPMENT 

GENERAL, ARRANGEMENT

The equipment used for all the tests made after 1954 was essentially 

the same. Embankment models were located in an existing flume. 

Water is supplied to the flume from the laboratory's constant-head 

system. A gate valve in the supply line is used to regulate the discharge, and a hinged tailgate in the flume is used to regulate the 

tailwater level. The maximum discharge used in the tests was about 

6 cfs (^=2 cfs per ft in the model, corresponding to 54 cfs per ft in 

the prototype) . Figure 7 shows the general 'arrangement of the experimental equipment.

690-195 O - 64 - 4
From constant-head tank

-Valve

Baffles

Width of flume = 3 ft

1:9 -scale Imodel

ij Standard location of .1.75ft 

upstream piezometers^"'

^C\\\\\X\\\\\\\\\\\\\X\\XXXJ1

FIGURE 7. Arrangement of experimental eq
DISCHARGE OVER EMBANKMENT-SHAPED WEIRS

FLUME

A21

The flume is 3 feet wide, 3 feet deep, and about 30 feet long. The 

average inside width at the position of the embankment is 3.010 feet. 

The flume is constructed of steel, with three 5-foot-long glass panels 

on each side. Adjustable rails on the top of the flume walls provide 

a level track for instrument carriages. The bottom-hinged steel tailgate is operated with a winch, which is mounted on the top of the 

flume. Figure 8 is a photograph of the test section of the flume with 

a model installed.

FIGURE 8. View of l:9-scale model embankment In test flume.

Water enters the flume from a 12-inch pipe, which discharges vertically into the deep forebay section of the flume. Baffles required to 

produce a uniform flow upstream from the models consist of chainlink wire fencing at the inlet, a low weir, 2 corrugated-metal cribs, 2 

expanded-metal screens, and a surface float.

EMBANKMENT MODELS

The basic structure for all models was a framework of exterior-grade 

plywood and aluminum angles. The pavement and shoulder surfaces 

were made of ^4-inch aluminum plate, and the embankment and berm 

surfaces were made of %-inch plywood. False floors both upstream 

rand downstream from the embankment model were used to vary the 

height of the embankment. The downstream floor was made of plywood; the upstream floor was made of aluminum plate.

When the original 1:9-scale model was built in 1954, sharp lines 

marked the intersections of the embankment, shoulder, and pavement 

surfaces. The crown line was marked by the intersection of the separate plates which comprised the two lanes of the pavement. Subsequent use and repeated polishing of the metal plates resulted in a slight 

rounding of the intersections. However, comparisons of similar tests
A22 FLOW OF WATER OVER DAMS AND WEIRS

made at different times during the period from 1954 to 1959 showed 

no effects which could be attributed specifically to the Founding. It is 

significant that the model with slightly rounded surface intersections 

is doubtless a better representation of the prototype than a model built 

in strict accordance with the design specifications.

Variations in the slopes of the pavement and shoulder surfaces were 

accomplished by placing metal shims between the surface plates and 

the supporting angles. For model K, a piece of 16-gage sheet metal 

was rolled to form a smooth transition between the upstream embankment and the shoulder. A %-inch bronze rod was fastened to the 

downstream edge of the downstream shoulder for model L. For model 

M, berms were built on a framework of wood and covered with 

plywood.

Figure 9 shows details and dimensions of the basic embankment 

design (model A). Figure 10 shows special details of models K, L, 

and M. A photograph of a typical l:9-scale model in the flume is 

shown in figure 8.

For models A through M, all exposed model surfaces were smooth. 

Aluminum surfaces were polished, and they were kept waxed to prevent pitting by corrosion. Plywood surfaces were sanded and painted.

For models AA, AB, AC, and KA the embankment surfaces were 

made rough. Two methods were used. For models AA and KA, a 

piece of new 14- by 18-mesh bronze window screen was stretched taut 

over the entire surface of the model. The diameter of the wire from 

which the screen was made was 0.011 inch.

For models AB and AC the roughness elements consisted of No. 9 

lead birdshot. The shot was 0.080 inch in diameter and very nearly 

spherical. The distribution density on the model was approximately 

75 shot per square inch. The birdshot was attached to the model surfaces with varnish. Two coats of varnish were brushed on the surfaces 

and allowed to become tacky, then the birdshot was sprinkled on the 

surfaces. The varnish was allowed to dry at least 48 hours, then a thin 

coat of varnish was sprayed on and allowed to dry at least 24 hours. 

For model AB the birdshot was applied to the embankment slopes and 

shoulder surfaces only. For model AC it was applied to the entire 

model surface. Figure 11 shows a photograph of model AC and a 

closeup of the birdshot-roughened surface.

DISCHARGE MEASUREMENTS

The discharge for all the tests was determined from measurements 

made with the laboratory's semiautomatic weighing system. Flow 

from the downstream end of the flume drops directly into the weighing 

tank, which is located on the floor below the flume. The weighing sys-
= l in. in 9ft =0.014

Ss = 4^ in. in 6 ft =0.062

Crown line 

Symmetrical about centerline

FIGURE 9. Basic embankment design. All dimensions as in prototype.

fe00
A24 FLOW OF WATER OVER DAMS AND WEIRS

5S = variable (rounded) 

5S = 0.062 (basic design)

A. Model K (rounded upstream shoulder)

\\\V/\\ Cv/V

Sp = 0.014

|5 =0.062 "g-in. rod (model) 

= 0.50

ff. Model L (trip rod on downstream shoulder)

Crown line

5s = 0.062

Se = 0.50

Symmetrical about centerline 

C. Model M (berms on embankment slopes)

FIGURE 10. Special design variations (models K, L, and M). All dimensions as in prototype.

tern includes automatic controls to move the diversion car, to open and 

close the tank valve, and to start and stop an electric time-interval 

clock. The overall accuracy of the equipment is believed to be such 

that discharge measurements are accurate within % of 1 percent. 

Figure 12 shows the downstream end of the test flume (upper floor) 

and the weighing-tank apparatus (lower floor).

PIEZOMETRIC-HEAD AND PROFILE MEASUREMENTS

Headwater and tailwater piezometric levels were generally measured with hook gages mounted over stilling wells connected to pairs of 

floor piezometers. Some measurements were made with a precise 

manometer. An engineer's transit and a special light-weight target 

rod were used to zero the hook gages. The average of several readings
DISCHARGE OVER EMBANKMENT-SHAPED WEIRS A25

A.- Looking downstream at model

B. Cloeeup of birdehot roughneee 

FIGURE 11. Model AC with birdshot roughness.
A26 FLOW OF WATER OVER DAMS AND WEIRS

FIGURE 12. Weighing-tank system for discharge measurements.

on the crown line was used to establish the reference level for gage 

zero.

Piezometers for most of the headwater measurements were located 

in the aluminum false floor, 1.75 feet (corresponding to 15.8 feet in the 

prototype) upstream from the intersection of the false floor and the 

embankment slope. For some tests, piezometers located farther upstream were used, but the measurements were subsequently adjusted, 

if necessary, to correspond with measurements made at the standard 

location.

Stilling wells were made of transparent plastic pipe. They were 

attached to the side of the flume and located close to the crown line 

to avoid any influence of laboratory-floor deflection. Headwater and 

tailw^ater levels are believed to be accurate within 0.001 foot.
DISCHARGE OVER EMBANKMENT-SHAPED WEIRS A27

Water-surface profiles and the depth of flow at the crown line were 

measured with a point gage mounted on a movable carriage (fig. 13). 

The headwater level, as determined by the hook gage, was used to zero

FIGURE 13. Point gage (left) and pitot-static tube (right).

690-195 O - 64 - 5
A28 FLOW OF WATER OVER DAMS AND WEIRS

the point gage. Longitudinal stationing for the point-gage measurements was obtained from a steel tape located adjacent to the carriage 

rail on the top of the flume. The accuracy of point-gage measurements depends on the instability of the water surface, but it is believed 

that the measurements were accurate within 0.002 foot.

VELOCITY MEASUREMENTS

Equipment required especially for boundary-layer measurements 

consisted principally of velocity-measuring instruments and precision 

manometers. Velocities outside the boundary layer were measured 

with a pitot-static tube. This tube, shown on the right in figure 13, 

has an outside diameter of %6 inch. Its stagnation and static openings 

are so located that the coefficient of the tube is very nearly 1.0. The 

tube was fastened to the staff of a hook gage, which was mounted on a 

movable carriage. A scale on the top rail of the flume, indexed to the 

carriage, and a scale on the hook-gage staff provided for accurate 

determination of the position of the pitot-static tube relative to the 

model. A pivoted connection between the gage staff and the tube permitted the tip leg of the tube to be adjusted so that it was always parallel to the flow at the point of measurement.

The air-water manometer used with the pitot-static tube is shown in 

figure 14. The manometer is of the zero-displacement type, with backlighted stainless-steel needles in glass tubes for accurate positioning 

of the menisci. Differential heads can be measured accurately within 

0.001 foot with this manometer.

The stagnation tube used for boundary-layer velocity measurements 

is shown in figure 15. The tip of this tube was made from a 22-gage 

(0.028-inch outside diameter) stainless-steel hypodermic needle. The 

vertical leg is a brass tube, ys inch in outside diameter, backed up 

with a streamlined brass bar. The stagnation tube is mounted on a 

hook-gage staff, and a dial-type displacement gage is attached to the 

staff to provide for the accurate determination of the position of the 

tube. The smallest division on the dial scale is 0.001 inch. The 

stagnation-tube staff was provided with a pivot mount, in order that 

velocity traverses could be taken on sections which were perpendicular 

to the boundary at all stations.

The gage zero for the stagnation-tube displacement gage was determined at each measuring station by setting the dial scale to read zero 

when the bottom of the tube was placed against the surface of the 

model without bearing any of the weight of the tube assembly. The 

bottom of the tube was accurately located in this position by means 

of a horizontal light beam focused through the glass wall of the flume 

from a position opposite the observer and at an elevation just above 

the surface. Thus, the tube could be positioned very sensitively by
DISCHARGE OVER EMBANKMENT-SHAPED WEIRS A29

FIGURE 14. Manometer for the pitot-static tube.

lowering it until no light could be seen between the bottom of the tube 

and the surface of the model.

The manometer used for the stagnation-tube measurements is shown 

in figure 16. The manometer is essentially a zero-displacement point 

gage. It consists of >a black-lighted needle in a small glass tube, 

mounted on a frame which utilizes a vernier caliper for level determinations. Piezometric levels can be measured accurately within 0.001 

inch with the manometer.

The zero reading for velocity determinations with the stagnation 

tube was determined by comparison with the differential-head reading
A30 FLOW OF WATER OVER DAMS AND WEIRS

FIGURE 15. Stagnation tube for boundary-layer velocity measurements.
DISCHARGE OVER EMBANKMENT-SHAPED WEIRS A31

FIGURE 16. Manometer for the stagnation tube.
A32 FLOW OF WATER OVER DAMS AND WEIRS

obtained with the pitot-static tube in a uniform-flow zone outside the 

boundary layer at the same station. Thus, the absolute accuracy of 

the stagnation-tube velocity measurements depends on the accuracy of 

the pitot-static tube measurements.

SCOPE OF EMBANKMENT-DESIGN VARIATIONS

Scale-model tests were made on 17 variations of the basic embankment design. Data used in the report were obtained from tests made by 

Prawel, Davidian, and Emmett. Included in these tests, which were 

made after 1956, are duplications of test conditions investigated before 

1956.

The scope of the design variations involved in the tests is given in 

table 1. All models described in table 1 were built to a scale of 1: 9. 

The corresponding unit-discharge (q) scale is 1: 27. Model A is the 

basic design, which is shown in prototype dimensions in figure 9. The 

design for model A was suggested by the Georgia State Highway 

Department in 1947. At that time it was typical of designs being 

used for secondary asphaltic-pavement two-lane highways. It is believed that subsequent changes in design standards, at least as they 

concern hydraulic characteristics, are adequately represented by the 

design variations described in table 1.

Models A-l through A-A are different versions of the basic design. 

Each model represents a minor reconstruction or refurbishing which 

followed tests on some of the design variations. The purpose of 

models B, C, and D was to demonstrate the influence of h/P by comparison with model A. Thus, models otherwise identical with model 

A were tested with a full range of heads and with values of P equal 

to one-fourth, one-half, and three-fourths of the value of P for the 

basic design.

To demonstrate the influence of pavement cross slope (/Sp ) , models 

otherwise identical with model A were tested with 4 different values 

of /Sp. 2 larger and 2 smaller than /Sp for the basic design. Similarly, 

the influence of shoulder slope (/Ss ) was demonstrated with two 

models. For one of these, fis was equal to Sp for model A. For the 

other, fSs w^as somewhat larger than jSs for the basic design.

Model K was built especially for tests concerned with the influence 

of the boundary layer. It involved a rounded intersection between 

the upstream embankment slope and shoulder. Model L was identical 

with the basic design except for a tripwire located on the downstream 

edge of the downstream shoulder. The purpose of this variation is 

described subsequently. Model M was designed to simulate the influence of the berm on both slopes of the embankment tested by 

Yarnell and Nagler (1930). Otherwise, model M was identical with
DISCHARGE OVER EMBANKMENT-SHAPED WEIRS A33

TABLE 1. Summary of designs tested, l:9-scale models 

[Asterisk (*) indicates that shape detail differs from basic design]

Model

A-1 

A-2 

A-3 

A-4 

B 

C 

D 

E 

F 

G 

H 

I 

J K-1 

K-2 

L 

M AA-1

AA-2 

AB

AC 

KA

Investigator

..... do..- 

Emmett--~

 do... 

-.-do--- 

 do___ 

... ..do-.- 

 -do__ 

 do..- 

 -do.-- 

 do... 

Davidian 

 do.. 

 do_.-..-

Davidian--_ 

Emmett- 

 .do.- 

 do.. 

Davidian. -.

Height, P (feet)

Prototype

10.5 

10.5 

10.5 

10.5 

*7.88 

*5.25 

*2.62 

10.5 

10.5 

10.5 

10.5 

10.5 

10.5 

10.5 

10.5 

10.5 

10.5 

10.5

10.5 

10.5

10.5 

10.5

Model

1.17 

1.17 

1 17 

1.17 

*.875 

*.583 

*.292 

1.17 

1.17 

1.17 

1.17 

1.17 

1.17 

1.17 

1.17 

1.17 

1.17 

1.17

1.17 

1.17

1.17 

1.17

Pavement cross 

slope, Sp

Inches: 

feeti

1.5:9 

1.5:9 

1.5:9 

1.5:9 

1.5:9 

1.5:9 

1.5:9 

*0:9 

*.9:9 

*2.2:9 

*2.8:9 

1.5:9 

1.5:9 

1.5:9 

1.5:9 

1.5:9 

1.5:9 

1.5:9

1.5:9 

1.5:9

1.5:9 

1.5:9

Nondimensional

0.014 

.014 

.014 

.014 

.014 

.014 

.014 *.ooo

*.008 

*.020 

*.026 

.014 

.014 

.014 

.014 

.014 

.014 

.014

.014 

.014

.014 

.014

Shoulder slope, 

Ss

Inches: 

feeti

4.5:6 

4.5:6 

4.5:6 

4.5:6 

4.5:6 

4.5:6 

4.5:6 

4.5:6 

4.5:6 

4.5:6 

4.5:6 

*1.0:6 

*5.7:6

4.5:6 

4.5:6 

4.5:6

4.5:6 

4.5:6

4.5:6

Nondimensional

0.062 

.062 

.062 

.062 

.062 

.062 

.062 

.062 

.062 

.062 

.062 

*.014 

*.079

.062 

.062 

.062

.062 

.062

.062

Surface 

roughness

 do... ..

.....do..........

.....do.. ........

 do.. 

. do.. ...... ..

 do... ...

 do_. ........

 do-.. 

.... -do-._ 

..... do....... 

 do-- 

 do... ....

 do... 

 do..- -

..... do..- 

 do-.- 

Window screen 

(all surfaces).

Birdshot (except on pavement) . 

Birdshot (all 

surfaces). 

Window screen 

(all surfaces) .

Remarks

Basic design. 

Do. 

Do. 

Do. 

Effect of P. 

Do. 

Do. 

Effect of Sp. 

Do. 

Do. 

Do. 

Effect of Ss. 

Do. (2) 

(2) 

(3) 

(4)

(2)

' Dimensions given in prototype units (see fig. 9).

2 Rounded transition between upstream embankment and shoulder surfaces.

3 Trip wire on downstream edge of downstream shoulder. (See fig. 10B.)

4 Berm on embankment slopes. (See fig. 10C.)

(Seeflg. 10,4.)

model A. Details of design for models K, L, and M are shown on 

figure 10.

The remaining four design variations shown in table 1 were intended to demonstrate the influence of surface roughness on the discharge characteristics of an embankment. These models involve 3 

kinds of roughness on model design A and 1 on model design K.

SIGNIFICANCE OF ROUGHNESS VARIATIONS

Only four different degrees or patterns of surface roughness (including "smooth") were involved in all the model variations described 

in table 1. Furthermore, the simple k/h ratio used to represent relative roughness in equation 10 is inadequate to describe the roughness 

variations tested. Nevertheless, the tests made with rough-surfaced 

models are expected to result in a general understanding of the influence of roughness on the coefficient of discharge and other important flow characteristics.

Model A, the basis of comparison for all other designs, included 

smooth surfaces on all parts of the embankment and roadway. In 

terms of the prototype, the characteristics revealed by this model are
A34 FLOW OF WATER OVER DAMS AND WEIRS

believed to simulate reasonably accurately the characteristics of a 

smooth, paved roadway in good repair.

Models AA and KA were rough-surfaced models. The roughness 

consisted of wire screen, a type of roughness used previously by Bauer 

(1954) in a related investigation. Uniform-flow tests made by Bauer, 

using a slightly different kind of screen, indicated that the effective 

roughness could be compared with rough concrete in the prototype.

Models AB and AC featured a relatively large, granular-type 

roughness, consisting of birdshot cemented to the surface in a random 

pattern (see fig. 11). This variety of roughness is believed to simulate a reasonable maximum prototype roughness. For model AC the 

birdshot was applied to the entire model surface. For model AB the 

birdshot was applied only to the embankment slopes and shoulders. 

Model AB should give an indication of the influence of rough shoulders bordering a smooth pavement.

Roughness investigations for open-channel flows are usually complicated by the fact that the depth of flow varies with the discharge. 

Therefore, a ratio like k/h or k/y, unlike the corresponding relativeroughness ratio used for pipe flow, varies with discharge as well as 

roughness. Also, because the Reynolds number for the mean flow is 

R=Fy/v=<?/v, values of R for the prototype are in a higher range 

than values of R in the model for corresponding values of geometric 

ratios such as h/P and h/L. Furthermore, the nature of prototype 

roughness, especially embankment and shoulder roughness, is extremely variable and very difficult to simulate in the laboratory. It 

is believed that a comprehensive investigation of the influence of 

roughness will require, in addition to small-scale model tests, a carefully controlled program of tests on full-scale models or prototypes.

INFLUENCE OF EMBANKMENT FORM AND ROUGHNESS

OBJECTIVE

The laboratory investigations had two main objectives. One was 

the experimental determination of the relationship between embankment form and roughness and some of the more important discharge 

characteristics. The second was the theoretical and experimental 

definition of the relationship between free-flow discharge and the 

boundary layer on the roadway. The first objective is discussed in 

this part of the report.

The data needed to accomplish the first objective are mostly those 

needed to define the coefficient of discharge and the characteristic 

flow patterns for free and submerged flow. The analysis is empirical, 

but it is guided by the results of the dimensional analysis. A summary 

of the results of the tests described in this part of the report is given 

in table 3 in the section "Experimental Data."
DISCHARGE OVER EMBANKMENT-SHAPED WEIRS A35 

ANALYSIS OF RESULTS

COEFFICIENT OF DISCHARGE FOR FREE FLOW

Tests made to determine the coefficient of discharge for free flow 

involved the measurement of the head and discharge. The discharge 

coefficient, 67, was computed from equation 8, wherein H-i is the total 

head referred to the crown line (see fig. 5),

< 23)

and q is the total measured discharge divided by the width of the flume 

at the crown line,

2=|- (24)

Figures 17, 18, and 19 show the coefficient of discharge for all the 

designs tested. In these figures, values of C are plotted as a function 

of both h and h/L, where h is the piezometric head (as measured in 

the model) and L is the total width of pavement and two shoulders. 

The two abscissa scales are independently significant. Thus, A. is a 

scale of reference for the influence of boundary resistance and "scale 

effect," because h is directly proportional to the Reynolds number. 

On the other hand, h/L is a scale of reference for form effects, including the effect of curvilinear flow at the control section.

Figure 17 shows the coefficient of discharge for the basic design,

3.2

3.0-

2.6

x Model A-2 

+ Model A-3 

O Model A-4

O.I 0.2. u.3 0.4 0.5 0.6 0.7 0.8 

Values of ft, in feet

0.10 0.15 0.20 0.25 

Values of 'h/L

FIGUKE 17. Coefficient of discharge for free flow; basic design, model A.

model A. The curve, visually fitted to the plotted points, shows an 

initial trend for C to increase with h and approach asymptotically the 

ideal value (3.09). When h exceeds about 0.5 feet, however, the curve

690-195 O - 64 - 6
A36 FLOW OF WATER OVER DAMS AND WEIRS

inflects, and C begins to increase more rapidly, eventually exceeding 

3.09. As implied by the overlapping scales of h and k/L, the inflection 

in the C curve is believed to be related to the gradual change from 

dominant boundary-resistance influence to dominant boundary-form 

influences.

Figure 18 shows the coefficient of discharge for all the smoothsurfaced models. The several parts of the figure show, by comparison 

with the basic design (model A), the independent effects of A/P, $p , 

and /Ss, as well as the special features represented by models K, L, 

andM.

Figure 18^1 shows the effect of P and, indirectly, the influence of 

h/P. The figure shows the results of tests made with models B, C, 

and D, as well as the curve for model A from figure 17. Considering 

the probable experimental error, which can be expected to increase as 

P decreases (the result of waves and surges in the approach channel), 

figure 18^1 shows no distinct, systematic relationship between C and P. 

Consequently, because the tests cover approximately the same range of 

h values for each of the four P values represented, it is concluded 

that C is virtually independent of h/P.

Figure 18B shows the influence of pavement cross slope on the coefficient of discharge for free flow. The results of tests on models 

A, E, F, G, and H show a poorly defined but systematic relationship 

between C and Sp. However, the average deviation from the model A 

curve is less than 1 percent for all models except model H, for which 

Sp was nearly twice the value specified for the basic design. Thus, for 

purposes of practical application, the correlation between C and Sp is 

believed to be insignificant.

Figure ISO shows the influence of shoulder slope, $8 , based on the 

results of tests on models I, J, and K-2, as well as model A. Model 

K-2 is included because it simulates the prototype condition of a 

rounded or variable-slope shoulder which is tangent to both the embankment slope and the pavement. The results of the tests shown on 

figure 18 C are somewhat contradictory, because the maximum consistent deviation from the model A curve is shown by the model which 

most nearly resembles the basic design. Again, however, the average 

deviation from the model A curve is not more than 1 percent for any 

of the design variations represented. Thus, for practical purposes, C 

is believed to be independent of S8 .

In view of the results shown on figures 18 (A, B, C} , it is not surprising to find that the special design variations represented by models 

L (tripwire on downstream shoulder) and M (berm on both embankment slopes) have an insignificant influence on the coefficient of discharge for free flow. This conclusion is substantiated by the test 

results shown on figure 18D.
<o 3.0

o

f 2.8

i i i i i i i i i i i i

a^^ ^^r-^ ^^'

G&J&- "S^^^T °

^---~~"~"~~~~'~~'~~^

° B

 Model A (P=l. 17 ft), A Model C (P = 0.583 ft) 

curve from fig. 17

0 Model B (P = 0.875 ft) EJ Model D (P = 0.292 ft)

i i i i i i i i i iii

 1 i 

U 

30^ X__-

L^^

2>8 " 

P 

0 ^ 

t-G0 O.I 0.2 0.3 0.4 0.5 0.6 0.7 ^'"0 O.I 

Volues of h, in feet 

A. Influence of h/p B. Influence of pav

0.10 0.15 0.20 cross slope, Sp 

Values of h/L 

<o 3.0

 5

01

a>

> 2.8

l i i i i i i ^

^~**^*'^

curve from fig. 17 

B Model K-2 (rounded -

O Model 1 (S5 = 0.014) shoulder)

i i i i i i i i i iii

3.0

2.8

_ 

^-^"^^ ^

_

i 

°" eO O.I 0.2 0.3 0.4 0.5 0.6 0.7 """0 O.I 

Values of /?, in feet 

C. Influence of shoulder D. Influence of spe

slope, Ss 0,10 0.15 0.20 design variation

Values of h/L 

FIGURE 18. Coefficient of discharge for free flow; smooth-s
ASS FLOW OF WATER OVER DAMS AND WEIRS

3.2

<o 3.0

2.6,

Model A (all surfaces smooth), curve

from fig. 17

Model AA-2 (screen-wire roughness) 

Head datum at middepth of screen-wire at

I____I____I____I____I____i j____|____I

, O.I 0.2

Screen-wire roughness 

on all surfaces

0.3 0.4 0.5 

Volues of h, in feet

0.6 0.7

0.10 0.15 0.20 

Values of h/L

on shoulder

and embankment slope

0.10 0.15 0.20 

Values of h/L

3.2

3.0

2.8

2.6

2.4

l l i i l l l i i i i

Datum at top of birdshot at crown line

Datum at middepth of birdshot

Q Model A (all surfaces smooth), curve 

from fig. 17

xxDatum at e CD Model AC (birdshot on all surfaces) 

smooth 

surface

i i i i

O.I 0.2

Birdshot roughness 

on all surfaces

0.3 0.4 0.5 

Values of h, in feet

0.6 0.7

0.10 0.15 0.20 

Values of h/L

0.25

<o 3.0

<t> 

| 2.8

2.6

I

1 i 1 1 1 1 1 1 1 l l 1 1 

O

from fig. 17 

O Model AB (birdshot on shoulders and 

embankment slopes)

Head datum at smooth surface at crown line 

1 1 l 1 1 1 1 1 1 1 1 l l

D O.I 0.2 0.3 0.4 0.5 0.6

a _. . , , Values of /?, in feet 3. Birdshot roughness

1 I

i i

0.7 0.

0.25

0.8

0.25

FIGURE 19. Coefficient of discharge for free flow; rough-surfaced models.
DISCHARGE OVER EMBANKMENT-SHAPED WEIRS A39

Figure 19 shows the influence of roughness on the coefficient of discharge. As in figure 18, the curve for model A is used as a basis for 

comparison. Figure 19^4 shows that screen-wire roughness over the 

entire model surface (model AA-2) causes the coefficients to be consistently about 1 percent lower than the coefficients for model A. It 

is noted in the figure that C is computed on the basis of the assumption 

that the head datum is at the level of the middepth of the screen wire 

at the crown line. Alternate assumptions might have placed the datum 

at the level of the top of the screen or at the level of the smooth surface 

of the model. As the thickness of the screen fabric is approximately 

0.002 foot, the maximum difference in heads computed on the basis of 

the different assumptions would have been only 0.002 foot. Thus, the 

corresponding difference in computed values of C would have been 

negligible except at very small values of h. The question of head 

datum relative to screen-wire thickness is obviously meaningless in 

terms of the prototype.

Figure 19# shows the influence of a relatively large, granular-type 

(birdshot) roughness on the shoulders and embankment slopes. Because the pavement was smooth, as in model A, the head datum was 

the level of the smooth model surface at the crown line. The coefficient of discharge for model AB appears to differ very little from 

that for the basic design.

Figure \$C shows the results of tests on model AC, with birdshot 

roughness on all model surfaces. The results are presented in a form 

which shows the influence of alternate assumptions regarding the 

datum for head computations. In this instance, because the diameter of the birdshot was nearly O.OOY foot, the effect of the alternative 

assumptions on computed values of C is appreciable. A dashed curve 

is drawn through the points which correspond to the datum at middepth of the birdshot.

COEFFICIENT OF DISCHARGE FOR SUBMERGED FLOW

In the previous discussion of submerged flow (p. 18), it was emphasized that the coefficient of discharge for submerged flow is independently related to all the variables involved in free flow, plus the 

submergence ratio, t/Ji. It was concluded that C for submergedflow tests, like C for free-flow tests, should be computed from equation 8.

Figure 20 shows C plotted as a function of t/h for models A-2, B, 

C, and D. Different symbols are used to identify the plotted points 

which correspond to different constant values of P. It is apparent 

from the figure that the relationship between C and t/h is independently correlated with P. For purposes of application to the 

prototype, the P variable in figure 20 should be expressed as a non-
A40 FLOW OF WATER OVER DAMS AND WEIRS

3.2

2.8

2.0

1.6

1.2

A 

Q

X* H X X

Model A-2 (P= I. 17ft)

A

Model B (^ = 0.875ft)

0 

Model C (P=0. 583ft)

X

Model D (P = 0. 292ft)

As x 

O A B 

O

0.80 0.84 0.88 0.92 0.96 1.00

Values of f/h

FIGURE 20. Coefficient of discharge for submerged flow; influence of h/P, with t/h as the submergence ratio.

dimensional ratio. Unfortunately, however, the most likely parameter for this purpose, A, varies independently with q and t/h. 

Therefore, the ratio h/P is incapable of defining a single family of 

curves on a figure such as figure 20. The problem was conveniently 

resolved by changing the definition of the submergence ratio. Thus, 

when the submergence ratio is defined as t/H-^ instead of t/h, the 

effect of P is virtually eliminated. This conclusion is substantiated 

by a comparison of figures 20 and 21.

Figure 21 shows a considerable scatter of plotted points but no 

systematic correlation with P. In general, points showing the maximum deviation from the average curve represent tests made with 

larger values of h/P. It is significant that waves and surges in the 

approach channel cause increasing difficulties and errors of measurement as h/P increases. Subsequent demonstrations of the sensitivity of C'subin to small irregularities on the roadway are more reason
DISCHARGE OVER EMBANKMENT-SHAPED WEIRS A41

3.2

2.8

2.4

1.6

1.2

0.80

Average for smoothsurfaced models

Average for model A, 

curve from fig.22

o 

Model A-2(/>=l.l7ft)

A

Model B (P =0.875 ft)

a

Model C (P =0.583 ft)

x 

Model D (P =0.292 ft)

0.84 0.88 0.92 

Values of

0.96 1.00

FIGURE 21. Coefficient of discharge for submerged flow; influence ofh/P, with t/Hi as the submergence ratio.

to accept the scatter of points in figure 21 as evidence of normal 

experimental error.

The data shown on figures 20 and 21 represent the same tests and, 

therefore, the same range of discharges as were used for the free-flow 

coefficients shown on figure ISA. In figure 18J., as for all free-flow 

tests, the coefficient is clearly related to h (or q). It seems reasonable to expect that the rate of flow will be similarly involved in the 

relationship between Csubm and t/H±. This conclusion is readily 

tested on the basis of the data shown on figure 22. Here, for two 

versions of model A, different rates of flow are identified with different symbols. The obvious conclusion to be drawn from figure 22 is 

that the effect of q is insignificant.

For purposes of comparison with other model designs, an average 

curve for model A is fitted to the data shown on figure 22. It is observed that the plotted points do not deviate from the curve by an
A42

3.2

2.8

2 4

<* 

o

to

O

> 2.0

1.6

1.2

FLOW OF WATER OVER DAMS AND WEIRS

1 1 1 1 1 1 1 1 1

Q1 ^""^^M^^ .-Average for all 

o "xX^^ smooth-surfaced

^v-J-v models

Average for model A *\ \

\ A 

B\ \ -

\ \A

Model A-2 Model A-3 \\

o x ckA - c7=O.I43cfs (7=0.221 cfs 'T\

q =0.222 cfs g= 0.476 cfs \^y 0

Q T A

<7=0.45I cfs q- 1.012 cfs ^ 

© j_ ®\\

(7=0.65 cfs <7= 1.53 cfs ^1

B \\ -

q = 1.14 cfs \j^

I I I 1 i i i i i

0.80 0.84 0.88 0.92

Values of////!

0.96 1.00

FIGURE 22. Coefficient of discharge for submerged flow; basic design, model A.

amount in excess of the probable experimental accuracy. Some 

uncertainty may exist, however, regarding the value of O represented 

by the incipient-submergence end of the curve. Here, at least, it 

might be expected that the previously observed influence of q on 

Ctree would require consideration.

It is recalled that the variation of tffree with h shown on figures 

17,18, and 19 is related to the influence of boundary resistance at low 

heads and to the influence of boundary-form effects (principally flow 

curvature) at higher heads. Characteristics related to heads smaller 

than 0.5 foot are seldom likely to be significant in terms of the prototype. Furthermore, when t/Hv is 0.84 or over, as it is when incipient 

submergence occurs, flow at the control section is very nearly uniform 

(see fig. 25), and, regardless of the head, flow curvature is not likely 

to have a significant influence on C. Thus, in figure 22 and all subsequent plots of C versus t/H^ the value of <7free used to define the 

incipient-submergence end of the average curves is the value at the
DISCHARGE OVER EMBANKMENT-SHAPED WEIRS A43

point of inflection on the corresponding curve in figures 17, 18, and 

19. The 'average curve for model A is shown on figure 21 as well as 

figures 22 through 28. Also shown on those figures is an average 

curve for all smooth-surfaced models (A through M).

Figure 23 shows the results of submerged-flow tests made with 

models A. E, F, G, and H, all with different values of Sp. The plotted 

points are remarkably close to the average curve for smooth-surfaced 

models. They show no significant correlation with Sp.

3.;

2.8

2.4

> 2.0

1.6

1.2

Average for smooth 

surfaced models

Model A (Sp =0.014), 

curve from fig. 22

Model E (5/7 =0.000)

A

Model F (Sp =0.008)

D

Model G (Sp =0.020)

x 

Model H (5^ =0.026)

I i i i i

0.80 0.84 0.88 0.92 0.96 1.00

Values of t/H

FIGURE 23. Coefficient of discharge for submerged flow; influence of pavement cross slope, Sp.

Figure 24 shows the influence of shoulder slope, $s, based on tests 

made with models A, I, J, and K-2. Points representing model I deviate considerably from the average curve. However, the value of 

Ss used in this model is smaller than that ordinarily used in practice. 

Considerable scatter characterizes the results of tests made with the 

rounded-shoulder model (K-2). The average of the scattered points 

agrees reasonably well with the average curve for all smooth-surfaced

690-195 O - 64 - 7
A44

3.2 

2.8

PLOW OF WATER OVER DAMS AND WEIRS

O

> 2.0

1.6

1.2

_ Average for smoothsurfaced models

Model A (Ss =0.062), 

curve from fig. 22

G

Model I (Ss =0.014)

A

Model J (Ss = 0.079)

Model K-2 (rounded 

shoulder)

0.80 0.84 0.88 0.92 

Values of

0.96 1.00

FIGURE 24. Coefficient of discharge for submerged flow; Influence of shoulder slope, Ss .

models. The magnitude of the scatter is believed to be consistent with 

the experimental accuracy to be expected.

The influence of the special design variations, models K-2, L, and 

M, is shown in figure 25. Maximum deviation from the average curve 

is shown by models K-2 (discussed above in connection with fig. 24) 

and L. The results of tests on model L, with a tripwire on the downstream shoulder (see fig. 10 #), are particularly significant. These 

results lead to the conclusion that the influence of small obstructions 

on the downstream side of the roadway, particularly at the downstream end of the shoulder, is greater than the influence of any of 

the major geometric parameters previously considered.

Figures 26, 27, and 28 show the influence of surface roughness on 

the submerged-flow discharge coefficient for models AA-2, AB, and 

AC. As in the preceding figures, the curve for model A and the 

average curve for smooth-surfaced models is shown for comparison.
DISCHARGE OVER EMBANKMENT-SHAPED WEIRS A45

2.8

ZA

> 2.0

1.6

1.2

0.80

Average for smoothsurfaced models

Model A (basic design\ 

curve from fig.22

o

Model K-2 (rounded 

shoulder)

Model L (trip wire on 

downstream shoulder)

Model M (berm on 

embankment slopes)

0.84 0.88 0.92 0.96 1.00

Values of ////,

FIGURE 25. Coefficient of discharge for submerged flow; influence of special design variations.

It is apparent that the effect of roughness is to decrease the coefficient 

of discharge and to cause the inception of submergence at lower 

values of t/Hi. It is also apparent, from comparison of figures 27 

and 28, that roughness on the shoulders and embankment slopes is 

nearly as effective as roughness distributed uniformly over all surfaces of the model. Because the birdshot roughness used for models 

AB and AC is believed to simulate a maximum prototype roughness, 

the curve showing the results of tests on model AC, figure 28, is believed to represent a reasonable maximum influence of roughness on 

^subm- The points showing considerable scatter on this figure correspond to small discharges and small heads and consequently large 

relative errors in computed values of C and

INCIPIENT SUBMERGENCE

The transition from free flow to submerged flow has been described 

as incipient submergence. In the laboratory test procedure, the tail-
A46

3.2 

2.8

^ 2.4

o 

</>

CD

JD

£ 2.0

1.6

FLOW OF WATER OVER DAMS AND WEIRS

I I I I I I T I I I I I I

Average for smoothsurfaced models

Model A (all surfaces smooth) 

Curve from fig. 22

Model AA-2 (screen-wire roughness) 

Head datum at middepth of screen 

at crown line

J____I____I____I____I____I

0.72 0.76 0.80 0.84 0.88 

Values of t/H\

0.92 0.96 1.00

FIGTJEE 26. Coefficient of discharge for submerged flow; influence of screen-wire roughness on all surfaces.

3.2

2.8

2.4

2.0

1.6

1 i i i i i i i i i l i r

o

Average for smoothsurfaced models

Model A (all surfaces smooth), 

curve from fig. 22

Model AC(birdshot on all surfaces) 

Head datum at middepth of' bird shot 

at crown line

0.72 0.76 0.80 0,84 0.88 0.92 0.96 1.00

Values of t/H±

FIGUEE 27. Coefficient of discharge for submerged flow; influence of birdshot roughness on shoulders

and embankment slopes.
DISCHARGE OVER EMBANKMENT-SHAPED WEIRS A47

3.2

2.4

> 2.0

1.6

n i i r

Average for smoothsurfaced models

Model A (all surfaces smooth), 

curve from fig. 22

 o 

Model AB (birdshot on shoulders

and embankment slopes) 

Head datum at smooth surface 

at crown line

1.21__ 

0.72

J____I____|____L

0.76 0.80 0.84 0.88 0.92 0.96 LOO

Values of f/Hi

FIGURE 28. Coefficient of discharge for submerged flow; influence of birdshot roughness on all surfaces.

water level corresponding to incipient submergence for a given discharge was determined by gradually raising the tailwater and observing the tailwater level at which the headwater began to rise. The 

results of tests made to define the incipient-submergence tailwater 

level for the full range of model discharge are shown in the upper 

half of figures 29 through 37, The average curves for model A are 

shown for comparison on figures 32 through 37.

Figures 29 and 30 show a comparison of alternative dependentvariable (ordinate) scales used to define incipient submergence. The 

variables used are the tailwater ratios t/Ji (fig. 29) and t/H-i (fig. 30), 

and the tests used for the comparison are those involving different 

values of P (models A, B, C, and D). Neither ratio shows a significant correlation with P, but the scatter of test points is somewhat 

smaller when t/H^ is used. Furthermore, because t/H-t was used in 

the presentation of Csnbm data (figs. 21 through 28), it is used in subsequent figures showing incipient-submergence and transition-range 

characteristics.

The overlapping abscissa scales used for figures 29 through 37 are 

Ji and h/L. It has been explained that the two scales are independently significant, because h is a scale of reference for the boundarylayer influence and h/-L is a scale of reference for form influence. 

The test results shown on figure 30 indicate that incipient submerg-
A48 FLOW OF WATER OVER DAMS AND WEIRS

1.0 

0.8

n fi

i i i i i i i i i i i i i

i , i i i i i i i i i i i

A. Incipient submergence

. Upper limit of transition range

 i i i i i i 

Model A-4(P=1.17 ft), 

A Model B(P=0.875 ft) 

Model C(P = 0.583 ft) 

_x_ Model D (P=0.292 ft)

0.3 0.4 0.5 

Values of h, in feet

0.10 0.15 0.20

Values of h/L 

C. Lower limit of transition range

0.25

FIGTJEE 29. Incipient submergence and free-flow transition range; Influence oi h/P, with t/h as the submergence ratio.

ence occurs at t/Hi=0.84i when h is greater than about 0.4 feet. Somewhat higher values of t/Hi are indicated for lower values of h. It is 

reasonable to believe that the increase in t/Hi is related to the influence of boundary resistance.

Figure 31 shows the results of tests on three versions of the basic 

design. The average curves obtained from the figure are shown as 

the characteristics of model A in the subsequent figures in this series.

Figs. 32, 33, and 34 show the results of tests involving different 

values of Sp (fig. 32) and Ss (fig. 33) and the special design variations 

represented by models K-2, L, and M (fig. 34). From these figures 

it is apparent that none of the design variations causes an appreciable 

change in the incipient-submergence characteristics of the smoothsurfaced models. All the data show remarkably good agreement with 

the results of tests on the basic design. A little scatter of test points
DISCHARGE OVER EMBANKMENT-SHAPED WEIRS A49

i i i r i r i i i i I

A. Incipient submergence

B. Upper limit of transition range

 ° Model A-4(P= 1.17 ft)

A Model B(P=0.875 ft)

m Model C(P=0.583 ft)

 X Model D (P=0.292 ft)

0.4 0.5 

Values of /?, in feet

0.10 0.15 0.20

Values of h/L 

C. Lower limit of transition range

0.25

FIGUEE 30. Incipient submergence and free-flow transition range; influence of h/P, with t/Hi as the submergence ratio

at small values of h is believed to be insignificant, because it is recognized that small values of h correspond to small values of t and relatively large experimental errors in t/Hl.

Figures 35, 36, and 37 show the influence of surface roughness. The 

results of tests on models AA-2 (fig. 35), AB (fig. 36), and AC (fig. 

37) indicate that values of t/ffj. for incipient submergence are smaller 

for the rough-surfaced models than for the smooth-surfaced models 

represented by model A. Remarkably, the largest deviation from the 

model A curve is shown for the screen-roughened model (model AA-2, 

fig. 35), and the smallest deviation is shown for the model which is 

completely covered with birdshot (model AC, fig. 37). Furthermore, 

for model AA-2 the incipient-submergence tailwater ratio varies with 

h over the full range of test values. For the birdshot-roughened 

models, as for the smooth-surfaced models, the tailwater ratio at in-
A50 FLOW OF WATER OVER DAMS AND WEIRS

A-2 A-3 A-4

O 0 

Incipient submergence

0.3 0.4 0.5 

Values af /?, in feet

0.6 0.7

0.10 0.15 0.20 

Values of h/L

0.8

0.25

FIGURE 31. incipient submergence and free-flow transition range; basic design, model A.

1.0 

0.8 

0.6

. Incipient submergence

0.4 0.5 

Values of h, in feet

0.10 0.15 0.20

Values of h/L 

C. Lower limit of transition range

0.25

FIGUKE 32. Incipient submergence and free-flow transition range; influence of pavement cross-slope, Sp.
DISCHARGE OVER EMBANKMENT-SHAPED WEIRS A51

i.o

0.8 

0.6 

1.0 

0.8

^0.6 

 S 0.4

-2 0.8 

o

>

0.6

0.4 

0.2 

0.0

-0.2

-0.4

O.I

mo

A. Incipient submergence

I T 1 I I I

I I I I I I I I I I I I I

B. Upper limit of transition range

 Model A (Ss =0.062), curve from fig. 31

o Model I (Ss = 0.014)

A Model J(SS = 0.079)

Q Model K-2 (rounded shoulder)

0.2 0.3 0.4 0.5 

Values of h, in feet

0.6 0.7

0.10 0.15 0.20

Values of h/L 

C. Lower limit of transition range

0.8

0.25

FIGURE 33. Incipient submergence and free-flow transition range; influence of shoulder slope, S«.

cipient submergence is constant for heads greater than about 0.4 feet. 

However, the effect of lower heads is to decrease the ratio for roughsurfaced models and to increase it for the smooth-surfaced models.

FREE-FLOW TRANSITION RANOE

The free-flow transition range is the range of tailwater levels within 

which a given discharge may produce either a plunging flow or a 

surface flow on the downstream side of the embankment. The upper 

and lower limits of the transition range were determined visually by 

gradually raising and lowering the tailwater and observing the level 

at which the transitions took place.

The results of tests made to define the free-flow transition range 

are shown in the middle and lower parts of figures 29 through 37.

690-195 O - 64 - 8
A52 FLOW OF WATER OVER DAMS AND WEIRS

1.0

0.8

0.6

1.0

0.8

0.6

£0.4

: 0.2 i

-3 0.8 

$

0.6 

0.4 

0.2 

0.0 

-0.2

-0.4

0.1

I____I____I____I

A. Incipient submergence

i i i i i i i r

i I____I I

B. Upper limit of transition range

i i r

 Model A (basic design), A Model L (trip wire on - 

curves from fig. 31 downstream shoulder)

© Model K-2 (rounded H Model M (berm on 

shoulder) embankment slopes)

0.2 0.3 0.4 0.5 0.6 

Values of h, in feet

0.7 0.8

0.10 0.15 0.20 

Values of h/L

0.25

C. Lower limit of transition range 

FIGUEE 34. Incipient submergence and free-flow transition range; influence of special design variations.

Figure 30, by comparison with figure 29, shows a slight advantage in 

using t/H-i to describe the tailwater levels at which the upper and 

lower limits occur. In particular, the difference in tailwater levels 

which reflects the influence of P on the lower limit is smaller when 

t/H-i is used. The data show no significant correlation with P for the 

upper limit.

Figure 31 shows the results of tests on three versions of the basic 

design. The curves obtained from the figure are shown for comparison on the subsequent figures in this series.

Figures 32 and 33 show the results of tests made with different 

values of Sp and Ss. In general, these data show good agreement with 

the curves obtained from model A. A minor correlation with Sp is 

not believed to be significant in view of the results shown on figure 34.

Figure 34 shows the results of tests made on the smooth-surfaced
DISCHARGE OVER EMBANKMENT-SHAPED WEIRS A53

1.0 

0.8

0.6 

0.4

0.8 

. 0.6

£

 S 0-4

</>

o>

"I 0.2

0.8 

0.6 

0.4 

0.2 

0.0

-0.2 

0.1

J 1 1 \ i l i l i

-i L i i i i i i i i i

. Incipient submergence

"l I

B. Upper limit of transition range

I I i i i i i i

~ " 0"

__-©-

o/

0.2 0.3

Model A (all surfaces smooth), curves

from fig. 31 

__ Model AA-2 screen-wire roughness

Head dafum of middepf/i of screen wire of 

crown line

I l l i l i l l

0.4 0.5 

Values of h, in feet

0.6 0.7 0.8

0.10 0.15 0.20

Values of h/L 

C. Lower limit of transition range

0.25

FIGUEE 35. Incipient submergence and free-flow transition range; influence of screen-wire roughness on

all surfaces.

models with special design variations. One of these, model L, with a 

small tripwire located at the downstream edge of the downstream 

shoulder, was especially designed to demonstrate the sensitivity of 

the transition-range limits to obstructions on the downstream side of 

the embankment. From the results of the tests on model L it is obvious that the transition limits cannot be defined with certainty for 

many natural prototype conditions. This conclusion is substantiated 

by the results of tests on model M, the model with berms on the embankment slopes. As would be expected, however, tests on model 

K-2, with the rounded upstream shoulder, agree with the results of 

tests on the basic design.

Figures 35, 36, and 37 show the influence of roughness on the 

transition-range limits. The results are remarkably similar for the
A54 FLOW OF WATER OVER DAMS AND WEIRS

1.0 

0.8 

0.6 

0.4 

1.0 

0.8 

0.6

!8 0.2

_2

CO

> 0.8 

0.6 

0.4 

0.2 

0.0

-0.2

-0.4

J> ___£> ~o- -0--

A. Incipient submergence

n r

B. Upper limit of transition range

"i i i i i i i i i i r

.£>--,.-13"

.--0--" _--©-____0 

o--' Model A (all surfaces smooth), curves

from fig. 31

Model AB (birdshot on shoulders and 

embankment slopes)

Head dafum af smooth surface at crown line

0.1 0.2 0.3 0.4 0.5 

Values of h, in feet

0.6 0.7 0.8

0.10 0.15 0.20

Values of h/L 

C. Lower limit of transition range

0.25

FIGUBE 36. Incipient submergence and free-flow transition range; influence of birdshot roughness on 

shoulder and embankment slopes.

three conditions of roughness represented by models AA-2, AB, and 

AC. In general, roughness tends to lower the tailwater levels for 

both the upper and lower limits. The curves are not well defined for 

small values of h because of the larger experimental errors which characterize this condition.

DEPTH AT CROWN LINE FOR FREE FLOW

Figures 38, 39, and 40 show the results of tests made to define the 

depth of flow at the crown line, y0, in terms of the theoretical critical 

depth for uniform flow, yc , where

(25)

Values of y0 used for this purpose were obtained us the average of 

several point-gage measurements of depth at the crown line. Because
DISCHARGE OVER EMBANKMENT-SHAPED WEIRS A55

i.o

0.8

0.6

___---® a o cr -° fi-°- o 

A. Incipient submergence

0.8

0.6

^0.4

£0.2 

o>

_3

> 0.8 

0.6 

0.4 

0.2 

0.0

-0.2

-0.4 

O.I

S. Upper limit of transition range

I I I I I I I I I

Model A (all surfaces smooth), curves 

from fig. 31

 O Model AC (birdshot on all surfaces)

Head datum at middepth of birdshot at crown

0 fine

j___i___i___i___i___i___i___i___i___i___i i i

0.2 0.3 0.4 0.5 

Values of h, in feet

0.6 0.7 0.8

0.10 0.15 0.20

Values of h/L 

C. Lower limit of transition range

0.25

FIGURE 37. Incipient submergence and free-flow transition range; influence of birdshot roughness on all

surfaces.

3

o

<D

1

\.<L 

I.I 

1.0

0.9

0.8

0.7

0.6 

0

i i i i i

^-"K A 0

X

Model A-2

Model A- 3

O

Model A-4

0 O.I 0.2

i i i i i i i i i i

* w 43-x -13-+^^^ -

-

i i i i i i i i i i

0.3 0.4 0.5 0.6 0.7 0

Values of /),in feet

0.10 0.15 0.20 

Values of h/L

0.25

FIGURE 38. Depth at crown line for free flow; basic design, model A.
\.<L

, ,

<^

=>? 1.0

0

£ 0.9

z)

> 0.8

0.7

i i i i i i i i i i i ii

.

ft

- 13 UA ^ * ~~ *b -_&^_-

 Model A (P= 1.17 ft), curve from fig. 38

0 Model B(P=0.875 ft) 

A Model C (P = 0. 583 ft)

H Model D(P= 0.292 ft)

1 ,C1.1

1.0

0.9

0.8

0.7

1 

O

_ 

____- *> ^

~

 Mod

cu

0 Mod

0.0 O.I 0.2 0.3 0.4 0.5 0.6 0.7 O.O O.i 

Values of h,\n feet 

A. Influence of h/ p L , \ B. Influence of pa

0.10 0.15 0.20 cross slope, Sp 

Values of h/L

1.2

^ I.I

\

^ 1.0

o

i 0.9

0.8 

O.7

i i i i i i i ii 1 i

G

 - A~~° ^ ~ -0 -3 D-- - ,____

a Q ^oCr £> _

O Model 1 (ss = 0014) Model A (Ss =0.062), 

A Model J (Ss = 0.079) curve from fig. 38 

H Model K-2 (rounded shoulder)

i i i i i i i iii ii

1.2 i 

41 

10 L fr

0.9

0.8 

0.7

 Model A (b

curve fr

O Model K-2 

0.0 O.I 0.2 0.3 0.4 0.5 0.6 0.7 0.0 O.I 

Values of h, in feet 

C. Influence of shoulder slope, Ss | ( : D. Influence of spe

OJO 0.15 0.20 debign variation

Values of h/L

FIGURE 39. Depth at crown line for free flow; smooth-sur
DISCHARGE OVER EMBANKMENT-SHAPED WEIRS A57

1.1

1.0

0.9 

OR

I I i

o o

 Model A (all s 

curve fr

i i i

I 1 ' I 

G O

jrfaces smooth), 

om fig. 38

I I I I I i i i

 a 2___

O Model AA-2 (screen-wire roughness) 

Head datum at middepth of

screen-w;re at crown line

1 I 1 I i I I I

-J?

0.0 0.1 0.2

A Screen-wire roughness 

on all surfaces

0.3 0.4 0.5 

Values of h, in feet

0.6 0.7

0.10 0.15 0.20 

Values of h/L

0.25

Values /yof0c 

o >- 

bo ioU

i i i i i i

- -er^-3 ©

- Model A (all surfaces smooth), 

curve from fig. 38

i i i i i i

i i i i i i i i

0 Model AB (birdshot on shoulders 

and embankment slopes) 

Head datum at smooth surface at crown 

ill. line .ill

0.0 -9,1 0.2 0.3 0.4 0.5 

Values of h, in feet 

B Qirdshot roughness on shoulder 

and embankment slopes

0.6 0.7

0.10 0.15 0.20 

Values of h/L

0.25

<o 

£0.9

0.8 

0

I 1 I i I 

0

 _ U. _(iL_ .

I I

_ Model A (all surfaces smooth), 

curve from fig. 38

i i i i i i i

0 0.1 0.2

C Birdshot roughness on 

all surfaces

I I 

<3~ o--

1 1 1 1 !

O Model AC (birdshot on all surfaces) 

Head datum at middepth of birdshot at 

! | crown line ( ( (

0.3 0.4 

Values of h, in feet

0.10

0.5 0.6

0.15

0.7

i

0.20 0.25

Values of h/L 

FIGURE 40. Depth at crown line for free flow; rough-surfaced models.

the roadway level as well as the water-surface level changes rapidly 

in the vicinity of the crown line, the measurements are not expected 

to be extremely accurate. Nevertheless, the results shown on the figures are remarkably consistent.

Figure 38 shows the results of tests on the basic design, and the 

average curve obtained from that figure is shown on figures 39 and 

40. The results of all tests on smooth-surfaced models agree well with 

the model A curve. The maximum deviation from the condition 

y0 =yc is about 5 percent. This is believed to be adequate confirmation 

of the assumption that critical-flow control occurs at the crown line.

Characteristics associated with roughness on the roadway are believed to be related to the thickness of the boundary layer. Thus, a 

small and not unexpected increase in y0/yc characterizes the results 

of tests on the rough-surfaced models, particularly model AC. Model
A58 FLOW OF WATER OVER DAMS AND WEIRS

AB, with roughness confined to shoulders and embankment slopes, 

gave results which agree substantially with those obtained from the 

basic, smooth-surfaced model.

For all designs and flow conditions tested, the test results give 

reasonably good confirmation of the assumption that critical depth 

occurs at or very near the crown line.

SUMMARY AND EVALUATION

Tests on 17 different embankment designs indicate that the discharge characteristics are nearly independent of embankment shape. 

The designs tested were modifications of a basic design which is typical 

of two-lane paved highways. Two features, embankment slope and 

roadway width, were identical for all designs. On the basis of the 

test results, it is believed that embankment slope is insignificant except 

as it affects the roller on the downstream side. The roller can be expected to have a small influence on the tailwater levels corresponding 

to incipient submergence and the free-flow transition range. The 

width of the roadway is believed to be significant only to the extent 

that it, as well as the discharge and the surface roughness, governs 

the head loss and the thickness of the boundary layer at the control 

section. These are effects which will have to be correlated with 

boundary-layer studies and full-scale prototype tests.

Embankment height, pavement cross-slope, and shoulder slope are 

insignificant in relation to free-flow discharge characteristics because 

the embankment comprises a critical-flow control section. Thus, because critical depth occurs very nearly at the crown line for all flows 

and shapes tested (see figs. 38, 39, and 40), boundary conditions upstream from the crown line have no influence on the free-flow headdischarge relationship. Similarly, because the effect of downstream 

disturbances cannot be transmitted upstream, through the section of 

critical flow, boundary conditions downstream from the crown line 

have no influence on the free-flow coefficient.

Surface roughness has a small but systematic influence on the principal flow characteristics. The effects associated with roughness in 

the model are difficult to interpret in terms of natural prototype roughness. Nevertheless,. the relative influence of a maximum range of 

prototype roughnesses is believed to be indicated by the results of 

the model tests. It is emphasized, furthermore, that the smoothsurfaced models are believed to be adequately representative of paved 

roadways in good condition.

Figure 41 shows a summary of the most important results of the 

tests made to determine the free-flow coefficient of discharge. A single 

curve represents all smooth-surfaced models (plus model AB, which 

was roughened with birdshot on the embankment slopes and shoul-
DISCHARGE OVER EMBANKMENT-SHAPED WEIRS A59

3.2

3.1

<o 3.0 

"o 

£ 2- 9

> 2.8 

2.7

2.6l_

0.0

Ail smooth-surfaced models Birdshot on all surfaces.

________ (model AC) 

Screen-wire roughness(model AA-2) Head datum at m/ddepth 

Head datum at middepth of of roughness 

roughness

O.I 0.2 0.3 0.4 0.5 

Values of h , in feet

O.6 0.7

0.10 O.I5 

Values of h/L

0.20 0.25

FIGUBE 41. Summary of the coefficient of discharge for free flow.

ders). Different curves show the results of tests on rough-surfaced 

models AA-2 and AC.

The overlapping abscissa scales shown on figure 41 are independently related to the flow pattern except in the middle range of values. 

In the lower range of values of h, boundary resistance dominates the 

flow pattern, and h/L is insignificant (except as L is involved in the 

loss of energy on the roadway). Conversely, in the upper range of 

values of h/L, boundary form and flow curvature dominate the flow 

pattern, and h is insignificant.

It is important to observe that the influence associated with small 

values of h is a "scale" effect; that is, it is related to the absolute value 

of h, whether it be a model or a prototype value. This does not imply 

that model and prototype coefficients will be identical for identical 

values of h, however, because the characteristics of the boundary layer 

at the control section are also related to the absolute value of L. 

Nevertheless, on the basis of the model tests, it is believed that effects 

associated with values of h less than 0.5 feet seldom would be significant in terms of the prototype. Furthermore, it is observed that the 

larger values of h/L shown in figure 41 are in excess of the values 

usually to be expected in the prototype.

Figure 42 shows a summary of the results of the tests made to 

determine the coefficient of discharge for submerged flow. It is noteworthy that the dependent variable on figure 39 is the simple coefficient of discharge (eq 9) and not, as usual, the ratio of the submerged-flow coefficient to a corresponding free-flow coefficient. This 

procedure is justified on the basis of the observation that the submerged-flow characteristics are independently related to the parameters which govern free flow. It is also noteworthy that the abscissa 

scale in figure 42 is the ratio of the downstream (tailwater) piezometric head to the upstream (headwater) total-energy head. This
A60 FLOW OF WATER OVER DAMS AND WEIRS

3.21 i i i i i i i i i i i 

All smooth-surfaced models

Screen-wire roughness ( model AA-2)

2.8 

2.6

<o 2.4

"o

K 2.2

Head datum at middepth of roughness 

> 2.0

Birds hot on shoulders and embankment slopes 

(model AB)

i Birdshot on all surfaces ( model AC) 

Head datum at middepth of roughness

1.4 

1.2

0.72 0.76 0.80 0.84 0.88 0.92 0.96 1.00

Values of t/H^

FIGUBE 42. Summary of the coefficient of discharge for submerged flow.

ratio, unlike the commonly used ratio of piezometric heads, results in 

a relationship with Oaubm which is virtually free of correlation with 

embankment height.

A single curve on figure 42 represents the submerged-flow characteristics of all the smooth-surfaced designs. Whereas the results of 

some of the tests deviated considerably from this smooth curve, the 

results of other tests showed conclusively that the relationship is 

extremely sensitive to roughness, irregularities, or obstructions on 

the downstream side of the embankment. Thus, in view of many 

uncertainties regarding the occurrence and effect of natural prototype 

features such as guardrails, windrowed gravel, and vegetation, it is 

believed to be impractical to seek a very accurate, general solution for 

submerged flow.

The influence of roughness on #8Ubm is shown by the three dashed 

curves on figure 42. These curves are believed to indicate the maximum range of the influence of roughness, although it is difficult to 

relate the model-test results to natural prototype conditions.

Figure 43 shows a summary of the results of tests made to determine the tailwater levels corresponding to incipient submergence and 

the free-flow transition range. The dependent variable in figure 43, 

t/Hi, is the same as the independent variable used in figure 42. The 

overlapping abscissa scales are the same as the corresponding scales 

in figure 41.
DISCHARGE OVER EMBANKMENT-SHAPED WEIRS A61

Figure 43 indicates that one curve can be used with sufficient 

accuracy to describe the incipient-submergence and upper-limit tailwater ratios for all smooth-surfaced designs. A correlation with P 

shown for the lower-limit tailwater ratio is indicated on figure 43(7

1.0

0.8

0.6 

0.4

1.0 

0.8 

0.6

0.4 

0.2

1.0 

0.8 

0.6 

0.4 

0.2 

0.0

-0.2

-0.4

-0.6

-0.8

O.I

i i i i \ i r

i i i i i i

A. Incipient submergence

i i r i r i T

B. Upper limit of transition range

Model D (P-262ft, 

prototype)

All smooth-surfaced models

A

(/> = !0.5ft, 

prototype)

Screent-wire roughness ( model A A-2) 

Head datum at middepth of roughness

____

Birdshot on shoulders and embankment slopes- 

(model AB)

Birdshot on all surfaces ( mo del AC) 

Head datum at. middepth of roughness

Note P- 10.5ft for all rough-surfaced models

0.2 0.3 0.4 0.5 

Values of /?, in feet

I__________________L__

0.6 0.7

0.10 0.15

Values of h/L 

C. Lower limit of transition range

0.20

FIGURE 43. Summary of incipient submergence and free-flow transition range.
A62 FLOW OF WATER OVER DAMS AND WEIRS

by limiting curves for the values of P tested. It is observed that the 

influence of P was not investigated separately for the rough-surfaced 

designs. Curves showing the results of tests on the rough-surfaced 

models are shown on the figure.

It should be emphasized tha the total-energy head, H^ is involved 

in the basic discharge equation (eq 8) as well as the tail water ratio, 

t/Hi. Also, whereas the solution for discharge requires a trial solution for Hi, the use of h instead of #1 will ordinarily result in a very 

small error in the first-trial solution.

It is pertinent in this summary and evaluation of the model test 

resultsj:o observe that, under certain circumstances, the coefficients 

and ratios used to describe the results are critically influenced by the 

location of the headwater- and tailwater-level measuring sections. 

In particular, for high values of h/P, the piezometric level used to 

determine h is influenced by head losses in the approach channel and 

flow curvature in the vicinity of the embankment. Thus, in order 

for the results to be applied accurately to similar model or prototype 

embankments, the measuring sections should correspond in location to 

the gage positions shown on figure 7.

INFLUENCE OF THE BOUNDARY LAYER ON FREE-FLOW 

CHARACTERISTICS

OBJECTIVE

In the preceding pages, dimensional reasoning and a simple discharge equation were used in an empirical analysis of laboratory tests 

of embankment models. The results provide a practical solution for 

the discharge characteristics of a variety of embankment forms and 

surface roughnesses. Like all empirical solutions, this solution is 

limited in usefulness by the scope of the tests on which it is based.

The second major objective of the investigation was the theoretical 

and experimental definition of the relationship between free-flow 

discharge and the boundary layer on the roadway. Here attention is 

restricted to free flow because the boundary layer is significant in relation to discharge control only when critical flow occurs on the 

roadway. The results of the studies made to accomplish the second 

objective are described in the following pages.

The ultimate goal of this part of the investigation could be described as the development of a general, analytical equation of discharge for free flow. More realistically, however, it is to test the 

validity of the approximate discharge equation (eq 19) in which 

8q and A are measures of the influence of boundary resistance. The 

minimum goal is a better understanding of some of the factors which 

govern .he flow pattern. An incidental benefit is a considerable 

amount of data on boundary layers in accelerated, free-surface flows.
DISCHARGE OVER EMBANKMENT-SHAPED WEIRS A63

DEFINITION OP 5 a AND X

The two quantities which represent the influence of boundary resistance in equation 19 <are 8q and A. The displacement thickness, 8g, is a 

measure of the influence of the boundary layer of the continuity equation. From equation 11 (p. 17) , for a boundary-layer flow such as that 

illustrated in figure 6,

g= rudz=(y-S,)U, (26) 

Jo

from which,

f=l jV tO dz= l-g dz, (27) 

or, because u=U when 0 exceeds the nominal boundary-layer thickness

From equation 28 it is apparent that 8q depends on the velocity distribution in the boundary layer.

For intermediate values of the Reynolds number, others (Bauer, 

1954; Halbronn, 1954; Delleur, 1957) have concluded that the velocity 

distribution in partially developed boundary-layer flows in open channels can be described by an equation of the power form,

u ,*v (29)

in which n is an exponent which must be evaluated by experiment. If 

equation 29 is substituted in equation 28, integration yields a simple 

equation for 8?,

a«=(-Tr) 5 > (3°)

which, of course, is applicable only when the velocity distribution can 

be described with an equation of the power form.

The velocity-head factor, A, is a measure of the influence of boundary resistance on the one-dimensional energy equation. It is defined 

by the relation.

X=JZ!_a Yl, (31) 

2g 2g

which is illustrated in figure 6. In this equation aF2/(2<7) is the true 

average-velocity head, which is given by the equation

v 1 r«

=~- v*dz. (32) 

2gq Jo
A64 FLOW OF WATER OVER DAMS AND WEIRS

From equations 31 and 32 and the continuity relationship (eq 26),

_ 1 fv 2 2

2.<7<Z Jo 

or, from equations 26 and 33,

X=l _ I I jj\ * VTT) \^z- (^

(It is of incidental interest that the integral in equation 34 is equivalent 

to 8e , the "energy thickness" of the boundary layer. Thus,

It is now apparent that A, unlike 8q depends on q and y as well as the 

velocity distribution.

In equation 34, the ratio which precedes the integral is proportional 

to a Froude number. When the flow is critical, the value of the 

ratio is approximately 1/2. Therefore, at the crown line, where the 

flow is very nearly critical for all conditions tested (see figures 38, 

39, and 40), an adequate approximation is

or, because u U when z is greater than 8,

It is emphasized that the coefficient 1/2 in equations 35 and 36 restricts their use to the critical-flow section.

For the conditions which warrant the assumption of an n- power 

velocity distribution (eq 29) , equation 36 gives

» (37)

or, from equation 30, (38)

BOUNDARY LAYERS IN ACCELERATED MOTION

The most promising solution for the growth of the boundary layer 

on the roadway is based on von Karman's semi-analytical equation for 

boundary layers in accelerated motion (Schlicting, 1955). The equa-
DISCHARGE OVER EMBANKMENT-SHAPED WEIRS A65

tion is derived from an application of the momentum principle. It 

presumes a hydrostatic normal-pressure distribution, and it ignores 

the momentum of the turbulence. It has been substantiated reasonably 

well for flat plates and circular pipes.

The von Karman equation can be written in the form

75^' (39) pU2 ds

in which, in addition to the symbols previously defined, TO is the shear 

stress at the boundary, s is distance along the boundary in the direction 

of mean motion, and 8m is the "momentum thickness" of the boundary 

layer. Defined by the deficiency of momentum flux which results 

from the formation of the boundary layer, 8m is

/* x / \

/ 0 /ti / /)/ \

Sm =\ ^(l-£)dz. (40) 

Jo -U \ U/

For an fi-power velocity distribution (eq 29), equation 40 gives

which, with equation 30, gives

(42>

From equation 42 it is apparent that 8m as well as A (eq 38) can be 

defined in terms of 8q . For convenience, therefore, the von Karman 

equation can be converted to the form

dS, /2n±3\

~

\2U2 ds } { }

which, of course, is limited by the assumption that the velocity can 

be described with an equation of the power form.

The left-hand member of equations 39 and 43 is a nondimensional 

shear coefficient, c/. In general, cf is a function of the boundary-layer 

Reynolds number and the relative roughness,

TO (AA\ (44)

in which v=^/p is the kinematic viscosity of the fluid. Approximations for Cf have been derived from analogies with uniform flow in
A66 FLOW OF WATER OVER DAMS AND WEIRS

pipes. One such approximation, for smooth boundaries only, is the 

equation identified with Blasius,

/rr*\-o.25 

C/=0.0225 } > (45)

which is based on the assumption of a 1/7-power velocity distribution. 

Equation 45 has been substantiated in application to smooth, flat plates 

and an intermediate range of Reynolds numbers. Corresponding 

equations for* rough boundaries have been based on the von Karman 

logarithmic velocity-distribution equation.

As an implicit equation for the displacement-thickness gradient 

(dBq/ds), equation 43 is limited mainly by the assumption of an 

n-power velocity distribution. However, its integration, to get 8q as 

a function of s, depends on experimental evaluation of n, U. and 

dU/ds, as well as TO . Thus, the most promising means of obtaining a 

general solution for 8q is fraught with potential obstacles. The information needed to appraise these obstacles must be obtained from measurements of boundary-layer characteristics under a wide variety of 

boundary and flow conditions.

BOUNDARY-LAYER MEASUREMENTS

PURPOSE AND SCOPE

The primary purpose of the boundary-layer measurements made as 

a part of this investigation was to test the validity of the discharge 

equation (eq 19) which contains 8q and A. A simple verification of 

equation 19 could have been accomplished with measurements of the 

velocity distribution at the crown line only. Aside from substantiating certain assumptions regarding the influence of boundary resistance, however, verification by direct determination of the crown-line 

values of 8q and A would lead to nothing more than an empirical solution for q. This solution, like simpler ones involving only the coefficient of discharge (eq 7 or 8), would be limited in usefulness by the 

range of conditions actually reproduced in the laboratory. On the 

other hand, if equation 19 is valid, the ideal solution for q would involve the computation of 8q and A from a general equation which describes the growth of the boundary layer on the roadway. The ideal 

solution would be applicable to roadways of all widths, embankments 

of all shapes- within reason, and a full, practical range of discharges. 

With this ideal as a possible result, the experiments included sufficient 

vertical velocity traverses and piezometric profile measurements to 

define the boundary layer between the upstream edge of the upstream 

shoulder and a point downstream from the control section near the 

crown line.
DISCHARGE OVER EMBANKMENT-SHAPED WEIRS A67

Tests were made on four different models, A-l, K-l, AA-1 and KA. 

The scope of the tests is shown in table 2, and a summary of the 

velocity measurements is shown in table 4 in the section "Experimental Data." Data used in this part of the report were obtained 

from tests made 'by Davidian. To a large extent, the tests duplicate 

and confirm the results of tests made earlier by Sigurdsson.

TABLE 2. Scope of boundary-layer tests made by Davidian1

Model 

design

A-l_____

K-1--...

AA-1____ 

KA___.__

Test 

No.

1

2 

3 

4 

5 

1

2

3 

4 

5 

1 

2 

1

2

Head, A, 

in model 

(ft)

0.084

. 183 

.301 

. 475 

. 632 

. 083

. 183

. 301 

.475 

.632 

. 183 

. 476 

. 183

.475

Discharge, g, 

in model 

(cfs per ft)

0.071

. 234 

. 503 

1.01 

1. 58 

.070

. 238

.511 

1. 02 

1. 59 

. 225 

. 991 

. 226

. 991

Number of 

velocity 

traverses

8

8 

10 

10 

10 

5

6

5 

6 

6 

10 

10 

6

6

Remarks

Basis design 2 , smooth surface.

Basic design, smooth surface. 

Do. 

Do. 

Do. 

Rounded shoulder 3 , smooth

surface. 

Rounded shoulder, smooth surface. 

Do. 

Do. 

Do. 

Basic design, screen roughness. 

Do. 

Rounded shoulder, screen

roughness. 

Do.

1 Davidian, Jacob, 1959, Influence of the boundary layer on embankment-shaped weirs: Georgia Inst. 

Technology, Master's degree thesis, 97 p., 38 figs.

2 See fig. 9 for details of basic design.

3 Rounded transition between upstream embankment and shoulder surfaces, as shown in fig. 10,4. All 

other shape details as in fig. 9.

Design details of the models are given in table 1 and figures 9 and 

10. Model A-l is the basic design. Model AA-1 is the same embankment section, but with screen roughness added. Details regarding the screen and the method of use are given on page 21. Model K 

is a design especially created for the boundary-layer tests. As shown 

in figure 1(L4, it involved a rounded transition between the upstream 

embankment and shoulder surfaces. The purpose of the rounding 

was to prevent flow separation at ohe upstream edge of the shoulder. 

Tests were also made on this embankment design with screen roughness added (model KA).

SUMMARY OF RESULTS

The basic data for this part of the report are the results of the 

velocity measurements which are summarized in table 4. Typical 

data illustrating the growth of the boundary layer on the upstream 

side of the roadway are shown in figures 44 and 45. Figure 44 shows 

the results of tests on models A-l and AA-1, each with a discharge
Sta. Sta.

Sta. Sta.

00060.10 0.25 0.40

Sta. 

1.00

23| 

Sta. 0.67

A Model A-l, test 4

I 234 

Values of u

Values of u, 

Sta. 0.67

B Model AA-I, test 2

FIGURE 44. Velocity distribution in boundary layer on upstream 

discharge.
Sta.

0.20 °- 60

a> 111

* \\\

^ 11

Flow N 1 111

> "5 5 J\

<n 111

a> 

\

."

'

>---~l 23 1 234 !

-^-"i \

^ ! Sta 0.67

i

__ 

V

Sta -°-00 A Model K-

Sta. Sta. 

0.20 0.60

<s |0 °CQ

-" 71

Flow x 1 OrfH

* N 1 1 1 1 YU

;

t

^*~~~ \ 23 1 234 j

^ i Sta.0.67

Sta. 0.00

r5

V

fi Model KA

FIGURE 45. Velocity distribution in boundary layer 

typical disc
A70 FLOW OF WATER OVER DAMS AND WEIRS

(q) of approximately 1 cubic foot per second per foot. Figure 45 

shows the results of similar tests on models K-l and KA.

The boundary layer is indicated in figures 44 and 45 by velocity 

profiles which 'are superposed on a silhouette of the embankment cross 

section. The location of the traverse section for each velocity profile 

is designated by a station number which is a measure of its distance 

from the nominal intersection of the upstream embankment slope 

and shoulder surface. The crown line for all models is at station 1.67. 

Thus, station 0 for the models with rounded shoulders (models K-l

0.001

0.50.6 0.8 1.0 1.5 2.0 3.0 

Values of u, in feet per second

A. Basic embankment design, model A-1

.0 1.5 2.0 3.0 4.0 

Values of u, in feet 

per second

B. Basic design with screen 

roughness on all surfaces, 

model AA-I

FIGURE 46. Boundary-layer velocity distribution near crown line (station 1.65), models A-1 and AA-1.
DISCHARGE OVER EMBANKMENT-SHAPED WEIRS A71

and KA) is on the roundings, at a distance 1.67 feet upstream from 

the crown line. Measurements were made at station 1.65 instead of 

station 1.67 in order to avoid the influence of the separation zone 

which occurs on the downstream side of the crown line.

Figures 46 and 47 show velocity profiles at station 1.65 for all the 

tests made on models A-l, AA-1, K-l, and KA. The profiles, like 

those in figures 44 and 45, are plotted on logarithmic coordinates.

0.001

0.6 0.8 1.0 1.5 2.0 3.0 4.0 

Values of u, in feet per second

A. Rounded upstream shoulder, 

smooth surfaces, model K-l

1.0 1.5 2.0 3.0 4.0 

Values of u, in feet

per second

B. Rounded shoulder, screen 

roughness on all surfaces, 

model KA

FIGURE 47. Boundary-layer velocity distribution near crown line (station 1.65), models K-l and KA.
A72 FLOW OF WATER OVER DAMS AND WEIRS

Straight lines are fitted to the measurements in the boundary layer, 

and the intersections of the straight lines with the vertical lines 

signifying the magnitude of U are marked as the outer limit of the 

nominal boundary layer. The fit of the straight lines with the points 

in the boundary layer is an indication of the validity of the assumption that the velocity distribution can be described with an equation 

of the power form, as in equation 29,

(29)

and the slope of the straight line is a measure of n in that equation.

Values of 8 and l/n obtained from figures 46 and 4Y are shown 

plotted as a function of h in figure 48. (Here l/n is used to avoid 

fractions.)

Values of 8 and l/n obtained from figures 44 and 45 by the same 

procedure are shown plotted in figure 49 as a function of s.

With some exceptions, notably at stations near the upstream edge 

of the shoulder, the velocity distribution in the boundary layer appears to be reasonably approximated by an equation of the power

I I I I I I I 1 I I 

A. Nominal boundary-layer thickness , 6

S- --sL Mode MA -1 ~~ ~~ 

/ 

Q ____I____I____i____i____I____i____I____i____I____I____I____L

O.I 0.2 0.3 0.4 0.5

Values of /?, in feet 

B. Quantity \/n in power equation

0.6

FIGURE 48. Boundary layer near crown line (station 1.65).
Flow

60

50

40

30

20-

DISCHARGE OVER EMBANKMENT-SHAPED WEIRS

-Origin of 5 distance

A73

o 0 

Model A-l , test 4 Model AA-I, test 2

Model K-l, test 4 Mode! KA , test 2

Flow,

12

10

0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

Values of 5, in feet 

A. Nominal boundary-layer thickness,8

-Origin of 5 distance Crown line

O Q

Model A-l, test 4 Model AA-I , test 2

A X

Model K-l, test 4 Model KA , test 2

0.2 0.4 0.6 0.8 1.0 1.2 1.4

Values of 5, in feet

B. Quantity \/n in power equation

1.6 1.8 2.0

FIGURE 49. Boundary layer on upstream side of roadway for a typical discharge.

form. However, as shown in figures 48 and 49, 8 and \/n are not 

clearly defined as functions of s and h (or q).

For moderately large discharges, the boundary-layer thickness 

grows most rapidly in the vicinity of the shoulder. The rate of 

growth near the crown line is very small. Values \/n are smaller 

for the rough-surfaced models (AA-1 and KA) than for the smoothsurfaced models (A-l and K-l), except at small values of h (fig.
A74 FLOW OF WATER OVER DAMS AND WEIRS

485). For larger values of #, the smootH-surfaced models show 

values of l/n approaching the value 7, which is commonly used for 

smooth pipes and flat plates.

Values of 8 for the smooth-surfaced models indicate an expected 

decrease in boundary-layer thickness with rounding of the upstream 

shoulder. This result is attributed to the elimination of the separation 

zone and the consequent displacement of the boundary layer at the 

upstream edge of the shoulder. However, the largest values of 8 

shown on figure 49J. are those which were measured on the roundshouldered model when it was covered with screen-wire roughness. 

In general, of course, the effect of roughness is to increase the thickness of the boundary layer. It is suggested that the contradiction 

which is related to the combined influence of rounding the roughness 

might be the result of a difference in tautness of the screen wire used 

on models AA-1 and KA.

In view of the inconsistencies in the data shown in figures 48 and 

49, no attempt has been made to draw smooth curves through the 

plotted points.

COMPUTATION OF C BASED ON BOUNDARY-LAYER 

MEASUREMENTS

Verification of the analysis which led to equation 19 requires that 

values of 8q and A at the crown line be determined from the boundarylayer velocity measurements. Two methods of evaluating 8q and A 

were used. One method, which was evolved from the assumption of 

an n-power velocity distribution, is suggested by equations 30 and 38, 

which require prior evaluation of n from the velocity profiles shown in 

figures 46 and 47. The results based on this method of computation 

are shown by the solid symbols in figure 50. Here the quantity (8q A), 

which appears in equation 19, is plotted as a function of h.

The second and most accurate method of evaluating 8q and A requires integration of equations 28 and 36. This was accomplished as 

a numeral approximation. Values of u/U were plotted on rectilinear 

graph paper, smooth curves were drawn through the points, mean 

values of u/U were read from the smooth curves, and the integrals 

represented by equations 28 and 36 were evaluated as the summation 

of finite increments. Values of (8q \) computed by this method are 

shown in figure 50 with open symbols.

Values of (8q A) computed by the two different methods agree 

reasonably well. The paucity of data did not permit accurate definition of the relation between (8ri ~\) and h for the rough-surfaced 

models, but a constant value of (8q -\) =0.001+ foot was quite well 

defined for the two smooth-surfaced models. For the purpose of 

testing the validity of equation 19, a straight-line relationship between 

h and (8q A) was assumed for the rough-surfaced models. The
DISCHARGE OVER EMBANKMENT-SHAPED WEIRS A75

+- 5

0)

= 4 

o

T

.^2

1 I I I I I I I I I I I I

...... .. j , .. , Note: Open symbols for sq and x

o . Model A-l o . Model AA-1 obtained from numerical?ntegra .

.A Model K-l o» Model KA

Rough-surfaced models AA-1, KA

tion; solid symbols for Sq and x 

obtained from power-equation 

approximation.

mooth-surfaced models A-l, K-l

0.1 0.2 0.3 0.4 

Values of h, in feet

0.5 0.6 0.7

FIGURE 50. Values of (Sq X) from boundary-layer velocity measurements.

straight line is shown in figure 50. It is observed that the unit divisions on the (Sq A) scale in figure 50 represent differences of only 

0.001 foot, and that the approximation represented by the straight line 

is not critical in determining the validity of equation 19.

Figure 51 shows a comparison of experimentally determined and 

computed values of the coefficient of discharge for the smooth- and 

rough-surfaced models. The experimentally determined values were 

obtained from the summary curves in figure 41. The computed values 

were determined from equation 20,

(20)

in which C is the coefficient of discharge defined by equation 19. 

Values of (Sq A) were determined from figure 50.

The magnitude of the discrepancies between the experimental and 

computed curves in figure 51 is a measure of the validity of equation 20. The disparity of values of G is reasonably small in the middle 

range of values of h. For smaller values of 7i, computed values of C 

are larger than experimental values. For the largest values of A, computed values of G are smaller. The difference in G values at small 

values of h can be attributed, perhaps, to a fault in the analytical 

treatment of boundary resistance. The difference at larger values of h 

is believed to be associated with boundary-form effects or, specifically, 

flow curvature at the control section. Thus, this difference is believed 

to be related to the magnitude of li/L rather than the magnitude of h. 

The analysis which led to equations 19 and 20 did not involve the
A76 FLOW OF WATER OVER DAMS AND WEIRS

3.2

3.1

,- 2.8

T I I I I 1 I 

Computed, models A-l and K-l

3.0 ____

models (fig. 41 ) 

2.9

Experimental, all smooth-surfaced

I I i I I I I I I I L

A Smooth-surfaced models

3.1

3.0

1 1 1 I 1 1 I I I 1 I I I 1 1 Computed, models AA-I ond KA C=3i°9

^,~^~~~'~' Experimental, model AA-2 (fig. 41) 

2.9

2.8

) O.I 0.2 0.3 0.4

Values of h,

i

0.5

in feet

i

0.6 0.7

i

0.8

0.10 0.15 0.20 025

Values of h/L 

B Models with screen-wire roughness

FIGTJBE 61. Values of C computed from equation 20 and figure 50.

influence of flow curvature. Consequently, it is not surprising that 

discrepancies due to this influence are indicated by the comparison 

shown in figure 51.

The discrepancies between experimental and computed values of C, 

especially at small values of /i, were somewhate greater than was expected. Therefore, a review of the analysis is pertinent.

MODIFIED DISCHARGE EQUATION

In the process of evolving equations 19 and 20 from the one-dimensional energy equation (p. 17), the term HL, which represents the loss 

in head between the headwater measuring section and the crown line, 

was assumed to be negligible. It was particularly convenient to 

neglect HL because it is not independently related to the discharge 

characteristics of the embankment. Furthermore, it is difficult to 

evaluate, because it depends on the location of the measuring section 

and the resistance characteristics of the upstream channel.

For the model tests reported herein the headwater was measured at 

a section which was relatively close to the embankment. Thus, HL 

was very small. Nevertheless, it is recognized that neglecting HL has 

the effect of making values of C computed from equation 20 larger 

than they would be if fIL were considered. Moreover, the relative 

effect of neglecting HL increases as h decreases. These observations 

suggest that neglecting IIL is a possible cause of the disparity of computed and experimental values of C at lower values of h in figure 51.
DISCHARGE OVER EMBANKMENT-SHAPED WEIRS A77

Not only is HL difficult to compute, but it is also difficult to measure 

accurately in the laboratory. From the magnitude of the discrepancies 

shown in figure 51 it is apparent that the effect attributable to HL is 

small and, therefore, that HL in the model tests was a very small quantity. Thus, when it is computed as the difference between total-head 

quantities determined from pitot-tube and point-gage measurements, 

the experimental error in HL is likely to be excessive. This conclusion 

was confirmed by attempts to determine HL from the model test data.

As an alternative and admittedly empirical method of handling HL 

in equation 14, it is observed that A is another small term in the 

equation and that the assumption of equal magnitudes of HL and A 

would result in their mutual elimination from subsequent equations. 

The resulting counterparts of equations 19 and 20 involve the relative 

magnitude of 8^ alone:

g=3.09 #, 

and,

(47)

Equation 47, like equation 20, is readily tested by comparison with the 

experimentally determined values of C.

Values of 8q corresponding to the values of (8f/ A) in figure 50 are 

shown in figure 52. The relationship between 8^ and h is not well 

defined, but the straight-line approximations shown on the figure are 

adequate to test the validity of equation 47.

Figure 53 shows a comparison of experimentally determined values 

of C from figure 41 with computed values of C from equation 47. At 

small values of A, for which the influence of boundary resistance is a 

maximum, the comparable curves are in substantial agreement. Discrepancies at large values of J\/L demonstrate, again, the influence 

which has been attributed to flow curvature at the control section. 

Thus, for the conditions represented by the models, it is concluded 

that the influence of boundary resistance is effectively accounted for in 

equations 46 and 47. It is emphasized that the test conditions include 

the location of the headwater gage, and that the effect of this condition 

is reflected in the experimentally determined values of C.

INFLUENCE OF THE SIDEWALL BOUNDARY LAYER

Values of q used to determine C from the model test results were 

computed from equation 24,

2=' (24)
A78 

ii

10

9

8

4-> _

Q> 7

QJ

c

 ~ 6 

o. i

^5

o 4 

(/>

<D

__3

> 3

2

1

FLOW OF WATER OVER DAMS AND WEIRS

I I i 

° Model A-l 

° Model AA-1 

A * Model K-l 

o » Model KA

Note: Open symbol for Sq obtained 

from numerical integration; solid 

symbol for &q obtained from powerequation approximation

i i i i \ i i f i r

Rough-surfaced models AA-1, KA

Model K-l

j__i__i__i__i i i j_ J____I____|____I

0.1 0.2 0.3 0.4 

Values of h, in feet

0.5 0.6 0.7

FIGURE 52. Values of Sg from boundary-layer velocity measurements.

3.2 

3.1 

3.0 

2.9

3.2 

3.1 

3.0

2.9 

2.8

_ Computed, model K~ Experimental, all smooth-surfaced 3.09

0

models (fig. 41 K________- ""' 2^±

"Computed, model A-|

A Smooth-surfaced models

1 ! 1 O.I 0.2 0.3 0.4 0.5 

Values of /?, in feet

0.6 0.7 0.8

0.10 0.15 0.20

Values of h/L 

B. Models with screen-wire roughness

0.25

FIGURE 53. Values of C computed from equation 47 and figure 52.
DISCHARGE OVER EMBANKMENT-SHAPED WEIRS A79

in which B is the average width of the test flume in the vicinity of the 

control section. It is recognized that this procedure ignores the effect 

of the boundary layer which occurs on both sides of the flume. Thus, 

if 8'q is defined as the displacement thickness on the side walls, at the 

control section, a better definition of q is

The experiments made for this investigation did not include measurements of 8'q . It might be assumed, however, that values of 8'q are 

commensurate with values of Sq measured for the round-shouldered, 

smooth-surfaced model. (The walls of the flume are plate glass.) 

Thus, a reasonable approximation of an average value of S'a, corresponding to the average value of Sq for model K-l on figure 52, is 

0.003 foot. The corresponding error in the computation of q (and C] 

is indicated by the ratio

Q

B _3.010-2(0.003)

Q ~ 3.010 =0.998. (49)

Therefore, in the experimentally determined values of C used in the 

report, the average relative error due to neglecting the sidewall 

boundary layer is estimated to be approximately 0.2 percent. This is 

less than the experimental error to be expected.

GENERAL SOLUTION FOB 5 a

Equation 47 has been substantiated as an effective means of accounting for the influence of the boundary layer on free discharge. However, the verification shown in figure 53 is based on measured 

crown-line values of Sq . Practical use of equation 47 requires a convenient, general solution for 8q. Anything less than this would result 

in an empirical discharge solution of limited usefulness and considerably less convenience than that which involves the simple, experimentally determined coefficient of discharge.

Equation 43 was proposed as a basis for a general solution for &q . 

Potential obstacles to its use are associated with the evaluation of TO 

(or <?/),?},£/, and dU/dx. It is now possible to appraise these obstacles 

on the basis of the boundary-layer measurements. The following 

observations are concerned with the practicality of integrating equation 43 to obtain 8q as a function of s:

1. Equation 43 involves the assumption that the velocity distribution 

in the boundary layer can be described with an equation of the
A80 FLOW OF WATER OVER DAMS AND WEIRS

power form. In general, the measurements confirm the use of 

this assumption for rough-surfaced as well as smooth-surfaced 

models. Values of n are not clearly defined as functions of s and 

h (or q). Nevertheless, for all but the smallest values of A, and 

for virtually the full distance upstream from the crown line, 

figures 48B and 49B support the use of a different, approximate, 

constant value of n for each of the models tested. Whereas the 

assumption of constant values of n facilitates the integration of 

equation 43 for specific models, the substantial variation of n 

with shape and roughness is an obstacle to a general solution.

2. Figures 44, 45, and 49A show that a secondary layer of considerable thickness exists at station 0 (at the nominal intersection of 

the upstream shoulder and embankment surfaces). The data 

show that the thickness of the layer at station 0 varies with model 

design and discharge, and that it is sensitive to the occurrence of 

a separation zone immediately downstream from station 0. These 

observations confirm the assumption that the effective origin of 

the boundary layer is upstream from station 0. Although the 

data are inadequate to provide a general method of accurately 

determining the location of the origin, it is observed that the 

thickness of the layer changes very rapidly in the vicinity of 

station 0 and very slowly in the vicinity of the crown line. Consequently, the use of an approximate origin probably would result 

in a relatively small error in crown-line values of 8q computed 

from equation 43.

3. The experimental data show that the boundary layer is displaced 

by the separation zone which occurs at station 0, especially in 

models A-l and AA-1. The thickness of the layer downstream 

from the point of reattachment is increased as a result of this displacement, but the increase, as indicated by a comparison of 

boundary layers on models A-l and K-l, diminishes with distance from station 0. The influence on crown-line values of 8q is 

not believed to be substantial.

4. The boundary-layer measurements do not provide a direct means 

of evaluating the effects of curvilinear flow in relation to the 

assumptions made in deriving equation 43. This influence remains an unknown but probably minor source of error in the 

application of the equation to the computation of V

5. Integration of equation 43 requires evaluation of the relationship 

between U and -<?. Figures 54 and 55 show the experimentally 

determined values from the boundary-layer tests. For each test
DISCHARGE OVER EMBANKMENT-SHAPED WEIRS

N Origin of s distance Crown line

A81

CO 

CD

J3

O

4.5

4.0

-3.5

3.0

2.5

2.0

1.5

1.0

0.5

3.5

3.0

2.5

2.0

1.5

1.0

0.5

0.0

xJ.Model A-l

0.0 0.4 0.8 1.2 1.6 

Val ues of s, in feet 

B. M o d e I A A - 1

FIGURE 54. Velocity outside the boundary layer , model of basic design.

2.0
A82

1

o

Q.

O) 

O)

M 

J ~B

>

4.5

4.0

3.5

3.0

2.5

2.0

1.5

1.0

0.5

3.5

3.0

2.5

2.0

1.5

1.0

0.5

0.0

FLOW OF WATER OVER DAMS AND WEIRS

^

i____I____i____i____I____I

A Model K-l

J____I____I____I____I____I____I J____I____I

0.0 0.4 0.8 1.2 1.6 2.0 

Valu es of s , in feet

B. Model KA

FIGURE 55. Velocity outside the boundary layer, model with rounded upstream shoulder.
DISCHARGE OVER EMBANKMENT-SHAPED WEIRS A83

the relationship is described approximately by an equation in the 

straight-line form,

U=U0+m8, (50)

in which U0 is the velocity outside the boundary layer at station 

0, and m is the slope of a straight line fitted to the plotted points, 

Comparison of the data shown in figures 54 and 55 indicates 

that m varies with the shape of the upstream shoulder, whereas 

it is virtually independent of discharge and embankment roughness. (Thus, m is approximately 0.5 for most tests on models 

A-l and AA-1 and 0.3 for tests 011 models K-l and KA.) The 

quantity £70 , of course, varies with the discharge. It also varies 

with the shape of the upstream shoulders; but it, too, is virtually 

independent of embankment roughness. The form of equation 50 

does not preclude integration of equation 43. However, because 

U0 varies with discharge, the computation of q by means of 

equations 43 and 46 would require a tedious successiveapproximations procedure.

6. The principal effects of embankment roughness are related to its 

influence on cf in equation 44 and and the exponent n in the 

velocity-distribution equation. Equation 43 does not account 

specifically for the influence of roughness, and the experimental 

data do not provide a general means of evaluating that influence 

in terms of a nondimensional relative-roughness parameter. 

Consequently, it is impossible to use the model test data to determine the effect of roughness in the prototype. 

The foregoing observations lead to the conclusion that a discharge 

equation which depends on equation 43 for the crown-line value of 8q 

is neither general nor practical. In summary, the decisive obstacles 

revealed by the experimental data are: (a) n varies substantially 

with both boundary form and roughness and, less critically, with s 

an<# q; (b) U0 varies with boundary form and discharge; and (c) m 

varies with boundary form. Furthermore, the data are inadequate to 

provide a general solution for r0 in terms of discharge and relative 

roughness. Whereas the combination of these obstacles precludes a 

satisfactory general solution for q, it neither disproves nor proves the 

validity of equation 43 as applied to embankment-shaped weirs. For 

academic interest, at least, the validity test is a logical, terminal objective of the boundary-layer measurements.

On the assumption that the Blasius equation for cf (page 66) is 

applicable to the smooth-surfaced models, equations 30 and 45 can be 

substituted in equation 43 to give

$+030 GW %
A84 FLOW OF WATER OVER DAMS AND WEIRS

which is an implicit equation for the displacement-thickness gradient 

(d8q/ds), in terms of the local values of n, £7, 8a, and dU/ds at distance s from the origin of the boundary layer. If equation 50 is 

substituted for £7, equation 51 can be integrated, whence

m* 1

in which,

#i=0.028(2n+l)

and,

(54)

The tests selected for verification of equation 52 are test 4, model 

A^l, and test 4, model K-l, tests which previously were used to represent the smooth-surfaced models in figure 49. The experimental data 

required for the evaluation of equation 52 were obtained from figures 

48, 49, 54, and 55. For comparison, with computed values of 8q from 

equation 52, measured values of 8 from figure 49 were converted to 

values of 8Q by means of equation 30, using average values of n. Values 

of £70, m, and 1/n used in equation 52, and values of n used in the 

conversion of 8 are shown on figure 56.

Figure 56 shows a comparison of the computed and experimentally 

determined values of 8f/ . The s-origin for the values computed from 

equation 52' was assumed to correspond with station 0 for the boundary - 

Jayer measurements. For this condition, experimental values of 8q 

compare favorably with the computed values at the crown-line station. 

Experimental values are larger than computed values at upstream 

stations, and values for model A-l show the effect of the separation 

zone at station 0. A small s-distance displacement of the computed 

curve would cause it to show good agreement with the K-l curve 

over a large part of the roadway. This displacement could be defined 

as the distance from station 0 to the effective origin of the boundary 

layer for that test. However, it is observed that only the crown-line 

value of 8q is involved in the discharge equation (eq. 46), and the 

crown-line value is best defined by the computed curve when its origin 

is* at station 0, as in figure 56.

The comparison shown on figure 56 is limited evidence of the validity of the analysis which led to equations 43 and 52. Nothing more 

is expected from this phase of the 'investigation, which previously 

was acknowledged to be of academic interest only.
DISCHARGE OVER EMBANKMENT-SHAPED WEIRS A85

00
A86 FLOW OF WATER OVER DAMS AND WEIRS 

COMPUTATION OF C FOB THE PROTOTYPE

The search for a general solution for boundary-layer growth was 

previously (p. 79) related to the need for a means of computing the 

crown-line value of 8q for prototype embankments of various forms, 

widths, and roughnesses. Equation 52 already has been described as 

impractical as a general solution for the boundary layer. Furthermore, the approximate confirmation of model values shown on figure 

56 is limited to a small range of model conditions. Nevertheless, this 

limited evidence of the validity of equation 52 is encouragement for 

an attempt to compute G for prototype embankments similar to the 

basic, smooth-surfaced model.

Prototype values of 8q were computed from equation 52 with 

m=0.3, l/n=7, and values of Z70 equal to the square root of the prototype-model length ratio (9:1) times the corresponding model values 

of Z70 from figure 55A. Using these values of 8g, values of G were 

computed from equation 47. The results are shown by the solid line 

in figure 57.

'o

rj

O.tL

3.1

3.0 

2.9 

2.8

2.7 

? R

i i i i i i i i i i i i i

-

Computed for prototype from 

equations 47and 52

From tests on smooth-surfaced 

models (fig. 4!)

1 1 1 2345 1 1 1 I 1 1 1 1 1 !..._

Values of /?, in feet (prototype)

0.14 0.18 0.22 

Values of h/L

FIGURE 57. Computed prototype values and measured model values of C, smooth-surfaced embankments.

Considering the limitations of the data on model boundary layers 

and the lack of data on model-prototype conformity, the solid-line 

curve shown in figure 57 is of doubtful accuracy. Nevertheless, it is 

reasonable to describe the curve as an estimate of the relative influence, of the boundary layer on a 9 :l-scale prototype which corresponds in design to the smooth-surfaced models.
DISCHARGE OVER EMBANKMENT-SHAPED WEIRS A87

Shown for comparison in figure 57 is the curve from figure 41 

which is the average for all smooth-surfaced models. Actually, the 

model curve is shown in two parts. On the left, the dashed-line 

curve covers the range of values of head in which the influence of 

boundary resistance is dominant in the model. (The corresponding 

range of heads for a 9 :l-scale prototype is represented by the solidline curve in the middle.) On the right, the dashed-line curve covers 

the range of h/L values in which the influence of flow curvature was 

dominant in the model. The prototype counterpart of this curve can 

be expected to be similar to the model curve. However, it cannot be 

predicted accurately without experimental data from tests on models 

of different size (scale) or from tests on prototype embankments.

CONCLUSIONS

From the results of the model tests summarized on page 58, it is concluded that the coefficient of discharge for free flow is primarily a 

function of head, roadway roughness, and the head-width ratio 

(h/L). The coefficient of discharge for submerged flow is primarily 

a function of the submergence ratio (t/Hi) and the roughness of the 

roadway surface. These conclusions are substantiated by the summary curves in figures 41 and 42. In general, values of t/Hi corresponding to incipient submergence and the free-flow transition-range 

limits vary with head, roadway roughness, and h/L. This conclusion 

is substantiated by the curves shown in figure 43.

For intermediate values of head, the coefficient of discharge for 

free flow is nearly equal to the ideal value (3.09) which corresponds 

to the assumption of critical-depth control at the crown. For smaller 

values of head, the influence of boundary resistance causes the coefficient to be smaller than the ideal value. For larger values of head 

(or, actually, larger values of A/Z), the influence of flow curvature 

at the control section causes the coefficient to be larger than the ideal 

value.

The influence of boundary resistance on the free-flow coefficient of 

discharge is related to the relative thickness of the boundary layer 

at the crown line of the embankment. The growth of the boundary 

layer and, therefore, the thickness of the boundary layer at the crown 

line can be computed approximately with empirical equations which 

are based on well-known general equations for turbulent boundary 

layers in accelerated motion. Consequently, the coefficient of discharge for free-flow, in the range of low and intermediate values of 

head, can be computed with equations which involve boundary-layer 

parameters. However, the equations are not sufficiently general and 

convenient to be practical. Therefore, the most practical solution for 

free-flow discharge is that which is based on the simple equation of
A88 FLOW OF WATER OVER DAMS AND WEIRS

discharge and experimentally determined coefficients which are related directly to the basic geometric and flow variables. This is the 

solution which makes use of the curves in figure 41.

The ultimate goal of this research is a satisfactory solution for the 

discharge characteristics of a full, practical variety of prototype embankments. The model tests show that the most significant characteristics of both free and submerged flow are virtually independent of 

embankment shape and relative height (h/P). However, the influence of boundary resistance is appreciable, and it depends on roadway width as well as roughness. The model test data are not adequate to define accurately the boundary-resistance effects for prototype-size embankments. For a more accurate evaluation of these 

effects, the results of the model studies should be correlated with a 

limited number of prototype tests.

REFERENCES CITED

Bauer, W. J., 1954, Turbulent boundary layer on steep slopes: Am. Soc. Civil

Engineers Trans., v. 119, p. 1212. 

Delleur, J. W., 1957, The boundary layer development in open channels: Am. Soc.

Civil Engineers Proc.. Journal of the Engineering Mechanics Div.. EMI,

January, Paper 1138. 

Halbronn, G., 1954, Turbulent boundary layer on steep slopes [discussion of

paper by Bauer (1954)] : Am. Soc. Civil Engineers Trans., v. 119, p. 1234. 

Kindsvater, C. E., 1957, Flood-erosion protection for highway fills [discussion

of paper by Chesley J. Posey I : Am. Soc. Civil Engineers Trans.. v. 122,

p. 548. 

Schlichting, Hermann, 3955. Boundary layer theory: New York. Pergamon Press,

535 p. 

Tracy, H. J., 1957, Discharge characteristics of broad-crested weirs: U.S. Geol.

Survey Circ. 397, 15 p., 11 figs. 

Yarnell, D. L., and Nagler, F. A., 1930, Flow of flood water over railway and

highway embankments:" Public Roads, v. 11, no. 2 (April), p. 30.
EXPERIMENTAL DATA

TABLE 3. Summary of data for discharge characteristics l

[F, free flow, tailwater below lower limit of transition range; LL, freeflow, tailwater at lower limit of transition range; UL, free flow, tailwater at upper limit of transition range; IS, incipient submergence; S, 

submerged flow]

Run

No.

? (cfs

per ft)

t

(feet)

h

(feet)

Ho

(feet)

C Remarks Run

No.

q (cfs

per ft)

t

(feet)

h

(feet)

Ho

(feet)

C Remarks

MODEL A-2

Testl

[2/ 0=0.120ft,'2/,/|/c=1.04]

1- ..

2......

3......

4.....

0.222

.222

.222

.222

0.118

.155

.179

0.175

.174

.175

.191

0.175

.174

.175

.191

3.03

3.05

3.03

2.65

F

UL

IS

S

5......

7

8 

.222

.222

.222

.222

.204

.233

.258

.052

.212

.239

.262

.175

.212

.239

.262

.175

2.27

1.90

1.65

3.03

S

S

S

LL

Test 2

[2/,=0.194 ft, »./»«= 1-04]

1- 

2 .

3......

4......

6 

0.451

.451

.451

.451

.451

0.196

.231

.271

.315

0.278

.278

278

.295

.332

0.280

.280

.280

.296

.333

3.05

3.05

3.05

2.79

2.34

F

UL

IS

S

S

6... 

7

8 ...

9_ 

10.. 

.451

.451

.451

.451

.451

.350

.381

.409

.443

.129

.363

.390

.418

.450

.278

.364

.391

.419

.451

.280

2.05

1.84

1.66

1.49

3.05

S

S

S

S

LL

Tests

 = 1.03]

I......

2.. .

3......

4. ....

5......

0.650

.650

.650

.650

.650

0.258

.295

.324

.388

0.353

.353

.353

.361

.411

0.356

.356

.356

.364

.414

3.06

3.06

3.06

2.96

2.44

F

UL

IS

S

S

6....-

7. ..

8 -

9. .

10.....

.650

.650

.650

.650

.650

.418

.461

.498

.543

.164

.438

.476

.511

.553

.353

.440

.478

.513

.555

.356

2.23

1.96

1.77

1.57

3.06

S

S

S

S

LL

Test 4

[y0=0.089 ft, y o/yc= 1.03]

1.

2-____ _

3

4......

0.143

.143

.143

.143

0.076

.111

.149

0.132

.132

.132

.156

0.132

.132

.132

.156

2.98

2.98

2.98

2.32

F

UL

IS

S

5 . .

6......

7--..

.143

.143

.143

.119

.178

.012

.134

.181

.132

.134

.181

.132

2.91

1.86

2.98

S

8

LL

Tests

[y,=OA5l ft, y<,lv c =0.977]

L. .

2......

1.78

1.78 0.543

0.647

.674

0.688

.688

3.12

3.12

F

UL

3

4......

1.78

1.78

.561

.414

.674

.672

.688

.687

3.12

3.12

IS

LL

See footnote at end of table, p. A106.

A89
A90 FLOW OF WATER OVER DAMS AND WEIRS

TABLE 3. Summary of data for discharge characteristics Continued

Run 

No.

?(cfs 

per ft)

t 

(feet)

A

(feet)

Ho

(feet)

C Remarks Run

No.

<?(cfs 

per ft)

t

(feet)

A 

(feet)

H0

(feet)

C Remarks

MODEL A-2 Continued 

Test 6

l

2... ...

3 

4 

5 

1.14 

1.14 

1.14 

1.14 

1.14

0.399 

.417 

.459 

.499

0.510 

.510 

.510 

.516 

.542

0.517 

.517 

.517 

.523 

.550

3.06 

3.06 

3.06 

3.01 

2.80

F 

UL 

IS

S 

S

6 

7 

8

9 

10. -

1.14 

1.14 

1.14 

1.14 

1.14

.534 

.584 

.637 

.689 

.289

.570 

.612 

.660 

.706 

.510

.576 

.618 

.667 

.712 

.517

2.60 

2.34 

2.09 

1.90 

3.06

S 

S 

S

s

LL

MODEL A-3 

Testl

[y . =0.122 ft, y./tf.= 1.06]

l._ .

4..- ...

0.221 

.221 

.221 

.221

0.117 

.153 

.191

0.174 

.174 

.174 

.202

0.174 

.174 

.174 

.2D2

3.04 

3.04 

3.04 

2.43

F 

UL 

IS

S

5 .

6 

7 

8 

.221 

.221 

.221 

.221

.233

.260 

.305 

.047

.239 

.264 

.307 

.174

.239 

.264 

.307 

.174

1.89 

1.63 

1.30 

3.04

S 

S

s

LL

Test 2

1 . ..

2.. .

3... ...

4......

5......

0.476

.476

.476

.476

.476

Q.216

.249

.278

.312

0.289

.289

.289

.304

.331

0.291

.291

.291

.306

.333

3.04

3.04

3.04

2.82

2.48

F

UL

IS

S

s

6... 

7 

8 

9. 

.476

.476

.476

.476

.359

.410

.463

.132

.372

.418

.469

.289

.374

.420

.470

.291

2.09

1.75

1.48

3.04

S

S

S

LL

TestS

1.03]

1 

2.. .

3 _.

4 

1.01

1.01

1.01

1.01

0.387

.404

.468

0.472

.473

.473

.505

0.478

.479

.479

.511

3.06

3.06

3.06

2.77

F

UL

IS

S

5 

6 

7_ 

1.01

1.01

1.01

.520

.610

.274

.547

.628

.472

.553

.633

.478

2.46

2.01

3.06

S

S

LL

Test 4

1 

2......

3_._ 

1.53

1.53

1.53

0.501

.531

0.617

.618

.618

0.629

.635

.630

3.08

3.07

3.07

F

UL

IS

4___ ...

5 

6 

1.53

1 53

1.53

.611

.682

.388

.655

.709

.617

.666

.720

.629

2.82

2.51

3.08

S

S

LL

TestS

[y.-0.471 ft, ya/y

1 

3... ...

1.89 

1.89 0.549

0.701 

.702

0.718 

.719

3.12 

3.11

F 

UL

3 . 

4 .

1.89 

1.89

.608 

.463

.702 

.701

.719 

.718

3.11 

3.12

IS 

LL

MODEL A-4 

Testl

0.037 ft, ?./».=1.001

1 

2 

0.0402 

.0402 0.001

0.058 

.060

0.058 

.060

2.91 

2.77

F

UL

3. -

4 .

0402 

.0402

.018 

-.089

.061 

.058

.061 

.058

2.70 

2.91

IS 

LL

Test 2

[y.=0.078ft,y,,/y.=1.04]

I .

2... ...

0.117 

.117 0.073

0.117 

.119

0.117 

.119

2.93 

2.86

F 

UL

3 

4... ._.

.117 

.117

.106 

-.005

.120 

.117

.120 

.117

2.82 

2.93

IS 

LL
DISCHARGE OVER EMBANKMENT-SHAPED WEIRS A91 

TABLE 3. Summary of data for discharge characteristics Continued

Run

No.

?(cfs 

per ft) (feet)

h 

(feet)

Ho

(feet)

C Remarks Run

No.

g (cfs 

per ft)

t

(feet)

k 

(feet)

Ha

(feet)

C Remarks

MODEL A-4 Continued 

Tests

[yo=0.140ft, t/o/t/c=1.03]

1 ...

2 ...

0.284

.284 0.146

0.208

.210

0.208

.210

2.99

2.95

F

UL

3 ...

4......

.284

.284

.180

.067

.211

.208

.211

.208

2.93

2.99

IS

LL

Test 4

[00=0.209 ft, t/o/t/<,=1.06]

l.-_ 

2 ...

0.498

.498 0.218

0.299

301

0.300

.302

3.03

3.00

F

UL

3 ...

4. ..

.498

.498

.260

.117

.303

.299

.304

.300

2.97

3.03

IS

LL

Tests 

[tf.=0.271 ft, ?./?«-1.03]

1 

2 .

0.765

.765 0.315

0.394

.397

0.397

.400

3.06

3.02

F

UL

3.. .

4......

.765

.765

.345

.209

.399

.394

.402

.397

3.00

3.06

IS

LL

Test 6 

[00=0.330 ft, Vo/Vc=1.02]

2 

1.05

1.05 0.397

0.483

.486

0.489

.492

3.06

3.04

F

UL

3 

4

1.05

1.05

.417

.259

.487

.483

.493

.489

3.03

3.06

IS

LL

Test?

[1/0=0.390 ft,

1 . 1.39 

2...... 1.39 0.46(

0.577 0.586 3.09 

. 580 . 589 3. 07

F 

UL

l/o/l/o=0.999]

3...... 1.39 .504 . 

4__.-_- 1.39 .354 .

581 .590 3.06 IS 

576 .586 3.09 LL

Tests

[t/o=0.435 ft, l/<,/l/c=0.986]

1... 

2......

1.66 

1.66 0.53D

0.648 

.651

0.651 

.664

3.09 

3.07

F 

UL

3.. 

4. ..

1.66 

1.66

.558 

.402

.652 

.648

.665 

.661

3.06 

3.09

IS 

LL

Tests

=0.474 ft, t/o/t/c=0.976]

1... ...

2. -

1.92 

1.92 0.588

0.711 

.714

0.727 

.73J

3.10 

3.08

F 

UL

3. ..

4... 

1.92 

1.92

.623 

.457

.716 

.711

.732

.727

3.07 

3.10

IS 

LL

Test 10

1 

2......

3......

0. 0592

.172

.277

0.07

.14

.2J2

0.074

.148

.203

2.97

3.03

3.02

F

F

F

4. -.

5.... -

6......

.386

1.59

1.88

.252

.628

.702

.253

.640

.718

3.04

3. It

3.01

F

F

F

Test 11

1. ..

2......

3 ...

4......

5......

6......

0.0343

.0900

.187

.323

.451

.674

0.050

.097

.156

.224

.278

.362

0.050

.097

.15o

.225

.250

.366

3.02

2.97

3.02

3.02

3.04

3.05

F

F

F

F

F

F

7 --.

8 ...

9... 

10- 

11. ...

.888

1.12

1.40

1.71

1.93

.435

.505

.581

.656

.710

.440

.512

.59i

.670

.726

3.04

3.01

3.01

3.11

3.11

F

F

F

F

F
A92 FLOW OF WATER , OVER DAMS AND WEIRS

TABLE 3. Summary of data for discharge characteristics Continued

Bun

No.

"g (cfs 

per ft)

t

(feet)

k

(feet)

Ha

(feet)

C Remarks Bun

No.

g (cfs 

per ft)

t 

(feet)

h

(feet)

Ho

(feet)

<? Remarks

MODEL A-4 Continued 

Test 12

1 

2. 

3... ...

4. 

5 

0.0335 

.109 

.228 

.375 

.640

0.050 

,110 

.179 

.248 

.352

0.050 

.110 

.179 

.250 

.354

2.95 

2.99 

3.00 

3.01 

3.04

F 

F 

F 

F 

F

1 

6......

7_ 

8 ...

9_ ..

.967 

1.28 

1.61 

1.91

.460 

.550 

.636 

.708

.465 

.559 

.648 

.725

3.05 

3.07 

3.08 

3.09

F 

F 

F 

F

MODEL B

Test 1

[y0=0.096 ft, y0/?«=1.02]

1... 

2 ...

3......

4_ ..

0.163

.163

.163

.163

0.097

.126

.133

.150

0.142

.142

.145

.158

0.142

.142

.145

1W

3.02

3.02

2.93

2 CO

F, UL

IS

S

s

5 ...

6_ _.

7..... .

.163

.163

.163

.182

.218

.005

.186

.221

.142

.186

.221

.142

2.02

1.56

3.02

S

S

LL

Test 2

[y.,-0.131 ft, Wtf .-1.02]

1. ~

2_.. 

3.._. _.

0.259

.259

.259

0.148

.167

.182

0.194

.194

.198

0.195

.195

.199

3.01

3.01

2.92

F, UL

IS

S

4

5 ...

6 ...

.259

.259

.259

.197

.217

.070

.210

.227

.194

.211

.223

.195

2.67

2.38

3.01

S

S

LL

Tests

1

2......

3... ... 

4...... 

5...... 

6......

0.506 

.506 

.506 

.506 

.506 

.506

0.233 

.260 

.294 

.314 

.336 

.365

0.300 

.300 

.316 

.332 

.352 

.377

0.303 

.303 

.319 

.335 

.355 

.380

3.03 

3.03 

2.81 

2,61 

2.40 

2.16

F, UL 

IS

S 

S

s 

s

7 

8 

9 

10 __ . 

11 

.506 

.506 

.506 

.506 

.506

.389 

.408 

.438 

.464 

.155

.399 

.417 

.445 

.470 

.300

.402 

.419 

.447 

.472 

.303

1.99 

1.86 

1.69 

1.56 

3.03

S 

S 

S

s

LL

Test 4

[00=0.256 ft, v,lvc =l.Q3\

I..-.. 

2...... 

3...... 

4.. .

5......

0.706 

.706 

.706 

.706 

.706

0.291 

,321 

.379 

.437

.475

0.371 

.371 

.400 

.451 

.486

0.376 

.376 

.405 

.455 

.490

3.06 

3.06 

2.74 

2.30 

2.06

F, UL 

IS

S

s 

s

6...... 

7 ... 

8 

9......

.706 

.706 

.706 

.706

.521 

.579 

.631 

.216

.529 

.585 

.636 

.371

.533 

.589 

.639 

.376

1.81 

1.56 

1.38 

3.06

S 

S

s

LL

Tests

[^=0.353 ft, VolVc=1.02]

I......

2...... 

3......

4...... 

5......

1.16 

1.16 

1.16 

1.16 

1.16

0.405 

.432

.485 

.527 

.558

0.515 

.515 

.529 

.559 

.584

0.526 

.526 

.540 

.569 

.594

3.03 

3.03 

2.92 

2.69 

2.53

F, UL 

IS

s 

s 

s

6...... 

7 ... 

8...-.-

1.16 

1.16 

1.16 

1.16

.608 

.677 

.747 

.327

.628 

.692 

.756 

.515

.637 

.701 

.764 

.526

2.27 

1.97 

1.73 

3.03

S

S

s

LL

Tests

[^0=0.477 ft, j/ 0/y c =0.981]

1 .... 

2___ J __

1,92 

1.92

0. 582 

.603

0.703 

.703

0.726 

.726

3.11 

3.11

F, UL 

IS

8~--~ 1.92 .513 .702 .725 3.12 LL

Test 7

I.. . 

2......

1,48 

.618

0.601 

.342

0.617 

.346

3.06 

3.04

F 

F

3 

4......

1.30 

1.84

.551 

.683

.564 

.705

3.06 

3.11

F 

F
DISCHARGE OVER EMBANKMENT-SHAPED WEIRS A93

TABLE 3. Summary of data for discharge characteristics Continued

Run

No.

?(cfs 

per ft)

t 

(feet)

h 

(feet)

Ho

(feet)

C Remarks Run

No.

q (cfs 

per ft)

t 

(feet)

h

(feet)

Ho

(feet)

C Remarks

MODEL C 

Testl

=0.108 ft, 2/«,/2/«=1.02]

2.....

3.....

0.197

.197

.197

0.112

.147

.182

0.161

.161

.189

0.162

.162

.190

3.01

3.01

2.37

F, UL 

IS

.197

.197

.197

.214

.243

.048

.218

.246

.161

.219

.247

.162

1.92

1.60

3.01 LL

Test 2

U/.=0.216 ft, Voly^l.04]

1 

2......

3......

4......

0.534 

.534 

.534 

.534

0.226 

.270 

.296 

.325

0.307 

.307 

.315 

.339

0.313 

.313 

.320 

.344

3.06 

3.06 

2.94 

2.64

F, UL 

IS

S

s

5......

6_ ._

7.. .

8......

.534 

.534 

.534 

.534

.370 

.419 

.458 

.175

.378 

.425 

.461 

.307

.383 

.429 

.465 

.313

2.26 

1.90 

1.68 

3.06

S 

S

s

LL

Tests

[y o=0.257 ft, y,/y e **1.02]

1 -.

2... ...

3......

4......

0.715 

.715 

.715 

.715

0.283 

.337 

.367 

.401

0.371 

.371

.388 

418

0.380 

.380 

.396 

.426

3.06 

3.06 

2.86 

2.57

F, UL 

IS

S

s

5. ..

6._.-_

7

8. ..

.715 

.715 

.715 

.715

.440 

.500 

.579 

.228

.452 

.507 

.583 

.370

.459 

.514 

589 

.379

2.30 

1.94 

1.58 

3.06

S 

S

s

LL

Test 4

U/«,=0.358 ft, y 0/Vc=1.02]

1-- 

2..... 

3.-... 

4._...

1.18 

1.18 

1.18 

1.18

0.417 

.452 

.487 

.541

0.510 

.510 

.519 

.561

0.528 

.528 

.537 

.578

3.08 

3.08 

3.01 

2.70

F, UL 

IS

S

s

5-.--. 

6-. 

7.- 

1.18 

1.18 

1.18

.592 

.663 

.359

.606 

.671 

.510

.621 

.685 

.528

2.41 

2.09 

3.08

S 

S 

LL

TestS

U/ 0 =0.468 ft, Vo/V, =0.977]

I.. 

2.....

1.88 

1.88

0.549 

.593

0.682 

.682

0.716 

.716

3.10 

3.10

F, UL 

IS

3---.. 1.88 .501 .682 7.16 3.10 LL

Test 6

1--.-. 1.47 0.586 0.610 3.08 F 2 0.873 .426 .438 3.02 F

MODEL D

Testl

[2/ 0 =0.067 ft, V,lv,

I--.-.

2__...

3_..__

0.0959

.0959

.0959

0. 087

.109

0.101

.101

.113

0.102

.102

.114

2.94

2.94

2.49

F

IS

S

4- .

5_.__.

.0959

.0959

.150

.105

.153

.111

.154

.112

1.59

2.56

S

S

Test 2

[3/0=0.094 ft, y 0/V,.=l.02]

1

2. .

3-. 

4._...

0 159

.159

.159

.159

0. 116

.134

.147

0.141

.141

.145

.154

0.143

.143

.147

.156

2.93

2.93

2.81

2.57

F

IS

S

S

5---_-

6---..

7- -._

.159

. 159

. 159

.160

.171

. 183

.165

.175

.187

.167

.177

.189

2.33

2.13

1.94

S

S

S
A94 FLOW OF WATER OVER DAMS AND WEIRS

TABLE 3. Summary of data for discharge characteristics Continued

Run

No.

g (cfs 

per ft)

t

(feet)

A

(feet)

Ho

(feet)

C Remarks Run

No.

g (cfs 

per ft)

t

(feet)

h

(feet)

H0

(feet)

C Remarks

MODEL D Continued 

Tests

[y0=0.135ft, y./y.=1.02]

I.....

2..... 

3.-...

4.....

0.272 

.272 

.272 

.272

0.166 

.181 

.191

0.196 

.196 

.197 

.202

0.201 

.201 

.202 

.207

3.02 

3.02 

3.00 

2.89

F 

IS

S 

S

5-. 

6.....

7.....

.272 

.272 

.272

.207 

.220 

.240

.215 

.226 

.243

.220 

.230 

.247

2.65 

2.46 

2.22

S 

S 

S

Test 4

foe =0.216 ft, y./y.=1.05]

1-..

2... .

3-....

4.....

0.525 

.525 

.525 

.526

0.270 

.282 

.310

0.300 

.300 

.303 

.323

0.312 

.312 

.315 

.334

3.01 

3.01 

2.97 

2.72

F 

IS

S 

S

5.....

6.....

7.....

8-....

.525 

.525 

.525 

.525

.328 

.355 

.375 

.396

.338 

.362 

.381 

.400

.349 

.372 

.391

.408

2.55 

2.31 

2.15 

2.02

s 

s 

s 

s

TestS

[y. =0.265 ft, y,/y«=1.01]

1. ....

2.....

3.....

4.....

0.763

.763

.763

.763

0.335

.342

.351

0.378

.378

.379

.380

O QAQ

OQO

.399

.400

3 04

3.04

3.02

3.01

F

TQ

s

s

5 .-

6.. 

7.. 

8-. 

.763

.763

.763

.763

.374

.400

.446

.488

.394

.415

.454

.493

.413

.433

.470

.508

2.87

2.68

2.36

2.11

S

S

S

s

Tests

=0.367 ft, yalv,=1.01J

1 

2 

3......

1.25

1.25

1.25

0.475

.498

0.527

.527

.536

0.563

.563

.571

2.95

2.95

2.88

F

IS

S

4_._ 

6 

6 _.

1.25

1.25

1.25

.520

.547

.591

.551

.571

.605

.585

.604

.635

2.78

2.66

2.46

S

S

S

Test 7

1 ...

2......

3 ...

4......

6 ...

6......

7. ..

8 ...

9......

10.....

11.. 

0.288

.288

.288

.490

.490

.490

.793

.793

.793

1.29

1.29

0.138

.189

.101

.206

.261

.181

.301

.357

.275

.423

.469

0.203

.203

.203

.285

.285

.285

.385

.385

.385

.523

.523

0.208

.208

.208

.296

.296

.296

.406

.406

.406

.562

.562

3.03

3.03

3.03

3.04

3.04

3.04

3.06

3.06

3.06

3.07

3.07

F.UL

IS

LL

F,UL

IS

LL

F,UL

IS

LL

F,UL

IS

12 

13 

14. -

15 

16 

17 

18 

19 

20 

21 .-

1.29

1.97

1.97

1.97

.159

.159

.159

.970

1.58

.777

.409

.581

.625

.549

.087

.116

.050

.523

.676

.676

.676

.141

.141

.141

.437

.590

.380

.562

.741

.741

.741

.143

.143

.143

.464

.640

.401

3.07

3.09

3.09

3.09

2.94

2.94

2.94

3.07

3.08

3.06

LL

F.UL

IS

LL

IS

IS

LL

F

F

 F

MODEL E 

Test 1

I... ...

2......

3... ...

4......

0.175

.175

.175

.175

0.121

.125

.163

0.151

.155

.155

.172

0.151

.155

.155

.172

2.97

2.86

2.86

2.45

F

UL

IS

S

5 

6_ _.

7 

.175

.175

.175

.187

.208

.054

.192

2.11

.152

.192

.211

.152

2.08

1.80

2.95

S

S

LL

Test 2

[y0=0.147ft, ya/

1 

2 

3 .

4_ _.

.0.292

.292

.292

.292

0.159

.171

.195

0.211

.213

.213

.216

0.212

.214

.214

.217

2.99

2.95

2.95

2.89

F

UL

IS

S

5 

6 

7 ...

8 

.292

.292

.292

.292

.225

.252

.285

.095

.238

.260

:290

.212

.239

.261

.291

.213

2.50

2.19

1.86

2.97

S

S

S

LL
DISCHARGE OVER EMBANKMENT-SHAPED WEIRS A95

TABLE 3. Summary of data for discharge characteristics Continued

Run

No.

ff(cfs 

per ft)

t

(feet)

h

(feet)

H0

(feet)

C Remarks Run 

No.

q (cfs 

per ft)

t 

(feet)

h

(feet)

Ho

(feet)

C Remarks

MODEL E Continued 

TestS

[2/o=0.203 ft, 2/o/2/c=1.03]

1 

2 

3 

4... ...

5. ..

0.496

.496

.496

.496

.496

0.226

.240

.261

.306

0.296

.299

.299

.301

.325

0.298

.301

.301

.303

.327

3.05

3.00

3.00

2.98

2.66

F

UL

IS

S

s

6.. _

7 

8 

9 .

.496

.496

.496

.496

.361

.415

.467

.152

.373

.421

.471

.297

.375

.423

.473

.299

2.16

1.80

1.53

3.04

S

S

S

LL

Test 4

[j/o=0.289 ft, 2/»/2/*=1.01]

1 ...

2 .__

3 ...

4......

5... ...

0.871

.871

.871

.871

.871

0.341

.367

.399

4KQ

0.430

.432

.432

.442

407

.437

.437

.447

Ann

3.04

3.01

3.01

2.92

2.53

F

UL

IS

S

S

6......

7 ...

8 -

9... 

.871

.871

.871

.871

.539

.578

.673

.245

.553

.688

.680

.431

.557

.592

.684

.436

2.09

1.91

1.54

3.02

S

S

S

LL

Tests

=0.351 ft, 2/o/2/ e=0.982]

1 ...

2......

3 ...

4......

5... ...

122

1 22

1.22

1.22

1.22

0.434

.459

.474

.507

0.538

.539

.539

.543

.557

0.546

K4.7

.547

.551

.565

3

m

3.00

3.00

2.97

2 Of*

F

UL

IS

S

S

6 .

7... 

9 

1.22

1.22

1.22

1.22

.541

.616

.672

.326

.580

641

.690

.539

.588

.648

.697

.547

2.70

2.33

2.09

3.00

S

S

s

LL

Test 6

, 2/o/2/«=0.968]

1 

2......

1.92 

1.92 0.603

0.711 

.714

0.728 

.731

3.09 

3.07

F 

UL

3. 

4... 

1.92 

1.92

.607 

.475

.714 

.714

.731 

.731

3.07 

3.07

IS 

LL

Test 7 

[2/o=0.415 ft, 2/o/2/«=0.965]

1 ... 1.60 0.635 0.648 3.06 F

MODEL F 

Test 1

[2/o=0.106 ft, 2/o/2/c=l-04]

1 ...

2 . .

3......

4. ..

0.183

.183

.183

.183

0.116

.127

.152

0.155

.156

.156

.165

0.155

.156

.156

.165

3.00

2.97

2.97

2.73

F

UL

IS

S

5 ...

6 _.-

7 

.183

.183

.183

.184

.234

.040

.190

.237

.156

.190

.237

.156

2.21

1.59

2.97

S

S

LL

Test 2

0.138ft, 2/o/2/c=1.02]

1 

2......

3......

4......

5......

0.281

.281

.281

.281

.281

0.150

.169

.199

.229

0.205

.206

.206

.215

.240

0.206

.207

.207

.216

.241

3.01

2.99

2.99

2.80

2.38

F

UL

IS

S

S

6 

7 ...

8 

9 ...

.281

.281

.281

.281

.265

.299

.328

.076

.271

.304

.331

.206

.272

.305

.332

.207

1.98

1.67

1.47

2.99

S

S

S

LL

Tests

1 ...

2 ...

3 ...

4......

5 

6 

0.484

.484

.484

.484

.484

.484

0.217

.240

.246

.264

.285

0.291

.293

.293

.294

.297

.309

0.293

.295

.295

.296

.299

.311

3.06

3.02

3.02

3.01

2.96

2.80

F

UL

IS

S

S

s

7 -

8 

9 ...

10 -

11 .

12 

.484

.484

.484

.484

.484

.484

.325

.338

.381

.428

.491

.142

.342

.352

.390

.435

.495

.292

.344

.354

.392

.436

.496

.294

2.40

2.30

1.98

1.68

1.38

3.04

S

S

S

s

s

LL
A96 FLOW OF WATER OVER DAMS AND WEIRS

TABLE 3. Summary of data for discharge characteristics Continued

Run

No.

g(cfs 

per ft)

t 

(feet)

h

(feet)

H,

(feet)

C Remarks Run

No.

g (cfs 

per ft)

t

(feet)

h

(feet)

Ho

(feet)

C Remarks

MODEL F Continued 

Test 4

1 _ ..

2-. 

3 

4 

5 

6... 

0.841

.841

.841

.841

.841

.841

0.335

.346

.373

.393

.427

0.420

.421

.421

.423

.433

.458

0.425

.426

.426

.428

.438

.462

3.04

3.03

3.03

3.01

2.99

2.67

F

UL

IS8'

S

S

7 

8 

g

10...-

11 

12-...

.841

.841

Sdl

841

.841

.841

.446

.492

MQ

.592

.646

.245

.472

.512

554

.602

.655

.421

.476

.516

.558

.606

660

426

2.56

2.27

2.02

1 78

1.57

3.03

S

S

S

S

S

LL

Tests

1 

2......

3 ...

4 ...

1.21

1.21

1.21

1.21

0.432

.444

.464

0.532

.534

.534

.535

0.540

.542

.542

.543

3.05

3.03

3.03

3.02

F

UL

IS

S

5 ...

6. 

1.21

1.21

1.21

1.21

.542

.614

.676

.329

.579

.608

.691

.532

.587

.645

.697

.540

2.69

2.34

2.08

3.05

S

S

S

LL

Test 6

i 

2 ..

1.88

1.88 0.594

0.701

.703

0.717

71Q

3.09

3.08

F

UL 4 ..

1.88

1 RR

.601

.477

.703

.701

71Q

.717

3.08

3.09

IS

LL

Test? 

[^=0.409 ft, ^.=0.979]

1 1.53 0.617 0.629 3.07 F

MODEL G 

Test 1

[^= 0.084 ft, y,/».=1.04]

1 ...

2......

3......

0.130

.130

.130

0.071

.107

0.122

.122

.122

0.122

.122

.122

3.05

3.05

3.05

F

UL

IS

4. 

5.. .

6 .....

.130

.130

.130

.127

.157

-.001

132

.160

.122

132

.160

.122

2.71

2.03

3.05

S

S

LL

Test 2

1 

2......

3......

4... 

5- .

0.282

.282

.282

- 282

.282

0.140

.178

..197

.217

0.203

.203

.203

.212

.229

0.204

.204

.204

.213

.230

3.06

3.06

3.06

2.87

2.56

F

UL

IS

S

S

6 

7 

8 

9 

.282

.282

.282

.282

,244

.273

.300

.069

.251

.278

.304

.203

.252

.279

.305

.204

2.23

1.91

1.68

3.06

S

S

S

LL

TestS

foo=0.189 ft, tfo/j

1 

2.. 

3._____

4 

5.. 

0.447

.447

.447

.447

.447

0.199

.243

.257

.288

0.277

.277

.277

.282

.303

0.278

.278

278

.284

.304

3.04

3.04

3.04

2.96

2.66

F

UL

IS

S

S

6. ..

7... 

8 

.9 

10 

.447

.447

.447

.447

.447

.326

.368

.409

.456

.114

338

.376

.415

.459

.277

.339

.377

.416

.460

.278

2.26

1.93

1.66

1.43

3.04

S

S

S

S

LL

Test 4

1 ...

2 .

3 

4 

5-___.

0.789

.789

.789

.789

.789

0.311

.341

.363

.393

0.402

.402

.402

.406

.424

0.406

.406

.406

.410

.428

3.05

3.05

3.05

3.01

2.82

F

UL

IS

S

S

6 

7 

8 

9......

10.....

.789

.789

.789

.789

.789

.427

.459

.538

.616

.213

.451

.477

.549

.623

.402

.455

.481

.552

.626

.406

2.57

2.37

1.92

1.59

3.05

S

S

S

S

LL
DISCHARGE OVER EMBANKMENT-SHAPED WEIRS A97

TABLE 3. Summary of data for discharge characteristics Continued

Run

No.

q (cfs 

per ft)

t

(feet)

h 

(feet)

Ho

(feet)

C Remarks Run

No.

q (cfs 

per ft)

t

(feet)

h 

(feet)

H0

(feet)

C Remarks

MODEL G Continued 

TestS

1 .

2......

3......

4......

5......

1.17

1.17

1.17

1.17

1.17

0.418

.447

.457

0.520

.520

.520

.523

.530

0.528

.528

.528

.531

.538

3.06

3.06

3.06

3.03

2 07

F

UL

IS

S

S

6......

7 

8 

9 .

1.17

1.17

1.17

1.17

.553

.607

.660

.309

.583

.676

.520

KQfl

.528

2.58

2.32

2.08

3.06

S

S

S

LL

Test 6

[^=0.474 ft, y 0/^=0.980]

i ...

2......

1.91

1.91 0.592

0.701

.702

0.718

.719

3.14

3.13

F

UL

3... 1.91

1.91

.598

.462

.702

.702

719

71 Q

3.13

3.13

IS

LL

Test?

[^=0.411 ft, y0/

l 1.48 | 0.602 0.613 3.08 F

MODEL H 

Testl

1......

2......

3......

4......

0.123

.123

.123

.123

0.065

.074

.084

0.118

.118

.118

.119

0.118

.118

.118

119

3.02

3.02

3.02

2.98

F

UL

IS

5 _

7. 

.123

.123

.123

.123

.109

.128

.142

-.017

.120

.131

.144

.118

.120

.131

.144

.118

2.94

2.58

2.24

3.02

S

S

S

LL

Test 2

feo=0.127ft, Jf 0/Jf e=1.02]

1 

2......

3 

4......

5 ...

0.247

.247

.247

.247

.247

0.129

.145

.161

.181

0.186

.187

.187

.188

.193

0.187

.188

.188

.198

.194

3.07

3.05

3.05

3.02

2.91

F

UL

IS

S

S

6......

7 

8 

9......

.247

.247

.247

.247

.199

.219

.255

.048

.208

.226

.259

.186

.209

.227

.260

.187

2.60

2.30

1.87

3.07

S

S

S

LL

Tests

=0.187 ft, VolV- =1.02]

1 .

2......

3......

4......

5.... ..

6......

0.446

.446

.446

.446

.446

.446

0.196

.223

.232

.250

.269

0.274

.275

.275

.276

.278

.290

0.276

.277

.277

.278

.280

.292

3.08

3.06

3.06

3.05

3.02

2.83

F

UL

IS

S

S

S

7 ...

8 

9 

10. 

11 

.446

.446

.446

.446

.446

.312

.345

.380

.427

.108

.325

.355

.387

.431

.274

.326

.356

.388

.432

.276

2.39

2.10

1.84

1.57

3.08

S

S

S

S

LL

Test 4

=0.277 ft, yjy, 1.02]

1- 

2 __

3 

4......

5......

0.799

.799

.799

.799

.799

0.309

.347

.353

.378

0.404

.405

.405

.406

.415

0.408

.409

.409

.410

.419

3.06

3.05

3.05

3.04

2.95

F

UL

IS

S

S

6. 

7 

8. _ -.

9 -

10 

.799

.799

.799

.799

.799

.431

.488

.539

.591

.198

.454

.504

.550

.598

.404

.458

.508

.553

.601

.408

2.58

2.21

1.94

1.71

3.06

S

S

S

S

LL

TestS

[^=0.353 ft, Voh =1.01]

1 

2. __

3......

4_ __

1.17

1.17

1.17

1.17

0.426

.442

.465

0.517

.520

.520

.523

0. 525

.528

.528

.531

3.09

3.07

3.07

3.04

F

UL

IS

S

5... ...

6 .

7 

8 ...

1.17

1.17

1.17

1.17

.534

.599

.647

.291

.568

.622

.665

.518

.575

.629

.672

.526

2.69

2.36

2.14

3.08

S

S

S

LL
A98 FLOW OF WATER OVER DAMS AND WEIRS

TABLE 3. Summary of data for discharge characteristics Continued

Run

No.

g(efs 

per ft)

t 

(feet)

h 

(feet)

H0

(feet)

C Remarks Run

No.

q (efs 

per ft)

t 

(feet)

A

(feet)

H0

(feet)

C Remarks

MODEL H Continued 

Test 6

[^,,=0.463 ft, y,ly«=0.977]

1 

2._.___

1.84

1.84 0,580

0.682

.687

0.698

.702

3.16

3.12

F

TJL

______

4... ...

1 ad. 1. ore

1.84

.583

.446

.687

.685

.702

.700

3.12

3.14

TQlib

LL

Test?

fo0=0.416 ft, tf«/tf .=0.998]

1 1.52 0.606 0.618 3.14

1 F

MODEL I 

Testl

=0.135 ft, y<,/y c --

1 ...

2......

3......

4......

0.266

.266

.266

.266

0.130

.161

.185

0.196

.197

.208

0.197

.198

198

!209

3.05

3.03

3.03

2 79

F

UL

IS

S

5 

6 

7 

g

.266

.266

.266

.266

.219

.270

.355

.065

.235

.279

.259

.197

.236

.280

.360

.198

2.33

1.80

1.23

3.03

S

S

S

LL

Test 2

1 _ ...

2......

3......

4.....

0.573 

.573 

.573 

.573

0.236 

.265 

.303

0.-325 

.326 

.326 

.342

0.327 

.328 

.328 

.344

3.06 

3.04 

3.04 

2.83

F 

UL 

IS

S

5... ...

6 

7 

Q

.573 

.573 

.573 

.573

.348 

.400 

.551 

.164

.373

.418 

.558 

.326

.375 

.420 

.560 

.328

2.49 

2.10 

1.37 

3.04

S 

S 

S 

LL

TestS

[j/ 0=0.345 ft, ye/y e=lM]

1 

2.. ....

3......

4......

5 __

1.01

1.01

1.01

1.01

1.01

0.380

.395

.434

.479

0.473

.475

.475

.491

.521

0.479

.481

.481

.497

.527

3.05

3.03

3.03

2.89

2.64

F

UL

IS

S

S

6 

7 

8 

9 

1.01

1.01

1.01

1.01

.552

.615

.689

.265

.584

.635

.704

.473

.589

.640

.709

.479

2.24

1.98

1.70

3.05

S

S

S

LL

Test 4

=0.404 ft, tf./?.=0.978]

1 ...

2......

3......

4......

1.50

1.50

1.50

1.50

0.489

.496

.538

0.608

.610

.610

.617

0.620

.622

.622

.628

3.08

3.07

3.07

3.02

F

UL

IS

S

5 --

6.... ..

7... 

1.50

1.50

1.50

.593

.673

.389

.650

.711

.609

.661

.721

.620

2.80

2.45

3.08

S

S

LL

TestS

[^.=0.467 ft, j/0/2^=0.967]

I......

2... 

1.90 

1.90 0.585

0.704 

.705

0.721 

.722

3.11 

3.10

F 

UL

3 

4......

1.90 

1.90

.600 

.473

.705 

.704

.722 

.721

3.10 

3.11

IS 

LL

MODEL J 

Test 1

0=0.111 ft, tfo/tf.

1- .

2......

3. .

4.....:

0.205

.205

.205

.205

0.128

.148

.171

0.164

.164

.164

.182

0.164

.164

.164

.182

3.08

3.08

3.08

2.62

F

UL

IS

S

5 ...

6... 

7 ...

.205

.205

.205

.202

.239

.032

.209

.242

.164

.209

.242

.164

2.14

1.72

3.08

S

S

LL
DISCHARGE OVER EMBANKMENT-SHAPED WEIRS A99

TABLE 3. Summary of data for discharge characteristics Continued

Run 

No.

g (cfs I t 

per ft) (feet)

h

(feet)

Ha

(feet)

C Remarks Run 

No.

q (cfs 

per ft)

* 

(feet)

h

(feet)

Ha

(feet)

C Remarks

MODEL J Continued 

Test 2

[y0=0.210 ft, y a/y e=l.05]

1 

3_..-__

5 ...

0.508 

.508 

.508 

.508 

.508

0.228 

.245 

.259 

.286

0.298 

.298 

.298 

.299 

.312

0.300 

.300 

.300 

.301 

.314

3.10 

3.10 

3.10 

3.08 

2.89

F 

UL

IS 

S

s

6 ...

7 ...

8 ...

9 ...

.508 

.508 

.508 

.508

.328 

.407 

.483

.127

.346

.415 

.489 

.298

.348 

.417 

.491 

.300

2.48 

1.89 

1.48 

3.10

S 

S 

S 

LL

TestS

[y«=0.321 ft, y«/y«~1.02]

1 

2 ...

3......

4__-_.

1.01 

1.01 

1.01 

1.01

0.374 

.402 

.452

0.467 

.468 

.468 

.491

0.473 

.474 

.474 

.497

3.10 

3.09 

3.09 

2.88

F 

UL

IS 

S

5 -..

6 ...

7 

1.01 

1.01 

1.01

.533 

.619 

.246

.558 

.636 

.467

.563 

.641 

.473

2.38 

1.96 

3.10

S 

S 

LL

Test 4

[j<<,=0.399 ft, j» 0/2^=0.982]

1... _

2......

3.......

4......

1.47

1.47

1.47

1.47

0.493

.503

.547

0.594

.597

.597

.606

0.605

.608

.608

.617

3.12

3.10

3.10

3.03

F

UL

IS

S

5... _

6... ...

7 ...

1.47

1.47

1.47

.591

.673

.346

.634

.700

.595

.645

.710

.606

2.84

2.46

3.11

S

S

LL

TestS

[y.=0.471 ft, y a/y ,=0.967]

1 ...

2......

1.93 

1.93 0.588

0.699 

.704

0.716 

.721

3.18 

3.15

F 

UL

3... 

4......

1.93 

1.93

.609 

.451

.704 

.701

.721 

.718

3.15 

3.17

IS 

LL

MODEL K-2 

Testl

tya=0.111 ft, ^.=0.974]

!-.. 

2.....

3.....

4.....

5.....

6.....

0.219

.219

.219

.219

.219

.219

0.121

.141

.148

.166

.207

0.172

.172

.172

.173

.178

.214

0.172

.172

.172

.173

.178

.214

3.06

3.06

3.06

3.04

2.91

2.21

F

UL

IS

S

S

s

7.- 

8.. 

9-.--.

10-.-

11....

12-.-

.219

.219

.219

.219

.219

.219

.238

.251

.284

.202

.184

.015

.243

.255

.287

.208

.190

.172

.243

.255

.287

.208

.190

.172

1.83

1.70

1.42

2.30

2.64

3.06

S

S

S

s

s

LL

Test 2

[y 0=Q.144 ft, 2^/2^=0.9721

1. ....

2-- 

3__ 

4...-

5-.--

0.324

.324

.324

.324

.324

0.174

.194

.196

.201

0.223

.233

.223

.224

.225

0.224

.224

.224

.225

.226

3.06

3.06

3.06

3.04

3.02

F

UL

IS

S

S

6-- 

7.. 

8.-...

9-- 

.324

.324

.324

.324

.238

.279

.314

.086

.252

.286

.320

.223

.253

.287

.320

.224

2.55

2.11

1.79

3.06

S

S

S

LL

Test3

=0.208 ft, 2/0/2^=1.001

1 --.-

2..-..

3__ 

4 ....

5___._

6-- 

0.538

.538

.538

.538

.538

.538

0.241

.269

.274

.302

.344

0.312

.312

.312

.314

.328

.363

0.314

.314

.314

.316

.330

.364

3.06

3.06

3.06

3.03

2.83

2.45

F

UL

IS

S

s

s

7-. 

8--.-

9-. 

10---

11-.-

.538

.538

.538

.538

.538

.395

.445

.482

.541

.155

.408

.454

.490

.548

.312

.409

.455

491

.548

.314

2.06

1.75

1.56

1.32

3.06

S

S

S

s

LL
A100 FLOW OP WATER OVER DAMS AND WEIRS

TABLE 3, Summary of data for discharge characteristics Continued

Run 

No.

?(efs 

per ft)

t

(feet)

A

(feet)

Ha

(feet)

C Remarks Run 

No.

g (cfs 

per ft)

t 

(feet)

A

(feet)

Ha

(feet)

C Remarks

MODEL K-2 Continued 

Teat 4

[#,,=0.283 ft, y a/y e=l.04]

!_. 

2.. 

3.....

4:....

5.. 

6.. 

0.805

- .805

.805

.805

.805 

:805

0.307

- .849

.386

.437 

.463

0.407

.407

-.407

.424

.462

tSR

0.411

.411

.411

.428

.466

AQQ

3.05

3.05

3.05

2.87

2.53 

2.36

F

UL

IS

S

S 

S

.7.. 

8-...-

9. .

10..-

11....

. 10

.805

.805

.865

.805

.805 

.805

,493

.525

.556

.593

.621 

.206

.511

.542

.570

.606

.631 

:407

.515

.546

.573

.609

.634 

.411

2.18

1.99

1.86

1.69

1.59 

3.05

S

S

S

S

S 

LL

Tests

1-- 

2.. 

3.....

4.. ...

5. __

1.20 

1.20 . 

1 ; 20 

1.20 

1.20

0.413 

.433 

.480 

.528

ff,524 

.524

-. 524 

.534 

:570

0.532 

.532 

.532

-. 542 

.577

3.09 

3.09 

.3:09 

3.00

2.74

F 

UL 

IS

S 

S

6-. 

7.. 

9.....

1.20 

1.20 

1.20 

1.20

.581 

.642 

.702 

.297

.612 

.664 

.719 

.523

.619 

.670 

.725 

.531

2.46 

2.19 

1.94 

3.10

S 

S 

S 

LL

Test6

I-. 

2-..-;-

1.60 

1.60. 0.510

0.633 

.633

0.645 

.645

3.09 

3.09

F 

TIL

3.- 

4-. 

1.60 

1.60

.549 

.388

.633 

.632

.645 

.644

3.09 

3.10

IS 

LL

MODEL L

Testl

= 1.04]

1---..

2.- 

-3_-.._

0.183

.183.

.183

0.043

.133

0.155

.155

.155

0. 155

.155

.155

'3.00

3.00

3.00

F

UL

IS

4_--.^

5-. 

6.- 

.183

.183

.183

.169

.209

-.006

,179

.212

.155

.179

.212

.155

2.42

1.88

3.00

S

S

LL

Test 2

1. _

2 ...

3_._ 

4......

0.524

.524

.524

.524

.0. 142

:.250

.280

0.307

.307

.307

.317

0.309

.309

-309

.319

"3.05

3.05

3.05

2.91

F

UL

IS

S

5 ...

6.._._.

7 :..

8... ...

.524

.524

. .524

.524

.323

.440

.505

.072

.349

.451

.513

.307

.351

.453

.515

.309

2.52

1.72

1.42

3.05

S

S

S

LL

Tests

=0.328 ftr »./»,-1.04]

I......

2. .:

3. .

4......

5 ...

1.01

1.01

1.01

1.01

1.01

0.252

t396

.434

.455

0.472

.472

.472

.492

.504

0.478

.478

.478

.498

.510

3.06

3.06

3.06

2.88

2,78

F

UL

IS

S

S

6... _

7 . 

8 ...

9 

10.-...

1.01

1.01

1.01

1.01

1.01

.512

.573

.650

.740

.187

.549

.600

.467

.752

.472

.555

.605

.672

.756

.478

2.45

2.15

1.84

1.54

3.06

S

S

S

S

LL

Test 4

 [»,r-0.411-ft, »«/».«* 1.00]

1 

2 

3......

4... _..

1.49

1.49

1.49

1.49

0.364

.513

.579

0.605

.605

.606

.640

 0. 616

.616

.617

.651

3.07

3.07

3.06

2.84

F

UL

IS

S

5 

6... 

7 

1.49

1:49

1.49

.633

-.680

.299

.680

-.717

.605

.682

.727

.616

2.64

2.40

3.07

S

S

LL

TestS

[3/o=0.467 ft, 2/<,/J/c=0.983]

I......

2... _

1.86 

1.86 0.448

0,693 

.693

0.709 

.709

311 

3.11

F 

. UL

3 

4 

1.86 

1.86

.590 

.375

.694 

.693

.7-10 

.709

3.10 

3.11

IS 

LL
DISCHARGE OVER EMBANKMENT-SHAPED WEIRS AlOl 

TABLE 3. Summary of data, for discharge cJiaracteristics Continued

Run 

No.

q (cfs 

per ft) (feet)

A 

(feet)

H0

(feet)

C Remarks Run 

No.

q (cfs 

per ft)

t 

(feet)

A 

(feet)

H0

(feet)

C Remarks

MODEL M

Test 1

1... ...

2 _

3 . _.

4 ...

0.190

.190

.190

.190

0.101

.140

.160

0.155

.157

.157

.171

0.155

.157

.157

.171

3.10

3.04

3.04

2.68

F

UL

IS

S

5......

6......

7 ...

8... ...

.190

.190

.190

.190

.182

.217

.247

.049

.189

.221

.250

.157

.189

.221

.250

.157

2.30

1.82

1.52

3.04

S

S

S

LL

1 

2......

3......

4......

5......

0.581

.581

.581

.581

.581

0.227

.278

.296

.323

0.327

.328

.328

.331

.348

0.330

.331

.331

.334

350

[

3.07

3.06

3.06

3.02

2.80

Te

^0 =0.230 ft

F

UL

IS

S

S

8t2 

, Vo/Ve=l

6

7......

8- ..

9 

10.. 

.05]

.581

.581

.581

.581

.581

.362

.399

.443

.499

.189

.379

.412

.452

.506

.328

.381

.414

.454

.508

.331

2.47

2.18

1.90

1.61

3.06

S

S

S

S

LL

Test3 

[^.=0.334 ft, ydyc= 1.03]

1 ...

2......

3... 

5 ...

1.04 

1.04 

1.04 

1.04 

1.04

0.356 

.400 

.430 

.471

0.482 

.483 

.483 

.485 

.506

0.488 

.490 

.490 

.492 

.512

3.05 

3.04 

3.04 

3.02 

2.84

F 

UL 

IS

S 

S

6. 

7 ...

8......

9 ...

1.04 

1.04 

1.04 

1.04

.510 

.554 

.615 

.330

.537 

.574 

.629 

.482

.543 

.580 

.634 

.488

2.60 

2.36 

2.06 

3.05

S 

S 

S 

LL

Test 4

[^.=0.407 ft, p0/Zfc=0.981]

1 ...

2...... 

3...... 

4......

1.51 

1.51 

1.51 

1.51

0.496 

.526 

.565

0.610 

.611 

.611 

616

0.622 

.623 

.623 

.628

3.08 

3.08 

3.08 

3.04

F 

UL 

IS

S

5_..-_.

6......

7 

1.51 

1.51 

1.51

.620 

.669 

.475

.650 

.689 

.611

.661 

.700 

.623

2.81 

2.58 

3.08

S 

S 

LL

Test 5

[^.=0.481 ft, Vo/V ,=0.985]

1... ... 

2... 

1.94 

1.94 0.603

0.710 

.712

0.727 

.729

3.13 

3.12

F

UL

3...... 

4. ..

1.94 

1.94

.621 

.555

.712 

712

.729 

729

3.12 

3.12

IS 

LL

MODEL AA-2 » 

Testl

to.-0.032 ft, Vo/Vc= 1.03]

1 

2......

3......

0.0306

.0306

.0306

-0.011

.016

0.048

.049

.050

0.048

.049

.050

2.96

2.88

2.78

F

UL

IS

4. ..

5......

6......

.0306

.0306

.0306

.068

.123

-.062

.072

.126

.048

.072

.126

.048

1.60

0.689

2.96

S

S

LL

Test 2

1 ...

2. ..

3... 

4... ...

0.0814

.0814

.0814

.0814

0.008

.039

.082

0.091

.091

.092

.095

0.091

.091

.092

.095

2.98

2.98

2.94

2.80

F

UL

IS

S

5 

6.... ..

7 ...

8 ...

.0184

.0814

.0814

.0814

.130

.184

.232

-.059

.135

.187

234

.091

.135

.187

.234

.091

1.73

1.06

0.756

2.98

S

S

S

LL

Test3

[00=0.110 ft, y,l

1 .-

2......

3......

4......

0.185

.185

.185

.185

0.055

.096

.181

0.157

.157

.158

.192

0.157

.157

.158

.192

2.98

2.98

2.95

2.20

F

UL

IS

S

5 

6 ...

7 

8... ...

.185

.185

.185

.185

.235

.296

.336

-.019

.241

.300

.339

.157

.241

.300

.339

.157

1.56

1.12

0.937

2.98

S

S

S

LL

See footnote at end of table, p. A106.
A102 FLOW OF WATER OVER DAMS AND WEIRS

TABLE 3. Summary of data for discharge characteristics Continued

Eun 

No.

g(cfs 

per ft) (feet)

ft

(feet)

H0

(feet)

C Eemarks Run 

No.

q (cfs 

per ft)

t 

(feet)

ft

(feet)

H0

(feet)

C Remarks

MODEL AA-2 Continued 

Test 4

1... ...

2... ...

3. -

4 

0.304 

.304 

.304 

.304

0.096 

.142 

.185

0.218 

.220 

.221 

.223

0.218 

.220 

.221 

.223

2.98 

2.94 

2.92 

2.88

F 

UL

IS

S

5 ...

6... ...

7-. .

8... ...

.304 

.304 

.304 

.304

.221 

.305 

.412 

.026

.250 

.323 

.425 

.219

.250 

.323 

.425 

.219

2.43 

1.65 

1.10 

2.96

8 

8 

8 

LL

TestS

[» .-0.199 ft, y a/y,=lM]

1 

2... ...

3... ...

4. -

0.504 

.504 

.504 

.504

0.158 

.206 

.250

0.304 

.305 

.306 

.307

0.305 

.306 

.307 

.308

2.99 

2.98 

2.96 

2.95

F 

UL

IS

S

5 

6... ...

7.. .

.504 

.504 

.504 

.504

.343 

.418 

.479 

.046

.366 

432 

.490 

.304

.367 

.433 

.491

2.27 

1.77 

1.46 

2.99

8 

8 

8 

LL

Test6 

[» .-0.260 ft, y,ly e=1.05]

1 ...

3 

4... 

5......

0.750 

.750 

.750 

.750 

.750

0.233 

.291 

.347 

.407

0.392 

.393 

.394 

.403 

.441

0.395 

.396 

.397 

.406 

.444

3.02 

3.01 

3.00 

2.90 

2.53

F 

UL

IS

S 

S

6... ...

7... 

9 ...

.750 

.750 

.750 

.750

.452 

.487 

.530 

.116

.479 

.507 

.546 

.392

.482 

.510 

.549 

.395

2.24 

2.06 

1.84 

3.02

S 

8 

S 

LL

Test?

1.. -

2... 

3... 

4... 

5 

1.11

1.11

1.11

Lll

1.11

0.318

.357

.395

.467

0.503

.504

.505

.506

.526

0.510

.511

.512

.513

.532

3.04

3.03

3.02

3.01

2.84

F

UL

IS

S

S

6......

8......

9--.-_.

1.11

1.11

1.11

1.11

.521

.567

.620

.186

.563

.600

.645

.503

.569

.606

.651

.510

2.57

2.34

2.10

3.04

S

S

S

LL

TestS

fo,,=0.412 ft, y,/y. lM]

1--. 

2... 

3 ...

4_.____

5... 

1.50

1.50

1.50

1.50

1.50

0.402

.443

.498

.560

0.611

.612

.613

.616

.630

0.622

.623

.624

.627

.641

3.06

3.05

3.04

3.02

2.92

F

UL

IS

S

S

6_._ 

7... ...

8......

9 

1.50

1.50

1.50

1.50

.624

.679

.706

.274

.673

.715

.739

.611

.684

.726

.750

.622

2.65

2.42

2.31

3.06

S

S

S

LL

TestS

1 ......

2......

1.72 

1.72 0.449

0.668 

.670

0.682 

.684

3.06 

3.05

F 

UL

3 ...

4......

1.72 

1.72

.503 

.333

.671 

.668

.685 

.682

3.04 

3.06

IS 

LL

Test 10

[1^=0.499 ft, ^,.=0.9821

1 - 

2......

2.06 

2.06 0.515

0.744 

.746

0.762 

.764

3.09 

3.08

F 

UL

3 

4 ...

2.06 

2.06

.577 

.405

.747 

.744

.765 

.762

3.07 

3.09

IS 

LL

MODEL AB a 

Testl

fo«,=0.038ft, jr0/1^=1.001

1-. -

2......

3 

4......

0.0421

.0421

.0421

.0421

0.000

.036

.070

0.058

.058

.059

.073

0.058

.058

.059

.073

3.05

3.05

2.98

2.16

F

UL

IS

S

5... ...

6... 

7......

8......

.0421

.0421

.0421

.0421

.096

.124

.159

-.105

.099

.126

.162

.058

.099

.126

.162

.058

1.36

0.948

.649

3.05

S

S

S

LL

See footnote at end of table, p. A106.
DISCHARGE OVER EMBANKMENT-SHAPED WEIRS A103 

TABLE 3. Summary of data fw discharge characteristics Continued

Run

No.

q (efs 

per ft)

t 

(feet)

h

(feet)

Ho

(feet)

C Remarks Run 

No.

<? (cfs 

per ft)

t 

(feet)

h 

(feet)

H0

(feet)

C Remarks

MODEL As Continued

Test 2

1 

2

3......

4... ...

5......

0.117

.117

.117

.117

.117

0.036

.067

.101

.125

0.115

.117

.118

.119

.135

0.115

.117

.118

.119

.135

3.02

2.95

2.91

2.87

2.37

F

UL

18

S

S

6 

7......

8... 

9......

.117

.117

.117

.117

.147

.166

.199

-.092

.154

.172

.203

.115

.154

.172

.203

.115

1.95

1.65

1.29

3.02

S

S

S

LL

Tests

yo/yc=i.02]

l.--_.

2__ 

3.....

4.....

0.262 

.262 

.262 

.262 

.262

0.092 

.136 

.169 

.201

0.195 

.197 

.198 

.201 

.221

0.195 

.197 

.198 

.201 

.221

3.04 

3.00 

2.97 

2.91 

2.52

F 

UL 

18

S 

S

6.....

7-. 

g

.262 

.262 

.262 

.262

.234 

.273 

.305 

-.093

.249 

.283 

.313 

.195

.249 

.283 

.313 

.195

2.11 

1.74 

1.50 

3.04

S 

S 

S 

LL

Test 4

[00 =0.189 ft, 0o/0t=1.03J

I.....

2.....

3.. 

4.....

5.. 

0.446 

.446 

.446 

.446 

.446

0.162 

.232 

.269 

.297

0.279 

.281 

.282 

.300 

.322

0.280 

.282 

.283 

.301 

.323

3.01 

2.98 

2.96 

2.70 

2.43

F 

UL 

IS

S 

8

6.. 

7.-...

8.- 

9....-

.446 

.446 

.446 

.446

.339 

.365 

.415 

.035

.357 

.380 

.427 

.279

.358 

.381 

.428 

.280

2.08 

1.90 

1.60 

3.01

S 

S 

S 

LL

Tests

[0o=0.245 ft, 0o/0«=1.04j

1.....

2.....

3-. 

4.....

5---_.

0.649

.649

.649

.649

.649

0.218

.286

.314

.341

0.356

.358

.359

.363

.382

0.358

.360

.361

.365

.384

3.03

3.00

2.99

2.94

2.73

F

UL

IS

S

S

6.....

7. .

8__ 

9.- 

.649

.649

.649

.649

.370

.408

.450

.095

.402

.433

.470

.356

.404

.435

.472

.358

2.53

2.26

2.00

3.03

S

S

S

LL

Test6

[8fo=0.308 ft, ya/V, =1.03]

1-- 

2.. ...

3.....

4.....

5-. 

0.930

.930

.930

.930

.930

0.285

.356

.382

.412

0.449

.450

.451

.453

.467

0.454

.455

.456

.458

.472

3.04

3.03

3.02

3.00

2.87

F

UL

IS

S

S

6. -

7. .-

8-- 

9.- 

.930

.930

.930

.930

.444

.489

.535

.141

.488

.524

.561

.449

.493

.528

.565

.454

2.69

2.42

2.19

3.04

S

S

S

LL

Test?

[y0=0.354 ft, VolV =0.995]

1.. 

2.....

3---..

4.....

5.. 

1.21

1.21

1.21

1.21

1.21

0.356

.409

.452

.480

0.532

.533

.534

.537

.546

0.540

.541

.542

.544

.553

3.06

3.05

3.04

3.02

2.94

F

UL

IS

S

S

6-.._-

7--._.

8-- 

9---..

1.21

1.21

1.21

1 21

.512

.564

.604

.221

.567

.604

.637

.532

.571

.611

.644

.540

2.78

2.54

2.35

3.06

S

S

S

LL

Tests

[y0=0.409 ft, Vo/V =0.970]

1__ 

2.-...

3.....

4.....

5.....

1.56

1.56

1.56

1.56

1.56

0.430

.491

.520

.561

0.627

.628

.629

.631

.639

0.638

.639

.640

.642

.650

3.06

3.05

3.04

3.03

2.97

F

UL

IS

S

S

6

7-. 

8-. 

9.. 

1.56

1.56

1.56

1.56

.619

.664

.711

.295

.675

.710

.758

.627

.686

.720

.768

.638

2.75

2.55

2.32

3.06

S

S

8

LL
A104 FLOW OF WATER OVER DAMS AND WEIRS

TABLE 3. Summary of data for discharge characteristics Continued

Run 

No.

g(cfs 

pet ft)

t

(feet)

h

(feet)

H0

(feet)

C Remarks Run 

No.

g (cfs 

per ft)

t

(feet)

h

(feet)

H0

(feet)

C Remarks

MODEL AB Continued

Test 9

[#,,==0.459 ft, P<,/J'«=0.987J

1 ...

2 ......

3 ...

1.87

1.87

1.87

0.492

.555

0.705

.706

.708

0.720

.721

.723

3.06

3.05

Q (\A O. Vrz

F

TIL

IS

4......

5 

1.87

1.87

.615

.376

.712

.705

.727

.720

3.01

3.06

8

LL

Test 10

[#,,= 0.057 ft, #,,/#c=1.02]

1 __ 

2... ...

0.0748 

.0748 0.017

0.087 

.088

0.087 

.088

2.93

2.88

F 

UL

3 _

4......

.0748 

.0748

.054 

 .141

.089 

.087

.089 

.087

2.84 

2.93

IS 

LL

Test 11

[#,,==0.098 ft, #0/#e=1.03]

I... ...

2......

0.167 

.167 0.068

0.146 

.148

0.146 

.148

3.00 

2.94

F 

UL

3 

4 .

.167 

.167

.100 

 .156

.149 

.146

.149 

.146

2.91 

3.00

IS 

LL

Test 12

1 

2... ...

3... ...

4... ...

5 .. . 

6... 

7 ... 

8 ... 

9 . 

10 

11 

12..... 

13 

14.....

1.31

1.55 

1.83 

.0400 

.0400 

.0400 

.0400 

.0969 

.0969 

.0969 

.0969 

.193 

.193 

.193

-.010 

.027 

-.116

.024 

.059 

-.115

.062 

.104

0.556 

.624 

.692 

.056 

.057 

.058 

.056 

.101 

.102 

.103 

.101 

.160 

.161 

.162

0.565 

.635 

.707 

.056 

.057 

.058 

.056 

.101 

.102 

.103 

.101 

.160 

.161 

.162

3.08 

3.06 

3.08 

3.07 

2.99 

2.90 

3.07 

3.04 

2.99 

2.95 

3.04 

3.02 

2.99 

2.96

F 

F 

F 

F 

UL 

IS 

LL 

F 

UL 

IS . LL 

F 

UL 

IS

15 

16 

17 

18 

19 

20 

21 

22 

23.. _ 

24 

25 

26 

27.- 

.193 

.297 

.297 

.297 

.297 

.362 

.362 

362 

.362 

1.49 

1.61 

1.77 

2.02

-.134

.104 

..ISO 

-.059

.127 

.170 

-.005

.160 

.213 

.214 

.215 

.213 

.243 

.244 

.245 

.243 

.611 

.642 

.682 

.741

.160 

.213 

.214 

.215 

.213 

.244 

.245 

.246 

.244 

.622 

.654 

.696 

.758

3.02 

3.02 

3.00 

2.98 

3.02 

3.01 

3.00 

2.98 

.3.01 

3.04 

3.05 

3.05 

3.06

LL 

F 

UL 

IS 

LL 

F 

UL 

IS 

LL 

F 

F 

F 

F

MODEL AC * 

Testl

[^,,=0.038 ft, ya/yc=!M]

1 ...

2......

3 .

4......

0.0346

.0346

.0346

.0346

-0.021

.003

.045

0.052

.053

.054

.057

0.052

.053

.054

.057

2.90

2.82

2.75

2.53

F

UL

IS

S

5... 

6 

7 

8 

.0346

.0346

.0346

.0346

.074

.098

.129

-.074

.076

.098

.129

.052

.076

.098

.129

.052

1.64

1.12

0.746

2.90

S

S

S

LL

Test 2

[#,,=0.068 ft, #<,/#«=1.12]

1 - .

2 ___

3. .

4......

0.0844

.0844

.0844

.0844

0.011

.043

.074

0.096

.097

.098

.099

0.096

.097

.098

.099

2.83

2.78

2.74

2.70

F

UL

IS

S

5_ 

6 

7 

8 

.0844

.0844

.0844

.0844

.107

.140

.185

-.129

.116

.145

.187

.096

.116

.145

.187

.096

2.13

1.52

1.04

2.83

S

S

S

LL

Tests

[#,,=0.100 ft, #<,/# e=1.08]

1 

2... _

3 

4......

5 

0.159

.159

.159

.159

.159

0.048

.107

.135

.168

0. 145

.147

.148

.155

.179

0.145

.147

.148

.155

.179

2.86

2.80

2.78

2.59

2.09

F

UL

IS

S

S

6 .

7 ...

8 .

9 

.159

.159

.159

.159

.204

.231

.264

-.098

.212

.237

.268

.145

.212

.237

.268

.145

1.62

1.37

1.14

2.86

S

S

S

LL

See footnote at end of table, p. A106.
DISCHARGE OVER EMBANKMENT-SHAPED WEIRS A105 

TABLE 3. Summary of data for discharge characteristics Continued

Run

No.

?(cfs 

per ft)

t

(feet)

h

(feet)

H,

(feet)

C Remarks Run

No.

q (cfs 

per ft) (feet)

h 

(feet)

H0

(feet)

C Remarks

MODEL AC Continued 

Test 4

[5/0=0.137 ft, £/ /£/ = 1.09]

j

2......

3... ._.

4... _

5......

0.253 

.253 

.253 

.253 

.253

0.081 

.135 

.171 

.212

0.196 

.198 

.199 

.203 

.229

0.197 

.199 

.200 

.204 

.230

2.90 

2.85 

2.83 

2.75 

2.30

F 

UL 

IS

S 

S

6... ...

7......

8 ...

Q

.253 

.253 

.253 

.253

.241 

.278 

.319 

-.048

.254 

.287 

.327 

.197

.255 

.288 

.328 

.198

1.97 

1.64 

1.35

2.88

S 

S 

S 

LL

Tests

[j/«=0.175 ft, jr«/jre=1.06]

1 

2......

3_..._.

4......

5_ _.

0.379

.379

.379

.379

.379

0.131

.199

997

9(U

0.256

.258

.259

.266

OQQ

0.257

.259

.260

OA7

9on

2.90

2.87

2.85

2 7J.

9 d.9

F

UL

IS

S

Q8

6 ...

7... ...

8 _

.379

.379

.379

17Q

.319

.379

.427

-.001

.335

.390

.436

.257

.336

.391

.437

.258

1.94

1.55

1.31

2.89

S

S

S

LL

Test 6

[5/0=0.226 ft, yjy. =1.08]

1 

2......

3 

4......

5 ...

0.541

.541

.541

.541

.541

0.174

.254

.304

.343

0.320

.321

.322

.340

.368

0.322

.323

.324

.342

.370

2.96

2.94

2.93

2.70

2.40

F

UL

IS

S

S

6 

8 ...

9... ...

.541

.541

.541

.541

.379

.431

.477

.044

.398

.445

.489

.319

.400

.447

.491

.321

2,14

1.81

1.57

2.97

S

S

S

LL

Test 7 

[£,,=0.257 ft, y 0/y<,=lM]

1 

2... ...

3......

4... ...

5 

0.678

.678

.678

.678

.678

0.212

.280

.329

.368

0.370

.371

.372

.381

.405

0.373

.374

.375

.384

.408

2.97

2.96

2.95

2.85

2.60

F

UL

IS

S

S

6 ...

7 

8 ...

9... 

.678

.678

.678

.678

.406

.452

.489

.075

.434

.473

.506

.370

.437

.476

.509

.373

2.35

2.07

1.87

2.97

3

S

S

LL

Tests 

[?.=0.30&ft, y,!ye =1.05]

1... ...

2......

3... 

4_.____

0.890

.890

.890

.890

0.264

.329

.370

0.440

.442

.443

.446

0.445

.447

.448

.451

3.00

2.98

2.97

2.94

F

UL

IS

S

6... 

6 

7 

8... 

9... 

.890

.890

890

.890

.890

.416

.452

.492

.528

.076

.466

.491

.523

.553

.441

.471

.496

.528

.557

.446

2.75

2.55

2.32

2.14

2.99

3

S

S

S

LL

Test9

[jr.-0. =1.05]

1 ......

2......

3... 

4......

1.14

1.14

1.14

1.14

0.338

.405

.468

0.517

.519

.520

.535

0.524

.526

.527

.542

3.00

2.99

2.98

2.86

F

UL

IS

S

6 

6... ...

7......

8_ ..

9 

1.14

1.14

1.14

1.14

1.14

.549

.598

.657

.702

.176

.590

.630

.684

.724

518

.597

.637

.690

.730

.525

2.47

2.25

1.99

1.83

3.00

S

S

S

s.

LL

Test 10

1J1.-ISW

1..... .

2......

3......

4......

5......

1.39

1.39

1.39

1.39

1.39

0.401

.479

.551

.611

0.586

.588

.590

.613

.657

0.596

.598

.600

.620

.666

3.01

3.00

2.98

2.84

2.55

F

UL

IS

S

S

6 

7 ...

8... ...

9......

1.39

1.39

1.39

1.39

.666

.721

.775

.251

.701

.750

.798

.586

.710

.759

.806

.596

2.32

2.10

1.92

3.01

S

S

S

LL
A106 FLOW OF WATER OVER DAMS AND WEIRS

TABLE 3. Summary of data for discharge characteristics Continued

Run

No.

?(cfs 

per ft)

t

(feet)

h

(feet)

H,

(feet)

C Remarks Run

No.

q (cfs 

per ft)

t

(feet)

h

(feet)

H0

(feet)

C Remarks

MODEL AC Continued 

Test 11

[y,,=0.426 ft, VolVe=l.02}

1..-.- 1.53

1.53 0.421

0.623

.625

0.635

.637

3.03

3.02

F

UL

3.._ 

4......

1.53

1.53

.492

.303

.626

.623

.638

.635

3.01

3.03

IS

LL

Test 12

[y,,=0.445 ft, VolV, =1.01]

1 .-,- 1.65

1:65' 0.427

0,654

656

0.667

- . 669

3.03

3.02

F

UL 4... _

1.65

1.65

.461

.323

.657

654

.670

.667

3.01

3.03

IS

LL

.Test 13

.=0.471 ft, yjy =0.999]

1 

2

1.84

1.84 0..464

0.700

.703

(K715

.718

0 f\A

o no UL

3_._ 

4......

1 84

1.84

too

.352

7(Vi

.701

719

.716

o m

3.03

IS

LL

. Test 14

=0.993]

1.. - 1.95 A "TOO 0 7Afi q no

1 Model dimensions and other- details are given in table 1.

2 Head datum takea at mid-depth of screen on crown line.

3 Head datum iaken at smooth surface on crown line. 

< Head datum taken at mid*depth of shot on crown line.

- TABLE 4. Summary *of*data for boundary-flayer velocity distribution

Station ? 

 (feet)

2 '(feet 

X103)

u 

(fps)

U

(fps)

Station * 

(feet)

2? (feet xio") u 

(fps)

U 

 (fps)

Station * 

(feet)

z s (feet 

XIO3)

u 

(fps)

U

(fps)

MODEL A-l 

Testl

0.006

.006

.006

.006

.006

.006

.006

.006

.006

.006

.006

.10

.10

.10

.10

.10

.10

.10

.25

.25

.25

.25

.25

.25

1.15

1.57

1.98

2.40

2.82

4.48

9.48

12.8

14.9

17.8

26.2

1.15

1.98

2.82

4,48

6.57

9.48

18.0

1.15

1.98

2.82

4.48

6.57

9.48

-0.264

.073

.264

.394

.532

.532

.532

.532

.518

.512

.532

.401

.469

.543

.543

.558

.567

.567

-.073

.351

.469

.603

.625

.621

0.532

.532

.532

.532

.532

.532

.532

.532

.532

.532

.532

.567

.567

.567

.567

.567

.567

.567

.629

.629

.629

.629

.629

.629

.25

.40

.40

.40

.40

.40

.40

.40

.40

.65

.65

.65

.65

.65

.65

.65

.65

1.05

1.05

1.95

1.05

1.05

1.05

1.05

17.8

1.15

1.98

2.82

4.48

6.57

9.48

17.8

26.2

1.15

1.98

2.82

4.48

6.57

9.48

17.8

26.2

1.15

1.98

2-82

4.48

6.57

9.48

17.8

.629

-.057

.359

.474

.576

.717

.717

, .717

.724

-.433

.264

.420

.532

.743

.825

.832

.828

-.057

.414

.558

.751

.918

.964

.961

.J629

.720

.720

.720

'.720

.720

.720

.720

.720-

.830

.830

.830

.830

.830

.830

.830

.830

.961

.961

.961

.961

.961

.961

.961

1.05

1.05

1.65

1.65

1.65

1.65

1.65

1.65

1.65

1.65

1.65

1.65

1.85

1.85

1.85

1.85

1.85

1.85

. 1.85

1.85

1.85

1.85

26.2

34.5

1.15

1.98

2.82

4.48

6.57

9.48

13.7

17.8

26.2

34.5

1.15

1.98

3.23

5.32

7.82

10.3

13.7

17.8

26.2

34.5

.961

.961

.625

.687

.779

1.003

1.186

1.276

1.298

1.309

1.309

1.309

.955

1.008

1.128

1.393

1.505

1.522

1.543

1.543

1.543

1.543

.961

.961

1.309

1.309

1.309

1.309

1.309

1.309

1.309

1.309

1.309

1.309

1,543

1.543

1.543

1.543

1.543

1.543

1.543

1.543

1.543

1.543

See footnotes at end of table, p. A114.
DISCHARGE OVER EMBANKMENT-SHAPED WEIRS A107 

TABLE 4. Summary of data for boundary-layer velocity distribution Continued

Station 3 

(feet)

03 (feet

X103)

u 

(fps)

U 

(fps)

Station 2 

(feet)

z s (feet 

X103)

u 

(fps)

U 

(fps)

Station 3 

(feet)

z 3 (feet 

X103)

u 

(fps)

U 

(fps)

MODEL A-l Continued 

Test 2

.006

.006

.006

.006

.006

.006

.006

.006

.10

.10

.10

.10

.10

.10

.10

.10

.10

.25

.25

.25

.25

.25

.25

.25

.25

.25

1.15

1.98

2.82

4.48

6.57

9.48

17.8

26.2

1.15

1.98

2.82

4.48

6.57

9.48

17.8

26.2

34.5

1.15

1.98

2.82

4.48

6.57

9.48

17.8

26.2

34. 5

.451

.533

.918

1.098

1.151

1.151

1.151

1.151

.613

.733

.819

1.006

1.111

1.158

1.158

1.158

1.149

.809

.938

.994

1.069

1.153

1.204

1.260

1.258

1.258

1.151

1.151

1.151

1.151

1.151

1.151

1.151

1.151

1.158

1.158

1.158

1.158

1.158

1.158

1.158

1.158

1.158

1.258

1.258

1.258

1.258

1.258

1.258

1.258

1.258

1.258

.40

.40

.40

.40

.40

.40

.40

.40

.40

.65

.65

.65

.65

.65

.65

.65

.65

.65

.65

.65

1.05

1.05

1.05

1.05

1.05

1.05

1.15

1.98

2.82

4.48

6.57

9.48

17.8

26.2

34.5

1.15

1.98

3.23

5.73

9.48

15.7

22.0

28.2

34.5

51.2

59.5

1.15

1.98

3.65

7.82

13.7

20.3

.469

.691

.930

1.089

1.123

1.239

1.282

1.302

1.302

.765

.988

1.148

1.269

1.346

1.420

1.437

1.441

1.448

1.451

1.455

.702

.776

1.014

1.284

1.413

1 455

1.302

1.302

1.302

1.302

1.302

1.302

1.302

1.302

1.302

1.455

1.455

1.455

1.455

1.455

1.455

1.455

1.455

1.455

1.455

1.455

1.520

1.520

1.520

1.520

1.520

1.520

1.05

1.05

1.05

1.65

1.65

1.65

1.65

1.65

1.65

1.65

1.65

1.65

1.65

2.00

2.00

2.00

2.00

2 00

2.00

2.00

2.00

2.00

2.00

2.00

27.0

34.5

51.2

1.15

1.98

3.65

7.82

13.7

20.3

27.0

34.5

51.2

59.5

1.15

1.98

3.65

7.82

13.7

20.3

27.0

34.5

42.8

51.2

59.5

1.508

1.513

1.520

1.172

1.194

1.458

1.707

1.834

1.907

1.933

1.945

1.950

1.954

1.593

1.637

1.849

2.030

2.089

2.125

2.172

2.188

2.212

2.212

2.212

1.520

1.520

1.520

1.954

1.954

1.954

1.954

1.954

1.954

1.954

1.954

1.954

1.954

2.212

2.212

2.212

2.212

2.212

2.212

2.212

2.122

2.212

2.212

2.212

TestS

.006

.006

.006

.006

.006

.006

.006

.006

.10

.10

.10

.10

.10

.10

.10

.10

.10

.25

.25

.25

.25

.25

.25

.25

.25

.25

.40

.40

.40

.40

.40

.40

.40

.40

.40

1.15

1.98

2.82

4.48

6.57

9.48

17.8

26.2

1.15

1.98

2.82

4.48

6.57

9.48

17.8

26.2

34.5

1.15

1.98

4.48

7.82

12 0

17.8

26.2

34.5

47.0

1.15

1.98

4.48

7.82

12.0

17.8

26.2

34.5

42.8

.753

1.365

1.563

1.682

1.712

1.733

1.733

1.733

1.156

1.203

1.308

1.404

1.530

1.688

1.736

1.736

1.732

1.106

1.186

1.453

1.559

1.657

1.746

1.772

1.772

1.772

1.195

1.260

1.552

1.628

1.716

1.808

1.860

1.864

1.867

1.733

1.733

1.733

1.733

1.733

1.733

1.733

1.733

1.736

1.736

1.736

1.736

1.736

1.736

1.736

1.736

1.736

1.772

1.772

1.772

1.772

1.772

1.772

1.772

1.772

1.772

1.867

1.867

1.867

1.867

1.867

1.867

1.867

1.867

1.867

.40

.65

.65

.65

.65

.65

.65

.65

.65

.65

.65

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.30

1.30

1.30

1.30

1.30

1.30

1.30

1.30

1.30

1.30

1.30

1.65

1.65

51.2

1.15

1.98

4.48

7.82

12.0

17.8

26.2

34.5

42.8

51.2

1.15

1.98

5.32

9.48

13.7

17.8

26.2

34.5

42.8

51.2

1.15

1.98

5.32

9.48

13.7

17.8

26.2

34.5

42.8

51.2

59.5

1.15

1.98

1.867

1.275

1.407

1.658

1.784

1.866

1.945

2.003

2.030

2.030

2.032

1.314

1.494

1.786

1.915

2.015

2.090

2.150

2.172

2.172

2.172

1.254

1.474

1.797

1.904

2.024

2.097

2.180

2.220

2.222

2.226

2.226

1.504

1.710

1.867

2.032

2.032

2.032

2.032

2.032

2.032

2.032

2.032

2.032

2.032

2.172

2.172

2.172

2.172

2.172

2.172

2.172

2.172

2.172

2.172

2.226

2.226

2.226

2.226

2.226

2.226

2.226

2.226

2.226

2.226

2.226

2.475

2.475

1.65

1.65

1.65

1.65

1.65

1.65

1.65

1.65

1.65

1.65

1.85

1.85

1.85

1.85

1.85

1.85

1.85

1.85

1.85

1.85

1.85

1.85

2.00

2.00

2 00

2.00

2.00

2.00

2.00

2.00

2.00

2.00

2.00

2.00

4.48

7.40

10.3

15.3

22.0

26.2

42.8

51.2

59.5

67.8

1.15

1.98

4.48

7.40

10.3

15.3

22.0

26.2

34.5

42.8

51.2

59.5

1.15

1.98

4.48

7.40

10.3

15.3

22.0

26.2

34.5

42.8

51.2

59.5

1.943

2.050

2.125

2.240

2.327

2.384

2.410

2.470

2.475

2. 475

1.598

1.806

2.060

2.196

2.295

2.420

2.500

2.560

2.580

2.620

2.624

2.624

1.722

1.803

2.176

2.302

2.394

2.532

2.620

2.684

2.702

2.755

2.755

2.759

2.475

2.475

2.475

2.475

2.475

2.475

2.475

2.475

2.475

2.475

2.624

2.624

2.624

2.624

2.624

2.624

2.624

2.624

2.624

2.624

2.624

2.624

2.759

2.759

2.759

2.759

2.759

2.759

2.759

2.759

2.759

2.759

2.759

2.759

See footnotes at end of table, p. A114.
A108 FLOW OF WATER OVER DAMS AND WEIRS

TABLE 4. Summary of data for boundary^layer velocity distribution Continued

Station 2 

(feet)

08 (feet 

X10>)

U"

(fps)

17

(fps)

Station 2 

(feet)

z s (feet 

X103)

U

(fps)

U 

(fps)

Station 3 

(feet)

z a (feet 

X103)

U

(fps)

U 

(fps)

MODEL A-l Continued

Test 4

.006

.006

.006

.006

-.006

.006

.006

.006

.006

.10

.10

.10

.10

.10

.10

.10

.10

.10

.10

.10

.25

.25

.25

.25

.25

.25

.25

.25

.25

.25

.40

.40

.40

.40

.40

.40

.40

.40

.40

.40

.40

.40

.65

.65

1.15

1.98

2.82

- 4.48

6.57

9.48

17-.8

26.2

38.7

1.15

1.98

2.82

4.48

5.32

6.57

7.40

9.48

17.8

26.2

,42.8

1. 15

1.98

3.23

6.15

9-.4S

13.7

 17.8

26.2

34.5

51.2

1.15

1.98

3.23

6.15

9.48

13.7

17.8

22.0

26.2

34.5

42.8

51.2

1.15

1.98

-1.718

-1.222

1.920

2.292

2.376

2.424

2.447

^.450

2.452

1.379

1.547

1.637

1.776

1.892

2.016

2.067

2.255

2,413

2,426

 2.426

1.562

1.706

1.836

2.007

2.139

:2.277

2.377

2.458

2-.45S

2.468

1.631

1.777

1.948

2.-106

2.204

2:347

2.424

2. 475.

2. 616

2.545.

2.545

2.545

1.779

2.020

-2 452

2.452

2.452

2.452

2/452

2.452

2.452

2.452

2.452

2.426

2.426

-2.426

2.426

2.426

2.426

2.426

2.426

2.426

2.426

2-.426

2.458

2.458

2.458

2.458

2.458

2.458

2.458

2.458

2.458

2.458

2.545

2.545

2.545

2.545

2.545

2.545

2.545

2.545

2.545

2.545

2.545

2.545

. 2. 752

2.752

.65

.65

.65

.65

.65

.65

.65

.65

.65

.65

.65

1.00

1.00

1.00

1.00

1.00

,1.00

1.00

l.-OO

1.00

1:00

1.00

1.00

1.00

1.30

1.30

1.30

1.30

1.30

1.30

1.30

1.30

1.30

T:30

1.30

1.30

1.30

1.30

1.30

1.65

1.65

1.65

1.65

3.23

6.15

9.48

13.7

17.8

22.0

26.2

34.5 .

42.8

51.2

59.5

1.15

1.98

2.82

6rl5

4.48

13.7

17.8

22.0

26.2

34.5

 42.8

51.2

59:5

1.15

1.98

2.82

6.15

 9.48"

13.7

 17.8

19.9

22.0

26:2

34,5

 42.8

51.2

59.5

67.8

1.15

1.98

2.82

.6: 15

2,163

2.331

2.436

2.538

2.627

>2.676

2:706

"2. 740

2.750

- 2. 752

2.752

1.757'

1.946

2.075

. 2. 312

2.464

2.610

2:707

2.765

2. 811

.2.843

2.862

2.862-

2.862

1?828

1.963

2.092

2.354

2.503

2.652

2.736

2.794

, 2.812

2.854

2.917

2.968

2.976

2.976

2.976

1.982

2.006

2.220

2.536

2.752

2.752

2.752

% 752

2.752

2.752

2.752

.2.752

2.752

2. 752

2.752

2.862

2.862

2.862

2.862

2.862

2.862

2.862

2.862

2:862

2.862

2.862

2.862

2.862

2-976

2.976

2.976

2.976

2,976

2.976

2.976

2.976

Z976

2.976

.2. 976

2.976

2.976

2.976

2.976

3.210

"3.210

3.210

2.210

1.65

1,65

1.65

1.65

1.65

L65

1.65

L65

1.65

1.65

1.65

1.85

1.85

.1.85

1.85

1.85

1.85

,1.85

1.85

1.85

1.85

1.85

L86 '

1.85

1.85

1.85

1.85

2.00

2.00

2.00

2.00

2.00

2.00

-2.00

2.00

2.00

2.00

2.00

2.00

2.00

2.00

2.00

2.00

9.48

13.7

17.8

22.0

26.2

34.5

42.8

51.2

59.5

67.8

76.2

1.15

1.98

2.82

6.15

9.48

13.7

17.2

22.0

26.2

34.5

42.8

51.2

59.5

67.8

76.2

92.8

1.15

1.98

2.82

6.15

9.48

13.7

17.8

22.0

26.2

34.5

42.8

51.2

59.5

67.8

76.2

92.8

2.682

2.851

2.947

3.040

3.104

3.160

3.192

3.198

3.210

3.210

3.210

2.047

2.252

2.382

2.623

2.728

2.947

3.058

3.142

3.203

3,260

3.290

3.292

3.297

3.297

3.297

3.297

2.045

2.270

2.394

2.643

2.810

2.976

3.106

3.194

3.250

3.333

3. 361

3.372

3.382

3.385

3.385

3.385

3.210

2.210

3.210

3.210

3.210

2.210

3.210

3.210

3.210

3.210

3.210

3.297

3.297

3.297

3.297

3.297

3.297

3.297

3.297

3.297

3.297

3.397

3.297

3.297

3.297

3.297

3.297

3.385

3.385

3.385

3.385

3.385

3.385

3.385

3.385

3.385

3.385

3.385

3.385

3.385

3.385

3.385

3.385

'See footnotes at end of table, p. A114.
DISCHARGE OVER EMBANKMENT-SHAPED WEIRS A109

TABLE 4. Summary of data for boundary-layer velocity distribution Continued

Station « 

(feet)

2 '(feet

X103)

u 

(fps)

U

(fps)

Station * 

(feet)

2 a (feet 

X1Q3)

u 

(fps)

U

(fps)

Station * 

(feet)

z s (feet 

X103)

u 

(fps)

17

(fps)

MODEL A-l Continued 

Tests

.006

.006

.006

.006

.006

.006

.006

.006

.006

.006

.006

.10

.10

.10

.10

.10

.10

.10

.10

.10

.10

.10

.25

.25

.25

.25

.25

.25

.25

.25

.25

.25

.25

.25

.25

.40

.40

.40

.40

.40

1.15

2.82

4.48

6.98

7.82

9.48

13.7

17.8

26.2

34.5

47.0

1.15

3.23

5.32

9.48

11.6

13.7

15.7

17.8

26.2

38.7

51.2

1.15

3.23

5.32

9.48

11.6

13.7

15.7

17.8

22.0

26.2

34.5

47.0

59.5

5.32

9.48

12.0

14.9

17.8

-2. 124

2.108

2.710

2.824

2.854

2.862

2.904

2.934

2.936

2.936

2.937

1.678

2.028

2.256

2.653

2.747

2.816

2.840

2.850

2.881

2.883

2.883

1.640

2.072

2.236

2.466

2.576

2.704

2.728

2.787

2.819

2.847

2.868

2.872

2.872

2.336

2.575

2.678

2.763

2.842

2.937

2.937

2.937

2.937

2.937

2.937

2.937

2.937

2.937

2.937

2.937

2.883

2.883

2.883

2.883

2.883

2.883

2.883

2.883

2.883

2.883

2.883

2.872

2.872

2.872

2.872

2.872

2.872

2.872

2.872

2.872

2.872

2.872

2.872

2.872

2.997

2.997

2.997

2.997

2.997

.40

.40

.40

.40

.40

.40

.65

.65

.65

.65

.65

.65

.65

.65

.65

.65

.65

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.30

1.3C

1.30

1.30

1.30

1.30

1.30

1.30

1.30

1.30

1.30

1.30

22.0

26.2

34.5

42.8

59.5

76.2

5.32

9.48

12.8

16.2

19.9

24.1

28.2

34.5

42.8

59.5

76.2

5.32

9.48

12.8

16.2

19.9

24.1

28.2

34.5

42.8

59.5

76.2

5.32

9.48

12.8

16.2

19.9

24.1

28.2

34.5

42.8

59.5

76.2

84.5

2.908

2.933

2 982

2! 997

2.997

2.997

2.618

2.840

2.922

2.992

3.080

3.124

3.162

3.204

3.230

3.232

3.232

2.740

2.974

3.074

3.190

3.273

3.320

3.353

3.384

3.402

3.428

3.428

2.794

3.070

3.192

3.290

3.372

3.427

3.456

3.510

3.527

3.561

3.562

3.562

2.997

2.997

2.997

2.997

2.997

2.997

3.232

3.232

3.232

3.232

3.232

3.232

3.232

3,232

3.232

3.232

3.232

3.428

3.428

3.428

3.428

3.428

3.428

3.428

3.428

3.428

3.428

3.428

3.562

3.562

3.562

3.562

3.562

3.562

3.562

3.562

3.562

3.562

3.562

3.562

1.65

1.65

1.65

1.65

1.65

1.65

1.65

1.65

1.65

1.65

1.65

1.65

1.65

1.85

1.85

1.85

1.85

1.85

1.85

1.85

1.85

1.85

1.85

1.85

1.85

1.85

2.00

2.00

2.00

2.00

2.00

2.00

2.00

2.00

2.00

2.00

2.00

2.00

2.00

5.32

9.48

12.8

16.2

19.9

24.1

28.2

34.5

42.8

51.2

59.5

76.2

92.8

5.32

9.48

12.8

16.2

19.9

24.1

28.2

34.5

42.8

51.2

59.5

76.2

92.8

5.32

9.48

12.8

16.2

19.9

24.1

28.2

34.5

42.8

51.2

59.5

76.2

92.8

3.008

3.261

3.377

3.498

3.600

3.642

3.676

3.732

3.752

3.782

3.815

3.815

3.815

2.973

3.255

3.364

3.558

3.657

3.708

3.754

3.805

3.848

3.858

3.876

3.876

3.880

3.026

3.300

3.455

3.608

3.715

3.784

3.834

3.881

3.906

3.928

3.955

3.955

3.955

3.815

3.815

3.815

3.815

3.815

3.815

3.815

3.815

3.815

3.815

3.815

3.815

3.815

3.880

3.880

3.880

3.880

3.880

3.880

3.880

3.880

3.880

3.880

3.880

3.880

3.880

3.955

3.955

3.955

3.955

3.955

3.955

3.955

3.955

3.955

3.955

3.955

3.955

3.955

See footnotes at end of table, p. A114.
A110 FLOW OF WATER OVER DAMS AND WEIRS

TABIJS 4. Summary of data for boundary-layer velocity distribution Continued

Station 2

(feet)

z s (feet xio3)u

(fps)

U

(fps)

Station 2

(feet)

2» (feet

X103)

u

(fps)

U

(fps)

Station 2

(feet)

2 '(feet

X103)

u

(fps)

U

(fps)

MODEL K-1

Testl

.20

.20

.20

.20

.20

.20

.20

.20

.20

.20

.20

.20

.20

.20

.60

.60

.60

60

.60

.60

.60

.60

.60

.60

.60

.60

.60

.60

1.15

1.98

2.82

3.65

4.48

5.32

6.15

6.98

7.82

8.65

9.48

10.3

12.8

17.0

1.15

1.98

2.82

3.65

4.48

5.32

6.15

6.98

7.82

8.65

9.48

10.3

11.2

12.8

.242

.319

.439

.528

.599

.650

.690

.735

.750

.756

.756

.764

.767

.774

.254

.351

.451

.528

.585

.642

.702

.757

.788

.809

.828

.853

.853

.857

.774

.774

.774

.774

.774

.774

.774

.774

.774

.774

.774

.774

.774

.774

.869

.869

.869

.869

.869

.869

.869

.869

.869

.869

.869

.869

.869

.869

.60

.60

.60

.60

1.10

1.10

1.10

1.10

1.10

1.10

1.10

1.10

1.10

1.10

1.10

1.10

1.10

1.10

1.10

1.10

1.10

1.10

1.65

1.65

1.65

1.65

1.65

14.5

17.0

22.0

26.2

1.98

2.82

3.65

4.48

5.32

6.15

6.98

7.82

8.65

9.48

10.3

11.2

12.0

12.8

14.5

17.8

22.0

26.2

1.15

2.40

3.65

4.90

6.15

.857

.860

.863

.869

.552

.616

.659

.721

.784

.850

.878.

.902

.926

.951

.960

.960

.965

.968

.968

.976

.976

.979

.671

.795

1.034

1.204

1.254

.869

.869

.869

.869

.979

.979

.979

.979

.979

.979

.979

.979

.979

.979

.979

.979

.979

.979

.979

.979

.979

.979

1.361

1.361

1.361

1.361

1.361

1.65

i.65

1.65

1.65

1.65

1.65

1.65

1.65

1.65

1.65

1.65

1.80

1.80

1.80

1.80

1.80

1.80

1.80

1.80

1.80

1.80

1.80

1.80

1.80

1.80

1.80

1.80

6.98

7.82

8.65

9.48

10.3

11.2

12.8

14.5

17.8

22.0

26.2

1.15

2.40

3.65

4.90

6.15

6.98

7.82

8.65

9.48

10.3

11.2

12.8

14.5

17.8

22.0

26.2

1.290

1.316

1.345

1.349

1.351

1.351

1.351

1.352

1.357

1.359

1.361

.663

.971

1.200

1.376

1.416

1.452

1.465

1.481

1.499

1.507

1.507

1.507

1.508

1.511

1.513

1.516

1.361

1.361

1.361

1.361

1.361

1.361

1.361

1.361

1.361

1.361

1.361

1.516

1.516

1.516

1.516

1.516

1.516

1.516

1.516

1.516

1.516

1.516

1.516

1.516

1.516

1.516

1.516

Test 2

.20

.20

.20

.20

.20

.20

.20

.20

.20

.20

.20

.20

.20

.20

.60

.60

.60

.60

.60

.60

.60

.60

.60

.60

.60

.60

.60

.60

.60

:eo

1.10

1.10

1.10

1.10

1.10

1.10

1.10

1.10

1.15

1.98

2.82

3.65

4.48

5.32

6.15

6.98

7.82

8.65

9.48

13.7.

17.8

26.2

1.15

1.98

2.82

3.65

.4.48

5.32

.6.57

7.82

9.48

1ft 3

11.2

12.8

14.9

17.8

22.0

26.2

1.15

1.98

a 23

5.32

6.57

7.15

7.82

8.40

.401

.542

.746

.923

1.060

1.174

-1.250

1.302

1.336

1.341

1.341

1.349

1.347

1.347

.401

.552

.778

.926

1.054

1.156.

1.269

'1,358

-1.414

1.437

1.437

1,437

1.437

1.441

1.442

1.444

.603

.702

.976

1.287

1.408

1.448

1.490

1.526

1.347

1.347

1.347

1.347

1.347

1.347

1.347

1.347

1.347

1.347

1.347

1.347

1.347

1.347

1.444

1.444

1.444

1.444

1.444

1.444

1.444

1.444

1.444

1.444

1.444

 1.444

1.444

1.444

1.444

1.444

1.638

1.638

1.638

1.638

1.638

1.638

1.638

1.638

1.10

1.10

1.10

1.10

1.10

1.10

1.10

1.10

1.10

1.10

1.10

1.10

1.10

1.10

1.10

1.10

1.40

1.40

1.40

1.40

 1.40

1.40

1.40

1.40

1.40

1.40

1.40

1.40

1.40

1.40

1.40

1.40

1.40

1.40

1.65

1.65

1.65

9.07

9.65

10.3

10.7

11.2

11.6

12.0

12.7

13.2

13.8

14.5

15.3

16.2

17.8

22.0

26.2

1.15

1.98

2.82

3.65

4.48

5.32

6.15

6.98

7.82

8.65

9.48

10.3

11.2

12.8

14.5

17.0

21.2 .

26.2

1,15

3.23

4.48

1.551

1.566

1.586

1.599

1.598

1.602

1.602

1.620

1.607

1.620

1.635

1.628

1.635

1.638

1.638

1.638

.774

.968

1. 114

1.304

1.411

l;495

1.574

1.630

1.674

1.706

, 1.732

1.750

1.750

1.750

. 1.752

1.753

1. 755 .

1.756

.937

1.409

1.644

1.638

1.638

1.638

1.638

1.638

1.638

1.638

1.638

1.638

1.638

1.638

1.638

1.638

1.638

1.638

1.638

1.756

1.756

1.756

1.756

1.756

1.756

1.756

1.756

1.756

1.756

1.756

1.756

.1.756

1.756

1.756

1.756

1.756

1.759

1.993

1.993

1.993

1.65

1.65

1.65

1.65

1.65

1.65

1.65

1.65

1.65

1.65

1.65

1.65

1.65

1.65

1.65

1.65

1.65

1.80

1.80

1.80

1.80

1.80

1.80

1.80-

1.80

1.80

1.80

1.80

1.80

1.80

1.80

1.80

1.80

1.80

1.80

1.80

1.80

5.73

6.98

8.23

9.07

9.90

10.3

10.7

11.2

11.6

12.0

12.8

14.9

17.0

19.5

22.0

26.2

30.3

1.15

1.98

2.82

3.65

4.48

5.32

6.15

6.98

7.82

8.65

9.48

10.3

11.2

12.0

12.8

14.5

16.2

17.8

22.0

26.2

1.769

1.872

1.912

1.942

1.958

1.962

1.965

1.965

1.971

1.976

1.979

1.980

1.985

1.985

1.993

1.992

1.993

.929

1.165

1.428

1.618

1.774

1.882

1.938

1.994

2.024

2.045

2.068

2.067

2.081

2.089

2.089

2.093

2.093

2.093

2.095

2.095

1.993

1.993

1.993

1.993

1.993

1.993

1.993

1.993

1.993

1.993

1.993

L993

1.993

1.993

1.993

1.993

1.993

2.095

2.095

2.095

2.095

2.095

2.095

2.095

2.095

2.095

2.095

2.095

2.905

2.095

2.095

2.095

2.095

2.095

2.095

2.095

2.095

See footnotes at end of table, p. A114.
DISCHARGE OVER EMBANKMENT-SHAPED WEIRS Alll 

TABLE 4. Summary of data for boundary-layer velocity distribution Continued

Station * 

(feet)

z* (feet 

X1Q3)

u 

(fps)

U 

(fps)

Station 2 

(feet)

z»(feet 

X1Q3)

u 

(fps)

U 

(fps)

Station » 

(feet)

z«(feet 

X103)

u 

(fps)

C7 

(fps)

MODEL K-l Continued 

Test3

.20

.20

.20

.20

.20

.20

.20

.20

.20

.20

.20

.20

.60

.60

.60

.60

.60

.60

.60

.60

.60

.60

.60

.60

.60

1.10

1.15

2.82

3.65

4.48

5.32

6.15

7.82

9.48

11.2

13.7

17.8

22.0

1.15

2.82

4.48

6.15

7.82

9.48

11.2

12.8

14.5

17.8

22.0

26.2

34.5

1.15

.501

1.110

1.503

1.642

1.825

1.837

1.892

1.927

1.935

1.962

1.962

1.962

.571

1.153

1.450

1.707

1.804

1.890

1.935

1.949

1.955

1.970

1.976

1.976

1.977

1.117

1.962

1.962

1.962

1.962

1.962

1.962

1.962

1.962

1.962

1.962

1.962

1.962

1.977

1.977

1.977

1.977

1.977

1.977

1.977

1.977

1.977

1.977

1.977

1.977

1.977

2.221

1.10

1.10

1.10

1.10

1.10

1.10

1.10

1.10

1.10

1.10

1.10

1.10

1.10

1.10

1.10

1.65

1.65

1.65

1.65

1.65

1.65

1.65

1.65

1.65

1.65

2.82

4.48

6.15

7.82

9.48

11.2

12.8

14.5

16.2

17.8

19.9

22.0

26.2

30.3

34.5

1.15

2.82

4.90

6.98

8.65

10.3

12.0

13.7

15.3

17.0

1.562

1.789

1.956

2.040

2.120

2.135

2.145

2.174

2.193

2.205

2.213

2.220

2.221

2.221

2.221

1.368

1.905

2.130

2.253

2.305

2.352

2.390

2.416

2.448

2.476

2.221

2.221

2.221

2.221

2.221

2.221

2.221

2.221

2.221

2.221

2.221

2.221

2.221

2.221

2.221

2.524

2.524

2.524

2.524

2.524

2.524

2.524

2.524

2.524

2.524

1.65

1.65

1.65

1.65

1.65

1.65

1.65

1.80

1.80

1.80

1.80

1.80

1.80

1.80

1.80

1.80

1.80

1.80

1.80

1.80

1.80

1.80

1.80

1.80

1.80

18.7

20.3

22.0

24.1

26.2

30.3

34.5

1.15

2.82

4.90

6.98

8.65

11.2

12.8

14.5

16.2

17.8

19.9

22.0

24.1

26.2

30.3

34.5

38.7

42.8

2.502

2.511

2.520

2.522

2.523

2.524

2.524

1.531

1.942

2.152

2.270

2.356

2. 419

2.455

2.490

2.514

2.523

2.545

2.562

2.568

2.574

2.580

2.582

2.583

2.583

2.524

2.524

2.524

2.524

2.524

2.524

2.524

2.583

2.583

2.583

2.583

2.583

2.583

2.583

2.583

2.583

2.583

2.583

2.583

2.583

2.583

2.583

2.583

2.583

2.583

Test 4

.20

.20

.20

.20

.20

.20

.20

.20

.20

.20

.20

.20

.20

.20

.20

.20

.20

.20

.20

.20

.60

.60

.60

.60

.60

.60

.60

.60

.60

.60

.60

.60

1.15

1.98

2.82

3.65

4.48

5.32

6.15

6.98

7.82

8.65

9.48

10.3

11.2

12.8

14.5

16.2

17.8

22.0

26.2

34.5

1.15

1.98

3.23

5.32

7.40

9.48

11.6

13.7

15.7

17.8

19.9

22.0

1.736

1.814

1.998

2.130

2.225

2.280

2.345

2.400

2.453

2.500

2.524

2.540

2.560

2.600

2.616

2.626

2.640

2.640

2.640

2.640

1.743

1.980

2.184

2.336

2.442

2.537

2.624

2.690

2.744

2.768

2.802

2.824

2.640

2.640

2.640

2.640

2.640

2.640

2.640

2.640

2.640

2.640

2.640

2.640

2.640

2.640

2.640

2.640

2.640

2.640

2.640

2.640

2.840

2.840

2.840

2.840

2.840

2.840

2.840

2.840

2.840

2.840

2.840

2.840

.60

.60

.60

.60

.60

1.10

1.10

1.10

1.10

1.10

1.10

1.10

1.10

1.10

1.10

1.10

1.10

1.10

1.10

1.10

1.10

1.10

1.40

1.40

1.40

1.40

1.40

1.40

1.40

1.40

1.40

1.40

24.1

26.2

30.3

32.4

34.5

1.15

3.23

5.32

7.40

9.48

11.6

13.7

15.7

17.8

19.9

22.0

24.1

26.2

28.2

30.3

34.5

42.8

1.15

5.32

9.48

13.7

17.8

21.2

24.5

27.8

31.2

33.7

2.830

2.835

2.836

2.838

2.840

1.694

2.032

2.182

2.300

2.416

2.484

2.580

2.655

2.704

2.744

2.795

2.835

2.860

2.885

2.897

2.898

2.900

1.737

2.245

2.405

2.585

2.694

2.973

2.842

2.893

2.940

2.955

2.840

3.840

2.840

2.840

2.840

2.900

2.900

2.900

2.900

2.900

2.900

2.900

2.900

2.900

2.900

2.900

2.900

2.900

2.900

2.900

2.900

2 900

2.975

2.975

2.975

2.975

2.975

2.975

2.975

2.975

2.975

2.975

1.40

1.40

1.40

1.40

1.40

1.40

1.65

1.65

1.65

1.65

1.65

1.65

1.65

1.65

1.65

1.65

1.65

1.65

1.65

1.65

1.65

1.80

1.80

1.80

1.80

1.80

1.80

1.80

1.80

1.80

1.80

36.2

38.7

40.7

42.8

51.2

59.5

1.15

5.32

9.48

13.7

17.8

22.0

26.2

29.5

32.8

36.2

38.7

41.2

44.5

51.2

59.5

1.15

3.23

5.32

11.6

17.8

24.1

30.3

34.5

42.8

51.2

2.962

2.970

2.970

2.973

2.973

2.975

1.842

2.376

2.573

2.723

2.840

2.934

3.002

3.036

3.062

3.080

3.086

3.093

3.098

3.102

3.103

2.302

2.573

2.684

2.850

3.010

3.120

3.170

3.190

3.192

3.193

2.975

2.975

2.975

2.975

2.975

2.975

3.103

3.103

3.103

3.103

3.103

3.103

3.103

3.103

3.103

3.103

3.103

3.103

3.103

3.103

3.103

3.193

3.193

3.193

3.193

3.193

3.193

3.193

3.193

3.193

3.193

See footnotes at end of table, p. A114.
A112 FLOW OF WATER OVER DAMS AND WEIRS

TABLE 4. Summary of data for boundary-layer velocity distribution Continued

Station 2 

(feet)

z« (feet 

X103)

u 

(fps)

U 

(fps)

Station * 

(feet)

z«(feet 

X103)

u 

(fps)

U 

(fps)

Station * 

(feet)

z«(feet

X103)

u 

(fps)

U 

(fps)

MODEL K-l Continued 

TestS

.20

.20

.20

.20

.20

.20

.20

.20

.20

.20

.20

.20

.20

.60

.60

.60

.60

.60

.60

.60

.60

.60

.60

.60

.60

1.10

1.15

2.82

4.48

6.15

6.98

7.82

9.48

11.2

14,5

17.0

17.8

22.0

26.2

1.15

3.23

5.32

8.65

12 0

15.3

18.7

22.0

25.3

28.7

34.5

42.8

1.15

2.200

2.503

2.643

2.736

2.786

2.826

2.903

2.960

3.036

3.054

3.057

3.060

3.060

2.345

2.662

2.803

2.964

3.080

3.165

3.216

3.256

3.265

3.265

3.269

3.269

2.228

3.060

3.060

3.060

3.060

3.060

3.060

3.060

3.060

3.060

3.060

3.060

3.060

3.060

3.269

3.269

3.269

3.269

3.269

3.269

3.269

3.269

3.269

3.269

3.269

3.269

3.562

1.10

1.10

1.10

1.10

1.10

1.10

1.10

1.10

1.10

1.10

1.10

1.10

1.40

1.40

1.40

1.40

1.40

1.40

1.40

1.40

1.40

1.40

1.40

1.40

1.40

1.65

3.65

7.40

11.2

15.7

19.9

24.1

28.2

32.4

36.6

42.8

51.2

59.5

1.15

3.65

7.40

13.7

19.9

26.2

32.4

38.7

42.8

47.0

51.2

59.5

67.8

1.15

2.722

2.987

3.194

3.338

3.416

3.475

3.512

3.530

3.548

3.560

5.562

3.562

2.342

2.825

3.113

3.402

3.562

3.643

3.698

3.722

3.743

3.747

3.747

3.747

3.747

2.507

3.562

3.562

3.562

3.562

3.562

3.562

3.562

3.562

3.562

3.562

3.562

3.562

3.747

3.747

3.747

3.747

3.747

3.747

3.747

3.747

3.747

3.747

3.747

3.747

3.747

3.893

1.65

1.65

1.65

1.65

1.65

1.65

1.65

1.65

1.65

1.65

1.65

1.80

1.80

1.80

1.80

1.80

1.80

1.80

1.80

1.80

1.80

1.80

1.80

1.80

1.80

3.65

7.40

13.7

19.9

26.2

32.4

38.7

44.9

51.2

59.5

67.8

1.15

3.65

7.40

13.7

20.3

27.0

33.7

40.3

47.0

53.7

60.3

67.0

73.7

80.3

2.900

3.222

3.526

3.700

3.795

3.840

3.876

3.886

3.890

3.893

3.893

2.383

2.855

3.183

3.533

3.713

3.818

3.862

3.892

3.918

3.920

3.922

3.922

3.923

3.923

3.893

3.893

3.893

3.893

5.893

3.893

3.893

3.893

3.893

3.893

3.893

3.923

3.923

3.923

3.923

3.923

3.923

3.923

3.923

3.923

3.923

3.923

3.923

3.923

3.923

MODEL AA-1

Testl

.006

.006

.006

.006

.006

.006

.096

.006

.006

.006

.10

.10

.10

.10

.10

.10

.10

.10

.10

.10

.25

.25

.25

.25

.25

.25

.25

.25

.25

.25

.25

.40

.40

.40

.40

.40

.40

.40

.40

.40

1.15

1.98

2.82

3.65

4.48

5.32

6.57

9.48

17.8

26.2

1.15

1.98

2.82

3.65

4.48

5.32

6.57

9.48

17.8

26.2

1.15

1.98

3.65

5.32

7.40

9.48

13.7

17.8

26.2

34.5

42.8

1.15

1.98

3.65

5.32

7.40

9.48

13.7

17.8

26.2

-.728

.841

.940

.974

1.003

1.026

1.091

1.091

1.092

1.089

.713

.794

.828

.847

.866

.885

.979

1.030

1.142

1.142

.702

.735

.818

.897

.946

.985

1.080

1.133

1.180

1.180

1.180

.666

.746

.828

.943

1.011

1.032

1.118

1.147

1.211

1.091

1.091

1.091

1.091

1.091

1.091

1.091

1.091

1.091

1.091

1.142

1.142

1.142

1.142

1.142

1.142

1.142

1.142

1.142

1.142

1.180

1.180

1.180

1.180

1.180

1.180

1.180

1.180

1.180

1.180

1.180

1.242

1.242

1.242

1.242

1.242

1.242

1.242

1.242

1.242

.40

.40

.40

.65

.65

.65

.65

.65

.65

.65

.65

.65

.65

.65

.65

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.30

1.30

1.30

1.30

1.30

1.30

1.30

1.30

1.30

1.30

1.30

1.30

34.5

42.8

51.2

1.15

1.98

3.65

5.32

7.40

9.48

13.7

17.8

26.2

34.5

42.8

51.2

1.15

1.98

3.65

5.32

7.40

9.48

13.7

17.8

26.2

34.5

42.8

51.2

1.15

1.98

3.65

5.32

7.40

9.48

13.7

17.8

26.2

34.5

42.8

51.2

1.242

1.242

1.242

.775

.878

1.037

1.124

1.168

1.233

1.303

1.342

1.375

1.415

1.416

1.416

.625

.690

.835

.960

1.029

1.150

1.278

1.384

1.454

1.509

1.509

1.509

.753

.854

1.019

1.116

1.204

1.254

1.371

1.467

1.585

1.654

1.654

1.654

1.242

1.242

1.242

1.416

1.416

1.416

1.416

1.416

1.416

1.416

1.416

1.416

1.416

1.416

1.416

1.509

1.509

1.509

1.509

1.509

1.509

1.509

1.509

1.509

1.509

1.509

1.509

1.654

1.654

1.654

1.654

1.654

1.654

1.654

1.654

1.654

1.654

1.654

1.654

1.65

1.65

1.65

1.65

1.65

1.65

1.65

1.65

1.65

.65

.65

.65

.65

.85

.85

.85

.85

.85

.85

.85

.85

.85

1.85

1.85

1.85

1.85

2.00

2.00

2.00

2.00

2.00

2.00

2.00

2.00

2.00

2.00

2.00

2.00

2.00

1.15

1.98

3.65

5.32

7.40

9.48

13.7

17.8

26.2

34.5

42.8

51.2

59.5

1.15

1.98

3.65

5.32

7.40

9.48

13.7

17.8

26.2

34.5

42.8

51.2

59.5

1.15

1.98

3.65

5.32

7.40

9.48

13.7

17.8

26.2

34.5

42.8

51.2

59.5

.990

1.140

1.299

1.384

1.475

1.549

1.660

1.756

1.872

1.926

1.942

1.942

1.942

1.068

1.156

1.352

1.569

1.582

1.682

1.818

1.916

2.040

2.088

2.10

2. 107

2.108

1.163

1.292

1.412

1.459

1.654

1.779

1.936

2.052

2.179

2.234

2.260

2.260

2.259

1.942

1.942

1.942

1.942

1.942

1.942

1.942

1.942

1.942

1.942

1.942

1.942

1.942

2.108

2.108

2.108

2.108

2.108

2.108

2.108

2.108

2.108

2.108

2.108

2.108

2.108

2.260

2.260

2.260

2.260

2.260

2.260

2.260

2.260

2.260

2.260

2.260

2.260

2.260

JSee footnotes at end of table, p. >AflJl4.
DISCHARGE OVER EMBANKMENT-SHAPED WEIRS A113

TABLE 4. Summary of data for boundary-layer velocity distribution Continued

Station 2 

(feet)

z a (feet 

X103)

u 

(fps)

17 

(fps)

Station 2 

(feet)

2 '(feet 

XW

u 

(fps)

U

(fps)

Station 2 

(feet)

2' (feet 

X10')

u 

(fps)

U 

(fps)

MODEL AA-1 Continued 

Test 2

.006

.006

.006

.006

.006

.006

.006

.006

.006

.006

.006

.006

.006

.006

.10

.10

.10

.10

.10

.10

.10

.10

.10

.10

.10

.10

.10

.10

.10

.25

.25

.25

.25

.25

.25

.25

.25

.25

.25

.25

.25

.25

.40

.40

1.15

1.98

2.82

3.65

4.48

5.32

6.57

7.82

9.48

11.6

13.7

17.8

26.2

38.7

1.15

1.98

2.82

3.65

4.48

5.32

6.57

7.40

9.48

11. C

13.7

17.8

26.2

34.5

42.8

1.15

2.82

4.48

6.15

7.82

9.48

11.6

13.7

15.7

17.8

26.2

34.5

51.2

1.15

2.82

1.411

1.581

1.722

1.834

1.935

2.028

2.119

2.236

2.290

2.343

2.388

2.430

2.437

2.437

.850

.960

1.052

1.128

1.216

1.322

1.433

1.545

1.772

2.016

2.196

2.325

2.393

2.393

2.393

1.075

1.252

1.412

1.472

1.568

1.676

1.814

1.935

2.060

2.160

2.380

2.395

2.395

1.322

1.452

2.437

2.437

2.437

2.437

2.437

2.437

2.437

2.437

2.437

2.437

2.437

2.437

2.437

2.437

2.393

2.393

2.393

2.393

2.393

2.393

2.393

2.393

2.393

2.393

2.393

2.393

2.393

2.393

2.393

2.395

2.395

2.395

2.395

2.395

2.395

2.395

2.395

2.395

2.395

2.395

2.395

2.395

2.524

2.524

.40

.40

.40

.40

.40

.40

.40

.40

.40

.40

.40

.40

.40

.65

.65

.65

.65

.65

.65

.65

.65

.65

.65

.65

.65

.65

.65

.65

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.30

1.30

5.32

7.40

9.48

11.6

13.7

15.7

17.8

22.0

26.2

34.5

42.8

51.2

59.5

1.15

4.48

8.65

11.6

13.7

15.7

17.8

22.0

26.2

30.3

34.5

38.7

42.8

51.2

59.5

1.15

5.32

9.48

13.7

17.8

22.0

26.2

30.3

34.5

38.7

42.8

51.2

59.5

1.15

5.32

1.645

1.770

1.867

1.964

2.054

2.136

2.220

2.343

2.450

2.508

2.524

2.525

2.524

1.362

1.756

2.030

2.164

2.257

2.330

2.397

2.525

2.618

2.665

2.698

2.714

2.714

2.714

2.714

1.193

1.665

1.964

2.210

2.360

2.502

2.630

2.680

2.730

2.745

2.788

2.788

2.788

1.470

1.916

2.524

2.524

2.524

2.524

2.524

2.524

2.524

2.524

2.524

2.524

2.524

2.524

2.524

2.714

2.714

2.714

2.714

2.714

2.714

2.714

2.714

2.714

2.714

2.714

2.714

2.714

2.714

2.714

2.788

2.788

2.788

2.788

2.788

2.788

2.788

2.788

2.788

2.788

2.788

2.788

2.788

2.976

2.976

1.30

1.30

1.30

1.30

1.30

1.30

1.30

1.30

1.30

1.30

1.30

1.65

1.65

1.65

1.65

1.65

1.65

1.65

1.65

1.65

1.65

1.65

1.65

1.65

1.85

1.85

1.85

1.85

1.85

1.85

1.85

1.85

1.85

1.85

1.85

1.85

1.85

2.00

2.00

2.00

2.00

2.00

2.00

9.48

13.7

17.8

22.0

26.2

30.3

34.5

38.7

42.8

51.2

59.5

1.15

5.32

11.6

17.8

22.0

26.2

30.3

34.5

38.7

42.8

51.2

59.5

76.2

1.15

5.32

11.6

17.8

24.1

30.3

34.5

38.7

42.8

47.0

51.2

59.5

76.2

1.15

5.32

11.6

17.8

24.1

30.3

2.160

2.362

2.510

2.652

2.776

2.842

2.890

2.936

2.962

2.976

2.976

1.490

1.983

2.352

2.360

2.768

2.888

2.980

3.060

3.110

3.135

3.162

3.163

3.163

1.459

2.011

2.421

2.701

2.877

3.045

3.133

3.180

3.240

3.256

3.272

3.277

3.277

1.573

2.061

2.462

2.765

2.977

3.138

2.976

2.976

2.976

2.976

2.976

2.976

2.976

2.976

2.976

2.976

2.976

3.163

3.163

3.163

3.163

3.163

3.163

3.163

3.163

3.163

3.163

3.163

3.163

3.163

3.277

3.277

3.277

3.277

3.277

3.277

3.277

3.277

3.277

3.277

3.277

3.277

3.277

3.368

3.368

3.368

3.368

3.368

3.368

MODEL KA

Test 1

.20

.20

.20

.20

.20

.20

.20

.20

.20

.20

.20

.20

.20

.60

.60

.60

.60

.60

.60

.60

.60

.60

.60

.60

.60

.60

.60

1.10

1.10

1.15

2.82

4.48

6.15

7.82

9.48

11.2

12.8

14.5

16.2

17.8

26.2

34.5

1.15

2.82

4.90

6.98

9.07

11.2

13.2

15.3

17.4

19.5

22.0

26.2

34.5

42.8

1.15

3.23

.872

1.013

1.102

1.172

1.234

1.282

1.298

1.312

1.323

1.323

1.323

1.323

1.323

.638

.937

1.082

1.145

1.186

1.228

1.262

1.302

1.341

1.377

1.409

1.440

1.450

1.451

.815

1.008

1.323

1.323

1.323

1.323

1.323

1.323

1.323

1.323

1.323

1.323

1.323

1.323

1.323

1.451

1.451

1.451

1.451

1.451

1.451

1.451

1.451

1.451

1.451

1.451

1.451

1.451

1.451

1.562

1.562

1.10

1.10

1.10

1.10

1.10

1.10

1.10

1.10

1.10

1.10

1.10

1.10

1.40

1.40

1.40

1.40

1.40

1.40

1.40

1.40

1.40

1.40

1.40

1.40

1.40

1.40

1.40

1.40

1.65

5.32

8.65

12.0

15.3

18.7

22.0

25.3

28.7

32.0

35.3

38.7

42.8

1.15

3.23

5.32

8.65

10.7

12.8

15.7

17.8

22.8

27.8

32.8

37.8

42.8

51.2

59.5

76.2

1.15

1.109

1.228

1.308

1.376

1.448

1.492

1.518

1.540

1.555

1.562

1.562

1.562

.899

1.127

1.253

1.378

1.439

1.480

1.541

1.574

1.636

1.678

1.708

1.723

1.733

1.733

1.733

1.733

1.168

1.562

1.562

1.562

1. 562

1.562

1.562

1.562

1.562

1.562

1.562

1.562

1.562

1.733

1.733

1.733

1.733

1.733

1.733

1.733

1.733

1.733

1.733

1.733

1.733

1.733

1.733

1.733

1.733

1.954

1.65

1.65

1.65

1.65

1.65

1.65

1.65

1.65

1.65

1.65

1.65

1.65

1.65

1.65

1.80

1.80

1.80

1.80

1.80

1.80

1.80

1.80

1.80

1.80

1.80

1.80

1.80

1.80

1.80

3.23

5.32

8.65

12.8

17.8

22.8

27.8

32.8

37.8

42.8

47.0

55.3

63.7

76.2

1.15

3.23

5.32

8.65

12.8

17.8

22.8

28.7

34.5

40.3

46.2

52.0

59.5

67.8

80.3

1.416

1.531

1.650

1.756

1.842

1.886

1.924

1.942

1.950

1.954

1.954

1.954

1.954

1.954

1.246

1.464

1.620

1.760

1.879

1.948

2.002

2.047

2.080

2.093

2.093

2.093

2.093

2.094

2.094

1.954

1.954

1.954

1.954

1.954

1.954

1.954

1.954

1.954

1.954

1.954

1.954

1.954

1.954

2.094

2.094

2.094

2.094

2.094

2.094

2.094

2.094

2.094

2.094

2.094

2.094

2.094

2.094

2.094

 See footnotes at end of table, p.
A114 FLOW OF WATER OVER DAMS AND WEIRS

TABLE 4. Summary of data for boundary-layer velocity distribution Continued

Station 2 

(feet)

z s (feet 

XW)

u 

(fps)

U

(fps)

Station 2 

(feet)

z s (feet 

X 103)

u 

(fps)

U 

(fps)

Station 2 

(feet)

z s (feet

X103)

u 

(fps)

U

(fps)

MODEL KA Continued 

Test 2

.20

.20

.20

.20

.20

.20

.20

.20

.20

.20

.20

.20

.20

.20

.20

.60

.60

.60

.60

.60

.60

.60

.60

.60

.60

.60

1.16

2.82

4.48

6.15

7.82

9.48

11.6

16.7

19.1

22.0

28.2

34.6

40.7

47.0

55.3

1.16

3.23

7.40

13.7

19.9

26.2

32.4

38.7

44.9

61.2

59.6

1.021

1.234

1.380

1.519

1.630

1.764

1.903

2.166

2.318

2.436

2.522

2.663

2.566

2.566

2.566

1.360

1.615

1.896

2.177

2.404

2.563

2.648

2.685

2.708

2.712

2.712

2.566

2.566

2.566

2.566

2.566

2.566

2.566

2.566

2.566

2.566

2.566

2.566

2.566

2.566

2.566

2.713

2.713

2.713

2.713

2.713

2.713

2.713

2.713

2.713

2.713

2.713

.60

1.10

1.10

1.10

1.10

1.10

1.10

1.10

1.10

1.10

1.10

1.10

1.10

1.10

1.10

1.40

1.40

1.40

1.40

1.40

1.40

1.40

1.40

1.40

.1.40

1.40

67.8

1.15

3.23

7.40

13.7

19.9

26.2

32.4

38.7

44.9

51.2

59.5

67.8

76.2

84.5

1.15

4.48

9.48

15.7

22.0

30.3

38.7

47.0

55.3

63.7

' 72.0

2.713

1.018

1.279

1.604

1.970

2.195

2.368

2.502

2.642

2.710

2.743

2.766

2.774

2.774

2.774

1.302

1.667

1.951

2.190

2.378

2.568

2.688

2.782

2.857

2.875

2.875

2.713

2.774

2.774

2.774

2.774

2.774

2.774

2.774

2.774

2.774

2.774

2.774

2.774

2.774

2.774

2.875

2.875

2.875

2.875

2.875

2.875

2.875

2.875

2.875

2.875

2.875

1.40

1.65

1.65

1.65

1.65

1.65

1.65

1.65

1.65

1.65

1.65

1.65

1.65

1.65

1.80

1.80

1.80

1.80

1.80

1.80

1.80

1.80

1.80

1.80

1.80

1.80

80.3

1.15

5.32

9.48

15.7

22.0

30.3

38.7

47.0

55.3

63.7

72.0

80.3

88.7

1.15

5.32

9.48

17.8

26.2

38.7

51.2

63.7

72.0

80.3

88.7

101.2

2.875

1.495

1.972

2.200

2.452

2.604

2.762

2.912

2.982

3.006

3.024

3.040

3.042

3.042

1.412

1.933

2.203

2.543

2.756

2.968

3.076

3.123

3.140

3.143

3.143

3.143

2.875

3.042

3.042

3.042

3.042

3.042

3.042

3.042

3.042

3.042

3.042

3.042

3.042

3.042

3.143

3.143

3.143

3.143

3.143

3.143

3.143

3.143

3.143

3.143

3.143

3.143

1 See table 2 for values of head (ft) and discharge (q).

2 Distance in feet (model) from upstream edge of upstream shoulder. Crown line is at station 1.67 feet for 

all models. 

8 Distance measured perpendicularly from roadway surface to position of pitot tube.

 fr U.S. GOVERNMENT PRINTING OFFICE : 1964 O 690-195