Energy Losses

Associated With

Abrupt Enlargements in Pipes

With special reference to the influence of boundary roughness

By CARL E. KINDSVATER

RIVER HYDRAULICS

GEOLOGICAL SURVEY WATER-SUPPLY PAPER 1369-B

UNITED STATES GOVERNMENT PRINTING OFFICE, WASHINGTON : 1961
UNITED STATES DEPARTMENT OF THE INTERIOR 

STEWART L. UDALL Secretary

GEOLOGICAL SURVEY 

Thomas B. Nolan, Director

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

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

Page

Abstract________________________________________________________ 53

Introduction. _ ____________________________________________________ 53

Acknowledgments ---_---______-_______-__________-_-___-__----_-__ 54

Preliminary considerations-__________________________________________ 55

Review of previous research_______________________________________ 56

Laboratory investigation.__________________________________________ 57

Experimental equipment.______________________________________ 57

General__________________________________________________ 57

Nozzles __________________________________________________ 58

Test pipes.__-________________-_--__-_-___-___-__-__--_-_- 58

Pipe entrance for resistance tests.___________________________ 60

Rough sleeve_____________________________________________ 61

Experimental method and analysis of results ______________________ 61

Normal-resistance tests_--____-____-_______-___-__-_-__--__- 61

Enlargement tests-----_---------_--_-------------------__- 61

Evaluation of results_________________________________________ 64

Conclusions.__ __---__-______-__-_________-____________--____--_--- 72

References cited______________-_-___-_____-_____-___-_______-_-_--_ 75

ILLUSTRATIONS

PLATE 3. Equipment used for laboratory investigation. A, Laboratory 

arrangement; B, nozzles, with downstream view of typical 

assembly; C, upstream view of typical nozzle assembly; 

D, expanded-metal sleeve, side view; E, expanded-metal Page 

sleeve, end view.________________________________ Facing 58

FIGURE 1. Arrangement of laboratory equipment____________________ 57

2. Details of nozzle construction and installation.______________ 59

3. Summary of normal-resistance tests______________________ 62

4. Summary of enlargement tests, nozzle l_-___-____-__---__-_ 65

5. Summary of enlargement tests, nozzle 2_ ___________________ 66

6. Summary of enlargement tests, nozzle 3_____._____-----__-_ 67

7. Summary of enlargement tests, nozzle 4____________________ 68

8. Summary of enlargement tests, nozzle 5____--__-_-_---_--_- 69

9. Summary of enlargement tests, nozzle 6___----_-_-------_-- 70

in
IV CONTENTS

TABLES

Page

TABLE 1. Scope of enlargement tests-------------------------------- 63

2. Summary of test results, series A___________________--_-__-_ 72

3. Summary of test results, series B_______________-___________ 74

4. Summary of test results, series C____________----___-------- 74

5. Summary of test results, series D__.___.___________---____ 75
RIVER HYDRAULICS

ENERGY LOSSES ASSOCIATED WITH ABRUPT 

ENLARGEMENTS IN PIPES

By CARL E. KINDSVATER 1

ABSTRACT

This report describes the results of a simple experimental investigation of the 

flow of water through abrupt, concentric enlargements in circular pipes. Particular attention is paid to the influence of pipe-wall roughness and to methods of 

computing the energy loss resulting from enlargements. The investigation grew 

out of a laboratory study of the hydraulics of bridge waterways, which resulted 

in conjecture regarding the influence of boundary roughness on the expanding 

flow downstream from open-channel constrictions. To avoid experimental 

complications caused by free-surface characteristics of open-channel flow, this 

phase of the study was converted to an investigation of enlargements in pipes.

In the conventional procedure for computing the total loss of energy in a pipe 

system containing an enlargement, a special enlargement loss is added to the 

normal resistance loss in the adjoining pipes. The enlargement loss is computed 

from the well-known Borda-Carnot equation, and the resistance loss is computed 

from a formula such as the Darcy-Weisbach formula. The procedure is based on 

several questionable assumptions.

A total of 115 tests was made with 6 different enlargement ratios and 3 degrees 

of pipe-wall roughness. A special series of tests was made with a rough sleeve in 

the separation zone at the beginning of the enlargement.

Nondimensional total-loss coefficients derived from the tests are compared with 

coefficients obtained by the conventional computation procedure. From this 

comparison, it is concluded that the conventional method of computing the 

energy loss in pipes is adequate for practical use. It is apparent that pipe-wall 

roughness does not have an independent influence on the diffusion process.

INTRODUCTION

Since 1951 the U.S. Geological Survey has been engaged in an 

intensive study of the hydraulics of bridge waterways. Of primary 

importance to the study was a laboratory investigation of elementary 

width constrictions in simple channels (Kindsvater and Carter, 1955; 

Kindsvater, Carter, and Tracy, 1953; Tracy and Carter, 1955). 

Although the bridge waterway was used to define the scope of the

1 Regents Professor of Civil Engineering, Georgia Institute of Technology, Atlanta, Qa.; consultant to 

the U.S. Geological Survey.
54 RIVER HYDRAULICS

laboratory research, an exhaustive study of the almost unlimited 

number of natural boundary conditions was obviously impractical. 

Consequently, many secondary or unusual variables were excluded 

from the scope of the investigation.

One phase of the bridge-waterway investigation included a detailed 

consideration of the abruptly enlarged stream downstream from the 

constriction. Related to this was considerable conjecture regarding 

the influence of channel roughness on the diffusion process. Because 

it was believed to be of secondary importance, and because a laboratory investigation of this question would involve special equipment 

and instrumentation, it was excluded from the scope of the principal 

investigation. Subsequently, however, it was selected by graduate 

students as a suitable topic for research.

In its simplest form, the problem which had been extracted from the 

bridge-waterway investigation was the question of the effect of 

boundary roughness on the flow downstream from abrupt enlargements in open channels. Of particular practical interest was the 

influence of boundary roughness on the total energy loss resulting 

from the enlargement. Preliminary analysis revealed that a laboratory investigation of the problem would be unnecessarily complicated 

by certain free-surface characteristics of open-channel enlargements. 

Thus, the research which is the basis for this report was concerned 

exclusively with abrupt enlargements in circular pipes.

Because of the dominant interest in the energy-loss characteristics of 

enlargements, the method of investigation selected for the laboratory 

studies involved simply the measurement of total head loss for a 

variety of enlargement ratios, Reynolds numbers, and pipe-wall 

roughnesses. However, it was hoped that variations in energy loss 

could be correlated with the experimental variables to give information regarding the flow mechanism.

ACKNOWLEDGMENTS

All of the experimental work was done in the Hydraulics Laboratory, 

School of Civil Engineering, Georgia Institute of Technology, Atlanta, 

Ga. The principal investigators were graduate students in the School 

of Civil Engineering. The first investigator was Trafton W. Fleetwood, Jr.,2 whose work revealed inadequacies in equipment and 

instrumentation. The second investigation, with improved equipment, was performed by Benjamin L. Kittle.3 Kittle's laboratory tests were limited to enlargements in a smooth pipe. His 

apparatus was subsequently used by Fred H. Ruggles on a special

s Fleetwood, T. W., 1955, Abrupt enlargements in smooth and rough pipes: Georgia Inst. Technology 

Master's degree thesis, 62 p., 28 figs.

3 Kittle, B. L., 1956, Abrupt enlargements in circular pipes: Georgia Inst. Technology Master's 

degree thesis, 59 p., 22 figs.
FLOW THROUGH ABRUPT ENLARGEMENTS IN PIPES 55

problem which included tests on pipes with two degrees of sand 

roughness. The experimental data used for this report are the data 

obtained by Kittle and Ruggles. Fleetwood's work was partly supported by the U.S. Geological Survey. Others who assisted in the 

acquisition of miscellaneous data and in the preparation of the report 

include Geological Survey engineers Jacob Davidian, Frederick 

Kilpatrick, and William Emmett. Homer Bates, laboratory technician, constructed the nozzles and special instruments used in the 

investigation. All work was carried out under the direction of the 

writer.

PRELIMINARY CONSIDERATIONS

The purpose of this investigation was to study the flow of water 

through abrupt concentric enlargements in smooth and rough circular 

pipes, with particular attention to the total energy loss which results. 

In the conventional procedure for computing the total energy loss in a 

pipe containing an enlargement, a special enlargement loss is added to 

the normal resistance loss in the adjoining pipes. The resistance loss 

is computed from one of several formulas which involve coefficients 

derived from laboratory studies of uniform flow. One of the most 

common is the Darcy-Weisbach formula,

LV2

in which Hf is the head loss (loss of energy in ft-lbs per lb),/ is a resistance coefficient which depends generally on wall roughness and the 

pipe Reynolds number, L is the length of the pipe, D is the diameter 

of the pipe, and V is the average velocity in the pipe cross section. 

The enlargement loss is computed from the Borda-Oarnot equation,

in which HB is the head loss attributed to the enlargement, KB is a 

coefficient usually taken to be unity if the enlargement is abrupt, FI 

and AI are the average velocity and area, respectively, at a cross 

section upstream from the enlargement, and V2 and A2 are corresponding quantities at a section downstream from the enlargement. The 

classic derivation of the Borda-Oarnot equation, which is based on the 

one-dimensional energy and momentum equations, is shown in nearly 

every textbook on elementary fluid mechanics.

A critical review of the conventional computation procedure results 

in the following observations : 

1 . The use of a uniform-flow equation to evaluate the resistance loss

on both sides of the enlargement is in contradiction with the fact
56 RIVER HYDRAUMOS

that the flow is nonuniform for a considerable distance downstream 

from the enlargement.

2. The tangential force due to boundary shear is neglected in the 

derivation of the Borda-Carnot equation. Thus, variations in wall 

roughness are meaningless in terms of that equation.

3. The Borda-Carnot equation does not take into consideration the 

effect of nonuniform velocity distribution on the energy and momentum flux at the terminal cross sections.

4. Actually, there is no physical basis for the assumption that the total 

energy loss can be separated into independent components. 

It is apparent that any errors related to these observations must be

either negligible or compensating if the conventional computation

procedure is to be applied successfully to a full range of practical

conditions.

REVIEW OF PREVIOUS RESEARCH

The technical literature of hydraulics contains the results of several 

investigations of abrupt enlargements in pipes. A brief review of 

selected references is pertinent as background for this investigation.

Notable among those who attempted to test the accuracy of the 

Borda-Carnot equation are Gibson (1910, 1912) and Archer (1913). 

Both of these investigators used the customary procedure for separating the enlargement loss from the total loss in the test pipe. Neither 

took into account the influence of upstream flow conditions, the 

Reynolds number of the flow, or the relative roughness of the 

boundaries.

Gibson's tests were made with small, smooth pipes covering a range 

of area ratios, A2/Ai, from 2.25 to 10.96. He concluded that the coefficient KB (eq 2) "increases slightly with the ratio of enlargement 

and, in pipes with the same ratio of enlargement, is greater the smaller 

the pipe" (Gibson, 1952, p. 91). Values of KB recommended by 

Gibson varied from 0.95 (A2/A1 =2.25) to 1.04 (A2/Ai=lQ.9ty.

Archer's tests with small brass pipes covered a range of values of 

AZ/A! from 1.45 to 9.32. He concluded that the Borda-Carnot equation gave results which were too small for low velocities and small 

values of A2/Ai and too large for high velocities and large values of 

A2/Ai. From a questionable extrapolation of his test results, he presented a table of recommended values of KB which varied from 1.22 

(A2 /A!=1.25, Vi=2 ft per sec) to 0.75 (A2/Ai=°°, V1 =80 ft per sec). 

Thus, Archer contradicts Gibson's conclusions. According to King 

(1954, chap. 6, p. 17), Archer's results were substantiated by tests 

made at the University of Michigan.

Schiitt (1926) concluded that the conventional procedure (with 

KB=l.O in the Borda-Carnot equation) would yield results accurate
FLOW THROUGH ABRUPT ENLARGEMENTS IN PIPES 57

to within 1 percent. Schiitt's tests covered a comparatively small 

range of discharges and four enlargement ratios varying from 2.8 to 

8.8. His equipment was generally superior. It is noted, for example, 

that he used short machined nozzles at the entrance to the enlargement, and the size of his pipe (about 6-inch diameter) was larger than 

that used by most other investigators. However, the significance 

of his conclusions is doubtful in view of a questionable method used 

to adjust values of AI for contraction beyond the end of the nozzles. 

A more recent study by Kalinske (1947) was concerned primarily 

with the mechanism of energy transioimation in enlargements. The 

results of three tests on a single abrupt enlargement (A2/Ai=3) included information on the distribution of velocity and pressure and 

the rate of energy dissipation in the turbulent wake downstream from 

the enlargement. From his tests Kalinske concluded that the principal energy loss occurs at the interface between the entering jet and 

the eddying fluid in the separation zone. He also demonstrated that 

the velocity distribution upstream from the enlargement has considerable influence on the energy loss in the enlargement. A valuable 

adjunct to his report are the comments and references contained in 

the published discussions.

LABORATORY INVESTIGATION

EXPERIMENTAL EQUIPMENT

GENERAL

Experiments made as a part of this investigation were performed in 

the hydraulics laboratory of the Georgia Institute of Technology. 

The arrangement of laboratory equipment used for the tests is shown 

in figure 1 and plate 3.

'Expanded-metal screen

,. Straightening vanes

,'3-in. diameter test pipe (22 ft long) ^Regulating valve

11 i______i

-Piezometers

12-in. diameter approach pipe (57 in. long) To weighing tank 

FIGURE 1. Arrangement of laboratory equipment.

The test reach consisted of a short nozzle at the entrance to the 

enlargement and a 22-ft length of 3-inch pipe downstream from the 

enlargement. The approach to the test section consisted of a 57-inch 

length of 12-inch pipe equipped with straightening vanes and baffles
58 RIVER HYDRAULICS

at the upstream end to reduce velocity nonuniformities at the entrance 

to the enlargement. The rate of flow was controlled with a gate 

valve located in the 3-inch pipe downstream from the test section.

The connecting pipe was arranged so that the laboratory's constanthead recirculating system could be used for most discharges. For 

the larger rates of flow the constant-head tank was bypassed with a 

pipeline connected directly to the pumps. The maximum discharge 

used in the tests was 1.33 cfs.

Electrically operated weighing-tank equipment was used for the 

measurement of discharges. Piezometric-head differences were 

measured with precision manometers.

NOZZLES

Nonuniformities in the entering stream were reduced by the use of 

short, smooth, machined aluminum nozzles instead of pipes upstream 

from the enlargement. Six nozzles were used to obtain a representative coverage of a reasonable range of enlargement ratios.

Plate 3B,C shows photographs of the nozzle and a typical entrance 

assembly. Figure 2 shows details of nozzle construction and 

installation.

Installed in the test section, a nozzle was mounted on a flange 

plate which was drilled to match the adjoining flanges on the approach 

pipe and the test pipe. A collar was affixed to the upstream end of 

the nozzle to provide a plane surface tangent to the elliptical curve 

of the nozzle entrance. Each nozzle was equipped with four piezometers. The piezometers were equally spaced on the circumference of 

the cylindrical throat of the nozzle, in a plane which bisected the 

length of the cylindrical surface.

Great care was taken in the construction of the nozzles, particularly 

to ensure that the throats were perfect cylinders. The nozzles were 

made from solid aluminum stock. They were highly polished after 

machining. The largest nozzle was 4 inches long. Smaller nozzles 

were shorter, but the cylindrical part was at least 1 inch long. The 

finished diameter of the throat was determined as the average of 

several measurements with an inside micrometer.

TEST PIPES

The pipe used as the downstream part of the test section for all 

the experiments was a 22-ft length of 3-inch-diameter extruded-aluminum pipe. It is shown in plate 3-4. For an earlier investigation the 

pipe had been ground with a cylinder hone to ensure smoothness and 

to make it as nearly circular in cross section as possible. Four piezometers were located at each of 14 sections along the length of the pipe. 

Sections near the enlargement were more closely spaced than the
GEOLOGICAL, SURVEY WATEK-SUPPLY PAPER 13(59 PLATE

A. Laboratory arrangement.

B. Nozzles, with downstream view of typical assembly. C. Upstream view of typical 

nozzle assembly.

D. Expanded-metal sleeve, side view.

EgUIPMENT USED FOR LABORATORY INVESTIGATION

E. Expanded-metal sleeve, end 

view.
PLOW THROUGH ABRUPT ENLARGEMENTS IN PIPES 59

Nozzle 

no.

1

2

3

4

5

6

Nozzle dimensions (inches)

o

4

4

4

4

4

4

b

2.508

2.009

1.499

1.201

1.004

.752

c

0.84

.67

.50

.50

.50

.50

d

1.67

1.33

1.00

1.00

1.00

1.00

e

4.17

3.33

2.50

2.20

2.00

1.75

B

FIGURE 2. Details of nozzle construction and Installation. A, entrance assembly; B, nozzle.
60 RIVER HYDRAULICS

others. The piezometers were one-sixteenth of an inch in diameter, 

drilled perpendicular to the pipe wall, and honed to prevent burrs at 

the inner pipe-wall orifice. The four piezometers at each section were 

connected through a manifold to the manometer.

For the smooth-wall tests the test-pipe diameter was measured with 

a cylinder gage calibrated to read thousandths of an inch. Diameter 

measurements were made on 2 perpendicular axes at 60 different sections over the length of the pipe. The average diameter determined 

in this manner was 3.057 inches.

For the rough-wall tests, screened crushed-granite sand was glued 

with varnish to the inner surface of the test pipe. The details of the 

shop procedures are not pertinent here, but it should be acknowledged 

that it is a frustrating experience which involves many unsuccessful 

trials. The difficulty results from the requirement that the sand 

coating be attached uniformly and firmly to a relatively thin coat of 

varnish. Two degrees of roughness were produced by this method. 

For the first (sand-roughness I, test-series C), the sand used was of 

such size and shape that it passed a No. 12 Taylor sieve and was 

retained on a No. 14 sieve. For the second (sand-roughness II, 

test-series D), the sand used was such that it passed a No. 6 sieve 

and was retained on a No. 8 sieve. Using the average sieve opening 

to indicate the nominal sand-grain size, the corresponding grain 

sizes were 0.060 inch for series C and 0.112 inch for series D.

In order to determine the effective diameter of the sand-roughened 

pipes, the disassembled pipe was capped, sealed, and weighed dry, 

then filled with water and weighed again. The volume of water 

determined in this manner was used to compute the diameter of the 

pipe. The average diameter was 3.024 inches for the series-C pipe 

and 2.976 inches for the series-D pipe. From the corresponding 

sand-grain sizes and pipe diameters, the relative roughness was 

0.020 for series C and 0.038 for series D.

It is observed that the sand used to provide pipe-wall roughness 

for this investigation was angular and irregular in shape, quite unlike 

the standard sand used in the classic tests by Nikuradse (1950). 

Furthermore, the sand coat was not covered with a seal coat of varnish 

whereas Nikuradse did use a seal coat. Consequently, for equal 

values of the relative-roughness ratio, the effective roughness of 

the sand used for this investigation was considerably greater than that 

used by Nikuradse.

PIPE ENTRANCE FOR RESISTANCE TESTS

A smooth, round entrance was used for the tests made to determine 

the normal resistance characteristics of the test pipes. The entrance 

piece was fabricated from wood, sanded smooth, and painted. The
FLOW THROUGH ABRUPT ENLARGEMENTS IN PIPE© 61

cylindrical throat was carefully matched to the end of the test pipe 

to prevent disturbance due to flow separation.

ROUGH SL/EEVE

For series-B tests a 6-inch long sleeve of deformed expanded metal 

was placed at the entrance to the smooth-walled test pipe. The sleeve 

was made from a single piece of %-inch by 18-gage unflattened expanded metal. Parts of the metal fabric were cut and bent inward 

approximately one-fourth inch to produce an exaggerated wall 

roughness at the beginning of the enlargement. Photographs of the 

sleeve are shown in plate 3D,E.

EXPERIMENTAL METHOD AND ANALYSIS OF RESULTS

NORMAL-RESISTANCE TESTS

The uniform-flow resistance characteristics of the test'pipes were 

determined as a basis for arbitrarily separating the normal resistance 

loss from the total measured loss, for the abrupt-enlargement tests. 

The tests involved measurement of the discharge and the piezometric 

profile in the test reach. They covered a range of values of the 

Reynolds number which was limited by the available water supply.

Values of the hydraulic gradient used to compute the resistance 

coefficient,/, in the Darcy-Weisbach formula (eq 1) were determined 

as the slopes of lines fitted to the downstream, straight parts of the 

piezometric profiles. The length of the test pipes was 80 times their 

diameter, and the entrance to the test section was designed to prevent 

disturbance in the flow. For the conditions of the tests, the results 

of experiments by Shapiro and Smith (1948) and others indicate that 

normal values of the resistance coefficient were attained in the downstream part of the pipe.

Computed values of/were plotted as functions of the pipe Reynolds 

number [R2 = (F2A)/"]. Trends indicated by Colebrook-White transition curves were used as a guide in drawing smooth curves through the 

plotted points. A summary of the results of the resistance tests is 

shown in figure 3. The curves shown were used in the reduction and 

analysis of the enlargement tests.

ENLARGEMENT TESTS

The scope of the enlargement tests is indicated by the summary 

shown in table 1. A total of 115 tests was made. The enlargement 

ratios tested ranged from 1.5 to 16.5, and the pipe Reynolds numbers 

were mainly in the range between 10* and 106 .

566151 61 2
U.l«t

0.12

v. 0.10

I-"

z

UI 

0

It Q08

UI

o

0 

UJ

£ 0.06

S 

tn

V) 

UI

K 0.04 

0.02 

0

i i ^ Oi i i i i n _

o 

o 

0 

0

 2__ - 

 «r>- - n

°?

Sand-rough

/Sand-rough

^ -o 

Smooth pipe

I04 2 3 456789 I05 

PIPE REYNOLDS NUMBER, 

FIGURE 3. Summary of normal-resistanc
FLOW THROUGH ABRUPT ENLARGEMENTS IN PIPES 63 

TABLE 1. Scope of enlargement tests

Series

A 

B

C 

D

Test Nos.

1-56 

57-74

75-100 

101-115

Condition of downstream 

pipe

Smooth 

Smooth, with sleeve

Sand-roughness I 

Sand-roughness II

Nozzles tested 

(No.)

1,2,3,4,5,6

2 A A

2, 3, 4, 5, 6 

1, 2, 4, 5, 6

Experimenter

B. M. Kittle 

_____do__.___-

F. H. Ruggles 

.....do...____

Summary 

of results 

(table No.)

2 

3

4 

5

The test procedure consisted of measuring the discharge and determining the piezometric profile for each setup. From the plotted 

and smoothed profiles, values of a total-loss coefficient were computed 

for each test. The coefficient, CL, is defined by the equation

2g_HL

(3)

29

in which hi and Vi are the piezometric head and average velocity, 

respectively, in the throat of the nozzle, and A2 and V2 are the corresponding quantities at a section arbitrarily taken to be at a distance of 

25 pipe diameters downstream from the beginning of the enlargement. 

Thus, HL is the nominal total head loss in the first 25 pipe diameters 

downstream from the enlargement.

The length of the test reach used to evaluate HL is not critical. It 

should be sufficient to contain most of the nonuniform flow resulting 

from the enlargement, but it should not be so great as to make the nominal enlargement loss, HB (eq 2), excessively small in comparison with 

Hf (eq 1). The 25-diameters length was selected for several reasons. 

The measured hydraulic gradients at the end of the 25-diameters 

reach compared favorably with the corresponding gradients for uniform flow as determined from the normal-resistance tests. Furthermore, Fleetwood 4 showed that the velocity distribution at that 

section was very nearly normal in comparison with velocities computed 

from the Karman-Prandtl equations for uniform, turbulent flow near 

smooth boundaries. From his own tests, Kalinske (1947) concluded 

that the disturbance caused by the enlargement was virtually dissipated within a reach of 17 diameters, whereas Schutt (1926) concluded 

that uniform flow was restored within 8 diameters. It is concluded 

that the 25-diameters reach is adequate for the purpose of this investigation.

4 See footnote 2, p. 54.
64 RIVER HYDRAULICS

The principal results of the enlargement tests are shown in tables 2 

through 5. In these tables, Cf is a nondimensional resistance coefficient which is derived from equation 1,

O=^=/ (4) 0/ V J D w

For the nominal 25-diameters reach, Cf is simply 25/.

The coefficient CB in tables 2 through 5 is an enlargement-loss 

coefficient which is obtained from equation 2. For abrupt enlargements, KB in that equation is theoretically equal to 1.0. Thus,

2g

In the conventional procedure for computing the total head loss 

resulting from an abrupt enlargement, Hf is added to HB. Therefore, 

to obtain a computed total-loss coefficient which is comparable with 

CL, Cf was added to CB. It follows that the ratio of CL to (CB+Cf) 

is a direct nondimensional measure of the adequacy of the conventional computation procedure.

Figures 4 through 9 show CLl(CB -\-Cf) plotted as a function of R2 

from the experimental data obtained in this investigation. Different 

symbols are used to identify the test series described in table 1. All 

the tests for a given nozzle are shown on a single figure. Small variations in the enlargement ratio result from differences in the pipe diameter caused by the addition of sand (as roughness) to the inner 

surface of the test pipe.

EVALUATION OF RESULTS

The results of the laboratory tests made for this investigation were 

expected to yield answers to the following questions :

1. Is the conventional procedure for computing the total energy loss 

adequate for practical purposes?

2. Is there any correlation between the accuracy of the conventional 

procedure and the roughness of the downstream pipe wall?

3. Can the inadequacies of the conventional procedure, if any, be 

correlated with observed flow characteristics?
I.It 

1.12 

1.10

1.08 

1.06 

1.04 

'1.02

0.98 

0.96 

0.94 

0.92

Q90 

rtfia

0

A

O

O

o

A

O

0 

0 

EXPL

Symbol 

0 

A 

10 4 5 6 7 8 9 10 

PIPE REYNOLDS NUMB

FIGURE 4. Summary of enlargement tests, no
1.10 

1.08 

1.06 

1.04

<o~l.02 +

(^1.00 

0.98 

096 

0.94 

0-92 

oan

o

 n 0o

A

>> 1 X

AO

D ( OO

3

>< 

X 

X D :

-O

0

3

EXPLA

Symbol 

0 

X 

D 

A 

4 5 6 7 8 9 10 

PIPE REYNOLDS NUMBE

FIGURE 5. Summary of enlargement tests, no
I.IU

1.08 

1.06 

1.04 

1.02 

+ 1.00

Q)

^ 0.98

0.96 

0.94 

0.92

OQO

0

D

o

O

n 

0 

t-Ar L

Symbol 

0 

D 

10'

4 5 6 7 8 9 10° 

PIPE REYNOLDS NUMB

FIGURE 6. Summary of enlargement tests, n
I.IU 

1.08 

1.06 

1.04

1.02 

1.00

0.96 

0.94

0.92 

nqn

O

 D 

X

A -'

O

>

n

O

A

D 

O O

X X

A

456789 10 

PIPE REYNOLDS NUMB

FIGURE 7. Summary of enlargement tests, no
I.IU

1.08 

1.06 

1.04

1.02 

tf 

+ 1.00

^0.98 

0.96 

0.94

0.92 

nan

A

O

D

D0 A fc

C 

O

1

M

AC >

EXPL

Symbol 

0 

D 

A 

10 4 5 6 7 8 9 10 

PIPE REYNOLDS NUMBE

FIGUEE 8. Summary of enlargement tests, no
I.IU

I.V/O

I.UO

1.04

1.02

1.00

0.98

.570

0.94 

0.92

r\ an

A

a

D

O

X

A

O -»P 

X

5 6 7 8 9 10' 1 2 3 4

PIPE REYNOLDS NUMBER, 

FIGURE 9. Summary of enlargement tests, noz
FLOW THROUGH ABRUPT EJSTLARGEMENTS IN PIPES 71

To obtain the answers to these questions, the experimental results 

shown on figures 4-9 were evaluated in the light of the following facts:

1. In the practical application of the conventional procedure for computing HL, the resistance coefficient/ and the areas A\ and A2 are., 

associated with commercial pipes of various kinds and conditions. 

Areas are seldom known exactly, and resistance coefficients can only 

be estimated.

2. It is generally agreed that 5 percent is a maximum probable accuracy 

for computed values of Hf based on estimated values of/.

3. Computed values of HB are critically influenced by the accuracy of 

measurement of A2 and AI. Furthermore, when A2/Ai is large, the 

enlargement loss is a dominant part of HL in short pipe systems.

4. Kalinske and others have shown that upstream flow conditions have 

a large influence on the flow in an abrupt enlargement. In this investigation the effect of upstream disturbances, velocity distribution, and similar variables was eliminated by the use of short, smooth 

nozzles at the entrance to the enlargement. Thus, the independent 

influence of upstream flow conditions is not apparent in the results 

of the laboratory tests.

5. Comparison of values of CL and Cf with values of CB should take into 

consideration the fact that values of CL and Cf are directly related 

to the length of the test reach used to evaluate the total head loss 

and the nominal resistance loss. On the other hand, CB is independent of pipe length. Values of the ratio CLl(CB -\-Cf) are influenced slightly by the length of the test reach, but the generality 

of the conclusions based on that ratio is unaffected.

6. The relative accuracy of the experimental results expressed in terms 

of CL/(CB-\-Cf) increases as A2/Ai increases. This conclusion is supported by the fact that the spread in plotted points in figure 9 

(A2/Ai = l5.7 to 16.5) is appreciably less than that in figure 4 

(A2/A1=lAl to 1.49). The trend indicated by this comparison 

is substantiated by figures 5 through 8.

In preliminary reports on the results of this investigation, certain 

conclusions were drawn from an analysis of residual quantities such as 

(CL Cf) and (CL CB). Considering the fact that neither Cf nor CB 

can be assumed to be exact, conclusions based on the residuals are 

believed to be misleading. One conclusion resulted in the decision 

to make the series-B tests. Thus, from the results of some early tests 

it was indicated that pipe-wall roughness might cause a reduction in 

the residual enlargement loss (CL Cf}. In explanation, it was suggested that wall roughness might retard and, therefore, reduce the 

energy requirements of the major eddy which occurs in the separation 

zone at the entrance to the enlargement. To test this hypothesis, a 

short, rough sleeve (pi. 3 D, E) was placed in the region occupied by the
72 RIVER HYDRAULICS

eddy. The downstream pipe was otherwise the same as it was for 

test-series A. The preliminary conclusion was not substantiated by 

the results of the series B tests shown on figures 5, 7, and 9.

CONCLUSIONS

From a critical analysis of the experimental results shown on figures 

4 through 9, with attention to the criteria described in the preceding 

section and the limitations imposed by the conditions of the tests, it is 

concluded that the conventional method of computing the total energy 

loss in a pipe system containing an abrupt enlargement gives results 

which are in good agreement with measured losses. The experiments 

revealed no independent influence of pipe-wall roughness which could 

be associated specifically with the diffusion process. It is observed, 

however, that the experimental method used for this investigation is 

not well suited to a study of the flow mechanism.

The extent to which these conclusions are transferrable to openchannel enlargements is conjectural. Nevertheless, the results are 

believed to substantiate the original decision to exclude the subject 

of this study from the scope of the bridge-waterway investigation.

TABLE 2. Summary of test results, series A

Test No.

!-_-_____-__

2

3 _- _

4___________

5-__________

6__ _____

?___________

8_._________

9___._______

10__________

11____._____

12__________

13__________

14__________

15__________

16.__-_.____

17-____.____

18__._______

19_____.____

20-__-___.__

21______

22_________

23__________

24____ _

25___.______

A2

Ai

1.49

1.49

1 4Q

1. 49

1 4Q

1 4Q

1. 49

1. 49

1 4Q

1 4Q

1 4Q

1. 49

1. 49

1 4Q

1.49

1.49

2. 32

2. 32

2.32

2. 32

2.32

2.32

2. 32

2. 32

2. 32

RsXIO-*

30. 5

22. 1

14. 3

11.0

7. 33

5. 35

3. 76

35. 5

66. 2

30. 1

18. 1

12.4

4. fifi

2.82

35. 2

ac o

59.0

45. 4

21. 0

10. 7

6

4.c

1 A O

32. 1

5. 60

3. 71

CL

0. 660

. 661

. 700

. 710

. 742

. 750

. 821

. 661

. 607

. 640

. 659

. 692

707

. 765

. 640

. 590

2.01

2.07

2.09

2. 21

2.24

2. 16

2.09

2. 16

2. 24

cf

0. 375

.400

. 440

. 463

.503

.538

. 578

. 365

. 335

.378

.418

ACQ

.555

.613

. 365

. 335

. 338

. 353

.405

. 473

. 518

.438

. 373

.533

. 580

CB

0.237

. 237

.237

.237

. 237

. 237

.237

. 237

. 237

. 237

. 237

.237

. 237

. 237

. 237

. 237

1.74

1.74

1.74

1.74

1. 74

1.74

1. 74

1. 74

1. 74

CB+C/

0.612

. 637

.677

.700

.740

. 775

.815

.602

.572

.615

. 655

. 690

. 792

.850

.602

. 572

2.07

2.09

2. 14

2. 21

2. 25

2. 17

2. 11

2.27

2. 32

CL

CB+C/

1.079

1.037

1.034

1.014

1.003

.968

1.007

1.097

1.062

1. 041

1.006

1.003

. 994

.900

1.063

1.032

. 970

. 991

.977

1.001

. 994

. 994

. 991

. 952

. 968
FLOW THROUGH ABRUPT ENLARGEMENTS IN PIPES 73

TABLE 2. Summary of test results, series A Continued

Test No.

26___. - __

27___. _..__.

28 

29____--____

30-_________

31 

32__._______

33---_._ 

34-___.__._.

35 

36-_--______

37__.__._.__

38__________

39__._______

40_.________

41 

42_.____.._.

43. _________

44_______ _

45_-_-_-_._.

46___-____._

47 

48 

49__-_______

50___. ______

51_________.

52._________

53___.______

54_-_---____

55___. ______

56__________

At

Ai

2. 32

2.32

2.32

2. 32

2.32

2.32

2. 32

2.32

2. 32

4. 16

4. 16

4. 16

4. 16

4. 16

6. 48

6.48

6.48

6.48

6.48

6.48

6.48

9.28

9.28

9. 28

9.28

9.28

9. 28

16. 5

16. 5

16. 5

16. 5

R2 X10-<

1. 95

26. 6

6.56

10. 1

7.03

15. 2

3. 53

31. 3

59. 8

18.4

11. 2

7. 76

5. 40

23. 1

12.2

5. 46

Q 2Q

8.00

3. 88

1. 80

14.3

1. 18

3.41

5.28

7. 31

9. 90

21. 2

2. 15

3. 77

4.34

5. 44

CL

2.32

2. 06

2. 15

2. 15

2.27

2.09

2. 21

2. 03

2.06

10.2

10.3

10.2

10. 4

10. 2

30. 1

30. 2

30. 1

30. 1

30.2

30.4

29. 9

68. 9

68. 3

68.3

68.0

67. 8

68. 1

239

238

237

236

c/

0. 663

. 385

. 515

.473

. 508

. 435

.585

. 375

.338

. 418

. 438

. 498

.538

.405

. 455

. 535

4QO

49*1

. 575

. 672

. 440

.735

.583

. 540

. 505

.475

.405

. 635

. 578

.560

.535

CB

1.74

1.74

1. 74

1. 74

1.74

1.74

1. 74

1.74

1. 74

9.99

9. 99

9. 99

Q QQ

9. 99

30. 1

30. 1

30. 1

30. 1

30. 1

30. 1

30.1

68. 5

68.5

68.5

68. 5

68.5

68. 5

241

241

241

241

CB+Cf

2. 40

2. 12

2. 25

2.21

2. 24

2. 17

2.32

2. 11

2.07

10. 4

10. 4

10.5

10. 5

10.4

30. 5

30.6

30.6

30.6

30. 6

30.7

30.5

69. 2

69. 1

69. 0

69. 0

69.0

69. 0

242

242

242

242

CL

Ce+C,

0.967

. 972

.955

.974

1.012

. 963

.952

.962

. 994

. 982

.985

. 977

. 983

. 982

.986

.989

.985

. 985

. 987

. 989

. 981

.995

. 989

. 989

.986

. 982

.987

.989

.985

. 981

. 976
74 RIVER HYDRAULICS 

TABLE 3. Summary of test results, series B

Test No.

57__________

58--------_-

59__________

60_-_. ______

61 

62_--_______

63_-_-______

64-_________

65 

66 

67__________

68 

69 

70__-_-__-__

71_-_--_____

72 

73 

74__-_ ______

A,

Ai

2.32

2.32

2.32

2.32

2. 32

2.32

2.32

2.32

2.32

2. 32

2.32

6. 48

6.48

6.48

6.48

16. 5

16. 5

16. 5

JBaXlO-*

25. 6

20.2

12 Q

10.0

5

4.C

5.00

3.51

29. 6

26. 1

10. 1

4 98

U q

3. 98

1 °>Q

14. 2

4.26

2. 85

5. 18

CL

2

00

2.34

2.35

2. 31

9 °.Q

2. 41

2.46

2.35

2.34

2.34

2. 44

9Q t\

29.5

29.6

29.4

235

235

233

Cf

O OOQ

.400

44Q

.473

co«

K4.K

E1R7

.378

.388

.472

. 545

. 456

.473

.710

. 440

.550

. 613

. 540

CB

1.74

1. 74

1.74

1.74

1.74

1.74

1.74

1.74

1. 74

1.74

1.74

30. 1

30. 1

30. 1

30. 1

241

241

241

CB+Cf

2. 12

2. 14

2. 18

2. 21

2. 27

2. 28

2.32

2. 11

2. 12

2. 21

2.28

30.5

30.5

30.8

30. 5

242

242

242

CL

CB+Cf

1.096

1.096

1.076

1.046

1.053

1.057

1.059

1. 112

1. 103

1.060

1.071

.967

. 965

. 963

. 963

.971

.972

.963

TABLE 4. Summary of test results, series C

Test No.

75 

76-__-___--_

77_-________

78 

79__________

80---______-

81-_________

82___.-_____

83-.._______

84___. ______

85-_________

86_________.

87_______-._

88----______

89-_-__.____

90---____.-_

91---.______

92__________

93__________

94__________

95__________

96__________

97-.-__-____

98----_.____

99 ..

100_________

At

Ai

2. 26

2. 26

2.26

2.26

2.26

4.06

4.06

4. 06

4. 06

4.06

6.33

6.33

6.33

6.33

6. 33

6. 33

9. 05

9.05

9.05

9.05

9.05

16. 1

16. 1

16. 1

16. 1

16. 1

JBzXlO-*

49. 6

o en

6. 21

12. 6

21. 9

40. 7

1.88

4.23

8. 82

16. 9

1. 56

4 82

10. 6

24.2

29.6

8.04

20.4

1. 59

2. 88

5. 16

7.94

12.0

01 n

1.77

3. 57

4.27

/~*T

3.24

3.00

3.20

3. 18

3.28

11. 1

10. 6

11.0

11.0

11.0

29. 2

on Q

29. 5

29. 4

on a

29.6

67. 2

65.3

66. 1

65.6

65. 9

227

00 K

224

226

225

Cf

1.40

1.42

1.42

1. 40

1.40

1.40

1.45

1. 42

1. 40

1. 40

1. 45

- 1.42

1.40

1.40

1.40

1.40

1. 40

1. 45

1. 42

1.42

1.40

1.40

1. 50

1. 45

1.42

1.42

CB

1. 59

1.59

1. 59

1.59

1. 59

Q 37

9.37

9.37

9.37

9.37

28.4

28.4

28.4

28.4

28.4

28. 4

64. 8

64.8

64.8

64.8

64. 8

229

229

229

229

229

CB+Cf

2. 99

3.01

3.01

2. 99

2. 99

10.8

10. 8

10. 8

10.8

10. 8

29.8

29. 8

29. 8

29.8

29. 8

29. 8

66.2

66.3

66. 2

66. 2

66.2

230

231

231

230

230

CL

CB+Cf

1.085

. 998

1.064

1.064

1.096

1. 032

.976

1.015

1. 024

1.020

.980

.983

. 992

.987

. 994

. 996

1. 015

.985

.998

. 991

.995

. 985

.978

.973

. 980

.977
FLOW THROUGH ABRUPT ENLARGEMENTS IN PIPE'S 75

TABLE 5. Summary of test results, series D

Test No.

101__ _______

102... ______

103_________

104_._______

105--------.

106____-_.__

107-.__---__

108---------

109._. --__-.

110.-__--___

lll.____ _ _

112_-____-._

113__----__.

114_________

115______ _

A,

Ai

1.41

1.41

1.41

2. 20

2. 20

2.20

6. 16

6. 16

6. 16

8.82

8.82

8.82

15.7

15.7

15.7

«aX10-*

17.4

4 16

10.4

14 1

4 24

5.93

10.4

1.77

5. 10

9.40

1.78

3.74

6. 26

.625

3.42

CL

3.96

3.65

400

5.95

5.29

468

28.5

28.2

28. 1

61.7

63.9

62.2

222

224

220

ct

3.40

3.40

3.40

3.40

3.40

3.40

3.40

3.40

3.40

3.40

3.40

3.40

3.40

3. 42

3.40

CB

0. 171

. 171

. 171

1 AK

1.45

1.45

26.7

26.7

26.7

59.6

59.6

59.6

217

217

217

CB+C{

3.57

3.57

3.57

485

485

4. 85

30. 1

30. 1

30. 1

63.0

63.0

63.0

220

220

220

CL

CB+C,

1. 110

1.022

1. 120

1.227

1.091

.965

.948

. 938

. 933

.979

1.014

.988

1. 006

1.015

1.001

REFERENCES CITED

Archer, W. H., 1913, Experimental determination of loss of head due to sudden

enlargement in circular pipes: Am. Soc. Civil Engineers Trans., v. 76, p. 999. 

Gibson, A. H., 1910, On the resistance to flow of water through pipes or passages

having divergent boundaries: Royal Soc. [Edinburgh] Trans., v. 48, pt. 1,

no. 5, p. 97. 

 1912, Conversion of kinetic to potential energy in the flow of water

through passages having divergent boundaries: Engineering [London], v. 93,

p. 205.

1952, Hydraulics and its applications: London, Constable, 5th ed.

Kalinske, A. A., 1947, Conversion of kinetic to potential energy in flow expansion: 

Am. Soc. Civil Engineers Trans., v. Ill, p. 355.

Kindsvater, C. E., Carter, R. W., and Tracey, H. J., 1953, Computation of peak 

discharge at contractions: U.S. Geol. Survey Circ. 284, 35 p., 34 figs.

Kindsvater, C. E., and Carter, R. W., 1955, Tranquil flow through open-channel 

constructions: Am. Soc. Civil Engineers Trans., v. 120, p. 955.

King, H. W., 1954, Handbook of hydraulics: New York, McGraw-Hill, 4th ed., 

chap. 6, p. 17.

Nikuradse, J., 1950, Law of flow in rough pipes: Natl. Advisory Comm. Aeronautics Tech. Memo. 1292 (translation), 62 p., 21 figs.

Schiitt, Herman, 1926, Versuche zur Bestimmung der Energieverluste bei plotzlicher Rohrweiterung: Hydraul. Inst. Tech. Hochschule Munchen Mitt., 

v. 1., p. 42.

Shapiro, A. H., and Smith, R. D., 1948, Friction coefficients in the inlet length 

of smooth round tubes: Natl. Advisory Comm. Aeronautics Tech. Note 

1785, 44 p., 15 figs.

Tracy, H. J., and Carter, R. W., 1955, Backwater effects of open-channel constrictions: Am. Soc. Civil Engineers Trans., v. 120, p. 993.

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