I L L I N 0NO
S
UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN
PRODUCTION NOTE
University of Illinois at
Urbana-Champaign Library
Large-scale Digitization Project, 2007.
UNIVERSITY OF ILLINOIS ENGINEERING EXPERIMENT STATION
Bulletin Series No. 404
GRADUALLY VARIED FLOW IN UNIFORM CHANNELS
ON MILD SLOPES
MING LEE
former Research Graduate
Assistant
HAROLD E. BABBITT
Professor of Sanitary
Engineering
E. ROBERT BAUMANN
Research Associate in
Civil Engineering
Published by the University of Illinois, Urbana
3050-11-52-51186 ..URS
t: p.Es. =:
ABSTRACT
The theory of varied flow was first postulated in a complete
and comprehensive manner by J. M. Belanger. The fundamental
principles were so well covered that little has since been added
to modify the differential equation of gradually varied flow in
its original form. All subsequent investigations have been con-
fined chiefly to the methods of solution of the differential
equation by direct, approximate, and graphical integration.
In this bulletin three well-known methods of integration
are discussed with special reference to their advantages and
limitations, and a new method is introduced. Recent investiga-
tions have tended to favor direct integration in order to dis-
pense with the cumbersome succession of steps required by the
approximate and graphical methods. Following the current trend,
a more convenient procedure is introduced in which limitations on
the existing methods of direct integration are removed, but
embodying at the same time all their advantages. The practical
application of the proposed method to the determination of
surface profiles of flow in various channels is given. In the
special cases of the rectangular and circular channels charts
have been prepared to facilitate the profile computations.
The validity of the proposed method is determined by comparing
the theoretical results with actual profile observation under
various conditions of flow in experimental flumes. The compari-
son of test data is also extended to the theoretical results
computed by the more widely accepted methods.
CONTENTS
I. INTRODUCTION 9
I. Purpose and Scope of Research 9
2. Notation 10
II. STEADY FLOW IN OPEN CHANNELS 12
3. Uniform Flow and Varied Flow 12
4. Energy Relations in Varied Flow 12
5. Evaluation of Resistance Losses in Varied Flow 15
III. EQUATION OF GRADUALLY VARIED FLOW IN UNIFORM CHANNELS 17
6. Formulation of the Equation of Varied Flow 17
7. Limitations of Applicability of.the Varied
Flow Equation 17
8. Specific Energy, Critical Flow, and Critical
Slope 18
9. Classification of Surface Profiles 20
IV. INTEGRATION OF THE GRADUALLY VARIED-FLOW EQUATION:
EXISTING METHODS 23
10. General Review 23
II. Integration of Equation of Varied Flow 24
V. INTEGRATION OF THE GRADUALLY VARIED-FLOW EQUATION:
PROPOSED METHOD 31
12. Assumptions 31
13. Derivation of Proposed Equation 32
14. Computation Procedure 37
15. Special Charts 47
16. Practical Application 50
17. Summary 67
CONTENTS (concluded)
VI. LABORATORY INVESTIGATIONS 68
18. Purpose and Scope. 68
19. Apparatus 68
20. Experimental Procedure 74
21. Results and Discussion 77
22. Comparison of Actual and Theoretical Profiles 80
VII. CONCLUSIONS 87
VIII. SUMMARY 88
APPENDIX: BIBLIOGRAPHY 90
FIGURES
I. Surface Profile of Gradually Varied Flow 13
2. Evaluation of dA/dy 14
3. Evaluation of dh/dx 15
4. Longitudinal Profiles of M, and M2 Curves 21
5. Solution of Gradually Varied Flow Equation by Graphical
Integration 26
6. Logarithmic Plot of Conveyance K Against Depth y for the
10-ft Rectangular Channel 41
7. Graphical Solution of the Equation of Gradually Varied Flow
in Broad Rectangular Channels 52
8. Graphical Solution of the Equation of Gradually Varied Flow
in Rectangular Channels 54
9. Logarithmic Plot of the Area A and of the Product AR2/3
Against Depth y for Circular Channels 56
10. Graphical Solution of the Equation of Gradually Varied Flow
in Circular Channels 58
II. Fall-Discharge Curve in Example 6 62
12a. Delivery of a Canal 64
12b. Delivery of the 10-ft Rectangular Channel in Example 7 64
13. Plan and Elevation of Experimental Flume 69
14. End View of Channel Showing the Tail-Water Gate 71
15. Point Gage 72
16. Side View of Experimental Channel and Supports 73
17. Elevation above Datum 79
18. Backwater and Dropdown Measurements in Rectangular Channel:
Comparison of Test Results with Results Computed by
Various Methods 81
19. BacKwater and Dropdown Measurements in Trapezoidal Channel:
Comparison of Test Results with Results Computed by
Various Methods 83
20. Backwater and Dropdown Measurements in Circular Channel:
Comparison of Test Results with Results Computed by
Various Methods 85
TABLES
1. Types of Surface Profiles in Uniform Channels
2. Values of the Varied Flow Function B( ,02): A02
A2 A2
3. Values of the Varied Flow Function B( -) 02 >
M A2 A2
4. Hydraulic Properties of the 10-ft Rectangular Channel
5. Values of the Exponents N and M for Rectangular Channels of
Various Widths
6. Values of x --O for Various Values of r and C1: Broad
YO 1
Rectangular Channels
So
7. Values of x - for Various Values of r and C,: Rectangular
YO
Channels of Finite Width
SS
Values of x -- for Various Values of r and CI: Circular Chi
YO0
Hydraulic Properties of Experimental Channels
Comparison of Test Data in Rectangular Channel with Results
Computed by Various Methods
Comparison of Test Data in Trapezoidal Channel with Results
Computed by Various Methods
Comparison of Test Data in Circular Channel with Results
innels
Computed by Various Methods
I. INTRODUCTION
The study of the hydraulics of flow in open channels is not
always subject to analysis in an exact or rigorous manner,
because of the large number of fundamental variables involved.
Under these circumstances it becomes necessary to consider only
the effect of those variables having a predominant influence upon
the flow. Indeed, the consideration of predominant forces not
only simplifies the solution of many practical problems, but also
permits the use of models in the analysis of flow characteristics
of prototype hydraulic structures. That the results obtained
from such simplification cannot be regarded as better than ap-
proximate is evident, but frequently the limitations imposed by
this simplification are compatible with the desired accuracy of
results in practice.
In the case of steady flow in open channels the forces present
are those due to gravity, inertia, viscosity, elasticity, and
surface tension, the last two being inconsequential except in
small models. Gravity forces are generally taken into considera-
tion, while on the other hand either the viscosity or inertia
forces may be neglected in the solution of many practical prob-
lems in open-channel flow. These problems involve either one of
two broad classes of motion: rapidly varied flow in which gravity
and inertia are the predominant forces; and gradually varied
flow, in which the flow pattern is determined primarily by grav-
ity and viscosity forces. Rapidly varied flow is usually associ-
ated with rapid or abrupt changes in the depth of flow occurring
in a comparatively short distance, such as the hydraulic drop or
the hydraulic jump. Gradually varied flow, on the other hand, is
characterized by the absence of a disrupted surface over a con-
siderable reach of channel, the changes of depth being so gradual
that the flow may be considered essentially parallel. Most of
the problems in gradually varied flow are concerned with the
determination of surface profiles of flow in both natural and
uniform channels. The present work is confined mainly to this
phase of open-channel flow.
I. Purpose and Scope of Research
The purpose of this research was to develop a method of
computing surface profiles of gradually varied flow in uniform
channels whereby the results could be obtained more directly
10 ILLINOIS ENGINEERING EXPERIMENT STATION
than are possible with the existing methods. To determine the
accuracy of the proposed method, profiles of flow were recorded
in experimental channels of rectangular, trapezoidal, and circu-
lar cross-sections under various conditions of flow. The test
data are compared with the results computed by the proposed
method and also by other methods that are in use. This investi-
gation was confined mainly to backwater and downdrop curves in
uniform channels on mild slopes, inasmuch as they form the more
important classes of surface profiles encountered in practice.
2. Notation
The nomenclature used in this bulletin conforms essentially
to that prepared by the Committee of the American Standards Asso-
ciation on Letter Symbols for hydraulics, approved in January,
1942. The symbols, defined when they first appear, are summa-
rized below for reference.
A: total cross-section area
6: width of channel
b,: width of water surface
C, Co: coefficient of flow (Chezy)
Co: exponent in Mononobe's functions
Cm: constant in the proposed varied-flow
equation to account for nonuniform
velocity distribution
C1: a parameter in the proposed varied-
flow equation
D: diameter of circular conduit
E: energy per unit weight
E,: mean energy per unit weight
ES: specific energy
G: Q2 (K)2 b
gA3 K0
g: gravitational acceleration
h: potential head = y + Z
K: conveyance of a cross-section
k: a constant in the proposed varied-flow
equation to account for eddy losses
1: exponent in the wetted perimeter-depth
relationship
Bul. 404. FLOW IN MILD, UNIFORM CHANNELS II
M: exponent in the area-depth relationship
N: hydraulic exponent
n: coefficient of roughness (Kutter and
Manning)
p,: wetted perimeter of cross-sectional area
Q: discharge rate of flow
R: hydraulic radius of cross-sectional area
r: ratio of depths, y/yo
r : ratio, yc/yo
S: slope of energy grade line
SC: critical slope
S : friction slope
So: slope of channel bed, tan a0
V: mean velocity, Q/A
x: horizontal length
y: vertical depth from bottom of channel
y,: Belanger critical depth
yo: depth for uniform flow
Z: elevation above datum
a0: angle of an inclined channel bed
measured from the horizontal
B: Bakhmeteff varied-flow function
A: used with x, y, r, B to indicate the
difference between two particular
values
4: function of
Bg: Bresse backwater function
1', P2: Mononobe backwater function
T1', 2: Mononobe dropdown function
Except for the classification of surface profiles, the sub-
scripts 1 and 2 are used to refer, respectively, to upstream and
downstream sections; critical conditions are designated by the
subscript c, and uniform conditions by the subscript 0.
II. STEADY FLOW IN OPEN CHANNELS
3. Uniform Flow and Varied Flow
Flow in a channel is called uniform when certain elements
related to the flow and the channel remain constant from section
to section. These elements are the depth of flow, the cross-
sectional area of flow, the velocity of flow and the slope of the
channel. Strictly speaking then, uniform flow can take place
only in prismatic channels of constant slope. Although varied
or nonuniform flow is the rule in nature, it can also be made to
take place in prismatic channels.
This bulletin deals primarily with certain phenomena connected
with varied flow. Varied flow must also be distinguished from
variable flow. In varied flow the velocity and depth, while
varying from section to section, do not vary with time. In
variable flow, however, these elements vary with time as in the
case of waves. There are two main types of varied flow - one in
which the depth of the water increases gradually downstream, and
the other in which the depth of the water decreases gradually
downstream. The frst type gives rise to a surface profile curve
which is called a backwater curve. The second type results in a
dropdown curve.
It will sometimes happen that the conditions of flow are such
that there may be a sudden break in the surface of the backwater
curve. We then get what is called a "hydraulic jump." Similarly
a sudden break in the surface of a dropdown curve gives rise
to a "hydraulic drop."
4. Energy Relations in Varied Flow
In order to derive the equation of gradually varied flow, it
is first necessary to review the various energy relations exist-
ing in uniform channels during the gradually varied flow. Con-
sider a liquid flowing in a prismatic channel at an angle a0 and
having a constant slope So = tan a, (Fig. 1). Let it be assumed
that the velocity at any particular section is uniform over the
cross section. Then, at any cross section, if Em equals the mean
energy per unit weight of liquid,
E = h + (i)
S 2g
Bul. 404. FLOW IN MILD, UNIFORM CHANNELS
where h = y + Z and V2/2g represents the average kinetic energy
of the stream.
Differentiating with respect to x, the displacement coordinate,
dE dh d V2
_ + --+ (2)
dx dx dx 2g
Consider the flow to be uniform; i.e., V is constant. Then Eq. 2
becomes
dE dh
M - (3)
dx dx
Equation 3 states that in uniform flow, the rate of change of
energy is equal to the rate of change of the surface above the
AT
2
--4-T
Fig. I. Surface Profile of Gradually Varied Flow
datum. It should be remembered that dh/dx will be numerically
negative, since the change of energy in the downstream direction
dh
is negative. Since in uniform flow, - = -So, the slope of the
channel,
dE
dx" - -SO (4)
From Eq. 4 it may be seen that in uniform flow the work done by
gravity is just sufficient to overcome the resistances of the
channel.
In uniform flow the liquid flows at a constant depth for a
given rate of discharge Q. This "normal depth" is designated by
the symbol yo. It should be remembered that for a given channel
and a given discharge, yo is one of the characteristics of the
channel.
Returning to Eq. 2 for the general case of nonuniform flow, V
will be a function of x. At any depth of flow
ILLINOIS ENGINEERING EXPERIMENT STATION
V2 = ( 2
where A is a function of y and Q is constant.
with respect to x
d d Q2 dA dy
dx dx A - A dy dx
Differentiating
Fig. 2. Evaluation of dA/dy
Substituting Eq. 5 into Eq. 2 gives
dE dh Q2 dA dy
dx dx gA3 dy dx
To evaluate dA/dy, consider Fig. 2, the cross-section of
prismatic channel with a survace width b . Clearly,
dA = b dy
dA
or - = b
dy
Substitution of Eq. 7 into Eq. 6 yields
dE dh Q2 dy
dx dx wgA3 dx
Next, dh/dx can be evaluated in terms of y and x if in Fig. 3
the slope is considered so flat that sin ao can be assumed equal
to the slope So and cos ao can be assumed unity, then
dh = Sodx - dy
dh dy
dx = S0 - dx (9)
Noting that dh/dx is negative, if Eq. 9 is substituted into Eq. 8
Bul. 404. FLOW IN MILD, UNIFORM CHANNELS
dEm dy Q2 dy
dx dx gA3 dx (10)
It is now necessary to find an expression for the resistance loss
in varied flow dEm/dx, in order to solve Eq. 10
5. Evaluation of Resistance Losses in Varied Flow
While it is probably true that the resistance losses of varied
flow follow a somewhat different law than that of uniform flow,
in most practical cases where the change in depth is gradual, it
is close to the truth if the following assumption is made: the
rate of resistance losses in varied flow at any section of depth
y is identical with the rate of loss which would take place if
Fig. 3. Evaluation of dh/dx
the flow in the same channel were uniform with the same discharge
and at the same depth. Uniform flow under these conditions can
take place only in a channel whose bottom slope is different from
the bottom slope of the channel in which the gradually varied
flow is occurring. The bottom slope of the uniform flow channel
which will give the desired rate of loss due to frictional resis-
tances can be obtained from any empirical equation of uniform
flow. This resistance slope Sf can be obtained, for example,
from the Manning formula(11 in which the foot is used as the unit
length.
n 2 V 2 n 2 Q2
S f 1.486 R2/3 1.486 A2R4/3 (lla)
In this case the frictional losses Sf are inversely proportional
to some function of the depth of flow. For developmental pur-
poses Eq. lla is used in the form
16 ILLINOIS ENGINEERING EXPERIMENT STATION
S Q2
f/ K2 (lib)
where K is called the "conveyance" of the channel and is a func-
tion of depth. The meaning of Eq. llb should be clearly under-
stood. It states that for a given channel with a given discharge
there is a certain slope Sf which is just sufficient to overcome
the resistances. The bottom slope So of the gradually varied
fow channel may be greater or less than Sf. The frictional re-
sistance Sf is numerically equal to dEm/dx but is different in
sign, since the former is a rate of loss of energy whereas the
latter is expressed as a rate of change of energy. [tsing the
relationships expressed in Eqs. 10 and 11, the equation of grad-
ually varied flow in uniform channels can be formulated.
III. EQUATION OF GRADUALLY VARIED FLOW
IN UNIFORM CHANNELS
6. Formulation of the Equation of Varied Flow
Substituting -Sf from Eq. 11 for dEl/dx in Eq. 10 gives
dy Q2 dy
-Sf = -SO + --- b A (12a)
f 0 dx w gA3 dx (12a)
or transposing
dy So - S f
(12b)
dx (1- bQ2
gA3
and
b Q2
(1- --)
dx -= dy (13)
SO - Sf
Equation 13 expresses the change in x in terms of the change in
y and functions of y and is the required differential equation
for nonuniform flow in prismatic channels. Although Eq. 13 is a
first-order differential equation with variables x and y sepa-
rated, its solution cannot be obtained by quadrature. For this
reason various simplifying assumptions have to be made, and dif-
ferent authorities have suggested different assumptions, more or
less useful from a practical standpoint. Before describing these
solutions it is necessary to consider briefly the limitations
of Eq. 13.
7. Limitations of Applicability of the Varied-Flow Equation
The equation of gradually varied flow was derived from certain
simplifying assumptions; therefore, it may not be valid beyond
the premises upon which the derivation is based. These assump-
tions are as follows:
(a) Steady flow in uniform channels on flat slopes
(b) hydrostatic distribution of pressure intensity
over a normal section
ILLINOIS ENGINEERING EXPERIMENT STATION
(c) Uniform distribution of velocity across the
entire section of flow.
The applicability of Eq. 13 is necessarily limited to flat
slopes because in the derivation 2J, sin a0 is assumed to be
equal to tan a0 (So) and cos a0 and cos2 a0 are assumed to be
unity. In other words, the depth of flow was assumed to be the
same whether the vertical or normal direction was used. Similar-
ly, the distance along the bottom of the inclined channel was
assumed to be the same as its horizontal projection. For a
channel slope of 1:10 the value of cos2 a0 is 0.99 resulting in
an error of only 1 percent.
Strictly speaking, hydrostatic distribution of pressure
intensity is attained only when the normal component of accel-
eration is zero. This condition imposes the restrictions that
the stream lines be neither divergent nor curvilinear31'. For
gradually varied flow in uniform channels, these conditions are
essentially met since the changes in depth are so gradual that
the flow may be considered substantially parallel. If the flow
involves rapid changes in depth or direction, the acceleration
effects cannot be neglected and the equation of gradually varied
flow is not applicable.
The assumption of uniform velocity distribution generally
involves little error, particularly in cases where the velocity
head is small compared with the depth, as is the case in grad-
ually varied flow on mild slopes. This is due to the fact that
the velocity-correction factor is actually very close to unity(4.
Values of this factor under various conditions have been found to
vary between 1.01 and 1.12, averaging about 1.05. If the veloc-
ity distribution deviates substantially from unity, then the
velocity-correction factor must be considered in the derivation
of the gradually varied-flow equation.
8. Specific Energy, Critical Flow, and Critical Slope
In order to obtain a better concept of the relationship
between So and S, Bernoulli's equation for expressing the
conditions between two cross-sections of a stream, 1 and 2,
may be stated in the form
(P.E..), + (K.E.)1 = (P.E.)2 + (K.E.)2 +(losses)1.2 (14)
In the equation, P.E. stands for the potential energy head, h;
K.E.. for the kinetic energy head, V2/2g; and the losses are as-
sumed to be equal to the frictional head losses between sections
1 and 2. Transposing
Bul. 404. FLOW IN MILD, UNIFORM CHANNELS
(P.E.)1 - (P.E.)2 - (losses)1-2 = (K.E.)2 - (K.E.), (15)
For a length of channel dx, we may write this relationship as
So dx - Sf dx = (K.E.)2 - (K.E. ) (16)
Equation 16 indicates that if So is greater than Sf, the work
done by gravity is greater than that required to overcome fric-
tion and that some energy will be added to the stream. If, how-
ever, Sf is greater than So, some energy will be withdrawn from
the store of energy in the stream. The former condition will be
represented by a rise of the water surface downstream and the
latter by a fall of the water surface downstream.
These relations can be made clearer by the concept of "speci-
fic energy" of the flow, E.. If we refer to the bed of the stream
as datum, the specific energy of the stream at any section is:
V2
E, = y + 2g(17)
where y is the depth of flow at the section. The distinction
between Eqs. 1 and 17 is important. Equation 1 is a measure of
the mean energy of the stream above any datum; Eq. 17 is a
measure of the specific energy of the stream where the bottom
of the channel is used as the datum. In uniform flow, Es is con-
stant. In varied flow, E, will change from section to section.
For dEs/dx positive, the specific energy increases downstream.
This means that the work of gravity exceeds that required to
overcome resistance losses, and the excess adds to the specific
energy. If the work of gravity is insufficient to overcome the
frictional resistances, part of the specific energy is used to
make up the difference.
By plotting E. (Eq. 17) against y it may be shown that for any
channel with a given discharge, there is a depth y for which E.
is a minimum. This depth is called the critical depth, y,. A
given discharge Q may flow in a channel at various depths, y.
For each value of y there corresponds a value of the specific
energy given by Eq. 17. But under no circumstances will the
content of specific energy fall below a minimum value of E., and
this minimum is attained at the critical depth, yc. Substituting
Q/A for V in Eq. 17, differentiating with respect to y, and
equating to zero gives
dE Q2 dA
dy--- - - 0
dy gAT dy
ILLINOIS ENGINEERING EXPERIMENT STATION
Since dA- = b, (Eq. 7),
dy
dE 6 Q2
--dEs= - Q2 = 0
dy jA 0
(18)
(19)
A3 Q2
b g
W g
In other words, the depth of flow for a given discharge is criti-
cal for the conditions expressed in Eq. 19. For a given dis-
charge, the slope that will make the flow critical at the normal
depth is called the critical slope and is designated Sc.
9. Classification of Surface Profiles
Longitudinal profiles of flow in open channels may be conven-
iently classified according to the bed slope So and the depth of
flow y in the following manner. 15
Table I
Types of Surface Profiles in
Uniform Channels
Slope Designation Relation of Depth, y,
to Uniform and
Critical Depths
Adverse, A2 Y0 > y > Yc
So < 0 A3 Y0 > Yc> y
Horizontal, H2 Y0 > y > Yc
SO = 0 H3 yo > Yc > y
Mild, M1 Y > YO > Yc
SO < Sc > 0 M2 Y0 > y > Yc
M3 YO > Yc > y
Critical, CI y > YC = YO
SO = Sc > 0 C3 Y0 = c > y
Steep S1 Y > Yc > Y0
S0 > Sc > 0 S2 YC > y > Y0
S3 Yc > Y0 > y
Bul. 404. FLOW IN MILD, UNIFORM CHANNELS
If SO < 0, the channel is termed adverse A; if SO = 0, it
is horizontal H; and if So > 0, it is called sustaining and is
further classified by the degree of slope. Uniform motion in
open channels of sustaining slopes may proceed in the tranquil,
critical, or rapid state depending on whether the uniform depth
of flow Yo is greater than, equal to, or less than the critical
depth, y,. Sustaining slopes producing uniform motion in the
tranquil, critical, and rapid states are said to be, respectively
mild M, critical C, and steep S.
Fig. 4. Longitudinal Profiles of M1 and M2 Curves
If the actual depth of flow y lies above both the uniform-
depth and critical-depth lines drawn parallel to the bottom of
the channel, it is of type 1; if between these lines it is of
type 2; and if below both lines, it is of type 3. The twelve
types of surface profile are outlined in Table I.
Of these surface profiles, the more common types are those on
mild slopes M1 and M2 (See Fig. 4). The former refers to the
portion of the water surface lying above the uniform depth, and
the latter to that lying between the uniform and critical depths.
The MNi curve, commonly known as the "backwater" curve, is pro-
duced when the downstream depth of flow is raised above the uni-
form depth by a dam or other obstruction. The M2, or "dropdown"
curve, results when the downstream depth is lowered below the
uniform depth. The upstream and downstream limits of the M, and
M2 curves are reached when y = yo, y = co; and y = yo, y = yc,
respectively. The conditions of flow existing at these limits
may be found by consideration of the gradually varied-flow equa-
tion. That is:
ILLINOIS ENGINEERING EXPERIMENT STATION
dy SO - Sf
dx 01 b Q2 (12b)
gA3
At the upstream limit, y = yo; the friction slope S, and that
of the channel So are identical; therefore, the left side of
Eq. 12b becomes zero. This condition indicates that both the M1
and M2 curves approach this limit asymptotically. Theoretically,
the length of these curves extends to infinity. For practical
purposes, however, these curves may be assumed to end at a depth
slightly above (for the M, curve) or below (for the M2 curve) the
uniform depth.
At the downstream limit for the M1 curve, y = o, the friction
slope Sf becomes zero, while the denominator of the right side of
Eq. 12b has a magnitude of unity. The left side of the equation,
consequently, is equal to So. Thus, the M1 curve approaches the
downstream limit horizontally.
At the downstream limit of the M2 curve, y = yc, the denonina-
tor of the right side of the equation becomes zero, so that the
left side approaches negative infinity as a limit. Consequently,
the M2 curve should end downward in a vertical direction.
The shapes of the M1- and M2-types of surface profile are
shown in Fig. 4 in greatly reduced horizontal scale. It should
be emphasized that the limiting conditions serve only the useful
purpose of establishing the general shape of the different types
of surface curves. Before many of the limits are reached, the
assumption of parallel flow may no longer be valid. Equation 13,
then, cannot be correctly applied to investigate the existing
state of flow under those conditions.
IV. INTEGRATION OF THE GRADUALLY VARIED-FLOW EQUATION:
EXISTING METHODS
10. General Review
The theory of varied flow was first postulated in a complete
and comprehensive manner by J. N. Belanger (61 His paper, pub-
lished in 1828, "Essai sur la solution numerique de quelques
problemes, relatives au movement permanent des eaux courantes,"
contains, in part, the general differential equation for gradual-
ly varied flow and a method of solution by successive approxima-
tions. The fundamental principles were so well covered that
little has since been added to modify the differential equation
in its original form. All subsequent investigations have been
confined mainly to the methods of solution of the gradually varied
flow equation by direct, approximate, and graphical integration.
The earlier attempts to obtain an analytical solution by
direct integration of the differential equation were restricted
to flow in channels of special form. The case of a rectangular
channel of great width was treated by Dupuit (1848) and in a
somewhat different manner by Ruhlmann (1880), both ignoring the
effect due to changes of velocity.161 The same case was pre-
sented in complete form by Bresse (1860) and subsequently by
Grashof. t7 These methods of integration were based on the Chezy
equation with a constant coefficient in evaluating the friction
slope. More recent solutions, based on a variable coefficient in
the Chezy equation, were due to Schaffernak (1913), Ehrenberger
(1914), Baticle (1921), hozeny (1928), Schoklitsch (1920), 7 and
Gunder (1943). t" The case of a broad parabolic channel was
treated by Tolkmitt (1898). " The first attempt to arrive at a
solution suitable for any type of cross-section was begun by
Bakhmeteff (1932), 19 and subsequently presented in a more
complete form by Mononobe (1936). 17
In the solution of the differential equation for gradually
varied flow by approximate integration, the channel is first
divided into several reaches and the depth at the end of each
reach determined by a trial-and-error procedure. A more direct
solution was given by Husted (1924), "" in which the lengths of
successive reaches were found for assumed increments of depth.
Since the results are obtained by a series of successive computa-
tions, this method is generally known as the "step method."
ILLINOIS ENGINEERING EXPERIMENT STATION
The gradually varied-flow equation may also be solved by
graphical integration. (11,123 The graphical methods involve
plotting suitable variables in rectangular coordinates in such a
manner that the distance along the channel is given by the area
under the curve. The area is found by means of a planimeter.
All three methods, direct, approximate and graphical integra-
tion, are used to a varying extent in computing surface profiles
of flow in uniform channels.
II. Integration of the Equation of Varied Flow
Several methods, more or less exact, are available for the
integration of Eq. 13. Only the more important ones are de-
scribed briefly here. These are the methods of graphical integra-
tion, the step method, and the direct integration methods of
Bakhmeteff and Mononobe. For discussing and comparing these
methods it is convenient to express Eq. 13 in a different form.
From Eq. 13
b Q2
dx = dy
So - s
b Q2
(1
SO dx = dy (20)
S
so
But
S Q2
f A2C2R4/3
Q2
SO= A2Co2R4/3
where R is the hydraulic radius at normal depth y and R0 is the
hydraulic radius at any depth of flow yo. By letting
ACR2/3 =K
AoCRo 2/ 3 = Ko
where K is called the conveyance of the channel, then Sf and So
become
Bul. 404. FLOW IN MILD, UNIFORM CHANNELS
Q2
Sf -K2
SQ2
SO Ko 2
02
And so
S K2
s o 17 -
G Q2 ()
G = (K- ) 2
Substituting Fqs. 21 and 22 into Eq. 20 gives
(1-G) (KI )2
SO dx = K dy
K
1 - (-0)2
K
Next, let
Equation 23 is a useful form of Eq.
for discussing the various methods
of varied flow.
13 and is used as the basis
of integrating the equation
a. Graphical Integration.
This may be called an exact method, since no simplifying
assumptions are involved. Equation 23 may be written in the form
1 - G
Sodx = dy + dy
S 1 (24)
Integrating Eq. 24 between two sections x1 and x2 at which the
depths are yl and y2 gives
x2 x
j dx
x1
-[f dy +
so 1
Ax = x2-x1 = - [(Y2 - y
0
y2 1 G- G y
J- -- -- dy]
yl (KL)2 - 1
Ko
Y2 1 - G
+l (- dy2
1 (K)2
(21)
(22)
(23)
(25)
ILLINOIS ENGINEERING EXPERIMENT STATION
K2
To evaluate the integral in Eq. 25, the reciprocal of-- -1
is plotted as a function of (l-G)y in Fig. 5. K0
For any known values of yl and y, the shaded area in Fig. 5,
obtained by a planimeter or analytically, gives the value of this
integral and the distance Ax may be calculated. By assuming
various successive values of y, the required curve is obtained
to any degree of accuracy desired.
(- G)g -:-
Fig. 5. Solution of Gradually Varied Flow Equation by Graphical Integration
b. The Step 'lethod
This method of integration is widely used and is the simplest
to perform. Equation 13 may be written
b Q2
(1 - -
Ax = g Ay (26)
so - Sf
b Q2
Provided Ay is small, an approximation of (1- --) and Sf can
be obtained by evaluating these quantities as the mean of their
values at the two extremes of Ax. In other words, if y, is the
depth at xl, and y2 is the depth at x1 + Ax, then these quanti-
ties can be evaluated by using a value of y = (y1 + y2)/2. Since
by = y2 - Y1, the right side of Eq. 26 is completely known, and
the length of the reach Ax can be determined. By taking success-
ive increments of Ay, the complete surface profile may be plot-
ted. This method gives surprisingly accurate results.
Bul. 404. FLOW IN MILD, UNIFORM CHANNELS
c. Methods of Direct Integration
The Method of Bakhmeteff.' 9' BakhmetefT developed a method for
the integration of the varied-flow equation based on the use
of a "varied-flow function" derived from two assumptions:
(1) The conveyance of a channel K = ACR2/3, may be approxi-
mated within a limited range of depths by the relationship
K2 A22R4/3 = a constant x yN (27a)
so that
K A2C2R4/3
A2c 2 4/3
0o Ao 0 oo YO
where N is called the "hydraulic exponent."
(2) G = Q2-- (-K)2 and changes very slowly in any practical
(23 Ko
case so that for a reasonably short range of integration, it may
be approximated by an average constant value.
Using these assumptions and setting y/yo = r and dy = Y dr,
Fq. 24 can be expanded and integrated from a section where x = xj,
y = yJ to a section where x = x2, y = Y2 resulting in
Ax = x2 - x1 =-- [(r2-rl) - (1-G) 2 (28)
r dr
The integral J0 l-rN is designated as B(N,r), the Bakhmeteff
form as
Ax = x2 - xI = _0[(r2-rl) - (1-G)[B(N,r2) -B(N,rl)] (29)
The values of the varied-flow function covering a wide range of
values of N and r have been computed and presented in tabular
form 91 to facilitate practical profile computations. In order
to determine the surface profile in any given case using Eq. 29,
the following step procedure is used:
(1) determine the class of curve by evaluating y, and yo
(2) determine the value of the "hydraulic exponent" N by
means of a logarithmic plot of Eq. 27a.
(3) plot the value of (1 - G) against y so that the average
ILLINOIS ENGINEERING EXPERIMENT STATION
value of (1 - G) between the assumed depths of flow
may be evaluated
(4) obtain the proper value of B(N,r) from the tables for
the terminal values of r
(5) substitute the required values in Eq. 29 and compute
the values of Ax.
The Method of Mononobe. '7 Mononobe developed a method for
integrating the varied-flow equation based on the assumption that
both the area A and the wetted perimeter p, may be expressed as
monomial functions of the depth y. That is:
p = a constant x yl (30)
A2 = a constant x yM (31)
In Eq. 30 and 31 the exponents I and N will generally vary
slightly with the depth y; in these cases, however, their average
values in the range of depths under consideration may be used.
Mononobe has shown that the above empirical relationships are
applicable to many types of cross-sections in common use. Using
these relationships and Manning's friction-slope formula (Eq
lib) it can be shown that
S (y )4/31-5/3M
so YO
By substituting this relationship and other relationships derived
from the assumptions in Eq. ??, Mononobe developed the following
equation of gradually varied flow:
Cy 2 C V0 2m r2 rCO--1
Ax = -x = r- dr - 0f2 r- dr] (32)
s2 r S 1co0 2gyo 1 rCo-1
in which r = y/yo, and Co = 5/3M - 4/31. Mononobe further sim-
plified Eq. 30 by using
CO
r
Ql(r) = f .oo -1dr (33)
r r
2t(r) = fl. ool- dr (34)
r O-1
These are referred to as Mononobe's "backwater functions." Upon
substitution, Eq. 30 reduces to
Bul. 404. FLOW IN MILD, UNIFORM CHANNELS
Ax = x2-x = Y-{ 1(r2) - $1(rl) - V 2(r-2(r)]} (35)
So 2gyo
The value of P2 depends only on the values of Co and r,
whereas the value of D2 is a function of Co, M and r. The values
of the backwater functions have been computed for various values
of Co, M and r and are available in graphical form.7)' These
charts are for use only in the case of backwater curves since
the values of 4, and t2 are given for corresponding values of r
greater than unity.
For convenience in integration in the cases of the dropdown
curve, the variable is taken to be - instead of r. Accordingly,
Eq. 32 is modified to r
1-2 M-1
Ax = -[f1 lr2 O d(-) - 12M /r2 d(1)] (36)
So i r; -i (-f)0 -1
and Mononobe's dropdown functions are defined by
S1~.001 oo 7o_ ) (37)
1 M-1
- r d(- ) (38)
2 1.001 Co_
so that Eq. 36 may be written as
y 1 1 V 2M 1
Ax - ( 2y0 27 2 2(--)]} (39)
As is the case with the backwater functions, the dropdown
function 1, depends only on the value of Co and 1i whereas Y2 is
a function of C0, M and---, The values of the dropdown functions
corresponding to various values of Co, M and-1, are given by
Mononobe in graphical form. 18) r
ILLINOIS ENGINEERING EXPERIMENT STATION
In applying Mononobe's method to profile computations it will
be found that logarithmic plots of the wetted perimeter against
depth do not approximate straight lines very well; therefore, an
error in the results may be introduced by assuming the wetted
perimeter to be a monomial function of the depth. The greatest
drawback in this method, however, lies in the inherent error due
to interpolation and the difficulty in using the charts to find
the values of the dropdown and the backwater functions. It will
be noted from Eqs. 35 and 39 that the values of these functions
are multiplied by the term yo/S0 in computing the distance along
the channel Ax. The value of yo/S0 is generally large, so that a
slight error due to interpolation may result in an appreciable
error in the computed distance along the channel. For this
reason, it has been suggested that the values of the varied-flow
functions be given to the fourth decimal place in order to keep
the error due to interpolation to a minimum. The graphical
charts prepared by Mononobe are drawn on such a scale that the
values of the backwater and dropdown functions can be read only
to the second decimal place. It would appear that unless the
charts are drawn to a much larger scale, the presentation of the
values of these functions in tabular form is to be preferred.
V. INTEGRATION OF THE GRADUALLY VARIED-FLOW EQUATION:
PROPOSED METHOD
12. Assumptions
In the foregoing discussion it is pointed out that assumptions
are necessary to simplify the solution of the gradually varied-
flow equation by direct integration. The equation is inte-
grated, in each case, between the range of depths in which the
simplifying assumptions are valid. Thus, the number of steps
required in the integration procedure for the whole range of
depths under consideration will depend on the limiting conditions
of each set of assumptions. For instance in the "step method,"
the assumption that the mean friction slope is equal to the mean
of the friction slopes at the ends of a reach is valid only for a
small increment of depth. Accordingly, a comparatively large
number of steps is required in the computation procedure. In the
method of Bakhmeteff the empirical relationship between the con-
veyance and depth holds for a wide range of depths; therefore,
for those cases involving insignificant e~ffect due to velocity
head changes, the computations may generally be made in one step.
If the effect of velocity head change is not negligible on the
other hand, the limitation imposed by assuming G to be constant
makes it necessary to confine the computations to small incre-
ments of depth. It is evident that the assumptions made must
conform to the actual conditions over a wide range of limits if
the method were to yield reasonably accurate results without
recourse to a large number of intermediate steps.
In the light of the studies made by Bakhmeteff(9' and Mono-
nobe,(71 it appears that both the area and the conveyance can be
expressed as monomial functions of the depth over a considerable
range for the channel sections in common use. These assumptions
are made in the proposed method of integrating the gradually
varied-flow equation, namely
K2 = a constant x yN (27a)
A2 = a constant x yM (31)
Equations 27a and 31 are only approximate; some cross-sections
satisfy these empirical relationships more closely than others.
ILLINOIS ENGINEERING EXPERIMENT STATION
However, even in channels not well adapted to the monomial
approximations over their entire range, the surface curves
usually cover only a limited range so that the actual variation
of the hydraulic elements conforms closely to these empirical re-
lationships. In most cases it will be possible to choose values
of the constants and the exponents that will make Eqs. 27a and 31
express the values of K and A with sufficient accuracy over a
wide range of depths. If greater accuracy in results is desired,
the whole range of depths under consideration may be subdivided
to determine the values of the exponents separately for each
interval. However, this procedure will hardly be required in
making practical profile computations.
It will be noted that the values of the exponents N and M are
characteristics only of the given cross-section. As such, they
can be determined once and for all for a given channel. Hence, a
saving of time as well as labor of computations may be effected
in those problems in which a large number of surface curves
are to be computed for the same channel under a wide variation
of discharge.
13. Derivation of Proposed Equation
The differential equation of gradually varied flow may be
solved with the aid of the simplifying assumptions by starting
from Eq. 13; i.e.,
b Q2 Q2 dA Q2
( 1- ) (1- gA ) dy + -d(-)
dx = dy = dy =
so-Sf so-SSf so-Sf
dy 2g A2
- So-Sf +so.S
W= 7 So-Sf
Multiplying both sides of the equation by S0/y0 gives
So Yo 2gyo
-0 dx = +
Yo 1-- 1-- i (40)
So so
By means of Eq. 31 and 27
SL. (0)2= (YO)N
S, K y
Bul. 404. FLOW IN MILD, UNIFORM CHANNELS
Therefore, Eq. 40 becomes
1- (YO N
y
Q2 1
d d-)
+ 2gyO A2
1-(YO )N
y
(41)
(YO )N
[1 + ] d(-)
1- ( .) Yo
y
= [1 + - l]d(y--)
yN
(- ) -1
yo
A 2
2 d(2
22gyo 02 A
oo I.(YO)N
y
V2
2gyo
A 2
d(A )
A2
1-(YO )N
y
substituting into Eq. 41 gives
Sdx = [1 +
Yo
-1 yo
A( 2
2gyo 1-(o)N
Y
and integration yields
so yo
YO Y
+ J N
)-
2
.-J
2gyo 1-(o)
y
Y dx=
Yo
Since
1- (YO)A
y
Q2 d(I
2gy0
1- (Yo)
y
(42a)
+ constant
(42b)
ILLINOIS ENGINEERING EXPERIMENT STATION
dr
Designating as before y/yo = r and B(N,r) = J----_ - Eq. 42b
becomes
A 2
Sx = r - B(N,r) + --- + constant (43)
Y0 2gyo 1.-(Y0)N
y
The integral on the right side of Eq. 43 may be reduced to the
form of the Bakhmeteff varied-flow function with the aid of Eq. 31,
from which it can be deduced that
(YO)N = [(YO)M]N/M= [Ao2] N/M
y y A2
Therefore A 2 A 2
A2 A2
A j- (44)
1-(Y0)N 1- (0 )N/M
y A2
Now, the integral on the right side of Eq. 44 is in exactly
the same form as Bakhmeteff's varied-flow function as defined
previously. In this case the exponent is N/M and the variable is
AO2/A2, therefore
A2 N A2
- 2 B(,-)
1_(A2 )N/M M'A2 (45)
A2
and Eq. 43 becomes
x =YO[r-B(N,r) + - B(-,IA0)] (46)
So 2gyo M A2
in which the constant of integration has been intentionally
omitted because Eq. 46 is used only to determine the difference
in the values of x between any two given sections.
For the downstream section at which r = r2, and A = A2,
y V 2 N A02
x o [r -B(N,r2) + 2 -B(-- )]
2 S2 2 2gy M A22 (47a)
Bul. 404. FLOW IN MILD, UNIFORM CHANNELS
and for the upstream section at which r = rl, and A = A1,
y V2 N A2
x = [r1 -B(N,rl) + - B(-,- o )] (47b)
so 2gyo M A12
Using the A notation and subtracting gives
Ax = x2-x1 =0 [Ar -AB(N,r) + oAB ( N )] (48)
So 2gyo0 M'A2
In applying Eq. 48 to practical profile computations, it is
necessary to know the values of both of these varied-flow func-
tions. Bakhmeteff has computed the first function for various
values of r in the range of N lying between 2.8 and 5.4. 9) For
the common types of practical channel sections the value of the
exponent N/M generally lies between 1.0 and 2.0 which is less
than the range covered by Bakhmeteff's tables. Hence a new set
of tables has to be computed for evaluating the second varied-
flow function in this range of N/M, and for appropriate values of
the variable A 2/A2. This has been done with the results pre-
sented in Tables 2 and 3 in which the values of the function are
given for sufficiently small intervals of both the variable and
the exponent to permit interpolation. The values of the varied-
flow function for the range of AO2/A2 between 0 and 0.990 are
given in Table 2, and for the range between 1.01 and 20 in
Table 3. In Table 3 all the values shown have been increased by
3 to prevent the values from changing signs. This procedure is
permissible since the relative values of the functions, which are
used in the computations, remain unaffected.
The proposed method, as well as that of Mononobe, differs from
Bakhmetef's method essentially i'n the treatment of the velocity
head changes. In Bakhmeteff's method the factor G, reflecting
these changes, is assumed to be constant. The limits of integra-
tion should, therefore, be confined to small increments of depth
inasmuch as the factor G is a variable with respect to the depth.
In the proposed method G is, in effect, regarded as a function of
the depth so that the integration limits may be generally taken
over the entire range of depths considered. For these cases, the
proposed method will afford a more direct computation procedure
whereby the results can be obtained without recourse to succes-
sive steps.
The integral embodying the effect of changes of velocity head
in the proposed method (Eq. 45) is reduced to the form of the
36 ILLINOIS ENGINEERING EXPERIMENT STATION
varied-flow function defined by Bakhmeteff. This form is preferred
in order to avoid the necessary adoption of a new term, as is
done in the method of Mononobe. In fact, both Mononobe's back-
water and dropdown functions are reducible to the same form as
Bakhmeteff's varied-flow function. It seems that little, if
anything, is gained by defining them in their present form,
especially when the resulting integrals I2 and T2 are expressed
in terms of three parameters. The integral in Eq. 45, on the
other hand, is dependent upon only two parameters, N/M and
Table 2
NA 2 A2
Values of the Varied Flow Function B(-,-°- ): -A <1
MA A2
N/MI
1.00 1.20 1.33 1.50 1.67 2.00
0.000 0.000 0.000 0.000 0.000 0.000
0.051 0.050 0.050 0.050 0.050 0.050
0.105 0.103 0.102 0.101 0.100 0.100
0.163 0.159 0.156 0.153 0.151 0.151
0.223 0.216 0.212 0.207 0.203 0.203
0.288 0.275 0.269 0.263 0.259 0.255
0.357 0.337 0.329 0.322 0.318 0.310
0.431 0.407 0.395 0.383 0.375 0.365
0.511 0.480 0.464 0.448 0.437 0.424
0.598 0.556 0.536 0.517 0.504 0.485
0.693 0.642 0.617 0.592 0.574 0.549
0.777 0.714 0.684 0.655 0.634 0.604
0.868 0.804 0.759 0.725 0.700 0.663
0.968 0.884 0.842 0.799 0.767 0.725
1.079 0.979 0.930 0.881 0.845 0.793
1.204 1.086 1.029 0.973 0.931 0.867
1.309 1.172 1.109 1.049 1.005 0.929
1.427 1.281 1.208 1.134 1.078 0.996
1.561 1.387 1.305 1.226 1.168 1.071
1.715 1.525 1.430 1.333 1.260 1.157
1.897 1.670 1.566 1.467 1.393 1.256
2.120 1.857 1.736 1.619 1.532 1.376
2.303 2.019 1.880 1.741 1.636 1.472
2.408 2.103 1.959 1.817 1.710 1.528
2.526 2.201 2.046 1.894 1.780 1.589
2.659 2.314 2.148 1.984 1.860 1.658
2.813 2.469 2.285 2.088 1.934 1.738
2.996 2.598 2.405 2.212 2.066 1.832
3.219 2.824 2.604 2.361 2.169 1.946
3.507 3.039 2.801 2.554 2.364 2.092
3.912 3.323 3.066 2.832 2.661 2.298
4.200 3.610 3.318 3.021 2.793 2.443
4.605 4.044 3.699 3.294 2.966 2.647
A02/A2
0.000
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.45
0.50
0.54
0.58
0.62
0.66
0.70
0.73
0.76
0.79
0.82
0.85
0.88
0.90
0.91
0.92
0.93
0.94
0.95
0.96
0.97
0.98
0.985
0.99
Bul. 404. FLOW IN MILD, UNIFORM CHANNELS
A0 2/A2; consequently, Tables 2 and 3 representing the values of
this integral are more simple to use than the graphical charts
prepared by Mononobei'' for the P2 and y2 functions.
14. Computation Procedure
In applying Eq. 48 to profile computations, it should be borne
in mind that the downstream conditions constitute the upper limit
of integration. The computations should be made with strict
observation to sign. If the location of the depths is assumed
correctly for the upper and lower limits, the distance between
any two sections Ax will always appear as positive.*
To compute the surface profile of flow in a given channel
when the rate of discharge is known, the following procedure
may be used:
a. Determine, from the given conditions, the points of
control from which the computations may start, and the type of
surface curve that will develop.
b. Determine the values of the exponents N and M in Eqs. 27a
and 31 for the entire range of depths covered by the surface
curve. This can be done algebraically by substituting into these
equations a pair of simultaneous values of the depths at the ends
of the range under consideration and the corresponding values of
the conveyance or the area. It can also be done graphically by
plotting on logarithmic scale the conveyance and area as ordi-
nates and the corresponding values of the depth as abscissas.
A straight line is then drawn to follow the points as closely as
possible. The slope of this line is one-half the value of the
exponent. The algebraic method corresponds to fitting a straight
line through the initial and final points on the logarithmic
plot; it has the disadvantage of not revealing the degree of
approximation for the intermediate points. For this reason, the
graphical method is more likely to give the average values of the
exponents by following all the points as closely as possible.
c. Compute the distance between any two given sections
according to Eq. 48. The necessary parameters in finding the
N A02
values of the two varied-flow functions are (N, r), and (,-A2-).
The value of N will generally be between 2.8 and 4.0 and for this
range the tables by Bakhmeteff19' should be consulted. For the
range of the exponent N/M between 1.0 and 2.0, Table 2 or 3 may
be used, depending on the value of the variable. In the event
*While either x, or both x2 and xI may have negative signs. Ax - x2 - x1 will
always be positive.
Table 3*
Values of the Varied Flow Function B(- , - ):
M A2
A2 > >
N/IM
1.00 1.20 1.33 1.50 1.67 2.00
1.01
1.02
1.03
1.04
1.05
1.06
1.07
1.08
1.09
1.10
1.15
1.20
1.25
1.30
1.40
1.50
1.60
1.70
1.80
1.90
2.00
2.10
2.20
2.30
2.40
2.50
2.60
2.70
2.80
2.90
3.00
3.10
3.20
3.30
3.40
3.50
3.60
3.70
3.80
3.90
4.00
4.10
4.20
4.30
4.40
7.605
6.912
6.507
6.219
5.996
5.813
5.659
5.526
5.408
5.303
4.897
4.609
4.386
4.204
3.916
3.693
3.511
3.357
3.223
3.105
3.000
2.905
2.818
2.738
2.663
2.594
2.530
2.469
2.412
2.358
2.307
2.258
2.211
2.167
2.124
2.084
2.044
2.007
1.970
1.935
1.901
1.869
1.837
1.806
1.776
6.944
6.367
6.033
5.793
5.609
5.456
5.328
5.219
5.122
5.045
4.699
4.462
4.286
4.143
3.901
3.724
3.579
3.457
3.358
3.263
3.182
3.108
3.041
2.983
2.928
2.875
2.826
2.782
2.738
2.700
2.662
2.626
2.593
2.560
2.531
2.503
2.471
2.455
2.419
2.397
2.372
2.348
2.327
2.307
2.285
6.621
6.102
5.801
5.585
5.420
5.283
5.169
5.070
4.983
4.913
4.607
4.394
4.236
4.109
3.897
3.741
3.615
3.509
3.423
3.341
3.272
3.209
3.152
3.102
3.056
3.012
2.971
2.934
2.898
2.866
2.835
2.805
2.778
2.751
2.727
2.704
2.679
2.658
2.637
2.619
2.599
2.580
2.563
2.547
2.529
6.295
5.835
5.566
5.376
5.229
5.109
5.008
4.921
4.843
4.775
4.512
4.329
4.187
4.073
3.896
3.761
3.654
3.564
3.488
3.422
3.364
3.312
3.265
3.223
3.185
3.149
3.117
3.087
3.059
3.032
3.008
2.985
2.963
2.942
2.923
2.904
2.887
2.870
2.854
2.839
2.824
2.810
2.797
2.784
2.771
6.046
5.632
5.387
5.217
5.084
4.977
4.886
4.808
4.737
4.669
4.441
4.282
4.151
4.046
3.896
3.778
3.687
3.608
3.538
3.485
3.436
3.392
3.353
3.316
3.283
3.254
3.229
3.204
3.181
3.158
3.139
3.121
3.103
3.086
3.071
3.054
3.043
3.029
3.017
3.003
2.993
2.982
2.972
-2.960
2.951
*All values in this table have been increased by 3 to prevent the numbers from being
negative.
5.652
5.308
5.107
4.966
4.857
4.768
4.694
4.629
4.573
4.522
4.331
4.199
4.099
4.019
3.896
3.805
3.733
3.675
3.626
3.585
3.549
3.518
3.491
3.466
3.444
3.424
3.405
3.389
3.374
3.360
3.347
3.335
3.323
3.313
3.303
3.294
3.285
3.277
3.269
3.262
3.255
3.249
3.243
3.237
3.231
AO2/A2
Table 3, (concluded)
N/M
1.00 1.20 1.33 1.50 1.67 2.00
4.50
4.60
4.70
4.80
4.90
5.00
5.20
5.40
5.60
5.80
6.00
6.20
6.40
6.60
6.80
7.00
7.20
7.40
7.60
7.80
8.00
8.20
8.40
8.60
8.80
9.00
9.50
10.00
10.50
11.00
11.50
12.00
12.50
13.00
13.50
14.00
14.50
15.00
15.50
16.00
16.50
17.00
17.50
18.00
19.00
20.00
1.747
1.719
1.692
1.665
1.639
1.614
1.565
1.518
1.474
1.431
1.391
1.351
1.314
1.277
1.242
1.208
1.175
1.144
1.113
1.083
1.054
1.026
0.998
0.972
0.946
0.921
0.860
0.803
0.749
0.697.
0.649
0.602
0.558
0.515
0.474
0.435
0.397
0.361
0.326
0.292
0.259
0.227
0.197
0.167
0.110
0.056
2.265
2.248
2.227
2.208
2.190
2.174
2.141
2.110
2.081
2.052
2.024
2.001
1.973
1.949
1.928
1.906
1.886
1.865
1.846
1.826
1.809
1.789
1.773
1.757
1.740
1.724
1.687
1.654
1.619
1.590
1.560
1.535
1.506
1.481
1.459
1.436
1.414
1.393
1.374
1.356
1.335
1.319
1.301
1.286
1.254
1.224
2.514
2.500
2.484
2.469
2.455
2.442
2.416
2.392
2.370
2.347
2.326
2.308
2.287
2.269
2.253
2.236
2.221
2.205
2.191
2.176
2.164
2.149
2.137
2.125
2.113
2.101
2.074
2.050
2.025
2.005
1.983
1.965
1.945
1.927
1.912
1.896
1.881
1.866
1.854
1.840
1.827
1.816
1.804
1.794
1.773
1.753
2.760
2.748
2.737
2.726
2.716
2.706
2.687
2.669
2.652
2.636
2.621
2.607
2.594
2.581
2.569
2.557
2.546
2.535
2.525
2.515
2.506
2.497
2.488
2.479
2.471
2.463
2.445
2.428
2.412
2.398
2.384
2.371
2.359
2.348
2.338
2.328
2.318
2.309
2.301
2.292
2.285
2.277
2.270
2.263
2.251
2.239
2.942
2.931
2.925
2.916
2.910
2.901
2.887
2.873
2.858
2.848
2.837
2.825
2.818
2.808
2.799
2.789
2.781
2.774
2.766
2.759
2.752
2.747
2.739
2.732
2.727
2.722
2.709
2.696
2.686
2.674
2.665
2.654
2.648
2.642
2.634
2.627
2.620
2.615
2.608
2.602
2.598
2.591
2.588
2.581
2.574
2.566
3.226
3.221
3.216
3.211
3.207
3.203
3.195
3.187
3.180
3.174
3.168
3.163
3.158
3.153
3.148
3.144
3.140
3.136
3.132
3.129
3.126
3.123
3.120
3.117
3.114
3.112
3.106
3.100
3.096
3.091
3.087
3.083
3.080
3.077
3.074
3.072
3.069
3.067
3.065
3.063
3.061
3.059
3.057
3.056
3.053
3.050
A2 2
ILLINOIS ENGINEERING EXPERIMENT STATION
that either the exponent or the variable lies between the values
shown in these tables, the value of the varied-flow function may
be found by straight-line interpolation.
This computation procedure is illustrated by the following
examples.
a. Example 1.
Let it be desired to find the distance along a rectangular
10-ft-wide channel between the depths y2 = 7.0 ft and yl = 4.5 ft
when the rate of discharge is 136.0 cu ft per sec. The slope of
the channel So is 0.0004, and the value of n, the roughness
factor in Manning's equation, may be taken as 0.015.
The uniform depth, yo, computed from Manning's equation is
4.0 ft for the given rate of discharge. The critical depth y. is
computed to be 1.79 ft. The profile of flow, therefore, belongs
to the case of backwater, or the M -type of surface curve.
Table 4
Hydraulic Properties of the 10-ft Rectangular Channel
Wetted Hydraulic
Depth Area Perimeter Radius Conveyance
y A p, R R2/3 K
2.0 20 14.0 1.43 1.28 2,540
3.0 30 16.0 1.87 1.52 4,530
4.0 40 18.0 2.22 1.71 6,800
5.0 50 20.0 2.50 1.85 9,190
6.0 60 22.0 2.72 1.96 11,630
7.0 70 24.0 2.92 2.05 14,200
8.0 80 26.0 3.08 9.13 16,900
The hydraulic properties of the 10-ft rectangular channel are
shown in Table 4, in which the conveyance K is computed according
to Eq. lla. The value of the hydraulic exponent N is found from
the logarithmic plot of conveyance-depth shown in Fig. 6. The
slope of the line is 1.40; therefore the hydraulic exponent N has
a value of 2.80. The value of M in Eq. 31 is 2.0, since the area
is directly proportional to the depth for channels of rectangular
cross-section.
The given data are
Yo = 4.0 = 10,000
So 0.0004
V02 (3.4)2
2gyo 64.4 x 4
Therefore
Bul. 404. FLOW IN MILD, UNIFORM CHANNELS
r =1 - - 1. 125
yo 4.0
r2. 2 7.0 = 1.750
2 y 4.0
0
Ar - r2 - r 1 = 0.625
From the tables prepared by Bakhmeteff,
B(2.8,1.125) = 0.705
B(2.8,1.75) = 0.212
hence
AB(N,r) = -0.493
The parameters necessary for using
Table 2 are
8 000
8000
6 000
4 000
2 000
/ ....
If
Depth "y"
Fig. 6. Logarithmic Plot of Conveyance K Against Depth y
for the 10-ft Rectangular Channel
7
/
Pd
7
7/
9
/00
! 2
7
?//7
6 " -
3 4 5
8 7
I
7
ILLINOIS ENGINEERING EXPERIMENT STATION
NA 2 = 1.40
M 2.0
A 2 402
-0 = - = 0.79
A2 452
AO2 402
2 i 0 - - 0.33
from which
B(1.4,0.79) = 1.273
B(1.4,0.33) = 0.365
Therefore
N A0
--ABý ) = -0.908
Substituting these values into Eq. 48 gives
Ax = x2 - x1 = 10,000 [0.625 + 0.493 - 0.045 (0.908)]
= 10,770 ft
b. Example 2.
Determine the distance along the rectangular channel in the
previous example between the depth y2 = 2.0 ft and yl = 3.6 ft.
The profile of flow in this case obviously belongs to the dropdown,
or M2-type of surface. The given data are
Yo = 10,000
so
V2
0 = 0.045
2gyo
Y1 3.6
r -o 4.0 0.90
Y2 2.0
r=O= = 0.50
Therefore
Ar = r2 - r1 = -0.40
Bul. 404. FLOW IN MILD, UNIFORM CHANNELS 43
From Bakhmeteff's tables,
B(2.8,0.90) = 1.253
B(2.8,0.50) = 0.521
hence
AB(N,r) = -0.732
The parameters necessary for using Table 3 are
A02 402
= - = 1.24
A12 362
A02 402
0 . - 4.0
A 2 202
N - 1.40
M
from which
B(1.4,1.24) = 4.246
B(1.4,4.00) = 2.692
Therefore
AB(_- -) = -1.554
A A2
Substituting these values into Eq. 48 gives
Ax = x - x1 = 10,000 [-0.40 + 0.732 - 0.045 (1.554)]
= 2620 ft
c. Example 3.
Let it be desired to determine the extent of the dropdown
curve in a 10-ft circular conduit (n = 0.015) laid on a slope of
0.001 when the rate of flow is 305 cu ft per sec. The upper
limit of the curve may be taken as 0.95 of the uniform depth.
The uniform and critical depths are first computed for the
given rate of flow and are found to be 6.0 ft and 4.11 ft respec-
tively. The values of the exponents N and M may be determined
according to Eqs. 27a and 31 as follows:
K 2 N A 2 0M
0 ' 0 0 0
ILLINOIS ENGINEERING EXPERIMENT STATION
Taking logarithms of both sides of the equations yields
K A
log( o log(T-)
N = 2 ; M = 2
log(--) log(--)
The exponents may be found by substituting into these equations a
pair of simultaneous values of the depth at the ends of the range
under consideration and the corresponding values of the convey-
ance or area.
In the example above when y = 0.95 x 6.0 = 5.7 ft
K = 8900; A = 46.20
and when yo = 4.11 ft
K0 = 5060; A0o 30.42
Therefore
log8900
N 2 506.7 = 3.50
log
log46.2
M = 2 30.42 = 2.58
log5.7
4.11
and
N_ = 1.355
M
The required distance along the channel may be determined
according to the scheme of computations in the following table.
Applying the results in the table to Eq. 48 yields,
Ax x2 - x - 6.0 [(0.685 - 0.95) - (0.733 - 1.369)
2 - 1 0.001
+ (6.2)2 (2.986 - 4.65)]
64.4 x 6
= 1230 ft
Bul. 404. FLOW IN MILD, UNIFORM CHANNELS
Computations for Dropdown Curve in 10-ft Circular Channel
Upstream Section, yl = 5.7 ft Downstream Section, y2 = 4.11 ft
r1 = L = 0.95 r2 = -2 = 0.685
yo yo
B(N,r1) = 1.369 B(N,r2) = 0.733
A0 2 (49.2)2 A02 (49.2)2
- - 1.14 - = 2.63
A12 (46.2)2 A22 (30.42)2
N A 2 N A2
B(') = 4.650 B( A) = 2.986
d. Example 4.
Supposing the 10-ft circular channel in the previous example
is preceded by a chute in which the fow of 305 cu ft per sec
takes place at a uniform depth of 2.0 ft, what will be the length
of the surface curve along the channel?
The depth of flow at the junction between the circular channel
and the chute will be taken as 2.0 ft. Since the flow must
cross the critical depth of 4.11 ft to attain a uniform depth of
6.0 ft in the circular conduit, the surface curve will end in a
hydraulic jump. The problem, then, is to determine the length of
the M3-type surface curve that lies between the limiting depths
of 2.0 ft and the lower conjugate depth of 2.7 ft at the foot of
the jump. The computations are shown in the following table,
assuming the same values for N and M as in Example 3.
Substituting these results into Eq. 48 gives
Computations for the Length of the M3-Type Surface Curve
in the 10-ft Circular Channel
Upstream Section, y, = 2.0 ft A02 (49.2)2
_-_ - = 19.60
yl A12 (11.18)2
r1 -- = 0.333
Y02
yo
B(Nrl) A0.334 B
B(Nr1) - 0.334 B()''- = 1.846
ILLINOIS ENGINEERING EXPERIMENT STATION
Downstream Section, y2 = 2.7 ft A0 2 (49.2)2
A22 (17.11)2
2 = N =I = 2.205
B(N,r2) = 0.456 M'A22
Ax = X2 - 1 = 6.0 [(0.45 - 0.333) - (0.456 - 0.334)
1 0.001
+ (6.2)2 (2.205 - 1.846)]
64.4 x 6
= 186 ft
The application of the proposed method to profile computations
has been illustrated by solving examples involving the three
types of surface curves of flow in uniform channels on mild
slopes. In all the foregoing computations the velocity distri-
bution is assumed to be unity, and no account is taken of the
eddy losses accompanying retarded flow as in the case of the
backwater and M3-type of surface profile. Equation 48 may be
modified to include the effect of nonuniform velocity distribution
as well as the eddy losses, which are usually estimated as
some percentage of the change of velocity head between adjacent
sections. The modified form of the equation, following the
procedure in deriving Eq, 48, becomes
y V02 N A02
Ax =S-[Ar - AB(N,r) + (Cm - k) 2g- AB('---)] (49)
SO 2gyo M A2
in which the correction factor for eddy losses k and the momentum
factor C are assumed to be constant from section to section.
It should be noted that these factors tend to offset each other
due to their opposite sign; the net effect, generally, is to
prolong the length of the M, curves and shorten the M3-type of
surface curves. For instance, using values of C = 1.05 and
k = 0.10 (assumed) in the solution of Example 1 would yield a
result of 10,870 ft instead of 10,770 ft; and in Example 4, the
length of the M3 curve would be 174 ft instead of 186 ft. These
results show only a small deviation from those computed by means
Bul. 404. FLOW IN MILD, UNIFORM CHANNELS
of Eq. 48. In the event, however, that the corrections for non-
uniform velocity distribution and eddy losses are large enough
to be significant, the computations may be made in accordance
with Eq. 49.
15. Special Charts
In the foregoing examples the distance along the channel is
computed between two sections of known depths for a given rate of
flow. In other types of problems it may be required to determine
the depth of flow at the end of a given reach of channel. When
Ax or y is to be computed over a wide variation of the discharge,
much of the subsequent labor of computation can be avoided by
preparing an auxiliary curve for the solution of the gradually
varied-flow equation. This can be done by plotting xSA0 against
r according to Eq. 46* YO
So V 2 A 2
X- = r - B(N,r) - B(A-. --)
YO 2gy0 M Af
= r - B(N,r) - 2gy -~,rM
In the above equation the values of the exponents N and M are
determined by the given channel section; therefore, it is possi-
ble to compute the values of x So corresponding to the various
V2 yO
v0
values of 2gy0 and r for the channel under consideration.
Such a plot is shown in Figs. 8a and 10a, in which xzo is
y0
plotted against the variable r and a parameter C1 for various
values of the exponents N and M. It is shown in the following
discussion that the particular values of the exponents used for
these figures correspond, respectively, to the special cases of
channels with rectangular and circular cross-sections.
*From Eq. 31
A 2
A0 (. )M r-=
72 Y0
ILLINOIS ENGINEERING EXPERIMENT STATION
a. Rectangular Channels.
For channels of rectangular cross-sections the area is direct-
ly proportional to the depth; therefore, the exponent M has a
value of 2.0 according to Eq. 31. The hydraulic exponent N may
be determined in accordance with Eqs. lla and 27a.
K = 1.486 AR2/3 = a constant x yN/2
n
Thus it can be seen that the value of N can be determined either
by a logarithmic plot of the conveyance against depth as ex-
plained previously, or of the product AR 2/3 against depth as
shown in Fig. 9. In the latter case, the roughness factor n need
not be taken into consideration in computing N.
Table 5
Values of the Exponents M and N for Rectangular
Channels of Various Widths
Hydraulic Range of Depths
Width, b Radius, R M N
ft ft From To
b << y R ... .... 2.0 2.0
5.0 T 1.0 8.0 2.0 2.60
10.0 R= 2.0 12.0 2.0 2.70
20.0 4.0 14.0 2.0 2.80
30.0 4.0 16.0 2.0 2.90
b >> y R y ... .... 2.0 3.33
For rectangular channels of broad cross-section the hydraulic
radius R is roughly proportional to the depth. Therefore, the
product AR2/3 varies approximately as the 5/3 power of the depth,
in which case N is equal to 3.33. The minimum value of N is
attained in rectangular channels whose width is small compared
to the depth, for which the hydraulic radius remains relatively
constant. Consequently, N is equal to 2.0 in the case of rect-
angular channels with deep and narrow cross-section. For the
intermediate cases N lies between these limiting values.
Values of N determined from a logarithmic plot of the product
AR2/3 against depth y for the various widths of channel are shown
in Table 5. The average value of N is found in each case for the
range of depths stated.
Bul. 404. FLOW IN MILD, UNIFORM CHANNELS
From the results in Table 5 it can be seen that the exponents
N and M for a broad rectangular channel are, respectively, 3.33
and 2.0. Using these particular values of the exponents, the
So
values of x-S calculated from Eq. 46 are shown in Table 6 for
YO
various values of r > 1 and for the parameter C1. The results
are also presented graphically in Fig. 7, in which the value of
x- s-is separated into two parts, [r - B(N,r)] and ClB(-, r-M),
for convenience in plotting.
S
Similarly the values of x -S for N and M equal to 2.8 and 2.0,
yo
corresponding to an average width channel, are computed in Table
7 and plotted in Fig. 8. The values for r >. 1 are in Fig. 8a
where again the components [r - B(N,r)] and C1B(-N,r"M) are used
M
for convenience. In Fig. 8b are the values for r < 1. The
application of Fig. 8a and 8b is recommended for rectangular
channels between 10 and 30 ft wide since the hydraulic exponent
in this range deviates only slightly from the average value of
2.8 used in the computations.
b. Circular Channels.
In the case of circular channels the exponents M and N may be
found by a logarithmic plot of the area and the product AR2/3,
respectively, against the depth. This has been done in Fig. 9
for 5, 10, and 20-ft circular conduits in a range of depths
between 0.2 and 0.7 of the diameter. The plot shows the slope of
the lines to be remarkably consistent with one another, the ex-
ponent N having an average value of 3.60 and M, 2.56. Applying
these values of the exponents to Eq. 46, it is possible to con-
struct the auxiliary curve for solving the equation of gradually
varied flow in circular channels. Table 8 shows the computations
of x - for various values of the parameter C1 and variable r.
YO
The results for r >. 1 are plotted in Fig. 10a in which the value
of x- is again separated into the two parts. Figure 10b shows
yo
the curves for r < 1. The curves in Fig. 10a and 10b may be
used to determine the surface profiles of flow in circular channels
between 5.0 and 20 ft in diameter.
ILLINOIS ENGINEERING EXPERIMENT STATION
In applying the charts to profile determinations it should be
noted that the curves possess several characteristic features.
First, all the variables appear as ratios; the resulting plot is
therefore dimensionless. Second, the use of the curves is not
confined to any one channel roughness since the exponents N and M
are determined only by the shape of the channel section. Third,
the curves for r < 1.0 (Figs. 8b, 10b) possess a maximum point
corresponding to the value of rc at the critical depth. Since
this point constitutes the lower limit of the dropdown curve, the
portion of the auxiliary curve comprising values of r greater
than r is to be used in connection the the M2-type surface
curve; the remaining part of the curve, less than re, is to be
used for the M3-type surface curve. In the event that a given
value of x- - is satisfied by more than one value of r, the
y0
required result can be obtained by considering the physical
conditions of flow.
16. Practical Application of the Charts
The special charts will be found to effect a saving of time
and labor of computations in those problems involving a wide
variation of discharge, as shown in the following examples.
a. Example 5
A natural channel, whose cross-section may be taken as a broad
rectangle, is laid on an average slope of 0.0004. Uniform flow
is attained at a depth of 4.0 ft when the discharge is found to
be 12.0 cu ft per sec per unit width of channel. Find the dis-
tance between two sections, 6 and 9 ft deep respectively, when
the discharge is 15 cu ft per sec per unit width, assuming: (a)
the effect of nonuniform velocity distribution and eddy losses to
be negligible; (b) the effect of velocity head changes to be
negligible; and (c) C- = 1.10, k = 0.20.
The uniform depth for q = 15.0 cu ft per sec may be found
as follows:
Q0 q0 K S0 y0
Therefore
f 3/5 15 3/5
Y ' y(-) - 4() 4.57 ft
q 12
Hence
V 02 1 Q2 1 1 q2
2gy0 .- .2 -03- 0.0365
2gyo 2g b2 y03 2g yo3
Bul. 404. FLOW IN MILD, UNIFORM CHANNELS
.4' C(O CO 4S eq CO .4 '
.4 co e In co c, .4
.3 t- C\ eq . t-- o^ 4
o o o '4* .^ .- »-4 c^
o 9 o 9c 9 9 9
ý C0 0 0 0 0
%a O C
v4 ln O\ al m 0
%0 t- 0 4 cc
0q a, Co t- 4n
C 9 C 9
o 9 9 0 0
o 0 0 9 9
o o o o o
Co d CD
CD C J 4 CI= t- -4 o
.4 4- VO tI CS4 . ^ '~ C'O .4 ^' ^O eqCS O
In \C t- -0 0 q, 1 - o\ m- V1 C^ M .4 o
\W t- .4 o .4 'o o \ mo Q' o - c 0 .4
o o 9 9 9 9 9 9 9 9 9 9 9 9
2 ~ ~ I C4 .l i2 9t' '^ ' o ^ n ^ ^
M , . M -4 &n eq eq rO m 1- co t- V,
*rt ~ e .4 t- o^ ( - o^ .4 u i O> eq c .4 a?
0 ~ ~ ~ C 0 10^rS M N C C> 10
CD C> .4 c C Cq 0- - > 9 0 C > 0
o o o o o o 9 9 9 9 o 9 o9
o o o o o 9 9 o 9 9 o o 9 9
0 ..
:9,9
I-
1--
CC
CO
M-
0 II
oo i
- 0
». CM.
0 11
> .
0 c
0
L.
CA
o (U0
4- 01
o §
<0 4.'
= (U
(U
10
£
- r 'C (1 W) Ce
9> c%3 .4 C4m
ei t- GT\ -4
-4 *-4 -4 4 l
n a * co .4 O
cq m4 O C
O~ 'C In .4 co eq eq eq -4
-4 9 9 9 9 9 9 9 9 9
o 9 9 9 9 9 9 9 9 9
9 9 9 9 9 9 9 9 9 9
.e t- 00
t- CO '-
9 9 9
0i de O .4 .4 CO
Int .4 CO eq .0 '"
9n 9^ 9o 94 9-
In %r. m%
CO C t- %Cl In3 . ?
9 9 9 9 9 9 9
9 9 -4 4 - 4 C
9 In .4* r-4 co e
*-4 eq .4 * I n *O C
0( 7 0% -4 m n 0q c0
In 1-4 .4 (1 w% % 0
Ci eq -4 c- 0 9 9
9 0 > 0 0C C4 -
C. t- v4 C40 e ' t -4
Sdd Gddo
9 9! 9 9 9
q eq eq c o co Co o
cl C4 c 4 M cm m 0ý
9
9
.4
9
9
t- nC> o eo C>c , 6 Do C
OOOOO4.- OOOOO4 000 000-*- 000M
00000ifr- 00000HO 000 000 000
0
0
0
0
0
0
F..
,03
I.
a; a
=3-
--
* II
(1
00
o i
L 4)
CB -
^
O 3=
L cc
Io-
01
- I.
(U (
°°D D 88 Q 0
0 0 0
(-4
OOCn '- 0OOt
00000 0000
t- 0 000 00 -
Ln L 0 ýCD n %Q4
a '.4o - 40 <
000 000
00'-t 'i»-'0 00
-4-4-4 -4-4- C .4000F
I- M Co " t--
FI CM '- C0"i
0T 0? I? 00 0
OOOOO
0(0In ONcC) \C cIC
t n %0'IC (OOCm
C. C0> 0
000 *-41-4»- .'4C104
F- . O '-m °\
o- E 0 00 - 0 00 0 t- o
cF4tc6=4 0000 o 000 00 o 0o
4.4 d d - D C OOC a0
' Lm t- GO un
C\ (\ 0> tn
OT- In eT I-T
oooooI
ILLINOIS ENGINEERING EXPERIMENT STATION
¼o
Fig. 8. Graphical Solution of the Equation of
1.5
1.0
0.6
0
1.
Bul. 404. FLOW IN MILD, UNIFORM CHANNELS
Gradually Varied Flow in Rectangular Channels
ILLINOIS ENGINEERING EXPERIMENT STATION
The distance along, the channel may be found with the aid of
Fig. 8. The parameters required in using the chart are r and C1.
In Example 5(a),
V0 2
C1 = 2-y° =0.0365
r 1 . . 1.31
Yo 4.57
Applying these values of the parameters to Fig. 8 yields
r - B(N,r) = 1.030
CIB(-,r"M) = 0.025
M
Depth y
Fig. 9. Logarithmic Plot of the Area A and of the Product AR2/3
Against Depth y for Circular Channels
co -e
0, 0,
o t0\- 4' m0
C00 C C 0 0 0 Ci
0~o 0 0 00
0C' m 0000a, l m 0
o0o 0o0o 0oo 0
SCO- 0 "t " t-0 C " 0 0
-0t- n o 1oC4 coCw
iooo r oo o0
o--4- 0000 00
000 0000 000
1-t "f 10 -i Uyci m- "M m< co Uin ý Co i U
0-000^ 0000 000 000 .4.---14 4O'. 0
0 0 ~ ' 0.C 0 0 0 0 0 0 0000 000- -< l
0o 0 o oo0 0 t-0 0 t- t-00 0 0 0
c o o o c d 171o-i C:o oodo oo
O I t leCm
Cm t- %r0 c0
00 0-i 0
IT0IT0
o0 o C C' o o0 o
-4-4-4 o o -4 -44-
0000 000c ^ -^o
II
C4
Ii
r
c0
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0
I..
cc
0
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0
0
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t.
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C.0
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0
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H
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u
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0000 000
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04C404'1 »~-141»4S 000
0000 0000 000
00CO Vt-- S 0 t-4 0 %C v
Sa , \C C\e)4'0, 04 04
In n WCO 4 m44
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0ýOM-t cc!*- COQ9 4 »-* c
D 0000 000
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oooc ^'C o -po ooo
as w t- 1n t- < t-C 4 m0
0stC- O t-. CO t 0 Ci
0000 0*000 000
ulV 0 UtfSW IC\ i--lp
04 co o t- \> 0 !4cco
t-o-
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000 -4-4 -4 -4 ....404(M
C\,(-s0 o(CO4 5-.C1
'CO MC! C C4-. ! 000
000 0000 000
000 0000 000
mn tn ON t- t- \0 t- \0 1-S
S-SO2 0000 000
O^i C! CM0 lcý
CO C)0-4.- - 000
000 0000 000
5-'00 0 000 000
U, if n 0So I o
000 0000 000
c ~Min mo^^c coo~l
dio 01-4di
O-- 4 m -4- 40t* 04
' 4 5 5 l l 4
ND ^ - ,o-'ý Wo O "-0 r <4 %C %r M t-- t- --. oo %, %- en^
00 tI Cl cn w C4, en n en Cs 1- 0 a, m0 - C0 0
mco(No : C oo-ý -i r--i l ntý co *-! C! "ý ý C o I'll
c^0JC 0J0r«.4 .-1i4.i4 000 0000 000
0000 0000oo 00 00 0000 000
moo co d 1oo ad0o 0 ood =ýC>C C
\C
0
0
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to co
S4)
0 0.
4) Q
00
0 U)
34.
0
.5 3
c .
00
L ..
00
0) 0
0) »
0)-
00
0 1
c -C
0
S-0
C0.
(0 0:
C -)
LA.<»
ILLINOIS ENGINEERING EXPERIMENT STATION
average of the slopes at the ends of a section in the step
method. The error, however, is likely to be small because in the
first two methods the values of the exponents and G are relative-
ly constant in the range of depths covered by the observed
profiles (see Table 9) while in the third method a large number
of steps is used to give better accuracy.
(2) Erroneous assumptions in derivation of varied-flow equa-
tion. In solving the differential equation of gradually varied
flow by graphical integration no assumption is made in the integ-
ration procedure; therefore any discrepancy between the results
of the graphical method and the test data, other than that due to
experimental error, may be taken as evidence of the fallacy of
the assumptions made in deriving the varied-flow equation. These
assumptions are:
(a) hydrostatic distribution of pressure intensity
(b) uniform velocity distribution; i.e., C. = 1.0
(c) losses confined mainly to friction, computed
as if the flow were uniform under identical
conditions of depth, discharge, and boundary
roughness.
Of these three assumptions the first is inconsequential except
in the immediate vicinity of the critical section. It seems
unlikely that this assumption would lead to any appreciable error
inasmuch as the dropdown measurements were not extended to the
critical depth. The error in the results is probably due pri-
marily to the second two assumptions.
The effects of the second assumption may be determined by a
consideration of the gradually varied-flow Eq. 13:
b Q2 b Q2
(1 ) C "
gA3 gA3
dx = dy - dy (50)
SO - Sf So - Sf
The effect of taking velocity distribution into consideration is
to increase the value of the second term on the right. It will
be noted that the first and second terms are always opposite in
sign, the latter being negative, for the M1- and M2-types of
surface profiles. Therefore the effect of nonuniform velocity
distribution is to shorten the distance along the channel between
two given depths. It follows then that in the case of the back-
water profile the actual curve will lie above the theoretical
one, resulting in depths larger than those computed by the graph-
ical method; while for the dropdown curve the actual profile will
fall below the theoretical resulting in actual depths which are
Bul. 404. FLOW IN MILD, UNIFORM CHANNELS 83
0.
t 0 o
0- xc
Cu §
00
am
o
- 0
• o 0a
0 -
CO
Nc -
SC-O
L,, ,>
40 0
SC.
< - .0
K
+' e
* + CC
c:
2 >*
tS 00**
e e
U/,'w-ig J/.oaq selUalu u/ uo//.l_
ILLINOIS ENGINEERING EXPERIMENT STATION
smaller than those computed by assuming uniform distribution of
velocity (i.e., C, = 1.0).
Finally, the assumption that the losses in gradually varied
flow are the same as for uniform flow may lead to erroneous re-
sults. This is due to the fact that the flow in the laboratory
tests is subject to an additional loss from turbulence, caused by
the water flowing past the joints of the channel sections. The
effect of this additional loss is to increase the friction slope
S in Eq. 50, with the results that the difference between the
two slopes, Sf and So, becomes smaller in the case of the back-
water and larger in the case of the dropdown curves. Consequent-
ly, the distance along the channel between two given sections
will be increased for the backwater and reduced for the dropdown
profiles. In either case the actual profile will lie below the
theoretical curve, yielding depths smaller than those computed by
neglecting turbulent losses.
Thus it can be seen that in the backwater case the effects of
nonuniform velocity distribution and turbulence tend to offset
each other. The net result will depend upon which of these
effects is predominant. For the backwater determination in the
circular channel (Fig. 20a), the test data fall consistently
below the theoretical curve computed by the graphical method.
This is probably due to the losses caused by turbulence, a factor
which is omitted in the theoretical profile computations. For
the rectangular and trapezoidal channels (Figs. 18 and 19a), some
of the actual data are greater than the theoretical values. This
discrepancy may be due to the effect of the nonuniform distribu-
tion of the velocity, which is neglected in the graphical method.
In the dropdown case the effects both of nonuniform velocity
distribution and turbulence tend to shorten the actual profile
from the theoretical curve. Therefore the observed depth at a
given section should be smaller than that computed by the graphi-
cal method. This can be seen from the dropdown measurements in
the trapezoidal and circular channels, shown in Tables 11 and 12
(or Figs. 19c,d and 20c,d). The greater discrepancy in results
for the dropdown curves may be caused, partly at least, by the
effects of nonuniform distribution of velocity and turbulent
losses, both of which produce errors of similar sign.
It should be emphasized that the maximum deviation of the test
data from the theoretical values computed by the graphical method
is reasonably small, (of the order of 0.12 in.). This close
agreement may be taken as evidence that the assumptions in the
gradually varied-flow equation did not result in any substantial
discrepancy between the theoretical and the observed profiles.
Bul. 404. FLOW IN MILD, UNIFORM CHANNELS 85
K
N
N
U
S0
0) 0
C C
m 0
L -
c M
00
L •
0 41
4.
CO
-S
O --
.0"
0o«
*~ 10
seai_u! u/ uo0/1iO-.0e
ILLINOIS ENGINEERING EXPERIMENT STATION
The comparison between the experimental data and the results
of the graphical, step, proposed, and Bakhmeteff methods, given
in Tables 10, 11 and 12, show the degree of accuracy of the
computations in that order. Both the graphical and the step
methods are likely to yield more precise results because of the
large number of intermediate steps taken in computing the surface
profiles. In the proposed and Bakhmeteff methods the results are
not affected by the number of intermediate steps, which are used
only to compute the elevations along the channel. In fact, these
methods tend to become less accurate when applied to short
reaches of channel, since in these cases a corresponding error
due to interpolation of the varied-flow functions yields a wider
discrepancy of results. The deviation in the proposed method is
probably caused by interpolation; however, the error is reason-
ably small in view of the close agreement between the actual
profiles and those computed by the proposed method. In general,
there is only a slight difference between the results of the
proposed and Bakhmeteff methods. This is due to the fact that
the observed profiles are confined to a limited range of depths,
in which the actual variation of the factor G is small. Under
these conditions it makes little difference whether G is assumed
to be constant as in Bakhmeteff's method, or considered as a
variable with depth as in the proposed method.
VII. CONCLUSIONS
The accuracy of the proposed method is determined by comparing
the computed surface curves with the observed profiles in various
experimental channels of rectangular, trapezoidal, and circular
cross-sections. The test data are also compared with the results
computed by the other existing methods. The method of Mononobe
is omitted because his graphical charts cannot be read with the
desired degree of accuracy. From the results of the experimental
investigations, the following conclusions are drawn:
(1) No substantial error is introduced by the assumptions
made in the gradually varied-flow equation as indicated
by the close agreement between the observed profiles
and those computed by the graphical method.
(2) The surface curves computed by the proposed method
agree reasonably well with the observed profiles so
that it may be expected that the proposed method can be
applied to practical computations with reasonable
accuracy.
(3) The computations of surface profiles by the graphical,
step, proposed, and Bakhmeteff methods yield only
slightly different results, the order of accuracy
being the same as the order of the above list.
It must be emphasized that the steps used in making the compu-
tations were so small that the graphical and step methods yielded
results which were more accurate than those obtained by the pro-
posed method. In practice, however, the proposed method will be
decidedly more convenient to use inasmuch as the computations
need not proceed by successive increments. The results of the
experiments indicate that the precision of the proposed method
falls well within the degree of refinement warranted in most
practical profile computations.
VIII. SUMMARY
The basic assumptions leading to the equation of gradually
varied flow are discussed at some length. The simplification is
based upon the one-dimensional method of analysis, in which the
flow is considered as being essentially parallel. The differ-
ential equation for gradually varied flow is derived from the
simplified equation of momentum, in contrast to the more common
procedure of deriving it from energy considerations. The differ-
ence between the two forms of the equation is explained. It is
concluded that the form derived from the momentum principle is
more consistent with the empirical expressions for the eval-
uation of the friction slope since the latter expressions are
essentially momentum equations.
The solution of the differential equation for gradually varied
flow by graphical, approximate and direct integration is discussed
with regard to the advantages and limitations of each method.
The choice of the method to be used is decided by the type of
problem under investigation. Theoretically, the graphical method
should yield results of the greatest accuracy inasmuch as no
assumption is made in the integration procedure. The work in-
volved in this method, however, is quite tedious so that it has
not been widely used. In many practical problems the method
of approximate integration or the step method may be used with
sufficient accuracy. The step method is limited by the fact that
computations must be made by small increments of depth in order
to satisfy the assumptions made in evaluating the mean friction
slope within each interval. The objection is not serious, how-
ever, in solving the type of problems in which only a few surface
curves have to be computed and elevations are needed all along
the channel. In another type of problem, on the other hand, such
as the determination of the delivery in a channel connecting two
reservoirs, a large number of surface curves are required for a
larger number of discharge rates. Under these circumstances, the
use of the direct integration methods will prove to be of decided
advantage in the time and labor of computations saved.
The progress of the various direct integration methods repre-
sent the trend in developing a solution of the gradually varied-
flow equation to dispense with the large number of intermediate
steps required by approximate integration. The earlier attempts
Bul. 404. FLOW IN MILD, UNIFORM CHANNELS
are incomplete in that either the effect due to velocity changes
is neglected or the friction slope is evaluated by means of the
Chezy equation with a constant coefficient, the use of which
has been outmoded. Moreover, these methods are based on some
idealized channel cross-section of simple geometric shapes.
These simplifications may lead to a substantial deviation between
the computed and observed profiles, as shown by the experiments
conducted by Mononobe. For closer agreement between the theore-
tical and actual results, the actual shape of the channel must be
considered, and the friction slope should be evaluated by a more
suitable friction formula than that of Chezy.
By using an empirical relationship between the conveyance
and the depth, Bakhmeteff was able to derive a more general
solution in which the effect due to the shape of the channel may
be taken into consideration, and the friction slope may be evalu-
ated by any suitable friction formula. his method is convenient
when applied to the problems in which the effect of velocity head
changes is insignificant. Otherwise, the correction that must be
made to take the velocity head changes into account necessarily
destroys the convenience of his method. In the method of Monono-
be the computation procedure is more direct; unfortunately it
is marred by the difficulty of using his graphical charts and
the inherent approximations that are likely to result due to
interpolation of the varied-flow functions.
These difficulties are avoided in the proposed method which
is, in effect, an extension of Bakhmeteff's method to cover those
cases involving significant effects due to velocity head changes.
The integral, embodying these changes, is reduced to the form of
the varied-flow function defined by Bakhmeteff. The values of
the integral have been computed over a wide range, sufficient to
meet the conditions encountered in most practical profile compu-
tations. The convenience of the proposed method is illustrated
by solving several typical examples.
APPENDIX: BIBLIOGRAPHY
1. King, H. W., Hand Book of Hydraulics, 3rd Edition, pp. 265-78.
2. Lee, Ming, "Steady Gradually Varied Flow in Uniform Channels on Mild
Slopes," Thesis, University of Illinois (1947), pp. 11-21.
3. Bakhmeteff, B. A., Hydraulics of Open Channel Flow, 1st Edition,
pp. 28-9.
4. O'Brien, M. P., "Velocity-Head Correction for Hydraulic Flow," Engin-
eering News Record, Vol. 113 (1934), p. 214
5. Woodward, S. M. and Posey, C. S., Hydraulics of Steady Flow in Open
Channels, 1st Edition, pp. 63-71.
6. Bakhmeteff, B. A., Hydraulics of Open Channel Flow, 1st Edition, Appen-
dix I, p. 299.
7. Mononobe, N., "Backwater and Dropdown Curves for Uniform Channels,"
Trans. A.S.C.E., No. 103 (1938), pp. 950-1001.
8. Gunder, D. F., "Profile Curves for Open Channel Flow," Trans. A.S.C.E.,
No. 108 (1943), pp. 481-506.
9. Bakhmeteff, B. A., Hydraulics of Open Channel Flow, 1st Edition,
Chapter 8.
10. Husted, A. G., "New Method of Computing Backwater and Dropdown Curves,"
Engineering News Record, Vol. 92 (1924), p. 719.
11. Thomas, H. A., "Hydraulics of Flood Movements in Rivers," Engineering
Bulletin, Carnegie Institute of Technology, (1934), p. 13.
12. Woodward, S. M. and Posey, C. S., Hydraulics of Steady Flow in Open
Channels, 1st Edition, pp. 75-9.
13. Mitchell, W. D. and Barron, E. C., "The Backwater Profile for Steady
Flow in a Rectangular Channel and Its Significance in the Stage-Fall-
Discharge Relation," Thesis, University of Illinois (1946).