River Hydraulics

GEOLOGICAL SURVEY WATER-SUPPLY PAPER 1369

This water-supply paper was published as 

separate chapters A-E
UNITED STATES DEPARTMENT OF THE INTERIOR 

STEWART L. UDALL, Secretary

GEOLOGICAL SURVEY 

Thomas B. Nolan, Director
CONTENTS

[The letters in parentheses designate separately published chapters]

Page

(A) Selected topics of fluid mechanics, by Carl E. Kindsvater_______ 1

(B) Energy losses associated with abrupt enlargements in pipes, with 

special reference to the influence of boundary roughness, by Carl E. 

Kindsvater ___________________________________ 53

(C) Surges in natural stream channels, by S. B. Rantz ___ 77

(D) Flow through openings in width constrictions, by Jacob Davidian,

P. H. Carrigan, Jr., and John Shen____________________ 91

(E) Stream gaging control structure for the Rio Grande conveyance channel near Bernardo, New Mexico, by D. D. Harris and E. V. 

Richardson ___________________________________ 123

O
Selected Topics 

Of Fluid Mechanics

GEOLOGICAL SURVEY WATER-SUPPLY PAPER 1369-A
Selected Topics 

Of Fluid Mechanics

By CARL E. KINDSVATER

RIVER HYDRAULICS

GEOLOGICAL SURVEY WATER-SUPPLY PAPER 1369-A

A discussion of the concepts of 

fluid mechanics which are the 

foundations of river hydraulics

UNITED STATES GOVERNMENT PRINTING OFFICE, WASHINGTON : 1958
UNITED STATES DEPARTMENT OF THE INTERIOR 

FRED A. SEATON, Secretary

GEOLOGICAL SURVEY 

Thomas B. Nolan, Director

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

Washington 25, D. C. - Price 55 cents
CONTENTS

Page

Abstract- ___________________________________...._-___------------__.. 1

Introduction._____________________________________________________ 2

Technical symbols, terms, and basic concepts ______________________ 2

Letter symbols and units____-______________------------_------- 2

Boundary conditions.______________________________________ 2

Fluid properties.__________________________________________ 3

Motion (flow) properties__________________________________ 3

Force properties_____________-__________-__-------_-------- 4

Ratios and coefRcients________________________-_-_-_-_---__ 4

Mathematical symbols___________-________------------_-------- 5

Basic concepts of mechanics______-_____________-__-_-------_-_- 5

Basic concepts of fluid mechanics________________________________ 7

Equations of motion_______________________________________________ 10

Fundamental equation of motion________________________________ 10

External forces in fluid motion._________________________________ 10

Kinematics of fluid motion___________________________________ 11

Equations for a simple fluid motion______________________________ 12

Hydrostatics.. _______ ___ ____ 16

Piezometry___________________________________________________ 18

One-dimensional equation___________________________-_-_----___ 20

Equation of eontmuity___________-_________--___--_----_-----_- 22

Discharge equation.___________________________________________ 23

Flow net.-_______--____---_-----____----_----_------________ 24

The experimental method______________________________________ 28

Influence of shear____________________________________________ 29

Experiments and the theory of similitude.________________________ 29

Influence of a free surface___________________________-__-----___ 32

Recapitulation ________________________________________________ 33

Example.__________________________________________________ 34

The energy equation_________________ 37

Review of the Bernoulli equation________________________________ 37

General energy equation______________________________________ 40

The momentum equation______________ ____ _ 42

Uniform and gradually varied flow_______________________________ 46

Characteristics of uniform flow_______________________-__________ 46

The Chezy equation_____________________________________-_____ 47

Alternate derivation of the Chezy equation _______________________ 48

Gradually varied flow in open channels_________________________ 50

ILLUSTRATIONS

[Plates follow p. 52]

PLATE 1. Flow through a two-dimensional orifice, piezometers in pipes

and open channels, and three-dimensional orifices. 

2. Typical two-dimensional flow nets, two-dimensional sluice gate, 

forces acting on a fluid mass, and uniform flow in pipes and 

open channels.

XXX
RIVER HYDRAULICS

SELECTED TOPICS OF FLUID MECHANICS

By CARL E. KINDSVATER*

ABSTRACT

The fundamental equations of fluid mechanics are specific expressions of the 

principles of motion which are ascribed to Isaac Newton. Thus, the equations 

which form the framework of applied fluid mechanics or hydraulics are, in addition to the equation of continuity, the Newtonian equations of energy and 

momentum. These basic relationships are also the foundations of river hydraulics.

The fundamental equations are developed in this report with sufficient rigor 

to support critical examinations of their applicability to most problems met by 

hydraulic engineers of the Water Resources Division of the United States Geological Survey. Physical concepts are emphasized, and mathematical procedures 

are the simplest consistent with the specific requirements of the derivations. 

In lieu of numerical examples, analogies, and alternative procedures, this treatment stresses a brief methodical exposition of the essential principles.

An important objective of this report is to prepare the user to read the literature 

of the science. Thus, it begins with a basic vocabulary of technical symbols, 

terms, and concepts. Throughout, emphasis is placed on the language of modern 

fluid mechanics as it pertains to hydraulic engineering. The basic differential 

and integral equations of simple fluid motion are derived, and these equations are, 

in turn, used to describe the essential characteristics of hydrostatics and piezometry. The one-dimensional equations of continuity and motion are defined 

and are used to derive the general discharge equation. The flow net is described 

as a means of demonstrating significant characteristics of two-dimensional irrotational flow patterns. A typical flow net is examined in detail.

The influence of fluid viscosity is described as an obstacle to the derivation of 

general, integral equations of motion. It is observed that the part played by 

viscosity is one which is usually dependent on experimental evaluation. It 

follows that the dimensionless ratios known as the Euler, Froude, Reynolds, 

Weber, and Cauchy numbers are defined as essential tools for interpreting and 

using experimental data. The derivations of the energy and momentum equations are treated in detail. One-dimensional equations for steady nonuniform 

flow are developed, and the restrictions applicable to the equations are emphasized.

Conditions of uniform and gradually varied flow are discussed, and the origin 

of the Chezy equation is examined in relation to both the energy and the momentum equations. The inadequacy of all uniform-flow equations as a means of 

describing gradually varied flow is explained. Thus, one of the definitive problems 

of river hydraulics is analyzed in the light of present knowledge.

 Regents-Professor of Civil Engineering, Georgia Institute of Technology, Atlanta, Oa.; consultant to tbe 

U. S. Geological Surrey.

1
2 RIVER HYDRAULICS

INTRODUCTION

This report is the outgrowth of a series of short schools conducted 

during the spring and summer of 1953 for engineers of the Surface 

Water Branch, Water Resources Division, U. S. Geological Survey. 

The topics considered are essentially the same as the topics selected 

for inclusion in the schools. However, in order that they might serve 

better as a guide and outline for informal study, the arrangement of 

the writer's original lecture notes has been considerably altered.

The purpose of the report, like the purpose of the schools which 

inspired it, is to build a simple but strong framework of the fundamentals of fluid mechanics. It is believed that this framework is 

capable of supporting a detailed analysis of most of the practical 

problems met by the engineers of the Geological Survey.

It is hoped that the least, accomplishment of this work will be to 

inspire the reader with the confidence and desire to read more of the 

recent and current technical literature of modern fluid mechanics.

TECHNICAL SYMBOLS, TERMS, AND BASIC CONCEPTS

The material in this section is the basic language of fluid mechanics. 

The reader is urged to become thoroughly acquainted with these 

symbols, terms, and fundamental concepts before reading the subsequent text.

LETTER SYMBOLS AND UNITS

BOUNDARY CONDITIONS

A Area; sq ft or ft2.

6, B Width of section; ft.

D Diameter of a circular section; ft.

L A length, as defined; ft.

n Direction or distance perpendicular to streamlines in

the s, n plane; as a radial distance it is measured

positively toward the center of curvature; ft. 

P Wetted perimeter or perimeter of the wetted cross

section; ft. 

R Hydraulic radius; ratio of the area to the wetted

perimeter; ft. 

r Radius of a circular arc or radial distance; as a

distance it is measured positively from the center

of curvature; ft. 

s Direction or distance along or parallel to streamlines;

ft. 

S Hydraulic "slope"; in uniform-flow equations it is

the flow-distance gradient of either the total energy

head or the piezometric head.
SELECTED TOPICS OF FLUID MECHANICS

V

m

y

7 (gamma) 

P (rho)

M (mu) 

<r (sigma) 

e (epsilon)

H

Q

t,T

v

Volume; cu ft or ft3.

Direction or distance perpendicular to streamlines in

the s, m plane and at right angles to the n direction;

ft. 

A direction or distance; as a plane coordinate it is

usually horizontal; ft. 

A direction or distance; as a plane coordinate it is

usually vertical; in open channels it is the depth of

flow referred to the average bottom level; ft. 

Elevation of a point referred to a horizontal datum;

ft.

FLUID PROPERTIES

Specific or unit weight; weight per unit volume; Ib

per cu ft or lb/ft3 . 

Mass density; ma'ss per unit volume; also, ratio of

specific weight to acceleration of gravity; slugs per

cu ft, slugs/ft3, or Ib-sec2/ft4. 

Viscosity; dynamic viscosity; lb-sec/ft2 . 

Surface tension; lb/ft. 

Modulus of elasticity; psf or lb/ft2 .

MOTION (FLOW) PROPERTIES

Acceleration; rate of change of velocity with respect 

to time; a vector quantity; ft/sec2.

The acceleration due to gravitational force, usually 

taken to be essentially constant in magnitude 

(32.2) and fixed in direction (vertically downward); 

ft/sec2.

Piezometric head; in general, equivalent to pressure 

head plus elevation; for uniform flow in open 

channels, it is equivalent to the elevation of the 

free surface; ft.

Total energy head; energy in ft-lb/lb of fluid flowing; 

ft.

Volume rate of flow per unit width; discharge per 

foot of width; cfs per ft or ft3/sec/ft.

Volume rate of flow; total discharge at a cross section; cfs or ft3/sec

An interval of time; sec.

Velocity at a point; instantaneous time rate of displacement of a particle or point; a vector quantity 

as distinguished from speed, which is scalar; fps 

or ft/sec.
4 RIVER HYDRAULICS

V Average velocity in a cross section; ratio of total discharge to the normal area of a cross section; fps or

ft/sec.

FORCE PROPERTIES

F Force; a vector quantity; Ib.

M Mass; ratio of weight to the acceleration due togravity; a scalar quantity; slugs.

p Pressure intensify; normal (pressure) force per unit 

area; a scalar quantity; psf or lb/ft2.

W Weight; total force due to gravity; a vector quantity, 

always vertical; Ib.

T (tau) Shear stress; tangential (shear) force per unit area; 

psf or lb/ft2.

it (eta) Eddy viscosity; apparent viscosity due to turbulence; lb-sec/ft5.

RATIOS AND COEFFICIENTS

C Any coefficient; usually dimensionless.

Ce Coefficient of contraction; ratio of area of contracted

stream to gross area of opening; dimensionless. 

Cen Coefficient in the Chezy equation; has dimensions of

V#, units of ft^/sec.

Cd Coefficient of discharge; dimensionless. 

CL Energy-loss coefficient; dimensionless. 

CT Shear-force coefficient; dimensionless. 

K A constant. 

n Coefficient in the Chezy-Manning equation; has

dimensions of RH, units of ft*. 

a (alpha) Velocity-head coefficient; ratio of the true average

velocity head in a cross section to the velocity

head computed on the basis of the average velocity

in the section; dimensionless. 

j8 (beta) Momentum coefficient; ratio of true rate of transfer

of momentum (momentum flux) at a cross section

to the momentum ,flux computed on the basis of

the average velocity in the section; dimensionless. 

E Euler number; ratio of a unit inertia! reaction to a

unit pressure force; a basic flow parameter. 

F Froude number; ratio of a unit inertial reaction to a

unit weight force; a number which varies inversely

with the relative influence of weight on the flow

pattern; dimensionless.
SELECTED TOPICS OF FLUID MECHANICS 5

R Reynolds number; ratio of a unit inertial reaction to 

a unit viscous shear force; a measure of the relative 

influence of viscosity; dimensionless.

W Weber number; ratio of a unit inertial reaction to a 

unit surface-tension force; a measure of the relative 

influence of surface tension; dimensionless.

C Cauchy number; ratio of a unit inertial reaction to a 

unit elastic force; a measure of the relative influence of elasticity; dimensionless.

MATHEMATICAL SYMBOLS

A (delta) Change in value; final value minus initial value;

increment.

S (sigma) Summation. 

<* Proportional to ; varies as. 

** Approximately equal. 

H-» Vector addition.

BASIC CONCEPTS OF MECHANICS

Newton's laws of motion. Engineering mechanics, which includes 

the mechanics of solid and rigid bodies as well as fluids, is based on 

Newton's hypotheses or "laws" concerning the behavior of masses 

acted upon by external forces. The fundamental proposition is 

stated in the second law of motion: a mass acted upon by an unbalanced external force or force system has an acceleration which is 

proportional to and in the direction of the resultant force. This 

proposition is expressed algebraically in the familiar relation 2F<x.Ma, 

in which both F and a are recognized as vector quantities. In a 

rectilinear coordinate system, for example, *2Fx<x.Max, 2Fy<x.Mav, and

The first law is: A mass remains at rest or moves with constant 

velocity in a straight line until acted upon by an unbalanced resultant 

external force; or, in accordance with the second law, if S,F=0, then 

a=0. The first law, as applied to fluid mechanics, describes both 

hydrostatic and uniform-flow conditions.

The third law is: A mass acted upon by another mass exerts an 

equal, opposite, and simultaneous reaction. If "2F is defined as the 

resultant external force acting on a mass, then Ma is the inertial 

reaction of the mass.

It is apparent that Newton's second law of motion provides a 

quantitative expression for the relationship between force, mass, and 

acceleration which is the basis for all equations of motion in engineering mechanics,
6 RIVER HYDRAULICS

Dimensions. The dimension of a quantity is its classification according to a system of independent physical categories. Because 

Newton's second law is a complete description of a mass in motion, 

the fundamental dimensions are the dimensions of the quantities involved in a physical statement of the law. Thus, the fundamental 

dimensions are mass, length, time, and force (M, L, T, and F) from 

the relationship F<xMa<xML/T2. Quantities such as acceleration, 

power^ and momentum are derived quantities; their dimensions are 

expressed in terms of the fundamental dimensions. Because the 4 

fundamental dimensions are related by means of the Newtonian equation, an adequate dimensional system need involve only 3 dimensions. Commonly used dimensional systems involve F, It, and T, or 

M, L, and T.

Units of measure. For each of the fundamental dimensions there 

is an accepted unit of measure. Thus, L is measured in feet,, f in 

pounds, T in seconds, and M in slugs. Only three units nee<| be involved in a complete dimensional system, however. For example, 

slugs need not be used in the F, L, T system. The units of derived 

quantities such as acceleration, power, and momentum are expressed 

only in terms of the units of the fundamental dimensions,

The American engineering system of units is an absolute system; 

that is, in the basic relationship Foc(ML/T2)=K(ML/T*), the proportionality constant, K, is taken to be unity. Thus, a force of 1 pound 

produces in a mass of 1 slug an acceleration of 1 foot per second per 

second. It follows that, expressed in the absolute system of units, 

Newton's second law can be written 2F=Ma.

Kinematics. Kinematics deals with the description of motion 

without respect to the forces which cause or influence this motion, 

Kinematics involves only time and space.

Dynamics. Dynamics deals with the effect of forces in causing or 

changing the motion of a mass. It is further subdivided into statics 

and kinetics.

Statics. Motions described under this heading correspond to the 

condition of zero acceleration in the directions considered.

Kinetics. Kinetics is that branch of dynamics which deals with the 

accelerated motion of a mass under the influence of unbalanced external forces.

Work. Work is the product of a force and the distance over which 

the force acts in accomplishing motion. Work is a scalar quantity; 

it does not involve the time during which the force acts. Work is 

measured in foot-pounds (ft-lb).

Power. Power is the time rate of doing work. The power delivered to a moving body is proportional to the product of the force 

producing the motion and the velocity of the body in the direction of
SELECTED TOPICS OF FLUID MECHANICS 7

the motion. Power is a scalar quantity. The unit of power is footpounds per second (ft-lb per sec) or horsepower. One horsepower is 

equivalent to 550 ft-lb per sec.

Energy. A body upon which work is accomplished acquires thereby 

the capacity to do an equal amount of work. The capacity to do 

work is called energy. Thus, energy is a measure of work, and vice 

versa. Energy is a scalar quantity. The unit of both work and 

energy is the foot-pound (ft-lb).

Momentum. Linear momentum is the product of mass and velocity. 

It is a vector quantity, and its unit is the pound-second (Ib-sec).

Impulse. Linear impulse is the product of a force and the time 

during which the force acts. It is a vector quantity. From Newton's second law, impulse accomplishes and is equivalent to the change 

in momentum experienced by the mass acted upon. The unit of impulse is, therefore, the same as that of momentum, the pound-second 

(Ib-sec).

BASIC CONCEPTS OF FLUID MECHANICS

Continuity and discharge. Fluid mass can be neither created nodestroyed by virtue of its motion. Thus, in a steady fluid flow ber 

tween solid boundaries or through a streamtube, the rate of passage 

of fluid mass through successive cross sections is constant. If the 

fluid be one of negligible compressibility (constant density), the rate 

of passage of fluid volume past successive sections is also constant. 

This is the simple basis for the continuity principle. The principle 

is expressed, for example, in the equation <2=.AV==constant, where 

 Q is the volume rate of flow, or discharge, in cubic feet per second and 

V is the average velocity normal to the surface represented by the 

area (A). In a section of nonuniform velocity, the total discharge is 

the summation (or integral) of the discharges through the elementary 

areas comprising the total area of the section.

Uniform and steady motion. Velocity in a fluid motion is a vector 

quantity and it is a function of both time and space. When the 

velocity remains constant with respect to time, the flow is described 

as steady. Most problems in engineering hydraulics involve steady 

motion.

When the velocity is constant with respect to distance in the direction of motion, the flow is described as uniform. Uniform flow occurs 

in long pipes. It seldom occurs in natural open channels.

Rotational and irrotational motion. The distinction between rotational and irrotational motion is essentially mathematical. The most 

significant characteristics of an irrotational flow are those usually 

associated with the classical "ideal" or "frictionless" fluid (a fluid 

without compressibility or viscosity). It is pertinent to the subse-
8 RIVER HYDRAULICS

quent use of the flow net in these notes to observe that irrotationality 

is corequisite to the existence of a flow net.

Laminar and turbulent motion. Laminar fluid motion is characterized by a steady, translatory movement of all particles (small elements) 

of the fluid. Turbulence imposed on a laminar flow is eventually 

damped out by the viscous shear forces. Because viscous forces are 

large relative to inertial reactions in laminar motion, this condition 

corresponds to low values of the Reynolds number.

Turbulent motion is characterized by an erratic, small-scale eddying 

motion which is superposed on the translatory movement of the fluid 

mass. Local disturbances are rapidly dispersed throughout a turbulent fluid flow because the viscous shear forces are small relative to 

the inertial reactions resulting from the turbulence. Thus, turbulent 

flow corresponds to high values of the Reynolds number. For most 

practical purposes, motion which is turbulent is described in terms of 

the 'average translatory velocity at a point or in a cross section.

Critical, tranquil, and rapid motion. It the mean total energy head 

referred to the bottom of an open channel is a minimum for a given 

discharge, the flow is described as critical. When critical flow occurs 

in a straight, rectangular, or nearly rectangular channel in which the 

velocity in the cross section is essentially constant, the average velocity 

is theoretically equal to the velocity of a small gravity wave, and the 

Froude number is equal to unity. The flow is described as tranquil if 

the average velocity is less than the critical value. If the average 

velocity is greater than the critical value, the flow is described as 

rapid.

Shear and pressure. The total force on any surface within a fluid 

in motion can be resolved into tangential and normal components. 

The tangential component of the total force on the surface is the shear 

force (FT). The limit of the ratio of the shear force to the area of the 

surface as the area approaches zero is the shear stress at a point (T).

The normal component of the total force on an area is the pressure 

force (Fp). The limit of the ratio of the pressure force to the area 

as the area approaches zero is the pressure intensity (p) at a point. 

Pressure intensity is a scalar quantity.

A fluid at rest with respect to its boundaries is in a state of zero 

shear stress throughout. It follows that pressure changes imposed 

at one point in a fluid mass at rest are transmitted undiminished to 

all points in the fluid mass (Pascal's principle). Furthermore, because 

shear stresses cannot exist in a fluid at rest, the resultant force on any 

surface within the fluid is a pressure force, and it is normal to the 

surface.

Viscous and turbulent shear. The viscous-shear force (FJ is a consequence of internal stresses which resist fluid deformation.
SELECTED TOPICS OP FLUID MECHANICS 9

relative movement of the fluid on adjacent stream surfaces in a laminar 

flow results in a viscous-shear stress (TM) . In a motion of this kind, TV is 

a function of the fluid viscosity and the velocity "gradient", a property of the flow defined as the rate of change of the tangential velocity 

with respect to distance perpendicular to the motion. The viscousshear stress at any point in the flow pattern, therefore, is TM = 

fj,(dvs/dn), where ju is the dynamic viscosity of the fluid and dvjdn is the 

velocity gradient at the point. It is significant that TM is independent 

of fluid density and pressure.

The total shear force (FT) acting on a fluid in turbulent motion is 

the sum of the viscous shear (F^ and the turbulent shear (FJ. Turbulent shear is a consequence of the momentum exchange between 

small masses of fluid moved transversely by the action of the turbulence eddies. Unlike viscous-shear stress, turbulent-shear stress (T,) 

is a function of fluid density, and it is independent of fluid viscosity.

It can be shown that the turbulent-shear force acting on a fluid 

element is directly proportional to its inertial reaction. It follows 

that the Reynolds number is proportional to the ratio of turbulent 

shear to viscous shear, or R oc T,/TM . The influence of viscous shear is 

negligible in fully developed turbulence.

Boundary layer. The boundary layer was originally conceived as a 

means of correlating real fluid motion with the theoretical behavior 

of the hydrodynamicist's ideal fluid. Thus, Prandtl suggested that 

in flows at high Reynolds numbers the effect of fluid viscosity could 

be assumed to be limited to a narrow region near the confining boundaries. This concept is of fundamental importance in the analysis of 

flow around bodies immersed in relatively large fluid spaces. It is 

also essential to an understanding of certain phenomena associated 

with the establishment of uniform flow at entrances to conduits or 

following local disturbances in an otherwise uniform conduit. In 

regions of fully established flow, as in long pipes and open channels, 

the boundary layer is virtually the entire cross section of the flow.

Streamlines and streamtubes. A streamline is a line drawn through 

a flow in such a manner that at every point along its length it is tangent 

to the velocity vector. It follows that there can be no flow across a 

streamline; in this sense, the streamline imposes a boundary condition 

on the adjacent flow.

A streamtube is a surface formed entirely of streamlines; in other 

words, it is the surface defined by the streamlines passing through a 

closed curve. As the boundary of a streamtube is comprised of 

streamlines, there can be no velocity and, therefore, no flow through 

the walls of a streamtube. Any solid boundary or separation surface 

which effectively guides the flow has the properties of a streamtube 

surface.
10 RIVER HYDRAULICS

Stagnation and separation. A stagnation point in fluid motion is 

defined as a point of zero velocity. When boundary conditions cause 

a flow to undergo an abrupt convergence, a tendency for stagnation 

occurs at a point on the boundary. In a sharp, concave boundary 

corner, or on the nose of a blunt obstruction, for example, the flow 

along the boundary would apparently experience an instantaneous 

change in direction of motion. As this implies an infinite normal 

acceleration if the tangential velocity were anything but zero, stagnation occurs.

When flow boundaries diverge abruptly, as in sudden enlargements 

or following blunt obstructions, the boundary ceases to guide the 

flow effectively and separation occurs. Thus, the apparent requirement that the fluid undergo an infinite acceleration at the boundary 

discontinuity does not, in this example, result in stagnation. Instead, 

the live stream separates from the boundary and follows a "freestream" surface bounded by eddying fluid. As a consequence of 

viscosity, such eddy-filled separation zones also occur in corners near 

regions of boundary convergence, thereby preventing stagnation.

EQUATIONS OF MOTION

FUNDAMENTAL EQUATION OF MOTION

The principal objective of fluid mechanics is a mathematical 

description of fluid motion. Newton's second law is the basis for 

equations of fluid motion. The sense of Newton's proposition is 

contained in the algebraic expression l 2FccMa=KMa. This is 

a vector relationship which states that the resultant of all external 

forces acting in a particular direction on a mass is proportional to 

the product of the mass and its acceleration in the direction of the 

resultant force. If the quantities involved in this expression are 

evaluated in accordance with an absolute system of units, then K 1.0, 

or 'LF Ma. In the Cartesian space system, for example, 2Fx=Max, 

and 2Fg

EXTERNAL FORCES IN FLUID MOTION

The independent forces which influence fluid motion are the consequence of 1 property of state (pressure) and 4 physical properties 

of the fluid (density, viscosity, surface tension, and elasticity). An 

additional force, the apparent shear force due to fluid turbulence 

(-F7,), is not recognized as an independent force because it is directly 

proportional to the inertial characteristics of the flow.

Pressure, or pressure intensity, is a scalar quantity which describes

i Except where emphasis Is Intended, the symbols and technical terms defined In the preceding section 

are used in the subsequent text without further explanation.
SELECTED TOPICS OF FLUID MECHANICS 11

a particular kind of fluid stress. Pressure is the cause of a resultant 

external force (Fp) only when it varies in the direction of motion.

Density is involved in fluid motion in two distinctly different ways. 

Thus, density accounts for an accelerative force (Fg) which is the 

component of the weight (W) in the direction considered. The definitive equation, W=Mg=pVg, is a particular form of Newton's second 

law in which p is the density and g is described as the acceleration due 

to gravity. On the other hand, the total inertial reaction due to the 

resultant of all the external forces, which might or might not include 

a component of the weight force, also involves the density; that is,

Viscosity is a fluid property which accounts for a shear force 

whenever fluids are in motion with respect to their boundaries. 

Viscous shear is the result of molecular forces which oppose fluid 

deformation. In turbulent fluid motion the total shear force (FT) 

is the sum of F? and the turbulent shear force (F,) .

Surface tension is the name given to certain manifestations of 

molecular forces which occur at the interface between different 

substances; for example, between a free jet of water and the surrounding air. The surface-tension force (Fa) is ordinarily of appreciable 

magnitude only when the other forces involved are extremely and 

unusually small.

Elasticity, or forces due to elastic deformation (Ft), are involved 

primarily in the unsteady flow of liquids and in the flow of gases under 

high pressure gradients. Neither of these problems is pertinent to 

this discussion.

KINEMATICS OF FLUID MOTION

A mathematical description of fluid motion necessarily involves the 

space-time characteristics which are known as velocity and acceleration.

Velocity is the time rate of displacement of a mass or a point. For 

example, if As is a small displacement or distance traversed along a 

streamline in a short time interval (M) , and if vs is the average tangential velocity in the interval on that part of the streamline, then 

vs=As/M. It follows that the instantaneous tangential velocity at 

a point on a streamline is vs=^dsjdt.

Velocity is a vector quantity. Thus, the total or resultant velocity 

at a point is the vector sum of its components on the various vector 

axes. From the definition of a streamline, the total velocity in the 

"natural" or s, n, m system of coordinates is vt, and the normal components, vn and vm, are zero. For many purposes, however, the 

Cartesian or x, y, z system of describing motion in three-dimensional 

space is preferred. Thus, in the Cartesian system v=vx -+*
12 RIVER HYDRAULICS

VV+V+^j where -f-> indicates vector addition and vx=dx/dt, 

vv=dy/dt, and vt=dz/dt.

Acceleration is time rate of change of velocity. As this discussion 

is concerned with steady motion only, the variation of the velocity 

at a point, with respect to time, is always zero. Thus, in the general 

definition a=dv/dt, the change in velocity represented by dv is the 

change which occurs over the^ distance traversed during the time 

interval dt. It follows that the total acceleration in steady motion 

is described as convective acceleration.

In the natural system of coordinates, the total acceleration includes 

both tangential and normal components. Thus, as dvs/.dt is the tangential acceleration. Similarly, an=dvn/dt and am dvm/dt are the orthogonal components of the normal acceleration. In the Cartesian 

system the total convective acceleration is comprised of the components ax dvx/dt, av=dvv/dt, and az dvg/dt.

For subsequent use, the tangential and normal accelerations in 

two-dimensional motion (am =0) are more conveniently expressed

dvt ds dvt ds dvs a,=-j -j- , ~ji Vt -j-)

at ds rig at as

or, from the calculus,

a" ==Vs ~ds == 2~ds" (1)

From any mechanics or physics text, the normal (centripetal) 

acceleration at a point on a streamline is

an= dt = Gfe" Jf*Vt ~ds =~r~' (2)

in which r is the radius of curvature of the streamline at the point. 

It is significant that equation 1 describes at as a measure of the rate 

of change of the magnitude of vs, whereas equation 2 describes an as 

a measure of the change in direction of vt. Equation 2 also explains 

the anomalous situation which appears to exist when it is observed 

that an is not necessarily equal to zero even though vn is always zero 

on a streamline.

EQUATIONS FOB A SIMPLE FLUID MOTION

The next logical step in writing an equation of fluid motion is to 

evaluate all of the forces acting on a typical accelerated fluid mass 

and to write an equation of the form

(3)
SELECTED TOPICS OF FLUID MECHANICS 13

in each of the coordinate directions. First, however, in view of the 

purpose of this discussion, the problem can be simplified by omitting 

further consideration of the relatively minor surface-tension and 

elastic forces, F, and Ff, respectively. Furthermore, in order to demonstrate certain basic principles without the complication of the most 

troublesome of all forces those due to viscosity and turbulence 

a kind of motion in which the influence of F, is negligible is initially 

selected for analysis.

The flow of a fluid from a large quiescent reservoir through a small orifice is shown in plate 1 (a-c). It is assumed that the orifice discharges 

into another reservoir containing a fluid of the same density in order 

that certain independent influences of fluid weight can also be ignored. 

Sufficient, generality and considerable convenience is achieved in this 

example by assuming that the flow is two-dimensional in the s, n plane. 

Thus, the only accelerations involved are aa and an, and the only 

forces involved are those which have components acting parallel and 

perpendicular to the streamlines. The natural system of coordinates 

is ideally suited to this analysis.

Plate 1 (b) shows a typical small mass of the fluid in motion. For 

convenience it is shown as a prismatic element of volume (As AA). 

Its axis is a streamline and it is as small as necessary in order that its 

motion can be described in terms of the motion of its mass center. In 

plate 1 (c), showing the same element as a free body, the pressure force 

on each end of the element is represented by the product of the average pressure and the end area. The weight force in the direction of 

motion (Fg) is shown as the s component of the total weight of the 

element (W). It should be apparent that any errors resulting from 

neglecting area, velocity, or pressure variations over the small volume 

(As &A) are of a second order of magnitude and that they disappear 

when the volume of the element is allowed to approach the infinitesimal (ds dA).

In the kind of motion represented by plate 1 (a-c) the influence of the 

total shear force (FT) is negligible. This follows from the fact that 

motion in the reservoir is assumed to be nonturbulent and that the 

velocity of the fluid relative to the boundaries is negligible except 

immediately adjacent to the opening. Thus, neither viscous- nor 

turbulent-shear forces are appreciable. Flow patterns of this kind 

are essentially similar to the theoretical flow patterns described in 

hydrodynamics for an ideal fluid. Thus, for the motion pattern 

described,

2F.=F,-}-*Fg=Ma., (4)

in which Fp and Fg are the s components of the forces due to pressure 

and weight, respectively.
14 RIVER HYDRAULICS

From plate 1 (c), the resultant pressure force is

Fp=p AA-(p+Ap)AA= Ap AA.

If it is assumed that the pressure intensity increases at a uniform rate 

(dp/ds) in the direction of motion, then Ap= (dp/ds) As is the total 

change in pressure over the distance As. Substituting and simplifying,

F^-^CAsAA). (5)

The weight of the element is y(AsAA), The s component of the 

weight is, from plate 1 (c),

dz Fg= -y(As A A) sin 6= -y(As AA) ^- (6)

CLS

The mass of the element is p(As AA), and the inertial reaction due 

to Fp and Fg is *

Mas=p(As AA)as. (7)

Combining equations 5, 6, and 7 with equation 4,

-^ (As AA)-y ^ (As AA)=pas(As AA). (8) 

as as

'Reducing this to a force-per-unit-volume relationship by dividing by 

the volume (As AA) of the element,

dp dz -

This equation shows that acceleration in the direction of motion results only when there is a gradient (a change with respect to distance) 

of the sum p-\-yz in that direction. It shows also that when dz/ds Q, 

which is true of any motion in a horizontal plane, the pressure gradient 

( dp/ds) is the sole accelerative force per unit volume, and it is 

exactly equal to the inertial reaction per unit volume (pas). It indicates furthermore that when dp/ds 0, then y(dz/ds)=pas, or 

 dz/ds=as/g, equations which, for example, describe flow along any 

free (constant-pressure) surface; thus, when ds= dz, as in free fall, 

a,s=g, or the total .acceleration is equal to the acceleration due to 

gravity. Finally, it indicates that when as=0, the pressure gradient 

is dp/ds= y(dz/ds), or, in the distance ds, dp= ydz; this describes 

the condition of hydrostatic pressure distribution.
SELECTED TOPICS OF FLUID MECHANICS 15

* T 2

If as=7j f- from equation 1 is substituted in equation 9 and the 

a ds

terms rearranged,

From this equation, the s rate of change of the quantity within parentheses, at any point on the streamline, is zero.

If equation 10 is multiplied by the differential distance (ds),

Q, (11)

which shows that the differential change in the parenthetical quantity 

over the differential distance (ds) is also zero. It follows that the 

quantity within the parentheses is a constant along the streamline. 

Thus,

iH-rz+^f-W, (12)

which is the integral of equation 11, is a relationship between pressure, 

elevation, and velocity in which K is the constant of integration for a 

particular streamline. At successive points along this line, therefore,

av* pv£

^+A+rBlSSI__J+ft+7gl| etc. (13)

Equation 13 can be written in an alternate "head" form, known 

more familiarly to civil-hydraulic engineers as the Bernoulli equation. 

Dividing by the constant specific weight of the liquid (7) and noting 

that yp=

The algebraic operations which led from the Newtonian equation 

to equation 13 involved dividing a force by a volume and multiplying 

by a distance. Thus, the units of every term in the equation are 

foot-pounds per cubic foot. It follows that the units of every term 

in equation 14 are foot-pounds per pound or, simply, feet.

Equations 13 and 14 are both properly described as Bernoulli 

equations. They are relationships between integral values of pressure, elevation, and velocity at successive points on a single streamline. In order to develop similar relationships between points on 

adjacent streamlines it is necessary that the foregoing analysis be 

repeated with the axis of the definitive fluid element oriented in the
16 RIVER HYDRAULICS

n direction. As a result, a differential equation is obtained which is 

the counterpart of equation 9:

However, when the equivalent expression for an in terms of the 

velocity (vs) is substituted, the similarity in the two developments 

ceases. Thus, an=vs2/r from equation 2. When this is substituted 

in equation 15, it appears to yield a nonintegrable equation. Returning to equation 2, however, let an=vs (dvnfds). Adding and subtracting the quantity vs (dvs/dn) ,

dvn . dvs dvg dvs . /dvn dv

From hydrodynamics, the quantity within the parentheses in the 

last term is a measure of rotationality. Irrotational flow corresponds 

to the condition that (dvjds) (dvs/dri)=Q. Whereas this distinction 

between rotational and irrotational flow is essentially mathematical, 

it is significant that hydrodynamics also shows that the absence of 

viscous shear is corequisite to irrotationality. Thus, because shear 

forces are negligible here, it can be assumed that the flow is irrotational, and, therefore,

dvs 1 dvs2 , . an=vs -7-=o -7 (16) dn 2 dn ^

When this is substituted in equation 15, the modified equation 

leads to differential relationships which are the counterparts of equations 10 and 11 and integral relationships which are identical with the 

Bernoulli equations (12, 13, and 14). Thus, when the shear forces are 

negligible, the same relationships between velocity, pressure, and 

elevation will apply between adjacent streamlines as apply along 

single streamlines. In other words, the constant of integration in 

equation* 12 is the same for all streamlines and the Bernoulli equation 

is applicable to all points in an irrotational or frictionless flow pattern.

Several important applications of the equations of motion, including 

hydrostatics, piezometry, the one-dimensional equation, and the discharge equation, are discussed in the following sections.

HYDROSTATICS

When only pressure and weight forces are effective in a fluid motion, 

as in the preceding example, the condition described as hydrostatic 

pressure distribution occurs along any line on which the acceleration 

at all points is zero. Thus, with as=0 in equation 9, Irydrostatie
SELECTED TOPICS OP FLUID MECHANICS 17

pressure distribution in the s direction implies that the pressure in an 

incompressible fluid (7 is constant) varies inversely with the elevation along a streamline. Similarly, from equation 15, pressure distribution is hydrostatic along lines normal to the streamlines when 

an=0. This also means, of course, that pressure is not hydrostatically 

distributed in the n direction if the streamlines are curved. It follows 

that hydrostatic pressure distribution occurs throughout a fluid in 

motion only if shear forces are negligible and the flow is truly uniform; 

that is, if a,=0 and a»=0.

However, when the shear force (FT] cannot be ignored, it should be 

apparent from plate 1 (c) that the pressure must vary with the influence 

of this force as well as with elevation along the streamline. Thus, in 

general, pressures are hydrostatically distributed in the s direction of 

a uniform motion only when the relative velocity of fluid and boundary 

is so small that the shear force is negligible. On the other hand, 

pressures are hydrostatically distributed in the n direction as long as 

an=0. This follows from the fact that, from the definition of a 

streamline, the average normal velocity (w») is zero and, as a consequence, the net shear force in the n direction is always zero.

Hydrostatics, as the term is generally used in technical literature, 

refers to the behavior of fluids, particularly liquids, at rest. The 

equations derived for a simple fluid motion are also applicable to hydrostatics. For this application, as there is no flow and consequently 

there are no streamlines, the s direction is arbitrarily defined as any 

direction. From equation 9, when the fluid is at rest the acceleration 

in all directions is zero, and the equation

defines the condition of hydrostatic pressure distribution throughout 

the fluid. Thus, along any line, in a distance ds,

dp= ydz. (18)

In horizontal planes, dz=Q, whence, from equation 18, dp=0. 

Therefore, one of the basic principles of hydrostatics is that the 

pressure is the same at all points on horizontal planes in a fluid at rest.

When 7 is constant, which is sensibly true for liquids, integration of 

equation 18 yields

Za)' (19)

Equation 19 is apparently a particular form of the Bernoulli equation 

(13). It is also the basic gage-pressure equation if conditions at point 1 

are defined as the conditions on a liquid surface open to the atmos-
18 RIVER HYDRAULICS

phere. Thus, if pi=0 (gage) and if 21 22=2/2 is the vertical depth of 

point 2 below the free surface, then #2 =7j/2- It follows that the gage 

pressure at any point in a continuous body of liquid at rest is

p=yy, (20)

in which y is the vertical depth of the point below the real or imaginary 

free surface of the liquid.

PIEZOMETBY

A piezometer is a small hole located in a surface which comprises a 

fluid boundary. The purpose of the piezometer is to provide a means 

of measuring the fluid pressure at the point where the hole is located. 

In its definitive application to a liquid, as shown in plate 1 (d), the piezometer is connected to an open tube called a manometer. When the 

pressure in the liquid at the piezometer is exactly equal to the pressure due to the weight of the liquid in the manometer column, the 

liquid in the manometer is at rest. Thus, the pressure at the point 

of attachment can be computed from the gage-pressure relationship 

(eq 20). When a piezometer is used to measure the pressure on the 

boundary of a flowing fluid, the piezometer orifice must be small and 

very carefully made in order that flow along the boundary is not 

disturbed by the presence of the piezometer.

The level of the liquid in a simple open-tube manometer is appropriately called the piezometric level. As its height above the level of 

the piezometer (y) is equal to pfy from equation 20, this distance is 

described as the pressure head at the point of attachment. The 

elevation of the point with respect to a horizontal datum is called its 

elevation head. Thus, the vertical distance between the piezometric 

level in the manometer and the elevation datum is described as the 

piezometric head at the point where the piezometer is located. From 

this definition, if the symbol h is used to denote the piezometric 

head at any point,

A=£+z. (21) 

Thus, from equation 15 (divided by 7),

~j~ r n 

dn\y / dn 7 g

This equation shows that the piezometric-head distribution in 

a section normal to the flow is proportional to the normal acceleration. 

The hydrostatic condition in the n direction occurs when an=0. For 

this condition the gradient of the piezometric head is zero; that is, 

the piezometric head is constant. Thus, when flow past a piezometer
SELECTED TOPICS OF FLUID MECHANICS 19

in a flow boundary is truly uniform, the piezometer indicates not only 

the piezometric head at the point of attachment but also the piezometric head at all points in the adjacent uniform flow. Under these 

circumstances, the piezometric level is independent of the elevation 

of the piezometer. These conclusions are illustrated in plate 1 (d).

When a liquid flows in an open channel, as in plate 1 (f), the upper 

boundary of the flow is both a stream surface and a piezometric surface. The gage pressure over the upper stream surface, regardless of 

its configuration or velocity, is zero. Thus, a piezometer located on 

the boundary of a uniform flow in an open channel indicates not only 

the piezometric level but also the level of the flowing liquid at the 

point of attachment. It follows that, if the piezometric datum is 

defined as the bottom of the channel, a piezometer located anywhere in the cross section indicates the depth of the liquid in the 

channel at, that section.

When a piezometer is located on the upstream end of a small object 

placed in a fluid flow, stagnation occurs over the piezometer. As 

viscosity has little influence on this occurrence, the Bernoulli equation 

is applicable between the stagnation point and any point on the same 

streamline in the undisturbed flow upstream from the obstruction. 

Two such points (on streamline AB) in a uniform pipe flow are designated points 1 and 2 in plate 1 (e). If 02=0, and h= (p/y) -\-z is substituted in equation 13, the piezometric head at the stagnation point 

is

A2=Ai+^- (23)

In plate 1 (e), the piezometer at point 2 is connected to an open 

manometer tube. The stagnation pressure on the nose of the tube 

causes liquid in the tube to rise to a height equal to p2/y above the 

level of the piezometer or h2 above the elevation datum. If the flow 

in the pipe is uniform at this section, a second piezometer located 

at any point on the boundary indicates the piezometric head in the 

undisturbed flow at point 1. Thus, from equation 23, the velocity 

head at point 1 is indicated as the difference in the piezometric levels 

in the two manometers. It follows that the velocity in the undisturbed flow on streamline AB is

). (24)

A tube such as that in plate 1 (e) which indicates the stagnationpressure head is called a stagnation or pitot tube. It is apparent that 

the pitot tube provides an accurate means of measuring velocities in 

fluid motion. Its application to open-channel flows is illustrated in 

plate 1 (g).
20 RIVER HYDRAULICS

The velocity in a normal section through a uniform fluid flow 

between fixed boundaries varies with distance from the boundary. 

This is incidentally a consequence of the shear stresses which result 

from the relative motion of a viscous fluid and a solid boundary. 

When a pitot tube is used to traverse such a section, the piezometric 

level indicated by the tube varies from one streamline to the next. 

However, the piezometric level indicated by all wall piezometers is 

the same as long as an 0 at the section. Thus, only the pitot-tube 

manometer indications are affected by the velocity variations in 

the section. These conclusions are illustrated for an open channel in 

plate 1 (h).

The equations of motion have been used to demonstrate some of the 

basic principles of piezometry in uniform fluid motion. These equations are equally valuable as aids in the interpretation of piezometric 

measurements in nonuniform flows. For example, piezometers are 

often located at various points on bridge piers or abutments as a 

means of measuring the average water-surface level hi the bridge 

opening. If a piezometer is located downstream from the disturbed 

flow near the entrance to the constricted waterway, and if the flow 

across the entire section normal to the piezometer station is essentially parallel to the face of the pier or abutment on which the piezometer is located, the measurement will be quite satisfactory as an 

indication of the average water-surface level at this section.

Piezometers located near the upstream end of the structure, however, are generally unsatisfactory. Thus, piezometers located on 

the upstream face of a pier are ordinarily in the vicinity of a stagnation zone, and measurements are influenced to an indeterminate 

degree by separation upstream from the pier nose. The so-called 

superelevation of the water curving around the nose of a pier is 

evidence of nonhydrostatic pressure distribution due to normal 

acceleration in the horizontal plane. Similarly, vertical accelerations 

in this zone account for nonhydrostatic pressure distributions in 

vertical planes. Therefore, piezometers located on the walls of a 

pier just downstream from the nose indicate neither the mean watersurface level in the cross section nor the level adjacent to the pier.

It should be apparent that short piers, circular piers, piers with 

irregular boundaries, webbed pile bents, or piers alined obliquely 

with the flow are unsatisfactory as locations for stage-recorder gage 

intakes unless they are calibrated in the field or hi the laboratory.

ONE-DIMENSIONAL EQUATION

The piezometric head has been shown to be constant in any normal 

section over which an is everywhere zero (eq 22). From equation 

16, which applies only when the shear forces are negligible, the normal
SELECTED TOPICS OF FLUID MECHANICS 21

acceleration is proportional to the n gradient of the tangential velocity. 

Thus, if a«=0, it follows that dv,,fdn~Q, or vs v is a constant. It may 

be concluded, therefore, that at all points over a normal section both 

the tangential velocity and the piezometric head are constant when 

FT and an are negligible. Two sections which satisfy these conditions 

are shown in plate 1 (i) .

Plate 1 (i), like plate 1 (a), represents the flow of a fluid from a large 

reservoir through a small orifice into another reservoir. Section 1 

in the figure is a hemispherical surface which is concentric with the 

center of the orifice. Streamlines passing through this surface are 

radial lines ; therefore they satisfy the requirement that the streamlines be without curvature if an be zero. As the influence of the 

shear forces is negligible in this kind of flow pattern, it is concluded 

that the piezometric head (hi) and the tangential velocity (PI) are 

constant at all points on section 1. Section 2 is a section through the 

parallel boundaries of the jet, downstream from the nonuniform flow 

in the immediate vicinity of the orifice. As the streamlines in this 

section are straight, an=0 and the velocity and piezometric head are 

constant at all points. It follows that every term in an equation 

of the Bernoulli form

applies to any and all streamlines between sections 1 and 2 in plate 1 (i). 

For the circumstances illustrated by plate 1 (i), it has been determined that the velocity at all points in section 1 is equal to v\. This 

velocity, therefore, can be represented by the symbol V\, for V has 

been defined as the average velocity in a total cross section. Similarly, 

here, 02=V2 . When these substitutions are made in equation 25, the 

resulting equation,

(26)

can be described as an equation of motion for a typical streamline or 

an equation of motion for a streamtube which envelops the total 

flow.

Plate 1 (j) shows an orifice flow pattern which differs in some very 

important respects from the flow pattern shown in plate 1 (i). In 

plate 1 (j) a diaphragm orifice is located in a pipe. Section 2 is again 

located in the parallel jet. Thus, as in the previous example, the 

velocity and the piezometric head at all points in section 2 are essentially constant. Section 1 is normal to the axis of the pipe and 

far enough upstream from the orifice to be removed from the zone 

of streamline curvature. Thus, as the streamlines are all straight
22 RIVER HYDRAULICS

and perpendicular to section 1, the normal acceleration is everywhere 

zero, and the piezometric head is constant. However, the influence 

of the shear forces on the velocity distribution in section 1 can no 

longer be ignored. As the velocity of a viscous fluid relative to its 

fixed boundaries becomes appreciable, shear forces cause the fluid 

near the boundary to be retarded. As illustrated in plate 1 (j), therefore, the velocities in section 1 vary with distance from the boundary.

Another effect due to the influence of shear is shown in plate 1 (j). 

As there can be no slippage at the boundaries, the velocity of the 

contact fluid is always zero. On the other hand, flow approaching 

the upstream corners must decelerate. The apparent conflict between the condition of zero velocity at the boundary and the tendency 

for deceleration along the boundary is resolved by separation and the 

formation of an eddy zone in the corner region. The result is a 

modification of the flow pattern and a depreciation of flow energy. 

These matters obviously cannot be considered in detail at this time 

because the simplified equations of motion were derived on the condition that the influence of shear forces was negligible.

For many practical purposes, it is permissible to ignore certain 

effects of viscous and turbulent shear. For a pattern such as that in 

plate 1 (j), for example, it is generally sufficient to ignore the nonuniformity of tangential velocities across the normal sections and to 

acknowledge that equation 26 is a good approximation. Under such 

circumstances, because it ignores the influence of transverse velocity 

variations, equation 26 is called a one-dimensional equation of motion.

EQUATION OP CONTINUITY

Plate 1 (k) shows a typical streamtube in a steady fluid motion. The 

average tangential velocity at any section is vg , and the area of a 

section normal to the axis of the tube at any station is AAn. The 

product vs &An has already been defined as discharge, or volume rate 

of flow, a quantity which has no counterpart in rigid-body mechanics. 

The dimensions of discharge indicate that it is a measure of the time 

rate of passage of fluid volume through the streamtube. The product 

of discharge and density, therefore, is a measure of the rate of passage 

of fluid mass. By definition, the flow normal to the streamtube surfaces is everywhere zero. Thus, because fluid mass can be neither 

created nor destroyed, in a steady fluid motion the mass rate of flow 

(pvt &An) is constant at successive sections, or, (pvs AAn)i=(pvs A^.^. 

For a liquid flow, furthermore, the density is essentially constant. 

Thus, the volume rate of flow (A<2) at successive sections is also 

constant, or

AQ=(». *An\=(v, A4.),. (27)
SELECTED TOPICS OF FLUID MECHANICS 23

The discharge (Q) through a total cross section between fixed 

boundaries is the summation of the discharges through the component 

subsections, or

(28)

in which V is the average velocity and A is the total area of the section. 

Thus, because the fixed boundaries comprise limiting stream surfaces, 

from equations 27 and 28, the total discharge is constant at successive 

sections, and

(29)

Equation 29 is the one-dimensional equation of continuity.

DISCHARGE EQUATION

Equation 29 can be applied, for example, to sections 1 and 2 hi 

plate 1 (j) . It can also be solved simultaneously with equation 26, 

which is an approximate equation of motion for the same flow pattern. 

The result,

(30)

is a discharge equation hi terms of area and piezometric head. As is 

true of equation 26, equation 30 is applicable only when the influence 

of the shear forces is small.

When the discharge equation is applied to a flow pattern such as 

that shown in plate 1 (j), the area at section 2 is most conveniently 

expressed hi terms of the area (A) of the orifice and the coefficient of 

contraction (<7e ). By definition, and as illustrated in plate 1 (1),

A2=CcA. (31)

Thus, from equations 30 and 31, a common form of the discharge equation for a convergent boundary is

Q.= . V2gft.-k) (32)

or

^), (33)
24 RIVER HYDRAULICS

in which C& is the coefficient of discharge. From equations 32 and 33,

PLOW NET

The flow net is a pattern of streamlines and "potential" lines which 

indicates by its configuration the velocity distribution in a fluid flow. 

Flow nets can be constructed only in accordance with the following 

restrictions: The flow to be represented is a steady flow; the flow 

pattern is independent of the influence of fluid viscosity, surface tension, and elasticity; the flow pattern is independent of the influence 

of fluid weight (this does not preclude flow nets for completely enclosed 

flows in vertical planes nor certain kinds of free-surface flows). 

Within the limits of the restrictions above, flow nets can be constructed 

for flows involving free-stream surfaces as well as those which are 

effectively guided by rigid boundaries. A flow net is unique for the 

given boundary conditions, regardless of scale (size) or any assumptions regarding fluid density, velocity, elevation, or pressure.

From hydrodynamics it is known that the equations of the lines 

which comprise the flow net can be determined for many two-dimensional flows and certain axisymmetric three-dimensional flows. From 

hydrodynamics it is also known that the requirement of negligible 

fluid viscosity is corequisite to the mathematical restriction of irrotationality.

Truly irrotational flow, of course, is impossible; however, as already 

observed, there are some kinds of fluid motion in which the influence 

of the complicating fluid properties is negligible. For such flows, 

usually involving rapidly accelerating motion and a completely 

enclosed or submerged flow pattern, the flow net affords a powerful 

means of analysis.

Whereas certain flow nets can be plotted as families of curves whose 

equations are developed by mathematical techniques, many others 

can be determined only by graphic or arithmetic successive-approximation procedures. Electric analogies, laminar-flow analogies, and 

other laboratory techniques facilitate the approximation procedures. 

Flow nets involving free-stream surfaces are more difficult to obtain 

than nets for totally enclosed flows. However, measurement of the 

free-stream profile under carefully controlled laboratory conditions is 

an expedient.

It is apparent that, in general, flow nets are not easily obtained. 

It will be made equally apparent, however, that a flow net once drawn
SELECTED TOPICS OF FLUID MECHANICS 25

for a particular boundary shape affords a universal solution for many 

fundamental flow characteristics. Some typical two-dimensional 

flow nets are shown on plate 2 (a-h) .

One of the definitive characteristics of a two-dimensional flow net 

is that the streamlines are so spaced that the discharge between all 

adjacent streamlines in the net is a constant. Furthermore, from the 

equation of continuity, the discharge past successive sections in each 

of the^two-dimensional streamtubes is also constant. In plate 2 (d), v 

is the average velocity and Aft is the corresponding normal distance 

between any pan1 of streamlines in a typical flow net. Because 

Ag=0 An, and because Ag<,=A<fr=Ag2=A<fe in plate 2 (d), it follows that 

v0 kn0=Vi Afti=02 Aft2, etc., throughout the flow net. Thus, the relative 

velocity at any point in the flow pattern is indicated by the distance 

between streamlines at that point. For example, if v0 and An0 are 

described as reference characteristics in a particular flow net, the 

velocity at any other point is v=&q/&n=(v0 Aft0)/Aft, or

I

_o t

. . .. ) Bill/. V0 Aft v0 Aft! v0 Aft

In regions of flow where acceleration is appreciable, finite values of 

the normal distance between streamlines provide only an approximate 

measure of the mean velocity between the streamlines. Further 

subdivision of the net, however, affords a means of improving the 

accuracy of the measure. The limit of the subdivision procedure 

occurs when &n=dn. Then Aq dq=v dn is the constant infinitesimal 

discharge in each of an infinite number of streamtubes which make up 

the flow net. Actually, the decision regarding the number of streamlines to be drawn that is, the practical limit of subdivision is 

always based on a consideration of the labor involved, the accuracy 

attainable, and the accuracy required.

The potential lines on a flow net are a family of lines which are 

orthogonal (perpendicular) to the streamlines, including the effective 

boundaries. The potential lines are customarily drawn so that they 

are everywhere equ^al in spacing (As) to the spacing of the adjacent 

streamlines (An). Together, the conditions of perpendicularity and 

equivalent spacing of streamlines and potential lines govern the 

graphical, successive-approximations method of constructing a flow 

net.

In regions where the flow is uniform the meshes formed by pairs 

of streamlines and potential lines are squares. In regions of nonuniform flow, the meshes are imperfect squares, although individually 

their medians are equal and their corners form 90° angles. If the net 

is further subdivided, however, all of the component meshes approach 

perfect squares as Aft and As approach dn and ds, respectively, just as
26 RIVER HYDRAULICS

all curved lines approach straight lines as the length of line considered 

approaches zero.

Because in any local zone As is equal to Aw, it follows that As is a 

measure of the mean velocity between adjacent potential lines for the 

same reason that Aw is a measure of the mean velocity between adjacent streamlines. Thus, in a uniform-flow zone, An=As, and

v An0 As0 . /OQv =-7 =-T-^ etc. (36) 

V0 An As

In nonuniform zones, either An or As might be preferable as an approximate measure of the mean velocity, depending on the characteristics 

of the flow pattern and the purpose of the analysis. For example, 

along any streamline, including the boundary, the maximum and 

minimum velocities are best indicated by the potential-line spacing.

Typical velocity-distribution curves, determined from the flow net 

for a sluice gate, are shown on plate 2 (g). Here, velocities along the 

floor of the channel and along the face of the sluice gate are represented as a multiple of the downstream uniform velocity (F2). Thus, 

the velocity-distribution curves shown on plate 2 (g) are plots of th« 

relationship

in which v is the velocity at any point, As. is the distance between potential lines at that point, and As2 is the distance between potential lines 

in the downstream, uniform-flow zone.

It has been established that the velocity variation throughout an 

irrotational-flow pattern is related to the piezometric-head variation. 

It follows that the flow net provides a means of analyzing the piezometric-head distribution throughout any flow pattern for which a net 

can be drawn. From the Bernoulli equation (14), when it is applied 

between a reference point and a general point in any irrotational-flow 

pattern,

, , Vg2 V2

in which h=(p/y)-\-z is the piezometric head. By dividing both sides 

of this equation by the velocity head at the reference point and 

observing that

v_Ang _As0

v0 An As ' 

then ' (37)

v '
SELECTED TOPICS OF FLUID MECHANICS 27.

The left-hand member of equation 37, called the piezometric-head 

ratio, is dimensionless. It contains only one dependent variable (h), 

in addition to the reference values (h0 and v0 ). The terms on the righthand side of the equation indicate that the piezometric-head ratio is a 

function of the flow-net configuration alone. It has been established 

from hydrodynamics that, if one can be drawn, a flow net is unique for 

any given boundary conditions; in other words, there is only one flow 

net for a given boundary form. It follows that the piezometric-head 

ratio is determined for all points in a given flow pattern by the shape 

of its boundaries. The full significance of this conclusion is best illustrated by the observation that both the velocity ratio (eq 36) and the 

piezometric-head ratio are independent of the size of the boundary as 

well as of the fluid density, discharge, elevation, and pressure.

Typical piezometric-head distribution curves, corresponding to the 

velocity curves in plate 2 (g), are shown for the sluice gate in plate 2 (h). 

From equation 37, when it is applied between a general point and a 

point in the uniform flow downstream from the gate,

in which As2 , V22/2g, and hz are constants which, for any scale of 

drawing, fix the scale of the piezometric-head graph. Thus, the only 

variable in this equation is the distance As, which is the scaled distance between potential lines at the point where the piezometric head is 

A. For convenience, in plate 2 (h) the piezometric head is represented as 

a multiple of the gate opening (6); that is, h=Cb, where C is a function 

of a single variable (As).

Because the elevation of any point on the flow pattern shown in 

plate 2 (h) is readily scaled off the drawing, it follows that the pressure 

head at any point along the gate face or the floor is represented as the 

difference between the plotted piezometric head and the elevation. In 

this manner the flow net can be used to evaluate the total fluid-pressure 

force acting on any portion of the boundary.

It is emphasized that the sluice gate shown in plate 2 (f-h) represents 

a fluid motion in which the influence of viscosity is almost negligible. 

Also, it is observed that, in this example, the influence of fluid weight 

is confined to its accounting for a variation of pressure with elevation. 

In other words, this is the kind of flow pattern for which a flow net 

yields useful, practical information. Many such examples could be 

given.2

' Rouse, Hunter (ed.) 1960, Engineering hydraulics: New York, John Wiley <& Sons, Inc. See p. 22, 

33, 47-52, 534-642.
,28 RIVER HTDRAULICS

The coefficient of discharge for convergent-boundary flow patterns 

has been defined by the relationships

Q

C A.

/

The prerequisites for the applicability of this equation are essentially 

the same as those which determine whether the flow pattern can be 

represented by a flow net. In fact, a property of all flow nets that 

the streamline configuration is uniquely determinedly the form of 

the fixed boundaries leads to the conclusion that the second member 

of equation 34 involves only dimensionless characteristics of the flow 

boundary. In other words, Ce and, therefore, Cd, are functions of the 

boundary geometry alone. This conclusion is substantiated by the 

observation that the right-hand member in equation 34 is a particular 

form'of the piezometric-head ratio; that is,

s.

C*= / / . .= ,~> (38) V

ft'l" /»2 ~ET

where V is the average velocity through the gross orifice area and 

hi hz is the difference in piezometric heads between the two uniformflow regions designated sections 1 and 2. Thus, Cd has all the characteristics of any piezometric-head ratio, including those of being 

dependent on the form of the fixed boundaries and independent of 

boundary scale, fluid, density, discharge, elevation, and pressure.

THE EXPERIMENTAL METHOD

A simple kind of motion which involved mainly the forces due to 

pressure gradient and fluid weight was examined in. the jpreceding 

section. The discharge of an incompressible fluid through a plate 

orifice between adjacent large reservoirs was described as a practical 

example of this kind of motion. In general, of course, additional 

forces must be considered. Thus, when a fluid flows between fixed 

boundaries, viscous and turbulent shear stresses are involved. On 

the other hand, when accelerated motion with a free surface occurs, 

certain independent effects due to fluid weight must be considered.

Unfortunately, flow patterns which depend on the combined effects 

of several fluid-property forces are seldom subject to complete mathematical description. Thus, a full complement of expanded equations 

of motion, counterparts of the equations for a simple fluid motion
SELECTED TOPICS OF FLUID MECHANICS 29

derived in the preceding section, is not forthcoming. In this section 

a method of analysis which provides an indirect evaluation of the 

relative influence of the fluid-property forces will be considered.

INFLUENCE OF SHEAR

The effects of viscous and turbulent shear are prominent in most 

fluid-flow phenomena. They are manifested, usually, by their influence on the energy of flow and th.3 external aspects of the flow pattern. 

Thus, as a consequence of viscous shear, heat is generated within a 

moving fluid and the energy of the flow system is thereby degraded. 

On the other hand, she'ar stresses are primarily responsible for the 

occurrence of boundary drag forces, separation zones, eddies, and 

various phenomena associated with the boundary layer.

In the absence of comprehensive equations of motion, practical 

solutions for many problems in engineering hydraulics involve empirical adjustments of the simple equations of motion derived in the 

preceding section. Thus, an understanding of the complex part 

played by the shear forces depends, in general, on an understanding 

of the techniques and limitations of dimensional analysis and experiment.

EXPERIMENTS AND THE THEORY OF SIMILITUDE

The principal purpose of experiment is to determine the relative 

influence of each of the variables which govern the occurrence of a 

phenomenon. For occurrences lacking a general analytical solution, 

however, the identification of the independent variables is often as 

obscure as the evaluation of their influence. Frequently, therefore, 

an auxiliary purpose of experimental observations is to isolate the 

effective variables. Many practical problems have been satisfactorily solved by semi-empirical methods based on experimental data. 

However, it is emphasized that empirical solutions lack generality by 

their failure to recognize the exact functional relationship between 

the primary variables.

The variables involved in fluid motion are contained in one of 

following categories: boundary conditions; kinematic or flow characteristics, including acceleration and velocity (or discharge); and 

dynamic characteristics, or the forces which cause motion. One of 

the most powerful tools of modern fluid mechanics is that which utilizes 

dimensionless ratios to describe boundary geometry, flow characteristics, and force characteristics. The dimensionless-ratio technique 

provides a means of correlating similar phenomena without restriction 

as to scale, flow, or fluid properties. It is fundamental to the planning 

and execution of experimental programs, to the analysis of experimental data, and to the application of solutions which are based on 

experimentation.
30 RIVER HYDRAULICS

The general Newtonian equation of motion for fluids has already 

been expressed in the symbolic form (eq 3), FP+*Fg+*FT -\-*F,-\-*Fe= 

Ma.

Plate 2 (i-1) illustrates the significance of such an equation when applied to the motion of a small element of fluid wherein the effects of 

Fff and Ft are negligible. Thus, in plate 2 (j),

F,+*Fg+>Fr=Ma. (39)

The size of the polygon which represents this force system depends 

on the scale selected for the drawing and the relative magnitude of 

the forces involved. It follows that the force picture at a corresponding point in a completely similar flow pattern might be represented 

to the same scale by plate 2 (k). However, if every term in equation 39 

were divided by the resultant inertial reaction (Ma),

F ++F F 1 ' (4e>

the force systems shown by the similar polygons in plate 2 (j, k) 

might both be represented by the unit polygon shown in plate 2 (1). 

At corresponding points in all truly similar flow patterns such dimensionless force polygons are identical.

The magnitude of each of the ratios in equation 40 is a relative 

measure of the part played by a particular accelerative force. Each 

of the ratios, therefore, has obvious significance as a criterion of 

similarity. For example, the first ratio is a measure of the influence 

of the pressure force relative to the total inertial reaction. It is 

evident that when this and each of the other ratios are fixed, the force 

pattern and, therefore, the motion pattern, are determined. It 

follows that two or more flow patterns are completely similar when 

their boundaries are geometrically similar and the corresponding force 

ratios are identical. Thus, dynamic similarity is independent of the; 

size of the boundaries, the magnitudes of the velocities, or the physical 

properties of the fluids involved.

Equation 40 and plate 2 (1) have been used to demonstrate the significance of dimensionless force ratios as a means of describing a particular 

fluid motion. Equivalent ratios, which are proportional to the reciprocals of the square roots of the ratios in equation 40, are widely 

accepted as general criteria of similarity. The first of these is the 

Euler-number (E),

» £= (41>
SELECTED TOPICS OF FLUID MECHANICS 31

in which V is a typical average velocity and Ap is a significant pressure difference. The Euler number is derived from the pressureforce ratio, the first term in equation 40. Thus, the Euler number 

is a measure of the relative influence of the pressure force.

Similarly, a ratio which is a measure of the influence of the gravity 

force and which is derived from the second term in equation 40 is 

called the Froude number (F). For free-surf aee flows, the accepted 

form of this ratio is

(42)

in which L is a significant length. 

In the third term of equation 40,

FT Fu . Fn fll,r\ 4. 4. TT irr JfTT' WH-Constant, Ma Ma Ma Ma

in which FJMa is a constant because turbulent shear forces are pro 

portional to the total inertia! reaction. It follows that

. (43) Ma

The third term in equation 43 represents the relative influence of the 

independent viscous shear force. The accepted form of the ratio 

which is derived from this term is called the Reynolds number (R) ,

(44)

Similar ratios which have been formulated to describe the surfacetension and elastic forces are called the Weber and Cauchy (or Mach) 

numbers, respectively.

Inasmuch as the Euler, Froude, and Reynolds numbers are proportional to tjie dimensionless ratios in equation 43, it follows that complete similarity of two or more flow patterns will prevail when their 

boundaries are geometrically similar and their Euler, Froude, and 

Reynolds numbers are equal.

Of the ratios in equation 43, the Euler number (E) has special 

significance as a general flow parameter. Because it involves no fluid 

property other than density, E is the sole criterion of similarity for 

motion which, from hydrodynamics, is described as ideal. Thus, for 

an enclosed flow in a horizontal plane with negligible influence due to 

shear, inertial reactions are a function of pressure forces alone. For 

any given boundary conditions, therefore, E, representing a ratio of
32 RIVER HYDRAULICS

a typical inertial reaction to a typical pressure force, is a constant. In 

other words, for ideal-fluid motion, E is a function of boundary conditions alone. On the other hand, if forces other than pressure are 

partly responsible for the total inertial reaction, the magnitude of E 

reflects the relative influence of the other forces. As all the other 

ratios are related to the fluid properties, the Euler number is appropriately selected as a dependent ratio. This leads to the statement 

of a general principle of similitude: Values of E for two or more flow 

patterns are identical, and the patterns, therefore, are exactly similar 

if their boundary conditions are similar and if the corresponding 

values of the fluid-property parameters, such as F and R, are equal. 

A significant example of the Euler number is the coefficient of 

discharge described on pages 24 and 28. Other examples are the 

uniform-flow resistance coefficients, various minor-loss coefficients, 

drag-force coefficients, etc. From equation 38,

£

A V ==E. (45)

/2

i Pa)

To all those conclusions concerning Cd which were derived previously 

from the piezometric-head ratio analogy and the flow-net characteristics, there are now added the limitations imposed by the influence 

of the fluid-property forces. Thus C&, obviously an important flow 

parameter, is now defined as a function of not only the boundary 

geometry but also the several fluid-property parameters derived in 

this section.3

INFLUENCE OF A FREE SURFACE

Certain kinds of free-surface flow patterns, such as the sluice gate 

(pi. 2, f-h) and the simple measuring weir, are free of independent 

weight-force effects. In other words, for those flow patterns which 

are uniquely determined by the fixed boundaries, the only influence 

of fluid weight is to account for a variation of pressure with elevation.

Under other conditions, as illustrated by the brink at the end of a 

sloping channel or the orifice at the end of a pipe, the weight force 

can have an independent influence on the flow pattern. For example, 

when a fluid discharges from an'orifice into a space occupied by 

another fluid of different density, the trajectory of the jet and many 

related characteristics of the flow pattern depend on the velocity of 

approach, the specific weight of the fluid flowing, and the specific 

weight of the surrounding fluid, as well as the boundary geometry.

* Rouse, Hunter, 1946, Elementary mechanics of fluids: New York, John WUey & Sons, Inc. The reader 

Is urged to study the sequence of Ct curves represented by figures 31, 62,86, and 129*
SELECTED TOPICS OF FLUID MECHANICS 33

The effect of the weight force in this example is to deflect the jet 

upward or downward, depending on its effective weight or buoyancy 

relative to the. surrounding fluid. The relative magnitude of this 

effect depends on the ratio of the inertia of the jet to the weight force. 

Such a ratio, of course, is the Froude number. However, for an 

application of this kind, F must be written in the more general form,

(46)

in which AY is the difference in specific weights of the two fluids involved and p is the density of the fluid flowing.

Practical problems related to free-surface flows are seldom subject 

to complete analytical solution. By means of the Froude criterion, 

however, along with the other dimensionless force ratios described 

previously, satisfactory solutions based on experiment can be 

achieved.

RECAPITULATION

A preceding section was principally concerned with a simple kind 

of fluid motion in which the only effective forces were those due topressure difference and weight. The main objective of that section 

was to derive equations which constitute a mathematical description 

of the motion. The most important equations derived were the 

Bernoulli equation, which defines the relationship between velocities, 

pressures, and elevations throughout the flow pattern; the onedimensional equation of motion, which describes the variation in the 

average flow characteristics between sections of essentially uniform 

motion; the one-dimensional equation of continuity; and the discharge equation. In addition, the flow net was introduced as a 

valuable analytical tool and a means of demonstrating the significance 

of the equations of motion.

It is now acknowledged that neither the simple equations of motion 

nor the flow net are theoretically compatible with the occurrence of 

shear forces. It is also acknowledged that, in general, it is not possible to derive expanded equations of motion which adequately define 

the influence of shear and certain free-surface flow characteristics due 

to fluid weight. Thus, for many problems involving appreciable effects due to viscosity, weight, or any other fluid property, solutions 

based on field or laboratory experiments must be sought. For this 

purpose the dimensionless-ratio technique of specifying boundary 

characteristics (a length ratio, usually), flow characteristics (a coefficient of discharge, piezometric-head ratio, efficiency, energy-loss coefficient, or some other form of the Euler number), and fluid-property
34 RIVER HYDRAULICS

force characteristics (the Froude, Reynolds, Weber, and Cauchy 

numbers) has been described. Flow-pattern descriptions based on 

dimensionless ratios are completely independent of restrictions regarding scale, discharge, pressure, or fluid properties. Thus, dimensionless ratios are the basis for the general principle of similitude 

which governs all experimental investigations.

EXAMPLE

The manner in which the principle of similitude can be used to guide 

the planning and execution of an experimental program is demonstrated by the following example. It is proposed to make laboratory 

tests on a diaphragm orifice. The purpose of the tests is to determine 

the discharge characteristics of an 8-inch thick-plate metering orifice 

in a 16-inch pipeline carrying crude oil. The laboratory tests are to 

be made in a 4-inch pipe carrying water. The principal objective of 

the investigation is to define the coefficient of discharge (Ca) for a full 

range of oil-pipeline (prototype) flows. This objective is incidental, 

of course, to the requirement that the flow pattern in the pipeline be 

exactly simulated by the flow pattern created in the experimental 

setup.

The scale ratio of lengths in this example is determined by the ratio 

of pipe diameters, 1:4. In order to achieve geometric similarity, 

therefore, the laboratory orifice must be one-fourth as large as the 

pipeline orifice, or 2 inches hi diameter. Similarly, all other length 

characteristics, such as plate thickness, rounding radius, and piezometer locations, must be reproduced in accordance with the 1:4 

length-scale ratio.

In order to obtain completely similar flow patterns in the vicinity 

of the geometrically similar orifices, the relative influence of each of 

the effective independent forces must be the same. In this example, 

viscous shear forces are involved in the occurrence of the boundary 

layer upstream from the orifice and in all the consequent effects described on page 22. Because the flow pattern is entirely enclosed, 

however, the weight force does not have an independent influence. 

Surface-tensiQn forces are absent because there are no fluid interfaces, 

and elastic forces are negligible because both of the fluids involved are 

liquids.

It follows that, for similar boundary conditions, pressure differences 

are functions of viscous shear forces and accelerations. Therefore, 

ratios of the Euler variety, including the coefficient of discharge, are 

functions of the Reynolds number alone. In other words, complete 

similarity of the flow pattern (including turbulence) will be achieved 

and the discharge coefficients for the two orifices will be equal when 

their Reynolds numbers are equal. It is this condition which de-
SELECTED TOPICS OF FLUID MECHANICS 35

termines the range of laboratory test conditions required to define C& 

for a full range of pipeline discharges.

Thus, for similar flow patterns and equal coefficients of discharge,

R«> ( water) = (oU)j

(VLP\ }

\ M /w \ M /o

in which V is usually defined as the average velocity based on the 

gross area of the orifice, and L is the diameter of the orifice. From 

this relationship, the ratio of velocities for similar flow conditions is

vw /LO \/PO \/HW\ ( 7. (47)

and, because discharge is proportional to the product of a velocity 

and an area (or the square of a length) ,

(giB_ A-w V w _ /jLuA '&-~Ajr0 -\j:J p.

Therefore, the discharge of water in the laboratory setup required to 

simulate a given rate of flow in the pipeline can be expressed in terms 

of the length ratio, the density ratio, and the viscosity ratio, or,

« {©(£)©> (48)

In this example, LWIL0=Y^ Furthermore, the density and viscosity 

ratios are virtually fixed because, for given fluids, these properties 

can be altered only slightly (by changes in temperature). Thus, 

equation 48 is of the form Quj=K Q0, and K is essentially constant.

The example would have been considerably complicated if the 

metering orifice had been located at the end of the pipeline. Because 

the orifice would then discharge freely into the atmosphere, fluid 

weight would be added to the independent variables already considered. Thus, the Froude number would be an additional criterion 

governing the performance of the laboratory investigation. Under 

certain conditions this added requirement makes complete similarity 

impossible to achieve.

The difficulty of satisfying more than one fluid-property criterion 

is easily demonstrated. From the requirement that the Froude 

numbers be equal for similar flow patterns,

LA /JLA
36 RIVER HYDRAULICS

or,

whence,

~o- (so)

For similarity with respect to viscous forces as well as the weight 

force, equation 50 must be equivalent to equation 48. Solving thetwo equations simultaneously,

or,

(51)

Equation 51 places a definite restriction on the physical properties 

of the fluids involved in the investigation. In fact, assuming that 

the length ratio and the experimental fluid are fixed within narrow 

limits by practical considerations, equation 51 imposes an impossible 

condition on the investigation. Thus, because of the small range in 

magnitude represented by the viscosities and densities of ordinary 

fluids, it is a conclusion of general significance that it is impossible 

to satisfy simultaneously more than one of the fluid-property criteria.

Fortunately, the apparent difficulty in attempting to satisfy all 

of the similitude criteria is not a hopeless barrier to experimental 

investigations. It is recalled that the Reynolds number, for example,. 

is proportional to the ratio of a typical inertial reaction to a typical 

viscous-shear force. The relative influence of F^ , therefore, diminishes 

as R increases. Experiments show that the effect of viscosity on 

CA for most sharp-edged orifices is negligible at values of R above a 

critical value of 100,000 * (not an unusually high value, as it corresponds, for example, to the discharge of water at 70° F at 6 feet 

per second through a 2-inch orifice). Similarly, the influence of theweight force diminishes as the Froude number increases, becoming 

negligible at a comparatively low critical value of F.6

In the range wherein variations in the magnitude of a given fluidproperty criterion produces negligible changes in the flow pattern,

* Bouse, Hunter, 1946, Elementary mechanics of fluids: New York, John Wiley & Sons, Inc. See fig. 

129, p. 256. 

« Ibid, fig. 52, p. 105.
SELECTED TOPICS OF FLUID MECHANICS 37

that criterion can be ignored as a condition for similitude. Thus, it 

is possible that neither viscous forces nor weight forces have an appreciable effect on the coefficient of discharge within the full range of the 

pipeline flows. If this were true, the coefficient of discharge would be 

a function of boundary conditions alone, and the laboratory program 

to determine Cd for the metering orifice might actually consist of a 

single test. However, that test could not be made with assurance 

until a series of tests had established the lower critical values for both 

F and R. It is apparent, therefore, that the fluid-property parameters 

are useful not only as similitude criteria but also as means of defining 

the effective range of influence of the various fluid-property forces. 

The practical theory of model testing makes use of the fact that a 

single force can have a dominant, although not an exclusive, influence 

on the flow pattern.

THE ENERGY EQUATION

REVIEW OF THE BERNOULLI EQUATION

One of the fundamental concepts of fluid mechanics and the basis 

for many of the practical formulas of engineering hydraulics is the 

energy equation. To bring a detailed consideration of the energy 

concept into this discussion requires first a review of certain steps 

taken in the derivation of the equations for a simple fluid motion.

Plate 1 (c) shows the small element of fluid which, on page 13, was 

considered to be undergoing accelerated motion because of a combination of pressure and weight forces. It is pertinent now to discuss 

certain characteristics of this element which make it possible to 

'associate with it some of the familiar concepts of particle mechanics. 

The elementary volume of fluid shown in plate 1 (c) was assvmed to be 

large enough to be represented by steady-motion characteristics hi a 

turbulent flow. On the other hand, it was assumed to be small 

enough that pressure forces could be accurately expressed in terms of 

a mean pressure intensity, pressure gradients could be assumed constant over the length of the element, and velocity characteristics could 

be described in terms of the velocity at the mass center of the element. 

Thus, as the volume of the element (As &A) was,allowed to approach 

the differential quantity ds dA, the motion characteristics of the 

element were described as the steady-motion characteristics at a point. 

They might also be described as the characteristics of a fluid particle.

Because fluid flow is a continuous process, it is often convenient to 

consider the fluid particle as the instantaneous image of the moving 

fluid contained in a particular space in the flow pattern. Thus, 

equations of motion for a fluid particle can be related to the rate of
38 RIVER HYDRAULICS

flow through the space designated as its location. The volume of the 

element, for example, is

dY =ds dA=v dt dA=dQ dt (52)

in which v is the mean velocity in the differential length (ds) of the 

infinitesimal streamtube on which the element is located. It follows 

that the weight of a fluid particle (dW) can be expressed as

dW=ydY=ydQdt. (53)

An equation for the simple fluid motion represented by plate 1 (a-c) 

is equation 8,

 Y(As AA)=pa,(As AA). (8)

Collecting terms in the left member of this equation, substituting the 

equivalent expression for a» from equation I,6 and replacing As AA 

with the differential volume (ds dA),

dA. (54)

Reviewing the steps taken in obtaining equations 8 and 54 from 

Newton's fundamental equation, it appears that they both represent, 

on the left, the resultant of the external forces acting on the fluid 

element in the s direction, and, on the right, the total inertial reaction 

in the same direction.

Continuing with the particle analogy, if dW is substituted for 

7 ds dA in equation 54, and terms are rearranged,

Equation 55 states simply that the s gradient (rate of change with 

respect to s distance) of the quantity in parentheses is zero. If the 

gradient is zero, it follows that the differential of the quantity in 

parentheses is zero over a small distance (ds), or,

(S6)

The process of determining the differential of the quantity in parentheses consisted of multiplying the gradient by the distance (ds). It

* The subscript on v. in the definition of a, will henceforth be omitted. By definition, vn is equal to zero, 

and v, is therefore equal to tbe total velocity, c.
SELECTED TOPICS OF FLUID MECHANICS 39

is this operation which converts the vector force-inertia relationships 

in the preceding equations into a scalar work-energy relationship 

(eq 56). The conversion might have been more apparent had the 

operation been performed while the differential equation was in the 

form shown by equation 54. In that procedure, multiplying the 

left-hand term, a force, by ds converts it to a measure of the work 

performed by the external forces during the time interval (dt) represented by the average velocity (v) and the distance (ds). Concurrently, the right-hand term, [d(pv*l2)] ds dA~d[dW(i^l2g)], is a measure of the change in kinetic energy of the particle whose weight is dW. 

Equation 56, therefore, is an expression of the familiar work-energy 

principle of mechanics. A convenient interpretation of the concept 

of work-energy equivalence and an examination of the units involved 

in equation 56 lead to the practice of referring to the parenthetical 

quantity as the total energy per pound of the fluid flowing. Thus, 

equation 56 is interpreted to mean that the total energy of the particle 

of weight (dW) remains unchanged over the distance interval (ds). 

It follows that the quantity in parentheses is a constant during the 

interval, or,

(57)

Equation 57 is simply the integral (along the streamline) of equation 56. When it is divided by dW it becomes the familiar Bernoulli 

equation,

/» » n \

-H, (58)

in which H, the constant of integration, is the total energy per pound 

of fluid flowing.

In review, the Bernoulli equation is a description of a simple fluid 

motion along a streamline. In its customary form (eq 58), it states 

that the sum of three quantities, collectively defined as the total 

energy per pound of fluid (H), remains constant during the motion. 

Individually, the quantities which constitute H may be either a 

measure of work performed on the fluid or a change in its mechanical 

energy. Thus, the Bernoulli equation states that the total energy or 

"work potential" of 1 pound of fluid in motion along a streamline 

remains constant regardless of transformations due to changes in 

velocity, pressure, or elevation.

It has already been established (p. 16) that the Bernoulli equation 

could be assumed to apply to an entire flow pattern as long as fluid 

viscosity is ignored. However, the practical objective is a general 

equation of motion which includes terms to account for the influence 

of viscous and turbulent shear.
40 RIVEH HYDRAULICS

GENERAL ENERGY EQUATION

As observed previously the influence of viscous and turbulent shear 

is manifested by its effect on the energy of flow and by its effect on 

the external aspects of the flow pattern. Unfortunately, general 

equations of motion which describe these effects are not yet available. 

Thus, a practicable general energy equation is based on a simple 

modification of the procedure leading to the Bernoulli equation.

The equation of motion for a single streamline (eq 56), modified to 

acknowledge the loss of energy, takes the form

(59)

in which dH is the differential loss in total energy head in the distance 

ds. From equation 53 the term dW in equation 59 is equivalent to 

y dQ dt, in which dQ is the differential flow through the elementary 

streamtube which contains and defines the particle represented by 

dW. Because dQ is constant in a steady flow, equation 59 is readily 

integrated with respect to distance along the streamline. Thus, between any two points designated 1 and 2 on a streamline,

(60)

In order to obtain an equation which describes the total flow 

between enclosing boundaries, it is necessary to integrate equation 

60 over the total cross-sectional areas which contain points 1 and 2. 

The difficulty of performing this operation is made apparent by 

recalling that dW y ds dA yv dt dA. It is evident that integration 

with respect to area must take into account the fact that the velocity, 

in general, is variable over the cross section. Therefore, unless the 

velocity is essentially constant or is analytically defined as a function 

of area, the-indicated integration is impossible. The assumption of 

constant velocity across a section between solid boundaries would be 

inconsistent with the known effects of viscosity. On the other hand, 

explicit analytical relationships between velocity and area are impossible to formulate except for a few classic examples of uniform 

motion.

Not only does variable velocity hinder integration over total cross 

sections, but, as a result of boundary nonuniformities, a nonhydrostatic pressure variation may present a similar obstacle. However, 

if it is assumed that the integration can be performed over a section 

of parallel flow, which is usually possible, the pressure distribution 

can be assumed to be hydrostatic that is, the sum of the quantities 

pjy and z is constant and this obstacle is removed. Thus, if y dQ dt is
SELECTED TOPICS OF FLUID MECHANICS 41

:substituted for dW in equation 60, and if both sides of the equation 

.are divided by dt, integration with respect to the discharge (Q) in the 

total area gives

Incidentally, the foregoing division by dt converts the work-energy 

relationship into a power relationship. Thus, each of the expressions 

in brackets on the left-hand side of equation 61 is a measure of the 

total power of the flowing fluid. The expression on the right-hand 

tside, therefore, is a measure of the power "lost" between the designated

 cross sections.

If equation 61 is divided by the quantity <2y, it again becomes an

-energy equation, and the units of every term in the equation are 

ioot-pounds per pound (or feet) of the fluid flowing. Thus,

Defining the right-hand member of this equation as the , average 

«energy (head) loss (HL) between sections 1 and 2, and substituting

 the piezometric-head symbol (h) for (p/y)-\-z,

-* -*« (62)

JEquation 62 is a general energy equation which is restricted only by 

the assumption of hydrostatic pressure distribution at the designated

 cross sections. The head-loss term (HL) ordinarily requires evaluation on the basis of experiment.

In many practical applications the integration involved in the 

velocity-head terms in equation 62 can be approximated by a numerical summation. Thus,

Ifrom this, the average kinetic energy or velocity head in a total cross 

^section can be defined as the discharge-weighted mean of the velocity 

Jieads in its constituent subsections. Each subsection is taken to 

represent a zone of uniform velocity. This process is conveniently 

adapted to the practice of determining discharge by means of currentaneter or pitot-tube velocity traverses.
42 RIVER HYDRAUUCS

The minor importance of the kinetic-energy terms- in certain applications of the general energy equation does not justify involved 

methods of accounting for the velocity variation in a cross section. 

For those applications it is customary to approximate the true average 

kinetic energy in terms of the average velocity in the section (V). 

The result is a modification of equation 62 which is another form of 

the one-dimensional equation of motion (eq 26),

The error inherent in the velocity-head terms of equation 64 is often 

acknowledged by means of a coefficient (a),

a, of course, is not an independent expression of the influence of 

velocity nonuniformity. It can be evaluated only in terms of the 

integral or summation definitions of the true average velocity head, 

equations 63 and 65. Thus,

«=^ J (66)

Equation 65, better than any other, demonstrates that the main 

objective of this discussion has been accomplished. The result is a 

general energy equation involving separate terms to account for both 

effects of shear a as a measure of its influence on the flow pattern, 

and HL as a measure of its energy-dissipating effect.

THE MOMENTUM 'EQUATION

Numerous problems arise in applied fluid mechanics which are not 

readily or completely defined by the energy equation. The solution 

of many of these problems can be accomplished by means of an independent form of the equation of motion which does not require 

knowledge of the energy losses due to internal viscous shear. This 

is known as the momentum equation. As both the momentum equation and the energy equation are derived from Newton's fundamental 

equation of motion, it is necessary to examine the derivations carefully in order to establish their independence.

A common starting point for the derivation of both the energy 

equation and the momentum equation is the differential equation 

given on page 38,

{54)
SELECTED TOPICS OP FLUID MECHANICS 43

Equation 54 is a statement of Newton's second law as applied to the 

accelerated motion of a fluid element. The left-hand member, for 

the conditions represented by plate 1 (a-c), is the resultant accelerative 

force, and the right-hand side is the total inertial reaction in the 

direction of the total-velocity vector. Equation 54 is a vector equation because force and inertial reaction are vector quantities.

It has been demonstrated that the work-energy significance of the 

Bernoulli equation resulted from the fact that, in its derivation, both 

sides of equation 54 were multiplied by the distance (ds). This operation converted force to work and inertial reaction to change in kinetic 

energy. Both work and energy are scalar quantities. Thus, the 

Bernoulli equation resulted from the integration of a work-energy 

equation over a streamline distance without regard for the change hi 

direction of the s distance.

It is pertinent to recall that the effects of the shear forces were 

ignored hi the derivation of equation 54. Subsequently, however, in 

the derivation of the general energy equation, a term which would 

compensate for the work done by the internal viscous shear was 

included.

In the derivation of the momentum equation, it is convenient to 

replace the left-hand member of equation 54 with one symbol which 

represents the resultant of all external forces. Thus, in the form

<M, (67)

the quantity dFa is assumed to include the boundary shear force as 

well as the boundary pressure force and the weight force. It does not 

include internal shear forces, because these are mutually canceling. 

Apparently, therefore, the independence of the energy and momentum 

equations is related to the fact that the first involves the internal 

viscous shear forces and the second involves the external boundary 

shear force.7

From the calculus, d^/ds=2v(dv/ds'), where dvfds is the gradient 

of the total velocity in the s direction. When this is substituted in 

equation 67,

p rWI j j A /dv\ j JA £ -j- I <k dA=pv [ j- ) ds dA, 2[_dsj \ds/

or,

dF,=pv dv dA=p dQ dv, (68)

in which dQ is the rate of flow in the infinitesimal streamtube which 

contains the fluid element and dv is the change in velocity over the

1 The boundary shear force Is not considered in the derivation of the work-energy equations because the 

relative velocity of the fluid at the boundary is zero, and therefore the work done by the shear force is also
44 RIVER HYDRAULICS

differential distance (ds) which is the length of the element. Equation 

68 is still a force-inertia equation. Therefore, it is still a vector 

equation.

Because motion along a streamline can involve a change in the 

direction as well as the magnitude of the force and velocity terms in 

equation 68, an adequate algebraic description of accelerated motion 

involves similar equations in each of the fixed coordinate directions. 

Thus,

dFx=p dQ dvx, (69a)

dFv=p dQ dvv, (69b) 

and dFs=pdQdvs. (69c)

The quantity p dQ dv in equations 68 and 69 has been described 

as the inertia! reaction of an elementary fluid particle. The product 

p dQ is equivalent to dM/dt, where dM is the mass of the particle. If 

this substitution is made in the x form of equation 69, for example,

P dQdvx=jt (Mvx\ (70)

the right-hand member acquires a new significance. In this form it 

is commonly described as the rate of change of x momentum or the 

change of z-momentum flux in the flow distance ds. This is the origin 

of the name "momentum equation."

For practical applications the momentum equation must be extended to a finite reach and a finite cross section. The operations 

required for this extension involve double integration of the differential equation of motion first with respect to distance along the 

streamline and then with respect to the area of the cross section. 

Similar operations were involved in the derivation of the general energy 

equation. This tune, however, a vector relationship is involved, and 

the integration procedures must be performed for each of the significant coordinate directions. In the following steps the x direction has 

been selected as typical.

Because dQ, in equation 69 is constant, integration with respect to 

distance is readily accomplished. Thus, if the 3-distance integral of 

dFx is represented by &FX,

A^=p dQ vf2 -p dQ vXl . (71)

Because all internal forces are canceling, &FX is simply the resultant 

of the external forces on the finite reach of streamtube between 

sections 1 and 2.
SELECTED TOPICS OF FLUID MECHANICS 45

The next step is to integrate equation 71 over the total areas at 

sections 1 and 2. Substituting dQ=v dA,

r CA* CAi |A*>=pJ vXzv2 dA2 -p\ VsfidAi, (72)

it is apparent that integration depends again (see page 40) on an 

integrable relationship between velocity and area. Thus, a general 

solution of equation 82 is prevented by the fact that explicit analytical 

relationships between v and A are seldom attainable. The left-hand 

member, however, can be simplified to 27^, a symbol which represents 

the resultant of the external forces acting in the x direction on the 

total volume of fluid between sections 1 and 2. Thus, general momentum equations in the fixed coordinate directions are

r

CQ 

Vx2 dQ p I vxi dQj (73a)

S*V=P JX <Z£-P JX dQ, (73b) 

and

2F,=p {Qv,2 dQ-p f X dQ- (73c)

The integrations implied by equation 73 can be approximated by 

the numerical summation

(74)

in which the quantity pvx A<2 represents the momentum flux through 

a constituent subsection assumed to be located in a zone of essentially 

uniform velocity.

Further paralleling the various forms of the general energy equation, a typical one-dimensional form of the momentum equation is

ZF^QpfaV^-QphV^, (75)

in which V is the average velocity and 0 is a coefficient which accounts 

for the nonuniformity of velocities in each cross section. From 

equations 73, 74, and 75,

The energy equation and the momentum equation describe the 

same state of motion, and they are derived from the same differential
46 RIVER HYDRAtfl/ICS

equation. They are independently useful, however, because the 

energy equation is independent of the external boundary shear, and 

the momentum equation is independent of the energy losses due to 

internal viscous shear. Thus, equations 65 and 75 can be shown to 

be equivalent only when the relationship between the boundary shear 

force and the internal, energy-dissipating viscous shear forces is 

explicit. This is true only when the flow is truly uniform (see below). 

Many complex phenomena (such as the hydraulic jump, abrupt 

enlargements, and short transitions, all involving rapidly accelerated 

motion) are readily and accurately described by means of the momentum equation simply because the boundaries involved are short and 

the boundary shear forces can be ignored. The energy equation can 

be applied to these phenomena, too, but it includes a term of primary 

importance which compensates for the work done by the internal 

viscous shear forces. Ordinarily, the energy-loss term is evaluated 

by experimental means. However, because of their peculiar independence in these phenomena, the energy and the momentum equations can be solved simultaneously to yield analytical solutions for 

tne energy loss.

UNIFORM AND GRADUALLT VARIED FLOW 

CHARACTERISTICS OF UNIFORM FLOW

In most of the foregoing discussion, generality was maintained by 

assuming that the flow under consideration was accelerated. The 

specific problem of uniform flow, illustrated in plate 2 (m, n), will be 

examined next,

By definition, the total convective acceleration at all points in a 

uniform flow is zero; in other words, a,=0 and aB=0. It follows 

that the velocity is constant along each streamline. Because velocity 

is a vector quantity, this implies that the streamlines are straight 

lines and, therefore, that the boundaries are prismatic.

Uniformity does not require that the velocities in a transverse 

section be equal. However, it does require that the transverse 

velocity distribution, which is a function of boundary drag and 

internal shear forces, be identical at all successive sections. As a 

corollary to the last requirement, uniformity implies that the rate of 

loss of energy is also constant. It follows that the average velocity 

and the velocity-head coefficient (a) are the same at successive sections. 

Thus, from the one-dimensional energy equation (65),

Jn-h^-M^Ht, (77)

which states that, as a distinctive characteristic of uniform flow, the 

loss of piezometric head (or the negative change of piezometric head) 

is equal to the loss of total energy head.
SELECTED TOPICS OF FLUID MECHANICS 47

Because the velocity pattern is uniform, the momentum coefficient 

(j8) and the momentum flux are also constant. Thus, from equation 

75,

S -0. (78)

Equation 78 states that, in a uniform fluid motion, the resultant of 

the external forces is zero.

Most of the uniform-flow characteristics described above are illustrated in plate 2 (m, n) for both open and enclosed conduits. It is 

apparent that the most distinctive characteristic of uniform flow in 

open channels is the requirement that the axis of the channel be parallel to the energy and piezometric grade lines. This is because the 

piezometric line and the upper, free surface of the flow are coincident.

THE CHEZY EQUATION

The Chezy equation is the basis for many empirical equations for 

uniform flow in both open and enclosed conduits. One of its derivatives, the Manning equation, is widely accepted in the U. S. Geological 

Survey as a practical formula for open channels, ^.s an equation of 

motion, the Chezy equation must be either a form of the energy 

equation (77) or the momentum equation (78). In fact, as will be 

shown, it can be considered to be both.

Most frequently, perhaps, the Chezy equation is derived from the 

one-dimensional momentum equation. Thus, from equation 78, the 

resultant of the external forces acting on a free body of fluid in uniform 

motion is zero, or,

0. (79)

In this equation s denotes the direction of the total velocity vector. 

The effective forces in the s direction are Fv, the force due to boundary 

pressure; Fg, the force due to weight; and Fr, the boundary shear force. 

From page 14 the sum of the pressure and weight forces on an 

element of volume (As &A) can be expressed as a function of the piezometric gradient. Thus,

bs AA,

UO

or, over the finite volume of length L and area A,

(80)

From equations 79 and 80, considering the fact that the boundary 

shear force is negative in the s direction,

(81)
48 RIVER HYDRAULICS

Dimensional analysis, substantiated by experiment, indicates that 

FT can be expressed by the relationship

Fr^CrpPLV2, (82)

in which <7r is a dimensionless drag-force coefficient, a form of the 

Euler number and a function of the Reynolds number and boundary 

geometry, including roughness. P is the perimeter of the wetted cross 

section.

From equations 81 and 82,

-Ah=Cr - LV2=Cr > (83)

in which R=A/P is the hydraulic radius of the wetted cross section. 

If equation 83 is solved for velocity,

(84>

The quantity M/i in this equation is the L gradient of the piezometric head. This quantity is unfortunately described as the hydraulic "slope" ($). For uniform flow in open channels, of course, 

S is also equivalent to the free-surface elevation gradient and the totalenergy head gradient. Thus, if S is substituted for the piezometric 

head gradient and <7cft is substituted for the coefficient radical in theright-hand member of equation 84,

(85>

This is the Chezy equation. 

From equations 84 and 85,

It is apparent that <7cft, unlike Cr, is not a dimensionless coefficient. It 

is, nevertheless, a drag-force coefficient and a function of the Reynoldsnumber and boundary geometry.

ALTERNATE DERIVATION OF THE CHEZY EQUATION

An alternate derivation of the Chezy equation is based on the onedimensional energy equation. In equation 77 the head-loss term 

(Hi) represents the amount of work done by the external forces in. 

overcoming the viscous resistance to fluid deformation. Thus, 

whether the flow be laminar or turbulent, uniform or nonuniform, theenergy equation must contain a term which represents the flow energy
SELECTED TOPICS OF FLUID MECHANICS 49

dissipated by the internal viscous shear forces. It is customary to 

express this loss of total energy as a function of the mean kinetic 

energy of flow, or, in general,

(87)

where CL is a dimensionless energy-loss coefficient which, like Cr, is a 

form of the Euler number and a function of the Reynolds number and 

boundary geometry. Unlike CT , of course, CL is not a drag-force 

coefficient.

From equation 77, as one of the most distinctive characteristics of 

uniform flow, Ah=HL . It follows that, for this condition, the independent momentum and energy relationships expressed by equations 

83 and 87 are equivalent, or,

from which,

From equations 86 and 89, therefore,

from which it is concluded that the Chezy coefficient is proportional 

to either a drag-force coefficient or an energy-loss coefficient. Thus, 

the Chezy equation can be regarded as a form of either the momentum 

equation or the energy equation.

It is pertinept t& observe that 0 is not included in tfae.bonmdorydrag equation (82) and a is not included in the energy-loss equation 

(87). Thus, the influence of velocity distribution is apparently 

ignored in both of the alternate derivations of the Chezy equation. 

This procedure is justified by the observation that ft and a are functions of the Reynolds number and boundary geometry variables 

which also govern the value of CT in equation 82 and CL in equation 87. 

Thus, the Chezy coefficient takes full account of the influence of 

velocity distribution because it depends on the variables which govern 

the normal velocity pattern in uniform flow.

If a velocity nonuniformity is caused by an added variable, such as 

a local boundary irregularity, a and ft will depend on this local condition as well as the normal resistance characteristics of the channel. 

For this condition, of -course, the flow would not be uniform, the
50 RIVER HYDRAULICS

derivations leading to the Chezy equation would be invalid, and the 

question of including a and /8 in the Chezy equation is not pertinent.

GRADUALLY VARIED FLOW IN OPEN CHANNELS

Truly uniform flow seldom occurs in open channels. Thus, most 

practical problems involving both artificial and natural channels are 

concerned with either rapidly varied or gradually varied nonuniform 

motion. In either kind of flow, the nonuniformity may represent 

either an acceleration or a deceleration. Accelerated fluid motion is 

characteristically stable, but decelerated motion, especially if it is 

rapidly varied, is usually accompanied by separation, extreme turbulence, and, therefore, a large energy loss.

The distinction between rapidly varied and gradually varied flow 

is conveniently associated with the relative magnitudes of the quantities HL and FT in the energy and momentum equations, respectively. 

A rapidly varied flow pattern, by definition, occurs in a relatively 

short length of channel. Thus, the boundary shear force (FT) is 

usually negligible, and the momentum equation, free of this troublesome term, provides an adequate solution for many rapidly varied 

flow problems. If the flow is accelerating as well as rapidly varied, 

the energy loss might also be negligible. For" this condition the energy 

equation provides useful solutions. On the other hand, if the flow is 

rapidly varied and decelerating, the energy loss is usually of major 

importance, even when the boundary shear is negligible.

Gradually varied flow, by definition, occupies a comparatively long 

length of channel. Thus, both the boundary shear force and the 

energy loss are significantly related to the flow pattern. The two are 

explicitly related to each other only in the limiting condition of uniform 

flow, as described on page 49.

Between the extremes of rapidly varied flow arid uniform flow, 

gradually varied flow in open channels is a complex function of many 

variables. Neither the boundary-shear mechanism nor the,energyloss mechanism for this kind of flow is fully understood. In fact> 

neither F* nor HL can be satisfactorily evaluated.

In view of the foregoing arguments, it would appear to be ridiculous 

to propose that the Chezy equation, or any other uniform-flow equa^- 

tion, could be used to describe gradually varied flow in open channels. 

Nevertheless, lacking: a .better solution, it is common practice to 

assume that a gradually varied nonuniform flow reach can be represented as the sum of constituent subreaches within which the flow is 

essentially uniform'. It is apparent that such reasoning leads to the 

conclusion that the energy-loss and boundary-shear mechanisms .are 

the same in nonuniform flow as in uniform flow. This must surely be 

incorrect and increasingly so «,s the nonuniformity increases.
SELECTED TOPICS OF FLUID MECHANICS 51

In general, the velocity distribution as well as the average velocity 

is different at successive cross sections in nonuniform flow. Thus, if 

the one-dimensional energy equation is applied to a gradually varied 

flow reach, the velocity-head coefficients at the end sections (04 and 

o2 ) are different. In addition, if an equation like the Chezy equation 

is used to evaluate the total head loss (HL) in a nonuniform reach it 

must be adjusted to account for the effect of channel shape and for the 

effect of using average values of the conveyance characteristics at the 

ends of the reach. In other words, if uniform-flow equations are to 

be used satisfactorily to describe a reach of gradually varied flow they 

must be adjusted to take into consideration the effect of various kinds 

and degrees of nonuniformity. These adjustments are ordinarily 

included in estimates of the Chezy coefficient, and the estimates are 

J>ased on analyses of field and laboratory measurements made in a 

variety of open channels; Needless to say, the procedure is not 

dependable.

Application of the one-dimensional momentum equation (75) to a 

gradually varied flow is also complicated by changes in the velocity 

distribution at successive sections. Thus, the momentum coefficients 

(j8i and j82 ) are different, and the value of the coefficient CT in equation 

82 must be adjusted to account for the variations in the velocity 

pattern within the reach. Evaluation of boundary pressure forces is 

also complicated by the fact that the boundary is not prismatic. There 

is clearly no advantage in applying the momentum equation instead 

of the energy equation to a gradually varied flow.

U. S. GOVERNMENT PRINTING OFFICE : 1958 O 483667