ENCYCLOPEDIA OF ENVIRONMENTAL SCIENCE AND ENGINEERING - WATER FLOWPROPERTIES OF FLUIDS pot

The determination of the friction factor, f, depends on the flow regime, that is, whether the flow is laminar, critical, transitional, smooth, turbulent or rough fully turbulent.. Lamin

PROPERTIES OF FLUIDS

The fluid properties most commonly encountered in water

flow problems are presented in the following paragraphs

The International System of units is used throughout the

dis-cussion unless otherwise stated to the contrary

The unit of mass, m, is the kilogram (kg) A mass of one

kg will be accelerated by a force of one newton at the rate of

1 m per sec 2

The density, r, of a fluid is its mass per unit volume and

is expressed in kilograms per cubic meter

The specific weight, g, is the weight per unit volume

and denotes the gravitational force on a unit volume of fluid

and is expressed in newtons per cubic meter

Fluid density and specific weight are related by the

expression:

in which g is the acceleration due to gravity

The specific gravity of a fluid is found by dividing its

density by the density of pure water at 4⬚C

The relative shearing force required to deform a fluid

gives a measure of the viscosity of the fluid An increase

in temperature causes a decrease in viscosity of a liquid

and vice versa Consider the space between two parallel

plate remains at rest while the upper plate moves with

velocity V under an applied force The velocity of the fluid

particles will range from V at the top boundary to zero at

the bottom as they will assume the same velocity as the

boundary in which they are in contact Experiments have

demonstrated that the shear stress, t, is directly

propor-tional to the rate of deformation, d u /d y Mathematically,

this can be written as:

d

u

Equation (2) is known as Newton’s equation of viscosity

The constant of proportionality, µ, in newton-second per

square meter (N-s/m 2 ), is termed the coefficient of viscosity,

the dynamic viscosity or the absolute viscosity

The kinematic viscosity, v, is defined as the ratio of the

coefficient of viscosity to the density and is expressed in

A more proper term for surface tension, s, would be surface

energy Surface tension is a liquid surface phenomenon and is caused by the relative forces of cohesion, the attrac-tion of liquid molecules for each other, and adhesion, the attraction of liquid molecules for the molecules of another liquid or solid Surface tension has the units of newtons per meter (N/ m) When a liquid surface is in contact with a solid,

a contact angel u, greater than 90⬚ results with depression of the liquid surface if the liquid does not “wet” the tube such

as mercury and glass If the solid boundary has a greater attraction for a liquid molecule than the surrounding liquid molecules, then the contact angle is less than 90⬚C and the liquid is said to “wet” the wall leading to a capillary rise as

in the case of water and glass

in the preceding paragraphs for a few common fluids

PRESSURE FLOW

Friction Formulae Darcy-Weishbach ’ s Equation The Darcy-Weishbach formula

was first proposed empirically but later found by dimensional reasoning to have a rational basis:

D

in which f ⫽ friction factor, L ⫽ pipe length, V ⫽ mean velocity, D ⫽ diameter, h ⫽ head loss, g ⫽ acceleration due

to gravity

Equation (4) was derived for circulation sections flowing full and the equation itself is dimensionally homogeneous It can be extended to other cross-sections provided these shapes are not too different from circular; in this case, the equation

has to be transformed by using the hydraulic radius, R, instead

of the diameter, D:

R

Table 1 gives the values of the fluid properties discussed

plates (Figure 1) which is filled with fluid; the bottom

where r ⫽ D /4 for flow at full bore The use of Darcy’s

equation in the form given by Eq (5) is sometimes extended

to open channel flow

The determination of the friction factor, f, depends on

the flow regime, that is, whether the flow is laminar, critical,

transitional, smooth, turbulent or rough fully turbulent

Laminar Flow Consider the mean pipe velocity, V, as

given by Hagen-Poiseuille’s equation for laminar flow:

m

SD

in which S ⫽ energy slope

Combining Eq (4) with Eq (6) and noting that S ⫽ Hf/L,

g ⫽ rg, and nmⲐr, the friction factor is given by:

f

e

⫽64

where Re⫽ nDⲐg is the Reynolds number Eq (7) can be

used for all pipe roughness as the friction factor in

lami-nar flow is independent of the wall protuberances and is

inversely proportional to the Reynolds number The energy

loss varies directly as the mean pipe velocity in laminar flow

which persists up to a Reynolds number of about 2000

The velocity profile, which has a parabolic

distribu-tion, can be obtained from Hagen-Poiseuille’s equation The

velocity, u, at any radius, r, of the pipe of diameter, D, is

given by:

m

2 2

r

⎛

⎝⎜

⎞

At the centre line, the velocity is a maximum:

umax⫽g SD

m

2

The mean velocity is:

umean⫽(umax)/2⫽ SD

32

2

g

Critical Flow From a Reynolds number of about 2000 and

extending to 4000 lies a critical zone where the flow may be either laminar or turbulent The flow regime is unstable and

no equation adequately describes it

Smooth Turbulent For pipes fabricated from hydraulically

smooth materials such as copper, plexiglass and glass, the flow is smooth turbulent for a Reynolds number exceeding

4000 The von Karman-Nikuradse smooth pipe equation is:

1

f ⫽ log (R e f)⫺ (11)

Equation (11) indicates that the friction factor depends on the fluid properties and deceases with increasing Reynolds number

pipes artificially roughened with uniform sand grains The results were fitted to the theory of Prandtl–Karman to give the well known rough-pipe equation:

1

TABLE 1 Fluid properties

Fluid

Temperature

⬚C

Mass density

Specific weight

Dynamic viscosity

Kinematic viscosity

Surface tension N/m

Moving plate

Applied force

V

dy

u+dy u du

u = 0

Stationary plate y

FIGURE 1 Fluid shear

Equation (12) states that beyond a certain Reynolds number,

when the flow is fully turbulent, the friction factor is

influ-enced only by the relative roughness,␧ⲐD and independent of

the Reynolds number

follow either the smooth pipe or rough pipe equations

Colebrook and White proposed a transitional flow equation

which would be asymptotic to both:

1 2

3 7

2 51

10

f

e D

R e f

/

⎛

⎝

Equation (13) approaches the smooth pipe equation for low

and the rough pipe equation for high values of the Reynolds

number respectively Unlike Nikuradse’s ␧, which represents

the actual height of the sand grains, the ␧ of Colebrook—

White’s equation is not an actual roughness dimension but a

representative height describing the roughness projections It is

referred to as the equivalent sand-grain diameter since the

fric-tion loss it represents is the same as the equivalent sand-grain

diameter; Table 2 gives experimentally observed values:

TABLE 2 Equivalent sand-grain diameter

Riveted steel 9.14 Rough concrete 3.05 Smooth concrete 0.31

Moody Diagram (Moody, 1944.) The Moody Diagram

(Figure 2) summarises and solves graphically the four fric-tion factor equafric-tions Eqs (7), (11), (12), (13) as well as delineating the zones of the various flow regimes The line separating transitional and fully turbulent flow is given by Rouse’s equation:

1 200

f

R D

e

Mannning ’ s Equation The Manning equation, although

originally developed for open channel flow, has often been extended for use in pressure conduits The equation is usu-ally favored for rough textured material (rough concrete, unlike rock tunnels) and cross-sections that are not circular (rectangular, horseshoe) It is most commonly given in the form:

v

⫽ 1 2 3 / 1 2 /

, (15)

in which N ⫽ roughness coefficient Equation (15) can also

be transformed to:

R

V

f ⫽19 6

2

2

4 3

2

A R

FIGURE 2 Pipe friction factors

Smooth pipes

0.01

0.02 0.03 0.04 0.05 0.06

turbulent Fully

0.0001 0.004 0.001 0.002 0.004

The dimension of the roughness coefficient, N, is frequently

taken as L 1/6 as the equation by itself is not dimensionally

homogenous Table 3 provides values of Manning’s N for the

more widely used pipe materials

Hazen-William ’ s Equation This equation is used mainly in

sanitary engineering

V ⫽ 0.36 CR 0.63 S 0.54 , (18)

where C ⫽ roughness coefficient Typical values of the

Hazen-William’s C is given in Table 4

Energy Losses Due to Cross-Sectional Changes,

Bends and Valves

Cross-Sectional Changes

Expansion Energy loss in an expansion is principally a

form loss:

D

V

2

1 2

2 2 1 2

⎛

⎝⎜

⎞

⎠⎟

⎡

⎣

⎢

⎢

⎤

⎦

⎥

in which D ⫽ diameter of the conduit, and subscripts 1 and

2 denote upstream and downstream values From Eq (19), if

D 2 is very large compared to D 1 , such as the discharge into a

reservoir, the entire velocity head is lost

Contraction The contraction loss equation can be expres sed

in terms of the downstream velocity, V 2, as:

H C

V

c

con⫽ 1⫺1

2

2 2

⎛

⎝⎜

⎞

Typical values of the coefficient of contraction, C c , are given

in Table 5

Entrance Loss The head loss at the entrance of a conduit

can be compared to that of a short tube:

h C

V

ent⫽ 1 ⫺1

2

2 2

⎛

hent⫽KendV2

in which C ⫽ coefficient of discharge, K ⫽ entrance loss coefficient Typical values of C and K are given in Table 6

Transition In gradual contractions and expansions, the

lead losses are calculated in terms of the difference of veloc-ity heads in the upstream and downstream pipes:

Gradual contraction: h tc⫽K te V2 ⫺V

2 1 2

2g 2g

⎛

⎝⎜

⎞

⎠⎟, (23)

Gradual expansion: h tc⫽K te V2⫺V2

2g 2g

⎛

⎝⎜

⎞

⎠⎟ (24)

K tc values vary from 0.1 to 0.5 for gradual to sudden

contrac-tions Values of K tc range from 0.03 to 0.80 for flare angles of

2⬚ to 60⬚

TABLE 4

Hazen-William’s C

Riveted steel 110

Vitrified clay 110

TABLE 3

Normal values of Manning’s N

Corrugated metal 0.024

Concrete, unfinished 0.014 Vitrified clay 0.014

TABLE 5 Coefficient of contraction

TABLE 6

Bends The effect of the presence of bends is to induce

secondary flow currents which are responsible for the

addi-tional energy dissipation:

h b⫽K b V

2

The bend loss coefficient, K b , depends on the ratio of the

bend radius, r, to the pipe diameter, d, as well as the bend

angel For a 90⬚ bend and r / d ratio varying from 1 to 12,

values of K b range from 0.20 to 0.07

Gates and Gate Valves The gate and gate valve loss can

be expressed as:

h g⫽K g V2

The value of the loss coefficient, K g , for gates depends on

a variety of factors The value of K g for the case having the

bottom and sides of the jet suppressed ranges from 0.5 to 1.0

for typical values of K g for gate valves see Table 7

Energy-discharge Relation

In pressure conduit flow, the water is transmitted through a closed boundary conveying structure without a free surface

Figure 3 illustrates graphically the various forms of energy losses which could take place within the conduit The follow-ing energy relation can be written:

in which h end ⫽ entrance loss, h tc ⫽ transition loss, h f ⫽ skin

friction loss If H denotes the total head required to produce the discharge and h v represents the existing velocity head,

H ⫽ h l ⫹ h v (29) Writing Eq (29) in terms of the velocity heads and their respective loss coefficients,

l

v

2 2

2 2 2

2

2

g

ent 1 2 tc 2 2 1 2

⎝⎜

⎞

⎠⎟

⎡

⎣

⎢

⎢

V22

2g

⎤

⎦

⎥,

(30)

where K v ⫽ combined velocity head and exit loss coefficient

By the continuity equation:

A V1 1⫽A V2 2 (31) and

A

V

1 2 2 2 1 2 2

Equation (30) could be expressed as,

V

A A

f L

l

v

⫽

2 2

2 2

2 1

2

2 2 1 2

2

2

g

⎛

⎝⎜

⎞

⎛

⎝

⎠

⎟

⎡⎡

⎣

⎢

⎢

⎤

⎦

⎥

⎥ (32)

in which

A A

f L

2 1

2

2 2 1 2

2

1

2

⎛

⎝⎜

⎞

⎛

⎝⎜

⎞

⎠⎟

⎡

⎣

⎢

⎢

⎤

⎦

⎥

⎥

V

1 2

⫽

2 2 2 1

1 2

2

2 2

/

g ,

(35)

TABLE 7

K g for gate values

Fully open 0.2 3

4 open 1.3 1

2 open 5.5 1

4 open 24.0

Exit Loss In general the entire velocity head is lost at

exit and the exit loss coefficient, K e is unity in the equation:

h e⫽K e V2

hent htc

TEL

Transition

H

hl

–V2/2g

hv p/y

FIGURE 3 Energy relations

in whichC⫽1C l1 2

/ /

is the discharge coefficient Equation (35) can be readily extended to multiple conduits in parallel

Pipe Networks

Introduction The Hardy Cross method is most suitably

adapted to the resolution of pipe networks The statement of

the problem resolves itself into:

1) the method of balancing heads is directly

appli-cable if the discharges at inlets and outlets are known,

2) the method of balancing flows is very suitable if

the heads at inlets and outlets are known

It is assumed that:

a) sizes, lengths and roughness of pipes in the system

are given, b) law governing friction loss and flow for each pipe

is known, c) equations for losses in junctions, bends, and other

minor losses are known These relations are most conveniently expressed in terms of equivalent lengths of pipes

The objectives of the analysis are:

a) to determine the flow distribution in the individual

pipes of the network, b) to compute the pressure elevation heads at the

junctions

In applying the Hardy Cross Method, two sets of

condi-tions have to be satisfied:

a) the total change in pressure head along any closed

circuit is zero:

H⫽ 0,

b) the total discharge arriving at any nodal point

equals the total flow leaving it:

Q⫽ 0

For the pressure head change in any closed path, the

clock-wise positive sign convention is used

For the discharge continuity requirement at a nodal point,

the inward flow positive sign convention is adopted

The friction head loss equation is used in the form:

H⫽rQ2

(38) Using Darcy’s formula:

D

f L

2

8

⎛

⎝⎜

⎞

and

D

⫽ 82 5

Method of Balancing Heads Based on the condition required

by Eq (36), the following equations for any closed pipe loop results (Figure 4):

0

where Q 0 ⫽ assumed flow in the circuit for any one pipe,

⌬ Q ⫽ required flow correction Expanding Eq (39) and

approximating by retaining only the first two terms, the flow correction ⌬ Q, can be expressed as:

rQ

2

0

2

∑

Method of Balancing Flows Utilising the continuity

require-ment at a pipe junction as given by Eq (37), the head correction, ⌬ H, at anodal point is given by the equation:

⌬H

H r H

H r

⫽

⎛

⎝⎜ ⎞⎠⎟

⎛

⎝⎜ ⎞⎠⎟

∑

∑

1 2

1 2

1 2

/

In both Eq (40) and (41), the proper sign conventions must

be used in the numerators

OPEN CHANNEL FLOW

Introduction

Open channel flow refers to that class of water discharge in which the water flows with a free surface The stream flow

is said to be steady if the discharge does not vary with time

If the discharge is time dependent, the water flow is termed

unsteady Uniform flow refers to the case in which the mean

velocities at any cross-section of the stream are identical; if these mean velocities vary from one cross-section to another,

the flow is considered non-uniform Steady uniform flow

requires the conveyance section of the stream channel to

be prismatic Where the water surface profile is controlled

FIGURE 4 Pipe network

principally by channel friction, this phenomenon is known as

gradually varied flow For the type of flow in which the water

surface changes substantially within a very short channel

length due to a sudden variation in bed slope or cross-section,

this category is referred to as rapidly varied flow

Open channels fabricated from concrete are often

rect-angular or trapezoidal in shape Canals excavated in erodible

material have trapezoidal cross-sections Although sewer

pipes are closed sections, they are still considered as open

channels so long as they are not flowing full; these

cross-sections are usually circular

Channel Friction Equation

The most widely used open channel friction formula is the

Manning equation as mentioned earlier in pressure flow:

Q

A R

⫽ Q

2 2

Manning’s equation in hydraulic engineering is used for fully

turbulent flow and, as such, the values of Manning’s N apply

to this flow regime

In a natural tortuous stream channel, the mean value

of Manning’s N can be obtained from the following

considerations:

1) estimate an equivalent basic N s , for a straight

chan-nel of that material,

2) select modifying values of N m for non-uniform

roughness, irregularity, variation in shape of cross-section, vegetation, and meandering,

3) sum the basic, N s together with the modifying

values to obtain the total mean N

Normal values of Manning’s N for straight channels

and various modifying values are given in Table 8 The total

mean N value for the channel is obtained from the relation:

N⫽N s⫹∑N m (43)

Energy Principles

In deriving the energy relationships for open channel flow, the following assumptions are normally used:

1) a uniform velocity distribution over the section is assumed, that is, the velocity coefficient,

a, in the velocity head term, aV 2 /2 g, is taken as

unity In practice, the value of ␣ depends on the

shape of the stream channel and has an average value of about 1.02 which makes this assumption sufficiently valid

2) streamlines are essentially parallel, 3) channel slopes are small

Consider the water particle of mass, m, and of weight, W

(Figure 5) The elevation and pressure energies of the

parti-cle are Wh 1 ; and Wh 2 respectively Thus, the potential energy

of the water particles is, W ( h 1 ⫹ h 2 ) and is independent of its elevation over the flow cross-section As the kinetic energy

is WV 2 /2 g the total energy of the water particles, e is:

e⫽W h1⫹ ⫹h2 V

2

2g

⎛

⎝⎜

⎞

Z⫹ ⫽ ⫹D h1 h2 (45)

and noting that the total flow passing the cross-section is gQ

the total energy of the water passing the cross-section per

second, E t is given by:

E t⫽g Q Z⫹ ⫹D V

2

2g

⎛

⎝⎜

⎞

TABLE 8

Values of Manning’s N Basic N S for straight channels

Changes in shape 0.005 to 0.020

V1/2g2

V1 D1 D h2

L

TEL

hf V2/2g2

h1

Z2

FIGURE 5 Energy principles

Thus, the energy per unit weight of water passing the

cross-section per second, H is:

H⫹ ⫹ ⫹Z D V2

The term, z ⫹ D ⫹ V2Ⲑ2g is known as the total head or total

energy level (TEL); the latter name is used here The slope

of the total energy level line is the energy gradient or friction

slope and gives the rate of energy dissipation in the flow

The energies at Sections 1 and 2 are related by the

expression:

A R

1 2

2

2 4

in which the Manning equation is used to calculate the

fric-tion slope and the mean values for the flow area, A, and the

hydraulic radius, R, are to be used

Flow Regimes

Critical Flow The specific energy, E, is defined as the total

head referred to the channel bottom (Figure 6):

A

⫽ ⫹ 12

Differentiating Eq (47) with respect to D and equating the

derivative to zero to obtain its minimum value,

d d

d d

E D

Q A

A D

2 3

Noting that d A ⫽ T d D, Eq (50) becomes:

T

Equation (51) is the fundamental equation for critical flow

and is applicable to all shapes of cross-sections

If the mean depth of the flow section is defined as

D m ⫽ A/T, substitution of this relation into Eq (49) would

give the significant expressions:

V c2 D m

and

V D

c

m

At critical flow, Eq (52) demonstrates that the velocity head equals one-half the mean depth and Eq (53) indicates that the Froude number equals unity

Specific Energy Diagram for Rectangular Channel For a rectangular channel, Q ⫽ qB in which q ⫽ discharge per unit width, B ⫽ channel width, and Eq (49) becomes,

D

A plot of Eq (54) for any given constant unit discharge gives

Figure 7, which is known as the specific energy diagram The

TEL

T

V2/2g

FIGURE 6 Derivation of critical flow

45° line

E = D

q2 Supercritical

Subcritical

FIGURE 7 Specific energy diagram

following flow regimes could be defined with reference to the

specific energy diagram

Subcritical flow denotes tranquil flow in which the

Froude number and the mean velocity are less than unity and

the celerity of the gravity wave respectively

Critical flow represents the discharge phenomenon

where: (1) for a constant specific energy, the discharge is a

maximum; (2) the specific energy is a minimum for a

con-stant discharge; (3) the critical velocity equals the celerity of

a small gravity wave; (4) the Froude number equals unity;

(5) the critical depth is also the depth of minimum

pressure-momentum force

Supercritical flow which is also known as shooting

or rapid flow, is that state of water flow where the Froude

number and mean velocity exceed unity and the speed of

transmission of a surface wave respectively

Based on the equations developed earlier for critical

flow, particular formulae can be derived for a rectangular

section:

D c⫽ q c

g

⎛

⎝⎜

⎞

⎠⎟

1 3 /

(55)

D c⫽2 E c

D c⫽V c

2

V D

c

c

Flow Transition The concept of the normal depth is an

impor-tant parameter in the study of flow transition For a given

channel and any fixed discharge, uniform flow will occur at

one unique depth It is the depth attained in a long channel

when the component of gravity force is just balanced by the

frictional resistance of the channel

When the normal and critical depth are equal, the flow

is critical and the bed and energy slopes are the same

The channel bed then has a critical slope The bed slope

is termed mild when the normal depth exceeds the critical

depth; the bed slope is then less than the critical energy

slope and the flow regime is subcritical When the normal

depth lies below the critical depth and, hence, the bed slope

is greater than the critical energy slope, the channel slope

is considered to be steep and the supercritical flow regime

prevails

When water makes a transition from a channel with a

mild slope to another with a steep slope, or vice versa, the

flow passes through the critical depth close to the junction

of the two channels The section in which the water depth is

critical defines a channel control The weir acts as a control

when water flows over it as critical depth is attained there

Hydraulic Jump A hydraulic jump occurs when supercritical

flow makes a transition to subcritical flow A common

occur-rence of a hydraulic jump takes place at the base of a chute

spillway Figure 8 shows the energy momentum and depth relations for a hydraulic jump and also defines the symbols

to be used

In developing the equations for the hydraulic jump, the following assumptions are used:

1) the bed slope is considered small and neglected, 2) frictional resistance along the bed and sides of the channel are omitted

Consider the control volume between Sections (1) and (2)

Applying the impulse-momentum principle:

F1⫺ ⫽F2 g Q V2⫺V1

or

F1⫹g QV1⫽ ⫹F2 g QV2

in which F 1 and F 1 denote the hydrostatic forces at sections 1

and 2 respectively The term (F ⫹ g/gQV) is given the name

pressure-momentum force Let A ⫽ flow area, y– ⫽ distance

of centroid of flow area from surface; then they hydrostatic force ⫽ gAy– and Eq (60) can be written as:

Q A

1 1 2 1

2 1 2 2

Equation (60) states the condition for the formation of a hydraulic jump and suggests a graphical solution A plot of

Eq (60) for any fixed discharge is shown in Figure 8 For any given up-stream supercritical water depth, which is usually known such as at the toe of a spillway, the subcritical hydraulic

jump depth or sequent depth can be obtained from the graph

For a rectangular cross-section channel and utilising the continuity relation:

D

F

FLOW

FIGURE 8 Hydraulic jump relations

Equation (6) can be written as:

D

2 1

1 2

1

where F1⫽ / (V1 gD1) is the upstream Froude number

The energy loss E j across the hydraulic jump on a

horizontal floor can be obtained by coming Eqs (61), (62)

and the energy equation:

2 2 2 2

to give:

D D

j⫽( 2⫺ 1)

3

1 2

The head loss E j is graphically shown in the specific energy

Surface Water Profiles

Non-uniform Differential Equation Using the notation given

in Figure 9, the energy relations can be expressed as:

2g d ( d ) 2g d2g

⎛

⎝⎜

⎞

⎠⎟ (66)

dL dD V /

⫺

(

2

2g)

(67) d

d

E

For a finite length, ⌬ L, Eq (67) becomes:

⫺

,

2

2

2 2 2 4 3

⎛

⎝⎜

⎞

⎠⎟

(69)

where the Manning equation is used to calculate the energy slope Flow computation must start at a control section where all the flow parameters are known The calculation proceeds upstream for subcritical and downstream for super-critical flow In Eq (69), the solution of the reach length,

⌬ L, is direct if the immediately upstream depth, D 2 , is given

a value If ⌬ L is given a value, D 2 has to be solved by trial

This method of computing surface water profiles is suitable for regular channels

Classification of Flow Profiles Twelve distinct types of

non- uniform profiles have been systematically classified

1) Firstly, the curves are identified according to bed slopes as mild (M), steep (S), horizontal (H), criti-cal (C) and adverse (A)

2) Secondly, numbers are assigned to flow regions

The numerical 1 refers to actual flow depths

exceeding both critical ( D c ) and normal ( D ) depths

For flow depths less than both critical and normal, the number 3 is affixed to it The numeral 2 is for depths intermediate between critical and normal

Water Profiles in Irregular Channels The river channel has conveying overbank flow Let Q ⫽ total flow, Q c ⫽ central

channel discharge, Q l ⫽ left overbank flow,

Q r ⫽ right overbank flow The continuity condition requires that:

By Manning’s Equation:

Q

⫽ 1 2 3 / 1 2 / ⫹ 1 2 3 / 1 2 / ⫹ 1 2 3 / 1 2 /

(71)

The energy slope, S, has been taken as the same for Q c ,

Q l , Q r ; this assumption seemed to be justified in practice

Due to different channel roughness, vegetative and other

obstructions, Manning’s N for the three flow panels would

i dL idL

D

(D + dD)

V2 2g

+ d (V2 2g )

S dL V2

2g

FIGURE 9 Non-uniform flow derivation

(Figure 10)

and flow diagrams (Figure 8)

to be divided into panels (Figure 11) with the side panels