11

Water Flow in Open Channels

The Islamic University of Gaza

Faculty of Engineering

Civil Engineering Department

Hydraulics - ECIV 3322

Chapter 6Chapter 6

22

Introduction

An open channel is a duct in which the liquid flows

with a free surface.

Open channel hydraulics is of great importance in civil

engineers, it deals with flows having a free surface,

for example:

• Channels constructed for water supply, irrigation,

drainage, and

• Sewers, culverts, and

• Tunnels flowing partially full; and

• Natural streams and rivers.

3

Pipe Flow and Open Channel Flow

Pipe Flow

• The liquid completely fills the pipe and flow under pressure.

• The flow in a pipe takes place due to difference of pressure (pressure gradient),

• The flow in a closed conduit is not necessarily a pipe flow.

Open Channel Flow

• Flow takes place due to the slope of the channel bed (due to gravity).

• The flow must be classified as open channel flow if the liquid has a free surface.

4

Pipe Flow Open Channel Flow

5

For Pipe flow (Fig. a):• The hydraulic gradient line (HGL) is the sum of the elevation and the

pressure head (connecting the water surfaces in piezometers).• The energy gradient line (EGL) is the sum of the HGL and velocity

head.• The amount of energy loss when the liquid flows from section 1 to

section 2 is indicated by hL.

For open channel flow (Fig. b):• The hydraulic gradient line (HGL) corresponds to the water surface

line (WSL); where it subjected to only atmospheric pressure which is commonly referred to as the zero pressure reference.

• The energy gradient line (EGL) is the sum of the HGL and velocity head.

• The amount of energy loss when the liquid flows from section 1 to section 2 is indicated by hL. For uniform flow in an open channel, this drop in the EGL is equal to the drop in the channel bed.

6

6.1 Classifications of Open Channel Flow

Classification based on the time criterion:1. Steady Flow (time independent)

(discharge and water depth do not change with time)

2. Unsteady Flow (time dependent)

(discharge and water depth at any section change with time)

Classification based on the space criterion:1. Uniform flow (are mostly steady) (discharge and water depth remains the same at every section in

the channel)

2. Non-uniform Flow

(discharge and water depth change at any section in the channel)

7

Non-uniform flow is also called varied flow ( the flow in which the water depth and or discharge change along the length of the channel), it can be further classified as:

• Gradually varied flow (GVF) where the depth of the flow changes gradually along the length of the channel.

• Rapidly varied flow (RVF) where the depth of flow changes suddenly over a small length of the channel.

8

a) Uniform flow are mostly steady b) Unsteady uniform flows are very rare in nature

c) Steady varied flow

(over a spillway crest)

d) Unsteady varied flow (flood wave)

e) Unsteady varied flow (tidal surge)

9

10

6.2 Uniform Flow in Open Channel

Uniform flow in an open channel must satisfy the following main features:

1. The water depth y, flow area A, discharge Q, and the velocity distribution V at all sections throughout the entire channel length must remain constant.

2. The slope of the energy gradient line (Se), the water surface slope (Sws), and the channel bed slope (S0) are equal.

Se = Sws = S0

11

This is possible when the gravity force (W sin component equal the resistance to the flow (Ff)

0

2

0 ..)( SP

A

KVPLKVALS

alityproportionoftcnsK

channelofareaunitperforceresisting

PLKVPLF

ALSALW

S

endsatforcescHydrostatiFF

FFFW

f

f

tan

,

)(

sin)(sin

tansin

221

0sin

0

2

0

0

0

21

ehSRCV

tconsChezyK

C tan

RadiusHydraulicP

ARh

The Chezy Formula

12

• Rh = hydraulic radius or hydraulic mean depth

P

A

perimeterwetted

areawettedflowofareaRh

)(

• C = Chezy coefficient (Chezy’s resistance factor), m1/2/s, varies in relation of both the conditions of channel and flow.

• Manning derived the following empirical relation:

where n = Manning’s coefficient for the channel roughness

See the next table for typical values of n.

Cn

Rh1 1 6/

13

14

Manning’s formula• Substituting into Chezy equation, we obtain the Manning’s formula for

uniform flow:

eh SRn

V 3/21

eh SRAn

VAQ 3/21OR

• Q in m3/sec,• V in m/sec,• Rh in m,• Se in (m/m),• n is dimensionless

Where:

15

Example 1

1.5

m

3.0m

2

1

Open channel of width = 3m as shown, bed slope = 1:5000, d=1.5m find the flow rate using Manning equation, n=0.025.

sVAQ

V

P

AR

P

A

SRn

V

h

eh

/m 84.49538.0

m/s 538.050001927.0

025.0

1

927.0708.9

9

9.708 35.132

m 95.1935.0

1

3

3

2

22

2

3

2

16

Example 2

The cross section of an open channel is a trapezoid with a bottom width of 4 m and side slopes 1:2, calculate the discharge if the depth of water is 1.5 m and bed slope = 1/1600. Take Chezy constant C = 50.

17

Example 2Open channel as shown, bed slope = 69:1584, find the flow rate using Chezy equation, take C=35.

18

sVAQ

V

P

AR

P

A

SRCV

h

eh

/m 84.11352.1627.0

m/s 7.01584

69.0917.035

917.018.177

52.162

m 177.18 04.552.28.166.38.115072.0

m 52.16215072.06.32

52.272.08.1652.2

2

04.552.2

3

2222

2

19

6.3 Hydraulic Efficiency of open channel sections

Based on their existence, an open channel can be natural or artificial:

• Natural channels: such as streams, rivers, valleys, etc. These are generally irregular in shape, alignment and roughness of the surface.

• Artificial channels: built for some specific purpose, such as irrigation, water supply, wastewater, water power development, and rain collection channels. These are regular in shape and alignment with uniform roughness of the boundary surface.

20

Based on their shape, an open channel can be prismatic or non-prismatic:

• Prismatic channels: the cross section is uniform and the bed slop is constant.

• Non-prismatic channels: when either the cross section or the slope (or both) change, the channel is referred to as non-prismatic. It is obvious that only artificial channel can be prismatic.

• The most common shapes of prismatic channels are rectangular, parabolic, triangular, trapezoidal and circular; see the next figure.

21

Most common shapes of prismatic channels

22

• Most economical section is called the best hydraulic

section or most efficient section as the discharge,

passing through a given cross-sectional area A, slope of

the bed S0 and a resistance coefficient, is maximum.

• Hence the discharge Q will be maximum when the wetted perimeter P is minimum.

PconstS

P

ACASRCAVAQ eeh

1*.

23

Economical Rectangular Channel

D,BA BD2P

D

A 2DP

0 dD

dP

222 2 0 2 D

DB

D

A

D

A

dD

dP

D

B 2 2

B D

P should be minimum for a given area;

D

D

DD

DD

DB

DB

P

ARh 4

2

22

2

2

2

So, the rectangular channel will be most economical when either:

the depth of the flow is half the width, or

the hydraulic radius is half the depth of flow.

2

DRh

24

Economical Trapezoidal Channel

D)Dn(BA 212 nDBP

Dn D

A B

212 nD ) Dn D

A(P

0 dD

dP 0 12 2

2nn

D

A

dD

dPn

D

An12

22

D

DnBn

D

DnD)(Bn

2 12

22

2

Dn2Bn1D 2

or

P B B n D B n D 2 2 ( )

RA

P

B n D D

B n Dh

( )

( )2R

Dh 2

25

Other criteria for economic Trapezoidal section

• When a semi-circle is drawn with the trapezoidal center, O, on the water surface and radius equal to the depth of flow, D, the three sides of the channel are tangential to the semi-circle”.

• To prove this condition, using the figure shown, we have:

OF OM B n DB

n D sin ( ) sin ( ) sin 1

22

2

sin

OFB

n D2

using triangle KMN, we have:

sin

MK

MN

D

D n1 2

26

OFB n D

n

( )2

1 2

D nB n D

12

22

using equation to replace the numerator , we obtain:

OFD n

n

1

1

2

2OF D

Thus, if a semi-circle is drawn with O as center and radius equal to the depth of flow D, the three sides of a most

economical trapezoidal section will be tangential to the semi-circle.

27

The best side slope for Trapezoidal section

when n 1

3

60

B D n n 2 1 2( )

2

21 2 DnB

nD

)(22 DnBDnBBP BA

Dn D

A

Dn D D n n 2 1 2( )

DA

n n

2

22 1

28

Now, from equations: P B n D 2( )

BA

Dn D

PA

D2

PA

DA n n2 2 24 4 2 1 ( ) ( )squaring both sides

0dn

dP2 4 1 2 12

1

2PdP

dnA n n

[( ) * ( ) ]

2

11

2

n

n 22 14 nn n

1

31 3

1n tan 60

The best side slope is at 60o to the horizontal, i.e.; of all trapezoidal sections a half hexagon is most economical. However, because of constructional difficulties, it may not be practical to adopt the most economical side slopes

29

Circular section

2sin

8

4

22 ddA

drP 2

In the case of circular channels, the area of the flow cannot be maintained constant. Indeed, the cross-sectional area A and the wetted

perimeter P both do not depend on D but they depend on the angle.

Referring to the figure shown, we can determine the wetted

perimeter P and the area of flow A as follows:

Thus in case of circular channels, for most economical section, two

separate conditions are obtained:

1. Condition for maximum discharge, and

2. Condition for maximum velocity.

30

1. Condition for Maximum Discharge for Circular Section:

154 D d0 95.

(Using Manning’s formula) 151 D d0 94.

2. Condition for Maximum Velocity for Circular Section:

(Using the Chezy formula)

Q AV A C R S CA

PSh

3

Q C SA

P2 2

3

dQ

d0

V C R S CA

PSh V C S

A

P2 2

dV

d0

128 75. D d081.

31

Variation of flow and velocity with depth in circular pipes

32

6.4 Energy Principle in Open Channel Flow

g

VyZEnergyTotal

2

2

The total energy of a flowing liquid per unit weight is given by:

If the channel bed is taken as the datum, then the total energy per unit weight will be:

g

VyEspecific 2

2

Specific energy (Es) of a flowing liquid in a channel is defined as

energy per unit weight of the liquid measured from the channel bed as

datum. It is a very useful concept in the study of open channel flow.

33

E yV

gE Es p k

2

2Ep = potential energy of flow = y

Ek = kinetic energy of flow = g

V

2

2

E yQ

g As

2

22Valid for any cross section

Specific Energy Curve:

It is defined as the

curve which shows the

variation of specific

energy (Es ) with depth

of flow y.

34

Specific Energy Curve (Rectangular channel)

Consider a rectangular channel in which a constant discharge

q = discharge per unit width = = constant ( since Q and B are constants)

Q

B

y

q

yB

Q

A

QV

kps EEyg

qyE

2

2

2pE

EK

EP Es

yc

35

Sub-critical, critical, and supercritical flow

The criterion used in this classification is what is known by Froude number, Fr, which is the measure of the relative effects of inertia forces to gravity force:

h

rDg

VF

T

A

Width SurfaceWater

Area) (Wetted Flow of AreahD

T

T

Fr Flow

Fr < 1 Sub-critical

1 = Fr Critical

Fr >1 Supercritical

gA

TQFr 3

22

V = mean velocity of flow of water,

Dh = hydraulic depth of the channel

36

cyE min

E y and ys1 1 1 '

Referring to the energy curve, the following features can be observed:

1. The depth of flow at point C is referred to as critical depth, yc.

(It is defined as that depth of flow of liquid at which the specific energy is

minimum, The flow that corresponds to this point is called

critical flow (Fr = 1.0).

2. For values of Es greater than Emin , there are two corresponding depths.

One depth is greater than the critical depth and the other is smaller then

the critical depth, for example;

These two depths for a given specific energy are called the alternate depths.

y yc

y yc

3. If the flow depth

In this case Es increases as y increases.

In this case Es decreases as y increases.

the flow is said to be sub-critical (Fr < 1.0).

4. If the flow depth the flow is said to be super-critical (Fr > 1.0).

37

38

Critical depth, yc for rectangular channel

0dy

dE

2

2

2 yg

qyEs

Critical depth, yc , is defined as that depth of flow of liquid at which the

specific energy is minimum, Emin,

The mathematical expression for critical depth is obtained by differentiating

energy equation with respect to y and equating the result to zero;

d

dyy

q

g y

q

g y( ) ( )

2

2

2

321

2

20

1 02

33

2

q

g yy

q

g

31

2

g

qyc

39

Critical velocity, Vc for rectangular channel

,2

3

g

qyc

y

q

yB

Q

A

QV

Vq

ycc

OR g

yVy cc

c

223 V g yc c

r

c

c Fyg

V

1

40

Minimum Specific Energy in terms of critical depth

E yq

g yc

cmin

2

22

g

qyc

23

E yy

cc

min 2

Eyc

min 3

2y

Ec

2

3min

OR

41

Critical depth, yc , for Non- Rectangular Channels

0dy

dEs 0)(2

21)

2(

3

2

2

2

dy

dA

Ag

Q

Ag

Qy

dy

d

(constant discharge is assumed) 1 02

3 Q

g A

dA

dy( )

dA/dy = the rate of increase of area with respect to y = T (top width).

OR

013

2

Ag

TQ Q

g

A

T

2 3

condition must be satisfied for the flow at the critical depth.

Recalling that T

ADh

Q

gA Dh

22

The equation may also be written in terms of velocity V

g

Dh2

2 2

The velocity head is equal to one-half the hydraulic depth for critical flow.

42

2

2

2 Ag

QyEs E y

A

Ts 2

This equation represents the critical state

E yA

Tc c 1

2( )OR

The general equation for the specific energy in critical state applicable to channels of all shapes.

Trapezoidal section

Circular section Triangle section

Rectangular section

EB n y y

B n ycc c

c

( )

( )

3 5

2 2

Ed d

c

21

16

2 2( cos )

( sin )

sin

E yc c5

4

2

3 cc

yE

43

Constant Specific Energy

The specific energy was varied and the discharge was assumed to be

constant. Let us now consider the case in which the specific energy is

kept constant and the discharge Q is varied.

2

2

2 Ag

QyEs Q A g E ys 2 ( )

Q A g E y gA E gA ys s2 2 2 22 2 2 ( ) ( )

The discharge will maximum if dQ

dy0

QdQ

dyg E A

dA

dyg y A

dA

dyAs

2 2 2 2 2( ) ( )

dA/dy = T 02)2(2)2(2 2 gAyATgATEg s

44

4 4 2 0E T yT As

AyET s )(2 E yA

Ts 2

but E yQ

g As

2

22

T

Ay

Ag

Qy

22 2

2 Q

g

A

T

2 3

Thus for a given specific energy, the discharge in a given channel is a

maximum when the flow is in the critical state. The depth corresponding

to the maximum discharge is the critical depth.

45

6.5 Hydraulic Jump • A hydraulic jump occurs when flow changes from a supercritical flow

(unstable) to a sub-critical flow (stable). • There is a sudden rise in water level at the point where the hydraulic

jump occurs.• Rollers (eddies) of turbulent water form at this point. These rollers cause

dissipation of energy.

•A hydraulic jump occurs in practice at the toe of a dam or below a sluice gate

where the velocity is very high.

46

General Expression for Hydraulic Jump:

In the analysis of hydraulic jumps, the following assumptions are made:

(1) The length of hydraulic jump is small. Consequently, the loss of head

due to friction is negligible.

(2) The flow is uniform and pressure distribution is due to hydrostatic

before and after the jump.

(3) The slope of the bed of the channel is very small, so that the

component of the weight of the fluid in the direction of the flow is

neglected.

47

Location of hydraulic jump

Generally, a hydraulic jump occurs when the flow changes from supercritical to subcritical flow.

The most typical cases for the location of hydraulic jump are:1. Jump below a sluice gate.2. Jump at the toe of a spillway.3. Jump at a glacis.

(glacis is the name given to sloping floors provided in hydraulic structures.)

48

•The net force in the direction of flow = the rate of change of moment in that direction

)( 12 VVg

Q

The net force in the direction of the flow, neglecting frictional resistance and the component of weight of water in the direction of flow,

R = F1 - F2 .

Therefore, the impulse-moment yields

F FQ

gV V1 2 2 1

( )

Where F1 and F2 are the pressure forces at section 1 and 2, respectively.

A y A y

Q

gV V1 1 2 2 2 1 ( )

A y A yQ

g A A1 1 2 2

2

2 1

1 1 ( )

Q

gAA y

Q

gAA y

2

11 1

2

22 2

y = the distance from the water surface to the centroid of the flow area

49

Comments:

• This is the general equation governing the hydraulic jump for any

shape of channel.

• The sum of two terms is called specific force (M). So, the equation can

be written as:

M1 = M2

• This equation shows that the specific force before the hydraulic jump

is equal to that after the jump.

Q

gAA y

Q

gAA y

2

11 1

2

22 2

50

Hydraulic Jump in Rectangular Channels

A B y1 1 yy

11

2 A B y2 2 y

y2

2

2

Q

g B yBy

y Q

g B yBy

y2

11

12

22

2

2 2 ( )( ) ( )( )

Q

gAA y

Q

gAA y

2

11 1

2

22 2

qQ

B

q

g

y y

y y

y y22 1

1 2

22

12

2

2 2

1 2 2 1q

gy y y y ( )

y y y yq

g2 12

22

1

220

using

, we get

51

yy y q

g y21 1

2 2

12 2

2

yy y q

g y12 2

2 2

22 2

2

This is a quadratic equation, the solution of which may be written as:

where y1 is the initial depth and y2 is called the conjugate depth. Both are called

conjugate depths.

These equations can be used to get the various characteristics of hydraulic jump.

y

y

q

g y2

1

2

13

1

21 1

8

y

y

q

g y1

2

2

23

1

21 1

8

52

yq

gc3

2

3

11

2 8112

1

y

y

y

y c

y

y

y

yc1

2 2

31

21 1 8

2

1

1

2 8112

1F

y

y

y

yF1

2221

21 1 8

1

11 yg

VF

FV

g y2

2

2

But for rectangular channels, we have

Therefore,

These equations can also be written in terms of Froude’s number as:

53

H E E EL 1 2

E yq

g ys

2

22

H yq

g yy

q

g yL

1

2

12 2

2

232 2

Due to the turbulent flow in hydraulic jump, a dissipation (loss) of energy occurs:

Where, E = specific energy

For rectangular channels:

hence,

Head Loss in a hydraulic jump (HL):

After simplifying, we obtain

21

3

12

4

)(

yy

yyHE L

54

h y yj 2 1

L hj j6

Height of hydraulic jump (hj):

The difference of depths before and after the jump is known as the

height of the jump,

Length of hydraulic jump (Lj):

The distance between the front face of the jump to a point on the

downstream where the rollers (eddies) terminate and the flow becomes

uniform is known as the length of the hydraulic jump. The length of the

jump varies from 5 to 7 times its height. An average value is usually

taken:

55

6.6 Gradually Varied Flow

• Non-uniform flow is a flow for which the depth of flow is varied.

• This varied flow can be either Gradually varied flow (GVF) or

Rapidly varied flow (RVF).

• Such situations occur when:

- control structures are used in the channel or,

- when any obstruction is found in the channel,

- when a sharp change in the channel slope takes place.

56

Classification of Channel-Bed Slopes

The slope of the channel bed is very important in determining the

characteristics of the flow.

Let

• S0 : the slope of the channel bed ,

• Sc : the critical slope or the slope of the channel that

sustains a given discharge (Q) as uniform flow at the critical

depth (yc).

• yn is is the normal depth when the discharge Q flows as

uniform flow on slope S0.

57

S S or y yc n c0

S S or y yc n c0

S S or y yc n c0

S0 0 0 .

S negative0

The slope of the channel bed can be classified as:

1) Critical Slope C : the bottom slope of the channel is equal to the critical slope.

2) Mild Slope M : the bottom slope of the channel is less than the critical slope.

3) Steep Slope S : the bottom slope of the channel is greater than the critical slope.

4) Horizontal Slope H : the bottom slope of the channel is equal to zero.

5) Adverse Slope A : the bottom slope of the channel rises in the direction of the flow (slope is opposite to direction of flow).

58

59

Classification of Flow Profiles (water surface profiles)

• The surface curves of water are called flow profiles (or water surface

profiles).

• The shape of water surface profiles is mainly determined by the slope of

the channel bed So.

• For a given discharge, the normal depth yn and the critical depth yc

may be calculated. Then the following steps are followed to classify the

flow profiles:

1- A line parallel to the channel bottom with a height of yn is drawn and

is designated as the normal depth line (N.D.L.)

2- A line parallel to the channel bottom with a height of yc is drawn and is

designated as the critical depth line (C.D.L.)

3- The vertical space in a longitudinal section is divided into 3 zones

using the two lines drawn in steps 1 & 2 (see the next figure)

60

4- Depending upon the zone and the slope of the bed, the water profiles

are classified into 13 types as follows:

(a) Mild slope curves

M1 , M2 , M3 .

(b) Steep slope curves

S1 , S2 , S3 .

(c) Critical slope curves

C1 , C2 , C3 .

(d) Horizontal slope curves H2 , H3 .

(e) Averse slope curves

A2 , A3 .

In all these curves, the letter indicates the slope type and the subscript

indicates the zone. For example S2 curve occurs in the zone 2 of the

steep slope.

61

Flow Profiles in Mild slope

Flow Profiles in Steep slope

62

Flow Profiles in Critical slope

Flow Profiles in Horizontal slope

Flow Profiles in Adverse slope

63

Dynamic Equation of Gradually Varied Flow

Objective: get the relationship between the water surface slope and other

characteristics of flow.

The following assumptions are made in the derivation of the equation

1. The flow is steady.

2. The streamlines are practically parallel (true when the variation in

depth along the direction of flow is very gradual). Thus the hydrostatic

distribution of pressure is assumed over the section.

3. The loss of head at any section, due to friction, is equal to that in the

corresponding uniform flow with the same depth and flow

characteristics. (Manning’s formula may be used to calculate the slope

of the energy line)

4. The slope of the channel is small.

5. The channel is prismatic.

6. The velocity distribution across the section is fixed.

7. The roughness coefficient is constant in the reach.

.

64

H Z yV

g

2

2

dH

dx

dZ

dx

dy

dx

d

dx

V

g

2

2

Consider the profile of a gradually varied flow in a small length dx of an open channel the channel as shown in the figure below.

The total head (H) at any section is given by:

Taking x-axis along the bed of the channel and differentiating the equation with respect to x:

65

S S

dy

dx

d

dx

V

gf 0

2

2

dy

dx

dy

dy

d

dx

V

gS S f

2

02dy

dx

d

dy

V

gS S f1

2

2

0

dy

dx

S S

d

dy

V

g

f

0

2

12

• dH/dx = the slope of the energy line (Sf).

• dZ/dx = the bed slope (S0) .

Therefore,

Multiplying the velocity term by dy/dy and transposing, we get

or

This Equation is known as the dynamic equation of gradually varied flow. It gives the variation of depth (y) with respect to the distance along the bottom of

the channel (x).

66

dy

dx

S S

Q T

g A

f

02

31

dy

dx

dE dx

Q T

g A

/

12

3

The dynamic equation can be expressed in terms of the discharge Q:

The dynamic equation also can be expressed in terms of the specific energy E :

67

dy

dx0

dy

dxpositive

dy

dxnegative

Depending upon the type of flow, dy/dx may take the values:

The slope of the water surface is equal to the bottom slope. (the water surface is parallel to the channel bed)

or the flow is uniform.

The slope of the water surface is less than the bottom slope (S0) . (The water surface rises in the direction of flow) or the

profile obtained is called the backwater curve.

The slope of the water surface is greater than the bottom slope. (The water surface falls in direction of flow) or the profile obtained is called the draw-down curve.

(a)

(c)

(b)

68

Notice that the slope of water surface with respect to horizontal (Sw) is different

from the slope of water surface with respect to the bottom of the channel (dy/dx).

A relationship between the two slopes can be obtained:

Sbc

ab

cd bd

abw

sin

Scd

ad

cd

ab0 sin

• Consider a small length dx of the open channel.

• The line ab shows the free surface,

• The line ad is drawn parallel to the bottom at a slope of S0

with the horizontal.• The line ac is horizontal.

Let be the angle which the bottom makes with the horizontal. Thus

The water surface slope (Sw) is given by

69

dy

dx

bd

ad

bd

ab

dx

dySSw 0

dy

dxS Sw 0

The slope of the water surface with respect to the channel bottom is given by

This equation can be used to calculate the water surface slope with respect to horizontal.

70

Water Profile Computations (Gradually Varied Flow)

• Engineers often require to know the distance up to which a surface

profile of a gradually varied flow will extend.

• To accomplish this we have to integrate the dynamic equation of

gradually varied flow, so to obtain the values of y at different locations

of x along the channel bed.

•The figure below gives a sketch of calculating the M1 curve over a

given weir.

71

Direct Step Method

• One of the most important method used to compute the water profiles is the direct step method.

• In this method, the channel is divided into short intervals and the computation of surface profiles is carried out step by step from one section to another.

For prismatic channels: Consider a short length of channel, dx , as shown in the figure.

dx

72

S dx yV

gy

V

gS dxf0 1

12

222

2 2

S dx E E S dxf0 1 2

dxE E

S S f

2 1

0

Applying Bernoulli’s equation between section 1 and 2 , we write:

or

or

where E1 and E2 are the specific energies at section 1 and,

respectively.

This equation will be used to compute the water profile curves.

73

12 yy

12 yy

Vn

R S f1 12 3

1

1 /

Vn

R S f2 22 3

2

1 /

SS S

fmf f

1 2

2

The following steps summarize the direct step method:1. Calculate the specific energy at section where depth is known.

For example at section 1-1, find E1, where the depth is known (y1). This

section is usually a control section.

2. Assume an appropriate value of the depth y2 at the other end of the small

reach. Note that:

if the profile is a rising curve and,

3. Calculate the specific energy (E2) at section 2-2 for the assumed depth (y2).

4. Calculate the slope of the energy line (Sf) at sections 1-1 and 2-2 using

Manning’s formula

and

And the average slope in reach is calculated

if the profile is a falling curve.

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L dxE E

S S fm1

2 1

0,2

LE E

SS Sf f

1 22 1

01 2

2

,

nnLLLL ,13,22,1 .......

5. Compute the length of the curve between section 1-1 and 2-2

or

6. Now, we know the depth at section 2-2, assume the depth at the next section, say

3-3. Then repeat the procedure to find the length L2,3.

7. Repeating the procedure, the total length of the curve may be obtained. Thus

where (n-1) is the number of intervals into which the channel is divided.