Gradually Varied Flow


Introduction:

In practice, it becomes necessary to change the channel section or

bottom slope and use of transition and control structures like sluice

gate, weir etc in the channel as a result of which the flow in the channel

becomes varied or non-uniform between two uniform states of flow. Varied

flow may be of two types: gradually varied flow and rapidly varied flow.

In gradually varied flow, the water depth and flow velocity vary gradually

along the channel length. The streamlines are practically parallel so that

there is no appreciable acceleration component normal to the direction of

flow and the pressure distribution over the section is hydrostatic.

Since in gradually varied flow, the depth of flow changes gradually, to

produce a significant change in depth, long channel lengths are usually

involved in the analysis of gradually varied flow. Consequently, the frictional

losses, which are proportional to the channel length, play a significant role

in determining the flow characteristics and must be included.


Introduction:

Example: Flow behind a dam and flow upstream of a sluice gate or weir

are examples of gradually varied flow.

The analysis of gradually varied flow involves the assumption that the

frictional losses in gradually varied flow are not significantly different from

those in uniform flow. By virtue of this assumption, the friction slope in

gradually varied flow is computed using the uniform flow formulae.

EquationManning

RA

Qn

R

VnS f ...........

3

4

2

22

3

4

22

EquationChezyRAC

Q

RC

VS f ...........

22

2

2

2


Governing Equation: At any channel section, the total energy is given by

)1....(..........2

2

g

VyZH b

Differentiating equation (i) with respect to x yields,

)2....(..........2

2

g

V

dx

d

dx

dy

dx

dZ

dx

dH b

fSdx

dH

0Sdx

dZb

dx

dy

TgA

V

dx

dy

dy

dA

Ag

Q

dx

dy

gA

Q

dy

d

g

V

dx

d

/

2

222

2

3

2

2

22

dx

dyFr

dx

dy

gD

V

g

V

dx

d.

2

222


Governing Equation cont’d:

dx

dyFr

dx

dySS f .2

0

fSSFrdx

dy 0

2)1( )3....(..........)1( 2

0

Fr

SS

dx

dy f

From equation (ii),

This is the basic differential equation of steady gradually varied flow and is

also known as the dynamic equation of steady gradually varied flow. It

represents the slope of the water surface with respect to channel bottom and

gives the variation of y in the x direction.

The water surface curve or profile represents a backwater curve when the

depth of flow increases in the direction of flow (dy/dx>0) and a drawdown

curve when the depth of flow decreases in the direction of flow (dy/dx<0).

When dy/dx=0, the water surface is parallel to the channel bottom and the

flow is uniform.


Other forms of the dynamic equation of gradually varied flow:

The section factor and the section factor for critical flow computation can

be expressed, respectively,

DAZDAZ 22 g

QZ

g

QZ cc

22

2

22

2

2

2

2

FrgD

V

gD

V

DgA

Q

Z

Zc

Where Z represents the numerical value of A√D to be computed for

discharge at the actual depth y of the gradually varied flow,

Zc represents the section factor to be computed for discharge as if the flow

in the channel is critical.


Where

K = numerical value of conveyance to be computed for Q at actual depth y of

the gradually varied flow.

Kn = conveyance to be computed for Q as if flow in the channel is uniform.

2

2

K

QSSKQ ff 2

2

0

nK

QS

2

2

0 K

K

S

Snf

From equation (3),

2

2

2

2

02

00

1

1

1

1

Z

Z

K

K

SFr

S

S

Sdx

dy

c

nf

M

cc

M yCZandyCZ 1

2

1

2

N

nn

N yCKandyCK 2

2

2

2

Now, for a given channel section, the section factor and the conveyance are

functions of the depth of flow, i.e.


Where C1 and C2 are the co-efficients and M and N are the hydraulic

exponents for critical flow computation and uniform flow computation,

respectively.

N

nn

y

y

K

K

2

2 M

cc

y

y

Z

Z

2

2

M

c

N

n

y

y

y

y

Sdx

dy

1

1

0

For a (wide) rectangular channel, M = 3 and when Chezy formula is used

to compute Sf, N = 3 and hence,

)4....(..........

1

1

3

3

0

y

y

y

y

Sdx

dy

c

n


Equation (4) is known as the Belanger equation. When Manning formula

if used to compute Sf, N = 3.33 and hence,

)5....(..........

1

1

3

3

10

0

y

y

y

y

Sdx

dy

c

n

Equations (5) and (6) can be used to compute gradually varied flow in

wide rectangular channel.

The differential equation of gradually varied flow can also be expressed

in terms of the specific energy, E.

g

VyE

2

2

EZg

VyZH bb

2

2

)6....(..........0 fb

b SSdx

dZ

dx

dHZH

dx

d

dx

dE


Characteristics and classification of Flow Profiles:

Types of bottom slopes:

The channel bottom slopes are conveniently classified as:

•Sustainable or positive slope (S0>0)

•Non-sustainable slope (S0<=0).

A positive slope is the slope for which the channel bottom falls in the

direction of flow. It may be:

•Mild (S0<Sc, yn>yc)

•Critical (S0 = Sc, yn = yc)

•Steep (S0>Sc, yn<yc).

Uniform flow can occur in positive slope channels only. In a mild slope

channel the uniform flow is subcritical, in a critical slope channel the

uniform flow is critical and in a steep slope channel the uniform flow is

supercritical.


A non-sustaining slope is the slope for which the channel bottom does not

fall in the direction of flow. It may be:

•Horizontal (S0 = 0)

•Adverse (S0<0).

A horizontal slope is a zero slope. An adverse slope is a negative slope

for which the channel bottom rises in the direction of flow.

M Mild

C Critical

S Steep

H Horizontal

A Adverse.

The symbols that are used for the bed slopes:


Types of Flow Profiles:

For the given discharge and channel conditions, the normal depth line

(NDL) and the critical depth line (CDL) divide the space above the

channel into three zones:

Zone 1: Space above upper line (y>yn, y>yc)

Zone 2: Space between two lines (yn>y>yc, or yc>y>yn)

Zone 3: Space between channel bed and lower line (y<yn, y<yc)

The three zones are designated by 1 to 3 starting from the top. The flow

profiles are classified according to the channel slope and the zone in

which the flow profile lies.

The name of a flow profile includes the symbol used for the channel

slope followed by the zone number. Thus, the name of the flow profile

which lies in Zone 1 of a mild slope channel is M1.


M1 M2 M3

C1 C2 C3

S1 S2 S3

- H2 H3

- A2 A3

The flow profiles H1 and A1 are physically not possible. Thus the flow

profiles may be classified into 13 different types designated as:

Of the 13 flow profiles, 12 are for gradually varied flow and one (C2) is for

uniform flow. The general characteristics of these flow profiles are given in

Table 6.1.


Slope Zone Design. Range Type of Curve Type of Flow

Horizont

al

S0 = 0

1 - - - -

2 H2 y>yc Drawdown Subcritical

3 H3 yc>y Backwater Supercritical

Mild

0<S0<Sc

yn>yc

1 M1 y>yn>yc Backwater Subcritical

2 M2 yn>y>yc Drawdown Subcritical

3 M3 yn>yc>y Backwater Supercritical

Critical

S0=Sc>0

yn=yc

1 C1 y>yc=yn Backwater Subcritical

2 C2 yc=y=yn Parallel to channel bed Uniform Critical

3 C3 yc=yn>y Backwater Supercritical

Steep

S0>Sc>0

yn<yc

1 S1 y>yc>yn Backwater Subcritical

2 S2 yc>y>yn Drawdown Supercritical

3 S3 yc>yn>y Backwater Supercritical

Adverse

S0<0

1 - - - -

2 A2 y>yc Drawdown Subcritical

3 A3 yc>y Backwater Supercritical

Behavior of Flow Profiles at Specific Depths:

(i) When y→yn, dy/dx→0, i.e. the flow profile

approaches the NDL tangentially.

(ii) When y→yc, dy/dx→∞, i.e. the flow profile becomes

vertical in crossing the critical depth line (CDL).

This indicates a hydraulic jump if the depth changes suddenly from a

lower value to a higher value or a hydraulic drop if the depth changes

abruptly from a higher value to a lower value. In both cases the flow

becomes rapidly varied and the theory of gradually varied flow does not

apply.

(iii) When y→∞, dy/dx→S0, i.e. the flow profile tends to be horizontal.

Note that the slope of the channel bottom S0 is determined with respect to

the horizontal line, whereas the slope of the flow profile dy/dx is

determined with respect to the channel bottom. Therefore, the flow

profile, whose slope is dy/dx = S0 with respect to channel bottom,

must be horizontal.

M

c

N

n

y

y

y

y

Sdx

dy

1

1

0

Behavior of Flow Profiles at Specific Depths cont’d:

(iv) As y→0, dy/dx→∞/∞. For wide rectangular

channel M = 3, and if Chezy formula is used to

compute Sf, N = 3, then3

3

0

1

1

y

y

y

y

Sdx

dy

c

n

When Manning formula is used to compute Sf, N = 10/3,

then3

3

10

0

1

1

y

y

y

y

Sdx

dy

c

n

Thus for wide channel and y→0, using Chezy formula,

3

3

033

33

03

3

0

1

1

c

n

c

n

c

n

y

yS

yy

yyS

y

y

y

y

Sdx

dy

And for wide channel and y→0, using Manning formula,

3

13

3

10

0

3

10

3

33

3

103

10

03

3

10

0

1

1

1

yy

yS

y

y

yy

yyS

y

y

y

y

Sdx

dy

c

n

c

n

c

n

Behavior of Flow Profiles at Specific Depths cont’d:

Thus the theoretical behavior of the flow profile at or near y = 0 depends

on the type of uniform flow formula used in the computation. However, this

result is not of much practical importance since zero depth does never

occur.

General Procedure for sketching Qualitative

Flow Profiles:

………………………………………………….

Flow Profiles in mild slope channel (S0>0, yn>yc):

M

c

N

n

y

y

y

y

Sdx

dy

1

1

0

0, ),y>y>(y :1 Zone cn

dx

dyve

dx

dy

0, ),y>y>(y :2 Zone cn

dx

dyve

dx

dy

0, y),y>(y :3 Zone cn

dx

dyve

dx

dy

Flow Profiles at Mild Slope Channel:

M1 profile:

The water surface profile in zone 1, designated as M1, is a backwater

curve and represents subcritical flow. At the upstream boundary (y→yn,

dy/dx→0), the profile is tangential to the NDL and at the downstream

boundary (y→∞, dy/dx→S0), the profile asymptotically approaches a

horizontal line. It may be noted that the water surface in an M1 profile

falls in the d/s direction and approaches its horizontal asymptote from

above.

Occurrence of M1 profile:

The M1 profile occurs (i) behind a dam, (ii) u/s of a sluice gate in a mild

slope channel, (iii) when a long mild slope channel ends in a reservoir to a

depth greater than the normal depth and (iv) when a mild channel is

followed by a milder channel. The M1 profiles may be very long compared

to other flow profiles. In rivers and canals, the M1 profiles may extend

considerable distance before merging with the normal depth. Since the

slope of most rivers and canals is mild, the M1 profile is the most

important flow profile from the practical point of view.

M

c

N

n

y

y

y

yS

dx

dy110

Flow Profiles at Mild Slope Channel cont’d:

M2 profile:

The M2 drawdown curve in zone 2 is tangential to the NDL at its u/s

boundary (as y→yn, dy/dx→0) and normal to the CDL (as y→yc,

dy/dx→∞), indicating a hydraulic drop at its d/s boundary.


The M2 profile can occur (i) at a free overfall, (ii) when a mild slope is

followed by a steeper mild or critical or steep slope, (iv) when a mild

slope channel ends in a reservoir to a depth less than the normal depth

and (v) at the u/s side of a sudden enlargement of a channel section.

M

c

N

n

y

y

y

y

Sdx

dy

1

1

0

Flow Profiles at Mild Slope Channel cont’d:

M3 profile:

The M3 backwater profile in zone 3 starts theoretically from the channel

bottom at its u/s end (y→0) and terminates in a hydraulic jump at its d/s

boundary (y→yc, dy/dx→∞).


The M3 profile occurs d/s of a sluice gate in a mild slope channel and

when a supercritical flow enters a mild slope channel. The M3 profiles are

relatively shorter than M1 and M2 profile.

M

c

N

n

y

y

y

yS

dx

dy110

EqnChezyy

yS

yy

yyS

y

y

y

y

Sdx

dy

c

n

c

n

c

n

.......

1

1

3

3

033

33

03

3

0

EqnManning

yy

yS

y

y

yy

yyS

y

y

y

y

Sdx

dy

c

n

c

n

c

n

.......1

1

1

3

13

3

10

0

3

10

3

33

3

103

10

03

3

10

0

Flow Profiles in steep slope channel (S0>Sc>0, yn>yc):

S1 profile:

The S1 backwater profile begins with a hydraulic jump at its u/s boundary

(y→yc, dy/dx→∞) and tends to be horizontal at the d/s boundary (y→∞,

dy/dx→S0). The water surface rises in the d/s direction and approaches its

horizontal asymptote from below.

Occurrence of S1 profile:

The S1 profile occurs behind a dam or u/s of a sluice gate built in a

steep channel and in a steep channel ending in a reservoir to a depth

more than the critical depth.

M

c

N

n

y

y

y

y

Sdx

dy

1

1

0

Flow Profiles in Steep Slope Channel (S0>Sc>0, yn>yc) cont’d:

M

c

N

n

y

y

y

y

Sdx

dy

1

1

0

S2 profile:

The S2 drawdown curve starts from the CDL (y→yc, dy/dx→∞) with a

vertical slope at its u/s boundary and is tangential to the NDL at the d/s

end (y→yn, dy/dx→0). It is usually very short or acts like a transition

between a hydraulic drop and uniform flow.


The S2 profile can occur d/s of an enlargement of a channel section and

also d/s of a transition of slope from mild to steep or steep to steeper.

Flow Profiles in Steep Slope Channel (S0>Sc>0, yn>yc) cont’d:

M

c

N

n

y

y

y

y

Sdx

dy

1

1

0

S3 profile:

The S3 backwater curve starts from the channel bottom (y→0,

dy/dx→∞/∞) at its u/s boundary and approaches the NDL tangentially

(y→yn, dy/dx→0).


The S3 profile may occur d/s of a sluice gate on a steep slope or at a

transition between steep slope and milder steep slope.

Flow Profiles in Critical Slope Channel (S0 = Sc>0, yn = yc):

M

c

N

n

y

y

y

y

Sdx

dy

1

1

0

0, ),yy>(y :1 Zone(i) nc

dx

dyve

dx

dy

0, y),y(y :3 Zone(ii) nc

dx

dyve

dx

dy

In a critical slope channel, NDL and CDL coincide since yn = yc.

Therefore, zone 2 and the C2 profile which satisfy the condition yn=y=yc

also coincide with the NDL and CDL. The C2 profile thus represents

uniform critical flow and may occur in a long prismatic critical slope

channel. But it is not a profile of gradually varied flow.

The C1 backwater profile in zone 1 starts from the yn = yc line and tends

to be horizontal at d/s.

The C3 backwater profile in zone 3 starts from the channel bottom and

meets the yn = yc line at its d/s end.

Flow Profiles in Critical Slope Channel (S0=Sc>0, yn=yc) cont’d:

For a wide rectangular channel, M = 3, using Chezy formula for computing

Sf, N = 3.

Using the condition yn = yc, we get

So, the C1 and C3 profiles in a wide channel are horizontal. For

channels which are not wide, generally,

So, using the condition yn = yc,

This indicates that C1 and C3 profiles are approximately horizontal for

channels other than wide ones.

3

3

0

1

1

y

y

y

y

Sdx

dy

c

n

0Sdx

dy

NM

0Sdx

dy

Flow Profiles in Critical Slope/Horizontal Channel :

Occurrence of C1 and C3 profiles:

The C1 profile may occur (i) u/s of a sluice gate on a critical slope or, (ii)

when a critical slope is followed by a mild or horizontal or adverse slope.

The C3 profile may occur (i) at d/s of a sluice gate in a critical slope

channel, (ii) at the transition between steep and critical slopes. The

critical slope profiles are very rare.

Flow Profiles in Horizontal channels (S0 = 0, yn = ∞):

For a horizontal channel (S0 = 0), Kn = ∞, or yn = ∞, and therefore, zone 1

and an H1 profile satisfying the condition y>yn>yc are not physically

possible.

0SKQ n

Flow Profiles in Horizontal channels (S0 = 0, yn = ∞) cont’d:

2

0

1 Fr

SS

dx

dy f

2

2

2

FrZ

Zc 2

2

K

QS f

MyCZ 1

2 M

cc yCZ 1

2

M

c

f

y

y

K

Q

Fr

S

dx

dy

11

0 2

2

2

0, ),y>(y :2 Zone(i) c

dx

dyve

dx

dy

0, ),y(y :3 Zone(ii) c

dx

dyve

dx

dy

The H2 drawdown profile has a horizontal asymptote at its u/s end

(y→∞) and ends in a hydraulic drop at its d/s end (y→yc, dy/dx→∞). It may

occur on a horizontal slope u/s of a free overfall.

The H3 backwater profile, which is similar to the M3 profile, is obtained

d/s of sluice gates on a horizontal channel. The horizontal slope

profiles may be considered to be the limiting cases of the mild slope

profiles when the channel becomes horizontal.

Flow Profiles in Adverse Slope channels (S0 < 0, yn = imaginary):

0SKQ n

For an adverse slope channel (S0 < 0), Kn2 = -ve, and yn is imaginary.

Therefore, zone 1 and A1 profile are not physically possible.

2

0

1 Fr

SS

dx

dy f

2

2

2

FrZ

Zc MyCZ 1

2

M

cc yCZ 1

2

M

c

fff

y

y

SS

Fr

SS

Fr

SS

dx

dy

111

0

2

0

2

0

0, ),y>(y :2 Zone(i) c

dx

dyve

dx

dy

0, ),y(y :3 Zone(ii) c

dx

dyve

dx

dy

The A2 and A3 profiles are similar to H2 and H3 profiles and are very

rare. Only short lengths of adverse slope profiles may be expected to

occur in practice.

Flow Profiles, summarized:

Thus, it is evident that the profiles in zone 1 (i.e., M1, S1 and C1) and

zone 3 (i.e., M3, S3, C3, H3 and A3) are backwater curves and those

in zone 2 (i.e., M2, S2, H2 and A2) are drawdown curves.

All the profiles in zones 1 and 2, except the S2 profile, represent

subcritical flow and those in zone 3 and the S2 profile represent

supercritical flow.

Flow Profiles in Serial Arrangement of channels:

When two or more prismatic channels of the same cross-section but with

different bottom slopes are combined and carry the same discharge, the

following procedure for the analysis of flow profile is to be adopted:

(i) Draw the channel profile. Plot CDL and NDL, if any, in each channel.

(ii) Locate all possible Control Sections at which the depth is known.

(iii) Starting from the known depth, which may be the normal depth or

the depth at a control section, draw the possible flow profiles in the

channels.

Flow Profiles in Serial Arrangement of channels cont’d:

The following points must be noted in connection with the flow profiles in

a number of channels:

(i) The critical depth yc will be the same for all the channels, since it does

not depend on the channel bottom slope, S0.

(ii) The normal depth, yn will be different in different channels. Since yn is

inversely related to the channel bottom slope S0, the normal depth yn for a

channel will be higher if S0 is lower and vice versa.

(iii) Flow u/s of a control must be subcritical and d/s of a control must be

supercritical. The control itself locates the subcritical flow profile u/s of it

and the supercritical flow profile d/s of it. In fact, the gradually varied flow

profiles are the results of interaction between the flow and the control (s).

Flow Profiles in Serial Arrangement of channels cont’d:

(iv) When the flow changes from subcritical to supercritical, a hydraulic

drop usually forms. On the other hand, when the flow changes from

supercritical to subcritical state, a hydraulic jump usually forms.

(v) Under normal condition, the flow in a long straight prismatic channel

having positive slope is taken to be uniform. Therefore, flow beyond the

influence of a control or transition in a mild, critical or steep slope channel

will be uniform, i.e., at the normal depth yn.

(vi) Under normal condition, the flow in a long horizontal or adverse slope

channel is subcritical. Therefore, the flow beyond the influence of a control

or transition in a horizontal or adverse slope channel will be zone 2, i.e., the

flow profile will be H2 in a horizontal channel and A2 in an adverse slope

channel.

Specific Examples:

1. Mild Slope Channel followed by Steep Slope Channel (Qualitative

Flow Profile):

Obviously, the flow is uniform both u/s and d/s of the transition. The five

possible flow profiles (1 to 5), over the transition are shown in fig.

Profile 1: Flow is uniform in the entire mild slope channel and the flow

passes through an S1 profile in zone 1 and an S2 profile in zone 2 of the

steep slope channel. But S1 must be a backwater curve, not a

drawdown curve.

Profile 5: It consists of an M2 profile in zone 2 and an M3 profile in zone 3

of the mild slope channel. But M3 profile must be a backwater curve.

We come to the conclusion, the flow profile marked 3 consisting of an M2

drawdown profile in zone 2 of mild slope channel and an S2 drawdown

profile in zone 2 of steep slope channel, are the only acceptable flow

profiles.

Obviously, the flow changes from subcritical in the mild slope channel to

supercritical in the steep slope channel which is possible only through a

hydraulic drop.

Specific Examples cont’d:2. Steep Slope Channel followed by a Mild Slope Channel (Qualitative Profile):

The uniform flow is supercritical in the u/s channel and subcritical in the d/s

channel. The change in the flow state from supercritical to subcritical can only

occur through a hydraulic jump. The location of the jump depends on the

relative magnitude of the two slopes.

If the magnitudes of the two slopes are such that jump is formed in the steep

slope channel, an S1 profile is formed in the steep slope channel and the flow

is uniform just from the beginning of the mild slope channel.

Now, if the slope of the d/s mild slope channel is gradually increased, the NDL

for the mild slope channel lowers and the jump gradually moves d/s and finally

the jump forms on the mild slope channel. In this situation, the flow is uniform in

the entire u/s channel, an M3 profile is formed in the d/s channel and the flow

eventually returns to uniform state after hydraulic jump in the mild slope

channel.

A hydraulic jump usually occurs when a steep slope channel is followed either

by a mild or a horizontal or an adverse slope channel. The jump forms either in

the u/s channel or in the d/s channel but such a situation does never occur that

part of the jump forms in the u/s channel and the rest of the jump forms in the

d/s channel.

Specific Examples cont’d:

3. Free Overfall at the end of a Mild Slope Channel (Qualitative Flow

Profile):

Suppose there is a free overfall at the end of a mild slope channel. The flow

is uniform in the channel far u/s of the free overfall. The water surface falls

as a result of the free overfall, an M2 profile develops immediately u/s of

the free overfall and the critical depth occurs just u/s of the brink.

Different Water Levels d/s of a Mild Slope Channel:

Three positions of the water level, yd d/s of a mild slope channel are shown

in fig. Obviously,

(i) When yd>yn, an M1 profile is formed

(ii) When yn>yd>yc, an M2 profile is formed

(iii) When yn>yc>yd, the situation is similar to a free overfall and an M2

profile is formed.

Specific Examples cont’d:

4. Sluice gate in a Steep Slope Channel:

The flow in the channel far u/s of the sluice gate is uniform and

supercritical. The presence of the sluice gate, which is a control,

changes the flow from supercritical to subcritical with the formation of a

hydraulic jump and an S1 profile is formed. An S3 profile representing

supercritical flow is formed d/s of the sluice gate through which the water

surface joins the NDL d/s.

5. Overflow Weir in a Mild Slope Channel:

The flow in the channel far u/s of the weir is uniform and subcritical. The

depth above the weir is approximately equal to the critical depth, i.e.,

critical section occurs just u/s of the weir. Therefore, an M1 profile is

formed u/s of the weir.

Example 1:

A trapezoidal channel with b = 6 m, n = 0.025, z = 2, and S0 = 0.001 carries a

discharge of 30 m3/s. At a certain section A of the channel, the depth of flow is 1.50 m.

(i) Determine the type of channel slope.

(ii) Determine the type of flow profile.

(iii) If at another section B, the depth of flow is 1.70 m, state whether section B is

located u/s or d/s of section A.

Soln:

y

(m)

Comme

nt

1 8 10.472 0.764 6.685

2 20 14.944 1.338 24.288

1.97 19.582 14.810 1.322 23.589

1.98 19.721 14.855 1.328 23.821 Closest

)(

26

2

2

m

yy

yyzbA

)(

*526

12 2

m

y

zybP

)(m

P

AR

3

2

AR

717.23

001.0

30025.0

3

2

n

nnn

S

nQRA

myn 98.1

137.1012.181.9

30

g

QZc

Example 1 cont’d:

y (m) Comment

1 8 10.000 0.800 7.155

2 20 14.000 1.429 23.905

1.2 10.080 10.800 0.933 9.738

1.3 11.180 11.200 0.998 11.170

1.22 10.297 10.880 0.946 10.017

1.23 10.406 10.920 0.953 10.158 Closest

)(

26

2

2

m

yy

yyzbA

)(

*46

2

m

y

yzbT

)(m

T

AD DAZ

Here, yn = 1.98 m, yc = 1.23 m.

yA = 1.50 m, yB = 1.70 m.

(i) yn>yc, the channel is of mild slope.

(ii) yn>yA>yc, the profile is M2.

(iii) Since the M2 profile is a

drawdown curve and yB = 1.70 m, so

Section B is located u/s of Section A.

137.1012.181.9

30

g

QZc

Example 2:

A rectangular channel 8 m wide and having α = 1.12 and n = 0.025 has

three reaches arranged serially. The bottom slopes of these reaches are

0.0035, 0.006 and 0.0085, respectively. For a discharge of 30 m3/s in this

channel, sketch the resulting flow profiles.

Solution:

mgb

Qyc 17.1

881.9

3012.13

2

2

32

2

Since the critical slope is the slope for which flow in the channel is both

uniform and critical, hence yn = yc = 1.17 m.

236.917.1*8 mbyA

mybP 34.1017.1*282

mP

AR 905.0

34.10

36.9

00733.0

905.036.9

30025.0

2

3

2

2

3

2

AR

nQSc

Thus the bottom slopes of the three reaches are mild, steeper mild and

steep, respectively as shown in sketch……..

Gradually Varied Flow

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