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

02

dy

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

0

2

31

dy

dx

dE dx

Q T

g A

/

1

2

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

dx 0

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 sinq

• 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 q 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

dySS

w

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

12yy

12yy

Vn

R S f1 12 3

1

1 /

Vn

R S f2 22 3

2

1 /

SS S

fm

f 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.

74

L dxE E

S S fm1

2 1

0,2

L

E 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.