Lecture 08

Date: 23-02-2020

Week Corresponding COs

4 Steady-gradually varied flow (GVF). CO-1

5 GVF: Flow profiles. CO-1

6 Computation of GVF. CO-1

7 Computation of GVF. CO-1

TEACHING PLAN

CO-1. Analyze and solve the GVF profiles, and determine the flow profiles.

Corresponding POs: PO-2; PO-3

PO-2. Problem analysis.

PO-3. Design/development of Solution.

GRADUALLY VARIED FLOW (GVF)

Steady and Non-Steady Flow

Unsteady

Steady

Depth

Time

Uniform and Non-Uniform Flow

V1 V2

A1 A2

V1

A1 V2

A2

Uniform Flow Non-Uniform Flow

V1 = V2

A1 = A2

INTRODUCTION

In a long straight prismatic channel in which no transition or

control structure is present, the flow tends to be uniform.

In practice, however, however, it becomes necessary to

change the channel section or bottom slope and use transition

and control structures like sluice gates, weirs etc. in the

channel as a result of which the flow in the channel usually

becomes varied or non-uniform between two uniform states of

flow.

Varied flow way be either gradually varied or rapidly varied.

In gradually varied flow the water depth and flow velocity vary

gradually along the channel length (h/x 0, U/x 0).

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 channel 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 dominating role in determining the

flow characteristics and must be included.

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 friction 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 a uniform flow formula, i.e.

RAC

Q

RC

US f 22

2

2

2

3/42

22

3/4

22

RA

Qn

R

UnS f

GOVERNING EQUATIONS

)3.6.........(2

2

g

UhzH b

GOVERNING EQUATIONS

)3.6.........(2

2

g

UhzH b

)4.6...(..........2

2

g

U

dx

d

dx

dh

dx

dz

dx

dH b

Differentiating Eq. (6.3) with respect to x yields

)7.6.(..........

222

22

3

2

22

2

22

dx

dhFr

dx

dh

gD

U

dx

dh

dh

dA

gA

Q

dx

dhA

dh

d

g

Q

dx

dh

gA

Q

dh

d

g

U

dx

d

GOVERNING EQUATIONS

f

dHS

dx

0bdz

Sdx

)4.6...(..........2

2

g

U

dx

d

dx

dh

dx

dz

dx

dH b

)5.6......(fSdx

dH

)6.6......(..........0Sdx

dzb

22 ...........(6.7)

2

d U dhFr

dx g dx

)8.6........(1 2

0

Fr

SS

dx

dh f

where dA/dh = B and A/B = D have been used. Using Eqs. (6.5) to (6.7), Eq. (6.4)

becomes

Equation (6.8) is the basic differential equation of steady gradually varied flow and is

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

DAZ

/g

QZc

22

2

2

2

2

/Fr

gD

U

DgA

Q

Z

Zc

2

2

0

nK

QS

2

2

K

QS f

2

2

0 K

K

S

Snf

2

2

0)/(1

)/(1

ZZ

KKS

dx

dh

c

n

0

21

fS Sdh

dx Fr

By using the above relationship one find

2

2

0)/(1

)/(1

ZZ

KKS

dx

dh

c

n

Z2 = C1hM

Zc2 = C1hc

M

)18.6.......()/(1

)/(10 M

c

N

n

hh

hhS

dx

dh

Chezy formula

M = 3

N = 3 and

Manning formula

M = 3

N = 10/3Belanger equation

K2 = C2hN

Kn2 = C2hn

N

2( / )

M

cc

hZ Z

h

2( / )

N

nn

hK K

h

?

?

?

n

c

h

h

h

?

?

?

n

c

h

h

h