Gradually Varied Unsteady Flow.pdf

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39.1 Gradually Varied Unsteady Flow

Gradually varied unsteady flow occurs when the flow variables such as the flow depth

and velocity do not change rapidly in time and space. Such flows are very common in

rivers during floods and in canals during the period of slow variation in gate opening or

closure. Typically two flow variables, such as the flow depth and velocity or thedischarge and depth, define the flow conditions at a channel section. Two governing

equations, known as Saint Venant equations, are used to descrine the spatial and

temporal variation of the above two flow variables. These equations are based on the

application of conservation of mass and momentum principles to a stationary control

volume such as shown in Figure. 39.1.

39.2 Assumptions

Following assumptions are made in the derivation of the Saint Venant equations:

The pressure distribution in the vertical direction at any cross section is

hydrostatic.

The channel bottom slope is small.

The velocity is uniform within a cross section.

The channel is prismatic.

Steady state resistance laws are applicable under unsteady conditions.

There is no lateral inflow or outflow.

39.3 Derivation

Consider unsteady flow in a channel as shown in fig 39.1. Consider a control volume of

length x as shown in this figure.


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A

C.G

y__

V1A1

y1

y1

___

V1A1

y1

y1

___ Flow

Bed

S0

x1 x2

x

1 2

x

Water Surface

Fig. 39.1: Definition sketch for derivation of St.Venant equations

The control volume in Fig. 39.1 has fixed boundaries. The Reynolds transport theorem

is applied to derive the continuity and momentum equations.

Continuity Equation

Based on the Reynolds transport theorem and treating water as an incompressible fluid,

Continuity equation for the control volume in Fig. 39.1 can be written as

( )2

2 2 1 1

1

0 39.1+ =x

x

dAdx AV AV

dt

in which A = flow area, V = flow velocity and subscripts 1 and 2 indicate flow variables

at sections 1 and 2, respectively.

Application of Leibritz's theorem to the first term on the left hand side of the above

equation, followed by the application of mean value theorem yields

( ) ( )2 1 2 2 1 1 0 39.2

+ =

Ax x AV AV

t

It may be noted that both A and

A

tare assumed continous with respect to both x and t.

Similarly, treating AV and

VA

tas continous with respect to x and t, and letting

2 1 = x x x tend to zero, one can get


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( )0 39.3

+ =

A AV

t x

Noting that flow rate, Q = AV.

( )0 39.4

+ =

A Q

t x

Equation (39.4) is the continuity equation in the "Conservation form". For prismatic

channels in which the top width, T is a continous function of the flow depth, y, Eq. (39.4)

may be written as

( )

( )

=0 39.5

or

T 0 39.6

+

+ =

dA y Q

dy t x

y Qt x

Substitution of Q = VA in Eq. 39.6 and subsequent simplification leads to

( )+V =0 39.7

+

y A V y

t T x x

Momentum Equation

Based on the Reynolds transport theorem, momentum equation for the control volume

in fig. 39.1 can be written as

( )2

1

2 2Re 2 2 1 1 39.8 = +

x

s

x

dF V Adx V A V A

dt

in which ResF = resultant force acting on the control volume in the direction of flow. As in

the case of continuity equation, application of Leibritz theorem and mean value theorem

to Eq. 39.8 leads to

( )( )

( ) ( )2Re 39.9

= +

s

AVFAV

x t x

Noting that flow rate Q = AV,


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( )( ) ( )Re 39.10

= +

sF Q QVx t x

Resultant force ResF on the control volume is evaluated as follows.

Channel is assumed to be prismatic. Therefore, forces do not arise due to

changes in cross section.

Waves set up by the wind action are not considered here. Therefore, shear

stress on the flow surface due to wind is neglected.

Open channel flows in canals, streams and rivers are considered. Flows in large

water bodies such as estuaries and oceans are not considered here. Therefore,

Coriolis forces are neglected.

Net force on Control volume comprises of

(i) pressure force at section - 1 (See Fig. 39.1), (ii) pressure force at section - 2, (iii)

Component of weight of water in the flow direction and (iv) the frictional force due to

shear between water and the channel sides and the channel bottom. These forces are

evaluated as follows.

Pressure forces at sections 1 & 2 are given by

( )1 1 1 39.11=F gA y ,

1y = depth to the centroid of area A1.

( )2 2 2 39.12=F gA y

2y = depth to the centroid of area A2.

1F acts in the positive x direction while 2F acts in the negative x direction.

Component of weight of water in the direction of flow =

( )2

3 0

1

39.13= x

x

F g AS dx


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Frictional force

= ( )2

4

1

39.14= x

f

x

F g AS dx

in which S0 = channel bottom slope and Sf= friction slope. Friction slope or the slope of

the energy gradient line to overcome friction may be estimated using any friction loss

equation such as the Manning equation. F3 acts in the positive x-direction while F4 acts

in the negative x-direction.

Substitution of equations for forces in Eq. (39.10) leads to

( )( ) ( ) ( )

1 21 2 20 39.15

+ = +

f

g A y A y QgA S S AV

x t t

Or

( ) ( ) ( ) ( )0 39.16

+ = +

f

QQV gAy gA S S

t t x

Or

( ) ( ) ( )0 39.17

+ + =

f

QQV gAy gA S S

t x

Equation (39.17) is the momentum equation in the conservation form. For any crosssection in which the top width, T is a continous function of flow depth, y

( )( )

( )

2

y 0

1+ T -Ay

2lim 39.18

+ =

A y y y

Ayy y

Neglecting higher order terms,

( ) ( )

( ) ( ) ( )

39.19

39.20

=

= =

Ay Ay

and

y ygAy g Ay gA

x y x x

Substitution of Eq. (39.20) in Eq. (39.17) leads to


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( ) ( )0 39.21

+ + =

f

Q QV ygA gA S S

t x x

Substitution of Q = AV into Eq. (39.21), subsequent expansion of terms, and further

simplification using continuity equation leads to

( ) ( )0V 39.22 + + = fV V yg g S St x x

Equation (39.22) is usually referred to as the "Dynamic Equation". In this equation, the

first term on the left hand side represents the local acceleration, the second term

represents the convective acceleration and the third term represents the pressure

gradient. The first term on the right hand side represents weight component (effect of

channel slope) while the second term represents the resistance effect due to shear

between the water and the channel surface. For steady, non-uniform flows, local

acceleration is zero and Eq. (39.22) reduces to

( )2

0

V39.23

2g

+ =

f

dy S S

dx.

Substitution of Q = AV leads to

( )

2

02

2

03

2

03

0

2

3

Q

A 2

Q-

gA

1

39.24

1

+ =

+ =

= =

=

f

f

f

f

dy S S

dx g

dA dyor S S

dx dx

dy Q Tor S S

dx gA

S Sdyor

Q TdxgA

Equation (39.24) is nothing but equation for steady gradually varied flow when the

energy correction factor 1 = .


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For steady, uniform flows, local and convective acceleration are zero and the flow

depth, y does not vary with x. Therefore, Eq. (39.22) reduces to

( )0 0 39.25 =fS S

Flood routing problem is defined as: given (i) the channel characteristics (slope, shape

parameters, roughness coefficient) and (ii) the flood discharge or the stage hydrograph

at an upstream section, determine the flood discharge and the stage hydrographs at

any downstream section. This is same as solving for the temporal and spatial variations

of Q and y given the (i) channel characteristics, (ii) initial conditions (Q and y at all

points in the channel at t = 0) and (iii) Boundary condtions (Q or y variation at x = 0 for

all t).

Flood routing based on the solution of complete equations for mass and momentum

conservation (Eqs. 39.7 and 39.22) is termed as "Dynamic Routing".

Flood rating in which the first two terms (acceleration terms) on the left hand side of

Eqs. 39.22 are negelected is termed as "Zero-Inertia Routing".

Flood routing in which equations 39.7 and 39.25 are solved together is termed as

"Kinematic Wave Routing". Many times Zero - Inertia Routing and Kinematic Wave

Routing methods are adopted to avoid computational difficulties.

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Gradually Varied Unsteady Flow.pdf