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Hydraulics

Prof. B.S. Murty

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 the discharge 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.

Indian Institute of Technology Madras

Hydraulics

Prof. B.S. Murty

1

2

Water Surface V1 A1 ___ y1 y1

V1 A1 ___ y1 y1 x x1

Flow

A C.G

__ y

Bed S0 x2

x

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

d dt

x2

x1

Adx

+ A2V2 - AV1 = 0 1

( 39.1)

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

A ( x2 - x1 ) + A2V2 - AV1 = 0 1 t

It may be noted that both A and Similarly, treating AV and

( 39.2 )

VA as continous with respect to x and t, and letting t x = x2 - x1 tend to zero, one can get

A are assumed continous with respect to both x and t. t

Indian Institute of Technology Madras

Hydraulics

Prof. B.S. Murty

A AV + =0 t x

Noting that flow rate, Q = AV.

( 39.3)

A Q + =0 t x

( 39.4 )

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 dA y Q =0 + dy t x or T

( 39.5) ( 39.6 )

y Q + =0 t x

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

y A V y + V =0 + t T x x

( 39.7 )

Momentum Equation

Based on the Reynolds transport theorem, momentum equation for the control volume in fig. 39.1 can be written as

FRe s

d 2 = V A dx + V22 A2 - V12 A1 dt x1

x

( 39.8)

in which FRe s = 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

( AV ) FRe s = + ( AV 2 ) ( x ) t x

( 39.9 )

Noting that flow rate Q = AV,

Indian Institute of Technology Madras

Hydraulics

Prof. B.S. Murty

FRe s Q = + ( QV ) ( x ) t x

( 39.10 )

Resultant force FRe s 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

F1 = gA1 y1

y1 = depth to the centroid of area A1.

( 39.11) ,

F2 = gA2 y 2

y 2 = depth to the centroid of area A2.

( 39.12 )

F1 acts in the positive x direction while F2 acts in the negative x direction.

Component of weight of water in the direction of flow =

F3 = g AS0 dx

x1

x2

( 39.13)

Indian Institute of Technology Madras

Hydraulics

Prof. B.S. Murty

Frictional force = F4 = g AS f dx

x1 x2

( 39.14 )

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

g A1 y1 - A2 y 2 x

(

) + gA S

(

0

- Sf ) =

Q + ( AV 2 ) t t

( 39.15)

Or

Q + ( QV ) = - gAy + gA ( S0 - S f t t x

(

)

)

( 39.16 )

Or

Q + QV + gAy = gA ( S0 - S f ) t x

(

)

( 39.17 )

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

1 2 A y + y + 2 Ty -Ay Ay = lim y 0 y y

( )

(

)

( 39.18)

Neglecting higher order terms,

Ay = A y and y y = gA gAy = g Ay x y x x

( )

( 39.19 ) ( 39.20 )

(

)

( )

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

Indian Institute of Technology Madras

Hydraulics

Prof. B.S. Murty

Q QV y + + gA = gA ( S0 - S f ) t x x

( 39.21)

Substitution of Q = AV into Eq. (39.21), subsequent expansion of terms, and further simplification using continuity equation leads to

V V y + V + g = g ( S0 - S f ) t x x

( 39.22 )

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

d V2 + y = S0 - S f dx 2g

( 39.23) .

Substitution of Q = AV leads to

d Q2 + y = S0 - S f 2 dx A 2 g 2 Q dA dy + = S0 - S f gA 3 dx dx dy Q 2T = 1 - = S0 - S f dx gA3 dy S0 - S f = Q 2T dx 1- 3 gA

or or or

( 39.24 )

Equation (39.24) is nothing but equation for steady gradually varied flow when the energy correction factor = 1 .

Indian Institute of Technology Madras

Hydraulics

Prof. B.S. Murty

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

S0 - S f = 0

( 39.25 )

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.

Indian Institute of Technology Madras

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