1.4 computation of gradually varied flow

67
1.4 Computation of Gradually Varied Flow The computation of gradually-varied flow profiles involves basically the solution of dynamic equation of gradually varied flow. The main objective of computation is to determine the shape of flow profile. Broadly classified, there are three methods of computation; namely: 1. The graphical-integration method, 2. The direct-integration method, 3. Step method. The graphical-integration method is to integrate the . dynamic equation of gradually varied flow by a graphical procedure. There are various graphical integration methods. The best one is the Ezra Method.

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Page 1: 1.4 Computation of Gradually Varied Flow

1.4 Computation of Gradually Varied Flow

• The computation of gradually-varied flow profiles involves basically the solution of dynamic equation of gradually varied flow. The main objective of computation is to determine the shape of flow profile.

• Broadly classified, there are three methods of computation; namely:

1. The graphical-integration method,

2. The direct-integration method,

3. Step method.

The graphical-integration method is to integrate the . dynamic equation of gradually varied flow by a graphical procedure. There are various graphical integration methods. The best one is the Ezra Method.

Page 2: 1.4 Computation of Gradually Varied Flow

The direct-integration method: Thje differentia equation of GVF can not be expressed explicitly in terms of y for all types of flow cross section; hence a direct and exact integration of the equation is practically impossible. In this method, the channel length under consideration is divided into short reaches, and the integration is carried out by short range steps.

• The step method: In general, for step methods, the channel is divided into short reaches. The computation is carried step by step from one end of the reach to the other.

• There is a great variety of step methods. Some methods appear superior to others in certain respects, but no one method has been found to be the best in all application. The most commonly ısed step methods are:

1. Direct-Step Method,

2. Standart-step Method.

Page 3: 1.4 Computation of Gradually Varied Flow

1.4.1 Direct-Integration Methods

We have seen that the flow equation

is true for all forms of channel section, provided that the Froude number

Fr is properly defined by the equation:

and the velocity coefficient, α = 1, channel slope q is small enough so

that cos θ =1.

We now rewrite certain other elements of this equation with the aim of

examining the possibility of a direct integration. It is convenient to use

here the conveyance K and the section factor Z.

21 r

fo

F

SS

dx

dy

3

222

A

T

g

Q

A

T

g

VFr

Page 4: 1.4 Computation of Gradually Varied Flow

The Conveyance of a channel section, K:

If a large number of calculations are to be made, it is convenient to

introduce the concept of “conveyance” of a channel in order to calculate

the discharge. The “conveyance” of a channel indicated by the symbol K

and defined by the equation

This equation can be used to compute the conveyance when the

discharge and slope of the channel are given.

When the Chézy formula is used:

where c is the Chézy’s resistance factor. Similarly when the Manning

formula is used

21/KSQ S

QK or

3/2ARn

1K

2/1CARK

Page 5: 1.4 Computation of Gradually Varied Flow

• When the geometry of the water area and resistance factor or

roughness coefficient are given,

One of the above formula can be used to calculate K. Since the

Manning formula is used extensively in most of the problems, in

following discussion the second expression will be used. Either K

alone or the product Kn can be tabulated or plotted as a function of

depth for any given channel section: the resulting tables or curves

can then be used as a permanent reference, which will immediately

yields values of depth for a given Q, S and n. This conveyance factor

concept is widely used for uniform flow computation.

Since the conveyance K is a function of the depth of flow y, it may be

assumed that:

where

C1 = coefficient, and

N = a parameter called hydraulic exponent

NyCK 1

2

Page 6: 1.4 Computation of Gradually Varied Flow

Taking the logarithms of both sides of above eq. And then differentiating

with respect to y:

On the other hand: from Manning’s Eq.

Taking the logarithm of both sides and differentiating with respect to y:

12 nCnyNnK

3/2ARn

1K

dy

nKdy2N

y2

N

dy

nKd

nnnAnRnK 3

2

dy

dA

A

1

dy

dR

R3

2

dy

nKd

Page 7: 1.4 Computation of Gradually Varied Flow

• The derivative of hydraulic radius with respect to y:

dy

dP

P

R

P

T

dy

dA

P

A

dy

dA

P

1

dy

dR2

A

T

dy

dP

P

1

3

2

A

T

3

2

A

T

dy

dP

P

R

P

T

R

1

3

2

dy

nKd

dy

nKdy2N

y2

N

dy

nKd

dy

dPR2T5

A3

1

dy

nKd

dy

dPR2T5

A3

y2N

Page 8: 1.4 Computation of Gradually Varied Flow

• This is the general Eq. for the hydraulic exponent N. If the channel

cross section is known N can be calculated accordingly provided that

the derivative dP/dy can be evaluated. For most channels, except for

channels with abrupt changes in cross-sectional form and for closed

conduits with gradually closing top, a logarithmic plot of K as ordinate

against the depth as abscissa will appear approximately as straight

line. Thus if any two points with coordinates (K1, y1) and (K2, y2) are

taken from the straight line, the approximate value of N may be

computed by the following Eq.

21

21

y/ylog

K/Klog2N

Lny

LnK

y1 y2

K1

K2

Page 9: 1.4 Computation of Gradually Varied Flow

• For wide rectangular channels:

• The Chézy Equation gives the value of K as:

• On the other hand the Manning Equation gives the value of K as:

yR

3NybCyybCRACK 322222222

310

111 3102

2

3422342

2

2

/

///

N

ybn

yybn

RAn

K

Page 10: 1.4 Computation of Gradually Varied Flow

The Section Factor: Z

• The Section Factor: Z is especially used for critical flow computation.

However it becomes useful to transform the GVF equation into a

form which can be integrated directly.

• We now consider the Froude number

• Since this term equals unity at critical flow, then

• Ac, Tc are the values of A and T at critical flow..

3

22

rA

T

g

QF

c

3

c

2

T

A

g

Q

Page 11: 1.4 Computation of Gradually Varied Flow

• By definition we introduce the concept of section factor as

• The section factor for critical flow becomes

• or for critical flow only.

Since the section factor z is a function of depth, the equation

indicates that there is only one possible critical depth for maintaining

the given discharge in a channel and similarly that, when the depth is

fixed, there can be only one discharge that maintains the critical flow

and makes the depth critical in that given channel section.

T

AZ

32 DA

T

AAZ 222 or DAZ and

g

Q

T

AZ

c

cc

232

g

QZ c

Page 12: 1.4 Computation of Gradually Varied Flow

• Since the section factor z is a function of the depth of flow y, it may

be assumed that

Where C2 is a coefficient and M is a parameter called the hydraulic

exponent. Taking the logarithms on both sides of above equation and

then differentiating with respect to y:

Now taking the logarithms on both sides of Eq.

MyCZ 2

2

T

AZ

32

nyd

nZd

dy

nZdyM

y

M

dy

nZd

22

2

nTnAnZ 32

Take the derivative with respect to y:

Page 13: 1.4 Computation of Gradually Varied Flow

• Then M becomes::

• This is a general equation for the hydraulic exponent M, which is a

function of the channel section and the depth of flow.

• For a given channel section M can be computed directly from this

expression, provided that the derivative dT/dy can be evaluated.

However, approximate values of M for any channel section may be

obtained from the following equation:

dy

dT

TT

Ady

dT

Tdy

dA

Ady

nZd

2

1

2

3

2

1

2

3

dy

dT

T

AT3

A

yM

Page 14: 1.4 Computation of Gradually Varied Flow

where z1 and z2 are section factors for any two depths y1 and y2 of

the given section, i.e.:

For rectangular sections: A=by, and T=b, hence section factor

becomes:

21

212yy

ZZM

/log

/log

1

3

12

1T

AZ

2

3

22

2T

AZ and

33233

2 Mybb

ybZ

Page 15: 1.4 Computation of Gradually Varied Flow

• Now, coming back to the computation of gradually varied flow; the

term, (S0-Sf )may be written as

• On the other hand and for uniform flow

• Therefore

• And hence

• Also, it is assumed that and

0

00 1S

SSSS f

f

fS

QK

0

0S

QK

2

2

22

22

0 K

K

KQ

KQ

S

S o

o

f /

/

2

2

00 1K

KSSS o

f

NyCK 1 NyCK 010

Page 16: 1.4 Computation of Gradually Varied Flow

• Therefore

• On the other hand:

• because for critical flow:

• Also it is assumed that

N

oof

y

yS

K

KSSS 11 02

2

00

2

2

2

2

3

22 1

Z

Z

Zg

Q

A

T

g

QF c

r

g

QZ c

22

MyCZ 2

2

Page 17: 1.4 Computation of Gradually Varied Flow

Therefore Froude number can be written as:

Finally the Gradually Varied Flow Eq. takes the form:

M

cM

Mc

ry

y

yC

yCF

2

22

M

c

N

o

r

f

y

y

y

y

SF

SS

dx

dy

1

1

102

0

Page 18: 1.4 Computation of Gradually Varied Flow

Bresse’s Method:

• Applicable only to wide rectangular channels. Assumptions:

• Wide rectangular channel,

• Chézy’s formula is applicable and Chézy’s C is constant.

• From Chezy’s eq’n: , A=by

• Therefore

Therefore, the GVF Equation becomes:

Manipulating this equation we can integrate it.

RCAK yR

3NybCyybCRACK 322222222

3Mybb

yb

T

AZ 32

3332

3

c

3

o

o

y

y1

y

y1

Sdx

dy

Page 19: 1.4 Computation of Gradually Varied Flow

Adding and subtracting

Rearrange it and multiply and

divide by

dy

y

y1

y

y1

dxS3

o

3

c

o

dy

y

y1

y

y

y

y

y

y1

dxS3

o

3

c

3

o

3

o

o

3

0

y

y

dy

y

y

y

y

.

y

y1

y

y

y

y

1dxS3

o

3

o

3

o

3

o

3

c

o

3

0

y

y

Page 20: 1.4 Computation of Gradually Varied Flow

or

Let change variables

Integrating

dy

1y

y

1y

y

1dxS3

o

3

o

c

o

dy

y

y1

y

y1

1dxS3

o

3

o

c

o

duydyy

yU o

o

duyU1

y

y1

1dxS o3

3

o

c

o

cu

du

y

yuyxS

o

coo

3

3

11

Page 21: 1.4 Computation of Gradually Varied Flow

Let f be the integral of

Therefore, the distance between any two

section becomes:

31 u

duf

12

3

3

1

1

1

6

1

1

1

2

2

3

uu

uun

u

dutanf

12

3

1212 1 ffo

c

o

o

y

yuu

S

yxxL

Page 22: 1.4 Computation of Gradually Varied Flow

Bakhmeteff Method

• Bakhmeteff improved Bresse’s method as follows:

• Let

Therefore Then we can write that:

On the other hand, we can wite that:

Hence dx becomes:

2

r

o

f

o2

r

fo

F1

s

s1

sF1

ss

dx

dy

o

fr

S

SF 2

o

f

o

f

o

S

S

S

S

Sdx

dy

1

1

dy

S

S

S

S

Sdx

f

f

0

0

0 1

11

N

of

y

y

S

S

0

Page 23: 1.4 Computation of Gradually Varied Flow

Add and subtract

and rearrange as:

N

o

N

o

o

y

y

y

y

s

dydx

1

1

N

o

N

o

N

o

N

o

o y

y

y

y

y

y

y

y

s

dydx

1

1

N

y

y

0

dy

y

y

y

y

dxSN

o

N

o

1

1

10

Page 24: 1.4 Computation of Gradually Varied Flow

Now multiple and divide by

and rearrange as:

Let

dy

y

y

y

y

y

y

y

y

dxSN

o

N

o

N

o

N

o

1

1

10

N

y

y

0

dy

y

ydxS

N

o

1

110

duydyy

yU o

o

Page 25: 1.4 Computation of Gradually Varied Flow

• Integrating

Therefore, the distance between any two

section becomes:

dy

uS

ydx

N

1

11

0

0

Nu

duu

S

yx

11

0

0

N,uFN,uF1uus

yxxL 212

o

o

12

Page 26: 1.4 Computation of Gradually Varied Flow

• Here we are assuming that β is constant

and hence

Therefore becomes:

3

2

0

3

2

0

2

gA

TQ

S

S

gA

TQ

S

SF

f

f

r

fRSCAQ RAC

QSf 22

2

P

T

g

CS

gP

TCS

gA

TQ

Q

RACS 2

0

2

0

2

2

2

22

0

Page 27: 1.4 Computation of Gradually Varied Flow

• If T/b is constant, then β will be constant.

• For example, for a rectangular channel,

T=b, and P=b+2y, hence

• And T/b becomes constant only for wide

rectangular channel.

b/y21

1

b

y21b

b

y2b

b

P

T

Page 28: 1.4 Computation of Gradually Varied Flow

• For a triangular channel: T=2zy

1z

Tyz12P 2

constant

22 112

2

z

z

yz

zy

P

T

Page 29: 1.4 Computation of Gradually Varied Flow

• The integral in the equation is designated

by F(u,N), that is

• Varied-Flow Function

• The values of F(u,N) have been obtained

numerically, and are given in tabular form

for N ranging from 2.2 to 9.8.

u

o

Nu1

duN,uF

Page 30: 1.4 Computation of Gradually Varied Flow

Example on Bakhmeteff Method: Water is taken from a lake by a triangular channel,with side slope

of 1V:2H. The channel has a bottom slope of 0.01, and a Manning’s

roughness coefficient of 0.014. The lake level is 2.0 m above the

channel entrance, and the channel ends with a free fall. Determine :

1.The discharge in the channel,

2.The water-surface profile and the length of it by using the direct-

integration method.

2 m

n=0.014, S0=0.01

1

2

X

Page 31: 1.4 Computation of Gradually Varied Flow

a) To determine the discharge, assume that the channel slope is

steep. Then at the head of steep slope, the depth is critical depth.

Hence 2=Emin

2

AD 4y,T ,

5

AR ,52P ,2 2 y

T

y

PyyA

/m 14.34Q

& slope steep m .y m .y .y

..

14.34 n

AQ

:depth normal the compute slet' /m 14.34Q

..

.

g

Q m ..A

m. .y

3

c0

/

0

//

0

3

22

c

C

s

y

yyS

P

A

s

xT

Ax

yy

yD

yE

o

o

c

c

CC

CC

CC

6122517161

01052

2

0140

2

9720614

125125612

61⇒4

5

422

38

32

0

2

0

2

0

32

0

332

Page 32: 1.4 Computation of Gradually Varied Flow

2. The water-surface profile and the length of it by using the direct-

integration method.

• The water surface profile will be S2 type, and the depth of flow will

change between the limits: 1.01y0≤y≤yc.

• Since T/P is constant for triangular channels, we can use

Bakhmeteff’s method. Let’s compute the value of :

803485819

94664010

59564

54760

2

2

0

2

0

...

...

.C

m 5.48P m, 4.9T m, .R

m 3.0A SRCAQ

0

2

0000

x

xx

P

T

g

CS

P

T

g

CS

Page 33: 1.4 Computation of Gradually Varied Flow

• Let’s obtain the value of N

33353

16

525

24523

2

253

2

2

.

52yP , 5

yR ,

N

xy

yxyx

yN

dy

dPRT

A

yN

N,uFN,uF1uus

yxxL 212

o

o

12

Page 34: 1.4 Computation of Gradually Varied Flow

• To obtain the values of F(u,N) we have to use tables with

interpolation.

• Interpolation:

• For u=1.30 and N=5 → F(1.30, 5)=0.100 and

• For u=1.32 and N=5 → F(1.32, 5)=0.093

• Hence by interpolation:

• For u=1.31 and N=5 → F(1.31, 5)=0.0965

y (m) u=y/y0 F(u,5.333)

1.6 1.31 0.0811

1.237 1.01 0.622

D -0.296 0.5411

Page 35: 1.4 Computation of Gradually Varied Flow

• Similarly:

• For u=1.30 and N=5.4 → F(1.30, 5.4)=0.081 and

• For u=1.32 and N=5.4 → F(1.32, 5.4)=0.093

• Hence by interpolation:

• For u=1.31 and N=5.333 → F(1.31, 5)=0.0811

• Therefore, the length of S2 profile is 150 m.

m .....

.

,,

34149541108312960010

2251

1 21212

L

NuFNuFuus

yxxL

o

o

Page 36: 1.4 Computation of Gradually Varied Flow

Ven Te Chow Method

• Ven Te Chow improved Bakhmeteff’s method for all types of cross

sections as follows:

The gradually-varied flow equation can be written as

multiply by both numerator and

denominator by and second term in numerator by

M

c

N

o

o

y

y

y

y

Sdx

dy

1

1

dyyy

yy

Sdx

N

o

M

c

o /

/

1

11

N

oy

y

dyyy

yyyyyyyy

Sdx

N

o

M

o

M

c

N

o

N

o

o

1

1 0

/

////

M

oy

y

0

Page 37: 1.4 Computation of Gradually Varied Flow

Now let then above equation becomes:

:

Adding and subtracting 1 to the numerator, dx becomes::

dy

yy

y

y

y

y

y

y

y

y

Sdx

N

o

N

o

M

o

M

o

c

N

o

o

/1

1

duydyy

yu o

o

duu

uuy

y

s

ydx

N

NMN

M

o

c

o

o

1

Page 38: 1.4 Computation of Gradually Varied Flow

Integrate

duu

uy

yu

S

ydx

N

MN

M

o

cN

o

o

1

11

duu

u

y

y

uS

ydx

N

MNM

o

cN

o

o

11

11

Rearrange to obtain

.constduu1

u

y

y

u1

duu

s

yx

u

o

u

o

N

MNM

o

c

N

o

o

Page 39: 1.4 Computation of Gradually Varied Flow

• The first integral on the RHS of above eq. is designated by or

The second integral may also be expressed in the form of the

varied-flow function. Let

and

FunctionFlow - Varied,

u

o

Nu

duNuF

1

/ JNuv 1MN

NJ

NJvu / dvvN

Jdu N

NJ

v

o

J

v

o

J

N

NJMN

N

Ju

o

N

MN

v

dv

N

Jdv

v

v

N

Jdu

u

u

111

Page 40: 1.4 Computation of Gradually Varied Flow

• Therefore:

or

const.

u

o

v

o

J

M

o

cN

o

o

v

dv

N

J

y

y

u

duu

S

yx

11

const. ,,

JvF

y

y

N

JNuFu

S

yx

M

o

c

o

o

NuFNuFuuS

yxxL

o

o ,, 121212

JvFJvFy

y

N

JM

o

c12

,

Page 41: 1.4 Computation of Gradually Varied Flow

• Where the subscripts 1 and 2 refer to sections 1 and 2 respectively.

This eq. contains varied-flow functions, and its solution can be

simplified by the use of the varied-flow-function table. This table

gives values of F(u,N) for N ranging from 2.2 to 9.8. Replacing values

of u and N by corresponding values of .F(v,J).

• In computing a flow profile, first the flow in the channel is analyzed,

and the channel is divided into a number of reaches. Then the length

of each reach is computed by the above Eq. from known or assumed

depths at the ends of the reach. The procedure of the computation is

as follows:

1. Compute the normal depth yo and critical depth yc from the given

values of Q and so, n

oo ySRn

AQ 32 /

c3

2

y1A

T

g

Q

Page 42: 1.4 Computation of Gradually Varied Flow

2. Determine the hydraulic exponents N and M for an estimated

average depth of flow in the reach under consideration. It is assumed

that the channel section under consideration has approximately

constant hydraulic exponents

• If any two depths in the section are known then

• N and M can be computed from above expressions or use the

general formulas if dP/dy and dT/dy can be evaluated.

3. Compute J by

21

21

y/ylog

K/Klog2N

21

21

y/ylog

z/zlog2M

1MN

NJ

Page 43: 1.4 Computation of Gradually Varied Flow

5. Compute values and at the two end

sections of the reach

6. From the varied-flow-function table, find values of F(u,N) and F(v,J).

7. Compute the length of reach by the given equation obtained above.

• In doing so a table can be prepared as follows

0yyu

JNuv /

y u v F(u,N) F(v,J)

y1

y2

y3

D

Page 44: 1.4 Computation of Gradually Varied Flow

Example on Ven Te Chow’s Method:

• Water flows at a uniform depth of 3 m. in a trapezoidal channel. The

trapezoidal section has a bottom width of 5 m., and side slope of

1H:1V. The channel has a bottom slope of 0.001, and a Manning’s

roughness coefficient of 0.013. The channel ends with a free fall.

Determine the water-surface profile and the length of it by using the

direct-integration method.

b=5 m

1

1

y0=3 m

S0=0.001, n=0.013

Page 45: 1.4 Computation of Gradually Varied Flow

First, determine the discharge, and then the type of the slope:

To determine critical depth yc

0

3/2 SRn

AQ

222 m 2435x3 zybyA

m. 46.13232512 2 xzybP

sP

/m 860.001 1.783 0.013

24Q m 1.783

AR 32/3

m 6.2yerror and trial

92.75381.9

86

25

5

c

23232

by

y

yy

T

A

g

Q

c

cc

c

c

y0=3 m > yc=2.6 m Therefore it is mild slope. Hence M2 profile occurs.

The depth changes between the limits yc = 2.6 m ≤ y ≤ 0.99y0 = 2.97 m

Page 46: 1.4 Computation of Gradually Varied Flow

• Let’s compute the hydraulic exponents

T

AA=Z

yy

ln

ZZ

ln

2=M Rn

A=K

yy

ln

KK

ln

2=N

1

2

1

2

3/2

1

2

1

2

JvFJvF

N

J

y

yNuFNuFuu

S

yxxL

M

c ,,,, 12

0

1212

0

012

N/J

0

u v y

yu

Page 47: 1.4 Computation of Gradually Varied Flow

y (m) A (m2) P (m) R (m) T (m) K Z N M J

2,60 19,76 12,35 1,60 10,2 2078,9 27,50 3,71 3,55 3,19

2,97 23,67 13,40 1,77 10,94 2660,7 34,82

Computation of Hydraulic Exponents N and M, n=0.013

Computation of Length of M2 profile by using

Direct-Integration Method in one step

y (m) u v F(u,3.7) F(v,3.2) L (m)

2,60 0,87 0,85 1,05 1,043 -341,781

2,97 0,99 0,99 1,7875 2,017

L=342 m

y0=3.0 m , yc=2.6 m, n=0.013, S0=0.001, N=3.7, M=3.6, J=3.2

Page 48: 1.4 Computation of Gradually Varied Flow

1.4.2 Step Methods for Uniform and Irregular

Channels

The Direct Step Method (DSM)• In direct step method, distance is calculated from the depth.

• It is only applicable to prismatic channels. The energy equation

between two sections is:

• Assume that a1=a2=1 z1-z2=S0Dx, Then above

equation can be written as:

• Solving for Dx:

xSg

Vyz

g

Vyz f D

22

2

2222

2

1111 aa

xSEExS f DD 210

fo SS

EEx

D 12

Page 49: 1.4 Computation of Gradually Varied Flow

• The distance between two sections can be calculated from:

• The slope of energy-grade line can be computed from

Manning’s Equation as:

fofofo SS

g

vy

SS

E

SS

EEx

D

D

D

2

2

12

a

34

22

/R

VnSf 212

1fff SSS

Page 50: 1.4 Computation of Gradually Varied Flow

Consider the M2 profile in a channel section. Suppose that we want to

calculate the length of M2-profile for a given discharge Q, and channel

section.

• We know that the depth will be changing between the limits yc≤y <y0.

• On the other hand the distance between two sections can be written as:

NDL

CDL

y01yc

O

A

M2

yc

Page 51: 1.4 Computation of Gradually Varied Flow

and

• Start from a control section. In this case the control section is the downstream section where critical flow occurs.

• Nominate a series values of y, between the range of yc≤y <y0

• Calculate the values of R, V, E, and Sf corresponding to these assumed depths.

• Calculate Δx for each interval between successive values of y

fofofo SS

g

vy

S

E

S

EEx

-

2

S-S-

-

2

12

D

D

D

a

( )2f1ff S+S2

1=S 3/4

22

fR

Vn=S

Page 52: 1.4 Computation of Gradually Varied Flow

• NOTE:

• If flow is subcritical computation is from downstream towards

upstream.

• If flow is supercritical computation is from upstream towards

downstream.

• If one of the depth is uniform depth for the section under

consideration, then 1% off value of normal depth y0 must be taken.

For example for profiles like M1 and S2 where normal depth is

approached asymptotically from above 1.01 y0, and for profiles such

as M2 and S3, where normal depth is approached asymptotically

from belove 0.99 y0 must be taken.

Page 53: 1.4 Computation of Gradually Varied Flow

y

(m)

A

(m2)

P

(m)

R=A/P

(m)

U

(m/s)

E

(m)

Sf Dx

(m)

x

(m)

Yc

Y1

Y2

y3

fS

The best thing is to prepare a table as follows:

Page 54: 1.4 Computation of Gradually Varied Flow

The Standard-Step Method

In this method, the depth is calculated from distance.

Applicable to both nonprismatic and prismatic channels. It is a trial and

error process

•Assume y2, compute H2:

•Also compute:

•Where

•Compare H2 to H2’:

•if then assumed y2 is O.K.

•But, if , then assume another value of y2

g2

VyzH

2

2222

( ) xΔS+S2

1+H=h+H=H′ 2f1f1f12

xΔS2

1=h ff

22 HH

12 H′H ≠

Page 55: 1.4 Computation of Gradually Varied Flow

Improve your initial estimate by this amount:

For natural rivers, instead of depth y, it is preferable to use the height h

of the water level above some fixed datum. This height, h=z+y, is

known as the STAGE. Hence total head at a section can be written as:

xR

s

2

3Fr1

Hy

2

e2

2

e

2

2 D

D22 -HHHewhere

g2

Vα+h=

g2

Vα+z+y=H

22

Page 56: 1.4 Computation of Gradually Varied Flow

• In nonprismatic channels, the hydraulic elements are no langer

independent of the distance along the channel

• In natural channels, it is generally necessary to conduct a field

survey to collect data required at all sections considered in the

computation.

• The computation is carried on by steps from station to station where

the hydraulic characteristics have been determined. In such cases

the distance between stations is given, and the procedure is to

determine the depth of flow at the stations.

Page 57: 1.4 Computation of Gradually Varied Flow

Step Method-Divided Channels

• In this section we consider cases where the channel cross section is divided into distinct

regions having distinct flow characteristics.

• The most common example of this situation is the one shown in Fig. below, the case of

overbank flow, when the flow over the berms has a different depths and the surface possible

a different roughness, from those existing in the main channel.

Page 58: 1.4 Computation of Gradually Varied Flow

• If the channel is straight the water-surface level will remain substantially constant over the

whole section of flow, since the hydrostatic pressure must remain constant along any

horizontal line drawn across the section.

• However, the distinct regions of flow shown in Figure will almost certainly have different

velocities and velocity heads; the problem then is to define a total head H applicable to the

entire cross section.

• The solution is to use the energy coefficient, a; the total head line then extends across the

whole water surface, a distance of above it, as shown in Fig. above. This total head line,

and any head losses deduced from it, are assumed to be applicable to the section as a

whole, and also to each of the individual subsections.

• The algebra required is conveniently handled by the use of the conveyance K. The energy

coefficient , a, is defined by the equation

• …………………………………... Eq.(1)

• Where subscripts 1, 2, 3,… indicate the distinct subsections of the flow

a2

1

3

1

3

1

2

1

1

3

m

1

3

1

A

Q

Q

A

A

A

Page 59: 1.4 Computation of Gradually Varied Flow

Now since we are assuming that the same value of friction slope Sf applies to each

subsection, it follows from the definition of the conveyance K , that

……………………………… Eq.(2)

And therefore that

…………… Eq.(3)

1

1

n

n

3

3

2

2

1

1

K

Q

K

Q...

K

Q

K

Q

K

Q

a2

1

3

1

3

1

2

1

A

K

K

A

Page 60: 1.4 Computation of Gradually Varied Flow

Since each element of Eq. (2) is equal to (Sf)1/2 , it follows that

………………………. Eq.(4)2

1

2

2

1

1

f

K

Q

K

Q

S

Where Q is the total flow. Thus a and Sf, the two factors which are of critical importance in the tabulation,

can be calculated without explicitly evaluating the discharges etc. The values of K which are to be inserted

in Eqs. (3) and 4) are obtained from the Manning equation as

an

3/21AR

nK

We now have all the information required to proceed with the computation.

For a typical flood-level computation shown in Fig.(1), the necessary tabulation is as follows

It is seen that at each major section a separate line is devoted to each subsection.

These lines are labeled M.C. (main channel) and L.B. (left berm) in column 2.

Again is assumed that a plan of each section is available, from which, given the water level, A and P

can be determined for each subsection.

The liquid interface between zones 1 and 2, shown as a broken line in Fig. (1), is not

normally included in estimates of the wetted perimeter P because the shear stress along

this interface is so much lower than at the solid boundaries. ,

The calculation proceeds upstream from known conditions at river km 1.

Page 61: 1.4 Computation of Gradually Varied Flow

River

Km

Sub

sec

Stage

,h

(m)

A

(m2)

P

(m)

R

(m)

R(2/3) n K K3/A2

a Vm(m/s)

H

(m)

(Sf)av

Dx

(m)

hf H’

(m)

1 M.C

L.B 1.32

Total …….. ……. ……..

2 M.C 1.25

L.B

Total

3

Page 62: 1.4 Computation of Gradually Varied Flow

1.5 The Discharge Problem

(Side-Channel Spillways)

2/3

ax xCHQ

A

A

x

L

A side-channel carries water away

from an overflow spillway in a channel

parallel to the spillway crest as

shown in figure.

The discharge over the entire width (L) of

an overflow spillway can be determined by

Eq.(1), and the discharge through any

section of the side channel at a distance x

from the upstream end of

the channel is

Eq.(1)

Page 63: 1.4 Computation of Gradually Varied Flow

The side-channel spillway must provide a slope steep enough to carry away the accumulating

flow in the channel. However, a minimum slope and depth at each point along the channel is

desired in order to minimize construction costs. For this reason, an accurate water surface

profile for the maximum design discharge is important in the side-channel spillway design.

The flow profile in the side channel cannot be analyzed by the energy principle

(i.e., gradually-varied flow profile) because of the highly turbulent flow conditions

that cause excess energy loss in the channel.

However, an analysis based on the momentum between two adjacent sections,

a distance of ∆x apart, in the side channel,

Eq.(2)

where r is the density of water, V is the average velocity, and Q is the discharge

at the upstream section. The symbol ∆ signifies the incremental change at

the adjacent downstream section.

QVVVQQF rDDr

Page 64: 1.4 Computation of Gradually Varied Flow

The forces represented on the left-hand side of Eq.(2) usually include the weight component

of the water body between the two sections in the direction of the flow ,

the unbalanced hydrostatic forces

and a friction force, Ff, on the channel bottom. Here A is the water cross-sectional area, is the distance

between the centroid of the area and the water surface, and q is the angle of the channel slope.

The momentum equation may thus be written as

Eq.(3)

qDr sinxgA

θyyAAgρθygAρ cosΔΔcos

y

fFθyyAAgρygAρθxgAρ cosΔΔsinΔ QVVVQQ rDDr

Let for a reasonably small angle, and ,

and ; the above equation may be simplified to

Eq.(4)

θSo sin 212121 / ,Δ , VVQQAVVVQQ

xASF ff D

xSxSQ

QVV

QQg

VVQy f0

1

2

21

211 DD

DD

D

Page 65: 1.4 Computation of Gradually Varied Flow

Where Dy is the change in water surface elevation between the two sections.

This equation is used to compute the water surface profile in the side channel.

The first term on the right-hand side represents the change in water surface

elevation between the two sections resulting from the impact loss caused by

the water falling into the channel. The middle term represents the change from

the bottom slope, and the last term represents the change caused by friction

in the channel. Relating the water surface profile to a horizontal datum, we may write

Eq.(5)

xSQ

QVV

QQg

VVQxSyz f

1

2

21

2110 D

DD

DDD

21 QQ 0Q DNote that when or when , Eq.(5) reduces to

Eq.(6)

which is the energy equation for constant discharge in an open channel as derived in

Computation of Gradually-Varied Flow.

xSg2

V

g2

Vz f

2

1

2

2 D

D

Page 66: 1.4 Computation of Gradually Varied Flow

Example

A 6-m overflow spillway discharge water into a side-channel spillway with a horizontal bottom slope. If

the overflow spillway (C=2.043 m1/2/s) is under a head of 1.28 meter, determine the depth change from

the end of the side channel (after it has collected all of the water from the overflow spillway) to a point

1.5 meter upstream. The concrete (n=0.013) side channel is rectangular with a 3-m bottom. The water

passes through critical depth at the end of the side channel.

Solution

The flow at the end of the side channel (Eq.(1)) is

The flow at a point 1.5 meter upstream (Eq.(1)) is

Solving for critical depth at the end of the channel, we have

sec/ 75.1728.16043.2 32/32/3mHCLQ a

sec/ 31.1328.1043.25.4 32/32/3mHxCQ a

m 528.181.9/92.5/

//m 92.53/75.17/

3/123/12

3

gqy

msbQq

c

Page 67: 1.4 Computation of Gradually Varied Flow

DX Dy y A=3y Q (m3/s) V (m/s) Q1+Q2 V1+V2 DQ DV R=3y/(3+2y) Sf Dy

0 1.528 4.584 17.75 3.872

1.5 -0.3 1.828 5.484 13.31 2.427 31.06 6.299 4.44 1.445 0.824 0.001289 -0.755

DX Dy y A Q V Q1+Q2 V1+V2 DQ DV R Sf Dy

0 1.528 4.584 17.75 3.872

1.5 -0.755 2.283 6.849 13.31 1.943 31.06 5.816 4.44 1.929 0.905 0.000729 -0.819

DX Dy y A Q V Q1+Q2 V1+V2 DQ DV R Sf Dy

0 1.528 4.584 17.75 3.872

1.5 -0.819 2.347 7.041 13.31 1.890 31.06 5.763 4.44 1.982 0.915 0.00068 -0.825

DX Dy y A Q V Q1+Q2 V1+V2 DQ DV R Sf Dy

0 1.528 4.584 17.75 3.872

1.5 -0.825 2.353 7.059 13.31 1.886 31.06 5.758 4.44 1.987 0.916 0.000675 -0.826

DX Dy y A Q V Q1+Q2 V1+V2 DQ DV R Sf Dy

0 1.528 4.584 17.75 3.872

1.5 -0.826 2.354 7.062 13.31 1.885 31.06 5.757 4.44 1.987 0.916 0.000675 -0.826

The solution method employs a finite-difference solution scheme (Eq.(4)) and an iterative processs

can be employed to compute the upstream depth. The upstream depth (or depth change, ∆y) is

estimated, and Eq.(4) is solved for a depth change. The two depth changes are compared, and a

new estimate is made if they are not nearly equal. Table... displays the solution. Because side-

channel spillway profile computations involve implicit equations, computer algebra software (e.g.,

Mathcad, Maple, and Mathematic) or spreadsheet programs will prove very helpful.