lecture13 gradually varied flow1 2

7/30/2019 Lecture13 Gradually Varied Flow1 2

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Gradually Varied Flow I+II

Hydromechanics VVR090


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Gradually Varied Flow

Depth of flow varies with longitudinal distance.

Occurs upstream and downstream control sections.

Governing equation:

21

=

o fS Sdy

dx Fr

(previously Sf= 0 was studied)


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Derivation of Governing Equation

Total energy:

2

2uH z y

g= + +

Differentiating with respect to distance:

( )2 / 2= + +

d u gdH dz dy

dx dx dx dx


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=

=

f

o

dH

Sdx

dzSdx

For a given flow rate:

( )2 2 22

3 3

/ 2d u g Q dA dy Q T dy dyFr

dx gA dy dx gA dx dx

= = =

(slope of energy grade line)

(bottom slope)

21

=

o fS Sdy

dx Fr Resulting equation:


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Definition of Water Surface Slope

Water surface slope dy/dx is defined with respect tothe channel bottom.

Hydrostatic pressure distr ibution is assumed

(streamlines should be reasonably straight and parallel).


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The head loss for a specific reach is equal to the

head loss in the reach for a uniform flow having the

same R and u. Manning equation yields.

The slope of the channel is small

No air entrainment

Fixed velocity distribution

Resistance coefficient constant in the reach under

consideration

2 2

4 / 3f

n uS

R=

Assumptions made when solving the gradually varied flow

equation:


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Classification of Gradually Varied Flow Profiles

The following condit ions prevail:

Ify < yN, then Sf > So

Ify > yN, then Sf < So

IfFr> 1, then y < yc

IfFr< 1, then y > yc

IfSf = So, then y = yN


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Water surface profi les may be classif ied with respect to:

the channel slope

the relationship between y, yN, and yc.

Profile categories:

M (mild) 0 < So < Sc

S (steep) So > Sc > 0

C (critical) So = Sc

A (adverse) So < 0


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Gradually Varied Flow

Profile Classification I


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Gradually Varied Flow Profile Classification II


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Mild Slope (M-Profiles)

Profile types:

1: y > yN > yc => So > Sf and Fr< 1

=> dy/dx > 0

2: yN > y > yc => So < Sf and Fr< 1

=> dy/dx < 0

3: yN > yC > y => So < Sf and Fr> 1=> dy/dx > 0

0 < So < Sc


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Steep Slope (S-Profiles)

Profile types:

1: y > yc > yN => So > Sf and Fr< 1

=> dy/dx > 0

2: yc > y > yN => So > Sf and Fr> 1

=> dy/dx < 0

3: yc > yN > y => So < Sf and Fr> 1=> dy/dx > 0

0 < Sc < So


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Final Form of Water Surface Profile

1. y , Sf 0, Fr 0, and dy/dx So

2. y yN, Sf So, and dy/dx 0

3. y yc, Fr 1, and dy/dx

21

=

o fS Sdy

dx Fr

Asymptotic conditions:


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Transition from Subcrit ical to Supercritical Flow


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Transition from Supercritical to Subcrit ical Flow


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Example: Flow into a Channel from a Reservoir


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Flow Controls

determine the depth in channel either upstream or

downstream such points.

usually feature a change from subcritical to supercrit icalflow

occur at physical barriers, for example, sluice gates,

dams, weirs, drop structures, or changes in channelslope

Locations in the channel where the relationship between the

water depth and flow rate is known (or controllable).

Controls:


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Strategy for Analysis of Open Channel Flow

1. Start at control points2. Proceed upstream or downstream depending on

whether subcritical or supercritical flow occurs,

respectively

Typical approach in the analysis:


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Computation of Gradually Varied Flow

21

=

o fS Sdy

dx Fr Governing equation:

Solutions must begin at a control section and proceedin the direction in which the control operates.

Gradually varied flow may approach uniform flow

asymptotically, but from a practical point of view a

reasonable definition of convergence is applied.


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Uniform Channel

Prismatic channel with constant slope and resistance coefficient.

Apply energy equation over a small distance Dx:

2

2o f

d uy S S

dx g

+ =

Express the equation in difference form:

( )2

2o f

uy S S x

g

+ =


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Over the short distance Dx assume that Manningsequation is suitable to describe the frictional losses (S

f

):

2 2

4 / 3f

n uS

R

=

The equation to be solved may be written:

( )

( )

2

2 2 4 / 3

/ 2

/o mean

y u gx

S n u R

+ =


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Dx i

Reach i

x

y i y i+1

( ) ( )

( )

2 2

1

2 2 4 / 3

1/ 2

/ 2 / 2

/

i ii

o i

y u g y u gx

S n u R

+

+

+ + =

All quantities known at i. Assume yi+1

and computeDx i (u i+1 given by the continuity equation).

u i

u i+1


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Example 6.1

A trapezoidal channel with b = 6.1 m, n = 0.025, z = 2, and So =

0.001 carries a discharge of 28 m3/s. If this channel terminates in

a free overfall, determine the gradually varied flow profile by the

step method.

b = 6.1 m

2

1yN


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Solution:

Compute normal water depth.

( )

( )

2 / 3

2

2

1

2 1

2 1

o

N N

N

N N

N

Q AR S n

A b zy y

P b y z

b zy yR

b y z

=

= += + +

+

= + +

yN = 1.91 m


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Compute crit ical water depth:

( )

1/

2

c c c

c c c

c

u QFr

gD A gA T

A b zy y

T b zy

= = =

= +

= +

yc = 1.14 m

yN > y > yc

Mild slope (yN > yc)M2 profile


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Table for step calculation:

y A P R u u2

/2g Sf Sfav Dx S (Dx)1.14 9.55 11.20 0.85 2.93 0.438 0.00670.0058 3 3

1.24 10.64 11.64 0.91 2.63 0.353 0.0049

0.0044 9.3 12.3

1.32 11.54 12.00 0.96 2.43 0.300 0.0039

and so on( ) ( )2 2

1

, 1/ 2

/ 2 / 2i i

i

o f i

y u g y u gx

S S

+

+

+ + =

( ), 1/ 2 , 1 ,1

2f i f i f i

S S S+ += +

2 2

4 / 3f

n uS

R=


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Other Solution Methods

Problem with the step method is that the water depths isobtained at arbitrary locations (i.e., the water depth is not

calculated at fixed x-locations).

By direct integration of the governing equation this problemcan be circumvented.

Different approaches for direct integration:semi-analytic

trial-and-error

finite difference


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Semi-Analytic Approach

Find solution in terms of closed-form functions (integrals).

Employ suitable approximations to these functions or

some look-up tables.

Approach OK for channels with constant properties.

(for more information, see French)


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Trial-and-Error Approach

Well-suited for computations in non-prismatic channels.

Channel properties (e.g., resistance coefficient and

shape) are a function of longitudinal distance.

Depth is obtained at specific x-locations.

Apply energy equation between two stations locatedD

xapart (z is the elevation of the water surface):

2

2 2

1 2

1 2

2

2 2

f e

f e

uz S x hg

u uz z S x h

g g

+ =

+ = + + +

he: eddy losses


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Equation is solved by trial-and-error (from 2 to 1):

1.Assume y1 u1 (continuity equation)2. Compute S

f

(and he

, if needed)

3. Compute y1 from governing equation. If this value agrees

with the assumed y1, the solution has been found.

Otherwise continue calculations.

Estimate of frictional losses:

( )1 212

f f fS S S= +


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Example 6.4

A trapezoidal channel with b = 20 ft, n = 0.025, z = 2, and So =

0.001 carries a discharge of 1000 ft3/s. If this channel terminates

in a free overfall and there are no eddy losses, determine the

gradually varied flow profile by the trial-and-error step method.

b = 20 ft

2

1yN


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Solution Table

Stn. z y A u u2/2g H1 R Sf Sfav Dx hf H2

0 103.74 3.74 103 9.71 1.46 105.20 2.81 0.00670 105.20

116 104.62 4.50 130 7.69 0.92 105.54 3.24 0.00347 0.00509 116 0.590 105.79

105.02 4.90 146 6.85 0.73 105.75 3.48 0.00251 0.00461 116 0.535 105.73

355 105.56 5.20 158 6.33 0.62 106.18 3.65 0.00201 0.00226 239 0.540 106.27

105.93 5.32 173 5.78 0.52 106.45 3.85 0.00156 0.00204 239 0.724 106.47

745 106.34 5.60 175 5.71 0.51 106.85 3.89 0.00150 0.00153 490 1.14 107.59

106.96 6.21 201 4.98 0.385 107.34 4.21 0.00103 0.00130 490 0.97 107.42


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Finite Difference Approach

Suitable for application on a computer (small length stepsDx might be needed).Can be applied for completely arbitrary channel

configurations and properties.

A range of numerical approaches are available to solve the

governing equations based on finite differences.

The equation is written in difference form and solved in terms

ofy:

( )

2

2 o fu

y S S xg

+ =


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Examples of Gradually Varied Flow

Flow in channel between two reservoirs (lakes):

1. Steep slope, low downstream water level

2. Steep slope, high downstream water level

3. Mild slope, long channel

4. Mild slope, short channel

5. Sluice gate located in the channel


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Steep Slope, Low Downstream Water Level

Critical section at inflow to channel. Normal water depth

occurs some distance downstream in the channel with Fr> 1(yN < ycr). A hydraulic jump develops before water is

discharged to the downstream lake.

Q in the channel depends on H1 and critical section.

Critical

section Hydraulic

jump

Lake

Lake


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Steep Slope, High Downstream Water Level

Downstream water level is high enough to cause dammingeffects to the upstream lake. No critical section occurs in the

inflow section. y > ycr > yN in the channel.

Q depends on H1 and H2.

No critical section

Fr < 1 in the channel,

although it is steep

LakeLake


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Mild Slope, Long Channel

Mild slope and long channel implies that normal water depth

occurs with yN > ycr. Normal water depth is also attained in

the inflow section to the channel. Non-uniform flow develops

in the downstream part of the channel before discharge tothe lake.

Q depends on H1 and yN in the inflow section.

LakeLake

uniform flow non-uniform flow

Normal water depth


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Mild Slope, Short Channel

A short channel implies that normal water depth will not

occur and y > yN > ycr. Non-uniform flow develops in the

entire channel because of the downstream effects of the

lake.

Q depends on H1 and H2.

Lake Lake

Non-uniform flow


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Sluice Gate Located in the Channel

Sluice gate cause damming upstream affecting inflow

from lake. Discharge from sluice gate depends on

upstream water surface elevation over gate opening.

Supercrit ical flow occurs downstream the gate,

followed by a hydraulic jump before the downstream

lake is encountered.

Q depends on H1 and sluice gate properties.

Jump

Sluice gate (Q a

function ofy)Lake

Lake


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Calculation Procedure for Some Gradually

Varied Flows

1. Flow from a reservoir to a long, steeply sloping channel

2. Flow from a reservoir to a long, mildly sloping channel

3. Flow from a reservoir to a short, mildly sloping channel

where a downstream water level affects the flow in the

channel

4. Flow from a reservoir to a short, steeply sloping channel

where a downstream water level affects the flow in the

channel


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Lake

Flow from a Reservoir to a Long, Steeply

Sloping channel

Critical section occurs in inflow section. Employ energy

equation from lake surface to inflow section.

2

12

1

crcr

cr

cr

uH y

g

uFrgy

= +

= =


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Flow from a Reservoir to a Long, Mildly

Sloping Channel

uniform flow non-uniform flow

Lake

Lake

Normal depth occurs in inflow section. Employ energy equation

from lake surface to inf low section.

2

1

2 / 3 1/ 2

2

1

NN

N N o

uH y

g

u R Sn

= +

=


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Flow from Reservoir to Short, Mildly Sloping Channel;

Downstream Water Level Affects Flow in Channel

Downstream lake water level affects inflow from upstream

lake. Non-uniform flow prevails. Q depends on H1 and H2.

Assume Q = Q1. Do a step calculation from downstreamlake water level to inflow section. Employ energy equation

from inflow section to upstrem lake water level. H1 is

regarded as unknown. Calculate for a new flow Q2 which

gives a new upstream lake water level.

Lake Lake

non-uniform flow


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Make a plot ofH1 as a function ofQ.

Determine the correct Q based on the actual upstream lake

water level H1.


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Flow from Reservoir to Short, Steeply Sloping Channel;

Downstream Water Level Affects Flow in Channel

LakeLake

Non-uniform flow Hydraulic

Jump

Non-uniform

flow

Critical section at inflow to channel. Make a step calculation

from upstream lake and downstream lake. The hydraulic jump

occur where the jump equation is satisfied.


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Hydraulic jump is assumed to have negligible spatial

extension.

( )

( )

22

1

1

212

2

1

1 8 12

1 1 8 12

y

Fry

y Fry

= +

= +

lecture13 gradually varied flow1 2

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