Open Channel Flows- Fluid Mechanics

7/29/2019 Open Channel Flows- Fluid Mechanics

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OPEN-CHANNEL FLOW Ch-10/Ch-13

Neary/S041


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Introduction/Application

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Pressure vs. Open-channel flow

pressure flow

(pressure driven)

open-channel flow

(typ. gravity driven)

free-water surface

?

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Section Parameters

Reach Length =L [L]= arbitrary horizontal distance between twocross-sections

Bed slope = So [L/L] = -dz/dx = -(z2-z1)/L = tanq

Flow depth =y [L] = vertical distance from channel bottom to freewater surface

Depth of flow sec =d [L] = flow depth normal to flow, d=ycosq

Top width = T[L] = Width at free-surface

Flow area = A[L2] = cross-sectional area normal to flow direction

Wetted perimeter= P[L] = length of channel boundary in contactwith water

Hydraulic radius = R[L] = A/P

Hydraulic depth = D[L] = A/T

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Types of Flow

Variation in Time?

Steady flow velocity and depth same with time

Unsteady flow velocity and depth change with time

Variation in Space?

Uniform flow velocity and depth same at every cross-section

Non-uniform flow velocity and depth vary between cross-

sectionsLaminar and Turbulent ?

Laminarflow appears to be as a movement of thin layers on

top of each other

Turbulent flow appears to be packets of liquid move in

irregular paths5


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Uniform Open Channel Flow

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Velocity Distribution

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Types of Flow

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The Froude Number is a dimensionless parameter proportional to the ratio of

the inertia force on an element of fluid to the weight of the fluid element -

proportional to the inertial force divided by gravitational force.The Froude Number can be expressed as

Fr = v / (g L)1/2

where

Fr = Froude number

v = velocity of flow

g = gravity

L = characteristic length

The Froude Number is relevant in fluid dynamic problems where the weight of

the fluid is an important force.

FROUDE Number

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The denominator of Froude No has the dimensions of

velocity and represents the speed co of a small

disturbance (wave) in still liquid. The Froude No may be defined as the ratio of the flow

speed to wave speed.

Fr = v / co

It is similar to Mach No, which is the ratio of fluid

velocity to speed of sound.

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Flow Classification by Froude Number

The Froude number is an important parameter that governs the

character of flow in open channels.

The flow is classified as

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At low flow velocities (Fr1), a small disturbance cannot travel

upstream (in fact, the wave is washed downstream at a velocity of V - c0)

and thus the upstream conditions cannot be influenced by the downstream

conditions. This is called rapid or supercritical flow, and the flow in this

case is controlled by the upstream conditions.

A surface wave travels upstream when Fr1, and appears frozen on the surface when Fr =1.

The surface wave speed increases with flow depth y, and thus a surface

disturbance propagates much faster in deep channels than it does in

shallow ones.

Fr = v / (g L)1/2

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Governing Equations

Momentum Eqt.

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Speed of surface wave

An important parameter in the study of

open-channel flow is the wave speed c0,

which is the speed at which a surfacedisturbance travels through a liquid.

Consider a long, wide channel that

initially contains a still liquid of height

y. One end of the channel is moved withspeed V, generating a surface wave of

height y propagating at a speed of c0

into the still liquid, as shown in Fig.

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Speed of surface wave

The steady-flow mass & force

balance for this control volume of

width b can be expressed as

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Combining continuity and momentum

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Therefore, the speed of infinitesimal surface waves is

proportional to the square root of liquid depth. Again note that

this analysis is valid only for shallow water bodies, such as

those encountered in open channels. Otherwise, the wave

speed is independent of liquid depth for deep bodies of water,

such as the oceans.

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This phenomenon can occur downstream of a sluice gate.

The liquid approaches the gate with a subcritical velocity.

If upstream liquid level is sufficiently high, it will acceleratethe liquid to a supercritical level as it passes through the gate.

If the downstream section of the channel is not sufficiently

sloped down, it cannot maintain this supercritical velocity.

The liquid jumps up (called Hydraulic Jump) to a higherlevel with a larger cross-sectional area, and thus to a lower

subcritical velocity.

The flow in rivers, canals, and irrigation systems is typically

subcritical. But the flow past sluice gates and spillways is

typically supercritical.23


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Uniform Flow: The Chezy Formula

Uniform flow:

Long straight channel

Constant slopeConstant channel cross section.

Water depth is constant at y = yn

Velocity is constant at V = V0.

Let the slope be S0 = tan is angle the bottom makes with

horizontal, considered positive for downhill

flow.

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fhzg

Vz

g

V 2

2

21

2

1

22

LSzzh of 21

For fully developed flow, the

Darcy-Weisbach relation holds

g

V

D

fLh o

h

f2

2

hh RD 4

21

21

)()8( oho SR

f

gV

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The above equations are called Chezy Formulas.

The quantity C, called the Chezy coefficient, varies from

about 60 (ft1/2/s) for small rough channels to 160 (ft1/2/s) forlarge smooth channels (30 to 90 (m1/2/s) in SI units).

21

)8(

f

gC

21

)( oho SRCV

21

)( ohSRCAQ

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Uniform Flow: The Manning Roughness Correlation

Over the past century a great deal of hydraulics research has

been devoted to the correlation of the Chezy coefficient with

the roughness, shape, and slope of various open channels.Ganguillet and Kutter in 1869

Manning in1889

Bazin in 1897

Powell in 1950

Here we confine our treatment to Mannings correlation, the

most popular.

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1 12 2

1 2.512.0log

3.7 Red

d

f f

Since typical channels are large and rough, we would generally

use the fully rough turbulent-flow limit.2)

8.14log0.2(

hRf

2

%84

)]log(03.22.1[ d

Rf h

A special case, for rocky channel beds, is recommended. d84%

is the size for which 84 percent of the rocks are smaller.

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In tests with real channels, Manning found that the

Chezy coefficient C increased approximately as the sixth

root of the channel size. He proposed the simple formula

n

R

f

gC h

61

21

)8(

where n is a roughness parameter, is a unit conversion factor. = 1.0 SI units; =1.486 BG units

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2/13/2)]([

0.1)/( oho SmRnsmV

The Manning formula for uniform-flow velocity is thus

2/13/2)]([486.1

)/( oho Sf tRn

sf tV

S0 is dimensionless, and n is taken to be the same in both systems.

2/13/2

oho SARn

AVQ

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EXAMPLE 10.1

A finished-concrete 8-ft-wide rectangular channel has a bed slope

of 0.5 and a water depth of 4 ft. Predict the uniform flow rate in

ft3/s.

Solution

From Table 10.1, for finished concrete, n = 0.012.

The slope S0 = tan 0.5 = 0.00873.For depth y = 4 ft and width b = 8 ft, the geometric properties

are

A=by=(8 ft)(4 ft)=32 ft2 ; P =b+2y=8+2(4)=16 ft;

Rh=A/P=32(ft2)/16(ft)=2.0 ft; Dh= 4Rh=8.0 ft;From Mannings formula in BG units, the estimated flow rate is

Q=(1.486/n) A Rh2/3S01/2 =1.486/2 (32 ft2)(2.0 ft)2/3(0.00873)1/2

=590 ft3/s.

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Uniform Flow in a Partly Full Circular Pipe

Consider a partially full pipe in uniform flow.

)2

2sin(2

q

q RA qRP 2 )2

2sin1(2

q

RRh

2/13/2)]2

2sin1(

2[ oo SR

nV

q )

2

2sin(2

qq RVQ o

For a given n and slope S0, wemay plot these two relations

versus . There are two

different maxima, as follows:

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2/13/2

max 718.0 oSRn

V

2/13/8129.2 oSRn

Q

= 128.73 degree; y=0.813D

= 151.21 degree; y=0.938D

The maximum velocity is 14 percent

more than the velocity when runningfull, and similarly the maximum

discharge is 8 percent more.

Uniform Flow in a Partly Full Circular Pipe

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2/13/2)]2

2sin1(

2[ oo SR

nV

q

)2

2sin

(

2 qq

RVQ o

Assignment-03 Due Date: 22-03-2013

Take R=Your Class Number

So=0.01/R

n=Glass, Asphalt, Cast irona) Generate Vo verses , Q verses tables.

b) Plot Vo verses , Q verses .

You may use computer softwares like Excel,

Matlab, C etc.

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Efficient Trapezoidal Uniform-Flow Channels

The simplicity of Mannings formulation enables us to

analyze channel flows to determine the most efficient low-

resistance sections for given conditions. The most common problem is that of maximizing Rh for

a given flow area and discharge.

Since, maximizing Rh for a given A is the same as

minimizing the wetted perimeter. There is no general solution for arbitrary cross sections,

but an analysis of the trapezoid section will show the basic

results.

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Efficient Trapezoidal Uniform-Flow Channels

Consider the generalized trapezoid

ofangle as in Fig. For a given

side angle , the flow area is2ybyA q cot

2/12 )1(22 ybWbP

2/12 )1(2 yyy

AP

To minimize P, evaluate dP/dy for constant A and and set

equal to zero. The result is

])1(2[ 2/122 yA yyP 2)1(4 2/12 yRh2

1

For any angle , the most efficient cross section for uniform flow

occurs when the hydraulic radius is half the depth.39


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Since a rectangle is a trapezoid with = 0, the most efficient

rectangular section is such that

To find the correct depth y, these relations must be solved in

conjunction with Mannings flow-rate formulas for the given

discharge Q.

22yA yP 4 yRh2

1

Efficient Rectangular Uniform-Flow Channels

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Best Trapezoid Angle

yyP 2)1(4 2/12

Above equations are valid for anyvalue of . What is the best value

of for a given depth and area?

To answer this question, evaluate

dP/d with A and y held

constant. The result is

2/12 )1(2 o60

3

1cot

2/1 q

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Thus the very best trapezoid section is half a

hexagon.

Similar calculations with a circular channel sectionrunning partially full show best efficiency for a

semicircle, y=1/2D.

In fact, the semicircle is the best of all possible

channel sections.

Best Trapezoid Angle

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Numerical Problems

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Numerical Problems

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Numerical Problems

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Numerical Problems

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Numerical Problems

Open Channel Flows- Fluid Mechanics

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