HYDRAULICS HYDRAULICS BAA 2723BAA 2723

FLOW IN OPEN CHANNELFLOW IN OPEN CHANNEL

INTRODUCTIONINTRODUCTION Open ChannelOpen Channel

A conduit in which a liquid flows with a A conduit in which a liquid flows with a free surfacefree surface

any flow path with a free surface, which any flow path with a free surface, which means that the flow path is open to the means that the flow path is open to the atmosphere atmosphere

Open channel hydraulicsOpen channel hydraulics The study of the physics of fluids flow in The study of the physics of fluids flow in

conveyances in which the following fluids conveyances in which the following fluids forms a forms a free surfacefree surface and is driven by and is driven by gravitygravity

Types of ChannelTypes of Channel There are There are 2 types2 types of open channel; natural of open channel; natural

and artificialand artificial Natural open channel Natural open channel are are riversrivers, , creekscreeks

and .... (have irregular cross section)and .... (have irregular cross section)• All channels which have been developed by All channels which have been developed by

natural processes and have not been significant natural processes and have not been significant improved by humansimproved by humans

Artificial open channel Artificial open channel (human construction) (human construction) are are flumesflumes and and canalscanals..

• All channels which have been developed by human All channels which have been developed by human effortsefforts

• Within the broad category of artificial, open channel are Within the broad category of artificial, open channel are following subdivisionsfollowing subdivisions

Types of FlowTypes of Flow Open channel flow can be classified into Open channel flow can be classified into

many types and described in various way.many types and described in various way. The types of flow encountered in open The types of flow encountered in open

channel are classified with respect to channel are classified with respect to time, spacetime, space, , viscosityviscosity, , densitydensity and gravity. and gravity.

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Geometric Characteristics of ChannelsParameter Symbol Definition

Depth of flow y Depth from the bottom of channel to water surface

Depth of section d Depth of flow, normal to water flowing

Stage z Distance from water surface to channel's bed

Upper width T Width of water surface

Area A Area of cross section normal to water flowing

Wetted parameter P

Hydraulics radius R Ratio of area to wetted parameter, R=A/P

Hydraulics depth D Ratio of area to upper width, D = A/T

d = y cos angle between slope of channel and channel's bed

If too small, y d

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Free SurfaceFree Surface Essentially an interface between two fluids of Essentially an interface between two fluids of

different densitydifferent density An interface between the moving liquid and An interface between the moving liquid and

overlying fluid medium and will have constant overlying fluid medium and will have constant pressurepressure

In the case of atmosphere, the density of air is In the case of atmosphere, the density of air is much lower than the density for liquid such as much lower than the density for liquid such as water. In addition the pressure is constant.water. In addition the pressure is constant.

In the case of flowing liquid, the motion is usually In the case of flowing liquid, the motion is usually caused by gravitational effects and the pressure caused by gravitational effects and the pressure distribution within the fluid is generally distribution within the fluid is generally hydrostatic (flows are almost turbulent and hydrostatic (flows are almost turbulent and unaffected by surface tension).unaffected by surface tension).

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Category of artificial open channelCategory of artificial open channel PrismaticPrismatic: A prismatic channel has both a constant cross-: A prismatic channel has both a constant cross-

sectional shape and bottom slope. Channels which do not meet sectional shape and bottom slope. Channels which do not meet this criteria are termed this criteria are termed non prismaticnon prismatic..

CanalCanal: the term canal refer to a rather long channels may be : the term canal refer to a rather long channels may be either unlined or lined with concrete, cement, grass, wood, either unlined or lined with concrete, cement, grass, wood, bituminous materials or artificial membrane.bituminous materials or artificial membrane.

FlumeFlume: In practice, the term refers to a channels built above the : In practice, the term refers to a channels built above the ground surface to convey a flow across a depression. Flumes ground surface to convey a flow across a depression. Flumes are usually constructed of wood, metal, masonry or concrete. are usually constructed of wood, metal, masonry or concrete. The term flumes is also applied to laboratory channels The term flumes is also applied to laboratory channels constructed for basic and applied research.constructed for basic and applied research.

Chute & DropChute & Drop: A chute is a channel having a steep slope. A : A chute is a channel having a steep slope. A drop channel also has a steep slope but is much shorter than a drop channel also has a steep slope but is much shorter than a chute.chute.

CulvertCulvert: A culvert flowing only partially full is an open channel : A culvert flowing only partially full is an open channel primarily used to convey a flow under highways, railroad primarily used to convey a flow under highways, railroad embankments or runways.embankments or runways.

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River & Creek

Natural Open Channel

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• All natural channels generally have varying cross-sections and consequently are non prismatic.• A nonprismatic channel varies in both the cross-sectional shape and bed slope between any two selected points along the channel length

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Prismatic

Rideau canalsRideau canals

Welland canalWelland canal

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Canals

Flumes

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Chute & Drop

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Culvert

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TimeTime

SpaceSpace

Flow in Open Channel

Steady Flow

Uniform Flow Non Uniform Flow

Rapidly varied Flow Gradually Varied Flow

Unsteady Flow

Back NextThe following classification is made according to The following classification is made according to the change in the change in flow depthflow depth with respect to with respect to timetime and and spacespace..

The primary criteria of classification is the variation of the The primary criteria of classification is the variation of the depth of flow depth of flow yy in time, in time, tt and space, and space, xx..

TimeTime a flow can be classified as being: a flow can be classified as being:

• SteadySteady - which implies that the depth and velocity do not - which implies that the depth and velocity do not change with time (change with time (y/y/tt = 0 = 0))

• UnsteadyUnsteady - which implies that the depth and velocity vary with - which implies that the depth and velocity vary with time (time (y/y/tt ≠ 0 ≠ 0))

SpaceSpace a flow can be classified as being: a flow can be classified as being:

• UniformUniform – if the depth and velocity of flow do not vary with – if the depth and velocity of flow do not vary with distance (distance (y/y/x x = 0= 0))

• Non uniform (varied flow)Non uniform (varied flow) - if the depth and velocity vary with - if the depth and velocity vary with distance (distance (y/y/x x ≠ 0≠ 0))

Rapidly variedRapidly varied – the depth of flow changes rapidly over a – the depth of flow changes rapidly over a relatively short distance such as is the case with relatively short distance such as is the case with hydraulic jumphydraulic jump

Gradually variedGradually varied (GVF)(GVF) – the depth of flow changes rather slow – the depth of flow changes rather slow with distance such as is the case of a reservoir behind the damwith distance such as is the case of a reservoir behind the dam

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Steady and Non-Steady FlowSteady and Non-Steady Flow

Unsteady

Steady

Depth, y

Time, tback

Uniform and Non-Uniform FlowUniform and Non-Uniform Flow

V1 V2

A1 A2V1A1 V2

A2

Uniform Flow Non-Uniform Flowback

ViscosityViscosity Depending on the Depending on the ratio of the inertial forces to the viscous forcesratio of the inertial forces to the viscous forces, a flow may be , a flow may be

classified as laminar, transitional or turbulentclassified as laminar, transitional or turbulent The basis for this classification is a dimensionless parameter known as the The basis for this classification is a dimensionless parameter known as the

Reynolds numberReynolds number

RRee= vL/= vL/v v = characteristic velocity of flow, = characteristic velocity of flow,

often taken as the average velocity of flowoften taken as the average velocity of flowL = characteristic lengthL = characteristic length = kinematics viscosity = = kinematics viscosity = // = dynamic viscosity= dynamic viscosity = density= density

In open channel flow, the characteristic length commonly used is the hydraulic radius, In open channel flow, the characteristic length commonly used is the hydraulic radius, RR which is which is the ratio of the flow area, the ratio of the flow area, AA to the wetted perimeter, to the wetted perimeter, PP..

RRe e < 2000< 2000 Laminar flowLaminar flow2000 < R2000 < Re e <4000 <4000 Transitional flowTransitional flow4000 < R4000 < Ree Turbulent flowTurbulent flow Next

Laminar flow Laminar flow one in which the viscous forces are so large relative one in which the viscous forces are so large relative

to the inertial forces that the flow is dominated by the to the inertial forces that the flow is dominated by the viscous forcesviscous forces

In such a flow, the fluid particles move along definite, In such a flow, the fluid particles move along definite, smooth paths in a coherent fashionsmooth paths in a coherent fashion

Transitional flowTransitional flow One which can be classified as neither laminar nor One which can be classified as neither laminar nor

turbulent.turbulent. In open channel flow, the characteristic length In open channel flow, the characteristic length

commonlycommonly Turbulent flowTurbulent flow

The inertial forces are large relative to the viscous The inertial forces are large relative to the viscous forces; hence, the inertial forces dominate the forces; hence, the inertial forces dominate the situationsituation

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Turbulent Flow

DensityDensity Flow are classified as homogeneous or stratified on the Flow are classified as homogeneous or stratified on the

basis of the variation of density within the flow.basis of the variation of density within the flow. Homogeneous Homogeneous – All spatial dimensions the density of – All spatial dimensions the density of

flow is constantflow is constant Stratified Stratified – The density of the flow varies in any direction– The density of the flow varies in any direction The absence of a density gradient in most natural open-The absence of a density gradient in most natural open-

channel flows demonstrates that either the velocity of channel flows demonstrates that either the velocity of flow is sufficient to completely mix the flow with respect flow is sufficient to completely mix the flow with respect to density or that the phenomena which tend to induce to density or that the phenomena which tend to induce density gradients are unimportant.density gradients are unimportant.

The importance of density stratification is that when The importance of density stratification is that when stable density stratification exists, i.e., density increase stable density stratification exists, i.e., density increase with depth or lighter fluid overlies heavier fluid, the with depth or lighter fluid overlies heavier fluid, the effectiveness of turbulence as mixing mechanism is effectiveness of turbulence as mixing mechanism is reduced.reduced.

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A commonly accepted measurement of the A commonly accepted measurement of the strength of the density stratification is the strength of the density stratification is the gradient Richardson numbergradient Richardson number

RRii = = g (g (//y)y) ((v/v/y)y)22

Where;Where; gg = acceleration of gravity = acceleration of gravity = fluid density= fluid densityy = vertical coordinatey = vertical coordinate

v/v/y = gradient of velocity in vertical directiony = gradient of velocity in vertical direction //y = gradient of density in vertical directiony = gradient of density in vertical direction

When When v/v/y y is small relative to is small relative to //y , y , RRi i is large and the is large and the stratification is stable.stratification is stable.

When When v/v/y y is large relative tois large relative to //y, Ry, Rii is small and Riis small and Ri 0, the flow system approaches a homogenous or natural 0, the flow system approaches a homogenous or natural condition.condition.

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GravityGravity Depending on the Depending on the magnitude of the ratio of magnitude of the ratio of

inertial forces to gravity forcesinertial forces to gravity forces, a flow is , a flow is classified as subcritical, critical or supercritical.classified as subcritical, critical or supercritical.

The parameter on which this classification is The parameter on which this classification is based is known as the based is known as the Froude Number:Froude Number:

Fr = Fr = vv (gL)(gL)1/21/2

Where;Where; v = a characteristic velocity of flow v = a characteristic velocity of flow L = a characteristic of lengthL = a characteristic of length = hydraulic depth (D) = A/T= hydraulic depth (D) = A/TA = flow areaA = flow areaT = width of free surfaceT = width of free surface

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If If Fr = 1Fr = 1, the flow is in a , the flow is in a criticalcritical state with the state with the inertial and the gravitational forces in inertial and the gravitational forces in equilibrium.equilibrium.

If If Fr < 1Fr < 1, the flow is in a , the flow is in a subcritical subcritical state and the state and the gravitational forces are dominant.gravitational forces are dominant.

If If Fr > 1Fr > 1, the flow is in , the flow is in supercriticalsupercritical state and the state and the inertial forces are dominant.inertial forces are dominant.

The denominator of the Froude number is the The denominator of the Froude number is the celerity of an elementary gravity wave in shallow celerity of an elementary gravity wave in shallow water.water.

Through theory of Mechanics of Wave;Through theory of Mechanics of Wave;c = √gyc = √gy

Where Where c = celerityc = celerityg = gravityg = gravityyy d = the depth f flow which is a value d = the depth f flow which is a value

assumption if the channel assumption if the channel is wideis wide Next

With this observation, the following interpretation With this observation, the following interpretation can be applied to the subcritical and supercritical can be applied to the subcritical and supercritical of flow:of flow: When the flow is When the flow is subcriticalsubcritical, , F<1F<1, the velocity of flow , the velocity of flow

is is less thanless than the celerity of an elementary gravity wave. the celerity of an elementary gravity wave. Therefore, such a wave can propagate upstream Therefore, such a wave can propagate upstream against the flow and upstream areas are against the flow and upstream areas are in hydraulic in hydraulic communication with the downstream areascommunication with the downstream areas..

When the flow is When the flow is supercriticalsupercritical, , F>1F>1, the velocity of , the velocity of flow is flow is greater thangreater than the celerity of an elementary the celerity of an elementary gravity wave. Therefore, such a wave can propagate gravity wave. Therefore, such a wave can propagate upstream against the flow and the upstream areas of upstream against the flow and the upstream areas of the channel are the channel are not in hydraulic communication with not in hydraulic communication with the downstream areasthe downstream areas..

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ExampleExamplei.i. Water flows at a depth of 1.2 m in a trapezoidal Water flows at a depth of 1.2 m in a trapezoidal

channel with a bottom width of 3.05 m and side channel with a bottom width of 3.05 m and side slopes of 1V:3H. If the discharge is 11.33 mslopes of 1V:3H. If the discharge is 11.33 m33/s, /s, determine the velocity, hydraulic depth, and determine the velocity, hydraulic depth, and Froude number. Froude number.

ii.ii. Then, If the Froude number is 1.2 for the same Then, If the Froude number is 1.2 for the same hydraulic depth as before, compute the velocity hydraulic depth as before, compute the velocity and discharge in the channel. and discharge in the channel.