Open Channel Flow Over a Permeable River Bed
Chapter One
AIMย ANDย OBJECTIVEย OFย THEย STUDY
Riverย asย anย exampleย ofย openย channelย flowย isย veryย important;ย theyย provideย usย with food; sometimes source of drinking water; source of energy; transportationย etc.
The main purpose of this study is to provide further insight into open channelย flow over permeable beds and to enhance the understanding of the effect of bedย permeabilityย onย theย meanย andย instantaneousย flow,ย asย wellย asย otherย flowย parametersย suchย asย viscosity,ย pressureย gradientย andย heightย ofย theย channel.
CHAPTERย TWO
LITERATUREย REVIEW
As a result of importance of fluid flow in open channels a lot of work has beenย done on open channel flows but not much has been done on open channel flowย throughย porousย media.ย Weย alsoย haveย muchย literatureย inย porousย media.
This section of our report aims at discussing the various research work done inย recent times with regard to open channel flow through porous media. Accordingย toย Rajputย (1998),ย fluidย mayย beย definedย asย aย substanceย whichย isย capableย ofย flowing, or a substance which deforms continuously when subjected to externalย shearingย force.ย Fromย theย aboveย definitionย itย isย clearย thatย fluidย hasย theย characteristics of conforming to the shape of the containing vessel. Hence fluidย has no definite shape and can undergo a deformation when a small amount ofย shear force is exerted on it. When fluid flows in aย channel it takes the shape ofย the channel. An open channel is defined as a passage in which liquid flows withย its upper surface exposed to the atmosphere, Rajput (1998). In order words openย channelย flowsย areย characterizedย byย aย freeย surfaceย whichย isย exposedย toย theย atmosphere. The pressure on this boundary thus remains approximately constantย irrespective of any changes in the water depth and the flow velocity. In openย channelsย theย flowย isย dueย toย gravity;ย thusย theย flowย conditionsย areย greatlyย influenced byย theย slopeย ofย theย channel.
These free-surface flows occur commonly in engineering practice, and includeย both natural channels and artificial channels. The natural channels include largeย scaleย geophysicalย flowsย suchย asย rivers,ย streams,ย estuaries,ย etcย whichย haveย irregular sections of varying shapes, developed in a natural way, while artificialย channelsย suchย asย irrigationalย channels,ย drainageย channelsย andย sewersย haveย cross-sectionsย withย regularย geometricalย shapesย (whichย usuallyย remainย sameย throughoutย theย lengthย ofย theย channel).
In general, channel flows may be steady or unsteady; uniform or varied; laminarย or turbulent. Rajput(1998). The flow in an open channel may be laminar orย turbulentย dependingย uponย theย valueย ofย Reynoldsย numberย definedย as
Re=ย rVR
m where Re is Reynolds number, ย ย ย ย ย ย is fluid density, ย ย ย is average velocity of flowย inย theย channel,ย Rย isย theย hydraulicย radiusย (definedย asย theย ratioย ofย theย areaย ofย flowย toย theย wettedย perimeter)ย andย isย theย dynamicย viscosity ofย the fluid.
Whenย Re < 500 flow is laminarย Reย >ย 2000ย flowย isย turbulent
500ย <ย Reย <ย 2000ย flow isย transitional.
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The velocity of stream flow is controlled primarily by gravity and friction. Gravity pulls the water from high elevations at the riverโs source to low c elevations at the riverโsย mouth.ย Theย steeper the elevationย gradient,ย the greaterย the velocity. Friction tends toย holdย the water in place. Thus friction occurs inย two places. First there is frictional drag between one layer of water with thoseย above and below it. Second, there is frictional drag between theย water andย theย bed and sides of the channel through which the water flows. The interaction ofย frictionalย andย gravitationalย forces createsย aย boundaryย layer,ย where theย forceย induced by gravity is retardedย by forces induced by friction. The change inย velocity between the bed and the top of the boundary layer is referred to as aย velocityย gradient.
The manner by which water flows in this boundary layer changes from lowerย velocitiesย upย streamย toย higherย velocitiesย downstream.ย Initiallyย whereย waterย movesย slowly,ย onlyย theย lowerย mostย layersย ofย theย waterย nearย theย streambedย showย a velocity gradient. Viscous forces between water layers and between the lowestย water layer andย streambedย cause aย decrease in velocity ofย theseย layers towardsย the streambed. Each layer moves in parallel horizontal layers. This is calledย laminarย flow.ย Theย boundaryย layerย undergoingย laminarย flowย atย firstย thickensย down flow. But as the velocity increases water layers become unstable [i.e. aย vertical component becomes part of the direction of flow]. First the layers ofย water become more wave-like, rising and falling in theย water column. Finally,ย the vertical component of flow result in the formation of eddies.
CHAPTERย THREE
THEORY OFย CHANNELย FLOW
FLUID
A fluid is a substance which deforms continuously when subjected to externalย shearingย force.ย Inย particular,ย weย areย interestedย inย waterย flowย overย anย openย channel.
CHARACTERISTICSย OFย Aย FLUID
- It has nodefinite shape of its own, but conforms to the shape of theย containing
- A small amount of shear force exerted on a liquid/fluid will cause it toundergo a deformation which is continuous as long as the force continuesย to be
PROPERTIESย OFย FLUID
Theย propertiesย ofย waterย areย ofย muchย importanceย becauseย theย subjectย ofย hydraulicsย isย mainlyย concernedย withย it.
Someย importantย propertiesย ofย waterย are:ย density,ย specificย gravity,ย viscosity,ย vapour pressure, cohesion, adhesion, surface tension, capillarity, compressibilityย etc.ย fluidย flowย isย affectedย byย myriadย ofย factorsย orย variablesย suchย as:
- Velocity of flow
- Flowrate
- Density of fluid
- Temperature of the fluid
- Compressibility of the fluid
- Time of flow
- Pressure gradient
- Viscosity/kinematics viscosity
- Shear stress
- Volume of fluid
- Mass of fluid
- Gravity
- Inertia force
- Cohesion
- Adhesion
- Surface tension
- Capillarity
Velocityย ofย flow:
Theย flowย velocityย u ย ย ofย aย fluidย isย aย vectorย field
uย =ย uย (x,ย t)
which gives the velocity of an element of fluid at a position xย and time t . The flow velocity of a fluid effectively describes everything about the motion of a fluid. For the present work, we suppose that the velocity profile is the same at all distances downย theย channel;ย thatย isย allย x.ย Thisย velocityย profile
uย (ย y)
also known as distributionย ofย velocityย isย aย functionย ofย yย only.ย Thusย uย (x,ย y)ย ยปย uย (ย y) as will be seen inย chapterย fourย ofย thisย work.ย Thisย meansย thatย theย velocityย isย dependentย onย depthย andย notย onย distanceย fromย theย source.
CHAPTERย FOUR
ย MODELEQUATIONย ONย OPENย CHANNELย FLOWย OVERย Aย PERMEABLEย BED
Fluid motions in porous media are governed by the same fundamental laws that govern their motion in solid medium. These laws are based on the conservation of mass, momentum and energy. From a practical stand point, these basic laws are not applied directly to the problems of flows in porous media; instead, a semi empirical approach is used where Darcyโs law is employed instead of the momentum equation.
CHAPTERย FIVE
ย ANALYSIS ANDย DISCUSSIONย OFย RESULTS
Theย followingย dataย wereย generatedย fromย theย analyticalย solutionsย presentedย hereย inย chapterย four
= cosh
+ sinh
–ย Gk (solution1).
mf
Inย tableย 5.1ย below,ย velocityย ()ย wasย computedย withย variedย valuesย ofย permeabilityย (k),ย whileย heightย ofย channelย (y),ย porosityย (fย ),ย pressure
gradientย (G)ย andย viscosityย (ย )ย wereย allย keptย constant.ย Weย alsoย notedย thatย ifย velocityย isย inย cm/s,ย viscosityย inย centipoisesย andย pressureย gradientย inย atm/cm,ย thenย theย unitย ofย k is Darcy.
CONCLUSIONS
Open channel flow over a permeable riverbed is studied by analytical solutionย within the fully developedย region. Solutionย (1) is forย channel flow over theย permeable bed.
Solutionย (2)ย isย forย openย channelย flow,ย whileย solutionย (3)ย isย obtainedย byย substituting the value of velocity vector in porous media, into the solution forย open channel flow. It is observed that by decreasing the permeability in Darcy,ย due to increase in resistance against the flow in the porous region, the differenceย between the maximum of velocity profile in Porous region and open channelย increases. This is as was predicted physically. It is also shown that there is aย penetration of flow from porous media to open channel due to resistance againstย flowย inย theย porousย mediaย andย thisย penetrationย increaseย byย decreasingย theย permeability. This penetration adjusts the mass flow in open and porous regionsย in order to have identical pressure difference in X-direction for both the openย channel andย the porous media.ย In theย other hand,ย when permeability is high,ย there is a penetration from the open channel to porous media due to decrease inย resistance against flow. Hence high permeability accounts for rivers meanderingย up and down. This becomes obvious in desert areas, where the river beds areย sandyย andย drought is aย problem.
As the level of a desert stream drops, places where it was at the bottom of itsย โwaveโ dry up, and all you can see is sand in the river bed. But if you walkย downstreamย a way, you will comeย toย where it was at the top ofย its โwaveโ andย you can still see water in that section. Water is flowing in both places, of course.ย Whereย youย sawย itย wasย dry,ย itย isย nowย flowingย onlyย underground.
RECOMMENDATIONS
Whenย theย permeabilityย ofย aย riverย isย high,ย floodingย inย thatย areaย isย notย catastrophic.ย Rainfallย andย snowย meltย willย causeย temporaryย andย gradualย flooding.
Createย โhardโย orย impermeable,ย surfacesย likeย roads,ย parkingย lots,ย sidewalks,ย patios, and roofs. This is called urban runoff. Instead of soaking into the groundย andย rechargingย theย aquifer,ย rainfallย beginsย toย simplyย rushย overย theseย hardย surfaces,ย runningย directlyย intoย streams.
The resultย is that aquifer levels goย down,ย lowering the level ofย the streams,ย whileย runย offย duringย rainfallย increasesย creatingย sudden,ย unseasonal flooding.
Wet lands like swamps and marshes shouldย not be filledย with dirt.ย Withoutย wetlands,ย rainย waterย entersย theย riverย directlyย andย causesย catastrophicย flooding.
Damsย orย weirsย mayย beย builtย toย controlย theย flow,ย storeย waterย orย extractsย energy.
The management of river is a continuous activity as rivers tend to โundoโ theย modifications madeย byย people.
REEFERENCES
- ย Batchelor, G.K (1967). An Introduction to fluid Dynamics. Cambridgeย Universityย Pressย Newย Yorkย Pp.ย 223-224.
- Calvert,ย J.B.ย (2003).ย Openย channelย flowย http://mysite.du..edu/-ย etuttle/tech/opench.htm.
- Howย aย riverย flowsย โย http://chamisa.freeshell.org//htm.Fromย Wikipedia,ย theย freeย encyclopaedia.ย Accessedย 24thย September,ย 2002.
- Murray R. Spiege (1983). Advanced Mathematics for Engineers andย Scientists.ย Mcย Graw-Hillย Bookย Company, Singaporeย Pp.ย 74-83.
- Nabovatiย A.,ย Sousa,ย A.C.M.ย (2007).ย Fluidย simulationย inย Randomย Porous Media at Pore level using the Lattice Boltzmann Method. Journal ofย Engineeringย Scienceย andย Technologyย pp. 226-237.
- Obikwelu,ย D.O.ย Nย (2003).ย Differentialย Equationsย andย Application.
- AMDI-Nelย publishersย Ogidi,ย Nigeria.pp.60-61
- Parvazinia, M. Nassehi, V.and Wakeman, R.J. (2006) Multiscale finiteย element modelling of laminar steady flow through highly permeable media.ย Chemicalย Engineeringย Science,ย pp583-596
- Radiom, M. Khakbaz, M. and Kermani, M.J (2007). Numerical and Analytical Calculation of Air flow through an open channel linked to a porous media. Department of Mechanical Engineering, Amirkabir University of Technology (Tehran Polytechnic) Tehran, Iran, 15875-4413.
- Rajput, R.K. (1998). Fluid Mechanics and Hydraulic Machines. Schandย andย Company,ย Ramย Nagar,ย Newย Dehi.pp.3-29,867-871.