Fluid Dynamics detailed ppt for Civil Eg

This is very much detailed

1. Fluid Dynamics
Dr. JanhabiMeher

2. Boundary Layer
Theory
Real fluid-
viscosity
Velocity
gradient
Shear stress
Boundary
layer
development
No-slip
condition

3. THICKNESS OF BOUNDARY LAYER
•Nominalthickness(δ):v=0.99V
•Thethicknessoftheboundarylayerisarbitrarilydefinedas
thatdistancefromtheboundarysurfaceinwhichthe
velocityreaches99%ofthevelocityofthemainstream.
•Displacementthickness(δ*):
•Thedistancebywhichtheboundarysurfacewouldhaveto
bedisplacedoutwardssothatthetotalactualdischarge
wouldbesameasthatofanideal(orfrictionless)fluidpast
thedisplacedboundary.
•Thedistancebywhichtheexternalstreamlinesareshifted
ordisplacedoutwardsowingtotheformationofthe
boundarylayer.

4. Reduced discharge with boundary layer Q=AV
Original discharge with out boundary layerQ1=A1V=π*R1^2
A1-A=A2=π*R2^2=> displacement thickness=R2-R1

5. •Mass=density*volume=ρ*A*V*t
•Momentum=Mass*velocity=
ρ*A*V*t*V=ρAV^2*t
•Momentum flux=M1-M2/t=
ρAV^2*t/t=ρAV^2=ρ(πr^2)V^2
•Energy=1/2*mass*velocity^2=
½*ρ*A*V*t*V^2=1/2*ρAV^3*t
•Energy flux=E1-E2/t=1/2*ρAV^3*t/t=
½*ρAV^3

6. THICKNESS OF BOUNDARY LAYER
•Momentumthickness(θ):
•Thedistancefromtheactualboundarysurfacesuch
thatthemomentumfluxcorrespondingtothemain
streamvelocityVthroughthisdistanceθisequalto
thedeficiencyorlossinmomentumduetothe
boundarylayerformation.
•Energythickness(δE):
•Thedistancefromtheactualboundarysurfacesuch
thattheenergyfluxcorrespondingtothemainstream
velocityVthroughthisdistanceΔeisequaltothe
deficiencyorlossofenergyduetotheboundarylayer
formation.

7. BOUNDARY LAYER CHARACTERISTICS
•Thevariousfactorswhichinfluencethethicknessofthe
boundarylayerformingalongaflatsmoothplatearenoted
below.
•Theboundarylayerthicknessincreasesasthedistance
fromtheleadingedgeincreases.
•Theboundarylayerthicknessdecreaseswiththeincrease
inthevelocityofflowoftheapproachingstreamoffluid.
•Greateristhekinematicviscosityofthefluidgreateristhe
boundarylayerthickness.
•Theboundarylayerthicknessisconsiderablyaffectedby
thepressuregradient(∂p/∂x)inthedirectionofflow.

8. BOUNDARY LAYER CHARACTERISTICS
•Inthecaseofaflatplateplacedina
streamofuniformvelocityV,thepressure
mayalsobeassumedtobeuniformi.e.,
(∂p/∂x)=0.
•However,ifthepressuregradientis
negativeasinthecaseofaconverging
flow,theresultingpressureforceactsin
thedirectionofflowanditacceleratesthe
retardedfluidintheboundarylayer.As
suchtheboundarylayergrowthis
retardedinthepresenceofnegative
pressuregradient.
•Ontheotherhandifthepressuregradient
ispositiveasinthecaseofdivergentflow
thefluidintheboundarylayerisfurther
deceleratedandhenceassistsin
thickeningoftheboundarylayer.Inthe
latercasebackflowandboundarylayer
separationmaybecaused.

9. BOUNDARY LAYER CHARACTERISTICS
•Laminarboundarylayer:parabolicvelocitydistribution
•Turbulentboundarylayer:logarithmicvelocity
distribution
•Laminarsub-layer:linearvelocitydistribution
•ThevalueofRexatwhichtheboundarylayermay
changefromlaminartoturbulentvariesfrom3×10^5
to6×10^5.
•However,changeofboundarylayerfromlaminarto
turbulentisaffectedbyseveralfactorssuchas
disturbanceintheapproachingflow,roughnessofthe
plate,platecurvature,pressuregradientandintensity
andscaleofturbulence.

10. •1
st
flow=Re=2000 (laminar flow at leading edge of
smooth plate)
•2
nd
flow=Re=10000 (turbulent flow at leading edge of
smooth plate)
•The boundary layer thickness of 1
st
flow would be more
as compared to that of the 2
nd
flow.
•1
st
flow=Re=2000 (laminar flow at leading edge of
smooth plate)
•2
nd
flow=Re=2000 (turbulent flow at leading edge of
rough plate)
•The boundary layer thickness of 1
st
flow would be more
as compared to that of the 2
nd
flow.

11. BOUNDARY LAYER EQUATIONS
•Theequationsof
continuityandmotion
forthesteadyflowof
anincompressible,in-
viscidfluidintwo
dimensionswithout
bodyforcesare

12. Prandtl’sBoundary Layer Equations
•Nowifaviscousfluidis
consideredthentheequation
ofcontinuitywillbe
unchanged,butinthe
equationsofmotionadditional
termswillbeintroduceddue
toviscousstresses.
•Theonlyviscousstressτthat
needbeconsideredisthat
actinginthedirectionparallel
totheplate.
•Thesecondequationof
motionisunchangedbythe
argumentthattheshearstress
isactinginthexdirectiononly.

13. Navier–Stokes equations
•ThePrandtl’sboundarylayerequationscanalsobe
deriveddirectlyfromtheNavier–Stokesequationsof
motionwhichareinfactthebasicequationsofmotion
fortheflowofviscousfluids.
•TheanalysisinvolvingtheuseofNavier–Stokes
equationsismoreaccurateandcomplete,sinceinthis
alltheviscousstressesareincluded.However,for
derivingthePrandtl’sboundarylayerequationsfrom
theNavier-Stokesequationsofmotiontheorderof
magnitudeofeachofthetermsoftheseequationsis
determinedandthetermsofsmallerorderof
magnitudeareneglected.

14. Euler’s equations
•Sinceoutsidetheboundarylayerthefluidmaybe
treatedasinviscid(ornon-viscous),theEuler’s
equationsofmotionmaybeapplied.
•Intheregionoutsidetheboundarylayerv=0
andu=Vthefreestreamorambientvelocityof
theapproachingstream.
•Integrationoftheaboveequationleadstothe
Bernoulli’sequationatanysection.

15. Von Karman’s Momentum Integral
Equation of Boundary Layer
•Expressestherelationthat
mustexistbetweentheoverall
rateoffluxofmomentum
acrossasectionofthe
boundarylayer,theshear
stressattheboundarysurface
andthepressuregradientin
thedirectionofflow.
•Formsthebasisfor
approximatemethodsof
solvingboundarylayer
problems.
•Itisappliedtobothlaminaras
wellasturbulentboundary
layers.

16. Von Karman’s Momentum Integral
Equation of Boundary Layer
•ThenetrateofmassflowacrossDFandAE,
outofAEFD
•Therateoftransportofmomentuminthex
directionacrossDFminustherateoftransport
ofmomentuminthexdirectionacrossAEis
•Therateoftransportofmomentuminthex
directionacrossEFoutofAEFDis

17. Von Karman’s Momentum Integral
Equation of Boundary Layer
•Thusequatingthenetincreaseintherateof
transportofmomentumtothesumofthe
forcesactinginthexdirection,wehave
•Dividingboththesidesoftheaboveequation
byδxandtakingthelimitδx→0,weget

18. Laminar Boundary Layer
•Blasius’sexactanalyticalsolutionoftheboundary
layerthickness
•Blasius’sexactanalyticalsolutionoftheshearstress
canbeobtainedas
•ThetotalhorizontalforceFD(orskinfrictiondrag)
actingononesideoftheplateonwhichlaminar
boundarylayerisdevelopedcanbeobtainedas
inwhichBisthewidthoftheplateandListhelengthofthe
plate.

19. Laminar Boundary Layer
•TheaveragedragcoefficientCfmaybe
obtainedas
•Furtherfromtheexactanalyticalsolutionof
theboundarylayerequationsbyBlasiusthe
expressionsforthedisplacementthickness
andthemomentumthicknessmaybe
obtainedas

20. Laminar Boundary Layer
•Ifitisassumedthatthevelocitydistributionacrossa
sectionoftheboundarylayerislinearwithyupto
theedgeoftheboundarylayer,then
•Furtherlocaldragcoefficientcfmaybeobtainedas
•SkinfrictiondragFDononesideoftheplatehaving
laminarboundarylayerisobtainedas
whereBandLarethewidthandthelengthofthe
platerespectively.

21. Laminar Boundary Layer
•TheaveragedragcoefficientCfmaybe
obtainedas
•Themomentumthicknessisgivenby
•Thedisplacementthicknessisgivenby

22. Turbulent Boundary Layer
•Ifitisassumedthatthevelocitydistribution
acrossasectionoftheboundarylayeris
exponentialwithyuptotheedgeofthe
boundarylayer,then
•Furtherlocaldragcoefficientcfmaybe
obtainedas
•TheaveragedragcoefficientCfmaybe
obtainedas

23. LAMINAR SUBLAYER
•Iftheplateisverysmooth,eveninthezoneof
turbulentboundarylayer,thereexistsaverythin
layerimmediatelyadjacenttotheboundary,in
whichtheflowislaminar.Thisthinlayeris
commonlyknownaslaminarsublayer,andits
thicknessisrepresentedbyδ’.
•Nikuradse’sexperimentalstudieshaveshownthat
inwhichV*isknownasshearorfrictionvelocity.

24. Boundary Layer on Rough Surface
•Foraroughplateifkistheaverageheightof
roughnessprojectionsonthesurfaceoftheplateandδ
isthethicknessoftheboundarylayer,thentherelative
roughness(k/δ)isasignificantparameterindicating
thebehaviouroftheboundarysurface.
•Forkremainingconstant,(k/δ)decreasesalongthe
platebecauseδincreasesinthedownstreamdirection.
•Asaresultthefrontportionoftheplatewillbehave
differentlyfromitsrearportionasfarastheinfluence
ofroughnessondragisconcerned.

25. Boundary Layer on Rough Surface
•Thelimitsbetweenthreeregimesofasurface
aredeterminedbythevalueofa
dimensionlessroughnessparameter.
inwhichksisequivalentsandgrainroughnessdefinedas
thatvalueoftheroughnesswhichwouldofferthesame
resistancetotheflowpasttheplateasthatduetothe
actualroughnessonthesurfaceoftheplate.

26. Boundary Layer on Rough Surface
•In the completely rough regime the local drag
coefficient cfand the average drag coefficient
Cfare given by the following expressions.

27. Separation of Boundary Layer
•Withthepressureincreasinginthe
directionofflowi.e.,withpositive(or
adverse)pressuregradient,the
boundarylayerthickensrapidly.
•Theadversepressuregradientplus
theboundarysheardecreasesthe
momentumintheboundarylayer
andiftheybothactoverasufficient
distancetheycausethefluidinthe
boundarylayertocometoresti.e.,
theretardedfluidparticles,cannot,
ingeneralpenetratetoofarintothe
regionofincreasedpressureowingto
theirsmallkineticenergy.
•Thus,theboundarylayerisdeflected
sidewaysfromtheboundary,
separatesfromitandmovesintothe
mainstream.Thisphenomenonis
calledseparation.

28. Methods of Controlling Boundary
Layer
•Theflowinadivergentpassageordiffuser
isanotherexampleinwhichthe
separationoftheflowmaybecauseddue
toadversepressuregradientprevailing
thereunlesstheangleofdivergenceis
verysmall.
•Sincetheseparationoftheboundarylayer
givesrisetoadditionalresistancetoflow,
attemptsshouldbemadetoavoid
separationbysomemeans.
•Theseparationmaybeavoidedby
adoptingsuitablemethodofcontrolling
theboundarylayersuchasmotionofsolid
boundary,accelerationoffluidin
boundarylayer,suctionoffluidfrom
boundarylayer.
•Alsobydevelopingsuchboundaryshapes
forwhichtheseparationwillbeassmall
aspossible,i.e.bystreamliningthebody
shapesisanothermethodtoavoid
separation.

29. Effect of Turbulence on Boundary
Layer
•Separationoccurswithbothlaminarandturbulentboundary
layers,butlaminarboundarylayerismoresusceptibleto
earlierseparationthanturbulentboundarylayer.
•Thisissobecauseinalaminarboundarylayertheincreaseof
velocitywithdistancefromtheboundarysurfaceislessrapid,
andtheadversepressuregradientcanmorerapidlyhaltthe
slowmovingfluidclosetotheboundarysurface.
•Ontheotherhandinaturbulentboundarylayerthevelocity
distributionismuchmoreuniformthaninalaminarboundary
layerbecauseofintenselateralmixing.
•Asaresultrelativelyhighervelocityprevailswithinaturbulent
boundarylayer,whichreducestendencyofseparation.

30. Laminar Flow
•Inasteadyuniformlaminar
flowthepressuregradientin
thedirectionofflowisequal
totheshearstressgradientin
thenormaldirection.
•Furtherforsteadyuniform
flow,sinceaccelerationis
absent,itisapparentthatthe
pressuregradient(∂p/∂x)is
independentofyandthe
shearstressgradient(∂τ/∂y)is
independentofx.

31. STEADY LAMINAR FLOW IN CIRCULAR
PIPES—HAGEN POISEUILLE
LAW
•The summation of all
forces in the x-direction
must be equal to zero.

32. STEADY LAMINAR FLOW IN CIRCULAR
PIPES—HAGEN POISEUILLE
LAW
•Theshearstressτvaries
linearlyalongtheradius
ofthepipe.Atthe
centreofthepipesince
r=0,theshearstressτ
iszeroandatthepipe
wall,sincer=Rthe
shear stressis
maximumdenotedasτ0.

33. Laminar flow through annulus
•Afluidelementhavingashape
ofsmallconcentriccylindrical
sleeveoflengthdxand
thicknessdrconsideredata
radialdistancerischosenas
freebody.
•Theforcesactingonthefluid
elementinthedirectionof
flowarenormalpressure
forcesovertheendareasand
shearforcesoverinnerand
outercurvedsurfacesofthe
cylindricalelement.

34. Laminar flow between parallel flat
plates-both plates at rest

35. Laminar flow between parallel plates-
both plates at rest
Theshearstressvarieslinearlywiththe
distancefromtheboundary.Ithasthesame
maximumvalueateitherboundary(i.e.,aty=
0ory=B)anddecreaseslinearlywiththe
distancefromtheboundary,withtheresult
thatitisequaltozeroatthecentreline
betweenthetwoplatesi.e.,aty=B/2.

36. Laminar flow between parallel flat
plates-one plate moving and other at
rest
Thislinearvelocitydistributioncaseisknown
assimpleCouettefloworsimpleshearflow.

37. Laminar flow between parallel flat
plates-one plate moving and other at
rest
•Theshearstressvaries
linearlywiththe
distancefromthe
boundary.

38. Turbulent Flow
•Thevelocitydistributionin
turbulentflowisrelatively
uniformandthevelocityprofile
ofturbulentflowismuchflatter
thanthecorrespondinglaminar
flowparabolaforthesamemean
velocity.Itbecomesevenflatter
withincreasingReynoldsnumber.
•Inthecaseofturbulentflowthe
velocityfluctuationsinfluencethe
meanmotioninsuchawaythat
anadditionalshear(orfrictional)
resistancetoflowiscaused.This
shearstressproducedin
turbulentflowisinadditionto
theviscousshearstressanditis
termedasturbulentshearstress.

39. Turbulent Flow
•Thelengthofpipex,fromthe
entranceofthepipeupto
sectionAA,isthelength
requiredfortheestablishment
offullydevelopedlaminarflow
orturbulentflowinthepipe.
•Experimentshaveshownthat
forlaminarflow(x/D)isa
functionofReynoldsnumber
Re(=ΡVD/µ).
•Experimentshaveshownthat
forturbulentflow(x/D)isa
NOTafunctionofReynolds
numberRe(=ΡVD/µ).

40. Turbulent Flow
•Forlaminarflowinapipe,laminarboundarylayerwillbedevelopedfor
theentirelengthofthepipeandatasectionthicknessoftheboundary
layerwillbecomeequaltotheradiusofpipe.Thelengthofpipex,from
theentranceofthepipeuptothatsection,isthereforethelengthrequired
fortheestablishmentoffullydevelopedlaminarflowinthepipe.
•Iftheflowinapipeisturbulent,forasmalldistancefromtheentrance
section,laminarboundarylayerwillbeformed,whichwillchangeto
turbulentboundarylayerbeforethethicknessoftheboundarylayer
becomesequaltotheradiusofthepipe.However,insomecases,ifthe
pipeisroughandtheintensityofturbulenceoftheincomingflowis
high,fromtheentrancesectionitselfturbulentboundarylayermaybe
formed.
•Sinceaturbulentflowhaslogarithmicvelocitydistribution,itismuch
moreuniformandhencethelengthofpipex,requiredforthe
establishmentoffullydevelopedturbulentflowinapipeisrelativelyless.

41. Fluid Flow Around Submerged
Objects–Drag and Lift
•Theforceexertedbythefluidonthemoving
bodymayingeneralbeinclinedtothe
directionofmotion,andhenceithasa
componentinthedirectionofmotionaswell
asoneperpendiculartothedirectionof
motion.
•Thecomponentofthisforceinthedirection
ofmotioniscalledthedragFD,andthe
componentperpendiculartothedirectionof
motioniscalledtheliftFL.
•However,forasymmetricalbody,suchas
sphereoracylinder,facingtheflow
symmetrically,thereisnoliftandthusthe
totalforceexertedbythefluidisequaltothe
dragonthebody.
•Further,itisknownfromtheprinciplesof
hydrodynamicsthatforasymmetricalbody
movingthroughanidealfluid(i.e.,havingno
viscosity)atauniformvelocity,thepressure
distributionaroundthebodyissymmetrical
andhencetheresultantforceactingonthe
bodyiszero.

42. Fluid Flow Around Submerged
Objects–Drag and Lift
•Thesumofthecomponentsofthe
shearforcesinthedirectionofflow
offluidiscalledthefrictiondragFDf.
•Similarlythesumofthecomponents
ofthepressureforcesinthedirection
ofthefluidmotioniscalledthe
pressuredragFDp.
•ThetotaldragFDactingonthebody
isthereforeequaltothesumofthe
frictiondragandthepressuredrag.
•Theliftonthebodyisgivenbythe
summationofthecomponentofthe
shearandthepressureforcesacting
overtheentiresurfaceofthebodyin
thedirectionperpendiculartothe
directionofthefluidmotion.

43. Fluid Flow Around Submerged
Objects–Drag and Lift
•Forabodymovingthroughafluidofmassdensityρ,atauniform
velocityV,themathematicalexpressionsforthecalculationofthe
dragandtheliftmayalsobewrittenasfollows:
•IntheaboveexpressionsCDandCLareknownasthedragandthe
liftcoefficientsrespectively,bothofwhicharedimensionless.
•TheareaAisacharacteristicarea,whichisusuallytakenaseither
thelargestprojectedareaoftheimmersedbody;ortheprojected
areaoftheimmersedbodyonaplaneperpendiculartothe
directionofflowoffluid.
•Theterm(ρV^2/2)isthedynamicpressureoftheflowingfluid.

44. Types of Drags due to Viscosity
Real fluid-
viscosity
Boundary layer
development
and No-slip
condition
Velocity
gradient
Shear stress
Surface drag
due to friction
Pressure drag due to
deformation of fluid
particles
Deformation drag = surface drag +
pressure drag (at low Reynolds number)
Separation
of flow
Form drag due to
development of
wake
Deformation drag=surface drag + form
drag(at high Reynolds number)

45. Types of Drags
•Therelativemagnitudeofthetwo
componentsofthetotaldragviz.,
frictiondragandpressuredrag/form
drag,dependsontheshapeandthe
positionoftheimmersedbody,and
theflowandthefluidcharacteristics.
•Thusifathinflatplateisheld
immersedinafluid,paralleltothe
directionofflowthepressuredrag
practicallyequaltozero.Assuchin
thiscasethetotaldragisequaltothe
frictiondrag.
•Ontheotherhand,ifthesameplate
isheldperpendiculartotheflow,the
frictiondragispracticallyequalto
zeroandthetotaldragisduetothe
pressuredifferencebetweenthe
upstreamanddownstreamsidesof
theplate.

46. Types of Drags
•Inthecaseofadiscoraplateheldnormal
totheflowtheseparationoftheflowwill
takeplaceattheedges.Averywidewake
isdevelopedonthedownstreamsideof
thedisctobeexposedtoazonehaving
pressureconsiderablybelowthatonthe
upstreamside.Theresultisthatalarge
pressureorformdragisexertedonthe
plate.
•Thewakeincaseofwellroundedbodiesis
smallerthanthatofthedisc.Assuchthe
pressureorfromdragofthesphereis
considerablysmallerthanthatofthedisc.
•Forawellstreamlinedbodythe
separationoccursonlyatthedownstream
end.Assuchthewakeinthiscaseis
extremelysmall.Hencethepressureor
formdragofsuchobjectsisverysmall
fractionofthatofthedisc.

47. Types of Drags
Form drag of well
stream-lined bodies <
Form drag of well
rounded bodies < Form
drag of objects having
sharp edges
Surface drag of well
stream-lined bodies >
Surface drag of well
rounded bodies >
Surface drag of objects
having sharp edges
Total drag of well
stream-lined bodies <
Total drag of well
rounded bodies < Total
drag of objects having
sharp edges

48. DRAG ON A FLAT PLATE
•Inthecaseofaflatplate
thedragcoefficientCDisa
functionofReonlyatlow
andmoderatevaluesofRe.
•However,asthevalueofRe
exceeds10^3,CDassumesa
constantvalueofabout2.0.
•Areductioninthevalueof
CDhoweveroccursifthe
ratioofthelengthLofthe
platetoitswidthBisnot
verylarge.ThevalueofCD
decreasesasthelengthof
theplateisreduced.

49. DRAG ON A FLAT PLATE
•Forathinflatplateheldperpendiculartotheflow,fordifferent
valuesoftheratiooflengthtobreadth(L/B)oftheplate,thevalues
ofCD,althoughremainsindependentofReynoldsnumberforRe>
1000,itvariesmarkedlywiththe(L/B)ratiooftheplate.
•ThelimitingvalueofCDisequaltoabout2.0foraninfinitelylong
flatplateheldperpendiculartotheflowwhichistwodimensional
incharacter.
•However,becauseoftheflowinthecaseofacirculardischeld
perpendiculartotheflowbeingthree-dimensionalincharacterthe
limitingvalueofCDisonlyabout1.1.

50. DRAG ON AN AIRFOIL
•ThecoefficientofdragCDinthecaseofanairfoil
willnecessarilydependonReanditsshape.
•Furtherinthecaseofanairfoil,asuddendrop
inthevalueofCD,cannotbeexpected,as
becauseofthestreamliningofthebody,the
separationoccursonlyattheextremerearof
thebody,resultinginasmallwakeand
consequentlysmallpressuredrag.
•However,inthecaseofasphereoracylinder
whentheboundarylayerchangesfromlaminar
toturbulent,duetowhichthewakesizealters
significantlytovarythecontributionofthe
pressuredragtothetotaldrag,asuddendropin
thevalueofCDcanbeexpected.
•Thepressuredistributionaroundanairfoil,
basedupontheactualmeasurementandthe
theoreticalirrotationalflowanalysisshowsthat
thetwopressuredistributionsagreeverywell
exceptattherearend,wheretheseparation
mayoccur.

51. DRAG ON A SPHERE
•Thepressuredifferenceisequalto
zeroonaccountofsymmetrical
pressuredistributionforanidealfluid
flowingpastasphere.
•Itthereforemeansthatthereisno
formdrag.Furthermore,sincethere
isnoviscosity,thereisneither
deformationdragnorfrictiondrag.
•Fortheflowofrealfluidofaninfinite
extent,pastasphereatReynolds
numbersaslowas0.2,accordingto
G.G.StokesthetotaldragFD=3πµVD.
•Outofthistotaldragtwo-thirdsis
contributedbysurfacedragandone-
thirdbythepressuredrag(developed
onaccountofdeformationoffluid
particles),thatissurfacedrag=2/3(FD
)=2πµVDandpressuredragdueto
deformation=1/3(FD)=πµVD.

52. DRAG ON A SPHERE
•WiththeincreaseintheReynolds
numbertheseparationofthe
boundarylayerhoweverbeginsfrom
thedownstreamstagnationpointand
thepointsofseparationmovefurther
forwardtowardsupstreamdirection
asReincreases.
•Onaccountoftheshiftingofthe
separationpointsconsiderably
towardstheupstreamside,avery
widewakeisformedontherearof
thesphere,whichresultsinalarge
contributionofthepressureor(form)
drag(about95%)inthetotaldragas
comparedwitharelativelysmall
magnitudeofskinfrictiondrag
(about5%).
•Becauseofthis,intherangeof
Reynoldsnumberfrom10^3to10^5,
CDbecomesmoreorless
independentofReynoldsnumber.
However,inthisrangeofRe,CD,
increasesslightlyfrom0.4to0.5only.

53. DRAG ON A SPHERE
•SinceuptoRe<3×10^5theboundarylayermaybe
consideredtobelaminar,thepressuredistributionaround
thesphereontheupstreamsideuptothepointsof
separationisalmostthesameasobtainedbyidealfluid
theory.However,beyondthepointsofseparation,the
pressuredistributionisaltogetherdifferentfromthat
obtainedbytheidealfluidtheory.
•WithafurtherincreaseintheReynoldsnumber,atRe≥3
×10^5,theboundarylayerbecomesturbulentwhichcan
travelfurtherdownstreamwithoutseparation.Assuchin
thiscasethepointsofseparationshiftconsiderablytothe
downstreamside.Duetothisshiftinthepointsof
separationthesizeofthewakeontherearofthesphereis
reducedandthevalueofCDdropssharplyfrom0.5to0.2.

54. Effect of Turbulence on Drag
Coefficient
•Itisthusseenthatwhentheboundarylayerchangesfromlaminartoturbulent
thedragcoefficientisgreatlyreduced.Thetransitionoftheboundarylayerfrom
laminartoturbulenttakesplaceatcriticalReynoldsnumberwhichhowever
decreaseswiththeincreaseintheturbulenceoftheoncomingfluidandthe
increaseinthesurfaceroughness.Moreover,byincreasingthesurfaceroughness
thepointoftransitionoftheboundarylayerfromlaminartoturbulentisshifted
towardstheupstreamside.Thisearlytransitionoftheboundarylayerfrom
laminartoturbulentontheboundarysurfacewouldresultinshiftingthepointsof
separationtothedownstreamsidewhichinturnwouldresultinthereductionof
thesizeofthewakeandhencethereductionofthedragandthedragcoefficient.
•WithturbulentboundarylayerandReequalto4.35×10^5alsothepressure
distributionaroundthesphereontheupstreamsideandmoreorlessuptothe
pointsofseparationisalmostthesameasobtainedbyidealfluidtheory.Beyond
thepointsofseparationthepressuredistributionisconsiderablychanged.
However,onaccountoftheboundarylayerbeingturbulentinthiscasethesizeof
thewakeisreducedandthepressureinthewakeisslightlypositive.

55. DRAG ON A CIRCULAR DISC
•ForthecirculardiscthereisreductioninCD
onlyatlowReynoldsnumbersduetoviscous
effects,andthepositionofseparationpoints
aretowardsthedownstreamsideofthedisc.
•ForthecirculardiscthereisnoreductioninCD,
because,exceptatlowReynoldsnumbers,CDis
independentofviscouseffects,andthe
positionofseparationpointsisfixedatthe
sharpedgesofthedisc.

56. DRAG ON A CYLINDER
•ForaninfinitelylongcylinderofradiusR,lyingwithitsaxisperpendicular
tothedirectionofflowinauniformstreamoffluidofinfiniteextent
havingvelocityofflowV,alsoifitisassumedthatthefluidflowingpast
thecylinderisideali.e.,non-viscous,thentheflowpatternwillbe
symmetrical.
•Becauseinthiscasethepressuredistributionaroundthecylinderis
symmetricalaboutthemid-section.Consequentlythedragonthecylinder
iszero.
•Thevaluesofthevelocitycomponentsatanypointonthesurfaceofthe
cylindermaybeexpressedasVr=0;andVθ=2Vsinθ.
•Itthusfollowsthattheresultantvelocityvatanypointonthesurfaceof
thecylinderisalongthetangenttothecylinderanditisgivenbyv=2Vsin
θ.

57. DRAG ON A CYLINDER
•However,duetotheviscosity
possessedbytherealfluidflowing
pastacylindertheactualpressure
distributionaroundthecylinderis
considerablymodified.
•Aslongastheboundarylayeris
laminarthepointsofseparationare
locatedontheupstreamhalfportion
ofthecylinder,butwhenthe
boundarylayerbecomesturbulent,
thepointsofseparationshiftfarther
downstreamtowardstherearofthe
cylinder.
•Furthermore,the pressure
distributiondiagramsarounda
cylinderaresimilartothosearounda
sphere.Buttheflowpatternbehinda
cylinderisaltogetherdifferentfrom
thatbehindasphere.

58. DRAG ON A CYLINDER
•Forthecaseofflowpasta
cylinderatverylow
valuesofReynolds
number(sayRe<0.5),
Lambhasobtainedthat
thedragcoefficientCDof
acylinderisalso
approximatelyinversely
proportionaltoRe.
•Evidentlyatsuchlow
valuesofResurfacedrag
accountsforalargepart
ofthetotaldrag.

59. DRAG ON A CYLINDER
•AstheReynoldsnumberincreasestheflowpatternwithrespecttoanaxis
perpendiculartothedirectionofflowbecomesunsymmetrical.Itison
accountofthefactthatinthewakedevelopedjustbehindthecylinder,a
moreorlessorderlyseriesofvortices(whichalternateinpositionabout
thecentreline)aredeveloped.
•AtRerangingfromabout2to30,veryweakvorticesareformedonthe
downstreamofthecylinder.Itis,therefore,theinitialstageforthe
developmentofthewake.Inthewakeregionthereexistsaflowinthe
oppositedirectionalongtheaxisofthewake,butintheouterportionthe
flowisinthegeneraldirectionofmotion.
•AtRerangingfromabout40to70thewakeaswellasthepairofvortices
becomequitedistinct.
•WithafurtherincreaseinthevalueofRethevorticesbecomemoreand
moreelongatedinthedirectionofflow,andatReequaltoabout90these
vorticesbecomesymmetrical,theyleavethecylinderandslowlymovein
thedownstreamdirection.

60. Karman Vortex Trails
•Experimentshaveshownthatwhenthe
Reynoldsnumberexceedsabout30,the
twovorticesformedatthepointsof
separationelongatetosuchanextentthat
theybecomeunstableandarewashed
down.SinceVonKarmanwasfirsttostudy
andanalyzethestabilityoftheseregular
vortextrailsbehindacylinder,thesetrails
ofvorticesarecommonlyknownas
“Karmanvortextrails”(orKarmanvortex
street).
•Thetwopossiblevortexconfigurationsin
theKarmanvortextrailsassuggestedby
KarmanareasshowninFig.18.9.
However,atheoreticalanalysisofthe
stabilityofvorticesofthesetwo
configurationsrevealedthatthe
symmetricalconfigurationofvortexpair
showninFig.18.9(a)isnotatallstable.
•Thereforeitmaybeconcludedthatasthe
fluidflowspastacylinder,analternate
sheddingofthevorticesoccurs,because
thisistheonlystabletypeofpattern
whichmaybedevelopedfortheKarman
vortextrails.
Infactsuchvorticesarealwaysshedwhenany
twodimensionalbluffbodyisheldinastream
andtheflowseparates.Thussameisthecase
withlongflatplatesheldnormaltothe
directionofflowofstreamoffluidforwhich
howevertheReynoldsnumberisgreaterthan
103.

61. DRAG ON A CYLINDER
•AstheReynoldsnumberincreasesthecontributionofthepressuredragin
thetotaldragincreasesfromaboutonethirdofthetotalatsmall
ReynoldsnumberstoabouthalfofthetotaldragastheReynoldsnumber
increasesandvorticesbegintoform.
•Thecontributionofthepressuredraginthetotaldragfurtherincreasesto
aboutthreequartersofthetotalatReequaltoabout200,whenthe
Karmanvortexstreetiswellestablished.
•AthighvaluesofRethevariationofCDwithReforacylinderfollowsa
patternsimilartothatforasphere.Thedragcoefficientforacylinder
reachesaminimumvalueofabout0.95atRe=2000andthenthereis
slightriseto1.2forRe=3×10^4duetotheincreasingturbulenceinthe
wakeandalsothewideningofthewakeastheseparationpointsgradually
advanceupstream.

62. DRAG ON A CYLINDER
AtRe=2×10^5theboundarylayerwhichwas
uptonowlaminar,becomesturbulentbefore
separationandthereforethereisadropinthe
valueofCDfrom1.20toabout0.3.
However,withafurtherincreaseinRewhere
CDispracticallyindependentofRedueto
relativelysmallviscouseffects,thevalueofCD
increasesgraduallyfrom0.3toabout0.7over
theapproximaterangeof5×10^5<Re<3×
10^6.

63. Effect of Free Surface on Drag
•Thegravityforcesbecomepredominantwhentheobjectis
lyingattheinterface(orcommonsurface)betweenthetwo
fluidsofdifferentdensities.Insuchcasesforthe
geometricallysimilarbodiesthedragcoefficientwill
dependonbothReandFr.
•Inthecaseofshipsmovingonthewatersurface,thetotal
draginthecaseofashipisthesumofskinfrictiondrag,
pressure(orform)drag,andthedragduetosurfacewaves.
•Thedifferencebetweenthetotaldragandtheskinfriction
dragiscommonlyknownasresidualdrag.Sincethe
formationofsurfacewavesisassociatedwithgravitational
action,theresidualdragisessentiallyafunctionofFroude
number,whichmay,however,bedeterminedbyship
modelstudies.

64. EFFECT OF COMPRESSIBILITY ON
DRAG
•Theelasticforcesbecomesignificantwhenthevelocityof
flowapproachesthevelocityofsoundinthatfluid,andthe
dragcoefficientbecomesafunctionofMachnumberMa.
•WiththeincreaseinthevalueofMachnumberthe
compressibilityoffluidhassignificantinfluenceontheflow
characteristicsaswellasthedrag.
•Inacompressibleflowadditionalforcesaretransmitted
throughthefluidbytheshockwavesproduced.
•Theshockwavesareessentiallytheelasticwaves(or
pressurewaves)sphericalinform,whichtravelatthe
velocityofsound,andtheseareproducedinthevicinityof
theobjectimmersedinacompressiblefluidflowingpast
theobject.

65. EFFECT OF COMPRESSIBILITY ON
DRAG
•Suchshockwaveswithconical
wavefrontareobservedto
extendbackwardsfromthe
leadingedgeoftheobjectsin
thecaseofsupersonicflows
pastimmersedobject.
•Anabruptchangeofpressure
alwaysoccursacrosssucha
shockwavewhichproduces
thedrag.
•Inadditionthedragalso
resultsfromtheenergy
dissipatedintheshockwaves
aswellastheskinfrictionand
theflowseparationeffects.

66. EFFECT OF SHAPE OF THE OBJECT ON
DRAG
•AthighvaluesofMachnumber,
thedragispractically
independentofReynoldsnumber
whereasatlowvaluesofMach
number,Reynoldsnumberisthe
significantparameter.
•Itmaybenotedfromthese
curvesthatthenumericalvalues
ofCDdropsteadilywhenthenose
ofthebodybecomessuccessively
morepointed,therearofthe
bodyremainingunchangedinall
cases.
•Thisissobecauseinsupersonic
flowasharppointednosecreates
anarrowshockwavefrontwhich
tendstominimizethedrag.

67. EFFECT OF SHAPE OF THE OBJECT ON
DRAG
•Fromtheabovediscussionitisobservedthatfor
minimumdraginsupersonicflowthebodyshouldhave
asharpforwardedge,orconicalnose,andtheshapeof
therearendisofsecondaryimportance.
•Thisrequirementis,however,thereverseofthatfor
subsonicflows,forwhichasstatedearlierthedragis
theleastforastreamlinedbodywelltaperedatthe
rearandroundedatthefront.
•Thusabodywellstreamlinedforsubsonicflowsmay
bepoorlyshapedforsupersonicflowsandviceversa.

68. DEVELOPMENT OF LIFT ON
IMMERSED BODIES
•Whenthebodyissymmetrical
withrespecttoitsaxisandso
locatedthatitsaxisisparallelto
thedirectionofmotion,thenthe
resultantforceexertedbythe
fluidonthebodyisinthe
directionofmotion,andinsucha
casetheliftiszero.
•However,iftheaxisofsymmetry
ofthebodymakesananglewith
thedirectionofmotion,the
resultantforceactingonthebody
willhavealiftcomponent.
•Aliftisexertedonacylinderlying
inauniformflowwhena
circulationissuperimposedon
theuniformflowfield.

69. DEVELOPMENT OF LIFT ON A
CYLINDER
•Thevelocityvofthecompositeflowonthe
surfaceofthecylinderisv=2Vsinθ+Γ/2πR.
•ThepositionofthestagnationpointsS1andS2on
thesurfaceofthecylinder,,maybedetermined
byconsideringv=0andsolvingforsinθwhich
givessinθ=–Γ/4πRV.
•Thenegativesignintheaboveexpressionforsin
θindicatesthatangleθisequalto–θor(180+
θ),sinceinboththesecasessinθisnegative.

70. DEVELOPMENT OF LIFT ON A
CYLINDER
•TheliftdFLactingonanelementary
surfaceareaofthecylinder(LRdθ)is
givenbydFL=–(LRdθ)psinθin
whichListhelengthofthecylinder.
•Thenegativesignhasbeen
introducedbecausethepressure
forceisalwaysdirectedtowardsthe
surface,andhenceforsinθbeing
positiveitscomponentisnegative
beingintheverticaldownward
direction.
•ThetotalliftFLexertedonthecylinder
isobtainedasFL=ρVLΓ.Thisequation
iscommonlyknownasKutta-
Joukowskiequation.
•TheliftcoefficientCLmaybe
expressedasCL=Γ/RVwhereAisthe
projectedareawhichisequalto2RL.
•Theliftcoefficientmayalsobe
expressedasCL=2πvc/V.
The phenomenon of the lift produced by
circulation around a circular cross-section
placed in a uniform stream of fluid, was first
investigated experimentally by a German
physicist H.G. Magnus in 1852. As such it is
commonly known as Magnus effect.

71. DEVELOPMENT OF LIFT ON A
CYLINDER
•Fromthisplotitcanbeseenthatto
produceagivenliftcoefficienttheactual
velocityrequiredisgreaterthantwicethat
requiredtheoretically.
•Furthermore,astheperipheralvelocityvc
increasestoaboutfourtimesthevelocity
offlowoffluidV,theliftcoefficientCL
approachesamaximumvalueofabout
9.0ascomparedwiththetheoretical
maximumvalueofabout12.6.
•Inadditiontothis,thedragcoefficientCD
alsovarieswithvelocityratio(vc/V)from
about1.0forsmallvelocityratiostoabout
5atavelocityratioequalto4.
•For(vc/V)≈1.0thevalueofthedrag
coefficientCDisminimumandafterwards
itincreasessteeplywithanincreaseinthe
valueof(vc/V).
Ifthecylinderisrelativelyshorti.e.,ithaslengthto
diameterratio(L/D)<10thenthereisconsiderable
effectofflowarounditsendswhichappreciably
reducestheliftcoefficient.Forexample,whenthe
ratio(L/D)=5,theliftcoefficientCLisabouthalfof
thatforalongercylinderhavingratio(L/D)>10.

72. DEVELOPMENT OF LIFT ON AN
AIRFOIL
•Symmetryofanairfoilis
generallycharacterizedbythe
chordlengthcandtheangle
ofattackα.
•Forasymmetricalairfoilthe
chordlinecoincideswithits
axisofsymmetry.
•Theoveralllengthofanairfoil
(inthedirectionperpendicular
tothecross-section)istermed
asitsspan.
•TheratioofspanLandchord
lengthcofanairfoil(i.e.,L/c)
isknownasaspectratio.

73. DEVELOPMENT OF LIFT ON AN
AIRFOIL
•Joukowskishowedthatthepatternof
flowroundacircularcylindercould
beusedtodeducetheflowpattern
aroundabodyofanyothershapeby
amathematicalprocesscalled
‘conformaltransformation.’
•AssuchitmaybenotedthatKutta-
Joukowskiequation,though
developedforacylinderofcircular
cross-section,isfoundtoholdgood
foracylinderhavingacross-section
ofanyshapeprovidedthereis
circulationaroundit.
•Fromthetheoreticalanalysisithas
beenfoundthatbyproperly
adjustingthecirculationitispossible
toobtaintheflowpatternsuchthat
thestreamlineatthetrailingendof
theairfoilistangentialtoit.

74. DEVELOPMENT OF LIFT ON AN
AIRFOIL
•ThecirculationΓrequiredtodo
thishasbeenfoundanalytically
asΓ=πcVsinα,wherecischord
length,Visuniformvelocityof
flowandαistheangleofattack.
•SotheliftFLontheairfoilofspan
LbecomesFL=ρVLΓ=ρVL(πcV
sinα)=πcLρV^2sinα.
•TheliftcoefficientCLmaybe
expressedasCL=2πsinα,where
Aistheareaoftheprojectionof
theairfoilonaplane
perpendiculartoitscross-section,
whichinthecaseofanairfoilof
spanLandchordlengthcisequal
to(cL).

75. DEVELOPMENT OF CIRCULATION
AROUND AN AIRFOIL
•Whenauniformstreamofrealfluidflowspastanairfoil,theninitiallythe
flowpatternissameasthatforanidealfluidflowingpastanairfoili.e.,an
irrotationalflowpatterndevelops.
•Whentheboundarylayerseparatesfromthelowersurfaceatthetrailing
edgeandontheuppersurfaceaflowisinducedfromthestagnationpoint
towardsthetrailingedge.Thisflowisintheoppositedirectiontothatof
theidealfluidwhichresultsintheformationofaneddycalledstarting
vortexontheuppersurfaceintheregionofthetrailingedge.
•Whenthestartingvortexleavestheairfoil,itgeneratesanequaland
oppositecirculationroundtheairfoil.Inthisway,thenetcirculationround
thecurveAremainszero.
•Nowinordertocounterbalancethecounterclockwisecirculationsofthe
startingvortex,aclockwisecirculationofthesamestrengthmustbeset
uparoundtheairfoil,sothatthesumofthecirculationaroundthecurveA
iszeroinaccordancewithThomson’stheorem.Theclockwisecirculation
aroundtheairfoilisusuallyknownasboundarycirculation.

76. DEVELOPMENT OF LIFT ON AN
AIRFOIL
•Aconstantcirculationaroundtheairfoilresultsin
settingupofaconstantliftontheairfoil.
•Furthermoreduetotheviscosity,forarealfluid
theactualliftexertedonanairfoilissomewhat
lessthanthatobtainedbyKutta—Joukowski
equation.
•Thedragandliftofanairfoilmayalsobe
expressedas
inwhichtheareaAisrepresentedbytheproduct
ofthelengthorspanLandthechordlengthcof
theairfoil.

77. DEVELOPMENT OF LIFT ON AN
AIRFOIL
•Theactualflowpatterndevelopedforthe
flowofrealfluidpastanairfoilisvery
muchsimilartothatforanirrotational
flowforsmallvaluesofangleα’.
•Withincreaseinthevalueofangleα’to
about10°theliftcoefficientattainsa
maximumvalueandforα’>10°thereis
decreaseinthevalueofliftcoefficient.
•Thereductionintheliftcoefficientwith
increaseintheangleofattackbeyondthe
stallingangleisduetotheseparationof
flowatsomepointontheuppersideof
theairfoil.
•Theconditioninwhichtheflowseparates
frompracticallythewholeoftheupper
surfaceoftheairfoilisknownasstall.
•Thedragcoefficientofanairfoilvaries
littlewiththeangleα’anditisonlyatthe
highervaluesoftheangleα’,sincethe
flowseparates,thedragcoefficientis
slightlyincreased.

78. Effect of Fluid Compressibility on the
Lift on an Airfoil
•Insubsoniccompressibleflowtheeffectoffluid
compressibilityistocauseanincreaseinthecoefficientof
liftCLatagivenangleofattackα.
•Insubsoniccompressibleflowpastthinsymmetricalairfoil
ofinfinitespanandsmallangleofattack,theliftcoefficient
isincreasedbyafactor(1–Ma^2)^(–1/2).
•Whenthefreestreamvelocityisincreasedfromasubsonic
tosupersonicrange,duetotheformationofshockwaves,
thereisadecreaseinthevalueofCLwiththeincreasein
thevalueofMa.
•Forsupersonicflowpastthinsymmetricalairfoilsof
infinitesspanandsmallangleofattacktheliftcoefficientis
decreasedbyafactor(Ma^2-1)^(–1/2).

79. INDUCED DRAG ON AN AIRFOIL OF
FINITE LENGTH
•IfanairfoilofafinitespanorlengthL
isplacedinafluidstream,thenthe
flowoffluidalsotakesplacealong
thetwoends,onaccountofwhich
boththedragwillbeincreased,as
induceddragisexertedwhichisin
additiontothenormaldragexerted
onanairfoilofinfinitespan.
•TheinduceddragFDimayalsobe
expressedas
whereCDiisthecoefficientof
induceddrag.
•Byassuminganellipticaldistribution
ofliftonanairfoiloffinitespan,
Prandtlhasobtainedthefollowing
approximateexpressionforthe
coefficientofinduceddragas
where(L/c)istheaspectratioofthe
airfoil.
Theflowaroundtheendsofashortcylinder
alsocausesaninduceddraginadditiontothe
normaldrag.Theinduceddragforashort
cylinderiscausedinthesamemannerasin
thecaseofanairfoiloffinitelength.

80. POLAR DIAGRAM FOR LIFT AND DRAG
OF AN AIRFOIL
•Theplotsindicatingthevariationof
thecoefficientswiththeangleof
attackαorangleα’maybeshownby
asinglecurveknownasthepolar
diagram,whichwasdevelopedby
Prandtl.
•InthisdiagramtheliftcoefficientCLis
plottedagainstthedragcoefficient
CD.
•Thehorizontalinterceptbetweenthe
twocurveswillrepresentthedrag
coefficientCD0forflowaroundan
airfoilofinfinitelength.
•Sincetheinducedangleofattackαiis
usuallysmall,theliftcoefficientis
sameforairfoilsoffiniteandinfinite
span.

81. Dimensional Analysis
•Dimensionalanalysishelpsindeterminingasystematic
arrangementofthevariablesinthephysicalrelationshipand
combiningdimensionalvariablestoformnon-dimensional
parameters.
•Inthestudyoffluidmechanicsthedimensionalanalysishasbeen
foundtobeusefulinbothanalyticalandexperimental
investigations.Someoftheusesofdimensionalanalysisare
Testingthedimensionalhomogeneityofanyequationoffluid
motion.
Derivingequationsexpressedintermsofnon-dimensional
parameterstoshowtherelativesignificanceofeachparameter.
Planningmodeltestsandpresentingexperimentalresultsina
systematicmannerintermsofnon-dimensionalparameters;thus
makingitpossibletoanalyzethecomplexfluidflowphenomenon.

82. DIMENSIONAL HOMOGENEITY
•Fourier’sprincipleofdimensionalhomogeneitystates
thatanequationwhichexpressesaphysical
phenomenonoffluidflowmustbealgebraicallycorrect
anddimensionallyhomogeneous.
•Adimensionallyhomogeneousequationhasthe
uniquecharacteristicofbeingindependentoftheunits
chosenformeasurement.
•Adimensionallyhomogeneousequationhasthe
advantagethatitisalwayspossibletoreducea
dimensionallyhomogeneousequationtoanon-
dimensionalform.

83. Rayleigh Method
(used when the number of variables is less)
•Thefollowingtwomethodsofdimensionalanalysisaregenerally
used:
RayleighMethod:Inthismethodafunctionalrelationshipofsome
variablesisexpressedintheformofanexponentialequationwhich
mustbedimensionallyhomogeneous.
ThusifXissomefunctionofvariablesX1,X2,X3…Xn;thefunctional
equationcanbewrittenas
whichmaybeexpressedas
inwhichCisadimensionlessconstantwhichmaybedetermined
eitherfromthephysicalcharacteristicsoftheproblemorfrom
experimentalmeasurements.
Theexponentsa,b,c…,…narethenevaluatedonthebasisthatthe
equationisdimensionallyhomogeneous.Thedimensionless
parametersarethenformedbygroupingtogetherthevariables
withlikepowers.

84. Buckingham π-Method
(used when the number of variables is more)
•Buckinghamπ-Method:TheBuckingham’sπ-theorem
statesthatiftherearendimensionalvariablesinvolvedina
phenomenon,whichcanbecompletelydescribedbym
fundamentalquantitiesordimensions,andarerelatedbya
dimensionallyhomogeneousequation,thenthe
relationshipamongthenquantitiescanalwaysbe
expressedintermsofexactly(n–m)dimensionlessand
independentπterms.
Mathematically,ifanyvariableQ1dependsonthe
independentvariables,Q2,Q3,Q4………Qn;thefunctional
equationmaybewrittenas
whichcanbetransformedto whereCisa
dimensionlessconstant.

85. Buckingham π-Method
Inaccordancewiththeπ-theorem,anon-dimensional
equationcanthusbeobtainedas
whereineachdimensionlessπ-termisformedbycombiningm
variablesoutofthetotalnvariableswithoneofthe
remaining(n–m)variables.
Thusthedifferentπ-termsmaybeestablishedas
inwhicheachindividualequationisdimensionlessandthe
exponentsa,b,c,d……metc.,aredeterminedbyconsidering
dimensionalhomogeneityforeachequationsothateachπ-
termisdimensionless.

86. Buckingham π-Method
Thefinalgeneralequationforthephenomenon
maythenbeobtainedbyexpressinganyoneof
theπ-termsasafunctionoftheothersas
Thesemvariableswhichappearrepeatedlyin
eachoftheπ-terms,areconsequentlycalled
repeatingvariablesandarechosenfromamong
thevariablessuchthattheytogetherinvolveall
themfundamentalquantities(ordimensions)
andtheythemselvesdonotformadimensionless
parameter.

87. Repeating Variables
•Therepeatingvariablesshouldbesuchthat
Noneofthemisdimensionless.
Notwovariableshavethesamedimensions.
Theythemselvesdonotformadimensionless
parameter.
Allthemfundamentaldimensionsareincluded
collectivelyinthem.
Moreover,asfaraspossiblethedependentvariable
shouldnotbetakenasarepeatingvariableas
otherwiseitwillnotbepossibletoobtainanexplicit
relationship.

88. Dimensional Matrix Approach
for Repeating Variables
•InsomecasestheBuckingham’sπtheoremdoesnotholdgoodfor
determiningthenumberofthedimensionlessgroupsinacompletesetof
variables.
•Assuchinordertoobtainthenumberofdimensionlessgroupsintowhich
acompletesetofvariablesmaybegrouped,anothermethodbasedon
dimensionalmatrixapproachmaybeadoptedwhichisasfollows:
Listallthevariablesinvolvedinthephenomenonandpreparea
dimensionalmatrixofthevariables,whichisnothingbutdisplayofthe
exponentsofthedimensionsofthevariablesbyatabulararrangement.
Findtherankofthedimensionalmatrix.Therankofamatrixissaidtobe
r,ifitcontainsanonzerodeterminantoforderrandifalldeterminantsof
ordergreaterthanrthatthematrixcontainsareequaltozero.
Thenumberofdimensionlessgroupsofπ-termsisequalto(n–r),thatis,
totalnumberofvariablesnminustherankofthedimensionalmatrixr.

89. SUPERFLUOUS AND OMITTED
VARIABLES
•Oftenvariablesmaybeincludedthatreallydo
nothaveanyeffectonthephenomenon.Such
variablesareknownassuperfluousvariables.
•Theinclusionofthesevariableswillresultinthe
appearanceoftoomanytermsinthefinal
equation,therebymakingthewholeanalysis
unnecessarilycomplicated.
•Thesesuperfluousvariablescanbeeliminated
fromtheanalysisonlyonthebasisoftheresults
obtainedfromtheexperimentalinvestigationof
theproblems.

90. SUPERFLUOUS AND OMITTED
VARIABLES
•Asetofexperimentscannowbecarriedouttodetermineifavariable
reallyaffectsthephenomenon.
•Forthispurposetheotherπ-termsmustbekeptconstantwhiletheπ-
termcontainingthatvariableisvaried.
•Aplotofoneoftheπ-termagainsttheπ-termcontainingthatvariable
preparedfromtheexperimentalobservationsmaythenindicateno
variationoftheπ-termcontainingthatvariableshowingtherebythatthe
variableisirrelevanttothisproblem.
•If,onthecontrary,anyofthepertinentvariablethatmayactually
influencethephenomenon,isomittedatthebeginning,theanalysismay
leadtoanincomplete,orevenerroneousconclusions.
•Thiswillhoweverbeindicatedbythefinalexperimentalplotofπ-terms
whichwillshowascatterofthepointswhichisnotduetoexperimental
error.Insuchcasesasearchmustthenbemadefortheomittedvariable,a
newdimensionalanalysismadeandarevisedplottingcarriedout.

91. SUPERFLUOUS AND OMITTED
VARIABLES
•Acommonmistakemaybeonaccountoftheomissionof
certainvariablesthathaveapracticallyconstantvalue.
Thesevariablesaresometimesessentialbecausethey
combinewithotheractivevariablestoformdimensionless
parameters.
•Itisthusevidentthatthemethodofdimensionalanalysis
doesnotgiveanyclueastothecorrectnessoftheselection
oftherelevantvariablesofaparticularphenomenon.
•Themethodofdimensionalanalysismerelytellshowthe
variablesshouldbegroupedsothatfromtheexperimental
investigationitmaybedecidedwhichofthevariablesare
importantones.Thisishowever,themainlimitationofthe
dimensionalanalysis.

92. USE OF DIMENSIONAL ANALYSIS
Dimensional analysis is
extremely useful in reducing
the number of variables in a
problem by formulating the
variables involved in any
problem into non-dimensional
parameters.
The reduction of the number of
variables in a problem provides
a systematic scheme for
planning laboratory tests and
also permits the presentation of
experimental results in a more
concise and useful form.

93. USE OF DIMENSIONAL ANALYSIS
•Supposeithasbeenfoundbythe
dimensionalanalysisthatsomeofthe
variablesinfluencinganyphenomenon,
areinterconnectedbysayfournon-
dimensionalgroupsπ1,π2,π3,andπ4
suchthatπ1=f(π2,π3,π4).
•Anexperimentalinvestigationwillprovide
certaindatawhichmaybepresentedasa
setofgroupsofπ1againstπ2witha
numberofcurvesfordifferentvaluesof
π4andeachgraphbeingforacertain
valueofπ3.
•Inthiswaytheeffectofeverygroupis
presentedseparately.Therewillofcourse
bemanysetsofgraphsifthenumberof
relevantnon-dimensionalπgroupsis
large.
•Ifonegroup,sayπ4isirrelevant,thenall
thecurvesfordifferentvaluesofπ4in
eachgraphwillcoincideleavingonlyπ1,
π2andπ3asrelevantgroups.

94. DIMENSIONAL ANALYSIS OF DRAG
AND LIFT

95. DIMENSIONAL ANALYSIS OF DRAG
AND LIFT
Usually it is not possible to
predict the total drag on a
body merely by analytical
methods. As such in almost all
the cases the general practice
is to determine the total drag
on a body experimentally.
The planning of the
experiments and the analysis
of the results obtained from
the experiments may
conveniently be carried out in
terms of the dimensional
analysis of the problem.

96. EXPERIMENTAL INVESTIGATION
•Forthepurposeoffindingout,inadvance,howthestructure
orthemachinewouldbehavewhenitisactuallyconstructed
theengineershavetoresorttoexperimentalinvestigation.
•Experimentstocheckperformanceofthestructureorthe
machinearealsonecessitatedinthecaseoftheproblems
whichcannotbesolvedcompletelysimplybytheoretical
analysis.
•Onthebasisofthefinalresultsobtainedfromthemodeltests
thedesignoftheprototypemaybemodifiedandalsoitmay
bepossibletopredictthebehavioroftheprototype.
•However,themodeltestresultscanbeutilizedtoobtainin
advancetheusefulinformationabouttheperformanceofthe
prototypeonlyifthereexistsacompletesimilaritybetween
themodelandprototype.

97. Geometric Similarity
•Thereareingeneralthreetypesofsimilaritiestobeestablishedfor
completesimilaritytoexistbetweenthemodelanditsprototype.
Theseare:
GeometricSimilarity
KinematicSimilarity
DynamicSimilarity.
•GeometricSimilarity:
Geometricsimilarityexistsbetweenthemodelandtheprototypeif
theratiosofcorrespondinglengthdimensionsinthemodelandthe
prototypeareequal.Sucharatioisdefinedasscaleratioe.g.length
scaleratio,areascaleratioandvolumescaleratioetc.
Itwillthusbeobservedthatifthemodelandtheprototypeare
geometricallysimilar,bymerechangeofscaletheycanbe
superimposed.

98. Kinematic Similarity
•KinematicSimilarity:
Kinematicsimilarityexistsbetweenthemodelandtheprototypeif
thepathsofthehomologousmovingparticlesaregeometrically
similar,andiftheratiosofthevelocitiesaswellasaccelerationof
thehomologousparticlesareequal.Sucharatioisdefinedasscale
ratioe.g.timescaleratio,velocityscaleratio,accelerationscale
ratioanddischargescaleratioetc.
Kinematicsimilaritycanbeattainedifflownetsforthemodeland
theprototypearegeometricallysimilar,whichinturnmeansthatby
merechangeofscalethetwocanbesuperimposed.
•Dynamic Similarity:
Dynamic similarity exists between the model and the prototype
which are geometrically and kinematicallysimilar if the ratio of all
the forces acting at homologous points in the two flow systems of
the model and the prototype are equal.

99. Dynamic Similarity
Forcompletedynamic
similaritytoexistbetweenthe
modelanditsprototype,the
ratioofinertiaforcesofthe
twosystemsmustbeequalto
theratiooftheresultant
forces.Thusthefollowing
relationbetweentheforces
actingonmodelandprototype
develops:
Inadditiontotheabovenotedcondition
forcompletedynamicsimilarity,theratio
oftheinertiaforcesofthetwosystems
mustalsobeequaltotheratioof
individualcomponentforcesi.e.,the
followingrelationshipswillbedeveloped:

100. Complete Similarity or
Complete Similitude
•Itmaythusbementionedthatwhenthetwosystemsare
geometrically,kinematicallyanddynamicallysimilar,then
theyaresaidtobecompletelysimilarorcomplete
similitudeexistsbetweenthetwosystems.
•However,asstatedearlierdynamicsimilarityimplies
geometricandkinematicsimilaritiesandhenceiftwo
systemsaredynamicallysimilar,theymaybesaidtobe
completelysimilar.
•Moreover,forcompletesimilitudetoexistbetweenthetwo
systemsviz.,modelandprototype,thedimensionless
numbersorπterms,formedoutofthecompletesetof
variablesinvolvedinthatphenomenon,mustbeequal.

101. FORCE RATIOS–DIMENSIONLESS
NUMBERS
•InertiaForceRatio:
•Inertia-ViscousForceRatio:
Thisnon-dimensionalratio(ρVL/μ)or
(VL/υ)iscalled‘Reynoldsnumber’(Reor
NR).
•Inertia-GravityForceRatio:
Thesquarerootofthisratioisknownas
‘Froudenumber’(FrorNF).
•Inertia-PressureForceRatio:
Thesquarerootofthisratioisknownas
‘Euler’snumber’(EuorNE).
•Inertia-ElasticityForceRatio:
Thesquarerootofthisratioisknownas
‘Machnumber’(MaorNM).
•Inertia-SurfaceTensionForceRatio:
Thesquarerootofthisratioisknownas
‘Webernumber’(WeorNW).

102. SIMILARITY LAWS
•Theresultsobtainedfromthemodeltestsmaybe
transferredtotheprototypebytheuseofmodellaws
whichmaybedevelopedfromtheprinciplesofdynamic
similarity..
•Inthederivationofthesemodellaws,ithasbeenassumed
thatforequalvaluesofthedimensionlessparametersthe
correspondingflowpatterninmodelanditsprototypeare
similar.
ReynoldsModelLaw:Fortheflowswhereinadditionto
inertia,viscousforceistheonlyotherpredominantforce,
thesimilarityofflowinthemodelanditsprototypecanbe
establishediftheReynoldsnumberissameforboththe
systems.

103. SIMILARITY LAWS
FroudeModelLaw:Whentheforceofgravitycanbeconsideredto
betheonlypredominantforcewhichcontrolsthemotionin
additiontotheforceofinertia,thesimilarityoftheflowinanytwo
suchsystemscanbeestablishediftheFroudenumberforboththe
systemsisthesame.
EulerModelLaw.Inafluidsystemwheresuppliedpressuresare
thecontrollingforcesinadditiontotheinertiaforceandtheother
forcesareeitherentirelyabsentorareinsignificant,thedynamic
similarityisobtainedbyequatingtheEulernumberforboththe
modelanditsprototype.

104. SIMILARITY LAWS
MachModelLaw.Ifinanyphenomenononlytheforces
resultingfromelasticcompressionaresignificantin
additiontoinertiaandallotherforcesmaybeneglected,
thenthedynamicsimilaritybetweenthemodelandits
prototypemaybeachievedbyequatingtheMachnumber
forboththesystems.
WeberModelLaw.Whensurfacetensioneffects
predominateinadditiontoinertiaforcethepertinent
similitudelawisobtainedbyequatingtheWebernumber
forthemodelanditsprototype.

105. TYPES OF MODELS
•Ingeneralhydraulicmodelscanbeclassifiedundertwobroadcategories:
UndistortedModels:
Anundistortedmodelisthatwhichisgeometricallysimilartoits
prototype,thatis,thescaleratiosforcorrespondinglineardimensionsof
themodelanditsprototypearesame.
Sincethebasicconditionofperfectsimilitudeissatisfied,predictioninthe
caseofsuchmodelsisrelativelyeasyandmanyoftheresultsobtained
fromthemodeltestscanbetransferreddirectlytotheprototype.
DistortedModels:
Distortedmodelsarethoseinwhichoneormoretermsofthemodelare
notidenticalwiththeircounterpartsintheprototype.
Sincethebasicconditionofperfectsimilitudeisnotsatisfied,theresults
obtainedwiththehelpofadistortedmodelareliabletodistortionand
havemorequalitativevalueonly.
Adistortedmodelmayhaveeithergeometricaldistortion,ormaterial
distortion,ordistortionofhydraulicquantitiesoracombinationofthese.

106. TYPES OF MODELS
Thefollowingaresomeofthereasonsforadoptingdistortedmodels:
Tomaintainaccuracyinverticalmeasurements.
Tomaintainturbulentflow.
Toobtainsuitablebedmaterialanditsadequatemovement.
Toobtainsuitableroughnesscondition.
Toaccommodatetheavailablefacilitiessuchasspace,money,watersupplyand
time.
Themeritsofdistortedmodelsmaybesummedupasfollows:
Theverticalexaggerationresultsinsteeperwatersurfaceslopesandmagnification
ofwaveheightsinmodels,whichcanthereforebemeasuredeasilyandaccurately.
Duetoexaggeratedslopes,theReynoldsnumbersofamodelisconsiderably
increasedandthesurfaceresistanceislowered.Thisassistsinsimulationofthe
flowconditionsinthemodelandtheprototype.
Incaseofdistortedmodelssufficienttractiveforcecanbedevelopedtoproduce
adequatebedmovementwithareasonablysmallmodel.
Modelsizecanbesufficientlyreducedbyitsdistortion,therebyitsoperationis
simplifiedandalsocostisloweredconsiderably.

107. TYPES OF MODELS
Besidestheadvantagesaccruingfromdistortionasindicatedabove,thereare
certainlimitationsofdistortedmodels,whichareaslistedbelow:
Themagnitudeanddistributionofvelocitiesareincorrectlyreproducedbecause
verticalexaggerationcausesdistortionoflateraldistributionofvelocityandkinetic
energy.
Thepressuresmaynotbecorrectlyreproducedinmagnitudeanddirection.
Someoftheflowdetailsmaynotbecorrectlyreproducedbecausedistortion
increaseslongitudinalslopesofmodelstreamsthustendingtoupsetflowregime
atapointwhereartificialmodelroughnessisrequiredtorestoreit.
Slopesofriverbends,earthcutsanddikesareoftensosteepthattheycannotbe
mouldedsatisfactorilyinsandorothererodiblematerial.
Amodelwavemaydifferintypeandpossiblyinactionfromthatoftheprototype.
Thereisanunfavourablepsychologicaleffectontheobserver.
Althoughdistortedmodelshaveanumberoflimitations,yetifjudicious
allowancesaremadeintheinterpretationoftheresultsobtainedfromsuch
models,usefulinformationcanbeobtained,whichisnotpossibleotherwise.

108. SCALE EFFECT IN MODELS
•Ifcompletesimilitudedoesnotexistbetweenamodelanditsprototype
therewillbesomediscrepancybetweentheresultsobtainedfromthe
modeltestsandthosewhichwillbeindicatedbytheprototypeafterits
construction.Thisdiscrepancyordisturbinginfluenceiscalledscaleeffect.
•Inthecaseofcertainproblemsifseveralforceshavepredominance,the
completesimilitudewillbeensuredonlyifallthepertinentmodellaws
aresimultaneouslysatisfied.However,asindicatedbelow,itisquite
difficulttosatisfyallthemodellawsinvolvedinthephenomenonand
henceinsuchcasescompletesimilaritycannotbeachieved.
•Undersuchcircumstancesthevariableswhichmaybeconsideredtohave
secondaryinfluenceonthephenomenonareneglected,sothatthe
numberofthemodellawstobesatisfiedisreduced.Butbyneglecting
thesevariablessomediscrepancyorscaleeffectwouldbedeveloped
betweentheresultsobtainedfromthemodeltestsandthoseofthe
prototype.

109. SCALE EFFECT IN MODELS
•Thescaleeffectmayalsobedevelopedincaseswheretheforces
whichhavepracticallynoeffectonthebehavioroftheprototype,
significantlyaffectedthebehaviorofitsmodel.
•Oftenitmaynotbepossibletocorrectlysimulatealltheconditions
(e.g.,roughness)inthemodel,asthatoftheprototype.Thismay
alsoresultindevelopingscaleeffectifanyoftheseconditionshasa
pronouncedeffectonthephenomenon.
•Inordertodetectthepresenceofsuchdisturbinginfluencesthe
proposedworkmaybetriedinmodelswithdifferentscalesandthe
resultingscaleeffectsjudgedfromthecomparativeresultsso
obtained.
•Besidesthistheobservationscollectedonmodelsconstructedto
differentscaleswillalsoprovideanempiricalrelationshipbetween
scaleeffectandsizeofmodel,whichmaybeutilizedtocorrectthe
resultsofthemodeltests.