1

By By

Dr. Dr. AjitAjit PratapPratap Singh,Singh,

Civil Engineering Department,Civil Engineering Department,

BITS, PilaniBITS, Pilani--333031333031

Lift and DragLift and Drag

IntroductionIntroduction

TheThe typestypes ofof problemsproblems encounteredencountered inin flowflowaroundaround submergedsubmerged bodiesbodies areare

�� FluidFluid flowingflowing aroundaround stationarystationary submergedsubmerged objectobject..

�� ObjectObject movingmoving throughthrough aa largelarge massmass ofof stationarystationary fluidfluid..

�� BothBoth thethe objectobject && fluidfluid inin motionmotion..

The force exerted by the fluid The force exerted by the fluid on the body may in on the body may in general be inclined in the general be inclined in the direction of the body or direction of the body or motionmotion

These can beThese can be

�� Along the direction of Along the direction of motionmotion

�� Perpendicular to the Perpendicular to the direction of motiondirection of motion

So the forces are called So the forces are called drag and lift force drag and lift force respectively & denoted by respectively & denoted by FFd d & F& FLL

FR

FL

FD

direction of

fluid flow

z

x

For a symmetrical body moving through an ideal fluid For a symmetrical body moving through an ideal fluid (sphere )(sphere )

�� having no viscosityhaving no viscosity

�� At uniform velocityAt uniform velocity

�� Pressure distribution is symmetricalPressure distribution is symmetrical

�� Net force acting on the body is zero.Net force acting on the body is zero.

But it is observed that body experience a resistive force But it is observed that body experience a resistive force while moving at uniform velocity in the real fluid.while moving at uniform velocity in the real fluid.

So it can be concluded thatSo it can be concluded that viscosityviscosity of the fluids is of the fluids is responsible for causing Drag on body.responsible for causing Drag on body.

θ

FD

FL

F

V

τ dA

p dA

Forces on an element of surface of an immersed body

A body held Stationary in a Stream of Real A body held Stationary in a Stream of Real Fluid Moving at Uniform VelocityFluid Moving at Uniform Velocity

2

AnalysisAnalysis

Let us consider a body held stationary in a fluid velocity of Let us consider a body held stationary in a fluid velocity of fluid be Vfluid be VForce acting on a small area dA can beForce acting on a small area dA can be

1. Along the direction tangential to the surface = 1. Along the direction tangential to the surface = ττdA dA =shear force=shear force

2.Normal to the surface = pdA = Pressure force2.Normal to the surface = pdA = Pressure force

So total drag on the surface is given by the summation So total drag on the surface is given by the summation over the entire surfaceover the entire surface

which is 1. Friction drag which is 1. Friction drag FFdf df = = ∫∫ ττdA cosdA cosθθ2.Pressure drag 2.Pressure drag FFdpdp = = ∫∫ pdA sinpdA sinθθ

Total DragTotal Drag FFd d = F= Fdf df + F+ Fdpdp

Types of DragTypes of Drag

1.1. SurfaceSurface oror frictionfriction dragdrag

2.2. DeformationDeformation dragdrag

3.3. FormForm dragdrag oror PressurePressure dragdragtoto viscosityviscosity

TheThe existenceexistence ofof viscosityviscosity forfor thetherealreal fluidsfluids isis mainlymainlyresponsibleresponsible forfor causingcausing dragdragonon thethe bodiesbodies..

Separation and WakesSeparation and Wakes

�� SeparationSeparation oftenoften occursoccurs atat sharpsharpcornerscorners

�� fluidfluid can’tcan’t accelerateaccelerate toto gogo aroundaround aa sharpsharpcornercorner

�� VelocitiesVelocities inin thethe WakeWake areare ____________(relative(relative toto thethe freefree streamstream velocity)velocity)

�� PressurePressure inin thethe WakeWake isis relativelyrelatively________________ (determined(determined byby thethe pressurepressureinin thethe adjacentadjacent flow)flow)

small

constant

Lift Lift TheThe LiftLift onon thethe bodybody isis givengiven byby thethe summationsummation ofof thethe componentcomponent ofof thetheshearshear andand pressurepressure forcesforces actingacting overover thethe entireentire surfacesurface ofof thethe bodybody inin thethedirectiondirection perpendicularperpendicular toto thethe directiondirection ofof thethe fluidfluid motionmotion::

FFLL == ∫∫ ττdAdA sinsinθθ ++∫∫ pp dAdA coscosθθ

ForFor aa bodybody movingmoving throughthrough aa largelarge massmass ofof fluid,fluid, ItIt cancan alsoalso bebe givengiven asas

FFDD== CCDDAAρρVV22//22 FFLL=C=CLLAAρρVV22//22

wherewhere

CCD,D, CCLL coefficientcoefficient ofof dragdrag && liftlift..

A=A= areaarea isis characteristiccharacteristic areaarea

ρρ =mass=mass densitydensity ofof fluidfluid

GenerallyGenerally AA isis representedrepresented inin termsterms ofof lengthlength takentaken asas LL22

Factors affecting drag and liftFactors affecting drag and lift

�� Shape of immersed bodyShape of immersed body

�� Position of immersed bodyPosition of immersed body

�� Flow of fluidFlow of fluid

�� Fluid characteristicsFluid characteristics

3

Shear and Pressure Forces: Horizontal and Shear and Pressure Forces: Horizontal and Vertical ComponentsVertical Components

( )dApD ∫ += θτθ cossinF 0

( )dApL ∫ −= θτθ sincosF 0

lift

drag

U

Parallel to the approach velocity

Normal to the approach velocity

�

2

2U

ACF dd

ρ=

2

2U

ACF LL

ρ=

A defined as projected

area _______ to force!normalnormal

drag

liftp < p0

negative pressure

p < p0

negative pressure

p > p0 positive pressurep > p0 positive pressure

Shear and Pressure ForcesShear and Pressure Forces

�� Shear forcesShear forces

�� viscous drag, frictional drag, or skin frictionviscous drag, frictional drag, or skin friction

�� caused by shear between the fluid and the caused by shear between the fluid and the solid surfacesolid surface

�� function of ___________and ______of function of ___________and ______of object object

�� Pressure forcesPressure forces

�� pressure drag or form dragpressure drag or form drag

�� caused by _____________from the bodycaused by _____________from the body

�� function of area normal to the flowfunction of area normal to the flow

surface area length

flow separation

ConsiderConsider aa cylindercylinder havinghaving radiusradius R,R, axisaxisperpendicularperpendicular toto flow,flow, rr isis thethe radialradial distancedistance ofof anyanypointpoint.. VelocityVelocity ofof flowflow v,v, θθ isis thethe angularangular distancedistanceofof thethe pointpoint fromfrom frontfront oror rearrear stagnationstagnation pointpoint..

�� Drag on cylinderDrag on cylinder

�� FluidFluid flowingflowing pastpast thethe cylindercylinder isis idealideal ii..ee.. nonnon viscous,viscous,flowflow patternpattern willwill bebe symmetricalsymmetrical..,, whichwhich isis representedrepresentedbyby velocityvelocity potentialpotential ΦΦ andand streamstream functionfunction ψψ isis givengivenbyby

θcosR

r V 2

+=Φ

r

θψ sin2

−=

r

RrV

and rr

Vr

Vr ∂∂

=∂∂

=φφ

θ

TheThe velocityvelocity components,Vcomponents,Vrr andand VVθθ atat anyany pointpoint inin thethe

flowflow fieldfield maymay bebe obtainedobtained asas..

ByBy substitutingsubstituting rr == RR wewe cancan findfind outout thethe resultantresultant

velocityvelocity whichwhich isis givengiven byby v=v= 22VsinθVsinθ ..TheThe pressurepressure pp

atat anyany pointpoint onon thethe cylindercylinder isis givengiven byby BernoulliBernoulli

equationequation

θcos12

2

−−=

r

RVVr

θθ sin12

2

+=

r

RVV

220 ρv

2

1ρV

2

1pp −+=

4

�� SoSo byby substitutingsubstituting thethe valuevalue ofof vv inin thethe equationequation wewe cancanfindfind pp asas

��

�� EquationEquation isis independentindependent ofof signsign ofof sinθsinθ.. PressurePressuredistributiondistribution isis symmetricalsymmetrical aboutabout thethe midmid--sectionsection ..

�� dragdrag onon thethe cylindercylinder isis 00..

θρρ 2220 sin2

2

1VVpp −+=

Real FluidReal Fluid�� DueDue toto thethe viscosityviscosity byby fluid,fluid, thethe pressurepressure distributiondistribution isis

modifiedmodified..�� LetLet thinthin circularcircular cylindercylinder ofof infiniteinfinite length,length, placedplaced

transverselytransversely inin aa fluidfluid streamstream�� NoteNote thatthat forfor aa givengiven cylindercylinder ofof aa givengiven diameterdiameter

immersedimmersed inin aa givengiven fluidfluid thethe ReynoldsReynolds numbernumber isisdirectlydirectly proportionalproportional toto thethe velocityvelocity andand thereforetherefore thethevariationvariation withwith ReRe NoNo couldcould bebe imaginedimagined asas thethe variationvariationwithwith velocityvelocity forfor aa givengiven cylindercylinder..

�� InIn thisthis case,case, asas longlong asas thethe boundaryboundary layerlayer isis laminar,laminar, thethepointpoint ofof separationseparation areare locatedlocated onon thethe u/su/s halfhalf portionportion ofofthethe cylinder,cylinder, butbut whenwhen thethe boundaryboundary layerlayer becomesbecomesturbulent,turbulent, thethe pointpoint ofof separationseparation shiftshift fartherfarther d/sd/s towardstowardsthethe rearrear ofof thethe cylindercylinder..

�� TheThe pressurepressure distributiondistribution diagramsdiagrams areare similarsimilar toto thatthat ofofspheresphere..

�� flowflow patternpattern behindbehind thethe cylindercylinder isis differentdifferent fromfrom thatthatbehindbehind aa spheresphere..

�� ForFor smallsmall velocitiesvelocities ofof flowflow (R(Ree << 00..55),), TheThe inertiainertia forcesforcesareare negligiblenegligible andand thethe streamlinesstreamlines areare similarsimilar toto thatthat ofofanan idealideal fluidfluid.. TheThe pressurepressure dragdrag isis negligiblenegligible andand thetheprofileprofile dragdrag consistsconsists mainlymainly ofof skinskin frictionfriction.. TheThe dragdrag isisproportionalproportional toto thethe velocityvelocity andand CCDD isis inverselyinverselyproportionalproportional ReynoldsReynolds numbernumber

�� ByBy thethe increaseincrease inin ReynoldsReynolds numbernumber thethe flowflow patternpatternww..rr..tt anan axisaxis perpendicularperpendicular toto thethe directiondirection ofof flowflowbecomesbecomes unsymmetricalunsymmetrical.. Why?Why?

�� BecauseBecause inin thethe wakewake developeddeveloped justjust behindbehind thethe cylindercylinderaa moremore oror lessless orderlyorderly seriesseries ofof vortices,vortices, whichwhich alternatealternatethethe positionposition aboutabout thethe centercenter line,line, areare developeddeveloped..

Vortex SheddingVortex Shedding

Vortex SheddingVortex Shedding

�� Vortices are shed alternately Vortices are shed alternately from each side of a cylinderfrom each side of a cylinder

�� The separation point and The separation point and thus the resultant drag force thus the resultant drag force oscillateoscillate

�� Dimensionless frequency of Dimensionless frequency of shedding given by Strouhal shedding given by Strouhal number Snumber S

�� S is approximately 0.2 over a S is approximately 0.2 over a wide range of Reynolds wide range of Reynolds numbers (100 numbers (100 -- 1,000,000)1,000,000)

�� IfIf velocityvelocity increases,increases, soso thatthat thethe ReRe rangingranging fromfrom 22 toto 3030,,boundaryboundary layerlayer separatesseparates atat twotwo pointspoints SS andand SS andand twotwo veryveryweakweak eddieseddies (vortices)(vortices) areare formedformed onon thethe d/sd/s ofof thethe cylinder,cylinder,whichwhich rotaterotate inin oppositeopposite directionsdirections.. thisthis isis thethe initialinitial stagestage forforthethe developmentdevelopment ofof thethe wakewake..

�� TheThe twotwo eddieseddies remainremain moremore oror lessless fixedfixed inin positionposition andand thethemainmain streamlinesstreamlines remainremain closeclose behindbehind themthem keepingkeeping thethelengthlength ofof wakewake limitedlimited..

�� AtAt RRee rangingranging fromfrom 4040 toto7070 wakewake asas wellwell asas pairpair ofof vorticesvortices arearedistinctdistinct andand aa periodicperiodic oscillationoscillation ofof thethe wakewake isis observedobserved

�� ByBy furtherfurther increaseincrease inin thethe valuevalue ofof RRee thethe vorticesvortices becomebecomemoremore && moremore elongatedelongated inin thethe directiondirection ofof flowflow..

�� AtAt RRee equalequal toto aboutabout 9090 thesethese vorticesvortices becomebecome symmetricalsymmetricalandand separateseparate awayaway fromfrom thethe cylindercylinder && slowlyslowly movemove inin thethe d/sd/sdirectiondirection.. TheThe eddieseddies breakbreak awayaway alternativelyalternatively fromfrom thethe twotwosidessides ofof thethe cylindercylinder andand washedwashed d/sd/s..

5

�� ThisThis processprocess getsgets intensifiedintensified withwith increaseincrease inin ReRe andand thethesheddingshedding ofof eddieseddies isis continuouscontinuous andand asas aa result,result, twotwodifferentdifferent rowsrows ofof vorticesvortices areare formedformed inin thethe wakewake..

�� TheThe centercenter ofof vortexvortex inin aa rowrow lieslies atat aa pointpoint midwaymidwaybetweenbetween thethe centerscenters ofof consecutiveconsecutive vorticesvortices inin thethe otherotherrowrow.. ThisThis arrangementarrangement ofof vorticesvortices isis knownknown asas vortexvortexstreetstreet oror VonVon KarmanKarman VortexVortex streetstreet..

�� TheThe periodicperiodic sheddingshedding ofof vorticesvortices fromfrom thethe twotwo sidessides ofofthethe cylindercylinder producesproduces alternatingalternating laterallateral forcesforces thatthat maymaycausecause aa forcedforced vibrationvibration ofof cylindercylinder atat thethe samesamefrequencyfrequency..

�� When the frequency of the vortex shedding is close to When the frequency of the vortex shedding is close to the natural frequency of the wires, a typical singing the natural frequency of the wires, a typical singing sound is produced. The frequency of the vortex shedding sound is produced. The frequency of the vortex shedding is given byis given by

5

e

e

102R250for R

19.710.198

U

fd×<<

−=

∞

L.H.S is known as The Strouhal Number which is adimensionless value and useful for analyzingoscillating, unsteady fluid flow dynamics problems

and f = oscillation frequency, d or L = relevant

length scale, v = relevant velocity scale

Initial instability causes KH vortices

KH vortices amalgamate to form large scale vortices

Large scale vortices impinge on body

Karman type shedding

(symmetric mode -interaction

with mirror vortex)

Karman type shedding in reattaching flows, illustrative example (leading edge of blunt cylinder)

�� AtAt highhigh RRee (say(say RRee == 101044),), thethe vorticesvortices disappeardisappear andand aahighlyhighly turbulentturbulent wakewake isis formedformed .. ThisThis leadsleads toto ananincreasedincreased valuevalue ofof CCDD andand thethe skinskin frictionfriction dragdrag isisnegligiblenegligible inin comparisoncomparison toto thethe pressurepressure dragdrag..

�� TheThe boundaryboundary layerlayer onon thethe cylindercylinder isis laminarlaminar upup toto ReRe ==22 xx 101055 andand dependingdepending uponupon thethe intensityintensity ofof thethe freefreestreamstream turbulence,turbulence, itit changeschanges toto turbulentturbulent bdrybdry layerlayerbeforebefore separationseparation..

�� WhenWhen thethe pointspoints ofof separationseparation movemove furtherfurther d/s,d/s, thethewakewake becomesbecomes narrowernarrower andand therethere isis substantialsubstantial dropdrop ininthethe valuevalue ofof CCDD.. TheThe criticalcritical valuevalue ofof RRee atat whichwhich thethevaluevalue ofof CCDD decreasesdecreases dependdepend uponupon thethe degreedegree ofofturbulenceturbulence inin thethe mainmain flowflow andand uponupon thethe roughnessroughness ofofthethe surfacesurface upstreamupstream ofof thethe pointpoint ofof separationseparation..

Effect of Boundary Layer Effect of Boundary Layer TransitionTransition

Ideal (non

viscous) fluid

Real (viscous)

fluid: laminar

boundary layer

Real (viscous)

fluid: turbulent

boundary layer

No shear!No shear!

6

Problem 1Problem 1

TheThe electricalelectrical transmissiontransmission towers,towers, 1010 mm highhigh areare fixedfixed 400400mm apartapart toto supportsupport 1616 cables,cables, eacheach 22 cmcm inin diadia.. IfIf aa 100100kmphkmph windwind isis blowingblowing transeverselytranseversely acrossacross thethe cables,cables,makemake calculationscalculations forfor thethe totaltotal forceforce toto whichwhich eacheach towertowerwouldwould bebe subjectedsubjected andand thethe momentmoment actingacting atat thethe basebase ofofeacheach towertower.. AssumeAssume airair densitydensity ρρ == 11..22 kg/mkg/m33 andand dynamicdynamicviscosityviscosity µµ == 11..6565 xx 1010--55 NN--sec/msec/m22.. AssumeAssume therethere isis nonointerferenceinterference betweenbetween thethe wireswires andand taketake dragdrag coefficientscoefficientsasas CCdd == 00..9595 forfor 101033 << ReRe << 101044 andand CCdd == 11..22 forfor 101044 << ReRe <<101055.. WouldWould thethe cablescables bebe subjectedsubjected toto selfself inducedinducedvibrationsvibrations andand ifif soso calculatecalculate thethe frequencyfrequency ofof vortexvortexsheddingshedding..

Problem 1Problem 1

TheThe transmissiontransmission wireswires areare ofof 55 mmmm inindiameterdiameter.. CalculateCalculate thethe frequencyfrequency ofof thethevortexvortex sheddingshedding whenwhen thethe windwind isis blowingblowing atataa speedspeed ofof 6060 km/hkm/h acrossacross thethe wireswires..AssumeAssume airair densitydensity ρρ == 11..22 kg/mkg/m33 andanddynamicdynamic viscosityviscosity µµ == 11..88 xx 1010--55 NN--sec/msec/m22

Problem 2Problem 2

AA chimneychimney inin aa streamstream powerpower plantplant isis 4040 mmhighhigh.. TheThe diameterdiameter atat thethe basebase isis 44..55 mm andanditit graduallygradually reducesreduces toto 22..55 mm atat thethe toptop..CalculateCalculate thethe bendingbending momentmoment atat thethe basebaseofof thethe chimneychimney whenwhen windwind speedspeed isis 6060 km/hkm/h..ρρ== 11..22 kg/mkg/m33;; µµ == 11..99 xx 1010--55 NN--s/ms/m22..

Drag on a sphereDrag on a sphere

LetLet usus considerconsider flowflow inin aa idealideal fluidfluid..

ObservationObservation::

�� ViscosityViscosity ofof flowingflowing fluidfluid isis absentabsent

�� FlowFlow patternpattern andand pressurepressure distributiondistribution isis symmetricalsymmetrical onon thethefrontfront andand rearrear ofof thethe spheresphere..

�� HighestHighest intensityintensity ofof pressurepressure occursoccurs atat stagnationstagnation pointspoints..

�� IntensityIntensity isis lowestlowest aroundaround thethe circumferencecircumference atat rightright anglesangles totothethe flowflow..

�� PressurePressure isis 00 duedue toto symmetrysymmetry.. SoSo nono pressurepressure dragdrag..

ConclusionConclusion:: NoNo dragdrag inin casecase ofof flowflow inin idealideal fluidfluid..

Flow in Real fluidFlow in Real fluid

�� ViscosityViscosity inin flowingflowing fluidfluid isis possessedpossessed.. ResultingResultingPressurePressure distributiondistribution isis entirelyentirely differentdifferent..

�� ViscousViscous forcesforces areare moremore predominantpredominant thanthan inertiainertia forcesforces..

�� PressurePressure distributiondistribution isis differentdifferent comparedcompared toto idealideal fluidsfluids..�� IfIf thethe velocityvelocity isis veryvery lowlow soso thatthat ReynoldsReynolds numbernumber isis

veryvery lowlow ((00..22))..�� ViscousViscous forcesforces areare moremore predominantpredominant thanthan inertiainertia forcesforces..

TotalTotal DragDrag isis givengiven byby FFDD== 33πµVDπµVDwherewhere µ=µ= viscosityviscosity

V=VelocityV=Velocity ofof flowingflowing fluidfluid..D=D= DiameterDiameter ofof thethe SphereSphere..

7

Skin Friction is given by 2/3 of FSkin Friction is given by 2/3 of FDD i.e. 2i.e. 2ππµVDµVDPressure Drag is given by 1/3 of FPressure Drag is given by 1/3 of FD D i.e. i.e. ππµVDµVDCoeff of drag can be found as CCoeff of drag can be found as CDD= 24/R= 24/Re.e.

The above equation is known as stokes law equation. The above equation is known as stokes law equation. These are satisfied for RThese are satisfied for Ree≤ 0.2.≤ 0.2.

Due to proximity of boundaries the resistance to motion is Due to proximity of boundaries the resistance to motion is increased , so drag coeff. is given byincreased , so drag coeff. is given by

+=

1

1.2124

D

D

RC

e

d

Where DWhere D1 1 is smallest lateral is smallest lateral dimension of the containerdimension of the container

WhatWhat wewe havehave seenseen tilltill nownow thatthat thethe aboveabove equationequation arearevalidvalid forfor Re≤Re≤ 00..22.. ButBut SwedishSwedish PhysicistPhysicist OseenOseen gavegaveequationequation whichwhich isis validvalid Re<Re<11..

+=e

e

DR

RC

16

31

24

Effect of Reynolds numberEffect of Reynolds number�� ByBy increaseincrease ofof RRee thethe viscosityviscosity isis reducedreduced inin thethe

predominantpredominant areaarea..�� ItIt isis restrictedrestricted toto aa veryvery smallsmall zonezone ofof boundaryboundary layerlayer

formedformed closedclosed toto thethe spheresphere..�� AA separationseparation ofof boundaryboundary layerlayer beginsbegins fromfrom d/sd/s toto u/su/s

andand pointpoint ofof separationseparation movemove furtherfurther forwardforward towardstowardsupstreamupstream asas RRee increasesincreases untiluntil RRee≈≈ 10001000..

�� AA moremore oror lessless stablestable positionposition forfor thethe pointpoint ofof separationseparationisis achievedachieved whichwhich isis aboutabout 808000 fromfrom thethe upstreamupstreamstagnationstagnation pointpoint..

�� AA largelarge wakewake isis producedproduced ..�� ItIt resultsresults inin (form)(form) dragdrag aboutabout 9595%%.. asas comparedcompared toto skinskin

frictionfriction dragdrag whichwhich isis 55%% ofof totaltotal dragdrag..�� CCDD isis independentindependent inin thethe rangerange ofof 101033 toto 101055 ofof RRee..ButBut CCDD

increasesincreases slightlyslightly fromfrom00..44 toto00..55 inin thisthis rangerange ofof RRee

�� UptoUpto RRee<<33××101055 boundaryboundary maymay bebe consideredconsidered toto bebelaminarlaminar andand thethe pressurepressure distributiondistribution aroundaround thethe spheresphereonon thethe U/SU/S sideside uptoupto thethe pointspoints ofof separationseparation isis almostalmostthethe samesame asas IdealIdeal fluidfluid..

�� RRee>>33××101055,boundary,boundary layerlayer becomesbecomes turbulentturbulent andand InIn this,this,pointpoint ofof separationseparation shiftshift toto thethe D/SD/S..

�� TheThe pointspoints ofof separationseparation shiftshift considerablyconsiderably andand arearelocatedlocated atat aboutabout isis aboutabout 11011000 fromfrom thethe upstreamupstreamstagnationstagnation pointpoint..

�� TheThe valuevalue ofof CCDD dropsdrops fromfrom 00..55 TOTO 00..22 ,, ii..ee.. dragdragcoefficientcoefficient isis reducedreduced whenwhen flowflow changeschanges fromfrom laminarlaminar tototurbulentturbulent..

Flow separationFlow separation

�� FrontFront--toto--rear asymmetry of forces rear asymmetry of forces results in high drag on the sphereresults in high drag on the sphere

Flow pattern varies with ReFlow pattern varies with Re

8

Flow pattern varies with ReFlow pattern varies with Re Flow pattern varies with ReFlow pattern varies with Re

Flow pattern varies with ReFlow pattern varies with ReEffect of Turbulence Levels on Effect of Turbulence Levels on

DragDrag�� Flow over a sphere: (a) Reynolds number Flow over a sphere: (a) Reynolds number

= 15,000; (b) Reynolds number = 30,000, = 15,000; (b) Reynolds number = 30,000,

Point of separationPoint of separation

Causes boundary layer to become turbulentCauses boundary layer to become turbulent

9

Drag on a Golf BallDrag on a Golf Ball

DRAGDRAG ONON AA GOLFGOLF BALLBALL comescomes mainlymainly fromfrompressurepressure dragdrag.. TheThe onlyonly practicalpractical wayway ofofreducingreducing pressurepressure dragdrag isis toto designdesign thethe ballball sosothatthat thethe pointpoint ofof separationseparation movesmoves backbackfurtherfurther onon thethe ballball.. TheThe golfgolf ball'sball's dimplesdimplesincreaseincrease thethe turbulenceturbulence inin thethe boundaryboundary layer,layer,increaseincrease thethe ______________ ofof thethe boundaryboundary layer,layer,andand delaydelay thethe onsetonset ofof separationseparation.. TheThe effecteffectisis plottedplotted inin thethe chart,chart, whichwhich showsshows thatthat forforReynoldsReynolds numbersnumbers achievableachievable byby hittinghitting thetheballball withwith aa club,club, thethe coefficientcoefficient ofof dragdrag isis muchmuchlowerlower forfor thethe dimpleddimpled ballball..

inertiainertia

Drag on a Flat PlateDrag on a Flat Plate

Plate held parallel to flowPlate held parallel to flow

�� Total drag= friction dragTotal drag= friction drag

�� Formation of boundary layerFormation of boundary layer

�� Magnitude depends upon the boundary layerMagnitude depends upon the boundary layer

�� Inertia forces are predominantInertia forces are predominant

direction of fluid

flow

FR

FD

F

L

Forces on a flat

surface

z

x

Flat Plate:Flat Plate:StreamlinesStreamlines

U

0 1

2

3

4

−=−=

2

0

2

2

21U

pp

U

vC p

ρPoint v Cp p

1

2

3

4

0 1<U >0>U <0

>p0

>p0

<p0

<p0

Points outside boundary layer!

Perpendicular to the flowPerpendicular to the flow

�� For ideal fluid flow pattern will be symmetrical on u/s & For ideal fluid flow pattern will be symmetrical on u/s & d/s.d/s.

�� Pressure distribution will be symmetrical( no drag).Pressure distribution will be symmetrical( no drag).

�� For real fluid flow pattern & Pressure distribution are For real fluid flow pattern & Pressure distribution are different.different.

Significance of CSignificance of CDD

�� CCDD is function of Ris function of Ree at low & moderate value.at low & moderate value.

�� RRee> 1000, C> 1000, CDD=0.2 ( constant)=0.2 ( constant)

�� CCDD is independent at ,Ris independent at ,Ree> 1000.> 1000.

�� If L/B is not large , CIf L/B is not large , CDD is reduced. is reduced.

10

Flat Plate Drag CoefficientsFlat Plate Drag Coefficients

0.001

0.01

1000

0

1000

00

1000

000

1000

0000

1000

0000

0

1000

0000

00

1000

0000

000

Rel

Ul

n=

l

e

DfC

1 x 10-3

5 x 10-4

2 x 10-4

1 x 10-4

5 x 10-5

2 x 10-5

1 x 10-5

5 x 10-6

2 x 10-6

1 x 10-6

( )[ ]2.5

1.89 1.62log /DfC le-

= -

( )0.5

1.328

ReDf

l

C =( )[ ]

2.58

0.455 1700

Relog ReDf

ll

C = -

( )[ ]2.58

0.455

log ReDf

l

C =

0.20.072ReDf l

C -=

Lift on CylinderLift on Cylinder

IdealIdeal FluidFluid�� WhenWhen thethe bodybody isis symmetricalsymmetrical withwith respectrespect toto itsits axisaxis

andand soso locatedlocated thatthat itsits axisaxis isis parallelparallel toto thethe directiondirection ofofmotion,motion, thenthen thethe resultantresultant forceforce exertedexerted byby fluidfluid onon thethebodybody isis inin thethe directiondirection ofof motion,motion, andand inin suchsuch aa casecase thetheliftlift isis zerozero..

IdealIdeal FluidFluid�� LetLet anan idealideal fluidfluid flowingflowing pastpast aa cylindercylinder ofof radiusradius RR withwith

aa uniformuniform velocityvelocity ofof fluidfluid VV..�� FlowFlow patternpattern willwill bebe symmetricalsymmetrical aboutabout bothboth axesaxes..�� ResultantResultant velocityvelocity vv atat anyany pointpoint onon thethe surfacesurface == 22VsinVsinθθ�� PressurePressure distributiondistribution areare identicalidentical (no(no lift)lift)

Spinning SpheresSpinning Spheres

�� What happens to the separation points What happens to the separation points if we start spinning the sphere?if we start spinning the sphere?

LIFT!LIFT!

In real fluidIn real fluid

�� ConsiderConsider aa aa constantconstant circulationcirculation ГГ�� FlowFlow patternpattern consistconsist ofof aa SeriesSeries ofof streamstream lineslines..�� PeripheralPeripheral velocityvelocity VVcc=Г/=Г/22πRπR ..�� SuperimposingSuperimposing thethe 22 casescases..�� FlowFlow patternpattern willwill bebe unsymmetricalunsymmetrical..

�� TheThe postionpostion ofof stagnationstagnation pointspoints SS11 andand SS22 onon thethesurfacesurface ofof thethe cylindercylinder maymay bebe obtainedobtained bybyconsideringconsidering vv == 00 andand solvingsolving forfor SinSinθθ asas

Rvv πθ 2/sin2 Γ+=

4πRV

ΓSinθ

×−=

11

�� VelocityVelocity isis higherhigher atat upper,upper, lowerlower atat lowerlower..

�� PressurePressure isis higherhigher atat upperupper andand lowerlower atat lowerlower..

�� ForceForce willwill bebe exertedexerted perpendicularlyperpendicularly toto thethe motionmotion..

�� pressure=pressure=

( )[ ]220 Γ/2πR2vsinθv0.5ρpp +−+=

( )[ ][ ] dθ sinθΓ/2πR2vsinθv0.5pLRF

2π

0

220L ∫ +−+−=

RV

Γ

ρV2

1

2RLρVLΓ

ρV2

1

AFC coeff.Lift ρVLΓF

22

LLL ===⇒=

�� ForFor thethe motionmotion ofof aa boatboat inin aa streamstream thethe coefficientcoefficient ofofdragdrag isis foundfound toto bebe 00..3030.. IfIf afterafter polishingpolishing thethe hullhull ofof thetheboatboat thethe coefficientcoefficient ofof dragdrag decreasesdecreases toto 00..2525 andand thethesamesame drivingdriving powerpower isis utilized,utilized, findfind thethe percentagepercentage inin thethespeedspeed attainedattained..

�� TheThe cupcup anemometeranemometer isis placedplaced inin windwind blowingblowing atat aavelocityvelocity ofof 5454 km/hrkm/hr.. IgnoringIgnoring bearingbearing friction,friction, findfind thethespeedspeed ofof rotationrotation.. TakeTake cdcd == 00..3434 forfor thethe convexconvex facefaceandand 11..3333 forfor concaveconcave faceface.. AssumeAssume itit rotatesrotates uniformlyuniformly..

ProblemProblem

�� AA cylindercylinder rotatesrotates clockwiseclockwise atat 300300 rpmrpm aboutabout itsits axisaxiswhichwhich isis perpendicularperpendicular toto thethe airair streamstream havinghaving aa velocityvelocityofof 22 m/secm/sec.. TheThe cylindercylinder isis 22mm inin diadia andand 1010 mm longlong..DetermineDetermine (a)(a) circulationcirculation (b)(b) TheThe theoreticaltheoretical liftlift forceforce perperunitunit lengthlength (c)(c) thethe positionposition ofof stagnationstagnation pointspoints.. (d)(d) TheTheactualactual lifelife andand dragdrag andand thethe resultantresultant forceforce onon thethecylindercylinder.. TakeTake densitydensity ofof airair 11..2424 kg/mkg/m33.. AlsoAlso assumeassumevvcc/V=/V=11..5757 forfor ccDD == 00..6565 andand ccLL == 33..4040 findfind alsoalso thethe speedspeedofof rotationrotation ofof thethe cylindercylinder whichwhich yieldsyields onlyonly aa singesingestagnationstagnation pointpoint..

�� CalculateCalculate thethe diadia ofof aa parachuteparachute toto bebe usedused forfor droppingdroppingandand objectobject weightingweighting 10001000 NN soso thatthat thethe maximummaximumterminalterminal velocityvelocity ofof droppingdropping isis 55 m/secm/sec.. TheThe drugdrugcoefficientcoefficient forfor parachuteparachute whichwhich maymay bebe treatedtreated asashemisphericalhemispherical isis 11..33.. TakeTake densitydensity ofof airair 11..216216 kg/mkg/m33..

�� InIn investigatinginvestigating thethe possibilitypossibility ofof usingusing rotorsrotors inin placeplace ofofairplaneairplane wingswings itit isis assumedassumed thatthat eacheach ofof thethe twotwo rotatingrotatingcylinderscylinders wouldwould havehave aa diameterdiameter ofof 11..00 mm andand lengthlength 44..00mm.. IfIf thethe weightweight ofof thethe entireentire planeplane isis 8080,,000000 N,N,determinedetermine thethe speedspeed ofof rotationrotation ofof rotorsrotors whichwhich willwillsupportsupport thisthis loadload atat aa 250250 km/hrkm/hr crushingcrushing speedspeed.. UseUse figfig1818..1717 forfor determiningdetermining thethe dragdrag andand liftlift coefficientcoefficient.. AlsoAlsodeterminedetermine thethe powerpower requiredrequired toto overcomeovercome thethe rotorrotordragdrag.. TakeTake densitydensity ofof airair 11..208208 kg/mkg/m33..

Thank YouThank You

12

Dimensional AnalysisDimensional Analysis

Let us consider a objectLet us consider a object

Characteristic length LCharacteristic length L

Moving with a velocity VMoving with a velocity V

Mass density of fluid Mass density of fluid ρρViscosityViscosity

Modulus of Elasticity KModulus of Elasticity K

Drag FDrag FDD = f= f11( L,( L, ρρ, , µµ ,V,K,g, ,V,K,g, ηη ))

Lift Lift FFL L = f= f 22(L,(L, ρρ, , µµ ,V,K,g, ,V,K,g, ηη ))

where where ηη =dimensionless shape factor=dimensionless shape factor

FFDD/ (0.5/ (0.5ρρLL22VV22) =f) =f33 [[ηη, (, (ρρVL)/ VL)/ µµ ,V/,V/√√ (K/ (K/ ρρ),V/ ),V/ √√gL]gL]

FL/ (0.5ρL2V2) =f4 [η, (ρVL)/ µ ,V/√ (K/ ρ),V/ √gL]

The left hand side of the above equations are CD and CL

lift & drag force depend on

Shape factor η

Reynolds number Re= (ρVL)/ µ

Mach number Ma=( V/√ (K/ ρ),

Froude Number Fr= V/( √gL).

The above describes

The geometry of object

effect of viscosity

effect of elasticity

effect of gravity respectively