IncompFluidMech Lecture 04 FluidStatics

FluidStaticsAE225IncompressibleFluidMechanicsAniruddhaSinhaDepartmentofAerospaceEngineeringIITBombayIntroductiontouidstaticsFluidstatics(colloquiallycalledhydrostatics)isthestudyofuidsatrestHydrostaticsisimportantinthefollowingareasofAerospaceEngineering Airships Ships AtmosphereFluidmechanicsinvolves Density Pressure TemperatureSincevelocityiszero,shearstrainrateiszeroandhenceshearstressiszero,irrespectiveofcoecientofviscosityThisreducesthecomplexityoftheuidmechanicsdrastically1 / 33PressureAirBalloonwallGasFictitiouswallGasGasGasAir Animaginarysurfaceseparatinguidcanhaveforcesactingoneachofitsside Pressureisthenormalforceperunitareaatagivenpointactingonagivenplanewithintheuid-massofinterest Pressurevariesfrompointtopoint Doespressurevarydependingonorientationoftheplanepassingthroughthepoint?2 / 33FluidparticleFBDinhydrostaticsConsiderauidparticlewithinamassofuidatrestTherearenoshearstressessincethevelocityiszeroeverywhere;theonlyexternalforcesactingonthewedgeshapedparticleare: pressure,and weightofuidwithintheparticle.xpy(xz)pz(xy)ps(xs)yg(xyz)/2zsFluidxyzFree-bodydiagramforauidparticlewithinauidmassParticleissmallenoughsothatpressuresareconstantovereachsurface3 / 33PascalslawNewtonslawiny-dir.: pyxz psxs sin = xyz2ay,Newtonslawinz-dir.: pzxy psxs cos g xyz2= xyz2azwhereay&azareaccelerationsiny&zdirections,respectivelyFromgeometry,y= s cos andz= s sin ,sothatpy ps= ayx/2, pz ps= (az+ g) x/2.SinceRHSvanishesforavanishinglysmallparticle,py= pz= psButthisresultisforanarbitrary,soweconcludethatpx= py= pz.Pascalslaw: Thepressureatapointinauidatrest,orinmotion,isindependentofthedirectionaslongastherearenoshearstressesForuidsinmotion(withshearstresses),normalstressatapoint(whichcorrespondstopressureinhydrostatics)isnotnecessarilysameinalldirectionsthenpressureisdenedastheaverageofanythreemutuallyorthogonalnormalstressesatthepoint4 / 33Pressurevariationinauidx

p pyy2

xz

p pzz2

xyz

p +pyy2

xz

p +pzz2

xyg(xyz)/2(x, y, z)xyzySurfaceandbodyforcesactingonauidparticlewithoutshearConsiderarectangularuidparticlecenteredatanarbitrarypoint(x, y, z),wherethepressureispTaylorseriesapproximationgivestheresultantsurfaceforceduetothevariationofpressureinthey-directionasFy,pressure=_p pyy2_xz _p +pyy2_xz= py xyzSimilarly,Fx,pressure= px xyz, Fz,pressure= pz xyz5 / 33GoverningequationforuidinabsenceofshearNetsurfaceforceactingonuidparticleduetopressurevariationsis_Fxi+ Fyj + Fzk_pressure= _pxi+pyj +pzk_xyzOr, Fpressure= p (xyz)Netsurfaceforceduetopressureperunitvolumeontheuidparticleisfpressure= pNetbodyforceduetogravityperunitvolumeisfbody= gNewtons2ndlawappliedtotheuidparticlegivesaccelerationaasa = fpressure + fbody= p + gWithusualchoiceofcoordinatesystemwherez-directionisup,wehavep gk= aGeneralequationofmotionforauidinabsenceofshearingstresses6 / 33PressurevariationinauidatrestInhydrostatics,uidaccelerationisidenticallyzero,a = 0,sop= gk= kSpecicgravityofauid,:= giscommonlyusedinhydrostaticsRemark: Bydenitionof operator, pisperpendiculareverywheretosurfacesofconstantpThus,uidinhydrostaticequilibriumwillalignitsconstantpressuresurfaceseverywherenormaltothelocal-gravityvectorIftheuidisaliquid,itsfreesurface,beingatatmosphericpressure,willbenormaltolocalgravityhorizontalIncomponentform,px= 0,py= 0,pz= g= Thus,hydrostaticpressureisafunctionofzalone,anddpdz= g= 7 / 33Pressurevariationinanincompressibleuid Assumeconstant(waterengineeringproblems) Assumeconstantg(variationofgisnegligibleformanyproblems)Then,integrating(dp/dz= g)gives_p2p1dp= g_z2z1dz =p2p1= g(z2z1) = ghxyzz2z1h = z2z1p2p1Pressuremustincreasewithdepthtoholduptheuidaboveith =(p1p2)/(g) iscalledpressureheadInterpretedasheightofacolumnofuidofspecicgravitygrequiredtogiveapressuredierencep1p28 / 33PressureinliquidsatrestwithafreesurfaceThefreesurfaceofaliquidisaconvenientreference,ifitexistsReferencepressurep0willtypicallybetheatmosphericpressurepatmPressureatanydepthhbelowthefreesurfaceisp= gh + p0Thepressureinahomogeneous,incompressibleuidatrestdependsonthedepthoftheuidrelativetosomereferenceplane,anditisnotinuencedbythesizeorshapeofthetankorcontainerinwhichtheuidisheld9 / 33PressureingasesatrestGases,beingcompressible,haveavariabledensitythatshouldbeconsideredintheintegrationof(dp/dz= g)However,gaseshavemuchlowerdensitycomparedtoliquids,sothatthepressuregradientintheverticaldirectioniscorrespondinglysmallWecanneglecttheeectofelevationchangesonthepressureingasesintanks,pipes,andsoforthinwhichthedistancesinvolvedaresmallIfheightvariationislarge( 1000feet),variationbecomesimportantTheidealgaslaw(forcompressiblegases)is =pRTwhich,whencombinedwiththehydrostaticequationresultsindpdz= gpRTIntegrationafterseparationofvariablesyields_p2p1dpp= gR_z2z1dzTThevariationoftemperaturewithelevationisrequiredtocompletethis10 / 33PressureincompressiblegasesisothermalIfthetemperaturehasaconstantvalueT0overtherangez1toz2(isothermalconditions),thenp2= p1 exp_g (z2z1)RT0_Evenfora10,000ftaltitudechange,dierenceinconstant-temperature(isothermal)andconstant-density(incompressible)resultsarerelativelyminorDensityvariationisalsogivenby2= 1 exp_g (z2z1)RT0_11 / 33PressureincompressiblegaseslineartemperatureTemperaturemaydecreasewithaltitudeataconstantlapserateofovertherangez1toz2;i.e.,T (z) = T0 + (z z0) ,wherethealtitudeismeasuredfromsomereferencelevelz0(typicallymeansealevel)wherethetemperatureisT0Integrationthengives,p= p0_1 + (z z0)T0_g/RDensityvariationisgivenby = 0_1 + (z z0)T0_g/R112 / 33Standardatmosphere Atmosphericconditionschangedaytodayandacrossseasons Standardatmospherewasrstdevelopedinthe1920stocodifyrepresentativeatmosphericconditions Itdoesnotchangewithday,dateortime Itisanidealizedrepresentationofmiddle-latitude,year-aroundmeanconditionsoftheearthsatmosphere ThecurrentlyacceptedStandardatmosphereisbasedonareportpublishedin1962andupdatedin1976 InIndiaweuseIndianStandardAtmosphere13 / 33Internationalstandardatmosphere(ISA)TemperatureT,KAltitudeh,kmTroposphereTropopauseStratopauseStratosphere56.5 C,11km= 6.5 C/km15 CMesosphereMesopause56.5 C,20km2.5 C,47km2.5 C,51km44.5 C,32km=2.8 C/km86.5 C,85km=1 C/km= 2.8 C/km58.5 C,71km= 2 C/km14 / 33RelevantpressuresanddensitiesinatmosphereAltitude T,C P,bar ,kg/m3Relevance0 15 1 1.22510 -50 0.261 0.412Civilian aircraft yatthisaltitude20 -56.5 0.054 0.088Military aircraft yatthisaltitude50 -2.5 6.7 1048.7 104Air-breathing vehi-clescantyatthisaltitude15 / 33TransmissionofuidpressureTherequiredequalityofpressureatequalelevationsthroughoutasystemisimportantfortheoperationofhydraulicjacks,lifts,andpresses,aswellashydrauliccontrolsonaircraftandothertypeofheavymachineryF1= pA1, F2= pA2, F2=A1A2F1Thetransmissionofuidpressurethroughoutastationaryuidistheprincipleuponwhichmany hydraulic devices are based16 / 33Measurementofpressureconventions Absolutepressureismeasuredw.r.t. perfectvacuum Gaugepressureismeasuredw.r.t. localatmosphericpressurepgauge= pabsolutepatmosphere Ifthepressurebeingmeasuredisexpectedtobebelowatmosphericpressure,thenvacuumpressureisusedpvacuum= patmospherepabsolute17 / 33BarometryMeasurementofatmosphericpressureisusuallywithamercurybarometerThetubeisinitiallylledwithmercuryandthenturnedupsidedownwithopenendinthemercurycontainerpatm= Hg gh + pvapourFormercury,pvapouris0.16Pa,andisthereforeneglectedAtmosphericpressureisconventionallyspeciedasheightofmercurycolumn,hStandardatmosphere(101325Pa,1bar)is760mmofHg18 / 33ManometryexampleU-tubemanometerasaowmeterThevolumerateofow,Q,throughapipecanbedeterminedusingaownozzlewithinthepipe. ThenozzlecreatesapressuredropalongthepipewhichisgivenbyQ= KpApB,whereKisaconstantdependingonthepipeandnozzlesize. Obtainanexpressionforthepressuredropintermsoftheparametersshown.AccountforthepressurestartatA,moveverticallyupwardto(1),switchto(2)(samepressure),switchto(3)(samepressure),moveupwardto(4),switchto(5)(samepressure),moveverticallydowntoBpA1h12h2 + 1(h1 + h2) = pB=pApB= (21)h219 / 33ManometryexampleU-tubemanometerasaowmeterThevolumerateofow,Q,throughapipecanbedeterminedusingaownozzlewithinthepipe. ThenozzlecreatesapressuredropalongthepipewhichisgivenbyQ= KpApB,whereKisaconstantdependingonthepipeandnozzlesize. Obtainanexpressionforthepressuredropintermsoftheparametersshown.AccountforthepressurestartatA,moveverticallyupwardto(1),switchto(2)(samepressure),switchto(3)(samepressure),moveupwardto(4),switchto(5)(samepressure),moveverticallydowntoBpA1h12h2 + 1(h1 + h2) = pB=pApB= (21)h219 / 33ManometryWhenpressuresaresmall,theycanbemeasuredbymeasuringheight/depthofliquidcolumninverticalorinclinedtubesPiezometertubepApatm= 1h1SimpleU-tubemanometerpApatm= 2h21h1DierentialU-tubemanometerpApB= 2h2 + 3h31h1Piezometerisonlyusefulifpressureinaliquidisabovepatm20 / 33InclinedtubemanometerenhancedprecisionTomeasuresmallpressurechanges,aninclined-tubemanometerisfrequentlyusedOnelegofthemanometerisinclinedatanangle,andthedierentialreading2ismeasuredalongtheinclinedtubepA + 1h1= pB+ 2

2 sin + 3h3Neglectingthecontributionsofgascolumnsofheightsh1&h3,pApB= 2

2 sin Thus,forsmall,thesamepressuredierenceismagniedtolarger2valueforgreateraccuracyinreading21 / 33HydrostaticforceonaplanesurfaceAsurfacesubmergedinauiddevelopshydrostaticpressureonitHydrostatic forces are important in design of ships, dams, storage tanks, etc.PressureisconstantonthebottomwallasheightisnotchangingPressureischangingonsidewallsasheightischanging22 / 33Hydrostaticforceonaninclinedplanesurfaceh(x, y)hc=h/ sin Centroid,cxyCPNetresultantforce,F=pcASideviewPlanviewFreesurfacep=patmp=patmdA=dxdyLiquidofsp. gravityAssumptions: Liquiddensityisconstant&pressureonlowersideispatm23 / 33Hydrostaticforceonaninclinedplanesurface(contd.)NethydrostaticforceontheplateisF=_pgaugedA =_hdA = _hdA = sin _dAsinceh = sin ,andisconstantalongtheplateSlantdistancefromthefreesurfacetothecentroidoftheplateisc=1A_dAsothatF= c sin A = hcA = pc,gaugeAThus,thenethydrostaticforceisindependentoftheparticularshapeoftheplateexceptfor thedepthofthecentroidoftheplatew.r.t. thefreesurfacehc,and theareaoftheplate24 / 33CenterofpressureforaninclinedplanesurfacePointofactionofnethydrostaticforceiscalledcenterofpressureNethydrostaticmoment(duetoF)aboutthexaxisthruthecentroidisFyCP=_pgaugeydA =_hydA = _ sin ydA= sin _(c y) ydA = sin c_ydA sin _y2dABut,theintegral_ydAinrsttermisidentically0bydenitionofcentroidWithIxxc:=_y2dAbeingtheareamomentofinertiaoftheplateaboutitscentroidalxaxis,computedintheplaneoftheplate,wehaveyCP= Ixxc sin hcA= Ixxc sin hcASinceallindividualquantitiesinthelastexpressionarepositive,centerofpressureisalwaysbelowthecentroidfortheinclinedatpateFor a given plate, CP approaches centroid as hcincreases and/or decreasesWithradiusofgyrationr2gx:= Ixxc/A,wealsohaveyCP= r2gx sin _hc25 / 33Centerofpressureforaninclinedplanesurface(contd.)Tondthex-coordinateofthecenterofpressure,xCP,wecalculatethemomentabouttheyaxisthruthecentroidxCP=_pgaugexdAF= sin _(c y) xdAhCGA= Ixyc sin hcAwhereIxyc:=_xydAistheareaproductmomentofinertiaoftheplateaboutitscentroidalx yaxesPlateswithareadistributedsymmetricallyaboutanaxisintheslantdirectionwillhaveIxyc= 0,sothattheirxCPwillbe026 / 33Momentsofinertiaofcommongeometries27 / 33BuoyancyforceBuoyancyforceFBonuidactingdownwardisFB= F2F1 +W= (h2h1)A {(h2h1) A V }= VThisisArchimedesprincipleBuoyancyforceFBonbodyactsupwardFBpassesthrucentroidofdisplacedvolume,calledcenterofbuoyancySameresultsapplytofully-submergedandoatingbodies,aslongassp.gravityofoutsideuidcanbeneglected28 / 33Stabilityoffullysubmergedbodies29 / 33StabilityofoatingbodiesStableconguration(e.g.,bargesthatridelowinthewater)Unstableconguration(e.g.,tallslenderbodies)30 / 33Rigidbodymotionofauid Theuidmassmovesasifitwerearigidbody Althoughuidelementsareinmotion,thereisnorelativemotionbetweenthem Thustherearenoshearstressesandonlypressureforcesareactingsoalthoughuidisnotstatic,hydrostaticanalysisissuitable Forrigidbodymotion,accelerationmustbeconstant Inpractice,thecontainermustbemovedwithconstantaccelerationandonemustbepatienttilltransients(sloshing)dieout Thegeneralequationofmotioninthiscaseisp= _gk + a_ UsingCartesiancoordinates,px= ax, py= ay, pz= g+ az31 / 33LinearrigidbodymotionofauidLetuidacceleratewithconstantaccelerationa = axi+ azkThedierentialpressureisgivenbydp=px dx +py dy+pz dz= axdx (g+ az) dzThefreesurfaceis,bydenition,acteduponbyuniformatmosphericpressure,sodp= 0thereatThustheequationforthefreesurface(oranyisocontourofpressure)isdzdx= axg+ az= tan a=0gax=0, az =0axp/azgazaxxz32 / 33RigidbodyrotationCentripetalaccelerationonuidparticleisinradialdirection,ar= r 2Pressuregradientincylindricalcoordinatesisp= p/r er+ r1p/ e + p/z ezThus,rigidbodymotionequationgivesdierentialpressureasdp= r 2dr gdzThefreesurface(oranyotherconstant-pressuresurface)isthengivenbydz/dr= r 2/g =z= 0.52r2/g+ constant (aparaboloid)33 / 33