IntroductiontoFluidMechanicsChapter6Incompressible

InviscidFlow1MainTopicsMomentumEquationforFrictionlessFlow:Euler’sEquationEuler’sEquationinStreamlineCoordinatesBernoulliEquation–IntegrationofEuler’sEquationAlongaStreamlineforSteadyFlowTheBernoulliEquationInterpretedasanEnergyEquationEnergyGradeLineandHydraulicGradeLine2MOMENTUMEQUATIONEQUATIONFORFRICTIONLESSFLOW:EULER’SEQUATIONEuler’sEquationThisequationstatesthatforaninviscidfluid

thechangeinmomentumofafluidparticle

iscausedbythebody(assumedtobegravityonly)andthenetpressureforce.3EULER’SEQUATION

INRECTANGULARCOORDINATESSYSTEM4EULER’SEQUATION

INCYLINDRICALCOORDINATESSYSTEM5EULER’SEQUATION

INSTREAMLINECOORDINATES(I)ApplyingNewton’ssecondlawinthestreamwise

(thes)directiontothefluidelementofvolumedsdndx,thenneglectingviscousforcesweobtaindzds6EULER’SEQUATION

INSTREAMLINECOORDINATES(II)Euler’sequationinthestreamwisedirectionwiththez-axisdirectedverticallyupwardsisForsteadyflow,andneglectingbodyforce,Euler’sequationinthestreamwisedirectionreducesto7EULER’SEQUATION

INSTREAMLINECOORDINATES(III)ApplyingNewton’ssecondlawinadirectionnormaltothestreamwise,thenneglectingviscousforces,weobtainForsteadyflow,theEuler’sequationnormaltoastreamlinebecomeInahorizontalplaneRVanzgnpcosgnpdndxdsadndxdscosgdsdx2dnnppdsdx2dnnpp2nn-==??-??-=-??-T=-÷???è???+-÷???è???-rrrbrrbrdzdsdndzgdndxds8SUMMARY:EULER’SEQUATIONForsteadyflow,theEuler’sequationinthestreamwisedirectionreducestoForsteadyflow,theEuler’sequationnormaltoastreamlinebecome9EULER’SEQUATION

Indication?Adecreaseinvelocityisaccompaniedbyanincreaseinpressureandconversely.Theonlyforceexperiencedbytheparticleisthenetpressureforce,sotheparticleacceleratestowardlow-pressureregionsanddecelerateswhenapproachinghigh-pressureregions.Pressureincreasesinthedirectionoutwardfromthecenterofcurvatureofthestreamwise.Theonlyforceexperiencedbytheparticleisthenetpressureforce,thepressurefieldcreatesthecentripetalacceleration.??10BERNOULLIEQUATION

AlongaStreamlineforSteadyFlow(I)Euler’sequationforsteadyflow

alongastreamlineisIntegrationofthisequationgives11BERNOULLIEQUATION

AlongaStreamlineforSteadyFlow(II)ForthespecialcaseofincompressibleflowRestrictions:Steadyflow.Incompressibleflow.Frictionlessflow.Flowalongastreamline.BERNOULLIEQUATION12Static,Stagnation,andDynamicPressure(I)Thepressure,p,whichhaveusedinderivingtheBernoulliequation,isthethermodynamicpressure;itiscommonlycalledthestaticpressure.Thestaticpressure?isthepressureseenbythefluidparticleasitmoves.Itismeasuredinaflowingfluidusingawallpressure“tap”,orastaticpressureprobe.Asmallhole,drilledcarefullyinthewall.13Static,Stagnation,andDynamicPressure(II)Neglectingelevationdifference,theBernoulliequationbecomesThestagnationpressureisobtainedwhenaflowingfluidisdeceleratedtozerospeedbyafrictionlessprocess.StagnationpressureDynamicpressurewhere14StaticandStagnationPressureMeasurementPitot-staticTube1516BERNOULLIEQUATION

ApplicationsTheBernoulliequationcanbeappliedbetweenanytwopointsonastreamlineprovidedthattheotherthreerestrictionsaresatisfied.Theresultis.Restrictions:Steadyflow.Incompressibleflow.Frictionlessflow.Flowalongastreamline.1718192021BERNOULLIEQUATION

ASANENERGYEQUATION(I)Considersteadyflowintheabsenceofshearforce.Wechooseacontrolvolumeboundedbystreamlinesalongitsperiphery.Thebasicequation:RESTRICTIONSNoShaftWorkNoShearForceWorkNoOtherWorkSteadyFlowUniformFlowandProperties22BERNOULLIEQUATION

ASANENERGYEQUATION(II)Underthoserestrictions,thebasicequationsbecomeUnderthoserestrictions,thecontinuityequation

Also23BERNOULLIEQUATION

ASANENERGYEQUATION(III)Thus,theenergyequationWithadditionalassumptionofincompressibleflow,,theenergyequationbecomesWithfurtherrestriction,,theenergyequationbecomes….24BERNOULLIEQUATION

ASANENERGYEQUATION(IV)RESTRICTIONS:NoShaftWorkNoShearForceWorkNoOtherWorkSteadyFlowUniformFlowandPropertiesIncompressibleFlowu2–u1–dQ/dm=02526ENERGYGRADELINEandHYDRAULICGRADELINE(I)Forsteady,frictionless,incompressibleflowalongastreamlinewithoutlossofmechanicalenergy,theenergyequationcanbedividedbyginordertorepresentthemechanicalenergylevelofaflowgraphically.Theheadduetolocalstaticpressure(pressureenergy)Theheadduetolocaldynamicpressure(kineticenergy)Theelevationhead(potentialenergy)Thetotalheadfortheflow27InterpretationofBernoulli’sEquation28ENERGYGRADELINEandHYDRAULICGRADELINE(II)EnergyGradeLine(EGLorEL):representsthetotalheadheight.HydraulicGradLine(HGL)height:representsthesumoftheelevationandstaticpressureheads.ThedifferenceinheightsbetweentheEGLandtheHGLrepresentsthedynamic(