Theintroductionofprinciples
Asmanymechanicalproblemsaredifficulttosolvebymathematicalmethods,theymustbestudiedthroughexperiments.However,thedirectexperimentalmethodhasgreatlimitations.Theexperimentalresultsareonlyapplicabletocertainspecificconditionsandarenotofuniversalsignificance.Therefore,evenifthecostishuge,itisdifficulttorevealthephysicalnatureofthephenomenonanddescribetherelationshipbetweenthevariousquantities.Regularrelationship.Therearemanyphenomenathatarenotsuitablefordirectexperimentation.Forexample,theairplaneistoolargetodirectlystudytheflightproblemoftheairplaneprototypeinthewindtunnel;andtheprototypeoftheinsectistoosmalltoconductdirectexperimentsinthewindtunnel;besides,directexperimentationThemethodcanonlyderivetheregularrelationshipbetweenindividualquantities,anditisdifficulttograsptheessenceofthephenomenon.Weprefertousereducedairplanemodelsorenlargedinsectmodelsforresearch.Thenthequestionwearemostconcernedaboutiswhetherthephysicalphenomenondescribedfromtheexperimentalresultsofthemodelcantrulyreproducetheoriginalphysicalphenomenon?Iftheaccuratequantitativedataobtainedfromthemodelexperimentcanaccuratelyrepresenttheflowphenomenonofthecorrespondingprototype,thefollowingsimilaritiesmustbesatisfiedbetweenthemodelandtheprototype.
Similarityoverview
(1)Geometricsimilarity
Geometricsimilaritymeansthatthemodelhasthesameshapeasitsprototype,butthesizecanbedifferent,andallcorrespondinglineardimensionsareproportional.Thelineardimensionsherecanbediameter,length,roughness,etc.Ifthesubscriptspandmareusedtorepresenttheprototypeandthemodelrespectively,then
ThelinearproportionalconstantcanbeexpressedasCl=lp/lm
TheareaproportionalconstantcanbeexpressedasCa=Ap/Am=Cl^2
ThevolumeratioconstantcanbeexpressedasCv=Vp/Vm=Cl^3
(2)Similarmotion
Similarmotionmeansthatfordifferentflowphenomena,thecorrespondingvelocityandaccelerationdirectionsatallcorrespondingpointsintheflowfieldareconsistent,Andtheratiosareequal,thatistosay,twoflowswithsimilarmotion,theirstreamlinesandflowspectraaregeometricallysimilar.
ThespeedproportionalconstantcanbeexpressedasCv=Vp/Vm;
Sincethedimensionoftimeisl/V,thetimeproportionalconstantisCt=tp/tm=(lp/Vp)/(lm/Vm)=Cl/Cv
TheaccelerationproportionalconstantCa=ap/am=Cv/Ct=CI/Ct^2
(3)PowersimilarityPowersimilaritymeansthatthevariousforcesactingondifferentflowphenomenaatcorrespondingpositionsonthefluid,Forexample,gravity,pressure,viscousforce,elasticforce,etc.,theirdirectionscorrespondtothesame,andtheratioofmagnitudesisequal,thatistosay,twoflowswithsimilardynamics,theforcepolygonformedbyeachforceactingatthecorrespondingpositiononthefluidisgeometricsimilar.
Generallyspeaking,theforcesactingonthefluidelementincludegravityFg,pressurePp,viscousforceFv,elasticforceFeandsurfacetensionFt.Ifthefluidismovingatacceleration(deceleration),afteraddingtheinertialforceFi,theaboveforceswillformaforcepolygon,soFg+Fp+Fv+Fe+Ft+Fi=0.
Ofcourse,inmanypracticalproblems,theabove-mentionedforcesarenotequallyimportant.Sometimessomeforcesmaynotexistoraresosmallthattheyarenegligible,suchasFeandFt,asshowninthefigure.Ifintwoflowphenomenasatisfyinggeometricalsimilarityandsimilarmotion,theforcesactingonanyfluidelementareFg,Fp,Fv,Fi,etc.,then,iftheseforcesmeetthefollowingconditions,thetwophenomenaaresaidtobedynamics.similar.
Thepowerproportionalconstantcanbeexpressedas:Cf=Fgp/Fgm=Fpp/Fpm=Fvp/Fvm=Fip/Fim=…
Whentheabovesimilarconditionsaremet,twoflowphenomena(Orflowfield)issimilarinmechanics.Amongthethreesimilarconditions,geometricsimilarityistheprerequisiteandbasisformotionsimilarityanddynamicsimilarity,dynamicsimilarityistheleadingfactorofflowsimilarity,andmotionsimilarityisonlyarepresentationofgeometricsimilarityanddynamicsimilarity;thethreearecloselyrelated,andoneismissing.Notpossible.
Similaritycriterion
Intheory,anyflowisuniquelydeterminedbythebasicdifferentialequationthatcontrolstheflowandthecorrespondingsolutionconditions.Fortwosimilarflowphenomena,inordertoensurethattheyfollowthesameobjectivelaw,theirdifferentialequationsshouldbethesame.Thisisageneralsolutionforsimilarflows;inaddition,aspecificsolutionforaspecificflowisrequired,anditssingle-valuedconditionmustalsoberequired.resemblance.Theseuniqueconditionsinclude:
(1)Initialconditions,whichrefertothedistributionofphysicalquantitiessuchasflowvelocityandpressureatthebeginningoftheunsteadyflowproblem;Thisconditionisnotrequiredforsteadyflow.
(2)Boundaryconditions,refertothephysicalquantitiessuchasflowvelocityandpressureontheboundaryofthestudiedsystem(suchasinlet,outletandwall,etc.)Distribution.
(3)Geometricconditions,refertothegeometricshape,positionandsurfaceroughnessofthesystemsurface.
(4)Physicalconditions,refertothetypeandphysicalpropertiesofthefluidinthesystem,suchasdensity,viscosity,etc.
Therefore,ifthetwoflowsaresimilar,theyareconsideredasthesingularityconditions.Theratiooftheinertialforceactingonthetwosystemstotheotherforcesshouldbecorrespondinglyequal.Inthefluidmechanicsproblem,ifthereareallthesixforcesmentionedabove,andthedynamicsaresimilar,theproportionsofthefollowingforcesmustbeequal.
Theratioofinertialforcetopressure(orpressuredifference):Fi/Fp
Theratioofinertialforcetogravity:Fi/fg
InertialforceandfrictionForceratio:Fi/Fv
Theratioofinertialforcetoelasticforce:Fi/Fe
Theratioofinertialforcetosurfacetension:Fi/Ft
Theabovefiveformulasrespectivelyintroducefivedimensionlessnumbers,whichareinorder:
1)EulernumberEu=2Δp/(ρ·V^2),forexample,laterOftenusedtoexpressthepressurecoefficientofthesurfacepressuredistribution,aswellastheliftcoefficientanddragcoefficient.Physically,Euler'snumberrepresentsthemagnituderatiobetweentheinertialforceandthepressuregradient.
2)FroudenumberFr=V/sqrt(l·g),inphysics,Froudenumberrepresentsthemagnituderatiobetweeninertialforceandgravity,Isadimensionlessquantitythatcharacterizestheflowrate.
3)ReynoldsnumberRe=Vl/υ,inphysics,theReynoldsnumberrepresentsthemagnituderatiobetweentheinertialforceandtheviscousforceinasimilarflow,theflowRenumberSmall,meansthatthemagnitudeofviscousfrictionismuchlargerthanthatofinertialforce,sotheeffectofinertialforcecanbeignored;conversely,alargeRenumbermeansthatinertialforceplaysamajorrole,soitcanberegardedasnon-viscousFluidhandling.
4)MachnumberMa=V/c,inphysics,Machnumberrepresentsthemagnituderatiobetweeninertialforceandelasticforce,andisameasureofgascompressibility,Usuallyusedtoindicatetheflyingspeedoftheaircraftortheflowspeedoftheairflow.
5)WebernumberWe,physically,theWebernumberrepresentsthemagnituderatiobetweeninertialforceandsurfacetension.
ItcanbeseenthatEu,Fr,Re,MaandWearealldimensionlessnumbers,whicharecalledsimilaritycriterionorsimilaritycriterioninthesimilaritytheory.b>,theyarethebasisforjudgingwhethertwophenomenaaresimilar.Therefore,forphenomenathataresimilartoeachother,thevalueofthesimilaritycriterionofthesamenamemustbeequal.Conversely,iftwoflowingsingle-valueconditionsaresimilar,andthevaluesofthesimilaritycriteriaofthesamenamecomposedofsingle-valuedconditionsareequal,thetwophenomenamustbesimilar.
Detailedsimilarityprinciple
Thefirsttheoremofsimilarity
Twosimilarflowphenomenabelongtothesametypeofphysicalphenomenon,andtheyshouldDescribedbythesamemathematicalandphysicalequations.Thegeometricconditionsoftheflowphenomenon(theboundaryshapeandsizeoftheflowfield),physicalconditions(fluiddensity,viscosity,etc.),boundaryconditions(distributionofphysicalquantitiesontheflowfieldboundary,suchasvelocitydistribution,pressuredistribution,etc.)Thereareinitialconditions(thephysicalquantitydistributionateachpointintheflowfieldattheinitialtimeoftheselectedstudy)mustbesimilar.Theseconditionsarecollectivelyreferredtoassinglevalueconditions.Asmentionedabove,thetwoflowphenomenaaremechanicallysimilar,andthephysicalquantitiesatthecorrespondingpointsinspaceandthecorrespondinginstantaneousphysicalquantitiesareincertainproportionstoeachother,andthesephysicalquantitiesmustsatisfythesamedifferentialequations.Therefore,theproportionalcoefficientsofthequantitiesareSimilarmultiplescannotbearbitrary,butrestricteachother.
Tosumup,theconclusioncanbedrawn:Physicalphenomenathataresimilartoeachothermustobeythesameobjectivelaws.Ifthelawscanbeexpressedbyequations,thephysicalequationsmustbeexactlythesameandcorrespondingSimilarcriteriaThevaluesmustbeequal.Thisissimilartothefirsttheorem.Itisworthpointingoutthatthesimilaritycriterionofaphysicalphenomenonatdifferentmomentsanddifferentspatiallocationshasdifferentvalues,whilephysicalphenomenathataresimilartoeachotherhavethesamevaluesimilaritycriterionatthecorrespondingtimeandatthecorrespondingpoint.Therefore,similarityThecriterionisnotconstant.
Thesecondtheoremofsimilarity
Onlywhentheexperimentalmodelissimilartotheresearchobjectitsimulatescantheresultsoftheexperimentbeappliedtotheresearchobject.Tojudgewhethertwophenomenaaresimilar,itisoftenimpossibletojudgewhetherthedistributionofthephysicalquantityinthecorrespondingtimeandspacemaintainsthesameratio.Forexample,theflowfieldofamodelairplaneinawindtunnelissimilartotheflowfieldofanactualflyingairplane.Often,onlytheincomingflowvelocityinthefarfrontoftheairplaneisknown,buttheflowfielddistributionneartheairplaneisnotknown.Therefore,thetwocannotbejudgedbasedonsimilardefinitions.Aretheysimilar.
Twophysicalphenomenaaresimilarandmustbethesamekindofphysicalphenomena.Therefore,thedifferentialequationsdescribingphysicalphenomenamustbethesame.Thisisthefirstnecessaryconditionforsimilarphenomena.
Similarsinglevalueconditionsarethesecondnecessaryconditionforsimilarphysicalphenomena.Becausetherearemanysimilarphenomenathatobeythesamedifferentialequations,single-valuedconditionscansinglelydistinguishtheresearchobjectfromcountlessmultiplephenomena.Mathematically,itisthedefinitesolutionconditionthatmakesthedifferentialequationshaveauniquesolution.
Thesimilaritycriterioncomposedofphysicalquantitiesinthesinglevalueconditionisequaltothephenomenonofsimilaritythethirdnecessarycondition.
Converselyspeaking,whentheybelongtothesametypeofphysicalphenomenonandthesinglevalueconditionsaresimilar,thetwophenomenahavethecorrespondingrelationshipbetweentimeandspaceandthesamephysicalquantityconnectedwithtimeandspace.Ifthecorrespondingsimilaritycriteriaareequal,Andmaintainthesameratioofphysicalquantitiesatthecorrespondingtimeandspacepoints,whichalsoensuresthesimilarityofthetwophysicalphenomena.
Tosumup,similarconditionscanbeexpressedas:Forthesametypeofphysicalphenomenon,whensinglevaluetheconditionsaresimilarandconsistofsinglevalueWhenthesimilaritycriterionofthephysicalquantitycompositionintheconditioncorrespondstothesame,thenthesephenomenamustbesimilar.Thisisthesecondtheoremofsimilarity,whichisasufficientandnecessaryconditionforjudgingwhethertwophysicalphenomenaaresimilar.
PrinciplesandExperiments
Similarprinciplesanddimensionalanalysismethodshavesolvedaseriesofproblemsinmodelexperiments.
Tocarryoutamodeltest,firstencounterhowtodesignthemodelandhowtochoosethemediumintheflowofthemodeltoensurethatitissimilartotheprototype(physical)flow.Accordingtothesecondtheoremofsimilarity,thedesignmodelandtheselectionmediummustmakethesingle-valuedconditionssimilar,andthesimilaritycriteriacomposedofthephysicalquantitiesinthesingle-valuedconditionsareequalinvalue.
Whatphysicalquantitiesneedtobemeasuredduringthetestandhowtodealwiththetestdatainordertoreflecttheobjectiveessence?Thefirsttheoremofsimilaritystatesthatphenomenathataresimilartoeachothermusthaveasimilaritycriterionofequalvalues.Therefore,somephysicalquantitiescontainedineachsimilaritycriterionshouldbedeterminedintheexperiment,andtheyshouldbesortedintosimilaritycriterion.
Howtoorganizethemodeltestresultstofindtheregularity,sothatitcanbepromotedandappliedtotheprototypeflow?ItcanbeseenfromtheΠtheoremthattherelationshipbetweenvariousvariablesdescribingacertainphysicalphenomenoncanbeexpressedasarelativelysmallnumberofdimensionlessΠexpressions,andeachdimensionlessΠhasdifferentsimilaritycriteria,andthefunctionalrelationshipbetweenthemisalsocalledIsthecriterionequation.Forphenomenasimilartoeachother,theircriterionequationsarealsothesame.Therefore,thetestresultsshouldbesortedintotherelationshipbetweensimilarcriteria,whichcanbepromotedandappliedtotheprototype.
Reynoldsnumbersimilaritymethod
Inordertobetterexplaintheapplicationofthesimilarityprinciple,thefollowingintroducesanapproximatemodelmethod:Reynoldsnumbersimilaritymethod
TherearemanypracticalFlow,theyaremainlyaffectedbyviscousforce,pressureandinertialforce.Forexample,ifthefluidflowsinapipewithafullcross-section,sincethereisnofreesurface,thereisnosurfacetensioneffect,sotheWesimilaritycriterioncanbeignored;gravitydoesnotaffecttheflowfield,sotheFrsimilaritycriterioncanbeignored;iftheflowvelocityisverylowcomparedtothespeedofsound,Thecompressibilityeffectcanalsobeneglected,thatis,itisnotnecessarytoconsidertheMasimilaritycriterion.Thesameistrueforthelow-speedairflowaroundtheobjectortheelasticforceonthefluidaroundthesubmarineindeepwaterandthecorrespondingwaterflow(thereisnosurfacewaveformationatthistime).
Fromthepointofviewofmechanicalsimilarity,iftwoflowfieldshavethesamedirectionandthesamemagnitudeofforceactingonthecorrespondingpoints,thedynamicsaresimilar.Inthecaseofconsideringonlythethreeforcesofviscousforce,pressureandinertialforce,inordertomaketheforcetrianglesimilar,itonlyneedstosatisfythatthetwosidesareproportionalandtheincludedanglesareequal,thatis,theinertiaofthemodelflowatthecorrespondingpointsTheforceandtheviscousforceareinthesameproportionsastheinertialforceandtheviscousforceactingontheflowofobjects.Therefore,aslongasthecorrespondingpointmeetstheReynoldsnumberequal.Fromamoregeneralsimilaritytheorem,iftwoflowsaresimilar,thenumberofsimilaritycriteriacorrespondstothesame,andthesimilaritycriterionequationderivedfromtheΠtheoremisalsothesame.Amongthe(nk)similaritycriteria,(nk-1)istheindependentsimilaritycriterion,ordecisivesimilaritycriterion(equivalenttotheindependentvariableofthefunction),andoneisthenon-independentsimilaritycriterionorthenon-deterministicsimilaritycriterion(equivalenttoThedependentvariableofthefunction).Fortheflowsituationthatonlyconsiderstheeffectofviscousforce,pressureandinertialforce,theReynoldscriterionandothercriteriarelatedtogeometricdimensionsareregardedasindependentcriteria,andtheEulercriterionisanon-independentcriterion.
Underthepremiseofgeometricsimilarity,thedecisivecriterionforsimilarityofflowphenomenaisonlytheReynoldscriterion,andthesimilaritythatthemodeltestmustcomplywithiscalledReynoldsphaselikeness.