Similarity Principle

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,andthevalues​​ofthesimilaritycriteriaofthesamenamecomposedofsingle-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,thephysicalequationsmustbeexactlythesameandcorrespondingSimilarcriteriaThevalues​​mustbeequal.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.