TheIntroDuctionOfPrinciples
AsmanymechanicalproblemesardefifculttosolveByMathematicalMethods, те са най -вечери.Въпреки това, thedirectexperimentalmethodhasgreateplimitations.Theexperimentalresultsareonlyapplicabletectistenspecificificconditionsandarenotofuniversaliancy.Следователно, evenifthe -estishuge, itisdifficultToreVealthephysicalnatureofthephenomenonanddescribtheterelationshipbetweEnteEntaryquantities -quantities.Редовна връзка.Therearemanyphenomenatatarenotsuitablefordirectexperimentation.Forexample,theairplaneistoolargetodirectlystudytheflightproblemoftheairplaneprototypeinthewindtunnel;andtheprototypeoftheinsectistoosmalltoconductdirectexperimentsinthewindtunnel;besides,directexperimentationThemethodcanonlyderivetheregularrelationshipbetweenindividualquantities,anditisdifficulttograsptheessenceofthephenomenon.WeprefertouseredUcedAirplanemodelsorenlargedinsectmodelsforresearch.Thenthequestionwearemostconcernedaboutiswhetherthephysicalphenomenondescribedfromtheexperimentalresultsofthemodelcantrulyreproducetheoriginalphysicalphenomenon?Iftheaccuratequantitativedataobtainedfromthemodelexperimentcanaccuratelyrepresenttheflowphenomenonofthecorrespondingprototype,thefollowingподобенitiesmustbesatisfiedbetweenthemodelandtheprototype.
Преглед на сходството
(1)Geometricподобенity
Геометричъристичен emaensTheModelhasthesameshapeasitsitsprototype, ButteSizeCanbedifferent, и AllCorReponspondingLineArdimensionsProportortal.Thelineardimensionsherecanbediameter, дължина, грапавост и т.н..IftheSubscriptspandmareusedtorepresenttheprototypeandthemodelrespective, тогава
Thelinearproportionalconstantcanbeexpressedascl = lp/lm
Theareaproportionalconstantcanbeexpressedasca = ap/am = cl^2
Thevolumeratioconstantcanbeexpressedascv = vp/vm = cl^3
(2)Similarmotion
Similarmotionmeansthatfordifferentflowphenomena, theCorrepspondingVelocityAndAccelerationDirectionsatallCorreplogningPointSinTheFlowFieldARECONSISTENT, andtheratiosAreequal, thatistosay, twoflowswithподобенmotion, streamlinesandfectrahaaregeometritysimition.
ThespeedproportionalconstantcanbeexpressedASCV = vp/vm;
Sincethedimensionftimeisl/v, thetimeProportoralConstantisct = tp/tm = (lp/vp)/(lm/vm) = cl/cv
Theaccelerationproportionalconstantca = ap/am = cv/ct = ci/ct^2
(3)PowerподобенityPowerподобенitymeansthatthevariousforcesactingondifferentflowphenomenaatcorrespondingpositionsonthefluid,Forexample,gravity,pressure,viscousforce,elasticforce,etc., тяхното насочване corredponspotothesame, и theratioofmagnitudesisequal, thatistosay, twoflowswithподобенdynamics, theforcepolygonformedbyeachforceactingatthecorredingpositiononthefluidisgeometricsipricalilarylilarilarymilarilaililarymilarilaililarypricподобенilarymilarilarythecorredingpositiononthefluidisgeometricsipricalilarilar.
Като цяло, theforcectingonthefluidelementincludegravityfg, pressurePP, viscousforcefv, elasticforcefeandsurfacetensionft.IfthefluidismovingataCceleration (забавяне), след това дадване на вентилаториалнифи, theaboveforceswillformaforcepolygon, sofg+fp+fv+fe+ft+fi = 0.
Разбира се, inmanypracticalproblems, theabove-mentionedforcesarenotequallyemportant.Someenimessomeforcesmaynotexistoraresosmallthattheyareneglible, suchasfandft, asshownInthefigure.IfintwoflowphenomenaSatsfyinggeometricalподобенityandsImilarMotion, theforsactingonanyfluidelementarefg, fp, fv, fi и др.., тогава, ifTheSeforcesMeetTheFollowingConditions, thetwophenomenaaresaidtobedynamics.подобен.
Thepowerproportionalconstantcanbeexpressedas: cf = fgp/fgm = fpp/fpm = fvp/fvm = fip/fim =…
Whentheaboveподобенconditionsaremet,twoflowphenomena(Orflowfield)isподобенinmechanics.Amongthethreeподобенconditions,geometricподобенityistheprerequisiteandbasisformotionподобенityanddynamicподобенity,dynamicподобенityistheleadingfactorofflowподобенity,andmotionподобенityisonlyarepresentationofgeometricподобенityanddynamicподобенity;thethreearecloselyrelated,andoneismissing.Невъзможно.
Критерий за сходство
Intheory, anyflowisuniquelydeterminedbythebasicdifferentialequationhatcontrolstheflowandthecorrepondingsolutionsolutionsitionsconditions.Fortwoподобенflowphenomena,inordertoensurethattheyfollowthesameobjectivelaw,theirdifferentialequationsshouldbethesame.Thisisageneralsolutionforподобенflows;inaddition,aspecificsolutionforaspecificflowisrequired,anditssingle-valuedconditionmustalsoberequired.прилика.Thenuniqueconditionsinclude:
(1)Initialconditions,whichrefertothedistributionofphysicalquantitiessuchasflowvelocityandpressureatthebeginningoftheunsteadyflowproblem;Thisconditionisnotrequiredforsteadyflow.
(2)Boundaryconditions,refertothephysicalquantitiessuchasflowvelocityandpressureontheboundaryofthestudiedsystem(suchasinlet,outletandwall,etc.) Разпределение.
(3)Geometricconditions,refertothegeometricshape,positionandsurfaceroughnessofthesystemsurface.
(4)Physicalconditions,refertothetypeandphysicalpropertiesofthefluidinthesystem,suchasdensity,viscosity,etc.
Therefore,ifthetwoflowsareподобен,theyareconsideredasthesingularityconditions.Theratiooftheinertialforceactingonthetwosystemstotheotherforcesshouldbecorpordleslequalequal.Inthefluidmechanicsproblem,ifthereareallthesixforcesmentionedabove,andthedynamicsareподобен,theproportionsofthefollowingforcesmustbeequal.
Theratioofinertialforcetopressure (Orpressuredference): fi/fp
Teratioofinertialforcetogravity: FI/FG
Съотношение на инерционната сила и силата на триене: за/телевизия
Theratioofinertialforcetoelasticforce: fi/fe
Съотношението на инерционната сила към напрежението на повърхността: към/ft
TheabovefiveformulaSrespective inTroducefivedimensionsnessnumbers, които Aareinorder:
1)EulernumberEu=2Δp/(ρ·V^2),forexample,laterOftenusedtoexpressthepressurecoefficientofthesurfacepressuredistribution,aswellastheliftcoefficientanddragcoefficient.Физически, Euler'sNumberReprestsTheMagnituderatioBetWeEntHeinertialForCeandThePressuregradient.
2)FroudenumberFr=V/sqrt(l·g),inphysics,Froudenumberrepresentsthemagnituderatiobetweeninertialforceandgravity,Isadimensionlessquantitythatcharacterizestheflowrate.
3)ReynoldsnumberRe=Vl/υ,inphysics,theReynoldsnumberrepresentsthemagnituderatiobetweentheinertialforceandtheviscousforceinaподобенflow,theflowRenumberSmall,meansthatthemagnitudeofviscousfrictionismuchlargerthanthatofinertialforce,sotheeffectofinertialforcecanbeignored;conversely,alargeRenumbermeansthatinertialforceplaysamajorrole,soitcanberegardedasnon-viscousFluidhandling.
4)MachnumberMa=V/c,inphysics,Machnumberrepresentsthemagnituderatiobetweeninertialforceandelasticforce,andisameasureofgascompressibility,Usuallyusedtoindicatetheflyingspeedoftheaircraftortheflowspeedoftheairflow.
5)WebernumberWe,physically,theWebernumberrepresentsthemagnituderatiobetweeninertialforceandsurfacetension.
ItcanbeseenthatEu,Fr,Re,MaandWearealldimensionlessnumbers,whicharecalledподобенitycriterionorподобенitycriterionintheподобенitytheory.b>,theyarethebasisforjudgingwhethertwophenomenaareподобен.Therefore,forphenomenathatareподобенtoeachother,thevalueoftheподобенitycriterionofthesamenamemustbeequal.Conversely,iftwoflowingsingle-valueconditionsareподобен,andthevaluesoftheподобенitycriteriaofthesamenamecomposedofsingle-valuedconditionsareequal,thetwophenomenamustbeподобен.
Detailedподобенityprinciple
Thefirsttheoremofподобенity
Twoподобенflowphenomenabelongtothesametypeofphysicalphenomenon,andtheyshouldDescribedbythesamemathematicalandphysicalequations.Thegeometricconditionsoftheflowphenomenon (theboundaryshapeandisizeoftheflowfield), физически условия (течност, вискозитет и др..), BoundaryConditions (DistributionOfphysicalquantitiesontheflowfieldboundary, SuchAsvelocityDistribution, Pressureduredibution и др..)Thereareinitialconditions(thephysicalquantitydistributionateachpointintheflowfieldattheinitialtimeoftheselectedstudy)mustbeподобен.Определенията от тях сареклективно се отнасят доноасингълвалуазионни.Asmentionedabove,thetwoflowphenomenaaremechanicallyподобен,andthephysicalquantitiesatthecorrespondingpointsinspaceandthecorrespondinginstantaneousphysicalquantitiesareincertainproportionstoeachother,andthesephysicalquantitiesmustsatisfythesamedifferentialequations.Следователно, theproportionalcoefficientsofthequantitiesaresimilarmultiplescannotbearbarbary, butrestricteChother.
Tosumup,theconclusioncanbedrawn:Physicalphenomenathatareподобенtoeachothermustobeythesameobjectivelaws.Ifthelawscanbeexpressedbyequations,thephysicalequationsmustbeexactlythesameandcorrespondingSimilarcriteriaThevaluesmustbeequal.Thisisподобенtothefirsttheorem.Itisworthpointingoutthattheподобенitycriterionofaphysicalphenomenonatdifferentmomentsanddifferentspatiallocationshasdifferentvalues,whilephysicalphenomenathatareподобенtoeachotherhavethesamevalueподобенitycriterionatthecorrespondingtimeandatthecorrespondingpoint.Therefore,подобенityThecriterionisnotconstant.
Thesecondtheoremofподобенity
Onlywhentheexperimentalmodelisподобенtotheresearchobjectitsimulatescantheresultsoftheexperimentbeappliedtotheresearchobject.Tojudgewhethertwophenomenaareподобен,itisoftenimpossibletojudgewhetherthedistributionofthephysicalquantityinthecorrespondingtimeandspacemaintainsthesameratio.Forexample,theflowfieldofamodelairplaneinawindtunnelisподобенtotheflowfieldofanactualflyingairplane.Често само theincomingflowvelocityinthefarfrontoftheairplaneis known, buttheflowfielddistributionneartheairplaneisnotknown.Therefore,thetwocannotbejudgedbasedonподобенdefinitions.Aretheyподобен.
Twophysicalphenomenaareподобенandmustbethesamekindofphysicalphenomena.Следователно, thedifferentialequationsdescribingphysicalphenomenamustbethesame.Thisisthefirstnecessaryconditionforподобенphenomena.
Similarsinglevalueconditionsarethesecondnecessaryconditionforподобенphysicalphenomena.Becausetherearemanyподобенphenomenathatobeythesamedifferentialequations,single-valuedconditionscansinglelydistinguishtheresearchobjectfromcountlessmultiplephenomena.Математически, itisthedefinitesolutioncondition thatmakeshedifferentialequationsheauniquesolution.
Theподобенitycriterioncomposedofphysicalquantitiesinthesinglevalueconditionisequaltothephenomenonofподобенitythethirdnecessarycondition.
Converselyspeaking,whentheybelongtothesametypeofphysicalphenomenonandthesinglevalueconditionsareподобен,thetwophenomenahavethecorrespondingrelationshipbetweentimeandspaceandthesamephysicalquantityconnectedwithtimeandspace.Ifthecorrespondingподобенitycriteriaareequal,Andmaintainthesameratioofphysicalquantitiesatthecorrespondingtimeandspacepoints,whichalsoensurestheподобенityofthetwophysicalphenomena.
Tosumup,подобенconditionscanbeexpressedas:Forthesametypeofphysicalphenomenon,whensinglevaluetheconditionsareподобенandconsistofsinglevalueWhentheподобенitycriterionofthephysicalquantitycompositionintheconditioncorrespondstothesame,thenthesephenomenamustbeподобен.Thisisthesecondtheoremofподобенity,whichisasufficientandnecessaryconditionforjudgingwhethertwophysicalphenomenaareподобен.
Принципна и сексуални процеси
Подобни principlesanddimensionalanalysismethodshavesolvedaseriesofproblemsinmodelexperiments.
Tocarryoutamodeltest,firstencounterhowtodesignthemodelandhowtochoosethemediumintheflowofthemodeltoensurethatitisподобенtotheprototype(physical)flow.Accordingtothesecondtheoremofподобенity,thedesignmodelandtheselectionmediummustmakethesingle-valuedconditionsподобен,andtheподобенitycriteriacomposedofthephysicalquantitiesinthesingle-valuedconditionsareequalinvalue.
Whatphysicalquantitiesneedtobemeasuredduringthetestandhowtodealwiththetestdatainordertoreflecttheobjectiveessence?Thefirsttheoremofподобенitystatesthatphenomenathatareподобенtoeachothermusthaveaподобенitycriterionofequalvalues.Therefore,somephysicalquantitiescontainedineachподобенitycriterionshouldbedeterminedintheexperiment,andtheyshouldbesortedintoподобенitycriterion.
Howtoorganizethemodeltestresultstofindtheregularity,sothatitcanbepromotedandappliedtotheprototypeflow?ItcanbeseenfromtheΠtheoremthattherelationshipbetweenvariousvariablesdescribingacertainphysicalphenomenoncanbeexpressedasarelativelysmallnumberofdimensionlessΠexpressions,andeachdimensionlessΠhasdifferentподобенitycriteria,andthefunctionalrelationshipbetweenthemisalsocalledIsthecriterionequation.Forphenomenaподобенtoeachother,theircriterionequationsarealsothesame.Therefore,thetestresultsshouldbesortedintotherelationshipbetweenподобенcriteria,whichcanbepromotedandappliedtotheprototype.
Reynoldsnumberподобенitymethod
Inordertobetterexplaintheapplicationoftheподобенityprinciple,thefollowingintroducesanapproximatemodelmethod:Reynoldsnumberподобенitymethod
ThereareManypracticalflow, теременонистиран ByViscousforce, pressUreandInertialforce.Forexample,ifthefluidflowsinapipewithafullcross-section,sincethereisnofreesurface,thereisnosurfacetensioneffect,sotheWeподобенitycriterioncanbeignored;gravitydoesnotaffecttheflowfield,sotheFrподобенitycriterioncanbeignored;iftheflowvelocityisverylowcomparedtothespeedofsound,Thecompressibilityeffectcanalsobeneglected,thatis,itisnotnecessarytoconsidertheMaподобенitycriterion.Thesameistrueforthelow-speedairflowAroundtheobjectortheelasticforceonthefluidaroundthesubmarineindeepwaterandthecorredingwaterflow (след товаоздрав, нормист)).
Fromthepointofviewofmechanicalподобенity,iftwoflowfieldshavethesamedirectionandthesamemagnitudeofforceactingonthecorrespondingpoints,thedynamicsareподобен.Inthecaseofconsideringonlythethreeforcesofviscousforce,pressureandinertialforce,inordertomaketheforcetriangleподобен,itonlyneedstosatisfythatthetwosidesareproportionalandtheincludedanglesareequal,thatis,theinertiaofthemodelflowatthecorrespondingpointsTheforceandtheviscousforceareinthesameproportionsastheinertialforceandtheviscousforceactingontheflowofobjects.Следователно, aslongasthecorredingpointpointmeetsthereynoldsnumberequal.Fromamoregeneralподобенitytheorem,iftwoflowsareподобен,thenumberofподобенitycriteriacorrespondstothesame,andtheподобенitycriterionequationderivedfromtheΠtheoremisalsothesame.Amongthe(nk)подобенitycriteria,(nk-1)istheindependentподобенitycriterion,ordecisiveподобенitycriterion(equivalenttotheindependentvariableofthefunction),andoneisthenon-independentподобенitycriterionorthenon-deterministicподобенitycriterion(equivalenttoThedependentvariableofthefunction).Fortheflowsituationthatonlyconsiderstheeffectofviscousforce,pressureandinertialforce,theReynoldscriterionandothercriteriarelatedtogeometricdimensionsareregardedasindependentcriteria,andtheEulercriterionisanon-independentcriterion.
Underthepremiseofgeometricподобенity,thedecisivecriterionforподобенityofflowphenomenaisonlytheReynoldscriterion,andtheподобенitythatthemodeltestmustcomplywithiscalledReynoldsphaselikeness.