柱坐标系和球坐标系下N-S方程的直接推导Word版

Derivationof3DEulerandNavier-StokesEquationsinCylindricalCoordinatesContents1.Derivationof3DEulerEquationinCylindricalcoordinates2.DerivationofEulerEquationinCylindricalcoordinatesmovingatintangentialdirection3.Derivationof3DNavier-StokesEquationinCylindricalCoordinates1.Derivationof3DEulerEquationinCylindricalcoordinatesEulerEquationinCartesiancoordinates(1.1)WhereConservativeflowvariablesInviscid/convectivefluxinxdirectionInviscid/convectivefluxinydirectioninviscid/convectivefluxinzdirectionAndtheirspecificdefinitionsareasfollows,,,TotalenthalpySomerelationshipWewanttoperformthefollowingcoordinatestransformationBecauseAccordingtoCramer’sruler,wehave(1.2.1)(1.2.2)WhereSimilartotheabove(1.2.3)(1.2.4)Inaddition,thefollowingrelationsholdbetweencylindricalcoordinateandCartesiancoordinate,,,,,(1.3)(1.4.1)(1.4.2)DerivationMultiplyingthebothsideofequation(1.1)byandapplyingequalities(1.4.1)and(1.4.2)gives,(1.5)Differentiatingthefollowingw.r.t.timegives,,(1.6.1)(1.6.2)Expandingthetermandapplyingtherelationships(1.6)yields,(1.7.1)Expandingthetermandapplyingtherelationships(1.6)yields,(1.7.2)Substitutingrelationships(1.7)intoequation(1.5)andrearranginggives,(1.8)Aswecanseefromexpressions(1.7),themomentumequationsinradialandtangentialdirectionscontainvelocitiesinCartesiancoordinate;weneedtoreplacethemwithcorrespondingvariablesincylindricalcoordinate.Writingdownthemomentumequationsinradialandtangentialdirectionsasfollows,(1.9.1)(1.9.2)Multiplying(1.9.1)byand(1.9.2)by,thensummingupandapplyingexpressions(1.6)andrearrangingyields(1.10.1)Multiplying(a)byand(b)by,thensummingupandapplyingexpressions(1.6)yields,(1.10.2)Replacing(1.10)with(1.9)andrearrangingequation(1.8)gives(1.11)Where,,,,Note:differentfromEulerequationinCartesiancoordinates,theEulerequationincylindricalcoordinatescontainssourcetermsfrommomentumequationsinradialandtangentialequations.2.DerivationofEulerEquationinCylindricalcoordinatesmovingatintangentialdirectionWhere,,,,,,,,,,,Thenequation(1.11)canbewrittenasfollows(2.1)WhereEquation(2.1)adoptsrotatingcoordinatesbutthevariablesaremeasuredinabsolutecylindricalcoordinates.3.Derivationof3DNavier-StokesEquationinCylindricalCoordinates3DNavier-StokesEquationsinCartesiancoordinates(3.1)Where,,,,,,,,,,，,,Inthefollowingderivation,onlyviscoustermswillbederivedfromCartesiancoordinatestocylindricalcoordinates,thoseinviscidtermshavingbeenderivedinsection1willbenotrepeated.Replacingwithgives(3.2.1)Replacingwithgives(3.2.2)Multiplyingequation(3.1)by,theviscoustermsaregivesasfollows(omittingthenegativesignbeforeitfromsimplicity),(3.3)(3.4.1),(3.4.2),(3.4.3),(3.4.4)(3.4.5)(3.4.6)Expandingexpression(3.3)gives,(3.5)=〉(3.6.1)=〉(3.6.2)(3.6.3)(3.7.1)DivergenceinCartesianCoordinates(3.7.2)Divergenceincylindricalcoordinates(3.7.3)(3.8.1)(3.8.2)(3.9.1)(3.9.2)Aswecanseefromtheabovethatviscoustermsinexpression(3.5)forthemomentumequationinaxial/xdirectionandenergyequationcanbeexpressedinvariablesincylindricalcoordinates,whiletheviscoustermsin(3.5)formomentumequationsinradialandtangentialdirectionsstillcontainvariablesinCartesiancoordinates.Similarmanipulationto(1.10)willbeadoptedinthefollowing.Writingouttheviscoustermsformomentumequationsinradialandtangentialcoordinatesasfollows,(3.10.1)(3.10.2)Multiplying(3.10.1)byandmultiplying(3.10.2)by,thensummingupandrearranginggives,(3.11.1)Multiplying(3.10.1)byandmultiplying(3.10.2)by,thensummingupandrearranginggives,(3.11.2)(3.12.1)(3.12.2)(3.12.3)(3.12.4)Substituting(3.6.1),(3.6.2)and(3.12)intoexpressions(3.11)andrearrangingyields,(3.13.1)(3.13.2)Makinguseofexpressions(3.4.1),(3.6.1),(3.6.2),(3.8.1),(3.8.2),(3.9.1),(3.9.2),(3.13.1)and(3.13.2),wecangetthefinalexpressionof3DNavier-StokesEquationincylindricalcoordinatesasfollows,3DNavier-StokesEquationincylindricalcoordinates，,,,，,，Ifthemomentofmomentumequationisadoptedtoreplacethetangentialmomentumequation,itsexpressionwillbesimpler.Nowforthemomentequation,thereisnosourceterm.，,,,，,，For2Daxisymmetricflowfield,thetangentialmomentumequationormomentequationcanbeomittedasfollows,2Daxisymmetricequationincylindricalcoordinate，,,,，，，（注：可编辑下载，若有不当之处，请指正，谢谢!）