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1、Derivationof3DEulerandNavier-StokesEquationsinCylindricalCoordinatesContents1.Derivationof3DEulerEquationinCylindricalcoordinates2.DerivationofEulerEquationinCylindricalcoordinatesmovingatintangentialdirection3.Derivationof3DNavier-StokesEquationinCylindricalCoordinates1.Derivationof
2、3DEulerEquationinCylindricalcoordinatesEulerEquationinCartesiancoordinates(1.1)WhereàConservativeflowvariablesàInviscid/convectivefluxinxdirectionàInviscid/convectivefluxinydirectionàinviscid/convectivefluxinzdirectionAndtheirspecificdefinitionsareasfollows,,,àTotalenthalpySomerelati
3、onshipWewanttoperformthefollowingcoordinatestransformationBecauseAccordingtoCramer’sruler,wehave(1.2.1)(1.2.2)Where第13页，共13页Similartotheabove(1.2.3)(1.2.4)Inaddition,thefollowingrelationsholdbetweencylindricalcoordinateandCartesiancoordinate,,,,,(1.3)(1.4.1)(1.4.2)DerivationMultiplyi
4、ngthebothsideofequation(1.1)byandapplyingequalities(1.4.1)and(1.4.2)gives,(1.5)Differentiatingthefollowingw.r.t.timegives第13页，共13页,,(1.6.1)(1.6.2)Expandingthetermandapplyingtherelationships(1.6)yields,(1.7.1)Expandingthetermandapplyingtherelationships(1.6)yields,(1.7.2)Substitutingre
5、lationships(1.7)intoequation(1.5)andrearranginggives,(1.8)Aswecanseefromexpressions(1.7),themomentumequationsinradialandtangentialdirectionscontainvelocitiesinCartesiancoordinate;weneedtoreplacethemwithcorrespondingvariablesincylindricalcoordinate.Writingdownthemomentumequationsinrad
6、ialandtangentialdirectionsasfollows,(1.9.1)第13页，共13页(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)andrearrangingequati
7、on(1.8)gives(1.11)Where,,,,Note:differentfromEulerequationinCartesiancoordinates,theEulerequationincylindricalcoordinatescontainssourcetermsfrommomentumequationsinradialandtangentialequations.2.DerivationofEulerEquationinCylindricalcoordinatesmovingatintangentialdirectionWhere,,,,,,,
8、,,,,Thenequa
