柱坐标系和球坐标系下NS方程的直接推导英文版.docx

柱坐标系和球坐标系下NS方程的直接推导英文版.docx

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柱坐标系和球坐标系下NS方程的直接推导英文版

Derivationof3DEulerandNavier-StokesEquationsinCylindricalCoordinates

Contents

1.Derivationof3DEulerEquationinCylindricalcoordinates

2.DerivationofEulerEquationinCylindricalcoordinatesmovingatintangentialdirection

3.Derivationof3DNavier-StokesEquationinCylindricalCoordinates

1.Derivationof3DEulerEquationinCylindricalcoordinates

EulerEquationinCartesiancoordinates

（1.1）

Where

→Conservativeflowvariables

→Inviscid/convectivefluxinxdirection

→Inviscid/convectivefluxinydirection

→inviscid/convectivefluxinzdirection

Andtheirspecificdefinitionsareasfollows

,,

→Totalenthalpy

Somerelationship

Wewanttoperformthefollowingcoordinatestransformation

Because

AccordingtoCramer’sruler,wehave

（

（

Where

Similartotheabove

（（

Inaddition,thefollowingrelationsholdbetweencylindricalcoordinateandCartesiancoordinate

,,,（1.3）

（

（

Derivation

Multiplyingthebothsideofequation（1.1）byandapplyingequalities（and（gives,

（1.5）

Differentiatingthefollowingtimegives

（（

Expandingthetermandapplyingtherelationships（1.6）yields,

（

Expandingthetermandapplyingtherelationships（1.6）yields,

（

Substitutingrelationships（1.7）intoequation（1.5）andrearranginggives,

（1.8）

Aswecanseefromexpressions（1.7）,themomentumequationsinradialandtangentialdirectionscontainvelocitiesinCartesiancoordinate;weneedtoreplacethemwithcorrespondingvariablesincylindricalcoordinate.Writingdownthemomentumequationsinradialandtangentialdirectionsasfollows,

（

（

Multiplying（byand（by,thensummingupandapplyingexpressions（1.6）andrearrangingyields

（

Multiplying（a）byand（b）by,thensummingupandapplyingexpressions（1.6）yields,

（

Replacing（1.10）with（1.9）andrearrangingequation（1.8）gives

（1.11）

Where

,,,

Note:

differentfromEulerequationinCartesiancoordinates,theEulerequationincylindricalcoordinatescontainssourcetermsfrommomentumequationsinradialandtangentialequations.

2.DerivationofEulerEquationinCylindricalcoordinatesmovingatintangentialdirection

Where

,,,,,

,,

Thenequation（1.11）canbewrittenasfollows

（2.1）Where

Equation（2.1）adoptsrotatingcoordinatesbutthevariablesaremeasuredinabsolutecylindricalcoordinates.

3.Derivationof3DNavier-StokesEquationinCylindricalCoordinates

3DNavier-StokesEquationsinCartesiancoordinates

（3.1）

Where

,,

,

,,

,，,,

Inthefollowingderivation,onlyviscoustermswillbederivedfromCartesiancoordinatestocylindricalcoordinates,thoseinviscidtermshavingbeenderivedinsection1willbenotrepeated.

Replacingwithgives

（

Replacingwithgives

（

Multiplyingequation（3.1）by,theviscoustermsaregivesasfollows（omittingthenegativesignbeforeitfromsimplicity）,

（3.3）

（

（,（

（

（

（

Expandingexpression（3.3）gives,

（3.5）

=〉（

=〉

（

（

（

DivergenceinCartesianCoordinates

（

Divergenceincylindricalcoordinates

（

（

（

（

（

Aswecanseefromtheabovethatviscoustermsinexpression（3.5）forthemomentumequationinaxial/xdirectionandenergyequationcanbeexpressedinvariablesincylindricalcoordinates,whiletheviscoustermsin（3.5）formomentumequationsinradialandtangentialdirectionsstillcontainvariablesinCartesiancoordinates.Similarmanipulationto（1.10）willbeadoptedinthefollowing.

Writingouttheviscoustermsformomentumequationsinradialandtangentialcoordinatesasfollows,

（

（

Multiplying（byandmultiplying（by,thensummingupandrearranginggives,

（

Multiplying（byandmultiplying（by,thensummingupandrearranginggives,

（

（

（

（

（

Substituting（,（and（3.12）intoexpressions（3.11）andrearrangingyields,

（

（

Makinguseofexpressions（,（,（,（,（,（,（,

（and（,wecangetthefinalexpressionof3DNavier-StokesEquationincylindricalcoordinatesasfollows,

3DNavier-StokesEquationincylindricalcoordinates

，,,,

，,

，

Ifthemomentofmomentumequationisadoptedtoreplacethetangentialmomentumequation,itsexpressionwillbesimpler.Nowforthemomentequation,thereisnosourceterm.

，,,,

，,

，

For2Daxisymmetricflowfield,thetangentialmomentumequationormomentequationcanbeomittedasfollows,

2Daxisymmetricequationincylindricalcoordinate

，,,,

，，

，