柱坐标系和球坐标系下NS方程的直接推导.docx
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柱坐标系和球坐标系下NS方程的直接推导.docx
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柱坐标系和球坐标系下NS方程的直接推导
Derivationof3DEulerandNavier-StokesEquations
inCylindricalCoordinates
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
(1.2.1)
(1.2.2)
Where
Similartotheabove
(1.2.3)(1.2.4)
Inaddition,thefollowingrelationsholdbetweencylindricalcoordinateandCartesiancoordinate
,,,(1.3)
(1.4.1)
(1.4.2)
Derivation
Multiplyingthebothsideofequation(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.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
(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,
2Daxisymmetricequationincylindricalcoordin
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