柱坐標(biāo)系和球坐標(biāo)系下N-S方程的直接推導(dǎo)

1、Derivationof3DEulerandNavier-StokesEquationsinCylindricalCoordinatesContents1. Derivationof3DEulerEquationinCylindricalcoordinatesintangentialdirection2. DerivationofEulerEquationinCylindricalcoordinatesmovingat3. Derivationof3DNavier-StokesEquationinCylindricalCoordinates1. Derivationof3DEulerEquat

2、ioninCylindricalcoordinatesEulerEquationinCartesiancoordinatesUEF£0txyz(1.1)WhereUConservativeflowvariablesEInviscid/convectivefluxinxdirectionFInviscid/convectivefluxinydirectionGinviscid/convectivefluxinzdirectionAndtheirspecificdefinitionsareasfollowsECvTuuvvwwuuvuwuHuCpTuvvvpwvHvuuvvwwuwvww

3、wpHwHTotalenthalpySomerelationshipWewanttoperformthefollowingcoordinatestransformationx,y,zBecausex,rAccordingtoCramer'sruler,wehaveWhereSimilartotheabove(1.2.1)(1.2.2)1zJr(1.2.3)1yy(1.2.4)JrInaddition,thefollowingrelationsholdbetweencylindricalcoordinateandCartesiancoordinateyrcos,zrsincosyzsin

4、,rsin,rcos,(1.3)FFrFFzFzJyryyrrcossinrsinrcosFrcosrFsinDerivationGGrGJzrzz一G一GrrGrsinGcosrMultiplyingthebothsideofequation(1.1)bygives,Ur-tUr-t0ExEr-xEr-xFGU,EJJ-yztxFrcos-FsinrFrcosGrsinrDifferentiatingthefollowingw.r.t.timegivesyrcos,zrsindydtdrcosdtddzdr.rsin,sindtdtdt(1.4.2)Jandapplyingequalitie

5、s(1.4.1)and(1.4.2)j£yGrsinrGcos(1.5)GcosFsindrcosdtdydtdrv,dtd,rdtdzv,wdtvcoswsinvr(1.6.1)vsinwcosv(1.6.2)ExpandingthetermFrcosGrsinandapplyingtherelationships(1.6)yields,FrcosGrsinvcosuvcosvuvvvprcoswvHvwsinwsinrvvcoswsinpcosrwvcoswsinpsinHvcoswsinwuwvwrsinwwpHwvuvrvvrpcosrGrwvrpsinHvr(1.7.1)E

6、xpandingthetermGcosFsinandapplyingtherelationships(1.6)yields,wuwGcosFsinvwcoswwpHwvsinwcosuvsinwcosvvsinwcospsinwvsinwcospcosHvsinwcosvuvvvpsinwvHv(1.7.2)vuvvvpsinFwvpcosHvSubstitutingrelationships(1.7)intoequation(1.5)andrearranginggives,GcosFsin(1.8)UErrFrcosGrsintxrUEFrGr八rr0txrAswecanseefromexp

7、ressions(1.7),themomentumequationsinradialandtangentialdirectionscontainvelocitiesinCartesiancoordinate;weneedtoreplacethemwithcorrespondingvariablesincylindricalcoordinate.Writingdownthemomentumequationsinradialandtangentialdirectionsasfollows,vurxrvvrpcosvvpsin0(1.9.1)wurxrwvrpsinwvpcos0(1.9.2)Mul

8、tiplying(1.9.1)bycosand(1.9.2)bysin,thensummingupandapplyingexpressions(1.6)andrearrangingyieldsMultiplying(a)byvvvvsinvrupsinpsinrvrvrpcossinwvwvand(b)bycossinpcos(1.10.1)pcoscos,thensummingupandapplyingexpressions(1.6)yields,VvurvvrPvvptxrvvpsinsinwvpcoscosvvpsincoswvpcossinrVrV(1.10.2)Replacing(1

9、.10)with(1.9)andrearrangingequation(1.8)givesrG(1.11)Whereuupvu,FuvuvrvVrvuvrvHuHvvrvrpHvr0vvrNote:differentfromEulerequationinCartesiancoordinates,theEulerequationincylindricalcoordinatescontainssourcetermsfrommomentumequationsinradialandtangentialequations.in2. DerivationofEulerEquationinCylindric

10、alcoordinatesmovingattangentialdirection,r,tx,r,tWherex,ttUrGThenequation(1.11)canbewrittenasfollowsU_EtxFUrrGSrrr(2.1)02vprWhereSvvrr00Equation(2.1)adoptsrotatingcoordinatesbutthevariablesaremeasuredinabsolutecylindricalcoordinates.3. Derivationof3DNavier-StokesEquationinCylindricalCoordinates3DNav

11、ier-StokesEquationsinCartesiancoordinatesUEVxFVyGVz(3.1)WherexxXXyxzxxxvyxzxqxuuvuwuHuxyyyyxxyxyyyzyuvvvpwvHvzyqyzxuwvwwwHwuxzyzxzxzyzzzwzzqzyyzz2zyzzyqxT,qyxT,qzykzInthefollowingderivation,onlyviscoustermswillbederivedfromCartesiancoordinatestocylindricalcoordinates,thoseinviscidtermshavingbeenderi

12、vedFsininsection1willbenotrepeated.FrcosReplacingFwithVygivesjM-VyrcosyrVysinJGrsinzrGcosReplacingGwithvxgives一VzrsinrVzcos(3.2.2)Multiplyingequation(3.1)byJ,theviscoustermsaregivesasfollows(omittingthenegativesignbeforeitfromsimplicity),VxJxVyxxyxxyVyrcosVysinVzrsinrVzcosurcosyyzzyzVyrcosVzrsinVzco

13、sVysinzy(3.3)1vrcosvsin1wrsinwcosrrrrrvrvrrruwzxxzzx,(3.4.2)iv1ursinxrrvuwyxzvrcosvsinu1wrsinwcosrxrrvrcosvsinu1wrsinwcosrxrrwvuzyxwrsinwcos1vrcosvsinrrrw_vyzwrcoswsin1vrsinvcosrrrwrcosvrsinwsinvcosuvwXyz(3.4.1)usinucos,(3.4.4)2x1(3.4.6)r221r21r21r(3.4.5)u2x2xr,(3.4.3)VxrxxrVyrcosVzrsin一VzcosVysin0x

14、ycos0xzsinrxxxyxrryyzycoscosyzsinzzsinuzxxxvyxwzxq：xuxycoswzycosxzsinzzsinVyycosqycosyzsinqzsin0xysinxzcosyysinyzcosuxysinzysinxzCOsVzzcosyysinyzcos(3.5)wzySinzzcosqysinqzcosrrxyCOSxzsinr1urcosusinV1ursinucosw-sinxrrrcosxrrr1ur_11Vr=rrruxruvrrrxxrrxurVrx(3.6.1)xySinxzCOS1urcosusinVsinx1ursinucoswcos

15、=rrrrx_u_vrxuV(3.6.2)xxrxv1urcosrr1urcosrr1urcosvrr1urvruvrrr1uu一rvrvrruuvrvrryxwzxusinvw1ursinucoswxrrxusinvursinucoswv一vwwxrxursinrwusinucosA/1vvwwvw2xuv1vvvrvr2xvvxvvxvvxvvrxuvruvVr一一vrxrxvrrxvxuvwu1vrcosvsin1一wrsin一一wcosxyzxrrrru1vrcoswrsin-vsinwcos(3.7.1)xrru1vrvr-xrr(3.6.3)DivergenceinCartesia

16、nCoordinatesuvwVxyzDivergenceincylindricalcoordinates(3.7.2)(3.7.3)vyycosyzsinwzycoszzsinyzvsinwcosvyycoswzzsinVurvrxrr1wrcosvrsinwsinvcos.一vsinrrwcosv2w2vcos2Vwsin2一Vy3z31wrcosvrsinwsinvcos.一vsinwcosrr2vcos1vrcosrrrvrrvsin1wrsin2wsinrrvr2v2vrvr2vvvrV3wcosrvvrvvr2、,一-vr2-Vvrrrrr3yysinyzcoszysinzzcos

17、yzvcoswrcosvrsinwsinvcosrcV2Vcw22-wcos2y3z3wrcosvrsinwsinvcoswsinvsinvcosrvsinV1ryywCoSzzwsinvcoswsinVv2vsinvrcosvsin2wcoswrsinwcosVrVrVrVrVrVrVrVrVrVrrvrvrvrv2v2vVr1vvwwVvVvVrVvqycosqzsink-cosyksinzk1rTrcosk1rqysinqzcosinyTrcosv2rTsinkrTcoszTsinqr(3.8.2)2-rVrcosk1rTrsinTcossin(3.9.1)sinTrsinTcos(3.

18、9.2)coskrAswecanseefromtheabovethatviscoustermsinexpression(3.5)forthemomentumequationinaxial/xdirectionandenergyequationcanbeexpressedinvariablesincylindricalcoordinates,whiletheviscoustermsin(3.5)formomentumequationsinradialandtangentialdirectionsstillcontainvariablesinCartesiancoordinates.Similar

19、manipulationto(1.10)willbeadoptedinthefollowing.Multiplying(3.10.1)bysin,thensummingupandrearrangingWritingouttheviscoustermsformomentumequationsinradialandtangentialcoordinatesasfollows,r-yxryyCoSyzSinyySinyzCoS(3.10.1)xrzxrzyCoSzzsinzysinzzcos(3.10.2)rxrcosandmultiplying(3.10.2)bygives,Multiplying

20、(3.10.1)bysinandmultiplying(3.10.2)bycosthensummingupandryxCoSzxsinryycosyzSincoszycoszzSinsin1xrwSinyzcoscoszysinzzcossinyy(3.11.1)sinyysinyzCoSzysinzzcoscosrearranginggives,yxSinzxCOSryyCOSyzsinsinzycosrxryySinyzCOSsinzysinzzcoscoszzcossinyzSincoszycoszzsinsinincosyycoscoszzsinsinwrcosvrsinwsinvco

21、srv2-cw2-Vcoscos2-Vy3z3wrcosvrsinwsinvcosrv-w.sin2coscos-V2sinyz3wrcosvrsinwsinvcos2zy12sinr12sinr1r2sin2yySinzySinyyCoSyzcoscos-2vrcosrrsincossincoscosvsincoscos2wrsinr一wrcossincosrwsinvcos.sinvrsincoszzsincos(3.11.2)sincosvsinwcossinsinvrcoscoscoswrsinsinsincoscoswcos一.一.sinsin27VV(3.12.1)3rvrvrru

22、vrrryycosyzsinsinzycoszzsincoszycoscossinsiny%sincos1 wrcosvrsinwsinvcos一coscossinsinrrv2w22 V2Vsincosy3z3sinsinsinsinvw2一一sincosyz1 wrcosvrsinwsinvcos一coscosrr2 wrcosvrsinwsinvcos一coscosrr2-vrcosvsinrr一wrsinrwcossincos2v(3.12.2)yysinyzcoscoszysinzzcossin1_rvrrvvrvrrrryysinyzcoscoszysinzzcossin(3.12

23、.3)yysinyzcossinzysinzzcoscosyysinyzcossinzysinsinsinyysinzzcosyzcoscosVcoscossinzysinzzcoscosyzcossincv.2sinysinwcoszcos2yzcossin21一vrcosrrvsinsinsin21wrsinrrwcoscoscos(3.12.4)c1wrcosvrsinwsinvcos.3 一cossinrrVrSubstituting(3.6.1),(3.6.2)and(3.12)intoexpressions(3.11)andrearrangingyields,4r廣rrx(3.13.1)xrrr-x'rr(3.13.2)xr(3.13.1)and(3.13.2),wecangetthefinalexpressionof3DNavier-StokesEquationincy