柱坐標系和球坐標系下N-S方程的直接推導(dǎo)

1、脅爲 Derivati on of 3D Euler and Navier-Stokes Equati ons in Cyli ndrical Coord in atesCon te nts1. Derivation of 3D Euler Equation in Cylindrical coordinates2. Derivati on of Euler Equati on in Cyli ndrical coord in ates movi ng atin tangen tial directio n 3. Derivati on of 3D Navier-Stokes Equati on

2、 in Cyli ndrical Coord in ates1. Derivatio n of 3D Euler Equati on in Cyli ndrical coord in atesEuler Equati on in Cartesia n coord in ates(1.1)WhereCon servative flow variablesInv iscid/c onv ective flux in x directi onInv iscid/c onv ective flux in y directi on inv iscid/c onv ective flux in z dir

3、ecti onAnd their specific definitions are as followsE CvTuuuuuvuwv ,Evv wwvu,Fvv p,Gvwwuwvww pHuHvHwCpTuu vvwwHTotal en thalpySome relati on shipWe want to perform the followi ng coord in ates tran sformatio nx, y,zx, ,rBecause y r z y r z rLAyAccording to Cramer s ru柜hawery1Jz (1.2.1)WhereSimilar t

4、o the aboveyy r-薊爲討斤彳y iy 0i丄心y zJy zyyz(1.2.3)J rz y z J ry zIn additi on, the followi ng relati ons hold betwee n cyli ndrical coord in ate and Cartesia n coordi nater cos , zr sin-cos rz . sin rr sinr cos,(1.3)cossinr sinr cosFr cosF sin(1.4.2)(1.4.1)Grsin rGcosDerivati onMultiplying the both sid

5、e of equation (1.1) by J and applying equalities (1.4.1) and (1.4.2) gives,脅爲 -,UEFG,U,E,F一 GJ -JJJJtxyztxyzU r -tE r -xrFr cos-F sinrGrsin一-GcosU rtE rxrFr cosGrsin-G cos F sin0(1.5)Differentiating the following w.r.t. time gives y rcos ,z rsindydrr sinddzdr .dcosJsinr cos -dtdtdtdtdtdtdydrddzv,Vr,

6、rv ,1wdtdtdtdtvcoswsi nvr(1.6.1)vsinwcos vExpa nding the termFr cos Gr sinand appl ying the relatio nships (1.6) yields,vwuvuwFr cos Grsinvv p r cosvw r sinwvww pHvHwvcoswsi nvru vcoswsi nuvrr v vcos wsinpcosrvvr pcosrGrw vcos wsinpsinwvr psi nH vcoswsi nHvrExpa nding the term Gcos(1.7.1)(1.6.2)w uw

7、v uvGcosF sinvw cosvv p sinww pwvHwHvvsi nwcosvu vsi nwcosuvvvsin wcospsinvvpsi nFwvsin wcospcoswv pcosH vsi nwcosHvF sin and appl ying the relati on ships (1.6) yields,(1.7.2)Substitut ing relati on ships (1.7) into equati on (1.5) and rearra nging gives,旳爲訥葉右-Fr cos Gr sin rrGrGcos F sin(1.8)As we

8、 can see from expressions (1.7), the momentum equations in radial and tangential directionscontain velocities in Cartesia n coord in ate; we n eed to replace them with corresp onding variables in cyli ndrical coord in ate. Writ ing dow n the mome ntum equati ons in radial and tangen tial directi ons

9、 as follows,vu r -xr vvrp cosvv psin0 (1.9.1)wu r -xr wvr psinwv pcos0 (1.9.2)Multiplying (1.9.1) by cos yieldsrGVrvrur uvpvvtxrvvpsi ncoswvpcossin(1.10.1)vvpsi nsinwvpcoscosv vpand (b) bycos,the n sum ming up and appl ying expressi ons (1.6) yields,vv ur v vrpv vptxrvvp sinsinwvp coscos(1.10.2)vvp

10、sincoswvp cossinvrvrrReplacing (1.10) with (1.9) and rearranging equation (1.8) givesand (1.9.2) bysin , then summing up and applying expressions (1.6) and rearrangingMultipl ying (a) by sin(1.11)uvuvpHvHvWhere脅爲訥葉彳Note: differe nt from Euler equatio n in Cartesia n coord in ates, the Euler equati o

11、n in cyli ndrical coordi nates contains source terms from mome ntum equati ons in radial and tangen tial equati ons.in tangen tial directio n2. Derivati on of Euler Equati on in Cyli ndrical coord in ates movi ng atx, ,r,t x, ,r ,tWhererr ,rt , x x,ttxrr, tt ,x rx,ttxttr ttxtttJrr rrxrtr00tr000rxtrx

12、trxtJxr xxxxtx00000x0UUUrGrGFFEEttr rrJrrrJxxThen equation (1.11) can be written as followsUUEFrGQtx rr rS02v pUEF U rr GrS(2.1)Where Sv vrtxrr rEquati on (2.1) adopts rotat ing coord in ates but the variables are measured in absolute cyli ndrical coord in ates.3. Derivatio n of 3D Navier-Stokes Equ

13、ati on in Cyli ndrical Coordin ates3D Navier-Stokes Equati ons in Cartesia n coord in atesU E VxFVyGVz(3.1)Where旳 u/rrtf -XXyx22-uvwxx3xyz2vuwyy 32-yxzzxu xx v yx w zx qxyzzyuv ,EwEwv，qxyzuupuvuwvu,Fvv p,GvwwuwvwwpHuHvHw00xyxzyy,Vzyzzyzzu xyvyywzy qyu xzvyzw zzqzuvw，Vyyxxyuv, zxxzyxvvw z22 - 3zzV2 )

14、T，qyxIn the followi ng derivati on, only viscous terms will be derived from Cartesia n coord in ates to cyli ndrical coordi nates, those in viscid terms hav ing bee n derived in secti on 1 will be not repeated.Fr cosF sinReplacing F with Vy givesVyJVyr cosy rVy sinReplacing G with Vx givesMultiplyin

15、g equation (3.1) bysimplicity),-Gr sin rG cos (3.2.1)J 乂Vzr sinz rVz cos(322)J , the viscous terms are gives as follows (omitting the negative sign before it fromVx r -xVx r -Vyr cos rVy sin一Vzr sinrVz cos Vyr cosVzr sin Vz cosVy sinrx脅爲w/m(xxyxxyur cosyy1vrcosvsin1wr sinwcosrrrrrvrvr rruwzxxzzxJ(3.

16、4.2)1ur sinvu cosxrruwxzvr cosvsi nu1wr sinwcosrxrrvr cosvsi nu1wr sinwcosrxrryz(341)usin,(344)vu21r21rv2 - y2上x2丄x,(3.4.3)wzz(3.4.5)yzzy(3.3)21rwr sinwcosvr cosvsinwr coswsinvr sinvcos(3.4.6)wr cosvr sinwsinvcos0Expa nding expressi on (3.3) gives,Vyr cos rVzr sinVz cosVysin00xy cosxzSinxxryy cosyz

17、sinyxrzy coszz sinzxuxy cosxzSinvyy cosyz sinxxv yxw zxqxw zy coszzSinqy cosqz sinusinxyr 一 xxxz cosyysinyz coszysinzz cos(3.5)xysinxz cosyy sinyz COSzy sinzz cosqy sinqz cos旳 u/rrtf -u1vVxycosxzSinV yxur cosu sinxySinV cosxxz cosurVrVrxrrxVrur sin(361)u cosw . sinx=zxurur cosusinsinurs inucosw= cos

18、xcosu sinur sinu cosur cosu sinursinu cos(3.6.2)ur cos rurVru rvru-VrruvrVrVrrxvDiverge neein工sinu sinu cosvvwwVrvvVrvrVr(3.6.3)vrcosvrcosvsinwrsin-wr sin r rwcosvsinwcos(3.7.1)rvrCartesia n Coord in ates(3.7.2)Diverge nee in cyli ndrical coordi nates爲rvrx r r(3.7.3)yy cosyzSi nwzy coszzsinyz vsin w

19、cosvyy coswzzSin1wr cosvr s inwsi nvcosrrv2w 2vcos2Vws in 2 y3z 31wr cosvr s inwsi nvcosrr1vr cosvsi n2vcos2wsi nvsinVvsin1rrwcoswcoswrsinwcos2 vr r 3rvrvvr2vrvr2v vvrrryy sinVrVr(3.8.1)yz coszysinzzcosyz vcoswr cosvr s inwsi nvcosrv2w 2、2 -Vwcos 2 -y3z 3wr cosvr si nwsi nvcoswsinvsinzzvcosrvsin1ryy

20、 wcoswsinvcoswsinVv2vsinvr cosvsin2wcoswr sinwcosvrvrvrvrvrrv1 vvwwVvrvvrvrVvrv2vVvrvvr2v2-r(3.8.2)vrvr2V3Vv3Vvvvr rv旳 u/rrtf -,TTkcos k-sinyz,1Tr cosT sink-cosrr,1Tr十,Tk -Tkqrrrrqy si nqz cos,TTksinkcosyzTr cosT sink-sinrrqycosqzs in1 T kqr,1 Tr sin T cos .(3.9.1)ksinr r,1 Tr sin T cos(3.9.2)kcosr

21、rAs we can see from the above that viscous terms in expressi on (3.5) for the mome ntum equati on in axial/x direct ion and en ergy equati on can be expressed in variables in cyli ndrical coord in ates, while the viscous terms in (3.5) for mome ntum equati ons in radial and tangen tial directi ons s

22、till contain variables in Cartesia n coord in ates. Similar manipulation to (1.10) will be adopted in the following.Writi ng out the viscous terms for mome ntum equati ons in radial and tangen tial coordi nates as follows,yx rxr yy cosyzSi nyy Sinyz coS(3.10.1)rr zxxr zy coszzSinzySinzz cos(3.10.2)r

23、Multiplying (3.10.1) by cos and multiplying (3.10.2) by sin , then summing up and rearranging gives,yx cosZX sinxr yy cos yzsin cos zy cos zzsin sinryysinyz cos coszySin zz cos sin(3.11.1)yy sin yz cos sin zySin zz cos cosMultiplying (3.10.1) by sin and multiplying (3.10.2) by cos , then summing up

24、and rearranging gives,yxsin zxcos r yy cosyzsinsin zy cos zzsin cosxr(3.11.2)yyS inyzcos sinzy s in zz cos cosyy sinyz cos coszyS in zz cos sin脅爲w/m(yy COSyz SECOSzzs insin2 zy sin cos1wr cosvr si nwsi nvcosrrv2 、，2 、，2Vcos cos2 -Vy3z31wr cosvr si nwsi nvcosrrvw .2 x/2cos cos2 sinsinVyz31wr cosvr si

25、 nwsi nvcosrr12vrcosvsincoscos 2rrzy COSyycos coszzsin sin2sinsin2sin2sincossincoscoswr sin r2 x/wcos sin sinV32 1wr cos sin cosvr s insin cosvrcos cos coswr sin sin sinrrwsi nvcosvsi nwcossincoscos cos -sin sinV (3.12.1)21rvr2rrvr32Vr2V2r33yycosyzsinzycoscossinVVruvvr2 -rrrxrrsinzy coszz sin cossin

26、yyzz sin coscos1wr cosvr sinwsi nvcosrra v2c w22V2V siny3z 31wr cosvr sinwsi nvcosrr1wr cosvr sinwsi nvcosrrcoscoscos1rvrcos rvsinwrsin rcoscoscossin sinsin sinsin sin2vw .sin cosyzwcossin cos1rvr r2v(3.12.2)r2(3.12.3)yy sinyz coScos脅爲wZ葉右 zy sinzz cosSinyy sinyz cos coszySinzz cos sinyySinyy sinv2

27、sinyyz cos sinzySinzz cosCOSyySinyz cossinzySinzz coscosyz cossinzySinzz coscossin21 vr cos r rwr cossin sinc w2 coszvr sin2zV cos cos2 yz cossin2cosvsi nsin sinwsin vcos2 yz cos sin21 wr sinr rw cos cos cosV (3.12.4)cos sinr rr(3.13.1)Substituting (3.6.1), (3.6.2) and (3.12) into expressions (3.11)

28、 and rearranging yields,rx r -x(3.13.2)Making use of expressions (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), we can get the final expressi on of 3D Navier-Stokes Equati on in cyli ndrical coord in ates as follows,3D Navier-Stokes Equati on in cyli ndrical coord in atesuvvuuu puvuvrv , Ev u , Fv