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1、柱坐系和球坐系下NS方程的直接推导Derivation of 3D Euler and Navier-Stokes Equations in Cylindrical Coordinates Contents 1. Derivation of 3D Euler Equation in Cylindrical coordinates 2. Derivation of Euler Equation in Cylindrical coordinates moving at w in tangential direction 3. Derivation of 3D Navier-Stokes Equat
2、ion in Cylindrical Coordinates 1. Derivation of 3D Euler Equation in Cylindrical coordinates Euler Equation in Cartesian coordinates UEFG+=0 (1.1) txyzWhere U Conservative flow variables E Inviscid/convective flux in x direction F Inviscid/convective flux in y direction G inviscid/convective flux in
3、 z direction And their specific definitions are as follows rrurvrwruruu+pruvruwU=rv,E=rvu,F=rvv+p,G=rvw rwrwurwvrww+prErHurHvrHwE=CvT+H Total enthalpy 1(uu+vv+ww) 2H=CpT+1(uu+vv+ww)=E+p 2rSome relationship We want to perform the following coordinates transformation (x,y,z)(x,q,r) Because ryrz+=1 yrz
4、rAccording to Cramers ruler, we have ryrz+=0 yqzq10r=yyryqzrzq=1z (1.2.1) zJqrzq第1页，共14页 y1ry (1.2.2) 0r1y=q=-yzzJqrryzqqWhere yj=ryzr zqqSimilar to the above qyqqyqzyr+zzr=0 yq+zq=1 0zy0rq1zry1=qqyy=-1zzJr (1.2.3) =qzyz=1yJr (1.2.4) ryrzryrzqqqqIn addition, the following relations hold between cyli
5、ndrical coordinate and Cartesian coordinate y=rcosq,z=rsinq yr=cosq,zr=sinq,yzq=-rsinq,q=rcosq, (1.3) FyzJy=JFrFqry+qyFzFz=rq-qrJ=sinqryr=cosqz-rsinqrcosq=r =rFzzq-qFr (1.4.1) qq=r(Frcosq)-q(Fsinq)JGGrGGyGyz=Jrz+qqz=-rq+qr=-yrGq+qGyr (1.4.2) =r(Grsinq)+q(Gcosq)Derivation Multiplying the both side of
6、 equation (1.1) by J and applying equalities (1.4.1) and (1.4.2) gives, 第2页，共14页 UEFGUEFGJ+=J+J+J+JtxyztxyzUE(Fsinq)+(Grsinq)+(Gcosq) (1.5) =r+r+(Frcosq)-txrqrq=rUt+rEx+r(Frcosq+Grsinq)+q(Gcosq-Fsinq)=0Differentiating the following w.r.t. time gives y=rcosq,z=rsinq dydrdqdzdrdt=dtcosq-rsinqdt,dt=dts
7、inq+rcosqdqdtdydt=v,drdt=v,rdqdzrdt=vq,dt=w vcosq+wsinq=vr (1.6.1) -vsinq+wcosq=vqExpanding the term (Frcosq+Grsinq) and applying the relationships (1.6) yields, rvrwruv(ruwFrcosq+Grsinq)=rvv+prcosq+rwvrvwrsinqrww+prHvrHwr(vcosq+wsinq)ru(vcosq+wsinq)rv (1.7.1) rruvr=rrv(vcosq+wsinq)+pcosq=rrvvr+pcos
8、qrw(vcosq+wsinq)+psinq=rGrrH(vcosq+wsinq)rwvr+psinqrHvrExpanding the term (Gcosq-Fsinq) and applying the relationships (1.6) yields, rwrvruwruv(Gcosq-Fsinq)=rvwcosq-rvv+psinqrww+prwvrHwrHvr(-vsinq+wcosq)rv (1.7.2) qru(-vsinq+wcosq)ruvq=rv(-vsinq+wcosq)-psinq=rvvq-psinq=rw(-vsinq+wcosq)+pcosqrwvFqrH(
9、-vsinq+wcosq)q+pcosqrHvqSubstituting relationships (1.7) into equation (1.5) and rearranging gives, 第3页，共14页 (1.6.2) UE(Gcosq-Fsinq)+r+(Frcosq+Grsinq)+txrq(1.8) UEFqrGr=r+r+=0txqrrAs we can see from expressions (1.7), the momentum equations in radial and tangential directions contain velocities in C
10、artesian coordinate; we need to replace them with corresponding variables in cylindrical coordinate. Writing down the momentum equations in radial and tangential directions as follows, rvrvur(rvvr+pcosq)(rvvq-psinq)r+r+=0 (1.9.1) txrqrwrwur(rwvr+psinq)(rwvq+pcosq)r+r+=0 (1.9.2) txrqMultiplying (1.9.
11、1) by cosq and (1.9.2) bysinq, then summing up and applying expressions (1.6) and rearranging yields rvrrvrur(rvrvr+p)rvrvqr+txrqcosqsinq=(rvvq-psinq)+(rwvq+pcosq) (1.10.1) qq=-(rvvq-psinq)sinq+(rwvq+pcosq)cosq=rvqvq+pMultiplying (a) by -sinq and (b) bycosq, then summing up and applying expressions
12、(1.6) yields, rvqrvqur(rvqvr+p)(rvqvq+p)+txrqsinqcosq (1.10.2) =-(rvvq-psinq)+(rwvq+pcosq)qq=-(rvvq-psinq)cosq-(rwvq+pcosq)sinqr=-rvrvqReplacing (1.10) with (1.9) and rearranging equation (1.8) gives UEFrG+=S (1.11) txrqrrWhere rvqrruruvqruruu+pU=rvq,E=rvqu,F=rvqvq+rvrvqrvrrvrurHvrErHuq00-rvqvr,S=pG
13、= r2rvq+prvrvr+prHvrr0rvrruvrrvqvr第4页，共14页 Note: different from Euler equation in Cartesian coordinates, the Euler equation in cylindrical coordinates contains source terms from momentum equations in radial and tangential equations. 2. Derivation of Euler Equation in Cylindrical coordinates moving a
14、t w in tangential direction (x,q,r,t)(x,q,r,t) Where r=r,q=q+wt,x=x,t=tr=r,q=q-wt,x=x,t=t rqxtrqxt=+=+trtqtxtttrrrqrxrtr, =0-w+0+=+0+0+0qtrrqxtrqxt=+=+qrqqqxqtqxrxqxxxtx, =0+0+0=0+0+0qxUUUrGrGFFEE, =-w=ttqrrrrrqrqxxThen equation (1.11) can be written as follows UUEFrG-w+=S tqxrqrr20rv+pqrWhere S=rvq
15、vr -r00UE(F-Uwr)rG+=S (2.1) txrqrrEquation (2.1) adopts rotating coordinates but the variables are measured in absolute cylindrical coordinates. 3. Derivation of 3D Navier-Stokes Equation in Cylindrical Coordinates 3D Navier-Stokes Equations in Cartesian coordinates U(E-Vx)(F-Vy)(G-Vz)+=0 (3.1) txyz
16、Where 第5页，共14页 rrurvrwruruu+pruvruwU=rv,E=rvu,F=rvv+p,G=rvw rwrwurwvrww+prErHurHvrHw000txxtxytxz,V=,V= tyxtyytyzVx=yztzxtzytzzut+vt+wt-qut+vt+wt-qut+vt+wt-qyxzxxyyzyyyzzzzxxxyxztxxv2uvwu2=m2-=m2-V,tyx3xyzx32vuwv2vuvuw=txy=my+x,tzx=txz=mz+x, 2wvuw2v,tzz=mtyy=m2-=m2-V2-=m2-V,3yxz3zyxz3y3wvTTTq=-k，, ty
17、z=tzy=m+q=-kq=-kyxzyzyxzIn the following derivation, only viscous terms will be derived from Cartesian coordinates to cylindrical coordinates, those inviscid terms having been derived in section 1 will be not repeated. Q JReplacing Fwith Vygives F(Fsinq) =(Frcosq)-yrq(Vysinq) J=(Vyrcosq)-yrqReplacin
18、g Gwith Vx gives VyGQ J(Gcosq) (3.2.1) =(Grsinq)+zrqJVz(Vzcosq) (3.2.2) =(Vzrsinq)+zrqMultiplying equation (3.1) by J, the viscous terms are gives as follows (omitting the negative sign before it from simplicity), VyVxVzJ+J+JxyzVx(Vysinq)+(Vzrsinq)+(Vzcosq) =r+(Vyrcosq)-xrqrqVx(Vzcosq-Vysinq)=r+(Vyr
19、cosq+Vzrsinq)+xrq第6页，共14页 (3.3) txx=m2-3xyz=uv2uvw2u1(vrcosq)(vsinq)1(wrsinq)(wcosq)m2-+- (3.4.1) 3xrrqrrq2u(rvr)vqm2-3xrrrqtyx=txy=my+x, (3.4.2) 1(urcosq)(usinq)v=m-+rqxr2vuwtzx=txz=muw+zx1(ursinq)(ucosq)w=m+rqxr, (3.4.3) tyy=m2-3yxz21(vrcosq)(vsinq)u1(wrsinq)(wcosq)=m2-+-,(3.4.4) 3rrqxrrq=21(vrcos
20、q)(vsinq)u1(wrsinq)(wcosq)m2-+-3rrqxrrq2wvutzz=m2-3zyx=21(wrsinq)(wcosq)1(vrcosq)(vsinq)um2+-3rrqrrqx (3.4.5) tyz=tzy=my+z1(wrcosq)(wsinq)1(vrsinq)(vcosq)=m-+ (3.4.6) rqrqrr1(wrcosq+vrsinq)(wsinq-vcosq)=m-rrqwvExpanding expression (3.3) gives, Vx(Vyrcosq+Vzrsinq)+(Vzcosq-Vysinq)+xrq00tcosq+tsinqxyxz
21、txxtyycosq+tyzsinqrtyx=r+tcosq+tsinqxrzyzztzxu(tcosq+tsinq)+v(tcosq+tsinq)xyxzyyyzut+vt+wt-qyxzxxxx+w(tcosq+tsinq)-qcosq-qsinqzyzzyz0-tsinq+tcosqxyxz-tyysinq+tyzcosq+-tzysinq+tzzcosqq(3.5) u(-tsinq+tcosq)+v(-tsinq+tcosq)xyxzyyyz+w(-tsinq+tcosq)+qsinq-qcosqzyzzyzr第7页，共14页 r(txycosq+txzsinq)r1(ursinq)
22、(ucosq)wr1(urcosq)(usinq)v=m-+cosq+m+sinqrrrrqxrqxr1(ur)vr=-u+mrrrx=ruvr+mrrx=txr=trx=m+r (3.6.1) rx(-txysinq+txzcosq)q= 1(ursinq)(ucosq)w1(urcosq)(usinq)v=-+cosq+sinq+m+-mqrrqrrqxx=uvq+mqrqxuvtxq=tqx=m+q (3.6.2) rqxvtyx+wtzx1(urcosq)(usinq)v1(ursinq)(ucosq)w=vm-+wm+rrqxrrqx=m=m=m=m1(urcosq)(usinq)v
23、w(ursinq)(ucosq)-v+wmw+vmrrqrqxx1(ur)(ur)(ucosq)1(vv+ww)(usinq)cosqv+sinqw-v+w+mrrqq2xr1(ur)u1(vqvq+vrvr)v-uv+v+mrrqrq2xr1uuvqvrrv+v+mv+mvrqqrrqxxruvuvvu=mvr+vq+mvqq+mvrrrqxxrvuvru=vrm+q+vqxxrrq=vrtrx+vqtqx(3.6.3) uvwu11(vsinq)()()+=+(vrcosq)-+wrsinq+wcosqxyzxrrqqrru1(3.7.1) (-vsinq+wcosq)=+(vrcosq+
24、wrsinq)+xrrq=u1v+(rvr)+qxrrqvuvwV=+ (3.7.2) xyz第8页，共14页 Divergence in Cartesian Coordinates Divergence in cylindrical coordinates vu(rvr)+vq (3.7.3) V=+xrrrqv(tyycosq+tyzsinq)+w(tzycosq+tzzsinq)=tyz(vsinq+wcosq)+vtyycosq+wtzzsinq1(wrcosq+vrsinq)(wsinq-vcosq)=m-(vsinq+wcosq)rrqvvv2w2+vcosqm2-V+wsinqm
25、2-Vz3y31(wrcosq+vrsinq)(wsinq-vcosq)=m-(vsinq+wcosq)rrqv1(vrcosq)(vsinq)1(wrsinq)(wcosq)2+2vcosqm-+2wsinqm+-vmVrrrqrrq3v1rvqrvrvr2=mvq+2vr+vq-2vrvr-2vqvq-vrmVrrrq3vrvqvrvqvr2=vqm+-+vrm2-Vrrrqrr3(3.8.1) =vqtqr+vrtrrv(-tyysinq+tyzcosq)+w(-tzysinq+tzzcosq)=tyz(vcosq-wsinq)-vsinqtyy+wcosqtzz1(wrcosq+vrs
26、inq)(wsinq-vcosq)=m-(vcosq-wsinq)rrqvvv2w2-vsinqm2-V+wcosqm2-Vy3z3v1(wrcosq+vrsinq)(wsinq-vcosq)2=m-(vcosq-wsinq)-mVvqrrq31(vrcosq)(vsinq)1(wrsinq)(wcosq)-2vsinqm-+2wcosqmrrqrrqv1(rvq)1(vqvq)1(vv+ww)2=mvr+-mVvqrr2q2q3vvq1(vqvq+vrvr)21(rvq)=mvr+vq+-3mVvqrrq2qvvv21(rvq)=mvr+2vqq+vrr-mVvqrrqq3vvr(rvq)v
27、q2=vrm+-mV+vqm2rrrqrq3vvrvqvq2vqvq2=vrm+-+-mV+vqm2rqrrrrq3vvqvr2vrvqvq=vrm+-+vqm2+-mVrqrrrqr3=vrtrq+vqtqq(3.8.2) 第9页，共14页 -qycosq-qzsinq=kTTcosq+ksinqyz(3.9.1) 1(Trcosq)(Tsinq)1(Trsinq)(Tcosq)=k-cosq+k+sinqrrqrrq1(Tr)T=k-T=k=-qrrrrqysinq-qzcosq=-kTTsinq+kcosqyz(3.9.2) 1(Trcosq)(Tsinq)1(Trsinq)(Tcosq
28、)=-k-sinq+k+cosqrrqrrq1T=k=-qqrqAs we can see from the above that viscous terms in expression (3.5) for the momentum equation in axial/x direction and energy equation can be expressed in variables in cylindrical coordinates, while the viscous terms in (3.5) for momentum equations in radial and tange
29、ntial directions still contain variables in Cartesian coordinates. Similar manipulation to (1.10) will be adopted in the following. Writing out the viscous terms for momentum equations in radial and tangential coordinates as follows, rtyxx+r(tyycosq+tyzsinq)r+(-tyysinq+tyzcosq)q (3.10.1) tzxr(tzycos
30、q+tzzsinq)(-tzysinq+tzzcosq)r+ (3.10.2) xrqMultiplying (3.10.1) by cosq and multiplying (3.10.2) by sinq, then summing up and rearranging gives, (tyxcosq+tzxsinq)r(tyycosq+tyzsinq)cosq+(tzycosq+tzzsinq)sinqr+xr(-tyysinq+tyzcosq)cosq+(-tzysinq+tzzcosq)sinq(3.11.1) +q-(-tyysinq+tyzcosq)sinq+(-tzysinq+
31、tzzcosq)cosqMultiplying (3.10.1) by -sinq and multiplying (3.10.2) by cosq, then summing up and rearranging gives, r+(-tyxsinq+tzxcosq)r-(tyycosq+tyzsinq)sinq+(tzycosq+tzzsinq)cosq+xr-(-tyysinq+tyzcosq)sinq+(-tzysinq+tzzcosq)cosq(3.11.2) q-(-tyysinq+tyzcosq)cosq-(-tzysinq+tzzcosq)sinq第10页，共14页 (tyyc
32、osq+tyzsinq)cosq+(tzycosq+tzzsinq)sinq=2tzysinqcosq+tyycosqcosq+tzzsinqsinq1(wrcosq+vrsinq)(wsinq-vcosq)=2m-sinqcosqrrqvvv2w2+m2-Vcosqcosq+m2-Vsinqsinqy3z31(wrcosq+vrsinq)(wsinq-vcosq)=2m-sinqcosqrrqvvw2+m2ycosqcosq+2zsinqsinq-m3V1(wrcosq+vrsinq)(wsinq-vcosq)=2m-sinqcosqrrqv12(vsinq)cosqcosq+2(wrsin
33、q)+(wcosq)sinqsinq-mV+m2(vrcosq)-rrqq3r()wrcosqsinqcosq+vrsinqsinqcosq+vrcosqcosqcosq+wrsinqsinqsinqv1r2=2m-mV(3.12.1) (vsinq)(wcosq)r(wsinq-vcosq)3-sinqcosq-cosqcosq+sinqsinqqqqv1rv2=2mr-vr-mVrr3v2vruvqvrvr2=2m-mV=m2-=trrr33rxrqr-(tyycosq+tyzsinq)sinq+(tzycosq+tzzsinq)cosq=tzy(cosqcosq-sinqsinq)+(-
34、tyy+tzz)sinqcosq1(wrcosq+vrsinq)(wsinq-vcosq)=m-(cosqcosq-sinqsinq)rrqvvv2w2+-m2-V+m2-Vsinqcosqy3z3vw1(wrcosq+vrsinq)(wsinq-vcosq)=m-(cosqcosq-sinqsinq)+2m-y+zsinqcosqrrq1(wrcosq+vrsinq)(wsinq-vcosq)=m-(cosqcosq-sinqsinq)rrq1(vsinq)()()+2m-(vrcosq)-+wrsinq+wcosqsinqcosqrrqqrv1rv=mq+r-2vqrrqvvv=mq+r-
35、q=trqrrqr(3.12.2) 第11页，共14页 (-tyysinq+tyzcosq)=qq(-tyysinq+tyzcosq)cosq+(-tzysinq+tzzcosq)sinqcosq+(-tzysinq+tzzcosq)sinqq-(-tyysinq+tyzcosq)sinq+(-tzysinq+tzzcosq)cosq=q-(3.12.3) vqvrvqmr+rq-r-(-tyysinq+tyzcosq)sinq+(-tzysinq+tzzcosq)cosqtrq=-tqqq-(-tyysinq+tyzcosq)sinq+(-tzysinq+tzzcosq)cosqvvv2w2
36、=m2y-3Vsinqsinq+m2z-3Vcosqcosq-2tyzcosqsinqvvw2=m2sinqsinq+2cosqcosq-V-2tyzcosqsinqyz3v112(vsinq)()()=m2(vrcosq)-sinqsinq+2wrsinq+wcosqcosqcosq-V (3.12.4) qrrq3rr1(wrcosq+vrsinq)(wsinq-vcosq)-2m-cosqsinqrrqv1vq2=2m+vr-mVrq3=tqqSubstituting (3.6.1), (3.6.2) and (3.12) into expressions (3.11) and rearranging yields, rrtrx(rtrr)trq+-tqq (3.13.1) xrqtqx(rtqr)tqq+trq (3.13.2) xrq Making use of expr
