GROUP 1. Run title and other preliminaries PHOTON USE AUTOPLOT file phi 5 cl msg MHD PLANE CHANNEL FLOW msg Hartmann number = 10 Reynolds number =100 msg Velocity (W1) profile msg Blue line --- PHOENICS solution msg crosses --- analytical solution da 1 w1;da 1 w1a col3 1;blb4 2 msg pressto continue pause msg press to end pause end END_USE DISPLAY TEXT(1D Laminar MHD Channel Flow TITLE DISPLAY This problem concerns the steady fully-developed laminar flow of an incompressible electrically-conducting fluid in the positive z-direction of a plane channel. A uniform magnetic field By is imposed normal to the walls and a current jx is induced in the fluid in the x-direction, together with a magnetic field Bz in the z-direction. The problem neglects end effects, secondary flows, Hall effect and ion-slip phenomena. ENDDIS The dimensionless momentum equation to be solved is: dp/dz + d/dy(dw/dy)/Re + Ha*Ha(K-w)/Re = 0 where y =y/yin z =z/yin w =w/win p=p/(rho*win**2) Re=win*yin/enul Ha=sig*(By*yin)**2/(rho1*enul) K =Ex/(win*By) Here, yin is the channel half width (m), By is the imposed magnetic flux density in the +ve y direction (volt.s/m**2), sig is the electric conductivity (ohm/m) and Ex is electric field intensity in the +ve x-direction (volt/m). The Hartmann number Ha represents the ratio of the electromagnetic forces to the viscous forces. The voltage ratio K is the ratio of the voltage to the open-circuit voltage. If K=0, this corresponds to a short-circuit condition. When K=1 the net current flow is zero, which is known as the open-circuit condition. This is the classical Hartmann problem. Further, if K < 1 the channel will act as a MHD pump, whereas if K > 1 the channel will act as a flowmeter. The analytical solution to this problem has been presented in 'Engineering Magnetohydrodynamics', Chapter 10, G.W.Sutton and A.Sherman, McGraw Hill, (1965). REAL(YIN,HIN,WIN);YIN=1.0;WIN=1.0 GROUP 2. Transience; time-step specification CARTES=T GROUP 4. Y-direction grid specification NY=40;GRDPWR(Y,NY,YIN,1.) GROUP 5. Z-direction grid specification ZWLAST=0.1 GROUP 7. Variables stored, solved & named SOLVE(W1);STORE(W1A) GROUP 8. Terms (in differential equations) & devices TERMS(W1,N,N,P,P,P,P) GROUP 9. Properties of the medium (or media) RHO1=1.;ENUT=0. REAL(REY,FLOWIN);REY=100.;ENUL=1./REY REAL(HA,KVOLT);HA=10.0;KVOLT=1.0 HA KVOLT REY GROUP 11. Initialization of variable or porosity fields FIINIT(W1)=WIN ** compute analytical solutions REAL(WA,GR,HAY);INTEGER(JJM1) DO JJ=1,NY +PATCH(IN:JJ:,INIVAL,1,NX,JJ,JJ,1,NZ,1,1) +GR=0.5*YFRAC(JJ) IF(JJ.NE.1) THEN +JJM1=JJ-1 +GR=YFRAC(JJM1)+0.5*(YFRAC(JJ)-YFRAC(JJM1)) ENDIF +GR=GR*YVLAST;HAY=GR*HA +WA=HA*(COSH(HA)-COSH(HAY))/(HA*COSH(HA)-SINH(HA)) +INIT(IN:JJ:,W1A,ZERO,WA) ENDDO GROUP 13. Boundary conditions and special sources PATCH(WALL,NWALL,1,NX,NY,NY,1,NZ,1,1) COVAL(WALL,W1,1.0,0.0) ** activate fully-developed pressure-drop calculation FDFSOL=T;USOURC=T FLOWIN=RHO1*WIN*YIN PATCH(FDFW1DP,VOLUME,1,NX,1,NY,1,NZ,1,1) COVAL(FDFW1DP,W1,FLOWIN,GRND1) PATCH(MHDFOR,VOLUME,1,NX,1,NY,1,NZ,1,1) COVAL(MHDFOR,W1,HA*HA/REY,KVOLT) GROUP 15. Termination of sweeps LSWEEP=5;LITHYD=8 GROUP 16. Termination of iterations RESREF(W1)=1.E-12*WIN*YIN GROUP 17. Under-relaxation devices REAL(DTF);DTF=10.0*ZWLAST/WIN RELAX(W1,FALSDT,DTF) GROUP 19. Data communicated by satellite to GROUND FLOWIN=RHO1*WIN*YIN GROUP 22. Spot-value print-out IYMON=NY;TSTSWP=-1 GROUP 23. Field print-out and plot control NPLT=1;NYPRIN=1;NZPRIN=1;NTPRIN=2 GROUP 24. Dumps for restarts