PHOTON USE AUTOPLOT file phi 5 cl msg Bingham-fluid pipe flow; Re = 10; Yield No = 2.5 msg msg W1 profile: Blue line from PHOENICS; crosses from analysis da 1 w1;da 1 w1a col3 1;blb4 2 pause END_USE DISPLAY GROUP 1. Run title and other preliminaries TEXT(Bingham-Fluid FD Lam Pipe Flow TITLE The problem concerns the steady fully-developed laminar flow of a Bingham-plastic non-Newtonian fluid in a pipe. This type of fluid remains rigid when the shearing stress is less than the yield stress tauy and flows somewhat like a Newtonian fluid when the shearing stress exceeds tauy. The apparent viscosity of such a fluid is given by the following two-paramter formula: enul = [n + tauy/(dw/dy)]/rho for tau > tauy enul = infinity for tau << tauy where n is the rigidity coefficient, and tau is the shear stress. Examples of fluids which behave as, or nearly as, Bingham plastics include water suspensions of clay, sewage sludge, some emulsions and thickened hydrocarbon greases. mesg(press RETURN to continue readvdu(nphi,int,nphi) For fully-developed flow the approximate analytical solution for the pressure drop is given by: dp/dz = 32.*rho*win**2*[1+Y/6]/Re/D where D is the pipe diameter, win the mean velocity, Re the Bingham Reynolds number, defined by: Re = rho*win*D/n and Y the yield number, defined by: Y = D*tauy/(win*n) The approximate analytical solution for the velocity profile is coded below in PIL and provided on the RESULT and PHI/PHIDA for comparison with the PHOENICS solution. mesg(press RETURN to continue readvdu(nphi,int,nphi) The problem is solved by use of the single-slab solver. ** GXPRPS=T activates ENUL coding via the file GXPRPS for which enddis BOOLEAN(GXPRPS);GXPRPS=F REAL(RIN,DIN,WIN,AIN,DPDZ);RIN=0.1;DIN=2.*RIN WIN=1.0;AIN=RIN*RIN/2. GROUP 2. Transience; time-step specification CARTES=F GROUP 4. Y-direction grid specification NY=20;GRDPWR(Y,NY,RIN,1.0) GROUP 5. Z-direction grid specification ZWLAST=0.1 GROUP 7. Variables stored, solved & named SOLVE(W1);STORE(W1A,VISL,BTAU,GEN1) GROUP 8. Terms (in differential equations) & devices TERMS(W1,N,N,P,P,P,P) GROUP 9. Properties of the medium (or media) RHO1=1.0;ENUT=0. REAL(REY,YIELDN,FLOWIN);REY=10.;YIELDN=2.5 ENULA=DIN*WIN/REY ENULB=YIELDN*WIN*ENULA/DIN ENUL=BINGHAM;DWDY=T DPDZ=32.*RHO1*WIN**2*(1.+YIELDN/6.)/REY/DIN DPDZ REY YIELDN GROUP 11. Initialization of variable or porosity fields FIINIT(W1)=WIN ** compute analytical solutions REAL(WA,GR,GRAT2,ACON,GRP);INTEGER(JJM1) ACON=YIELDN*WIN/DIN ACON GRP=2.*YIELDN*WIN**2*RHO1/REY/DPDZ GRP 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 IF(GR.LE.GRP) THEN + GR=GRP ENDIF +GRAT2=(GR/DIN)**2 +WA=8.*WIN*(1.+YIELDN/6.)*(0.25-GRAT2)-ACON*(RIN-GR) +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) FDFSOL=T;USOURC=T FLOWIN=RHO1*WIN*AIN PATCH(FDFW1DP,VOLUME,1,NX,1,NY,1,NZ,1,1) COVAL(FDFW1DP,W1,FLOWIN,GRND1) GROUP 15. Termination of sweeps LSWEEP=10;LITHYD=30 GROUP 16. Termination of iterations RESREF(W1)=1.E-12*DPDZ*ZWLAST*AIN GROUP 17. Under-relaxation devices REAL(DTF);DTF=100.*(YVLAST/NY)**2/ENULA RELAX(W1,FALSDT,DTF) GROUP 19. Data communicated by satellite to GROUND FLOWIN=RHO1*WIN*AIN GROUP 21. Print-out of variables OUTPUT(VISL,Y,N,N,Y,Y,Y) GROUP 22. Spot-value print-out IYMON=NY;TSTSWP=-1 GROUP 23. Field print-out and plot control NPLT=1;NYPRIN=1;NZPRIN=1 GROUP 24. Dumps for restarts ENULA ENULB IF(GXPRPS) THEN STORE(PRPS);FIINIT(PRPS)=33 ** mat no. rho enul cp kond expan ** 1 air CSG10=Q1 MATFLG=T;NMAT=1 33 1. GRND5 1. 1. 0 0.02 0.25 ENDIF DISTIL=T