GROUP 1. Run title and other preliminaries TEXT(KE_PIPE FLOW OF A BINGHAM FLUID: T305 DISPLAY The problem concerns the steady fully-developed turbulent flow of a Bingham-plastic non-Newtonian fluid. 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 turbulence is simulated by the high-Reynolds number k-e model plus wall functions, although the user might like to try the low-Reynolds-number k-e model because the laminar viscosity varies rapidly in the near-wall layer due to the rapid increase in the rate of strain as the wall is approached. ENDDIS 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. The flow may be characterised by Re the Bingham Reynolds number, defined by: Re = rho*win*D/n and Y the yield number, defined by: Y = D*tauy/(win*n) For sufficiently high Reynolds numbers, experimental data confirm that the friction factor f is independent of the Yield number. The data are fairly well correlated by the simplified Torrance relation: 1./sqrt(f) = 2.265*log10[0.5*Re*sqrt(f)] - 1.15 where f = 2*D*(dp/dz)/rho/win**2. The present calculations are for Re = 3.E4 and Y = 5.0 for which PHOENICS yields f = 0.0265 and the Torrance correlation yields f = 0.0239. The effect of grid size has not been investigated, and the user may like to try performing the calculation with the the low-Reynolds-number k-e model. AUTOPLOT USE file phi 5 da 1 w1 col9 1 msg Velocity (W1) profile msg Press RETURN to continue pause clear;da 1 ke col9 1 msg KE profile msg Press e to END ENDUSE REAL(FRIC,TKEIN,MIXL,EPSIN,RIN,DIN,WIN,AIN,DPDZ,YPLS1) REAL(DELT,US,REY,YIELDN,FLOWIN,KAPPA); REY=3.E4;YIELDN=5.0 RIN=0.1;DIN=2.*RIN;WIN=1.0;AIN=RIN*RIN/2.; RHO1=1.0 ENULA=DIN*WIN*RHO1/REY;ENULB=YIELDN*WIN*ENULA/DIN ** use Newtonian pipe-flow correlation to estimate f FRIC=1.1*(1./(1.82*LOG10(REY)-1.64)**2) DPDZ=FRIC*RHO1*WIN*WIN/(2.*DIN);US=WIN*(FRIC/8.)**0.5 FRIC DPDZ REY YIELDN GROUP 2. Transience; time-step specification CARTES=F GROUP 3. X-direction grid specification GROUP 4. Y-direction grid specification YPLS1=30;DELT=2.*YPLS1*ENULA/US;KAPPA=0.41 DELT=DELT/(1.-2.*SQRT(2.)*ENULB*YPLS1*KAPPA/US**1.5/WIN) NREGY=2; REGEXT(Y,RIN) IREGY=1;GRDPWR(Y,19,RIN-DELT,0.8);IREGY=2;GRDPWR(Y,1,DELT,1.0) GROUP 5. Z-direction grid specification ZWLAST=0.1 GROUP 6. Body-fitted coordinates or grid distortion GROUP 7. Variables stored, solved & named SOLVE(W1);STORE(VISL,BTAU,GENK) TURMOD(KEMODL);STORE(ENUT,LEN1) GROUP 8. Terms (in differential equations) & devices TERMS(W1,N,N,P,P,P,P);TERMS(KE,Y,N,P,P,P,P) TERMS(EP,Y,N,P,P,P,P) GROUP 9. Properties of the medium (or media) ENUL=BINGHAM;DWDY=T GROUP 10. Inter-phase-transfer processes and properties GROUP 11. Initialization of variable or porosity fields FIINIT(W1)=WIN;FRIC=0.001;TKEIN=0.25*WIN*WIN*FRIC MIXL=0.09*0.5*DIN;EPSIN=TKEIN**1.5/MIXL*0.1643 GROUP 12. Convection and diffusion adjustments GROUP 13. Boundary conditions and special sources PATCH(WALLN,NWALL,1,NX,NY,NY,1,NZ,1,1) COVAL(WALLN,W1,LOGLAW,0.0);COVAL(WALLN,KE,LOGLAW,LOGLAW) COVAL(WALLN,EP,LOGLAW,LOGLAW) ** activate fully-developed-flow solver 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 14. Downstream pressure for PARAB=.TRUE. GROUP 15. Termination of sweeps LSWEEP=10;LITHYD=50 GROUP 16. Termination of iterations RESREF(W1)=1.E-12*DPDZ*ZWLAST*AIN RESREF(KE)=RESREF(W1)*TKEIN; RESREF(EP)=RESREF(W1)*EPSIN GROUP 17. Under-relaxation devices REAL(DTF);DTF=ZWLAST/WIN; RELAX(W1,FALSDT,DTF) RELAX(KE,FALSDT,DTF); RELAX(EP,FALSDT,DTF) WALPRN=T GROUP 18. Limits on variables or increments to them GROUP 19. Data communicated by satellite to GROUND GROUP 20. Preliminary print-out 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=2;NYPRIN=1;NZPRIN=1 GROUP 24. Dumps for restarts