Web Two-phase Flow Analyses Using Interface Volume Tracking Tomoaki Kunugi Kyoto University 1) 2) 3)
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1 Web Two-phase Flow Analyses Using Interface Volume Tracking Tomoaki Kunugi Kyoto University kunugi@nucleng.kyoto-u.ac.jp 1) 2) 3) Lagrangian 4) MAC(Marker and Cell) 5) (VOF:Volume of Fluid ) Navier-Stokes SOLA-VOF 6) SOLA-VOF VOF 7) SOLA-VOF VOF VOF Donor-Acceptor
2 Web DeBar 8) KRAKEN Nichols & Hirt 9) VOF SLIC (Simple Linear Interface Calculation) Ashgriz 10) FLAIR Lafaurie 11) SURFER Youngs 12) PLIC (Piecewise Linear Interface Calculation) Kothe 3 PLIC TELLURIDE 13) Zaleski 14) PLIC Kelvin-Helmholtz Rayleigh-Taylor k Navier-Stokes PLIC :Interface Volume Tracking MARS (Multi-interface Advection and Reconstruction Solver) 15) SOLA-VOF SLIC Donor-acceptor PLIC 2 3 PLIC SLIC SLIC Real PLIC Fig.1Interface Reconstruction forsubstances Fig. 1 SLIC PLIC 109
3 Web MARS (15) VOF Donor-Acceptor ( f) δf ( f/ x) S(x) δf x δf = S( x) dx (1) x u t f S(x) S(x) C(x) S( x) = C( x) (2) C(x) S(x) MARS C(x) f C(x) 1 C ( x) = f x + (3) 2 Fig. 2 ( x, y) V (f <0) S(x) C(x) C(x) S(x) x u>0 x=1/2 [ tu S(x,t+ t)s(x,t)]/2] x 0 f 1 S( x) f ( x + x) +, 0 2 ( Q0 S( x) 1) (4) 1<f <0 Fig
4 Web y=1 x 2 y=1 x 1 f(x) y=0 x 1 y=0 x 2 0<f <1 1< f x 1 y=c(x) V y=s(x) Fluid 2 Fluid 1 y -1/2 1/2 x 0 x Fig. 2Normalized Computational CellFluid Area Fraction S(x) and Line-segment Function C(x) 15) sign = sign( f ) x1= sign 1, x2 = sign f 2 f 1 1 F Min ( 1, sign f = ), x = sign fgiven 2 F 2 1 x0 = x2 fgiven ( 1 + sign), for x1< x f x0 = sign fgiven, for x1 x x1 2 F 2 1 x0 = x2+ fgiven ( 1 sign) f 2, for x < x 1 (5) f given f (5) x 0 (4) S(x) f =0 S(x 0 +x)=f given ( ) f =± Donor-Acceptor 9) δf S(x)f (1) C(x) 111
5 Web (1) (2) (3) PLIC f (6) f f f + ( uf ) = f u (6) t (6) Filtering [(3) ] (Unsplit time integration) (Fractional Step Method) 16) (6) uf i δ (7) (8) ( m ) δ F = h f u (7) h+ 1 h n+ 1 m ( δ, + 1 δ, ) ( + 1 ) f = f F F f h u u h + 1 h h 1 h 1 h i, n 1 i, n 1 m m mi mi m i i i i (8) u i i N h = t / N Fractional Step Sub-cycling (1) f (2) f F (3)Fractional Step f 1 MARS MARS (x i, i=1,2,3) f(t,x i ) <f> M f M f n = 1. 0 (9) 112
6 Web f n n m f = f m (10) φ φ ( f mφ ) φ (11) = m Fractional step 2 Projection 17) m (6) f t mu m u ( f ) f 0 + = Navier-Stokes CFS (12) (18) F u 1 + ( u ) u= G ( P + τ F V ) (13) t ρ CSF F V = σκ f k ( ) ( ) T τ = µ u + u ρ ρ (14) V (15) σ ρ κ n 1 n κ = n-( n ) n n MARS (16) T ρcv M + u T = ( λ T M ) P( u) + Q (17) t C v λ Q MARS MARS Q Q=ρ W g L W g L g f 113
7 Web ). Fig. 3 60mm 15mm ( 55mm) ( 3.5mm) 2mm 1mm ρ water /ρ air =987 (0.1m/s) Fig. 4 t=0.4 Fig. 5 (f=0.5) Free Boundary Water 15mm 60mm Orifice Slit: 2mm Air Supply Air Plenum Bubble Column Fig. 3 Bubble Generation in a Bubble Column Fig. 4 Bubble Shapes and Vector Field at t=0.4s 114
8 Web Fig. 5Transient Behavior of Generated Bubbles through an Orifice in a Bubble Column 20). ( ) MARS Fig cm 4cm 9m/s 6cm Fig
9 Web U g,in =9 m/s Non-slip wall 0.04m Air Continuity 0.06m Water Non-slip wall 3 m Fig. 6Boundary Condition of Slug Flow Analysis in a Horizontal Duct Fig. 7Transient Behavior of Slug Flow in a Horizontal Duct. Fig Fig.9 116
10 Web Cold Wall (0 C) Adiabatic Wall Vapor (100.1 C) 27mm 7mm Water (99.9 C) Hot Wall (250 C) Fig.8Phase Change Simulation in an Enclosure Fig.9Transient Behavior of Boiling and Condensation in an Enclosure. Fig.10 15mm 4mm ( ) 10mm 5m/s Fig.11 (a)1ms (b)5ms (c)9ms (a) (b) (c) Fig.10Initial Configuration of Water-Droplet above the Water Thin Film for Simulation of Coronet Formation 117
11 Web (a) t= 1 ms (b) t= 5 ms (c) t= 9 ms Fig.11Transient Behavior of Coronet Formation 22). 18m/s Fig.12 4 MARS 118
12 Web Air Water Fig.12Snapshot of Turbulent Free Surface Flow 1) Yeung, R. W.: Numerical Methods in Free-Surface Flows, Annu. Rev. Fluid Mech., Vol. 14 (1982) pp ) Bonnerot, R., Jamet, P.: Numerical Computation of the Free Boundary for the Two-Dimensional Stefan Problem by Space-Time Finite Elements, J. Comput. Phys., Vol. 25 (1977) pp ) Ryskin, G., Leal, L. G.: Numerical Solution of Free-boundary Problems in Fluid Mechanics. Part 1. The Finite-Difference Technique, J. Fluid Mech., Vol. 148 (1984) pp ) Fritts, M. J., Boris, J. P.: The Lagrangian Solution of Transient Problems in Hydrodynamics using a Triangular Mesh, J. Comput. Phys., Vol. 31 (1979) pp ) Harlow, F. H., Welch, J. E.: Numerical Study of Large-Amplitude Free-Surface Motion, Physics of Fluids, Vol.9 (1966) pp ) C. W. Hirt, Nichols, D. B.: Volume of Fluid (VOF) Method for the Dynamics of Free Boundaries, J. Comput. Phys., Vol.39 (1981), pp ),,, :VOF,,Vol. 57,,No.539 (1991) pp ) DeBar, R.: Fundamentals of the KRAKEN code, Technical Report UCIR-760, Lawrence Livermore National Laboratory (1974) 9) Nichols, B. D., Hirt, C. W.: Methods for Calculating Multi- dimensional, Transient Free Surface Flows Past Bodies, Technical Report LA-UR , Los Alamos National Laboratory (1975) 10) Ashgriz, N., Poo, J. Y.: FLAIR: Flux Line-Segment Model for Advection and Interface Reconstruction, J. Comput. Phys., Vol.93 (1991) pp
13 Web ) Lafaurie, B., Nardone, C., Scardovelli, R., Zaleski, S. and Zanetti, G.: Modelling Merging and Fragmentation in Multiphase Flows with SURFER, J. Comput. Phys., Vol. 113 (1994) pp ) Youngs, D. L.: Numerical Methods for Fluid Dynamics, Academic Press (1982) pp ) Rider, W. J., Kothe, D. B., Reconstructing Volume Tracking, J. Comut. Phys., Vol.141, (1998) pp ) Zaleski, S., Li, J., Succi, S., Scardovelli, R. and Zanetti, G.: Direct Numerical Simulation of Flows with Interfaces, Proc. 2nd Int. Conf. Multiphase Flow 95, Kyoto, (1995) PT2. 15) :,,Vol.63,No.609 (1997) pp ), :MARS,8, (1997) pp ) Chorin, A. J., Numerical Solution of the Navier-Stokes Equations, Math. of Comput., Vol.22 (1968) pp ) Brackbill, J. U., Kothe, D. B. and Zemach, C.: A Continuum Method for Modeling Surface Tension, J. Comput. Phys., Vol.100 (1992) pp ) :, (1985), 20) Kunugi, T., Ose, Y., Banat, M.:Slug-Plug Flow Analyses of Stratified Flows in a Horizontal Duct by means of the MARS, Proc. 5 th ASME/JSME Joint Thermal Conf., CDROM AJTE (1999) 21) :MARS,34 (1997) pp ) Kunugi, T., Satake, S., Ose, Y.: Direct Numerical Simulation of Turbulent Free-Surface Flow, Turbulence and Shear Flow Phenomena-1, Begell House Inc. (1999) pp
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