Pseudo-compressibility 방법에서이상유동해석을위한 Level Set 방법의적용 Level Set Method Applied on Pseudo-compressibility Method for the Analysis of Two-phase Flow

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1 w w Áw œwz 7 «3 y, pp. 58~65, 25 9 Pseudo-compressibility 방법에서이상유동해석을위한 Level Set 방법의적용 Level Set Method Applied on Pseudo-compressibility Method for the Analysis of Two-phase Flow *Á½ *Á **Á ** Seung-Won Ihm*, Chongam Kim*, Jae-Seol Shim** and Dong-Young Lee** : Level Set w» wì w w ww. Level Set w l y w, w y ƒ wš e w. Level Set w w w w, pseudo-compressibility wì t. w ƒ w š t w š, ew w w wì w. w g t»s w w q w k w. w :, Level Set, pseudo-compressibility, š t,»s w, q Abstract : In order to analyze incompressible two-phase flow, Level Set method was applied on pseudocompressibility formulation. Level Set function is defined as a signed distance function from the phase interface, and gives the information of the each phase location and the geometric data to the flow. In this study, Level Set function transport equation was coupled with flow conservation equations, and owing to pseudo-compressibility technique we could solve the resultant vector equation iteratively. Two-phase flow analysis code was developed on general curvilinear coordinate, and numerical tests of bubble dynamics and surging wave problems demonstrate its capability successfully. Keywords : two-phase flow, Level Set method, pseudo-compressibility method, general curvilinear coordinate, bubble dynamics, surging wave.» (phase) w w œw w», œ w. w y w w, v r w w œ (cavita-tion) x, þƒ l (boiling) x (two-phase flow). e w w» w e w ƒ v w. w j w (Unverdi and Tryggvason, 992 ) ü w w w (Hirt and Nichols, 98; Osher and Sethian, 988 ) w, w x ƒ ƒ wš 3 y w z y š. ü w w t volume * w» wœœw (Corresponding author: Chongam Kim, School of Mechanical and Aerospace Engineering, Seoul National University, Seoul 5-744, Korea. chongam@ snu.ac.kr) **w w Áw œw (Coastal and Harbor Engineering Research Division, KORDI) 58

2 Pseudo-compressibility w w Level Set 59 of fluid(vof) Hirt and Nichols (98) w z š. w wš v w w, v w. wr Osher and Sethian(988) v ƒ y w w Level Set w. Level Set w l y w, w y wš, w œ. Level Set VOF w, VOF š v š, w ù š x y w (, 23). x ¾ Level Set w w t w, t projection pressure correction (Zhu and Sethian, 992; Osher and Fedkiw, 2). Level Set w w l Chorin(967) pseudo-compressibility wì t w. š t w w ƒ w w, g» s w w q w w w». u u l ùkü. t š Navier- Stokes w. ρ u u u p + µ 2 u + ρg + σκδn t» ρ, µ, ƒ, g ƒ l. t w () (2) σκδn σ t, κ( n) š, n w ùkü. δ w dirac delta w.» ƒ w w sx (hydrostatic equilibrium) k p ρ g š š ƒ w, w y r» w., p, ρ w sx k š, ρ ρ +ρ', p p + p' l (2) x. u u u t (3)» p'. 2.2 Pseudo-compressibility () (3) l w wì tš w, () w w w w»ƒ. Chorin(967) w š x(hyperbolic)» w ƒ (pseudo time) τ w. p β u ρ τ (4), (5) τ w w w (), (3) z š, ƒ w š x w w. (4) β ƒ, w y 2 w (Ok, 993). 2.3 Level Set Level Set w Φ š, w w, w ƒ. w Level Set w w (Sussman, 994). p' µ u ---g σκδn ρ ρ ρ ρ u p µ u u u ρ ---- g σκδn u τ ρ ρ ρ ρ t φ u φ t (3) (4) (5) (6) Pseudo-compressibility w» w ƒ ρ'

3 6 Á½ Á Á τ w w sww (6) x. (7) t e š, Φ y e w. w 2 2 ùkü, H ε e ùkü Heaviside w. ε e y w» w Ì, j».5. φ φ u φ τ t ρ ρ + ( ρ 2 ρ )H ε ( φ), µ µ + ( µ 2 µ )H ε ( φ) H ε ( φ) if φ < ε φ + ε sin( πφ ε) ε 2π if φ ε if φ > ε Fig. Level Set w e. w e φ š, w, w ƒ ƒ φ e ùkú. ù ƒ yw, (8) e e Ì ü w. (5) ùkù t w w n dirac delta w δ Level Set w φ l w. n( φ) φ φ (7) (8) (9) --( + cos( πφ ε) ) ε, if φ ε δ δ( φ) 2, otherwise () () Ì 2εü t w w (Brackbill et al., 992). wr (7) t ùƒ Level Set w w w» w»y t φ φ sign( φ )( φ ) τ j φ w. () () t» x. 4., 4.2 e tƒ ú, 4.3 e ƒ τƒ 5 ú () Ì ü g. 2.4 t x w w w ξ η š t w w. x-y t ξ η t y (Hoffmann, 2). ξ ξ ξ x y x η η x η y y ξ x, ξ y, η x, η y y metric š, ξ η. (4), (5) Level Set w (7) t lx ùkü. Q τ J Q E F ξ η S v + S g + S s S t p ρ( φ) u v φ uu, E -- J uu + + βu ξ x p ρ( φ) ξ y p ρ( φ) φu, (2) µ ( φ) u xx + u yy S ij Jρ( φ) v xx + v yy, S s Jρ( φ) τκ( φ)δ( φ)n x ( φ) τκ( φ)δ( φ)n y ( φ) Fig.. Numerical property in Level Set approach. S t J t u v φ

4 Pseudo-compressibility w w Level Set 6 S v w, S g w, S s t w ùkü, S t t w w 2 z w. J ξ x η y η x ξ y, U ξ x u + ξ y v, V η x u + η y v ƒƒ ξ,η const.. 3. ew 3. z w v E, F ü w, (2) (Hoffmann, 2). A B Q E F S J τ ξ η ξ η v + S g + S s S t (3)» QQ n+ -Q n š, n ƒ time level. n+ Q z w» w Q w t Yoon and Jameson (988) LU-SGS w. Jacobian w A E B F,, š, Q Q » w. ƒ w (steady) k ƒ, τ w (Hirsh, 989). 3.2 œ y (3) v E F Rogers and Kwak (99) t w. v š ù, w w Level Set w sww w. E ξ w v E w E ξ i + 2 E i 2 i+/2 v (Rogers and Kwak, 99). E i + 2 ± E i + 2 R n -- ( E i + E i + ) -- E + ( i + E 2 i + 2 ) 2 2 A ± ( Q) Q i + 2 (4) (5) Q e, e s³w. Jacobian w A. n A βξ x βξ y E Q ξ x U + ξ x u ξ x u -- J ξ y ξ x v U + ξ y v ξ x φ ξ y φ U (6) (5) A ± w š y w. A ± J --XΛ± X U U Λ U D U + D X DU ( + D) DU ( D) ξ y ξ y V ξ x D ξ y V + ξ x D ξ x ξ x V ξ y D ξ x V + ξ y D φ( ξ x + ξ y ) φ( ξ x + ξ y ) (7) Λ ƒ w A š, Λ ± š y ù. X š š l w š, D U. η w β( ξ x + ξ y ) v F w Jacobian w B w ξ η w ã. š œ y» w e 3 MUSCL w (Hirsh, 989). w e yƒ e y y w» w minmod van Leer w (limiter) w (Hirsch, 989). w r v 2 w., w (nonslip) w. 4.3 w ww, ñ (slip) w. wš, w. t» w v ƒ. 4.3 q q y j» w ƒ (sponge layer, or absorbing beach) w (Barone 23).

5 62 Á½ Á Á Q RQ ( ) ax ( )( Q Q ref ) τ + x L x ax ( ) a o L (8) x d q ¼ ƒ ¼ L w š, a 8 w. Q ref Q l». 3, x o L x x o 4. ew , t sww w g w» w»s w w.»s»s x p ƒ. 4.3 w q mw w g ƒ š t w š, w j š w w w w ƒ r. 4. Chang»s Chang et al.(996) Level Set w ƒ»sƒ w ww w w. Boussinesq ƒ w š, y w.»»s (.5,.35). š,»s (.5,.65).5.»s ƒƒ, t σ.5 t j skirt x w. Fig. 2 Chang et al.(996). ùkü š, Fig. 3 w.»s x ew y w. 4.2 Unverdi»s Unverdi et al.(992) w w ƒ ƒƒ 4 w w.»s.264,.2343 š, t σ.882 t j Fig. 4 Unverdi et al.(992) w, w. Fig. 5 w w.»s w»sƒ š f, t j w w. w Fig. 2. Bubble shape of Chang et al. at t,.,.2,.3,.4 sec. Fig. 3. Bubble shape of present approach at t.,.2,.3,.4 sec.(non-dimensional size)

6 Pseudo-compressibility w w Level Set 63 spherical cap xkƒ. Fig. 5 k w q. Fig. 4. Bubble shape (a) and indicator function (b) of Unverdi et al.(992). 4.3 w q w c*.34 m ¼ L m x U.849 m/s w q ƒ w w (Liou, 2). ew Fig. 6 w w U w. ƒ û» w w j, g t wš, wz w ƒ j w w w ƒ r» w kw. Liou(2) w w w w, w wš w ü Fig. 4 w w, w z ùkü indicator w ƒ ü û. Unverdi et al.(992)»s»s x w ww, Fig. 6. Surging wave problem. Fig. 5. Present result at every.2 sec. (non-dimensional size)

64 Á½ Á Á Fig. 9. Pressure around wave surface. Fig. 7. Computational region and grid. Fig. 8. Pressure variation due to hydrofoil. Fig.. Comparison of wave elevation.

226 kg/m kg/m 3 š, t 2 σ.882 kg/s w ( x, 992). Fig. 7 w 23 2 w š. w w. m w», Level Set w Ì ε.2m w. w Fig. 8~ r e y w ùkü. Fig. 8 w y, û ü š, z q ƒ w. Fig. 9 (» 325Pa) w.

7 64 Á½ Á Á Fig. 9. Pressure around wave surface. Fig. 7. Computational region and grid. Fig. 8. Pressure variation due to hydrofoil. Fig.. Comparison of wave elevation. VOF w (½, 22; x Á½, 23). œ» w w, Liou ƒ ƒ w. w x ƒ o α 5 NACA2 w. Froude Fr U gc, x *.567 w q ¼ λ2πfr m l 3.3 w ƒ 7.98m x m w. œ» ƒƒ kg/m kg/m 3 š, t 2 σ.882 kg/s w ( x, 992). Fig. 7 w 23 2 w š. w w. m w», Level Set w Ì ε.2m w. w Fig. 8~ r e y w ùkü. Fig. 8 w y, û ü š, z q ƒ w. Fig. 9 (» 325Pa) w. w -y w Fig.. Velocity vectors near the interface. ƒw ƒ w x z w y w. w Fig. xe Liou(2) w. Liou (2) ƒ x5 l w. w w w qx xe Liou(2) ew y w. q q ¼ x dw ew 2., l Fig. œ» y w Ì r.

8 Pseudo-compressibility w w Level Set w w Level Set w sww pseudo-com-pressibility w. wì Level Set w lx wì w t ý, w ƒ w t w. e» w ü t Jacobian w w š, š l w., t š w w g»s q w w. w œw ù sw ƒ», wz ù z sww ƒ v w q. w w ww w» y (KORDI PE292) w.. š x ½, Ÿy, ½ (22). n w q w q x w. w w Áw œwz, 4(2), 7-8. x (992). w»..,, ½ (23). w w Level Set» volume of fluid». w w œ wz w z, KSAS x, ½ (23). e w e. w w Áw œw z, 5(3), Barone, M.F. (23). Receptivity of compressible mixing layers. Ph.D. Dissertation, Stanford Univ., U.S.A. Brackbill, J.U., Kothe, D.B. and Zemach, C. (992). A continuum method for modeling surface tension. J. Comput. Phys.,, Chang, Y.C., Hou, T.Y., Merrian, B. and Osher, S. (996). A level set formulation of Eularian interface capturing methods for incompressible fluid flow. J. Comput. Phys., 24, Chorin, A.J. (967). A numerical method for solving incompressible viscous flow problems. J. Comput. Phys., 2, Hirsch, C. (989). Numerical Computation of Internal and External Flows. John Wiley & Sons, U.K. Hirt, C.W. and Nichols, B.D. (98). Volume of fluid (VOF) method for the dynamics of free boundaries. J. Comput. Phys., 39, Hoffmann, K.A. and Chiang, S.T. (2). Computational Fluid Dynamics, 4th Ed. Engineering Education System, Kansas, U.S.A. Liou, B.H. (2). Calculation of nonlinear free surface waves with a fully-implicit adaptive-grid method. Ph.D. Dissertation, Princeton Univ., U.S.A. Ok, H. (993). Development of an incompressible Navier- Stokes solver and its application to the calculation of separated flow. Ph.D. Dissertation, Univ. of Washington, U.S.A. Osher, S. and Fedkiw, R.P. (2). Level set methods: an overview and some recent results. J. Comput. Phys., 69, Osher, S. and Sethian, J.A. (988). Front propagation with curvature dependent speed: Algorithms based on Hamilton- Jacobi formulations. J. Comput. Phys., 79, Rogers, S.E. and Kwak, D. (99). Upwind differencing scheme for the time-accurate incompressible Navier-Stokes equations. AIAA J., 28(2), Sussman, M., Smereka, P. and Osher, S. (994). A level set approach for computing solutions to incompressible twophase flow. J. Comput. Phys., 4, Unverdi, S.O. and Tryggvason, G. (992). A front-tracking method for viscous, incompressible, multi-fluid flows. J. Comput. Phys.,, Yoon, S. and Jameson, A. (988). Lower-upper symmetric- Gauss-Seidel method for the Euler and Navier Stokes equations. AIAA J., 26(9), Zhu, J. and Sethian, J. (992). Projection methods coupled to level set interface techniques. J. Comput. Phys., 2, Received February 8, 25 Accepted July 26, 25

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