Application of CLEAR-VOF method to wave and flow simulations

Transcription

1 Water Scece ad Egeerg, 212, 5(1): do:1.3882/.ss e-mal: Applcato of CLEAR-OF method to wave ad flow smulatos Yg-we SUN, Ha-gu KANG* School of Hydraulc Egeerg, Faculty of Ifrastructure Egeerg, Dala Uversty of Techology, Dala 11623, P. R. Cha Abstract: A two-dmesoal umercal model based o the Naver-Stokes equatos ad computatoal Lagraga-Eulera advecto remap-volume of flud (CLEAR-OF) method was developed to smulate wave ad flow problems. The Naver-Stokes equatos were dscretzed wth a three-step fte elemet method that has a thrd-order accuracy. I the CLEAR-OF method, the OF fucto F was calculated the Lagraga maer ad allowed the complcated free surface to be accurately captured. The propagato of regular waves ad soltary waves over a flat bottom, ad shoalg ad breakg of soltary waves o two dfferet slopes were smulated wth ths model, ad the umercal results agreed wth expermetal data ad theoretcal solutos. A bechmark test of dam-collapse flow was also smulated wth a ustructured mesh, ad the capablty of the preset model for wave ad flow smulatos wth ustructured meshes, was verfed. The results show that the model s effectve for umercal smulato of wave ad flow problems wth both structured ad ustructured meshes. Key words: water wave; water flow; umercal smulato; CLEAR-OF; three-step fte elemet method 1 Itroducto Modelg usteady free surface flows s a mportat step study of water wave problems. At preset, the volume of flud (OF) method s the most popular free surface modelg method. But the orgal OF method, whch s based o the fte dfferece method, s lmted to problems wth complcated computato domas. I order to make up for ths lmtato, ovel OF methods have bee developed by may researchers. Jeog ad Yag (1998) mproved the method for calculatg the OF fucto ad preseted a umercal model that combed the OF method wth fte elemet aalyss ad could be appled to adaptve meshes. Löher et al. (26) developed a ew OF model, whch ca be operated o adaptve ustructured meshes to smulate the teractos betwee extreme waves ad three-dmesoal structures. Yag et al. (26) smulated terfacal flow wth the OF method o ustructured tragular meshes by combg a adaptve coupled level set. Ths work was supported by the Natoal Natural Scece Foudato of Cha (Grat No ). *Correspodg author (e-mal: hgkag@dlut.edu.c) Receved Oct. 24, 211; accepted Ja. 11, 212

2 Mecger ad Žu (211) preseted a varat of the OF model, whch s based o the pecewse lear terface calculato method ad ca be used o geeral movg meshes. Ashgrz et al. (24) preseted a ovel OF method, amely the computatoal Lagraga-Eulera advecto remap-of (CLEAR-OF) method. I ths method, the volume fracto of each elemet s calculated the Lagraga maer wth geometrc tools, ad the pecewse lear recostructo method s used to recostruct the terface. Wth the help of the fte elemet method, ths method ca be easly appled ustructured meshes. I ths study, a two-dmesoal umercal model was establshed by combg the Naver-Stokes equatos wth the CLEAR-OF method. The three-step fte elemet method (Jag ad Kawahara 1993), whch has a thrd-order accuracy, was used to dscretze the Naver-Stokes equatos. Usg ths model, the propagato of regular waves ad soltary waves over a flat bottom, ad shoalg ad breakg of soltary waves o two dfferet slopes were smulated. Numercal results were compared wth expermetal data ad theoretcal solutos. To verfy the performace of the model o ustructured meshes, a bechmark test of dam-collapse flow was also carred out. 2 Goverg equatos ad dscretzato 2.1 Goverg equatos The goverg equatos are the followg cotuty equato ad two-dmesoal Naver-Stokes equatos for compressble vscous flud: u = (1) x u u 1 p u u u = ve f (2) t x ρ x x x x where u s the velocty; the subscrpts ad represet coordate drectos; s the flud desty ad equals 1 kg/m 3 ; p s the pressure; f s the ut body force ad equals m/s 2 the x drecto ad 9.8 m/s 2 the y drecto, respectvely; ν s the effectve vscosty coeffcet ad ca be descrbed as ν = ν ν, where ν s the kematc vscosty ad equals e t m 2 /s, ad ν t s the turbulet vscosty. Here, the Smagorsky subgrd-scale turbulece model s employed: 2 u u u νt ( csδ) = x x x where cs s the Smagorsky costat ragg from.1 to.2, ad s the characterstc legth of the elemet. 2.2 Three-step fte elemet method The Naver-Stokes equatos are dscretzed wth the three-step fte elemet method 1 2 e (3) 68 Yg-we SUN et al. Water Scece ad Egeerg, Mar. 212, ol. 5, No. 1, 67-78

3 whch the tme-splttg dea s employed to dscretze the tme term. As show Eq. (4), the Taylor seres expaso of velocty u as a fucto of t ca be descrbed as u Δt u Δt u 4 u = u Δ t O 2 3 ( Δt ) (4) t 2 t 6 t where deotes the umber of tme steps. By approxmatg Eq. (4) to a thrd-order accuracy, the three-step formula for velocty u ca be wrtte as 13 Δt u u = u (5) 3 t Δt u u = u (6) 2 t 12 1 u u = u Δt By applyg the three-step formula to Eq. (2), the followg dscretzed Naver-Stokes equatos ca be obtaed: 1/3 u u 1 u p u u = u ν e f Δt 3 x ρ x x x x p 13 = u ν e f Δt 2 x ρ x x x x t u u u u u u u u p 12 = u ν e f Δt x ρ x x x x It ca be see from Eq. (1) that u u (7) (8) (9) (1) 1 p 1 should be obtaed advace to calculate u. By applyg dvergece to both sdes of Eq. (1) ad troducg the compressble costrat 1 u, =, the followg Posso equato of pressure s obtaed: p 1 u u 12 u u f = u ν e ρ x x t x x x x x x x Δ x By applyg the stadard Galerk fte elemet method, the weak form of the mometum equatos ad the Posso equato are derved: 1/3 u u φ 1 φ wφ d d d = wφ 3 γu γ u w p t x Δ ρ x (12) w φ φ γ νe u u d w f d γ φ x x x 12 u u φ φ wφ d d d = wφ 2 γu γ u w p t x Δ ρ x (13) w φ φ 13 γ νe u u d w f d γ φ x x x (11) Yg-we SUN et al. Water Scece ad Egeerg, Mar. 212, ol. 5, No. 1,

4 u u φ φ w d = w u u d w p d φ φ γ γ t x Δ ρ x (14) w φ φ 12 γ 12 1/2 νe u u d w f d γ φ x x x 1 w φ 1 1 φ w φ 1 2 φ p d w u d u u d w f d x x t φ x γ γ x x ρ = Δ x (15) where w s the weght fucto, s the volume of elemets, u s the velocty at ode drecto, u γ s the velocty at ode drecto, ad φ ad φ are the bass fuctos. It should be oted that the same order bass fuctos are used for the terpolato of veloctes ad pressures. I the preset study, the quadrlateral soparametrc elemet s used, so ad rage from 1 to 4, ad the bass fuctos are as follows: φ1 = ( 1 ζ )( 1 ψ), φ2 = ( 1 ζ )( 1 ψ), φ3 = ( 1 ζ )( 1 ψ), φ4 = ( 1 ζ )( 1 ψ) (16) where ζ ad ψ represet the local coordates of the stadard elemet ad rage from 1 to 1. The explct mometum equatos are solved usg the lumped-mass matrx method, ad the Posso equato s solved usg a precodtoed b-cougate gradet method. 3 CLEAR-OF method 3.1 Calculato of OF fucto I the orgal OF method, the OF fucto F s updated every tme step by solvg ts trasport equato. Sce the CLEAR-OF method s also appled Eulera fxed meshes, ad the volume of flud advecto s calculated the Lagraga maer wth geometrc tools, t s uecessary to solve the trasport equato. The procedure for calculatg the F fucto wth the CLEAR-OF method ca be dvded to three steps: (1) Costructo of flud polygos for elemets: I ths step, the vertces of flud polygos each elemet eed to be determed, as show Fg. 1. If F =, there s o flud polygo because o flud exsts ths elemet. If < F < 1, the elemet s partally flled wth flud: the odes a, b, ad d of the elemet are sde the flud polygo, ad odes e ad g are the tersectos of the terface ad the backgroud grd. The odes a, b, e, g, ad d costtute the vertces of the flud polygo. If F = 1, the elemet s fully flled wth the flud, ad the flud polygo s detcal to the backgroud grd. The odes a, b, c, ad d of the grd are also the vertces of the flud polygo. Fg. 1 ertces of flud polygo 7 Yg-we SUN et al. Water Scece ad Egeerg, Mar. 212, ol. 5, No. 1, 67-78

5 (2) Polygo movemet calculato: Oce a polygo s detfed, ts movemet s calculated the Lagraga maer. u x ad u y are the velocty compoets of the vertces of the flud polygo at the th tme step. After a tme step Δ t, the dsplacemets ad ew locatos of the vertces ca be obtaed by Eqs. (17) ad (18): Δ x = u Δt, Δ y = u Δ t (17) x y 1 1 x = x Δ x, y = y Δ y (18) As show Fg. 2, odes a, b, c, d, ad e Fg. 2(a) ad odes a 1, b 1, c 1, d 1, ad e 1 Fg. 2 (b) are the vertces of the flud polygo at the th tme step ad the (1)th tme step, respectvely. Fg. 2 Sketch of CLEAR-OF method (3) OF fucto calculato at a ew tme step: The ew flud polygo a 1 b 1 c 1 d 1 e 1 tersects wth the backgroud grds. A porto of the flud remas sde the home elemet, ad the rest eters ts eghborg elemets. I Fg. 2, the home elemet has eght eghborg elemets. The flud-covered areas the home elemet ad other eghborg elemets are calculated, ad the ew OF fucto F ca be derved by repeatg the procedure for all the flud polygos. 3.2 Iterface recostructo The pecewse lear recostructo method (Ashgrz et al. 24) s used to recostruct the terface. For a terface elemet I wth J eghbourg elemets, the ut ormal vector m of the le terface, expressed as m = mx my, ca be obtaed by m = F F (19) where s the dfferetal operator. F for elemet I s solved by the followg lear system: AI FI = b I (2) where J 2 J J δxik δxik δy IK δxikδfik K 1 dik K 1 d = = IK K 1 d = IK AI = J J 2 δxikδyik δy, bi = J δy IK IKδFIK, K= 1 dik K= 1 dik K = 1 dik ce ce ce ce 2 2 δ x = x x, δ y = y y, d = δ x δ y, δ FIK = FK FI IK K I IK K I IK IK IK Yg-we SUN et al. Water Scece ad Egeerg, Mar. 212, ol. 5, No. 1,

6 ce ce where (, ) x y s the locato of the cetrod of each elemet. The the terface equato ca be obtaed by where D s a costat. 4 Numercal results 4.1 Regular wave smulato ( ) gxy, = mx my D= (21) x A two-dmesoal umercal wave flume model was establshed. The doma was 1 m log ad.8 m hgh. The computatoal doma ad coordate system are show Fg. 3. The org of the x-axs was fxed at the flow boudary o the left sde ad the org of the y-axs was fxed at the bottom boudary. A mesh of 8 elemets the x drecto ad 15 elemets the y drecto was used. y Fg. 3 Sketch of computatoal doma for travelg wave O the left sde, the umercal wave geerato theory (Dog ad Huag 24) was adopted to geerate secod-order Stokes waves. Wth a psto-type wave board, the horzotal velocty vt () of the wave paddle satsfes ωa 3 1 vt () = ξ ωs ( ωt) cos 2 ( 2ωt) (22) h 14sh ( kh) 2 where s the crcular frequecy; h s the stll water depth (SWL); a s the wave ampltude; k a1 1 2kh s the wave umber; ad ξ =, where 1 = 1 The the tah ( kh ) 2 sh( 2kh ). secod-order Stokes wave s geerated, ad the wave surface ca be descrbed as 2 η H H cosh ( kh ) ( x, t) cos( kx ) cosh 3 ( 2 ) 2 cos2( ) 2 ω π = t kh kx t 8 L sh kh ω (23) ( ) where H s the wave heght, ad L s the wave legth. O the rght sde, a dampg layer boudary codto (Larse ad Dacy 1983) was adopted to absorb the wave eergy gradually. I the dampg layer, the velocty compoets are dvded by μ ( x). The dampg factor ( x) μ ( x) μ was descrbed as ( ) ( 1 Lw x λ ) 1 α ( Lw x) exp 2 2 l λ = 1 ( Lw x) > λ (24) 72 Yg-we SUN et al. Water Scece ad Egeerg, Mar. 212, ol. 5, No. 1, 67-78

7 where α s equal to 1.1, L w s the legth of the computatoal doma, ad s the thckess of the dampg layer ad s set as the wave legth. Fg. 4 ad Fg. 5 show the tme hstores of the smulated regular wave cases wth a stll water depth ( h ) of.35 m, wave heghts (H) of.1 m ad.12 m, ad wave perods (T) of 1.2 s ad 1.3 s, respectvely. After the frst three or four waves, the temporal curves of the smulated waves are very stable. The phases ad ampltudes of the waves geerated by the umercal model are good agreemet wth the theoretcal solutos. The ature of the secod-order Stokes wave wth the wave shape asymmetrcal to the stll water level s well demostrated the umercal results. Fg. 4 Temporal curves of wave free surface (H =.1 m, ad T = 1.2 s) Fg. 5 Temporal curves of wave free surface (H =.12 m, ad T = 1.3 s) 4.2 Soltary wave smulato To geerate a soltary wave, the followg dsplacemet of a psto-type wave maker (Gorg ad Rachle 198) s gve: where H ξ H = K( ct ξ) Kh (25) () t tah s the tal heght of the soltary wave, c s the wave speed ad ca be calculated Yg-we SUN et al. Water Scece ad Egeerg, Mar. 212, ol. 5, No. 1,

8 by c g( h H ) 3 =, ad K 3H ( 4h) =. The velocty of a wave maker s descrbed as follows: dξ vxt (, ) = = ch 2 ( ) { 2 sech K x ct h Hsech K( x ct) } dt (26) where x s the posto of the wave maker Propagato of soltary wave The cofgurato of the flume s smlar to the prevous case show Fg. 3. The flume s 18 m log ad.5 m deep. The doma s dvded by 1 44 elemets the x drecto ad 9 elemets the y drecto. A soltary wave for H h =.2, propagatg rghtward over a water surface wth a costat stll water depth of h =.3 m was smulated. Fg. 6 shows the soltary wave profle geerated by the preset umercal model usg Boussesq s theory at t = 2.5 s. The umercal result s show to be very close to the aalytcal soluto. Fg. 7 shows the soltary wave propagato at four momets the umercal flume. It ca be see that the soltary wave profle s stable a permaet form durg Fg. 6 Comparso of soltary wave profle propagato. The atteuato of the soltary wave heght durg the propagatg process was small. Accordg to Me s perturbato theory (Me 1989), the soltary wave heght durg propagato ca be determed by ν x H = H (27) 12 2 g h h Fg. 8 shows that H h calculated by the preset model agrees wth Me s perturbato theory. Fg. 7 Propagato of soltary waves Fg. 8 Atteuato of soltary wave umercal flume heght durg propagato Shoalg ad breakg of soltary waves o slopes The shoalg ad breakg of soltary waves o dfferet slopes were computed wth the 74 Yg-we SUN et al. Water Scece ad Egeerg, Mar. 212, ol. 5, No. 1, 67-78

9 preset model. Fg. 9 shows the computatoal doma. Fg. 9 Computatoal doma for soltary wave shoalg ad breakg o slopes I the frst case, the slope was 1:15, ad H h was.3. I the secod case, the slope was 1:35 ad H h was.2. Fgs. 1 ad 11 show the comparso of the umercal results of the preset method wth those of Grll et al. (1997), whch a expermetally valdated fully olear wave model was used to smulate the shoalg ad breakg processes of soltary waves. The coordates both the horzotal ad vertcal drectos were ormalzed by the stll water depth, ad t 1, t 2, t 3, ad t 4 represet four dfferet momets. It ca be see that the shoalg ad breakg processes the preset study were qute smlar to those of Grll et al. (1997) both cases. The breakg pots also matched well. The water togues the preset study were a lttle thcker tha those Grll et al. (1997). Fg. 1 Free surface profles of soltary wave shoalg ad breakg o 1:15 slope ( H h =.3 ) Fg. 11 Free surface profles of soltary wave shoalg ad breakg o 1:35 slope ( H h =.2 ) The maxmum wave ruup (R m ) of a soltary wave o a 1:1 slope was also vestgated ad a comparso of results s show Fg. 12. It ca be see that the results of the preset study le betwee those of Syolaks (1987) ad Mat ad Se (1999). Yg-we SUN et al. Water Scece ad Egeerg, Mar. 212, ol. 5, No. 1,

10 Fg. 12 Maxmum ruup of soltary waves o 1:1 slope 4.3 Dam-collapse flow smulato Dam-collapse flow smulato, whch s a classc-bechmark test for the verfcato of a free surface model, was coducted wth a ustructured mesh to verfy the performace of the CLEAR-OF method. As show Fg. 13, a tal water colum wth both the heght (h 1 ) ad wdth (h 2 ) of.55 m s determed o the left of the flume. The computatoal doma, whch s.3 m log ad.6 m hgh, s dvded by a ustructured quadrlateral mesh wth 2172 elemets ad 2313 odes. Fg. 13 Computatoal doma wth ustructured quadrlateral mesh Fg. 14 shows the evoluto of the free surface at the stats of t =.6 s, t =.8 s, t =.1 s, ad t =.12 s. It ca be see that the terface ca be captured by the preset model wth a ustructured mesh. Fg. 14 Evoluto of free surfaces 76 Yg-we SUN et al. Water Scece ad Egeerg, Mar. 212, ol. 5, No. 1, 67-78

11 Fg. 15 ad Fg. 16 show the comparsos of umercal results ad expermetal data (Mart ad Moyce 1952) for the varato of the water surface elevato Y o the left sde wall ad the posto of water frot X alog the bottom, respectvely. The dmesoless coordates are defed as * * * t t g h 1 X = X h2 h1, Y = Y h 1 (28) The comparsos of the umercal results ad expermetal data are farly acceptable, =, ( ) demostratg the valdty of the preset model o ustructured meshes. Fg. 15 arato of free surface elevato o left sde wall 5 Coclusos Fg. 16 arato of water frot alog bottom Based o the Naver-Stokes equatos ad the CLEAR-OF method, a two-dmesoal umercal model for wave ad flow smulatos was developed ths study. The three-step fte elemet method was used to dscretze the Naver-Stokes equatos. The model was used to smulate the propagato of regular waves ad soltary waves over a flat bottom, ad shoalg ad breakg of soltary waves o dfferet slopes, ad a bechmark test of dam-collapse flow wth a ustructured mesh was also carred out. The result shows that, compared wth the orgal OF method that could be oly appled structured meshes, the CLEAR-OF method ca be appled both structured ad ustructured meshes due to ts combato wth the fte elemet method. The preset model provdes a coveet way for smulatg wave ad flow problems wth complex boudares. Refereces Ashgrz, N., Barbat, T., ad Wag, G. 24. A computatoal Lagraga-Eulera advecto remap for free surface flows. Iteratoal Joural for Numercal Methods Fluds, 44(1), [do:1.12/fld.62] Dog, C. M., ad Huag, C. J. 24. Geerato ad propagato of water waves a two-dmesoal umercal vscous wave flume. Joural of Waterway, Port, Coastal ad Ocea Egeerg, 13(3), [do:1.161/(asce)733-95x(24)13:3(143)] Gorg, D., ad Rachle, F The geerato of log waves the laboratory. Proceedgs of the 17th Iteratoal Coastal Egeerg Coferece, New York: ASCE. Grll, S. T., Svedse, I. A., ad Subramaya, R Breakg crtero ad characterstc for soltary waves o slopes. Joural of Waterway, Port, Coastal ad Ocea Egeerg, 123(3), [do:1.161/ (ASCE)733-95X(1997)123:3(12)] Jeog, J. H., ad Yag, D. Y Fte elemet aalyss of traset flud flow wth free surface usg OF Yg-we SUN et al. Water Scece ad Egeerg, Mar. 212, ol. 5, No. 1,

12 (olume of Flud) method ad adaptve grd. Iteratoal Joural for Numercal Methods Fluds, 26(1), [do:1.12/(sici) ( )26:1<1127::aid-fld644>3..co;2-q] Jag, C. B., ad Kawahara, M The aalyss of usteady compressble flows by a three-step fte elemet method. Iteratoal Joural for Numercal Methods Fluds, 16(9), [do:1.12/ fld ] Larse, J., ad Dacy, H Ope boudares short wave smulatos: A ew approach. Coastal Egeerg, 7(3), [do:1.116/ (83)922-4] Löher, R., Yag, C., ad Oñate, E. 26. O the smulato of flows wth volet free surface moto. Computer Methods Appled Mechacs ad Egeerg, 195(41-43), [do:1.116/.cma ] Mat, S., ad Se, D Computato of soltary waves durg propagato ad ruup o a slope. Ocea Egeerg, 26(11), [do:1.116/s29-818(98)6-2] Mart, J. C., ad Moyce, W. J A expermetal study of the collapse of lqud colums o a rgd horzotal plae. Phlosophcal Trasacto of the Royal Socety of Lodo, Seres A, Mathematcal ad Physcal Sceces, 224(882), Me, C. C The Appled Dyamcs of Ocea Surface Waves. Sgapore Cty: World Scetfc. Mecger, J., ad Žu, I A PLIC OF method suted for adaptve movg grds. Joural of Computatoal Physcs, 23(3), [do:1.116/.cp ] Syolaks, C. E The ruup of soltary waves. Joural of Flud Mechacs, 185, [do:1.117/ S X] Yag, X. F., James, A. J., Lowegrub, J., Zheg, X. M., ad Crst,. 26. A adaptve coupled level-set/volume-of-flud terface capturg method for ustructured tragular grds. Joural of Computatoal Physcs, 217(2), [do:1.116/.cp ] 78 Yg-we SUN et al. Water Scece ad Egeerg, Mar. 212, ol. 5, No. 1, 67-78