Development of CISPH Method for Accurate Water-Surface Tracking in Plunging Breaker

Abstract

A Corrected Incompressible SPH (CISPH) method is proposed for accurate tracking of water surface in breaking wave and resultant splash-up. Corrective terms are derived based on a variational approach to ensure the angular momentum preservation of Incompressible SPH (ISPH) formulations. The proposed CISPH model is applied to simulate the breaking and post-breaking of solitary waves on a plane slope. The high precision of the CISPH model is confirmed through comparisons with experimental data of plunging breaker. The introduction of corrective terms significantly improves the capability and the accuracy of the ISPH model in the simulation of wave breaking and resultant splash-up. 1.‚Í ‚ ¶ ‚ß ‚É SPH-@(Lucy,1977)‚Í,‰F 'ˆ •¨-• Šw ‚Ì •ª-ì ‚Åˆ³ •k •«-¬ 'Ì ‚̉ð •Í ‚ð-Ú"I ‚AE ‚µ‚Ä ŠJ "-‚³ ‚ê ‚½ ‚ª,Žãˆ³Žãˆ³ •k •« ‚ðˆµ ‚¤WCSPH (Weakly Compressible SPH)-@(Monaghan, 1994)‚Ö ‚AE‰ü-Ç ‚³‚ê,•…-Ê "g ‚Ì ‰ð •Í ‚É ‚à"K-p ‚³‚ê ‚é ‚ae ‚¤ ‚É ‚È ‚Á ‚½.Monaghan•EKos (1999),Dalrymple•ERo-gers(2005)‚Í,WCSPH-@ ‚ð•…-Ê "g ‚ÌŽÎ-Ê 'k •ã ‰ß 'ö ‚É "K-p ‚µ‚Ä ‚¢ ‚é.‚³ ‚ç ‚É,SPH‚AE •À ‚Ô-± Žq-@ ‚Ì 'ã •\ Ši ‚Å ‚ ‚éMPS-@(Koshizuka•EOka, 1996)‚Ì "ñ ˆ³ •k •«-¬ 'Ì-p ‚Ì ƒA ƒ‹ ƒS ƒŠƒY ƒ€ ‚ðŽae ‚è"ü ‚ê ‚½"ñ ˆ³ •k •«-¬ 'Ì ‚Ì ‰ð •Í-@ ‚Å ‚ ‚éISPH(Incompressible SPH)-@ ‚ªShao•ELo (2003)‚É ‚ae ‚è'ñ ˆÄ ‚³‚ê,-{ Ši "I ‚È•…-Ê "g ‰ð •Í ‚Ö ‚Ì "K-p ‚ª ‰Â "\ ‚AE ‚È ‚Á ‚½(-á ‚¦ ‚Î,Shao•EGotoh, 2004). ISPH-@ ‚Í•Ó "g •E'k •ã ‰ß 'ö ‚É ŠÖ ‚µ‚ÄMPS-@ ‚AE"¯ 'ö "x ‚Ì •Ä OE» •« ‚ð-L ‚· ‚é Žè-@ ‚Å ‚ ‚é ‚ª,Šª ‚« "g OE^ •Ó "g ‚É ‚¨‚¯‚¨‚¯ ‚é splash-up‚Ì •Ä OE» •« ‚É ŠÖ ‚µ‚Ä •s •[ •ª ‚È "_ ‚ª Žc ‚³ ‚ê ‚Ä ‚¢ ‚é. CIP-@ ‚ÉŠî ‚Ã ‚-splash-up‚Ì OEv ŽZ(-á ‚¦ ‚Î,"n •" •E•² "OE, 1996)‚Å ‚Í,plunging point‚© ‚ç•Ä 'µ-ô ‚· ‚é'å ‹È-¦ •E"-'w ‚Ì•…-¬ ‚ª-¾-Ä ‚ÉOEv ŽZ ‚³‚ê ‚é‚ª,-± Žq-@(-á ‚¦ ‚Î,Gotoh•E Sakai, 1999)‚Å ‚Í,plunging point‚© ‚ç•Ä 'µ-ô OEã ‚É•… ‰ò •ª-ô ‚ª Šˆ"Šˆ"-‰» ‚µ‚Ä 'å ‹È-¦ •E"-'w ‚Ì•…-¬ ‚ªˆÛ‚ªˆÛ Ž• ‚³‚ê ‚È ‚¢.-{ •e ‚Å ‚Í,‚± ‚Ì ‚ae ‚¤ ‚ÈCIP-@ "™ ‚Ì •‚ •¸ "xEulerOE^ ƒ‚ ƒf ƒ‹ ‚AE"ä Šr ‚µ‚½ •ê • ‡ ‚Ì-± Žq-@ ‚Ì Žã "_ ‚É ‚ ‚¢ ‚Ä,(i)-¬ 'Ì ‰ž-Í ‚Ì "ñ "™ •û •« ‚Ì ‹L •q ‚AE(ii)-£ ŽU ‰» ‚É "º ‚¤‰^ "®-Ê ‚Ì •Û ' ¶ •« ‚Ì2‚ ‚Ì"_ ‚© ‚çOEŸ "¢ ‚µ,-± Žq ŠÔ'Š OEÝ •ì-p ƒ‚ ƒf ƒ‹ ‚É "à •ï ‚³ ‚ê ‚é-â 'è "_ ‚ð ‚Å ‚« ‚é OEÀ ‚èŠÈ•Ö ‚É•C •³ ‚· ‚é ‚½ ‚ß ‚Ì ƒ‚ ƒf ƒ‹ ‚Ì ‰ü-Ç,‚· ‚È ‚í ‚¿CISPH (Corrected Incompressi-ble SPH)-@ ‚ð'ñ ˆÄ ‚· ‚é. 2.ISPH-@ (1)ISPH-@ ‚Ì Šî 'b Ž®-± Žq-@ ‚Å ‚Í,Navier-StokesŽ® (1) (ƒÏ:-¬ 'Ì ‚Ì-§ "x,u:-¬ '¬ ƒx ƒN ƒgƒ‹,P:ˆ³-Í,T:"S •« ‰ž-Í ƒe ƒ" ƒ\ ƒ‹,g:•d-Í ‰Á '¬ "x)‚ð Šî 'b Ž® ‚AE‚· ‚é.•W •€ "I ‚ÈSPH-@(WCSPH-@)‚Å ‚Í,-z ‰ð-@ ‚É ‚ae ‚Á ‚ĈÀ‚ĈÀ 'è ‚µ‚½OEv ŽZ ‚ðŽÀ •s ‚· ‚é ‚½ ‚ß ‚É•l •H "S •« ‚ð-p ‚¢ ‚½ "™ •û "I"S •« •€ ‚ª "± "ü ‚³ ‚ê ‚é.‚± ‚ê ‚É 'Î ‚µ‚Ä,ISPH-@(Shao•ELo, 2003)‚Å ‚Í,˜c ‚Ý'¬ "x ƒe ƒ" ƒ\ ƒ‹ ‚ð OEv ŽZ ‚µ‚Ä"S •« ‚ð•] ‰¿ ‚· ‚é ‚Ì ‚Å,-¬ 'Ì ‰ž-Í ‚Ì "ñ "™ •û •« ‚ª •l-¶ ‚³ ‚ê ‚é.‚µ ‚© ‚µ, ISPH-@ ‚àplungingpoint‚© ‚ç•Ä 'µ-ô ‚· ‚é 'å ‹È-¦ •E"-'w ‚Ì •…-¬ ‚ð •[ •ª ‚É•Ä OE» ‚Å ‚«‚È ‚¢ ‚± ‚AE‚ª 'm ‚ç‚ê ‚Ä ‚¢ ‚é ‚Ì ‚Å (Shao,2006),‚à ‚¤ˆê‚¤ˆê •û ‚Ì-â 'è ‚· ‚È ‚í ‚¿-£ ŽU ‰» ‚É "º ‚¤ ‰^ "®-Ê ‚Ì •Û ' ¶ •« ‚É ŠÖ ‚µ ‚Ä,ISPH-@ ‚ð 'Î •Û ‚É •l Ž@ ‚ð •i ‚ß ‚é ‚± ‚AE ‚AE‚· ‚é.