WAVE BREAKING USING A ROLLER APPROACH IN A HYBRID FINITE-VOLUME FINITE-DIFFERENCE BOUSSINESQ-TYPE MODEL

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1 WAVE BREAKING USING A ROLLER APPROACH IN A HYBRID FINIE-VOLUME FINIE-DIFFERENCE BOUSSINESQ-YPE MODEL Benjamn alock 1, Rccardo Brgan 1 Rosara E. Musumec 2 A new scheme mplemenng a roller approach no a hybrd fne-volume fne-dfference Boussnesq-ype model s presened. he relevan mahemacs are oulned a numercal solver s descrbed. Predcons obaned from he model are valdaed agans physcal observaons, demonsrang he capables of he scheme o replcae he complex hydrodynamc processes ha occur durng wave breakng. he benefs of boh he hybrd scheme he roller approach are dscussed. he resuls llusrae he feasbly of modellng he breakng process wh a roaonal roller mehod n a fne-volume fne-dfference scheme show he obanable accuraces. Fnally, furher ess mprovemens o he model are proposed. Keywords: Boussnesq equaons; Fne-volume fne-dfference; Wave breakng INRODUCION Boussnesq-ype equaons (BEs allow numercal models o smulae wave propagaon from nermedae dephs o shallow waer. he onse of wave breakng n shallow waer requres ha schemes dsspae energy o accuraely replcae he hydrodynamcs across hs regon. Several approaches have prevously been adoped o represen wave breakng whn numercal models. Recenly, numerous schemes have ulsed one such approach where a swch from BEs o nonlnear shallow waer equaons (NSWEs s performed a smaller dephs. Here, dspersve erms are removed when he rao of he wave hegh o he local waer deph exceeds a hreshold value. hs has commonly been coupled wh hybrd fne-volume fne-dfference (FVFD schemes, such as he one used n he presen work (onell Pe, 2009; Sh e al., Good levels of sably have been observed when usng hs approach, he need for srong numercal flerng s negaed. Indeed, usng a fler o remove nose nroduced by hgher order erms has prevously led o sgnfcan numercal dsspaon whch has masked he physcal behavour of he sysem. Moreover, smulang breakng by removng hgher order erms provdes a less physcal descrpon of he hydrodynamcs reduces he accuracy wh whch he free surface profle s replcaed. he roller approach s anoher energy dsspaon model whch allows BEs o be used hroughou he breakng regon. When dervng BEs wh a poenal flow formulaon, roaon whn he flud s no longer resolved, bu greaer compuaonal effcency s ganed. Whle hs s accepable n deeper waer where he flow s more unform along he waer column, means energy dsspaon s no longer seen where he flow s more urbulen, as s he case durng wave breakng. Defnng a urbulen regon whn he flow allows he roaonal componen o be nroduced by njecng an approprae level of vorcy. Addonally, by deermnng he roaonal componen of he flow, horzonal velocy profles can be obaned. hs breakng approach provdes a more physcal descrpon of he flow behavour has prevously been combned wh fne-dfference (FD schemes (Veeramony Svendsen, 2000; Brgan e al., 2004; Musumec e al., Alhough he roller approach has no prevously been mplemened n a FVFD scheme (Brocchn, 2013, dong so yelds several benefs. he FVFD scheme manans good sably whou he need for flerng, whch ensures he flow dynamcs reproduced by he model are physcally grounded, no domnaed by numercal effecs. he breakng erms nroduced when usng he roller approach provde energy dsspaon based on physcal characerscs. By reanng he Boussnesq erms, he wave shape hydrodynamc behavour can be beer represened. GOVERNING EQUAIONS he presen scheme uses he weakly-nonlnear formulaon wh breakng erms esmaed usng a roaonal roller approach, as oulned by Veeramony Svendsen (2000 Musumec e al. (2005, s wren n conservave form o su he fne-volume (FV solver used by he model. hs requres 1 he Unversy of Nongham, Uned Kngdom 2 he Unversy of Caana, Ialy

2 2 COASAL ENGINEERING 2014 ζ s d ω s q Fgure 1. Injecon of vorcy along he lower edge of he roller regon. he equaons o be wren n erms of he oal waer deph, d, flow rae, q, (see Fg. 1 whch provdes wh ( S= Bgh 2 W= h 3 ζ x 3 W q h h q 3 x + 2 h + F x d x ( B+ 1 3 [ ] q F= 2 ζ x x 2 q 2 d + gd2 2 0 ( M x = S, (1 h 2 2 q x 2 3 ( P x 2, (2 (3 + D s + gd h. (4 x Here, denoes me, x he horzonal dmenson, h he sll waer level, g he acceleraon due o gravy ζ= d h (5 he free surface elevaon. he breakng erms conss of M= whch represens he convecve momenum erm, P= ζ h ζ ζ z h z h ( u 2 r ū 2 r dz, (6 (u r ū r dz dz dz, (7 whch represens he pressure due o vercal acceleraon caused by roaonal flow, he vscous shear sress, D s = d 2 x (ν u 2 r ζ+ ζ x x (ν u r ζ+ ( ν u r ζ ζ 2 x x x (ν ū 2 r d. (8 he erm u denoes he velocy, u r he roaonal velocy, ū r he deph averaged roaonal velocy,ν he eddy vscosy, z he vercal dmenson. he roaonal velocy s calculaed from he vorcy,ω, accordng o u r = z h ω dz. (9

3 COASAL ENGINEERING By dvdng he oal horzonal velocy no poenal roaonal componens, such ha he deph averaged velocy can be wren as where he dspersvy s gven by u=u p + u r, (10 ū=ū p + ū r +O ( µ 2, (11 µ=kh, (12 wh k denong he wavenumber. Snce u p s consan across he waer column for erms up oo ( µ 2, can be noed ha ū p = u p +O ( µ 2. (13 hs allows he oal velocy o be deermned by u=ū+u r ū r +O ( µ 2. (14 he model deermnes he dsrbuon evoluon of he vorcy usng he vorcy ranspor equaon: ω + u ω ( 2 x + w ω z =ν ω ω (15 x 2+ 2 z 2 A perurbaon approach s used o calculae a sem-analycal soluon o hs equaon, wh ω= Here,σrepresens a new coordnae sysem descrbed by he relaon G n sn (nπσ. (16 n=1 σ= h+z d ζ s, (17 where z=0 defnes he sll waer level ζ s he roller hckness. he boundary condons mposed on he vorcy equaon are ω σ=1 =ω s (18 a he lower edge of he roller a he bed (see Fg. 1. hs requres he defnon of he roller hckness, an njeced vorcy, ω σ=0 = 0 (19 ζ s = 0.78d c dc d ψ (1 ψ exp ( ψ, (20 gd ω s =ω m (d 2dc 3 + d c (1 ψ exp ( 40ψ, (21 whch are derved from hydraulc jump observaons due o her smlary wh breakng waves (Veeramony Svendsen, Here, he dephs a he cres oe of he roller are gven by d c d respecvely, ω m = 6 defnes he srengh of he njeced vorcy he roller coordnae sysem s defned by ψ= x x x x c (22 where x c x gve he locaons of he roller cres oe respecvely. he nal vorcy s saed o be zero, such ha ω =0 = 0. (23

4 4 COASAL ENGINEERING 2014 NUMERICAL SCHEME he hybrd mehod uses an FV scheme o solve he NSWE erms gven n Eq. (1. A each mesep, he conserved varable [ ] d W = (24 W s updaed accordng o W (n+1 = W (n + G (n. (25 Here, n ndcae spaal emporal FV cell ndces respecvely. W s deermned usng a rdagonal solver s defned by W = ( j k q 1 + (1+2k q ( j + k q +1, (26 where G (n k = j = h 6 x ( h x (27 ( B+ 1 ( 2 h. (28 3 x s calculaed usng an Adams-Bashforh-Moulon predcor-correcor scheme, where s used a he predcor sage G (n G (n = ( 23E (n 16E (n 1 12 = ( 9E (n E (n 5E (n E (n 2 + E (n 2 a he correcor sages. Calculaon of hese erms requres he FV flux erm he FD source erm o be deermned hen combned o gve E (n (29 (30 = S F(n x. (31 he correcor sage s repeaed unl successve seps produce suffcenly smlar values of such ha where Ũ (n+1 s he esmae of U (n+1 U (n+1 U = [ d q ], (32 Ũ (n+1 < 10 3, (33 U (n+1 obaned followng he prevous eraon. Fne-volume flux soluon he FV par of he scheme requres he soluon of he flux erm, F = F F 1 2, (34 wh he nercell fluxes obaned usng he HLL approxmae Remann solver (Haren e al., he flux a each cell boundary s evaluaed accordng o he saes of adjacen cells: F = F L 0 s L s R F L s L F R +s L s R (U R U L s R s L s L < 0< s R F R s R 0 (35

5 COASAL ENGINEERING he wave speeds of he lef rgh cells are gven by q s L = d (d +d gd 2d d > d q d (36 gd d d respecvely he fluxes by Here, q +1 s R = d +1 (d +d +1 gd 2d +1 d > d +1 q +1 d +1 (37 gd +1 d d +1 d = 1 4g F L = F R = q q 2 d + g 2 d2 q +1 q 2 +1 d +1 + g 2 d2 +1 (38. (39 ( gd + gd q +1+ q (40 2d 2d +1 Fnally, conserved varables are reconsruced a each cell boundary accordng o U L = U H (41 U R = U H +1, (42 where H defnes he lmed slope, deermned from he free surface profle flow rae, H = [ ζ q ], (43 usng a MUSCL-VD scheme (Leer, 1984; Yamamoo e al., 1998; oro, he cell slope s calculaed as ( H =φ l H 1, H , (44 where he van-leer lmer s defned by 2ab φ l (a, b= a+b ab>0 (45 0 ab 0 he lmed nercell jumps are calculaed as H = H Here, he nercell jump s he Mnmod lmer s defned by [ φm ( H 1 2, H + 1 2, H φm ( H+ 1 2, H + 3 2, H 1 2 +φ m ( H+ 3 2, H 1 2, H ]. (46 H = H +1 H (47 φ m (a, b, c=sgn (a max { 0, mn [ a,bsgn (a,csgn (a ]}. (48 Fne-dfference soluon Calculang he FD componen of he hybrd scheme s comparavely sraghforward. he source erm, S, gven n Eq. (31 nroduces he hgher order erms breakng erms gven n Eq. (4 no he model, wh dervaves calculaed o second-order accuracy.

6 6 COASAL ENGINEERING 2014 Wavemaker x=0 h 0 Fgure 2. Expermenal seup for model valdaon ess. l MODEL VALIDAION he capables of he model are assessed by repeang ess prevously performed n a laboraory, allowng he numercal resuls o be compared wh he expermenal daa colleced n each case. he ess conduced here are seleced n order o evaluae he breakng performance of he model n parcular. As such, he expermens performed by Hansen Svendsen (1979 are chosen o look n deal a he wave hegh decay n he surf-zone. he quany of vorcy njeced no he model s also assessed for he ess of Cox e al. (1995, allowng he resulng velocy profles o be examned alongsde physcal resuls. he smple unform slope shown n Fg. 2 s used n all cases, wh monochromac waves generaed offshore over a horzonal bahymery. he numercal doman ncludes a 5 m sponge layer a he offshore boundary nroduces waves no he sysem usng he nernal generaon mehod descrbed by We e al. (1999. he movng shorelne s hled nrnscally by he Remann solver. Hansen Svendsen (1979 ess he daa provded by he physcal expermens of Hansen Svendsen (1979 allows he nfluence of he breakng scheme on he wave hegh o be nvesgaed. he parameers used for hese ess are gven n able 1. he sll waer deph s h=6 m he dsance from he sar of he slope o he nal shorelne s l=12.33 m. Several wave generaon condons are seleced o provde a range of ess wh boh plungng spllng breakers. able 1. Wave parameers for Hansen Svendsen (1979 ess. es case Wave frequency, f (s 1 Wave hegh, H (m Breaker ype O Spllng Q Spllng R Plungng In order o quanfy he dsspaon nroduced by he breakng scheme, he model predcons are compared o dealed expermenal wave hegh daa. Fg. 3 presens he numercal resuls alongsde he laboraory daa. he resuls obaned when swchng o NSWE o replcae breakng are also provded o demonsrae he model performance n each case. he resuls show ha, when used n conjuncon wh a hybrd FVFD scheme, he roller mehod s capable of provdng a good predcon of he wave hegh decay due o breakng. he weakly-nonlnear naure of he model reduces he level of shoalng presen, bu consderaon of hs n calbraon of he breakng parameers ensures breakng naon s correcly locaed. he resulng dsspaon of energy s close o ha seen n he physcal ess. he NSWE breakng approach produces some oscllaons due o he dscrepancy nroduced when he scheme swches, bu provdes a smlar decrease n wave hegh. For spllng breakers, he model s able o reproduce he decay n wave hegh observed n he physcal expermens. he roller approach allows a greaer amoun of energy dsspaon o be nroduced han s possble wh he ranson o NSWEs. Swchng from BEs o NSWEs provdes a smple breakng mehod, bu reles solely on he nheren properes of he equaons, does no allow any calbraon o refne he amoun of roaonal flow nroduced no he sysem. he roller approach nsead adops physcal parameers, whch nfluence he energy removed from he flud.

7 COASAL ENGINEERING H (m es O H (m H (m es Q 10 2 es R x (m Fgure 3. Comparson of wave hegh predced by model smulang breakng usng he roaonal roller approach ( by swchng o NSWE (, expermenal resuls of Hansen Svendsen (1979 (. he plungng breakng presen durng es R hghlghs he lmaons of he roller mehod. he separaon of flow seen n such cases means mulple free surfaces exs, hereby lmng he applcably of a deph averaged scheme. he bulk of he urbulence s assumed o exs whn he roller regon, where here are hgh levels of ar enranmen, bu he enre waer column s reaed as connuous. Despe hs, he dsspaon produced means he model s sll able o provde a reasonable predcon of he wave hegh n he surf-zone. Cox e al. (1995 ess he expermens performed by Cox e al. (1995 provde daa on he free surface velocy under a breakng wave. he wave perod s = 2.2 s a sll waer deph of h= m s used, wh he shorelne l=14 m from he sar of he slope (Cox e al., Snce he breakng erms are calculaed from he vorcy predced by he roller approach, s mporan o sudy he dsrbuon of hs quany across he surf-zone. he evoluon of he roller he resulng vorcy durng breakng can be seen n Fg. 4. he assumpon of rroaonaly n he absence of breakng means no vorcy s presen whou a roller. As he wave propagaes no shallower waer, he creron for breakng naon s me a roller regon s defned. he njecon of vorcy along he lower edge of he roller s clearly vsble, wh he dsrbuon defned n Eq. (21 causng he majory o be focused a he oe. he energy dsspaed by hs roaonal moon leads o he reducon n wave hegh dscussed prevously. By acqurng a predcon of he vorcy whn he flud, s possble o calculae horzonal velocy profles usng Eq. (14. Fg. 5 llusraes hs velocy a several momens durng he breakng process. he larges shoreward velocy s seen a he cres of he wave, whle a smaller seaward velocy can be found a he rough. he peak veloces are found along he lower edge of he roller are seen o reduce as he breakng connues, snce much of he energy s dsspaed durng hs process. he dsrbuon magnude of he veloces obaned by he numercal model can be valdaed by comparng he resuls wh physcal observaons. Fg. 6 ncludes he veloces recorded by Cox e al. (1995 usng a laser Doppler velocmeer (LDV gves he equvalen gauge daa provded by he presen model. Before he breakng process has naed, he velocy s found o be unform along he

8 8 COASAL ENGINEERING = = ω (s 1 10 = x (m Fgure 4. Evoluon of roller ( vorcy durng breakng = = u (ms 1 = x (m Fgure 5. Evoluon of roller ( horzonal velocy durng breakng. 0.2

9 COASAL ENGINEERING x=4.2 m 0.8 z h x=9.0 m 0.6 z h x=10.2 m 0.2 u (ms 1 z h z h x=11.4 m Fgure 6. Free surface profle ( LDV measuremens of Cox e al. (1995 (lef alongsde roller ( horzonal velocy calculaed by model (rgh. waer column, wh some resdual varaon lef n he wake of a breakng wave due o he advecon of vorcy. Snce no LDV daa s avalable n he upper regon of he wave, he veloces near he roller canno be relably assessed, bu a good agreemen s seen across he res of he flow. he model s able o accuraely reproduce boh he wave shape horzonal velocy profles. hs suggess ha he quany dsrbuon of he vorcy njeced by he breakng model s accurae, nroducng an approprae level of roaonal flow creang breakng erms ha correcly dsspae energy. CONCLUSIONS AND FURHER WORK he presen work provdes a new model ncorporang a roller approach no a hybrd FVFD model. hs brngs several benefs over alernave mehods used o smulae wave breakng usng an FVFD scheme or a roller approach. he combned effec of complex breakng Boussnesq erms presen n prevous models has demed sgnfcan levels of flerng whch has nroduced undesrable numercal dsspaon no he scheme. hs masks he physcal behavour of he model lms he hydrodynamc mpac of he breakng erms. he added sably provded by he FV solver removes he need o fler d q a each mesep, allowng a more physcally realsc breakng mechansm provdng a robus model ha s able o relably smulae long wave sequences. Prevous models ulsng an FVFD scheme have removed hgher order Boussnesq erms n order o smulae breakng, causng he wave shape o become bore-lke. Insead, he presen model reans hese erms hroughou he breakng process, hereby esablshng a more physcal descrpon of he wave allowng a more realsc surface profle o be obaned. Velocy daa provded by he model when usng a roller mehod s parcularly valuable n he surfzone. hs nformaon reveals addonal deals abou he hydrodynamc processes occurrng n hs area whch sgnfcanly mpac morphodynamc changes n nearshore regons. Deph averaged equaons, such as hose used n he presen model, reduce he complexy of a numercal scheme under he assumpon

10 10 COASAL ENGINEERING 2014 of rroaonaly. By njecng vorcy, urbulen moon can sll be represened whn he flow whle mananng a reasonable level of compuaonal effcency. he execuon me of smulaons s of parcular neres when comparng models of hs ype. Whle he calculaon of breakng erms requres some addonal compuaonal work, hs dem s mnmsed by updang hese erms only a he end of he predcor sage. Breakng naon ermnaon, he roller geomery are all deermned from he spaal properes of he wave a he curren me sep. he breakng crera for breakng wh NSWEs s esablshed from hsorc wave profles, resulng n hgher memory requremens greaer compuaonal needs. Consequenly, he roller approach acheves comparable compuaonal performance o NSWE breakng. he resuls presened here demonsrae he model o be capable of producng resuls n good agreemen wh laboraory expermens. Several areas of furher developmen are suggesed by dscrepances beween he numercal daa physcal observaons. Exendng he model o nclude hgher order Boussnesq erms would provde a fully-nonlnear scheme wh mproved shoalng capables. he addon of a boom boundary layer model could also mprove accuracy. As reanng accurae wave shapes durng breakng ensures he model s sued o wave reformng, more complex bahymeres could also be used. ess nvolvng barred beaches can resul n breakng ermnaon renaon, would herefore be an effecve way o evaluae hs aspec of he model. ACKNOWLEDGEMENS hs work was suppored by EPSRC Career Acceleraon Fellowshp EP/ /1 he Ialan Mnser of Educaon, Unversy Research projec PRIN HYDROCAR (proocol no J2Y8M 003. REFERENCES Brgan, R., R. E. Musumec, G. Bello, M. Brocchn, E. Fo (2004. Boussnesq modelng of breakng waves: Descrpon of urbulence. Journal of Geophyscal Research 109.C07015, pp Brocchn, M. (2013. A reasoned overvew on Boussnesq-ype models: he nerplay beween physcs, mahemacs numercs. Proceedngs of he Royal Socey A: Mahemacal, Physcal Engneerng Scence Cox, D.., N. Kobayash, A. Okayasu (1995. Expermenal numercal modelng of surf zone hydrodynamcs. CACR Unversy of Delaware: Cener for Appled Coasal Research. Cox, D.., N. Kobayash, A. Okayasu (1996. Boom shear sress n he surf zone. Journal of Geophyscal Research: Oceans 101.C6, pp Hansen, J. B. I. A. Svendsen (1979. Regular waves n shoalng waer: expermenal daa. Seres paper No. 21. echncal Unversy of Denmark: Insue of Hydrodynamcs Hydraulc Engneerng. Haren, A., P. D. Lax, B. van Leer (1983. On Upsream Dfferencng Godunov-ype Schemes for Hyperbolc Conservaon Laws. SIAM Revew 25.1, pp Leer, B. van (1984. On he Relaon Beween he Upwnd-Dfferencng Schemes of Godunov, Engqus- Osher Roe. SIAM Journal on Scenfc Sascal Compung 5.1, pp Musumec, R. E., I. A. Svendsen, J. Veeramony (2005. he flow n he surf zone: a fully nonlnear Boussnesq-ype of approach. Coasal Engneerng 52.7, pp Sh, F., J.. Krby, J. C. Harrs, J. D. Geman, S.. Grll (2012. A hgh-order adapve me-seppng VD solver for Boussnesq modelng of breakng waves coasal nundaon. Ocean Modellng 43-44, pp onell, M. M. Pe (2009. Hybrd fne volume - fne dfference scheme for 2DH mproved Boussnesq equaons. Coasal Engneerng , pp oro, E. F. (2009. Remann Solvers Numercal Mehods for Flud Dynamcs. A Praccal Inroducon. 3rd ed. Sprnger-Verlag. Veeramony, J. I. A. Svendsen (2000. he flow n surf-zone waves. Coasal Engneerng , pp We, G., J.. Krby, A. Snha (1999. Generaon of waves n Boussnesq models usng a source funcon mehod. Coasal Engneerng 36.4, pp Yamamoo, S., S. Kano, H. Daguj (1998. An effcen CFD approach for smulang unseady hypersonc shock-shock nerference flows. Compuers&Fluds , pp