Multi-scale simulation with a hybrid Boussinesq-RANS hydrodynamic model

Transcription

1 INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS Int. J. Numer. Meth. Fluds (2009) Publshed onlne n Wley InterScence ( Mult-scale smulaton wth a hybrd Boussnesq-RANS hydrodynamc model K. I. Stanggang and P. J. Lynett, Coastal and Ocean Engneerng Dvson, Zachry Department of Cvl Engneerng, 3136 Texas A&M Unversty, College Staton, TX , U.S.A. SUMMARY A hybrd wave model s developed for smulaton of water wave propagaton from deep water to shorelne. The consttuent wave models are the rrotatonal, 1-D horzontal Boussnesq and 2-D vertcal Reynoldsaveraged Naver Stokes (RANS). The models are two-way coupled, and the nterface s placed at a locaton where turbulence s relatvely small. Boundary condtons on the nterfacng sde of each model are provded by ts counterpart model through data exchange. Pror to the exchange, a data transformaton step s carred out due to the dfferences n physcal varables and approxmatons employed n both models. The hybrd model s tested for both accuracy and speedup performance. Tests consstng of dealzed soltary and standng wave motons and wave overtoppng of nearshore structures show that: (1) the smulaton results of the current hybrd model compare well wth the dealzed data, expermental data, and pure RANS model results and (2) the hybrd model saves computatonal tme by a factor proportonal to the reducton n the sze of the RANS model doman. Fnally, a large-scale tsunam smulaton s provded for a numercal setup that s practcally unapproachable usng RANS model alone; not only does the hybrd model offer more rapd smulaton of relatvely small-scale problems, t provdes an opportunty to examne very large total domans wth the fne resoluton typcal of RANS smulatons. Copyrght q 2009 John Wley & Sons, Ltd. Receved 2 August 2008; Revsed 24 February 2009; Accepted 26 February 2009 KEY WORDS: Boussnesq model; breakng wave; coupled wave model; hybrd model; RANS model; soltary wave 1. INTRODUCTION Most of the currently avalable ocean wave models are developed based on a sngle set of governng equatons appled on the entre computatonal doman. Whle model tests have shown agreement Correspondence to: P. J. Lynett, Coastal and Ocean Engneerng Dvson, Zachry Department of Cvl Engneerng, 3136 Texas A&M Unversty, College Staton, TX , U.S.A. E-mal: plynett@cvl.tamu.edu Copyrght q 2009 John Wley & Sons, Ltd.

2 K. I. SITANGGANG AND P. J. LYNETT wth varous target scenaros, the applcablty of such models for more general purpose applcatons such as the smulaton of offshore-to-shorelne wave propagaton, whch ncludes both the nonbreakng and breakng wave processes, s stll lmted due to the physcal assumptons n the models. For nstance, depth-ntegrated equatons usng potental flow assumptons are one of the more commonly employed equatons/assumptons n exstng wave models. The mplementaton of such models on a certan doman s vald only f the flow regme s far from the hgh-ntensty turbulence area such as the nearshore breakng zone. Although the applcaton of such models wth the help of the ad hoc turbulence [1, 2] may be pushed further nearshore to nclude the wave-breakng processes, cauton should be taken for the mplementaton of a more general and complex 3-D nearshore bathymetry. Another set of equatons that are currently wdely used for wave modelng s the turbulence-closed Reynolds-averaged Naver Stokes (RANS) equatons. The models developed based on these equatons are well suted for breakng wave and wave structure smulatons [3, 4]. The mplementaton of ths model, however, s usually confned to the nearshore zone where a relatvely large number of grd ponts and fne mesh are needed to accurately capture the turbulence. Ths ncurs expensve computatonal effort. Hence, extendng the mplementaton of the model to a larger doman s often not practcally feasble. Ths paper descrbes the smultaneous use of two wave models that belong to the frst and second aforementoned wave models for wave smulaton n the ocean. The two models are the Boussnesq-equaton type and the RANS-equaton-based wave models. In ths study, the scope s lmted to couplng the 1-D Boussnesq and 2-D RANS wave models. To allow for applcaton on a wder range of wave nonlnearty, the fully nonlnear Boussnesq-equaton model [5, 6] s employed. For the second model, we use a RANS-based model wth a two-equaton turbulence closure scheme [3]. The two models are two-way coupled, and so act as f they are a sngle model workng on a contnuous doman. In the couplng mplementaton, the Boussnesq model s appled n the non-breakng zone and the RANS model n the breakng/hgh-turbulence zone. The two models share a common doman nterface for exchangng data, used as boundary condtons n the models. By couplng the two models, accurate large-scale wave smulaton usng a coarse grd and smple physcs n the deep-to-ntermedate water and fne grd and detaled physcs n the nearshore area s computatonally feasble. In summary, the coupled hybrd model brdges the two wdely used wave models. The rest of the paper s organzed as follows. Frst, we brefly explan the two consttuent wave models along wth ther numercal soluton procedures. Second, the couplng method and the data exchange for boundary condtons are explaned n detal. Thrd, we test the hybrd model aganst several scenaros for valdaton and relatve speedup. Fnally, a large-scale tsunam smulaton s provded for a numercal setup that s practcally unapproachable usng RANS model alone; demonstratng the potental of the hybrd model to effcently tackle mult-scale problems. 2. BOUSSINESQ-EQUATION WAVE MODEL 2.1. Governng equaton Consder wave propagaton from offshore to the shorelne as depcted n Fgure 1. Dependng on the wave condton, ths sea area s unequally dvded nto nonbreakng (offshore) and breakng (nearshore) zones. Turbulence s assumed to be small nsde the nonbreakng zone and s therefore neglected. Usng ths assumpton along wth the ncompressblty of water, the flow nsde the

3 MULTI-SCALE SIMULATION Fgure 1. Wave propagaton from offshore to shorelne (adapted from [7]). In the hybrd model, the 1-D Boussnesq equaton s employed to model the 2-D nonbreakng wave moton and the 2-D RANS equaton to model the 2-D wave-breakng moton. The boundary condton n each model s provded through data exchange on the nterface area. nonbreakng zone can be modeled usng the contnuty and Euler equatons: μ 2 u x +w z =0 (1) μ 2 u t +εμ 2 uu x +εwu z +μ 2 p x =0 (2) εw t +ε 2 uw x + ε2 μ 2 ww z +εp z +1=0 (3) The above equatons are dmensonless where the coordnate system, tme, velocty, and pressure are nondmensonalzed as x = x l, z = z h 0, t = gh0 l t, u = h 0 a 0 gh0 u, w= h2 0 a 0 l gh 0 w, p = p ρga 0 All varables wth subscrpt are the dmensonal forms. The varable g s gravty, h 0 s the characterstc depth, l and a 0 are wave length and typcal wave ampltude. Related wth the last three varables are the nonlnearty and frequency dsperson parameters, defned as ε=a 0 /h 0 and μ=h 0 /l, respectvely. For a sngle-valued free surface problem, (1) (3) can be vertcally ntegrated wth respect to z over the entre depth, η(x,t) h(x) (...)dz, to reduce the model from 2-D to a one horzontal dmenson

4 K. I. SITANGGANG AND P. J. LYNETT (1HD) problem. Ths requres the mposton of the dynamc, p =0, and knematc, w =η t +uη x, free surface boundary condtons at z =η, and zero normal velocty, w = uh x, at the bottom, z = h. From here, certan assumptons are made to derve the Boussnesq equatons. Frst, t s assumed that the frequency dsperson s small, O(μ 2 ) 1. The wave nonlnearty, however, s not assumed to be small. Second, to perform the ntegraton the horzontal velocty s expressed n terms of the velocty at some reference level, z α,asusedn[8], va a Taylor seres expanson u(x, z,t)=u(x, z α,t)+(z z α )u z (x, z α,t)+ (z z α) 2 u zz (x, z α,t)+ (4) 2 Note that whle the assumptons made here permt the creaton of an extremely effcent wave model, they govern the lmts of applcaton of the model, and thus play an mportant role n the development of the coupled system. Retanng hgher-order terms n (4), as done n [6], and performng the ntegraton (wth some algebrac manpulatons) gves, n dmensonal form, the 1HD contnuty and momentum equaton u α t { [( 1 H H + (Hu α) t x x ( ) 1 T + 2 (η h) z α x + 1 u 2 α 2 x + x η +g x + { 1 t 2 z2 α { η t (T +ηs)+(z α η) ) S 6 (η2 ηh +h 2 ) 1 2 z2 α x ]} =0 (5) S x +z T α x x ( T u α x ) (z2 α η2 ) ( )} 1 2 η2 S +ηt ( ) S u α x (T +ηs)2 }=0 (6) where u α s the velocty at the reference level, H =h +η, S = u α / x,andt = (hu α )/ x + h/ t. The use of z α n the ntegraton has resulted n a frequency dsperson term, the thrd term n (5), whch would not be produced f we had to use the depth-averaged velocty as the velocty varable [9]. Ths varable s set to z α = 0.531h, whereh s the depth, to obtan the best ft between the lnear dsperson of the model and the exact dsperson for a wde range of water depths [8]. The lnear-dsperson accuracy lmt of the above equaton s near kh=3.0, where k s the wave number and the nonlnearty s accurate up to kh=1.0 [5]. As wll be explaned later, the Boussnesq model passes the vertcal profles of the veloctes, u(z) and w(z), to the RANS model n the data exchange between the two models. Although the 2-D problem has been converted to the 1HD, ths profles are obtanable from the orgnal 2-D equatons. The former can be obtaned from the z-ntegraton of the rrotatonalty condton, w x u z =0: u =u α 1 2 (z2 z 2 α )u αxx (z z α )(hu α ) xx (7)

5 MULTI-SCALE SIMULATION whle the latter s found from (1): In both dervatons, hgher-order term, O(μ 4 ), s truncated. w = zu αx (hu α ) x (8) 2.2. Numercal soluton Equatons (5) and (6) are coupled, frst order n tme dfferental equatons for the free surface elevaton, η, and reference velocty, u α. A few methods are avalable to ntegrate the equatons n tme, such as the Runge Kutta, the Rchardson extrapolaton, and the predctor corrector methods. Owng to the complexty of the equatons, we choose the last method for solvng the equatons. One varant of ths method s the hgher-order Adams Bashforth Moulton predctor corrector [10], whch has good stablty propertes. For comparson, we also have mplemented a Runge Kutta method, whch demonstrates nstablty for most of the cases tested. We conjecture that the Adams Bashforth Moulton predctor corrector s the better choce for solvng (5) and (6), and s also the approach most commonly found n the lterature for solvng these equatons (e.g. [6]). Followng terms arrangement used n [5] for better numercal stablty recasts (5) and (6) as where E = h t [(η+h)u α ] x η t = E(u α,η,h) (9) U t = F(u α,η,h) (10) +{(η+h)[( 1 6 (η2 ηh +h 2 ) 1 2 z2 α )S x +( 1 2 (η h) z α)t x ]} x (11) F = 1 2 (u2 α ) x gη x z α h xtt z αt h xt +(ηh tt ) x [E(ηS +T )] x [ 1 2 (z2 α η2 )u α S x ] x [(z α η)u α T x ] x 1 2 [(T +ηs)2 ] x (12) U =u α (z2 α η2 )u αxx +(z α η)(hu α ) xx η x [ηu αx +(hu α ) x ] (13) In the predctor step, the explct thrd-order Adams Bashforth method s used to solve (5) and (6): η n+1 =η n Δt(23En 16En 1 +5E n 2 ) (14) U n+1 =U n Δt(23Fn 16F n 1 +5F n 2 ) (15) where superscrpts ndcate the tme level and subscrpts represent the grd or spatal locaton. The mplct fourth-order Adams Moulton method s used n the corrector step: η n+1 U n+1 =η n Δt(9En+1 =U n Δt(9Fn+1 +19F n +19E n 5En 1 + E n 2 ) (16) 5F n 1 + F n 2 ) (17)

6 K. I. SITANGGANG AND P. J. LYNETT Algorthm 1. { Algorthm to numercally } solve the Boussnesq equatons. 1. Calculate η n+1 =1,..., N from (14). { } 2. Calculate U n+1 =1,..., N from (15). { } 3. Solve the trdagonal system of equatons (13) for u n+1 α =1,..., N. { } 4. Wth the new η n+1 and u n+1 α, calculate E n+1 =1,..., N from (11). { } 5. Calculate η n+1 =1,..., N from (16). { } 6. Wth the new η n+1 and u n+1 α, calculate F n+1 =1,..., N from (12). { } 7. Calculate U n+1 =1,..., N from (17). { } 8. Solve the trdagonal system of equatons (13) for u n+1 α =1,..., N. 9. Calculate N=1 (η n+1 ) old (η n+1 ) new r η = N=1 (η n+1 ) old and N=1 (u n+1 α ) old (u n+1 α ) new r u = N=1 (u n+1 α ) old where subscrpts old and new denote the correspondng prevous and current values of the teratve process f r η <10 4 and r u <10 4 then 11. Advance to the next tme level and repeat procedure untl end tme s reached. 12. else 13. Goto step: end f The frst-order spatal dervatves n (11) (13) are calculated usng fourth-order accurate spatal fnte dfference method, whle hgher dervatves are dfferenced to second-order accurate. Wth ths scheme, the calculaton of (14) (17) requres η and u α from fve ponts to the left and rght of pont-, creatng an 11-pont stencl. Note that both η and u α are calculated at the same locaton. Let there be N ponts n the doman, numbered 4,...,N +5 (Fgure 2), wth the extra fve ponts on the left and rght sdes of the doman actng as magnary ponts used to enforce the boundary condtons. Suppose that the free surface elevaton and velocty, {η m,um α = 4,...,N +5,m = n,n 1,n 2}, are gven and hence {E m, F m =1,...,N,m =n,n 1,n 2} can be calculated. Gven these ntal values, the Boussnesq equatons are solved usng the procedure presented n Algorthm 1. The corrector steps,.e. steps 4 to 8, requre boundary values on the left at ={ 4,...,0} and rght sde at ={N +1,...,N +5}. On the left sde the boundary values are provded usng the reflectve boundary condton combned wth the sponge layer [11] to damp out the ncomng and outgong waves. On the rght sde, the boundary values are provded by the RANS model, and wll be dscussed later n ths paper.

7 MULTI-SCALE SIMULATION Fgure 2. Boussnesq grd and RANS mesh. 3. RANS WAVE MODEL 3.1. Governng equaton The wave moton n the breakng zone (Fgure 1) should deally be modeled usng the Naver Stokes (N S) equatons. Ths drect numercal smulaton, however, s very expensve, requrng a huge number of computatonal grd ponts to resolve turbulence length scales, and s generally beyond the capablty of current computng technology for all but very small-scale studes [12]. A statstcal approach s commonly employed to perform a turbulence smulaton. Here, both the nstantaneous velocty and pressure are splt nto the mean and fluctuatng quanttes: u = u +u and p = p +p. The contnuty and momentum equatons are then tme-averaged, 1/(t 2 t 1 ) t 2 t 1 (...)dt, wheret 2 t 1 s long compared wth that of the turbulent moton, to get the tme-averaged RANS equatons that govern the mean flow and pressure: u x =0 (18) u + u j u = 1 p +g + 1 τ j u u j (19) t x j ρ x ρ x j x j where ρ s the densty and τ j s the stress due to molecular vscosty. Ths ntegraton produces the unknown correlaton between the fluctuatng veloctes, the last term of (19), whch represents the transport of momentum due to the fluctuatng moton. To close (19), a nonlnear eddy vscosty model s used; here the k ε model s employed (see [3, 13] for detals). Snce ths s a 2-D problem n the x- andz-drectons, the subscrpt wll have values 1 and 3 ndcatng both drectons, respectvely.

8 K. I. SITANGGANG AND P. J. LYNETT 3.2. Numercal soluton The 2-D RANS computatonal doman s dscretzed nto a rectangular mesh as shown n Fgure 2. Ths mesh employs the staggered grd system where the flow and pressure varables n a cell are located at dfferent locatons; the horzontal velocty, u, s located at the mdpont of the vertcal edge, the vertcal velocty, w, at the mdpont of the horzontal edge, and the pressure, p, F (volume of flud), k, andε n the center of the cell. Dfferent shades n the mesh are used to denote the physcal cells (whte) and the ghost cells (gray) added for the boundary condton mplementaton. The two-step projecton method, as mplemented n [14], s employed to solve (19), whch s tme-dscretzed as follows: ũ n+1 u n δt = u n u n j +g + τn j (20) x j x j u n+1 ũ n+1 = 1 p n+1 (21) δt ρ x For convenence, the are dropped and the sum of the molecular and turbulence stresses, the last two terms of (19), s smply wrtten as τ j. In the frst step, the projected velocty feld ũ n+1 s frst calculated wthout takng the pressure p n+1 nto account. In the second step, (21) and (18) are combned to get the pressure Posson equaton, whch can be solved for the pressure p n+1 : x ( 1 p n+1 ) ρ n = 1 ũ n+1 (22) x δt x Wth ũ and p n+1 obtaned from the prevous steps, the solenodal velocty feld can be calculated from (21) and accordngly usng the k ε equatons, the knetc energy k and the energy dsspaton ε may be calculated. The solenodal velocty calculated n the two-step projecton method s later used to determne the new volume of flud dstrbuton over the mesh va the advecton equaton: F t +u F =0 (23) x Addtonal descrpton of the volume of flud method can be found n [14]. In summary, the computaton n the RANS model s gven n Algorthm 2. Suppose that n ths algorthm ntally we have all the dependent varables (u n,w n, p n, F n, k n,andε n ) n both the physcal and ghost cells. In Algorthm 2 all the varables along the boundary and n the ghost cells reflect the type of boundary condtons mposed on the assocated sdes of the doman. Relevant to the model couplng are the boundary values on the left sde of the doman overlappng wth the rght sde of the Boussnesq doman. On ths sde the nflow/outflow boundary condton, whch requres the specfcaton of the flow velocty, pressure, knetc energy, and energy dsspaton along the correspondng boundary/ghost cells, s mposed. Whle the specfcatons of the flow and turbulence

9 MULTI-SCALE SIMULATION Algorthm 2. Algorthm to numercally solve the RANS equatons. 1. Calculate { ũ n k, wn k =2,...,max 1, k =2,...,kmax 1} from (20). { } pk n+1 =2,...,max 1, k =2,...,kmax Solve (22) for { } 3. Calculate u n+1 k,w n+1 k =2,...,max 1, k =2,...,kmax 1 from (21). 4. { Solve the k ε equatons for } k n+1 k, ε n+1 k =2,...,max 1, k = 2,...,kmax 1. { } 5. Calculate Fk n+1 =2,...,max 1, k =2,...,kmax 1 from (23). 6. Check f tme step sze, δt, satsfes the Courant (C r ) stablty condton: δt C r Δx, u =1, 3, for all cells n the mesh. If δt > C r Δx u n some cells, set t to the smallest C r Δx. u 7. Repeat the procedure untl the end of the smulaton. boundary condtons are smply the mposton of the two varables on the boundary, ths s not the case wth the pressure boundary condton. The pressure along the boundary, whch s requred n step 2 of the algorthm, s gven by the Neumann boundary condton p n= 1/δt(u n+1 ũ) n, where n s the outward normal unt vector. It can be shown that wth ths boundary condton the lnear system of equatons resultng from the centered-dfference spatal dscretzaton of (22) n step 2 does not explctly depend on p n+1 and ũ, but rather on u n+1 n along the boundary (see, for nstance, [15]). Ths smply means that the system of equatons depends mplctly on p n+1 and ũ through the Neumann boundary condton above. Therefore, solvng the pressure Posson equaton does not requre the mposton of the pressure along the boundary. 4. RATIONALE OF MODEL COUPLING Suppose that the offshore-to-shorelne wave propagaton as depcted n Fgure 1 s to be smulated usng the two models presented earler, wth the Boussnesq model beng mplemented n the nonbreakng zone and the RANS model n the breakng zone. To perform these computatons, we should provde the boundary values on the sde of each model located on the nterface area. To do ths, the two subdomans must overlap so that one model can provde the other wth boundary values. Hence, from the computatonal pont of vew, the couplng of the Boussnesq and RANS models s possble provded that the data exchange scheme can be formulated, whch wll be explaned n detal n the next sectons. In general and from the physcs pont of vew, the two models cannot exchange approprate boundary values because both are derved under dfferent assumptons. For nstance, n the Boussnesq model the flow s treated as rrotatonal and nvscd whle n the RANS model t s rotatonal and vscous. Clearly the two dfferent flow propertes cannot be exchanged. Turbulence accounted for n the RANS s absent n the Boussnesq model, and therefore ths property, whch s one of the boundary values mentoned above, s mpossble to exchange between the two models. However, n some stuatons certan propertes of the physcal problem may be so small that they can be neglected n the calculaton wthout sacrfcng accuracy. If for nstance the nterface of the two models s located n the nonbreakng zone and the wave under consderaton satsfes the

10 K. I. SITANGGANG AND P. J. LYNETT assumptons (weak dsperson and turbulence) made n the Boussnesq model, some quanttes such as turbulence and rotatonalty n the RANS model become unmportant, and can be neglected. In such a case, data can be exchanged n a physcally consstent context between the models. Therefore, due to the dfference n physcs and assumptons n the two models, the nterface area must be located n the nonbreakng zone where the flow s relatvely rrotatonal, the turbulence s small, and the wave under consderaton must be relatvely long wth respect to the water depth. 5. COUPLING TECHNIQUE The Boussnesq and RANS model algorthms have been presented n detal n the prevous sectons. To perform the calculatons n each step of the algorthms, the nterface boundary values should be provded. Couplng the two models, and hence algorthms, wll provde these values. Snce some steps requre boundary values from the prevous tme steps and others from the current one, the two algorthms should be so connected that n proceedng from one step to the next the boundary values requred n one model are always avalable from the other. In Algorthm 1, the calculatons n steps 1 to 3 requre the boundary values from the prevous tme levels, whch are gven by the RANS model at the nterface. In Algorthm 2, step 1 requres boundary values and varables gven from the Boussnesq model at the nterface from the prevous tme level. The second step of ths algorthm requres u n+1 1k along the nterface boundary. Ths velocty can be obtaned from the Boussnesq model, whch calculates the new tme level velocty n step 3. Once the second step s fnshed, RANS steps 3 to 5 can be computed snce all of the computatons n these steps are explct. The corrector steps n the Boussnesq model can also now be calculated wth the new tme level boundary values obtaned from the RANS model. To summarze, the hybrd computaton s presented n Algorthm RANS BOUNDARY CONDITION In Algorthm 3 the vertcal profles of veloctes and the volume of flud on the frst column of the RANS model mesh are obtaned from the Boussnesq model. These varables, however, are not mmedately avalable from ths model. To obtan these varables, certan transformatons must be done wth η and u α. Fgure 3 depcts the hybrd grd ponts on the nterface. Shown n the fgure are the frst column of the RANS model mesh wth 18 computatonal cells and the grd ponts N 2, N 1, N, andn +1 of the Boussnesq model. Note that grd pont N of the Boussnesq model s algned wth the rght face of the frst column n the RANS mesh. To fnd the flud dstrbuton, F, n ths column, we frst determne η n the mddle of the column at x = x N δx/2 by quadratcally nterpolatng η N 2, η N 1,andη N : η = (x x N 2 )(x x N 1 ) (x N x N 2 )(x N x N 1 ) η N + (x x N 2 )(x x N ) (x N 1 x N 2 )(x N 1 x N ) η N 1 + (x x N 1 )(x x N ) (x N 2 x N 1 )(x N 2 x N ) η N 2 In a typcal smulaton, each column of the RANS model mesh may have tens or hundreds of computatonal cells.

11 MULTI-SCALE SIMULATION Algorthm 3. Hybrd algorthm. 1. whle t n <t end do 2. Calculate {η n+1 =1,...,N} from (14). 3. Calculate {U n+1 =1,...,N} from (15). 4. Solve the trdagonal system of equatons (13) for {u n+1 α 5. Calculate {ũ n k, wn k =2,...,max 1, k =2,...,kmax 1} from (20). 6. Solve (22) for {p n+1 step Calculate {u n+1 k,w n+1 8. Solve the k ε equatons for {k n+1 =1,...,N}. k =2,...,max 1, k =2,...,kmax 1}. Ths requres u n+1 k =2,...,max 1, k =2,...,kmax 1} from (21). k, ε n+1 k =2,...,max 1, k =2,...,kmax 1}. 9. Calculate {F n+1 k =2,...,max 1, k =2,...,kmax 1} from (23). 10. Wth the new η n+1 and u n+1 α, calculate {E n+1 =1,..., N} from (11). Ths requres {η n+1, u n+1 α = N +1,...,N +5}. 11. Calculate {η n+1 =1,...,N} from (16). 12. Wth the new η n+1 and u n+1 α, calculate {F n+1 =2,...,N} from (12). F n+1 1 s calculated based on the correspondng Boussnesq elevaton. 13. Calculate {U n+1 =1,...,N} from (17). 14. Solve the trdagonal system of equatons (13) for {u n Calculate and N=1 (η n+1 ) old (η n+1 ) new r η = N=1 (η n+1 ) old N=1 (u n+1 α ) old (u n+1 α ) new r u = N=1 (u n+1 α ) old α =1,...,N}. 16. f r u <10 4 and r η <10 4 then 17. Repeat the procedure untl the end of the smulaton. 18. else 19. Goto step: end f 21. Check f tme step sze, δt, satsfes the Courant (C r ) stablty condton: δt C r Δx u, =1, 3, for all cells n the mesh. If δt > C r Δx n some cells, set t to the smallest C r Δx u. 22. f δt s changed then 23. Interpolate the new η n, η n 1, η n 2, u n α, un 1 α 24. end f 25. t n+1 =t n +δt. 26. n =n end whle u,andu n 2 α 1k from at all nodes on the Boussnesq grd.

12 K. I. SITANGGANG AND P. J. LYNETT Fgure 3. Calculaton of the flud dstrbuton n column 1 of the RANS model mesh based on the Boussnesq model free surface elevaton. Then we search for the cell n whch η s located n the column. In the exemplfed fgure ths nterpolated free surface elevaton s located at cell 14. All the cells below t are full wth F =1 and the cells above t are empty wth F =0. For the free surface cell tself, F s calculated as the rato of the heght of the nterpolated free surface elevaton, δ, n the cell to the cell heght, δz 14 : F 14 = δ (24) δz 14 Equatons (7) and (8) are employed to calculate the vertcal structures of u along the rght face and w along the center of the frst column of the RANS model mesh. Employng the second-order centered fnte dfference formula on (7) at x N gves u N (z) = u αn 1 2 (z2 z 2 α ) N u αn+1 2u αn +u αn 1 Δx 2 (z z α ) N (hu α ) N+1 2(hu α ) N +(hu α ) N 1 Δx 2 (25) A smlar scheme s employed to dscretze (8) for the vertcal velocty profle. As the reference velocty grd pont s not algned wth the center of the RANS mesh where the vertcal velocty s located, pror to the calculaton, we frst determne u α at x N+1 = x N+1 δx/2 andx N 1 = x N 1 δx/2 by quadratc nterpolaton. Then the vertcal structure of the velocty s calculated at x N = x N δx/2: w x N (z)= z u αx N +1 u αx N 1 2Δx (hu α) x N +1 (hu α ) x N 1 2Δx (26)

13 MULTI-SCALE SIMULATION The nterface of the hybrd model wll be located at the low turbulence area. In dong so, the turbulence varables requred along the frst column of the mesh are small, as wll be demonstrated later n the model test smulaton. Accordngly, transfer of turbulence varables from the Boussnesq to the RANS model, as done n the mposton of the flow boundary condton, s not necessary n the hybrd model. Wth ths very weak assumpton of turbulence at the nterface, the employed condton at ths locaton for turbulence quanttes s zero horzontal gradent. 7. BOUSSINESQ BOUNDARY CONDITION The specfcaton of the Boussnesq boundary condton s the nverse procedure descrbed n the prevous secton. In ths procedure the reference velocty and the free surface elevaton are calculated based on the velocty, u, and the flud dstrbuton, F, n the RANS model. Ths calculaton s carred out on the fve Boussnesq ghost grd ponts from N +1 ton +5. In ths secton we descrbe the calculaton of the boundary condton on the grd pont N +1. The dentcal procedure apples to the rest of the ghost grd ponts. Fgure 4, a modfcaton of Fgure 3, depcts the RANS Boussnesq model grd system on the nterface area. As shown the two models employ dfferent grd szes. Wthout loss of generalty, the Boussnesq grd s depcted here larger than the RANS model grd, whch wll be typcally the case. To determne the free surface elevaton and the reference velocty, we should frst fnd the column n the RANS model mesh that algns Fgure 4. Calculaton of the Boussnesq model free surface elevaton and reference velocty based on the RANS model velocty and flud dstrbuton.

14 K. I. SITANGGANG AND P. J. LYNETT wth grd pont N +1 of the Boussnesq model,.e. column- n Fgure 4. Snce n general the center of the RANS column does not perfectly algn wth the Boussnesq grd pont, quadratc nterpolaton based on the RANS free surface elevatons n column 1,, and +1 semployed to determne the Boussnesq free surface elevaton at grd pont N +1. Note that, n the RANS model doman, the bottom of the lowest cell s assgned an elevaton of 0. Accordngly, the RANS free surface elevaton, as measured from the stll water level, n column- of the RANS model mesh s η =d d, whered s the heght of the water n column- and d the heght of the stll water level (the same for all columns) n the RANS model mesh. The reference velocty calculaton starts wth dentfyng the cell n the column of the RANS model mesh where the reference level s located. In Fgure 4 the reference level n column-, for nstance, s located at cell 8. To determne the reference velocty to be gven to the Boussnesq, a quadratc polynomal of the form u =β+θ(z z α )+γ(z 2 z 2 α ) (27) s ftted to the three neghborng veloctes u 7, u 8,andu 9. Comparng (27) and (7), t s apparent that the reference velocty s the zeroth-order coeffcent of the polynomal u α =β. A smlar procedure s employed to determne u α 1 and u α+1. Based on these three reference veloctes, u αn+1 s calculated va nterpolaton. It s reterated here that the Boussnesq model does not requre any boundary values for vertcal velocty or pressure. In the Boussnesq model, these are explct functons of the horzontal velocty and free surface elevaton. 8. HYBRID MODEL TEST To test the model for both the valdty and relatve speedup, we use the hybrd model to smulate the followng target scenaros: (1) Soltary wave propagaton. (2) Standng wave moton. (3) Snusodal wave overtoppng of a seawall. (4) Soltary wave overtoppng of a levee. (5) Hypothetcal large-scale tsunam smulaton. In the frst scenaro, we consder the propagaton of a soltary wave along a channel of constant depth, where as an mportant property, the wave heght s constant as the wave propagates. Ths scenaro s smulated usng the hybrd model to see how well the model produces ths property. In the second test, the hybrd model s used to generate a standng wave n a channel of constant depth. The wave s drven from one end of the channel toward a vertcal, reflectng wall. Ths wave s reflected back toward the orgn and thus two waves opposte n drectons meet on the nterface to create a standng wave. In the thrd smulaton, the model s run to smulate the wave overtoppng of a seawall. The numercal data collected n ths smulaton s the mass flux of the wave overtoppng across the top of the structure. In the fourth smulaton, we conduct the smulaton of the soltary wave overtoppng of a coastal structure. The calculated free surface elevaton, before and after nteracton wth a levee, s recorded and compared wth expermental data. In the last smulaton, a hypothetcal tsunam n deep water s generated. The hybrd model s employed to predct ts evoluton on the coast, ncludng nteracton wth a breakwater.

15 MULTI-SCALE SIMULATION (a) (b) Fgure 5. (a) Hybrd smulaton setup for soltary wave propagaton and (b) soltary wave form centered at the coordnate orgn and symmetrcal about the z axs. Gray area occupes 95% of the total water volume n the wave. To measure the relatve speedup of the hybrd computaton, the frst four scenaros are also run usng the RANS model n the whole doman, a pure RANS model smulaton. The CPU tmes used by the pure RANS model to compute those scenaros are compared wth the CPU tmes used n the hybrd model. Ths comparson gves the relatve speedup of the hybrd model Soltary wave propagaton The frst test smulates the propagaton of a soltary wave n a 0.5 m deep and m long channel. The doman s dvded nto two subdomans of equal lengths. The frst 50.0 m subdoman, x B,s occuped by the Boussnesq model and the second 50.0 m, x C, by the RANS model. The wave s generated n the Boussnesq doman and propagates to the RANS model doman (Fgure 5). To observe the behavor of the model wth respect to the wave heght varatons, two waves of dfferent nonlneartes, ε= H/h, and dspersveness, δ=h/l, are consdered n the test. Whle a soltary wave s unquely defned wth only ε, the dspersve parameter s ncluded here to both provde addtonal characterzaton of the wave and for use as a length scale. The soltary wave length, L, appearng n δ, s defned as the length of the symmetrcal regon n the soltary wave (gray-colored area n part (b) of Fgure 5), whch contans 95% of the total volume of water [16]. The ntal free surface elevaton and velocty of the soltary wave that are used to drve the model have the form and η= A 1 sech 2 [B(x Ct)]+A 2 sech 4 [B(x Ct)] (28) u = A sech 2 [B(x Ct)] (29) The two equatons are the analytcal soluton to the weakly nonlnear Boussnesq equaton [8]. The dervaton of the analytcal solutons and the determnaton of the coeffcents A, A 1, A 2, B, and C may be found n [17].

16 K. I. SITANGGANG AND P. J. LYNETT In the followng two soltary wave smulatons, the flud vscosty and the turbulence n the RANS model are set to zero. Hence both models are runnng under an nvscd (potental) flow condton, wth no means n the RANS model to generate vortcty. In the frst soltary wave smulaton, a small ampltude wave of heght 0.05 m ntally centered at x =10.0m s used. The length of ths wave n the 0.5 m deep channel s 6.64 m. The correspondng nonlnearty and dspersveness are ε=0.1 andδ=0.075, respectvely. Both the Boussnesq and RANS models use unform spatal grds. The Boussnesq grd sze s Δx =0.125m and the RANS grd szes are δx =0.0625m and δz =0.0175m. In ths smulaton constant tme step, δt =0.01s, whch corresponds to the Courant number C r =0.18 n the Boussnesq model doman and C r =0.35 n the RANS model doman, s employed. Here, the characterstc velocty used n the Courant number s the constant, lnear long wave speed. To save some computatonal tme, the RANS model starts ts calculaton when the free surface elevaton of the Boussnesq model on the nterface area (whch acts as the boundary condton n RANS model) exceeds a threshold, η threshold =10 5 m. For ths partcular smulaton, there are no calculatons n the RANS model doman for the frst 13.0 s of physcal tme or about 44.0% of the total tme. In a huge computatonal doman, partcularly for transent wave studes, ths procedure can save a sgnfcant amount of computatonal tme. Ths procedure s mplemented for all the smulatons n ths study. To see the evoluton of the solton as t propagates along the channel, several snapshots of the free surface elevaton, η, at dfferent tmes are captured at 6 s nterval (Fgure 6). The dstance between two consecutve soltons s 13.9 m. The solton moves at 13.9/6.0=2.32m/s. As the wave travels from the ntal locaton to x =80.0m, the wave heght s nvarant. The wave s correctly transmtted from the Boussnesq to the RANS doman. For comparson the smulaton s also carred out usng the Boussnesq model for the full doman 0 x 100.0m. The wave profles of ths smulaton are shown n dots n Fgure 6. Both the hybrd and full-boussnesq model wave profles agree very well. Note the small, oscllatory tal followng the solton at later tmes n both the hybrd and full-boussnesq results; ths s a common occurrence when usng the weakly nonlnear soltary wave soluton n the fully nonlnear model [6]. In the second smulaton, wave heght s ncreased to 0.15m. The wavelength s 4.3 m, shorter than the prevous wave, and thus the wave steepness s consderably larger here. The nonlnearty s ε=0.3 and the dspersveness s δ=0.12. For ths smulaton the spatal and temporal grds are smlar to the prevous soltary wave smulaton. The snapshots of the solton at 6.0 s temporal nterval are presented n Fgure 7. Here, the wave moves at a constant speed of 2.51m/s, faster than the prevous, smaller wave. The t =18.0s solton ndcates that there s a smooth transton as the wave enters the RANS model doman from the Boussnesq model doman. As the wave travels n the RANS model doman, however, the wave heght decays slowly as ndcated by snapshots at t =24.0s and t =30.0s. From t =24.0s to t =30.0s, there s a 2.9% wave heght reducton. Another smulaton usng the pure RANS model and employng the same wave and temporal/spatal grds confrms ths decay (Fgure 8). Ths fgure obvously shows that the wave undergoes dampng as t propagates from ts ntal poston to t =15.0 s poston. The wave at t =9.0 s, for nstance, decays from to m at t =15 s, a 3.2% wave heght reducton. As n the prevous case, we also perform the full- Boussnesq smulaton and the result s also presented n dots n Fgure 7. The fgure shows that the wave heght of the full-boussnesq run s nvarant n the course of the smulaton. From t =18.0 to 30.0 s, the solton n the hybrd model, as a result of beng smaller, lags slghtly behnd the full-boussnesq wave, whch s apparent n the close-up snapshot at t =30.0 s. The wave heght decay s due to numercal dsspaton n the RANS model.

17 MULTI-SCALE SIMULATION Fgure 6. Soltary wave (ε=0.1, δ=0.075) propagaton n a 0.5 m deep channel smulated usng the hybrd wave model. 0 x 50 s Boussnesq model, 50 x 100 s RANS model, and dots are the full-boussnesq model. Smlar smulatons usng the RANS model on the whole doman are carred out to measure the relatve speedups of the hybrd smulatons. The temporal and spatal grds for the pure RANS smulatons are smlar to RANS model grds n the hybrd smulatons. Hence, the total number of the RANS model horzontal computatonal grd ponts n these smulatons s twce as many as n the hybrd model, whle the number of vertcal grd ponts remans unchanged. For the hybrd model, the computatonal clock tme for the 0.05 m wave s s and for the 0.15 m wave s. The same smulatons usng the RANS model take and s for the 0.05 and 0.15 m waves, respectvely. These computatonal clock tmes correspond to 30.4 s smulaton and are run n

18 K. I. SITANGGANG AND P. J. LYNETT Fgure 7. Soltary wave (ε=0.3, δ=0.12) propagaton n a 0.5 m deep channel smulated usng the hybrd wave model. 0 x 50 s Boussnesq model, 50 x 100 s RANS model, and dots are the full-boussnesq model. a 3.0 GHz Pentum processor. The speedup factors ganed by the hybrd model are 4.0 for the frst wave and 4.2 for the second. In both the hybrd smulatons the RANS-start threshold s reached at 13.4 s (44.0% of the total) smulaton tme and takes 25.0 s (8.0% of the total) computatonal

19 MULTI-SCALE SIMULATION Fgure 8. Soltary wave (ε = 0.3, δ = 0.12) propagaton n a 0.5 m deep channel smulated usng the RANS model. clock tme. Ths comparson demonstrates that the threshold procedure can save, for these partcular cases, 44.0% effort to run RANS n the hybrd model from the begnnng Standng wave moton In the followng test, the hybrd wave model s used to smulate standng wave moton. The standng wave s created by supermposng two snusodal waves of the same heght, H, movngnthe opposte drectons n a channel of constant depth, h, as shown n Fgure 9. Here, the numercal channel s 72.0 m long, 0.5 m deep, and dvded nto two subdomans of equal lengths. The left

20 K. I. SITANGGANG AND P. J. LYNETT Fgure 9. Hybrd model setup for standng wave smulaton. subdoman, x B, s occuped by the Boussnesq model and the rght one, x R, by the RANS model. Both ends of the channel are mpermeable walls and thus act as reflectng boundares. Adjacent to the left boundary, a sponge layer s nstalled to damp out all the waves that enter the sponge layer area. For wave generaton, we employ the nternal source method [11]. Identcal snusodal waves wth heght, H, propagate from the source to both the left and rght drectons. The wave that propagates to the left s damped out by the sponge layer and the one to the rght wll propagate along the Boussnesq and the RANS domans. As the wave reaches the rght boundary n the RANS doman, t s reflected and propagates back toward the wave source and supermposes wth the ncomng wave to create the standng wave. In the frst test of ths smulaton, a relatvely small wave, of 0.01 m n heght wth a perod of 4.0 s, s generated 5.0 m to the rght of the left boundary. A 3.0 m sponge layer s located adjacent to the left boundary as the dampng mechansm. From the dsperson relatonshp, ths wave has a wavelength of 8.7 m. The nonlnearty and the dspersveness of ths wave are then ε=0.02 and δ=0.06, respectvely. The Boussnesq doman s dscretzed nto unform grd, Δx =0.08m. Smlarly, the RANS doman s unformly dscretzed both horzontally, δx =0.04m, and vertcally, δz =0.001m. The smulaton s run for s wth the constant tme step Δt =0.009s and zero vscosty and turbulence. Fgure 10 depcts the nstantaneous wave profles at sx dfferent nstants. The frst three profles show the wave propagatng to the rght and reachng the wall at t =31.0s. Wthn ths perod the wave n the channel s not contamnated by the reflected wave from the rght boundary and the heght s stll 0.01 m. Afterwards, the reflected wave starts propagatng n the channel and s supermposed wth the wave from the source. Snce the two waves are dentcal and 180 out of phase, ths superposton results n a standng wave n the channel wth heght 0.02 m,.e. twce the orgnal wave heght. In all the snapshots the wave profle on the nterface s smooth. Also, for comparson, the full-boussnesq model smulaton s carred out and the correspondng nstantaneous profles are plotted n the fgure n dots. The two smulatons are n good agreement. Although the wave source nput s lnear (sngle harmonc), the wave undergoes some nonlnear evoluton due to the nonlnear governng equatons employed n both the Boussnesq and RANS models. The hgher the nonlnearty, ε, the stronger the nteracton. In the frst case the nonlnearty s relatvty small and ts effect mght not be so obvous n the wave profles n Fgure 10. Lookng at the profle from a dfferent perspectve, for nstance n the frequency doman, the nonlnearty effect becomes apparent. Fgure 11 gves the ampltude spectrum of the tme seres of the free

21 MULTI-SCALE SIMULATION Fgure 10. Standng wave (ε = 0.02, δ = 0.06) moton n a 0.5 m deep channel smulated usng the hybrd model. 0 x 36 s Boussnesq model, 36 x 72 s RANS model, and dots are the full-boussnesq model. surface elevaton at x =38.0m. Ths partcular tme seres s recorded before the reflected wave reaches the recordng locaton. Ths spectrum clearly shows that the ntally monochromatc wave, n the course of the propagaton, transforms nto a polychromatc wave. In the spectrum, there are two dstnct spkes correspondng to f 1 =0.25Hz and f 2 =0.5Hz. Note that the frequency of the orgnal sgnal s f =0.25Hz. Here, the frst harmonc, f 1, nteracts wth tself resultng n the second harmonc, f 2 = f 1 + f 1. The nteracton of the two harmoncs, f 1 and f 2, s not too strong

22 K. I. SITANGGANG AND P. J. LYNETT Fgure 11. Ampltude spectrum of the free surface elevaton tme seres of the ε=0.02 standng wave smulaton. The tme seres s recorded at x =38.0 m before the reflected wave reaches ths locaton. n ths case as there s no other dstnct spke occurrng n the spectrum. In the next test, as the wave heght becomes hgher, the effect of nonlnearty s stronger. In the second test, the wave heght s ncreased to 0.05 m, wth a nonlnearty of ε=0.1. The model setup remans the same as n the prevous test. Fgure 12 shows snapshots of the wave profles as the wave propagates n the channel. Here, a smooth transton s agan observed on the nterface of the two models. Although there s very slght dscrepancy between the hybrd and full-boussnesq profles, n general the two smulatons show good agreement. The wave profle dffers from the profle n the prevous test n two ways: a wder trough and an occurrence of a secondary crest on the trough, whch s due to the nonlnearty effect. The ampltude spectrum shown n Fgure 13 has three dstnct harmoncs: f 1, f 2,and f 3. The frst two harmoncs are at the same frequences as n the prevous case. The nteracton between the frst and second harmonc results n the thrd harmonc whose frequency s f 3 = f 1 + f 2 =0.75Hz. Ths harmonc s obvous here, not n the prevous test, snce the ampltudes of the nteracted harmoncs,.e. f 1 and f 2,are much hgher n ths test than the prevous one. In addton to the hybrd and full-boussnesq smulatons, we also smulate both cases of the standng wave moton usng the RANS model for speedup comparson. The RANS model s appled on the whole doman wth the same grd szes as n the hybrd-rans model. In the horzontal drecton the number of computatonal grd ponts s twce as many as the grd ponts n the hybrd-rans model mesh, and n the vertcal drecton both setups employ the same number of grd ponts. Therefore, n total, the number of grd ponts n the pure-rans model s twce the number of grd ponts n the hybrd-rans model. Both the pure-rans and hybrd smulatons are carred out n a 3.4 GHz Pentum processor for s smulaton tme. In the frst standng wave test, the hybrd model spends an average of s for one wave perod smulaton whle the pure-rans model takes about s. Hence, n the frst test we gan a 1.8 factor of speedup. In the second test, for one wave perod the hybrd and full-rans models spend and s, respectvely, and the ganed speedup s Flux of snusodal wave overtoppng As reported n [18] the Beach Eroson Board (BEB) conducted a laboratory experment n the Waterways Experment Staton of the Corps Engneers, at Vcksburg, Msssspp, to study wave

23 MULTI-SCALE SIMULATION Fgure 12. Standng wave (ε=0.1, δ=0.06) propagaton n a 0.5 m deep channel smulated usng the hybrd model. 0 x 36 s Boussnesq model, 36 x 72 s RANS model, and dots are the full-boussnesq model. run-up and overtoppng of shore structures. The experment was conducted n a concrete wave flume 36.6 m long, 1.52 m wde, and 1.52 m deep. The model was an undstorted scale model wth 1:17 length scale and 1:4.1 tme and velocty scales. A wave maker was used on the upstream sde of the flume for wave generaton. Downstream of the flume, shore structures of varous shapes (smooth slope, curve-faced wall, recurved wall, etc.) were bult. Behnd the structure a calbrated measurng tank was nstalled for collectng the overtoppng water. The water from the frst three

24 K. I. SITANGGANG AND P. J. LYNETT Fgure 13. Ampltude spectrum of the free surface elevaton tme seres of the ε=0.1 standng wave smulaton. The tme seres s recorded at x =38.0 m and before the reflected wave reaches ths locaton. Fgure 14. Expermental setup of the snusodal wave overtoppng by the Beach Eroson Board; fgure s not scaled. or four waves was dscarded to allow for the wave to attan a stable condton, after whch the water from sx or seven waves was collected n the tank for an overtoppng flux measurement. In ths study, numercal smulatons are undertaken of the BEB flux overtoppng experment usng the hybrd model whose setup s gven n Fgure 14. For the smulaton comparsons, the smooth structure data, as used by the prevous researchers [19, 20] n ther overtoppng studes, are employed here. In the numercal smulatons the wave s generated usng the snusodal wave source combned wth the sponge layer on the left boundary for dampng. Ths approach s dfferent from those used n [19, 20], where dependng on the Ursell number, U r, at the model boundary, h t, Stokes or Cnodal waves are used to drve the smulatons, although the wavemaker n the physcal

25 MULTI-SCALE SIMULATION Fgure 15. RANS computatonal mesh for hybrd smulaton of BEB snusodal wave overtoppng. experments created sngle harmonc waves only. In all the experments, the structure wth slope 1:s was fronted by a fxed 1:10 nclned floor. The doman s dvded nto the Boussnesq model subdoman, x B, and RANS model subdoman, x R. The model nterface s located near the toe of the 1:10 floor where turbulence s small and the wave does not yet break. In all the smulatons, ths nterface dvdes the doman nto the two subdomans wth rato x B /x R 7. The heght of the RANS doman, y R, s large enough to prevent the overtoppng wave from reachng the RANS model top boundary. For ths smulaton, the Boussnesq model doman s dscretzed nto a unform grd, Δx, and the RANS model doman s dscretzed, for effcency, nto nonunform horzontal grd, δx, and unform vertcal grd, δz, as depcted n Fgure 15. As shown n the fgure, the mesh s relatvely coarse near the left boundary and fner around the crest to allow for an accurate flux measurement. In addton for effcency, a dynamc tme step, δt, s employed n both the RANS and Boussnesq models; when the tme step changes, free surface and velocty values at the prevous tme levels are nterpolated or extrapolated to ft on a unform tme grd. The flux s computed at the back edge of the structure and gven by q = nt k max n=1 k=1 F n k un k δz kδt n (30) The ndces and k correspond to a cell n the column along whch the flux s computed. Snce some cells mght be not full, Equaton (30) ncludes the correspondng value of the volume of flud, Fk n. n =1 and nt are the tme ndces that correspond to the startng/endng tmes of the water collectng. As n the BEB experment, the smulaton s run under varous geometrcal setups wth dfferent combnatons of varables ncludng the offshore depth, h, the depth at the toe of the structure, h s, free board heght, h c, slope of the structure, s, wave heght, and perod. These varables, the expermental data, the computed hybrd model fluxes, and the publshed results from [19, 20] are presented n Table I. In general the computed fluxes are n good agreement (slght dfference) wth the expermental data and consstent wth the results of the prevous two researchers.

26 K. I. SITANGGANG AND P. J. LYNETT Table I. Expermental and smulated fluxes of the BEB snusodal wave overtoppng. h h s h c H T q data q KW q OTT q hyb Run (m) (m) (m) (m) (s) (m 2 /s) (m 2 /s) (m 2 /s) (m 2 /s) Runs 1 16 use s =3 and runs use s =1.5. Flux s presented n dmensonal form nstead of dmensonless form as n [19]. The slght dscrepances n these results are attrbuted to the small dfferences n the modeled physcs and numercal accuracy n the non-breakng part of the doman. Ths datum also demonstrates the relatvely wde varablty that can be found n publshed overtoppng predctons, partcularly for low overtoppng rates. As prevously explaned, the nterface of the model should be located such that the turbulence ntensty on the nterface s small. To provde nsght nto ths, n Fgure 16 the nstantaneous ntensty of the turbulent knetc energy, k, for run 1, s gven. Ths fgure shows that the ntensty of the turbulent knetc energy, k, s hgh near the structure compared wth other locatons. The knetc energy near the structure s roughly 10 2 m 2 n contrast to a value less than 10 2 m 2 on the nterface. As the wave approaches the structure, the wave heght to depth rato becomes so large that the wave breaks whle mpngng on the structure and hence releases knetc energy. Fgure 16 also ndcates that n the course of the smulaton the locaton of the turbulence hot spot remans close to the crest of the structure whle the nterface area s always low n knetc energy. Ths satsfes the requrement that the turbulence ntensty should be low on the nterface. To benchmark the smulaton tme, fve of the prevous smulatons are re-run usng the RANS model n the whole doman. Table II presents the run tmes per wave perod for both the hybrd and pure-rans wave models for the fve selected runs. In dscretzng the pure-rans model, the part of the hybrd doman where the RANS model s appled uses an dentcal mesh as n the hybrd case. Offshore of ths pont, where the Boussnesq model s used n the hybrd model, the pure- RANS model doman s unformly dscretzed wth a grd sze equal to the grd sze at the hybrd nterface locaton.

27 MULTI-SCALE SIMULATION Fgure 16. Turbulence knetc energy dstrbuton of the BEB snusodal wave overtoppng, run 1. The computatonal tmes of both the hybrd and pure-rans smulatons are summarzed n Table II. From ths table t s clear that the speedup due to use of the hybrd model s sgnfcant, rangng from a factor of near 10.0 (run 18) to over 17.0 (run 3 and 5). Ths large speedup s of course due to the smaller RANS mesh used n the hybrd model. However ths dfference s two fold: a sngle teraton of the Posson pressure solver requres less tme wth a smaller matrx and a smaller matrx wll converge n fewer teratons of the Posson solver.

28 K. I. SITANGGANG AND P. J. LYNETT Table II. Computaton tme per wave perod of the BEB snusodal wave overtoppng, runs 3, 5, 9, 18, and 19. A H and A R are the areas, the product of the number of grd ponts n the x- andz-drectons, nx nz, of the meshes employed n the hybrd and pure RANS models. Hybrd RANS nx nz t Flux nx nz t Flux Run (A H ) (s) (m 2 /s) (A R ) (s) (m 2 /sec) A R /A H Speedup Fgure 17. Expermental setup of the HR soltary wave overtoppng Free surface elevaton of soltary wave overtoppng In 1996, Hydraulc Research (HR) Wallngford n the U.K. performed an experment on soltary wave overtoppng of a breakwater. The expermental setup s gven n [20] and depcted here n Fgure 17. The wave flume used n ths experment was 40.0 m long and 0.5 m wde and flled wth water to h 1 =0.7 or0.6m seaward of the breakwater and h 2 =0.3 or0.2m behnd the breakwater. A breakwater wth 1:4 or 1:2 slope was bult at the rght end of the flume. Ths breakwater, whch s 0.5 m hgh and 0.16 m wde on the top, was fronted by a 1:50 nclned floor. To measure the free surface elevaton of the overtoppng water, a seres of wave gauges were nstalled on top of and behnd the breakwater. The frst gauge, WG-13, was located m behnd the leadng edge (A), the second, WG-14, and thrd, WG-15, gauges were nstalled and 0.11 m from the frst gage, respectvely. Dependng on whether the frst or second depth was used, the fourth gage,

29 MULTI-SCALE SIMULATION Table III. Wave heght and water depth of the HR soltary wave overtoppng. Test H (m) h 1 (m) h 2 (m) WG-16 (m) 4c7a c7b c7c c6a c6b c6c c6d Dstance from the backedge B n Fgure 17. WG-16, was located 0.72 or 1.1 m behnd the back edge (B) of the breakwater. The last gauge was fxed 0.44 m behnd the back toe (C) of the breakwater. The experment was conducted for several soltary wave heghts and water depths as gven n Table III. The smulaton setup s very smlar to the one used n the BEB smulaton where the unform Boussnesq grd s coupled wth the RANS nonunform x- and unform z-grds. The nterface dvdes the doman nto two segments, x B and x R, wth rato x B /x R 9. A dynamc tme step s also employed n the smulaton. In all tests the ntal locaton of the soltary wave s 10.0 m from the left boundary of the Boussnesq doman. The tme seres of the free surface elevaton of the hybrd smulatons and the expermental data are presented n Fgure 18. For comparson the numercal smulaton data from [20] (called OTT), usng a nonlnear shallow water wave equaton model, are also gven n the same fgures. In all the smulatons, the hybrd wave model shows a clear bas toward overpredctng the water elevaton on top of the structure. Ths s consstent wth the OTT smulatons, whch show an even larger bas. On the lee sde of the breakwater, the hybrd smulatons n general agree qute well wth the data, wth a remarked mprovement over the shallow water equaton-based OTT. The relatve speedup s also measured for ths case. The dscretzaton of the pure-rans doman s almost the same as n the prevous BEB dscretzaton. The only dfference s that to save computatonal tme n the pure-rans smulatons, the doman offshore of the nterface locaton of the hybrd model s dscretzed nonunformly, wth an ncreasngly coarse grd n the deeper water. The computaton tmes of both the hybrd and pure-rans smulatons are summarzed n Table IV. As wth the speedups n the BEB tests, the reducton n CPU tme shows a factor greater than the decrease n RANS doman sze. Ths, agan, s due to the Posson solver requrng fewer teratons to converge wth a smaller matrx sze Hypothetcal tsunam smulaton In many tsunam events, such as the devastatng tsunam that struck the West Sumatera Coast n 2004, the wave s generated by a sudden uplft or subsdence of the seafloor followng a massve tectonc earthquake. The vertcal dsplacement of the seafloor dsturbs the equlbrum of the water column above t and n consequence the water mass spreads as a long wave to attan a new gravtatonal equlbrum. In the last model applcaton, the hybrd wave model s used to smulate the propagaton of a tsunam-type wave from the open ocean to the coast. The smulaton setup s gven n Fgure 19. The doman conssts of a 1.0 km deep ocean connected wth a 1:50 seafloor. A 4.0 m elevaton

30 K. I. SITANGGANG AND P. J. LYNETT Fgure 18. Tme seres of the free surface elevaton of the HR soltary wave overtoppng. breakwater s placed along the coast n a shallow water depth of approxmately 3 m. For the hybrd smulaton, the Boussnesq model occupes 93.0% of the total horzontal doman length; the RANS model occupes only 7.0%, whch s located n the nearshore regon. The domans

31 MULTI-SCALE SIMULATION Table IV. Computaton tme of HR soltary wave overtoppng. Hybrd RANS Test nx ny (A H ) t (s) nx ny (A R ) t (s) A R /A H Speedup 4c7a c7b c7c c6a c6b c6c c6d Fgure 19. Smulaton setup of the hypothetcal tsunam generaton and propagaton. Fgure s not scaled. of the Boussnesq and RANS models are unformly dscretzed nto Δx =15.0m and δx =4.0m and δz =0.5m grds, respectvely. The smulaton s run wth a dynamc tme step for s of smulaton tme. The wave s generated km offshore by uplftng the sea surface to form a Gaussan shape wave wth zero ntal velocty. The generated wave has an offshore wavelength of roughly 10.0 km, and thus mght represent the leadng wave of a leadng elevaton tsunam. Fgure 20 shows snapshots of the wave at three dfferent tmes. At t =1070.0sthewavehasjust reached the detached breakwater wth a turbulent wave front that appears as an 18.0 m hgh wall of water movng at a speed of nearly 10m/s. The second snapshot n Fgure 21 gves the wave at t =1140.0s. Just 70.0 s after the wave reaches the breakwater, the coast s flooded up to 1.6 km nland. The average flow depth n the flooded area s 13.0 m, wth a 16.0 m/s average speed. The bottom part of the fgure shows the detal of the flow around the breakwater. At the upstream sde the flow moves 6.0 m/s and due to the breakwater actng as a sll, the velocty ncreases nearly three tmes to 17.0m/s at the leeward sde of the breakwater. In addton note the regons of separaton at the breakwater corners, ndcated by a relatvely low flud speed. Fgure 22 shows

32 K. I. SITANGGANG AND P. J. LYNETT Fgure 20. Snapshot of tsunam wave reachng the breakwater. Fgure 21. (Top) Snapshot of tsunam wave passng the detached breakwater and (bottom) close-up look of velocty near the breakwater. Fgure 22. Snapshot of tsunam wave nundatng the coast at the tme of maxmum run-up.