Internal Wave Maker for Navier-Stokes Equations in a Three-Dimensional Numerical Model

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1 Journal of Coastal Research SI ICS2011 (Proceedngs) Poland ISSN Internal Wave Maker for Naver-Stokes Equatons n a Three-Dmensonal Numercal Model T. Ha, J.W. Lee and Y.-S. Cho Dept. of Cvl and Dept. of Cvl and Envron. Eng., Envron. Eng., Hanyang Unv., Seoul Hanyang Unv., Seoul kevn4324@hanyang.ac.kr moonguy@hanyang.ac.kr Correspondng Author Dept. of Cvl and Envron. Eng., Hanyang Unv., Seoul ysc59@hanyang.ac.kr ABSTRACT Ha, T., Lee, J.W. and Cho, Y.-S., Internal Wave Maker for Naver-Stokes Equatons n a Three- Dmensonal Numercal Model. Journal of Coastal Research, SI 64 (Proceedngs of the 11th Internatonal Coastal Symposum),. Szczecn, Poland, ISSN Recently, numercal smulatons of water surface waves usng the Naver-Stokes equatons models have been ncreased snce those models are well known as one of the numercal models capable of overcomng the lmtatons of nonlnearty, dsperson and wave breakng. For the Naver-Stokes equatons model, an nternal wave maker usng the mass source functon has been employed to generate varous types of waves. However, the method has been appled to vertcally two-dmensonal numercal models and has not been expanded to three dmensons yet. In ths study, the nternal wave maker for the Naver-Stokes equatons s employed to generate target waves n the three-dmensonal numercal wave tank, NEWTANK. The model s frst verfed by applyng to smple numercal tests n two dmensons. The model s then appled to predct wave transformaton on a varable bathymetry n two dmensons and expanded to three dmensons. Numercal results are compared wth avalable laboratory measurements and flow characterstcs are analyzed n detal. ADDITIONAL INDEX WORDS: Soltary wave, Runup and rundown, Concal sland INTRODUCTION Many numercal models have been used to study wave behavors n coastal areas. The transformaton of mult-drectonal waves can be numercally smulated by solvng a boundary value problem. To solve the boundary value problem, the wave energy s requred to be generated precsely at the offshore boundary. Then, governng equatons, whch are generally partal dfferental equatons such as the shallow water equatons, the Boussnesq equatons and also the Naver-Stokes equatons, should be solved n both tme and space by usng approprate numercal technques. However, dfferent types of problems and some lmtaton are nduced when numercal models are employed to predct wave transformaton. One of the key problems occurrng n smulatng wave moton s re-reflecton to the computaton doman at the ncdent boundary. Many numercal technques have been developed to deal wth ths problem and the nternal wave generatng-absorbng boundary condton has been commonly used n numercal wave models. In depth-averaged equatons wave models, the method of characterstcs s generally employed to separate ncdent and reflected waves. For example, based on one-dmensonal shallow water equatons, Kobayash et al. (1987) developed an approxmate boundary condton for absorbng the reflected waves whle sendng n the ncdent waves at the same tme. Followng the approach, a smlar type of wave absorbng-generatng boundary condton s developed for a two-dmensonal shallow water equatons model. Snce these methods are based on shallow water assumptons, the accuracy problem has been occurred when the methods are appled to dealng wth short perod waves. All of these approaches assume the lnear superposton of ncdent and reflected waves at the wave generatng-absorbng boundary. Thus, these can only be employed to small ampltude waves. Furthermore, We and Krby (1995) found that ths type of boundary condton can lead to the accumulaton of errors that may eventually contamnate numercal results n the entre doman when a long tme smulaton s performed (Ln and Lu, 1999). Recently, numercal smulatons of water surface waves usng the Naver-Stokes equaton models have been common snce these models are much more superor than the depth-averaged wave models. The Naver-Stokes equaton-based models employ the VOF (volume of flud) method, therefore, those can manage nterfaces between multple phases,.e., water and ar. In addton to that, the models can solve the complete momentum equaton, resolvng vertcal components as well as nonlnear terms and vscous and turbulent stresses. Therefore, those models are known as one of the numercal models capable of overcomng the lmtatons of nonlnearty, dsperson and wave breakng, whch most of the conventonal wave models cannot do. Although applcaton of the model s stll n an early stage, studes on water surface waves by usng the Naver-Stokes equatons models employng the VOF scheme have been ncreasng (Cho and Yoon, 2009). It s even more dffcult to treat the wave absorbng-generatng boundary for a numercal model based on Naver-Stokes equatons or Reynolds averaged Naver-Stokes (RANS) equatons, snce the model should descrbe the vertcal velocty dstrbuton as well as the free surface dsplacement at the boundary (Myata 1986; Ln and Lu 1998). Therefore, the model s much more senstve to the errors from the boundary than the depth-averaged equatons models, such as the shallow water equatons model and the 511

2 Internal Wave Maker for the Naver-Stokes Equatons Model Boussnesq equatons model (Ln and Lu, 1999). For the Naver- Stokes or RANS equaton model wth the VOF method, Ln and Lu (1999) proposed the nternal wave maker usng mass source functon of the contnuty equaton and ths wave maker has been used to generate varous types of waves. The method has been appled n several numercal tests and succeeded n predctng wave transformaton n the numercal wave tank (Garca et al., 2004; Lara et al., 2006; Ln and Karunarathna, 2007). However, all tests are performed n two dmensons and have not been expanded to three dmensons yet. In ths study, the nternal wave maker for the Naver-Stokes equatons whch s usng mass source functon of the contnuty equaton s employed to generate targeted lnear and nonlnear waves n the three-dmensonal numercal model. By applyng the present method to the three-dmensonal numercal wave tank, NEWTANK, dfferent numercal tests are performed n two and three dmensons. NUMERICAL MODEL Mathematcal model The motons of an ncompressble flow are governed by the Naver-Stokes equatons and the contnuty equaton. u = 0 (1) x u 1 1 τ u + p j u = j + g + (2) t xj ρ x ρ xj where, j = 1, 2,3 for the three-dmensonal flows, ρ denotes the densty, p s the pressure, u and g represent th component of the velocty vector and the gravtatonal acceleraton, respectvely. The drect numercal smulaton (DNS) to NSE for turbulent flows at hgh Reynolds number s computatonally too expensve. As an alternatve, the large eddy smulaton (LES) approach (Deardorff, 1970), whch solves the large scale eddy motons accordng to the spatally averaged Naver-Stokes (SANS) equatons and models the small-scale turbulent fluctuatons, becomes attractve. In the LES approach, the top-hat space flter (Pope, 2000) s appled to the NSE and the resultng fltered equatons of motons are as: u = 0 (3) x r u uu 1 1 τ j p + = j + g + (4) t xj ρ x ρ xj where p denotes the fltered pressure, u and τj r represent fltered veloctes and the vscous stress of the fltered velocty feld, respectvely. The vscous stress terms are modelled by the Smagornsky SGS model (Smagornsky, 1963). Numercal solver for NSE In ths model, the governng equatons are solved by the fnte dfference method on a staggered grd system. A two-step projecton method (Chorn, 1968, 1969; Ln and Lu, 1998), whch has been proved to be very robust, s employed. The forward tme dfference method s used to dscretze the tme dervatve. The convecton terms are dscretzed by the combnaton of the central dfference method and upwnd method, whle only the central dfference method s employed to dscretze the pressure gradent terms and stress gradent terms. The VOF method s adopted to track the free surface. Detaled numercal technques are well revewed n the other paper (Lu, 2007) and not repeated here. Internal wave maker for the 2D NSE For the 2D Naver-Stokes equaton model wth the VOF method, Ln and Lu (1999) proposed the nternal wave maker usng mass source functon of the contnuty equaton. u w + = s ( xzt,, ) (5) x z where s( xzt,, ) denotes the nonzero mass source functon whch s defned n the source regon Ω. The source regon, Ω, s defned by a rectangular shape wth the x-axs drecton length, L s and the z- axs drecton length, H s. When the wdth of the source regon s small relatve to the target wavelength, the generated wave can be regarded to start from the secton where the center of the source regon s located. If we further assume that all of the mass ncrease or decrease ntroduced by the mass source functon contrbutes to the generaton of the target wave, the followng relatonshp between the source functon s( xzt,, ) and the expected tme hstory of free surface dsplacement η () t above the source regon can be derved (Ln and Lu, 1999). t s( x, z, t) dω dt = 2 Cη () t dt (6) 0 Ω 0 where C denotes phase velocty of the target wave. Internal wave maker for the 3D NSE The nternal wave maker proposed by Ln and Lu (1999) s employed n the 3D Naver-Stokes equatons model to generate a target wave n three dmensons. Eq. (5) can be extended to three dmensons as follows. u v w + + = s ( xyzt,,, ) (7) x y z If we assume the generated wave start from the secton where the center of the source regon s located and the drecton of the target wave propagaton s parallel, the source functon of eq. (7) can be regarded as the same functon usng n eq. (5). Therefore, we can easly generate several types of waves wthout modfyng orgnal functons or addng any terms on. At the outgong boundary a sponge layer s used to absorb the wave energy. The sponge layer s employed by addng a dampng term to each momentum equaton wthout pressure terms at the frst step of a two-step projecton method as follows: u u n u n u n u + u + v + w = υ u cu s (8) v v n v n v n v + u + v + w = υ v cv s (9) w w n w n w n w + u + v + w = υ w cw s (10) where ± n, ± n, ± n u v w represent ntermedate veloctes n each axs drecton and cs denotes the dampng coeffcent, whch s defned as cs = exp ( x/ xs) 1 exp() 1 1 nsde the sponge layer and set to zero outsde the sponge layer. The sponge layer s the nterval, 0 x xs, x s s a length of the sponge layer (L, 2008). NUMERICAL SIMULATION t 512

3 Ha et al. Table 1: Condtons of the numercal experments Items Components Water depth ( d ) 0.2m Wave Heght ( H ) 0.01m / 0.02m / 0.04m X-axs drecton 700 (non-unform: 0.02m-0.06m) Cell Y-axs drecton 10 (unform: 0.02m) Z-axs drecton 65 (unform: 0.004m) Total 455,000 (18m 0.2m 0.26m) Soltary wave on a constant water depth Numercal experments are conducted to verfy the nternal wave maker n three dmensons. The soltary waves wth dfferent ratos of wave heghts to local water depth are generated by the nternal wave maker. Water depth s set to be constant and three dfferent wave heghts are employed. Table 1 shows condtons of the numercal experments. Fgure 1 represents comparson of water surface profles between numercal results and analytc solutons. As seen n the fgure, the numercal model wth the nternal wave maker quanttatvely matches well analytc solutons. Fgure 2 shows the free surface profles of nternally generated soltary waves for all three cases. Soltary waves are generated at the center of the numercal doman and propagated symmetrcally to left and rght boundares. By comparng to analytc solutons, numercal results are agreeable. Soltary wave runup and rundown on a steep slope Laboratory experments of soltary wave runup and rundown on a steep slope have been performed by Ln et al. (1999) and detaled nformaton s estmated by usng PIV (partcle mage velocmetry) system. The experments have been used to verfy free surface trackng methods because stable and contnuous runup and rundown are occurrng smultaneously. Numercal smulatons (a) H/d=0.05 (t=5.25sec) (b) H/d=0.10 (t=4.25sec) (c) H/d=0.20 (t=4.25sec) Fgure 2. Propagaton of nternally generated soltary waves usng the nternal wave maker are performed n a doman wth 4.49m x 6.99m and -0.16m z 0.11m. The computatonal doman s dscretzed by 350 unform grds n x drecton wth x=0.01m, whle n z drecton, 95 unform grds are used wth z=0.003m. The tme step sze s dynamcally adjusted durng the computaton to satsfy the stablty constrants. Table 2 shows condtons of the numercal experments and Fgure 3 gves the (a) t=2.5sec (b) t=4.5sec Fgure 1. Comparson of numercal results and analytc solutons of soltary wave propagaton on a constant depth (H/d=0.05) Fgure 3. Computatonal doman of the numercal model and the comparson of free surface dsplacement (t=5.68sec) 513

4 Internal Wave Maker for the Naver-Stokes Equatons Model Table 2: Condtons of the numercal experments Items Components Water depth ( d ) 0.16m Wave Heght ( H ) 0.027m X-axs drecton 350 (unform: 0.01m) Cell Y-axs drecton 10 (unform: 0.02m) Z-axs drecton 95 (unform: 0.003m) Total 332,500 (3.5m 0.2m 0.285m) sketch of the computatonal doman and the measurement free surface at t=5.68sec. Fgure 4 shows the free surface profles of soltary wave runup and rundown on a steep slope. By comparng wth laboratory observed data, numercal results represent well observed free surface profles. (b) t=5.00sec Soltary wave runup around a concal sland Laboratory experments of soltary wave runup around a concal sland were performed at the US Army Engneer Waterways Experment Staton, Coastal Engneerng Research Center (Lu et al., 1995). The research has been well known a benchmark (b) t=6.25sec (a) t=6.38sec (b) t=6.58sec (b) t=8.25sec (c) t=6.78sec (d) t=7.18sec (b) t=10.50sec (e) t=7.38sec Fgure 4. Soltary wave runup and rundown on a steep slope Fgure 5. Soltary runup around a concal sland problem of runup smulatons and provded a better understandng of the physcal phenomena, such as wave refracton, dffracton 514

5 Ha et al. and breakng around the sland. Therefore, the Naver-Stokes equatons model wth the nternal wave maker s appled to smulate the laboratory experment. Fgure 5 shows numercal smulatons around the sland. Once a soltary wave approaches the sland, large runup s occurred at the front sde of the sland. Whle a soltary wave passes the sland, wave refracton and dffracton are occurred and then transformed waves are concentrated at the lee sde. Numercal smulatons represent well the physcal phenomena. Therefore, the model can be regarded to smulate runup problems n three dmensons. A detaled quanttatve analyss between numercal results and observed data has not been performed, however, addtonal researches are requred to verfy the model. CONCLUDING REMARKS In ths study, the nternal wave maker proposed by Ln and Lu (1999) s employed n the 3D Naver-Stokes equatons model to generate a target wave. The soltary waves wth dfferent ratos of wave heghts to local water depth are generated by the nternal wave maker and compared wth correspondng analytc solutons. Then, the model s appled to nvestgate soltary wave runup and rundown n two and three dmensons. Despte of small dscrepancy, numercal results are reasonable and the nternal wave maker can be appled to generate targeted waves nternally n other numercal experments. The runup and rundown process of soltary waves around a varable topography have been attracted many researchers snce they are smlar to those of the tsunams on coastal areas. Most of them have been conducted usng depth-averaged numercal models or 2D Naver-Stokes equatons model and numercal approaches usng 3D Naver-Stokes equatons models have been barely performed. Although applcaton of the model s stll n an early stage, the model can be regarded as a prospectve effcent tool to analyze those problems. LITERATURE CITED Cho, J.W. and Yoon, S.B., Numercal smulaton usng momentum source wave-maker appled RANS equaton model, Coastal Engneerng, 56, Chorn, A.J., Numercal soluton of the Naver-Stokes equatons, Mathematcs of Computaton, 22, Chorn, A.J., On the convergence of dscrete approxmatons of the Naver-Stokes equatons, Mathematcs of Computaton, 23, Deardorff, J.W., A numercal study of three-dmensonal turbulent channel flow at large Reynolds numbers, Journal of Flud Mechancs, 41, Garca, N., Lara, J.L. and Losada, I.J., D numercal analyss of near-feld flow at low crested permeable breakwaters, Coastal Engneerng, 51, Kobayash, N., Otta, A.K. and Roy, I., Wave reflecton and runup on rough slopes, Journal of Waterway, Port, Coastal and Ocean Engneerng, 113(3), Lara, J.L., Garca, N. and Losada, I.J., RANS modellng appled to random wave nteracton wth submerged permeable structures, Coastal Engneerng, 53, L, B., A 3-D model based on Naver-Stokes equatons for regular and rregular water wave propagaton, Ocean Engneerng, 35, Ln, P., Chang, K.A and Lu, P.L.-F., Runup and rundown of soltary waves on slopng beaches, Journal of Waterway, Port, Coastal and Ocean Engneerng, 125, Ln, P. and Karunarathna, S.A.S., Numercal study of soltary wave nteracton wth porous breakwaters, Journal of Waterway, Port, Coastal and Ocean Engneerng, 133(5), Ln, P. and Lu, P.L.-F., A numercal study of breakng waves n the surf zone, Journal of Flud Mechancs, 359, Ln, P. and Lu, P.L.-F., Internal wave-maker for Naver- Stokes equatons models, Journal of Waterway, Port, Coastal and Ocean Engneerng, 125(4), Lu, D., Numercal modelng of three-dmensonal water waves and ther nteracton wth structures. Sngapore: Natonal Unversty of Sngapore, Ph.D. thess, 146p. Lu, P.L.-F., Cho, Y.-S., Brggs, M.J., Kanoglu, U. and Synolaks, C.E., Runup of soltary waves on a crcular sland, Journal of Flud Mechancs, 302, Myata, H., Fnte-dfference smulaton of breakng waves, Journal of Computatonal Physcs, 65, Pope, S.B., Turbulent Flows. Cambrdge Unversty Press, New York, USA. Smagornsky, J., General crculaton experments wth the prmtve equatons: I. The basc equatons, Monthly Weather Revew, 91, We, G. and Krby, J.T., Tme-dependent numercal code for extended Boussnesq equatons, Journal of Waterway, Port, Coastal and Ocean Engneerng, 121, ACKNOWLEDGEMENTS Ths research was supported by a grant from Countermeasure System aganst Typhoons and Tsunams n Harbor Zones Program funded by the Mnstry of Land, Transport and Martme Affars, Korea. 515

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