Recent Advancement in Modeling Seafloor Dissipative Mechanism in Shallow Water

China Oeean Enginering, Vol.22, No.1, pp.1- 10◎2008 China Ocean Press, ISSN 0890-5487Recent Advancement in Modeling Seafloor DissipativeMechanism in Shallow Water*Philip L.F. LIU*, b, 1 and Yong Sung PARK*●School of Ciril and Eimuironmental Enginering, Cornell Unirersiy,Ithaca, NY 14853, USAb Insitule of Hydrological and Oceanic Sciences，National Central University ,Jhongli 32001, China(Received 29 December 2007; accepted 15 February 2008)ABSTRACTIn this paper, the recent development in modeling seabed dissipative mechanisms in shallow water is reviewed.Specifically, laminar and turbulent boundary layer solutions as well as viscous mud flow solutions under tansient wavesare presented . These analytical solutions are compared with experimental data for both solitary waves and cnoidal waves.Very good agrement is obtained. 'The Bussinesq equations with boundary layer efects and he muddy scabed efecs arealso shown.Key words: seabed disipaire mechanisms; boundary layer ; Boussinesq equations1. IntroductionOn continental shelf and in coastal zone, wind waves interact with seafloor and a number of dy-namic physical processes will take place. Bathymetric variations affect both wave propagation directionand the spatial variation of wave heights, which is commonly known as wave refraction, diffraction andscattering (Mei and Liu, 1993). lregular ( random) bathymetric variations can also enhance wave re-flection (Mei and Li, 2004).Interaction between ocean waves and seafloor also leads to energy dissipation. Over a rigidseabed, shear flows inside a laminar or turbulent boundary layer contribute to significant wave attenua-tion over a distance of tens and hundreds wavelengths of propagation. If the seafloor is composed ofsediments, a variety of dissipation mechanisms associated with the sediment heology enhances wavedamping and modifies wave characteristics. For instance, the effects of percolation become important ifsalloor sediments consist of coarse sand or shingle. With finer sediment grain sizes, the sea bottombecomes deformable under wave loading and the effects of porelasticity need to be considered in esti-mating energy dissipation. When the sea bottom consists of silt or mud, the physical processes becomemore complex. Depending on the density of the mud, some may behave like a viscous fluid, while theothers behave as viscoelastic materials or viscoplastic materials. Based on the viscosimetric measure-ments of mud samples, Krone ( 1963) reported that for con(中国煤化Ieen 10 to 110* The research presented here has been supported by National SciendHC N M H Gesearch and Na-tional Sea Grant Program.1 Corresponding author. E-mail: pl3@ comell edu2Philip L.F. uU and Yong Sung PARK /China oean Enginering, 22(1), 2008, 1-10g/L, mud along the east and west coast of the United States behaves like a Bingham plastic. Similarmud beds have been reported in many coasts, rivers, estuaries, around the world (Healy et al.,2001; Shermnet and Stone, 2003).Extreme wave energy dissipation rates due to seafloor effects have been reported in the literature(e. g. Foristall and Reece, 1985). Gade ( 1958) reported that there is a location in the Gulf of Mexi-co, where the attenuation of ocean surface wave due to mud floor is so great that fishing boats routinelyuse it as an emergency harbor for protection during stomns. Off the coast of Surinam, more than 90%of incident wave energy is dissipated across the 20 km wide shallow mudflats ( Wells and Coleman,1981). Fnall, Mathew (1992) reported that off the coast of India, 95% of wave energy is dissipatedacross 1.1 km wide mudbanks.Within the framework of linear water wave theory, theoretical formulae for wave attenuation rateand the modifed dispersion relationship of simple harmonic progressive waves are available if theseafloor properties and rheology can be specifed a prioni (e.g. Liu, 1973; Dalrymple and Liu, 1978;Yamamoto et al., 1978; MacPherson, 1980; Mei and Liu, 1987; Wen and Liu, 1998; Ng, 2000;Liu and Chan, 2007a) . These formulae have been employed in wave evolution ( phase averaged) mod-els, such as the WAM model (e.g. Komen et al., 1994) and the SWAN model (Booij et al.,1999). Recently, using the SWAN model, Winterwerp et al. (2007) investigated the wave attenua-tion in the Guyana coastal system that is characterized by very thick deposits of Amazon mud. Thedamping rate obtained from Gade' s (1958) two-layer model was used in their SWAN model. They re-ported that the model predicted significant wave attenuation and the computed changes in wave energyspectrum qualitatively agreed with measurements in Surinam. We remark here that while some of thetheoretical results, e. g. percolation and porelastic theories, have been validated with experimentalresults, others have not. The lack of adequate facilities and instruments for measurements is one of themost important reasons for this shortcoming.In shallow water, waves are usually nonlinear and possibly dispersive. Linear wave theory is nolonger adequate. In recent years, Boussinesq-type (phase- resolving) models have been extensively de-veloped as practical wave dynamic models in both intermediate and shallow water (i.e. from the conti-nental shelf to the surf zone) (e.g. Wei and Kirby, 1995; Lynett and Liu, 2004). The phase- resolv-ing feature of the Boussinesq-type models is essential in better understanding and predicting nearshoreprocesses (Mei and Liu, 1993). On the other hand, very litle work has been done in formulating thedissipative effects of seafloor within the framework of Boussinesq-type equations. Very recently Kai-hatu, Sheremet and Holland (2007) presented a parabolic frequency domain model in which the effectsof a viscous muddy seafloor were included by adopting the dispersion relationship ( damping rate andfrequency shift) for each harmonic component. The dispersion relationship in their model was based onNg's (2000) work for a linear, simple harmonie wave with the assumption that the muddy seabed isvery thin. Although the frequency domain models are use中国煤化工ity slially endtemporally, which could be limiting in many cases.TYHCNMHGTo advance the predictive skills of wave models in the coastal egions where the seafloor are madePhilip L.-F. UU and Yong Sung PARK /China Oeun Engineering, 22(1), 2008, 1-103of fine sediments, mud or other dissipative materials, it is essential to fill the knowledge gap in under-standing the sea bottom dissipative processes under transient wave loadings. Once these mechanismsare properly described and understood, they need to be included in the Boussinesq-type wave models.2. Boussinesq Equations and Bottom Boundary LayerBased on the potential flow theory, the classical Boussinesq equations are well known and can beexpressed in dimensionless form as fllows (Peregrine, 1967):1望+ v .(HF)=o,(1)Jud: +ei.vu+εvH-号vv.() == 0(p*),(2)where H is the total water depth, u the depth-averaged horizontal velocity vector, E and μ2 are thesmall parameters representing the weak nonlinearity and frequency dispersion, respectively. It is welllknown that the continuity equation (1) is exact and the momentum equation (2) is derived with the as-sumption that 0(e) = 0(μ2). The classical Boussinesq equations have been expanded to includehigher order nonlinearity and frequency dispersion (e.g. Liu, 1994; Kirby et al., 1998).To include the effects of seafloor dissipative mechanisms, the dynamic coupling between wave mo-tions in the water column and the seafloor dynamics need to be considered. In the present engineeringpractice, however, it is customary to parameterize dissipative mechanisms and introduce a bttom fric-tion-like term in the momentum equation (2). In the case of a bottom boundary layer, for example,the traditional approach is to add a bottom shear-stress term in the momentum equation. The bottom-shear stres term is routinely modeled as a function of the free stream velocity u (i.e. quadratic law).However, using a rigorous perturbation analysis, Liu and Orfila (2004) dermonstrated that the leading-order effects of a laminar boundary layer appear as a convolution integral in the continuity equation,Eq. (1), i. e.+ v .(F)=。[号'"dr,(3)e Jlwhere a -2 denotes the Reynolds mumber based on the phase velocity and wavelength. The right-handside represents the boundary layer displacement effects. Furthermore the bottom stress can be writtenanalytically as:。=咖(s,)0。-dr.4)VI-τAccording to Eq. (4)，the bottom shear stress is a convolution integral, which is more weightedby the influence of the present time. It is also noted that the bottom shear stress is affected by the ac-celeration. It is clear that the frictional efect is not only ou中国煤化工hh1 velocity butalso has memory. Using Eqs. (2) and (3), Liu et al. (2Fon of a solitary .wave and found good agreement with laboratory measuremenhCNMHG! Paricle ImageVelocimetry (PIV), the velocity inside the laminar bottom boundary layer under a soliary wave has4Philip L. -F. uU and Yong Sung PARK /China Oeean Enginering, 22(1), 2008, 1-10been measured (Liu et al., 2007). The bottom shear stress is then directly calculated from the veloci-ty measurements. As shown in Fig. 1, the experimental data agree with the theory. The bottom shearstress changes sign as the free stream velocity starts to decelerate. This also suggests that flow reversaloccurs as the deceleration increases (see Fig. 2), which might lead to flow separation and tubulencefor high Reymolds mumber flows.(2)1.50~0.0.4 0.6 0.8(b)0.520.202二0.4.0.60.8=2. ()0.20.4_ 0.6_.8-0.5d1-0.5 0 0.31.2.50.2 00.2 0.4 0.6 0.8FTg. 1. Time history of botomn shear stress under a soli- Fig. 2. Vertical profiles of the dimensionless horizonaltary wave. Gircles are the experimental data,velocity inside and above the laminar boundarydash-dot line denotes the linear analytical solu-layer under solitary wave. Gircles are PIV mea-tion (4) and solid line represents the nonlinearsurements and solid line denotes the theoreticalnumerial solution (Liu et al., 2007).resuts. (a)-ξ= -0.22; (b)-ξ=0.06;(c)-ξ=0.34; (d)-ξ=0.63, whereE=x .- C: is the moving corlinate (Liu et al.,2007).Note that theoretical results by Liu and Orfila (2004) are valid for diferent kinds of wave loadingin shallow water. For example, Fig. 3 shows the comparison between the theoretical solution (4) andthe measured data by Sana et al. (2006) for a cnoidal wave. We note that the free-stream velocitiesare digitized from the laboratory data. The agreement between the analytical solution (4) and the ex-perimental data is excellent.Additional experiments have been perforned in the U-tube facility at Technical University of Den-mark to investigate the incipient and evolution of turbulent boundary layer under a solitary wave (Sumeret al., 2008) . The preliminary data show that as the Reynolds number increases, shear instability OC-curs during the flow reversal phase and vorticity tubes are generated. As the Reynolds number is fur-ther increased, turbulent fluctuations are recorded during the deceleration phase. The bottom shearstress was also measured. The phase lag between the bottom shear stress and the free stream velocity issmaller than that for the laminar boundary layer, reducingnhant 10adequate theory中国煤化工”is needed to describe the entire processes.1YHCNMHGPhilip L.-P. UU and Yong Sung PARK / China Oean Enginering, 22(1), 2008, 1-1053Fig. 3. Time history of the dimensionless bottomshear stress under cnoidal wave. Circlesare the experimental data by Sana et al.+0(2006) and solid line denotes the analyti-cal solution (4).22.12.2?2.32.42.52.62.72B2.93ξAssuming that the eddy viscosity for a fully developed turbulent boundary layer flow is a powerfunction of the elevation from the bed, Liu (2006) derived approximate analytical solution for the tur-bulent flow velocities under long waves. In particular, Prandl' s power-law formulas (Prandul, 1952)are used, which are valid for steady turbulent boundary layers up to Reynolds number 10 with negligi-ble pressure gradients. Certainly, flows of interest here are neither steady nor pressure-gradient-free.Nevertheless, Patel (1977) showed that the mean flow and turbulent intensity distributions are quiteinsensitive to the free- stream ocillations in his measurements with traveling waves. Moreover, Jensenet al. ( 1989) also observed log- law regions in ocillatory turbulent boundary layers, which were ingod agreement with the corresponding steady boundary layer flows. Thus one may expect that Liu' 8(2006) approximate solutions can capture some important physics in spite of the assumptions em-ployed. The analytical solutions are compared with experimental measurements by Sana et al. (2006)in Fig. 4. Indeed, generally good agreenent is observed.F 0.5-0.8-0.6-0.4-0.20 0.2 0.40.6 0.b)'1 -0.8-0.6-0.40.200.20.40.60.8 T后0.5C)--0.8-0.6-0.4-0.200.20.40.60.8(d).1-0.8-0.6-0.40.200.20.40.60.81Fg. 4. Vertical profiles of the dimensionlese horizontal velocity inside and above the tubulent boundarylayer un-der cnoidal-like wave. Circles are the experimental data中国煤化工line denotes thetheoretical resuts by Liu (2006). (a)-ξ=2.5; (b) .YHCNMH(ξ= 3.2, whereξ=%- Ct is the moving coordinate and wave crests are locatea att=u,土1,土山，●6Pilip L.-F. uu and Yong Sung PARK /China Oeean Enginering, 22(1), 2008, 1- 10We should point out that the vertical coordinate in Fig. 4 has been normalized by the boundarylayer thickness based on the effective riscosity that requires use of the friction velocity. Since the fric-tion velocity depends on the free stream velocity, Liu (2006) assigned a constant to simplify his analy-sis. In Fig. 4, the friction velocity has been assumed to be 2% of the maximum free-stream velocity.In other words, the efctive viscosity can be viewed as a ftting parameter characterizing the turbulentboundary layer flows. Therefore, the approximate model can be of more practical use when rigorous ex-perimental investigations can be performed to determine either a single number or a functional relation-ship to the parameter.3. Viscous Mud Flows Under A Long WaveIn this section, we demonstrate here the effects of viscous fluid mud on transient long wave propa-gation (Liu and Chan, 2007b). In the theoretical analysis, assuming that the viscosity of the fluidmud is much larger than that of water and mud layer thickness is much smaller than the wavelength,Liu and Chan (2007b) have derived analytically the horizontal and vertical velocity components (Um，wm) in the viscous fluid mud, subject to a prescribed fuid particle velocity at the water-mud inter-face, u(x, t), in the following dimensionless form:Um=Vm+Yu,withm(x,7,t) =- r民告)-rx(- 1r。3H[ er(二共(三0,et(土(2n鄂)]dt(s)V4(t-7) )V4(t-T)andwm(x,7,1) =-7●(m+yu})dn(6)in which erfc (* ) is the complimentary error function, r the density ratio between water and mud, ηthe normalized vertical boundary layer coordinate, and d the normalized mud layer thickness.On the water-mud interface (η =0)，the vertical velocity component becomes:uom(x,0,t) =- r{dv●u-」。{'8(V : u)V4(T-T) I(t-r)dr}，(7)Jτin which_d(-)= 1- o(c2:1]+-V4(t- 2)st(4(I- )){仁|[p(- (-=122- 2cp( -品照周)+ap( - (《n,2”]2nd中国煤化工)V4(t- τ)l| 2nerte/4(t -) <-YHCNMH)} (>)The vertical velocity along the water- mud interface creates the boundary layer displacement efectePhilip L.-F. UU and Yong Sung PARK / China Oean Engineering, 2(1), 2008, 1-107on the water column above. Therefore, the leading order effects of a mud layer also modify the continu-ity equation in a similar way as shown in Eq. (3). In tems of the depth averaged horizontal velocity,the Boussinesq equations can be derived as (Liu and Chan, 2007b):1望+v.[(H+r间= γ兰「"3又:“74(1-7) 1(t- r)dr+ 0(n*),(9)ε FtuJo Jψ+eiuτ .u+4vH-些V7(引)。: 0(u2).(10)It is noted that Liu and Chan (2007b) have presented the analytical ( perturbation) solutions forthe damping rate of a solitary wave over a viscous fuid mud, using the 1D version of the above Boussi-nesq equations.!|(@)后-0.1-0.1 .-0.050.050.1-0.2 00.2 0.40.6 0.F -0.向)-0.-0.2 00.20.4 0.60.|(C-0.2 0 0.20.40.60.8”()-0.T -0.05-0.2 0 0.2 0.4 0.6 0.82G-50.2 0.4 0.6 0.80.2 00.20.40.6 0.(m)-1000.2 0.4 0.6 0.L()(0)[(1)2”-0.2 00.20.4 0.60.8 i-1902 00.2 0.40.60.8Fig. 5. Vertical profiles of dimensionless horizontal velocity under solitary wave at diferent phases with ε=0.19ford=0.178(a~d), d=2(e~h), d=5(i~1) and d = 10(m~ p). Circles are experimental data,solid line denotes the analytical solution and dashed line represents the parabolic approximation. (a)-ξ= 0.33; (b)-ξ= -0.12; (c)-ξ=0.09; (d)-ξ=0.33; (e, i, m)-ξ= -0.3; (f,j, n) .-ξ=0.0; (g, k, o)-ξ=0.3; (h, I, p)-ξ=0.6(中国煤化工To ensure the validity of the approximations used in theCNMHGes of laboratoryexperiments has been conducted (Park et al., 2008). A clear slulcone ttuid ( LJow Lorming SYLGARD8Philip L. -F. uU and Yong Sung PARK /China Ocun Engiecering. 22(1), 2008, 1- 10(e)r -0.1-0.2-0.10.0-0.50.5(b)F820.10.2E -0.1[ (C)2L..5()2O"u。-2[()|(m)4-100向)oL(向)[(1)0[(P)r -20LFig. 6. Vertical profiles of dimensionless horizontal velocity under cnoidal wave at diferent phases with ε=0.19and m=0.9 ford=0.223(a~d), d=2(e~h), d=5(i~l) andd= 10 (m~p). (a, e, i, m)-ξ=2.5; (b, f,j, n) -ξ=2.75; (c, g, k, o) -ξ= 3.0; (d, h, l, p) -6=3.25.184 base fuid; γ-1= 1.05;Um= 5.24 x 10-3 m2s-' at 25°C) was used as the viscous mud whichis dynamically similar to top layer sediments on the Misisppi Delta reported by Gade (1958). Usingthe similar PIV technique used by liu et al. (2007), velocity fields inside and above the mud layerwere measured for d=0.178 (see Figs. 5a~ 5d), in which agreement between measured data andanalytical solutions is very good validating the most of simplifications and treatment of the boundaryconditions in the analytical approach.It tums out that d is an important parameter characterizing the mud flow regime. Time histories ofthe horizontal velocity showed that velocities at different elevations have the same direction whend <1and the velocity profile can be fitted by a parabola as shown in Figs. 5a ~ 5d. However, when thethickness of the mud layer is much larger than the bottom boundary layer thickness, the velocity profileappears to be a plug flow above a thin viscous layer. The中国煤化工es can be deler.mined theretically. More complex flows occur for d~ 0,YHCNMHGcurswhend≥1cannot be captured by the existing work (Mei and Liu, 1987), in which the velocity profile is assumedPhilip L. -F. uU and Yong Sung PARK /China Oean Enginering, 22(1), 2008, 1-10to be parabolic.Note that the analytical solution by Liu and Chan ( 2007b) can be readily applied to differentwave loadings in shallow water. For example, theoretical solutions for cnoidal waves with different val-ues ofd are plotted in Fig. 6. The flow patterms for cnoidal waves are very similar to those under asolitary wave.4. Concluding RemarksIn present study, it is showm that, by integration of the laboratory , analytical and numerical ap-proach, the fundamental understanding of the viscous fluid mud responses to transient wave loadingscan be achieved. Our final goal is to develop depth- integrated, phase resolving wave propagation mod-els, which will include the effects of muddy sealoor. The basic numerical algorithm for solving theBoussinesq-type wave equations has already been developed (Lynett and Liu, 2004) . The inclusion ofviscous boundary layer efects has also been implemented (Liu et al., 2006). Significant amount ofwork remains in coupling various types of mud flows and wave motions. The resulting models need tobe tested by using existing field data and new field data.ReferencesBooij, N., Ris, R. C. and Holthuijsen, L.. H, 1999. A third- generation wave model for coastal regions, 1. Modeldescripion and validation, J. Geophys. Res., 104, 7649 ~ 7666.Dalymple, R. A. and Liu, P. L.-F., 1978. 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Fffects of seafloor conditions on water wave damping, in; Free Surface Flous withViscosity (ed. by P. A. Tyvand) 145 ~ 178. Computational Mechanices Publications.Winterwerp, J. C., De Graf, R. F.. Groeneweg, J. and Luijendijk, A. P., 2007. Modeling of wave damping atGuyana mud coast, Coastal Engineering, 54(3): 249 ~ 261.Yamamoto, T., Koning, H. L., Sellmeigher, H. and Hjum, E. V., 1978. On the response of poro-elastic bed towater waves, J. Fluid Mech., 87, 193 ~ 206.中国煤化工YHCNMHG