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Numerical Model for Wave and Current-Induced Sediment Concentration for Cohesive Beaches

The current paper deals with the incorporation of the effect of electrochemical behaviour of cohesive sediments and of the wave friction factor in a numerical model in order to estimate the space and time variation of the cohesive sediment
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  18 Canadian Hydrotechnical Conference 18ième congrès spécialisé hydrotechnique canadienne Challenges for Water Resources Engineering in a Changing World Winnipeg, Manitoba, August 22 – 24, 2007 / 22 – 24 août 2007    1 Numerical Model for Wave and Current-Induced Sediment Concentration for Cohesive Beaches Soroush Sorourian 1 , Ioan Nistor  2 , Ronald Townsend 3   1 Graduate Student, Department of Civil Engineering, University of Ottawa, Canada 2  Assistant Professor, Department of Civil Engineering, University of Ottawa, Canada 3 Professor Emeritus, Department of Civil Engineering, University of Ottawa, Canada  Abstract: The current paper deals with the incorporation of the effect of electrochemical behaviour of cohesive sediments and of the wave friction factor in a numerical model in order to estimate the space and time variation of the cohesive sediment concentration. The processes that govern the behaviour of cohesive sediments differ significantly from those governing non-cohesive sediments. In addition to the physical complexities of cohesive sediment processes, chemical aspects must be considered as well. In this sense, the electrochemical behaviour of cohesive sediments plays an important role as a stabilizing force against the eroding shear force. On the other hand, cohesive sediments are important in terms of environmental impacts due to their association with contaminants. All of these factors point to the need for a multi-disciplinary approach in studies that seek a better understanding of cohesive sediment transport. Although cohesive sediment transport under current and wave action has been investigated by several researchers such as Ross and Mehta (1990), electrochemical resistance of cohesive sediments and the effect of wave friction factor have not been incorporated in previous models. The proposed model includes both of these phenomena. Simulated suspended sediment concentration profiles are compared with experimental data from Vermilion Bay, Louisiana. The comparisons indicate that the model performs satisfactorily, especially at the end of the simulation time when equilibrium has been achieved. 1. Introduction Cohesive sediments are characterized by the presence of attraction forces such as electrochemical forces which dominate the gravity force in resisting turbulence (CEM, 2002). Size, composition and plasticity are specific factors with which these sediments are characterized. The cohesive nature of sediment appears at a size of about 74 µm ( φ >3.76) (Dean and Dalrymple, 2002). Cohesive behaviour of fine sediments becomes very important for particle sizes less than 0.02 mm (Taki, 2001; Mehta and Li, 2003). However, even a small percentage of clay in the bed material is sufficient to make a soil cohesive (10%: Raudkivi, 1998; 25%: Torfs, 1997). The other important factor with respect to the effect of the strength of cohesive bonds is water chemistry, especially salinity. Even sand with particle size up to 120 µm exhibits cohesive behaviour in saltwater (Willis and Krishnappan, 2004). When the salinity exceeds a critical value, the size, density and strength of the flocs change as well (Ariathurani et al. 1977). Therefore, it is better to define cohesive sediment by behaviour rather than by size (CEM, 2002). There are a number of processes associated with the cohesive sediment transport that are schematically presented in Figure 1. One of the most important processes is erosion. Bed material is eroded from the structured bed as a result of excess shear stress and becomes suspended in the water. As the concentration of the suspended material increases, cohesive sediments tend to clog together and form   2sediment flocs. The size and specific weight of these flocs are functions of flow field that adds to the complexity of the study (van Leussen 1988; Krishnappan et al. 1992; Willis and Krishnappan, 2004). The eroded sediments become suspended in the water column and follow the water motion. Cohesive sediments are mostly transported by suspension because of their size (Dean and Darlrymple, 2002). When concentration of the suspended sediments becomes high (about 25 g/l), water becomes entrapped in the sediment deposit which prevents further deposition. This condition is called hindered settling. The whole mass of sediment-water mixture behaves like a poroelastic or viscoelastic fluid. This fluid then flows downhill due to gravity or wave forces in what is known as “fluid mud”. The fluid mud can either be entrained or deposited. During flow, friction between adjacent lower layers of mud generates up currents that can lead up to consolidation or resuspension of the sediments (Willis and Krishnappan, 2004). Hence, if cohesive sediments become consolidated, it is difficult to remove them because of their cohesion (Kamphuis, 2000). Figure 1: Cohesive sediment processes  A number of studies have been conducted for modeling cohesive sediments and mud transport in coastal areas (Sakakiyama and Bijker, 1989; Ross and Mehta, 1990; Shibayama and An, 1993; Li, 1996; Soltanpour et al., 2003). Since physical modeling is expensive and proper material cannot always be provided in order to properly duplicate the behaviour of cohesive sediments at smaller scale, numerical modeling is commonly used to simulate their transport. This paper follow the concepts developed by Mehta and Li (2003) while trying to incorporate the effect of electrochemical resistance of cohesive sediments into the model in order to incorporate some of the currently neglected physical behaviour of cohesive sediment and thus, increase the accuracy of the numerical model. 2. Cohesive Sediment Transport Model 2.1 Suspended Sediment Concentration Model The suspended sediment concentration is calculated based on the one-dimensional time-dependant model initially developed by Mehta and Li (2003). Simulations using this model assume the presence of several hydrodynamic conditions: current, waves or wave plus week current. A weak current is a condition in which wave-induced upward diffusion is modulated by the current, but the wave-induced flow field remains unchanged. Deth Suspended Sediment ConcentrationConsolidated Bed DepositionEntrainmen tWater Deposition Erosion Settling Diffusion mud c  Fluid Mud Deforming Bed   3The model treats sediment concentration in two different ways; fluid mud resuspension and bed resuspension. In the case of fluid mud, entrainment of fluid mud into the water column and its re-deposition can be simulated under conditions of waves or waves plus weak current. The existence of current is not considered for fluid mud, since physically, it does not occur. The resuspension and re-deposition of sediments from bed is simulated for current, waves, or waves plus weak current. The computational domain in the model is divided into two parts: (1) water layer   and ( 2) fluid mud or bed layer  . Assuming isotropic behaviour of sediment in all directions, this mass transport governing equation of sediment can be expressed as: [1] ( ) ( ) ( )( ) cwzzcDzycDyxcDxzwcyvcxuctc smmm ∂∂+⎟ ⎠ ⎞⎜⎝ ⎛ ∂∂∂∂+⎟⎟ ⎠ ⎞⎜⎜⎝ ⎛ ∂∂∂∂+⎟ ⎠ ⎞⎜⎝ ⎛ ∂∂∂∂=∂∂+∂∂+∂∂+∂∂ , where u , v , w  are instantaneous flow velocities in the x, y, z directions, respectively, c  represents sediment concentration, m D  represents the molecular diffusion coefficient, and s w  is the settling velocity.  After dividing each of the velocity and concentration terms into time-averaged values over one wave-period, and into oscillating parts and turbulent parts, applying the closure model with the eddy viscosity concept and neglecting the horizontal convection and diffusion terms in the mass transport governing equation, the wave-mean vertical settling-diffusion equation can be expressed as: [2] 0cwzc  ztc ss  =⎟ ⎠ ⎞⎜⎝ ⎛ +∂∂∂∂−∂∂ , where c  is the sediment time-averaged concentration in the water column and s   is the diffusion coefficient in the z-direction. The over-bar notation referring to the time-averaged concentration is omitted in the rest of the paper for simplicity. The boundary conditions of Eq. 2 are considered to be: •  The upper boundary condition states that there is no vertical sediment flux at the water surface ( ) hz  = . i.e.: [3] 0cwzc  hzss  =⎟ ⎠ ⎞⎜⎝ ⎛ +∂∂ = , •  The lower boundary condition states that the vertical sediment flux at the water-mud interface ( ) 0z  =  is equal to the net resuspension flux: [4] n0zss  Fcwzc   −=⎟ ⎠ ⎞⎜⎝ ⎛ +∂∂ = , in which n F is the resuspension flux. The settling velocity is expressed as (Hwang, 1989): [5] ( )( ) ⎪⎪⎩⎪⎪⎨⎧>+≤=+= 1  212  11sf   2121  11s cc  cc α ccw  cc α w 1111 , where 111   ,  , α and 1  are empirical constants, 1 c  is the minimum concentration for the flocculation settling region and sf  w  is the free settling velocity.   4The diffusion coefficient depends on the flow field and the sediment concentration gradient. Therefore, one can express it in terms of the neutral diffusion coefficient and of the stratification factor as: [6] Φ ns  = , in which sn   is the neutral diffusion coefficient for non-stratified flows and Φ  represents the density stratification correction factor which increasingly reduces the ratio, sns   , below unity, as the concentration gradient increases in the water column. In other words, the flow energy for particle entrainment decreases (Dyer, 1986). The density stratification correction factor is modeled using the Munk and Anderson (1948) formula as: [7] ( )  0  i0 R  α 11 Φ += , In which 0 α  and 0   are non-dimensional empirical coefficients that depend on the effect of suspended sediment on the turbulent mixing length. i R   is the gradient Richardson number and is expressed as: [8] ( ) 2i zuz ρρ gR  ∂∂∂∂−⎟⎟ ⎠ ⎞⎜⎜⎝ ⎛ = , where u is horizontal velocity, ρ is the fluid density and g  is the gravitational acceleration. For the neutral diffusion coefficient under wave motion, the expression of Hwang and Wang (1982) is adopted: [9] kh2sinhkzsinh  H α 222wnw  = , where H  is the wave height, w α  is a non-dimensional wave diffusion constant, T2 π  =  is the wave frequency, T  is the wave period, L2 π k   =  is the wave number and L  is the wave length. The neutral diffusion coefficient under current is expressed as (Vanoni, 1975): [10] ⎟ ⎠ ⎞⎜⎝ ⎛ −=  ∗ hz1z κ  u  nc , where κ   is the von Karman constant and ∗ u  is the friction velocity. For the case of fluid mud, the rate of entrainment is based on the concept of Li (1996) and is expressed as: [11] ( )  ( )( ) ⎪⎩⎪⎨⎧≥−<−−= ==− icig0zsicig0zsig1ig2ic4 bm n R R cw R R cwR R R  α u ρ F , where m ρ  is the density of fluid mud,  b u  is the amplitude of the horizontal velocity immediately outside the bottom boundary layer, 4 α  is a non-dimensional coefficient, ig R   is the global Richardson number and ic R   represents the critical value of the global Richardson number. The Richardson number is defined as:   5[12] 20mig ∆ ug ρρρ R  −= , where  ν T   =  is the thickness of wave boundary layer,  ν  is the kinematic viscosity of water, T  is the wave period and 0 ∆ u  is the magnitude of the maximum difference between velocities across the water-fluid mud interface that is obtained either from measurements or using the wave-mud interaction model, which is based on the work of Jiang (1993). 2.2 Wave-Mud Interaction Model In order to use the wave-mud interaction model, the rheological description of mud must be known. The model is capable of working with three rheological options. First, rheological model is three parameter standard solid developed by (Jiang, 1993). The general stress-strain relation is expressed as follows: [13]  2G  µG2G  µGG 221221 &&  +=++ , where   is the applied shear stress,   is the shear strain, 1 G  and 2 G  are the elastic moduli, 2 µ  is the mud dynamic viscosity, and the dots denote time derivatives. The three-parameter standard solid model can be simplified to become the Voigt model which is less accurate when ∞→ 1 G  .The equation becomes: [14]  2µ  2G  22  & += . If 0G 2  = , Eq. 14 further simplifies to approximate the behaviour of mud-water mixture in the form of a Newtonian fluid: [15]  2µ  2  & = . In the governing equations, convective acceleration terms for both water and mud layers are neglected. The governing equations for continuity and momentum are: [16] 0yvˆxuˆ  j j =∂∂+∂∂ , [17] ⎟⎟ ⎠ ⎞⎜⎜⎝ ⎛ ∂∂+∂∂+∂∂−=∂∂ 2 j22 j2 jej j j j zuˆxuˆ ρ µxPˆ ρ 1tuˆ , and [18] ⎟⎟ ⎠ ⎞⎜⎜⎝ ⎛ ∂∂+∂∂+∂∂−=∂∂ 2 j22 j2 jej j j j zvˆxvˆ ρ µxPˆ ρ 1tvˆ , where  j uˆ  and  j vˆ  are the horizontal and vertical components of the wave orbital velocity vector, respectively, ej µ  is the equivalent dynamic viscosity, subscript  j  represents the water layer, ( ) 1 j =  and mud layer ( ) 2 j =  and  j Pˆ  is the dynamic pressure.
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