175 5 Hydrodynamic Modeling When considering the currents in surface waters as diverse as rivers, lakes, estu- aries, and nearshore areas of the oceans, it is evident that large variations exist in the length and time scales describing these currents. Signicant length scales vary from the vertical dimensions of the microstructure of stratied ows (as lit- tle as a few centimeters) to the size of the basin (up to several hundred kilometers), and time scales vary from a few seconds to many years. Although in principle the equations of uid dynamics can describe the motions that include all these length and time scales, practical difculties prohibit the use of the full equations of motion for problems involving any but the smallest length and time scales. Because of this, considerable effort and ingenuity have been expended to approxi- mate these equations to obtain simpler equations and methods of solution. The result is that many different numerical models of currents in surface waters currently exist. The primary difference between these models is usually the different length and time scales that the investigator believes is signicant for the specic problem. For example, if the details of the ow in the vertical are not thought of as signicant, one can use a two-dimensional, vertically inte- grated model; this may be either steady state or time dependent, depending on whether the time variation is considered signicant. To investigate ows where vertical stratication due to temperature and/or salinity gradients is signicant but horizontal ows in a transverse direction are not, a two-dimensional, hori- zontally integrated model is relatively simple and may be useful. More complex three-dimensional, time-dependent models may be necessary when ow elds vary signicantly in all three directions and with time. In whatever numerical model chosen, for reasons of accuracy and stability, the grid sizes in both space and time must be smaller than the smallest space and time scales that are thought to be signicant. Fluid mechanics is a fascinating and diverse science; texts on the subject abound and should be consulted for descriptions of the fundamental processes and its many and diverse applications. Numerous texts also exist that emphasize civil engineering applications such as river ooding and control, the design of control structures, and the modeling of plumes from power plants or dredging operations. The present chapter is rather brief and does not discuss these subjects; its purpose is simply to give an overview of hydrodynamic modeling as applied to the transport of sediments and contaminants in surface waters. Most examples are rather elementary and are meant to illustrate interesting and signicant fea- tures of ows that affect sediment and contaminant transport. A few more com- plex examples are given to illustrate the present state of the art in hydrodynamic modeling. In the following chapters, additional applications of hydrodynamic © 2009 by Taylor & Francis Group, LLC 176 Sediment and Contaminant Transport in Surface Waters models are described as a complement to the analyses of specic sediment and contaminant transport problems. However, for a thorough understanding of the rich and fascinating eld of uid mechanics in surface waters, additional articles, texts, and conference proceedings should be consulted (e.g., Sorensen, 1978; Mei, 1983; Martin and McCutcheon, 1999; Spaulding, 2006). In the present chapter, the basic three-dimensional, time-dependent conserva- tion equations and boundary conditions that govern uid transport are presented rst. This is followed by brief discussions of eddy coefcients, the bottom shear stress due to currents and wave action, the surface stress due to winds, sigma coordinates, and the stability of the numerical difference equations. By integra- tion of the three-dimensional equations over the water depth, simpler and more computationally efcient, vertically integrated (or vertically averaged) models result and are the topic of Section 5.2. These reduced models are comparatively easy to analyze and require little computer time, but they do not give details of the vertical variation of the ow. For some problems of sediment and contami- nant transport, this detail is not necessary and a vertically integrated model is adequate and gives comparable results to those from a three-dimensional model. Two-dimensional, horizontally averaged models also have been developed and are useful for a qualitative understanding of the ow and for preliminary trans- port studies where vertical variations of the ow are signicant but where ow in one horizontal direction can be neglected (e.g., a thermally stratied lake or a salinity stratied estuary). The basic equations and an example of this type of model are presented in Section 5.3. Three-dimensional, time-dependent models and applications of these models are described in Section 5.4. In the modeling of sediment transport, an important parameter for erosion is the bottom shear stress; this stress is due not only to currents but also to wave action. A simple model of wave action and an application to Lake Erie are described in Section 5.5. 5.1 GENERAL CONSIDERATIONS IN THE MODELING OF CURRENTS 5.1.1 B ASIC EQUATIONS AND BOUNDARY CONDITIONS The basic equations used in the modeling of currents are the usual hydrodynamic equations for conservation of mass, momentum, and energy, plus an equation of state. In waters with variable salinity, a conservation equation for salinity is also needed. In sufciently general form for almost all modeling of surface waters, these equations are the following. Mass conservation:r t t  t t  t t  t t  RRRR t u x v y w z 0 (5.1) © 2009 by Taylor & Francis Group, LLC Hydrodynamic Modeling 177 Conservation of momentum in the x-direction:r t t  t t  t t  t t  t t  t t t t u t u u x v u y w u z fv p xx A u x H 1 R ¤¤ ¦ ¥ ³ µ ´  t t t t ¤ ¦ ¥ ³ µ ´  t t t t ¤ ¦ ¥ ³ µ ´ y A u yz A u z Hv (5.2) Conservation of momentum in the y-direction:r t t  t t  t t  t t  t t  t t t t v t u v x v v y w v z fu p yx A v x H 1 R ¤¤ ¦ ¥ ³ µ ´  t t t t ¤ ¦ ¥ ³ µ ´  t t t t ¤ ¦ ¥ ³ µ ´ y A v yz A v z Hv (5.3) Conservation of momentum in the z-direction:r t t  t t  t t  t t  t t  t t t t ¤ w t u w x v w y w w z p z g x A w x H 1 R ¦¦ ¥ ³ µ ´  t t t t ¤ ¦ ¥ ³ µ ´  t t t t ¤ ¦ ¥ ³ µ ´ y A w yz A w z Hv (5.4) Energy conservation:r t t  t t  t t  t t  t t t t ¤ ¦ ¥ ³ µ ´  t t T t u T x v T y w T zx K T xy K H HHvH T yz K T z S t t ¤ ¦ ¥ ³ µ ´  t t t t ¤ ¦ ¥ ³ µ ´  (5.5) Salinity conservation:r t t  t t  t t  t t  t t t t ¤ ¦ ¥ ³ µ ´  t t S t u S x v S y w S zx K S xy K H HHv S yz K S z t t ¤ ¦ ¥ ³ µ ´  t t t t ¤ ¦ ¥ ³ µ ´ (5.6) An equation of state:r S = S(p,S,T) (5.7) where u, v, and w are uid velocities in the x-, y-, and z-directions, respectively (z is positive upward); t is time; f is the Coriolis parameter, which is assumed constant; p is the pressure; S is the density; A H is the horizontal eddy viscosity and A v is the vertical eddy viscosity; K H is the horizontal eddy conductivity and K v is the vertical eddy conductivity; g is the acceleration due to gravity; T is the temperature; S is salinity; and S H is a heat source term. In this chapter, for sim- © 2009 by Taylor & Francis Group, LLC 178 Sediment and Contaminant Transport in Surface Waters plicity and following convention, the symbol S denotes the density of water; in other chapters, S denotes the bulk density of the sediment, whereas S w denotes the density of water. Several approximations are implicit in these equations. These are (1) eddy coefcients are used to account for the turbulent diffusion of momentum, energy, and salinity; and (2) the kinetic energy of the uid is small in comparison with the internal energy (which is proportional to the temperature) of the uid so that energy transport (Equation 5.5) is dominant and can be described by the transport of internal energy (temperature) alone. However, these equations are more general, and hence more complex and computer intensive, than usually necessary for most surface water dynamics. In most applications, they can be further simplied by making the following approx- imations: (1) vertical velocities are small in comparison with horizontal veloci- ties so that a hydrostatic approximation is valid, and (2) variations in density are small and can be neglected except in the buoyancy term in the vertical momen- tum equation (the Boussinesque approximation). With these approximations, the above equations reduce to the following: t t  t t  t t  u x v y w z 0 (5.8) t t  t t  t t  t t  t t  t t t t u t u x uv y uw z fv p xx A u r H 2 1 R xx y A u yz A u z Hv ¤ ¦ ¥ ³ µ ´  t t t t ¤ ¦ ¥ ³ µ ´  t t t t ¤ ¦ ¥ ³ µ ´ (5.9) t t  t t  t t  t t  t t  t t t t v t uv x v y wv z fu p yx A v r H 2 1 R xx y A v yz A v z Hv ¤ ¦ ¥ ³ µ ´  t t t t ¤ ¦ ¥ ³ µ ´  t t t t ¤ ¦ ¥ ³ µ ´ (5.10) t t  p z gR (5.11) t t  t t  t t  t t  t t t t ¤ ¦ ¥ ³ µ ´  t t T t uT x vT y wT zx K T xy K H HHvH T yz K T z S t t ¤ ¦ ¥ ³ µ ´  t t t t ¤ ¦ ¥ ³ µ ´  (5.12) t t  t t  t t  t t  t t t t ¤ ¦ ¥ ³ µ ´  t t S t uS x vS y wS zx K S xy K H HHv S yz K S z t t ¤ ¦ ¥ ³ µ ´  t t t t ¤ ¦ ¥ ³ µ ´ (5.13) © 2009 by Taylor & Francis Group, LLC Hydrodynamic Modeling 179 S = S(S,T) (5.14) where S r is a reference density. The appropriate boundary conditions depend on the particular problem to be solved. At the free surface, z = I(x,y,t), usual conditions include (1) the specica- tion of a shear stress due to the wind, RTRTA u z A v z vx w vy w t t  t t , (5.15) where T x w and T y w are the specied wind stresses in the x- and y-directions, respec- tively; (2) a kinematic condition on the free surface, t t  t t  t t  HHH t u x v y w0 (5.16) (3) the pressure is continuous across the water-air interface and therefore the uid pressure at the surface equals the local atmospheric pressure p a , p(x,y,I,t)=p a (5.17) and (4) a specication of the heat ux at the surface, qK T z HT T va  t t R () (5.18) where q is the energy ux, H is a surface heat transfer coefcient, and T a is the air temperature. At the bottom, the conditions are (1) those of no uid motion or a specication of shear stress in terms of either integrated mass ux or near- bottom velocity and (2) a specication of temperature or a specication of heat ux. Variations in these boundary conditions and additional boundary conditions are discussed in the following sections. 5.1.2 EDDY COEFFICIENTS The numerical grid sizes for a problem usually are determined from consider- ations of the physical detail desired and computer limitations. Once the grid size is chosen, it is implicitly assumed that all physical processes smaller than this can either be neglected or approximately described by turbulent uctuations. Turbu- lent uctuations usually manifest themselves in an apparent increase in the vis- cous stresses of the basic ow. These additional stresses are known as Reynolds stresses. The total stress is the sum of the Reynolds stress and the usual molecular viscous stress. In turbulent ow, the latter is comparatively small and therefore can be neglected in most cases. Analogous to the coefcients of molecular viscosity, © 2009 by Taylor & Francis Group, LLC 180 Sediment and Contaminant Transport in Surface Waters an eddy viscosity coefcient can be introduced (as has been done in the equations above) so that the shear stress is proportional to a velocity gradient. Similarly, an eddy diffusion coefcient can be introduced so that the heat and salinity uxes are proportional to temperature and salinity gradients, respectively. In turbulent ow, these coefcients are not properties of the uid as in laminar ow but depend on the ow itself, that is, on the processes generating the turbulence. The determination of these turbulent eddy coefcients is a signicant prob- lem in hydrodynamic modeling. Two approaches to this problem are (1) the use of semi-empirical algebraic equations to relate the eddy coefcients to the local ow conditions and (2) the use of turbulence theory to relate the eddy coefcients to the turbulence kinetic energy and a turbulence length scale by means of trans- port equations for these quantities. Both approaches depend on laboratory and eld measurements to quantify parameters that appear in the analyses. However, the use of turbulence theory, although more complex, is also more general and requires less adjustment of parameters for a specic problem. In the rst approach, semi-empirical algebraic relations are used that relate the eddy coefcients to processes (which generate turbulence) and to density changes (which reduce turbulence). Because the scale and intensity of the vertical and hori- zontal components of turbulence are generally quite different, it is convenient to consider these effects separately, as has been done in the equations presented above. The vertical eddy viscosity, A v , and vertical eddy diffusivity, K v , should in general vary throughout the system. Some of the more important generating processes of this vertical turbulence and causes for its variation include (1) the direct action of the wind stress and heat ux on the water surface, (2) the presence of vertical shear in currents due to horizontal pressure gradients, (3) the presence of internal waves, and (4) the effect of bottom irregularities and friction due to currents and waves. If the density of the water increases with depth, stability effects will reduce the intensity of the turbulence. These effects depend on the Richardson number, dened by Ri g z u z  t t t t ¤ ¦ ¥ ³ µ ´ R R 2 (5.19) where u is the mean horizontal velocity. Various empirical equations have been developed that relate the eddy viscosity coefcient and the Richardson number. A typical relation is that developed by Munk and Anderson (1948): AA Ri vv   0 12 110() / (5.20) where A v0 is the value of A v in a nonstratied ow. Values for A v0 are generally on the order of 1 to 50 cm 2 /s. © 2009 by Taylor & Francis Group, LLC Hydrodynamic Modeling 181 In a nonstratied ow, it is believed that the eddy diffusivity is approximately equal to the eddy viscosity. However, for a stratied ow, the mechanisms of turbulent transfer of momentum and heat are somewhat different, and this leads to different dependencies of these coefcients on the Richardson number. For example, a semi-empirical relation (Munk and Anderson, 1948) similar to Equa- tion 5.20 suggests that K v =K v0 (1 + 3.33 Ri) −3/2 (5.21) where K v0 is the value of the vertical eddy diffusivity in a nonstratied ow. Many additional relationships for A v and K v similar to the above two equations have been proposed and used. Horizontal viscosity coefcients are generally much greater than the vertical coefcients. It is found from experiments that the values of the horizontal vis- cosity coefcient increase with the scale  of the turbulent eddies. An empirical relation of this type is Aa H E 13 43// C (5.22) where a is a constant and F is the rate of energy dissipation (e.g., see Okubo, 1971; Csanady, 1973). Observations indicate values of 10 4 to 10 5 cm 2 /s for A H for the overall circulation in the Great Lakes (Hamblin, 1971), with smaller values indi- cated in the nearshore regions. In numerical models, A H and K H are often chosen as the minimum values required for numerical stability (see below). Implicit in the use of algebraic equations for the eddy coefcients is the assumption that the production and dissipation of turbulent mixing are in local equilibrium (Bedford, 1985). However, in many cases, this is not an accurate assumption and the transport of turbulence must be considered. For this purpose, equations have been developed that describe the transport of turbulent kinetic energy, k, and viscous dissipation, F. Alternately, because the viscous dissipation is proportional to k 3/4 /, where  is a turbulence length scale, a transport equation for  can be used instead of a transport equation for F. In either case, the resulting model for turbulence is generally known as a k-F (turbulence closure) model. An example of a k-F model that is widely used is the following (Mellor and Yamada, 1982; Galperin et al., 1988; Blumberg et al., 1992). In this model, the vertical mixing coefcients are given by AA KK vvM vvH   ˆ , ˆ UU (5.23) ˆ , ˆ AqS KqS vMvH CC (5.24) where q 2 /2 is the turbulent kinetic energy, S M and S H are stability functions dened by solutions to algebraic equations given by Mellor and Yamada (1982) as modi- ed by Galperin et al. (1988), and V M and V H are constants. The stability functions © 2009 by Taylor & Francis Group, LLC 182 Sediment and Contaminant Transport in Surface Waters account for the reduced and enhanced vertical mixing in stable and unstable verti- cally density-stratied systems in a manner similar to Equations 5.20 and 5.21. The variables q 2 and are determined from the following transport (or con- servation) equations: t t t t t t t t t t t t q t uq x vq y wq zz K q z q 22 2 2 2 () ()( ) ĐĐ â ă ả á ã t t Ô Ư Ơ à t t Ô Ư Ơ à Đ â ă ă ả á ã ã 2 22 A u z v z v 22 2 3 1 g K z q B F o vq R Rt t C (5.25) t t t t t t t t t t () ( ) ( ) ( )q t uq x vq y wq zz K 222 2 CCC C qq v q z EA u z v z t t Đ â ă ả á ã t t Ô Ư Ơ à t t Ô Ư Ơ à () 2 1 2 C C Đ â ă ă ả á ã ã t t ê ô ơ ạ ằ 2 3 1 g K z q B F o v R R W C (5.26) where K q =0.2q, the eddy diffusion coefcient for turbulent kinetic energy; F q and F represent horizontal diffusion of the turbulent kinetic energy and turbulence length scale and are parameterized in a manner analogous to Equation 5.22; W is a wall proximity function dened as CWK1 2 2 EL(/ ); (L) 1 =(I z) 1 +(h+z) 1 ; L is the von Karman constant; h is the water depth; I is the free surface elevation; and E 1 , E 2 , and B 1 are empirical constants set in the closure model. The above and similar k-F models have been used extensively, and the coef- cients appearing in them have been determined from laboratory experiments as well as from comparison of results of the numerical models with eld measurements. Applications of these models are described in the following sections and chapters. 5.1.3 BOTTOM SHEAR STRESS 5.1.3.1 Effects of Currents A shear stress is produced at the sediment-water interface due to physical interac- tions between the owing water and the bottom sediments. This stress depends in a nonlinear manner on the ow velocity and the bottom roughness. It has been investigated extensively and quantied approximately by means of laboratory experiments, eld tests, and model calibrations. Results for the shear stress are usually reported as ẩ = Sc f qq (5.27) â 2009 by Taylor & Francis Group, LLC Hydrodynamic Modeling 183 where c f is a coefcient of friction and is dimensionless; q is a near-bed, or refer- ence, velocity; and a bold symbol denotes a vector (i.e., the shear stress is a vector aligned with the ow velocity). The components of the stress can be written as U x = Sc f qu (5.28) U y = Sc f qv (5.29) where q=(u 2 +v 2 ) 1/2 . The coefcient of friction depends on the grain size of the sediment bed; sedi- ment bedforms such as mounds, ripples, and dunes; and any biota present. Typical values for c f are between 0.002 and 0.005. In the absence of site-specic informa- tion, a value of 0.003 often is chosen for smooth, cohesive sediment beds. A more accurate value for c f can be determined as an average over the entire sediment bed from calibration of modeled to measured currents when the latter are available. From turbulent ow theory and assuming a logarithmic velocity prole, it can be shown that the coefcient of friction can be determined from c h z f  ¤ ¦ ¥ ³ µ ´ K 2 0 2 2 ln (5.30) where L is von Karman’s constant (0.41), z 0 is the effective bottom roughness, and h is a reference distance or depth. For three-dimensional models or two- dimensional, horizontally integrated models, h is the thickness of the lowest layer of the numerical grid; for vertically integrated models, h is the water depth. From this formula and by assuming a value for z 0 , c f can be calculated and will vary locally as a function of depth and z 0 . However, the difculty with this formula is that an effective value for z 0 is difcult to determine accurately. This is often done by model calibration. For a more accurate determination of z 0 and c f , eld measurements of veloc- ity proles can be used (e.g., Cheng et al., 1999; Sea Engineering, 2004). As a specic example, consider the measurements and analyses made by the USGS and Sea Engineering, Inc., for the Fox River (Sea Engineering, 2004). For fully developed turbulent ow in a river, the near-bottom velocity prole is given by u u n z z  * K C 0 (5.31) where u * =(U/S) 1/2 . This is known as the universal logarithmic velocity prole, or law of the wall (e.g., Schlichting, 1955). It can be rewritten as © 2009 by Taylor & Francis Group, LLC 184 Sediment and Contaminant Transport in Surface Waters CCnz u unz K * 0 (5.32) Once u(z) is obtained from eld measurements and n z is plotted as a function of u, then the z intercept of the line given by the above equation is z 0 and the slope of the line is L/u * . To determine u(z), more than 100 vertical proles of current velocities were measured at various locations in the Fox during four separate ow events. From this, z 0 and u * were determined. Because U = Su * 2 , the coefcient of friction then can be calculated from Equation 5.27 as c U f avg  T R 2 (5.33) where U avg is the vertically averaged velocity. For each vertical prole, c f as deter- mined in this way is shown in Figure 5.1. Also shown is c f as determined from the law of the wall (Equation 5.30) with z 0 equal to 0.2 cm, the average from all the measurements. There is a reasonable t to the data except for shallow waters, where the law of the wall under-predicts the values of c f compared with measured data. This discrepancy may be due to the larger irregularities (bottom roughness) in the local bathymetry near shore; this would increase z 0 and hence c f . Because of this discrepancy, a semi-empirical equation given by c h f   0 004 119 128 . . exp( . ) (5.34) was chosen to approximate the data (Figure 5.1) and was later used in hydrody- namic modeling of the Fox River. 0.025 0.020 0.015 0.010 0.005 0 0 1 2 3 Eq. 5.30 Eq. 5.34 Data 4 5 6 7 8 Depth (m) C f 9 FIGURE 5.1 Bottom shear stress coefcient of friction, c f , as a function of depth from measurements, from the law of the wall with z 0 = 0.2 cm, and from Equation 5.34. (Source: From Sea Engineering, 2004.) © 2009 by Taylor & Francis Group, LLC [...]... depth, and U and V are vertically integrated velocities defined by U u dz (5. 53) v dz (5. 54) ho V ho where u and v are the velocities in the x- and y-directions, respectively, and z is the vertical coordinate Equations 5. 50 through 5. 52 are also often written in terms of the vertically averaged velocities, defined as U avg Vavg © 2009 by Taylor & Francis Group, LLC 1 h 1 h u dz U h (5. 55) v dz V h (5. 56)... water, a wind-shelter coefficient has been proposed (Cole and Buchak, 19 95) that has a range between 0 and 1, depending on the shape and size of the water body © 2009 by Taylor & Francis Group, LLC 188 Sediment and Contaminant Transport in Surface Waters 5. 1 .5 SIGMA COORDINATES In a Cartesian grid, the vertical distances between grid points depend only on z and not on x or y In a variable-depth basin, the... particular problem being studied © 2009 by Taylor & Francis Group, LLC 190 Sediment and Contaminant Transport in Surface Waters 5. 2 TWO-DIMENSIONAL, VERTICALLY INTEGRATED, TIME-DEPENDENT MODELS 5. 2.1 BASIC EQUATIONS AND APPROXIMATIONS Two-dimensional, vertically integrated conservation equations are obtained by integrating the three-dimensional equations of motion (Equations 5. 8 through 5. 14) over the water... to the wind was specified Kv Av u z w x , T z q Av (5. 69) v z w y (5. 70) At the sediment- water interface, the conditions were those of no fluid motion and either zero heat flux or continuity of heat flux and temperature across the sediment- water interface © 2009 by Taylor & Francis Group, LLC 198 Sediment and Contaminant Transport in Surface Waters 5. 3.2 TIME-DEPENDENT THERMAL STRATIFICATION IN LAKE... and Contaminant Transport in Surface Waters 60 Near Surface cm/s 50 40 30 20 10 0 60 280 2 85. 5 286 286 .5 287 287 .5 286 286 .5 Julian Day (1996) 287 287 .5 Near Bottom cm/s 50 40 30 20 10 0 280 2 85. 5 FIGURE 5. 14 Comparison of computed and measured currents near surface and near bottom at the Boeing station Practical salinity units = psu = ppt (Source: From Arega and Hayter, 2007.) Sound An investigation... the surface and near the bottom at the location in Elliot Bay where the steepest bottom slope (and largest induced currents) occurs are shown in Figure 5. 13 for numerical simulations with 5, 10, and 20 vertical layers For 5 and 10 layers, the induced velocities 6 Near Surface 5 cm/s 4 5 3 10 2 1 0 20 0 100 50 150 10 Near Bottom cm/s 8 5 6 10 4 20 2 0 0 100 50 150 Days FIGURE 5. 13 Currents near surface. .. stratification exists and the flow is similar to that in a constant-temperature basin © 2009 by Taylor & Francis Group, LLC 200 Sediment and Contaminant Transport in Surface Waters FIGURE 5. 10 Flow variables at 120 days Heat flux is decaying linearly and is zero at this time Winds are from left to right (a) Temperature distribution, (b) currents in the x-z plane (Source: From Heinrich et al., 1981 With... x x (5. 59) AH u x AH v x z z AV u z (5. 60) AV v z (5. 61) g (K H (5. 62) T x z Kv T z (5. 63) Hydrodynamic Modeling 197 = (T) (5. 64) v dx dz 0 (5. 65) R where R denotes the area of the two-dimensional cross-section in the x-z plane In the analysis, the horizontal eddy viscosity and eddy diffusion coefficients were assumed constant From previous one-dimensional analyses and in analogy to Equation 5. 21,... thermocline formation, © 2009 by Taylor & Francis Group, LLC 196 Sediment and Contaminant Transport in Surface Waters maintenance, and decay, and therefore the time scales of interest were weeks and months rather than hours or even days The purpose was to approximately reproduce and understand the main features of the observed spatial and temporal distributions of temperatures and currents in Lake... (Equations 5. 1 through 5. 5) must be used A schematic of the flow around a cylindrical bridge pier is shown in Figure 5. 17 The flow generally has a time-dependent surface at the water-air interface and a time-dependent surface at the sediment- water interface due to sediment scour The flow approaching the pier forms a bow wave at the free surface on the upstream side; further along the sides and back of . is salinity; and S H is a heat source term. In this chapter, for sim- © 2009 by Taylor & Francis Group, LLC 178 Sediment and Contaminant Transport in Surface Waters plicity and following convention,. Sediment and Contaminant Transport in Surface Waters 5. 1 .5 SIGMA COORDINATES In a Cartesian grid, the vertical distances between grid points depend only on z and not on x or y. In a variable-depth. specic sediment and contaminant transport problems. However, for a thorough understanding of the rich and fascinating eld of uid mechanics in surface waters, additional articles, texts, and