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A balanced treatment of secondary currents, turbulence and dispersion in a depth-integrated hydrodynamic and bed deformation model for channel bends

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A balanced treatment of secondary currents, turbulence and dispersion in a depth-integrated hydrodynamic and bed deformation model for channel bends
  A balanced treatment of secondary currents, turbulence and dispersion in adepth-integrated hydrodynamic and bed deformation model for channel bends Lorenzo Begnudelli a, * , Alessandro Valiani b , Brett F. Sanders c a Department of Civil Environmental Engineering, University of Trento, Trento 38100, Italy b Department of Engineering, University of Ferrara, Ferrara 44100, Italy c Department of Civil and Environmental Engineering, University of California, Irvine, CA 92697, United States a r t i c l e i n f o  Article history: Received 25 May 2009Received in revised form 6 October 2009Accepted 7 October 2009Available online 28 October 2009 Keywords: Bed morphologySediment transportChannel bendsDepth-integrated modelFinite volume method a b s t r a c t Thisworkdealswiththeformulationandnumerical implementationofatwo-dimensionalmathematicaland numerical model describing open channel hydrodynamics, sediment and/or scalar transport and riv-erbed evolution in curved channels. It is shown that a well balanced 2D model can predict flow features,sediment and scalar concentration, and bed elevation with an accuracy that is suitable for practical riverengineering. The term ‘‘balanced” implies that important physical processes are modeled with a similardegree of complexity and exhaustiveness. The starting point of the model formulation is the assumptionof self-similarity of vertical velocity profiles (horizontal velocities in the longitudinal and transversedirections), that are scaled by shear velocity and streamline curvature, both resolved by the model.The former is scaled by a bed-resistance coefficient that must be estimated or calibrated – as usual –onaapplication-specific basis, andthelatter is computed byanew, grid-based but grid orientation inde-pendent, scheme that acts on the discrete solution. All processes, including bottom shear, momentumdispersion, scalar dispersion, turbulent diffusion, bed load, and suspended load, are modeled using phys-icallybased,averagedvaluesofempiricalorsemi-empiricalconstants,andconsistentlywiththeassumedvelocity profiles (and bed-generated turbulence). Bed deformation modeling can be implemented witheitheranequilibriumornon-equilibriumformulationoftheExnerequation,dependingontheadaptationlength scale, which must be taken under consideration if significantly larger than the length scale of thespatialdiscretization. Thegoverningequationsaresolvedbyhigh-resolution, unstructured-gridGodunovmethod, which is elsewhere tested and shown to be reliable and second-order accurate. Application of the model to laboratory test cases, using standard parameter values and previously reported bed-resis-tance coefficients, gives results comparable to many 2D and 3D models previously applied to the samecases,mostpartofwhichbenefitfromcase-specificparametertuning.Thereareobviouslyintrinsiclimitsto the descriptive ability of 2D models in river modeling, but the results of this study point to the utilityand cost-effectiveness of a well-designed 2D model.   2009 Elsevier Ltd. All rights reserved. 1. Introduction Numerical modeling of flow and bed deformation in alluvialchannels is of primary importance in river engineering to simulatethe morphological changes and to predict the impact of hydraulicstructures on channel stability, navigability and habitat. To be suc-cessful in channel bends, numerical models must account forimportant three-dimensional (3D) flow features, in particular, aspiral motion where fluid parcels near the free surface deviate to-wards the outer bank, while parcels near the bed deviate towardstheinnerbank[8,9,16,26–28,33,34,46,47,67,97].Thecomponentof  thisspiral motioninthetransversedirectionaretermedsecondarycurrents and are responsible for a redistribution of longitudinalvelocitytowardstheouterbank,aswellasthenettransportofsed-iment towards the inner bank [43,46,47,52]. The 3D structure of  channelflowinbendshasmotivatedmany3Dmodeldevelopmentstudies to predict sediment transport and bed deformations[1,51,53,61,64,74,91].However,thecomputationalcostof3Dmod- els is high and often prohibitive for engineering analysis and de-sign studies. Moreover, the physical knowledge of most part of sediment transport processes is not established concerning 3D as-pects like pick-up fluxes evaluation, significant local bed slope ef-fects on particle entrainment, sediment hiding by different grainsizes,effectsofsedimentsuspensionduetoturbulentburstsratherthantomeanflowconvection, andso on. This has motivatedmanyresearchers to formulate depth-integrated, two-dimensional (2D)models [2,24,25,29,43–47,52,54,57,60,75,77,94–96]. The challenge 0309-1708/$ - see front matter    2009 Elsevier Ltd. All rights reserved.doi:10.1016/j.advwatres.2009.10.004 *  Corresponding author. Tel.: +39 0461 882629. E-mail address: (L. Begnudelli).Advances in Water Resources 33 (2010) 17–33 Contents lists available at ScienceDirect Advances in Water Resources journal homepage:  of 2D modeling is to reasonably depict 3D flow features that affectflow,sedimenttransport,andmorphologicalchange.Therearetwoaspects to this challenge. The first is to reconstruct 3D (horizontal)velocities from 2D flow attributes such as the depth, depth-aver-aged velocity, shear velocity, and 2D streamline curvature. Thiscan be accomplished by adopting one of several different verticalvelocity profiles for the longitudinal and transverse directions, en-abling key flow features such as the near-bed velocity to be esti-mated. The second is to account for the impact of 3D flowfeatures on 2D flow attributes which is addressed using so-calleddispersive terms that arise when convective non-linearities in the3D equations are vertically integrated [37,43,46–48,60]. The as-sumed vertical velocity profiles dictate the expression of theseterms, and are therefore dependent on 2D flow attributes. 2D flowmodels may also need to consider turbulence terms (i.e., Reynoldsstresses) particularly in the context of recirculating flows[2,7,37,44,54,57]. An overview of 1D, 2D, and 3D modeling ap-proaches for river dynamics is presented by Wu [93].There are several examples of 2D flow, mass transport, andmorphological models that carefully consider Reynolds stresseswith turbulence transport models, but ignore dispersive terms,and do not perform well in channel bends [54,57,73,94]. Hence,dispersive terms must be considered in modeling studies of chan-nel bends. However, the appropriate level of approximation is notclear.BernardandSchneider[15]andBernard[14]presentamodel that considers the transport of secondary-current vorticity, whichincreases due to streamline curvature and is dissipated by a num-beroffactors.Thisisthoughttobeimportantinchannelswithvar-iable curvature where there is a lag between curvature andsecondarycurrents. Asimpleralternativeistoassumealocal equi-librium between transverse pressure gradients and radial acceler-ation (e.g. [58]). When this approximation is adopted, it isreasonable to address Reynolds stresses with a similar level of approximation.Forexample,turbulenceintensitiesanddissipationmechanisms can be assumed to be locally controlled by bed shear,not transported as in 2D  k —   type models (e.g. [5,63]). Finally,upon formulation of a model that similarly approximates disper-siveandturbulencetermsanumericalsolverisrequiredwithgoodstability and low dissipation properties over a wide range of flowconditions. For example, Abad et al. [2] adopted the model byBernard[14]whichismathematicallyformulatedforchannelbendapplication, but the numerical solver limits the model to a re-stricted set of flow conditions, since the secondary flow correctiononthebedloaddoesnotseemsoaccurateforhigh-curvaturecases,whichin the opinion of Abad et al. [2] is due to the curvature eval-uation method used in the paper.Inthisstudywe developa balanced2Dmodel for hydrodynam-ics, mass transportandbedmorphologyforchannelbendsthatap-plies asimilar level of approximationrelative to theformulationof secondary currents, dispersive terms and turbulence, as describedabove, utilizes 3D flow structure information to model sedimenttransport, and adopts a stable and robust numerical solver. Theindependent validationof the simple treatment of dispersion, bothfor the passive scalar contaminant transport and the sedimenttransport dynamics in curved flows, is considered to be a reliablesupport to a self consistent mechanical scheme of the real physics.We are motivated by the diversity of previously reported modelswhichvaryconsiderablyincomplexitybutrarelyoffersignificantlybetterresults. Indeed, giventheuncertaintyinsedimentdynamics,the benefit of hydrodynamic model complexityshouldbe carefullyscrutinized.Hence,wearemotivatedtostreamlinemodelformula-tion, making it as simple as possible, and to minimize the numberof parameters that must be specified by user, and the associatedcalibration requirements. This is accomplished by adopting estab-lished values of model parameters wherever possible, and relyingheavily on a case-specific bed-resistance coefficient which engi-neersareaccustedtoestimatingorcalibratingonasite-specificba-sis. For example, the velocity profiles (vertical distribution of longitudinal andtransversevelocities) arescaled byshearvelocity,which is turn related to the chosen bed-resistance parameter, andstreamline curvature, which is computed by a new grid-basedschemethatactsonthediscretesolution.Asisdescribedinthefol-lowing sections, all processes of importance are modeled consis-tent with the assumed velocity profiles (and bed-generatedturbulence) including bottom shear, momentum dispersion, scalardispersion, turbulent diffusion, bed load, and suspended load. Themodel is evaluated using a set of laboratory channel bend testproblems with validation data. In addition, predictions by severalother 2D and 3D models are compared and provide insight intothe merit of the proposed formulation, its utility for engineeringanalysis and design purposes and its limitations. Accuracy relativeto the overall simplicity of the model is emphasized. Having inmind practical river engineering applications, we also stress theimmense challenge (perhaps even futility) of a 3D description of highly variable near-bed dynamics that bear on vorticity andpick-up capacity. According to the present experience, a well-de-signed and carefully verified 2D scheme that adopts analyticalmodels for critical 3D flow features is able to reproduce the essen-tial aspects of the flow field, solute and sediment transport, andbed deformation for a manageable computational cost. 2. Flow model By assuming a hydrostatic pressure distribution, the Reynolds-Averaged Navier–Stokes equations are integrated over the depthof a river (  z   direction) to obtain the following system of equationsthat conserve mass and balance momentum in the horizontaldirections,  x  and  y , @  U @  t   þ  @  E @   x  þ  @  G  @   y  ¼  S  ð 1 Þ where  U ¼ð h   uh   v  h Þ T  ;  h  is the depth,   u  and   v   are the depth-averaged velocity in the  x  and  y  directions, respectively, and  E ,  G  and  S  are defined as, E  ¼ h  uh  u 2 þ  12  gh 2 þ  h ð T   xx  þ  D  xx Þ h  u  v   þ  h ð T   yx  þ  D  yx Þ 264375 ð 2 Þ G   ¼ h  v  h  u  v   þ  h ð T   xy  þ  D  xy Þ h  v  2 þ  12  gh 2 þ  h ð T   yy  þ  D  yy Þ 264375 ð 3 Þ S  ¼ 0   gh @   z  b @   x    c  D  uU    gh @   z  b @   y    c  D  v  U  264375 ð 4 Þ where  g   is the acceleration due to gravity;  z  b  is the bed elevation; U   ¼  ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   u 2 þ  v  2 p   ;  c  D  is the bed drag coefficient, related to theManning’s coefficient  n M   from the relation  c  D  ¼  gn 2 M  h  1 = 3 and tothe Chézy coefficient  C   by  c  D  ¼  gC   2 ;  T   xx ;  T   xy ;  T   yx ;  T   yy  are depth-integratedReynoldsstresses; D  xx ;  D  xy ;  D  yx , D  yy  aredispersionterms. The most important source of momentum dissipation in rivermodeling is bottom shear which is modeled with a quadratic draglawabove.Aspatiallydistributedresistancecoefficientcaninprin-ciplebeusedwithoutcomplicationintheabovemodel, butforriv-er modeling the estimation of such distributions is rarely possiblewithout detailed knowledge of bed forms. Exceptions include thecase of flood inundation modeling where channel and overbanksurfaces are distinctly different. Nevertheless, note that bottomshear stress is computed locally by the model as  s o  ¼ q c  D U  2 , andthe shear velocity is simply  u   ¼ð c  D Þ 1 = 2 U  . The latter forms a key 18  L. Begnudelli et al./Advances in Water Resources 33 (2010) 17–33  component of our Reynolds stress and dispersion formulations de-scribed next.  2.1. Reynolds stress formulation Depth-integrated Reynolds stresses account for the transfer of energy to recirculating flows and should be modeled to predictplanform circulation [37]. Most models adopt the Boussinesq approximation to describe Reynolds stresses in terms of an eddyviscosity  m t   and gradients in depth-averaged velocities (e.g.[2,30,52,57]). This gives the Reynolds stresses as [55], T   xx  ¼  2 ð m t   þ m Þ @   u @   x  ð 5 Þ T   xy  ¼  T   yx  ¼ ð m t   þ m Þ  @   u @   y  þ  @   v  @   x    ð 6 Þ T   yy  ¼  2 ð m t   þ m Þ @   v  @   y  ð 7 Þ where m t   istheturbulenceeddyviscosityand m  isthemolecularvis-cosity which can be ignored in environmental applications as it isseveral orders of magnitude smaller [36]. Researchers have adopted models of varied complexity for theeddy viscosity. The simplest option is a constant eddy viscosity[37];algebraicclosureschemesoftheform m t    u  h areonlyslightlymore complicated, while  k —   turbulent transport models requiresolution of additional scalar transport equations [5,63]. Barbarusti et al. [7] report that the importance of turbulent transport modelsrelative to recirculation zone modeling diminishes in relativelyshallow conditions where bottom shear dominates turbulenceand dampens turbulence caused by lateral shear. In this limit, theeddy viscosity predicted by  k —   models converges to that of thealgebraicmodelunderuniformflowconditions[5,63].Furthermore, Barbarusti et al. [6] report experimental work on recirculationzones and subsequent modeling [7] shows that depth-integratedmodels of 2D recirculation zones are insensitive to the eddyviscosity when  c  D W  = h  >  0 : 1, where  W   represents the width of therecirculationzone measured in the transverse direction.Forthisstudyanalgebraicturbulencemodelisadoptedbecauseitstrikesanattractivebalancebetweenmodelcomplexityandper-formance. As indicated earlier, the distribution of shear stress iscomputedtoaccountforbedshear, soitistrivial tocomputeadis-tribution of   m t   ¼ a u  h  where  a  is a proportionality constant, set inthis study to 0.08 [7,63]. A consequence of this approximation is that poor results are to be expected in applications with recircula-tion zones if   c  D W  = h  >  0 : 1, in which case inclusion of turbulencetransport in the model should be considered.  2.2. Dispersion formulation Whereas Reynolds stresses account for energy transfer to circu-latingflows,dispersiontermsaccountforthetransferofenergyoutof circulating flows and are most important in channel bends[37,43,46–48,60]. Dispersion terms can be expressed as D  xx  ¼  1 h Z   h 0 ð u ð  z  Þ    u Þ 2 dz   ð 8 Þ D  xy  ¼  D  yx  ¼  1 h Z   h 0 ð u ð  z  Þ    u Þð v  ð  z  Þ    v  Þ dz   ð 9 Þ D  yy  ¼  1 h Z   h 0 ð v  ð  z  Þ    v  Þ 2 dz   ð 10 Þ where  u ð  z  Þ  and  v  ð  z  Þ  are the  x  and  y  components of the horizontalvelocity distribution, respectively. We note that  u ð  z  Þ  and  v  ð  z  Þ  arealso functions of   x ;  y , and  t  , as well as   u  and   v  , but this dependenceis not shown to simplify the presentation. The dispersion stresses can be evaluated analytically once avelocity distribution is adopted. For the streamwise componentprevious researchers have adopted power-law distributions[37,45,92], log-law distributions [30], and linear models [30,41,92], while linear models have been used extensively for the transverse component [41,45]. A power-law is adopted here for the longitudinal component and a linear model is adopted forthe transverse component. Indicating with  u ‘  and  u t   the velocitycomponents in longitudinal and transverse direction, respectively,the longitudinal velocity is then given by [58], u ‘ ð  z  Þ U   ¼  m þ 1 m z h   1 = m ð 11 Þ where m ¼ j U  = u  ¼ j ð 8 =  f  Þ 1 = 2 ¼ j C  =  g  1 = 2 ¼ j n M   g  1 = 2 = h 1 = 6 and j ¼ 0 : 41isthevonKarman’sconstant.Thetransversevelocityisgivenby[58], u t  ð  z  Þ v  s ¼  2  z h   1 ;  v  s  ¼  2 m  þ 12 j 2 mhRU   ð 12 Þ where v  s  representsthetransversevelocityatthefreesurfaceand R is the local radius of curvature. An important assumption of thetransverse velocity profile is that transverse pressure gradientsare in equilibrium with the radial acceleration [58]. This is a goodassumption when the radius of curvature remains constant in thestreamwise direction, but questionable when the radius of curva-turevaries.BernardandSchneider[15]derivedatransportequationfor streamwise vorticity that considers vorticity generation (due tocurvature) and dissipation mechanisms to estimate the dispersionterms.ThisisimplementedinthemodelSTREMR [14]whichhasre-cently been modified further by Abad et al. [2]. Another approachproposed by Ghamry and Steffler [40,41] involves the solution of  vertically averaged and moment (VAM) equations. In this approacha velocity distribution is adopted for  u ;  v  , and  w , the vertical com-ponent of velocity, as well as pressure  p . Each of these distributionsare modeled with parameters (i.e., moments) and separate equa-tions are solved for each. For example, the  x  component of velocityismodeledas u ¼ u o þ u 1 ð 2  z  = h  1 Þ andevolutionequationsarede-visedfromadepth-averagingprocesstopredict u o  and u 1 .Inall,thismethod requires the solution of a coupled system of 10 equations.The streamwise vorticity and the VAM methods represent alterna-tives to Eq. (12) that promise a more complete process description.However, eachof theserequiresthesolutionof additional transportequations which mandates greater computational effort. In addi-tion, given that a relatively simple algebraic turbulent closurescheme was adopted, for consistency a dispersion treatment of asimilar level of complexity may be justified. Integrationof Eqs. (8)–(10) usingthesevelocityprofilesleadsto[92], D ‘‘  ¼  U  2 m ð 2 þ  m Þ ;  D ‘ t   ¼  D t  ‘  ¼  v  s U  1 þ 2 m sign ½ R  ;  D tt   ¼  v  2 s 3  ð 13 Þ wherethesubscripts ‘ and t  havebeenusedtoemphasizethattheseexpressions apply to a coordinate system aligned with the longitu-dinal and transverse coordinate systems, respectively. Further,  R  istaken to be positive when currents bend clockwise along thestreamwise direction, while negative  R  implies the opposite. Totransformthedispersiontermsrelativeto  x  and  y  directions, arota-tional transformation can be applied as follows, D  xx  D  xy D  yx  D  yy    ¼  M ð u Þ D ‘‘  D ‘ t  D t  ‘  D tt    M T  ð u Þ ;  M ð u Þ ¼  cos u   sin u sin u  cos u   ð 14 Þ where u  is the angleof the depth-averagedhorizontal velocityvec-tor measuredcounter-clockwise fromthe  x  axis and  M ð u Þ accountsfor a counter-clockwise rotational transformation by an angle  u ,and  M T  represents the transpose of   M . L. Begnudelli et al./Advances in Water Resources 33 (2010) 17–33  19  We remark that this dispersion formulation requires no addi-tional parameters. The required input includes  U  ;  m ;  u  and  R ,and it is straightforward to compute all of these except  R  basedusing model predictions of   h ;   u ;   v   and the assumed resistanceparameter. Curvature evaluation is a unique problem facing 2Dmodels.In3D,the3Dflowfeaturesareresolvedsothereisnoneedto reconstruct velocities. In 1D, no attempt is made to model flowdirection. Focusing on 2D models, strategies to locally estimate  R have been devised for idealized cases where flow is aligned withthe curvilinear grid [23,34,43,45–47,73,79]. These cases have in-volved channels of constant width with bends of constant radiusor whose centerline is expressed by a sine-generated curve, sothe evaluation of the radius of curvature is straightforward. Themodel presented by Struiksma et al. [77] is applied to field cases[76] but it requires as input a piecewise constant value of   R . Manyresearchers have not even reported the method of evaluating  R [30,31,40,52,57]. Abad et al. [2] present a more general, grid- and rotation-insensitive method to evaluate  R  that appears suited topractical applications. However, Abad et al. [2] report that it isnot accurate in high-curvature cases. A new method is developedin this study and it is presented later in Section 6.Despite the relative simplicity and potential importance of dis-persion in channel bends, many researchers have presented flowand sediment transport models that have either ignored disper-sion, or claimed to have captured it through a manipulation of the Reynolds stresses [54,57,73,94]. One potential disadvantage to such models is the introduction of additional parameters thatmust be tuned to bring model predictions in line with observa-tions. On the other hand, it is also possible that these stresses aresimply too small in comparison to bottom shear to significantlyimpact flow conditions. Numerical tests presented later are de-signed, in part, to shed insight into this issue. Moreover, Godu-nov-based schemes utilized in this study provide an excellentframework to assess the importance of dispersion and Reynoldsstress terms because of their stability in the limit that these termsvanish.Previousnumericalmodelingstudiesofthisissuemayhaveutilized unrealistically large eddy diffusivities, for stability pur-poses, which masked the contribution of dispersion. 3. Scalar transport model Scalar transport must be considered to account for suspendedsediment andto predict the transport of dissolvedsubstances suchas contaminants, which are often sorbed to sediments. Scalartransport models are also required to solve  k —   type turbulenttransport equations, though this is not done in this study. For gen-erality, the scalar transport equations are presented here in termsof   N   arbitrary scalars,  c  i ;  i ¼ 1 ; . . . ; N   (e.g., suspended sediment,passive tracers, temperature, contaminant concentration) and areobtainedbyverticallyintegratingthe3DReynolds-Averagedtrans-portequations.Writteninadifferentialformsimilartothe2Dflowequations, the transport equations appear as, @  C @  t   þ  @  E C  @   x  þ  @  G  C  @   y  ¼  S C   ð 15 Þ where  C ¼½ h  c  1  h  c  2    h  c  N   T  is the vector of the depth-inte-grated concentrations;  S C   ¼½ s 1  s 2    s n  T  is a generalizedsource/sink term vector that accounts for non-conservative pro-cesses;  E C   and  G  C   are the depth-integrated fluxes in  x  and  y  direc-tion, respectively [3], E C   ¼ h  u  c  1    h ð T   x  c  1  þ  D  x  c  1 Þ h  u  c  2    h ð T   x  c  2  þ  D  x  c  2 Þ h  u  c  N     h ð T   x  c  N   þ  D  x  c  N  Þ 2666437775 ;  G  C   ¼ h  v   c  1    h ð T   y  c  1  þ  D  y  c  1 Þ h  v   c  2    h ð T   y  c  2  þ  D  y  c  2 Þ h  v   c  N     h ð T   y  c  N   þ  D  y  c  N  Þ 2666437775 ð 16 Þ where  T   accounts for Reynolds stresses and  D  accounts for disper-sion. Note that   c  i  represents a depth-averaged scalar concentration. Reynolds stresses account for scalar transport caused by turbu-lent velocityfluctuationsand Taylor [80] showed that these can bemodeled as a Fickian process in terms of a turbulent diffusivity,  e t  that is orders of magnitude larger than molecular diffusivities.Defining a turbulent Schmidt number  r t   ¼ m t  = e t  , the Reynoldsstresses can be modeled as, T   x  c  i T   y  c  i    ¼  m t  r t  @   c  i @   x @   c  i @   y " #  ð 17 Þ where the eddy viscosity is computed from the algebraic turbulentclosure previously described and the Schmidt number is as of yetundefined. Scalar dispersion like momentumdispersion is propertyofreduced-dimensionmodels,butscalardispersioncanbemodeledas a Fickianprocess [32,81,82]. Elder [32] formulated the longitudi- nal dispersion coefficient for a wide rectangular channel as: K  ‘  ¼  1 h Z   h 0 ð u ‘    U  Þ dz  Z   z  0 dz  e  z  Z   z  0 ð u ‘    U  Þ dz   ð 18 Þ and using  e  z  ð  z  Þ¼ j u   z   1   z h    and  u ‘ ð  z  Þ¼ U  þð u  = j Þð 1 þ log  z  = h Þ ,where  u ‘  is the longitudinal velocity, obtained K  ‘  ¼  a ‘ u  h ;  a l    5 : 86  ð 19 Þ Adopting instead a power-law (Eq. (11)), the final result becomesnearly identical. Transverse dispersion can be evaluated in a similar manner byconsidering thetransversevelocityprofile, followingthesame rea-soningadoptedbyElder [32]for longitudinal dispersion. Whenthetransverse dispersion is defined to be normal to the depth-aver-aged velocity, the depth-averaged transverse velocity is zero andthe dispersion coefficient can be computed as, K  t   ¼  1 h Z   h 0 u t  dz  Z   z  0 dz  e  z  Z   z  0 u t  dz   ð 20 Þ Fischer [35] adopted the tranverse velocity profile of Rozovskii[67] to obtain a transverse dispersion coefficient,  K  t  . Using thelinear distribution of velocity of Odgaard [58], Eq. (12) ,  K  t   is givenby, K  t   ¼  a t  u  h ;  a t   ¼  16 j v  s u    2 ¼ ð 2 m þ 1 Þ 2 24 j 7 hR   2 ð 21 Þ Yen[98]reportsthat m  variesfrom4to12, and m ¼ 7hasbeenuti-lized by a number of researchers to describe turbulent boundarylayers. Using Eq. (21), the values of   a t   corresponding to  m ¼ 7 anda range of channel curvatures become, R = h  ¼  50  )  a t   ¼  1 : 9 R = h  ¼  100  )  a t   ¼  0 : 48 R = h  ¼  200  )  a t   ¼  0 : 12 ð 22 Þ Such values are in general agreement with laboratory observations,forexampleWard[90]reportedvaluesof  a t   rangingfrom0.2to1.7.A similar approach has been followed by Holly and Usseglio-Polatera [42], who ignored the turbulent diffusion and simply set a t   ¼ 0 : 23 to account for transverse dispersion. However, this elim-inatesthesensitivityof   a t   to R  whichshouldberetainedforgeneralpurpose river modeling. Now, a rotational transformation is again used to obtain a dis-persion coefficient tensor for the case where flow is directed withan angle  u  measured counter-clockwise from the  x  axis, K  ¼ K   xx  K   xy K   yx  K   yy    ¼  M ð u Þ  K  ‘  00  K  t    M T  ð u Þ ð 23 Þ and the dispersive fluxes follow as, 20  L. Begnudelli et al./Advances in Water Resources 33 (2010) 17–33  D  x  c  i D  y  c  i    ¼ K   xx  K   xy K   yx  K   yy    @   c  i @   x @   c  i @   y " #  ð 24 Þ Note that the combined effects of Reynolds stresses and disper-sion can be modeled as follows, T   x  c  i  þ  D  x  c  i T   y  c  i  þ  D  y  c  i    ¼ K  0  xx  K  0  xy K  0  yx  K  0  yy " #  @   c  i @   x @   c  i @   y " #  ð 25 Þ where  K 0  ¼ K þð m = r t  Þ I  and  I  represents the unit tensor. Implementationoftheaboveexpressionsforturbulentdiffusionanddispersionispossiblewithlocalknowledgeof  U  ;  m ;  u ;  R ,and r t  . All of these except  r t   were used in closing the momentumbal-ance, and here  r t   is set to unity. A number of researchers haveviewed  r t   as a fitting parameter to bring numerical model predic-tions in line withobservations. For example, Ye and McCorquodale[94] adoptedthe2Dturbulencemodel of Rastogi and Rodi [63] but ignoreddispersive terms and comparedmodel predictions to labo-ratory observations of flow and mass transport by Chang [22] whoexperimented using a flume with two 90   bends in alternatingdirections. Results show that the 2D model captures depth-aver-aged velocities reasonable well, but performs poorly relative tomasstransportasaconsequenceofsecondaryflow.Hence,anunre-alistically small Schmidt number,  r c   ¼ 0 : 15, was required to com-pensate for the lack of dispersion. Duan [30] incorporatedmomentum dispersion into a 2D model and revisited the Chang[22]experiment;thiswasfoundtoimprovethevelocityprediction.However, Duan [30] also considered the mass transport observa-tions of Chang [22] but ignored the effect of scalar dispersion. Tobring predictions in line with observations, the Schmidt numberwasviewedasafittingparameterandanunrealisticallysmallvalue, r c   ¼ 0 : 02,wasagainrequired.Theseandotherstudiessuggestthat,insomesituations,itmaybepossibletoobtainreasonablygoodpre-dictions of water depth and velocity in meandering channels with2D models by considering Reynolds stresses and ignoring momen-tum dispersion. However, mass transport in meandering channelscannot be realistically predicted by models that ignore dispersion.  3.1. Magnitude of Reynolds stresses and dispersion terms inmomentum equation In order to better understand the relative importance of Rey-nolds stresses and dispersion terms in momentum equation (1),it is useful to evaluate the different terms as follows.For the Reynolds stresses we have (Eqs. (5)–(7)) hT   x i  x  j    m t   H U R      a C H R   ½ U  2 H   ½ O ð 10  1 Þ   O ð 10  2 Þ  H R   ½ U  2 H   ð 26 Þ where  ½ U   ;  ½ H   , and  ½ R   are representative values of velocity, flowdepth, and radius of curvature, respectively, and  m t   ¼ a U   H  , where a  is set in this study to 0.08 [7,63]. For the dispersion terms we have, considering the transversedispersion (Eq. (13)): hD tt   ¼  13 h v  2 s   ð 2 m  þ 1 Þ 2 12 j 4 m 2 H R   2 ½ U  2 H     O ð 10 1 Þ h i  H R   2 ½ U  2 H   ð 27 Þ and considering the longitudinal direction (Eq. (13)): hD ‘‘  ¼  hU  2 m ð 2 þ  m Þ  ½ U  2 H   m ð 2 þ  m Þ  ½ O ð 10  1 Þ   O ð 10  2 Þ½ U  2 H   ð 28 Þ Therefore, assuming  m  to range from 4 to 12, we can comparethe magnitude of the three terms for the three following cases: H  = R ½ O ð 10  3 Þ  (mild curvature),  H  = R ½ O ð 10  2 Þ  (strong curva-ture), and  H  = R ½ O ð 10  1 Þ (verystrong curvature, laboratory chan-nels). The values are reported in Table 1.According to this quite simplified analysis, transverse disper-sion is equally or slightly less important than Reynolds stressesfor mild curvatures and can probably be ignored for very mild cur-vatures, while it is equally or slightly more important than Rey-nolds stresses for strong curvatures. For very strong curvatures,than can be found mostly in laboratory channels, they becomethe most important term. On the other hand, the longitudinal dis-persion does not depends on the curvature, being generally 1 or 2orders of magnitude smaller than momentum flux  U  2 H  . 4. Bed deformation model The evolution of bed elevation is modeled by solving a massconservation equation for sediment, the Exner equation, whichcan be written as, ð 1   / Þ @   z  b @  t   þ  r   q b  ¼  M  D    M  E   ð 29 Þ where  /  represents the porosity of the bed,  q b  represents the near-bottomvolumetricfluxofsedimentperunitwidthinthehorizontaldirection (bed load), r   represents the 2D divergence operator,  M  D represents the volumetric flux of sediment from the suspended-load layer to the bed-load layer due to settling (deposition), and M  E   represents the volumetric flux from the bed-load layer to thesuspended-load layer caused by turbulence (entrainment). To solveEq. (29), one option is to adopt empirical models for  q b  and  M  E  , torelate  M  D  to a suspended sediment concentration by volume   c  through a settling velocity,  w s , and to solve Eq. (15) to predict sus-pended sediment, e.g.,   c  1  ¼  c   with  s 1  ¼ M  E   M  D . Another approachthat avoids use of a transport equation for suspended sediment isknown as an equilibrium formulation. The essence of the equilib-rium formulation is that horizontal volumetric flux of suspendedsediment per unit width  q s  is based only on local flow conditionsand bed properties. Furthermore, it is assumed that the rate of entrainment and deposition adjusts instantaneously to account forchanges in flow or bed properties that affect  q s . Mathematically,thisiswrittenintermsofthedivergenceofthesuspendedsedimentflux as follows, r   q s  ¼  M  E     M  D  ð 30 Þ In view of Eq. (15), this approach assumes that the accumulation,turbulent diffusion and dispersion terms are small in comparisonto the those appearing in Eq. (30). When Eq. (29) is added to Eq. (30) the result is, ð 1   / Þ @   z  b @  t   þ  r   ð q b  þ q s Þ ¼  0  ð 31 Þ The applicability of the equilibrium assumption can be testedby considering the length (or time) scales ( L s  or  T  s  ¼ L s =  u ) of theadaptation of the sediment suspensions to changes in bed condi-tions. The adaptationlengthscale L s  isbasedonlongitudinal trans-port and is defined as [62, Chapter 10, 91, Chapter 2], L s  ¼  h  u c w s ð 32 Þ  Table 1 Order of magnitude of Reynolds stresses and dispersion terms in momentumequations (1) for different values of the radius of curvature. H  = R ½ O ð 10  3 Þ  H  = R ½ O ð 10  2 Þ  H  = R ½ O ð 10  1 Þ½ hT  ij  = ½ U  2 H   ½ O ð 10  4 Þ O ð 10  5 Þ ½ O ð 10  3 Þ O ð 10  4 Þ ½ O ð 10  2 Þ O ð 10  3 Þ½ hD tt   = ½ U  2 H   ½ O ð 10  5 Þ ½ O ð 10  3 Þ ½ O ð 10  1 Þ½ hD ‘‘  = ½ U  2 H   ½ O ð 10  1 Þ O ð 10  2 Þ ½ O ð 10  1 Þ O ð 10  2 Þ ½ O ð 10  1 Þ O ð 10  2 Þ L. Begnudelli et al./Advances in Water Resources 33 (2010) 17–33  21
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