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Arch Comput Methods Eng (2007) 14: 303341DOI 10.1007/s11831-007-9009-3

Depth Averaged Modelling of Turbulent Shallow Water Flow withWet-Dry Fronts

Luis Cea Jernimo Puertas

Mara-Elena Vzquez-Cendn

Published online: 9 August 2007 CIMNE, Barcelona, Spain 2007

Abstract Depth averaged models are widely used in engi-

neering practice in order to model environmental flows inriver and coastal regions, as well as shallow flows in hy-draulic structures. This paper deals with depth averaged tur-bulence modelling. The most important and widely useddepth averaged turbulence models are reviewed and dis-cussed, and a depth averaged algebraic stress model is pre-sented. A finite volume model for solving the depth aver-aged shallow water equations coupled with several turbu-lence models is described with special attention to the mod-elling of wet-dry fronts. In order to asses the performanceof the model, several flows are modelled and the numericalresults are compared with experimental data.

1 Introduction

This paper is focused on the numerical modelling of quasi-2D free surface turbulent flows. The two-dimensional char-acter of a free surface flow is usually enforced by a hori-zontal length scale much larger than the vertical one, andby a velocity field quasi-homogeneous over the water depth.

L. Cea () J. PuertasDepartamento de Mtodos Matemticos y de Representacin,Universidad de A Corua, Campus de Elvia, 15071 A Corua,Spaine-mail: [email protected]

J. Puertase-mail: [email protected]

M.-E. Vzquez-CendnDepartamento de Matemtica Aplicada, Universidad de Santiagode Compostela, Campus Sur, 15782 Santiago de Compostela,Spaine-mail: [email protected]

Under these conditions the 3D Reynolds averaged Navier-

Stokes equations (3D-RANS) can be simplified in order toobtain the depth averaged shallow water equations. Here, theterm shallow refers to a small ratio between the vertical andhorizontal length scales. Shallow flows appear in many en-gineering applications, mainly in river and coastal engineer-ing, but also in certain hydraulic structures like open chan-nels or sedimentation tanks, just to cite some examples.

An important feature of free surface flows is that they areunbounded in space, being the limits of the spatial domainan unknown of the problem to be solved. Frequent problemsin which the limits of the fluid are unknown and unsteady in-clude, among others, tidal flow in estuaries and flood propa-gation in rivers. In those situations it is necessary to computea non-stationary wet-dry front, which is part of the solutionlooked for. In addition to the former considerations, and dueto the large length scales involved in hydraulic engineeringpractise, the Reynolds number is usually large enough forthe flow to be fully turbulent. Hence, in most environmen-tal hydraulic engineering studies it is necessary to deal withunsteady turbulent shallow water flow with wet-dry fronts,and that is what this paper deals with.

The choice of a specific numerical model in hydraulic en-gineering depends on the problem considered. The models

most commonly used nowadays for the computation of shal-low water flows solve either the 3D shallow water equations(3D-SWE) or the 2D shallow water equations (2D-SWE).Although its use is becoming more common due to the in-crease in computational power at the present time, the 3D-SWE are mainly used to model the flow in hydraulic struc-tures or in problems with a relatively simple geometry. Inthe modelling of realistic coastal regions and estuaries thecomputational cost of a 3D model becomes very expensive.First, because the spatial domain is usually extensive and

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complex, such that a large numerical mesh is needed. Sec-ond, because the flow is usually unsteady, specially in en-vironmental problems, and the presence of a non-stationarywet-dry front limits the maximum computational time stepin order to achieve accurate results. For these reasons depthaveraged models are still much more commonly used in en-gineering practise for the modelling of shallow flows. Thedepth averaged shallow water equations have been exten-sively used in order to model the flow in rivers and coastalregions [65, 81, 83, 98], in open channels and hydraulicstructures [11, 36, 62, 100], as well as in flooding and dryingproblems [1, 13, 16, 70].

The modelling of turbulence in shallow water flows hasnot been treated so profusely as in other fluid dynamicsareas. Some depth averaged turbulence models have beenproposed for the 2D-SWE. Those models derive from wellknown RANS turbulence models, including somehow theeffects of bed friction in the turbulence field. Special men-tion should be given to the depth averaged k model pro-posed by Rastogi and Rodi in 1978 [76], which was the first

depth averaged two-equation eddy viscosity model, and it isstill the most commonly used with the 2D-SWE when tur-bulent effects are included in the computation.

The treatment of wet-dry fronts in the 2D-SWE has beenstudied and improved in recent years by several researchers,achieving non-diffusive and stable numerical schemes [13,16, 21, 34, 46, 70]. However, the modelling of turbulentshallow flow with wet-dry fronts is seldom found in the lit-erature. We have intended to give a forward step in that di-rection by using several turbulence models in problems withwet-dry fronts.

An unstructured finite volume model for quasi-2D turbu-

lent free surface flow with wet-dry fronts is presented in thispaper. Several depth averaged turbulence models are consid-ered, including a depth averaged mixing length model andthe depth averaged k model of Rastogi and Rodi [76], withadditional limiters to the production of turbulence. A newdepth averaged algebraic stress model is proposed and com-pared with the k model.

Regarding the numerical schemes, an hybrid second-order/first-order scheme (first order in the water depth andsecond order in the unit discharge) is presented. The hybridscheme uses a second order discretisation for the two unitdischarge components, whilst keeping a first order discreti-

sation for the water depth. In such a way the numerical diffu-sion is considerably reduced, without a significant reductionon the numerical stability of the scheme. In order to avoidspurious oscillations of the free surface when the bathym-etry is irregular, an upwind discretisation of the bed slopesource term was implemented. This has proved to be morestable and accurate than a centred scheme [8, 38, 39].

In order to show the capabilities of depth averaged mod-els in the computation of turbulent flows, the finite vol-ume model developed has been used to model the flow in

a coastal estuary, in an open channel with a 90 bend andin two designs of vertical slot fishways. The estuary con-sidered has extensive flat marsh areas which flood and dryperiodically due to the tidal driven flow. This makes it pos-sible to test the turbulence models in the presence of wet-dry fronts in complex two-dimensional geometries with avery irregular bathimetry. The 90 bend has a simple geom-

etry, but at the same time it presents a recirculation bubble,which makes it well suited for testing and comparing differ-ent turbulence models. The flow in the fishway is highly tur-bulent and it has strong recirculation regions, which makes ita perfect test case for the turbulence models. All the numer-ical results are compared with extensive experimental data,which permits us to evaluate the degree to which the shal-low water hypotheses are fulfilled in the considered flows.This is important because, as it is pointed out by Lloyd andStansby [55], due to the assumptions made in the derivationof the shallow water equations, and considering the numeri-cal dissipation inherent to the numerical schemes, the accu-

racy of the results is problem dependent and usually uncer-tain.

The rest of this paper is organised as follows. Section 2summarises some previous theoretical, experimental and nu-merical studies about turbulent shallow water flows, as wellas some features which distinguish them from 3D flows. Thederivation hypotheses made in the 2D-SWE are summarisedand discussed, in order to understand the limitations of theequations we are working with, and to be able to asses criti-cally the numerical results obtained from them.

Section 3 includes some notions about turbulence mod-elling in shallow waters. The depth averaged mixing length

model as well as several versions of the depth averaged kmodel are presented and discussed briefly.

In order to account for non-isotropic turbulence withouta significant decrease in the numerical stability, a depth av-eraged algebraic stress model is proposed in Sect. 4 as anextension of the 2D algebraic stress model, with additionalterms which account for the production of turbulence due tobed friction.

In Sect. 5, a finite volume solver for the turbulent shal-low water equations is presented. The numerical schemesand discretisation techniques implemented in the solver are

described in detail, and some recent advances in this fieldare discussed.Section 6 is intended to show the results obtained with

the model in three specific real engineering applications: thetidal flow in the Crouch estuary (Essex, England), the openchannel flow around a 90 bend, and the turbulent flow intwo different designs of vertical slot fishways. The resultsobtained with the mixing length, the k and the algebraicstress models are discussed and compared with comprehen-sive experimental data.

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Depth Averaged Modelling of Turbulent Shallow Water Flow with Wet-Dry Fronts 305

Finally, the main conclusions and contributions of the pa-per are summarised in Sect. 7, and future research lines areproposed.

2 Numerical Modelling of Shallow Flows in Hydraulic

Engineering

2.1 Shallow Water Flows

The study of turbulent shallow water flows is strongly linkedwith 2D flow and 2D turbulence. Shallow water flows arethose in which the vertical characteristic length scale ismuch smaller than the horizontal one. The water depth lim-its the development of the 3D turbulent eddies in the ver-tical direction, enhancing the 2D character of the turbulentstructures with a characteristic length larger than the waterdepth. The possibility of distinguishing between quasi-2Dand 3D turbulent structures depends on the ratio between

the horizontal and the vertical turbulent length scales. The2D state is approached as the water depth diminishes, but,in any case, no matter how small the water depth is, there isalways some interactions between the quasi-2D and the 3Dstructures.

There are several studies which reveal that the large hor-izontal quasi-2D coherent structures play an important rolein shallow water flows. Plane turbulent jets in shallow wa-ters have been extensively studied experimentally by Dracoset al. [35], Giger et al. [42], Thomas et al. [86] and Chen etal. [26]. Shallow water wakes around circular cylinders andconical islands were studied by Lloyd et al. [5557] as well

as by Chen and Jirka [24, 25]. The homogeneous decayingturbulence produced by a grid in shallow flows was investi-gated by Uijttewaal and Jirka [91], who found the mergingof vortex which characterises the inverse energy transfer in2D turbulence. There are several studies by Uijttewaal et al.[90, 92]andbyChuetal.[27] about the shallowness and bedfriction effects in the development of 2D turbulent structuresin free surface mixing layers. The turbulence characteris-tics in open shallow channels were measured and analysedby Prooijen et al. [94]. Different generation mechanisms oflarge coherent structures in shallow jets, wakes and mixinglayers were proposed by Jirka [51].

All the previous theoretical and experimental studieshave revealed the critical role which large 2D turbulentstructures play in shallow water flows, as well as the impor-tance of the bottom and free surface boundary conditions,which confine the flow and modify the turbulence proper-ties. The no-slip bottom condition enhances the 3D turbu-lence production, while the slip condition at the free surfacepromotes 2D turbulence [92]. The confinement of the flowby the bottom and free surface prevents the generation ofvorticity by the vortex stretching mechanism, which makes

an important difference between quasi-2D shallow flows and3D flows over 2D geometries, since the vortex stretchingprocess appears only in the latter.

The 2D properties are not always present in shallow wa-ter flows. The stability of the large 2D structures depends onthe balance between the effect of horizontal shear, whichproduces 2D structures, the effect of shallowness, whichdamps the 3D instabilities and promotes 2D turbulence, andthe effect of bed friction and vertical shear, which dampand stabilise the large 2D eddies by generating 3D turbu-lence [28, 92].

Wolanski et al. [99] studied the wakes formed behind is-lands and decided to characterise the flow pattern by an is-land wake parameter P, which they defined as:

P = U H2

t,z D, (1)

where U is the free stream velocity, H is the water depth,D is the island characteristic diameter and t,z is the vertical

eddy viscosity, which can be approximated as:

t,z = 1/6ufH. (2)

From (2) and (1), we get:

P = 6U H2

ufH D= 6

U

uf

H

D. (3)

According to the study of Wolanski et al., when the islandwake parameter is very small (P 1) the flow is stable andno bubble appears in the wake, for P 1 a stable turbu-lent wake is formed behind the island, and for P 1 thebed friction becomes negligible and an unsteady wake ap-pears. Further theoretical and experimental studies on theshallow water flow around islands have been made by Chenand Jirka [24, 25]. In order to classify the shallow wake be-hind a cylinder they used a wake stability parameter S de-fined by Ingram and Chu [49] as:

S= 2cfD

H= 2

u2f

U2

D

H, (4)

where cf is the bed friction coefficient, defined as cf =b

U2= u

2f

U2, and b is the bed shear stress. Chen and Jirka

found that above a critical vertical Reynolds number ofRv = U H > 1500, the flow structures depend mainly on thewake stability parameter S, being quite insensitive to thehorizontal Reynolds number (Rh = U D ). This implies thateven if shallow flows can be considered as 2D in the hor-izontal plane, they should not be characterised by the hor-izontal Reynolds number (as it would be the case of 2Dflow), but the wake stability parameter should be used in-stead. For small values of the stability parameter (S < 0.2) avon Krmns vortex street appears behind the body. This is

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the case of a small bed friction and a large water depth. Asthe bed friction increases and the water depth diminishes,an unsteady bubble appears in the wake for values of thewake stability parameter in the range 0.2 < S < 0.5. Forlarger values ofS the wake is stable and the bubble becomessteady. Similar results were obtained by Lloyd et al. [55, 57]in flows around conical islands.

The role played by the stability parameter S in shallowflows can be explained by dimensional analysis of the shal-low water equations. Simple considerations about the char-acteristic scale of each term in the x-momentum equationgives:

hU2

x= gh zs

x b,i

+

x

( + v + h)h

U

x

,

(5)U2H

L gH

2

L u2f

U H

L2 ufH

U H

L2 U L U H

L2.

In (5) only the terms in x-direction have been consid-ered for simplicity in the notation. The total diffusion term

has been split in three terms which account respectively forviscous diffusion, turbulent diffusion due to 3D bed gener-ated turbulence, and turbulent diffusion due to 2D horizontalturbulent structures. The diffusivity coefficient for each dif-fusion term is proportional to , ufH and U L respectively.Considering that the convective term is of order (1), thesimilarity relations given by (5) can be expressed as:

1 gHU2

u2f

U2

L

H

LU uf

U

H

L 1. (6)

The following non-dimensional ratios have formed fromthe previous dimensional analysis:

gH

U2= 1

F2R

, cfL

H= S

2,

(7)

U L= 1

Rh,

cf

H

L= 1

Rv,t.

Considering usual values in hydraulic engineering, theorder of magnitude of the previous ratios can be estimatedas follows. The horizontal Reynolds number in environmen-tal flows is always much larger than Rh > 106, and so theviscous forces are negligible (except in the near wall re-gion). The Froude number FR is usually smaller than 1

and therefore, the hydrostatic pressure term is always im-portant and should always be considered. The other twonon-dimensional ratios involve the bed shear stress. In en-vironmental free surface flows the bed friction coefficientcf is generally in the range 0.0050.01 [24, 51, 90], andthe ratio H

Lis usually smaller than 0.1 in shallow flows.

Then the wake stability parameter is in general larger thanS > 0.1. The vertical turbulent Reynolds number is of theorder Rv,t 100 ( 1Rv,t 0.01) and thus, it has less impor-tance in the flow pattern than the wake stability parameter S.

This dimensional analysis confirms the results of Chen andJirka [24, 25] regarding the importance of the wake stabilityparameter in the analysis of shallow flows.

The results of Wolanski [99], Chen and Jirka [24, 25],and Lloyd [55, 57], show clearly that unlike in unboundedflows, where the flow patterns are classified according to theReynolds number, in shallow waters the flow patterns de-pend strongly on the water depth and on the bed friction.These parameters establish the differences between 2D un-bounded flow and 2D shallow water flow.

2.2 RANS Models for Free Surface Flow

The most general approach to model free surface flows, al-though not the most often used, is to compute the fully 3Dflow field with a specific treatment of the free surface bound-ary. The main drawback to a fully 3D approach is its com-putational cost, specially in environmental problems, wherethe size of the spatial domain is very large and there are

flow patterns of different length scales involved in the flow.For this reason, it is not yet efficient to use the fully 3D ap-proach in river engineering applications, as it is pointed outin recent works by Duan [36] and Minh-Duc [61]. Still, the3D approach can be used to compute the flow in hydraulicstructures. Olsen [66, 67] used a 3D-RANS model in orderto compute the flow in a spillway, where the vertical ve-locity is important, the extension of the spatial domain issmall, and the geometry is simple. Wu et al. [101] used thesame approach to compute the flow and sediment transportin open channels. Since 3D-RANS free surface models willnot be treated profusely in this paper, a brief description of

the current approaches is done in the following, in order toprovide the interested reader with some basic concepts andreferences for further reading.

The characteristic feature and main difficulty in 3D-RANS free surface models compared with models over afixed geometry is the tracking of the unsteady free surface.There exist several numerical techniques to compute the freesurface position in 3D computations. A rather general classi-fication distinguishes between mesh methods and meshlessmethods. Meshless methods use a Lagrangian formulationin order to compute the movement of fluid particles applyingNewtons Second Law. The most popular meshless method

nowadays is the Smoothed Particle Hydrodynamics (SPH)method. It has the advantage of being able to treat compli-cated free surface deformations, but it has problems with thecorrect modelling of boundaries.

Mesh methods can be classified in moving grid methodsand fixed grid methods. Moving grid methods use a La-grangian formulation in order to move the grid nodes andboundaries with the fluid [48, 89]. The free boundary iscomputed with a front tracking technique. A popular mov-ing grid method for small deformations of the free surface is

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Depth Averaged Modelling of Turbulent Shallow Water Flow with Wet-Dry Fronts 307

the Arbitrary Lagrangian-Eulerian method (ALE), in whichthe boundary of the grid moves with the fluid and the innernodes of the mesh move arbitrarily in order to reduce meshdistortion. The advantage of this approach is that the freesurface is resolved sharply, without diffusing it. The maindisadvantage is the computational cost, since the nodes ofthe mesh move at each iteration, and thus, the geometricproperties of the mesh need to be recomputed. Lagrangianmethods are mainly used when the movement of the freesurface is small, because otherwise it is necessary to add orremove some nodes from the mesh in order to avoid a largedistortion of the elements. In the latter case it is necessary toremesh the whole domain, which is time consuming.

Fixed grid methods are more commonly used. They canuse a fully Eulerian formulation (interface capturing) or acombined Eulerian-Lagrangian formulation (interface track-ing). Fixed grid methods introduce a volume fraction func-tion which defines if an element is completely wet, partiallywet or completely dry. Among the many Eulerian methods

which have been proposed, it should be mentioned the Vol-ume of Fluid (VOF) method and the level set method, al-though the latter one has some problems with mass conser-vation [41]. The VOF method is very popular, existing sev-eral formulations [58, 78]. Some VOF formulations solveonly for the liquid phase, using a constant pressure bound-ary condition at the free surface, while other VOF methodsneed to solve for both liquid and gas phases. Nonetheless,in its original formulation, proposed by Hirt and Nicholsin 1981 [44], the VOF method solves only for the liquidphase. The method solves a transport equation for a function which takes the value 1 in the regions filled with fluid, and

the value 0 elsewhere. At each time step the transport equa-tion for is solved in a fixed mesh, the free surface is recon-structed according to the values of at the mesh nodes, andsuitable boundary conditions are applied at the free bound-ary. In order to avoid loss of accuracy in the definition ofthe interface due to numerical diffusion when advecting thefunction , a sharp reconstruction of the interface must bedone at each time step. Several methods of reconstructionexist in the literature [3, 82], being that step an importantfeature of the method. The fully Eulerian methods capturethe interface using the function , that is why they are of-ten called interface capturing methods. On the other hand,

combined Eulerian-Lagrangian methods use a front-trackingtechnique (like Lagrangian methods) in order to follow thefree surface, although all the computations are done over afixed grid (like Eulerian methods). In this category fall theheight-function methods [63, 64] (which use a single-valuedheight function zs = f(x,y) in order to define the free sur-face), the surface marker methods [63, 64], and the volumemarker methods.

From the previous description, which does not include allthe existing methods for tracking the free interface, but only

the most popular ones, it is obvious that there are many pos-sibilities for computing the free surface in 3D models. De-pending on the problem considered some methods are moreefficient than others. In the particular case of environmentalfree surface flows the most suited methods are probably theheight-function methods for cases in which the free surfacecan be defined by a single-valued function zs = f(x,y).This can be usually done in environmental problems, withfew exceptions, as for instance if we are interested in mod-elling the shape of a breaking wave. In the case of a multi-valued free surface, the Volume of Fluid method in its orig-inal formulation (modelling only the fluid phase) is a sen-sible option, and it has been reported to produce satisfac-tory results for modelling the flow in spillways [44, 77].At the present time, no published applications of the VOFmethod for modelling the flow in rivers and coastal regionsare known to the authors.

In shallow water flows it is possible to simplify the 3D-RANS equations assuming an hydrostatic pressure distrib-ution. In such a case the vertical momentum equation is

simplified to the hydrostatic pressure equation, and there-fore, only the two horizontal momentum equations need tobe solved in a 3D mesh. The continuity equation is usedin order to compute the free surface level, which in turn,defines the hydrostatic pressure distribution. The numericalmesh is often built as a 2D horizontal mesh with several lay-ers in the vertical direction, being the number of layers de-pendent on the expected complexity of the vertical velocityprofile. This way of building the 3D mesh is not compul-sory, but it simplifies the mesh generation and also, it pro-duces a well oriented mesh for stratified flows, which arerather common in environmental problems. This approach

is called 3D shallow water computations, and it has beenused by Bijvelds et al. [10] to compute the turbulent flowin square harbours, by Lloyd and Stansby [55, 56] to modelthe shallow flow around conical islands, and by Stansby andLloyd [84, 85] to compute the wake around islands in oscil-latory laminar and turbulent shallow flow. All the previoussimulations were done over simple or simplified geometries.Nonetheless, some works have been done using the 3D shal-low water equations to model the free surface flow in envi-ronmental problems with complex geometries. In particular,we should cite the works by Miglio et al. [60] and by Casulliand Stelling [18]. The latter model was extended to non-

hydrostatic free surface flows by Casulli and Zanolli in [19].All of them use a height-function formulation in order totrack the free surface.

2.3 Depth Averaged Models for Free Surface Flow

Further simplifications can be done in order to derive thedepth averaged shallow water equations, also known asSt. Venant equations or 2D shallow water equations (2D-SWE), which are obtained after vertical integration of the

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3D shallow water equations (3D-SWE). The depth averagedformulation has been successfully applied to different prob-lems, obtaining quite accurate results with a relatively lowcomputational cost when compared with the 3D approach.It has also the advantage of being very robust for comput-ing accurately the water depth, even in unsteady problemswith free surface shocks, as it happens to be the case in hy-

draulic jumps and in dam break simulations. The treatmentof unsteady wet-dry fronts, which appear typically in coastalregions and in flooding problems, is also much simpler andstable than in the 3D approach. The 2D depth averaged for-mulation has been extensively used to model the dam breakproblem [7, 15], the propagation and runup of shallow waterwaves [21, 34, 46], flooding and drying problems [13, 16,17, 70], free surface flow in hydraulic structures [11, 23, 62],flow in rivers and estuaries [22, 33, 83, 97, 98, 103], flow incoastal regions [81], and sediment transport in open chan-nels and reservoirs [65, 100]. Another simplification stepcan still be given in order to obtain the 1D St. Venant equa-

tions, which can be applied to channels, hydraulic structuresor rivers, when the transversal effects are of little impor-tance.

Due to the physical assumptions done in the shallow wa-ter models, the accuracy of the results is problem dependent.Lloyd and Stansby [55] used both the 2D-SWE and the 3D-SWE to compute the shallow water flow around conical is-lands, and they found that in some cases the 2D model givesmore accurate results than the 3D model. The authors at-tributed those results to the fact that in a 2D model the verti-cal mixing is instantaneous, while in a 3D model it dependson the turbulence model used.

Regarding turbulence, simple models are often used inhydraulic and coastal engineering. The extension of the spa-tial domain and the different scales involved preclude theuse of fully 3D-LES and DNS techniques in practical prob-lems at the present time. The 2D features which are presentin most shallow water flows have induced Uittenbogaardand van Vossen to simulate the turbulent scales larger thanthe water depth, rather than modelling them [93]. This ap-proach leads to 2D-LES computations. This may seem con-tradictory with the idea of LES being always three dimen-sional, but it can be justified by using a filter size approx-

imately equal to the water depth, and by the fact that thehorizontal large scales have 2D characteristics. Of course,this approach is only appropriate when the water depth isextremely small, otherwise the filter size would not be ade-quate for LES purposes.

The 2D-SWE are often used without any turbulence mod-elling at all, in many cases directly neglecting the turbulentdiffusive forces [13, 34, 46] or using a constant eddy viscos-ity [62]. Two-equation eddy viscosity models, specially kmodels, are usually the more sophisticated models used with

the 2D-SWE. Uijttewaal and Tukker [92] studied the devel-opment of quasi-2D structures in a shallow free-surface mix-ing layer, concluding that the turbulence model used shouldaccount for both, the 3D turbulent structures created by bedfriction and the quasi-2D large structures originated by hor-izontal strain. Hence, the effects of bed friction in the turbu-lent kinetic energy must be included in the model. A classi-cal depth averaged turbulence model proposed by Rastogiand Rodi [76] is a depth averaged version of the famousk model of Jones and Launder [52], with additional sourceterms which account for the bed generated turbulence. Booij[12] proposed the modification of some constants in the bedfriction production terms of the Rastogi and Rodi model.Babarutsi and Chu [4] proposed a two-length-scale depth av-eraged version of the k model which accounts in a differ-ent way for the 3D bed generated turbulence. The same two-length-scale model, but with a slight modification on the dis-sipation equation, was used by Babarutsi and Chu in [5]. Allthese models will be presented and discussed in Sect. 3.

2.4 Mathematical Approximations in Depth AveragedModels

The mathematical derivation of the 2D-SWE can be foundin many hydrodynamic books, and has already been pre-sented by many authors. Minor differences appear from onederivation to another, but basically the process consists in as-suming an hydrostatic pressure distribution, integrating thehorizontal RANS equations over the water depth, applyingLeibnitzs rule, and using the kinematic free surface and bedsurface conditions. Several approximations are done throughthe mathematical derivation. It is important to have in mindthese approximations in order to understand the limitationsof the equations and to interpret the results obtained fromthem correctly. The approximations made are summarisedin the following:

Incompressible flow:The incompressible RANS equations are taken as the baseequations for the derivation. By assuming incompress-ible flow the density variations with the pressure gradientsare neglected, which is a reasonable hypothesis in waterflows.

Hydrostatic pressure:The hydrostatic pressure distribution is mainly a conse-quence of assuming a separation of the vertical and hor-izontal characteristic scales. This occurs when both thehorizontal length scale Ln and the horizontal velocityscale Un are larger than the vertical ones Hn, Wn, whichis a typical characteristic of quasi-2D flows. The defini-tion of the horizontal and vertical scales is not trivial, anddepends on the flow conditions and geometry. In a longshallow wave propagation problem the horizontal scale isgiven by the wave length, while the vertical scale is given

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by the water depth, since it is over those distances thatthe velocity and pressure changes occur. In some casesthe vertical length scale is given by the variations in thebed and free surface elevation rather than by the waterdepth, and therefore, the separation of scales condition(Hn/Ln 1) is actually a restriction on the free surfaceand bed slopes. Therefore, sometimes the 2D-SWE maybe applied to flows with a large water depth, assumingthat the bed and free surface slopes are small. Apart fromthe separation of scales, there are another two conditionswhich must be fulfilled in order to assure an hydrostaticpressure distribution: the Reynolds number must be muchlarger than 1, and the turbulence intensity must be smallerthan 1. Both conditions are usually fulfilled in shallowwater flows. Finally, it should be noticed that the fact ofassuming a separation of length scales is equivalent to ne-glect the vertical accelerations, but it does not mean thatthe vertical velocity is neglected.

Homogeneous behaviour over the water depth:This approximation implies that the values of both the ve-locity and the Reynolds stresses are almost independentof the vertical coordinate. This cannot be assumed as fre-quently as the hydrostatic pressure assumption. Verticalintegration over the water depth of the convective termsin the horizontal RANS equations gives:

hU

t+ hU

2

x+ hUV

y+ Duu

x+ Duv

y(8)

with

Duu

= zs

zb

u2dz, Duv=

zs

zb

uvdz,

U=zs

zb

udz, V =zs

zb

vdz,

u(x,y,z) = U(x,y) + u(x,y,z),v(x,y,z) = V(x,y) + v(x,y,z),

(9)

where h is the water depth, U and V are the depth aver-aged horizontal velocities, u and v are the deviation ofthe horizontal velocities with respect to the depth aver-aged velocities, zb is the bed elevation, and zs is the freesurface elevation.

The fluxes Duu and Duv are often called longitudinal andlateral dispersion stresses respectively. Their relative im-portance respect to the convective and turbulent stressterms depends on the magnitude of the velocities u andv. In the limit case of a complete uniform velocity pro-file over the water depth the dispersion terms vanish. In ageneral case, their value is strongly dependent on the ex-istence of secondary vertical currents, which appear typ-ically when the curvature effects in the velocity field areimportant. These secondary flows create non-uniformities

in the vertical profile of the horizontal velocities u and v.Rastogi and Rodi [76] computed the shallow water flow ina longitudinal channel with secondary flows due to buoy-ancy effects, and found that the bed friction tends to dampthe secondary flows, diminishing in this way the relativeimportance of the dispersion terms. Hence, a better agree-ment between a 2D and a 3D model is obtained for roughbeds rather than for smooth beds.The dispersion terms Duu and Duv are generally ne-glected in the depth averaged equations, assuming the fol-lowing approximation:

Duu

x+ Duv

y= 0. (10)

In recent works, Duan [36] and Lien et al. [54] proposedapproximated expressions for the dispersion terms, basedon experimental velocity profiles obtained in curved chan-nels. Both formulations are specially suitable for bendchannels for two reasons: first, due to the specific velocity

profile assumed over the water depth; and second, becausethey need the definition of a streamwise direction, as wellas an inner and outer bank which cannot be defined in ageneral geometry. Both formulations are quite recent and,to our knowledge, they have only been tested and usedin bend channels, but not in general flow conditions, dueto the two reasons mentioned above. An alternative wayto account for the dispersion terms is to increase the ef-fective eddy viscosity. However, in that case the value ofthe effective viscosity must be fixed on empirical basis,and it depends on the problem geometry, as well as on theflow conditions. Molls and Chaudry [62] use and effective

stress including the laminar stress, the turbulent stress andthe dispersion stress. The same idea is used by Minh-Ducet al. [61], who introduced a coefficient in the k modelin order to increase or decrease the eddy viscosity value.

Depth averaged viscous and turbulent stresses:Slightly different formulations of the viscous and turbu-lent diffusion terms can be found in the literature depend-ing on the authors preference. The most common formu-lation is given by:

xj ( + t)h

Ui

xj , i = 1, 2, (11)

but it is also common to see the diffusion term expressedas:

xj

( + t)

hUi

xj

, i = 1, 2, (12)

or even as:

h

xj

( + t)

Ui

xj

, i = 1, 2. (13)

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310 L. Cea et al.

Even though there are some authors who justify one or theanother formulation based on mathematical assumptionsand approximations, in practise (11) is the most extended,and to our criterion the most adequate, and therefore, ithas been the one adopted in this paper.It has been assumed in the previous expressions that theturbulent stresses are computed with an eddy viscosityturbulence model. In the case that an algebraic stressmodel or a Reynolds stress model was used, the shearstress source term would be given by:

xj

h

Ui

xj

xj

hui u

j

, i = 1, 2, (14)

where ui uj are the depth averaged Reynolds stresses.

2.5 The Depth Averaged Equations

With the former approximations, the turbulent depth aver-aged shallow water equations are obtained as:

h

t+ hUj

xj= 0, (15)

hUi

t+ hUi Uj

xj

= gh hxi

gh zbxi

b,i

+ xj

h

Ui

xj

hui u

j

xj, i = 1, 2.

Additional terms can be added to (15) in order to accountfor differences in atmospheric pressure and wind shear stressin the free surface. Those terms have not been included in(15) for clarity in the notation.

Leaving the turbulent terms apart, which must be com-puted with a suitable depth averaged turbulence model, the2D-SWE are a system of 3 partial differential equationswith 3 unknowns (U , V , h), which are defined over a two-dimensional spatial domain. This is an important reductionin the computational cost with respect to the original 3D-RANS equations, which are 4 equations defined over a 3D

spatial domain, with the additional inconvenience of the freesurface moving boundary.

3 Turbulence Modelling in Shallow Water Flows

3.1 Depth Averaged Turbulence Models for Shallow Flows

The depth averaged horizontal Reynolds stresses u2, uv,v2 appearing in the 2D-SWE need to be computed by means

of a depth averaged turbulence model, which are usually de-rived from RANS turbulence models by introducing in someway the effects of bed friction and shallowness in the turbu-lence field. The most simple turbulence models used withthe 2D-SWE are the depth averaged parabolic model andthe depth averaged mixing length model. Both of them con-sider an equilibrium state of turbulence, and compute theeddy viscosity with simple algebraic expressions assuming

a turbulent length scale dependent on the water depth. Bothare local models, in the sense that the eddy viscosity de-pends only on the local flow field, and it is not transportedwith the flow. The parabolic model accounts only for the tur-bulence generated by bed friction, while the mixing lengthmodel accounts also for the horizontal strain-rate produc-tion of turbulence. The parabolic model was used by Dingand Wang [33], Duan [36], and Hsieh and Yang [45] in thecomputation of several open channel flows. In order to ac-count for the production, transport, and dissipation of turbu-lence, Rastogi and Rodi [76] proposed a depth averaged kmodel for the 2D-SWE, which was used, among many oth-

ers, by Minh-Duc et al. [61] and Wu [100] to compute thesediment transport in open channels, and by Rameshwaranand Shiono [75] to model the flow in meandering channels.

3.2 The Depth Averaged Parabolic Eddy Viscosity Model

This is the simplest turbulence model for shallow flows. Itassumes an equilibrium state of the flow, in which the verti-cal shear stress and the hydrostatic pressure gradient are inperfect balance. By assuming uniform flow the convectionterms in the RANS equations can be neglected, and the x-momentum equation reduces to a balance between vertical

shear and pressure gradient:1

P

x= xz

z= t

u

z, (16)

where is the density of water, P = g(zs z) is the hydro-static pressure, zs is the free surface elevation, and t(z) isthe eddy viscosity (not its depth averaged value). Assuminga logarithmic velocity profile in depth, the model predicts aparabolic profile in the vertical coordinate for the eddy vis-cosity [20], which is given by:

t(z) = ufz1 z

h, (17)where uf is the bed friction velocity = 0.41 is the vonKarmans constant, and h is the water depth. The depth av-eraged eddy viscosity is obtained after vertical integrationof (17) over the water depth, which yields:

t =1

h

zszb

ufz

1 z

h

dz

= 16

ufh = 0.068ufh. (18)

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The parabolic eddy viscosity model does not account forthe effect of horizontal velocity gradients, but only for theturbulence generated by bed friction. It leads to very loweddy viscosity values when the turbulence production due tohorizontal shear is important. It does not account for trans-port and dissipation processes. Despite its simplicity, it hasbeen used by Hsieh and Yang [45], Lien et al. [54] andDuan [36] in the computation of simple channel flows. Itis also used by more complex models in order to account forthe turbulence produced by bed friction.

3.3 The Depth Averaged Mixing Length Model

This is a depth averaged version of the original mixinglength model. In order to account for both the horizontal andthe vertical production of turbulence, the total eddy viscos-ity is split into an horizontal ht and a vertical

vt component.

The horizontal eddy viscosity accounts for the turbulenceproduced by horizontal shear. It is computed as:

ht = l2s

2Sij Sij = l2s

2(S211 + S222 + 2S212), (19)

where ls is the characteristic horizontal turbulent lengthscale, and Sij is the horizontal mean strain-rate tensor com-puted from the depth averaged velocity as:

Sij =1

2

Ui

xj+ Uj

xi

. (20)

On the other hand, the vertical eddy viscosity is generatedby the vertical velocity gradient produced by bed friction.Therefore, it is computed from the parabolic eddy viscosity

model as:

vt =1

6ufh. (21)

The total eddy viscosity is evaluated from the horizontaland vertical values as:

t =

(ht )2 + (vt )2 = l2s

2Sij Sij +

1

6

ufh

l2s

2. (22)

In the inner part of the domain the turbulent length scalels is assumed to be dependent on the water depth, since it

is a restriction on the size of the turbulent eddies. In orderto estimate ls , a fictitious turbulent length derived from theparabolic eddy viscosity model is averaged over the verticalcoordinate, which yields [20]:

ls 0.267h. (23)

It should be remarked that, as it was pointed out inSect. 2, there might be 2D turbulent structures with a lslarger than the water depth. Therefore, the assumption of

ls dependent on the water depth can often lead to an under-estimation of t, specially in flows in which turbulence ismostly 2D.

Equation (23) is not valid near the walls, as it would pre-dict very large length scales. Instead, it is more correct to usethe wall distance as the turbulent length scale in the near wallregions. The final expression for the eddy viscosity given by

the model is:

t = l2s

2

U

x

2+ 2

V

y

2+

U

y+ V

x

2+

2.34uf

h

2,

ls = min(0.267h,dwall), (24)

where dwall is the distance to the nearest wall. In the ab-sence of horizontal velocity gradients, (24) and (18) areequal. Hence, the depth averaged mixing length model tendsto the parabolic eddy viscosity model when turbulence ismainly produced by bed friction. This is the case of flowswith a very low water depth, a relatively rough bed surface,

and small horizontal velocity gradients. The depth averagedmixing length model was used in channel flow computationsby Jia and Wang [50], among others.

3.4 k Models

3.4.1 The k Model of Rastogi and Rodi

The first k model for shallow water flows was proposed byRastogi and Rodi [76] as a depth averaged version for quasi-2D flows of the original k model of Jones and Launder

[52]. Instead of solving for the three-dimensional turbulenceand dissipation, it solves for their depth averaged value.The equations of the model are very similar to those of

the standard 2D k model. The main difference is that itis necessary to introduce a production term in order to ac-count for the production of turbulence due to bed friction.The equations of the model are given by:

hk

t+ Uj hk

xj=

xj

+ t

k

h

k

xj

+ hPk + hPkv h,

ht

+ Uj hxj

= xj

+ t

h xj

+ hc1

kPk + hPv hc2

2

k,

t = ck2

, Pk = 2t(S2uu + S2vv + 2S2uv), (25)

Pkv = cku3f

h, ck =

1

c1/2f

,

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312 L. Cea et al.

Pv = cu4f

h2, c = 3.6

c2c1/2

c3/4f

,

c = 0.09, c1 = 1.44, c2 = 1.92,k = 1.0, = 1.31,

where cf is the bed friction coefficient (u2f = cf|U|2). The

five constants of the model (c, k, , c1, c2) are assumedto have the same values as in the original k model. Theterm Pk accounts for the production of turbulent energy dueto horizontal velocity gradients, and it has the same expres-sion as in the 2D model. The source terms Pkv and Pv areresponsible for modelling the 3D turbulence generated bybed friction.

In uniform channel flow conditions turbulence is in anequilibrium state. In that case the production of turbulentkinetic energy is due to bed friction, and the k equationsreduce to:

Pkv = , Pv = c22

k . (26)

Using the expressions for Pkv and Pv given by (25), thevalues for the turbulent energy and dissipation predicted bythe model in uniform channel flow are given by:

ku =c2

cc2k u

2f =

|U|1/2u3/2f1.08

,

u = cku3f

h=

|U|u2fh

(27)

and the eddy viscosity is equal to:

t,u = cc3kc2

2

c2ufh = 0.08ufh (28)

which is a very similar value to that one given by the depthaveraged parabolic and mixing length models (see (18)). Onthe other hand, when the horizontal shear is much larger thanthe vertical shear, the vertical production terms are negligi-ble compared to the horizontal shear production (Pkv 0,Pv 0), and the model reduces to the standard 2D kmodel.

A slight modification of the coefficient c has been used

in recent works by Minh-Duc et al. [61]. The expressionused is given by:

c =1

(et)1/2c2c

1/2

c3/4f

, (29)

where t is a Prandtl/Schmidt number relating the eddy vis-cosity and diffusivity for the transport of scalars (t = 0.7 in[61]), and e is an adjustable empirical parameter. The con-stant 3.6 used in the definition ofc in (25), corresponds to a

value ofe = 0.11 in (29). In the results presented in this pa-per the original definition of c was used, since (29) wouldneed a previous calibration for each practical application.

3.4.2 The Modified Constants of Booij

The production terms Pkv and Pv in (25) depend on twocoefficients, ck and c. The value of these coefficients deter-

mine the turbulence level in uniform channel flow conditions(see (27)). Booij [12] proposed the following modification inthe value ofck and c:

cBk =1

10ck, c

B =

1

44c, (30)

where the superindex B refers to Booij. The effect of dimin-ishing the coefficients ck and c in such a way is that theproduction terms of both k and due to bed friction (Pkvand Pv ) are strongly reduced. Using the modified coeffi-cients of Booij to evaluate the turbulent kinetic energy anddissipation in uniform channel flow (see (27)) gives:

kBu = 0.44ku, Bu = 0.10u, Bt,u = 1.94t,u , (31)

where the values ku, u, t,u are those given by the originalconstants used by Rastogi and Rodi [76]. Hence, in uniformchannel flow conditions, the modification proposed by Booijreduces the turbulent kinetic energy by a factor of 0.44, butit almost duplicates the eddy viscosity. On the other hand,in flows dominated by horizontal shear the differences withthe Rastogi and Rodi model smear, because the term Pkvbecomes much smaller than Pk in (25).

3.4.3 The k Model of Babarutsi and Chu

Babarutsi and Chu [4] proposed a two-length-scale depthaveraged k model. The main difference with Rastogismodel is the way in which the effects of the 3D turbulenceproduced by bed friction are introduced. Babarutsi and Chupropose to split the total eddy viscosity into a 3D part and a2D part as:

t = 2Dt + 3Dt . (32)

The 3D eddy viscosity accounts for the small scalebed generated turbulence. The equation for 3Dt used by

Babarutsi and Chu is given by:3Dt = 0.08ufh (33)

which is the same value given by the model of Rastogi andRodi for uniform channel flow (see (28)). The 2D part iscomputed from k and , which account for the large-scale2D turbulence generated by horizontal shear, as:

2Dt = ck2

. (34)

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It should be remarked that the turbulent kinetic energy k

does not include the 3D bed generated turbulence. Hence,the transport equations used to compute k and are similarto those used in Rastogis model (see (25)), but the verticalproduction terms are zero (Pkv = Pv = 0) and a new termF is introduced in order to account for the transfer of en-ergy between the large 2D turbulent scales and the small 3Dscales. The equations for k

and

are given by:

hk

t+ Uj hk

xj=

xj

+ t

k

h

k

xj

+ h(Pk F) h,

h

t+ Uj h

xj(35)

= xj

+ t

h

xj

+hc1

k(Pk

(1

c3)F

)

hc22

k.

A value ofc3 = 0.8 is proposed in [4]. Note that in (35)the diffusion term is modelled with the total eddy viscosityt. On the other hand, the production term Pk is computedwith (25), but using only the 2D eddy viscosity 2Dt . All theconstants of the model are the same as in Rastogis model.The sink term F is associated with the negative work doneby the large scale turbulent fluctuations against the bed fric-tion, i.e. the bed friction takes out energy from the large 2Dturbulent structures. It is computed as:

F =cf

[u2(2U2

+V2)

+2uvU V

+v2(U2

+2V2)

]hU2 + V2 ,(36)

where the Reynolds stresses u2, uv, v2 are computed fromthe Boussinesq approximation using the large scale eddyviscosity 2Dt .

In uniform channel flow, the production of large scaleturbulence due to horizontal shear is zero (Pk = 0). Thelarge scale turbulent kinetic energy equation reduces to 0 =F . Therefore, all the large scale turbulent variablesare zero, i.e.:

ku =

0, u =

0, 2Dt,u =

0,

t,u = 3Dt = 0.08ufh,(37)

where the subindex u refers to uniform channel flow con-ditions. In this case all the turbulence is 3D, and generatedby bed friction. The total eddy viscosity is similar to thatone given by the model of Rastogi. On the other hand, whenthe bed friction velocity uf tends to zero, the flow is domi-nated by horizontal shear. In such a case the term F and the3D eddy viscosity may be neglected (F 0, 3Dt 0), all

the turbulence is 2D, and generated by horizontal shear. Theequations for k and reduce to the standard 2D k model.

This model has been used by Babarutsi et al. in [6], withthe constant c3 = 1, to compute shallow recirculating flowsdominated by friction. It has also been used by Babarutsiand Chu to model shallow mixing layers [5], where they ob-tained the same results with c3 = 1 and with c3 = 0.8.

3.4.4 Differences between the 3 Versions of the k Model

The main differences between the previous depth averagedk models appear in friction dominated flows. The modifi-cation of Booij reduces the bed generated turbulence, and asa consequence, the large scale turbulence as well as the totalturbulent diffusion of the model are increased. We have notfound concluding results showing that the modification pro-posed by Booij improves the results of the original model ofRastogi in general flow conditions. Furthermore, the workby Minh-Duc et al. [61] seems to show that the optimum

value of the constant c might be problem dependent.The version of Babarutsi and Chu is more interestingfrom a conceptual point of view, since it distinguishes be-tween two different turbulent length scales. The main differ-ences with Rastogis model appear in shear flows dominatedby friction. Both models converge to the standard 2D kmodel for zero bed friction. On the other hand, in uniformchannel flow, where all the turbulence is generated by bedfriction, both models give the same eddy viscosity value (see(28) and (37)). It is in horizontal shear layers dominated bybed friction where both models show some differences. Al-tai et al. [2] estimate that, in order to produce the modelled

effects introduced by the term F, the water depth should beapproximately 100 times smaller than the horizontal turbu-lent length scale, i.e. the effects of F are only significantin flows which are strongly dominated by friction and at thesame time have large 2D turbulent structures. Babarutsi etal. [6] compared the results given by the models of Babarutsiand Rastogi in a recirculating shallow water flow dominatedby friction. Despite the different eddy viscosity predicted bythe models, no significant differences in the mean flow field,neither in the recirculation length, were found, and there-fore, no conclusion was obtained regarding which modelperforms better. Both models were compared again by

Babarutsi and Chu [5] in order to model transverse mixinglayers in shallow flows dominated by friction. In that casethey found that the results given by the model of Babarutsiwere in better agreement with the experimental data.

All the results presented in this work have been obtainedwith the original depth averaged model of Rastogi and Rodi.The reason of using the original model of Rastogi and Rodiis that it has been used in many river flow and channel sim-ulations, showing a good behaviour [10, 61, 68, 75, 97, 100,102]. On the other hand, the versions of Booij and Babarutsi

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314 L. Cea et al.

have not been so extensively used. In any case, in the appli-cations presented in this work no differences are expected tobe found in the mean flow field obtained with the differentversions of the model. This is because the flow in the verti-cal slot fishway, as well as in the open channel 90 bend, isnot dominated by bed friction, while in the Crouch estuary,although the turbulence is mainly generated by bed friction,there are not strong horizontal shear stresses.

3.4.5 Limiter to the Production of Turbulent Kinetic Energy

It is well known that Boussinesq assumption does not workwell when the turbulence is strongly non-isotropic, as it hap-pens in swirling flows, in regions with strong adverse pres-sure gradients, or near stagnation points. The main cause ofthese problems is that the eddy viscosity is assumed to beisotropic. In depth averaged models, the non-isotropy be-tween the vertical and horizontal directions is somehow in-troduced in the equations by the different ways of account-ing for the production of 2D horizontal turbulence and 3Dturbulence generated by bed friction. However, it still re-mains the problem with non-isotropy in the horizontal plane.

It is well known that in stagnation points the produc-tion of turbulent kinetic energy is overpredicted by the kmodel, which gives much larger values of turbulent kineticenergy than the measured ones [59]. For this reason, a lim-iter in the production term is introduced in almost all theeddy viscosity models in order to avoid too large values ofthe turbulent energy in stagnation regions. Menter [59] lim-its the ratio between the production and dissipation of turbu-lence as:

P = min(Pk, cl), (38)where cl is a constant with a value between 10 and 20. Thislimiter is a rough estimation based on empirical observationswhich does not imply a good modelling of the stagnationregion, but it can improve the numerical results, avoiding anextremely large turbulence level which may propagate andaffect the solution in the whole numerical domain.

In this work the turbulent production limiter has been ap-plied independently to both, the horizontal shear productionand the bed friction production as:

Pk = min(Pk, cl),Pkv = min(Pkv , cl )

(39)

with a value ofcl = 10. The limit imposed in the bed frictionproduction Pkv is specially suited for problems with wet-dryfronts. The term Pkv can take quite large values when thewater depth is very small. In that case, the second limiter in(39) avoids instabilities in the solution, and allows the useof smaller values of the wet-dry tolerance parameter (seeSect. 5).

3.4.6 Realizability in the Eddy Viscosity Models

Another important drawback to Boussinesq assumption,which is somehow related to the excessive turbulence pre-diction in non-isotropic turbulence, is that it can producenegative values of the normal Reynolds stresses, speciallyin stagnation regions. This was pointed out by Durbin in[37]. In order to avoid negative normal stresses, a constraintshould be used in the eddy viscosity value. This procedurealso helps to avoid the excessive turbulent production nearstagnation regions.

In 2D flow, the constraint over the eddy viscosity whichassures that all the normal turbulent stresses remain positiveis given by:

t 0,min[0, max(wi rij , ij )], ifij < 0

with ij = wj wi (75)

with an analogous expression for j . In (75) rij is the dis-tance vector between the two nodes N

iand N

j. The gradi-

ents wi and wj are computed from the values ofw at thenodes (Ni , Ni1 , Ni2 ) and (Nj , Nj1 , Nj2 ) respectively [95](Fig. 2(a)). The limited slopes computed from (75) repro-duce the Minmod limiter [88].

The boundary integral of the convective flux in (73) isapproximated by the numerical flux ij as:

Lij

(Fxnx + Fyny ) dL ij (wL, wR, nij ) (76)

with

ij (wL, wR, nij ) =Z(wL, nij ) + Z(wR , nij )

2

12|Q(wL, wR , nij )|(wR wL),

Z = Fxnx + Fyny , |Q| = X|D|X1,

X = 0 1 1ny Ux + cnx Ux cnx

nx Uy + cny Uy cny

,

|D| = |1| 0 00 |2| 0

0 0 |3|

,

1 = nx Ux + ny Uy , 2 = 1 + c

n2x + n2y ,

3 = 1 c

n2x + n2y , (77)

where wL = wIj and wR = wiJ. In (7677) nij is a normalvector to the cell face Lij with the same length as the cellface. The matrices X, X1 and |D| are evaluated at the meanstate of Roe [80], which is given by:

h =

hLhR, c =

ghL + hR

2,

Ux =

hLUx,L +

hRUx,RhL +

hR

,

Uy =

hLUy,L +

hRUy,RhL

+

hR

.

(78)

5.4 Bed Slope Source Term

When the upwind scheme given by (7478) is used with acentred discretisation of the bed slope source term S, spuri-ous oscillations are generated under hydrostatic flow con-ditions. In order to avoid this unphysical oscillations, thebed slope source term must be discretised with an upwindscheme [9, 38, 39, 96]. A suitable upwind discretisation ofS which is free of spurious oscillations when used with thefirst order scheme of Roe is given by [9, 38]:

Si = SCi

1

Ai

jKi

d

,ij

2 |nij ||Q|ij Q1

ij Sij ,

SCi =1

Ai

jKi

d,ij2

|nij |Sij , (79)

Sij = ghi + hj

2

zb,j zb,id,ij

0nx,ij

ny,ij

,

where Si is the upwind discretisation of the bed slope sourceterm, SCi is a centred discretisation of the source term in the

cell Ci , Sij is a centred approximation of the source term at

the cell face Lij , Ai is the area of the cell Ci , and d,ij =rij nij is the projection of the distance between the nodes Niand Nj (dij ) on to the unit normal vector nij .

In order to prove the convenience of upwinding the bedslope source term, the first order scheme of Roe will be ap-plied to hydrostatic flow conditions. In the hydrostatic as-sumption, the discrete stationary equations reduce to:jKi

ij = Si Ai . (80)

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The numerical flux given by the scheme for the cell Ci isgiven by:

ij =Zi + Zj

2 1

2Xij |D|ij X1ij (wj wi ), (81)

where the subindices ij indicate Roes mean state, i.e. Xij =X(

wij ) and

|D

|ij

= |D

|(

wij ). Considering that in hydrosta-

tic conditions qx = qy = 0 in all cells, the mean state of Roeis reduced to:

h =

hi hj , c =

ghi + hj

2,

(82)qx = 0, qy = 0.

The values of the different vectors and matrices in (81)are given by:

Xij =0 1 1

ny,ij cij nx,ij cij nx,ijnx,ij cij ny,ij cij ny,ij ,

X1ij =1

2cij

0 2cij ny,ij 2cij nx,ijcij nx,ij ny,ij

cij nx,ij ny,ij

,

Dij = cij |nij |0 0 00 1 0

0 0 1

,

|D|ij = cij |nij |

0 0 00 1 00 0 1

,

D1ij =1

cij |nij |

0 0 00 1 0

0 0 1

,

Zi = |nij |

0

gh2i2 nx,ij

gh2i2 ny,ij

, wi =

hi0

0

.

(83)

Introducing expressions (83) into (81) gives, after somemathematical manipulation, the total flux for the cell Ci as:

jKi

ij =

jKi

|nij |2

gh2i + h2j

2

0nx,ijny,ij

jKi

|nij |2

cij (hj hi )10

0

, (84)

where the first addend accounts for the centred contributionand the second addend for the upwind contribution.

The discretisation of the bed slope term is obtained intro-ducing expressions (83) into (79), which yields:

Si Ai =

jKi|nij |

2g

hi + hj2

(zb,j zb,i ) 0nx,ij

ny,ij

+ jKi

|nij

|2 cij (zb,j zb,i )1

00 , (85)

where the first addend accounts for the centred contributionand the second addend for the upwind contribution.

Considering that in hydrostatic flow the water surface el-evation is constant (zb,i + hi = zb,j + hj ), it is straightfor-ward to show that the upwind contribution in the convectiveflux (second addend in (84)) and in the source term (secondaddend in (85)) are equal. The balance of the centred contri-butions follows directly considering that for any closed vol-ume the following property (see (86)) applies for both the x

and the y component of the normal vector:jKi

h2i nx,ij |nij | = h2i

jKinx,ij = 0. (86)

The previous argument is valid for the first order upwindscheme. If the second order extension of Roe scheme isused, the exact balance between the numerical flux and thebed slope source term is broken, even if a second order ex-trapolation is used to evaluate the bed elevation at the cellfaces. This is because the extrapolated value ofhi is differ-ent at each cell face, and therefore,jKi

h2Ij nx,ij |nij | = 0, (87)

where hIj is the extrapolated value of the flux from the nodeNi to the cell face Lij . For this reason, the centred con-tributions in the flux term discretisation (see (84)) and inthe source term discretisation (see (85)) do not balance anymore. There is actually a deficit in the source term given by:

Ri

= jKi g|nij |

2

h2Ij 0

nx,ij

ny,ij (88)and the following relation applies:

jKi

ij = Si Ai + Ri . (89)

We have found that a simple and efficient solution to thisproblem is to use a first order approximation for the waterdepth (hIj = hi ), and a second order approximation for the

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unit discharges. In this case the deficit Ri is zero:

Ri =

jKig |nij |

2h2i

0nx,ij

ny,ij

=

00

0

(90)

and the exact balance between flux and source is recovered.This approach leads to an hybrid scheme which is able to

compute the hydrostatic flow solution exactly, and at thesame time reduces in a very significant way the numericaldiffusion.

Hubbard and Garca-Navarro [47] proposed a high ordercorrection of the source term in order to obtain a fully sec-ond order scheme which gives an exact balance betweenconvective flux and bed slope in the hydrostatic case. Fol-lowing the ideas presented in [47], when using a second or-der scheme (88) can be replaced by the more general rela-tion:

Ri = jKi

[Fx,Ij nx,ij + Fy,Ij ny,ij ]

=

jKi[(Fx,Ij Fx,i )nx,ij + (Fy,Ij Fy,i )ny,ij ].

(91)

Equation (91) is actually a discretisation of the flux gra-dient within the node Ni and the face Lij . Hence, it can berewritten as:

Ri

= jKiFx

x +

Fy

y iI|nij |d,ij

2

=

jKiSiI|nij |

d,ij2

, (92)

where it has been considered that we want to maintain a bal-ance between the convective flux and the bed slope sourceterm in the hydrostatic case. The vector SiI in (92) accountsfor a discretisation of the bed slope term between the nodeNi and the face Lij . Having this in mind, and considering(89), the discretisation of the source term applied to the fi-nite volume defined in Fig. 1 is given by:

Si = Si +1

AiRi = Si

1

Ai

jKi

SiI|nij |d,ij

2

= Si 1

Ai

jKi

|nij |

d,ij2

ghi + hIj

22

zb,Ij zb,id,ij

0nx,ij

ny,ij

, (93)

Fig. 3 Discretisation of the diffusion term

where Si is the new discretisation of the source term. In(93), the upper-case subindices refer to extrapolated vari-ables at the cell faces, and the lower-case subindices referto node values. It is straightforward to show that the sourceterm Si balances exactly the convective flux in the hydrosta-tic case.

All of the applications presented in this paper have beencomputed with both the hybrid scheme (first order for the

water depth and second order for the unit discharges) aswell as with the fully second order scheme. No significantdifferences have been found in the results given by bothschemes, probably because the water depth gradients arenot large enough. On the other hand, the hybrid scheme hasbeen found to be much more stable than the fully second or-der scheme, especially when the bathymetry is irregular andthere are unsteady wet-dry fronts in the solution.

5.5 Turbulent and Viscous Diffusion

The diffusion term is discretised with a semi-implicit cen-

tred scheme. In the following we will refer only to thex-momentum equation, with analogous expressions for they-momentum equation. Integration of the diffusion term inthe x-momentum equation over the cell Ci gives:

Dtot =

Ci

x

eh

Ux

x

+

y

eh

Ux

y

dA

jKie,ij hij

Ux

xnx +

Ux

yny

ij

. (94)

When the eddy viscosity is large, it is convenient to im-plicit the main diagonal of the diffusion source term in order

to relax the stability condition over the time step. In orderto do so, following the ideas of Davidson [29], the total dif-fusive flux (Dtot) is split in two parts: an orthogonal dif-fusion (D) and a non-orthogonal diffusion (D), so thatDtot = D + D.

The gradient of the velocity at the cell face Lij is com-puted applying the Gauss theorem on the volume Aij , whichis defined by the shaded area in Fig. 3. The two edges of thevolume Aij which pass through the nodes Ni and Nj are de-fined by the same normal vector as the cell face Lij , i.e. nij .

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322 L. Cea et al.

The other two edges are parallel to the line which joins thenodes Ni and Nj , and are defined by the normal vector ij ,with modulus |ij | = |rij |, so that ij rij = 0. The area ofthe volume is given by Aij = |nij |d,ij = rij nij . With thesedefinitions, the gradient of Ux at the cell face Lij is evalu-ated as:Ux

x ij 1

Aij AijUx

x

dA

=

1

Aij L Ux nx dL 1

Aij(Ux,j nx,ij + Ux,B x,ij Ux,i nx,ij

Ux,V x,ij ) (95)with an analogous expression for Ux

y|ij . The following ex-

pression is obtained for the discrete diffusive flux at thecell Ci :

Dtot

jKie,ij hij

|nij |d,ij

(Ux,j Ux,i )

Dorthogonal+

jKie,ij hij

dij

d,ij(Ux,B Ux,V)(x,ij nx,ij + y,ij ny,ij )

Dnon-orthogonal

.

(96)

The non-orthogonal part (D||) is treated explicitly withthe rest of source terms, while the orthogonal part (D) issplit as:

D,ij = e,ij hij|nij |d,ij

(Ux,j Ux,i )

= D,ij Ux,j D

,ij

hi qx,i , (97)

where D,ij = e,ij hij|nij |

d,ij is the orthogonal diffusion coef-

ficient. In (97) all the variables are evaluated at time tn ex-cept the unit discharge qx,i , which is evaluated at time tn+1.In this way no additional computational cost is introduced,since there is no need to solve any system of equations. Inorthogonal meshes the vectors nij and ij are perpendicularand therefore, the non-orthogonal diffusion in (96) vanishes(D = 0).

5.6 Discretisation of the k Equations

The modelled k equations are written in vectorial form as:

t+ F,x

x+ F,y

y=

4m=1

Hm,

=

hk

h

, F,x =

hkUxhUx

= Ux,

F,y =

hkUyhUy

= Uy,

H1 =

xj

(( + tk

)h kxj

)

xj(( + t

)h

xj)

,

H2 =

min(2tSij Sij h, 10h)c1

k

2tSij Sij h

H3 =

min(cku3f, 10h)

cu4f

h

, H4 =

h

c2

2

kh

. (98)

The source terms Hm (m = 1, 4) account respectively forthe viscous and turbulent diffusion (H1), the production dueto horizontal velocity gradients (H2), the production due tobed friction (H3), and the dissipation rate (H4). The samesecond order discretisation scheme described for the meanflow equations is used for the k equations. However, inthis case the normal convective flux to the face Lij (Z,ij )depends linearly on the depth averaged velocity as:

Z,ij = (Ux nx + Uyny )ijij = Un,ijij . (99)At each cell face the numerical flux Z

,ij

is computedas:

Z,ij = Un,ijIj +iJ

2 1

2|Un,ij |(iJ Ij ), (100)

where Un,ij is a centred discretisation of the normal veloc-ity to the cell face, and iJ,Ij are linearly extrapolatedvalues from the cell nodes to the cell faces using (74).

All the source terms are discretised at the cell nodes usinga centred scheme. In order to reinforce the stability of thescheme and to help the turbulent quantities k and to remainpositive during the computation, the following semi-implicitlinearization of the source terms is used in the solver [29]:

H = HnNn+1 + HnP. (101)All the negative source terms are discretised in the form

HnNn+1, while the positive source terms are included in

HnP

. The production source terms H2 and H3 are alwayspositive and therefore, they are included in Hn

P. The dissi-

pation source term H4 is always negative, so it is discretisedas:

H4 =

hc2

2

kh

= (

k)n(kh)n+1

c2( k )n(h)n+1

. (102)

The diffusion term H1 can be positive or negative andthus, it is included in HnP or H

nN

n+1 depending on its sign.With these considerations the terms HnN and H

nP in (101) are

given by:

HnN = min

H1

, 0

n+

k

c2 k

n,

(103)HnP = max(H1, 0)n + Hn2 + Hn3.

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5.7 Wet-Dry Fronts

A fixed finite volume mesh is used to discretise the wholespatial domain, and the control volumes are allowed to wetand dry during the simulation in order to model unsteadywet-dry fronts. A wet-dry tolerance parameter (wd) is de-fined, such that if the water depth in a cell is lower than wdthe cell is considered to be dry. In the same way, if the waterdepth at the face Lij is lower than wd, the face is consid-ered to be dry and it does not participate in the calculation.The water depth is never forced to be zero, in order to keepthe mass conservation property of the scheme. The lowestvalue ofwd is desired in order to obtain accurate solutions.However, an excessively low value ofwd promotes numeri-cal instabilities and necessitates the use of a very small CFL,especially when dealing with a very irregular bathymetry.

The wet-dry tolerance parameter is also used in the tur-bulence models. In the dry cells (hi wd) the eddy viscos-ity is set to zero, and the turbulent energy and dissipationare fixed to a residual value (k and eps), which is severalorders of magnitude smaller than the minimum values ex-pected in the flow. At the same time the production termsare set to zero (Pkv = Pk = Pv = P = 0). When a cell iswet (hi > wd) but the water depth is very small, the verti-cal production term Pkv might become very large. In thosesituations the production limiter given by (39) helps to con-trol the production of turbulence, giving more stable results.In such a way, smaller values of wd can be used withoutproducing instabilities in the k model.

The fixed mesh approach needs a suitable wet-dry con-dition at the fluid interface which ensures the conservationof mass and momentum, and is at the same time not diffu-sive and free of spurious oscillations. Assuming a piecewiseconstant distribution of the bed elevation, two conditions areimposed at the wet-dry front: (1) redefinition of the bed ele-vation; (2) reflection condition.

The aim of redefining the bed elevation is to obtain anexact balance at the wet-dry front between the bed slope andthe hydrostatic pressure term for hydrostatic conditions. Asdetailed by Brufau in [14], if the bed slope is not redefinedspurious waves are generated at the front. If the wet-dryfront occurs between the cells Ci and Cj (assuming with-out loss of generality that the cell Ci is the wet one), themodified bed slope at the front (z

b,ij) is defined as [14]:

zb,ij =

hi hj if,zb,j zb,i otherwise,

hj wd and hi < zb,j zb,i . (104)

The treatment of the wet-dry fronts given by (104) givesthe exact hydrostatic flow solution for any bed elevationwithout diffusing the front, and without generating any spu-rious oscillations of the free surface.

(a) Bed redefinition zb,ij = hi . Reflection condition qn,ij = 0.

(b) No redefinition. No reflection condition.

Fig. 4 Wet-dry front

The reflection condition sets to zero the normal unit dis-charge at the cell face where the wet-dry front occurs (qn,ij ),which is the kinetic condition at a wall boundary:

qn,ij = qx,ij nx,ij + qy,ij ny,ij = 0. (105)Condition (105) only applies when hi < zb,j zb,i

(Fig. 4(a)). Setting the normal unit discharge to zero at theinterface is justified by the assumption of a piecewise con-stant bed elevation, which is similar to a representation ofthe bed by a set of small vertical walls. In this way thewet-dry front is only allowed to advance when the waterdepth in the wet cell is larger than the bed step between cells(hi > zb,j zb,i , Fig. 4(b)). It should be noticed that theunit discharge is not set to zero in the left cell Ci , but onlyat the face Lij when computing the convective flux. Condi-

tion (105) also assures that the convective transport ofk and at the wet-dry front is zero when hi < zb,j zb,i . In thiscase the diffusive flux ofk and over the face Lij is also setto zero.

6 Some Applications and Results

6.1 Open Channel Flow in a 90 Bend

Shallow flow in open channels is one potential applicationof the 2D-SWE. In the absence of abrupt obstacles which

might induce 3D flow features, the flow is usually welldescribed by the shallow water approximation. However,strong changes in the direction of the channel can produce2D recirculation eddies, as well as 3D flow patterns whichmight deteriorate the results given by the 2D-SWE. In thissection the free surface flow in an open channel with a 90bend is computed and compared with experimental velocitydata obtained by Bonillo [11].

The open channel has two rectilinear sections joined bya 90 bend (Fig. 5). The first section is 0.86 m wide with a

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324 L. Cea et al.

Fig. 5 Spatial domain in the 90 bend

flat bed. At the end of the first section, just before the bend,there is a small step with a change in the bed elevation ofz

b = 0.013 m. The second section is 0.72 m wide with a

flat bed.The water discharge in the channel is 29.5 l/s, the mean

water depth is around 0.175 m, and the horizontal velocitiesare of the order of 0.20 m/s. The ratio between water depthand horizontal length scale is approximately H /L 0.2,and the Froude number FR 0.15. The wake stability para-meter for this flow conditions can be evaluated using Man-ning formula as:

S= 2cfL

H= 2 gn

2

h0.33L

h 0.035, (106)

where g is the gravity acceleration, n is the Manning coef-ficient, h is the water depth, and L is the horizontal lengthscale, taken to be equal to the width of the channel. Althoughit is out of the scope of this work, it is worth to mentionthat the wake stability parameter could be used in order tocharacterise the properties of the recirculation eddy that isformed just downstream the bend, in the inner part of it. Af-ter the results of Chen and Jirka [24, 25] and Lloyd et al. [55,57], it is expected that the length of the recirculation region,its stability and steadiness will depend on the value of S. Inany case, in order to verify the former assertion a compre-hensive experimental campaign should be undertaken.

The most challenging point of the numerical simulation isthe correct prediction of the separated region which appearsin the inner corner of the bend. It is well known that RANSmodels usually fail to give accurate predictions of strongseparated flows over 3D geometries. However, in this casethe 2D forcing produced by the shallowness of the flow isexpected to reduce the 3D features of the flow field, helpingto improve the accuracy of the results. In order to correctlypredict the recirculation eddy, three depth averaged turbu-lence models have been used in the simulations: the mixing

(a) First order scheme.

(b) Second order scheme.

Fig. 6 Influence of the numerical scheme on the velocity field. Longi-tudinal velocity Vx (m/s) field

length model, the k model of Rastogi and Rodi, and the

depth averaged algebraic stress model presented in Sect. 4.All the models have been used with the original constants,without any modification or previous calibration.

It is very important to use an accurate numerical schemewith a low numerical diffusion. Otherwise the solution maybe too diffusive, and the velocity profiles too flattened. Inorder to check the influence of the numerical scheme on thesolution, two different numerical schemes have been used:the first order Roe scheme and its second order extension.The velocity fields shown in Fig. 6 have been obtained with-out using any turbulence model. Further, the effective vis-cosity was set to zero, and therefore, all the diffusion comes

from the numerical scheme. The first order scheme is toodiffusive, and thus, it is not able to generate a recirculationregion downstream the bend (Fig. 6(a)). On the other hand,the second order scheme, with a much lower numerical dif-fusion, produces a large recirculation bubble (Fig. 6(b)). Theless diffusive the numerical scheme is, the larger the recir-culation bubble is. Since the water surface elevation and thebed are almost flat, in this case the hybrid first/second or-der and the fully second order schemes produce exactly thesame results.

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Depth Averaged Modelling of Turbulent Shallow Water Flow with Wet-Dry Fronts 325

(a) Mesh 1. (b) Mesh 2.

(c) Mesh 3. (d) Mesh 4.

Fig. 7 Numerical meshes in the near bend region

A mesh convergence analysis has been done in four dif-ferent grids, with different mesh size near the walls andin the separated region. The spatial domain covered by themeshes is shown in Fig. 5. The finite volume mesh is gen-erated from a triangular mesh by the procedure presented inSect. 5. Figure 7 shows the near bend region of the four tri-angular meshes which have been used to build the respectivefinite volume meshes. The characteristics of both the trian-gular and the finite volume grids are shown in Table 1. The

main difference between them is the mesh size near the walland in the separated region. A refined grid is needed in orderto avoid an excessive numerical diffusion in this area and tobe able to resolve the recirculation bubble. The grid size inthe recirculation region differs approximately by a factor 2between each mesh.

The mesh convergence has been analysed according tothe separation length and the velocity profiles in the recircu-lation region, since it is there where the largest differencesbetween the four meshes appear. Table 2 shows the recir-

Table 1 Characteristics of the computational grids for the 90 bend.ywall : distance from the first inner node to the wall in the recirculationregion

Triangular mesh Finite volume mesh

Vertex Elements Volumes Faces ywall

M1 738 1296 2033 3888 0.031 m

M2 1697 3124 4820 9372 0.018 m

M3 2495 4712 7206 14136 0.008 m

M4 3003 5700 8702 17100 0.004 m

culation length computed with several meshes and turbu-lence models. The bubble width can be approximately de-termined from the experimental data, showing a very goodagreement with the numerical results given by the three tur-bulence models (Table 3).

The velocity field does not reach convergence until gridM3. The results are quite insensitive to further mesh refine-

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326 L. Cea et al.

ment. No significant differences appear between grids M3and M4 (Fig. 8(b,c)). On the other hand, the ML model

Table 2 Recirculation length computed with several meshes and tur-bulence models

ML k DASM

M1 0.38 m 0.43 m 0.35 m

M2 1.12 m 1.05 m 1.12 m

M3 1.46 m 1.10 m 1.09 m

M4 1.71 m 1.15 m 1.15 m

is more sensitive to mesh refinement, appearing still somedifferences between the velocity profiles obtained with thegrids M3 and M4 (Fig. 8(a)).

The comparison between the numerical results and theexperimental data will be made in the recirculation region,since the first section of the channel does not present any in-

Table 3 Recirculation width computed with mesh M4

Exp. ML k DASM

M4 0.13 m 0.12 m 0.11 m 0.12 m

(a) ML model. (b) k model.

(c) DASM.

Fig. 8 Mesh convergence in the 90 bend. Velocity profile Vx (m/s) at x = 1 m

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Depth Averaged Modelling of Turbulent Shallow Water Flow with Wet-Dry Fronts 327

(a) Experimental. (b) ML model.

(c) k model. (d) DASM.

Fig. 9 Velocity field Vx (m/s) in the 90 bend. Several turbulence models and experimental results. White colour accounts for positive Vx

teresting flow features. The following facts should be takeninto consideration when comparing the numerical and ex-

perimental results: (1) the numerical velocity is the depthaveraged velocity, while the experimental velocity is the hor-izontal velocity at approximately the mid-point between thebed of the channel and the free surface (zexp h/2); (2) ac-cording to the turbulence level and the number of instanta-neous samples which have been used to compute the sta-tistical estimators of the mean variables, the variability ofthe experimental mean velocity in the recirculation region isabout 10%.

It should be noticed that in uniform channel flow the ve-locity profile is quite homogeneous in the vertical direc-tion, and the velocity at zexp is approximately equal to the

depth averaged velocity. However, near the bend a verti-cal eddy generates in the flow, which tends to destroy thehomogeneity in the vertical direction. This has two maineffects. First, the differences between the depth averagedvelocity and the experimental velocity measured at zexpmay increase, depending on the actual vertical profile. Sec-ond, a modelling error is introduced in the depth averagedequations, which neglect the dispersion terms due to thenon-homogeneity of the vertical profiles. Despite these 3Dflow features appearing in the near bend region, the over-

all agreement between the experimental and the numericalfields is very satisfactory, as it is shown in Fig. 9. The ML

model overpredicts the recirculation region (white colourin Fig. 9). This is because the model does not account fortransport processes, and therefore, all the turbulence gener-ated in the inner corner of the bend is not convected, nei-ther diffused, downstream the bend, as it is in the DASMand k models. The size of the recirculation bubble com-puted with the DASM and k models shows a better agree-ment with the experimental one, although it is still slightlylarger.

A more detailed comparison of the experimental and nu-merical fields is done in cross sections downstream the bend.Considering the variability of the experimental data, the

agreement in the velocity is very satisfactory (Fig. 10). Fromthe cross section at x = 1.5 m (Fig. 10(a)), it seems that thelength of the recirculation bubble is overestimated by all themodels, specially by the ML model, which is the one whichpredicts the largest recirculation region. The cross sectionsat x = 0.8 m and x = 1.3 m show a small bump in the ex-perimental velocity profile near the outer wall, which seemsto propagate toward the wall. This bump occurs in the re-gion of maximum vertical velocity, which means that it isprobably generated by the secondary flow originated in the

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328 L. Cea et al.

(a) Section x = 1.5 m.

(b) Section x = 1.3 m.

(c) Section x = 0.8 m.Fig. 10 Longitudinal velocity Vx (m/s) at several cross sections in the90 bend

bend region. It