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Numerical modelling ofsubcritical open channel flowusing the K-E turbulencemodel and the penalty functionfinite element techniqueA. N. Puri and Chin Y. KuoDepartment of Civil Engineering, Virginia Polytechnic Institute and State University,Blacksburg, Virginia 24061, USA(Received September 1983)

A numerical model has been developed that employs the penalty functionfinite element technique to solve the vertically averaged hydrodynamicand turbulence model equations for a water body using isoparametricelements. The full elliptic forms of the equations are solved, thereby allow-ing recirculating flows to be calculated. Alternative momentum dispersionand turbulence closure models are proposed and evaluated by comparingmodel predictions with experimental data for strongly curved subcriticalopen channel flow. The results of these simulations indicate that thedepth-averaged two-equation k-e turbulence model yields excellent agree-ment with experimental observations. In addition, it appears that neithertne streamline curvature modification of the depth-averaged k-e model,nor the momentum dispersion models based on the assumption of heli-coidal flow in a curved channel, yield significant improvement in thepresent model predictions. Overall model predictions are found to be asgood as those of a more complex and restricted three-dimensional model.Key words: mathematical model, curved channel, turbulence, subcriticalflow, finite element, penalty function

Colnputational hydraulics usually involves the predictionof turbulent flows, which, due to their complexityand inherent three-dimensional nature, are extremelydifficult to predict theoretically. Predictions made withexperimental models are certainly useful, but they oftendo not provide a detailed description of the flow field,lack flexibility, and are expensive and time consuming.Consequently, it is not surprising that numerical models,with their inherent flexibility and relatively low cost, arein great demand as a predictive tool in the field of hydrau-lics. For example, computational models have been success-fully used to predict a wide range of engineering problemsassociated with a free surface. It is important to note thatmuch of this success is due to the introduction of thedepth-averaged equations of motion.These equations are obtained by averaging the three-dimensional, time-averaged equations of motion over thewater depth, thereby reducing the problem to that of

solving for two depth-averaged horizontal componentvelocities and water depth. However, the averaging pro-cess creates additional unknown terms, namely thebottom, wind and effective stresses, and adequate closureexpressions for these terms are required. The effectivestresses were defined by Kuipers and Vreugdenhil to con-sist of the depth-averaged viscous and turbulent stresses,and additional stresses which result from the depth-averag-ing of the nonlinear convective acceleration terms presentin the original three-dimensional equations. These stressesare often called momentum dispersion terms2

In an earlier paper by Puri and Kuo,~ a free surfacesteady-state hydrodynamic model based upon the penaltyfunction finite element method and utilizing a simplifiedone-equation turbulence closure was presented. The goal ofthe present study is to extend that model to include an ad-vanced two-equation turbulence model, and also examinedifferent momentum dispersion closure schemes. These

82 Appl. Math. Modelling, 1985, Vol. 9, April 0307- 904X/ 85/ 02082-07/ $03.000 1985 Butterworth & Co. (Publishers) Ltd.

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Subcritical open channel flow: N. Puri and C. Y. Kuodifferent closure schemes will be evaluated by comparingmodel predictions with experimental data for stronglycurved subcritical open channel flow.

Depth-averaged model

where* = ($ + fi2)12 (5)Cf = g/c2 = 1ICff (6)

In equation (6) cf and cff are empirical friction factors,and C is Chezys roughness coefficient.The governing steady-state depth-averaged equations ofmotion are: l-3 Depth-averaged Reynolds stress closure

Combining the Boussinesq eddy viscosity hypothesisConservation of mass with the depth-averaging procedure, the depth-averageda&h turbulent Reynolds stresss tensor may be written as:2p=oax, (1) i;ln = fi, a(Gnh) + a(&$)- -

Conservation of momentum ax, ax, 1a(&& h) a(h212) azb aF

ax, tg--!E=O

ax, +gh - +T~~+ax, ax, (2)where:g acceleration due to gravitym, n 1, 2 and repeated indices imply a summationvm two-dimensional depth-averaged velocity vector(& 5)Xm coordinate directions (x, 7)rbm components of bottom shear stress per unitmassfrnn components of depth-averaged effective stresstensor per unit mass

indicates a depth-averaged quantityThe depth-averaged effective stress tensor per unit mass,

as defined by Kuipers and Vreugdenhilr and Flokstra,4 maybe written as:h+zbj;nn= [v($+$)-iz

zh

where:-(urn - timm)(u, i$J dz (3)

(7)where :li depth-averaged kinetic energy per unit mass

=:a6mn Kronecker deltaVt depth-averaged turbulent eddy viscosity

The turbulence model that has found the widest appli-cation for a variety of two-dimensional flows is the so-called k-e turbulence model, which characterizes thelocal state of turbulence by two parameters: the turbulentkinetic energy per unit mass, k, and the rate of its dissipa-tion, E. A detailed description of this model and its appli-cations may be found in the literature.2>6-g The semi-empirical transport equations for the transport of the depth-averaged quantities k and c may be given by:2?g1o

a(kh) + a(zj,lih)c_~ at ax,

1VV?l horizontal turbulent velocity fluctuation where:V fluid viscosityzb elevation of channel bottom above an arbitrarydatumh water depth

For the present study, it is assumed that the viscousstresses are small compared to the turbulence effects, andare consequently neglected.

Closure modelsThe closure problem associated with the depth-averagedequations of motion requires the parameterization, in termsof known or inferred quantities, of: (1) the bottom shearstress, (2) the depth-averaged turbulent Reynolds stresses,and (3) the momentum dispersion terms.Bottom shear stress closure

The bottom shear stress per unit mass may be para-meterized by:2Y3$5rbm = cf crnq (4)

a( i h) a(zjmeh)at ax,

(8)

+P, (9)

p h a(c,h) l a(&$) a(c,h)Ix, ax, ax,ft2fit = cp-6

(10)(11)

and ci, c2, Sk, u, and cP are empirical constants. Theirvalues, adopted from Launder and SpaMing are: cr = 1.43,c2 = 1.92, cP = 0.09, ok = 1 O, (3, = 1.3. The source termsPk and P, account for the production mechanism resultingfrom the presence of a vertical boundary layer. Rastogi andRodi related these vertical production terms to the bottomshear stress, and obtained:

Pk = cku; (12)P, = c&/h (13)

where u, is the shear or friction velocity, and ck and cEare empirical constants. Using Laufers measured turbulentviscosity as empirical input, Ratogi and Rodi obtained:

1ck = ~fif (14)

Appl. Math. Modelling, 1985, Vol. 9, April 83

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Subcritical open channel f low: N. Puri and C. Y. Kuo

c =pqE (15)Curvature modif ication of k-e model

Since it is well known that streamline curvature stronglyinfluences turbulent transport in shear layers, Leschizinerand Rodi presented a curvature modification of thestandard k-e model. Using a simplified version of the(two-dimensional) algebr-sic stress model of Gibson, l2 theyobtained the curvature modification of the coefficient cc1as:

c,, = - k,k, 1 + Rk;k2,+& +t,z] (16)where

1-PkI= __(Y (17)k, = $( 1- CY /3)/a (18)

and R, = local radius of curvature of the streamline, v, andv, are the velocities in the streamline and normal to thestreamline directions respectively, and 1yand /3 are con-stants having values 1.5 and 0.6 respectively.13

The curvature modification of the depth-averaged k-emodel was then accomplished by replacing the two-dimensional quantities u, u, k, e-in equation (16) with thedepth-averaged quantities li, ir, k. 6.M omentum dispersion closur e

Previous attempts at formulating closure schemes forthe momentum dispersion terms have been based on theassumption of a simple secondary flow field which allowsthe depth variation of the horizontal velocity componentsto be modelled by known theoretical velocity distributions.Flokstra4 substituted power law velocity profiles into thethird term of equation (3) to derive a closed set of predic-tion equations. However, he failed to present numericalvalues for the coefficients in his equations, nor did hepresent computational results to support the general useof his model. The closure expressions derived by Flokstra4are: I,, = cweti 2arfizi + CQfi (19)

rxv=1Li2+((Y~-(11*)~ii,-cyl~2 (20)ryy = a21iZ + 2arfifi + a(# (21)

in which rXX, PYYand rX,, are the momentum dispersioncomponents of the (depth-averaged) effective stress tensorper unit mass. The coefficients, (Y~, = 0, 1, 2 were evaluatedin the present study as:

a0 = 0.02 CX~ 0.34(/z/R) 01~= 7.2(h/R) (22)where R is the radius of the curvature of the main stream-line. This model is here referred to as momentum dispersionmodel MDl.

An additional, simplified, model for the momentum dis-persion terms may be obtained by neglecting the normalcomponent of the three-dimensional velocity field. Follow-ing Flokstras4 approach, it is easy to show that:rxx= o.02ti2 (23)rxy= 0.02afi (24)rvr= 0.02i2 (25)

This model is referred to as a momentum dispersion modelMD2.

Computational methodThe computational method employed to obtain approximatenumerical solutions of the depth-averaged model equationsis the penalty function finite element technique describedby Reddy. 14,15Further details of the method and its appli-cation to depth-averaged open channel flows may be foundin Puriand Kuo3and Puri.g The hydrodynamic modeldescribed by Puri and Kuo3 was extended in the presentstudy to incorporate the depth-averaged k-e turbulencemodel by means of a staggered finite element meshscheme. The necessity for a staggered grid stems from theneed to decouple the solution of the mean flow equationsfrom the solution of the turbulence equations, and toassist in proper implementation of the boundary conditions.This decoupling then allows the mean flow equation set andthe turbulence equation set to be solved alternately ratherthan simultaneously, with control being transferred be-tween the two sets until overall convergence is achieved.This procedure was found to greatly reduce the possibilityof non-convergence of the equations.16 The staggered gridstructure is shown in Fi gure 1.

The velocity components at a wall were specified to bezero in accordance with a no-slip condition, while the wallboundary conditions for the turbulence quantities k and Ewere imposed using a modification of the wall functionmethod of Launder and Spalding.6 Consequently, for thenear-wall turbulence grid nodes, the following relations wereimposed:

%J = pkkff, = %JI~Cppkklf imY2

(26)

(27)where the subscript w denotes a near-wall value.

For the flow cases considered in the present study, thegradients of all the primary unknowns normal to theoutflow boundary were specified to be zero. Implementa-tion of the inflow boundary conditions on the velocityfield was achieved by assigning uniform inflow velocitiesfrom the known system discharge rate. Inflow boundaryconditions for the turbulence quantities were obtainedby following the approach specified by Rastogi and Rodi.

I I0

I II v I l .u,vI Ai,i,h---- __t --A-- __I

4c I 0I

I IFigure 7 Staggered mesh configurations showing interlockingvelocity and turbulence grids

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Subcr i t ical open channel f low: N. Puri and C. Y. KuoConsequently, the following inflow boundary conditionsmay be imposed:u:iiii = h (28)

ki = (0.0765~1*h~~/C~)~ (29)where i denotes an inflow value.

It is worth noting at this point that the governing modelequations presented in this study are fully elliptic, notparabolic, as the model equations of Leschziner and Rodiare. Consequently, it is possible to use the present modelto predict important flow phenomena such as flow separa-tion and recirculation.

Model simulationsAlthough there exists a large body of experimental datafor curved channel flows, suitable and detailed data forrectangular cross-sectioned channels having a strongcurvature (mean radius to width ratio of order 3 andbelow17) are relatively scarce. Of the available experi-mental studies, Rozovskiis18 data was found to be suit-able for model verification purposes, and his experimentalresults have been used by other authors7Yg for the pur-pose of evaluating their respective computational models.Consequently, various variants of the proposed model wereused to simulate Rozovskiis experimental flow condi-tions so as to evaluate the different Reynolds stress andmomentum dispersion closure schemes.

The channel consisted of a 180 bend, having a meanradius to width ratio, Ro/B, of 1 .O and straight inlet andoutlet reaches of length L/B = 3.75. The inlet depth towidth ratio, hi/B, was 0.075 and the Reynolds number, Rehad the value 15 600 associated with an inlet Froudenumber F = Uinlet/(ghi)12 = 0.03376, where Uinret is theinlet velocity. The friction coefficient was specified to bethe experimental value of cff = 366, and the finite elementmesh consisted of 10 elements in the transverse direction,10 elements in the longitudinal or main flow directionalong the straight reaches, and 17 elements in the circum-ferential direction along the curved reach. This finiteelement mesh was used for all the simulations described inthe present study.The simulations made with the depth-averaged hydro-dynamic-turbulence model are summarized in Table I.In addition, Model 5 was utilized to conduct a parametricstudy of some of the parameters governing flow in a curvedchannel, including Ro/B, L/B, cff, and finite element meshconfiguration. To allow comparison of the results, the inletdepth to width ratio, system discharge, and inlet Froudenumber were kept constant for this group of simulations.

Tabe 1 Model simulationsModel 1 CP value Momentum dispersionModel 1 0.09 NeglectedModel 2 Equation (16) NeglectedModel 3 0.09 Model MD1Model 4 0.09 Model MD2Model 5 Equation (16) Model MD3

654321

0

-1L?0F= -2

-3-4-5-6-7-6-E

- ExperImental data-Model 1

Outer bank

)_) I0 1 2 3 4 5 6 7 8 9 IOFigure 2 Comparison between predicted (Model I) and experi-mental water depths along channel inner and outer banks (SIB =normalized distance along centreline; n% =normalized water depth)

Results and discussionA comparison of experimental observations of normalizedwater depths along the channels inner and outer bankswith the predictions of Models, 1, 2,3 and 4 is shownin Figures 2,3,4 and 5 respectively. It is clear that thestandard hydrodynamic turbulence model (Model 1)results in excellent agreement with the experimental data(Figure 2), and is associated with a maximum error of 2.5%.It also appears that model predictions are not significantlyaltered by the curvature tnodification of Leschziner andRodi17 incorporated in Model 2, as may be seen fromFigure 3. In addition, Figure 4 shows that momentumdispersion model MD1 degrades the overall predictivecapability of Model 3, and is associated with a maximumerror of 4.5%. However, the simplified momentum disper-sion model MD2 incorporated in Model 4 clearly resultsin improved model predictions when compared to Model 3.

The failure of the curvature modifications may be due tothe following two reasons: (1) the modification was derivedfor two-dimensional and axisymmetric flows, and may notbe directly applicable to depth-averaged computations, and(2) while the tnodification proved to be of some significancewhen simulating the strongly recirculating flows examinedby Leschziner and Rodi,17 its significance may be diminishedwhen applied to subcritical curved channel flows that donot exhibit separation.

A possible explanation for the failure of momentumdispersion model MD1 may lie in the fact that the closureexpressions utilized in the model were derived assuminghelicoidal flow in a long gentle channel bend. Conse-quently, the normal velocity profile of van Bendegommay not be strictly applicable to the strongly curvedchannel flow present in Rozovskiis* experimental


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Subcr i t ical open channel f low: N. Puri and C. Y. Kuo6 - Exoerlmental data5432

10-1

-z0F -2-3-4-5-E

-7-E-2

S/BFigure 3 Comparison between predicted (Model 2) and experi-mental water depths along inner and outer banks

St-- ExperImental data5_-Model 34-3-2- Outer bankl-Q-

-1 --2--3 -

40 -4 -F -5--6--7 --8- Inner bank-9-

-lO--11 --12 -- 131-14h , , 1 1 I I I I I I0 1 2 3 4 5 6 7 8 9 IO

S/BFigure4 Comparison between predicted (Model 3) and experi-mental water depths along inner and outer banks

flume. Consequently, it may be advisable to entirelyeliminate this inappropriate normal velocity profile fromthe closure model when simulating strongly curved flow.This observation is borne out by the significant improve-

ment in the predictive capability of Model 4 over Model3, as may be seen, from Figures 4 and 5.Model 5 was found to result in an overall minimumdiscrepancy between model predictions and experimentaldata, and Figure 6 shows a comparison of Model 5 predic-

+ - ExperImental data-Model 45-

- 1 --20F -2

-3--4 --5-

Outer bank

0123456789 10 11S/BFigure 5 Comparison between predicted (Model 4) and experi-mental water depths along inner and outer banks

- ExperImental data2- D model (model 5 )---- 3-D model

6 t

Outer bank

-5--6-

-8--9-

-1oj , , \_I I I I I I I I0 1 2 3 4 5 6 7 8 9 10S/B

Figure6 Comparison between Model 5, three-dimensional modeland exoerimental data


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Subcr i t ical open channel f low: N. Puri and C. Y. Kuotions, experimental data, and predictions made with theparabolized three-dimensional model of Leschziner andRodi.17 It is clearly evident that the predictions of thedepth-averaged model are certainly as good as the predic-tions made with the more expensive, complex and restrictedparabolic three-dimensional model. In addition, it is im-par-tant to note that the depth-averaged model is far moreeconomical in computer storage and execution time thanthe three-dimensional model, despite the parabolic natureof the latter.Space limitations preclude presentation of all the resultsobtained from the parametric study, but the effect of thefriction coefficient and the radius of curvature on the flowmay be seen from Figures 7 and 8 respectively. Figure 7clearly indicates that increased roughness (a smaller cffvalue) results in a lower water surface elevation along theinner bank of the curved channel segment. Also, as ex-pected, the increased roughness results in increased fric-tional losses so that the water surface elevation at thechannel outlet is lower than the outlet elevation corre-sponding to smaller roughness values. Figure 8 shows thewater surface profile at the point of maximum curvaturein the channel curve (at 0 = 90). The figure shows that asmaller R,/B ratio results in a more convex water surfaceprofile, and also results in larger superelevation differencesbetween the channel inner and outer bank at 6 = 90.

ConclusionsBased on the results of simulations conducted. the follow.ing conclusions may be made:

The curvature modification adopted for the presentmodel appears to have little effect on the computation of

0 98

+- 0.92

0.89-0.88-0.87-0.86 -

0.85L , , , , , , , ,0 1 2 3 4 5 6 7 8R/BFigure 7 Normalized water depth profiles along channel innerbank for different cff values

1.01 r

0.98 -

0.96 -

092-0.91 -

0.90 -0 89-0.88 -:.::I , , , , , , ,

0 01 02 0.3 0.4 0.5 0.6 0.7 08 0.9 1.R/BFigure 8 Normalized water depth profiles at 0 = 90 for differentR,lB values

strongly curved subcritical open channel flows that do notexhibit separation.Momentum dispersion closure schemes computed using athree-dimensional normal velocity profile based upon heli-coidal flow in a wide, gentle, channel bend are not signifi-cant for the prediction of strongly curved open channelflows that do not exhibit separation.

The optimal computational model (Model 5) resultsin excellent agreement between model predictions andexperimental data. The maximum discrepancy betweenthe predicted and observed normalized water depth pro-files is approximately 2.5%, and the predicted andobserved velocity fields also appear to be in good agree-ment. Overall model predictions are as good as thosemade with a more expensive and restricted three-dimensional model.The simulation model is capable of predicting separationin subcritical strongly curved open channel flow due to theelliptic nature of the governing equations. However, caremust be taken to avoid simulating flows in which the localFroude number exceeds unity (as may happen along theinner bank of the channel curveg,21). In this case, crosswaves will form along the channel and the present modelcannot be applied.

The above conclusions indicate that the model issuperior to the previous computational models reportedin the literature, and the results of the parametric studyare useful in the design of optimal channel bend configura-tions.In conclusion, the following extensions and/or modifi-cations of the existing model should be useful: (a) incor-poration of pollutant transport module, (b) extension of


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Subcr i t ical open channel f low: N. Puri and C. Y. Kuothe steady-state model to solve time dependent problems,and (c) improvement of the computational method byreplacing the lower order four node finite element used inthe present study with a higher order eight node element.This modification would remove the need for a fine finiteelement mesh since the higher order element is moreaccurate. These modifications will greatly increase thegeneral applicability and usefulness of the model for flowpredictions.AcknowledgementsThis research was supported by National Science Founda-tion, CME-8004364. Dr J. N. Reddy provided a computerprogram which was modified for inclusion as a part of theentire model.References

Kuipers, J. and Vrengdenhil, C. B. Calculation of two-dimensional horizontal flow, Report Sl63-1, DelftHydraulics Laboratory, Delft, Netherlands, 1973Chapman, R. S. and Kuo, C. Y. Application of the two-equation k-c turbulence model to a two-dimensional, steady,free surface flow problem with separation, Proc. Symp. onRefined Model& g of Flow s. Vol. 2, Paris, France, 1982Puri, A. N. and Kuo, C. Y. Numerical modeling of fluid flowusing the penalty function finite-element technique, Proc.ASeE Hydraulic Conf., Cambridge, MA, 1983Flokstra. C. The closure problem for depth-averaged two-dimensidnal flows, Paper_AI O6,17th IA HR Congress,Baden-Baden, Germany, 1977Schlichting, H. Boundary layer theory, McGraw-Hill, NewYork, 1955Launder, B. E. and Spalding, D. B. The numerical calcula-tion of turbulent flows, Computer M ethods in Appl iedM echani cs and Engineeri ng, Vol . 3. 1914Ratogi, A. and Rodi, W. Prediction of heat and mass transferin open channels, J. Hy draul ics D iv ., ASCE 1978, 104 (HY3)

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Rodi, W. Turbulence models and their application inhydraulics: a state of the art review, presented by the IAHR-Section on Fundamentals of Division II Experimental andMathematical Fluid Dynamics, Delft, 1980Puri, A. N. Numerical analysis of subcritical open channel flowusing the penalty function finite element method, a DoctoralDissertation, Virginia Polytechnic Institute and State Univer-sity, Blacksburg, VA, 1983McGuirk, J. J. and Rodi, W. A depth-averaged mathematicalmodel for near field of side discharges into open channelflow. J. Flui d M echani cs 1978,86 (4)Leschziner, M. A. and Rodi, W. Calculation of annular andtwin parallel jets using various discretization schemes andturbulence ~ model variations, Trans. ASME, J. Flui dsEngng 1981, 103Gibson, M. M. An algebraic stress and heat-flux model forturbulent shear flow with streamline curvature, Int. J.Heat and M ass Transfer 1918, 21Launder,B. E., Pridden, C. H. and Sharma, B. I. The calcula-tion of turbulent boundary layers on spinning and curvedsurfaces. Tran s. ASME, J. Fl ui ds Ennnn 1977, 99 (1)Reddy, j. N. On penalty function methods in the finite-element analysis of flow problems, J. Numerical M ethodsin Fluids 1982, 2Reddy, J. N. Private communication, Virginia PolytechnicInstitute and State University, Blacksburg, VA, 1982Schamber, D. R. and Larock, B. E. Numerical analysis of flowin sedimentation basins, J. Hvdraulics Div., ASCE 1981, 107(HY5)Leschziner, M. A. and Rodi, W. Calculation of strongly curvedopen channel flow, J. Hyr aulics D iv., ASCE 1979, 105 (HY 10)iozovskii, I. L. Flow of water in bends of open channels,Academv of the Sciences of the Ukranian SSR, Kven, USSR(translation No. OTS 60-S 1133), Office of Technical Services,Washington, D.C., 1957de Vriend, H. J. A mathematical model of steady flow incurved open shallow channels, Report 76-1, Laboratory ofFluid Mechanics, Delft, Netherlands, 1976van Bendegom, L. Some consideration on river morphologyand river improvement, Delngerzieur 1940,.59 (B l-11)Soliman, M. M. and Tinney, E. R. Flow around 180 bendsin open rectangular channels, J. Hydr aulics Di v., ASCE 1968,94 (NY4)


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