Jiemin Zhan,Jianbo Zhang,Yejun Gong?Department of Applied Mechanics and Engineering,School of Engineering,Sun Yat-sen University,Guangdong 510275,China

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Numerical investigation of air-entrainment in skimming flow over stepped spillways

Jiemin Zhan,Jianbo Zhang,Yejun Gong?
Department of Applied Mechanics and Engineering,School of Engineering,Sun Yat-sen University,Guangdong 510275,China

h i g h l i g h t s

?Simulation is performed of the air-entrainment in skimming flow using the volume of fluid(VOF),mixture and Eulerian.?The VOF+large eddy simulation(LES)method is able to capture the splashing water droplets.

?The mixture+LES method predicts the inception of air entrainment most accurately.

?The Eulerian+Reynolds-averaged Navier-Stokes(RANS)method fails to capture the free-surface aeration.

a r t i c l ei n f o

Article history:

Received in revised form

29 March 2016

Accepted 30 March 2016

Available online 6 May 2016

Skimming flow

Stepped spillways

Air-entrainment

Large eddy simulation

As a widely used flood energy dissipator,the stepped spillway can significantly dissipate the kinetic or hydraulic energy due to the air-entrainment in skimming flow over the steps.The free-surface aeration involves the sharp deformation of the free surface and the complex turbulent shear flows.In this study, the volume of fluid(VOF),mixture,and Eulerian methods are utilized to simulate the air-entrainment by coupling with the Reynolds-averaged Navier-Stokes/large eddy simulation(RANS/LES)turbulence models.The free surface deformation,air volume fraction,pressure,and velocity are compared for the three different numerical methods.Only the Eulerian+RANS method fails to capture the free-surface aeration.The air volume fraction predicted by the VOF+LES method best matches the experimental measurement,while the mixture+LES method predicts the inception point of the air entrainment more accurately.

?2016 The Author(s).Published by Elsevier Ltd on behalf of The Chinese Society of Theoretical and Applied Mechanics.This is an open access article under the CC BY-NC-ND license(http:// creativecommons.org/licenses/by-nc-nd/4.0/).

The stepped spillway at the toe of a dam is one of the widely seen energy dissipation structures in hydro-power projects[1]. Energy of the flood is dissipated due to the air-entrainment in skimming flow,together with the generated vortexes on the steps[2].The experimental investigation in the laboratory is not able to capture the vortex structures in the flow.Hence, researchers have been devoted to the numerical investigation of the flow aeration in skimming flow,with the development of highperformance computers and parallel computation methods[3].

Limited by the computation condition,the earlier numerical studies of the stepped spillway overflow did not consider the coupling of the turbulence model[1].Then,the Reynolds-averaged Navier-Stokes(RANS)turbulence models,including the standard k-εmodel and the re-normalisation group(RNG)k-εmodel, are widely utilized in conjunction with the volume of fluid(VOF) method[4,5].The VOF+RANS method is able to capture the fre e surface in good agreement with the experimental results. However,the time averaged RANS method is not able to capture the fluctuating instantaneous flow characteristics.Different to RANS method,the large eddy simulation(LES)method resolves the spatial-filtered Navier-Stokes equations,such that it is able to capture the small scale eddies[6].

In this study,the LES sub-grid scale(SGS)Smagorinsky-Lilly model will be used to resolve the turbulent structures in skimming flow overthe stepped spillways using a commercial computational fluid dynamics(CFD)tool,ANSYS FLUENT[7]. Three multiphase models are available in FLUENT:the VOF model[8],the mixture model[9],and the Eulerian model[10]. The VOF method captures the gasliquid interface by calculating the volume fraction of water through each computing cell,but it does not reflect the phase interaction very well.The mixture method considers the interactions between phases(can be more than two)by introducing the relative velocity into the mixed momentumequation.ThemostcomplexEulerianmethodresolves the governing equations for each phase with coupled pressure andinterfaceinteraction.Moredetailsofthethreemultiphasemethods refer to Ref.[11].

Fig.1.Computation domain and boundary conditions[12].

Fig.2.(Color online)Air volume fraction distribution above the steps.Left:time averaged air volume fraction above No.4-10 steps;right:instantaneous air volume fraction above No.7-9 steps at 60 s.

Fig.3.Air volume fraction distribution on probes P7-P9 with distance y measured normal to the pseudo-bottom and Y90 the characteristic distance where the air volume fraction is 90%.

To compare the three typical multi-phase models,the skimming flow will be simulated using three different numerical methods with details shown in Table 1.Because the LES turbulence option is not allowed for the Eulerian method in FLUENT,we use the RNG k-εmodel for turbulence in the Eulerian case.The two fluids in the VOF model share a single set of momentum equations,the liquid phase volume fraction is resolved throughout the whole computation domain and the gas-liquid interface is build using the Geo-reconstruct method.For each multiphase method,pressure-velocity coupling is coordinated via the Pressure Implicit with the Splitting of Operators(PISO)scheme for LES or the Semi-Implicit Method for Pressure Linked(SIMPLE)scheme for RANS.Different combinations of the computation algorithms are tested for each method,and the best performers are listed in Table 1,where pressure staggering option(PRESTO)scheme calculates the''staggered''pressure using the discrete continuity balance for pressure discretization,quadratic upstream interpolation for convective kinematics(QUICK)is a quadratic-upwind differencingscheme[7].NotethatthevelocityfieldsresolvedbytheVOF andmixturemethodarethemixturevelocityoftheair-waterflow, while the Eulerian method predicts the fluid velocity instead.

As shown in Fig.1,the tested stepped spillway model is positioned 1.5 m from the inlet,0.4 m from the top boundary and 0.5 m from the right boundary.The stepped spillway includes ten identicalstepswithheighthandwidthl.Inaskimmingflow,water enters the inlet at a fixed mass flow rate qw,and then flows over the dam with a critical flow depth of dc.The mass flow rate qwis adjustedsuchthatdc/h=1.15.Then,airiscontinuouslyentrained andreleasedthefreesurfaceabovethesteps,andlastlytheaerated flow leaves the downstream outlet freely.

The skimming flow is simulated using the three different numerical methods listed in Table 1,and the calculated air volumefraction are compared in Fig.2.Obviously,the Eulerian method is not able to capture the air entrainment pattern as vividly as the other two methods.In experiment,the location of inception point of free-surface aeration is above the No.6 step[13].The simulated location of air entrainment is No.5.step for the VOF method and No.6 step for the mixture method.Though the VOF method predicts an earlier air entrainment,it is capable to capture the water splashing far from the steps,while the mixture method ignore the small size water drops,as shown by the instantaneously air volume fraction on the right side of Fig.2.

Table 1 Computation models.

Fig.4.Time averaged pressure and velocity distribution of the No.7-9 steps.Top:static pressure on the horizontal step edge,and the maximum points are labeled by solid circles;bottom:velocity magnitude on the horizontal line adjacent to the step edge with a distance of 1 mm,and the reattachment points are also labeled by solid circles.

Fig.5.(Color online)Vortex structures above the steps.Left:time averaged vorticity magnitude versus the velocity streamlines above No.4-10 steps;right:instantaneous velocity streamlines above No.7-9 steps at 60 s,where the reattachment points are labeled by red solid circles.

More details are shown in the air volume fraction distributions on probes P7-P9,as given in Fig.3,where the VOF case best matches the experimental data.The positions of the probes are shown in Fig.1.On each probe,the air volume fraction is monotonically increasing with the normalized distance y/Y90, where Y90 is the characteristic distance where the air volume fraction is 90%.For the mixture model,the predicted air volume fraction starts from a value much greater than the experimental measurement,because it is not able to accurately simulate the thin layer downstream of each step edge.For the Eulerian method,the air volume fraction suddenly increase from zero to a value near 1, indicating again its failure to capture the air entrainment.

The pressure and velocity distributions on the horizontal edges of the No.7-9 steps are shown in Fig.4.We observed two extreme points on each pressure curve.The pressure is the smallest in the flow recirculation region,and reaches the maximum at the reattachment point,which is a stagnation point with zero velocity, as in Fig.4.Note that this statement is not applicable to the Eulerian+RANS method,due to its weak capability of predicting vortex structure.

The reattachment point separates the flow recirculation region and the mixing layer downstream,as confirmed by the streamline plot in Fig.5.The VOF and mixture methods capture two or more vortexes inside the recirculation region,while the Eulerian+RANS method only predicts the primary vortex with comparatively smaller vorticity magnitude.Outside the recirculation region,thefree surface is disturbed by the interaction between the shear layer and the solid step edge.The resulting turbulent fluctuations can produce the free-surface aeration and spray generation.

In conclusion,coupled with the LES method,both the VOF and mixture methods are able to simulate the air entrainment in the skimming flow over the stepped spillway.Compared with the mixture method,the VOF method predicts one step earlier the inception point of the air entrainment,but it is able to capable to capture the small scale water drops or layers,such that a better agreement of the predicted air volume fraction with experimental data.The Eulerian+RANS method is not very suitableforthesimulationoffreesurfaceaeration.Additionally,we observed that the maximum pressure on each horizontal step edge is positioned exactly at the reattachment point,which separates the recirculation region and the mixing layer downstream.In the future,the relationship between the critical flow depth and the reattachment length will be further investigated.

Acknowledgments

ThisworkwassupportedbytheGuangdongSpecial ResearchFundofPublicWelfareandCapacityBuilding (2015A020216008)and the Special Program for Applied Research onSuperComputationoftheNSFC-GuangdongJointFund(thesecond phase).

Appendix A.Supplementary data

Supplementary material related to this article can be found online at http://dx.doi.org/10.1016/j.taml.2016.03.003.

References

[1]M.R.Chamani,N.Rajaratnam,Characteristics of skimming flow over stepped spillways,J.Hydraulic Eng.125(1999)361-368.

[2]J.H.Wu,B.Zhang,F.Ma,Inception point of air entrainment over stepped spillways,J.Hydrodynamics 25(2013)91-96.http://dx.doi.org/10.1016/S1001-6058(13)60342-X.

[3]A.Eghbalzadeh,M.Javan,Comparison of mixture and VOF models for numerical simulation of air centrainment in skimming flow over stepped spillways,Procedia Engineering 28(2012)657-660.

[4]Q.Chen,G.Dai,H.Liu,Volume of fluid model for turbulent numerical simulation of stepped spillway over flow,J.Hydraulic Eng.128(2002) 683-688.

[5]X.Cheng,Y.Chen,L.Luo,Numerical simulation of air-water two-phase flow over stepped spillway,Sci.in China Series E:Tech.Sci.49(2006) 674-684.

[6]Y.Gong,Large eddy simulation of dispersed multiphase flow,(Ph.D.thesis), Michigan Technological University,Michigan,USA,April 2012.

[7]ANSYS Inc.,ANSYS Fluent Version 15.0 User's Guide.

[8]C.Hirt,B.Nichols,Volume of fluid method for dynamics of free boundaries, J.Comput.Phys.39(1981)201-221.

[9]J.Sanyal,S.Vasquez,S.Roy,etal.,Numericalsimulationofgas-liquiddynamics in cylindrical bubble column reactors,Chem.Eng.Sci.54(1999)5071-5083.

[10]J.Chahed,V.Roig,L.Masbernat,Eulerian-Eulerian two-fluid model for turbulent gas-liquid bubbly flows,International J.Multiphase Flow 29(2003) 23-49.

[11]X.Cheng,X.Chen,Progress in numerical simulation of high entrained airwater two-phase flow,in:2012 Third International Conference on Digital Manufacturing and Automation(ICDMA),Guilin,China,2012,pp.626-629.

[12]C.Gonzalez,H.Chanson,Turbulence manipulation in aircwater flows on a stepped chute:An experimental study,Eur.J.Mech.B Fluids 27(2008) 388-C408.

[13]G.Carosi,H.Chanson,Air-water time and length scales in skimming flows on a stepped spillway.Application to the spray characterisation,Report No.CH59/06,Division of Civil Engineering,The University of Queensland, Brisbane,Australia,2006.

29 January 2016

?Corresponding author.

E-mail addresses:gongyj3@mail.sysu.edu.cn,yejungong@126.com(Y.Gong).

http://dx.doi.org/10.1016/j.taml.2016.03.003

2095-0349/?2016 The Author(s).Published by Elsevier Ltd on behalf of The Chinese Society of Theoretical and Applied Mechanics.This is an open access article under the CC BY-NC-ND license(http://creativecommons.org/licenses/by-nc-nd/4.0/).

*This article belongs to the Fluid Mechanics