DAM BREAK FLOW SIMULATION ON GRID

Transcription

1 Mterils Physics nd Mechnics 9 (2010) Received: Mrch 17, 2010 DAM BREAK FLOW SIMULATION ON GRID Arns Kčeniusks 1*, Rusln Pcevič 1 nd Toms Ktkevičius 2 1 Lbortory of Prllel Computing, Vilnius Gedimins Technicl University, Sulėtekio 11, LT-10223, Vilnius, Lithuni 2 Computing Centre, Vilnius Gedimins Technicl University, Sulėtekio 11, LT-10223, Vilnius, Lithuni *E-mil: rnk@vgtu.lt Abstrct. In this contribution we report on dm brek flow simultion on glite bsed grid infrstructure. The dm brek problem including breking wves is solved by the pseudoconcentrtion method improved by interfce shrpening technique. The developed interfce shrpening procedure helps to preserve interfce shrpness nd mss conservtion. The computed position of the leding edge of wter column hs been compred with the experimentl mesurements. 1. Introduction Dm brek flow hs been the subject of extensive reserch for long time [1]. The originl problem hs direct ppliction in the industril res of fluid mechnics nd environment protection. The breking wve phenomen occurring in some cses of dm brek problem includes it into the clss of complex pplictions such s solitry wve propgtion, tnk sloshing nd wter on ship deck simultion. Some experimentl mesurements were performed on the dm brek flow or collpse of liquid column problem [1-2]. Photogrphs showing the time evolution of the collpsing column s well s the wve returning fter hitting wll on the opposite side re vilble for the purpose of evluting the numericl methodology on the bsis of flow visuliztion. Mesurements of the exct interfce shpe re not vilble, but some secondry dt such s the reduction of the wter column height [2] cn be employed for quntittive comprison of the obtined results. Severl modifictions of the broken dm problem hve been extensively used s clssicl test cses for numericl simultion of free surfces nd moving interfces [3-4]. However, the universl, ccurte nd efficient numericl technique for breking wve simultion ttrcts close ttention of reserch community nd softwre developers. Over the pst 30 yers, reserchers hve put lot of effort into developing vrious numericl methods to simulte the moving interfce flows governed by the Nvier-Stokes equtions. All numericl methods for modelling of the moving interfce flows re bsed on interfce trcking pproch or interfce cpturing pproch. In former, the liquid region is subdivided by mesh, while ech cell is deformed ccording to the movement of the interfce nd computed velocities [5-6]. However, this pproch requires remeshing procedures to void of computtion filure due to serious distortion of cells or elements. These methods re unble to cope nturlly with interfce intercting with itself by folding or rupturing. In the interfce cpturing pproch, the mesh remins fixed nd moving interfce cn not be directly defined by the mesh nodes. Therefore, dditionl technique is necessry to define the res occupied by fluid or gs on either side of the interfce. The mrker-nd-cell method [3], the volume of fluid method [4] nd the level set method [7] re well known methods using the interfce cp- 2009, Institute of Problems of Mechnicl Engineering

2 Dm brek flow simultion on grid 97 turing pproch. These methods require no geometry mnipultions fter the mesh is generted nd cn be pplied to interfces of complex topology [8]. However, the loction of the interfce is not explicit nd, sometimes, the pproprite boundry conditions cnnot be prescribed with required ccurcy. A pseudo-concentrtion method [9] often used with the finite element method (FEM) is n interesting lterntive for the other interfce cpturing methods. This method uses pseudo-concentrtion function defined in the entire domin nd solves hyperbolic eqution to determine the moving interfce [10]. In the most cses, the pseudo-concentrtion method is more efficient thn the level set method, becuse it uses simpler front reconstruction techniques. The FEM hs become powerful tool for solving mny scientific nd engineering pplictions; therefore, the demnd for further investigtion of the ICT nd implementtion in commercil FEM codes is rpidly growing [11]. The min concerns with the interfce cpturing pproch hve been the extensive usge of computtionl resources nd sustining globl mss conservtion in long-time integrtions [12]. It ws observed tht numericl diffusion introduces norml motion proportionl to the locl curvture of the interfce, which leds to non-physicl mss trnsfer between the two fluids. The choice of the interfce shrpening procedure, the numericl schem nd numericl prmeters remin stte of the rt problem. In the present pper, the efficient interfce shrpening procedure preserving mss conservtion is pplied to the complex dm brek flow. Numericl experiments re crried out on glite bsed BlticGrid infrstructure providing the necessry computtionl resources. 2. Mthemticl model of the flow The lminr nd Newtonin flow of viscous nd incompressible fluids is described by the Nvier-Stokes equtions in the Eulerin reference frme: ui u σ i ij ρ + u j = ρ Fi +, (1) t x j x j ui = 0, (2) xi where ui re the velocity components; ρ is the density; F i re the grvity force components nd σ is stress tensor ij u u i j σ + ij = p δij + µ. (3) x j xi Here µ is the dynmic viscosity coefficient, p is the pressure nd δ ij is Kronecker delt. Slip boundry conditions for velocity re prescribed on rigid wlls: u i n i = 0, (4) here n i re the components of unit norml vector. This is usul choice of boundry conditions used for modelling of moving interfce flows. The zero stress boundry conditions re prescribed on the open upper boundry σ 0. (5) ij n j = The reference pressure is prescribed on the upper wll. The zero initil conditions re prescribed for the equtions (1-3) in the performed investigtion. The pseudo-concentrtion method [9] is developed for moving interfce flows using the Eulerin pproch nd the interfce cpturing ide. The pseudo-concentrtion function ϕ serves s mrker, identifying fluids A nd B with densities ρ A nd ρ B nd viscosities nd µ B. In this context, the density nd viscosity re defined s: ρ = ϕ ρ + ( 1 ϕ ρ, (6) A ) B µ A

3 98 ρ = ϕ µ ( µ, (7) A + 1 ϕ ) B Arns Kčeniusks, Rusln Pcevič, Toms Ktkevičius where ϕ = 1 for fluid A nd ϕ = 0 for fluid B. The evolution of the interfce is governed by the time dependent convection eqution: ϕ ϕ + u j. (8) t x j The velocity u i is obtined from the solution of the Nvier-Stokes equtions (1-3). The initil conditions defined on the entire solution domin should be prescribed for eqution (8). The spce-time Glerkin lest squres (GLS) finite element method [5] is pplied s generl-purpose computtionl pproch to solve the prtil differentil equtions (1 3, 8) with boundry conditions (4 5) pplied. Equl order biliner shpe functions re used for both the pressure nd velocity components s well s for the pseudo-concentrtion function. The stbiliztion nture of the formultion prevents numericl oscilltion of incompressible flows when equl-order interpoltion functions for velocity nd pressure re used nd preserves the consistency of the stndrd Glerkin method when dptive remeshing is performed. The detiled description of vritionl formultion nd stbiliztion prmeters cn be found in work [13]. 3. Interfce shrpening procedure The stndrd Glerkin formultion of the FEM yields oscilltory solutions when pplied to convection dominnt problems in conjunction with clssicl time stepping lgorithms. Conventionl stbiliztion techniques dd to the eqution (8) rtificil viscosity terms, introducing numericl diffusion nd reducing numericl oscilltions. In prcticl ppliction of the GLS stbilizing method to the complex problems governed by convective trnsport, some overshoots nd undershoots re observed [13]. In order to pply the developed interfce shrpening procedure, simpler limiter should be implemented: ϕ = min [ mx [ ϕ,0], 1]. (9) It removes the overshoots nd undershoots, preventing the numericl technique from unexpected incorrect vlues nd the loss of ccurcy. While the eqution (8) moves the interfce t correct velocity, the pseudo-concentrtion function my become significntly influenced by the numericl diffusion fter some period of time. It leds to interfce smering nd problems with mss conservtion. The pseudoconcentrtion function should be reconstructed in order to mintin interfce shrpness. In this work, the vlues of the pseudo-concentrtion function ϕ re replced by the vlues of the reconstructed function φ ccording to the following formul: 1 φ = c ϕ, 0 ϕ c, (10) 1 φ = 1 (1 c ) ( 1 ϕ ), c ϕ 1, (11) where the prmeter c represents mss conservtion level, while governs shrpness of the moving interfce. The implicit prmeter ns indictes how often this procedure should be pplied. Usully, the interfce thickness tends to grow, occupying wide bnd of finite elements. Frequent ppliction of the interfce shrpening procedure (10 11) with lrge vlues esily resolves this problem, but it cn distort the smoothness of the interfce. The moving front cn dpt to the FE mesh proceeding stircses. In this undesirble cse, the ccurcy of the interfce cpturing technique becomes directly limited by the mesh size. Thus, the vlues of prmeter ns, controlling interfce shrpness, should be chosen very crefully. Mss conservtion is very importnt issue of the interfce shrpening lgorithm. Insignificnt numericl errors, which result in slight non-physicl mss trnsfer between the two fluids, my led to significnt errors in long-term time integrtion of the problem. In order to

4 Dm brek flow simultion on grid 99 overcome this difficulty nd to preserve mss conservtion, the vlues of coefficient c should be computed considering precise mss distribution in the interfce region. At ny time, mss conservtion for fluid A in the solution domin Ω cn be described by the formul: M A = ρ A φ d Ω, (12) Ω where M A is the initil mss of the fluid A. The eqution for determining c cn be obtined by substituting φ from formuls (10 11) to eqution (12). Assuming tht is given nd constnt, the resulting eqution for mss conservtion cn be written s follows: 1 1 ϕ K1 c K2 (1 c ) = M A M A, (13) where coefficient K is defined on the nrrow bnd of the moving interfce 0 ϕ c : 1 K1 = ρ A ϕ d Ω. (14) Ω Coefficient K2 is computed on the remining prt of the interfce c ϕ 1 : K2 = ρ A (1 ϕ ) d Ω.. (15) Ω ϕ Coefficient M A represents the current mss of the fluid function c ϕ 1 : ϕ M A = ρ A d Ω. (16) Ω A, defined by the vlues of the ϕ Despite the fct tht only one fluid is explicitly presented in eqution (13), the described procedure conserves mss for ech fluid. In the solved problems, the fluid A is hevy (wter), while the fluid B is reltively light (ir). A good strting vlue for c is 0.5, which indictes stisfctory mss conservtion. The finl vlues of c re obtined solving one-dimensionl non-liner eqution (13). Modelling breking wves nd other extreme phenomen, the finl c vlue vries in reltively wide rnge from 0.2 to 0.8 [14]. After determining nd c, equtions (10-11) re stisfied t the node-level, nd the new vlue of the pseudoconcentrtion function is used to resume computtions. 4. Softwre deployment in grid The discussed numericl lgorithms hve been implemented in the code FEMTOOL [13], which llows implementtion of ny prtil differentil eqution with minor expenses. Time dependent problems re solved using spce-time finite elements. The order of shpe functions is determined by input nd is limited neither in spce nor in time. A given trnsient problem cn be solved in severl implicit time steps from one time level to the other or in one single implicit step for ll time levels. Spce-time finite element integrtion in time nd the high order shpe functions generted utomticlly mke FEMTOOL to be pplicble to complex strongly coupled problems of interest. Severl times FEMTOOL softwre ws deployed in glite bsed grid. The glite Worklod Mngement System (WMS) ntively supports the submission of MPI jobs, which re jobs composed of number of processes running on different Working Nodes (WN) in the sme Computing Element (CE). However, this support is still experimentl; therefore, running MPI pplictions on the glite grid requires significnt hnd-tuning for ech site. Some time go MPI services were very inconvenient for the clusters using distributed home file systems. It ws necessry to trnsfer executble nd dt files from the governing WN (zero number ssigned by MPI) to the other WNs, to implement dditionl checks for their presence nd to ensure tht ll result files were trnsferred bck to the governing WN. Any implementtion of MPI-2 stndrd ws not supported. The first FEMTOOL deployment relied

5 100 Arns Kčeniusks, Rusln Pcevič, Toms Ktkevičius on shell scripts consisted of pproximtely 100 lines nd required long preprtion time. Only experienced system dministrtor cn prepre nd test such scripts for new MPI pplictions. Improved FEMTOOL gridifiction ws bsed on the compilers g95 nd gcc s well s on the OpenMPI implementtion. FEMTOOL version 3.0 hs been deployed in BlticGrid-II testbed by using SGM (Softwre Grid Mnger) system. All softwre pckges re instlled in the predefined loction, which is specified by content of $VO_BGTUT_SW_DIR vrible. After successful instlltion, the SGM mrks the site in globl grid informtion system s cpble of running this prticulr ppliction. By use of this flg, clled " tg" the ordinry users my indicte which sites they wnt to use. The FEMTOOL tg is VO-blticgrid-A- ENG-FEMTOOL-3.0. Recent efforts mde in Interctive Europen Grid project (int.eu.grid) [15] improved sitution. The MPI-START scripts were developed by the MPI working group contining members from int.eu.grid nd EGEE. The incresed portbility nd flexibility ws chieved by working round hrd-coded constrints from the Resource Broker (RB) nd by off-loding much of the initilistion work to the MPI-START scripts. Two implementtions of MPI-1 stndrd (LAM nd MPICH) nd two implementtions of MPI-2 stndrd (OpenMPI nd MPICH2) re supported by the MPI-START. Using the MPI-START system requires the user to define wrpper script nd set of hooks. The MPI-START system then hndles most of the low-level detils of running the MPI job on prticulr grid cluster. The ltest FEMTOOL deployment in grid relies on the dvnced MPI-START system. The generic wrpper script nmed mpi-strt-wrpper.sh is used for setting environment, defining the executble, setting MPI implementtion nd inititing the MPI-START processing. The script mpi-hooks.sh contins scripts clled before nd fter the MPI executble is run. The pre_run_hook() function is clled before the MPI executble is strted, therefore, it is employed for compiltion of the FEMTOOL code. The function post_run_hook() is clled fter MPI executble is finished. It rchives the result files nd trnsfers them to Storge Element (SE). If grid cluster hs distributed home file system the results files should be copied to the governing WN by specil loop in the post_run_hook(). The discussed scripts re defined in the usul JDL file, which describes grid job. The job type must be MPICH while MPI implementtion OpenMPI is defined in the list of rguments. Despite of its nme, the importnt ttribute NodeNumber defines the number of CPUs required by the job. It is not possible to request more complicted topologies bsed on nodes nd CPUs in glite middlewre. MPI-START bsed FEMTOOL gridifiction nd deployment is more portble nd llows the user more flexibility. The obtined results were visulized by e-service VizLitG designed for convenient ccess nd efficient visuliztion of results produced by scientific computtions in BlticGrid- II. The client-server rchitecture of the grid visuliztion service is bsed on widely recognized web stndrds. VizLitG hs flexible environment nd remote instrumenttion of e- Service provided by Jv nd GlssFish ppliction server [16]. Dtsets re stored in prescribed HDF5 [17] files tht cn be downloded from different loctions. The visuliztion engine of the VizLitG is bsed on VTK toolkit [18]. VTK modules re enwrpped by Jv progrmming lnguge nd build the service running on the server. The visuliztion server runs on the specil UI nmed UIG (User Interfce for Grphics). User uthentifiction nd full dt trnsfer from SE is performed by glite mens. Moreover, remote instrumenttion of e-service provides for users flexible ccess of remote dt files locted in grid. Trnsfer of interctively selected prts of dtsets locted in experimentl SE insted of the whole dt files cn sve significnt mount of visuliztion time. GVid softwre [19] provides user full interctivity like ny VTK bsed ppliction runing on the UIG.

6 Dm brek flow simultion on grid Numericl results nd discussions Description of the dm brek problem. The geometry of the solution domin is shown in Fig. 1. The dimensions of the reservoir nd the wter column correspond to those used in the experiment crried out by Koshizuk et. l. [2]. The reservoir is mde of glss, with bse length of 0.584m. The wter column, with bse length of 0.146m nd the height of 0.292m (=0.146m), ws initilly supported on the right by verticl plte drwn up rpidly t time t=0.0s. The wter flls by grvity (g=9.81m/s 2 ), cting verticlly downwrds. The density of wter is ρ A =1000 kg/m 3, while the dynmic viscosity coefficient is µ A =0.01kg/(m s). The density of ir is tken to be ρ B =1kg/m 3, nd the dynmic viscosity coefficient is µ =0.0001kg/(m s). B open boundry 2 wter ir 4 Fig. 1. Geometry of dm brek problem. The slip boundry conditions (4) re pplied to the bottom nd sides of the reservoir. The stress boundry conditions (5) re prescribed on the upper open boundry. They my be chnged to fixed pressure nd zero norml grdients of the velocities. The computtions re performed on the structured finite element meshes of different resolution nd The investigted time intervl is t = [0.0; 1.0] s. The size of time step is t = for the finite element mesh. The number of time steps is equl to 600. The size of the time step for the finite element mesh is t = The number of the time steps used is equl to Breking wve phenomen nd vlidtion of numericl results. The breking wve phenomenon is simulted by the pseudo-concentrtion method nd the developed interfce shrpening technique. Grvity cuses the wter column on the left of the reservoir to seek the lowest possible level of potentil energy. Thus, the column will collpse nd eventully come to rest. The initil stges of the flow re dominted by inerti forces with viscous effects incresing s the wter comes to rest. On such lrge scle, the effect of surfce tension forces is insignificnt. The complexity of velocity fields, occurring t different stges of breking wve phenomen, cn be esily cptured using simple structured meshes.

7 102 Arns Kčeniusks, Rusln Pcevič, Toms Ktkevičius Fig. 2. Visuliztion of breking wve phenomen. z/ E1 E2 smll big t * Fig. 3. Quntittive comprison of the numericl results nd experimentl mesurements. Fig. 2 shows the screenshot of visuliztion e-service VizLitG illustrting the breking wve phenomenon. Computed velocity field is represented by thresholded glyphs. Moving interfce is cptured by the iso-countour. Grey colours illustrte the pressure field while the chrt plot shows vrition of the pressure t the bottom. To predict the behviour of the smll bubbles correctly is more difficult tsk. When t = 0.83 s, the bckwrd moving wve hs folded over nd smll mount of ir is trpped. However, in experiments, the ir is present in the form of smll bubbles. The current methodology hs been derived for shrp interfces; therefore, the mesh needs significnt refinement to resolution smller thn the bubble size. The numericl results hve been vlidted by the quntittive comprison with experimentl mesurements obtined for the erly stges of this experiment [1-2]. Non-dimensionl position of the leding edge of the collpsing wter column on the left wll versus nondimensionl time t * = t 2 g / is shown in Fig. 3. Two different sets of experimentl dt E1 nd E2 were presented by Mrtin nd Moyce, illustrting the difficulty to determine the exct position of the leding edge. A thin lyer of wter shoots over the bottom nd the rest of the bulk flow follow behind it. The ccurte numericl results hve been obtined by pplying intensive shrpening of the moving interfce ( ns = 1, = 2.0). Almost identicl vlues hve been obtined by using two structured finite element meshes of different resolution, nd (the curve smll nd the curve big, respectively). The numericl

8 Dm brek flow simultion on grid 103 experiments hve shown tht the extreme shrpening is necessry only t the beginning of the time intervl t = [0; 0.09] s. Thus, the ccurcy of the obtined results is strongly influenced by the numericl prmeters, but it is lmost independent of the resolution of the finite element mesh, if sufficiently dense FE meshes re used. Interfce shrpness nd mss conservtion. The quntittive comprison of the numericl results nd the experimentl mesurements hs shown tht the detiled nlysis of the interfce shrpening prmeters should be performed in order to develop n ccurte nd efficient interfce shrpening procedure. The shrpening frequency ns determines how often the interfce shrpening procedure should be pplied. Usully, interfce shrpening hs been performed t regulr time intervls (ech k time step). Fig. 4 shows time evolution of the totl number of nodes totnum, belonging to the interfce ( 0 < ϕ < 1). The vlue of prmeter is fixed nd equls to 1.5. The curve nsk0 illustrtes how the interfce grows without shrpening. It is obvious tht this process is drsticlly influenced by the numericl diffusion. On the contrry, the interfce shrpening performed t ech time step reduces the interfce thickness to one finite element (the curve nsk1). However, in this cse, the moving interfce cn lose its smoothness. The results obtined in shrpening the interfce less frequently re quite cceptble. The curve nsk10 visulizes the cse ns =10. () totnum nsk0 nsk10 nsk t, s m, kg (b) nsk0 nsk10 nsk10m t, s Fig. 4. Interfce shrpness () nd mss conservtion (b). Mss conservtion is one of the most importnt tsks for ny interfce shrpening procedure. Fig. 4b shows mss evolution in time for severl interfce shrpening cses. The time intervl t = [0.4; 0.7] s illustrtes the cse when wter is leving the computtionl domin nd is coming bck. In order to preserve the consistency between the flow physics nd the numericl techniques, mss correction is utomticlly switched off in this specil cse. All plotted curves re ctully of the sme chrcter, but quntittive results re quite different. The significnt mss loss is observed when interfce shrpening is not pplied (the curve nsk0). The interfce shrpening without mss correction t regulr time intervls (the curve nsk10, ns = 10) does not significntly reduce the mss loss. The ppliction of mss correction to regulr interfce shrpening (the curve nsk10m, ns = 10) significntly improves mss conservtion. 6. Conclusions In this pper, the dm brek problem hs been solved by the pseudo-concentrtion method nd the developed interfce shrpening technique. Dm brek flow simultion hs been efficiently performed on glite bsed BlticGrid infrstructure. The numericl pproch hs been vlidted by quntittive comprison with the experimentl mesurements. The computed position of the leding edge of the collpsing wter column hs been in good greement with the experimentl dt. The regulr interfce shrpening with mss correction is ble to pre-

9 104 Arns Kčeniusks, Rusln Pcevič, Toms Ktkevičius serve interfce shrpness nd mss conservtion. The ccurte numericl solution of the dm brek problem, including highly non-liner breking wves, proves tht the developed numericl technique is cpble of simulting moving interfces undergoing lrge topologicl chnges. Acknowledgments The work described in this pper is supported by the Europen Union through the FP7- INFRA : e-science Grid infrstructures contrct No project "Bltic Grid Second Phse (BlticGrid-II)" References [1] J.C. Mrtin, W.J. Moyce // Philos. Trns. Roy. Soc. London, A244 (1952) 312. [2] S. Koshizuk, H. Tmko, Y. Ok // J. Comput. Fluid Dynmics 4 (1995) 29. [3] F.H. Hrlow, J.E. Welch // J. Phys. Fluids, 8 (1965) [4] C.W. Hirt, B.D. Nichols // J. Comput. Physics 39 (1981) 201. [5] A. Msud, T.J.R. Hughes // Comput. Methods Appl. Mech. Eng. 146 (1997) 91. [6] F. Del Pin, E. Oñte, R. Aubry // Computers & Fluids 36 (2007) 27. [7] S. Osher, J.A. Sethin // J. Comput. Physics 79 (1988) 12. [8] R. Löhner, C. Yng, E. Oñte // Comput. Methods Appl. Mech. Eng. 195 (2006) [9] E. Thompson // Int. J. Numer. Methods Fluids 7 (1986) 749. [10] R.W. Lewis, K. Rvindrn // Int. J. Methods Eng. 47 (2000) 29. [11] T.E. Tezduyr // Computers & Fluids, 36 (2007) 207. [12] D.A. Di Pietro, S. Lo Forte nd N. Prolini // Appl. Numer. Mth. 56 (2006) [13] A. Kčeniusks nd P. Rutschmnn // Int. J. Informtic, 15 (2004) 363. [14] A. Kčeniusks // Mechnik, 69 (2008) 35. [15] int.eu.grid: [16] GlssFish: [17] HDF5: [18] VTK User's Guide Version 5, Inc. Kitwre. 5th Edition, [19] GVid: