Numerical Methods for Solving Turbulent Flows by Using Parallel Technologies

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1 Journal of Computer and Communcaton, 0,, -5 do:0.46/cc.0.00 Publhed Onlne February 0 ( Numercal Method for Solvng urbulent Flow by Ung Parallel echnologe Albek Iakhov Department Mechanc and Mathematc, al-farab Kazakh Natonal Unverty, Almaty, Kazakhtan Emal: alahov@mal.ru Receved 0 ABSRAC Parallel mplementaton of algorthm of numercal oluton of Naver-Stoke equaton for large eddy mulaton (LES) of turbulence preented n th reearch. he Dynamc Smagornky model appled for ub-grd mulaton of turbulence. he numercal algorthm wa worked out ung a cheme of plttng on phycal parameter. At the frt tage t uppoed that carryng over movement amount take place only due to convecton and dffuon. Intermedate feld of velocty determned by method of fractonal tep by ung homa algorthm (trdagnal matrx algorthm). At the econd tage found ntermedate feld of velocty ued for determnaton of the feld of preure. hree dmenonal Poon equaton for the feld of preure olved ung upper relaxaton method. Moreover varou way of geometrcal decompoton for parallel numercal oluton of three dmenonal Poon equaton are nvetgated. Keyword: Doman Decompoton; Parallel Computaton; Dynamc Smagornky Model; LES Approach. Introducton Mot flow occurrng n nature and n engneerng applcaton are turbulent. urbulent flow a flud moton that poee complex and eemngly random tructure at ome macrocopc cale of dynamcal mportance. he mot mportant phycal conequence of turbulence the enhancement of tranport procee. In turbulent flow, momentum, energy and partcle tranport rate greatly exceed the correpondng molecular tranport rate. urbulent flow exhbt much more mall-cale tructure than ther non-turbulent counterpart. In fact, th mall-cale tructure correlated wth enhanced turbulent tranport phenomena. Small-cale tructure telf evdence of enhanced tranport n the ene that mall cale develop from the degradaton of large-cale exctaton and are mantaned by energy tranport from one cale to another. Another mportant charactertc of turbulent flow ther apparent randomne and ntablty to mall perturbaton. Currently, there are three bac and commonly ued approache for mulaton of turbulent flow. Frt approach drect numercal mulaton (DNS) whch apple to olve Naver Stoke equaton, reolvng all the cale of moton, wth ntal and boundary condton approprate to the condered flow. Each mulaton produce a ngle realzaton of the flow. he DNS approach wa nfeable untl the 970 when computer of uffcent power became avalable. In DNS whole range of patal and temporal cale of the turbulence mut be reolved. All the patal cale of the turbulence mut be reolved n the computatonal meh, from the mallet dpatve cale (Kolmogorov mcrocale), up to the ntegral cale L, aocated wth the moton contanng mot of the knetc energy. Second approach large eddy mulaton (LES), the larger three dmenonal unteady turbulent moton are drectly repreented, a the effect of the maller-cale moton are modelled. In computatonal expene, LES le between Reynold-tre model and DNS. Becaue the large-cale unteady moton are repreented explctly, LES can be expected to be more accurate and relable than Reynold-tre model for flow n whch largecale unteadne gnfcant uch a the flow over bluff bode, whch nvolve unteady eparaton and vortex heddng. he computatonal cot of DNS hgh, and t ncreae a the cube of the Reynold number, o that DNS napplcable to hgh Reynold number flow. Nearly all of the computatonal effort n DNS expended on the mallet, dpatve moton, a the energy and anotropy are contaned predomnantly n the larger cale of moton. In LES, the dynamc of the large-cale moton are computed explctly, the nfluence of the maller cale beng repreented by mple model. hrd approach the Reynold-averaged Nav- Copyrght 0 ScRe.

2 Numercal Method for Solvng urbulent Flow by Ung Parallel echnologe er Stoke equaton (or RANS equaton) are tme-averaged equaton of moton for flud flow. he dea behnd the equaton Reynold decompoton, by an ntantaneou quantty decompoed nto t tme-averaged and fluctuatng quantte, an dea wa frt propoed by Oborne Reynold. he RANS equaton are prmarly ued to decrbe turbulent flow. hee equaton can be ued wth approxmaton baed on knowledge of the properte of flow turbulence to gve approxmate tme-averaged oluton to the Naver Stoke equaton.. Mathematcal Model Under the aumpton of ncompreble flow, the dmenonle governng equaton are a follow [,,7]: t u + p = + Re () = 0 ( =,,). () = u u u u he oluton of pread of flow n three dmenonal area were condered n th work. u velocty, p repreent the total preure. he Reynold number defned a Re = DV / ν (ν dynamc vcoty). Furthermore Cartean coordnate ytem employed, n whch z tream we drecton, x, y are n the lateral drecton. A for contructng model of turbulence we ued dynamc model of Smagornky, the followng the underlyng prncple of the dynamc model for extractng nformaton concernng a gven eddy-vcoty model va a double flterng n phycal pace. It worth to admt that the mot of the htorcal development have been done wth Smagornky model [6,9] = ν g (), = δ = Kroneker ymbol 0, ν = ( C ), g = +, ( ) / = x y z (4) / 4 C k C =, C =0.8 for a Kolmogorov contant of,4. π But the dynamc procedure apple n fact to the type of eddy vcote uch a thoe ued n the tructure-functon model. We tart wth regular LES correpondng to a bar-flter of wdth x, an operator aocatng an functon f ( x, t). hen we defne a econd tet flter tlde of large wdth x aocatng f ( x, t). So let u frt apply th flter product to the Naver-Stoke equaton. he ubgrd-cale tenor of the feld u obtaned from equaton (4) wth the replacement of the flter bar by the double flter and tlde flter: = u u u u (5) l = u u u u (6) Now we apply the tlde flter to equaton (4), whch lead to = u u u u (7) Addng equaton (6) and (7) and ung equaton (5), we obtan l = Further we ue Smagornky model expreon for the ubgrd tree related to the bar flter and tlde-flter to get A = C A = ( x) S S (8) We have to determne, the tre reultng from the flter product. h agan obtaned ung the Smagornky model, whch yeld = CB ( ) S S B = x (9) Copyrght 0 ScRe.

3 Numercal Method for Solvng urbulent Flow by Ung Parallel echnologe Subtractng (8) from (9) wth the ad of Germano dentty we get the followng l l = CB C A l l = CM M = B A (0) All the term of equaton (0) may now be determned by mean of u. Unfortunately, there are fve ndependent equaton for only one varable C, and thu the problem over determned. he frt oluton wa propoed by Germano to multply (0) tenorally by to get l S C = M S. Numercal Smulaton S he numercal oluton of ytem bult on the taggered grd wth the uage of the compact cheme for convectve term and cheme agant a tream of the econd type [5]. he cheme of plttng on phycal parameter ued for the oluton of turbulence problem [9-,4]: n u u n I. = ( u u ν u ), u II. p =, * u n + u III. = p. he frt tage olved by fractonal tep method n combnaton of homa algorthm (trdagnal matrx algorthm) [8,,]. hree dmenonal Poon equaton for preure feld ung an over - relaxaton method handled at the econd tage. hree dmenonal Poon equaton parallelzed by ung varou geometrcal decompoton (D, D and D). Geometrc decompoton of the grd area elected a the bac approach of parallelzaton. In th cae, there are three dfferent way of harng the value of the grd functon on the compute node one-dmenonal, two-dmenonal and three-dmenonal of the grd computng node [,4]. After a tage of decompoton, when performed on eparate data block for the contructon of a parallel algorthm, we proceed to relaton between the block, the calculaton whch wll be run parallel. Becaue of we ued an explct dfference cheme for computng the next approxmaton n the border node of each ubdoman neceary to know the value of the grd functon wth borderng neghborng proceor element. o accomplh th, n each compute node a fake edge for torng data from a neghborng computatonal node and arranged hpment of thee boundary value needed to enure the homogenety of the calculaton by explct formula. Sendng data done ung the procedure lbrary MPI. Let u turn to a prelmnary theoretcal analy of the effectvene of varou method of decompoton of the computatonal doman for th cae. We wll etmate the tme of the parallel program a the tme of content program, dvded by the number of proceor ued, calc = / p +. Whle plu the tme hpment p calc com hpment to dfferent way of decompoton can be approxmately expreed n term of the amount of bandwdth: D = t N x com end D t / end N x4 / p com = () D = com t / end N x6 / p N - dmenon of fnte-dfference problem, p number of computng node, t end - hppng tme of one number. Calculaton were performed on a cluter ytem URSA KazNU after al-farab on grd of and by ung up to 64 proceor. Reult of computatonal experment howed the preence of a good peed n olvng problem of th cla. hey are manly focued on over-tme hpment and tme calculaton for varou method of decompoton. In the frt tage we ued one overall program, the ze of array from run to run have not changed, each proceor element numberng of the array element tartng from cratch. Depte the fact that, n accordance wth the theoretcal analy of the D decompoton the bet opton for parallelzaton (Fgure ), computatonal experment have hown that better reult can be acheved ung D decompoton when the number of procee from 5 to 44 (Fgure ) On the ba of prelmnary theoretcal analy of the graph t mut have the followng pattern. Computaton tme wthout nterproceor communcaton cot at dfferent way of decompoton hould be approxmately the ame for the ame number of proceor and hrnk a Copyrght 0 ScRe.

4 4 Numercal Method for Solvng urbulent Flow by Ung Parallel echnologe calc / p. In realty, the calculated data (Fgure 4) ndcate that the ue of D decompoton on dfferent grd gve the mnmum cot for computaton and payment chedule dependng on the computaton tme on the number of proceor whch placed much hgher than calc / p. o explan thee reult there a need to pay attenton to the aumpton that were made durng the prelmnary theoretcal analy of the problem. Frtly, t wa aumed that regardle of how the dtrbuton of data on a ngle proceor element executed the ame amount of computatonal work, whch hould lead to dentcal tme-conumng. Secondly, we aumed that the tme pent on nterproceor hppng any order of the ame amount of data that doe not depend on ther electon from memory. o undertand what happen n realty, the next et of tet calculaton wa held. o ae the contency of frt admon wa condered when the program run n a ngle-proceor veron, and thu mulate dfferent way of geometrc data decompoton for the ame amount of computaton performed by each proceor. hu, for explct dfference method for olvng three dmenonal Poon equaton can be appled one-dmenonal, two and three-dmenonal decompoton, but the reult of tetng program have hown that the D decompoton doe not gan n tme compared wth the Fgure Computaton tme wthout conderng the cot of data tranfer for varou method of decompoton D decompoton, at leat for the number of proceor doe not exceed 50, and the D decompoton ha a more tme-conumng oftware mplementaton and the ue of D decompoton uffcent for the cale of the problem at the preent number of compute node. 4. etng reult of the numercal method Conderaton of a turbulent flow, whch located n the channel (Fgure ). Computaton were performed for the Reynold number Re = U m D / ν equal to 8000 defned baed on the et ax velocty. Alo the followng grd N x xn y xn z = 80x80x60 taken n the calculaton. he pread of flow n three dmenonal area decrbed n numercal oluton. Fgure 6 how ourface of pread flow n three dmenonal area at dfferent tme cale. Grd number 8x8x8 0 5 Speddup Number of proceor D D D Fgure Speed up for dfferent way to decompoe the computatonal doman tme, ec,4, 0,8 0,6 0,4 0, Grd number 8x8x Number of proceor D D D 5. Concluon he reult of numercal experment howed that the contructed mathematcal model of turbulence able to reproduce the charactertc feature of turbulent flow. he uage of dynamc Smagornky model allowed u to obtan good data for the tudy area. Applcaton n the calculaton of D decompoton gve 65% effcency n the ue of 5 compute node. Wth further ncreae n the number of compute node and 00 for the choen meh ze, a charactertc wa obtaned for problem of th cla effcency value around 45%. REFERENCES [] J.D. Jr. Anderon, Computatonal Flud Dynamc, New York: McGraw-Hll [] C.A. Fletcher, Computatonal echnque for Flud Dynamc, Vol : Specal echnque for Dfferental Flow Categore, Berln: Sprnger-Verlag [] G. E. Karnadak, Parallel Scentfc Computng n C++ and MPI. 000 [4] P. Pacheco. Parallel Programmng wth MPI, Morgan Copyrght 0 ScRe.

5 Numercal Method for Solvng urbulent Flow by Ung Parallel echnologe 5 Kaufmann [5] S.K. Lely. Compact fnte dfference cheme wth pectral-lke reoluton, J. Comp. Phy., 8, 99, pp [6] M. Leeur, O. Meta, P. Comte, Large-eddy mulaton of turbulence, Cambrdge unverty pre [7] R. Peyret, D. h. aylor, Computatonal Method for Flud Flow, New York: Berln: Sprnger-Verlag. 98. [8] N.N. Yanenko, he Method of Fractonal Step, New York: Sprnger-Verlag. In J.B.Bunch and D.J. Roe (ed.), Space Matrx Computaton, New York: Academc Pre [9] A. Iakhov, Large eddy mulaton of turbulent mxng by ung D decompoton method, J. Phy.: Conf. Ser. 8(4), 0405, 0. do: 0.088/ /8/4/ 0405 [0] B. Zhumagulov, A. Iakhov, Parallel mplementaton of numercal method for olvng turbulent flow, Vetnk NEA RK. (4), 0, pp. 4 [] A. Iakhov, Parallel algorthm for numercal oluton of three-dmenonal Poon equaton, Proceedng of world academy of cence, engneerng and technology 64, 0, pp [] A. Iakhov, Mathematcal modelng of the nfluence of hydrothermal procee n the water reervor, Proceedng of world academy of cence, engneerng and technology 69, 0, pp [] A. Iakhov, Mathematcal modellng of the nfluence of thermal power plant to the aquatc envronment by ung parallel technologe, AIP Conf. Proc. 499, 0, pp do: /0.06/ [4] A. Iakhov, Development of parallel algorthm for numercal oluton of three-dmenonal Poon equaton, Journal of Communcaton and Computer. Volume 9, Number 9, 0, pp Copyrght 0 ScRe.

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