Qualitative behavior of mixing phenomena - the case of axisymmetric extensional flows

Transcription

1 ATHEATICAL ETHODS COPUTATIOAL TECHIQUES O-LIEAR SYSTES ITELLIGET SYSTES Qualiaive behavior of miing phenomena - he case of aisymmeric eensional flows ADELA IOESCU DAIELA COA Deparmen of Applied Sciences and Environmen Proecion Universiy of Craiova Address AI Cuza Sr no 585 ROAIA Absrac: The problems of flow kinemaics are far from complee solving even in our days A modern heory recenly appeared in his field: he miing heory Is mahemaical mehods and echniques developed he significan relaion beween urbulence and chaos The urbulence is an imporan feaure of dynamic sysems wih few freedom degrees he so-called far from equilibrium sysems These are widespread beween he models of eciable media Sudying a miing for a flow implies he analysis of successive sreching and folding phenomena for is paricles he influence of parameers and iniial condiions In he previous works [4] he sudy of he D non-periodic models ehibied a quie complicaed behavior In agreemen wih eperimens [6] hey involved some significan evens - he so-called rare evens The variaion of parameers had a grea influence on he lengh and surface deformaions The D (periodic) case was simpler bu significan evens can issue for irraional values of he lengh and surface versors herefore a comparison wih D case would reveal new analyical and quaniaive feaures This paper brings ino aenion anoher D miing flow model namely he aisymmeric eensional flow [5] There is performed a qualiaive analysis of is behavior from he srech efficiency sandpoin handling modern appliances of APLE sof [] The recorded daa are used for furher saisical analysis ey-words: - Turbulen miing Sreching Folding Rare even Ineracive Plo Builder Inroducion The miing concep The urbulence is an imporan feaure of dynamic sysems wih few freedom degrees he socalled far from equilibrium sysems In his area wo imporan heories are disinguished: he ransiion heory from smooh laminar flows o chaoic flows characerisic o urbulence on one hand and saisic sudies of he complee urbulen sysems on he oher hand The saisical idea of flow is generally represened by he map: Φ ( ) Φ ( ) () In he coninuum mechanics he relaion () named flow is a diffeomorphism of class C k and i mus saisfy he relaion: ( ( )) i J de D Φ de () j where D denoes he derivaion wih respec o he reference configuraion in his case The relaion () implies wo paricles and which occupy he same posiion a a momen on-opological behavior (like break up for eample) is no allowed Wih respec o here is defined he basic measure of deformaion he deformaion gradien F namely [5]: T ( ) i F Φ Fij () j where denoes differeniaion wih respec o According o () F is non singular The basic measure for he deformaion wih respec o is he velociy gradien Afer defining he basic deformaion of a maerial filamen and he corresponding relaion for he area of an infiniesimal maerial surface we can define he basic deformaion measures: he lengh deformaion λ and surface deformaion η wih he relaions [5]: / λ C : η ( de F ) ( C : ) / (4) where C (F T F) is he Cauchy-Green deformaion ensor and he vecors are he orienaion versors in lengh and surface respecively defined by: d da (5) d da The scalar form for (4) used in pracice is: ISS: ISB:

2 ATHEATICAL ETHODS COPUTATIOAL TECHIQUES O-LIEAR SYSTES ITELLIGET SYSTES λ Cij i j η ( de F ) Cij i (6) j wih i j he condiion for he versors The deformaion ensor F and he associaed ensors C C - represen he basic quaniies in he deformaion analysis for he infiniesimal elemens In his framework he miing concep implies he sreching and folding of he maerial elemens If in an iniial locaion P here is a maerial filamen d and an area elemen da he specific lengh and surface deformaions are given by he relaions: D( lnλ ) D( lnη) D : mm v D : nn (7) D D where D is he deformaion ensor obained by decomposing he velociy gradien in is symmeric and non-symmeric par We say ha he flow Φ () has a good miing if he mean values D(lnλ)/D and D(lnη)/D are no decreasing o zero for any iniial posiion P and any iniial orienaions and As he above wo quaniies are bounded he deformaion efficiency can be naurally quanified Thus here is defined [5] he deformaion efficiency in lengh e λ e λ () of he maerial elemen d as: D( ln λ) / D eλ (8) / ( D : D) and similarly he deformaion efficiency in surface e η e η () of he area elemen da: in he case of an isochoric flow (he jacobian equal ) we have: D( lnη) / D eη (9) / D : D Recen resuls In previous works [4] here were realized elaboraed sudies boh for D and D miing models Saring from he widespread D basic flow [5]: G () G few cases were aken ino accoun wih some model perurbaions as follows: G () G and G () G G ( ) For boh models he analyic behavior from he miing efficiency was realized and i mus be noiced ha alhough he model had small perurbaions he behavior was very differen especially because of he irraional values of he lengh and surface versors oreover rare evens have issued sanding from he numeric/simulaion sandpoin This is a very special feaure which argued anoher imporan qualiaive analysis of D flow model Saring also from () he D flow model was creaed []: G G () c wih he same condiions: G where ccons represen he velociy The associaed eperimenal flow process was consiued by a vore insallaion [6] where a biological maerial he Spirulina Plaensis algae was voreed in special deermined condiions in a basic fluid waer The voreaion insallaion gained few imporan prizes and has very few large applicaions in imporan indusrial branches [6] The imporan fac ha mus be noiced is ha he numeric simulaions mached he eperimens amely here were considered 6 saisical cases (for he lengh and surface versors values and for he parameers G and ) and he numeric simulaions confirmed ha a some special irraional values of he versors especially of he surface versors he filamens of he biological maerial breake up This fac coincides wih he breake up of he program and he new resuled biological maerial has new properies differen from he iniial one [] This fac could give an imporan rend in few indusrial/bioechnological areas For he momen we focus on he analyic sudy and he imporance of he APLE insrumens The eensional flow model The model analysis Le us consider he aisymmeric eensional flow wih he following mahemaical ISS: ISB:

3 model [5]: (4) where -< < I is ineresing from wo basic sandpoins: he special symmeric form of he flow on one hand and he fac ha i has only sreching phenomena - because of is eensional feaure- on he oher hand If a he momen we have d d d d hen he soluion of he Cauchy problem associaed o (4) is: ep ep ep (5) Then he mari F and he Cauchy-Green ensor C are compued o be ep ep ep F (6) ep ep ep C (7) Due o is eensional form his flow model has only sreching phenomena Therefore in his framework he lineal srech given by (6) becomes: ep ep ln D D λ (8) I mus be noiced some symmery in he above relaionship This is characerisic o linear flows In wha follows we shall presen he numeric analysis of he behavior of he lineal srech (8) If in [] he analysis had wo sages one of solving he differenial equaion given by he form of D D λ ln wih a specific numeric APLE procedure dsolve [] and he ne sage of realizing poin plos of pair poins resuled in he firs sage in he presen case he work speed is significanly increased using an special APLE builder namely he ineracive plo builder [] This plo solves auomaically he required differenial equaion and hen plos he corresponding plo in is opimal form for a specific aim Le us noe ha his builder has also useful opions for parameers Few cases were aken ino accoun for he versors values and for he parameer Because of he symmery of he relaionship (8) he sign of he versors and some zero componen are no significan The seleced cases are as follows: a) ; b) ; c) ; d) 6 Combining wih wo cases for he parameer namely: i) 8; ii) 5 i resuls graphic cases labeled ai) aii)di) dii) as follows The plos were simulaed for ime unis and when he case for ime unis ATHEATICAL ETHODS COPUTATIOAL TECHIQUES O-LIEAR SYSTES ITELLIGET SYSTES ISS: ISB:

4 ATHEATICAL ETHODS COPUTATIOAL TECHIQUES O-LIEAR SYSTES ITELLIGET SYSTES Fig Fig4 Fig Fig5 Fig Fig6 ISS: ISB:

5 ATHEATICAL ETHODS COPUTATIOAL TECHIQUES O-LIEAR SYSTES ITELLIGET SYSTES Fig7 Fig Fig8 Fig Fig9 Fig ISS: ISB:

6 4 Comparison aspecs wih D general miing flow Le us refer o he general D miing model cons c c G G (9) This is he naural D version of he widespread D flow [5]: cons c G G wih he parameer G is posiive generally <G< The hree-dimensional version is obained by adding he velociy componen considered o be consan for he momen Alhough he model seems no complicaed he soluion of he Cauchy problem associaed o (9) is: ep ep ep ep c () wih he noaion G Therefore he deformaion ensor F and he associaed Cauchy-Green ensor C have quie comple relaions [] As said above in D miing case here where sudied very few saisical cases abou 6 cases all in discree ime unis Several value ses were aken ino accoun for he lengh and surface versors and correspondingly for he parameers [] In wha follows we shall presen few cases for he surface deformaion for beer comparing he behavior in he discree case I mus be noiced ha he discree case allows one o observe he proimiy or he disancing of he poins in he considered ime scale The figures are numbered according o he simulaion case as follows: a) Fig he case : 8 wih b) Fig 4 he case: 8 wih c) Fig 5- he case: 8 wih d) Fig 6 he case: wih e) Fig 7- he case: 8 wih Fig ATHEATICAL ETHODS COPUTATIOAL TECHIQUES O-LIEAR SYSTES ITELLIGET SYSTES ISS: ISB:

7 ATHEATICAL ETHODS COPUTATIOAL TECHIQUES O-LIEAR SYSTES ITELLIGET SYSTES Fig 4 Fig 5 Fig 6 5 Conclusion remarks Some imporan remarks issue from he above analysis: i Analyzing he above plos i can be observed ha nonlinear behavior prevails for he lineal srech of he aisymmeric eensional flow The irraional values of he versors have a grea influence on his behavior ii For he nonlinear behavior here were aken ino accoun also smaller ime momens ime unis As can be seen he plo changes is form when sudied in a small ime inerval which means i can be beer evaluaed he energy dissipaion of he eensional flow in his case iii I mus be noiced ha in he presen case he analysis is realized on coninuous ime unis comparing o [] where he analysis was realized in discree ime unis This is because he ineracive plo builder of APLE works wih his opion iv Alhough he sudied model has only sreching phenomena (and no sreching folding ) he influence of irraional values could be seen here also This is because he irraionals 5 ec could be inerpreed as random values hemselves The same happened for D model [] Thus a ne aim could be he sudy of he issue of random phenomena conjecure v Anoher fuure aim is o es and compare he procedures of APLE in solving and analyzing he efficiency of miing For eample he numeric procedure dsolve has wo oupu ypes lisprocedure and piecewise producing oupus for discree and coninuous ime respecively [] Choosing one or oher oupu lead o plos which have significan differen aspecs I can be easily observed when sudying he figures -7 The discree case is more difficul bu offers more accurae observaions his is he case wih he figures 6 and 7 where here issue he so-called rare evens meaning he breakup of he simulaion program Comparing o his siuaions he coninuous case can wase some ime momens of sudy This feaure is more perinen when observing ha he discree case were sudied on a larger scale 5 ime unis References: [] A C Hindmarsh RS Sepleman (eds) Odepack a sysemized collecion of ODE solvers orh Holland Amserdam 98 Fig 7 ISS: ISB:

8 ATHEATICAL ETHODS COPUTATIOAL TECHIQUES O-LIEAR SYSTES ITELLIGET SYSTES [] A Ionescu The srucural sabiliy of biological oscillaors Analyical conribuions PhD Thesis Polyechnic Universiy of Bucares [] A Ionescu Cosescu Compuaional aspecs in eciable media The case of vore phenomena In J of Compuers Communicaions and Conrol vol I(6) Suppl issue: Proceedings of ICCCC6 pp 8-84 [4] A Ionescu Cosescu The influence of parameers on he phaseporrai in he miing model In J of Compuers Communicaions and Conrol vol III(8) Suppl issue: Proceedings of IJCCCC8 pp -7 [5] J Oino The kinemaics of miing: sreching chaos and ranspor Cambridge Universiy Press 989 [6] S Savulescu Applicaions of muliple flows in a vore ube closed a one end Inernal Repors CCTE IAE (Insiue of Applied Ecology) Bucares 998 ISS: ISB: