Fictitious Boundary and Moving Mesh Methods for the Numerical Simulation of Particulate Flow

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1 Fcttous Boundary and Movng Mesh Methods for the Numercal Smulaton of Partculate Flow Model for artculate flow Numercal technques Examles Further mrovements Stefan Turek, Rahael Münster wth suort by Dan Anca, Otto Merka, Kamran Usman and Decheng Wan Insttut für Angewandte Mathematk, TU Dortmund htt:// htt:// Page 1

2 Flud (Rgd) Sold Interfaces Consder the flow of N sold artcles n a flud wth densty ρ and vscosty μ. Denote by Ω f (t) the doman occued by the flud at tme t, by Ω P (t) the doman occued by the artcle at tme t and let Ω Ω f U Ω P. Flud flow s modelled by the Naver-Stokes equatons n Ω f (t) : u ρ + u u σ f, u 0 t where σ s the total stress tensor n the flud hase, whch s defned as : σ T ( X, t) I + μ[ u + ( u) ] Page 2

3 Model for Partcle Moton (I) Moton of artcles s descrbed by the Newton-Euler equatons,.e., the translatonal veloctes U and angular veloctes ω of the -th artcle satsfy: M du dt dω ' F + F + ( ΔM ) g, I ( ) + ω I ω T. dt wth M the mass of the -th artcle ( 1,,N); I the moment of nerta tensor of the -th artcle; ΔM the mass dfference between the mass M and the mass of the flud occuyng the same volume. Page 3

4 Model for Partcle Moton (II) F T and are the hydrodynamc forces and the torque at mass center actng on the -th artcle: Γ F n d Γ, σ ( ) ( ) T X X σ n Γ dγ ' F and are the collson or agglomeraton forces. X Γ n s the oston of the center of gravty of the -th artcle; Ω the boundary of the -th artcle; s the unt normal vector on the boundary. Γ Page 4

5 Flud-Partcle Interacton No sl boundary condtons at nterface between artcles and flud.e., for any, the velocty u(x) s defned by: X Γ Γ u ( X ) U + ω ( X X ) X The oston of the -th artcle and ts angle are obtaned by ntegraton of the knematc equatons: θ dx dθ U, ω dt dt Page 5

Use a (adated) mesh that covers the whole doman where the flud may be

6 Classfcaton Exlct coulng n t flud n n+1 t force on sold sold t n+1 t flud Euleran aroach: fxed meshes ossble! Use a (adated) mesh that covers the whole doman where the flud may be resent. FEM-Fcttous Boundary Methods (FBM) Comutatonal mesh (can be) ndeendent of nternal obects Page 6

7 Force Calculaton wth FBM Hydrodynamc forces and torque actng on the -th artcle F n dγ, P σ T ( X X ) ( σ n ) P dγ FBM: 0/1 - reconstructon of the shae s only 1st order accurate local grd adatvty or algnment near nterface s ossble only averaged/ntegral quanttes are requred But: The FBM can only decde INSIDE or OUTSIDE Idea: Relace the surface ntegral by a volume ntegral Page 7

8 Calculaton of Hydrodynamc Forces α Defne auxlary functon as α α ( X ) for for X Ω X Ω Remark: everywhere excet at wall surface of the artcles, and equal to the normal vector defned on the global grd. n f n α Force actng on the wall surface of the artcles can be comuted by F Γ σ n dγ Ω T σ α dω T Ω wth (analogously for the torque) T Ω f Ω Page 8

Valdaton of Force Calculatons LEVEL 6 280.

9 Valdaton of Force Calculatons LEVEL elements LEVEL elements LEVEL elements Page 9

10 Oerator-Slttng Aroach The algorthm for t n t n+1 conssts of the followng 4 substes Flud velocty and ressure : Calculate hydrodynamc forces: Calculate velocty of artcles: Udate oston of artcles: ( n+ 1 n+ 1) ( n n u, BC Ω u ) NSE, u Ω f n +1 F g ( n F ) ( n u ) n n f 5. Algn new mesh Requred: effcent calculaton of hydrodynamc forces Requred: effcent treatment of artcle nteracton (?) Requred: fast (nonstatonary) Naver-Stokes solvers (!) Page 10

11 Grd Deformaton Method ( ) Idea : construct transformatonφ, x φ ξ, t wth det φ f local mesh area f ( ) 1 1. Comute montor functon f x, t > 0, f C and Ω f 1 ( x, t ) dx Ω, t [0,1] 2. Solve ( t [0,1]) Δ v 1, t f ( ξ t ) ( ξ, t ), v n Ω 0 3. Solve the ODE system φ t ( ξ, t ) f ( φ ( ξ, t ), t ) v ( φ ( ξ, t ), t ) new grd onts: x φ ( ξ,1) Grd deformaton reserves the (local) logcal structure of the grd Page 11

12 Numercal Examles Page 12

13 Generalzed Tensorroduct Meshes Page 13

14 Numercal Examles Vscous flow around a movng arfol (Glownsk) 1 Level 3 Level 4 Level Level 3 Level 4 Level 5 Angular Velocty (ω) Angle (θ) t t Page 14

15 Rotaton of an Arfol Wng Page 15

Lft-Off for Crcle Velocty (d w 0.1) Velocty (d w 1.0) d w 0.11 y 2 1.5 1 d w 0.31 d 0.

16 Lft-Off for Crcle Velocty (d w 0.1) Velocty (d w 1.0) d w 0.11 y d w 0.31 d 0.4 w d 0.6 w d w 0.8 d w t y of center of ball Page 16

17 Lft-Off for Ellse Velocty (d w 0.4) Velocty (d w 1.8) y d 0.21 w d w 0.4 d w 0.6 d w t y of center of ellse Page 17

18 Numercal Examles Kssng, Draftng, Thumblng Page 18

19 Numercal Examles Imact of heavy balls on 2000 small artcles ρ ρ ρ f bd s ρ ρ ρ f bd s ρ ρ ρ f bd s Page 19

20 Collson Models Theoretcally, t s mossble that smooth artcle-artcle collsons take lace n fnte tme n the contnuous system snce there are reulsve forces to revent these collsons n the case of vscous fluds. In ractce, however, artcles can contact or even overla each other n numercal smulatons snce the ga can become arbtrarly small due to unavodable numercal errors. ' F R ρ d R F F ' ' 0, d ε, d R d R + + R R. + ρ, R + R R + R + ρ d Page 20

21 Page 21 Page 21 Reulsve Force Collson Model For the artcle-artcle collsons (analogous for the artcle-wall collsons), the reulsve forces between artcles read: Handlng of small gas and contact between artcles Dealng wth overlang n numercal smulatons The total reulsve forces exerted on the -th artcle by the other artcles and the walls can be exressed as follows: ( )( ) ( )( ) P P P d R R X X d R R X X F, ' 2, ε ρ ε for for for +ρ + > R R d, ρ R R d R R, R R d + <, + N W P F F F 1,, '

22 Examles Page 22

23 Effcent Data Structures L elements d. o. f. s L elements d. o. f. s L elements d. o. f DEC/COMPAQ EV6, 833 MHz. s Requred: Effcent flow solver (for small Δ t )??? Page 23

24 A Many Body Lubrcaton Model In ths model, roosed by Maury, the lubrcaton forces for artcle artcle collsons (artcle wall collsons) are estmated by : F F κ ( D )[( x& x& ) e ] e κ ( D )[( x& x& ) e ] e, κ κ and vansh when the dstance D between two bodes s greater than a gven value, whch s taken equvalent to the sze of the body. d o ( ) ( ) ( ); κ d μ( 1/ d), κ d μ ln do / d th x s the mass centre of the body; e s the unt vector along xx and e s erendcular to e. Page 24

25 m Partcle moton: x '' Φ + F, ( ' ' x, x ),,, Ω f Ω '' I θ r Φ F, s the body force. ( ' ' x, x ).,, Ω x r D r x D o Numercal examle: Fg. 1. Partcle-Partcle and Partcle-Wall collson Page 25

26 Partcle Agglomeraton ' F R ρ d R F F ' ' 0, c ε, d R d R + + R R. + ρ, R + R R + R + ρ d F P 0-1 ' ε P 1 ε P ( X X )( R + R + ρ d ) ( X X )( R + R d ),, 2 for for for d, > R + R +ρ + R d R + R R, d < R + R, + ρ Page 26

27 Examles Page 27

28 3D Examles Page 28

29 Challenges Adatve tme steng + dynamcal adatve grd algnment/ale Coulng wth turbulence models. Deformable artcles/flud-structure nteracton. Analyss of vscoelastc effects. Benchmarkng and exermental valdaton for many artcles. Why tensorroduct-lke meshes and r-adatvty???. Page 29

30 Hardware-orented Numercs FEM for 8 Mll. unknowns on general doman, 1 CPU, Posson Problem n 2D Seed-u (logscale) Dramatc mrovement (factor 1000) due to better Numercs AND better data structures/ algorthms on 1 CPU Page 30

31 Numercs on secal hardware CELL multcore rocessor (PS3), 7 synergstc rocessng 3.2 GHz, 218 GFLOP/s, 3.2 GHz GPU (NVIDIA GTX 285): GHz, GHz memory bus (160 GB/s) 1.06 TFLOP/s UnConventonal Hgh Performance Comutng (UCHPC( UCHPC) Page 31

32 Examle: Sarse MV on TP Grd 40 GFLOP/s, 140 GB/s on GeForce GTX (1.4) GFLOP/s on 1 core of Xeon E5450 Page 32

33 Alternatve Forceless Model An alternatve model, based on the work of Patankar, allows to smulate the flow of rgd artcles n a flud wthout the exlct calculaton of the hydrodynamc forces! The general dea of the model can be summarzed as follows: 1. The Naver-Stokes equatons are solved everywhere wth dfferent denstes for the flud and the rgd body. 2. In a ostrocessng ste the soluton for the rgd body s roected from a flud moton onto the moton of a rgd body. 3. The rgd body s moved accordng to the velocty calculated n the ostrocessng ste. Start the next tme ste n+1 at Ste 1. Page 33

34 Flud Moton Model Consder the flow of a sold artcle wth densty ρ s n a flud wth densty ρ and f vscosty ν. Denote by Ω f (t) the doman occued by the flud at tme t, by Ω P (t) the area occued by the artcle at tme t and let Ω Ω f U Ω P. The flud flow s modelled by the Naver-Stokes equatons wth the ntermedate velocty ũ : u~ ρ t + ~ ~ ~ ( u ) u νδu + ρg, u 0 ~ ~ ρ ρ f ( n+ 1) n+ 1 1 H + ρ H s H n n Ω n Ω f P Page 34

35 Page 35 Page 35 Proecton onto Rgd Moton Ω + + P n s n d M 1 1 ~ u U ρ Ω Ω + + d M d P P n n / ~ r u r ω The quanttes that govern the moton of a rgd body are ts translatonal velocty u and ts angular velocty ω. In the roecton ste we calculate these quanttes from the soluton of the flud. The roecton onto rgd body moton s realzed as follows: The velocty nsde the rgd body n the next tme ste n+1 s then set to: n n n n P n r U u ω

36 Advantages of the New Method Summary and further caabltes of the new method: No exlct calculaton of the hydrodynamc forces. The calculaton of the artcle moton requres only the evaluaton of two ntegrals. Better ntegraton results by usng the enalty method. The model s caable of handlng non-rgd bodes or very comlcated geometres by addng a level-set functon Page 36

37 Penalty Method Usng the usual FBM-aroach we get a functon that s non-contnuous on an element. By usng a enalty aroach we can smooth ths functon to make t contnuous, whch wll gve more accurate results durng ntegraton. ρ 1 Ths s done by calculatng the area that flud and resectvely artcle occuy and settng the value for the element to: ρ 2 ρ Element a a Flud Element asold ρ 2 + ρ 1 a Element Page 37

38 Rotatng Dsc Examle Page 38

39 Examles Page 39

40 Partculate Flow Examle Page 40

41 Sedmentaton Examle Page 41

Particulate Flow Simulations with Complex Geometries using the Finite Element-Fictitious Boundary Method

Particulate Flow Simulations with Complex Geometries using the Finite Element-Fictitious Boundary Method Partculate Flow Smulatons wth Complex Geometres usng the Fnte Element-Fcttous Boundary Method Raphael Münster, Otto Merka, Stefan Turek Insttut für Angewandte Mathematk, TU Dortmund Page Related Multphase