The Euler-Lagrange Approach for Steady and Unsteady Flows. M. Sommerfeld. www-mvt.iw.uni-halle.de. Title. Zentrum für Ingenieurwissenschaften

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1 Tile The Eler-Lagrange Aroach for Seady and Unseady Flows Joseh-Lois Lagrange (736 83) M. Sommerfeld Leonard Eler ( ) Zenrm für Ingenierwissenschafen D Halle (Saale), Germany www-mv.iw.ni-halle.de

2 Conen of he Lecre Inrodcion discree aricle mehods (DM) Descriion of he Eler/Lagrange aroach aricle racking deails Seady and nseady coling Convergence behavior Examles of alicaion nseady aricle-laden swirling flow aricle disersion in a sirred vessel Conclsions/Olook

3 Hybrid CFD DM The nmerical calclaion of disersed wo-hase flows is ofen based on a coled hybrid aroach: CFD (flid flow) and DM (aricle hase) Flid Flow Eler aroach based on coninm assmion (fixed nmerical grid): DNS, LES, RANS or URANS aricle hase Lagrangian racking of a large nmber of aricles assming oin masses wih D << x (exce for collisions) Two-way coling Advanages: aricle size disribion deailed modelling

4 CFD-Mehods for Flid Flow For he nmerical calclaion of flid flows he following mehods are alied: nmerical mehods for laminar flows Direc nmerical simlaions (DNS) Energy conaining eddies Increased modelling reqiremens E L E S Large eddy simlaions (LES) Regime of dissiaion k / x / η Nmerical mehods based on he Reynolds-averaged conservaion eqaions (RANS) in connecion wih rblence modelling.

5 DM Mehods Discree aricle mehods (oin aricles) for flid-aricle sysems: DM wih sof-shere collisions Discree elemen mehods (DEM) m I β =,i,i d v d d ω d β = 50,i,i = j V ( F + F ) n,ij,ij + m + g ( ε) * ρg CD g ε 4 D ( ε) ε 2 j µ D g β V = i,ij,ij + ( ) ( ) g ε ( r F M T ) ρ D ε g ( ε) g,i Mlile aricle conacs (finie sized aricles) aricle moion cased by flid-dynamic, exernal and conac forces (Van der Waals, elecrosaic,. ) Conac model: sring, dasho, fricion slider Limied by he nmber of ossible aricles > 0.8 normal ε 0.8 Wen & Y for dile regime angenial Ergn for dense regime

6 DM Mehods 2 Discree aricle mehods (oin aricles) for flid-aricle sysems: DM wih hard-shere collisions DM wih all real aricles sandard Lagrangian racking (reresenaive aricles, arcels) Solve imlse eqaions for ranslaion and roaion aricle moion driven by flid-dynamic and exernal forces as well as binary collisions Only one insananeos collision of wo aricles a a ime (ime se: even driven) m deerminisic iner-aricle collisions sochasic iner-aricle collision model d d v β V,i,i = + I d x d d ω,i,i d =,i,i ( ) ( ) V m g g ε = T,i

7 DM Mehods 3 Remarks on he alicaion of DM: The aricles need o be considerably smaller han he nmerical grid The acion of he aricles shold be disribed o he neighboring cells Wiho iner-aricle collisions here is no mechanism o avoid aricle concenraions larger han he closes acking There shold be no limiaion in he alicaion of he Eler/Lagrange aroach as long as all he reqired hysics is modelled roerly!!!

8 CFD DM Calclaions Conservaion eqaions for he coninos hase wih momenm coling erms. Coniniy eqaion ( ε ρg ) + ε ρg g = 0 Momenm eqaion (Navier-Sokes) N ( ε ρg ) β V + ε ρ = ε ε τ + ε ρ ( ) ( ) g g g g g V = ε cell i Examles of alicaions for dense aricle-laden flows: CFD-DM (laminar): Hoomans e al. (996) wo-dimensional flidised bed CFD-DM (laminar): Helland e al (2002) wo-dimensional flidised bed CFD-DEM (laminar): Feng and Y 2007 binary mixre in flidisaion RANS-DEM: Zhao e al. (2008) rblen 2d soed bed LES-DEM: Zho e al. (2004) 2d simlaion of flidised bed

9 Examles of CFD-DM Calclaions RANS-DEM of soed bed (Zhao e al. 2008) widh 52mm, deh 5 mm, saic heigh 00 mm wo-dimensional flow simlaions (k-ε rblence model) serficial gas velociy.58 m/s glass beads: 2 mm Averaged gas velociy Trblen kineic energy

10 Eler/Lagrange Aroach The flid flow is calclaed by solving he Reynolds-averaged conservaion eqaions (seady or nseady) by acconing for he inflence of he aricles (sorce erms). Trblence models wih coling: k-ε rblence model Reynolds-sress model Two-way coling ieraions The Lagrangian aroach relies on racking a large nmber of reresenaive aricles (oin-mass) hrogh he flow field acconing for roaion and all relevan forces like: Models elemenary rocesses: rblen disersion aricle-rogh wall collision iner-aricle collisions drag force graviy/boyancy sli/shear lif sli/roaion lif orqe on he aricle aricle roeries and sorce erms resl from ensemble averaging for each conrol volme

11 Eler/Lagrange Aroach 2 Flid flow calclaions: Reynolds averaged (conservaion) Navier- Sokes eqaions (RANS) combined wih a siable rblence model. General form of he conservaion eqaions wih sorce erms (Lain ( σ φ) + ( σ U φ) = Γ + S S f x & Sommerfeld 202): σ = ( α )ρ f j f j x j φ x j φ + k-ε rblence model φ φ S φ S φ, Γ φ U i k U j U x Γ + ρ i j x i x i ε k C G G k ρε ε ( C ρε) G k U k 2 U U i j i = µ +, x j x i x j g i µ = C µ µ + µ S U i, S µ k, S µ ε, 2 k ρ ε µ + σ k µ + σ ε consans C µ = 0.09 C =.44 C 2 =.92 σ k =.0 σ ε =.0

12 Eler/Lagrange Aroach 3 The simlaion of he disersed hase by he Lagrangian aroach imlies he racking of a large nmber of aricles hrogh he calclaed flow field solving a se of ordinary differenial eqaions (Sommerfeld 996, 2008, 200). aricle locaion aricle velociy aricle anglar velociy d x d d d = m = Fi Hea and mass ransfer beween hases reqires he solion of wo addiional arial differenial eqaions for drole diameer and drole emerare (Sommerfeld e al. 993 a). drole size d D drole emerare d T 6 Q 2 m = = 2 3 d π ρ D d π ρ D c l The aricles are reaed as oin-masses and heir size ms be smaller han he dimensions of he nmerical grid. Each arcel consiss of a nmber of real aricles wih idenical roeries. Seqenial (saionary flows) and simlaneos (nseady flows) racking of he arcels. I d ω d = T l,l

13 µ ρ = τ D B 2 B B B c Re D 3 4 aricle Tracking Isses Inegraed version of he eqaion of moion o be solved nmerically: The Lagrangian ime se is adjsed dynamically along he aricle rajecory Crieria for he selecion of Lagrangian ime se (Sommerfeld 996, Lain and Sommerfeld 2008): ( ) τ + τ τ + = drag non L L new F ex ex ( ) K L cross L,, T, T min 0.25 τ τ = Residence ime in he conrol volme aricle resonse ime: Time scale of rblence µ ρ = τ D 2 c Re D 3 4 Collision ime: ( ) Rel K, D, n = f τ ( ) τ τ + τ τ τ + + = drag non L L 2 L L new F ex ex x x

14 aricle Tracking Isses 2 Random aricle injecion a inle bondary: inle lane rofile Nmber of real aricles in a arcel: ff gm, j A j N,k = m N m arcel ref Samling of he aricle saring osiion wih a niform disribed random nmber: y 0,k = + y RY Samling of he iniial aricle size from a rescribed or measred size disribion (cmlaive disribion fncion) y j,k j Samling of he aricle velociy comonens from a normal disribion fncion (mean vale and sandard deviaion inerolaed on aricle osiion): Generaion of he flid velociies seen by he aricle sing local mean vales and rblen flcaions from a normal disribion fncion..0.0 Q 0 (x i ) Q (x ) r i x i x i+ 0,k =, j x x + σ, j Selec RN Loo hrogh he cmlaive disribion Q 0 (x i ) > RN aricle size RU

15 aricle Tracking Isses 3 roeries of he aricle hase in a conrol volme (Decker 2005): aricle volme fracion: α aricle velociy: U,i = N n S π 3 = D,n N,n f s,n V 6 n S n= KV,i n= N,n f s,n n ref n f s,n = * n n aricle mean flcaing velociy: σ,i = N n S n= 2,i N,n f s,n n U 2,i Weighing facor: N n = S n= N,n f s,n n * n aricle mass flx: f,i = α ρ U,i * n In he deerminaion of he averaged aricle roeries i ms be insred ha enogh arcels have crossed he conrol volme. redced saisical ncerainy

16 Eler-Lagrange Coling Coling aroaches beween Elerian and Lagrangian calclaions Flly nseady calclaion Seady calclaion Ensemble averaging of sorce erms from insananeos aricle disribion Temoral and ensemble averaging of sorce erms from aricle rajecories Momenm sorces from all flid dynamic forces Momenm sorces from change of Reqires a large nmber of aricles aricle velociy along rajecories Reqires less aricle rajecories

17 Eler-Lagrange Coling 2 Temoral discreisaion of Eler/Lagrange aroach: Saionary Flows are solved sing he seady-sae formlaion of he conservaion eqaions ( T E = ): T L is deermined from he characerisic ime scales of aricle moion Seqenial or simlaneos racking of he arcels Ensemble and emoral averaging of sorce erms dring each coling ieraion Under-relaxaion is essenial o imrove convergence Unseady flows are calclaed on he basis of he ime-deenden conservaion eqaions ( T L T E ): Elererian nd Lagrangian rogramme modles are solved seqenially wih idenical ime ses Simlaneos aricle racking Saial and ensemble averaging of aricle hase roeries and sorce erms (no nder-relaxaion) wihin a comaional cell A large nmber of arcels are needed o obain reliable sorces

18 Eler-Lagrange Coling 3 Unseady flows wih T E > T L are also calclaed on he basis of he ime-deenden conservaion eqaions (Sommerfeld e al. 997; Liowsky and Sommerfeld 2005; Sommerfeld e al. 200): The selecion of T E deermines he emoral resolion of he flow Seqenial calclaion of boh hases wih differen ime ses The aricles see a flow field frozen over T E Simlaneos aricle racking Temoral and ensemble averaging of aricle hase roeries and sorce erms (no nder-relaxaion) wihin a comaional cell Elerian Calclaion E Flow field Sorces Lagrangian Calclaion L

19 Eler-Lagrange Coling 4 aricle hase sorce erms for k-ε rblence model: Samling of sorce erms along arcel rajecories calclaed from aricle velociy change (ensemble and ime averaging): Momenm eqaions, Crowe e al. 977: SU i, = V Trblen kineic energy: cv E k m k N k n n+ n ( ) k,i k,i g i ρ ρ L B L S k, = S i k,i S k,i = i S U i i S U i Dissiaion rae: ε Sε = Cε3 Sk, ε3 k ( C =..8),

20 Eler-Lagrange Coling 5 Grid generaion, Bondary condiions, Inle condiions Elerian ar Calclaion of he flid flow wiho aricle hase sorce erms Lagrangian ar Tracking of arcels wiho ineraricle collisions, Samling of aricle hase roeries and sorce erms Two-way coling aroach for saionary flows Under-relaxaion of sorce erms: S i+ φ = i i+ ( γ) Sφ + γ Sφ ( calclaed) Coling Ieraions no Elerian ar Calclaion of he flid flow wih aricle hase sorce erms: Converged Solion Solion wih a fixed nmber of ieraions Lagrangian ar Tracking of arcels wih iner-aricle collisions, Samling of aricle hase roeries and sorce erms Convergence wo-way coling Under-relaxaion of he sorce erms imroves convergence behavior!!! (Kohnen e al. 994) yes O: Flow field, aricle-hase saisics

21 Eler-Lagrange Coling 6 Temoral coling of Elerian and Lagrangian calclaions. Flid flow calclaed a fixed ime se E aricles are racked in a frozen flid field over E Dynamic calclaion of L for every Lagrangian ime se Indeenden ime seing for each aricle STs are wrien back o flid solver for he nex Elerian ime se

22 Horizonal ie Flow Convergence behavior of saionary aricleladen flow in he ie (seady-sae) (Lain & Sommerfeld 202) Normalised Residals [ - ] Two-way coling E-3 E-4 E-5 E-6 E Nmber of Ieraions [ - ] hor. velociy ver. velociy k-ε rblence model ie lengh: 6 m ie diameer: 63 mm U 0 = 20 m/s 30 μm aricles ρ = 2450 kg m 3 mass loading η =.0 R0 = 2.2 µm, γ =.4 For-way coling

23 Horizonal ie Flow 2 Convergence behavior of saionary aricle-laden flow in he ie: y / D [ - ] single-hase. ieraion 3. ieraion 7. ieraion 8. ieraion 0. ieraion 8. ieraion 27. ieraion rofile of sream-wise gas velociy.0 N ier 7 Two-Way U / U 0 [ - ] rofile of rblen kineic energy y / D [ - ] k / U 2 0 [ - ] single-hase. ieraion 3. ieraion 7. ieraion 8. ieraion 0. ieraion 8. ieraion 27. ieraion

24 Examles of Alicaion Unseady aricle-laden swirling flow Liowsky and Sommerfeld 2005; Sommerfeld e al. 200 aricle disersion in a sirred vessel (Decker and Sommerfeld 2000; Sommerfeld and Decker 2004)

25 Examles of Alicaion aricle searaion devices nemaic conveying

26 Thank yo very mch for yor aenion!!! We like o si on he aricles