Lubrication correction for the simulation of dense suspensions Application to solid propellants

Transcription

1 Lubrication correction for the simulation of dense suspensions Application to solid propellants Stany GALLIER 1,2, Laurent LOBRY 2, Elisabeth LEMAIRE 2, François PETERS 2 1 SAFRAN-SME, Centre de recherches du Bouchet, Vert le Petit 2 Laboratoire de Physique de la Matière Condensée, Université de Nice 22/11/2011 S.GALLIER et al. Workshop GISEC 33 - Pessac 22/11/11 1/ 18

2 Solid propellant : an heterogeneous material Scale : 10+1 m S.GALLIER et al. Workshop GISEC 33 - Pessac 22/11/11 2/ 18

3 Solid propellant : an heterogeneous material Scale : 10+1 m S.GALLIER et al. Scale : 10 1 m Workshop GISEC 33 - Pessac 22/11/11 2/ 18

4 Solid propellant : an heterogeneous material Scale : 10+1 m Scale : 10 1 m Scale : 10 3 m Scale : 10 4 m Particles (AP, aluminum,...) embedded in a rubber-like polymer Particle size µm Volume fraction > 70 % S.GALLIER et al. Workshop GISEC 33 - Pessac 22/11/11 2/ 18

5 Solid propellant : process Main process steps : - Mixing (particles in a liquid-phase polymer) - Casting - Curing Uncured solid propellant is a concentrated suspension Propellant casting S.GALLIER et al. Workshop GISEC 33 - Pessac 22/11/11 3/ 18

6 Solid propellant : microstructure Microscale ordering of particles (microstructure) is altered by process (mixing, casting, conveying,...) Homogeneous Isotropic S.GALLIER et al. Workshop GISEC 33 - Pessac 22/11/11 4/ 18

7 Solid propellant : microstructure Microscale ordering of particles (microstructure) is altered by process (mixing, casting, conveying,...) Homogeneous Isotropic Inhomogeneous (Migration) Inhomogeneous (Segregation) Homogeneous Anisotropic Microstructure does indeed have a signicant eect on propellant macroscale properties (e.g., combustion, mechanics, dielectric properties...) S.GALLIER et al. Workshop GISEC 33 - Pessac 22/11/11 4/ 18

8 Numerical simulations of suspensions Motivation Study the link : Process Microstructure Macroscale properties Development of a 3D microscale DNS (Direct Numerical Simulation) code Non-boundary-tted Cartesian mesh Fictitious domain method S.GALLIER et al. Workshop GISEC 33 - Pessac 22/11/11 5/ 18

9 Numerical simulations of suspensions Lagrange multiplier-free ctitious domain (Yu et al., 2007) u t ρ f Z ρ f Z P u = 0 in D (1) + u u = p ρ f + ν 2 u + λ in D (2) P u = U + Ω r in D p (3) λ dx = (ρ f ρ p) M[ du g] (4) ρ p dt r λ dx = (ρ f ρ p) ρ p ˆJ. dω dt + Ω (J.Ω) (5) Numerical method Finite dierencing - Second-order accurate in space Projection method S.GALLIER et al. Workshop GISEC 33 - Pessac 22/11/11 6/ 18

10 Lubrication forces Lubrication forces play a major role in dense suspensions Challenging to account for in DNS due to subgrid nature : - signicant for separation less than 10 2 a while grid spacing is 10 1 a S.GALLIER et al. Workshop GISEC 33 - Pessac 22/11/11 7/ 18

11 Lubrication forces Lubrication forces play a major role in dense suspensions Challenging to account for in DNS due to subgrid nature : - signicant for separation less than 10 2 a while grid spacing is 10 1 a Widely-used (though defective) approach in DNS consists in just adding the theoretical force F th lub(ξ) More rigorous way in Stokesian dynamics but hardly suits DNS framework S.GALLIER et al. Workshop GISEC 33 - Pessac 22/11/11 7/ 18

12 Lubrication correction method Novel approach based on the superposition : (R) = (D) + (L) S.GALLIER et al. Workshop GISEC 33 - Pessac 22/11/11 8/ 18

13 Lubrication correction method Novel approach based on the superposition : (R) = (D) + (L) Calculation steps : 1 Compute conguration (D) by DNS (lubrication-free = no model needed) 2 Superpose forces/torques : F (D) + F (L) = 0 3 Solve lubrication problem (L) Compute U (L) 4 Superpose velocities : U (R) = U (D) + U (L) S.GALLIER et al. Workshop GISEC 33 - Pessac 22/11/11 8/ 18

14 Step 1 : Compute rigid doublet ow Account for new (non-spherical) particle P = P 1 + P 2 (rigid dumbbell) DNS computation yields velocities U doublet and Ω doublet Forces/torques on each particle P k read : F (D),k = ρ f ZP k λ (D) dx T (D),k = ρ f ZP k r λ (D) dx S.GALLIER et al. Workshop GISEC 33 - Pessac 22/11/11 9/ 18

15 Step 2 : Solve lubrication problem Superpose forces and torques (F (L), T (L) ) = (F (D), T (D) ) Solve the lubrication problem (e.g., in mobility formulation) 0 U (L),1 1 2 A 11 A 12 (B 11 ) T (B ) T BU (L),2 Ω (L),1 A = µ 1 6A 12 A 22 (B 12 ) T (B 22 ) T 7 B 4B 11 B 12 C 11 C 12 Ω (L),2 B 21 B 22 C 12 C 22 From which (U (L), Ω (L) ) are deduced F (D),1 F (D),2 T (D),1 T (D),2 1 C A S.GALLIER et al. Workshop GISEC 33 - Pessac 22/11/11 10/ 18

16 Step 3 : Correct velocities Superpose velocities : Velocities in conguration (D) read : U (R) = U (D) + U (L) Ω (R) = Ω (D) + Ω (L) U (D),k = U doublet + Ω doublet (x k G x G,doublet ) Ω (D),k = Ω doublet S.GALLIER et al. Workshop GISEC 33 - Pessac 22/11/11 11/ 18

17 Stresslet correction Similarly, consider S (R),k = S (D),k + S (L),k with : - S (D),k computed from DNS code - S (L),k obtained by lubrication theory : 0! S (L)» 1 G 11 G 12 H 11 H 12 S (L) = B G 21 G 22 H 21 H 2 Isotropic and deviatoric parts corrected separately F (D),1 F (D),2 T (D),1 T (D),2 S.GALLIER et al. Workshop GISEC 33 - Pessac 22/11/11 12/ 18 1 C A

18 Two major upsides Correction relative to a rigid doublet conguration : Rigid doublet is the solution for ξ 0 Expected good behavior for near-contact (the correction becoming vanishingly small) Lubrication problem solved for a ow at rest : Rigorous even for non-linear ows (e.g., Poiseuille ows) In contrast, usual methods consider E in the lubrication problem, which is correct only for constant E S.GALLIER et al. Workshop GISEC 33 - Pessac 22/11/11 13/ 18

19 Validation : sheared rigid doublet Rigid doublet in a shear ow Creeping ow conditions L = 20a = a/5 Force (F (D) x,f (D) y ) En bloc rotation velocity Ω z,doublet S.GALLIER et al. Workshop GISEC 33 - Pessac 22/11/11 14/ 18

20 Two particles in a shear ow : velocities Theoretical solution (Batchelor, 1972) : U (1) U (2) = ɛ i i ijk ω j r k + r j E ij r k E jk {A(r) r i r j r 2 Ω i = ω i + C(r)ɛ ijk E kl r i r j r 2 + B(r)[δ ij r i r j r 2 ]} S.GALLIER et al. Workshop GISEC 33 - Pessac 22/11/11 15/ 18

21 Two particles in a shear ow : velocities Theoretical solution (Batchelor, 1972) : U (1) U (2) = ɛ i i ijk ω j r k + r j E ij r k E jk {A(r) r i r j r 2 Ω i = ω i + C(r)ɛ ijk E kl r i r j r 2 + B(r)[δ ij r i r j r 2 ]} No lubrication correction S.GALLIER et al. Workshop GISEC 33 - Pessac 22/11/11 15/ 18

22 Two particles in a shear ow : velocities Theoretical solution (Batchelor, 1972) : U (1) U (2) = ɛ i i ijk ω j r k + r j E ij r k E jk {A(r) r i r j r 2 Ω i = ω i + C(r)ɛ ijk E kl r i r j r 2 + B(r)[δ ij r i r j r 2 ]} No lubrication correction Lubrication-corrected S.GALLIER et al. Workshop GISEC 33 - Pessac 22/11/11 15/ 18

23 Two particles in a shear ow : relative trajectories S.GALLIER et al. Workshop GISEC 33 - Pessac 22/11/11 16/ 18

24 Two particles in a shear ow : relative trajectories Theoretical solutions : DaCunha & Hinch (1996) Relative trajectories Relative distance S.GALLIER et al. Workshop GISEC 33 - Pessac 22/11/11 17/ 18

25 Conclusions and perspectives Conclusions Development of a novel lubrication correction method Particularly suited for DNS methods Rigorous for non-linear ows Good behavior and accuracy in near-contact conditions Ongoing and future work Extension to many-particle systems Particle-wall interactions S.GALLIER et al. Workshop GISEC 33 - Pessac 22/11/11 18/ 18