Experience with DNS of particulate flow using a variant of the immersed boundary method

Transcription

1 Experience with DNS of particulate flow using a variant of the immersed boundary method Markus Uhlmann Numerical Simulation and Modeling Unit CIEMAT Madrid, Spain ECCOMAS CFD 2006

2 Motivation wide range of applications for particulate flow: 1 Introduction - pneumatic transport - fluidized beds - sedimentation, very complex dynamics: - formation of agglomerations - interaction between particles and turbulence, many open questions remain M Uhlmann 1

3 The Challenge of DNS 1 Introduction 2h example: turbulent plane channel flow - solid fraction: 05% 2πh - particle size: 1 10 wall units particles Re τ = 200 πh adapting grid is very costly use fixed grid/immersed boundary agglomerations long correlation times M Uhlmann 2

4 Objectives 1 Introduction 1 Method for DNS of Dilute Particulate Flow resolution of all turbulent flow scales resolution of fluid/solid interfaces efficiency, parallelism validated on standard test cases 2 Grid Convergence Study unsteady free motion in shear flow laminar & turbulent regime M Uhlmann 3

5 Immersed Boundary Method 2 Numerical Method t u = rhs + f direct forcing technique (Mohd-Yusof 1997) f = u(d) u n t rhs x ijk S M Uhlmann 4

6 Immersed Boundary Method 2 Numerical Method t u = rhs + f direct forcing technique (Mohd-Yusof 1997) f = u(d) u n t rhs x ijk S 1 define set S of forced points 2 determine velocity u (d) to be imposed M Uhlmann 5

7 Immersed Boundary Method 2 Numerical Method t u = rhs + f direct forcing technique (Mohd-Yusof 1997) f = u(d) u n t rhs x ijk S 1 define set S of forced points 2 determine velocity u (d) to be imposed assure temporal smoothness of forces M Uhlmann 6

8 Grid-Induced Perturbations 2 Numerical Method Taylor-Green flow in immersed circle max error x 10 3 linear interpolation shift D/ x=40 max error x 10 3 present method shift/ x M Uhlmann 7

9 Algorithm of Present Method distribute forcing points on the interface (X l ) 2 Numerical Method 1 preliminary velocity ũ = t rhs n + u n 2 compute the forcing 3 solve Navier-Stokes (a) interpolation: ũ δ h Ũ (b) F = (U rigid Ũ)/ t (c) distribution: F δ h f t u = rhs n+1/2 + f u = 0 M Uhlmann 8

10 Transfer: Regularized Dirac Function 2 Numerical Method Ũ(X l ) = i,j,k ũ(x ijk ) δ h (x ij X l ) x 3 (Peskin 1972,2002) compact support: 3 x δ 1 h (r) smooth: continuously differentiable second order precision conserves angular momentum r/ x M Uhlmann 9

11 Details of Numerical Framework 2 Numerical Method Navier-Stokes projection method 3-step Runge-Kutta & Crank-Nicholson finite-differences: second order, centered, staggered grid Particle Motion discretized Newton equations hydrodynamic force (torque) = sum forcing fluid coupling fluid-particles: explicit M Uhlmann 10

12 Parallel Implementation 2 Numerical Method Fluid Work 3D Cartesian domain decomposition Helmholtz: approximate factorization Poisson: multi-grid Particle Work particles assigned to processor holding fluid environment overlapping: share work with neighbor processors M Uhlmann 11

13 Convergence Study: Configuration 3 Results h Vertical plane channel flow one wall, one symmetry plane particle size D/h = 1/20 y z g x grid resolution: D/ x N L G G G G M Uhlmann 12

14 A) Laminar Flow Regime Single Particle trajectory 3 Results x h y/h Re b = 1000 box size: Ω/h = CF L = 025 terminal particle velocity: Re D, = 100 M Uhlmann 13

15 Visualization of the Near-Field 3 Results G 1 G 2 G 3 G 4 elapsed time t 2 = 098h/U b ; Re D = 614 M Uhlmann 14

16 Convergence Results 3 Results particle velocity fluid statistics G 1 G 2 G 3 G 4 4 max error 10 2 x 2 u u τ x/h y/h evaluated at t 2 accumulated up to t 2 M Uhlmann 15

17 B) Turbulent Flow Regime 4 Particles Re b = 2700; box size: Ω/h = 2 1 1; CF L = Results terminal particle velocity Re D, = 136; D + = 9 M Uhlmann 16

18 Convergence Results: Particle Velocity 3 Results streamwise wall-normal spanwise 15 G 1 G 2 G u p U b tu b /h tu b /h tu b /h max error = 008U b M Uhlmann 17

19 Eulerian Fluid Statistics 3 Results 4 streamwise G 1 1 wall-normal 1 spanwise G 2 3 G u i u τ y/h y/h max relative error below 7% y/h M Uhlmann 18

20 Conclusions 4 Conclusion 1 Established tool for DNS with interface resolution smoothness, stability, efficiency access to systems with great complexity 2 Grid refinement for unsteady motion in shear flow laminar: determined convergence with O( x 2 ) turbulent: sufficient resolution D/ x 13 guideline for future DNS M Uhlmann 19

21 Perspectives 4 Conclusion 1 Additional validation - comparison with spectral element solutions 2 Applications - pure sedimentation - vertical plane channel flow - sediment transport in horizontal channels M Uhlmann 20

22 4 Conclusion 4096 particles φ s = 0042 zoom M Uhlmann 21

23 Extra EXTRA M Uhlmann 22

24 Distribution of Points on a Sphere Platonic solid, sub-division repulsive point particles Extra M Uhlmann 23

25 Interpolation & Distribution Extra Euler Lagrange: Ũ(X l ) = i,j ũ(x ij ) δ h (x ij X l ) x 3 Lagrange Euler: f(x ij ) = l F(X l ) δ h (x ij X l ) V l M Uhlmann 24

26 Configuration Laminar Case Extra ρ p /ρ f = 417 St = D 2 ρ p /ρ f U b /(18νh) = 058 g D/Ub 2 = u c, = 2U b Re D, = u c, D/ν 100 M Uhlmann 25

27 Configuration Turbulent Case Extra ρ p /ρ f = St = D 2 ρ p /ρ f U b /(18νh) = 083 g D/Ub 2 = u c, = U b Re D, = u c, D/ν 136 M Uhlmann 26

28 Convergence Results: Angular Particle Velocity Extra 2 streamwise 2 wall-normal 2 spanwise ω c,i h U b tu b /h tu b /h tu b /h max difference between solution on G 1 and G 3 : 004U b /h 040U b /h 023U b /h M Uhlmann 27

29 Plot Legend Extra u = ±45u τ (positive, negative) intense vorticity in grey (λ 2 ) M Uhlmann 28

30 Particle Animation Extra u p +2U b, u p 2U b M Uhlmann 29

31 Efficiency of the Method Extra time step: CFL near theoretical limit 3 operation count: fluid O(N ( x N y N z ) particles O N p (D) 2 ) h timing: IBM p655 system, Power 4 D N x N y N z N p h nproc t exec [s] M Uhlmann 30

32 Extra References [1] J Kim, D Kim, and H Choi An immersed-boundary finite-volume method for simulations of flow in complex geometries J Comput Phys, 171: , 2001 [2] M Uhlmann An immersed boundary method with direct forcing for the simulation of particulate flows J Comput Phys, 209(2): , 2005 M Uhlmann 31