Application of the Immersed Boundary Method to particle-laden and bubbly flows

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1 Application of the Immersed Boundary Method to particle-laden and bubbly flows, B. Vowinckel, S. Schwarz, C. Santarelli, J. Fröhlich Institute of Fluid Mechanics TU Dresden, Germany EUROMECH Colloquium 549, Leiden, June 2013

2 Motivation - Particles [J. Gaffney, University of Minnesota, Bed load transport Important for multiple applications Fundamental research 2

3 Motivation - Particles Experiments at University of Aberdeen in progress Our goal: Analysis by direct numerical simulations Later: comparison with experiments 3

4 Motivation - Bubbles Our goals: Bubble - turbulence interaction Bubbles in MHD flows Formation of metal foams X-Ray measurements [Boden EPM 2009] Ar in GaInSn No magnetic field 4

5 1. Basic fluid solver and IBM 2. Collision modeling 3. Bed load transport 4. Bubble laden flows

6 PRIME (Phase-Resolving simulation Environment) Solver for continuous phase Incompressible Navier-Stokes equations S u t ρ ( uu ) + p = ν u + f 2 nd order Finite Volumes Staggered Cartesian grid Speedup with grid points solid: ideal, dash-dot: PRIME Explicit 3-step Runge-Kutta for convective terms Implicit Crank-Nicolson for diffusive terms E Highly sophisticated libraries PETSc for parallelization (MPI) HYPRE as solver (Poisson, Helmholtz Eq.) Scaleup with grid points per processor 6

7 Immersed boundary method [Uhlmann JCP 2005] Fluid solved on fixed Cartesian grid Fluid-solid interface represented by markers Phase coupling by volume forces Interpolation of velocities to marker points Direct forcing Spreading to grid points Regularized δ functions used U F = d Δt Γ U f F Regularized δ function [ Roma, 1999] Grid points and surface markers Properties good stability order of convergence nd Od Order 1 st Order L 2 - error of velocity, single sphere in channel flow 7

Angular momentum balance: I c dω p = ρ f r τ n S t ( ) d dt Γ + M c Runge Kutta (3 rd order) for time

8 Mobile particles [Uhlmann JCP 2005] [Kempe & Fröhlich JCP 2012] Linear momentum balance: m p du u dt p = ρ f Γ τ n d S + ( ρ p ρ ) V f p g + F c Collisions Pressure & viscous forces buoyancy Sphere with 874 marker points Angular momentum balance: I c dω p = ρ f r τ n S t ( ) d dt Γ + M c Runge Kutta (3 rd order) for time integration Parallel implementation in PRIME Master & slave strategy Particle with interface resolution in decomposed domain 8

Improved IBM g f u + = p V V t f p p f p Ω ρ ρ ρ d

density ratios Accurate imposition of BC s 9

9 Improved IBM g f u + = p V V t f p p f p Ω ρ ρ ρ d ) ( d d 8 ( ) = = ,, m m m m m k j i H φ φ φ α R Velocity u at L y /2, Re=10 Stable time integration for previously g p y inaccessible density ratios Accurate imposition of BC s 9 Velocity of buoyant & sedimenting particles [Kempe & Fröhlich, JCP, 2012]

10 1. Basic fluid solver and IBM 2. Collision modeling 3. Bed load transport 4. Bubble laden flows

11 Normal particle-wall collision with various collision models u p soft-sphere model repulsive potential, k n =1e6 experiment Repulsive potential Hard-sphere model Soft-sphere model repulsive potential, k n =1e4 Choice of coefficients? Time scale separation for hard-sphere model fluid and collision Surface distance vs. time Δt t Δt f c 100 Models not usable for large-scale l simulations i 11

12 Adaptive collision time model (ACTM) Purpose Avoid reduction of Δ t f Give correct restitution ratio e = dry u u out in Ideas Stretch collision in time to T c = 10 Δ t f Optimization to find coefficients ODE of collision process m p 2 d ζ n 3/ 2 + d + 2 n kn ζ n d t d ζ n d t = 0 ζ Physically exact ACTM Problem: given : u in, e dry, T c 0 r p find : d n, k n Fixed-point problem solved by Quasi-Newton scheme T c t linear approaches possible [Breugem 2010] 12

impact angle Sliding: Coulomb friction RS

t I m n, i n, i n, i n, i ( u t t ( t t p

13 Oblique collisions Oblique = normal + tangential [Joseph 2004] Normal ACTM Tangential force model (ATFM) Consider only sliding and rolling Critical local impact angle Sliding: Coulomb friction RS Ψ F col t = μ Ψ f in F = n g g cp t, in cp n, in Rolling: Compute force such that relative cp surface velocity is zero = 0 g t I m n, i n, i n, i n, i ( u t t ( t t p uq Rp ω p + ωq ) Δt ( I + m R ) col p p Ft = 2 2 p p p t 13

14 Lubrication modeling Local grid refinement not desired Under-resolved fluid in gap [Gondret, 2002] with lubrication model without model explicit expression for lubrication force [Cox & Brenner, 1967] F lub n R R p q 1 = 6π μ f gn ζ + n R p R q 2 Particle-wall collision, St = 27 Resolved and modeled contributions during a particle-wall collision 14

15 Adaptive collision model (ACM) Phase 1 Phase 2 Phase 3 U p,out U p,in Approach Surface contact Rebound Lubrication model Normal ACTM Tangential ATFM Lubrication model 15

Normal collisions Results [Kempe & Fröhlich

97 present Experiment [Gondret, 2002] present

simulation, right: experiment [Eames, 2000]

time step can be used Local grid refinement is

16 Normal collisions Results [Kempe & Fröhlich JFM 2012] e dry = 0 e dry = 0.97 present Experiment [Gondret, 2002] present Flow around sphere impacting on wall, Left: simulation, right: experiment [Eames, 2000] Steel spheres impacting on glass wall Fluid time step can be used Local grid refinement is avoided Good agreement with experiments St τ p ρ p u p D = = τ 9μ f f p 16

17 Oblique collisions Results [Kempe & Fröhlich JFM 2012] Effective angle of impact Effective angle of rebound Ψ Ψ in out = = g g cp t, in cp n, in g g cp t, out cp n, in cp g n, in cp g t, in cp g t, out Compared with experiment [Joseph, 2004] Steel spheres, smooth, μ f = 0.01 Ψ RS = 0.25 Glass spheres, rough, μ f = 0.15 Ψ RS = 0.95 present sliding Ψ RS rolling 17

18 Stretched collisions with multiple particles Results with static stiffness E Fluid E kin Significant influence of Δt Final particle positions Results with ACM E total /20 E Fluid E kin CFL = 1 (stretched) CFL = 0.02 (Hertz) Results independent of Δt 18

19 1. Basic fluid solver and IBM 2. Collision modeling 3. Bed load transport 4. Bubble laden flows

Short history DNS of sediment transport Uhlmann &

fluid points 560 particles, short runs, no

velocity yprofile for fluid Papista 2011 2D,

interface resolved 645 million fluid points 8696

20 Short history DNS of sediment transport Uhlmann & Fröhlich D, interface resolved 18 million fluid points 560 particles, short runs, no statistics Osanloo D, mass points Fixed velocity yprofile for fluid Papista D, interface resolved 200 particles Vowinckel D, interface resolved 645 million fluid points 8696 particles This project 3D, interface resolved 1.4 billion fluid points particles 20

21 Computational Setup H/D L x /H L z /H Re b Re τ D Two layers of particles: Fixed bed: single layer of hexagonal packing Sediment bed on top Free slip No slip Periodic in x- and z- direction 21

22 Numerics N x N y N z N tot D/Δx Δx + N proc true DNS all scales resolved 22

23 Computational Setup Case N p,fixed N p,mobil Θ /Θ crit Configuration Fix Two fixed layers Ref One layer of mobile particles FewPart Lower mass loading LowSh Lower mobility Mobility Shields parameter θ = 2 uτ ρ ( ρ p ρ )gd θ Smooth Transitional Probability of particle motion Rough Ref, FewPart D + LowSh 23

24 Case Ref Θ /Θ crit = 1.18 [Vowinckel et al. ICMF 2013] Spanwise oriented dunes Contour Iso-surfaces of fluctuations Particle u/u b u p / u τ 4 u' / U b = -0.3 u p / u τ < 4 u' '/U b = fixed 24

25 Case Ref Θ /Θ crit = 1.18 mean fluid velocity fluctuations Fix Ref FewPart LowSh 0.6 y / H 0.4 y / H y / H / U b / U b Mobile or resting Fixed bed / U 2 b higher intrusion, larger resistance, more turbulence 25

26 Less particles Θ /Θ crit = 1.18 Streamwise oriented, inactive ridges Contour Iso-surfaces of fluctuations Particle u/u b u p / u τ 4 u' / U b = -0.3 u p / u τ < 4 u' '/U b = fixed 26

27 Less particles Θ /Θ crit = 1.18 Streamwise oriented, inactive ridges Mobile Fixed 27

28 Heavier particles Θ /Θ crit = 0.75 Inactive plane bed with single eroded particles Contour Iso-surfaces of fluctuations Particle u/u b u p / u τ 4 u' / U b = -0.3 u p / u τ < 4 u' '/U b = fixed 28

29 Conclusions so far [Vowinckel et al. ICMF 2013] Two parameters: mass loading & mobility (=density) many & light particles: dunes with distance 12H few & light : inactive ridges many & heavy : closed plane bed Experiments [Dietrich et al. Nature 1989] Sediment patterns in agreement with experimental evidence 29

30 1. Basic fluid solver and IBM 2. Collision modeling 3. Bed load transport 4. Bubble laden flows

31 Bubble shape depends on regime Reynolds number [Clift et al. 1978] Re = u d p eq ν Eötvös number Eo = ρ p ρ f σ g d 2 eq 31

32 Representation of bubble shapes Spherical Particles as model for bubbles Ellipsoidal time dependent shape X(t) b X = a a b Spherical harmonics + m r ( φ, θ, t ) = a0 a ( t ) Y ( φ, θ ) 0 nm n ) n, m Exact evaluation of curvature, normal vector, volume Coupling of shape to fluid loads by local force blance (vanishing displacement energy) [Schwarz & Fröhlich ICNAAM 2012] [Bagha 1981] 32

33 Light particles in vertical turbulent channel flow Upward flow lenght x height x width = 2.2H 2H x H x 1.1H1H periodic in x- und z- direction Wall: no-slip Fluid Reynolds number Mesh for unladen flow 33

34 Particle parameters Fixed shape: oblate ellipsoid ρ p Density ratio: ρ f N p D eq /H D eq / a/b Void fr. Re p ~24 2 ~ 2 % ~214 34

35 Looking at transport of particles by the flow influence of particles on turbulence particle-particle interaction [Santarelli ICMF 2013] 35

36 Results: fluid phase mean streamwise velocity turbulent kinetic energy Symmetry loss and turbulence enhancement 36

37 Ascent of single bubble in liquid metal [Schwarz 2013] Argon bubble d eq = 4.6 mm in quiescent GaInSn: Eo = 2.5 Experiments by [Zhang et al. 2005] Cartesian grid: 84 Mio. Points 9093 Lagrangian surface markers Wobbling bubble shape 60 x 61.5 CPU hours Re Re t Unsteady rise velocity, ellipsoidal Comparison PRI Averaged 287 rise ME velocity 287 matches exp. data Underprediction 9 of 1standard deviation in Re Good agreement in oscillation frequency f σ Re / f ref 0 6 [Schwarz, 2011, 2013] 37

38 Single bubble in liquid metal with vertical magnetic field f = N ( j B) L N = σ el B 2 0 ρ u ref d eq j σ ( Φ + u B) j = 0 = e 2 Φ = ( u B) ) B y B y j Φ σ e electric current density j electric potentialj electric conductivity j Instant. helicity iso-contours N y = 0 N y = 1 N y = 0 N y = 1 More rectilinear bubble trajectory [Schwarz 2013] Reduced doscillation frequency & amplitude in Re Large bubbles rise faster, small ones slower Reduced wake vorticity with larger vortices being aligned with B 38

39 Thank you for your attention. Feedback and questions? Bubble Coalescence 13:10 13:30 S. Schwarz, S. Tschisgale and J. Fröhlich Bubble coalescence model for phase-resolving simulations using an Immersed Boundary Method 39

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

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 Motivation wide range