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, 17-19 June 2013
Motivation - Particles [J. Gaffney, University of Minnesota, www.youtube.com] http://ryanhanrahan.wordpress.com Bed load transport Important for multiple applications Fundamental research 2
Motivation - Particles Experiments at University of Aberdeen in progress Our goal: Analysis by direct numerical simulations Later: comparison with experiments 3
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
1. Basic fluid solver and IBM 2. Collision modeling 3. Bed load transport 4. Bubble laden flows
PRIME (Phase-Resolving simulation Environment) Solver for continuous phase Incompressible Navier-Stokes equations S u 1 + 2 t ρ ( uu ) + p = ν u + f 2 nd order Finite Volumes Staggered Cartesian grid Speedup with 2.65 10 8 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 1.3 10 5 grid points per processor 6
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 1.7 2 nd Od Order 1 st Order L 2 - error of velocity, single sphere in channel flow 7
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 ) ( d d 8 ( ) = = 8 1 8 1,, 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]
1. Basic fluid solver and IBM 2. Collision modeling 3. Bed load transport 4. Bubble laden flows
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
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
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
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
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 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
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
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
1. Basic fluid solver and IBM 2. Collision modeling 3. Bed load transport 4. Bubble laden flows
Short history DNS of sediment transport Uhlmann & Fröhlich 2006 3D, interface resolved 18 million fluid points 560 particles, short runs, no statistics Osanloo 2008 2D, mass points Fixed velocity yprofile for fluid Papista 2011 2D, interface resolved 200 particles Vowinckel 2012 3D, interface resolved 645 million fluid points 8696 particles This project 3D, interface resolved 1.4 billion fluid points 40500 particles 20
Computational Setup H/D L x /H L z /H Re b Re τ D + 9 24 6 2941 193 21 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
Numerics N x N y N z N tot D/Δx Δx + N proc 4800 240 1200 1.4 10 9 22.2 0.95 4048-16192 true DNS all scales resolved 22
Computational Setup Case N p,fixed N p,mobil Θ /Θ crit Configuration Fix 27000 0 0 Two fixed layers Ref 13500 13500 1.18 One layer of mobile particles FewPart 13500 6750 1.18 Lower mass loading LowSh 13500 13500 0.75 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
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 = 03 0.3 0 fixed 24
Case Ref Θ /Θ crit = 1.18 mean fluid velocity <u> fluctuations <u u > 1 1 0.8 1 0.8 0.6 0.8 0.6 Fix Ref FewPart LowSh 0.6 y / H 0.4 y / H 0.4 0.2 y / H 0.4 0 0.2-0.2 0 0.5 1 <u> / U b 0.2 0-0.2 0 0.5 1 <u> / U b Mobile or resting Fixed bed 0-0.2 0 0.02 0.04 0.06 <u u > / U 2 b higher intrusion, larger resistance, more turbulence 25
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 = 03 0.3 0 fixed 26
Less particles Θ /Θ crit = 1.18 Streamwise oriented, inactive ridges Mobile Fixed 27
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 = 03 0.3 0 fixed 28
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
1. Basic fluid solver and IBM 2. Collision modeling 3. Bed load transport 4. Bubble laden flows
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
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
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
Particle parameters Fixed shape: oblate ellipsoid ρ p 0 001 Density ratio: 0.001 ρ f N p D eq /H D eq / a/b Void fr. Re p 740 0.05172 ~24 2 ~ 2 % ~214 34
Looking at transport of particles by the flow influence of particles on turbulence particle-particle interaction [Santarelli ICMF 2013] 35
Results: fluid phase mean streamwise velocity turbulent kinetic energy Symmetry loss and turbulence enhancement 36
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 / 0.28 369 0.27 245 f ref 0 6 [Schwarz, 2011, 2013] 37
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
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