An introduction to the lattice Boltzmann method for multiphase flows with interfacially active components Roar Skartlien 2012
Multiphase flow a big research area gas/liquid/solids/particles combinations Some flows in nature Sedimentary flow (particles, turbulence, dense beds..) Bubbles in liquid (breaking waves, jets...) Industry Emulsions (oil/water droplets) Boiling (steam/water), oil/water/gas Solid particles or fibers in liquid Simulations and models on diffent scales MD < DPD < Lattice Boltzmann < Navier Stokes < Large scale models
Some multiphase flows
Lattice Boltzmann method (LBM): what is it? A multiphase Navier Stokes solver Solves the underlying Boltzmann equation instead of the Navier Stokes equations Turns out to be a numerically very efficient method for multiphase flows The Boltzmann equation governs the probability distribution function (PDF) of the fluid particles (function of position and particle velocities) The statistical moments of the PDF give the hydrodynamic quantities (same info as solving NS) The constituents are treated as fluids: continuous fields, not molecules
Why LBM, then? Easy to implement your own code Numerically efficient; suits 3D (important) CHEMISTRY: Relatively easy to implement interfacially/surface active components; surfactant Can handle complex solid boundaries
Some applications Emulsions in shear flow oil g water Generation of droplets by flow Rayleigh-Taylor instability Initially: water over oil
Boltzmann s equation (one for each fluid): f is the Probability Distribution Function of the particles in (x,v,t) space 7 dimensional space (3 space, 3 velocity, 1 time)!! Can handle, if clever! F: forces between fluids over the interfaces Collision term: This is where VISCOSITY originates In an equilibrium: f is the isotropic Maxwell-Boltzmann distribution (a symmetric Gaussian type of PDF)
Moments of f give the fluid dynamical quantities :
Phase space (v,x) for 1 space dimension X PDF shown as contours: V X Integration over V gives hydrodynamic quantities at chosen X
Brute force discretization: too many sampling points! V X
V1 Much better and sufficient: Coarse sampling in V (~10), fine in X (up to ~millions in 3D) PDF-populations ai Associated to the velocity points Vi: a1 V2 a2 V3 a3... X Populations ai are associated with the discrete velocities Vi
Clever coupling between velocity sampling and grid point separation: We choose that each population (ai) move with a discrete velocity (Vi), that fits the grid structure: Populations move exactly between neighbor grid points, over one timestep (also diagonally between nodes): No interpolation necessary Vi Basic algorithm is: move > collide/ mix all ai > move > collide
The lattice Boltzmann equation: move > collide > move > collide 1. The populations fa are updated by the collision term (right hand side) 2. Populations move to neighboring grid nodes (left hand side, 1.st term) 3. Collision step is repeated on all the grid nodes The collision term relaxes fa to the equilibrium value on a timescale Force can be incorporated here: Maxwell distribution, expanded for numerical reasons
3D and LBM LBM is extremely well suited for parallel processing (e.g., MPI) on many CPU s simultaneously Simple algortihm: data transfer only between neighbor gridpoints algebraic operations only (no differential equations to solve) no explicit interface tracking in the basic version of the method
Simplicity comes at a cost: Limitations in viscosity ratio, Reynolds number, velocities May then be numerically unstable if pushed : find flow regime where it works! May have mass diffusion over interfaces in some cases! The simulated fluid is COMPRESSIBLE (and there will be sound waves) compressibility can be made small by gentle driving NOTE: more sophisticated versions of LBM exist that overcome some of these restrictions
Interfacial chemistry in fluid dynamics Molecules with an affinity to the interface determine Interfacial tension (lowered with more surfactant) Interfacial stress variations (along the interface) Interfacial stability (hydrodynamic) Coalescence and breakup rates of droplets Droplet sizes Challenge: Hydrodynamics on large scales (relative to the molecular scale): Molecular scale inacessible in the same simulation Solution: Include surfactant as a continuum (fluid) that interacts with the two ordinary fluids (oil and water usually) Preserve most important effects from molecular surfactant structure: some approximations involved Can t handle / impractical to construct, continuum models of complex molecules
Simple LBM surfactant force model water oil Forces with adjustable strengths rotation e.g.span 80 Coarse graining Surfactant density Surfactant continuum model Dumbbell model: a very coarse grained amphiphile (representing e.g. Span or Tween) A dipole vector field, with a mass density: vectorial fluid Vector represents average direction over computational grid cell Force strengths control Solubility in either fluid Interfacial tension Diffusivity in solvents and on interface hydrophobic hydrophilic 15-Oct-13 17
DPD vs. LBM DPD: coarse grained molecules OIL LBM: continua surfactant density OIL WATER WATER OIL WATER DPD: Dissipative Particle Dynamics coarse grained, but still molecular detail LBM: all species are treated as CONTINUA SAME TYPE of forces act in both DPD and LBM Bottom line: This is equivalent, but less details emerge in the LBM results
Droplet coalescence Surfactant can oppose coalescence between droplets Litte or no flow, long timescales: short range electrostatic forces dominate, thin films between droplets Vigorous flow, shorter timescales: Flow induced Marangoni effect, thicker films between droplets LBM simulations can shed light on emulsions in a dynamic flow, with the Marangoni effect Not suited for short range molecular electrostatic forces
Marangoni effects Lowered interfacial tension where there is more surfactant on the interface Diverging flow induced by force: Force Did you observe the effect of a soap droplet on water (while doing the dishes)??
-Buoyant droplet settling on plane interface -Suppressed film drainage, by the Marangoni effect No S With Lower row: surfactant density Lower surfactant concentration here. time Delayed coalescence (lower row) with surfactant 15-Oct-13 21
Interfacial force opposing film draining Draining Force
Droplet coalescence in large 3D simulations; 256^3. water-in-oil type t=1000 t=10000
Interfacial area with time Increased surfactant activity (stronger surfactant forces): Interfacial area up at any given time (after coalescence begins) More and smaller drops
Rheology with emulsions in a shear flow What influences the emulsion viscosity? Droplets increase the viscosity, but how much depends on the flow conditions and the accompanying droplet shapes LBM simulations can be used for these studies Influence of surfactant
Emulsion in shear with surfactant: we observe a stabilizing Marangoni effect here as well (2D simulation) No S applied shear time W/S Droplets slide past each other with surfactant (lower row) 15-Oct-13 26
Large 3D simulations Weak surfactant forces Strong surfactant forces: Less coalescence, more breakup, more and smaller droplets Strong: IFT reduced with a factor of about 0.5 Strain at given time = 14 (horizontal stretched distance / vertical distance)
Effective viscosities & interfacial area (as function of droplet volume fraction) no surf w/surf w/surf no surf Surfactant: higher interfacial area, but still lower total interfacial stress! (at high volfract.) Mainly due to a reduced IFT More sensitive at higher volume fraction due to to preferrential concentration of surfactant on interface. More efficient reduction of interfacial stress. Tends to bi-continuous domains (no droplets) at volume fraction 0.5
Preferrential concentration of surfactant: reduced interfacial shear stress at high volume fraction (here: bi-continuous morphology) Mobility on interfaces: View in flow direction Surfactant (yellow) accumulates near higher curvature interfacial areas (red), and opposes high interfacial shear stresses there Stronger net effect in bicontinuous morphology, so shear viscosity is reduced more
Some final remarks on the LBM Very flexible modelling tool to study multiphase flow Computationally FAST Any wall geometry Some disadvantages that lead to limited parameter ranges: LBM is a research tool, more than an engineering tool (but try for yourself) Highly suited for hydrodynamic problems Rheology Emulsions Turbulence, etc Surfactant must be modelled as a continuum (fluid) too A suitable micro-macro link = meso scale model Offers a host of research possibilities Not suited to model effects of complex molecules Use MD or DPD to capture complex molecules:
Thank you for your attention! Acknowledgements: Collaborators on LBM simulations at IFE/NTNU (via the project FACE): E. Sollum (IFE, NILU) P. Meakin (IFE, UiO, INL) K. Furtado (IFE, MetOffice) A. Akselsen (NTNU) T. Kjeldby (NTNU) F. Fakharian (IFE MSc student) Current collaborators in Ugelstad Lab: Johan Sjöblom Brian Grimes Galina Rodionova Funding sources: NRC and oil companies: Statoil ASA, ConocoPhillips Scandinavia A/S, VetcoGray Scandinavia A/S, SPTgroup AS, FMC technologies, CD-adapco, and Shell Technology Norway AS
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For more details on the LBM without surfactant: Quick intro for thos who want to implement now : Lattice Boltzmann Modelling Sukop and Thorne More broad, for applications The lattice Boltzmann Method, S. Succi For details on LBM force model: contact me Short recap of basic LBM properties: + LBM is computationally fast, so large 3D is possible -/+ Simple algorithms - May not be robust (numelical stability issues), so limited parameter ranges + Easy to get started with your own code + Complex geometry easy to implement Many different versions/improvements of LBM exist (force version, phase field version, multiple relaxation time version), but all built on top of the Boltzmann equation
Exact relaxation (offset for a net macro velocity Intermadiate relaxation in general:
Velocity moments: now sums over velocity vectors
What about the forces, F? species direction Forces are given in terms of potentials strengths G and coupling depends monotonically on fluid densities Repulsive (oil/water) and/or attractive (van der Waals) depends on sign of G
What can the LBM do? Multiphase flow: Solid particles, liquid droplets, bubbles With surfactant species (e.g. soap) Complex porous media (any geometry) Turbulence as DNS or LES simulations Other: Liquid vapor Magnetic and electric fields + fluids Thermal convection
Parameters for this 3D case Viscosity ratio 1 and 0.3 (water in oil type) Viscosity ratio = 1.0: isolates contributions from interfacial stress Number of droplets ~ 100-500: sufficient number Parallel processing MPI - FORTRAN 90 10 000 time steps: 1-2 days on 8-12 processor workstation (quite fast!) 4.2 million gridpoints, 256 (streamwise) x 128 (vertical) x 128 (spanwise) Physical sizes (scaleable), e.g, Domain size ~ 1 cm (2.8 x 1.4 x 1.4 cm) Drop or filament size ~ 1 mm Shear rate ~ 10 (1/s) (Delta U = 0.1 m/s over 1 cm) Interf. Tens. ~ 20 mn/m Re ~ 100 (based on wall velocities). Laminar for single phase, But complicated fluctuating velocity field in the emulsion Pe ~1 (ratio between surfactant diffusion and advection in the interface)
Parameters 3D, shear Run over 10^4 timesteps (~1 sec. with scaling above), obtaining a strain = 15.6. Large deformations Initially, Ca below critical. Coalescence may bring it above critical, sometimes breakup. Re ~ 100 (based on wall velocities). Laminar for single phase, Complicated velocity field in the emulsion, may even have significant Reynolds stress Pe ~1 (ratio between surfactant diffusion and advection in the interface)
Capillary number, deformation, tilt angles as function of time (viscosity ratio 1.0, w/surfact.) Average capillary number increases (due to coalescence : larger drops) Larger drops: more deformation Droplets gradually align with the flow, so effective viscosity should decrease Figure by Andreas Akselsen
Effective shear viscosity with surfactant Increasing volume fraction Total shear viscosity Viscous stress Interfacial stress Surfactant contribution Unity viscosity ratio: interfacial stress is the main contributor Time variation explained by gradual tilting and stretching of droplets (interfaces) Late times: more aligned droplets, smaller effective viscosity
Strong vs weak surfactant forces, late times Unity viscosity ratio Reduced shear viscosity with strong surfactant, but only at higher volume fractions!
Experimental, polymer blends Affine deformation model T. Jansseune, I. Vinckier, P. Moldenaers, J. Mewis, J. Non-Newt. Fluid Mech. 99, 167-181
Some parameters coalsescence, 2D N x N=128 x 128 Length X [m]= 0.128 Spatial resolution [m]= 0.001 Time resolution [s]= 0.1 Viscosity phase 0 [m^2/s]= 1.66667e-006 (close to water) Viscosity phase 1 [m^2/s]= 1.66667e-006 Coupling parameters : g00, g11, g01, g0s, g1s, gss: 0.0 0.0 1.2 0.026-0.026-0.053
Setup for large 3D simulation Viscosity ratio = 0.3 (highest achievable for reliable results with current method) 256 x 256 x 256 = gridpoints, 10000 timesteps, 2-3 days simulation time, 30 Gb data MPI parallel processing O(1000) to O(100) droplets, decreases over time due to coalescence Volfrac = 0.25 Different surfactant concentrations
Some other results: New results of the stability properties of dynamic interfaces for the buoyant jet, and for droplet detachment (dripping faucet), presented at ICMF 2010, Florida (presented by G. Zarruk) Good feedback confirming the novelty of our work, and good publicity for FACE! Simulations and analysis by Espen Sollum, Roar Skartlien, Kalli Furtado, Paul Meakin, and summer student in 2009. 15-Oct-13 48