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

g

oil

water

Rayleigh-Taylor instability

Initially: water over oil

Generation of droplets by flow

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:

X

V

Integration over V gives hydrodynamic quantities at chosen X

Brute force discretization:

too many sampling points!

X

V

Much better and sufficient: Coarse sampling

in V (~10), fine in X (up to ~millions in 3D)

X

V1

Populations ”ai” are associated with the discrete velocities ”Vi”

a1

a2

a3 ....

PDF-populations ai

Associated to the

velocity points Vi:

V2

V3

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

Basic algorithm is:

move > collide/”mix” all ai > move > collide…

Vi

Maxwell distribution,

expanded for

numerical reasons

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

The lattice Boltzmann equation: move > collide > move > collide…

Force can be incorporated here:

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

15-Oct-13 17

Simple LBM

surfactant force model

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

Forces with

adjustable strengths

hydrophobic

hydrophilic

oil

water rotation

e.g.Span 80

Surfactant density

Coarse graining

DPD vs. LBM

• 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

LBM: continua

surfactant density

DPD: coarse grained

molecules

OIL

WATER OIL

OIL

WATER WATER

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)??

15-Oct-13 21

-Buoyant droplet settling on plane interface

-Suppressed film drainage, by the Marangoni effect

time

No S

With

Lower surfactant concentration here.

Delayed coalescence (lower row) with surfactant

Lower row: surfactant density

Force

Draining

Interfacial force opposing film draining

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

15-Oct-13 26

Emulsion in shear with surfactant:

Droplets slide past each other with surfactant (lower row)

time

applied shear No S

W/S

we observe a stabilizing Marangoni effect here as well (2D simulation)

Weak surfactant forces

Strong: IFT reduced with a factor

of about 0.5

Large 3D simulations

Strong surfactant forces:

Less coalescence, more breakup,

more and smaller droplets

Strain at given time = 14 (horizontal stretched distance / vertical distance)

Effective viscosities & interfacial area

(as function of droplet volume fraction)

w/surf

w/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

no surf

no surf

Preferrential concentration of surfactant: reduced interfacial shear

stress at high volume fraction (here: bi-continuous morphology)

Mobility on interfaces:

Surfactant (yellow)

accumulates near

higher curvature interfacial

areas (red),

and opposes high interfacial

shear stresses there

Stronger net effect in bi-

continuous morphology, so

shear viscosity is reduced

more

View in flow direction

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

END

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

Intermadiate relaxation

in general:

(offset for a net macro velocity)

Velocity moments:

now sums over velocity vectors

What about the forces, F ?

species direction

•Forces are given in terms of potentials and coupling

strengths G

• 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

Total shear

viscosity

Interfacial

stress

Surfactant contribution

Viscous

stress

Increasing volume fraction

•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

T. Jansseune, I. Vinckier, P. Moldenaers, J. Mewis, J. Non-Newt. Fluid Mech. 99, 167-181

Affine deformation model

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

15-Oct-13 48

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.