lbm, a useful tool for mesoscale modelling of single phase ... · that the lbm is a useful tool for...

2nd Micro and Nano Flows ConferenceWest London, UK, 1-2 September 2009

- 1 -

LBM, a Useful Tool for Mesoscale Modelling of Single Phase and MultiphaseFlow – the variety of applications and approaches at Nottingham

Yuying YAN*, Yingqing ZU, Bo DONG

* Corresponding author: Tel.: ++44 (0)115 951 3169; Fax: ++44 (0)115 951 3159;Email: [email protected]

School of the Built Environment, University of Nottingham, UK

Abstract Giving an overview of Nottingham group’s recent progress on numerical modelling andapproaches in developing and applying the lattice Boltzmann method (LBM), the paper tries to demonstratethat the LBM is a useful tool for mesoscale modelling of single phase and multiphase flow. The variety ofapplications of the LBM modelling is reported, which include single phase fluid flow and heat transferaround or across rotational cylinder of curved boundary, two-phase flow in mixing layer, electroosmoticallydriven flow in thin liquid layer, bubbles/drops flow and coalescence in conventional channels and inmicrochannels with confined boundary, liquid droplets in gas with relative large density ratio; viscousfingering phenomena of immiscible fluids displacement, and flow in porous media.

Keywords: LBM, multiphase flow, mixing layer, electroosmosis flow, drops, bubbles, coalescence, wetting,hydrophobic, large density ratio

1. IntroductionIn recent years, along with extensive

applications of conventional computationalfluids dynamics (CFD) to the study of fluidsflow and transport phenomena, the latticeBoltzmann method (LBM) has become anestablished numerical scheme for simulatingsingle phase and multiphase fluid flows. Thekey idea behind the LBM is to recover correctmacroscopic motion of fluids by incorporatingthe complicated physics of problems intosimplified microscopic models or mesoscalekinetic equations. In this method, kineticequations of particle velocity distributionfunctions are first solved; macroscopicquantities are then obtained by evaluatinghydrodynamic moments of the distributionfunction. This intrinsic feature enables theLBM to model phase segregation andinterfacial dynamics of multiphase flow, whichare difficult to handle by conventional CFD orthe molecular dynamics (MD) method. Sincethe foundation of the first lattice Boltzmannmodel for multiphase flow in 1991, a numberof multiphase and multi-component modelsbased on LBM have been introduced (Succi,2001). To date, the LBM has demonstrated a

significant potential and broad applicabilitywith many computational advantagesincluding the easy parallelisation and thesimplicity of programming (Chen & Doolen,1998). Based on the two-component lattice gasmethod, Gunstensen et al. (1991) proposed amulti-component model; Shan & Chen (1993)have initiated a pseudo-potential model whichhas the capacity of simulating multiphase andmulti component immiscible fluid flow ofmean-field interactions, and the model waslater analysed and evaluated carefully in Shanet al., 1994, 1995, 1996, and Martys et al.,2001). Meanwhile, Swift et al. (1995, 1996)proposed the free energy model, in which theVan der Waals formulation of quasi-localthermodynamics for two component fluid inthermodynamic equilibrium state is built. Later,a new LBM model called index model wassuggested by He et al. (1999) which use indexfunction to track the interface of multi-phaseflow with large density ratios. To overcome thedifficulties of dealing with large density ratioof two-phase flow, Inamuro et al. (2004)developed a LBM model based on theprojection method to predict the behaviours ofincompressible bubbles/particles in bulk liquid.

mailto:[email protected]


- 2 -

The method calculates two distributionfunctions of particle velocity to track theinterface and to predict velocities; thecorrected velocity field satisfying thecontinuity equation is obtained by solving thePoisson equation. On the above basis, Yan &Zu (2007a) developed the LBM model forincompressible two-phase (with large densityratio) flow on a partial wetting surface and themethod is applied to the present simulation.

A variety of applications of LBMmodelling has been attempted by the authors atNottingham since 2006; these are summarisedand reported in the following sections.

2. Fluid Flow and Heat Transfer acrossRotational Cylinders of Curved Boundary

Fluid flow around a rotating isothermalcylinder (or between two rotating cylinders) isa common occurrence in a variety of industrialprocesses, such as contact cylinder dryers inthe chemical process, food-processing, papermaking and the textile industries to thecylindrical cooling devices in the glass andplastics industries. Although this is asingle-phase flow problem as shown in Fig. 1,the LBM modeling of such flow is concernedwith effective treatment of moving and curvedboundaries, and considering heat transfer. Inthe numerical approach, a D2Q9 LBM modelwith multiple distribution functions wasemployed to simulate incompressible viscousthermal flow (Yan & Zu, 2008a).

Fig. 1: flow field set-up

An extrapolation method for dealing withthe curved boundaries of temperature andvelocity fields was developed based on Guo, etal., 2002; Mei, et al, 2002) and can achievesecond-order accuracy for both velocity andtemperature on the curved wall.

The modelling was well validated against

experiment. Useful results were obtained, suchas Fig. 2 shows the distributions of local Nunumber at Re=200, Pr=0.5, ]2/,2/[

for ratio of rotational velocity and inflowvelocity at k=0.0, 0.1 and 0.5, respectively.

Fig. 2: Distributions of local Nusselt numbersfor Re=200, Pr=0.5, ]2/,2/[ .

3. Two-phase Flow in Mixing LayerMixing layers can often be observed in

flow fields of many engineering applicationssuch as in combustion chambers, pre-mixers ofgas turbine combustors, chemical lasers,propulsion systems, flow reactors, micromixers, etc. Controlling the formation andevolution of the coherent structure in a mixinglayer can normally improve efficiencies ofcombustion, chemical reaction, etc. Certainflow features of mixing layers such as theinstability of flow, evolutions and interactionsof vortices layers have drawn attentions ofboth experimental and computational studies.

The index model (He, et al., 1999) wasemployed to study vortices merging in atwo-dimensional (based on D2Q9) two-phasespatial growing mixing layer (Yan and Zu,2006). With an initial velocity field whichconsists of a hyperbolic tangent profile asdefined in Fig. 3, phase distributions of threevortices merging with values of differentsurface tension were obtained.

Fig. 3 shows

Fig. 3

Fig. 4 shows the phase distribution of zerosurface tension compared with one of surfacetension coefficient at k=0.1. Fig. 5 shows

)2/tanh(1),( yRayxu

0),( yxv


- 3 -

interface distributions and the correspondingvortices contours with different values ofsurface tension.

Fig. 4: Phase distribution of three vorticesmerging

Fig. 5: the interface distribution

4. Electroosmotically Driven Flow in ThinLiquid Layer

The electro-osmotic flow near anearthworm body surface is a basicelectrokinetic phenomenon that takes placewhen the earthworm moves in moist soil. Theflow within a micro thin liquid layer near theearthworm’s body surface is induced by theelectric double layer (EDL) interaction. Themodelling (Zu & Yan, 2006; Yan, et al, 2007)incorporated the lattice Poisson method (LPM)for electric potential in the EDL and the LBM(D2Q9 and index model (He et al. 1999)) forfluid flow with external electric body force.Fig. 6 shows streamline evolution ofelectro-osmotically driven flow within the thinliquid layer near an earthworm body surface.The moving vortices shown may play a majorrole in anti- soil adhesion.

Fig. 6: Stream evolution near earthworm surface

5. Modelling of Viscous FingeringPhenomena of Immiscible Displacement

5.1 The displacement in a channelThe wettability of reservoir rocks plays an

important role in oil recovery industry, whichis recognised to be capable of affecting theproperties of fluid-rock interactions, such asresidual oil saturation, relative permeabilityand capillary pressure. Practically, wettabilityalternation can be achieved by injectingsurfactants, altering salinity, and increasingtemperature. In the oil industry, whensupercritical carbon dioxide (SCO2) is injectedinto a reservoir, the wettability of reservoirrocks can be changed due to a series ofreaction of SCO2, crude oil and rock materials.In the LBM modelling, we focus on the effectof wettability of rocks on viscous fingeringphenomenon which can account for thebreakthrough of the displacing fluid.Additionally, the effect of gravity is considered,even at the pore scale level, still has an impacton fluid distribution and its patterns.

A pseudo-potential D2Q9 model (Shan &Chen, 1993) was employed to the modelling.In addition, the formulation of fluid-solidinteraction (Martys and Chen, 1996) wasconsidered and applied. The effects ofCapillary number, Bond number, viscosityratio and the channel surface wettability on thefingering phenomenon are evaluated through aseries of numerical simulations.

Fig. 7 shows the effect of gravity on theoffsets of finger width and finger length underdifferent conditions of wettability in terms ofthe strength of fluid-solid interaction Gw1(Dong, et al., 2009).

Fig. 7: the effect of gravity on finger patterns

k=0

k=0.1

t=0

t=3Tp/8 X

Y

50 100 150 200

40

Line7:Gw1=-0.02

41 2

Line8:Gw1=-0.03

Line9:Gw1=-0.04

Line1:Gw1=0.04

Line2:Gw1=0.03

Line3:Gw1=0.02

Line4:Gw1=0.01

Line5:Gw1=0.0

Line6:Gw1=-0.01

3

6

5

7

9

8

(b)


- 4 -

5.2 The displacement in porous mediaThe pseudo-potential LBM D2Q9 model

(Shan & Chen, 1993) was employed for themodelling. The effect of capillary number,Bond number and the viscosity ratio of twoimmiscible fluids on viscous fingeringphenomena were simulated and discussed.Fig. 8 shows the interface positions at finalstage of the displacement process withdifferent Bond numbers; the interface front hasa tendency to move downwards under theaction of gravity.

Fig. 8: Final finger patterns (a) Bo = 2.79;(b) Bo = 5.58; (c) Bo = 8.37; (d) Bo = 11.16.

6. LBM Modelling of Bubbles and Dropleton Wetting Surface

A major work of LBM modelling atNottingham aiming to improve the suitabilityof LBM for simulating two-phase fluids flowof large ratios of densities and viscosities waspresented (Yan & Zu, 2007). In addition todeal with large density ratio or viscosity ratio,the LBM model can also deal with fluidsinteraction with solid surface of differentwettability or hydrophobicity (Yan, 2009); thismakes the LBM to be more attractive tool forsimulating two-phase flow problems inengineering and industrial processes.

A D3Q15 free energy lattice Boltzmannmethod is used for three-dimensionalmodeling. To simulate flows with large densityand/or viscosity ratio, a projection method(Inamuro et al., 2004) is employed. In themethod, two particle velocity distribution

functions are used. One is for calculating theorder parameter; the other is for calculating thepredicted velocity field without pressuregradient. The corrected velocity satisfying thecontinuity equation can be obtained by solvinga Poisson equation.

In order to realize the partial wetting onsolid surfaces to agree with the Cahn theory(Cahn, 1977), a wetting boundary condition isintroduced. According to Young’s law, whena liquid-gas interface meets a partial wettingsolid wall, the contact angle, w , measured in

the liquid, can be calculated from a balance ofsurface tension forces at the contact line as

2

)1()1(cos

2/32/3 w

where is the wetting potential and can berepresented by surface tensions SG , SL

and LG (Yan and Zu, 2007). To calculate

such surface tensions within a mean fieldframework, Cahn (1977) assumed that thefluid-solid interactions are sufficientlyshort-range so that they contribute a surfaceintegral to the total free energy of the system.To calculate wall-fluid surface tensions in aclosed form, a new form of free energy (ratherthan the van der Waals free energy used in thetraditional free energy model) is used (Yan &Zu, 2007). Based on such method reported, themotions of bubble and droplet with the effectsof confined boundary are simulated. The ratioof densities of two fluids are from 775 to 1000(for water to air), and the ratio of dynamicviscosities is of 50. The initial surface tensionbetween water and air is of

23 /101 skgLG and the gravitational

acceleration is set at 2/8.9 smg .

6.1 Bubbles flow and coalescenceThe modelling is concerned with the

motion of air bubbles surrounded by waterflow in a horizontal rectangular microchannel(Yan & Zu, 2008b). Initially, two air bubbleswith same diameter md 200 are placed

m300 apart in water inside a rectangular

channel of the length mLx 1200 , the width

and the height mLL zy 300 . The channel

(d)(c)

(a) (b)


- 5 -

has an inlet boundary on the left hand side ofthe channel and a free outflow boundary on theright hand side of the channel. The other foursides of the channel are no-slip solid walls.

The velocity distribution at the inletboundary is specified as,

;0),,0(

;0),,0(

;)/())((16),,0( 2

zyu

zyu

LLyzzLyLUzyu

z

y

zyzyx

where, U is the maximum value of),,0( zyu x .

The bubbles flow in the microchannel atReynolds number ( LxLUL /Re ) of 100

and Capillary numbers )/( LGLUCa of

0.33 is simulated. The evolution with time ofbubbles shapes and interaction are shown inFig. 9. It can be seen clearly that the bubblesmove in x-direction by the thrust force ofsurrounding water flow and meanwhile go upin y-direction due to the effect of buoyancyforce; and with time marching, the twobubbles can finally coalesce into a larger one.To focus only on the shape evolution of the leftbubble at the early stage, it is found that thelower part of the bubble moves more quicklyin x-direction than the upper part, which ismainly caused by the effects of velocityboundary layer near the solid wall of thechannel.

Fig.9: Evolution of bubble shapes

The velocity fields are obtained throughthe numerical modelling. For example, at t =2ms, the velocity distribution at different crosssections of y-z plane such as x = Lx / 2, 2Lx/3and 3Lx/4 , respectively, is shown in Fig. 10;where the solid line, the constant density line,

indicates the interface between the two phases.

Fig.10: Velocity field at cross section, t = 2ms,Re=100: (A: x=Lx/2 , B: x=2Lx/3 , C:x=3Lx/4 ).

Figs. 11(a) and (b) show the velocityvector and the vortices contours, respectively,at t= 2 ms, and at y = Ly / 2 on x-z plane. It canbe seen that the local distribution in coherentstructures is evident; the shape of thecoalescent bubbles is a result of the interactionbetween the fields of velocity and densityconcentration, and this is mainly affected bythe effects of buoyancy force of the bubbles(Mazzitelli, et al., 2003).

Fig. 11: Velocity vector and vortices contoursof coalescent bubble at y = Ly / 2 on x-z plane

for t = 2ms at Re = 100.

As both pressure and velocity distributionsacross the interface are normally excellentindicators of numerical stability for the LBMcalculations (Lee and Lin, 2005), Figs. 10 and11 have actually shown that the present LBMcan be used to obtain reasonable and stablevelocity fields. Indeed, similar to theconventional CFD, the numerical instabilitiesof the LBM for two-phase flow of large

0.5 ms 1.0 ms

1.5 ms 2.0 ms


- 6 -

density ratios are mainly caused by spuriousvelocities and/or the large oscillation of thepressure distribution across the phase interface.However, in the present method, the velocityand pressure are both corrected by solving anadditional Poisson equation after eachcollision-stream step. Such corrections are ableto ensure the velocity to satisfy the continuityequation and smooth pressure distributionseven across the interface, so that to ensure thenumerical stability.

6.2 Droplet spreading on partial wettingsurface

Water droplets in air spreading on a partialwetting wall are simulated (Yan & Zu, 2007).Fig. 12 shows how a small hemisphericalwater droplet evolves with time on aheterogeneous surface. A narrow hydrophobic

strip with width of ml 4106 is located atthe centreline of the surface where

6/5 w , and the other areas are occupied

by the hydrophilic surface with 6/ w .

The initial droplet which has a radius

mr 3105.1 is set at the centre of thewetting surface. As shown in the figure, thedroplet stretches over the area occupied by thehydrophilic surface in the early stages of flowevolution due to the adhesive force of thesurface. At the same time, the droplet rapidlycontracts inward along the hydrophobic strip.With the development of time, the dropletspreads further on the hydrophilic area, andmeanwhile contracts inward along thehydrophobic strip and finally breaks up intotwo smaller droplets. The newly formeddroplets continue spreading until anequilibrium state is reached.

Then, a single droplet spreading on aheterogeneous surface with intersectinghydrophobic strips is simulated. As shown inFig. 13, two cross hydrophobic strips

( 9/5 w ) with width of ml4109 are

located at the centreline of the square surface,the other areas are occupied by the hydrophilicsurface with 4/ w . Initially, the droplet

has a shape of spherical cap with radius of

mr 3102 and height of mh 3101 ,

and is set at the centre of the surface. Theshape evolution of the droplet with time isshown in Fig. 13. From the figure, it can beseen that the droplet symmetrically spreadsinto four hydrophilic sections with thedevelopment of time and finally reaches anequilibrium state with a shape of four-leavedflower.

st 0.0 st 14.0

st 152.0 st 154.0

Fig. 12: Snapshots of a droplet spreading on asurface with a hydrophobic strip.

st 0.0 st 03.0

st 05.0 st 09.0

Fig. 13: Snapshots of a droplet spreading on asurface with intersecting hydrophobic strips.

Finally, the evolution of a water dropletspreading on the heterogeneous surfaceconsisting of alternating and parallelhydrophilic strips with the width and the

steady-state contact angle of ml 4107 ,9/2 w and the hydrophobic strips with

ml 4105 and 9/5 w is considered.

Initially, the droplet has a shape of spherical

cap with radius mr 3102 and height

mh 3101 ; the centre of initial circularcontact line is set on the hydrophilic surfaceand the minimum distances form it to twoneighbouring hydrophobic strips are


- 7 -

m4102 and m4105 respectively. Thebehaviour of 3D droplet and the time evolutionof the contact are shown in Figs. 14 and 15respectively. It can be noted from the figuresthat, the droplet can finally reach a symmetricshape although the centre of the droplet isinitially set at a location other than thecentreline of any strip.

st 0.0 st 04.0

st 14.0 st 18.0

Fig. 14: Droplet spreading on a surfaceconsisting of alternating and parallel strips.

Fig. 15: Evolution of moving contact line on asurface with alternating and parallel strips.

As shown in Fig. 16, two cross sections,CS-I and CS-II, vertical to the heterogeneoussurface are defined. L1 and L2 show thecorresponding lines of intersection of CS-I andCS-II with the heterogeneous surfacerespectively.

Fig. 17 and 18 show the evolution of theinterface and the corresponding velocity fieldson cross section CS-I and CS-II respectively.From Fig. 17, it can be found that the dropletmoves to the left. This should be caused by theasymmetry of the wall surface tension withrespect to the centre line of the droplet. Asshown in Fig. 17, the hydrophilic areaoccupied by the droplet at the left side of L2 islarger then that at the right side of L2. Thismeans that the wall total surface tension at the

left part of the droplet is larger than right one.Therefore, the droplet tents to reach a steadystate when the centre of the droplet reaches acentreline of any strip if the heterogeneoussurface is horizontal. See Fig. 18, the crosssection connects with the hydrophilic strip ofthe surface. So, the contact angle is less then90 degree. Also, the interface and velocityfield are symmetric due to the symmetry of thesurface tension distribution.

Fig. 16: Definition of L1 and L2

(a) st 02.0

(b) st 12.0

(c) st 14.0

Fig. 17: Evolution of interface and thecorresponding velocity fields on CS-I.

(a) st 02.0

(b) st 12.0

(c) st 14.0

Fig. 18: Evolution of the interface and thecorresponding velocity fields on CS-II.

7. ConclusionsAn overview of the work at Nottingham on

developing and applying LBM has been


- 8 -

reported. So far, LBM has been successfullydeveloped or extended to solve fluid dynamicproblems including single phase fluid flow andheat transfer on curved and moving boundary,electroosmotically driven flow in thin liquidlayer, two-phase flow in mixing layer, viscousfingering phenomena of immiscible fluidsdisplacement, and flow in porous media. Inaddition, as a major work, a LBM withcapability of modelling interaction betweentwo phase fluid and partial wetting surfacewith relative large density ratio has beenreported. Using the method, bubbles/dropsflow and coalescence in micro-channels withconfined boundary and liquid droplets in gaswith large density ratio have been simulated.The results have shown that LBM is a usefultool for mesoscale modelling of single phaseand multiphase flow.

8. ReferencesChen, Shiyi, and Doolen, Gary D., 1998,

Lattice Boltzmann method for fluid flows, Ann.Rev. Fluid Mech. 30, 329-364.

Chen, W. L., Twu, M. C., and Pan, C.,2002, Gas-liquid two-phase flow inmicro-channels, Int. J. Multiph. Flow, 281235-1247.

Cahn, J. W., 1977. Critical-Point Wetting. J.Chem. Phys. 66, 3667-3672.

Dong, B., Yan, Y.Y., Li, W.Z., 2009.Numerical study of the effect of wettabilityalteration on viscous fingering phenomenon ofimmiscible fluids displacement in a channel.In: Proceedings of the 17th UK NationalConference on Computational Mechanics inEngineering, 69-72.

Gunstensen, A. K., Rothman, D. H.,Zaleski, S., and Zanetti, G., 1991, LatticeBoltzmann Model of Immiscible Fluids, Phys.Rev. A 43, 4320-4327.

He, X. Y., Chen, S. Y., and Zhang, R. Y.,1999, J. Comput. Phys. 152, 642-663.

Inamuro, T., Ogata, T., Tajima, S., andKonishi, N., 2004, J. Comput. Phys. 198,628-644.

Ji, C., Yan, Y.Y., 2008, A moleculardynamics simulation of liquid-vapour-solidsystem near triple-phase contact line of flowboiling in a microchannel. Applied Thermal

Engineering 28(2-3), 195-202.Lee, T., and Lin, C. L., 2005, J. Comput.

Phys. 206, 16-47.Martys, N.S., Chen, H., 1996, Physical

Review E, 53(1), 743.Mazzitelli, I. M., Lohse, D., and Toschi, F.,

2003, The effect of microbubbles ondeveloped turbulence, Phys. Fluids 15, L5-L8.

Shan, X.W., and Chen, H. D., 1993, Phys.Rev. E 47, 1815-1819.

Succi, S., 2001, The lattice Boltzmannequation for fluid dynamics and beyond,Clarendon Press, Oxford.

Swift, M. R., Orlandini, E., Osborn, W. R.,and Yeomans, J. M., 1996, Lattice Boltzmannsimulations of liquid-gas and binary fluidsystems, Phys. Rev. E, 54, pp 5041-5052.

Swift, M. R., Osborn, W. R., and Yeomans,J. M., 1995, Phys. Rev. Lett. 75, 830-833.

Yan, Y.Y., 2006, Lattice Boltzmannsimulation of vortices merging in a two-phasemixing layer. In: Advanced ComputationalMethods in Heat Transfer IX, 87-96.

Yan, Y.Y., et al. 2007. Proceedings of theInstitution of Mechanical Engineers, Part C, J.of Mechanical Engineering Science, 221(10),1201-1210.

Yan, Y.Y., Zu, Y.Q., 2007. J.Computational Physics, 227 (1), 763-775.

Yan, Y.Y., Zu, Y.Q., 2008a, Int. J. of Heatand Mass Transfer, 51(9-10), 2519-2536.

Yan, Y.Y., Zu, Y.Q., 2008b. 6th ASMEConference Darmstadt, ICNMM2008-62162.

Yan, Y.Y., 2009, Chinese Science Bulletin,54(4), 541-548.

Zu, Y.Q., Yan, Y.Y., 2006, Numericalsimulation of electroosmotic flow nearearthworm Surface, Journal of BionicEngineering, 3, 179-186.

lbm, a useful tool for mesoscale modelling of single phase ... · that the lbm is a useful tool for...

Documents