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Multiphase Flow Modelling

Dr. Gavin Tabor

Combustion – p.1

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What is multiphase flow

Multiphase flow is the flow of 2 (or more) immisciblefluids, or a fluid and a solid component. Examplesinclude :

• Solid particles in air• particulate polutants• coal dust combustion• particle separation• fluidized beds

• Liquid droplets in water/air• emulsions, food (eg. mayonaise)• diesel combustion

Combustion – p.2

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• Gas bubbles in liquid• bioreactors• food (eg. ice cream)

• Solid particles in water• slurry flow, hydrotransport, sedimentation

• Free surface flow• marine applications (ship design)• sloshing (tanks)• free surface channel flow

Combustion – p.3

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Physics questions

Two phase flow consists of a dispersed phase(droplets/particles/bubbles) intermingled in acontinuous phase (gas/liquid).

Continuous Phase

dispersedphase

. . . or a macroscopic interface

Combustion – p.4

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Physics questions

Several questions arise :

How big are the dispersed phase particles?(Also – shape, variations in size etc)

How dense are they? (express as a phase fraction –ratio of volumes occupied to total volume of cell)

How do they interact with each other?

How do they interact with the continuous phase?

Combustion – p.5

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Modelling techniques

We will look at 3 different approaches – suitable fordifferent flow regimes

1. Lagrangian particle tracking – follow individualparticles (or groups)

2. Eulerian 2 phase flow– treat both phases as fluids.

3. Free surface modelling – VOF

Combustion – p.6

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Particle tracking

Applicable for low phase fractions – usually solidparticles in air (eg. coal particle combustion) or fluiddroplets in air (eg. diesel spray combustion).

We know how to solve the NSE to find the motion ofthe continuous phase. Calculate the hydrodynamicforce on that particle, and apply NII to find itstragectory.

dvi

dt= f(ρf , ρp, d, µf , g, ui, u̇i, vi)

Combustion – p.7

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Equation of Motion

the Basset-Boussinesq-Oseen (BBO) model. Wecan integrate this numerically to provide thetrajectory for each particle.Note :

• Only valid for d � local turbulence length scale• Can be extended to include lift forces• In some regimes (eg. ρf/ρp ∼ 10−3) several terms

can be neglected

Combustion – p.8

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Stokes number

Important scaling parameter in fluid/particle flows :ratio of particle response time τR to characteristicfluid motion time τF :

St =τR

τF

For Stokes drag this can be evaluated :

St =ρpd

2U

18µL

for characteristic fluid length/velocity scales L, U .

Combustion – p.9

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Fluid/particle coupling

• St � 1 – particles follow flow exactly• St � 1 – particles unaffected by

continuous phase flow

If τF is a turbulent scale, then St � 1 implies theparticles will move with the turbulent motion.

However the turbulence will often be modelled(k − ε model). Need to introduce (often stochastic)model to account for effect of turbulence onparticles.

Combustion – p.10

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Particle/fluid coupling

Particles will also affect the turbulence– either enhance it or dampen it.

Distinguish between 1-way and 2-way coupling.• 1-way : fluid affects particles• 2-way : particles also affect fluid

As phase fraction increases, particles can also affectthe large-scale mean motion – Einstein correction tovisocsity.

Combustion – p.11

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Higher phase fractions

Individual particle tracking becomes impossible forhigh phase fractions of small particles – too manyparticles to track.

Track groups of particles – statistical approach.

Alternative : Eulerian two-phase flow modelling. Afluid is composed of particles (molecules) but we cantreat it as a continuum. Why not model the dispersedphase as a second fluid?

Apply this to : liquid/liquid, gas/liquid and solid/liquidflows.

Combustion – p.12

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Conditional averaging

Start by defining an indicator function γ, which takesthe values

γ(x) = 1 if x is in phase a, and

0 if not

We include this in our standard averaging operation.For a quantity φ :

αφa =1

∆t

∫γφ dt

Combustion – p.13

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Averaged NS Equations

Apply this averaging to the NSE, eg. momentumequation :

∂αua

∂t+ ∇.αuaua + ∇.αu′

au′a

= −1

ρa∇pa + ν∇2αua + interface terms

We also findαa + α 6a = 1

→ a set of equations for the continuous phase,and for the dispersed phase.

Combustion – p.14

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Eulerian 2-phase modelling

We can solve the continuous phase equations,plus the equation

αa + α 6a = 1

plus some model for the dispersed phase.

Dispersed phase model can be algebraic, orsolve the NSE for this phase.

Cond. averaging generates interface terms –represent effect of one phase on the other!

Combustion – p.15

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Points

• Modelling more complex, but providesmathematical framework to fit into. Eg. neednear-wall model

• Turbulence modelling required.• Continous phase – create phase-weighted

k − ε model• Dispersed phase – turbulence is some

fraction Ct of cont. phase• . . . but what is dispersed phase turbulence?

Combustion – p.16

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• Numerical problems, particularly with

αa + α 6a = 1

• Much wider range of applicability– can account for :• high phase fraction,• phase inversion,• droplet breakup/coalescence

Combustion – p.17

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Free surface flow

Free surface flow ≡ 2 immiscible fluids separated byan interface. Of importance in :

• Investigating bubble/droplet behaviour• Slug flow (very large bubbles)• Large-scale interfaces – ship wakes, ink jets,

channel flows etc.

4 basic methods used. We will look at one – VolumeOf Fluid (VOF).

Combustion – p.18

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Modelling overview

If the density ratio is large (air/water) :

1. Particle-based methods (SPH, cellular automata,lattice boltzmann)

All other methods solve NSE in both phases, andexplicitly follow the position of the interface :

3. Explicit parameterization of the surface

4. Create an indicator function – VOF methods

5. Create a general function G for which G = 0represents the interface – level set methods

Combustion – p.19

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VOF Method

In the VOF method the indicator function α takesvalue 1 in one phase and 0 in the other. We havea continuity equation :

∂α

∂t+ ∇.αu = Sα

and of courseαa + α 6a = 1

The velocity u comes from solving the NSE for themixture (not individual components).

Combustion – p.20

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Indicator function

The indicator function represents the position of theinterface and is advected by the flow.

1

1

1

1

1

1

11

1

1

1

1

1

11

1

1

1

11

1

0 0 0 0 0 0 00

0000

0 0

0

0.80.60.40.1

0.850.50.1

0.70.2

0.90.2

0.7

Computationally the interface smeared over 3-4 cells

Combustion – p.21

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Reconstruction

Reconstructing the interface not trivial :

1

1

1

1

1

1

11

1

1

1

1

1

11

1

1

1

11

1

0 0 0 0 0 0 00

0000

0 0

0

0.80.60.40.1

0.850.50.1

0.70.2

0.90.2

0.7

4 schemes in Fluent – geometric, donor/acceptor,Euler explicit and implicit.

Combustion – p.22

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Mixture

Given the phase fraction α, construct mixtureproperties as

µm = αµa + (1 − α)µb

Solve NSE for velocity, pressure

∂u

∂t+ ∇.uu = −

1

ρ∇p + νt∇

2u

and allocate this u to each component in each cell

Combustion – p.23

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Additional problems

Note that if there are large differences betweenproperties of the two phases, accuracy maybe limited.

May need to include surface tension. Importantgroups

Capillary number Ca =µU

σfor Re � 1

Weber number We =ρLU2

σfor Re � 1

Combustion – p.24