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Lecture 18 - Eulerian Flow Modeling

Applied Computational Fluid Dynamics

Instructor: André Bakker

http://www.bakker.org© André Bakker (2002-2006)© Fluent Inc. (2002)

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Contents

• Overview of Eulerian-Eulerian multiphase model.• Spatial averaging.• Conservation equations and constitutive laws.• Interphase forces.• Heat and mass transfer.• Discretization.• Solver basics.

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Eulerian-Eulerian multiphase - overview

• Used to model droplets or bubbles of secondary phase(s) dispersed in continuous fluid phase (primary phase).

• Allows for mixing and separation of phases.• Solves momentum, enthalpy, and continuity equations for each

phase and tracks volume fractions.• Uses a single pressure field for all phases.• Uses interphase drag coefficient.• Allows for virtual mass effect and lift forces.• Multiple species and homogeneous reactions in each phase.• Allows for heat and mass transfer between phases.• Can solve turbulence equations for each phase.

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Methodology

• A general multiphase system consists of interacting phases dispersed randomly in space and time. Interpenetrating continua.

• Methods:– Use of continuum theory and thermodynamical principles to

formulate the constitutive equations (ASMM).– Use of microstructural model in which macroscopic behavior is

inferred from particle interaction: Eulerian-Granular.– Use of averaging techniques and closure assumptions to model the

unknown quantities:• Space averaging with no time averaging.• Time averaging with no space averaging.

• Ensemble averaging with no space averaging.

• space/time or ensemble/space averaging: Eulerian-Eulerian.

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Two-fluid model (interpenetrating continua)

• Deductive approach:– Assume equations for each pure phase.– Average (homogenize) these equations.– Model the closure terms.

• Inductive approach:– Assume equations for interacting phases.– Model the closure terms.

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• Application of the general transport theorem to a property Ψkgives the general balance law and its jump condition:

• Continuity equation:

• Momentum equation:

0,0, === kkkk J ϕρψ &r

kkkkkkkkk FIPJur

&r ρϕτρψ =−== ,,

kkkkk Ju

tϕψψ&

rr =⋅∇+⋅∇+∂

∂0))((

,1=⋅+⋅−∑

≠=kkikiik

n

ikik Jnnuu

rrrψ

Spatial averaging: basic equations

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• Consider an elementary control volume dΩ bounded by the surface dS.

– Length scales:

– Volumes:

– Averaging volume and coordinate system:

tk ldlL ~>>>>

∑=

Ω=Ωn

ii tdd

1)(

2x1x

3x

pxr

i

k

kinr

Ωd

kdΩnr

1y2y

3y

Spatial averaging

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• Definitions:– Volume average Intrinsic phasic average.

– Volume phase fraction:

∫Ω

Ω+Ω

>=<id

dtyxfd

f '),(1 rr

∫Ω

Ω+Ω

=><kdk

k dtyxfd

f '),(1 rr

ΩΩ

=d

txd pkk

),(r

α 11

=∑=

n

k

kα kk ff ><>=< α

Space averaging: basic equations

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• For all k-volumes that are differentiables, (Gray and Lee (1977), Howes and Whitaker (1985)).

– Temporal derivative:

– Spatial derivative:

– Next, apply averaging to the conservation equations.

Sdnufd

ftt

fki

dS

ki

ki

'1 rr ⋅Ω

−><∂∂>=

∂∂< ∫

Sdnfd

ffkidS

ki

'1∫Ω

+><∇>=∇< v

><∂∂>≠

∂∂< f

tt

f

><∇>≠∇< ff

Space averaging: averaging theorems

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kkkkkkkkkkkkkkkkkkk FPuuut

><+><⋅+∇><−∇=><⋅+∇><∂∂ ρατααραρα rrr

SdnPuuud ki

dS

kkkkikk

ki

')(1 rrrr ⋅+−−Ω

+ ∫ τρ

∫ ⋅−Ω

−=><⋅+∇><∂∂

kidS

kkikkkkkkkkk Sdnuud

ut

')(1 rrrr ρραρα

Space averaging: conservation equations

• Continuity equation:

• Momentum equation:

mass transfer term

interaction term

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• Drag is caused by relative motion between phases.

• Commonly used drag models (fluid-fluid multiphase).– Schuh et al. (1989). – Schiller and Naumann (1935).– Morsi and Alexander (1972).– Schwarz and Turner (1988; for bubble columns).– Symmetric law.

• Many researchers devise and implement their own drag models for their specific systems.

,0)(1

=−∑=

n

ikiik uuKrr

ik

drag

kkik

fK

τρα=

i

kkik

d

µρτ18

2

=

Interphase forces

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Interphase forces: drag force models

Fluid-fluid drag functions

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

10 100 1000 10000 100000

Re

Cd

Schiller andNaumann

Schuh et al.

Morsi et Alexander

( )

>≤+

=1000Re44.0

1000ReRe15.0124 687.0

DC

(Re)are,,whereReRe 3212

321 faaa

aaaCD ++=

Schiller and Naumann

Schuh et al.

Morsi and Alexander

( )( )

>>≤+

≤<+=

2500Re4008.0

2500Re200Re/Re0135.0Re914.024

200Re0Re15.0124282.0

687.0

DC

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• Virtual mass effect: caused by relative acceleration between phases Drew and Lahey (1990).

– Virtual mass effect is significant when the second phase density is much smaller than the primary phase density (i.e., bubble column).

• Lift force: caused by the shearing effect of the fluid onto the particle Drew and Lahey (1990).

– Lift force usually insignificant compared to drag force except when the phases separate quickly and near boundaries.

∇⋅+∂∂−∇⋅+

∂∂

= )()(, sss

ff

f

fsvmfsvm uut

uuu

t

uCK

rrr

rrr

ρα

)()(, fsffsLfsk uuuCKrrr ×∇×−= ρα

Interphase forces: virtual mass and lift

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Modeling heat transfer

• Conservation equation of phase enthalpy.

kkkkkk

kkkkkkkk RuSPut

Phuh

t+∇+∇⋅+

∂∂=⋅∇+

∂∂ rrr

:)()()( αραρα

)()(1

ikkkiki

n

kkik hhmTTq −+−+∇− ∑

=

&r γ

Viscous dissipationterm

Originates fromwork term for

volume fractionchanges

Energy sourcese.g., radiation

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kkkkkk

kkkkkkkk RuSPut

Phuh

t+∇+∇⋅+

∂∂=⋅∇+

∂∂ rrr

:)()()( αραρα

)()(1

ikkkiki

n

kkik hhmTTq −+−+∇− ∑

=

&r γ

Interphase heattransfer term

Heat conduction

Energy transfer with mass transfer

Modeling heat transfer

• Conservation equation of phase enthalpy.

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• Assume Fourier’s law:

For granular flows κk is obtained from packed bed conductivity expression, Kuipers, Prins and Swaaij (1992).

• Near wall heat transfer is calculated as in single phase. • All standard boundary conditions for temperature can be

implemented for multiphase.

kkk Tq ∇−= καr

Heat conduction

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• The interphase heat transfer coefficient is given by.

• Granular Model (Gunn, 1978).

• Fluid-fluid model.

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6

i

iikik d

Nuακγ =

)PrRe7.01)(5107( 312.02 ++−= ffsNu αα3

17.02 PrRe)2.14.233.1( ff αα +−+

3.05.0 PrRe6.02+=iNu

k

ikikk duu

µρα −

=Rek

kkpC

κµ,Pr =

Interphase heat transfer

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• Conservation equation for the mass fraction of the species i in the phase k:

• Here is the diffusivity of the species in the mixture of the respective phase, is the rate of production/destruction of the species.

• Thermodynamic relations an state equations for the phase k are needed.

• When calculating mass transfer the shadow technique from Spalding is used to update diameter of the dispersed phase.

i

k

i

k

i

kkk

i

kkkk

i

kkk mmDmumt

&r +∇⋅∇=⋅∇+

∂∂

)()()( ραραρα

i

km

i

kDi

km&

Conservation of species

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• Evaporation and condensation: – For liquid temperatures ≥ saturation temperature, evaporation rate:

– For vapor temperatures ≤ saturation temperature, condensation rate:

– User specifies saturation temperature and, if desired, “time relaxation parameters” rl and rv (Wen Ho Lee (1979)).

• Unidirectional mass transfer, r is constant:

( )sat

satlllvv T

TTrm

−= ρα&

( )sat

vsatvvll T

TTrm

−= ρα&

1212 ραrm =&

Modeling mass transfer

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• Coupled solver algorithms (more coupling between phases).– Faster turn around and more stable numerics.

• High order discretization schemes for all phases.– More accurate results.

Implicit/Full EliminationAlgorithm v4.5

TDMA CoupledAlgorithm v4.5

Multiphase Flow SolutionAlgorithms

Only Eulerian/Eulerian model

Solution algorithms for multiphase flows

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• Momentum equation for primary and secondary phase:

• Elimination of secondary phase gives primary phase:

• Secondary phase has similar form.• Applicable to N phases.

( )

spppnbpnbpeffp

ssnbsnbs

ppnbpnbpss

p

bbuaua

buaka

kbuaua

ka

ka

,,,,

,,,,

++=

++

++=

++

∑

∑∑

( )( ) ssnbsnbssp

ppnbpnbspp

buaukauk

buaukuka

+=++−

+=−+

∑

∑

,,

,,

Full elimination algorithm

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• Discretized equations of primary and secondary phase are in matrix form:

• Results in a tri-diagonal matrix consisting of submatrices

• Closer coupling in each iteration gives faster convergence

BUU

b

b

u

u

a

a

u

u

kak

kka

nbnb

s

p

snb

pnb

snb

pnb

s

p

s

p

rrr+=

+

=

+−−+

∑∑

AA

,

,

,

,0

0

=

nnnn C

C

C

C

U

U

U

U

A

AA

AAA

AA

r

r

r

r

r

r

r

r

3

2

1

3

2

1

3332

232221

1211

000

00

0

00

Coupled TDMA-algorithm

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Solution algorithm for multiphase

• Typical algorithm:– Get initial and boundary conditions.– Perform time-step iteration.– Calculate primary and secondary phase velocities.– Calculate pressure correction and correct phase velocities, pressure

and phase fluxes. Pressure is shared by all phases.– Calculate volume fraction.– Calculate other scalars. If not converged go to step three.– Advance time step and go to step two.

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Solution guidelines

• All multiphase calculations: – Start with a single-phase calculation to establish broad flow patterns.

• Eulerian multiphase calculations:– Copy primary phase velocities to secondary phases.– Patch secondary volume fraction(s) as an initial condition.– Set normalizing density equal to physical density.– Compute a transient solution.– Use multigrid for pressure.

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Summary

• Eulerian-Eulerian is the most general multiphase flow model.• Separate flow field for each phase.• Applicable to all particle relaxation time scales.• Includes heat/mass exchange between phases.• Available in both structured and unstructured formulations.