1 lecture 18 - eulerian flow modeling applied computational fluid dynamics instructor: andré bakker...
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Lecture 18 - Eulerian Flow Modeling
Applied Computational Fluid Dynamics
Instructor: André Bakker
© André Bakker (2002)© 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 k gives the general balance law and its jump condition:
• Continuity equation:
• Momentum equation:
0,0, kkkk J
kkkkkkkkk FIPJu
,,
kkkkk Jut
0))((
,1
kkikiik
n
ikik Jnnuu
Spatial averaging: basic equations
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• Consider an elementary control volume d bounded by the surface dS.
– Length scales:
– Volumes:
– Averaging volume and coordinate system:
tk ldlL ~
n
ii tdd
1)(
2x1x
3x
pxi
k
kin
d
kdn
1y2y
3y
Spatial averaging
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• Definitions:– Volume average Intrinsic phasic average.
– Volume phase fraction:
id
dtyxfd
f '),(1
kdk
k dtyxfd
f '),(1
d
txd pkk
),(
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n
k
k kk ff
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
Sdnfd
ffkidS
ki
'1
f
tt
f
ff
Space averaging: averaging theorems
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kkkkkkkkkkkkkkkkkkk FPuuut
SdnPuuud ki
dS
kkkkikk
ki
')(1
kidS
kkikkkkkkkkk Sdnuud
ut
')(1
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 uuK
ik
drag
kkik
fK
i
kkik
d
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2
Interphase forces
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Interphase forces: drag force models
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
)()(, fsffsLfsk uuuCK
Interphase forces: virtual mass and lift
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Modeling heat transfer
• Conservation equation of phase enthalpy.
kkkkk
kkkkkkkkk RuSPu
t
Phuh
t
:)()()(
)()(1
ikkkiki
n
kkik hhmTTq
Viscous dissipationterm
Originates fromwork term for
volume fractionchanges
Energy sourcese.g., radiation
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kkkkkk
kkkkkkkk RuSPut
Phuh
t
:)()()(
)()(1
ikkkiki
n
kkik hhmTTq
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 k 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
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
Nu
)PrRe7.01)(5107( 312.02 ffsNu 3
17.02 PrRe)2.14.233.1( ff
3.05.0 PrRe6.02 iNu
k
ikikk duu
Rek
kkpC
,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
)()()(
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
sat
vsatvvll T
TTrm
1212 rm
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
AA
,
,
,
,
0
0
nnnn C
C
C
C
U
U
U
U
A
AA
AAA
AA
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.