double averaged turbulence modelling in eulerian multi
TRANSCRIPT
Jul-04 Dresden, Germany, 2004 1 ANSYS CFX
Double Averaged Turbulence Modelling in Eulerian Multi-
Phase Flows
Alan Burns1, Thomas Frank1, Ian Hamill1 and Jun-Mei Shi2.1ANSYS CFX and 2FZ-Rossendorf
Jul-04 Dresden, Germany, 2004 3 ANSYS CFX
Multi-Phase Turbulence Phenomena
• Continuous Phase Effects On Dispersed Phase Turbulence.– Simplest model for dilute
dispersed phase Reynolds stresses:
• Turbulent Dispersion.– Migration of dispersed
phase from regions of high to low void fraction.
– Continuous phase eddies capture dispersed phase particles by action of interfacial forces.
cd uCu ′=′ jcicjdid uuCuu ′′=′′ 2
νβσ
νν tc
td =2
1
C=νβσ
Jul-04 Dresden, Germany, 2004 4 ANSYS CFX
Multi-Phase Turbulence Phenomena
• Dispersed Phase Effects On Continuous Phase
Turbulence.
• Turbulence Enhancement
– Due to shear production in wakes behind particles.
• Turbulence Reduction
– Due to transfer of turbulence kinetic energy to dispersed phase kinetic energy by action of interfacial forces.
Jul-04 Dresden, Germany, 2004 5 ANSYS CFX
Averaging Procedures
• First Average = Phase Average
• Phase indicator function:• χα(x,t) = 1 if phase α is present, = 0, otherwise.
• Use ensemble-, time- or space-averaging to
define phase-averaged variables:
– ‘Volume Fraction’:
– Material Density
– Phase Averaged Transport Variable:
• Essentially Mass-Weighted Average
αα χ=r
ααα ρχρ r/=
ααα ρρχ /Φ=Φ
Jul-04 Dresden, Germany, 2004 6 ANSYS CFX
Averaging Procedures:
• Phase averaged Momentum and Continuity.
• Mαk = interfacial forces
• = Reynolds Stress like terms
• Phase induced turbulence, or full turbulence?
• Some researchers assume this represents full
turbulence, e.g. Kashiwa et al.
• We assume it represents phase-induced turbulence.
( ) ( )( )( ) kk
k
t
ikikki
i
k MBrx
PrUUr
xUr
tαααααααααααα ττρρ ++
∂
∂−=+−
∂
∂+
∂
∂
( ) ( ) 0=∂
∂+
∂
∂i
i
Urx
rt
ααααα ρρ
αααα ρτ ji
t
ik uu ′′−=
Jul-04 Dresden, Germany, 2004 7 ANSYS CFX
Averaging Procedures:
• Models for Phase-Induced Turbulence.
• Sato: Algebraic Eddy Viscosity:
• Kataoka and Serizawa (1989) derived exact transport
equations for kpi andε pi.
• Term identified for enhanced turbulence production:
• Exact once the terms for interfacial forces are closed.
∂
∂−
∂
∂+
∂
∂+−=
i
iij
i
j
j
ipitijpipiij
x
U
x
U
x
Uk δµδρτ
3
2,, dcpdcpit UUdrC −= ρµ µ,
( )αβαβα UUMP pik
rrr−⋅−=
,
Jul-04 Dresden, Germany, 2004 8 ANSYS CFX
Averaging Procedures
• Second Average = Time or Favre Average.
• Ensemble averaged phase equations are fully
space and time dependent.
• Hence, may apply a second time- average.
• Shear induced turbulence?
• Favre or Mass Weighted averaging is
favoured, as it leads to much fewer terms in
the averaged equations.
Jul-04 Dresden, Germany, 2004 9 ANSYS CFX
Favre Averaging
• Favre averaging of phase-averaged variables
is defined as follows:
• For constant density phases, reduces to a
volume fraction weighted average:
• Favre-averaged and time-averaged quantities
are related as follows:
αααα φ rr /~
=Φ
αααααα ρφρ rr /~
=Φ
ααααα φ rr /~
′′+Φ=Φ
Jul-04 Dresden, Germany, 2004 10 ANSYS CFX
Favre Averaging
• Time- and Favre- averaged velocities are related by:
• is fundamental to turbulent dispersion, as it
describes how phasic volume fractions are spread out by velocity fluctuations.
• Eddy-viscosity type turbulence models, employ eddy
diffusivity hypothesis (EDH):
• Turbulent Prandtl number is typically of order unity.
ααα uUUrrr
′′+=~
αααα ruru /rr′′=′′
α
α
ααα
σ
νrur
r
t ∇−=′′r
ααurr′′
Jul-04 Dresden, Germany, 2004 11 ANSYS CFX
Favre Averaging
• Time Averaged Continuity Equation
– Includes volume fraction-velocity correlation term.
– Yields additional diffusion term, if we employ the eddy diffusivity hypothesis.
• Favre Averaged Continuity Equation
– No extra terms.
– A mathematical simplification, not a physical one.
( ) ( )( ) 0=′′+∂
∂+
∂
∂ii
i
urrUx
rt
ααααααα ρρ
( ) ( ) 0~
=∂
∂+
∂
∂ααααα ρρ rU
xr
ti
i
Jul-04 Dresden, Germany, 2004 12 ANSYS CFX
Turbulent Dispersion Force
• Assume caused by interaction between turbulent eddies and inter-phase forces.
• Model using time average of fluctuating part of interphase momentum force.
• Restrict attention to drag force, assumed proportional to slip velocity and interfacial area density Aαβ.
• Assume Dαβ approximately constant as far as averaging procedure is concerned.
( ) ( )αβαβαβαβαβα UUADUUCMrrrrr
−=−=
Jul-04 Dresden, Germany, 2004 13 ANSYS CFX
Favre Averaged Drag Force
• Express the time averaged drag in terms of Favreaveraged velocities.
• Turbulent Dispersion Force (General Form):
• Applicable in this form to flows of arbitrary morphology, using arbitrary turbulence models.
• Modeled Form using EDH:
( ) TDMUUCM ααβαβα
rr+−=
~~
( )
′−′′−
′′−
′′−=−=
αβ
αβαβ
α
αα
β
ββαββα
A
uua
r
ur
r
urCMM
TDTD
rrrrrr
∇
−−
∇−
∇=−=
αβ
αβ
α
α
β
β
α
α
α
α
β
β
β
β
αββασ
ν
σ
ν
σ
ν
σ
ν
A
A
r
r
r
rCMM
A
t
A
t
r
t
r
tTDTDrr
Jul-04 Dresden, Germany, 2004 14 ANSYS CFX
Polydispersed Multi-Phase Flow
• Algebraic form of area density is known:
• Hence, area density-velocity correlations may be expressed in terms of volume fraction-velocity correlations:
• General Form:
• Eddy Diffusivity Hypothesis (EDH):
β
β
αβd
rA
6=
′−
′′=−=
β
αβ
α
αααββα
r
ur
r
urCMM
TDTD
rrrr
∇−
∇=−=
α
α
β
β
α
ααββα
σ
ν
r
r
r
rCMM
r
tTDTDrr
Jul-04 Dresden, Germany, 2004 15 ANSYS CFX
Dispersed Two-Phase Flow
• Further simplifications occur for two phases only:
• Modeled EDH form of the turbulent dispersion force reduces to a simple volume fraction gradient:
1=+ βα rr0=∇+∇ βα rr
α
βαα
ααββα
σ
νr
rrCMM
r
tTDTD ∇
+−=−=
11rr
Jul-04 Dresden, Germany, 2004 16 ANSYS CFX
Comparison With Other Models
• Imperial College Model
– Gosman, Lekakou, Politis, Issa, and Looney, AIChEJ 1992, Multidimensional Modeling of Turbulent Two-Phase Flows in Stirred Vessels
– Behzadi, Issa, and Rusche (ICMF 2001), Effects of turbulence on inter-phase forces in dispersed flow.
• Chalmers University Model
– Ljus (Ph. D. Thesis, 2000), On particle transport and turbulence modification in air-particle flows.
– Johansson, Magnesson, Rundqvist and Almstedt (ICMF 2001), Study of two gas-particle flows using Eulerian/Eulerian and two-fluid models.
• RPI Models
– Lopez de Bertodano, (Ph. D. Thesis,1992), Turbulent bubbly two-phase flow in a triangular duct,, RPI, New York, USA
– Moraga, Larreteguy, Drew, and Lahey (ICMF 2001), Assessment of turbulent dispersion models for bubbly flows.
Jul-04 Dresden, Germany, 2004 17 ANSYS CFX
Imperial College Model
• Idea of modeling turbulence dispersion force by Favreaveraging drag term was first proposed by Gosman et al (1992).
• Behzadi et al (ICMF 2001) also consider lift and virtual mass forces, but found them insignificant.
• Equivalent to our model in the dilute limit.
• Hence, validation reported by Gosman et al valid for FAD model
0→βr
Jul-04 Dresden, Germany, 2004 18 ANSYS CFX
Chalmers University Model
• Similar philosophy and derivation to our model.
• However, requires unconventionally low volume fraction Prandltnumbers, of order 0.001, to achieve reasonable agreement with experiment.
• Due to minor errors in analysis, confusing time-averaging with Favre Averaging.
• Equivalent to our model, if we identify:
• where
• Explains low values of σd1required to match gas-solid flow.
αββα ββ krMM TDTD ∇+∇=−=21
rr
1
1
d
t
r
C
σ
νβ β
β
αβ=1
2
d
r
σ
ρβ ββ=
νβ
α
σ
σσ r
d =1
νβ
αβ
σ
νν t
t =
Jul-04 Dresden, Germany, 2004 19 ANSYS CFX
RPI Models: Lopez de Bertodano
• CTD is a non-dimensional empirical constant. – CTD = 0.1 to 0.5 gave reasonable results for medium sized
bubbles in ellipsoidal particle regime (Lopez de Bertodano et al
1994a, 1994b).
• However, flow regimes involving small bubbles or small solid particles were found to require very different values of CTD, up to 500.
• Revised by Lopez de Bertodano (1999).– Proposed that CTD be expressed as a function of turbulent
Stokes number as follows:
αααβα ρ rkCMM TD
TDTD ∇−=−=rr
)1(
14/1
StStCCTD
+= µ
Jul-04 Dresden, Germany, 2004 20 ANSYS CFX
RPI Models: Lopez de Bertodano
• Compare with Favre Averaged Drag (FAD) model for dispersed 2-phase flow employing EVH:
• Substitute Eddy Viscosity Formula
• Equivalent to a Lopez de Bertodano model with variable empirical constant:
• Strong function of Stokes number, as expected.
α
βαα
ααββα
σ
νr
rrCMM
r
tTDTD ∇
+−=−=
11rr
ααµα εν /2kCt =
+=
+= 1
1
41.0
11
α
β
α
µ
βαα
α
α
αβ
α
µ
σερσ r
r
St
C
rr
kCCC
rr
TD
Jul-04 Dresden, Germany, 2004 21 ANSYS CFX
RPI Models: Carrica et al
• Requires dispersed phase volume fractions to obey a turbulent diffusion equation in limit where drag + turbulent dispersion balances body forces.
• β = 2,…,ND
• Equivalent to FAD+EDH model in the following limits:
– Two Phases Only
– Dilute Dispersed Phase.
• Satisfactory agreement found with DNS data for dilute bubbly flows, and for bubbly mixing layer (Moraga et al, ICMF 2001).
β
α
ααβ
β
αβ
σ
νρrUU
dCM
r
tD
TD ∇−−=rrr
4
3
∑=
−=ND
TDTDMM
2ββα
rr
Jul-04 Dresden, Germany, 2004 22 ANSYS CFX
Validation: Bubbly Flow in Vertical Pipe
• Uses Grace Drag Law
• SST turbulence model + Sato eddy viscosity.
• Compares FAD with RPI = constant
coefficient Lopez de Bertpdano model.
Jul-04 Dresden, Germany, 2004 23 ANSYS CFX
Liquid-Solid Flow in Mixing Vessel
• Wen Yu drag correlation for dense solids.
• SST turbulence + Sato eddy viscosity.
• Three solid lines are minimum, average and maximum values of CFD results, within region ±5mm from the data point.
• Dimension representative of size of conductivity probe.
Jul-04 Dresden, Germany, 2004 24 ANSYS CFX
Liquid-Solid Flow in Mixing Vessel
• Particle volume fractions underpredicted, though
correct trends are predicted.
• Similar results for 710 micron particles.
Jul-04 Dresden, Germany, 2004 25 ANSYS CFX
Turbulence Modulation
• Turbulence Enhancement.
• Due to turbulence production in wakes behind particles.
• Averaged out by first averaging procedure (phase averaging).
• Hence, must include in first averaged equations.
Jul-04 Dresden, Germany, 2004 26 ANSYS CFX
Turbulence Enhancement: Simple Models
• Sato: Treat particle-induced
and shear-induced turbulence separately.
• Algebraic Eddy Viscosity model for phase-induced turbulence:
• Modifed k-εεεε Models
• Lump particle-induced and shear-induced turbulence together.
• Add additional production terms to shear-induced k-eequations, e.g. Lee at al
dcpdcpit UUdrC −= ρµ µ,
( )2
αβαβα UUCPk
rr−=
α
α
αεα
εkPC
kP
1=
α
ααµα
ερµ
2
,,
kC sisit =
pitsitt ,, ααα µµµ +=
α
ααµα
ερµ
2k
Ct =
Jul-04 Dresden, Germany, 2004 27 ANSYS CFX
Turbulence Reduction
• Energy transferred from turbulent eddies to particles by acceleration of particles due to drag.
• Turbulence-drag interaction, like dispersion.
• Interaction with shear-induced turbulence, so
appears as additional source terms in 2nd averaged k-equation
ααα uMSk′⋅=rr
βββ uMSk′⋅=rr
Jul-04 Dresden, Germany, 2004 28 ANSYS CFX
Turbulence Reduction: Simple Model
• Chen-Wood: Consider drag only, and treat Cαβ as constant in averaging procedure:
• = velocity covariance
• Sum of sources is negative
• Hence, can only model turbulence reduction.
• Requires model for velocity covariance:
• Chen-Wood:
( ) ( ) ( )βαβαββββααβββααββ kkCuuuuCuUUCSk 2−=′′−′′=′⋅−=rrrrrrr
( ) ( ) ( )ααβαβααβααβαββαβα kkCuuuuCuUUCSk 2−=′′−′′=′⋅−=rrrrrrr
βααβ uuk ′′=rr
( ) 02
≤′−′−=+ βααββα uuCSS kk
rr
cd uCu ′=′ jcicjdic uuCuu ′′=′′
Jul-04 Dresden, Germany, 2004 29 ANSYS CFX
Turbulence Enhancement:
Proposed Double Averaged Approach
• Treat phase-induced and shear-induced separately, as in Sato model.
• Solve separate transport equations for phase-induced and shear-induced turbulence.
• k-l model for phase-induced turbulence: (Lopez de Bertodano et al)
• Choose
• Matches Sato:
• Choose time-scale τk,pi to match Kataoka-Serizawa production:
• Hence, τk,pi proportional to particle relaxation time.
• Time Averaged kpi equation introduces additional terms involving turbulence dispersion force.
• Hence, affected by volume fraction gradients.
( ) ( )pipi
piki
pi
pik
pit
pi
i
pi kkx
kkU
xk
t,,
,
,
,
,
,,
~~~
αααα
ααααατ
ρ
σ
µρρ −=
∂
∂−
∂
∂+
∂
∂∞
pitpipit lkC,,, αααµα ρµ =
2
,4
1αβα UUrk pi
rr−=∞ βα dl pit ∝
,
( )2
,
,
~
αβαβ
α
τ
ρUUC
k
pik
pirr
−=∞
dcpdcpit UUdrC −= ρµ µ,
Jul-04 Dresden, Germany, 2004 30 ANSYS CFX
Turbulence Reduction:
Proposed Double Average Approach
• As for Chen-Wood, but take area density fluctuations into account in averaging procedure.
• Hence, additional source terms in shear-induced k-equation:
• + additional terms
• Additional terms proportional to
• Hence, also affected by volume fraction gradients.
• Consider better models for velocity covariance, e.g. transport equation.
( ) ( )ααβαβαββαβαβα kkCuUUADSk 2−=′⋅−=rrr
( ) ααβαβσ
µrUUC
r
t ∇⋅−~~
Jul-04 Dresden, Germany, 2004 31 ANSYS CFX
Conclusions
• Double Averaging Approach yields a natural model for turbulence dispersion, with wide degree of universality.
• Implemented as default model for turbulence dispersion in CFX-5.7 (2004).
• Also produces potentially fruitful approaches to turbulence modulation.
• Topics for further investigation:– How is the model affected by taking into account non-linear
dependence of drag on slip velocity?
– How is the model affected by taking into account volume fraction dependence of the drag coefficient?
– second order closure models.
– separated flows.