1847 joliet way boulder, co 80305 [email protected] 304-816-6332 2014 multiphase flow science...
TRANSCRIPT
A Turbulence Model for Gas-Particle Flows with Favre-Averaging : III. Closures
Dr. Thomas J. O’BrienNETL-emeritus
1847 Joliet WayBoulder, CO [email protected]
2014 Multiphase Flow Science ConferenceNational Energy Technology Laboratory (NETL)Lakeview Golf Resort &Spa, Morgantown, WVAug. 5&7, 2014
What is multiphase turbulence?
Favre-like (phasic) average
Summary of averaged equations: continuity, momentum and granular temperature◦ Equations◦ Closures
Expansion of closure relations Turbulent kinetic energy equations
◦ Derivation◦ Closures
Future formal work◦ Boundary conditions◦ Thermal temperatures◦ Chemistry
Outline
Transient calculations are very costly and time consuming, although they are possible.
It is difficult to base scoping studies (design, process simulation) on transient calculations.
Coupling of CFD to process simulations requires a “lower order model,” faithfully reduced from the full transient model.
Multiphase flows are inherently transient … the effects of which must be captured faithfully by a turbulence model.
Steady-State Calculations
Most work published is based on modifications of gas phase turbulence theory… this probably won’t work for dense flows. Energy is in the granular fluctuations.
Current approach … turbulence is a property of a fluid, so a granular continuum theory has to be the starting point.
Utilize the concept of Favre-like (phasic) averaging, to reduce the closure “burden.”
General Comments
Single average approachAverage microscopic equations to obtain “two-fluid” equations describing interpenetrating continuum, characterized by: all functions of space and time.
Examine “fluctuations” of the microscopic variables about these two-fluid variables.
Double average approachAverage again to obtain mean behavior
Examine “fluctuations” about these, e.g.,
Multiphase Turbulence
...,,,,,, sgsgsg P UU
sgsgsg P ,,,,, UU
etc. ,,' gggggg UUU
Indicator function of fluid/solids phase:
Fluid/solids phase fraction:
Conditional average, based on the gas/solids phase: ,e.g., Turbulent fluctuations: , e.g.,
Single Average Approach
presentnot /0
present /1/ sg
sgI sg
sgsg I //
sgsgsg II
sg
ggiggi IuIU
gigigi Uuu
Favre & Favre-like Averaging
ggiggi UU ~Favre average: compressible gas
Favre-like or phase average: two-fluid model
ssssssissiggiggi UUUU ~,
~,
~
Favre, A., “Équations des gaz turbulents compressibles I.- Formes générales,” Journal de Mécanique 4, 361-390, 1965. Favre, A., “Formulation of the statistical equations of turbulent flows with variable density,” in Studies in Turbulence, T.B. Gatski, S. Sarkar and C.O. Speziale eds., Springer-Verlag, 324-341, 1992.
“The Favre velocity is a mean variable … that includes the velocity fluctuation correlated with density fluctuation. (The Favre velocity fluctuation is) … only the part uncorrelated with density.”
Besnard, D.C., and F.H. Harlow, “Turbulence in multiphase flow,” Int. J. Multiphase Flow 14, 679-699, 1988.
Physical Meaning of Phase-average Velocity
sisisi UUU~
sggisigs PUU ,,,,,
NotationDependent two-fluid variables:
Reynolds mean:
sg
gisi
gs
P
UU
,
,
,
ssssg
ggiggi
ssissi
gs
P
UU
UU
~,
~,
~,
Favre mean:
t,xfcn
xfcn xfcn
NotationDependent variables:
Reynolds decomposition
sss
ggg
gigigi
sisisi
ggg
sss
PPP
UUU
UUU
t,t,
xxx
sss
ggg
gigigi
sisisi
ggg
sss
PPP
UUU
UUU
t,t,
~
~
~
xxxFavre decomposition
t,xfcn
sggisigs PUU ,,,,,
Continuity Equations
0~
average
0
giggi
gg
giggi
gg
Uxt
Uxt
0~
average
0
sissi
ss
sissi
ss
Uxt
Uxt
Gas phase
Solids phase
Gas Phase Momentum Equations
igggsigijTgijji
gg
gjgiggj
gigg
igggsij
gij
i
gg
gigjggj
gigg
gIxx
P
UUx
Ut
gIxx
P
UUx
Ut
~
~~~
averageggjgiggijTεUUετ ~
Solids Phase Momentum Equations
issgsisijTsijji
gs
sjsissj
siss
issgsij
sij
i
gs
sisjssj
siss
gIxx
P
UUx
Ut
gIxx
P
UUx
Ut
~
~~~
averagessjsissijTεUUετ ~
Granular Temperature Equation
ssssj
sisij
i
ss
i
jTsssjsj
ssss
ssssj
sisij
i
ss
i
j
ssjsssss
Jx
U
xx
qUxt
Jx
U
xx
x
U
t
~~~
2
3~
2
3
average
2
3
2
3
sssjsjTUq ~
Turbulent Granular Temperature Flux vector
Relating Reynolds mean to phase-averaged mean:
Momentum equations:
Granular temperature equation:
sssisjTεUεq ~
ggjgiggijTεUUετ ~
ggiggi
gigi
UU
UU
~
ssss
ss
~
ssissi
sisi
UU
UU
~
Correlations
ssjsissijTεUUετ ~
gsiI
i
gg x
P
gij sij
i
ss x
sssJ
j
sisij x
U
igggsigijTgijji
gg
gjgiggj
gigg
gIxx
P
UUx
Ut
~
~~~
issgsisijTsijji
gs
sjsissj
siss
gIxx
P
UUx
Ut
~
~~~
Gas & Solids Phase Momentum Equations
Mean Phase Interaction Coefficient
2
2
2
,ˆ :nscorrelatio New
ˆ
~~ˆˆ
where
ˆ~~ˆ
:Average
,ˆˆˆ,ˆ
:form General
gsigs
sisgiggs
sigisgssigigssgsi
gsigigssigigssgsi
gsgsgssigigssgsi
I
UU
UUUUI
IUUUI
tUUI
xx
1. A fitting parameter, independent of space (and time):
2. Same function , but in terms of Reynolds-averaged variables:
3. Same function ,but in terms of Favre-averaged velocity variables:
Representaions for s ˆ
0ˆˆgsgs x
xUxUxx sgggsgsgs ,ˆˆˆ 1
xUxUxx sgggsgsgs
~~,ˆˆˆ 2
gs
gs
ssisggigsg
sg
UUxUxU
xUxU
~~
xUxUxx sgggsgsgs ,ˆˆˆ 1
ggs
sT
gg
gTsg
sss
sTg
gg
gTsgsg
1
~~
~~
~~
~~
xUxU
xUxUxUxU
sigigsigsigsigiggs UUUUfUUf where,,,ˆ
xUx
xUx
gsg
gsg
, :average and functions spatialsteady about Expand
andoffunctionalisˆ
gsjgsjgsj
gsggg
g
gsggsggsggs UU
U
ffff
gsggsg UU
UUUU
,,
,,,,ˆ
gsjgsjgsj
gsggg
g
gsggsggsggs UU
U
ffff
gsggsg UU
UUUU
,,
,,,,ˆ
Functional Taylor Series Expansion
Hrenya, C and J. Sinclair, “Effects of particle-phase turbulent in gas-solid flows,” AIChE J., 43, 853-869, 1997.
gsjgsjgsj
gsggg
g
gsggsggsggs UU
U
ffff
gsggsg UU
UUUU
,,
,,,,ˆ
xUxxUx gsggsg ,,For 2.)
xUxUxx sgggsgsgs ,ˆˆˆ 1
xUxxUx gsggsg
~,,For 3.)
gsjgsj
gsj
gsggsgsggsgs UU
Ugsg
~,ˆ~,ˆˆ
~,U
UU
Functional Taylor Series Expansion
required! nscorrelatio additional No
ˆ~,ˆ
~ˆ~,ˆˆˆ
~,
~,
4
s
sjg
g
gjg
gsj
gsgsggs
gsjgsjgsj
gsgsggsgsgs
UU
U
UUU
gsg
gsg
U
U
U
U
Recommended Form for gs
Wen & Yu
s ˆ
g
sggg
g
gggg
gs
d
d
UU
Re
1000Re,Re15.0118ˆ 65.2687.0
2
sg
gsigg
g
gsi
gs U
dU UU
65.1313.0Re837.0
36ˆ
Wen & Yu
s ˆ
required! nscorrelatio additional No
~~
Re
~~
~Re837.0
36
Re15.0118ˆ
65.1313.0
65.2687.0
24
g
sgggg
s
sjg
g
gjg
sg
gsigg
g
ggg
gs
d
UUU
d
d
UU
UU
Gas & Solids Phase Momentum Equations
issgsisijTsijji
gs
sjsissj
siss
gIxx
P
UUx
Ut
~
~~~
igggsigijTgijji
gg
gjgiggj
gigg
gIxx
P
UUx
Ut
~
~~~
i
gg x
P
Only the correlations are required.
Volume Fraction-Gas Pressure Correlations
i
gg
i
gg
i
gg x
P
x
P
x
P
i
gg
i
gs
i
gs x
P
x
P
x
P
igggsigijTgijji
gg
gjgiggj
gigg
gIxx
P
UUx
Ut
~
~~~
issgsisijTsijji
gs
sjsissj
siss
gIxx
P
UUx
Ut
~
~~~
Gas & Solids Phase Momentum Equations
No new correlation is required:
Mean molecular viscosity stress
ggijgggijgggijggij SSS UUU 2~
22
ggiggigigi UUUU ~
issgsisijTsijji
gs
sjsissj
siss
gIxx
P
UUx
Ut
~
~~~
igggsigijTgijji
gg
gjgiggj
gigg
gIxx
P
UUx
Ut
~
~~~
Gas & Solids Phase Momentum Equations
KTGF Granular Stress
sijsijisibssij SxUP 2
041 gP sssss
sspsssss geP 0212
~
2"
00'
00
'
000002
~
2
1
~
sssss
sssssssss
gg
ggg
ss
s
sssss d
gdggg
0'
00000
i
sib
i
sib
i
sib x
U
x
U
x
U
~
KTGF Granular Stress (cont.)
22321221
2121
~
4
1~
2
1~
2
1
~
2
1~
ssssssssss
sssss
ssb
bbb
bb
b
fff
ff
f
i
sisss
i
sisss
i
siss
i
siss
x
Uf
x
Uf
x
Uf
x
Uf
bb
bb
2121
21
~~
2
1
~
Kinetic Theory of Granular FlowGranular Temperature Equation
ssssj
sisij
i
ss
i
jTsssjsj
ssss
Jx
U
xx
qUxt
~~~
2
3~
2
3
jT
TssjsjT x
Uq
~
Pr~
~~
Turbulent Granular Temperature Flux vector
j
sisij x
U
Dilatation & Solenoidal Dissipation
s sss J
Specific Gas Phase Turbulence Kinetic Energy
gigiggg UUk 2
1~
ggggggggggggg kUU~~
2
1~
2
1
2
1 222 UU
The first term is the energy density of the gas flow due to thecombined mean motion and turbulent motion correlated with the void fraction.
The second term is the gas energy density due to the residual turblent motion, i.e., that not correlated with voidage fluctuations. Besnard, D.C., and F.H. Harlow, 1988.
??2
1
2
1 22 ggggggg U UU
Total kinetic energy of gas flow is nicely partitioned:
Specific Solids Phase Turbulence Kinetic Energy
sisisss UUk 2
1~
sssssssssssss kUU~~
2
1~
2
1
2
1 222 UU
The first term is the energy density of the granular flow due to thecombined mean motion and turbulent motion correlated with the solids fraction.
The second term is the solids energy density due to the residual turblent motion, i.e., that not correlated with solids fluctuations. Besnard, D.C., and F.H. Harlow, 1988.
Total kinetic energy of solids flow is nicely partitioned:
Gas Phase Turbulence Kinetic Energy
phasegranular thetoenergy kinetic e turbulencphase) (gas
of transfer of RatenDissipatio
nsfluctuatio pressure the todueenergy kinetic turbulence ofTransport
itself e turbulenc the todue energy kinetic turbulence ofTransport
gradient flowmean thefromenergy kinetic turbulence
of production of Rate
~
flowmean by the convection todue massunit per energy kinetic turbulence
of change of Rate
tystationari-non todue massunit per energy kinetic turbulence
of change of Rate
2
1~
~~~
gsigij
gijgig
i
ggig
i
ggig
gigigjggjj
gi
gijTgg
gjgggj
ggg
IUx
Ux
PU
x
PU
UUUxx
U
Ukx
kt
Solids Phase Turbulence Kinetic Energy
phase gas thetoenergy kinetic e turbulencphase) (solids
of transfer of Rate
)inelastic!(not nDissipatio
nsfluctuatio pressure the todueenergy kinetic turbulence ofTransport
/
itself e turbulenc the todue energy kinetic turbulence ofTransport
gradient flowmean thefromenergy kinetic turbulence
of production of Rate
~
flowmean by the convection todue massunit per energy kinetic turbulence
of change of Rate
tystationari-non todue massunit per energy kinetic turbulence
of change of Rate
/
2
1~
~~~
gsisij
sijsis
i
gsigs
i
gsis
sisisjssjj
sisijTss
sjsssj
sss
IUx
Ux
PU
x
PU
UUUxx
U
Ukx
kt
Kinetic energy density of the gas phase
Kinetic energy density of the solids phase
Total kinetic energy density of the flow
Kinetic Energy Density of the Flow
gigiggg UUK 2
1
sisisss UUK 2
1
sisissgigiggsgtot UUUUKKtK 2
1
2
1, x
gas phase specific turbulence kinetic energy
Average Kinetic Energy Density of the Gas Phase
ggggigiggg
gigiggigigg
gigiggigiggigigg
gigigigigggigiggg
kUUK
UUUU
UUUUUU
UUUUUUK
~~~
2
1
~~
2
1
~2
~~
2
1
~~
2
1
2
1
gigiggg UUk 2
1~… the energy density due to the combined mean and density-correlated turbulent motion,
… measures the residual turbulent motion.Besnard et al, 1992
solids phase specific turbulence kinetic energy
Average Kinetic Energy Density
ssssisiss
gigiggigiggsisisss
kUU
UUUUUUK
~
2
1~~
2
1
~~
2
1
2
1
sisisss UUk 2
1~
sssgggsisissgigiggsgtot kkUUUUKKK~~~~~~
2
1
Correlations
sisgig UU and
nscorrelation fluctuatio velocity -n fluctuatiofraction Volume
jT
TssjsjT
sjsissijTsgjgiggijTg
xUq
UUUU
~
Prand
,
stresses avreReynolds/F e turbulencphase gas/solid Specific
~
~~
~~
sigisgssigigsgsggss
gigigs
UUUU
UU
ˆand ,ˆ, ˆ-ˆ
,~
,ˆ
nscorrelatio related Drag
gij
gg Ux
P
,ncorrelatio related Stress
Closuresvelocity & volume fraction
(solids) gasfor number Prandtl turbulent-
sity eddy visco kinematic phase (solids) gas -
.
sign in oppositebut al,proportion are nscorrelatio These
and
model e turbulenctypeviscosity -eddyan Using
sg
tstg
gigs
g
tg
tssis
gs
tss
s
tssisg
g
tggig
UU
UU
Closuresvelocity relations
ggs
tsg
ss
tssisi
ggg
tggigi
ssissisiggiggigi
UU
UU
UUUUUU
1
~
~
,~
and~
identities theUsing
jT
TssjsjT x
Uq
~
Pr~
~~
Extract dissipation, εg/s
Closures for kg/s & εg/s
Look at energy cascade Equation for the full stress tensor, τ Compare in detail with Reynolds formalism Formulate drag terms for a chosen form
Include thermal temperatures Chemistry (contacting)– hopefully temperature
fluctuations will be small
Develop closure relationships … related to specific experimental data
(A NEVER ENDING TASK)
To Do List
Thank youQuestions?
[email protected], 304-816-6332, 1847 Joliet Way, Boulder, CO 80305… or Munich, Germany