1847 Joliet Way Boulder, CO 80305 [email protected] 304-816-6332 2014 Multiphase Flow Science...

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

~~


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.)


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


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

~

~~~


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

~

~~~


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

1847 Joliet Way Boulder, CO 80305 [email protected] 304-816-6332 2014 Multiphase Flow Science...

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