On the Modelling of Non-Reactive and Reactive Turbulent Combustor ...

NASA

Contractor

Report

4041

On the Modelling of Non-Reactive and Reactive Turbulent

Mohammad

GRANTS APRIL

NAG3-167 1987

Combustor

Nikjooy

and

and

NAG3-260

Ronald

Flows

M. C. So

NASA

Contractor

Report

4041

On the Modelling of Non-Reactive and Reactive Turbulent

Combustor

Mohammad

Nikjooy

Arizona

State

Tempe,

Arizona

Prepared NASA and

Advanced

Grants

NAG3-167

N/LSA and

Aeronautics

Space

Administration

Scientific and Technical Information Branch 1987

M.

C. So

University

Research

Defense

National

Ronald

for Lewis

under

and

Flows

Center Research and

Project NAG3-260

Agency

TABLE

OF

CONTENTS

Paoe

NOMENCLATURE SUMMARY Chapter

Chapter

..........................................

vii

.............................................. i:

2:

INTRODUCTION

xii

..............................

1

I.i

Background

............................

1.2

Objectives

............................

13

1.3

Outline

The

14

GOVERNING FLOWS 2.1

of

Report

EQUATIONS

1

.................

FOR

VARIABLE-DENSITY

..................................... Mean

Equations

In

17

Favre-Averaged

17

.....

Form 2.2

Reynolds

Equations

In

21

Favre-Averaged

Form 2.3

Modelling

of

The

Reynolds

Equations

2.3.1

Modelling

of

The

uiuj-Equation.

2.3.2

Modelling

of

The

uiS-Equation

2.3.3

The

Dissipation

Transport 2.3.4

The

Scalar

Transport

Rate

..........

..

23 23

.

31 33

Equation Fluctuation

........

35

Equation

iii

Pk6E BLANK NOT FILMS)

Paqe 2.4

Different

Levels

2.4.1

k-c

2.4.2

The

of

Model

Closure

Models

...

36

.....................

Algebraic

40

Stress/Flux

41

.....

Models 2.4.3 2.5

2.6

Reynolds

Near-Wall

Flow

2.5.1

Wall

2.5.2

Direct

Turbulent

Chapter

3:

4 :

Chemistry

Non-premixed for

PROCEDURE

3.1

Grid

and

3.2

False

3.3

Quadratic

4.2

Evaluation Models

The

55

......

Model

..

71 71

..........

Differencing

Without k-E

iv

Flow

Different

Combustor

Model

61

67

......................

Experimental

Calculations 4.2.1

For

72

Scheme

.

.........................

of for

52

Combustion

Sequence

Upwind

Basic

..........

Combustion

Premixed

Iteration

EVALUATIONS

49

.......................

Diffusion

The

Model

Finite

Model

4.1

Models

Rate

48

............

Combustion

NUMERICAL

MODEL

Calculation

Chemistry

for

48

.................

Non-premixed 2.6.2

47

.......

.............

Function

Fast

2.6.3

Models

Modelling

Combustion

2.6.1

Chapter

Stress

81

Fields

Closure

74

... ......

83 85

Flow Swirl Results

.........

85

4.2.2

The Algebraic

Stress

87

..........

Model Results

4.3

4.2.3

Reynolds

Stress

4.2.4

Conclusions

Model Results

.

...................

Model Evaluation

For

Calculations 4.3.1

The

k-c

4.3.2

The

Algebraic

96

Combustor With

Model

92

Flow

..

97

Swirl

Results Stress

......... Model

98

....

99

Results

4.4

4.3.3

Reynolds

4.3.4

Conclusions

Scalar

Transport and

4.5

4.4.2

Swirling

4.4.3

Conclusions of

Variable

4.5.1

k-_

5:

REACTING 5.1

Flow Flow

i01 i01

...........

Calculations

Constant

Model

Calculations

102

103 ....

................... Density

105 106

Modelling

107

Calculations

Versus

Algebraic

....

108

Model

Conclusions FLOW

.

...................

Density

Stress 4.5.2

Results

Comparison

Non-swirling

Extension

Model

Modelling

4.4.1

To

Chapter

Stress

...................

CALCULATIONS

Non-premixed

Combustion

v

................ ..............

109 179 182

paqe 5.1.1

Coaxial

5.1.2

The

Dilute

Model

Chapter

6:

References

5.2

Premixed

5.3

Conclusions

CONCLUDING

Jets

Swirl-Stabilized

...

Combustion

..................

6.1

Conclusions

6.2

Recommendations

AND

RECOMMENDATIONS

....

..........................

Flow

Equations

for

the

k-_

Appendix

B:

Turbulent

Flow

Equations

for

the

ASM

Appendix

C:

Algebraic

Stress

D:

Algebraic

Appendix

E:

Reynolds-Stress

The

Scalar

Coordinates G:

Boundary High

in

Model

235

Model

238

Axisymmetric

..

241

(x,r)

Appendix

Appendix

223 226

Turbulent

Closure

220 220

......................

A:

F:

187

194

............................................

Coordinates

182

190

..........................

Appendix

Appendix

.....

Combustor

REMARKS

Coordinates

Flux

Model

in

Closure

Axisymmetric in

......

Axisymmetric

...

244 245

(x,r) Flux

Transport

Closure

.........

251

(x,r) Conditions

Reynolds-Number

for

...................

Models

vi

L

Non-swirling

253

NOMENCLATURE

A

area

C I

C 2

C RI

C R2

C P

C

of

control

coefficient

in

volume

surface

modelled

form

of

coefficient

in

modelled

form

of

break-up

model,

Eq.

(2.71)

constant

in

eddy

break-up

model,

Eq.

(2.72)

specific

heat

constant

pressure

at

equation

(2.27)

coefficient

in

equation

(2.24)

coefficient

in

modelled

production

coefficient

in

modelled

destruction

C10

coefficient

in

equation

(2.22)

C20

coefficient

in

equation

(2.23)

C02

coefficient

in

equation

(2.25)

CD02

coefficient

in

equation

(2.25)

Cs0

coefficient

in

equation

(2.20)

E

activation

C

C _2

1

(2.13)

(2.14-2.16)

eddy

in

e

,Eqs.

in

coefficient

C

_.

constant

equations

#

Eq.

_j,2

in

C

, ij,1

coefficient

$

H

E

roughness

f

mixture

(2.18)

energy

parameter

fraction

vii

and

(2.19)

of

of

E,

E,

Eq.

Eq.

(2.24)

(2.24)

assumed

h

density

stagnation

H

heat

weighted

pdf

of

in

the

enthalpy

of

combustion

C

j

. ]

k

k I k

2

diffusion

flux

of

turbulent

kinetic

scalar

xi

direction

energy

pre-exponential

constant

for

Arrhenius

reaction

rate

pre-exponential

constant

for

Arrhenius

reaction

rate

characteristic

mixing

turbulence

length

scale

length

m

m

mass

P

static

p!

fluctuating

P..

ij P

k

fraction

of

pressure

pressure

production

rate

of

Reynolds

production

rate

of

turbulent

r

polar

R

universal

gas

turbulent

Reynolds

R

T

S..

ij

species

strain

t

time

T

averaging

radius

rate

in

stress

kinetic

axisymmetric

flows

constant

tensor,

number

Eq.

time

viii

k2/u_

(2.14)

tensor,

energy,

Eq.

(2.15)

Eq.(2.15)

T

temperature i

_2

instantaneous

velocity

streamwise

component

U

friction

U

Favre-averaged

_2

V

radial

of

velocity

of

azimuthal

X

streamwise

Y

distance

normall

stress

component

velocity

Reynolds

radial

component

W

(i=i,2,3)

(r_p).5

component

azimuthal

Reynolds

streamwise

Favre-averaged

_2

component

normall

stress

velocity

of

of

Reynolds

normall

Reynolds

stress

normall

stress

direction

from

wall

+

Y

dimensionless

Greek

Symbols

wall

coefficient

in

distance

modelled

form

of

_

,

Eq.

(2.16)

, Eq.

(2.16)

,

(2.16)

_J,2

coefficient

in

modelled

form

of

_. 1j,2

coefficient

Ffa[s e

in

gamma

function

false

(numerical)

Kronecker

modelled

form

diffusion

delta

lj

ix

of

_

ij,2

coefficient,

Eq.

Eq.

(3.2)

dissipation

rate

of

turbulent

dissipation

rate

tensor

kinetic

of

Reynolds

ij the

von

scalar

O

Karman

constant

fluctuation

Favre-averaged

dynamic

n ij II

of

scalar

viscosity

of

turbulent

viscosity,

kinematic

viscosity

pressure-strain

fluid

Eq.

of

(2.27)

fluid

term

fluctuating

velocity

part

of

9

ij,1

n

ij

mean

strain

part

of

ij,2

If.. ij

pressure-scalar-gradient

]-[i8,1

turbulence

IIi@, 2

mean

G

interaction

strain

density

correlation

part

of

of

part

_i6

fluid

turbulent

Prandtl

number

for

turbulent

Prandtl

number

for

k

O S

wall

T

shear

stress

w

viscous

T

stress

tensor

ij any

of

dependent

variable

X

k

Hi8

energy

stresses

Subscripts

E

value

at

EBU

denotes

fu

fuel

F

fuel

N

value

ox

oxidant

O

oxidant

pr

product

st

denotes

S

value

t

turbulent

W

value

node

eddy

east

of

break-up

P

model

stream

at

node

north

of

P

stream

stoichiometric

at

node

south

at

node

west

value

of

of

P

P

xi

SUMMARY A

numerical

axisymmetric

Closure

different

Reynolds equilibrium

is

of

one

the

other

fluxes. of

Two

presented. the

transport

In

chemistry

combustion models

models

suitable turbulence

are

show

to of

and

for

the

the

of

to

for

heat

rate

xii

high

also

transport

closure

which

fluxes,

while

the

scalar

for

and and

one

case

finite-rate

non-premixed

combustion.

chemistry

models,

constants.

a

are

non-premixed

release

and

and

scalar

examined

A

diffusion

scalar

Fast-

is

calculations

in

further

the

models

a

application

rate

be

of

equations

non-

models.

of

the

and

locally

second-moment

considered.

finite

two

closures for

by

stress

stress

Effects

is

achieved

flow

these

applied

need

coupling field

combustor

cases

promise

However,

realistic,

for

the

are

of

performance

models

two

is

pressure-strain

algebraic

addition,

combustors. more

the

swirl

algebraic

the

equations

reactive

without

algebraic

on

employes

solves

k-c,

closures.

different

One

premixed

Both

model

models

investigated.

solves

models:

of

and

equations

different

stress

pressure-strain

are

Reynolds

and

Performance

made

number

Reynolds

with

equilibrium four

also

Reynolds

using

levels

assuming

comparison low

the

closure.

and

non-reactive

flows

of

stress

analyzed

of

combustor

presented. three

study

gas

to effects

turbine which

establish on

are a the

CHAPTER

1

INTRODUCTION

i.i

BACKGROUND The

calculation

received to

of

considerable

different

attention

reasons.

Some

efficiency

combustors,

instruments,

rapid

of

fossil

fuel

in

of

growth

of

recent

these

which

industrial

societies.

As

combustion

modelling

has

a

a

result, been

an

higher

limitation

understanding

of

the

ecology

in

interest

in

increased

generated

due

measurement

technology,

jeopardize

is

for

of

better

has

This

demands

cost

and

flows

years.

are:

computer

resources,

formations

by

researchers

in

area. At

the

combusting

higher

pollutant

this

turbulent

first,

attention

complexities

scores reality,

of

of

fifty

complete

description are

equilibrium

composition

to

fifty

the of

(Lavoie,

chemical the the

of

the

gases

as

are

kinetic

gases

ai.,1970).

a

required

through

for

a

Typical

rocket

variation flow

In

reactions,

calculation in

of

involve

process.

accurate

they

handling which

et

species,

with

the

elementary

kinetically-influenced the

to

reactions,

hundred

of

concerned

given

chemical

one

composition by

only

species

to

ten

followed

real

individual

involving

examples

is

of

nozzle, of

the

the

the

nozzle.

Later,

two-

and three-dimensional

variations

in

O'Rouke,

the

are

1977; Griffin

"turbulence flow

time

taken

et al.,

models"

variations

permits

into

account

1978),

and the

realistic

and chemical

reactions.

In order

to construct

a comprehensive

numerical There

may be broadly

methods,

are

some serious

comprehensive

grouped

turbulence

molecular

transport

different,

and there

models,

model.

and are

chemical

of partial

differential

equations

mass, momentum, energy

and species

conservation.

and generally

they

procedures

because

introduced

by

processes.

For

Sanders

(1977),

splitting time

transport portion finite

of

example, a

difference

in

numerical to

associated

the

extremely

model

work

compensate

for

molecular

problem

scheme,

while

was the

include

are

by

known

coupled

reaction Dwyer

as

vastly

and

operator different

The fluid

split

of

scales

the

done

solved

set

by conventional

transport,

wave propagation.

with

would

This

in

the

a

vastly

that

physical

involved

in

process

solved

disparate

the

kinetics.

temperature

field

technique

with

a

flow

be easily

gradients

and unsteady of

combustion

cannot

steep

was used

scales

the

such

the

involve

describing

categories:

are

in

the

model,

associated

fields.

equations

of

of

involved

and density a set

of

reactions

gradients

Simulation

development

and chemical

Time scales

sharp

and

three

obstacles

and

(Butler

combustion

into

numerical

combustion

space

representation

patterns

problem

in

using reaction

chemical dynamics explicit terms

were solved Otey

by an ordinary

(1978)

has

methods.

His

chemical

flows

presented

work

difficulties

differential a review

retained

and

in

of

the

allowed

involved

equation different

numerical

essential

insight

the

technique.

features

into

the

of

such

solution

of

numerical a

complex

in

current

system. However,

among the

combustion

research

interaction

between

reactions place

(Smoot

in

motion

the

that thus

an

the

and

of

those fluid

the

allowing

instantaneous with

and

changes,etc,

volume

local

the

other

in

the

mixing

and to

reaction

associated

On

controls

reaction

the

products

can

of

In

or

take

turbulent

the

reacting

and

frequency

mixed

together,

turn,

the

local often

absorption,

the

of

chemical

themselves,

release impact

the

hand,

are

processes

role

reactions

time

proceed.

heat

and

Chemical

role

reactants

the

mechanics

level.

turbulence

questions

regarding

Hili,1983).

important

Local

each

are

molecular

plays

species.

most important

local

density

turbulent

fluid

mechanics.

are

In

practice,

the

so

complex

that

required

to

equations

possible.

reality

with

turbulent characteristics,

render

the

of

mechanics

various

These

ease

nature

fluid

and

modelling

solution assumptions

formulation

of

the

flow,

and

the

radiative

chemical

the

assumptions

of

the

which are

are

conservation

combine

concerned

flame, heat

kinetics

the transfer

physical with

the

combustion from

the

products

of combustions.

prediction

of

distribution

Improper

models

combustion

in

the

result

in

efficiency,

combustion

erroneous

temperature

system

and

pollutant

formations. Numerous devised

for

turbulence

constant

Lumley,1975a; to

flows

However,

one

of

models

for

identify,

knowledge

technique

rate

the to

a

of

the In

correct

to

evaluation

order

manner,

Reynolds

to

express

it

is

4

and

of

the

thus

the

quantities

is

the

mean formation

the

mean formation

convenient

decomposition

of

flow

functions

of

or

relating

associated

latter

in

presence

expressions

these

but

formation

the

to

not to

areas

of

due

are

difficult

these

concentrations, of

of the

(Pratt,1979

are

rate

non-linear

mean values allow

characterization

mechanisms

in

for

physically

reactions

rates

species

basis

or destruction

Analytical

highly

ai.,1975;

the

in the

not

species

been

and combustion.

constants

lies

et

devising

kinetic

time-mean

always

(Pratt,1979). in

the

reaction

and of

lies

chemical

molecular

are

insufficient

rate

to

problem

instantaneous

temperature

rates

due

(Borghi,1974).

quantities

in

of formation

proper

of

turbulence

problem

net

rate

provide

properties

flows

major

the

destruction

the

main

successfully

(Launder

These

and kinetic

the

obtaining

flows

variable

Although

known

have

reactive

species

;Bilger,1980). always

with the

of the time-mean molecular

properties

Reynolds,1976).

extension

valid

models

and

to

use

express

the the

Arrhenius

term

mean values

in

of

terms

the

scalars

cross-correlations. because

the

with

the

type

of

This

correlation

reactions Borghi

involving

third

combustion

scheme,

Therefore,

it

is

combustion

models

for

quite

to

scale is

To help

identify

interest turbulence

large

eddies.

of

This problem

turbulent

of

the

diffusion

in

realistic

reacting

was recognized

alternate

state

species

are present.

turbulent

paths art

of

flames

chemistry-turbulence

by

have been different has

been

for

different

interactions

types it

two hypothetical

time

scales:

the

turbulent

time

a typical

time

time

type

effects,

react

mixing

of this

the

scalars,

any

these

as to

For

on

seven terms,

the

identify

and the

defined

the

of these

different

reactions.

at least

closure

magnitude

(1976).

The effects are

of

convergent

chemical

moments of

and different

A review

by Bilger

that

and

Depending

considered.

problem.

suggested.

given

order be

clear

researchers

always

the

that

many equations

is

not

and on

estimated

to

a formidable

various

is

the

statistics

can be of comparable

considered

have

involving

higher

(Borghi,1974).

and lower

expansion

flow

series terms

(1974)

series

and their

mean quantities

involved,

the

of an infinite

time

completely scale

for This

scale time

scale.

is

reduction scale

the

its

chosen to

is

time

scale

reacting

be a typical

species

of

value.

The

fine-scale

turbulent

must be adequate

convenient time

equilibrium

by

chemical

reaction

The reaction

for to

of

breakup for

of

molecular

interaction scale

to take

then

molecular scale

is

the

level

is

the

contacted,

systems

time

the

the

reaction time

compared if

the

highly

should

and the

tt,

tr,

then

in

the

is the

local

of

the

effect

of

ignored.

the

turbulence

case is the calculated

However,

the

the

very

In

this

small,

turbulent on the

mean reaction

fluctuations. but

Thus,

significant

mean reaction Therefore,

mean reaction

rate

the

are

very

effects.

from

only

reaction

rates

local

produces

in

equal

mean variables.

a very

are

not

turbulence,

appreciable 6

error

(Pratt,

to the

it

has

number

suffeciently use

in

Although

limited

the

rate

only

has been used by many researchers,

be valid

the

of

than

fluctuations

the

When

relationship

are

presence

still

shown to

the

any variable

exist,

been

in turbulent

turbulence.

fluctuations

approximation

products.

reactions

slow

this

their

once

much greater

are

rate

approximation

scale,

rates

reaction

to

be

for

time

species,

reactions

reaction

special

compared

form

temperature

the very

case_.

reacting

to

to the

The reaction

to

account

to proceed

by examining

of

sensitive

though

temperature

these

chemical

time

unaware

can

mixing

time

scales.

changes

is

rate

is

even

for

fluctuations

mechanics,

reaction rate

can occur.

scale,

to

chemistry

fluid

reaction

incorporating

turbulent

case,

for

completely

two time

The turbulence

required

can be characterized

If

slow

(micromixing).

required

react

for

between these

the

time

before

to

Approaches

place

of slow

of 1979).

this

When the magnitude

as

kinetics for.

It

turbulent

fluctuations

this

area

as the

relating

to

flames.

Over

composition

the

This

turbulent

flow

products

extended

to

between

local

for

temperature

turbulent

very

If

the

by combustion

in

non-equilibrium, molecular

species

equivalence

ratio,

thermodynamic reaction

fraction

with

data

then

combustion

(Bilger,

process

(Libby 7

heat

hydrocarbon the

and the

composition

with

time

the

be

relationships

composition

to

for

base could

similar

scalar,

rate

in these

chemistry

mean

compared

a diffusion

significance

chemical

of

Bilger

observation

species

expeimental

associated

specific

Recently,

in

limited

flames

oxidation

micromixing

the

great

kinetically

calculating

short

accounted

the

not

the

conserved

reactions

be

of the mixture

to non-equilibrium

However,

scales

that

instantaneous

instantaneous

of

holds

turbulent

must

a given

were

a function

field.

chemical

experimental

a function

between

the

most need of

flame,

of

both

1983).

of

diffusion

observation

interactions

same order

interaction

region

He concluded

techniques

has the

important

broad

was only

seems to be only

extended

that

(Smoot and Hill, an

a

equilibrium.

high

area

scale,

the

has been identified

chemistry/turbulence

though

local

that

hydrocarbon

flames.

of

and the

reported

laminar,

is

time

advances

(1978)

scale

turbulent

is

research

time

the

researchers

even

reaction

and

the

statistical could

be

1980). release

fuels scale

in

the

have

time

of

the

and Williams,1980).

In

this

case,

species

are

can then applied in

the

mixed

assumption

process

is

rate

process.

Thus,

chemistry

mixed

exist

through

equilibrium

at

all

the

motions,

heat

loss

heat

release,

If

and heat

to the

surrounding

then

temperature

the

can

be

quantity.

conserved

scalar

or mixture

degree

of

simplifications,

the non-reacting

significant

weakness

concerning nitric require

the

oxide,

consideration

at flow

in

this

problem

of

finite

are rate

the

and the

of

a

to

the

single

identify these

reduced

(Bilger, is

with

With

is

made

conventional

point.

problem

fuel

of

defined

a

to

composition

terms

is

and emission

unburnt

are

same rate

cases,

are

proceed

compared

in

approach

oxidant

reactants

chemical

fraction

mixing

formation and

the

For these

reacting

equivalent

at

the local

and

assumptions

negligible

"mixedness"

in

reactions

determined

scalar

kinetic

concerned,

Once the

instantaneous

conserved

the

is

is

fuel

the

diffuse

micromixing

considered

the

enter

chemical

flame be

can be

reacting

the

the

same point.

turbulent

species

turbulent

Therefore,

instantaneously.

that

and

to

assumption

and oxidant

of

overall

enough

fuel

not

reacting

assumption

made that

and

equilibrium.

both

this

type be

as the

fast

instantaneous

can

limiting

as far

is

this

the

chemistry

where the

In

once

fast

Unfortunately

streams.

the

quickly

The

to situatioins

separate

cannot

occur

together.

be applied. only

flows,

reactions

that carbon

available. chemistry.

to

an

1980).

A

no details monoxide, All

these

However,

this

method

assumed,

is

but

chemical

is

quickly

equilibrium

to

mean

probability

density

and variance

be used to parameter

some

the of

their

However,

distribution

has

Despite direct

For

the

solution for

example,

Deardorf

rapid

(1974)

fuel

on

the

approach

determined

equations

can

Different

two-

by a number of

and

Jones

advances made in of

the flows

calculation

using

of pure

are

Naguib,1975

support

by

With

PDF. The mean

and tested

in

scalar

of

the

(1977)

;

betain

his

flames.

turbulent the

which

Lockwood

the

known.

The

transport

provided

of diffusion

for.

unknowm parameters.

;

the

turbulence

a two-parameter

evidence

been

is

of

(Spalding,1971

reactions

once

effects

scalar

mixture

1981).

intermittency

accounted

the

local

approximates

the

be

that

the

the

evaluated

PDF's have been proposed

calculation

that that

respective

the

Rhode,1975).

equations

of

(PDF)

conserved

determine

researchers

a

be

specifying

solutions

a function

and Smoot,

of

can

of

of

implies

be

fast

function

function

reactions involves

only

can

streams,

chemical

a

(Smith

incorporation

oxidant

only

cannot

sufficiently

some condition

density

appropriate

are

this

condition

The

rates

and thus

Physically,

proceed

from

even when equilibrium

reaction

ratio,

fraction.

adopted

the

composition

equivalance

or

sufficient

a

computer

time-dependent is

not of

sub-grid-scale

the

currently diurnal scheme

technology, conservation practical. cycle required

by a

week's

computing

time

7600 computer.

the

equations.

to the

time-mean

extra

the

whole

The conventional

time-averaged identical

using

In

terms

technique

this

instantaneous

variable,

the

way,

is

the

there

are

to

a CDC

solve

large

the

become

equations a

fluctuating

of

equations

form of the

but

involving

resources

only

in

number

of

components

(Borghi,

--r---

1974).

Terms

density

such

flow

For

variable

is

not

as

simplifies

and

flow,

the

However,

(1976)

indicate

that

Puiu j

momentum this

flux

averaging

can

averaging, density

be

quantities before

equations

are

equations,

except

In

the

deal

The

mean the

turbulence-field

by

form

to

the

the

same

circumvent how

problem.

Favre

In

Favre

instantaneous

partial

differential

uniform

density

flow

replace

the

ones in

Bilger

turbulent

out

variables

remaining

constant

of

to

the

p'

pare

by

the

way

this

resulting

p =0. that

the

be

points

with

Favre-averaged

density

a

weighted

in

can than

is

Reynolds-averaged

Furthermore, time

to are

identical

v

(1975;1976)

averaging.

conventional

the

applied

for cited

greater there

since

involving

those

Up

constant

assumed

terms

as

sometimes

Bilger

is

measurements -c-

pu---_. However,

difficulty'

it

to

such

A

greatly

thus,

some

and

appear.

often,

reduce

terms

etc.

problems

u i,

equations

flow.

as

u i,

more

with

density

order

,Up

these

density

correlated

neglected

Puiu j

(Favre,

the

1969).

equation

is

still

density. fluid

mechanics closure

area,

models

lO

by

a Mellor

survey and

of

the

Herring

mean(1973)

gives the

an excellent mathematical

Spalding

(1975)

problems

in

model the

discussion

of

modelling

of

gives

deficiencies

developments Different such

in

as

sub-grid-scale

which

advanced

turbulence

models

stress

stress/scalar

1975),

etc.

of the

In Reynolds

to

similar

approach

fluxes.

Models

flux

of

local

the

turbulent

processes

most

realistically

and

hence

suitable important

at for as

the

than

simpler

(Mellor 1976),

on the

;

non-

present

state

point 11

are

rates

are

and

models.

of

for

but

turbulent

components

applications.

starting

components

equations

and computationally

practical a

and

(Launder, 1979).

the

transport and flux

tested

developed

Schumann,1975

focus

equations

stress

thoroughly

modelling.

mean strain

turbulent

not

is

encouraged

closures

stress

used to determine

employing

better

k-E It

and Launder,

these

individual

potentially

the

(Hanjalic

1973,1975;

transport

is

on

have been

Gibson

closures, the

from their

1973.

eddy viscosity.

stress

related

All

in

and unsolved

have

closures

1976;

to

and shortcomings.

k-E model

turbulence

; Rodi,

nature

determined

advantages

scheme (Deardorff,

al.,

no longer

focuses

algebraic

and Yamada,1974

isotropic

He

Reynolds

Launder,1972),

Kwak et

modelling.

the

of

up

solved

more

classes

turbulence of

its

in

has been achieved

a discussion

turbulence

and enumerates

what

scalar for

simulate are

However, more

development

A

the the

therefore they

are

expensive, not

very

However,

they

are

deriving

algebraic

expressions

for

(Rodi,1976;

Mellor

expressions, for

most

cases

together

where

a

at

the

k-_

closures

when

forces

such

different

only

turbulent

motion

computer

in

capacity

is

for

the

turbulence

that

cannot

must

then

turbulence scale be

is

be

represented

passive

so

flow scalar

with by

a

that

the

Kim points

in

a

to

investigate

for

channel.

the

12

small-scale

chosen The

numerical small-scale

than

the

large

turbulence This

used numerical

used

the

small-scale

the

interaction

can

approach

three-dimensional

direct He

modelling computers

the

(1985)

the

time-dependent

models.

solving

solve

However,

the

the

body

completely

to

the

sub-grid-scale simple

of

present

model.

problem-dependent

stress

and

motions;

resolved

for

is

solve

less

problems. grid

scale

relatively

a

solution. to

They

simplicity

effect

resolve

any

needed.

simulation

numerical

be

hardly

Reynolds

the

The

to

sufficient

promising

computational

by

slow

are

Finally,

turbulence.

a

the

such

sufficient

is

approach

directly

that

numerical

rotation.

approximated

by

very

dependent

turbulent

for

large

much

turbulence

appears

accounting

too

equations

grid

of

and

the

modelling

and

model

fluxes

are

there

generality

equations

small

that

the

sub-grid-scale

too

and

extent,

buoyancy

seems

_ equations,

equation

to

and

It

some

comes

turbulence

the

are

to

it

Navier-Stokes

and

transport

with

as

k

problems

least

stresses

Yamada,1974).

with

full

model

turbulent

and

engineering

combine, of

the

time128x129x128

simulation temperature of

of as

the

wall-

a

layer

structure

short

and

drawback time

with

methods

give

approach

For

layer.

overview

(1979)

this

of

outer

this

to more

is

the

reason,

improving

(1979)

subgrid

modelling

details.

gives

The

a and

greatest

huge amount of computing

they

the

Herring

are

being

modelling

looked

upon

approximations

of

closures. In

averaged idea

and

Leslie

involved.

simpler

view

of

the

equations

to

to

the

average

variables

1979)

advocates

area.

variable

our

model.

flow, models

flows

variables

for

has

may be

present

objective

(1976,

of

constant-density

1977, model

done

is

by

standard

turbulent been

the

simply the

Bilger

of

much work

validity

density

density

hypothesis

not

density-weighted

turbulent

in a particular

Therefore, the

uniform

density

this

the

uniform-density

density-weighted

However,

establish

of

for

existing

non-uniform

substituting

similarity.

similarity those

has emerged that

adapted

1.2

the

introduction

Love

as

with

in

to

this

try

to

models

for

flows.

OBJECTIVES The main objectives

A. To evaluate model swirling include stress

for

of this

and identify

turbulent combustor

k-E model,

research

the most

momentum exchange flow

calculations.

algebraic

stress

models. 13

are:

general in

and efficient

swirling

and non-

The evaluations models

and full

will

Reynolds

B. TO provide models

and

efficient

examined

The

of and

existing identify

model

these

for

turbulent

scalar

the

general

most

swirling

and

flux and

non-swirling

fast

will

be

and

is

density

variable-density

applicability

models

is

finite

flows

demonstrated

by

will

be

reacting

chemistry

applied flows

models.

Their

to using

validity

detail.

effect

examined,

of

and

turbulence

above

non-premixed

rate

in

the

identified

and

examined

Finally,

for

measurements.

premixed

both

models

their

turbulence

field

and

flux

with

calculate

E.

evaluate

validity

comparison

D.

of the

flows.

The

is

to scalar

combustor

C.

a review

heat

the

models

release

validity for

on and

reacting

the

turbulent

extent flow

of

flow

constant-

calculations

is

assessed. 1.3

OUTLINE The

and

the

the flow

THE

REPORT

remainder

of

accompanying

In flows

OF

is

chapter posed

report

consists

of

five

chapters

appendices. 2,

more

density-weighted quantities.

the

the

problem

precisely

by

averaged The

of

introducing

equations

appearance

14

calculating

of

turbulent and

governing

turbulent

discussing the

transport

meanterms

in

these

equations

makes

turbulence

models.

introducing the

actual

review

of models

models

are

Section

2.5 considers

to

discussed

include

found

in

this

the

is

reactants,

cases

of

to

of

2.4;

models

that

always

than

is

wall.

Although

1% of because

it.

the

complexity.

a smooth

across

is

turbulence

region

importance

modelling

chapter

2.3

of the

of

the 50% of

Section

combustion

flow

2.6

the

turns

processes.

non-premixed

It

and

premixed

the

solution

respectively.

procedure

3

the

presents

adopted

governing

for

the

equations.

numerical

(false)

the

to reduce In

diffusion of

this

source

chapter

4,

models

(ASM)

for

the

the

and

model.

demonstrated

to

and determined the

which

and

non-linear

briefly

may or

solution.

discusses

may not

A scheme is

of

different

The results

correlation

coupled

chapter

effects

investigated. Having

of

seriously introduced

of error.

on three

swirling

details

highly

This

accuracy

strain

turned

of this

increasing

less

occurs

the

the

Chapter

affect

of

of

occupies

necessity

sections

number

significant

to

discuss

order

vicinity

in mean velocity

attention

will

of

in the

Reynolds

usually

it

change

low

the

The heart

the extension

immediate

region

domain,

the

the

in

apparent

performance

four

different

algebraic

non-swirling

the effect

of 15

of the

suitable the

stress

turbulent

are compared with

its

pressure-

the

flows

are

standard

k-E

pressure-strain

model,

Reynolds

closures

attention

stress

is

closure

(RSM). A comparison a low two

Reynolds

model

and

some

made using

the

are

calculations. of

two

algebraic of

to

scalar are

closure.

Finally,

k-_ model

are

results

premixed

flux

5

and

standard For

swirling

ASM and RSM are

best

for

field

calculations,

(AFM)

the

k-E

pressure-strain

on

are

full

flow effects

two

different

analysed

Reynolds

of three

model

non-swirling

and

the

stress/flux

different

variable-density

discusses

However,

in contrast the

case,

ASM and the

swirling

rates

are

applied

to

of

calculated eddy break-up

study

6

of

flows

work. 16

and

and

flames, rate

process from

the

rate

combustion.

premixed reaction

is

and

finite

non-premixed

flames only.

used and the Arrhenius

In mean

reaction

model.

summarises

and put

non-premixed

fast

finite

reaction

chapter

emerged from this

Both

to non-premixed

are the

results

models.

a two-step

Finally,

the

consideration

and also

future

model,

also

are discussed.

combustion

formation

with

of

the

models

models

made for

models

for

scalar

comparison

chemistry

rates

the

compared

Chapter

this

k-_

the

Effects

closures.

model

perform

and

on RSM are

compared with stress

of a high

closure.

models

pressure-scalar-gradient

results

require

stress

standard

As for

algebraic

the

are

diffusion

found

performance

diffusion

results

comparison

which

Reynolds

turbulent

and the

flows,

made of the

number

different

analysed

is

forward

the

main

conclusions

some recommendations

CHAPTER2 GOVERNINGEQUATIONS FOR VARIABLE-DENSITY FLOWS

2.1 MEAN EQUATIONS IN FAVRE-AVERAGEDFORM

In

this

section,

distribution These mass,

of

equations

the

mean

are

derived

momentum

Cartesian

tensor

and

+

aPui ax

=

equations flow from

scalar

notation

mass conservation

_P 8t

the

which

quantities the

which

govern

are

conservation can

be

the

presented. laws

expressed

of in

as:

:

o,

(2

i)

I

momentum

conservation

a5 i ~ Pa-t + Psi

scalar

a[ P_t

puj

0_.. +_;j ax;

a6 i = _ a_ ax--jj ox,

conservation

+

:

a_ _ axj

,

(2.2)

:

a_

(2.3)

Ox i

17

where

ui

is

the

x i direction, an

component

p

is

instantaneous

of

8

instantaneous

in

these

motions

numerical flow

flows

contain

of in

grid

computers. analysed

Reynolds,

the

flows

values two

unweighted

form

flows

or

Favre

(1969).

flow

of

the

be

grid

the

=

Ui

is

beyond

reason,

T

--_

OO

I

are For

used

be

t

18

density

=

averaging

0

the

density

suggested and

Ul

is

into

either

constant

by

and

the

capacity

separated

decomposition

dt

the

Following

used;

for

of

flow

variable

decomposition

ui

size

turbulent

ui

1

the

resolve

methods.

can

represnted

Lim

is

the

T+t

=

flux

Storing

where

Ui

is

to

smaller.

fluctuations.

decomposition

+

_

smaller

mesh

!

ui

the

much

order

quantities

unweighted

are

_ij

are

In

statistical

density-weighted

variables

the

which

points

conventionally

The

diffusion

even

this

instantaneous

types

is

domain.

to

For

their

pressure,

procedure,

using

and

static

and

motions

flow

many

in

tensor.

have

so

]i

velocity

direction,

numerical

would

normally

mean

the

a

at

present

xi

stress

variables

of

quantity,

the

extent

instantaneous

instantaneous

viscous

Turbulent the

the

scalar

scalar

than

of

by of

which

the

defined

density-weighted

decomposition

and averaging

are

as:

ui

= Ui + ui

where T+t

U i

=

Lim T-_

Since

by

over

The

the

the

averaging

for

stationary

are

the

after either the scalar

the

that

equations may

of be

is but

shown

long shorter

should

involve

ensemble

1970).

For

high

weighting density

continuity

, the and

as

19

and

not

and

In

the period

general,

averaging,

ensemble

to

be

of

but

averaging

number

Favre-averaged

conservation

the

vary.

Reynolds is

_i#O with

than

may

averaging

that

compared

quantities

time

or

written

T

be

flow

density

pressure

can

scales,

average

(Lumley,

it

time

time

flows,

noting

t

averaging

process

same

dt

_i_0,

turbulence which

_i ~~ P

I

oO

definition

pui=-p-_u'i. largest

_ i

flows,

applied forms

momentum

to of and

8_a- + 8 at 8_

the

I

z2

l ii/

_l; I

r",

I

_

I

i

I

I

I

o

z2

o

z2

o

zz

44

calculated

mean

•

_%% °

i

°

•

_1%

• t%

i

',,

%

•

._

._"

_

•

%o e %

,

o_

.

_, s #_

W

I 0

I 10

i 0

i 10 "Y='B U V

I 0 [M/St

i 0

10

i 10

-.2

COLD FLOX ---

FRST CHEMISTRT

.....

FINITE

RRTE CHEMISTRT

0

!,

I

i

I

I

I

i

I

0

I0

0

I0

0

IO

20

UV

Figure

5.9

Effect turbulent

of

heat shear

release stress

206

(M/S)

""2

on

the

(Lewis

calculated &

Smoot,

1981)

u.V/r"_

m/s ) O

70

R

o SW IRLING

_]

I00

_

240

o o

°2_'°'-__., _o

0

o o

o

o oo

'

o

o

oo

o

,

o

0

\:°

! 0

I 7

I 0

I 7

I 0

I 7

Figure

5.11

Comparison of calculated with measurements (Brum

208

/_

IA

I

I/o

I

_'°

If °

I1__

I £

_o \ o ) -

I_, o I_ o I Jo

II r" I to I |,,

I /'_ oOA'rA I IO (e,uf I |_ S,S_uEI

I 7

I 0

V/'_w

° "

I 0

I 7

I 0

15, ,,,_

I -

I 7

(m/s)

&

u_and Samuelsen

,1982)

I 0

I_FINIT1

,s,,

I 2'

I I,

I

i

I

=,

o

x t

Figure

5.10

t

tt

t

t

Comparison of calculated mean axial and tangential velocities with measurements (Brum & Samuelsen ,1982)

207

DILUT

;. ON RIR

SWIRL

O

-_

O

O

• ..

•

.

.!> ID

FUEL ''_

°lo

•

o

q

0

_'_

INJECTION l

I

l

l

I

l

l

l

I

0

15

0

1_5

0

15

0

15

3O

I

I

I

o

15

30

U_

•

Figure

5.12

1H/51--2

m

I

I

I

o

is

o

Comparison (Brum &

l

l._S

of measurements Samuelsen (1982),

209

with ASM)

_-_

calculations

m

L I 0

I !

I 2

I 3

I I;

I 5

X/R

Figure

5.13

Contour (Brum

&

plots of Samuelsen

210

unburnt ,1982)

fuel

I $

I 7

FINITE

- RATE

T -700" K 0 1600

m

1600 I000 0

_V A

2000

T- 700"

FAST

K

- RATE

t

!

!

I

r

I

I

I

0

I

2

3

4

,5

6

7

XIR

Figure

5.14

Contour inside

plots of temperature the combustor (Brum

211

&

distribution Samuelsen

,1982)

1900

FINITE

- RATE

FAST

- RATE

I

I

I

,I

i

1

1

I

O

I

2

3

4

,5

6

7

X/R

Figure

5.15

Contour (Brum

plots &

Samuelsen

212

of

mixture ,1982)

fraction

0.08 0 05 0.07 0.02

I

Figure

|

5.16

I

Contour

I

plots

of

I

CO 2

213

(Brum

I

& Samuelsen

I

I

,1982)

(c)

-_ i'kE;_iiEC; ......... Tm=294

=K;

Urn=7.5

O.25O. D.1 M/S;

"-JET

Uj=I35'M/S

INJECTOR

f PREMIXED PROPANE ALL

DIMENSIONS

IN

CM

AIR =1.0

Figure

5.17

Sketch

of

the

combustor

214

(McDannel

et

al.,

1982)

150 HOT

FLOW

COLD

FLOW

CALC. 120

_ DATA

(McDANNEL

ET

1983;

AL.

SAMUELSEN

D

1986)

9O

(.n 6O

o, 0,,`,6 _ / o o

/ /

/ &

.S

&

S

o

- 30

o

I 0

0.5

8

&

_-y_ z

I

I

I

1.0

1.5

2.0

I

I

I

2.5

3.0

3.5

X/D Figure

5.18

Opposed jet centerline velocity decay for and cold flow and their comparison with calculations (McDannel et al. ,1982)

215

hot

u/uj 0

0.5

0

05

i

0

0.5

I

i

2.13

X/0-1.86

0S

0

05

0

0.5

O

0.5

i

]

i

!

!

!

2.36

2.63

2.88

0

3.13

%

,/

I

I j

d:

C_D

FLOW

HOT

o-"s,_

/2

I

FLOW

/

/

DATA ( _cOANNEL T AL 1983; AMUI[LSEN )86]

I 0

/"

O

I

I I

! --: ;Z__..

!

CALC.

i

| I I I

t | I I I I

I

t

i

3.38

2.72 t I I

0.5

I

I

I

I !

I I

J

/

I

I

I I

I

10

I

0

I

10

I

0

|

10

I

0

;

_0

|

|

I

0

I0

0

I

10

l

0

I

10

k/U2m

Figure

5.19

A comparison of the profiles at different (McDannel et al. ,

216

axial velocity axial locations 1982)

I

I

!

0

I0

20

\ \ % i!

I

|

i

0

3.

0

i 3.

I 0

I

I 0

I 3.

i

3.

-UVIUIN-N2

\

\ \

\

I 3.

J 0

I 3. -UV/U

Figure

5.20

[N,.-2

A comparison of the turbulent profiles at different axial (McDannel et al. , 1982)

217

I 6.

shear stress locations.

I

I

3.

6.

CALC. T= 800 140Q

185o

DATA ET

1650

_--175o 1800

1850 1800 ITSO

(McDANNEL AL.

198:5)

T1800"

t

!

1

2.0

2.5

3.0

K

1650

I

3.5

!

1

I

4.0

4.5

5.0

X/D Figure

° K

5.21

Comparison temperature (McDannel

of the calculated contours inside et

al.

218

,

1982)

and measured combustor.

CALC. !.0

1.0

I -T I I I I I I I I I l I I

DATA

.0 (McDANNEL ET AL. 1985) CO CONCENTRATION CONTOURS !

!

2.0

2.5

!

:5.0

IN %

2

!

!

!

3.5

4.0

4.5

I

5.0

X/D Figure

5.22

Comparison contours (McDannel

of the calculated inside combustor. et al. , 1982)

219

and

measured

CO

CHAPTER CONCLUDING

Important in

i.

4

study

and

As

far

are

makes

as

The

k-_ of

region

the

For

Reynolds

provided

at

for

each

the

end

presents

of of

general

recommendations

mean just

of

non-swirling

model

gives

complex ASM

for

the

sections

each

section.

conclusions

further

predictions

prediction

well

as

good

of

work.

better

do

the

centerline

are

subject

for for

the

ASM's

better the

k-c

job model

recirculation to

conditions.

220

or

k-_

RSM

in

the

the

developing

the

far-field

prediction.

a

while

concerned,

flows.

However,

flows,

models

ASM

correlation flows.

a

is

any

combustor

combustor

velocities, of

as

provides

stress

description

field

swirling

swirling

tangential

boundary

the

performs

region

the

RECOMMENDATIONS

conclusions

therefore,

calculation

3.

5

AND

CONCLUSIONS

closure

2.

and

chapter,

this

6.1

specific

chapters

This

REMARKS

6

uncertainties

and of

the

predicting

gives zone from

full

a

good

although the

inlet

4. the

Low-Reynolds-number mean and turbulence

This is

model

model also observed

quantities

predicts

the

and

models.

5.

transport

Models

employing

turbulent

stress

turbulent

processes

potentially

computationally development,

they

applications. point

for

stresses

deriving and

with

engineering

It

should

model

be

competing

factors

phenomena

and

invalidate

It

the

k

the

and

c

simpler

at

the

models.

for as for

such

and present

however,

that

the

therefore

tested

expressions

seems

individual

are

suitable

important,

the

are

are state

practical a

starting turbulent

expressions

equations

all

used

sufficient

in for

problems.

predictions

contribution

very

algebraic

fluxes.

conjunction most

are

to

by

the

and

Hence,

not

which

simulate

thoroughly

are

They

for

of

region.

zone

components

expensive.

wall

completely

equations

not

more

missed

compared

are

near

realistically

general

they

the

estimate

recirculation

fluxes

more

more

However,

6.

and

a better

in

corner

experimentally

high-Reynolds-number

of

provides

from

noted

that

could ---

the be

obscured

boundary

numerical any

conclusions

of

effectiveness to

regarding

the

turbulence extent

by

oscillatory A

aforementioned

221

some

conditions, diffusion.

the

of

significant

factors superiority

tends

to or

inferiority is

of a given

a complex

Inlet

and

function

boundary

strongly

for

turbulence

mesh size

flows.

inadequacy

model

appropriately

of

and

are

aspect

also

importance

of

this

area.

must and

ratio.

model

It

be

is

in

cannot

apparent

preceeded

that

by

(I)

experimental

detailed

elimination

diffusion

and cell

turbulence

validation

(2)

Numerical

The in

configured

studies,

model.

conditions

swirling

compensate

turbulence

of

false

case

diffusion

considerations.

7.

Favre-averaging

isothermal, flows, of

the

technique

variable

density

turbulence

model

chemical

heat

is

a reasonable

flows. should

release

on

approach

However, also

the

for

include

the

Reynolds

for

reacting effects

stress/flux

components.

8. Two-step

reaction

gas turbine

combustors

model

for

However, flames that

determining they

to

have

establish

the major

species

scheme shows promise and is the to

preferred

over

combustion be

model

further constants

effects validated and rate

can be accurately

222

for

application

in

fast-chemistry in

combustors. with

simple

constants,

predicted.

so

6.2 RECOMMENDATIONS

I.

The

low-Reynolds-number

better

estimate

the

wall

to

transport

the there

E-equation

in

sufficiently this

to

directions al.

1979)

3.

The

the

and should

be developed

assumptions gradient

of

Hij,l

than

that

need in

of

a near

similar for

the

review

and

in many

developments. not

heel

The

to

be

As observed

for

most models. for

are promising

different

(Hanjalic

et

further.

the

pressure-strain

and

correlations

are

also

not

very

improvement.

Proposals

for

the

(1951)

about

the

this

equations

inhomogeneous

and Khaheh-Nouri,

coefficients

for

of Rotta

expansion

in

be improved.

processes

and

a

model

appears

or different

a series

(Lumely

form

length-scale

satisfactory

in

present

several

pressure-scalr

apply

further

is the Achilles

model

stresses

have shown to work well

is much room for

and should

to

models described

equation

use

Reynolds

provide

too.

k-c model,

its

to

low-Reynolds-number

universal

by many,

further

a

found

appropriate

equations

situations,

behavior

is

some of the

in particular

is

mean and the

It

develop

2. Although

Ideas

the

region.

approach scalar

of

closure

the 1974).

terms 223

flows

have

by including isotropic However,

in the

only

further

homogeneous optimization

expansion

on the

gone terms state of basis

of

available

indeed. Hij ,i should

experimental

It

data

seems unlikely

will

be made in

be placed

is

that the

a

very

any

near

on developing

difficult

serious

future

proposal

and

a better

task

that

for

emphasis

approximation

for

Hij,2"

4.

The

derivation

schemes holds improvement

and validation

the

for

strongly

5. The difficulty of

numerical closures

stable

and

Taylor

especilly

and fluid

numerical

important

the coupled

in

the

stages

behavior fluctuations

of

axial in the

model

calculations

free

testing Efforts

terms

should

to

retained

exist

higherfind

a

in

an

scheme that

continue.

case of reacting which

of

differencing

diffusion

of the

analyzing of

turbulence

flows

between

This

is

because the

of

chemical

processes.

solution

important

of

expansion)

in the

mechanical

the flows.

(order

non-linearities

6. The correct

early

demonstrating

restricted

order series

for

closure

flows.

recirculating

higher

can eleminate

swirling

has

in

higher-order

potential

in clearly errors

order

equivalent

greatest

of

the the

potential

chemical

reaction

variations

heat

turbulent

flames.

224

region

release

and obtaining of

diffusion

core

is

effects the

very at

correct

velocity/scalar

7.

An

intensive

especially

for

four-step

the

with

PDF

data

modelling

on

correlations

are

assumptions.

In

fuels fraction modelling

are

are

closure.

emphasis

needed

to

diffusion

to and

more assess

the

flames.

225

in

a

lack

or

idealized

application

of Many

constant-density, experiments

fluxes

experiments the

its

is

further

support

scheme

combustion.

scalar

addition,

relation

turbine

Therefore,

and

continued.

there

to

two-step

kinetic

be

similar

turbulent

required

of

gas

efforts,

model,

instances

to

assumptions

development

two-step

should

many

relevant

Reynods-stress more

and

approach

in

and

stress/flux

schemes,

Unfortunately,

quality

validation

algebraic

kinetic

conjunction

8.

submodel

to

and

with density

improve with

these different

density-mixture to

turbulence

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On incompressible, co-flowing stream.

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,

First

Simplified fuels

Velocity confined

391-416. Course

in

reaction flames.

in

APPENDIX A TURBULENT

The sections

FLOW

transport reduce

EOUATIONS

equations in

FOR

THE

k-_

presented

axisymmetric

MODEL

in

coordinates

the

previous

(x,r)

to

the

following CONTINUITYEQUATION :

I[0 r

X -

0 _]

b-_ (r_U)

+ _-(rpV)

o

(AI)

MOMENTUHEQUATION :

a _F(pu)

+- l r

a [_-_-(r_UU)+

a _F( r_VU)

] =_

ax + (A2)

r

Y-

_xx(2r#T

_)

+

_[r/_T(_

+ _)

3

c_x

(_k)

MOMENTUM :

_(a pV )

+

_Ir [a_._ ( rpUV

)

+

_--_-( a rpVV)]

=

(3r (gP

2/_T

V

-

r 2d"

P

W2 --

r

+

(A3)

-r

r_T(_-'x

+

_rr )]

+

_-r-(2r_T

235

_-r)

@

-

MOMENTUM :

o- _[_ _-(pW)

+

-r

(rpUW)

+

o r_V_];-_ -_

_-_(

r 2

P

r

(A4) k

-

TRANSPORTEQUATION:

a I [a _(r_Uk) _-T(pk) +- r

+ _(r_Vk) o ] : -pP_

_

+ (AS)

1 r

r(#a--_, k

L

-

+

_u)_--_-]+

_-_[r(_t Ok

+

_)_r]

TRANSPORTEQUATION:

O _-_-(p_)

[

+- 1r

O _-_(rpU

_)

+

8 rpV_) _7(

]- _ =

pCcl

Pk

2

-C

t2 p

•

+ (AS)

-

r

_ ) _-_]

+

t

t

TURBULENCE MODEL :

_

k2 -

_ut=

C

p

ak =

3 _

C_/Cs

(AT)

(A8)

2 O

=

c

(A9)

K

(c c2

-

Ccl)4C_

236

where

_T = _t+_

=

[

8V

2

V 2]

237

_)U

aV

2

aW

2

_

1

APPENDIX TURBULENT

The

the

EOUAT1ONS

governing

equations

in

cylindrical

assumption to

FLOW

the

B FOR

under

THE

the

coordinate

ASM

MODEL

gradient system

diffusion are

reduced

following

CONTINUITY EQUATION :

aT

_

_(r_U)

+ _(rpV)

: 0

(BI)

X-MOMENTUMEQUATION :

o

_-(

-

r

aP aT

+

_-_(rpUU)

+

o

] I[o ou.o.ou]

_-{(rpUV)

-

r

_-x-(r_le

_-x)+_'{r(r_4e

_-{)

S u

=

(B2)

where

r

_'x'(r_le

_'_'x )

_-r"r(r#4e

_-x )

-

_'_'x(3 pk)

-_-x'(3_ll

_x

) m

Y-MOMENTUMEQUATION:

_-(pV

OP ar where

)+

+

r1

_-{r (r#vv)

Sv

[a_-x(

r_4e

_-x) av

+

_--{r( r_2e a

O-F av] )

(B3)

238

=

svi_ =

r

o_ o

(r#4e

_-r)

la r

+

ov]1-2 v

_-r(r#2e

2

_-rr)

+

_pW

-

2#3 e

r2

au__, 2 .__ au__

c_r(3r#22

ax

) +

3

r

ax

m

m

@-MOMENTUM EQUATION :

o_w_+1[oo r_VW_] i[o ow_,o ,_w] _-t(

r

_(rpUW)+

_-_(

-

r

_(r#6e

a_

_(r#Use

_-r)

=

I

_5e

a Wr ) _-_(

i r

a c_r(_5e

-

W)

-P r VW

(B4)

k-TRANSPORT EQUATION :

(rpUk)+_-{r(

c_"--t r

=Pk

-PS

-

a a-_(#4k

-

ak) ar

1 r

r

_x-(r(_+/_lk)_-x)+_-{(r(/_+#2k)_-r-)

a ar(r_4k

ak _-_)

(B5)

_:-TRANSPORTEQUATION:

o_ i[o o c_t

CelP

-r

+

k

_(rpU$

Pk

-

BU _

m +uv

)+_(

#

Ce2

ks 2

rpV_)

] i[o -

a ax(_4e

r

_(

as _--{)

a_ a r(_+iu le)_-_)+_-{(r(#+#2e)_-

I r

a Or ( r_4e

a_ {)

as _-x)

(B6)

where

Pk=(UU

aU _

-+uv

aV _

_ +vv

8V _ _-_ +vw

@W -_-_ -vw

W --r +uw

aW _

+

-ww

V _) (BT)

239

I

J

2

---

all=(_k

ou

-auu)/2(_-_-

1 u o;v +

_(_

2 a22=C3/'k-p_vv)/2(O_

OV

2 .a33=(_,Pk-P_)/2(

v + 1 u o_ r S(_- _

-

1 U Op 3(_ Ox

_ OU OV uv/(_-_. +_-)

_12=-.o

_23=-P

+

- -vw/(or ow -

(BS) (Bg)

v o; + - _'-F) )

(BIO) (Bll)

w) r

(B12)

_ OW uw/(_x- x)

_31=-P

o;

O--_x_- _-r ))

(B13)

#le=#ll+#

(B14)

#2e=a22+_

(B]S)

#3e=_33+#

(BIB)

#4e=#12+P

(B17)

#5e=#23+#

(B18)

#6e=#31+#

(BIg)

p

k

m

_ik=Cs

--z

DLI

#2k=Cs

- k p --

VV

#4k=-C

s

(_2o) (B21)

- k p -z

-ku--

uv

(B22)

J (B23)

_2c =Cep #4e=_Ce_

k -z

m vv -k£

(B24)

UV

(B25)

240

APPENDIX ALGEBRAIC-STRESS

The

Reynolds

algebraic

p

2

i j-"

3

CLOSURE

IN

stresses

C

AXISYMMETRIC

are

COORDINATES

obtained

by

the

(x,r)

following

expression

B_i3z"

-

•

C 1

_(u.u.-1

2

j

3

_ijk)

2

-a(P.1j.-

2--

36ijPk-

a__u)

3uiuj

ax,r

(Cl) 2 _6ijPk

-#(Dij-

Constants

A

and

¥amada's

and

for

cylindrical

2 --

P22

=-2(uv

P33=_2(u

_

Pl3=-(u

B

are

_U_, _x,,, )

equal

to

_kS

1

equilibrium

model

A=O

coordinates

system,

mean

strain

A -k

ij .-

for

by

the

is

-

u iuj

Rodi's

and

B=I

production rate

model.

assigining

are

Mellor

A=O In of

expressed

=0.

i:)

(Pk-

and

B=Pk/z

the the

Reynolds as

aU _

+

-uv

au _-.{)

av _

+

-v2

aV --c%r -

-vw

W r )

(C3)

aW + _-_

-vw

aW + _-{

-w2

rV )

(C4)

---aw- + (uv_

2

j

obtained

aV +_-_ _-_-

PI2=-(u-2-

P23=-

and

and

Pll=-2(u

uiu.

model

the

stresses

2 -

_w

aw _-_ +

av _-_-

r +vjvr -uv

aw _-_ +

(C2)

-uw

W -r +

--_v +uw_-_ v u-w r

241

-uv

aU _-_

--_v +vw_-{

+

-v2

au _r )

-w 2 w r )

au + _-Q _-x +

_u _-_ _)

(C5)

(C6)

(c7)

OU 2 _ + _

Dll=-2(u

OU -_ + v2

D22=-2(uv

D33=2(

_

Wr

_z

OV _ + _w

OW _)

OV _r --+

_Yr)

-vw

(C8)

OH

(C9)

(cio)

%7)r

--

OW

--

Dl 2 =- (u z _+ OU

-uvsy+OY

-D23=-fw2.

OW _ +

OV OU _-_ _ + _-_ _ + _

DIS=-(Q2.

_W _ +

_-Q

uw _

_V

c3U

+ _-Q

V2

Eft +

aV

-+ vw

5-ff

V -r V

_U + _-Q

OU Ox

Sll

+ uv

OW

b-if )

(Cll)

(C12) --

W

UV

--) r

(cz3)

(C14)

OV

(cls)

S22=_ V

(czs)

s33=r

szs=.s(b-

OU 7 +

OV _)

(C17)

aW

W r)

(C18)

S23=-5(Or OW

sl3=- 5 (_-ffx)

(C19)

Pll

(c2o)

=0 •

P22=2v _ -P33=-2vw -P12=uw

P23=(w-Z

(c21)

_W r W -r

(C22)

W -r

(C23)

-

(C24)

vz)Wr 242

D P33=-uv

Pk :-u

-2_U _'_

W -r _ aU -uv(_-_r

(c25) #V + _-_x)

-_2aV _-_

_ +vw

W -r -

_ uw

aW

_-_ -vw

aW

---W2

V -r

(C26)

243

APPENDIX ALGEBRAIC

The

FLUX

turbulent

algebraic

IN

fluxes

AXISYMMETRIC

are

COORDIANRES

obtained

by

the

(x,r)

following

expressions

£

-C

MODEL

D

u--_ __2ae _

--uv

0o _-{ +(C28-i)(u0

--

aU _

-ve

+

0U _-_-r ) -

ue A_-_(Pk-£)=0 (DI)

-Cle

£

--

_

ve

+w-'S " w

-C le

+A

+

£ k

-

A

--uw

ae

a--;-

v-2

_e

_-_ +

(c2e-1)(ue

--

ev _-_

+

-ve

ov _--{ -

2w--_w)

ae _

(D2)

--vw

_e _

-+(C2e-l)(ue

_W _-_- +

-ve

_W _-_ +

-we

V --r +v8

A

is

to

is 1

W

_)

(D3)

z ) =0

Constant equal

uv

ve _-_(Pk-£)=O

-we

w6 _-_(Pk-

--

equal for

to Launder

0

for and

244

Mellor

and

Gibson'model.

Yammada's

model;

and

SITZ

_n--_

_ ds3(I+V)

+ _)J

-

-

xe ]

.s

(t._) : _._nn aot

• sCssCa_s ao$

Z o_

aot 0

pue o_

ienb_

sr

(OI'g'Z Isnb¢

sT

([['g'g "b3)

Y

aeCqs "b3)

ICpom

_ue_suo_

uoT_udTssTp

" "H

+

" "_

[¢pom

uozsn_t_p 'suoz]enb¢

z(_x

uoT_nqTa_sTpoa

ao$

_e_,)

I

pue

uoT_Bnb

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