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UNOERSTANDING I_JRBULENT SCALAR MIXING (lnalytical Services and Materials)

55

Unclas

p H1/34

0117772

--

.

D

_

_

=

_..

--

_._z_

T 7

--

L

°

- __.

=

w_

=

NASA

Contractor

Towards

Services

Hampton,

Virginia

Langley under

Mixing

_z Materials,

Inc.

for Research Contract

Center NAS1-18599

National Aeronautics Space Administration

and

Office of Management Scientific and Technical Information Program 1992

Scalar

S. Girimaji

Analytical

Prepared

4446

Understanding

Turbulent

Sharath

Report

Towards

understanding

scalar

turbulent

mixing Sharath A. S. & M. Inc.,

S. Girimaji Hampton,

Virginia

23666

Abstract In an effort

towards

understanding

turbulent

scalar

mixing,

we

study the effect of molecular mixing, first in isolation and then accounting for the effects of the velocity field. The chief motivation for this approach stems from the strong resemblance of the scalar probability density function (pdf) obtained from the scalar field evolving from the heat conduction equation and that evolving in a turbulent velocity different

field. However, the evolution for the two cases. We attempt

of the scalar dissipation is to account for these differ-

ences, which are due to the velocity field, using a Lagrangian analysis. After establishing the usefulness of this approach, the heat-conduction simulations (ItCS), in lieu of the more

frame we use expen-

sive direct numerical simulations (DNS), to study many of the less understood aspects of turbulent mixing. Comparison between the HCS data and available models are made whenever possible. It is established that the/3 pdf characterizes, quite well, the evolution of the scalar pdf during mixing from all types of non-premixed initial conditions.

1

Introduction

Diffusion and

of smoke

air in an aircraft

scalars

engineers

many

fices.

and

applications,

a detailed

scalar

(Pope

Pope

(1988)

turbulence.

sive scalar DNS also

and

apart

from

provide

model

(Chen,

strongly

model.

represents simplified

understanding

by DNS

data

of the

models

is the

study

equation

1991b). mixing

often of the

rather

use

process

than

mixing

Despite

The

(Girimaji

has

been

turbulent1992)

to formulate

importance recent

closure

1991a,

observations

these

These mixing,

1991)

leading

first

of pas-

1989).

mapping

Gao

of

density

studies

(Givi

as the

of practical

remains

used

DNS

1991,

emerged

of DNS

us the

of turbulent

models.

Pope

for scalar

for problems

(Girimaji

technique

has

direct

other

of the

(DNS)

pdf in constant

understanding

1989,

(pdf)

gave

reactions

mixing

and

model

of turbulent

A popular

of Burger's

Kraichnan

been

suf-

reaction,

simulations

henceforth)

chemical our

and

an instance

combustion

processes

enhancing

The/3-pdf

closure

with

bed for evaluating

Chen

validated

mixing

bulent

a test

mixing

have

of mixing

function

of the scalar

there

process.

by chemical

numerical

to as EP

in which

Environmental-

statistics

density

direct

of fuel

this mixing

is accompanied

The

mixing

instances

flow field.

basic

probability

then,

and

of numerous

of the

(referred

Since

atmosphere

in understanding

of the evolution

mixing

data,

two

mixing

1985).

the

in a turbulent

of the

close look at the details isotropic

but

a knowledge

when

is required and

are

diffused

understanding

Eswaran

into

alike are interested

However,

scalars

chimneys

engine

are convected

ists and For

from

such

as tur-

long strides,

our

incomplete.

in the

past

in related

Navier-Stokes).

to understand simpler In the

turbulent

systems same

vein,

(e.g.,

use

we first

study the evolution of initially-random scalar fields evolving according to the heat conductionequationrather than the full scalarevolution equation. Then weaccountfor the effectof the velocity field using a Lagrangianframe analysiswheretheseeffectsarerelated to the material-elementdeformation characteristicsof the turbulent velocity field. 1.1

Motivation

We study tion

the

evolution

of a scalar

field

¢(x, t) subject

to the

heat

conduc-

equation,

0¢

0_¢

--Ot = D--Ox_Ox_' rather

than

the

full evolution

equation,

0¢

where

D is the

field.

There

better

our

whether pdf

(F(¢))

several

understanding

the

scalar

02¢ = D--,OxiOxi

of Fickian a priori

turbulent

equation

¢ is the

conditional

probability-space

scalar

dissipation

same.

X(¢)

In either

to (Pope

t) is the such

a study

irrespective

(1) or (2), the case,

velocity

the

pdf

to of

scalarevolves

1985)

0_ aCa¢ value

X(¢)

For example,

to equation

field according

u(x,

to perform

mixing.

is the

(2) and

motivations

according

0F(¢) at where

diffusion

of turbulent

field evolves

evolution

in an isotropic

0¢

0--[ + ul--Oxi coefficient

are

(1)

(3)

{F(_/')x(_b)}' of the

is given

= D( ._-[¢ oxi(]xi

scalar

concentration

and

the

by

= ¢).

(4)

The notation (bit) is usedto denote the conditional expectation of respect

to r.

it is clear for the

For the

from

equation

diffusion

without

vation

that

the

pdf's

are

similar

quite

responding

values

principal

any

stage

of scalar

scalar

study

of the

scalar

rescaled

pdf,

to account

be assumed

of the

the velocity

of equation

a tilde

wavenumber

denotes vector

given

value

heat

in the

study

to be unity

can

study

is that

the

scalar

length

initial

scalar

easily

one

adopted

by

the

is, at shape

scale

and

tur-

we find

that

the

is quite

similar

to

by EP.

we first

scalar

understood

that

of variance,

study,

field in turbulent

cor-

That

length

decay

at the

scale.

variance,

to our

of equal-

here

shapes

equation

obser-

of scalar of EP

field obtained

As a prelude

our

case

be recalled

value

turbulent

be most

stages

DNS

of scalar

from

(for the

by the

conduction

of the velocity

field

(1)

It should

of the

field.

role

the

of the initial

the

pdf

velocity

from

DNS

comes

at various

characterized

from

scale-independent

equation

variance.

of the

At any

obtained

of the

obtained

is independent

0q_(k, t) Ot -

where

for this

condition)

decay

statistics. pdf

version

can

initial

are independent

pdf

to understand role

evolution

diffusivity

from

findings

pdf

Role

the

t can be simply

calculated

of scalar

scalar

the

the

motivation

to those

the

scalar

time

and

a posteriori

non-premixed

bulence

(3) that

coefficient

quantity,

of the

of analyzing

loss of generality.

A strong

of the

purpose

b with

from

attempt

mixing. the

The

spectral

(2),

v/L-T/_

the

rniffi(k-

Fourier

of magnitude

m,t)¢(m,t)dm-

transform k. From

of the

k2¢(k,t),

quantity

this equation

and

(5)

k is the

we can glean

that,

when the length scale of of the

scalar

field

(Is),

other

hand,

when

the

length

scale

smaller

is larger,

scales

until

the

the

physical

are

>> 1, the

effect

of the

velocity

velocity

are

diffusion

unaffected.

is such

velocity

latter

case

(l_,/Is 0), the

¢rnax < 1 and

¢,,i,_

values

are set

outside

In Figure

the

The

agreement

range

16

(¢,,,,,,

DNS

5, the

is compared

(a 2 > 0.05).

data

and

model,

of mixing

the DNS

of mixing

the

stages

and,

stages

between

fraction.

is probably

both

stages.

of mass

is

well with

latter

range

trend

M1 agrees

at the

of HCS

as a ratio

a 1283 simulation

intermediate

fraction.

decays

this

pdf and

accurate

model

consid-

dissipation

of the scalar

The

pdf

conditional

ratio

fraction

scalar

data

(when

to zero

the

of mass

for

(a = 0.1)

dissipation

HCS

X(¢)/X(0.5)

the early

range

Taking

is good

stage

and

as the

errors. DNS

early

of mass

the entire

and

conditional

DNS

to be statistically

comparison

dissipation

M2 over

The

Gaussianity

during

range

the

that

a 643 simulation.

DNS

a narrow

ditional and

from

fact

increases

the observed

HCS

the

over

of variance.

statistical

for the intermediate

of mixing,

the

with

between

adequate

constant

is minimum

due

agreement

and

is nearly

fraught

data

is only

normalized

con-

with

the

models

models

agree

with

During

the latter

is not Cmax).

as good. In the

The

middle

M1 HCS

stages HCS of the

range,the agreementis adequate,whereasneaxthe edges,it is poor. This disagreementnear the edgesof the range (¢mi,,,Cm_,)could be due to statistical error in HCS data. Nearthe edgesthe probability of _ is solow that at thesevaluesof massfraction there are very few samplesof scalar dissipation from which to computethe conditional dissipation. This statistical error makesit difficult to completely comprehendthe evolution of conditional dissipation at the edges.However,it can be surmisedfrom the earlier findings about Cm;,,(t)and ¢,_(t)

that the zero valuesof the conditional

dissipation migrate inwards (from 0 and 1) with increasedmixing. To circumvent the problem of too few statistical samplesat the edges of the scalar range, in Figure 6, we plot the fractional scalar dissipation, F{_)x(¢} £1

of the

HCS

data

is that

when

the statistical

dissipation

is weighted

'

quantity conditional Here,

the

agreement

between

is excellent.

The

is again

to the

scalar

due field.

yielding Apart dissipation than

the

the

scalar

jagged

With

smooth from

time,

along

suggests

that

nature

M1.

error

HCS of the

high

this

is high due the low probability,

the

advantage

by the

data

of very

the

of plotting

down

The

and

HCS-data

high

probability the

fl-pdf-based plot

at the

wavenumber

wavenumber

density

itself.

model earliest

components

components

M1 time

in the

die out

quickly,

the fractional

scalar

statistics. the

conditional

appears

model

the

presence

mathematical

is a physically

pdf

and

our

important

dissipation

(equation with

convenience

the efforts

3).

should

quantity.

alone, In fact,

probability

that the

density be

it offers,

directed

17

It is this determines

conditional function towards

quantity, the

rather

evolution

dissipation (equation modeling

of

always 27).

This

fractional

scalardissipation rather than conditional scalardissipation. Moreover,the valueof conditional dissipation outside the range (¢mi,_,Cm_,)is irrelevant. Our inferencesfrom Figures4 - 6 are 1. The conditional dissipation ratio, the

HCS,

2. The

and

normalized

M1 and ing.

model

M2 agree

too

large

is difficult always based

HCS

the statistical

to make

a valid

form

(as

¢ = 0.5 ¢,_,

both

that

the

to thin

down

(as suggested tend latter

fractional

with

is the

error

at the edges

the

more

dissipation

the early

For

normalized by

at long times

probable

scenario.

_

stages

of mix-

same

and

reason,

it

dissipation

Jiang

to a spike since

(Figure

of the

range

mapping-closure-

However,

mean

ratio,

this

asymptoting 1992).

DNS,

models

of the scalar

the

of O'Brien

time,

the

of the

conditional

suggested

by Girimaji

towards

X(¢)/X(0.5), during

model

from

well.

data

the

conditional-dissipation

if it tends

adequately

comparison.

whether

a constant

calculated

dissipation,

well with

to evaluate

has

M1 agree

conditional

At long times,

are

3. The

the

X(¢)/es,

HCS

1991),

or

shape

at

Cmi,, and

3), it appears

and

the

model

_$

M1 show the

2.3

excellent

scalar

agreement.

evolution,

Evolution

the

of scalar

Since

agreement

it is this quantity

that

determines

is of significance.

variance

and

mean

scalar

mean

scalar

dissipation

scales

are

dis-

sipation The

evolution

obtained

from

7. To precisely

of scalar HCS

variance

for various

understand

the

(a _) and initial

the

effect

length of the

18

velocity

field

plotted on the

(es)

in Figure evolution

of these

quantities,

of Eswaran between

and

the

HCS-es

we compare Pope

HCS

(1988,

and

decreases

a non-monotonic

scalar

dissipation

DNS

increases gradually

monotonic

behavior

is more

of HCS The

variance

above

discussed accounted

2.4

ities

of

scalar

gradient the

ks.

For all

However,

peaking,

then

more

for smaller DNS

mean

the mean

scalar

ks. The rate

DNS-es

DNS

rapidly.

decay

ks, the

the

For ks _< 4, the

After

the

biggest

of es.

as expected.

and

is that

Section,

scalar

This

non-

implication

of

is higher

than

is entirely

where

the

due

effects

to the of the

velocity

field,

velocity

is

field

are

initial

gradient

according

evolves

conditions

in the

according of these

scalar

functions.

to the

field,

In Figure

For

k8 = 16. times.

The

both

cases,

the

initial

8, the

Pope

(1988)

different.

evolution

approach

pdf

shown)

Eswaran

find that

normally

distributed.

This

log-normality

is discussed

in detail

in the

next

19

the

to discontinu-

field

is composed

flatness-factor

is, however, scalar

and

for ks = 8

Gaussian

is a consequence Section.

scalar

However,

is shown

moments

Gaussian.

Due

of the 0¢/0xl

(not

the

equation.

scalar-gradient

component the

equation,

heat-conduction

are quite

scalar-dissipation and

heat-conduction

to the fields

and

and

evolution

for small

which

of scalar-gradient

field

difference

pronounced

superskewness

long

12).

ks.

next

evolves

also

initial

of delta

11 and

The

data

for.

Pdf

If the

DNS

is in the

at first

phenomenon,

in the

the corresponding

in time,

evolution

for small

7 with

in the beginning.

decays,

that

data

evolution

dissipation

on the

Figures

monotonically

shows

this

Figure

dissipation of the

values far

at

from is log-

velocity

3

Accounting

In this Section, field

we attempt

on scalar

effect

effect

grangian field the

mixing.

simplifying

The

for the

assumptions

frame

Sections.

velocity

Let

the

(x,

velocity

coordinates

from

field t).

of the turbulent

previous

easily

Consider

U(X,

system

field

Section

velocity

are invoked

to

analysis.

as X, rather

coordinate

the

is most

field

of velocity

for the effect

in our

of reference

in a turbulent Eulerian

to account

Findings

of the

effects

the

t). than

accounted evolution

In this

Section,

x, as was

done

evolve

according

U[X(x,

t), t],

for in the of the

La-

scalar

we designate in the

previous

;_o

0X(x, t) cOt with

the

initial

-

(31)

condition,

x = x(o). The

velocity

Now,

field U(X,t)

let the

Eulerian

Lagrangian

system

coordinate according

evolves

system

according

coordinate

to the

system

X is nonstationary, following

(32)

curvilinear,

fluid particles,

the

Navier-Stokes

x be and

Cartesian. nonorthogonal.

scalar

concentration

equation. Then,

the In the

evolves

to cO¢(x, t) cOt -D

where

D is scalar

terms

of the

diffusivity.

Lagrangian

The

scalar

0¢(x,t) COt

cO2¢(X, t) cOXicOX_ '

molecular

diffusion

(33) can

be rewritten

in

derivatives:

-

COx_ coxb co2¢(x,t) D-_i cOX_ cOz_cOxb "

2O

(34)

The evolution equation of the pdf, ¢(x,

t) following

standard

methods

OG(¢,,t) Ot (In the (3)).

absence

the

(Pope

to the

Eulerian

It

the

time

value

findings

of scalar

that,

of the

can

only

the

concentration

be derived

using

the

velocity

field

pdf

modified

of the

concentration

shapes

velocity

pdf

to simplify has

during

a major the

(35)

to equation

scalar

Section

the

]¢ = ¢,)].

reduces

Lagrangian

previous

slightly

effect

equation

of the

while

averaged

scalar

field,

evolution,

for ks > 2 and the

the above

t) are identical.

F(¢,

scale

that

field,

pdf

found

field

scalar

Oxo Oxb O¢(x,t)O¢(x,t) Oxa Ozb

of the

was

unaltered implies

in an isotropic

isotropy

the

t), of the

1985):

of the velocity

Now we invoke (35).

particles

__r _m =-Do_p,O_,,tG_¢t't)(oXiOXi

Due

and

fluid

G(¢l,

Gt(¢t,

t)

equation effect

on

evolution

are

for ks _< 2. This,

we argue,

field

upon

is independent

conditioned

of the

scalar

value.

the That

is, we suggest

(0_o ozb O¢(x,t)O¢(x,t) ozo Oxb_O¢(x,t)O¢(x,t) _, ax, azo a_b I¢ = ¢,) _ (ax, ax," a_o axb I¢ = ¢,), (36) where fetched function

/°-_ °--_\ \ OXi

OXi

I

is independent

assumption,

since

of the initial

Subject

to the

OG(¢t,t) _ at _-D_ax,

the

scalar

above

of the value

scalar

of the

field. scalar

This

concentration

field, which is independent

assumption,

equation

is not

(35)

such

a far-

is a strong

of the velocity can be simplified

field. as

_____ Oxb 02 _Ozo 0¢(x, t) 0¢(x, t) I¢= ¢,)]. ox,'aCta¢,[a(f"t'( ax_ a_b (37)

21

Owing to the isotropy of the turbulencefield and the initial scalarfield, we can further simplify as follows: Ox_ Oxb _ =

(OX_Oxi'

(3s)

(-b-G_axe) - _obis).

Then,

Oa( ¢. t) Ot where

Xt(@)

is the

05 _ -(S>

Lagrangian

o_toc

conditional

= Now we suggest G(¢t,t) the

(and,

hence,

Lagrangian

field.

that

the

on

Rescaling

time

effect

pendent

seen

of the

scalar

of the

conduction

heat

be decoupled.

but

less so otherwise.

dissipation

dissipation

defined

as

[¢ = ¢')"

(40)

velocity

field on the

through

the

dissipation

scalar

evolution {S),

is independent

and

of the

of that

velocity

Z'

The

dissipation,

of a scalar

effects

these

coordinate

field evolving from

the

of molecular

simplifications

es.

time

T is inde-

velocity-field-independent

(1) alone,

the

(41)

in the

of a scalar

equation

Clearly,

dt,

evolution

field.

Thus,

can

scalar

pdf

dissipation

as the full problem.

Mean

the

velocity

scalar

(39)

as

that

conditional

of the is only

r = it is easily

scalar

Ox:

F(¢,t))

conditional

[G(¢t,t)xdCt)],

In isotropic

field can be approximated

Zt(¢l) under

same action

as (from

influence

initial and

will be accurate

turbulence,

the

the

is the

conditions

velocity

field

for ks > 2,

conditional

equations

36 and

3S) 0¢ 0¢ X(¢)

= (0---X, _-,

1¢ = ¢> = M 10

-4

I

10-5

l

I

0.04

O.O0

O.OB Time

102

101

.2 c

1

O°

t

_4

121

z_

0

4 M

D I

0 t_

A

I(1-I

a 4. 4

_ 44

8 io -2

+

o tO L/3

N N 10-3

10

4 0.00

I

I

I

0.04

I

0.08 Time

Figure 7i EvoluLion of IICS scalar variance and mean scalar dissipation tile case of # = 0.5. (Symbols same as in Figure 2.)

/44

for

18 16 &

&

&

&

6

&

_14

0 0

A

c 3

,0

_12 I

o

0

L

&

0

glO

0 O 0

_

8

O O _

A A

I _

6

5 o

4-

2

ooo o B 0

0_ 0.00

Elo o

rl

°°_°_°8

I

I

I

0.02

o o o

I

I

0.04

0.06

lime

Figure gradient

8:

Ev(,lution Odp/Oxl

Supcrskewncss:

of IlC, S flatness-factor (/t

--

0.5).

(Flatness

/x, _:, _- 16; 0 k, = 8.)

45

and factor:

super-skewness Q) k,

=

16;

of scalaro

k,

=

8.

5O

6

a 2 = 0.134

40

a 2 = 0.058

5 4

3e 3 20 2 '- \k%,. 10

1

0 ""--' 0.0

1

J

I,_==_

O.4

0 0.0

O.8

i 0.4

0.8

6

a 2 = 0.0 56

II=

5

:-_

4

C

c_3 "-

7_

CI U

ft.

0 0.0

I

0.4 Moss

0.8

Froclion

_,

#=0.,3

10

2O

8

16-

6

12-

4

8-

2

a 7 = 0.0005

4i

' 0.0

Figure represents

).4

of scalar data,

and

,)

0

0.8

9: Evolution ItCS

'

0.0

pdf, solid

F(¢), line

for the

represents/3-pdf

46

case

I

I

0.4

of in = 0.3. model

....

0.8

Dashed

calculations.

line

7 6 ,_

q,,,

/ ,'El

o

"4

.I/D

.6

q) to

2

1

I

0

I Illlll

I

I

]JHulll

1,

10-,3

10-4

0

Figure

I0:

Scalar

premixed

initial

0.20: (_(hezagon), model. /t = 0.40:

flatness-factor

I Ililll

I

(y

I

I Illl

10-I

10 0

2

vs.

conditions. O,

I

10-2

scalar

= 0.15:

]ICS; .... /3-pdf llC, S; - - -/3-pdf

variance

1:3, HCS; model. model.)

for

....

non-'equal

/3-pdf

/z = 0.30:

O,

model,

IICS;

nony =

--/3-pdf

60

f%

5O

t t l

O_

l (/1

4o

/

I:1

l

to C

El

! /

a,.

I

a) I

30

1

I:i

I /O

otrl

/"0

\

7

2O o ._.uo_d."--6'6-"

-

10

o

0

oO

o_

c_

"

0 10-4

Figure premixed

11:

Scalar initial

I I llllll

I

I

I lllll_

I0-3

vs.

(Symbols

scalar same

47

O"

(>... 0"_

I I IIIIII

10-2 (7 2

super-skewness conditions.

\ " "I_I

I llll

10-1

variance as Figure

10 0

for 10.)

non-equal

non-

2.0

_t

= 0.0056

5

1.6

__

4i 3

-._0.8

2

LL

1

0.4

o-:_ = 0.058

0.0

, 0.0 Moss

i O4

\"_"'. ,

Fraction

t

0 0,0

_z___ 0.8

0.4

I

0.8

_'J, /1.=0.5

12

20

a 2 = 0.0005

10

lo[

8 12 6 84 4-

2 O

0,0

Figure

12: Evolution model

of fractional

Dashed

line

') ).4

0.0

0.8

case of It = 0.3. fl-pdf

o

I

scalar

represents

dissipation, ItCS

calculations.

48

data

0.8

F(¢)X(¢)/e,, and

solid

line

for the represents

4.0 3.5 .3.0 2.5 t_

,,_ 2.0 1.5

1.0

0.5

0.0 0.0

0.2

0.4 Mass

Figure of/t

13: = 0.3.

Evolution HCS

of conditional

I 0.8

0.6

Froction0

scalar

data..

49

1.0

/_=0.3

dissipation,

X(4,)/¢,-,

for

the

case

12

12 10

a 2 = 0.201

8 6 4 2

o 0

0 0.0

0.4

0.8

0

0.0

4,0

0.4

0

0.8

3.0

..---,.

%3.5 2.5

I.L

_ 3.o

a_

= 0.130

_2

= 0.079

2.0

m 2.5 C o

2.0

-_

1.5 -

1.5 1.0 0.5

Q.

0.0 0.0

0.4 Moss

graclion

0.0 0.0

0.8 _,

tz=0.5

I

,

I

I

I

0,4 Mass

0.8

Fraction

1/_, #--0.5

0.4

0.8

3.0 2,5

7

_2 = 0.043

6 2.0 5 1.5

4

1.0

3 2

0.5 !. 0.0 0.0

Figure Case sents

14:

0,4

Evolution

0 0.0

0.8

of scalar

2 of subsection 4.2. Circles fl-pdf model calculations.

pdf,

F(¢,),

represent

5O

for partially-mixed tICS

data

and

initial solid

field,

line repre-

¢

4.Oh.

3.5

3.0

o,,o'b _.,

.. 0

2.5

_

_

U

I

vJ 2.0 In

¢1

_

1.5

1.0

0.5

0.0

|

I I IIIIIL__.]._t..I.L[|jI__I

lO-S

10-4

I IIllII

10-3

10-2

I

I I llIll

10-1

0 o

0 2 Figure

15:

initial 4.2,

flatness-factor

conditions. The

to cases the

Scalar

(O aud

solid

line

1 and

2.

corresponding

vs.

scalar

A represent

represents

tile

Q) ropresents mo(lel

HC$ /3-pdf

HCS

variance data

model

data

for

of Cases

partially-mixed

i and

calculations

of Case

2 of Section

corresponding

3, and

the

dashed

calculations.)

18 16

0

o_

0

0

0

14 o_. (/} r_ 0) C

12 10

,e, u'l

[

8 6 4 2 0

I

O-S

I I IIllll

I

I I IIIIII

10-4

I

initial

16:

Scalar

conditions.

super-skewness (Symbols

same

I

I I IIIiIl

10-2 o"

Figure

LJLIIIIII

10-3

vs.

I IIII

0 0

2

scalar

as Figure

51

I

10-1

variance 15.)

for

partially-mixed

line

REPORT P_k: r_ gathering

_

co_

04 kd'onna_on,

b_datl maJnlakl_

_lt.

BrAe

1204.

DOCUMENTATION

0ot 111_ _ ol _otrt_ltk_l ks 4mtJR_AId to athlK_ the d4Ma ne4l_. _nd con_olotir_ and t4wl_ng the kclud_;

Atilt.

suggeello_4 VA

_

2_2¢_-4302.

1. AGENCY USE ONLY (Leave b/ank,)

mduck_g

Ih,k; burden,

and to Ihe Offloe

TITLE

AND

| h_Jr _

to Washington

o( M_ma_

Form_prowd OMB No. 0704-0188

_ ml_OttN. |t_ud_ 0 11_ tkl_ 4_ inlor_M, Iovl. Send _

14eadq,.m,'tecs

_tcee.

and Budgeq. Papenvoqk

2. I:IEPORT DATE

July 4.

PAGE

1992

SUBTITLE

Dkeclorale Redu_lon

Project

kit rWvl41wk_ blltt_tk)414, s_chk_ r_N(b,t_ regatdk_g Ibis burdlk_ eltlmale 06' &ny othot for Inlormaflon

Opera=k)_;

(0704-01U).

Washln_on.

and R4potts. OC

1215

_ I_

soufc4l. o4 this

Je_wson

Davlg

20503.

3. REPORT TYPE AND DATES COVERED

Contractor Report 5.

FUNDING

NUMBERS

Towards Understanding Turbulent Scalar Mixing C NAS 1-18599 e. AUTHOR(S) Nll 505-62-40-06

Sharalh S. Girimaji

7. PERFORM=NO OROAN_.XTION NAME(S) AND,U>DRESS(ES) Analytical Services and Materials, Inc. 107 Research Drive Hampton, VA 23666

8. PERFORMING ORGANIZATION REPORT NUMBER

9. SPONSORING I MONITOR=NO AGENCy NAME(B) ANDADDRESS(ES) • National Aeronautics and Space Administration Langley Research Center Hampton, VA 23665-5225

10,

SPONSORING AGENCY

/ MONITORING

REPORT

NUMBER

NASA CR-4446

11. SUPPLEMENTARY NOTES Technical Monitor: J. Philip Drummond

12,. DIB'rRIBU'nON; AV,UL*BlUTY STATEMENT

12b. DISTRIBUTION CODE

Unclassified- Unlimited Subject Category 34

13. ABSTRACT (M_xlmum3OOw_)

In an effort towards understanding turbulent scalar mixing, we study the effect of molecular mixing, first in isolation and then accounting for the effects of the velocity lieU. The chief motivation for this approach stems from the strong resemblance of the scalar probability density function (pdf) obtained lrom the scalar field evolving from the heat conduction equation and that evolving in a turbulent velocity field. However, the evolution of the scalar dissipation is different for the two cases. We attempt to account for these differences, which are due to the veioctiy field, using a Lagrangian frame analysis. Afler establishing the usefulness of this approach, we use the heat-conduction simulations (HCS), in lieu of the more expensive direct numerical simulations (DNS), to study many of the less understood aspects of turbulent mixing. Comparison between the HCS data and available models are made whenever possible. It is established thai the Beta pdf characterizes, quite well, the evolution of the scalar pdf during mixing from all types of non-premixed initial conditions.

14. SUBJECT TERMS

15. NUMBER OF PAGES

turbulent-mixing models stochastic mixing

52 16. PRICE CODE

A04 17. SECURITY CLASSIFICATION OF REPORT

Unclassified NSN

7540-01-280-5500

18. SECURITY CLASSIFICATION OF THIS PAGE

lB. SECURITY CLASSIFICATION OF ABSTRACT

20. UMITATION OF ABSTRACT

Unclassified

Unlimited Standard F_rm 298 (Rev. 249) Pre_cdbed L_J- IG2

by A_NSI S_.

Z3g-18

NASA-Langley,

1992