Computational Modeling of Turbulent Transport

(1)

(2)

(3)

-:

B

!!3

U

OTH~;::

:

(

\VJterloopkunciiz Laboratorium Postbus 177 - DELFT ( by

anà

.John L. Luml ey

Bejan Khajeh-~ouri

The Pcrmsvlvan ia :::tate Unive r-sit.y Uni.versi t y Park, PennsyIvan.i a 168C:~.

---'-f-l-\'.

A rational c losur e techn ique is prcsented for the fi.r st allel second moment-cquations in a strati f i cd, cO!1taminated turbu Lerrt fIow, Fe,),}i);.'ing the applicat Ion of high Reynol dsyPec l et number appr oxi.rnat i.on s, Tel112,ilï:Ï.ag thil.'d mcmerits are expanded about the isotropie, homogeneons state. The . stratified, unconta!lünated case reduces t.o seventcen equat.ions in seven- .

teen unkriowns , Ot.her aut hors have suggestcd some of the ternrs geneyated, but SOJ:lehave been us ing the wrong terms, or the right tcrms fo r the wrong

reasons.

1be approximation

is

kinctic

-

th

e

orctic

(turbul

e

nc

e

!me

a

n

motion scales aSSUJ11CU sma l l , and turbu.lcn ce ne ar l y in equilibrium) .and resul~s in

a rel

a

xation

time,

and

in

generaliz

e

d

gradicnt.

trall

Spo

Tt

fürms;

. 0

ho~cvcr,

gr

aJi

cn

t

s o

f

ane

qu

c

ntity

can

pr

o

duc

e

fluxcs of

anothcr.

The model r e lat cs thc timc sca l e for return to Isorr opy Lo the Lagrang ian iilte'gral Urne sca lo (r cduc ing to ~:-tl:(:ory in <! homo gcnc ous par al l cl flow

(4)

-2-w

it

h

or

t

h

ogon

a

l

t

Cr.lpcraturc

gradicnt}.

S

O):!c

coeffic

i

e

nt

s ar

c

est

iril

a

t

C'd

,

an

d p

r

c

l

iminar

y co

mputations arc prcscnt

c

d

o

f thc uns

t

ratif

i

ed

2-D

t

u

r

b

u-l

e

nt

wak

e ;

onIy component

cn

ergi0s

n

ea

r th

e

c

nt

e

rlin

e

a

re

not

we

ll

r'cp

roduc

e

d

,

p

roba

b

Iy due

to thc

oini

ss

ion of

a

term w i

th

which t

emporary

comput

a

t

i

o

na l difficult

ics

we

re b

eing

cxper.i

en

c

d.

St

rat

if ied, corrt

a

min

a

t

e

d

3-D calcul

ati

ons app

e

a

l'

to

be practical.

e

II1bLodLLc-üoH

Inexpensive

semi-quantitative numerical

simulation

of

turbulent

transport

wout

d

have

many applications.

In

part.i cu

lar,

....

·hen

applied to

pollution dispersal in an urban atmospheric environment,

or in an estuary,

it

w6uld permit rational decision

making

by the

govcrnment bodies

involvcd.

There are,

in addition,

problems in oceanography

and meteorology

(such

as t}lermocline

form~tion) en which a reasonably realistic,

th6ugh

semi-empirical,

computational

technique could shed light.

Bc

fore attempting to construct such

a

method,

we

should

consider

two basic,

relatcd questions:

is the

method "'ithin our grasp concepttially

and

comput

ationa

tjyt

That is, do

KC

undcrstand turbulencc

,

...

e

lI enough

to

model it

w

i

th

accept abr

e

accur

a

cy , an

d

can

thc model provide

simulation

at

a

ccept

ab lo

co

s

t,

w

it

h .&ailablc

co

mp

utational

facili ties?

These are

ver}'

r'e

al qucstions

fo

r

,

although

we undc

r

s

t

anrl

a

great dcal

about

(5)

-3

-that- èo not e xcccd thc c apa c ity of the most .rcccnt generat i.on of computers

can e asiIy excccd thc ability (o r wil lingncss) of rca sonabl c men to pa)',

if corrput cr tine must be accounted for.

A groat deal of SllCCCSS has been had \'Iith direct numeric al simulation

of turbulcnce (Ors:ag and Patterson, 1972). This inv61ves rio dynamical

modeling. ' However , -Fox and Li lly (1972) have ShOl-.'D that su ch an approach rapidly receeds beyond economie reaçh as the Reynoids number increases. The next nast successful approach is, direct simulation with so-called sub

-grid scal e modeling (Deardorf f, 1973, for examp Lc) . Th is recogniZes that

it is economically impossible to carry in the computat ion t.he smallest scales at high Reyno l ds number s; the grid s ca le is made as small as is economically feasihle, anel the motion on scales below thc grid scalc is modeled, mak ing uso of the welLcsuppo rted propcr ty of turbu Lence dynami cs

(Tenn~kes

&

Lurnley, 1972) that the prccise nature of thc dissipative mechanisn does not influcnte the large scales of the motion, if the Reynol ds nurnbcr is high cnough.

Thc success of this mcthod depends not only on having a sufficiently

high Reyno l ds number to have an i.nertial subrange (sa that ene rgy-containing and dis sipative sc alcs are, to a first approximation,

dynami calJy related onIy by the va l ue -of thc spcct ral energy flux) but, also on

bcing ablc

to use a grid scalc that is small cnough to lie in thc i.nc rtia l subrangc . It suf fcr s froin thc d-is advantago, that onc

(6)

-4-cnscrsbl c, and a suf ficicnt number of independent runs must be made to obtain stabIe statistics; for exac?le, of the order of 200 runs to obtain lOO~accur acy in second order quantities.

10

date, this techniquc has been uscd primari Iy f.or flo·...s having a homogeneous direction ; in such

f'Jows ,' the numbcr of runs ncce s s ary to obtain stablc stat istics can be subs tant ia lly r cduced by spa cial avcraging in thc homogeneous direct ion .

If one is attempting to model the flow in 6 full)' three-dimcnsional ~ regi on s uch as an urb an environmen

t,

two f acts rap id ly bocome cl ear :

the number of points required to make even a crude model of the region precludes the use of a grid scale lying in the inertia1 subrange, and a single calculation is so expens ive as to preclude the possibi1ity of doing stati5tics on an ensemble cf them (since the1'e is na homogeneous direction for averaging). We must then use a grid scale lying.in the ener gy containing range, and comput e on Iy the statistica1 properties of

the turbulence. The so-callcd sUb-grid scale motions nO\\1 contain v ir tua I ly

all thc turbu lcnce , and the dynamical modeling bccomes much more critical.

e

In fact, since the entire influcncc of the turbulencc is boi ng computed through the moments, it is no longer correct to think of turbulence

quant ities as being sub-grid sca l e ; fo r exampl e, the scalc of the turbu-Lcncc is not now rclated to the grid s cale, but must be obtaincd from

dyn3~ical considcrations

.

~o good direct model of second order turbulence quantities exists. Thc oniy practical n.odcI is t.hc so-ca l lcd cddy di ffus ivity , or "K-j heoz-y"

(7)

-5

-model, which has been used with s on-- succc ss in sirnplc situa tior.s

(Tcnnck cs G Lumlcy, 19ï2) to prcdict f irst order quantitics. This model is known to fail, however, in si tuat ions which are rapidly chang ing in space or tinc, r\ number of autho rs , roa liz ing that a good model js dèsir-able but that good second-ordcr models arc not availabl e, have decidcd to carry the equations for second momcnts exactly. and model third order terms (Donaldson, 1972; ~~llor, 1973; Daly and Har10K, 1970; Jones and Launder, 1972; Ng and_Spalding, 1972). There is some justifi -ca~ion for this approach (~hich wc wil I follow); as wc shall show 1ater.

i

whiI e it is not poss i.ble to construct a rational model at s econd order, it is at third order. HOKcver, the model constructed still rests on a fa lIacy : kinetic theory con ccpt s are embodied, .i.mply irig tha t length and time s cales of the transpor ting mechanism (the t.urbul cncc) are small

relativc to length and time s cal es of the I:1C2.n moticn. This is, of course,

kno~n not to be the case for turbulcnce. There is thus an article of faith involved: if a crude assumption for second moments prediets first moments adequat eIy, pe rhaps a crude assumption for third moments w il l . predict second moments adequately.

Somcof these third order c losur e s chcmes are Lncomplete in the sense that thcy provide no epTediction for one of thc scales (which ma)' be tak cn to be equi valcnt to a lcn gt.h scalc) (DonaIds on , 1972; ~lellor,

1973). Othcrs (Daly and Ilarl ow, 1970; ;\g and Spa Id'i ng, 1972; Joncs and Laundcr , ]~)ï~) do provide :.l. supplc.acnt ary cquation equivalent to onc for

(8)

I

f

I

!

I·

t

I

f ~ I -6 -a Icngt.h s caï c, All of thcm, howcver, suffer from a basic flaw: thcy do not present any mcthod for generating thc models used f'or the third

order terms. Since thc rnodel s are constructed on an

ad-ho

c

:

basis, usually bcing required to have only the samc general. tensor character as the terDS modelcd, models arc occasiorially constrûctcd that behave

incorrect 1)' i';ith ReynoLds number (Corrsin, 1972); or as we shal l s ee, the right term is included for the wrong reason, or important terms are omitted.

1\'e will present here two related techniques whi ch make it poss i.bLe to

I,

-gcnez-at e, rrra 'consist cnt "and straight-fon:ard manner , models of all orders of the third momcnts, and of all order in Reynolds number. The technique is equalIy applic.abl e to str'at.Lf ication, to po l Iut ion dispersal, to chemical reactions, etc. Nany of the terms generated are e s s entia ljy those suggcstcd by ot.lie r authors on an ad-hoc basis. However , in the

case of the third-order transport terms, wewiLl find that i t is i ncon-sistent w ithin the model not to a Llow the flux of one second order quant i ty

to be produced by gradients of

ali.

the othcr s, much as a mol ecu lar flux

e

of s a lt can be produced

in

a liquid by a tcmperaturc gradient, and vice versa. This opens the possibil ity of up-gradient diffusion, an important process in atrnospheriê modeling. Unlike the situation in kinetic theory,

~herc the cross-diffusion coefficients are ordinarily smal!, the t urbu-Jcnt cross-diffusion coefficicnts mal' be substant iaI. Howcvcr , in the

(artificial) situation of constant cddy viscosity and constant structure,

(9)

-7-,

It is possible that ~e will co~cludc ulti~2tcly that these third

order c Io s u rc s , though within Dur r ca ch eo~putationally, a r-e , for s omc

purposes, inadequate models of thc sccond moments (although prclir.ünar)'

results, as

wc

shall sec later, appeal' quite favorable). It should not

be nccessary to point out, however, that we cannot reach a 'rational con-c l usion on this question unless He are sure that the cIosu re us ed is

e

and that all terms, arid onIy thosc tcrms, gcrier-ated by these principles

are us ed. "Otherwise, I'Je wiI'I riot know if (in unsatisfactory r-csu lt can be I

attributed to the omission of a vi tal term, the inclusion of an extraneous

bas cd

o

n

a

s

ma

ll number of explicitly statecl, readi ly graspcd principles,

one, or the use of an incorrect basic principle. Th€! model t.h at we wiLl present, in common with the other third order closures, contains many undetermined const.ants. It has long been part of thc folk w isdom in t.urbu lenca that a model ean be made to fit' a flow, gi ven suf fi.cLentIy

2 man)' constant s

I~hile thor-e is some justice to this , it is not quite

true. )\ model may b e incapabie of rcproduc ing a ce rta i.n quaI i tati ve

e

'

behavior, regardless of the values assigned to the constants . Nodel s \>Iith many constants can still

he

intcllectuall)' satisfactory, so long as t he constants arc not optimiz cd for cach flow, or group of fl ows, and so

long as the ph)'sical interpretation of thc constant is cleÇl.r.

It is i!!lj_lortant.that the valuos of thc constants governing e ach J?hysically

distinct effect be dct ermiried by computation in a si tuation in which that effect is not in f Iucnco.! by othcr s. Othcrwise, anc is in dang cr

(10)

-8-of 'adjustiilg the wrong constaat fo r the right reason. As an asidc, if this principle is appLi cd conscientiously, it quickIy bccomes clcar that

despite the ~ealth of experimental data collccted over thc past fcw decades, there :is a remarkabje dearth of well+documcnt cö e..e.CJlleJUcu!.U turbuient flows, in which one effect at a time is c~rcfully studied.

~

tt.«

H.{_o/t R~I/JlO.t.Ó

,~h

m

b

Vl

_A_p_p_ll.o _XÁ._{Jna:ti_o}11 _an_d

_ti

_i

_e.

_V_~!>_._{o'{_paT}_{"';_}_on_E

q

u

ari.o

11.6 K

In Tennekes _{and L}_um_l_e_{y (1972)}_, _it _is _ShO_h'

llhow orders of magnitude may he assigned _{to vario}_u_s _correl_a_tions _appearing

in the dyrramical

equat ions. Roughly: instantane:ollS quant; ities appear irig In t.he cor re lat icns are of two types, belonging eithcr to the encrgy containing range of

eddies, or to thc dissipation range. The former has characteristic

, o -,3 0 ,2

frequency u

/

Á..-

(wher e c

=

U

/

Á..-

,

3u is t.w ice the mean fluctuating energy q2, and

c

is the mean dissipation of energy per unit mass) whi Le the

, -1/2 t

latter has characteristic frequency u

FA

,

wher e À - 4lR.e: ' Rl = u l/v.

The correlation coefficient between two quantities from the same range' may usualJ.y be taken as uni ty, but the cocfficient b et.wecn t.v-;o quant ities,

each from a different range, is of the order of thc time scale ratio,

o

In addition, of course, wc may make usc of thc more fami Lia r fact

that dcr ivat i.vcs which arc cxt cr na I .to c(lrrel~tions corrcspond to scalcs in the cnerg)' cont aining range .. whil c dcrivatlvcs within thc cor rcIat ion

(11)

-9-cor

resp

ond t

o

d

i

s

i

p

a

tLon s

c

a

l

e

s ,

\\'c

h'i5h to

2pply

this

S

OTt of

r

C3son

ing to

e

v

e

r)" ter

m

appc

a

riu

g

r

n

t

he

e

qu

a

t

io

n

s

,

but

p

é

'

l"

t

icul

a

rl)'

to

thc cquation

s f

cr th

e

di

ssi

p

a

ti

o

n o

f

cn

e

r

e

)"

(

on

d of t

cm

p

e

rature

or

con-centr

a

tion

vari

3 ncc);

a

p

plicd to th

e

s

e

cqu

a

tio~

s

,

it is

p

a

r

t

icularl)'

productive,

b0c

a

use

the d)~

Bm

ics

of th

e

se

quantities is dom

i

nated

by

t

he

smal

l

s

cales

,

and

intc

l

'Bcts

onIy we

ak

l

y w.i

th

.thc

energy

containing

cddics.

Procce

d

ing in

this wa)', thc equation

for the mean dissipation

of cnergy

may

b

e

reduccd

(as is

done with the

.

(equivalent) vorticity

equation in Tennekes

&

Lumlcy,

_19ï2)

to thc

form

c

+ Ë .U. + (EU.) . ==

,J J J ,J

2

- 2v

u.

u. .u.

-

Zv

u.

.

u.

.

1., K 1.,J J, r~ 1.,KJ 1., KJ (1)

A

s is dis

c

ussed in Tennekes

&

Lumley

(1972),

the two terms on the

right

-1/2

a

re of order one but differ Dy ordcr

Rl

The

remaining terms are of

-1/2

o

rder

R,e_

Ot

her tcrms

(many

o

f

wh i

ch

appear .i

n th

e

full equations)

a

re of higher order

.

The first term on the right represents th

e

product ion

o

f veloeity gradicnts

by

stretching by fluctuélting

'

st

rain rate

,

whi

le the

s

econd represcnts

the destruction of these gradicnts by viscosity.

Seve

ral authors (Dal)"

and

Ha

rlow

J 19]0;

Jones

and Laundcr,

1972,

Reynol ds, 1970,

Ng

and Spalding

,

1972)

have retaincd the terms on the

1 eft-hand s

i

de

o

2 u. u.. U.

(12)

-10-and anothcr term of si.mil ar farm, cor rcctly feeling that thcre must be .

seme.source of dis s ipat ion. 1I0hE".'el', these. terms are of (relative

-1

order

Rl

'

sincc (ac is shown in !umley, 1970)

,

\) u. u . .

=

c(Ó •

/3

+ 0 (S .

À/

u ))

1,

~

1,J

KJ

.

K

J

(3)

~here SKj is the mean strain rate; since Ui is incompressible, on1)' thc second term contributes. Since a t~rm lik~ (3), namely \) Ui,KUj,K

. appc~r~. ~n ~!1~ )~eynol~s stress cquati on, IvC should mention here that

this term also has thc same behavior as (3) (c.f. Corr5in, 1972). 1t has been modeled by some authers (U?ly and Harlow, 1970; DonaIdson,' 1972) as. proportional to uiUj' vhercas the ratio of off-diagonal to diagonal

-1/2

terms must vanish as R.e. ' as shown by (3).

111eproper source of the product ion of dissipation is in the first term in the right hand side of (1). In the fo l l ow.ing section, wc wiLl app1y a forma1 procedure to obtain an unambiguous expre s sion for the right-hand side of (1), and we will find that wc obtain a term of farm

similar to that retcined b)' the authors mentioned in connection with (2), so th at in a sen~e th~y have been using the right term for the wrong reason. Befare going to the f'ormaI procedure, however, it \\'i11 be instructive to carr)' out a ph)'sical analysis ofthc right-hand sidc of

(1), to sec hal.. a production term can be r eta ined at infini te Reynol ds numbcr.

(13)

-11

-T

he

r

i

ght

hand sièe of

(

1 )

rcpre

s

euts a balance betwcen stretchin

g

and diss

i

pation

,

an

d

it

must be

p

ossi.b

le

for the

(relat

i

vcl)'

sm

a

l

l)

mismat

e

n

la

be of

either

sign

.

T

h

at

is

,

con

si

d

er the

st

retching

o

f a

s

i

n

g

l

e

v

o

r

t

ex t

o

eq

u

i

l

ib

r

iuni

:

i

f

thc st

rot

ch

i

ng

is su

dd

cn l

y

in

cr

e

a

sed

,

m

o

~en

t

a

r

i

l

y

t

hc

fi

rs

t t

erm wi

l

l d

o

minatc

th

e

s

econd

,

s

et

ting up

mor

e

v

o

rt

i

ci

ty

un

t

i

l

e

quilib

rium

i

s

ag

a

i

n

attai

ned

,

at a

hi

g

h

e

r

l

e

v

e

l.

I

f

t

nc st

i

e

t

ch

ing

is re

duc

e

d

,

the p

ro

c

ess is

r

e

ver

se

d

.

P

u

t

i

n

stat

is

t

i

ca

l

te

r

ms

, if

thc

s

p

e

c

t

r

a

l

e

n

er

g

y f

lu

x i

n

cr

case

s

, t

hc first

t

e

r

m

s

hou

ld

do

m

in

at

e t

n

e second

u

nti

l t

he

dis

~

i

pa

ti

o

n

h

a

s b

ee

n in

c

re

ase

d

t

o

mat

ch

t.h

e

flux,

a

nd vice vers

a

.

Th

e

re is

,

in

a

ddition

a

n

u

n

st

e

ady

e

ff

e

ct;

i

n

a fluc

t

u

ating

t

urb

u

l

en

t

v

e

lo

cit

y field, equili

br

i

u

m

is nev

e

r

at

t

ai

ned

,

sinc

e

the str

a

in

ra

t

e

cha

n

g

e

s bef

o

re it can be

a

ch

i

e

v

e

d

,

H

e

nce

,

t

h

e

r

c

i

s always a fluctu

a

t

i

n

g

mis

rna

teh

;

altho

u

g

h to fi

rs

t o

r

der

,

we

wo

uld

exp

e

ct this to av

e

r

a

ge to

z

e

r

o

,

we wou ld

e

xpect non

-

linear ef

fe

cts t

o

prbduc

e

~ s

m

all net lo

s

.

The response to these effec

t

s should depend on the tim

e

sc

a

le ratio.

The ti

m

e

scal

e

of the di

s

sip

a

tive eddi

e

s

is

(v/Ë) 1/2;

if the

t

u

rb

ulenc

e

were isotropic

an

d decaying, a time scale d

es

crip

ti

v

e

of t

ne

u

n

st

e

ady

2

-str

e

tc

h

i

ng w

ould b

e

q /

2

E. If th

e

rc i

s a

n in

put t

o

the

s

pe

c

tr

a

l flux

,

th

e

r

e

I\"ill

b

e a

noth

e

r ti

me

sc

a

le as

so

ci

ate

d with th

i

s in

p

ut

.

_{In a ho}

_mo

-gen

e

o

u

s

flo

w

,

th

e

i

n

p

u

t

i

s

c

hara

ct

e

r

i

ze

d b

y P, th

e pro

ductio

n

,

a

nd t

he

ti

m

e s

calc wi 11 o

e

q

2

/2P.

I

n

an Lnh

o

m

o

gc

ne

ous s

itu

a

ti

on , I

t

i

s mo

r

e

(14)

t I I I f I i ··12-~.

of P is transportcd. In the next s cction we wi LI obtain by formal

means an app rop riate e xprcs sion in the inhol:logcneous si tuation; hcr c wo

will retain P, which may be thbught of as an approximation for small lnhomogcnei ty. This is effecti vely what was done by Dal y and HarIow

(1970), Jones and Launder (1972) and Ngand Spalding (1972). The inverse

of the time scale thus may be written as (2Ë/q 2)F(P/C), wher'e F is an

unknown function. If the production (and hence thc anisotr-opy) is small, we'mal' expand to obtain (2€,

/q

2) (l-aP/E). It is, of course, not legitimate

,.

"to use such an cxpres sion.for va l ues of PI'¬ ::- 0(1); the expr e ssi.on wi l l

serve at least to prcdict qualitat i.ve behavior, however , s inc e it reverses

sign as we have reasoned it must.

The difference on the right hanel side of (1) might consequently be modeled as

EI

c

/v

{O + _b~2- _{(1 -}

_a=

P

₎

_.

₁

rv

c

_{+ O(J./}_R,_e₎_}

q2

E , (4)

wher e the 0 symboLizes the equal i ty of the terms at infini te Reynolds

number-. To second order in time, the direct response to the mean flow

distortions \\'ill appcat through Sij' etc.; these cannot appeal' to first

order becausc thc~ are of the wrong tensor rank; a scalar term is needed,

,

.

I

and can be made f'rom Sij onIy by aquadratic fo rm. Not e that thc terms

are the s ame ordcr

as

the others r eta i.ncd in the cquat ions bcc aus e the

1/2 ',' '. . _/ )1/2

(15)

I !

I

,

1 'j I

I

! !

I

-13-i.e

.

-

as the Reynolds nurnber increascs, the magnitude of each term gro~s, but the difference shrinks at the same rate.

In Lumlcy (1970), a similar ana Iys is was carried out, though less physical and more formal; consequentl)', although the order of the te~m obtairred there was correct, the fact that it should be r eve r sibLc was

missed. The general conclusion obtaincd there, regarding the continual

growth,of the length scale in a hornogencous flow is correct, ho~evcr, 50

.Iorig as a

I

1, as ma)' be eas iIy verified using (4)"in the an alysis therc. I

I'. -; 'As in -Lumley '(1970), one of 'the' coef Fi.cient s may be identified by

I

reference to honogerieous decay ; we find that

b

=

2 ₍₅₎

Thus , (1) becomes

c + € ,U, + (cu.) .

,J J J ,J 4a 'CP2 (6)

q

'I11Ïs is essentially the farm used by thc author referred to ab ove.

The temperature (or contaminant) dissipation equation ma)' be attacked

in exactly the same wa)'. Presuming that the Prandtl number is of order

one (a ve ry large or ver)' small vaIue can lead to the ret entLon or discard

of different terms) we obtain

Co

+ (;0 . U, + (cOu,) . = ,J J J ,J 2 - 2K 0

,

e

.u. . - 21::

e .. e

"

,J ,1 l,J ,1J -lJ (7) Y'

(16)

--l

t!-The

:

Ln

t

c

rpr-ct

a

ti

on

of

thc

t

erms

is exac

t

l

y t

he sa

m

e

,

and

t

he

dynami

ca

l

r

casoning is

the sa

m

e

.

Thc ti

m

e scalc ch

a

ractcri:ing

the sma

l

sea

l

es

(

_aga

_l

0

_{n presu~lng}

0)

_t

_Je

P

_ran

d

l

_t

}...

_nu

_re

_~

_e

_rt

_o

1 _)e

₀f..l

_o

_ruer

_unlty

0) _IS0 (/-;::'\

_v

_C₎1/2_;

th

e

ti

me s

c

ale

ch

aracteriz

i

ng the

f

l

uc

tu

ati

n

g

stre~c

h

i

n

g

i

s

q2

/2~,

and

t

hat c&

a

r

ac

t

er

izin

g

th

e

i

nput

to th

e s

p

e

c

t

r

a

I flu

x

is qZ

/2P

.

(ag

ain

,

w

e

wi

ll

ob

t

a

i

n a bet t

e

r e

xpre

s

ion t

han P

fOT

th

e

i

nput to

t

he s

p

e

c

t

ra

l flux

later

)

.

H

c

n

c

e

,

the for

m

is

'

v

e

ry

s

i

mi

l

a

r

t

o

(4

)

,

and we o

bta

in

v• (8)

gi

ving

ES + Ee .U. + (EeU.) .

=

,J J J ,J e:Ee Ee - 5 --2 + Sd"2" P

q

(9)

A

g

a

in

,

the r

e

lat

i

on bet

~ee

n th

e

co

effi

cients i

s

obt

a

in

e

d by

r

e

fe

rence t

o

h

o

m

ogen

é

ous d

e

c

a

y

d

ata

.

s

pe

cif

i

cal

l

y Gib

so

n

an

d

Sc

h

war-

z

(1963).

I

n

t

he

foll

ow

i

n

g sc

c

ti

on

J

we

wi

ll obt

a

i

n

by

fo

rmal

me

thod

s

L

m

pro

ve

d

app

roxi

mat

io

n

s

to (

6 )

a

n

d (

9 ).

-/_1d. _{(_'t}_{Cl ,}1 1/_._,_Cnit'J_l_."'"_d

0

I

f w

e

c

o

n

s

iste

n

tl

y

a

p

ly t

h

e R

e

yn

ol

c1s

/

Pe

clet num

be

r or

d

er of

m

a

g

n

i

-tu

d

e

a~3

1)"si

sgi

v

e

n h

e

r

e,

w

c

o

b

t

ajn

t

he s~

t o

f eq

G

a

t

io

ns

gi

v

c

n belaw

(17)

,15-(in addition to (1) an d (7), we are considcring here on1y vc locity and

- i

I

tcmpc ratur o; the trcat mcnt of a pas sive contaminant, or of an activc contaminant other th an tempcrat ur c, can be handled in cxac tLy thc s amc

\\"a)',

_We

_wri_t_e

_thc

_e_qllat_i_ons _{in t~e} _Boussin_e_sq _a_p_pr_o_xi_rn_a_t_i_on _{- Ph}_i11_i_p_s,

l

I

I-I

I

1966) "

u

,

+ U, ,U. + (u.u.) . ;:::- p

.L

o

+ ó .gO/T, U.. = 0 (10) 1 1,) J 1 J _,) ,1 0 31 0 1,1 \'. _f) + 0 .U. +. (eu.) .;::: 0 ,1 1 1 ,1 (11) u.u + U.. u_u 1 K 1,)) f~ + U_K_,₎.u._Ju.₁ + (ï:ï:tl₁ I() ,).U.) + (u.u u1 K ).) ,).

=

(uj;--:-

+ u.p )/p +

(ü-

ë

ê . + u. e ó )g/T + - 2Ëo. /3 (12) K,1 l,K 0 K 31 1 3K 0 lK

-A-e

2 +

2

0 .

U.e

+

e

2

.U

.

+

(

e

2u.) .

= -

2€e

,J J ,J ) J ,J (13) êu. + U .. êu, + 0- .u.u. + (eu.) .U. + (eu.u.) . :: 1 l.J J .J 1 ) 1 ,) J 1 J .) ~p + ,1 0

s

62' 'T 3i g/ 0 (14)

(18)

r

!

t

I

t

I

,

I

-16 -tempcru tur c . I'quat ions (10)-(1-1), plus (1) and (7) cons ti tut e thc s c t we wi ll cons idc r .

The averages are to he unelerstood as ensemble averages , so that the

cqu3tions can accomoclatc evolution of the turbulent field, or changing mean or bouridary conelitions . ln addition, such phcnomena as internal waves can be accomodated so long as the period is long comparcd to an)'

chara cteris t ie time of the turbulcncc, so th at there is no direct coupl ing.

-

\..

I~chave neglected K0,jj' VUi,jj, V(UiUK),jj' K82,jj' v(8ui,j),j and

.. ...

-

,

K(ui6,j) ,j which are of order R.e. • in their respective equations, and -1/2

VO,jUi,j and KUi,j8,j which are of order R.e. In the at mospher e,

. 1/2 3

t.ypic e l Iy Rl. - 10 , so that the neglect of these terms may be expcct ed to be an excellent approximation. We are a l so neglecting off-diagona1 .components of the last term in (12), in accord with (3).

Let us ·consider first the term

(IS)

in cquation (12). Part of this term is a transport term; let us subtract the tr acc, and consider

o

- (u P . + u.p )/p + 2(u.p) .0. /3p = A,·, sa)'

K ,I I, K 0 J ,] IK 0 IK (16)

(19)

-17

-of _lI_i_Llj' vUi alid

t

h

c

valLlc of

g

/T

o for all tjme

and

Sp:1CC,wc would know U₁· and

O·

,

having U₁·

a

nd

0

,

togC'thcr I,'i th

-_

_{Uj_}_ll_j_' _U_i_e_g_/_'f_o _an_d _c_, _e_q_u_a_ti_on

(12) woul d give us the transport term plus AiK. Thus it must be poss ibl c

to I...rite the sum of Ah and the transport term as a functional of uiUj' GUi' ~, g/To (the latter occurring w ith and without eUi)' Toe trac e of the transport term itse1f may be so written (by ?pplying tJle same reasoning

to the trace of equation

(

12 ),

si~cc Aii =

0 )

.

It

is surely a small step

to as sume that the w!wte transport term ma)' be so writ tcn , and hence that

AiK ma)' b evso wr itten.

t

A.

=

F. (il.u,

Elu., 2, giT}

IK IK 1 K I 0 (17)

where the functional ext ends over all space , and all earlier time. We

.have not included the direct ion of the gravity vector, since this is pa rt

of toe structure of the equations; information about the direct ion of

gravity must be forcecl to appear in UiUj and 8ui'

This far, no approximation is involved (except the assumption that

either of the third order terms in

(1

2 )

ma)' be written as

(1

7 )

if both

ma)' be). NOl';, we wish .to introduce the approximation of uieah: asL-0!JcJ.:,,'lCPU

(which .implies ll'c.aQ }.__'lfwmogelle)~tu) and quasi-steadiness. Both these con

-ccpt s ar-e k inctic thcory coriccpt s: th at tine and l ength -sc a l cs of Inhcmo -geneity arc large re1ative to time· and length scales of thc turbulcnce,

and that the turbulcnc c is ncar Ly in equilibrium (isotropie) . These are

(20)

known to bo poor dc scriptors of turbulenc c, ,equivalent to g r adicnt trans

-port conccpt s, Howevcr, we are apply ing t.hcrn here to third-order quant

r-ties, handli.ng the second-order oncs exact ly; it is hoped that the

predicticns ~ill be less sensitive to assumptions made at this level.

111cas sumpt i.on at least provides an exact model of a phys ic aI ly rcalLzabJc

proeess (something like a ver)' rari f ied gas), obt a i.nable f'r ornturbu lence

in a conccptually (though not physiqally) possibl~:way (by letting the

turbulent lcngth and time seales become short), 50 that ~ne

ma

r

hope that the predictions w iLl not be unphysical (i. e. - produeing negati ve ene rgy,

etc.) and should bear some qualitative res emblance to turbu lence . To implcment the approximation of weak anisotropy, we

w

ill expand

thc right hand side of (J7) in a funetional powei series (assuming fading

memory and limited awarenes s, as in Luml

e

y

,

1967); in keeping with quasi

-steadiness, we will negl~et time derivatives. Carrying out the expansion requires thc following steps: _e_xpr_e_ss _u,u_" _{as u}_"_u' _{- q}2,_ó_'_" / :_~ ₌ _{al")" sa)',}

1 ) 1 ) IJ

and q2, writi ng 6Ui .= bi' The funetional (17) is now a funetion of a symmetrie tensor aij' a vector bi' and three sca lar s , Ë, q2 and g/To' In

the isotropie limit aij' bi and g/To all vanish, as do all gradients.

2

Thus aij ,!~ woul d be a secend order term, and q ,i a first order term.

First, form thc functiona1 Taylor seriQs in the gradients of the arguments. Second, thc tensor cocfficient s ar e now functions of thc loeal val ues of aij' bi' ct c,; cxpr cs s thcm in invariant fortn , arranging them acco r ding to

(21)

I

!

I

_I

-1

9-Exparid in ro:':crs of g/To; finalLy, apply dimcnsional anal ysis to thc

coe ff icicnts.

The net result consists of terms of two types: thosc that woul d bc

present in a homogcncous flow, and corrections for .inhomogen eity .

'Through third order, the homogeneous terms are 2 A.. = - {èl + B II/q )

a:

,

+ B

C

a

"

IJ 1 IJ 2 IJ (18) \'. 2 2.'

.whe re aij - aiKa~d'; and II"=-'aijaji', 'thc"'seë"ond Irrvar iant .

T

=

cq

I

E

,

where c

ma

r

be eva1uated from the initia1 rate of return to isotropy (scc,

for example, Tucker and Rcynolds, 1968). Later, we wi Ll identify T as

the Lagrangian integra1 time scale, which permits approximate eva1uation

of c -

1 /

8.

The Lowes t order term, aijlT, ma)' be 'identified as that suggested

by,Rotta (1951). The form (18) is consistent with the observations of

Champagne, et. al. (1970) that thc principal axes of AiK and aiK wer e the

s ame, in a homogeneons flow. Ini tial Iy , i t had been hoped that f ir.st

order tcrms "'ould provide an adcquate model; howevcr, the second and

thlrd order terms provci to be neccssary, at least in the wake. Thc third

order tcrm speeds the return to Is otropy for 1arge anisotropy , whi1c the

sccond providcs s c.nc rcdi.stributi on in thc pre s cnc e of shcar , In thc

wakc , thc pcak of \\,2 is dirc ctly at tribut ahlc to thc sccond order term

')

on l y , whil c tho pc ak .in u- canno t be rcduc cd to rcas onablc proportions

(22)

-

20-Tlrrcugh th ir d order, thc te rrns I nvo l v ing fint de r ivat iv c s ar e

buoyancy corrc~tions: I I .1

I

,

f

I

i

2 2 B (giT )(b.o. + b 0.. - b.6. 2/3) q ./q 1 0 1 JK K IJ J IK ,J (19)

B

(giT )(b.o. + b é.. - bi ó• 2/3).-ë .jc 2 ~ 0 1 JK K IJ J J.K ,J ',...

where use has been made of thc fact that AiK = AKi' Aii :;: O. äls,o tcnsor iaI

h

'

"

approp riat

e

terrns in q2,iJ"

€

.

"

q7-•

-t

"

,lJ ,1 ,J There are 2 2 q .q " E:.E: . ,1 ,J ,1 ,J !

and gbi j/T ' the coefficients of the f irst five bcing first order

,. 0

functions of t::,q2 and a~ ~; the sixt h term, being a third order buoyancy ....J

i correct ion , h as a coefficient which is a function. only of

€

,

q2 Finall)',

2

there are third order tcrms in aiK,jl' aiK,jq,l and RiK,jE:,l' Kith coefficients in €,q2. It is not expected that all these terms wiH be equa l Iy important. In the (isothermai) wake, we found that only the terms in q~ij' C,ij and aiK,jl wer e dynarni.cal ly important.

Note that the type of second order terms suggestcd by Rotta (1951) and Rcynolds (1970) and others do not appear; our formalism excludcs

ex)

:

üc

U

dependenee on the mcnn ve loci ty profi Ie, and forces implicit depcnclcncc through sccónd order quantitics.

KOh', we may apply thc sar:e rcas oning to the terra in (14). ExaD~ning

(23)

-21 -- ;:Op

./

p

=.: ,1 0 2

F

.

{eu., u.u., ge /T ,

g

i

T

,

c} 11, 1 J 0 0 (20)

The dissipation E does not appear directi)', sincc (assundng high Rcynolds an d Peclet numbers) this equat ion has no dissi.pat ion tcrms. llowever , in the homogeneous case, eUi and lIiUj (and g/To) are not suff~cient to deteJ;'

-mine Ui and 0; some other quantity must be incluclcd to fix a length scale for the turbul ence , This is rcflected in the group in (19) being

di mensionaj Iy incomplete without

"E

(in the homogeneous case); that is, a non-trivial

fOl

m

can'bc-found only at an order higher than the first. We

will include

"E

,

noting that including redundant quantities ean do no

harm

.

Applying our expansion procedure, we find for thc homogencolls pal't (to third order)

- ep:-/p

,1 0 4 2 22 2 2

= -

{(17/16 + B1II/q + a

T

g 8 /q T )ó .. 2 0 1J (21) 2 2 4 + a a ..

/

q

+ .a a..

/

q }

b./T 3 1J 4 1J . J

whcrc T is the samc as in (18), and the coefficient has been evaluated

by re fercncc to homogeneous deca)' data aga i.n, That is, the critorion was

app licd that, in thc Li.mit of s mal l an

i

sotropy ,

L

e. - late in the dcc ay ,

,-- 0

OUi Iq2/8

2

shoul d dccay at the s amc rat e as Bij' The Iowcst order part of (21), bilT, h as been suggcstcd on all

ad

-

t

i

cc.

basis by Donaldson (1972).

Thc inholllogcncous tcrms ma)' cas i.Iy be gcnc rrrt cd ; through s ccond c rde r tÏ1(')' :11"(' (w ith numerical cceffici('HtS)

(24)

-22 -.-2c

/1

'

g~. . ,1 0' (qTge2'T/ 0),/•

,

1..J

T

2/

_(2/

_{,_}

( q) gG T JE . o ,1 (22)

These a

rc all

b

u

o

y

a

ncy c

or

r

e

c

t

ion

s

;

t

h

ir

d o

rder

t

e

r

ms

br

i

ng a

n

2

b

.

;

b ..

t: ;

b ..

q ; J.,JK 1,),K. 1,J,K 2 .

-:2

2

ge

.

e

.

./T;

gB.q

.

/

T

,1,J 0 ,1',).0 (23)

plus anisotrop~

correction

s

to the first and second

,

order terms

.

It

~ill

i

-

I

.

r .

;..

be intercsting to sec by calculation ~heth

e

r

it is

neccssary

to

retain

t

erms

vof

this-·oruer

.

The variolIs transport terms may be attac:ked 1n the same

way

.

Adopting

t

he

forms

(18) and (20),

we find

= _F_{. {}_~, _Ou., _E~ _EO_' _g_/_T_o_}

1 1 J 1 (24)

Fo

lLow

i

ng

e

xact

l

y

the

same

procedure,

we obt

a

iri

to first order

C,tt. b 1.

2 _ 2- 2 - _ 1/2

= -A (eTeq /2) .-1\ EeT c; .-,.\ ,,(q /3)TCe .+A (Eo/T) 611.

41 . ,1 42 ,1 4.. ,1 It5 1 (25)

so

t

hat

g

radi

cnts

oi

s

evciat

quant iti

cs

,

and

another

flux, can

produce a

flux

of

c

o

(25)

I

-2~- ..

Tbc s amc rc a s onin g wil l p.ro ducc

-

z

-e

ll, 1 ::

F

.

{

lï.ïl.""

,

êu . 1 1 J 1 c -8'

g

i

T }

_o (26) .so that (27) where T

_{e "}

_e

z_

₁

₈_c

_e

,_.,_'t..,,.._ ,..,c .......1....1.: .; ,......_..... .,

"""""lll..I\,..J,. V.J.. Cl.\.l\....J.. .l.\..I11a.~ cons t ants iiitrûuuct:u in (25) _a_U_G (27) _Ü.

somewhat startling. It is encour aging that we find Cl simpler resu lt when

we ex

a

m

i

n

e

u

.

u .

.

Thu

s

1, J '

au

.

u

.

1 J :: F..

{

ë

U:-,

1J 1

z-u, u"

e

giT , c, 1 J 0 gfT 0} (28)

which leads (to fiTst order) to

eu.u,

1 J

2"

=

A

T

q

e

ó.. giT

53' 1J~ 0 (29)

h'hct!1er or not it is ncccssa ry to car ry second order ter ms in any

parti.cu l ar case is n point that C3i1 bc s et tlcd in gener::!l only by comp

(26)

I

L

I,

-24 -fJuxcs, sincc-taking the divc i-gcnce raiscs the order by oue . Onc situat ion, howcve l', prov idcs [! c l car Deed fo r s econd ordcr tc rms: If : i I

. !

I

!

, thc first order terrns vanish .ideut icaLly in a particular physic.al pr ob lcm, In (28), i.f gravlty is not .important, thc transport of thcrma l flux

vanishes . Since this is suroly not truc, second order terrns are nccessary to provide an adequate model whcn stratification is wcak : .

eu,u, 1 J

=

i

! .(30) + .A

ss

Tq(3u-_,u,/q 2--0, ..)8 2giT 1 J IJ 0 2 -- -_ + A Cl T

re

Ou .) , + (Su . ) ,] 56 1 ,J J ,1

OnIy thc terms in AS4 and AS6 wil l remai.n if gravitational effects are relati veIy weak. The term in Ass simply corrects the term in AS3 for anisotropy. "/e ma)' gi ve a simple phys ic al interprctation to t.he terms in

AS3 and Ass: eg/To is thc buoyant acceleration~ and Tq the turbulent lcngth scale; hen ce, Tq8g/To is that part of the turbul ent energy that is cor re l at ed \\'Ïth the tcmperat ure f Iuc t.uat ion, The terms in AS4 and AS6

arc

simpl)' gradient transport.

Thc purc Iy ncchan ical transport tcrms are also r c lat ivc Iy simpl e.

For thc mcchan ica I dissipation we have.

CU,

1

=

F

_l

,

_IJ

{~,

_. ~,C· t>I'IT}'0' (31)

(27)

-2

5

-lcading to cu. = l. (32) In an exactly simil ar w ay, we find (u.u + 20. p/:)::;

Ju

.

l. K l.K 0 J

=

F.l.KJ.

(

i:ï~

l. J

,

u.l.6.,

E,

giT }0

(

33 )

.. wh{ch gives (to first order)

A

T(

q2

/

3)

[

0 .

8'0 11 l.K J-l-+ a(o.;o::..J K-l-p+o.{>öl.-l-KJ,.-l-)]q\ (u.u +

2

0 .

p

/

3p

Ju

.

=

l. K l.K 0 J 2 2

A12

T

(q

/3) [

0 .

0'0 +

b(o

.

.o

0 +

o.

0

.)]E 0 l.K J-l- 1J K-l- l.-l-KJ ,-l- (34)

whcre a and bare constants. In each case we have tried to choose signs

and numerical coefficients which wi11 result in the constants

Apq

being

positivc and of order one.

Just as we found for thc transport of therrnal flux, th ere is a

si tuat icn in which the f irst order transport of Reyno lds stress van ishcs ,

and sccond order tcrms a r c cons equcnt Iy ncccs sary. Spec i.fi.calLy, in a

sin~lc shcar U1(x2), in ~hich only·t~o-gradicnts are non-zero, u1uZu2

=

0

f rorn (3:». Since this is a dynnmi call y important quan tity , wc wil1 necd '

.

(28)

-2

6-omission of tllis transport, r csults .in undcsirablc, and unr6alistic,

I.

hyp c rb ol ic bch avior, with ste cp cninp fronts, etc.). The sccond order

to

rms are 2 . T(q /3)[a a .. + a (a.. + a .. ) . l;H~,J 21J,K KJ"l I \ + + ex ó• a .. + Cl (o.. a +\0 .a , )] 3 lK JP,P '+ lJ KP,p \ KJ lp,p

,

\

T

[B1ó.

a,.e.

+ Ba.

o

.

.i

+ (3 Có. .a.

.e.

+ Ö .a .

.e.

)

l.K J

2

IK J

3

\1.?_

k\

KJ 1

-

\

2

+

B

,

(

0, (,>a , +' Ö (,>a.. )] q 0 \ 4: 1.-t- KJ K-t-1J \~. \ \ " \ \ 2 . '. " \ T [y O. a.(,> + y a. 0'0 + Y (6. .a ().+ 6".a.o) ; ,1 IK jc . 2 lK j-e, 3 1J K-t-;. KJ 1-t-(35) "_\_. 1\ t• I

a

+ ."_,' .' ".:.' ₊ ,. Y4

16. ~a

.

+ ó

()

a

:

.

)]E ()

I 1-t- KJ K-t- 1J ,-t-I " ....ï:'::

\

"

\

Finally, we may apply t~e same procedure to obtain th~ forms for·the

\

.

\

right hand sidcs of (1) and (7). The right'hand\'side of

'

(l)

must be a ..

\

\ \

,

. \ \

.sca l ar functional of the s amc variables as (31). 'Completè to terms of ,

-, \ + \

\

.~..': third order,

thc

,

ri

gh

t h

a

Rd s

i

dc

of

(1)

is RIIS(1) :: ₍₃₆₎ s, \

\

-_. -

-

. .. ;:'---..

(29)

,

.

-D

-.

-and for (7) wc obtain (with the s an.e depcndency as (24)

rJIS(7) = S--u.r_olo_•2 - 'I +. d a .. b . b

./E

e

T

q

4 1) 1 ) d -

1/2

T

3 /2/

+

gE

e

T

q

5

°

(37) Ij 2 - 1/2 3 + db. b .

T

g/

Ee

q T } 7 1 1

°

• t \' .

!'ichave ac:cepted the coefficients dctermined fr omthe homogeneous deca)'

data. 111 is the third invariant of aij' IJJ

=

aijajKaKi'

If we look at (36); keeping only second order terms, Ke see that

what we sould have us ed in (6) and (9) in pl ace of P is IJ = a , ·a·· (wi t.h

1))1 .

a suitable coefficient to make the dimensions correct). It is reasonable

to as sociat e IJ 1.;1th the spe ct r a l flux, since the inequal ity of compcncnt s

jndicates that straining of the turbulence is taking place. ₁₁ _is _sa_ffic_thin_g

).ike P; if \\'C made thc simplistic K-theory approximation that aij 0:: Sij

"here Sij is the mean str ain rate, then P ex: IJ. In rea Iity , however ,

11

I

0 in rcgions of most flOKS where P vanishcs. Hence, we ~ill still

hnve product ion of

t

thc rc . The sarae reasoning applies to (37), where

it

i

s

a lso evident that , even to seco nd order. O\.1T physical reasoning

rC'sultcd in the neglcct of 3 numbcr of import~nt terms.

(30)

r

i

,

I I ~ f';._~.il'", ..

~

-

--

..

-

.. - .. _{-.._}

_

.

The evaIuat iou (o r eliminat ion) of most of these eocffieicnts will

have to await detail cd cal cul at icn of f l cws in which many mcasurcmcnt s .

exist.

A

few statecc~ts can bc made, however,

b

y

considcring simple floKs. First, consider a steady, hOQogcncous, isotropie turbulence with a. linear tCT:lperature gradient, and na gravit)' or

mean

veloeit)'. Thcn (14) reduee~ to (using (21))

e

'

.

ü:U":.16T/17 ::-.~- Su.

.,] 1] 1 (38)

This is just the Lagrangian transport farm for a passi.ve scalar. Hence, ev ident ly 16T

lli

may be identifieà \\'Îth the Lagrangian integraJ. time sca l e, TI1is scale has been estimated f'r om first prineiples (by a very erude teehnique) as

.il

3uI, and f'rora wake decay data as

.i12.

so

'

(see

- J3 ,2 2

Tennekes and Lumley , 1972, p. 229), where c = u

l

.i,

and 3u = q .

2

-1his gives roughly

T

=

q

lBc

.

If we consider a steady, homogeneous flow with Ui

=

(U1,3x3' 0, 0), UI ':

=

const., and El :: ex, and na gravi ty, \,-C do obtain "K-theory"

JJ .3 3 forJ:ls

ëu

cc_ 3 --2 Tu

o

16/17; 3 ,3 T (39)

-:---2

II

U

3 1,3 II U cc_ 1 3

Ta f irst ordcr , thc co cf ficicnts a re unit)'. Howcver , if wc keep OUT

(31)

· r

for ve ry weak shc ar , .it rapid Iy changes as the she a r beCO!!K'Smore intense (mcJsured in tcrms of

U'

T)

.

,

.

I

!

,

-2

9-an

d

thc anisotropy. _Thu_s_, _a_{l t}_h_c_ugh thc rat10 K~;/l\Hbeg ins at 1Î/16

If the first order rr.cdeI is applied to the constant stress laycr of

2

a neutral turbulent boundary layer, ~e obtain (using

T

= q

/

8 ~

)

--2 ti :: 1 2 --2 --2 2 __ . --2

=i:

1/2 q/2,u2 =u3 =q/4,c-uu/Cu u) 1 2 1 2 :: 1/2 (40)

~I

_u _u 2 _:: 2

*

r-;:;2 ~/ 2 =VL,U U :: 1 ~;

2/2'

Including higher order terms prec1udes a1gebraic ·evaluation. In addition, one may obtain arelation between A22and the cl of equation (36). In a similar wal', if a constant heat flux is added to the neutral constant stress Layer , a re l at ion may b e obt ained among thc constants A,t2, A411 and

Further evaluation of the constants will have to await further

computat ions. One general guideline has sugges ted i tsel f, however , which

provides at least an estimate of the magnitudes of some of thc constants,

nnd eliminates others: in a region of.constant eddy viscosity, and constant

st ructure (which e xi sts on ly conccpt uajjy) , the transport tcrrns should

reduce to the K-theory forms3 _Th_a_t _is _{(con s id}_e_r_i_ng _th_e _pur_e_ly _m_ech_a_ni_ca_l case), in such a region, writing Qij :: lliUj'

(32)

4 c c: q 2 Q. ,/CI := const. 1]" (41) - 30-which leads to 2q2. q

I

2 := ,1 e:,1

./

e:

,

(42) 2 If, for example, we are considering q

,K' then equa11)' good expressions are. 2 ₂ 2 .. q ; (q /2s) e

.

q QJ1Q.. /3 ₍₄₃₎ K'. K' _ 1J ,K

,

..

,

. ji pi

where Q is the inverse of Qij'

Q

Qij

=

transport of q2 to be a form like

p Ó, •

J Ne shou1d thus expect the

2

TQ

..

{Aq . 1J ,J 2 + B(q /2c)e: . . ,J 2 ji + Cq

Q

Q.. 13} 1J ,K (44)

whe re A + B +' C - 1. This expression ma)' be expanded, keeping on ly second order terms, to eliminate nany of the unknown constants .

For a first computation , we shoul d begin with an Lsothez-maL, mechanicaI

o

fIow; "hen thc var ious cons tant.s have been ev a l uated , if the r:::511]ts are sat isfa cto ry , we can pr occcd to a fl ow with ter.pe rat.ur c fl uctuat ions, but