A detailed derivation of conditioned equations for intermittent turbulent flows

A detailed derivation of conditioned equations for intermittent turbulent flows

IN H E A T A N D M A S S TRANSFER 0094-4548/81/060491-12502.00/0 Vol. 8, pp. 491-502, 1981 ©Pergamon Press Ltd. Printed in the United States A DETAIL...

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IN H E A T A N D M A S S TRANSFER 0094-4548/81/060491-12502.00/0 Vol. 8, pp. 491-502, 1981 ©Pergamon Press Ltd. Printed in the United States

A DETAILED DERIVATION

OF C O N D I T I O N E D

FOR I N T E R M I T T E N T

EQUATIONS

TURBULENT FLOWS

P. DUHAMEL L a b o r a t o i r e de Thermique du C.N.A.M. 292 rue St Martin 75141 Paris c4dex 03 France

(Ccmmmicated by J. Gosse)

Introduction Experimenters and analysis 2,3,4].

improved

techniques

of c h a r a c t e r i s t i c

These methods

properties

experiments specialized position

and non-turbulent)

equations

attitude

to theory

of c o n d i t i o n e d - a v e r a g e d

one is the closure

of these equations.

is used not only in the m o d e l i n g equations

of difficulties,

properties

and attempts

studies are built on two levels.

is the derivation

[10,12].

This is the result

(i) the treatment

experimental

results.

different

cal aspect m e t h o d i c a l l y

This trans-

is new

[10,11,

The first one the second

that intuition

zones,

from others

at the in-

(ii) the writing (iii) the lack

[11] who develops treats the first

but his formalism

the

of three kinds

of d i s c o n t i n u i t i e s

Dopazo

of

of two sta-

step but also in obtaining

of terms in a form compatible with m e a s u r e m e n t s , lism a p p a r e n t l y

[i,

to write too

equations,

It appears

terface between t u r b u l e n t / n o n - t u r b u l e n t of useful

flows

[8,9]. The numerous

for each state seem justified.

of the e x p e r i m e n t e r ' s

12]. T h e o r e t i c a l

conditioned

boundaries

carried out show the specific

(i.e. turbulent

in turbulent

sampling

are used more and more often in studies

zones near free and also material tes

of c o n d i t i o n e d

a for-

theoreti-

seems to be far from

that commonly used in turbulence. In the present

work a strict m e t h o d based on the use of ge-

neralized

functions

is applied.

different

ways of obtaining

It gives new insights

conditioned 491

equations

into the

(a comparison

492

P. Duhamel

between nal

present

results

and previous

in a form

1. General 1.1. The

Effect

familiar

part

tency

derivative,

to be made

of the flow

function

~1 HT This

HT(M,t)

function

L

is not

with T t domain

[14].

1-HT) , it is easy the turbulent For

example,

= for possible

intermit-

: t

from the S t e p - f u n c t i o n equation

the equation

controling

zone,

domain

(1.1)

i.e.

HTG

one may write

associated

by H T (resp. property (resp.

G in

(1-HT)G)

:

Y ] xj where

part

?xsb×S/ +gs of the F-function,

H T is derivated.

This

form and is otherwise

(due to d i s c o n t i n u i t y

on the r i g h t - h a n d

side of eq.

at

(1.2)

last

while term

F S stand appears

missing. interface)

function

as a specific

boun-

source-term. 1.2.

lized

thin free

~x3 ~x5

when F is a d i f f e r e n t i a l

S appears

The turbulent

an infinitely

at time

non-turbulent)

nt

The g e n e r a l i z e d

dif-

form which

zone T t. Classical

as

By m u l t i p l y i n g

the turbulent terms

section.

molecular

0 otherwise

different

D t

F T represents

dary

defined

in the turbulent

where

in this

to possess

if M e T t

to derive

(resp.

or d i f f e r e n t i a l

the turbulent

H T is then

or vectorial)

:

a is the constant

explicit

is supposed

limiting

(scalar

= F

of G and F is an algebraic

boundary,~t,

fi-

interface

transportable

as follows

DO _ a~G Dt ~xj

does not need

and provides

to experimenters.

of any

G, may be w r i t t e n

is the m a t e r i a l

fusivity

is made)

of t u r b u l e n t / n o n - t u r b u l e n t

balance

(1.1) D/Dt

formalism

basis

local

property,

Vol. 8, No. 6

Expression

Temporal

and

function

S must

of the boundary

spatial

derivatives

be expressed.

source-term of H T a p p e a r i n g

in genera-

Vol. 8, NO. 6

INTERMI~

Let-~be ry ~t'

~IJRB~

FL(Z~S

493

the unit normal vector at point P, tied on bounda-

pointing outwards, and let n t be the associated coordinate.

A point M out of (resp. in) the turbulent domain T t possesses a positive

(resp. negative) coordinate n t and if M coincides with

P its coordinate is nil. Consequently

nt(P t ) :

:

0

is the equation of the unsteady boundary ~t"

If ~ assigns veloci-

ty of point P, for a fixed point M one may write (1.3)

_~l-{,r_ _ V.n

n

FIw= o

-

S([q0

~t

:

.

This equality is directly deduced

Et

from fig. 1 and may be, otherwise

V

derived from the following steps similar to arguments appearing in ref [15]. Considering an elementary displacement of P one may write first

:

o=

bx~

~t FIG. 1

(1.4) O:

or

~nt

for

Then, owing t o ~ } ~ t : - $ ( ~ t (1.3). Known results yes

and using

eq.

:

m

~x~ (1.4), one obtains

[14] give expressions of spatial derivati-

:

(1.5)

8xi?x I = - ?x~ The material derivative

Dt

E 43

is

thus

:

49 4

P. Duhamel

Or, introducing

Vr, the velocity

tied on the boundary

Vol. 8, No. 6

of a fluid element

in a frame

at P,

(1.7)

DHT

= _ Vr " n 8(n 0

Dt 1.3.

Comparison with previous

Indentity

results.

(1.7.) may also be write

: ]

Dt Av

av-,o

is an arbitary

a v J~v

control volume.

If this volume

includes

Z~Z of i n t e r f a c e E t ,

it is easy to show

[14] that

thus an expression*

proposed by Dopazo

[11] is obtained

DH~_L~ Dt This

shows the connection

I av-- o £v between

a part

:

:

-V r has the f o r m a l i s m developed

by this

author and ours. Let t

be a time where the boundary is at a given point M. If the m interface movement is such that M effectively crosses the boundary during the time interval

are true

[tm,tm+dt[,

~t (~ , % / =

o

~ n~ ~ 6t It:tin

' and one may write

conditions

identity

o

,

:

$(ntl:$(t-tmV;~ntl Therefore

the following

:

or, w i t h ( 1 . 4 ) g ( n ¢ =

(1.7) becomes

S(t-t~/IV.~l

:

*This author, as Bouix [14], takes a - ~ n o r m a l p o i n t i n g towards turbulent domain, wherea~ we took the opposite. One must change the sign of dot product Vr.n to pass from one result to the other.

VOI. 8, NO. 6

~

(1.8)

%~]RH[K~V9 ~

DHT :_~r-7 S(t-tm]

IV EI

Dt This formula

DH~

posed

495

shows that Chevray

: ±

$(tmt~]

and Tutu

considered

[12] who intuitivily

a somewhat

pecular

pro-

case.

Dt 2. Instantaneous By applying ferent

equations

equations

the general

equations

rules developped

of local balances

specialized

Incompressible

conditioned

fluids

one easily

to fluid elements

in § 1 to the difobtains

conditional

of the turbulent

zone.

are considered.

2.1. Mass balance. Unspecialized The conditioned

continuity

equation

(2.1)

~(H~U~]

2.2. Momentum The stress properties

equation

is then

=

r~

is

:

:

~U~ : 0 ~x~

w~th

A

: -n]U~ ~(n 0

balance.

equation

and negligible

of motion

of the fluid with constant

body forces we are considering,

is

:

(2.2)

y ~x~.

Dt In the turbulent

zone one has

Dt

~

~xd~x ~

:

~x~

~x i ~x~

with

2.3. Thermal Putting but retaining

energy balance.

aside the rate of reversible the viscous

ning temperature

dissipation

is written

:

term,

work per unit volume the equation

gover-

496

P. Duhamel

nT Dt

Vol. 8, No. 6

_

o

~)x~bx3 ~C~ ~x~

where

If h y p o t h e s i s

of identical

intermittency

city

and t e m p e r a t u r e

is valid**

the t e m p e r a t u r e

zone

is governed

by

function

for velo-

in the turbulent

:

(2.~)

~x~x~

bt

~C~

~x~

-

with

2.4.

Kinetic

Similarly

energy

balance.

one derives

tion for the turbulent

zone

the conditional

kinetic

energy

equa-

:

(2.4) with

3 Calculation

3.1.

General

of boundary

averages.

basis.

The r i g h t - h a n d in the following

source-term

parts

general

of p r e c e d i n g

form

equations

may be w r i t t e n

:

**This h y p o t h e s i s is a c c e p t a b l e in case of small temperature differences. Body forces are then n e g l i g i b l e [16,17]. One may think that it is also valid in some cases of stratified flows [18].

Vol. 8, NO. 6

~

gj and f are

set for v e c t o r i a l

~

FL~gS

497

and scalar

functions

of space and

time. The flows tion on.

is a more

considered delicate

are not h o m o g e n e o u s

experimental

It is the r e a s o n why we p r e f e r

of d e l t a - D i r a c

and owing

to

function

:

~__

the form

Then by a p p l y i n g

to suppress

by w r i t i n g

first

and

spatial

spatial

derivatives

:

~t

dn t

dn~ ~t

:

formulae

of a p p e n d i x

we write

:

~t I~t

~t fo ~s t~e ~ t e ~ a c e tional

3.2.

Owing

cro~s~

expectation

of the right-hand

deriva-

than time d e r i v a t i -

-

~x~ S takes

operation

~e~e~

(point-average)

at

side disappears

Expression

of v a r i o u s

Results

of c a l c u l a t i o n s

to eq.

(1.4) we r e p l a c e d

a ~ ~lt=t [~, ~ ~ t.e c o n ~

in

the interface, stationary

boundary

are g i v e n ~t

The l a s t

conditions.

source-terms

in the s t a t i o n a r y

case.

by -nkV k.

~t a/ S o u r c e - t e r m

of mass

balance

:

= foF lt:t~{_InkVkjnJU~ ] In s t a t i o n a r y

flows

the p r o p e r t y

:

= o j easily

deduced

from eq.

(i.3)

is v e r i f i e d ,

U. = V.+ V . one may also w r i t e O O r0

:

term

consequently,

since

498

P. Duhamel

Vol. 8, No. 6

<.4~ > = fo Eit=t ~ {_ In~V~ n~Vkl 1 b/Source-term of momentum balance

A

e/Source-term

of

B

thermal

energy

A

d/Source-term

of

:

balance

C

energy

balance

A

<&>=gglt=t all dary nal into

these

values,

tionarity

I

l

terms

a property

account.

a weighting

D

k of

consonant

frequency.

One may n o t i c e inversely

:

Y

that

proportional

conditional with

we c o n s i d e r e d ,

t o mean c r o s s i n g

D

c

make u s e

hypothesis

:

B

kinetic

D

C

all

Libby

intuition. terms [10] the

of boun-

With the

are

took

conditional to

expectation also

these

properties

expectations

normal

sta-

proportio-

interface

show velo-

city.

3.3. Discussion Similar quantities are recognized in each source-term balance apart).

(mass

Some of them possess an obvious interpretation.

It is the case of the flow-rate density through the interface

(i-

dentified with letter A) or of the flux density through the interface

(letter B).

quantities of t y p e

letter

are diffusion terms resulting from the interaction between the geometric properties of the boundary and the spatial properties

of

field g. A supplementary flux (adding B-type flux) and an effect from the interface curvature, dive, appear in this term. Engulfment-type transport

(if present) may greatly contributes to this

last term because of the smallness of the curvature radius during

VOl. 8, NO. 6

this process.

INTERMI~

~

~

499

One will notice that B and C-type expressions may

be rearranged differently and, in the proposed form, B quantity is missing in the kinetic energy source. This absence has its origin in an exact balancing between this kinetic energy flux

(B-ty-

pe) and a term stemming from viscous dissipation.

Quantities of type

h t~\ &~t[~

:

)

(letter D)

have their origin in the interaction between boundary motion and dynamic properties of g field. These terms, owing to the presence of the normal celerity nk V k of the boundary are connected with expansion

(or contraction)

of the turbulent zone.

Lastly, the thermal energy equation possesses the supplementary source

:

~c~

C~

2

whose origin is in irreversible work done by the non-turbulent fluid on the turbulent one.

The preceding identity calls for two kinds of comments. First, the sudden variations of kinetic energy and vorticity through the interface, as observed by experimenters, may give significance to this source. It is not sure that even if dissipation is negligible in the turbulent zone and of course in the non-turbulent one, this boundary source term will be small. This remark may be applied to others terms like for instance the rate of reversible work wich has been neglected in thermal energy balance. This last term may be small everywhere except on free boundaries and so the A-type source of thermal energy equation becomes

:

_T(I_~P~. ~CpJ r~ The second remark concerns the supplementary source-term sign. Clearly, despite its origin (the dissipation vity of U£'t<~n~

~C<~U~/~x~),

positi-

is not ensure and, consequently this source is

not strictly a heat source. This fact is confirmed if we write a conditioned equation for the non-turbulent

zone. In this equation

500

P. Duhamel

the free sign,

boundary

source-term

consequently

one of the Therefore

this

interface, finite

if the

two equations source

of

(2.3)

source

appears

corresponds

it corresponds

is not

a possible

but non zero

Vol. 8, No. 6

connected

mechanism

with

to heat

to heat with

arising

the opposite generation

loss

dissipation

in physical

in

in the other. "in" the

interfaces

of

thickness. Conclusion

This work,

free

ing the various how to pass vantages

points

(or point

which

experimental

boundary

a necessity

for the next

are the d e t a i l e d operation:

to experimenters. and more

is reinforced. source-terms,

studies.

Such

stage

averages.

Thus,

may guide still

of m o d e l i n g

ad-

expression

crossing

fre-

the relevance

especially Besides,

studies,

It shows The real

conditional

and of a quantity:

analysis

on p o i n t - a v e r a g e s ,

is a way of unify-

to that used here.

presented

averaging)

conditional

experimental

in conditioned

of a statistical

are familiar

sion of various ture

the use

of intuition,

formalism

of the derivations

expectation

results

of view

from Dopazo's

of source-terms,

quency

from the use

of

those

showing

the full

expres-

the choice

of fu-

too fragmentary,

conditioned

are

equations.

Nomenclature a

Diffusivity

C

Specific

coefficient

heat

of some quantity.

at constant

pressure.

P 9

Thermal

E]t=t m

Conditional

k

Kinetic

fo

Interface

f,G,g,g~

Scalar

HT,HT(M,t)

Intermittency

m

diffusivity. expectation.

energy

per unit

crossing

mass.

frequency.

or vectorial

diffusing

or H e a v i s i d e

quantities.

function.

Source-term

in the c o n d i t i o n e d

Unit

vector

normal

pointing

nt t,t~

Normal

Tt U,U •

Turbulent

zone.

Galilean

velocity

of a fluid

V,Vj

Galilean

velocity

of the

Time,

continuity

outwards

coordinate. interface

crossing

time. element.

8

interface.

equation.

the turbulent

zone.

VOl. 8, No. 6

~

~

~

501

Relative velocity of a fluid element the interface).

Vr,Vrj

(frame tied on

Source-term in conditioned momentum equation. Thermal expansion coefficient.

§(n t )

Delta-Dirac generalized function. Source-term in conditioned kinetic energy equation. Kinematic viscosity. Density. Source-term in conditioned thermal energy equation.

Zt < >

Boundary of the turbulent

zone.

Stochastic average. Appendix One will calculate

:

< y > = < ~(m,t}~(n0> Let tm be a time such as nt(M,t m) : 0 and Then

[14]

~(~]:

s(t_tmVl~n,I ~t

We consider the joint probability functions

~nt ~ O ~t t = t m

density

f x~(X'@;

t) of random

×(m~t)= ~ ( M , t ) A ~ I and ~ = t-tm. ÷~

Then :

fx®(x,o/t)= fxle(Xo;O,t)f®(oit]=

But

=

One easily shows that,

fo(o/t)_ <

~nt

=

S(t_t~]>_fo

at

where fo is the average crossing frequency of interface at M. Therefore,



=

fO EIt=tm[~(M/t)/

~t]

.

In stationary and ergodic conditions this last result may be more easily obtained by time averaging. To calculate the average :

< Y'> : < 9(M,t) ) one writes

:

I%(.0)> 0t

502

P. Duhamel

and a p p l y i n g

In s t a t i o n a r y

previous

formula

conditions

Vol. 8, No. 6

one obtains

the first

:

right-hand

term disappears.

References [I]

S. Coorsin, A. Kistler, Naca

TN 3133, (1954).

[2]

R.E. Kaplan, J. Laufer, Proceedings

[3]

L.S.G. Kovasznay, V. Kibens, R.F. Blackwelder, J. Fluid p. 283, (1970).

[4]

I. Wygnanski, H.E. Fiedler, J. Fluid Mech.,41, p. 327 (1970).

[5]

L. Fulachier, J.P. Giovanangelli, M. Divas, L.S.G. Kovasznay, A. Favre, C.R. Acad. Sc., B, Vol. 278, p. 999, (1974).

12th Int. cong. Appl. Mech. p. 236, (1968). Mech. 41,

[6]

J.G. Kawall, J.F. Keffer, Physics of fluids, 22, n°1 p. 31, (1979).

[7]

C. Alcaraz, G. Charnay, J. Mathieu, C.R. Acad. Sc., B, 285, p.165, (1977).

[8]

T.R. Heidrick, S. Banerjee,

[9]

G. Comte-Bellot, J. Sabot, I. Saleh, Dynamic measurements in unsteady flows, Proceedings, p. 213, Skovlunde, Ed. (1979).

R.S. Azad, J. Fluid Mech., 82, p.705, (1977).

[10] P.A. Libby, J. Fluid Mech., 68, p. 273, (1975). [11] C. Dopazo, J. Fluid Mech., 81, p. 433, (1977). [12] R. Chevray, N.K. Tutu, J. Fluid Mech., 88, p. 133, (1978). [13] E.A. Spiegel, Physics of fluids, 15, n°8, p. 1372, (1972). [14] M. Bouix, les fonctions [email protected] ou distributions, Masson ed. (1964). [15] G. Tavera, H. Burnage, C.R. Acad. Sc., A, 284, p. 571, (1977). [16] M. Sunyach, J. Mathieu, Int. J. Heat and Mass Transfer, 12, p.1679,(1969). [17] K.R. Sreenivasan, R.A. Antonia, D. Britz, J. Fluid Mech., 94, p.745, (1979). [18] J.J. Lorenz, P.A. Howard, Trans. Asme, J. Heat Transfer, 101, p. 538, (1979).