IN H E A T A N D M A S S TRANSFER 00944548/81/06049112502.00/0 Vol. 8, pp. 491502, 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 nonturbulent)
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].
1HT) , 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
(1HT)G)
:
Y ] xj where
part
?xsb×S/ +gs of the Ffunction,
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
sourceterm. 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
nonturbulent)
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.
sourceterm 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
,
:
$(ntl:$(ttmV;~ntl Therefore
the following
:
or, w i t h ( 1 . 4 ) g ( n ¢ =
(1.7) becomes
S(tt~/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 :_~r7 S(ttm]
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
sourceterm
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 righthand
deriva
than time d e r i v a t i 

~x~ S takes
operation
~e~e~
(pointaverage)
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.
sourceterms
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/Sourceterm of momentum balance
A
e/Sourceterm
of
B
thermal
energy
A
d/Sourceterm
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 sourceterm balance apart).
(mass
Some of them possess an obvious interpretation.
It is the case of the flowrate 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 Btype flux) and an effect from the interface curvature, dive, appear in this term. Engulfmenttype 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 Ctype 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
(Bty
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 nonturbulent 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 nonturbulent 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 Atype source of thermal energy equation becomes
:
_T(I_~P~. ~CpJ r~ The second remark concerns the supplementary sourceterm 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 nonturbulent
zone. In this equation
500
P. Duhamel
the free sign,
boundary
sourceterm
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. sourceterms,
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 sourceterms,
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.
Sourceterm
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
Sourceterm in conditioned momentum equation. Thermal expansion coefficient.
§(n t )
DeltaDirac generalized function. Sourceterm in conditioned kinetic energy equation. Kinematic viscosity. Density. Sourceterm 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 ~ = ttm. ÷~
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
:
righthand
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. ComteBellot, 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).