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INT. ~ . HEAT MASS TRANSFER 0735-1933/83 $3.00 + .00 Vol. I0, pp. 435-439, 1983 ©Pergamon Press Ltd. Printed in the United States
TURBULENT REYNOLI~ AND PECLET N I ~ E R S RE-DEFINED L. Fulachier IMST, IA CNRS N ° 130, Universit6 d'Aix-Marseille, 13003 Marseille, France R. A. Antonia Department of Mechanical Engineering, University of Newcastle New South Wales, 2308, Australia
(C~l,LLlnicated by J. Gosse)
ABSTRACt It is suggested that the use of the fluctuating velocity vector u i may yield a more meaningful comparison between turbulent Reynolds and P6clet numbers.
Non-dimensional measures of the local strength of turbulence co~pared with the e f f e c t i v e n e s s of molecular action are given by the turbulent Reynolds number Rlu and P6clet number PlO convaonly defined as RIu
___h xu =
u2
--~
;
Pie
__h with u 2
=
~
~e
u2
--a
... (i)
__%
½
the rms longitudinal velocity, lu the Taylor microscale [u2 /(~u/~x) 2 ],
__~_
18 the Corrsin microscale [82 /(~8/~x) diffusivities respectively.
h ], v and ~ the momentum and thermal
It has already been suggested [i] that the use
of u 2 and lu (or to a lesser extent 18) is ambiguous in non-isotropic turbulence in view of the directional dependence of these quan__tities. While the replacement in (i) of u--2 by the turbulent energy -q2 (- ui2 ' where u i is the velocity fluctuation vector with components u, v, w) partially removes the ambiguity, a comparison of Rlu with Pie can only be as meaningful as the comparison between lu and le"
Fulachier [2] (also [3]) has found that a 435
436
L. Fulachier and R.A. Antonia
Vol. i0, No. 5
closer analogy exists in a turbulent boundary layer between a spectrum F q (defined below) corresponding to q2 and the temperature spectrum F@ than between the spectr~n of any of the velocity fluctuations and F@.
This anal-
ogy suggests that, for the purpose of comparing with %0, a more appropriate longitudinal velocity microscale should be based on the fluctuating velocity vector u i rather than on the longitudinal velocity u. For isotropic turbulence, the one-dimensional velocity spectrum Fu(kz) , where k I is the one-dimensional wavenumber, is related to the three-dimensional velocity spectrum E(k) according to [e.g. Hinze [4])
:
k
k2J i
Also
v~ ~v~)
:
--
--- ~wCk) w2
~i
:
~
I
+
dk
. . .
(3)
k2J
kl
E(k) is the three-dimensional spectr~n (k is the magnitude of the wavenumber vector) such that
E(k)
dk
=
q2 2
...
(4)
o
Relations (2) and (3) yield .
.
u 2 Fu(kl)
.
+
.
.
v 2 Fv(kl)
+
w 2 Fw(kl)
=
q2 Fq(kl)
=
2
dk k
1
The three-dimensional temperature spectrum E@(k) may be defined such that
i
E@ (k)
dk
=
@2 2
. . .
(s)
0
in analogy with (4). The relationship between the one-dimensional and three-
Vol. I0, No. 5
TURB~
REYNOLDS AND PECLET NLMBERS RE-DEFINED
437
dimensional temperature spectra is given by (Kovasznay et al [5])
-02 F0(kl)
=
i TEO(k)
2
dk
...(6)
kI The analogy between F 0 and Fq tends to imply correspondence between E and E@. Interestingly, Kovasznay et al [5] supposed that, for isotropic turbulence, the three-dimensional spectra E and E o were proportional.
Using yon K~rMn's
interpolation formula to estimate E(k) for small values of k, they deduced the relative behaviour of Fo and Fu for small values of k I . number range Fo was found to be smaller than Fu.
In this wave-
This relative behaviour
is in accord with measurements in a wide range of non-isotropic flows (Fulachier and Antonia [6] ). Analogous expressions can be written for the mean dissipation rates ¢ and e0 corresponding to q2/2 and 02/2 respectively.
For locally isotropic
turbulence,
¢o
:
~
e
:
iSv
~(~_f]2
. . . (7)
and
t~xJ
...
(8)
It follows, from the definition of ui, that
/T~-J
:
l~xJ
+
l~J
+
:
s
. . . (9)
since, by local isotropy,
---
--
2
Using (9), (8) can be re-written [~Ui] 2 =
3,~
l~-J
...
(10)
438
L. Fulachier and R.A. Antonia
Vol. i0, No. 5
in close analogy with (7). The use of (I0) may, in general, be preferable to (8) when local isotropy is not a good assL~ption. Equations (i0) and (7) can be re-written :
3~
q2 ... (ii) X2 q
c@
=
3~
2
... (12)
where the microscale Xq is defined as
q2/(~q/~x)2
The analogy between (ii) and (12) suggests that the longitudinal microscale Xq, while perhaps less easily measurable than Xu' may be more appropriate than Xu for the purpose of defining a turbulent Reynolds nunber and TABLE i Statistics of Velocity and Temperature Fields in Two Turbulent Shear Flows Quasi-Homogeneous Shear Flow [7] S-2
(~U/~X) 2
S-2
(~v/~x) e
S-2 m2s-2
Plane Jet [8]
14,100
18,960
21,300
28,440
23,400
28,440
0.475
0.417
m2s - 2
vT
0.165
0. 234
m2s-2
W2
0.248
0.251
K2
02
0.0156
2.16
K2m-2
(~@/~x) 2 k
1,200
2.3 x l0 s
m
-3
4.7 x 10 -3
Xo Xq
3.6 x 10-3
3.1 x 10 -3
3.9x10
3.5 x I0 -3
RXu
25O
190
U
m m
5.8xi0
-3
q2
Xq/~
229
205
q2
%8/~
155
i33
Vol. i0, NO. 5
%I~BULENT REYNOIDS AND PECLETN[94BERS RE-DEFINED
comparing it with the turbulent P4clet nunber.
_C~ these nt~bers may be q2
439
Suitable definitions of
__% lq/~ and q2
i@/~. The table contains statistics
of velocity and temperature fields measured in a quasi-homogeneous
shear
flow with a uniformmean temperature gradient (Tavoularis and Corrsin [7]) and on the centreline in the self-preserving region of a turbulent plane jet (experimental details may be found in Antonia et al [8]). The two sets of measurements were made at con~oarable values of RIu and Pl 0 •' significant departures from local isotropy were recorded, in both flows, for velocity and temperature fields.
It is clear from the Table that lq is much closer
to 18 than is Iu.
References 1.
S. Corrsin, in Han~uch der Phyeik VIII~2 (eds. S. Flfigge and C. T r u e s d e l l ) p.524, Springer, B e r l i n (1963).
2.
L. F u l a c h i e r , Th~se Docteur ~s Sciences Physiques, U n i v e r s i t 6 de Provence (1972).
3.
L. Fulaahier
4.
J . O. Hinze, Turbulence, McGraw-Hill, New York (1975).
S.
L. S. G. Kovasznay, 1263 (1949).
6.
L. F~lachier
and
and
R. Ikmms, J . Fluid Mech., 7_~7, 257 (1976).
M. S. Uberoi
and
S. Corrsin, Phys. Rev., 7_65,
R. A. Antonia, submitted to I n t . J . Heat Mass T r a n s f e r
(1983). 7.
S. Tavoularis and S. Corrsin, Pts. I and II, J. Fluid Mech., 104, 311, 349 (1981).
8.
R. A. Antonia, L. W. B. Browne, A. J. Chambers and S. Rajagopalan, Int. J. Heat Mass Transfer, 26, 41 (1983).
TURBULENT REYNOLI~ AND PECLET N I ~ E R S RE-DEFINED L. Fulachier IMST, IA CNRS N ° 130, Universit6 d'Aix-Marseille, 13003 Marseille, France R. A. Antonia Department of Mechanical Engineering, University of Newcastle New South Wales, 2308, Australia
(C~l,LLlnicated by J. Gosse)
ABSTRACt It is suggested that the use of the fluctuating velocity vector u i may yield a more meaningful comparison between turbulent Reynolds and P6clet numbers.
Non-dimensional measures of the local strength of turbulence co~pared with the e f f e c t i v e n e s s of molecular action are given by the turbulent Reynolds number Rlu and P6clet number PlO convaonly defined as RIu
___h xu =
u2
--~
;
Pie
__h with u 2
=
~
~e
u2
--a
... (i)
__%
½
the rms longitudinal velocity, lu the Taylor microscale [u2 /(~u/~x) 2 ],
__~_
18 the Corrsin microscale [82 /(~8/~x) diffusivities respectively.
h ], v and ~ the momentum and thermal
It has already been suggested [i] that the use
of u 2 and lu (or to a lesser extent 18) is ambiguous in non-isotropic turbulence in view of the directional dependence of these quan__tities. While the replacement in (i) of u--2 by the turbulent energy -q2 (- ui2 ' where u i is the velocity fluctuation vector with components u, v, w) partially removes the ambiguity, a comparison of Rlu with Pie can only be as meaningful as the comparison between lu and le"
Fulachier [2] (also [3]) has found that a 435
436
L. Fulachier and R.A. Antonia
Vol. i0, No. 5
closer analogy exists in a turbulent boundary layer between a spectrum F q (defined below) corresponding to q2 and the temperature spectrum F@ than between the spectr~n of any of the velocity fluctuations and F@.
This anal-
ogy suggests that, for the purpose of comparing with %0, a more appropriate longitudinal velocity microscale should be based on the fluctuating velocity vector u i rather than on the longitudinal velocity u. For isotropic turbulence, the one-dimensional velocity spectrum Fu(kz) , where k I is the one-dimensional wavenumber, is related to the three-dimensional velocity spectrum E(k) according to [e.g. Hinze [4])
:
k
k2J i
Also
v~ ~v~)
:
--
--- ~wCk) w2
~i
:
~
I
+
dk
. . .
(3)
k2J
kl
E(k) is the three-dimensional spectr~n (k is the magnitude of the wavenumber vector) such that
E(k)
dk
=
q2 2
...
(4)
o
Relations (2) and (3) yield .
.
u 2 Fu(kl)
.
+
.
.
v 2 Fv(kl)
+
w 2 Fw(kl)
=
q2 Fq(kl)
=
2
dk k
1
The three-dimensional temperature spectrum E@(k) may be defined such that
i
E@ (k)
dk
=
@2 2
. . .
(s)
0
in analogy with (4). The relationship between the one-dimensional and three-
Vol. I0, No. 5
TURB~
REYNOLDS AND PECLET NLMBERS RE-DEFINED
437
dimensional temperature spectra is given by (Kovasznay et al [5])
-02 F0(kl)
=
i TEO(k)
2
dk
...(6)
kI The analogy between F 0 and Fq tends to imply correspondence between E and E@. Interestingly, Kovasznay et al [5] supposed that, for isotropic turbulence, the three-dimensional spectra E and E o were proportional.
Using yon K~rMn's
interpolation formula to estimate E(k) for small values of k, they deduced the relative behaviour of Fo and Fu for small values of k I . number range Fo was found to be smaller than Fu.
In this wave-
This relative behaviour
is in accord with measurements in a wide range of non-isotropic flows (Fulachier and Antonia [6] ). Analogous expressions can be written for the mean dissipation rates ¢ and e0 corresponding to q2/2 and 02/2 respectively.
For locally isotropic
turbulence,
¢o
:
~
e
:
iSv
~(~_f]2
. . . (7)
and
t~xJ
...
(8)
It follows, from the definition of ui, that
/T~-J
:
l~xJ
+
l~J
+
:
s
. . . (9)
since, by local isotropy,
---
--
2
Using (9), (8) can be re-written [~Ui] 2 =
3,~
l~-J
...
(10)
438
L. Fulachier and R.A. Antonia
Vol. i0, No. 5
in close analogy with (7). The use of (I0) may, in general, be preferable to (8) when local isotropy is not a good assL~ption. Equations (i0) and (7) can be re-written :
3~
q2 ... (ii) X2 q
c@
=
3~
2
... (12)
where the microscale Xq is defined as
q2/(~q/~x)2
The analogy between (ii) and (12) suggests that the longitudinal microscale Xq, while perhaps less easily measurable than Xu' may be more appropriate than Xu for the purpose of defining a turbulent Reynolds nunber and TABLE i Statistics of Velocity and Temperature Fields in Two Turbulent Shear Flows Quasi-Homogeneous Shear Flow [7] S-2
(~U/~X) 2
S-2
(~v/~x) e
S-2 m2s-2
Plane Jet [8]
14,100
18,960
21,300
28,440
23,400
28,440
0.475
0.417
m2s - 2
vT
0.165
0. 234
m2s-2
W2
0.248
0.251
K2
02
0.0156
2.16
K2m-2
(~@/~x) 2 k
1,200
2.3 x l0 s
m
-3
4.7 x 10 -3
Xo Xq
3.6 x 10-3
3.1 x 10 -3
3.9x10
3.5 x I0 -3
RXu
25O
190
U
m m
5.8xi0
-3
q2
Xq/~
229
205
q2
%8/~
155
i33
Vol. i0, NO. 5
%I~BULENT REYNOIDS AND PECLETN[94BERS RE-DEFINED
comparing it with the turbulent P4clet nunber.
_C~ these nt~bers may be q2
439
Suitable definitions of
__% lq/~ and q2
i@/~. The table contains statistics
of velocity and temperature fields measured in a quasi-homogeneous
shear
flow with a uniformmean temperature gradient (Tavoularis and Corrsin [7]) and on the centreline in the self-preserving region of a turbulent plane jet (experimental details may be found in Antonia et al [8]). The two sets of measurements were made at con~oarable values of RIu and Pl 0 •' significant departures from local isotropy were recorded, in both flows, for velocity and temperature fields.
It is clear from the Table that lq is much closer
to 18 than is Iu.
References 1.
S. Corrsin, in Han~uch der Phyeik VIII~2 (eds. S. Flfigge and C. T r u e s d e l l ) p.524, Springer, B e r l i n (1963).
2.
L. F u l a c h i e r , Th~se Docteur ~s Sciences Physiques, U n i v e r s i t 6 de Provence (1972).
3.
L. Fulaahier
4.
J . O. Hinze, Turbulence, McGraw-Hill, New York (1975).
S.
L. S. G. Kovasznay, 1263 (1949).
6.
L. F~lachier
and
and
R. Ikmms, J . Fluid Mech., 7_~7, 257 (1976).
M. S. Uberoi
and
S. Corrsin, Phys. Rev., 7_65,
R. A. Antonia, submitted to I n t . J . Heat Mass T r a n s f e r
(1983). 7.
S. Tavoularis and S. Corrsin, Pts. I and II, J. Fluid Mech., 104, 311, 349 (1981).
8.
R. A. Antonia, L. W. B. Browne, A. J. Chambers and S. Rajagopalan, Int. J. Heat Mass Transfer, 26, 41 (1983).