paper 186 nima
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11th International Conference on Fluid Control, Measurements and VisualizationFLUCOME 2011 Paper No. 186
December 5-9, 2011, National Taiwan Ocean University, Keelung, Taiwan
1
SIMULTANEOUS MEASUREMENTS OF ENERGY AND TEMPERATURE DISSIPATION RATES IN
THE FAR FIELD OF A HEATED CYLINDER WAKE
N.M. SoumehsaraeiSchool of Mechanical and Chemical
EngineeringThe University of Western Australia
35 Stirling Highway,Crawley, WA,6009, Australia
Z. HaoCollege of Logistics Engineering,
Shanghai Maritime University,Shanghai
China 200135
T. ZhouSchool of Civil and Resource
EngineeringThe University of Western Australia
35 Stirling Highway,Crawley, WA, 6009,Australia
ABSTRACTUsing a probe consisting of four X hot wire probes and
two pairs of parallel cold wires, simultaneous measurements of
full energy and temperature dissipation rates, denoted as f
andf
, are obtained. The performance of the probe in
measuring velocity and temperature derivatives and vorticityhas been checked satisfactorily by comparing with isotropy
calculations. The differences between the present
measurements of the two dissipation rates and those obtainedby using isotropic relations are also discussed in terms of their
mean values and corresponding spectra. The correlations
between the two dissipation fields are also examined. The
present results show that the use ofiso and
iso underestimates
rr
ln,lnby about 20% in the scaling range. The large scatter of
rr
ln,lnin the literature indicates that this correlation
coefficient may depend on the magnitude of
R as well as the
nature or the initial conditions of the flows.
INTRODUCTIONThe turbulent energy dissipation rate )2(
ijijfss , the
temperature dissipation rate )(,, iif
k and the enstrophy
)(2
ii are important characteristics of the small-scale
properties of turbulence, where 2/)(,, ijjiij
uus is the rate of
strain; is the kinematic viscosity and ;/, jiji
xuu k is the
thermal diffusivity of the fluid, ii x /, and )3,2,1( ii is the vorticity component; subscript f denote the full
expressions of two dissipation rates. Measurements of eitherenergy or temperature dissipation rates have been made by a
number of researchers in different turbulent flows (e.g. [1-4]).
While it is fraught with difficulties to measure the above two
dissipation fields simultaneously, direct numerical simulations(DNS) complement experiments in some respects and provide
new information on both statistical and structural aspects of the
small-scale structures since they can resolve complete
information on three-dimensional vorticity and full energy and
temperature dissipation fields. The correlation of the two above
dissipation fields can therefore be analyzed using DNS results[5,6]. To our best knowledge, there are not many experimentalresults reported previously on the simultaneous measurements
of the full energy and temperature rates except the study by
Zhou et al [7] in a grid turbulent flow. The most common
method used to measure the two dissipation rates is employing
a single hot wire and single cold wire probes by invoking
Taylors hypothesis i.e.
1,1215 u
iso (1)
and2
1,3 k
iso (2)
However, significant differences of the above substitutions with
their full expressions have been reported (Zhou and Antonia
[8]; Hao et al. [9]).The first objective of this paper is to measure the full
energy and temperature dissipation rates simultaneously in acylinder wake by using a multiple hot and cold wire probe
(Figure 1), which contains eight hot wires and four cold wires
The performance of the probe will be assessed by comparingthe present results with those reported previously. On the basis
of these measurements, the correlations between the two
dissipation fields and the vorticity will be further examined
(second objective). These results will be compared with those
obtained preciously by using various approximations to the
dissipation rates so that appropriateness of theseapproximations can be assessed. Differences of the correlation
are compared when the full expressions for and or their
isotropic counterparts iso andiso are used.
EXPERIMENTAL SET UPThe experiments were conducted in a wind tunnel a
Nanyang Technological University with a cross-section of 1.2m
(width) 0.8m (height) and 2 m long. The free stream velocityU is about 3.6 m/s, corresponding to a Reynolds number Re
( /dU ) of 1520, or a Taylor microscale Reynolds number
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R of about 50, where mm35.6d is the diameter of the
brass cylinder. The measurement location is at 240/ dx . At
this location, the Kolmogorov length scale [ 4/13 )/( ]
is about 0.66 mm. The local turbulence intensity Uui /' is about
2%, favoring the use of Taylor's hypothesis (where a primedenotes rms value). A heating wire with diameter of about 0.5
mm was wrapped and inserted into a ceramic tube, which was
put inside the brass cylinder as a heating element. The
resistance of the heating wire is about 30. The wire washeated electrically with a power consumption of about 200W.The mean temperature increase T on the centerline of the
wake at the measurement location, relative to the ambient, is
about 1.0C. This value is small enough to avoid any buoyancy
effect and large enough to assure high signal-to-noise ratio.
A probe consisting of four X-probes and a pair of parallelcold wires (Figure 1) was used to approximate the energy and
temperature dissipation rates. The four X-probes also allow the
calculation of all components of the vorticity vector using the
measured velocity signals u1, u2,and u3. With the two pairs of
parallel cold wires (Fig.1), the three temperature derivatives
ixi(/ 1, 2, 3) can be measured simultaneously. The
included angle of the X-wire is about 100
. The separations x2andx3of the two X-probes in the opposite direction are 2mmand 2.7mm, respectively, and the separations x2c and x3c
between the opposite cold wires are about 2.5mm and 2.2 mm.
Figure 1: Sketches of the multi hot and cold wire probe.
(a) Side view; (b) Front view.
The hot and cold wires were etched from Wollaston (Pt-
10%Rh) wires. The active lengths are about 200dw and 800dw
for the hot and cold wires respectively (where dw is the
diameter of the wires and equals to 2.5 m for the hot wires and1.27 m for the cold wires). The hot wires were operated with
in-house constant temperature circuits at an overheat ratio of0.5. The cold wires were operated with constant current (0.1
mA) circuits. The probe was calibrated at the centerline of the
tunnel against a Pitot-static tube connected to a MKS Baratronpressure transducer. The yaw calibration was performed over
20C. The output signals from the anemometers were passedthrough buck and gain circuits and low-pass filtered at a
frequency fc of 800 Hz, which is close to the Kolmogorov
frequencyfK(U/2). The filtered signals were subsequently
sampled at a frequency fs = 2fc using a 16 bit A/D converter
The record duration was about 60 s.
RESULTS AND DISCUSSIONThe full experission for the mean dissipation rate
ijijss 2 can be written as
.22
2
222
2,33,21,33,1
1,22,1
2
2,3
2
3,2
2
1,3
2
3,1
2
1,2
2
2,1
2
3,3
2
2,2
2
1,1
uuuu
uuuu
uuuu
uuu
(3)
Except for the second and third terms all velocity derivatives on
the right hand side of Eq. (3) can be measured with the present
probe. The quantities 2,2u and 3,3u can, however, be estimated
by assuming incompressibility, viz.
1,1u + 2,2u + 3,3u = 0 (4)
or .42223,32,2
2
1,1
2
3,3
2
2,2 uuuuu (5)
Assuming homogeneity the last term on the right hand of Eq
(5) can be replaced by 2,33,2
4 uu since
2,33,2
uu = 3,32,2
uu . (6)
Substitution of Eqs. (6) and (5) into (3) then yields
.22
2
4
2,33,21,33,1
1,22,1
2
2,3
2
3,2
2
1,3
2
3,1
2
1,2
2
2,1
2
1,1
uuuu
uuuuu
uuuu
(7)
With the four X-wires the three vorticity components were also
obtained from the measured ,iu viz.
3
2
2
3
3,22,31 x
u
x
uuu
(8)
1
3
3
1
1,33,12x
u
x
uuu
(9)
2
1
1
2
2,11,23x
u
x
uuu
, (10)
where3
u and1
u in Eqs. (8) and (10) are velocity differences
between X-wires a and c, respectively (Fig. 1);2
u and1
u inEqs. (8) and (9) are velocity differences between X-wires b and
d, respectively. Derivatives in the x direction were estimated
using Taylors hypothesis, i.e. txU //1
. With the two
pairs of parallel cold wires (Fig. 1), the three temperatur
derivatives ixi (/ 1, 2, 3) can be measured. The ful
temperature dissipation rate can be obtained, viz.
.)/()/()/( 23
2
2
2
1 xxxk (12)
As for ,/1
xui
1/ x can be determined using Taylors
hypothesis.
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Basic performance checks of the probe
Measurements of and are not straightforward since the
velocity and temperature derivatives have to be resolved
adequately by using finite differences. It is therefore important
to ensure the performance of the probe before any further
analysis can be conducted. Fig.2 shows the distribution of the
root-mean-square values (indicated by a prime) of u across the
wake normalized by the maximum velocity deficit0
U and
wake half-width L. As it can be seen in Fig.2. the 'u values
from 4 different X-probes agree very well with each other and
also shows the same level of agreement with the results from
Hao et al. [ 9 ] measured using 2-X wire probes in the
streamwise direction ( results are not shown here).
Fig3. Shows the distributions of ' , normalized by the
temperature increase0
T at the centerline, across the wake
from the four cold wires. The differences of the rms values of
temperature fluctuations measured by two cold wires are small
enough to be regarded as the experimental uncertainty ( %4 ).The measured values of the vorticity and energy and
temperature dissipation rates have to be corrected due to the
effect of imperfect spatial resolution of the probe on
measurement of the velocity gradients. The spectra of thevelocity derivatives and the vorticity will be attenuated due tofine wire length and wire separation. This attenuation can be
corrected based on the local isotropy assumption. Details of the
correction procedures can be found in Zhu and Antonia [4]. Asa brief check on the performance of the probe, Figure 4 shows
the comparison of the measured vorticity spectra with
calculated based on isotropic assumption. For isotropic
turbulence, the spectra )(1k
i can be calculated from
1,1u (Antonia et al. [10]):
1
1,1
1,11
)(4)()(
11 k
u
u
caldk
k
kkk
(13)
and
1
11
111
)(
2)(
2
5)()( 1,1
1,132 k
kkkkk
u
u
calcal
dkk
kk
u
1
1,1
)(2
1
(14)
where1,1u
is the spectrum of the streamwise velocity
derivative. The agreement between measured and calculated
spectra is very good, especially for wavenumber larger than
0.05 except for )(11k
x over the wavenumber range of
0.010.05. In the figure, the asterisk * denotes normalization by
the Kolmogorov length scale , velocity scale /Ku and/ortemperature scale 2/1)/(
KKu ].
Fig.5 compares the rms values of1
,2
and3
obtained
in the present study with those previously for the wake flow.
These values are corrected for the spatial resolution of theprobe based on the corrected spectra shown in Figure 4 using
0
*
1
*
1
*2* )( dkki
c
i , where the superscript c represents
corrected. The rms values ofi
are normalized based on L
and U0. To account for the Reynolds number effect, Antonia e
al. [11] suggested that ReL ( /0LU ) should be included, i.e
0
'2/1'/)(Re UL
iLi
. The present rms values of the three
vorticity componentsc
i
' after correction for spatial resolution
are significantly smaller than those obtained by Antonia et al
[12] and Zhu and Antonia [4]. The present values of c
i
' agree
more favorably with those reported by Lfdahl et al. [13].
0
0.1
0.2
0.3
0 0.5 1.0 1.5 2.0 2.5
X-probe 1
X-probe 2
X-probe 3
X-probe 4
y/L
u'/U
0
Fig. 2. Rms values of velocity component fluctuationsmeasured using four X-probes across the wake.
0
0.1
0.2
0.3
0.4
0 0.5 1.0 1.5 2.0 2.5
cold wire 1
cold wire 2
cold wire 3
cold wire 4
y/L
'/T
0
Fig3.Rms valuses of temperature fluctuations across the wake
Fig4. Comparison of the measured vorticity spectra with their
calculated counterparts.
0.001
0.01
0.1
1
10
0.0001 0.001 0.01 0.1 1
*
z
(k1
*)
*
y
(k1
*)
*
x
(k1
*)
k1
*
*
x(k
1*),
* y
(
k1*
),
* z
(k1*
),
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0
0.04
0.08
0.12
0.16
0 0.5 1.0 1.5 2.0 2.5
Antonia et al. (1988)
Lofdahl et al. (1995)
Zhu and Antonia
Present
(a)
y/L
(ReL-1
/2)x'
L/U
0
0
0.04
0.08
0.12
0 0.5 1.0 1.5 2.0 2.5
Antonia et al. (1988)Zhu and Antonia (1999)
Lofdahl et al. (1995)
Present
(c)
y/L
(ReL-1
/2)z'
L/U
0
Figure 5. Distributions of rmsvalues of the three vorticitycomponents.
(a)x
; (b)y
; (c)z
.
Statistics of energy and temperature dissipation rates
The mean energy dissipation rates obtained using Eqs. (1) and
(7) are compared in Fig.6. All the velocity derivatives have been corrected according to procedures outlined by Zhu and
Antonia [4] to account for the effect of spatial resolution. Since
more derivative correlations are included in Eq. (7) than that in
Eq. (1), where isotropy is assumed, it is expected that the
former should provide more reasonable values for than when
iso(Eq. 1) is used. Figure 6 shows that the magnitude of
f
is about 10% larger than that ofiso over the range ofy/L 0.8, there agreement betweenf and
iso is
satisfactory, indicating that the one-dimensional surrogate can
represent the full mean energy dissipation rate properly. The
present values of f and iso are about 20~50% smallerthan those by Zhu and Antonia [4], Lofdahl et al [13] and
Browne et al. [14], which could be caused by the difference in
Reynolds numbers among different experiments.
The mean temperature dissipation rates obtained using Eqs.(2) and (12 ) are compared in Fig.7. The magnitude of
iso
and full
collapse across the whole wake, indicating
that the iso
is a good approximation to full
, at least in
terms of the mean values. This result seems in agreement with
that reported previously by Hao et al. [9] using other simplerapproximations.
0
0.01
0.02
0.03
0 0.5 1.0 1.5 2.0 2.5
Browne et al. (1988)
Lofdahl et al (1995)
Present (full
)
Present (iso
)
y/L
L/U
03
Figure 6. Distribution of energy
dissipation rates.
0
0.01
0.02
0 0.5 1.0 1.5 2.0 2.5
(iso)
(full)
y/L
L/(T2U
0)
Figure 7. Distribution oftemperature dissipation rates.
The results shown in Figures 6 and 7 seem to indicate that the
mean energy and temperature dissipation rates estimated using
the isotropy relations can represent the full dissipation rate
properly. However, this does not necessarily mean that they can
represent the full dissipation rates in other aspects. Fig. 8 (a,b)shows the spectra corresponding to the energy and temperature
dissipation rates, respectively, obtained on the wake centerline
using both the full expressions and the relations based on
isotropic assumption. These spectra, )(1k
f and )(
1k
f are
obtained by summing all the spectra of the velocity andtemperature derivatives involved in Eqs. (7) and (12)
respectively. Since the velocity derivative spectra are
normalized by the Kolmogorov scales, the areas under thevarious distributions of the energy dissipation rate should be
equal to 1. This is true both for )(1k
f and )(
1k
iso . The
distribution of )(1k
f shows significant departure from that of
)(1k
iso for 5.0
*
1k , even though the agreement between these
two distributions is reasonable for 5.0*1k . Since
)(15)( 1/1kk
xuiso , the spectrum for
iso is actually the same(multiplied by a factor of 15) as that of xu / while thespectrum of the full energy dissipation rate )(
1k
f is the sum
of the spectra of the velocity gradients involved in Eq. (7)
which are different from that of xu / . The departure o)(
1k
iso from that of )(
1k
f in Figure 8 is therefore no
surprising. This result indicates that the use of the instantaneous
values ofiso as a substitute of the instantaneous values of, as
usually used in experimental studies, may cause significanterrors, at least in the spectral domain. The satisfactory
agreement between )(1k
f and )(
1k
iso at high wavenumbers
indicates that local isotropy is satisfied by variousapproximations to energy dissipation rate, even at a relatively
low Reynolds number. The measured distribution )(1k
iso
departs significantly from )(1k
f [ )()()(
111 3,2,1,kkk
over the entire wavenumber range, indicating the
inappropriateness ofiso
as a substitute off
, at least in the
spectral domain.
10-2
10-1
100
101
10-4
10-3
10-2
10-1
100
*c
(full)
*c
(iso)
k*
1
*c
full
(k* 1),
*c
iso
(k* 1)
Figure 8. Comparison of the
spectra )(1k
f and )(
1k
iso
10-3
10-2
10-1
100
101
102
10-4
10-3
10-2
10-1
,iso
*c(k
1
*)
,full
*c(k
1
*)
k1
*
,iso
*c
(k1*),
,full
*c
(k1*
)
Fig. 9. Comparison of the spec
)(1k
f and )(
1k
iso
0
0.04
0.08
0.12
0 0.5 1.0 1.5 2.0 2.5
Lofdahl et al. (1995)
Antonia et al. (1988)
Zhu and Antonia (1999)
Present
(b)
y/L
(ReL-1
/2)y'
L/U
0
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To further examine the difference between the signals ofiso
andful or between
iso and
ful , the spectra of the above
quantities, denoted by )(1k
to differentiate with those shown
in Figures 8 and 9, which are obtained by summing or the
velocity (or temperature) gradients involved in their
expressions, calculated from their instantaneous signals areshown in Figure 10. These spectra are defined such that
1)(10 1
dkk
. The difference between )(
1k
iso and )(
1k
ful
or )( 1kiso and )( 1kful are apparent. It seems that there are
more contributions to )(1k
iso or )(
1k
iso than to )(
1k
ful or
)(1k
ful at high wavenumbers. This result suggests that the
isotropic substitutesiso and
iso are more related to small
scale structures thanful and
ful .
The correlation betweenr
ln andr
ln is important in
quantifying the behavior of the temperature structure function
n)( in the inertial range, perhaps in a more physical
sense, ascertaining the interdependence between the two
dissipative fields through their relation to the vortical structures
of the flow, where )()( 11 xrx is the temperaturestructure function, the subscript r means averaging over alength r. Van Atta [5 noted that
rr
ln,lnshould vary only
weakly with r across the inertial range. By using the present
probe, the energy and temperature dissipation rates can be
measured simultaneously and therefore, this correlation can bequantified. In the present study, an inertial range is not expected
due to the low
R (about 50). A scaling range instead, centered
on the value ofr* at which 1*3* )( ru is maximum can be
defined, where )()( xurxuu is the velocity increment in
the streamwise direction. The values ofrr
ln,ln
calculated
using either (ff
, ) or (isoiso
, ) on the wake centerline are
shown in Fig. 11. The magnitudes ofrr
ln,ln
obtained using
(ff
, ) and (isoiso
, ) increase with r in the scaling range,
with the former combination being about 20% larger than the
latter in the range 10030 * r . This finding suggests that forthe present flow, the isotropic substitutes (
isoiso , ) cannot
represent (ff
, ) properly. Comparing with other reported
data, the present magnitudes ofrr
ln,ln
are consistently larger
than that reported by Meneveau et al. [15] in a heated wake at
higher values of
R . This may suggest that the correlation
coefficientrr
ln,ln
decreases with the increase of
R in the
same flow, which is in contrast with the R dependence of
rr
ln,lnreported by the DNS data of Wang et al. [6] in a forced
box turbulence. The present measured values ofrr
ln,ln
differ
significantly from those for a heated turbulent jet reported by
Antonia and Van Atta [16] and the atmospheric surface layer
reported by Antonia and Chambers [17]. Meneveau et al. [15]
noted that the apparent discrepancy ofrr
ln,ln
may be due to
the influence of the hot wire on the cold wire signals. This
effect may be significant in a turbulent jet because of the high
turbulent intensity of the flow. However, this explanation maynot be adequate, especially when the separation between the ho
and cold wires is sufficient to avoid any interference from the
hot wire to the cold wire signals. The large scatter in Fig. 11
suggests that the magnitude ofrr
ln,ln
may depend on the
relations used to approximate and . A more likely
possibility is the effect of
R . Further, the dependence of
rr
ln,lnon the nature of the flow or the initial conditions
cannot be dismissed.
10-2
10-1
100
101
10-4
10-3
10-2
10-1
100
iso
(k1
*)
ful
(k1
*)
iso
(k1
*)
ful
(k1
*)
k1
*
(k1*
)
Figure 10. Spectra of the energy and temperature dissipation rates
calculated from the their instantaneous signals.
0.1
0.2
0.5
1
100
101
102
103
r*
ln
r,
lnr
Figure 11. Comparison of
rr
ln,lnmeasured on the wake
centerline with previous results. Present: , (ff
, ); +
(isoiso
, ). , Meneveau et al. [15]; , Antonia and Van Atta
[16];
, Antonia and Chambers [17]; , Wang et al. [6]. The
arrows indicate the scaling range.
CONCLUSIONS
A twelve-wire probe consisting of eight-hot wires and four
cold wires was used to measure the full energy and temperature
dissipation rates simultaneously in the present study. Theperformance of the probe in measuring the two dissipation rates
and also the three-component vorticity has been verified
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properly. While the isotropic approximations (isoiso
, ) can
represent the full expressions (ff
, ) properly in the mean
values, the spectral analysis shows that there is large departure
of the spectrum ofiso (or
iso ) from that of (or ) at low
wavenumbers, indicating that the use of the instantaneous
values ofiso (or iso ) as a substitute of the instantaneous values
off (or
f ), as usually used in experimental studies, may
cause significant errors, at least in the spectral domain. This is
supported by the spectra calculated from the instantaneoussignals of
iso (or
iso ) and
f ( or
f ). Previous experimental
results obtained usingiso and
iso yields rather disparate
values forrr
ln,ln
in different flows and at different
R . The
present results show that the use ofiso and
iso underestimates
rr
ln,lnby about 20% in the scaling range. The large scatter of
rr
ln,lnin the literature indicates that the correlation
coefficient may depend on the magnitude of
R as well as the
nature or the initial conditions of the flows.
REFERENCES
[1] K.R. Sreenivasan, R.A. Antonia 1997, The phenomenologyof small-scale turbulence. Ann Rev Fluid Mech 29, 435-
473.[2] L. Mydlarski, 2003, Mixed velocity-passive scalar statistics
in high-Reynolds-number turbulence, J Fluid Mech 475,
173-203.[3] G. Stolovitzky, P. Kailasnath, K.R. Sreenivasan, 1995,
Refined similarity hypotheses for passive scalars mixed by
turbulence. J Fluid Mech 297, 275-291.
[4] Y. Zhu, R.A. Antonia, I. Hosokawa, 1995, Refinedsimilarity hypothesis for turbulent velocity and temperature
fields. Phys Fluids 7, 1637-1648.[5] I. Hosokawa, S.I. Oide, K. Yamamoto, 1996, Isotropic
turbulence: Important differences between true dissipationrate and its one-dimensional surrogate. Phys Rev Lett 77,
4548-4551.
[6] L.P. Wang, S. Chen, J.G. Brasseur, 1999, Examination ofhypotheses in Kolmogorov refined turbulence theory
through high-resolution simulations, Part 2. Passive Scalar
Field. J Fluid Mech 400, 163-197.
[7] T. Zhou, R. A. Antonia, L. P. Chua, 2002, Performance of a probe for measuring turbulent energy and temperature
dissipation rates, Experiments in Fluids 33, 33434.
[8] T. Zhou, R.A. Antonia, 2000, Approximation for turbulent
energy and temperature dissipation rates in grid turbulence.Phys Fluids 12, 335-344.
[9] Z. Hao, T. Zhou, L.P. Chua, S.C.M. Yu, 2008,Approximations to energy and temperature dissipation ratesin the far field of a cylinder wake, Experimental Thermal
and Fluid Science 32, 791799.
[10] R.A. Antonia, Y. Zhu, H.S. Shafi, 1996, Lateral vorticitymeasurements in a turbulent wake. J Fluid Mech 323, 173-
200.
[11] R.A. Antonia, S. Rajagopalan, Y. Zhu, 1996, Scaling ofmean square vorticity in turbulent flows. Exp in Fluids20393-394.
[12] R.A. Antonia, L.W.B. Browne, D.A. Shah, 1988Characteristics of vorticity fluctuations in a turbulent wake
J Fluid Mech189, 349-365.
[13] L. Lfdahl, B. Johansson and J. Bergh,1995, Measuremen
of the enstrophy and the vorticity vector in a plane cylinder
wake using a spectral method. In: R Benzi (ed) Advances
in turbulence V. Kluwer Academic Publishers, Dordrechtpp 330334.
[14] L.W.B. Browne, R.A. Antonia, D.A. Shah, 1987, Turbulenenergy dissipation in a wake. J Fluid Mech 179, 307-326.
[15] C. Meneveau, K.R. Sreenivasan, P. Kailasnath, M. Fan1990, Joint multifractal measures: theory and applications
to turbulence. Phys Rev A 41, 894-913.
[16] R.A. Antonia, C.W. Van Atta, 1975, On the correlation between temperature and velocity dissipation fields in
heated turbulent jet. J Fluid Mech 67, 273-287.[17] R.A. Antonia, A.J. Chambers, 1980, Velocity and
temperature derivatives in the atmospheric surface layer
Boundary-Layer Meteorol18, 399-410.