RESEARCH ARTICLE Decay of passive-scalar fluctuations in slightly stretched grid turbulence S. K. Lee • A. Benaissa • L. Djenidi • P. Lavoie • R. A. Antonia Received: 8 November 2011 / Revised: 17 May 2012 / Accepted: 22 May 2012 / Published online: 14 June 2012 Ó Springer-Verlag 2012 Abstract Isotropic turbulence is closely approximated by stretching a grid flow through a short (1.36:1) secondary contraction. The flow is operated at small values of the Taylor microscale Reynolds number (about 25–55) and is slightly heated just downstream of the grid, so that the temperature serves as a passive scalar and the initial velocity/thermal length-scale ratio is about 1. For the same grid, the contraction reduces the skewness and kurtosis of the thermal fluctuations and their derivative. The thermal fluctuations and their mean dissipation rates follow a power-law rate of decay that depends on the geometry of the grid. Comparison with velocity measurements shows that, for three different grids, the ratio between the tem- perature and velocity power-law exponents closely mat- ches the velocity/thermal timescale ratio. For the present measurements, the timescale ratio is slightly larger than 1 but does not exceed 1.2, in accordance with the proposal by Corrsin (J Aeronaut Sci 18(6):417–423, 1951b). 1 Introduction The decay in grid-generated turbulence has been the sub- ject of extensive research since the work of Taylor (1935b). Grid turbulence is of interest because it represents a close approximation to homogeneous isotropic turbulence. The similarity analysis of this flow (e.g. Ka ´rma ´n and Howarth 1938; Dryden 1943; Batchelor 1953) allows testing of the concept of universal behaviour of turbulence. While it is well accepted that, in grid turbulence, the decay of the turbulent kinetic energy, q 02 (a prime denotes the root- mean-square value), follows a power law q 02 t n q , where n q \ 0, the actual value of n q has yet to be established. There are currently two different theories for predicting n q . The first theory by Batchelor and Proudman (1956) indi- cates n q =-10/7, and the second theory by Saffman (1967) predicts n q =-6/5. In reality, measurements of n q are rather sensitive to different grid flows, and this makes it difficult to test the theories. In fact, the variability of n q (from near -1 to -1.5 reported in literature) has led to the notion that n q may not be universal at finite Reynolds numbers (e.g. George 1992a, b; George et al. 2001). There are very few measurements, perhaps only those of Lavoie et al. (2007) and Krogstad and Davidson (2010), where large number of velocity data points are collected over a wide downstream range in an attempt to more accurately determine the decay rate (n q ). Lavoie (2006) has established that the variation in n q mainly arises from different initial conditions (i.e. grid geometry and Reynolds number) that affect the character- istics of the turbulence. These characteristics, for example, include the intensity of vortex shedding behind the grid, the anisotropy of the flow and the shape of the energy spec- trum. Strong vortex shedding and large anisotropy tend to shift the turbulent energy to lower wavenumbers and increase the magnitude of n q . By carefully modifying the grid, the intensity of vortex shedding may be reduced. To improve the isotropy of the flow, an effective method is to S. K. Lee (&) L. Djenidi R. A. Antonia School of Engineering, University of Newcastle, Newcastle, NSW 2308, Australia e-mail: [email protected]A. Benaissa Faculty of Engineering, Royal Military College of Canada, Kingston, ON K7K 7B4, Canada P. Lavoie Institute for Aerospace Studies, University of Toronto, Toronto, ON M3H 5T6, Canada 123 Exp Fluids (2012) 53:909–923 DOI 10.1007/s00348-012-1331-3
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RESEARCH ARTICLE
Decay of passive-scalar fluctuations in slightly stretchedgrid turbulence
S. K. Lee • A. Benaissa • L. Djenidi •
P. Lavoie • R. A. Antonia
Received: 8 November 2011 / Revised: 17 May 2012 / Accepted: 22 May 2012 / Published online: 14 June 2012
� Springer-Verlag 2012
Abstract Isotropic turbulence is closely approximated by
stretching a grid flow through a short (1.36:1) secondary
contraction. The flow is operated at small values of the
Taylor microscale Reynolds number (about 25–55) and is
slightly heated just downstream of the grid, so that the
temperature serves as a passive scalar and the initial
velocity/thermal length-scale ratio is about 1. For the same
grid, the contraction reduces the skewness and kurtosis of
the thermal fluctuations and their derivative. The thermal
fluctuations and their mean dissipation rates follow a
power-law rate of decay that depends on the geometry of
the grid. Comparison with velocity measurements shows
that, for three different grids, the ratio between the tem-
perature and velocity power-law exponents closely mat-
ches the velocity/thermal timescale ratio. For the present
measurements, the timescale ratio is slightly larger than 1
but does not exceed 1.2, in accordance with the proposal by
Corrsin (J Aeronaut Sci 18(6):417–423, 1951b).
1 Introduction
The decay in grid-generated turbulence has been the sub-
ject of extensive research since the work of Taylor (1935b).
Grid turbulence is of interest because it represents a close
approximation to homogeneous isotropic turbulence. The
similarity analysis of this flow (e.g. Karman and Howarth
1938; Dryden 1943; Batchelor 1953) allows testing of the
concept of universal behaviour of turbulence. While it is
well accepted that, in grid turbulence, the decay of the
turbulent kinetic energy, q02 (a prime denotes the root-
mean-square value), follows a power law q02� tnq , where
nq \ 0, the actual value of nq has yet to be established.
There are currently two different theories for predicting nq.
The first theory by Batchelor and Proudman (1956) indi-
cates nq = -10/7, and the second theory by Saffman
(1967) predicts nq = -6/5. In reality, measurements of nq
are rather sensitive to different grid flows, and this makes it
difficult to test the theories. In fact, the variability of nq
(from near -1 to -1.5 reported in literature) has led to the
notion that nq may not be universal at finite Reynolds
numbers (e.g. George 1992a, b; George et al. 2001). There
are very few measurements, perhaps only those of Lavoie
et al. (2007) and Krogstad and Davidson (2010), where
large number of velocity data points are collected over a
wide downstream range in an attempt to more accurately
determine the decay rate (nq).
Lavoie (2006) has established that the variation in nq
mainly arises from different initial conditions (i.e. grid
geometry and Reynolds number) that affect the character-
istics of the turbulence. These characteristics, for example,
include the intensity of vortex shedding behind the grid, the
anisotropy of the flow and the shape of the energy spec-
trum. Strong vortex shedding and large anisotropy tend to
shift the turbulent energy to lower wavenumbers and
increase the magnitude of nq. By carefully modifying the
grid, the intensity of vortex shedding may be reduced. To
improve the isotropy of the flow, an effective method is to
S. K. Lee (&) � L. Djenidi � R. A. Antonia
School of Engineering, University of Newcastle, Newcastle,
obtained with a grid of round bars at 36 % solidity that
closely matches the present grid Rd35. Inspection of Fig. 7
shows that the contraction keeps the magnitude of the
thermal-derivative skewness small (i.e. |Sqh/qx| \ 0.2) and
reduces the thermal-derivative kurtosis (Kqh/qx) by a factor
of (up to) about 3. For grid Sq35, the magnitudes for both
-Sqh/qx and Kqh/qx are smaller with the contraction than
with no contraction.
From the evidence above, we conclude that, for a fixed
grid (Sq35) with the (1.36:1) secondary contraction, the
passive scalar is more nearly isotropic at both the large
scales (h) and the small scales (qh/qx). Although Sh and Kh
are small (\0.3), the effect of grid geometry on the large
scales is not negligible. The small scales are much less
sensitive to the grid and, with the secondary contraction,
the improvement in isotropy is obvious, that is, the PDF
of ðoh=oxÞ=ðoh=oxÞ0 is more nearly Gaussian (Fig. 5), and
10-5
10-4
100
101
PDF(
ϑ)a
(1994); no contrac.
Sq35; with contrac.Sq35; no contrac.Tong and Warhaft
-12 -9 -6 -3 0 3 6 9 12
ϑ = (∂θ/∂x)/(∂θ/∂x)’
-0.4
0
0.4
ϑ3 × PD
F(ϑ)
Gaussian
b
Fig. 5 PDF for a square-bar grid flow with and without the (1.36:1)
contraction. The normal distributions are shown as dashed lines. The
data (open square) of Tong and Warhaft (1994) are for a square-bargrid of 34 % solidity; x/M = 62; RM = 9,700; Rk = 38
-0.1
0
0.1
Skew
ness
, Sθ
Rd35
(Rd44)
Rd44w
Sq35
No contraction; Sq35
10 20 50 100
Streamwise distance from the grid, tUo/M
0.1
0.2
0.3
Kur
tosi
s, K
θ
No contraction; Sq35
Rd35
Sq35
Rd44w
Fig. 6 The thermal skewness and kurtosis for grid flow with the
1.36:1 contraction (open square Sq35, open circle Rd35, opentriangle Rd44w) and with no contraction (filled square Sq35). For the
Rd44 data (dashed lines) of Mills et al. (1958) (a round-bar gridof 44 % solidity; x=M � 15! 80; RM � 7; 000; Rk � 19! 26;DT � 5 �C), the grid flow is with no contraction. The solid curvesare a visual guide
914 Exp Fluids (2012) 53:909–923
123
-Sqh/qx and Kqh/qx are closer to zero (Fig. 7). Moreover, the
new findings here should highlight a need for future studies
to explore different contraction geometries and their effects
on the large and small scales produced by a grid.
6 The power-law decay of scalar fluctuations
Since the previous section has established that the passive
scalar is nearly isotropic and, for mixing in isotropic tur-
bulence, Antonia et al. (2004) have shown that the scalar
transport equation has the solution:
h02
DT2� ðt � thoÞUo
M
� �nh
; ð10Þ
we shall apply this relation (10) to determine the decay rate
‘‘nh’’ of the scalar fluctuations.
Equation (10) is valid for thermal measurements
downstream of the regions of initially developing turbu-
lence and accelerated decay in the contraction (e.g.
Mills and Corrsin 1959; Warhaft 1980). The temperature
variance (h02) in the range 22. tUo=M. 110 is
shown in Fig. 8, where the time-averaged velocity
(Ucl/Uo & 1 ± 0.01) is independent of streamwise position
(see Fig. 3). For the range 16. tUo=M. 25, Lavoie (2006)
has established that each grid flow is approximately
homogeneous, where the spanwise distribution of the free-
stream velocity, U(y)/Ucl, is &1 ± 0.02; the distribution of
the turbulent kinetic energy, qðyÞ02=qcl02, is &1 ± 0.05. To
avoid possible effects of the duct exit, the temperature
measurements reported here stop at approximately one duct
width (about 12 M) short of the exit plane of the duct.
The analysis starts by selecting the virtual origin,
which makes the decay exponent independent of stream-
wise distance from the grid. The technique of curve fitting
the data points is similar to the least-square method
described by Mohamed and LaRue (1990) and Lavoie
et al. (2007). For each grid, there are a total of 27 data
points spaced at intervals that appear uniform when
plotted on a logarithmic scale; these measurements are
shown in Fig. 8.
In Fig. 9, the scalar decay exponent nh is plotted as a
function of ‘‘tstartUo/M’’—the position of the starting data
point used for the curve fitting. We increase tstartUo/M by
dropping one data point at a time, starting from the data
point nearest to the grid. For each curve fit, the same last
data point is used (i.e. tlastUo/M & 110). The minimum
number of data points used for curve fitting is 6.
Figures 9 and 10a show that the determination of nh
depends on the virtual origin. The optimal virtual origin
tohUo/M is selected such that the rms variation in nh is
minimum over the range 20 \ tstart Uo/M \ 90. At the
optimal value, the curve-fitting error for the thermal vari-
ance, rrðh02Þ, is &0.1 % (Fig. 10b). If the virtual origin is
placed at the grid, that is tohUo/M = 0, the error rrðh02Þ is no
more than 0.5 %. Table 1 is a summary of the curve-fitting
results. The width of the 95 % confidence limit for nh
is ±0.01. It is clear that once the optimal value of tohUo/M is
reached, nh becomes independent of the measurement
range (t - toh)Uo/M.
0
0.1
0.2−S
∂θ/∂
x
No contraction; Sq35
With (1.36:1) contraction
Sq35
Rd35Rd44w
10 20 50 100Streamwise distance from the grid, tU
o/M
1
2
4
6
K∂θ
/∂x
No contraction; Sq35
With (1.36:1) contraction
Sq35
Rd35Rd44w
Fig. 7 The thermal-derivative skewness and kurtosis for grid flow
with and with no contraction. The symbols are as defined in Fig. 6
10 20 50 100
Streamwise distance from the grid, (t−tθo)U
o/M
10-3
10-2
10-1
θ’2 /Δ
T2
tθo=0
tθo=0
tθo=0
θ’2∝(t−tθo)nθ
n = −1
Sq35
Rd35
tθoU
o/M=5.5
nθ=−1.19
nθ=−1.26
nθ=−1.22Rd44w
tθoU
o/M=6.5
tθoU
o/M=4.5
Fig. 8 Decay of a passive scalar downstream of each grid with and
without adjustment for virtual origin (tohUo/M). The top 2 set of data
are vertically offset by 1 and 2 decades. RM & 10,400
Exp Fluids (2012) 53:909–923 915
123
7 Mean dissipation rate and the Corrsin microscale
In this section, we determine the decay rate nh by analysing
the mean dissipation rate hvdi and the Corrsin microscale
kh. In Fig. 11, hvdi is approximated by the streamwise
decay rate of h02. The formula is taken from the statistical
analysis of grid turbulence by Zhou et al. (2002):
hvdiMUoDT2
¼ � 1
2
d h02=DT2� �d x=Mð Þ : ð11Þ
Each data point shown in Fig. 11 is calculated by using the
3-point centre-difference scheme and then averaged over
its two closest points. To avoid end effects due to the
scheme, the outer 2 points on each end of a batch of 27 data
points are removed. To obtain a power-law expression for
hvdi, we substitute (10) into (11) and use Taylor’s
hypothesis, viz.
hvdiMUoDT2
� � nh
2
t � tho� �
Uo
M
� �nh�1
: ð12Þ
In Fig. 11, the rms difference between the data points
and the power law (12) is no more than 0.5 %. Given that
hvdi, like h02, reasonably follows a power-law decay, we
may write the following expression for kh (after Corrsin
1951b; Monin and Yaglom 1975; George 1992a):
kh2
M2¼ 6j
M2
h02
hvdi¼ � 12j
nhMUo
t � tho� �
Uo
M: ð13Þ
Equation (13) shows that kh2 is a linearly increasing
function of (t - toh)Uo/M, and the measurements in
Fig. 12 support this. It follows that dk2h=dt should be
constant and this can be used to estimate nh, viz.
kh2
M t � tho� �
Uo
¼ � 12jnhMUo
: ð14Þ
The term on the left side of Eq. (14) is plotted in Fig. 13 as
a function of tUo/M. The virtual origin tohUo/M is selected
such that kh2=½M t � tho
� �Uo� and nh are constant for the full
range of measurements.
The results, summarised in Table 2 and Fig. 14, show
that the ‘‘lambda’’ method yields nearly the same virtual
origin (tohUo/M) and decay exponent (nh) as those obtained
by the ‘‘power-law’’ method. However, the lambda method
uses a centre-difference scheme, where the number of
useful data points are reduced from 27 to 23, which slightly
increases the width of the 95 % confidence limit for nh
from ±0.01 to ±0.02. At the optimal tohUo/M, the curve-
fitting error for the mean dissipation rate, rrðhvdiÞ, is
&0.2 %. If tohUo/M = 0, the error rrðhvdiÞ is no more than
0.5 %. Adjusting tohUo/M by ±0.5 changes nh by no more
than ±0.02.
8 Discussion on the decay rates
Table 3 provides a review summary of the available mea-
surements from the present wind tunnel. The results show
-1.3
-1.1Sq35
Incr
easi
ngde
cay
rate
10.0
tθoU
o/M = 0
5.5
-1.4
-1.2
Dec
ay e
xpon
ent f
or te
mpe
ratu
re f
luct
uatio
ns, n
θ
tθoU
o/M = 0
10.0 Rd35
6.5
10 20 50 100
tstart
Uo/M
-1.3
-1.1Rd44w10.0
tθoU
o/M = 0
4.5
Fig. 9 Decay exponent nh as a function of the starting position tstart
for the curve-fitting range ftstart ! tlastg and the virtual origin tohUo/
M. The position of the last data point is fixed at tlast Uo/M & 110. The
error bars are for 95 % confidence limits
20 50 100 150
tstart
Uo/M
0
0.5
σ r(θ’2 ),
(%
)
Grid tθoU
o/M
b
Sq35 0
Rd35 0Rd44w 0Sq35 5.5
Rd35 6.5Rd44w 4.5
0 2 4 6 8 10 12
tθoU
o/M
0
1
2
3
σ r(nθ),
(%
)
Rd44w
Rd35
Sq35
a
Fig. 10 a The root-mean-square (rms) variation in the decay
exponent (nh: 20 \ tstart Uo/M \ 100 in Fig. 9) as a function of
virtual origin tohUo/M. b The rms curve-fitting error for the scalar
variance (h02 in Fig. 8) as a function of tstart and toh for the curve-fitting
range ftstart ! tlastg, where tlast Uo/M & 110
916 Exp Fluids (2012) 53:909–923
123
that, for approximately the same Reynolds number (Rk) and
optimum virtual origin, grids Sq35 and Rd44w produce
very similar thermal decay rates; Rd35 produces the largest
magnitude of the thermal decay rate. With the contraction,
the effect of grid geometry is rather weak and the differ-
ence between nh produced by each grid is small (.0:07).
For the velocity fluctuations produced by the same grids
(i.e. Sq35, Rd35 and Rd44w), Lavoie et al. (2007) indi-
cated that the large-scale anisotropy tends to increase the
magnitude of nu (since Rd44w produces the most isotropic
turbulence with u02=w02 � 0:99); by using the secondary
contraction to improve the flow isotropy, nu is less
dependent on initial conditions. Table 3 shows that, for (up
to four) different grids, the difference between nu produced
by each grid is smaller with the 1.36:1 contraction (.0:12)
than with no contraction (.0:23).
Inspection of the results in Table 3 shows that the
magnitude of nh is generally larger than that of nu. This
trend (i.e. nhJnu) is observed in many studies of turbulent
mixing at low Reynolds numbers (RM & 103 and Rk & 35)
by heating the flow with the grid (e.g. Yeh and Van Atta
1973; Sepri 1976; Sreenivasan et al. 1980) or with a
mandoline downstream of the grid (e.g. Warhaft and
Lumley 1978; Warhaft 1980). The evidence supports
Table 1 Summary of curve fit
using the ‘‘power-law’’ method
The 95 % confidence limit for
nh is �0:01; rrðh02Þ is the rms
difference between the ‘‘log’’ of
the data and the ‘‘log’’ of Eq.
(10). The range of the curve fit
data is shown as (t - toh)Uo/M
Grid tohUo/M (t - to
h)Uo/M -nh rrðh02Þð%Þ
Sq35 0 22–110 1.35 0.25
35–80 1.35 0.08
Rd35 0 22–110 1.46 0.43
35–80 1.44 0.13
Rd44w 0 22–110 1.35 0.31
35–80 1.33 0.09
Sq35 5.5 17–110 1.19 0.08
35–110 1.20 0.04
35–80 1.20 0.05
Rd35 6.5 17–110 1.26 0.10
35–110 1.26 0.10
35–80 1.26 0.09
Rd44w 4.5 17–110 1.22 0.10
35–110 1.21 0.06
35–80 1.21 0.06
10 20 50 100
Streamwise distance from the grid, (t−tθo)U
o/M
10-6
10-5
10-4
10-3
10-2
<χd>M
/UoΔT
2
Sq35
Rd35
Rd44w
tθoU
o/M=6.5
tθoU
o/M=4.5
<χd>∝(t−t
θo)nθ−1
nθ−1 = −2.23
tθoU
o/M=5.5
nθ−1 = −2.27
nθ−1 = −2.21
tθo=0
tθo=0
tθo=0
Fig. 11 Mean dissipation rate of the passive scalar downstream of
each grid with and without adjustment for virtual origin (tohUo/M). The
top 2 set of data are vertically offset by 1 and 2 decades. RM & 10,400
10 20 50 100
Streamwise distance from the grid, (t−tθo)U
o/M
10-2
10-1
100
101
λ θ2 /M2
Rd44w
λθ2/M2∝(t−t
θo)/(−nθ)
Rd35
Sq35
tθoU
o/M=5.5
tθo=0
tθo=0
tθo=0
nθ=−1.21
tθoU
o/M=6.5
nθ=−1.27
tθoU
o/M=4.5
nθ=−1.23
Fig. 12 The Corrsin microscale downstream of each grid with and
without adjustment for virtual origin (tohUo/M). The top 2 set of data
are vertically offset by 1 and 2 decades. RM & 10,400
Exp Fluids (2012) 53:909–923 917
123
Mydlarski and Warhaft’s (1998) notion that the scalar and
velocity fields behave differently and that the difference
cannot be accounted for by the method of heating alone.
In the following Sects. 8.1 and 8.2, we discuss the dif-
ference between the scalar and velocity decay rates from
the perspective of the length-scale and timescale ratios for
small Reynolds and Peclet numbers. The ratios are
important parameters, for example, in the ‘‘calibration’’ of
numerical models to yield results that would match
experimental observations (e.g. Viswanathan and Pope
2008).
8.1 The scalar/velocity length-scale ratio
Durbin’s (1980) theory on turbulent dispersion, which is
extended from the early work of Taylor (1921, 1935a) on
the dispersion of heat from a (line) source in a turbulent air
stream, has since been adapted to model turbulent mixing
with multiple line sources and mandoline (e.g. Sawford and
Hunt 1986; Sawford 2004; Viswanathan and Pope 2008). It
is therefore fitting to provide here a brief summary of
Durbin’s (1982) findings on the scalar decay rate in iso-
tropic turbulence (in the context of length-scale ratio) and
to compare the present measurements with his model
results (reproduced in Fig. 15).
According to Durbin (1980, 1982), the rate of decay of
scalar fluctuations is largely determined by mixing due to
relative dispersion. The length scales or (by Taylor’s
hypothesis) timescales of both scalar and velocity fields are
necessary to describe the scalar decay rate nh. At low/finite
Reynolds numbers, both fields are transient and depend on
their initial scales. Durbin (1982) suggests that ‘‘the exis-
tence of two scales relaxes similarity constraints, so that a
universal decay law need not exist’’. His findings show
that, for isotropic turbulence, nh depends on lu;o=lh;oð.2:5Þ,the ratio between initial length scales for the velocity and
the scalar fluctuations. Figure 15 shows that, for the range
covered by measurements, this dependence is negligible
provided that lu,o/lh,o [ 2.5 (or lh,o/lu,o \ 0.4).
0.8
1.2
1.6
2.0 Sq35
tθoU
o/M=0
4.5
10.0
Incr
easi
ngde
cay
rate
0.8
1.2
1.6
2.0
(λθ2 /[M
(t−t
θ o)Uo])
×10
3
Rd35
tθoU
o/M=0
6.5
10.0
10 20 50 100
Streamwise distance from the grid, tUo/M
0.8
1.2
1.6
2.0 Rd44w
tθoU
o/M=0
5.5
10.0
Fig. 13 The Corrsin microscale as a function of streamwise position
and virtual origin. Equation (14) is used to obtain the decay rate nh
Table 2 Summary of curve fit using the ‘‘lambda’’ method
Grid tohUo/M (t - to
h)Uo/M -nh rrðh02Þ ð%Þ rrðhvdiÞ ð%Þ
Sq35 4.5 17–110 1.23 0.07 0.15
35–110 0.05 0.14
35–80 0.04 0.15
Rd35 6.5 17–110 1.27 0.09 0.17
35–110 0.10 0.15
35–80 0.09 0.16
Rd44w 5.5 17–110 1.21 0.08 0.21
35–110 0.07 0.17
35–80 0.07 0.18
The 95 % confidence limit for nh is ±0.02; rrðh02Þ is the rms dif-
ference between the ‘‘log’’ of the data and the ‘‘log’’ of Eq. (10);
rrðhvdiÞ is the rms difference between the ‘‘log’’ of the data and
the ‘‘log’’ of Eq. (11). The range of the curve fit data is shown as
(t - toh)Uo/M
20 50 100 150
tstart
Uo/M
0
0.5
σ r(<χ d>)
, (%
)
Grid tθoU
o/M
b
Sq35 0
Rd35 0Rd44w 0Sq35 4.5
Rd35 6.5Rd44w 5.5
8 100 2 4 6 12
tθoU
o/M
0
3
6
9
σ r(λθ2 ),
(%
)
Rd44w
Sq35
Rd35
a
Fig. 14 a The root-mean-square (rms) variation in the Corrsin
microscale (kh2: 20 \ tUo/M \ 100 in Fig. 13) as a function of virtual
origin tohUo/M. b The rms curve-fitting error for the mean dissipation
rate (hvdi in Fig. 11) as a function of tstart and toh for the curve-fitting
range ftstart ! tlastg, where tlastUo/M & 100
918 Exp Fluids (2012) 53:909–923
123
For the review experimental data shown in Fig. 15, the
grid flow is not stretched by a secondary contraction and
the spacing between the grid and the downstream mando-
line is large—up to 20 M (Warhaft and Lumley 1978) and
54 M (Sreenivasan et al. 1980). In Fig. 15, the decay rate
nh is determined by plotting the variance h02 as a function
of streamwise distance from the heat source (i.e. mando-
line). For large spacing between the grid and the mandoline
(J5M), Durbin (1982) demonstrated that, by replotting the
data versus distance from the mandoline rather than from
the grid, this slightly reduces the magnitude of nh. With the
present measurements shown in Fig. 15, the distance
between the grid and the mandoline is too small (1.5 M) to
produce a significant change in nh.
To allow direct comparison between temperature and
velocity for the present grids (Sq35, Rd35 and Rd44w) and
to compare with the review data in Fig. 15, we have
obtained simultaneous measurements of temperature (h)
and streamwise-velocity (u) fluctuations using cold and hot
wires. The cold wire is operated under the same condition
described in Sect. 4. The hot wire (diameter d & 2.50 lm;
length l & 200 d) is operated at constant temperature with
an overheat ratio of 1.5. The wires are parallel with
a (fixed) spanwise separation of 1 mm (&1.5–3.0
Kolmogorov lengths). For this test, a total of 9 points are
measured in the range 22. tUo=M. 110; by using
the same methods (Sect. 2) and procedure (Sects. 6 and 7)
with extension to velocity, we have determined that, at
90 % confidence limit, nh and nu are the same as those
reported in Table 3 (within ±0.03 for the decay exponents
and ±1 for the virtual origins).
For each present data point ‘‘9’’ shown in Fig. 15, the
length scales lh and lu are obtained by integrating the
(spatial) auto-correlation function for the temperature and
the streamwise-velocity fluctuations, respectively. This
method is the same as that described by Comte-Bellot and
Corrsin (1971) and Sreenivasan et al. (1980). The ratio of
length scales, lh/lu, is plotted as a function of tUo/M in
Fig. 16. In view of the scatter and to avoid extrapolation,
we have decided to estimate the initial length-scale ratio
lh,o/lu,o by taking the average value over the range of the
Table 3 Review of decay exponents for velocity (nu) and temperature (nh) from the same wind tunnel