Decaying grid turbulence is considered at low Reynolds number (Rλ ~ 50) for different initial conditions. Three different grid geometries are used. Heat is injected via a mandoline at a distance of 1.5 M from the grid. The amount of heating is such that temperature may be treated as a passive scalar. A small contraction (1.36:1) is added at a distance of 11M downstream of the grid. The power-law exponents for the scalar variance are compared with those for the turbulent kinetic energy. These exponents depend on the grid geometry. For the isotropic dissipation rate 〈χ〉iso, the power-law exponent agrees with that inferred from the temperature variance transport equation. Restricting the range of validity of the decay law affects the magnitudes of the origin and decay exponent. Secondorder temperature structure functions collapse when the normalization is based on the local temperature variance and the Corrsin microscale but the asymptotic form of this collapse depends on the initial conditions

Antonia, R. A.

Benaissa, A.

Djenidi, L.

Parker, R.

University of Queensland eSpace

16th Australasian Fluid Mechanics Conference 
Crown Plaza, Gold Coast, Australia 
2-7 December 2007 
 
Effect of Initial Conditions on the Scalar Decay in Grid Turbulence at Low Rλ 
 
 
A. Benaissa1, L. Djenidi2, R. A. Antonia2, R. Parker2 
 
1Department of Mechanical Engineering, Royal Military College of Canada, Kingston Ontario, Canada 
2
 Disciplines of Mechanical Engineering, University of Newcastle, NSW, 2308 Australia 
 
Abstract 
Decaying grid turbulence is considered at low Reynolds number 
(Rλ ~ 50) for different initial conditions. Three different grid 
geometries are used. Heat is injected via a mandoline at a 
distance of 1.5 M from the grid. The amount of heating is such 
that temperature may be treated as a passive scalar. A small 
contraction (1.36:1) is added at a distance of 11M downstream of 
the grid. The power-law exponents for the scalar variance are 
compared with those for the turbulent kinetic energy. These 
exponents depend on the grid geometry. 
 
For the isotropic dissipation rate 〈χ〉iso, the power-law exponent 
agrees with that inferred from the temperature variance transport 
equation. Restricting the range of validity of the decay law 
affects the magnitudes of the origin and decay exponent. Second-
order temperature structure functions collapse when the 
normalization is based on the local temperature variance and the 
Corrsin microscale but the asymptotic form of this collapse 
depends on the initial conditions.  
 
Introduction  
Grid turbulence has been extensively studied because it 
represents an approximation to homogenous isotropic turbulence. 
The similarity analysis of this flow has given a clear view of the 
concept of universal behaviour of turbulence. From the large 
amount of literature available on the subject (e.g. [1,2,3,4,5]) and 
in the recent work of Lavoie et al. [10], it has become clear that 
George’s [7] ideas about the influence of initial conditions on 
similarity need to be carefully taken into account.  
 
Lavoie et al. [8,9,10] clearly established that the decay law of 
mean turbulent energy depends on the organization of the flow 
downstream of the grid. The nature of this organization imposes a 
range of validity on the decay law which differs from grid to grid, 
at nominally the same Rλ.  
 
Different grids were used to study decay laws of different 
dynamic properties (the mean turbulent energy, the mean 
turbulent energy dissipation rate and the entropy). A contraction 
was also used to improve isotropy (mainly at the large scales), as 
described in [6] and [15]. The contraction reduced the anisotropy 
of the large scales and yielded an improved approximation to 
HIT. It also lessened the importance of initial conditions on the 
scaling range and its exponent.  
 
For the decay of the temperature variance, previous studies 
[1,3,17] have yielded different decay exponents, most likely 
reflecting differences in initial conditions, which have not always 
been adequately described. Zhou et al. [17] showed that the 
decay law exponent of 〈χ〉 (the mean dissipation rate of 〈θ 2〉/2) 
and  ( )23iso k xθχ ∂〈 〉 = ∂  are the same. 〈χ〉 is given by: 
2
0
2 ( )dχ φ
∞
= ∫κ k k k  (1) 
 
and the temperature variance <θ 2> is given by 
 
 2
0
( )dθ φ
∞
= ∫ k k   (2) 
 
George [7] showed that the equation for scalar spectrum satisfied 
similarity (at all scales) with 〈θ  2〉 as the relevant temperature 
scale and the Corrsin microscale scale λθ ~ (x)1/2 as the relevant 
length scale. The exponent n in the variance decay law depends 
on initial conditions. 
 
Introducing the notion of effective origin for the decay, we can 
write: 
 
2
0( / / )
nA x M x Mθθ〈 〉 = −  (3) 
 
and 2 06 ( )k x x
nUθ
λ −= −  (4)   
which leads to 2 0 2
6/ ( ) kM x x const
nUMθ
λ − = − =  (5) 
This latter relationship was used by Antonia et al. [3] to 
determine x0 and n over a wide range of x/M, between 30 and 80. 
 
In the present study, we analyse the effect of initial conditions on 
the decay of the temperature variance. Three different grids 
already used in Lavoie et al. are chosen [10] and only the Power 
Law Decay Range (PLDR), where the decay law is satisfied, is 
considered. Because only the test section with the secondary 
contraction is used, the following definition is adopted: 
 
x = tU0, where t is defined as in Comte-Bellot and Corrsin [6]. 
 
                   
0
( )
x dst
U s
= ∫  (6) 
 
Structure functions are also calculated and compared for two 
different grids.  
 
914
 
Figure1: Schematic of the wind tunnel (Lavoie et al. [10]) 
 
 
Figure2: Schematic of grid geometries (Lavoie et al. [10]) 
 
Experimental Conditions 
The flow is generated in an open ended wind tunnel with a square 
section (350mm×350mm, 2.4 m long) as shown in figure 1. The 
grids are located at the entrance of the working section and their 
specifications are summarized in table 1. Temperature 
measurements are carried out at a mean velocity of U = 6.4 m/s 
on the centreline of the working section downstream of a biplane 
grid heated mandoline similar to that of Warhaft and Lumley [16] 
and Sreenivasan et al. [14].  The mandoline, of mesh size Mθ=M, 
is located at a distance of 1.5 M downstream of the grid and is 
made from a 0.5 mm-diameter chromel-A wire. The heating is 
implemented as in Zhou et al. [17]; the mean temperature 
increase relative to ambient is sufficiently small (~2 °C) for  
temperature to be considered passive. The temperature probe  
consists of one wire operating in a constant current (equal to 0.1 
mA) mode. The wire is etched from Wollaston (Pt-10% Rh) to an 
active length of about 800dw (dw =0.63 μm). The output signals 
from the constant current circuit are digitized with a 12-bit 
analog-to-digital converter at a sampling frequency close to 2 fK  
(where fK =U/2πη is the Kolmogorov frequency) after the low-
pass filter cutoff frequency was set to be approximately equal to 
fK . Taylor’s hypothesis is applied to convert temporal increments 
to spatial increments. The focus is on how the temperature 
variance and its dissipation rate decay with the streamwise 
direction x for the different grid types. 
 
Grid Geometry M 
(mm) 
d 
(mm) 
σ RM 
Sq35 Square 24.76 4.76 0.35 10,564 
Rd35 Round 24.76 4.76 0.35 10,564 
Rd44w Round 24.76 6.35 0.44 10,564 
 
Table 1: Geometry of the grids and flow conditions. ( )2d d MMσ = −  is 
the grid solidity, and d is the size of the bars. RM is Reynolds number 
relative to the mesh size M. 
 
Experimental Results and Scalar Similarity Analysis  
 
Temperature variance decay as for mean turbulent energy can be 
written as in Zhou et al. [17]: 
 
2' nxx
T M M
θθ⎛ ⎞ ⎛ ⎞
∝ −⎜ ⎟ ⎜ ⎟⎜ ⎟Δ ⎝ ⎠⎝ ⎠
  (7) 
915
where xθ  is the distance between the grid and the mandoline. In 
our case, this distance was small and did not affect the scaling 
exponent n.  In figure 3, the temperature variance as a function of 
(x−x0)/M for x/M between 30 and 100 is plotted. A power law can 
be fitted to the data over a range extending to a distance of 100M 
from the grid. For Sq35, Zhou et al. [17] found n = −1.46 while 
Antonia et al. [3] obtained n = −1.37; in the case of the present 
study, the exponent was −1.11, It would seem that the difference 
was caused by the secondary contraction. Our exponents compare 
well with the results of Lavoie et al. [10] for the case with 
contraction.   
 
0.0001
0.0010
0.0100
10 100
Δ     Sq35 :     25 < x/M< 77   ;           n = -1.11
  Rd 35 :        25 < x/M< 77   ;        n = -1.23
        Rd44w:       25 < x/M< 77   ;       n = -1.05
<θ2>/ΔT2
(x -x0)/M
 
Figure 3: Variation of the normalized temperature variance with x/M for 
the 3 different grids and x0 = xθ. Straight lines are power laws fitted to the 
data. 
For decaying grid turbulence, the transport equations for 〈θ 2〉 can 
be simplified to: 
 
2 2
0
2 2
d d u
dt dx
θ θ χ
⎛ ⎞ ⎛ ⎞〈 〉 〈 〉
+ + 〈 〉 =⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
  (8) 
 ( ) ( ) ( )2 22k x y zθ θ θχ ⎧ ⎫⎪ ⎪∂ ∂ ∂〈 〉 = + +⎨ ⎬∂ ∂ ∂⎪ ⎪⎩ ⎭  (9) 
As stated in Zhou et al. [17], the second term of the left side 
equation (8) was about 200 times smaller the other term which 
leads to: 
 
2
0
2
d
dt
θ χ
⎛ ⎞〈 〉
+ 〈 〉 =⎜ ⎟⎜ ⎟⎝ ⎠
 (10) 
The streamwise decay of 〈χ〉 thus followed a power law with a 
decay exponent of n−1. To verify this, 〈χ〉 was plotted as a 
function of (x−x0)/M for the three grids in figure 4, the same as in 
figure 3. The agreement between the decay exponents of 〈χ〉 and 
〈θ 2〉 was good and reflected the validity of equation (10) over 
almost the entire range of measurements. 
 
From figures 3 and 4, one can notice the difference in the decay 
exponent between the grids which support the finding of Lavoie 
et al. [10] for the decay of mean turbulent energy. In this phase of 
decay, initial conditions affected the decay law.  
 
If a universal power law exists, its determination is not easy, 
particularly when it has to be determined over a specific PLDR. 
Because our data are limited to high values of tU0/M, a limitation 
imposed by the experimental setup, we have assumed that Lavoie 
et al.’s [10] determination of the PLDR also applies here. We 
argue that the similarity of the dynamic field has to be established 
before that of the scalar field. To correctly determine the power 
law exponent n, the origin x0 needs to be determined. With the 
PLDR, we now have 3 parameters to determine in order to 
estimate n. In our approach, we combine the method developed 
by Lavoie et al. [10], which is based on fitting to the mean 
turbulent energy q2 data, with that of Antonia et al. [3] which 
uses the linearity of λ2. This linearity is used to infer x0; as we 
can see from figure 5 (for grid Sq35), restricting the range to the 
PLDR yields a smaller value of n with a slightly different x0. 
0.001
0.010
0.100
1.000
10 100
<χ>M/UΔT2 
(x -x0)/M
Δ     Sq35 :     25 < x/M< 77   ;           n = -2.15
  Rd 35 :        25 < x/M< 77   ;       n = -2.26
        Rd44w:       25 < x/M< 77   ;      n = -2.08
 
Figure 4: Variation of normalized mean temperature dissipation rate with 
x/M for the 3 different grids (as in figure 3). 
To visualize the effect of restricting the range on the power law 
exponent, we present in figure 6 (for the same grid) 〈θ 2〉1/n as a 
function of (x − x0)/M as in Antonia et al. [3], the difference 
being that, in the present case, x0 is different from zero (see 
figure 5).  
0
0.00001
0.00002
0.00003
0.00004
0.00005
10 20 30 40 50 60 70 80
(x -x0)/M
λθ2 /Μ(x−x0)
   25 < x/M< 77   ;   x0 = 9  ;       n = -1.33
           35 < x/M< 77   ;   x0 = 10  ;      n = -1.28
 
Figure 5:  Ratio of λ2θ with (x−x0)/M for different ranges of x/M. Open 
symbols correspond to the data calculated with x0/M = 10 (with the PLDR 
from Lavoie [10]; the horizontal line corresponds to n= −1.28).  
0
20
40
60
80
100
10 30 50 70 90
<θ2>1/n
(x -x0)/M
  25 < x/M< 77   ;   x0 = 9  ;       n = -1.33
        35 < x/M< 77   ;   x0 =10  ;      n = -1.28
 
Figure 6:  Variation of 〈θ 2〉 1/n with (x−x0)/M obtained from figure 5 (grid 
Sq35). The straight lines through the origin are least squares regressions 
to the data. 
The two lines represent least-squares regressions to the data and 
crossing at the origin. In the lower range of x/M, the data are 
above the line and the opposite for the higher end. The collapse is 
better in the central region of the PLDR. For the 2 others grids, 
PLDR 
916
this approach is used over the PLDR and x0 and n are determined, 
the results are plotted in term of 〈χ 〉1/n-1 as a function of (x−x0)/M 
in figure 7. A good linear variation is observed for the 3 different 
grids with different slopes as observed earlier. These new 
exponents are different from the previous ones. 
 
For the 2 others grids, this approach is used over the PLDR and 
x0 and I are determined, the results are plotted in terms of  〈χ 〉1/n-1 
as a function of (x−x0)/M in figure 7. A good linear variation is 
observed for the 3 different grids with different slopes as 
observed earlier. These new exponents are different from the 
previous ones. The Sq35 grid gives an exponent comparable to 
Antonia et al. [3]. 
0
0.01
0.02
0.03
0.04
0.05
10 30 50 70
<χ>1/(1-n)
(x -x0)/M
   Rd44w  :     25 < x/M< 77   ;      n = -1.28
  Sq35 :            25 < x/M< 77   ;      n = -1.33
Δ        Rd 35 :       35 < x/M< 77   ;       n = -1.16
 
Figure 7: Variations of 〈χ〉 1/n-1 with (x−x0)/M, x0 and n are determined as 
in figure 6. The straight lines through the origin are linear regressions to 
the data 
 
Figure 8: Second-order structure functions of temperature for Rd35 and 
Sq35. 
 [ ]22 ( ) ( )x x xθ θ θΔ = + Δ −  (11) 
 
Second-order structure functions of temperature (Equation (11)) 
are represented in figure 8. They are normalized with temperature 
variances and separation is normalized by Corrsin’s length scale 
λθ. The collapse on the solid thick line, which represent the 
asymptotic state for grid Rd35, is attained at about x/M = 40. For 
Grid Sq35, the broken line represents the asymptotic curve 
reached also at about 40 M and it is different from grid Rd35. 
This is in agreement with Antonia et al. [3].  
0
500
1000
1500
2000
2500
3000
0.0 0.2 0.4 0.6 0.8 1.0
f2φ (f/fk)
f/fk
  Rd35 ;  x/M = 30
 
Figure 9: Modelling the dissipation spectrum. The solid line is the model 
and the broken line is the experimental data for grid rd35 at x/M = 30.  
 
Another approach for studying similarity is to calculate 
normalized spectra and examine their evolution with x/M. We 
chose to model the spectra as in Lemay et al. [11], viz. 
 
1.65( ( / ) ( / ) )( / ) k kn A f f B f f Ckf f f eθφ − + +∝  
 
with A= 1,97, B = −9.33, C = 7.81 and nθ = 1.42. 
 
The coefficient of proportionality is determined for each position 
to provide an adequate fit to the experimental data as in Figure 9. 
Fitting across the inertial and dissipative ranges is also used to 
calculate the isotropic dissipation rate. 
 
Conclusions 
Decaying grid turbulence at low Reynolds numbers (Rλ ~ 50) 
depends on initial conditions, irrespectively of whether the mean 
turbulent energy [5,9,10] or temperature variance is considered. 
Three different grid geometries are used with heat injected via a 
mandoline. The power-law exponents for the scalar variance 
compare are lower than previous results when they were 
considered over a large range of positions downstream of the grid 
(30M to 100M). This would seem to be caused by the secondary 
contraction. 
  
The power-law exponent for the isotropic dissipation rate 〈χ〉 
agrees with that inferred from the temperature variance transport 
equation, irrespectively of the grid used. Restricting the range of 
validity of the decay law affects the magnitudes of the effective 
origin and decay exponent. In this range, the contraction does not 
seem to affect the behaviour of the scalar as much as that of the 
turbulent energy. Second-order temperature structure functions 
collapsed when the temperature variance and the Corrsin 
microscale λθ are used for the normalization. The asymptotic 
states depend on the initial conditions.  
  
Acknowledgements 
 
The support of Natural Science and Engineering Research 
Council (NSERC) of Canada is gratefully acknowledged. We are 
also indebted to P. Lavoie for his contribution. 
 
<(Δθ)2>/<θ2> 
〈χ〉 1/n-1 
917
Nomenclature 
 
Aθ Constant of proportionality  
dw Cold wire diameter 
fk Kolmogorov frequency 
k Thermal conductivity 
M  Mesh size 
Mθ Distance between the grid and the Mandoline 
n Decay exponent for kinetic energy  
nθ Decay exponent for temperature variance 
Rλ Taylor microscale Reynolds number ( =  λu’/ν ) 
t  Time 
x0 Virtual origin 
U Mean velocity 
U0  Free stream velocity approaching the grid 
u’ Velocity fluctuation ( streamwise component) 
 
Symbols 
 
θ Temperature fluctuations 
χ Mean temperature dissipation rate 
K Wave number  
φ Power spectrum 
η Kolmogorov length scale 
σ Solidity ( ( )2d d MMσ = −  
λ Taylor microscale ( )//(' xuu ∂∂= ) 
 
Subscripts 
 
iso Isotropic 
k  Kolmogorov 
 
References 
 
[1] Antonia, R.A. & Orlandi, P., Similarity of decaying 
isotropic turbulence with a passive scalar, J. Fluid Mech., 
505, 2004, 123–151. 
[2]  Antonia, R.A., Ould-Rouis, M., Anselmet, F. & Zhu, Y., 
Analogy between predictions of Kolmogorov and Yaglom, 
J. Fluid Mech., 332, 1997, 395–409. 
[3] Antonia, R.A., Smalley, R.J, Zhou, T., Anselmet, F. and 
Danaila, L., Physical Review E, 69, 2004 016305. 
[4] Batchelor, G.K., The Theory of Homogeneous Turbulence, 
Cambridge University Press, 1953. 
[5] Burattini, P., Lavoie, P., Agrawal, A., Djenidi, L. & 
Antonia, R.A., On the Power Law of Decaying 
Homogeneous Isotropic Turbulence at Low Reynolds 
Number, Phys. Rev. E, 73, 2006, 066304. 
[6]  Comte-Bellot, G. & Corrsin, S. 1966 The Use of a 
Contraction to Improve the Isotropy of Grid-generated 
Turbulence, J. Fluid Mech., 25, 657–682. 
[7] George, W.K., The Decay of Homogeneous Isotropic 
Turbulence, Phys. Fluids, 4, 1992, 1492–1509. 
[8]  Lavoie, P., Burattini, P., Djenidi, L. and Antonia, R.A., 
Effect of Initial Conditions on Decaying Grid Turbulence at 
Low Rλ, Exp. Fluids, 39 (5), 2005, 865–874. 
[9]  Lavoie, P., Djenidi, L. & Antonia, R.A., Effect of Initial 
Conditions on the Generation of Coherent Structures in Grid 
Turbulence, in Whither Turbulence Prediction and Control 
Conference (ed. H. Choi) Seoul National University, 2006. 
[10] Lavoie, P., Djenidi, L. and Antonia, R.A., Effect of Initial 
Conditions in Decaying Turbulence Generated by Passive 
Grids, Under consideration for publication in J. Fluid Mech, 
2006. 
[11] Lemay, J, Benaissa A, Antonia R.A. Correction of Cold-
wire Response for Mean Temperature Dissipation Rate 
Measurements. Experimental Thermal and Fluid Science, 27 
2003, 133-143  
[12] Mohamed, M. S. & LaRue, J. 1990 The decay power law in 
grid-generated turbulence. J. Fluid Mech. 219, 195–214.
 Mohamed, M.S. & LaRue, J., The Decay Power Law in 
Grid-generated Turbulence, J. Fluid Mech. 219, 1990, 195–
214. 
[13]  Speziale, C.G. & Bernard, P.S., The Energy Decay in Self-
preserving Isotropic Turbulence Revisited, J. Fluid Mech., 
241, 1992, 645–667.Monin, A.S. & Yaglom, A.M., 
Statistical Fluid Mechanics, vol. 2 . MIT Press, 1975. 
[14] Sreenivasan K.R., Tavoularis S., Henry R., Corrsin S. 
Temperature Fluctuations and Scales in Grid-generated 
Turbulence. J. Fluid Mech, 100, 1980, 597-291. 
[15]  Uberoi, M.S., Effect of Wind-tunnel Contraction on Free-
stream Turbulence, J. Aero. Sci., 23, 1956, 754–764. 
[16] Warhaft, Z. and Lumley, J.L. An Experimental Study of the 
Decay of Temperature Fluctuations in Grid Generated 
Turbulence. J. Fluid Mech, 88, 1978, 659-684. 
[17] Zhou, T, R.A. Antonia, Chua L.P. Performance of a Probe 
for Measuring Turbulent Energy and Temperature 
Dissipation Rates. Experiment in Fluids, 33, 2002, 334-345. 
 
 
  
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Effect of Initial Conditions on the Scalar Decay in Grid Turbulence at Low Rλ

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Effect of Initial Conditions on the Scalar Decay in Grid Turbulence at Low Rλ

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