1. Analysis of the Decay of Moderately High Intensity, Nearly
Homogeneous, Isotropic Turbulent Flow
MEng in Aeronautical Engineering
Final Year Project 2015/2016
Department of Aeronautical Engineering, University of Glasgow
Department of Mechanical and Aerospace Engineering, University of California
Irvine
Lee Catherine Ramsay 1103072R
Supervisors:
Dr Craig White (University of Glasgow)
Professor John C. LaRue (University of California, Irvine)

2. 2
Acknowledgements
I would like to thank my supervisor, Professor John C. LaRue, for allowing me to use his lab facilities
for this study. I would also like to express my deepest appreciation to Mr Timothy Koster whose
patience and guidance made this project possible and also, Mr Pierre Vanderbecken who assisted with
the experimental process. Finally, I would also like to thank all of the other graduate and
undergraduate students, and interns within the laboratory for their additional support and kindness.

3. 3
Contents
Nomenclature ……………………………………......…………………….…………………….4
List of Figures………………………………………………...………………...………...............6
List of Tables ………………………………………………………..………...………................7
1. Abstract…………………………………………….…………………...…...………………..8
2. Introduction…………………………………………………………………..………………9
3. Background Theory ………………………………………………………………....……...10
3.1 Power Law Decay …..……………………………………………………………....10
3.2 Measures of Isotropy ………..……………………………………………………...13
3.3 Extension of the Power Law ………………………………………………..............14
3.4 Infinite Reynolds Number ………………………………………………….............15
4. Experimental Setup ………………………………………………………………………...16
4.1 Wind Tunnel ………………………………………………………………………..16
4.2 Turbulence Generator ……………………………………………………………...17
4.3 Homogeneity …………….…………………………………………………………20
4.4 Sensors ……………………………………………………………………..............22
4.5 Calibration Procedure ……………………………………………………………...24
5. Experimental Procedure ……………………………………………………………………27
5.1 Single Hotwire …..………………………………………………………….............27
5.2 Crosswire …………………………………………………………………………...27
5.3 Sampling Rates ……………………………………………………………………..27
6. Results ……………………………………………………………………………………...28
6.1 Validation …………………………………………………………………………..28
6.2 Single Hotwire Results ……………………………………………………………..35
6.3 Turbulent Flow Characteristics ……………………………………………..……...43
6.4 Dissipation …………………………………………………………..……………...49
7. Further Research……………………………………………………...……………...…...….53
8. Conclusion……………………………………………………....……...……..………...…..53
9. References...……………………………………………………..………...………………..54
Appendix A: MATLAB code for flattening the nearly isotropic range of u2
/(x/Mu-x0/Mu),
∈/(x/Mu-x0/Mu) and λ2
/(x/Mu-x0/Mu) …………………………….…………..…………...…57
Appendix B: MATLAB code for finding V0, nu, and n∈ ………………………………….……59
Appendix C: 6ms-1
data for V0, nu and n∈ obtained using MATLAB ……………………….…63

4. 4
Nomenclature
∈: Dissipation rate
∈*: Power law estimated dissipation rate
λ: Taylor length scale
η: Kolmogorov length scale
A: Downstream decay coefficient
Ak: First constant of King’s Law
Bk: Second constant of King’s Law
CTA: Constant Temperature Anemometry
D: Wire diameter
E: Voltage between anemometer prongs
E(k): Velocity power spectrum
In: Loitsianskii integral
K(): The Kurtosis of a PDF
𝑘: Wavenumber
L: Wire length
Lu: Integral length scale
n∈: Dissipation exponent
nu: Decay exponent
Mu: Active Grid mesh size
PDF: Probability Density Function
PRT: Platinum Resistance Thermometer
q2
: Turbulent kinetic energy
Rw: Resistance of the wire
Rλ: Taylor Reynolds number
S(): Skewness of a PDF
SHW: Single hotwire
T 𝑘 : Transfer power spectrum
U: Mean downstream velocity
u: Variance of downstream velocity
UV: u and v components of velocity
UW: u and w components of velocity
v: Variance of velocity along y-axis
V0: Virtual origin

5. 5
w: Variance of velocity along z-axis
x: Downstream distance measured with origin at the active grid
Xs: the estimated starting point of the u/v and u/w flat region
X0: Downstream position of the virtual origin
XW: cross-wire
XW-UV: Data obtained by UV configuration of crosswire
XW-UW: Data obtained by UW configuration of crosswire

6. 6
List of Figures
Figure 1: Closed-return wind tunnel, image courtesy of Selzer (2001) ……………….…….……..17
Figure 2: Active grid with rod mounted, square agitator flaps ………………………………..…....18
Figure 3: Layout and coordinate frame of the wind tunnel, Nguyen (2015) ……………….............19
Figure 4: (A) 𝑈, (B) 𝑢+, (C) 𝑆 𝑢 , (D) 𝜕𝑢/𝜕𝑡 + and (E) 𝑆 𝜕𝑢/𝜕𝑥 for x/Mu=142 and
z/Mu=0 for 4ms-1
………………………………………………………………………….21
Figure 5: (A) Side and (B) birds-eye view of crosswire anemometer ……………………………...22
Figure 6: (A) u2
/U2
(B) v2
/U2
(C) w2
/U2
plotted as functions of downstream position for
4ms-1
……………………………………………………………………………………...31
Figure 7: (A) u/w, (B) u/v and (C) v/w as a function of downstream location for 4ms-1
…..............32
Figure 8: (A) XW-UV decay of turbulent kinetic energy (B) XW-UW decay of turbulent
kinetic energy ………………………………………………………………….………...34
Figure 9: (A) u2
, (B) du2
and (C) ∈ as a function of downstream location for 4ms-1
……………...36
Figure 10: (A) u2
/U2
, (B) du2
/U2
and (C) ∈/U2
as a function of downstream location for
4ms-1
……………………………………………………………………………..............38
Figure 11: (A) S(u) and (B) K(u) as a function of downstream location for 4ms-1
………………...39
Figure 12: (A) S(du) and (B) K(du) as a function of downstream location for 4ms-1
……………...40
Figure 13: (A) λ2
and (B) λ2
*U as a function of downstream location for 4ms-1
………………...41
Figure 14: (A) λ2
and (B) λ2
*U as a function of downstream location for 6ms-1
………………...42
Figure 15: (A) u2
/(x/Mu-x0/Mu), (B) ∈/(x/Mu-x0/Mu) and (C) λ2
/(x/Mu-x0/Mu) as a function
of downstream location for 4ms-1
………………………………….................................44
Figure 16: (A) XW-UV and (B) XW-UW for ∈/∈* as a function of downstream location for
4ms-1
…………………………………………………………………………….............51

7. 7
List of Tables
Table 1: Decay coefficient, nu, as found by various research papers ...................................................11
Table 2: Summary of results from research using a linear fit of 𝜆+
= 𝑚(𝑥/𝑀5) + 𝐵 ………………12
Table 3: Summary of results from active grid studies ……………………………………….............13
Table 4: Details of user input for active grid …………………………………………………...........18
Table 5: Calibration velocities for single hotwire ……………………………………………...........25
Table 6: Calibration velocities for crosswire …………………………………………….….………25
Table 7: Crosswire calibration angles and velocities ………………………………………………26
Table 8: Corner frequency, fc and sampling rate for specific downstream locations ……….………..28
Table 9: Mean and standard deviation for raw variables …………………………………….............29
Table 10: Mean and standard deviations between calibrations and velocity components ……...........30
Table 11: Average ratios of turbulent fluctuations ………………………………………...…...........33
Table 12: Estimation of location of the start of nearly isotropic region …………………..………….35
Table 13: Flat range results for various isotropic region start points for 4ms-1
………..…..………….45
Table 14: Virtual origin, V0, decay exponent, nu, and dissipation exponent, n∈, as a function of
Start point x/Mu = 95 ……...………………………………………………………………...46
Table 15: Virtual origin, V0, decay exponent, nu, and dissipation exponent, n∈, as a function
of start point x/Mu = 100 ………………………………………………………………...46
Table 16: Virtual origin, V0, decay exponent, nu, and dissipation exponent, n∈, as a function
of start point x/Mu = 95 ………………………………………..………………………...47
Table 17: Normalized virtual origin, V0, decay exponent, nu, and dissipation exponent, n∈, as
a function of start point x/Mu = 95 …………………………….………………………...47
Table 18: Reduced single hotwire reference dataset comparison for virtual origin, V0, decay
exponent, nu, and dissipation exponent, n∈, as a function of start point
x/Mu = 95 …......................................................................................................................48
Table 19: Taylor Reynolds number and turbulence intensity ………………………………….…...49
Table 20: Average value and standard deviation of ∈/∈* results for isotropic region ………………52
Table 21: Average of ∈/∈* for multiple velocities and turbulent kinetic energy
configurations ……………………………………………………………...……….....…52

8. 8
1. Abstract
Decaying homogenous-isotropic turbulent flow, generated by an active grid, is assessed by means of
hotwire anemometry for mean velocities of 4ms-1
and 6ms-1
, with consideration of the power law to
describe the decay of the turbulent flow. The statistical data is validated and assessed by the data
obtained using a single hotwire anemometer, and finds the flow to be nearly isotropic downstream of
the active grid.
The results, whilst holding considerable uncertainty due to a less than sufficient number of data points
measured in the nearly isotropic range, find the virtual origin to be V0= 35.919 downstream of the
active grid, with a decay exponent of nu= -2.045, and a dissipation exponent, n∈ = -3.047. Results
found in this study also include several accurate descriptions of power law fit decay. However, the
lack of fully isotropic flow hinders some of the later results describing the nature of the flow. For
example, the ratio of dissipation computed from the time derivative of the velocity, ∈, to the
corresponding value computed using the power law decay for turbulent kinetic energy, ∈*, should be
1. Both crosswire velocity configurations, UV and UW, find this ratio to be less than 1 for 4ms-1
,
whilst for 6ms-1
the UV configuration generates a ratio of less than 1, and the UW configuration
produces a ratio greater than 1.

9. 9
2. Introduction
The study of turbulence has always been complicated, due to its non-linearity and the random nature
of the phenomenon. Despite its complexities, there are three major motivators for the study of
turbulent flow: most fluid flows are turbulent in nature; the transport and mixing of matter,
momentum and heat in flows are all highly important; and turbulence greatly influences the rates of
each of these processes.
Currently, there is no general analytical solution by which turbulent flows can be modelled. The
present knowledge surrounding turbulence has been derived from experimental results produced from
homogeneous and isotropic flows, where the nature of the describing equations is much easier to
compute. With the inclusion of several considerations such as similarity, order of magnitude analysis
and self-preservation, a basic understanding of turbulent flows has been widely achieved.
The most accurate representation of decaying homogeneous, isotropic turbulence to date can be found
in a wind tunnel behind a passive biplane grid consisting of rods and equally spaced flaps. However,
there are limitations to the passive grid which makes the use of one undesirable in this study. A
conventional passive grid limits the turbulence intensity of the flow to approximately 3% or less, with
a Taylors Reynolds number rarely exceeding 150. Therefore, it is much more effective for an active
grid to be used in this study. The active grid utilises rotating rods and square cross sections to achieve
a much greater turbulence intensity of approximately 20%, and a Taylor Reynolds number of up to
1000, Kang et al (2003).
Initiated by Tennekes & Lumley (1972), and later backed by several other analytical studies, it has
been found that the region of homogeneous-isotropic flow in the downstream decay of the turbulent
kinetic energy, and the Taylor micro-scale, can be described by a power law simply by applying
dimensional analysis to the governing terms in the turbulent kinetic energy equation.
The entire flow field downstream of a turbulence-producing grid can be divided into three distinct
regions. The first of which, nearest to the grid, is the developing region where the rod wakes are
merging, the flow is inhomogeneous, anisotropic and consequently, there is a production of turbulent
kinetic energy, Mohamed & LaRue (1990). Following on from this region, the flow becomes nearly
homogeneous, isotropic and locally isotropic and there is appreciable energy transfer from one wave
to another; it is in this region that the form of the power decay law used in this study is applicable.

10. 10
Lastly, the final period of decay is found to be the location furthest downstream of the grid, where
the viscous effects act directly on the large energy scales.
In this study, the form of the decay power used is only applicable to region two, and so, only data
from within that region will be used to determine the decay exponent, the decay coefficient and the
virtual origin. For this reason, criteria which relate to the identification of the positions downstream
of the active grid where the power law decay region begins and ends, and in particular, the location
at which the flow becomes nearly homogeneous and isotropic will be analysed by making use of hot
wire anemometry to obtain time-resolved velocity measurements in both the downstream and traverse
directions along the length of the wind tunnel test section.
3. Background Theory
3.1 Power Law Decay
The equation for the turbulent kinetic energy in homogeneous-isotropic flow, neglecting lower order
terms, is given as:
𝜖∗
= −
;
+
𝑈
<=>
<?
, (1)
Where 𝑈 is the mean velocity, u2
is the variance of the velocity defined as 𝑢+
≡ 𝑢 − 𝑈 +, ∈∗
is the
dissipation and x is the downstream distance measured with the origin at the active grid.
Applying dimensional analysis to the governing terms of the above equation was suggested by
Tennekes and Lumley in 1972 (pp 71-73) and led to the conclusion that the velocity variance, 𝑢+,
should follow a power law decay proportional to x-n
, whilst the Taylor length scale, along with other
length scales and variables, should increase in proportion to xn/2
. However, Tennekes and Lumley
(1972) used “crude” assumptions and estimates of the time scale of the energy transfer from large to
small scales and the decay time of the large eddies. Therefore, whilst the analysis suggests that n=1,
the experimental values found for n will most likely not equal 1.
Earlier studies by Kolmogorov (1941) and Saffman (1967) used an alternative method (2) based on
the Loitsianskii (1939) integral for the decay exponent which they believed was required to be
constant:

11. 11
𝐼C = 𝑟C
𝑢 𝑥 𝑢 𝑥 + 𝑟 𝑑𝑟 = 𝐶𝑜𝑛𝑠𝑡𝑎𝑛𝑡
K
L
, (2)
where 𝑛 denotes the order of the Loitsianskii integral, 𝑟 is the separation distance, 𝑢 is the
downstream velocity, and indicates the time average. In both studies, the velocity correlation is
assumed to be self-similar, i.e. 𝑢 𝑥 𝑢(𝑥 = 𝑟) = 𝑢+
𝑓(𝑟/𝐿), with the dissipation related to the
integral length scale, 𝐶O =
O P=
=>
Q
>
~𝐶𝑜𝑛𝑠𝑡𝑎𝑛𝑡, but differ over the order of Loitsianskii integral to be
held constant. Kolmogorov assumes that the fourth order integral is constant, leading to a decay
exponent of 10/7, whilst Saffman predicts that it is the second order integral that is held constant,
leading to a decay exponent of 6/5.
The above form given by eqn. (1) for the turbulent kinetic energy equation, used by both researchers,
concludes that for the downstream decay to accurately be described by a power law, the addition of
the virtual origin parameter is required. Thus, the form of the power decay of the velocity variance,
inclusive of the virtual origin, x0/Mu but otherwise referred to as V0, is given below, in equation (3).
=>
5>
= 𝐴
?
T=
−
?L
T=
UCV
, (3)
where 𝑥 is the position downstream of the grid, 𝑥L is the position of the virtual origin, uM is the
active grid mesh size, and 𝐴 is a coefficient dependent on initial conditions (George, 1992). The decay
coefficient, nu, is not constant and varies between studies, as illustrated in Table 1.
Researcher 𝑅X (x/M) nu Virtual origin, V0 Comments
Comte-Bellott & Corrsin
(1966)
N/A(30)
N/A(30)
1.300
1.270
2.0
2.0
𝑢+/𝑈+
longest linear
range
Mohamed & LaRue
(1990)
28.37 (40)
43.85 (40)
1.309
1.299
0
0
Least-square fit to
𝑢+/𝑈+
Antonia et al. (2003) N/A(40) 1.32 -0.177 𝜆+
/𝑀(𝑥 − 𝑥L) flat
Lavoie et al. (2005) 42 (40)
40 (60)
1.20
1.29
6.0
3.0
Constant decay
exponent
Table 1: Decay coefficient as found by various research papers

12. 12
Although the decay exponent is never equal to 1, table 1 does show promise for Saffman’s predicted
value of nu = 1.2, (1967). The differences are expected; each of the studies exhibit small differences
in initial conditions, different indicators for the isotropic range, different virtual origins and methods
by which to calculate the virtual origin.
George’s (1992) similarity analysis also lead to two noteworthy revelations:
1. The squared value of the Taylor length scale, 𝜆+
, in homogeneous-isotropic turbulent flow is
directly proportional to downstream position.
2. The decay exponent, nu, and the virtual origin, 𝑉L can be produced from a linear fit of 𝜆+
=
𝑚(𝑥/𝑀5) + 𝐵.
These findings were implemented by both George (1992) and later Antonia et al (2003), with their
results tabulated in Table 2.
Table 2: Summary of results from research using a linear fit of 𝜆+
= 𝑚(𝑥/𝑀5) + 𝐵
In their study, Antonia et al (2003) commented on the results, specifically that the linear fit approach
leads to the same values of 𝑉L and 𝑛= as the power decay law for the velocity variance.
With the above values recorded for passive grid studies, it is important for this study to consider the
values of the virtual origin and decay exponent for an active grid. A summary of the results from a
variety of studies, all of which utilize active grids, is given below in Table 3. The range of decay
exponent for the active grids is 1.21 to 1.43 and does not vary consistently with the value of Taylor
Reynolds number between studies. Although this contradicts one of the predictions made by George
(1992), it is acceptable to suggest that the inconsistencies in the behaviour of the decay exponent may
Researcher Turbulence
generation
𝑚 𝐵 Virtual origin, V0 𝑛=
George (1992) Passive
grid
0.00625
0.00314
-0.0284
-0.0281
4.54
8.95
-1.21
-1.20
Antonia et al. (2003) Passive
grid
3.961𝐸U`
7.012𝐸Ud -0.177 -1.32

13. 13
simply be a result of either the differences in the calculation of virtual origin or the true isotropic
conditions of the flow.
Table 3: Summary of results from active grid studies
3.2 Measure of Isotropy
Multiple indicators of isotropy in a flow are outlined in studies by Batchelor (1953), Mohamed &
LaRue (1990) and George (1992). Each of these measures are evaluated as a function of downstream
distance and aim to determine the isotropic region in the flow. Mohamed & LaRue provide the first
measure of isotropy for this study which states that the skewness of velocity fluctuations, 𝑆 𝑢 =
𝑢;/𝑢+
Q
>
, should be zero. They then outline a second indicator, based on Kolmogorov’s analysis
where 𝑀e =
f=
f?
e
/
f=
f?
+
g
>
= 𝐶𝑜𝑛𝑠𝑡𝑎𝑛𝑡. However, for the second indicator of isotropy, this study
will make use of Batchelor (1953), who stated that for a flow that is both locally isotropic and locally
similar, the skewness of the velocity derivative, 𝑆 𝜕𝑢/𝜕𝑥 = 𝜕𝑢/𝜕𝑥 ;/ 𝜕𝑢/𝜕𝑥 +
;/+
, will be
constant.
George (1992) illustrates a third measure of isotropy which determines 𝑆 𝜕𝑢/𝜕𝑥 𝑅X = 𝐶𝑜𝑛𝑠𝑡𝑎𝑛𝑡.
This analysis requires the assumption that the velocity power spectrum and the transfer power
spectrum can be described as functions of a similarity function, and that 𝑆 𝜕𝑢/𝜕𝑥 is of the following
form:
Researcher Turbulence
generation
𝑅X (x/M) nu Virtual origin,
V0
Notes
Makita & Sassa
(1991)
Active grid 387 (50) 1.43 -12.0 𝑥L determined by
least square fit
Mydlarski & Warhaft
(1996)
Active grid 319 (68) 1.21 0 𝑥L = 0 assumed
Kang et al. (2003) Active grid 676 (30) 1.25 0 𝑥L = 0 assumed
Mordant (2008) Active grid 240 (16) 1.24 0 𝑥L = 0 assumed

14. 14
𝑆 𝜕𝑢/𝜕𝑥 = −
; ;L
h`
𝑘+
𝑇 𝑘 𝑑𝑘
∞
L
/ 𝑘+
𝐸 𝑘 𝑑𝑘
∞
L
Q
>
, (4)
where 𝑇(𝑘) is the transfer power spectrum, 𝑘 is the wavenumber, and 𝐸(𝑘) is the velocity power
spectrum. However, the analysis of George (1992) is contradicted by the work of Van Atta & Antonia
(1980) which shows that 𝑆 𝜕𝑢/𝜕𝑥 is proportional to 𝑅X over a variety of turbulent flows, and so will
not be used in this study.
3.3 Extension of the Power Law
The knowledge that the downstream decay of the velocity variance or, equivalently, the turbulent
kinetic energy can be represented by a power law implies that more aspects of the flow may also be
described in such a way, such as the downstream decay of the dissipation and the downstream growth
of the length scales. In this section, the form of those power laws will be observed along with other
implications. A small revision of eqn. (1) leads to the following equation, (5), which relates the
dissipation rate to the time rate of decay of the turbulent kinetic energy:
𝜖 =
h
+
<j>
<k
=
5
+
<j>
<k
, (5)
The above form takes into account the isotropic assumption which follows from:
𝑞+
= 𝑢+
+ 𝑣+
+ 𝑤+
, (6)
where u² = v² = w², and so:
𝑞+
= 3𝑢+
, (7)
Substituting the power law form for the downstream variation of the velocity variance from eqn. (3)
into the left side of eqn. (1) leads to, after some simple algebra, a valid description of the downstream
decay of the dissipation:
𝜖∗
=
;
+
Cs5Q
Tt
?
Tt
−
?u
Tt
UCUh
, (8)

15. 15
By combining the power-law expressions for the downstream variance of the velocity variance and
the dissipation, as appropriate, with the defining equations for the Taylor length scale, the
Kolmogorov length scale, the integral length scale, and the Taylor Reynolds number, power law
expressions for the downstream variations of those quantities can be determined. The defining
equations for those quantities are as follows:
𝜆 =
hdv∗=>
O
L.d
, (9)
𝜂 =
vQ
O
L.+d
, (10)
𝑳 𝑼 =
𝒖 𝟐
𝟏.𝟓
𝝐
, (11)
R€ =
•>
u.‚
€
ƒ
, (12)
Explicit forms for the power law behaviour can be found by substituting equations (3) and (5) into
equations (7) to (12). Further manipulation will also provide the relationship between Lu and the
Taylor Reynolds scale; unless 𝑛= = 1, Lu is not directly proportional to λ.
3.4 Infinite Reynolds Number
Von Karman & Howarth (1938) propose that the decay exponent, nu, equals 1 when 𝑅X → ∞. Setting
𝑛 = 1 yields the following:
OTt
5Q
=
;s†
+
?
Tt
−
?u
Tt
U+
, (13)
X>5
vTt
= 10
?
Tt
−
?u
Tt
, (14)
‡ˆ5Q
vQTt
=
+
;s‰
?
Tt
−
?u
Tt
+
, (15)

16. 16
Pt
>
Tt
> =
`sŠ
‹
?
Tt
−
?u
Tt
, (16)
Υ
>
ŒŽt
= 10𝐴Œ•
, (17)
Equation (17) is equivalent to eqn. (69) in George’s similarity analysis (1992). Of note, the
dependence on the virtual origin in eqn. (17) disappears where 𝑛= = 1. Equation (15) is equivalent
to the equation noted in Batchelor (1947, p. 136, footnote).
Finally, for 𝑛 = 1, equations (14), (15), and (16) show that 𝜆, 𝐿5, and 𝜂 are directly proportional to
each other. The fact that 𝜆 is directly proportional to 𝐿5 is in agreement with George (1992) but, again
this only occurs when 𝑛= = 1. Setting 𝑛= = 1 yields the following:
𝜂+ 5Q
vQTV
L.d
=
h
hdLs
L.d X>5
vTV
, (18)
Pt
>
TV
> =
+s
`d
X>5
vTV
, (19)
Which show that at 𝑛= = 1, and only at this value for nu, the length scales are proportional to one
another.
4. Experimental Setup & Procedures
4.1 Wind Tunnel
The experimental study will be carried out in a closed-return wind tunnel at the University of
California, Irvine. The test section has a width of 0.61m, height of 0.91m and is 6m in length, with
divergent top and bottom walls to account for the boundary layer growth in the flow. The mean speed
is stable to within ±0.05ms-1
when the tunnel is free from apparatus. The flow in the wind tunnel is
also homogeneous to less than 1% across the test section and the area ratio for contraction section to
test section is 9.36:1 which decreases from 5.15m2
to 0.55m2
.

17. 17
Figure 1: Closed-return wind tunnel, Selzer (2001)
At the entrance of the wind tunnel the background turbulence intensity is measured as 0.17% and at
the exit, 0.22%. The inside of the wind tunnel is fitted with a traverse to displace the sensors upstream,
downstream along the length of the test section and vertically, though remains within ±1mm from the
centre of the tunnel in the vertical position. This vertical position is measured to a resolution of 5µm
using a USB interface attached to a LM10 Renishaw magnetic encoder.
4.2 Turbulence Generator
The active grid, built in-house, is based on the 1991 Makita design (Makita & Sassa, 1991) which
was implemented by Mydlarski and Warhoft (1996) during their study of “High Reynolds number
grid-generated wind tunnel turbulence”. The design consists of 12 vertical and 18 horizontal rods,
each with a 9.5mm diameter and a grid mesh size, Mu, of 50.4mm. Equally spaced along the rods are
187 square agitator flaps, 34.3mm in length and 1.55mm thick. The flaps are centre mounted along
the rods, as can be seen in Figure 2, which is not always common with other active grids used to
generate turbulent flow. The rods slide through an oil-coated brass sleeve bearing that allows them to
rotate smoothly. To maintain the experimental condition for each experiment, the active grid must be
oiled on a daily basis.

18. 18
Figure 2: Active grid with rod mounted, square agitator flaps
The motion of the rods is controlled by 30 Anaheim Automation 17MD102S-00 stepper motors, each
with a resolution of 200 steps per revolution. The control of the motors is divided evenly by two
Propeller Proto USB boards using a P8X32A-Q44 propeller chip manufactured by Parallex Inc.
(Rocklin, CA), which take four inputs from the server. It is these four inputs from the server that
define the parameters for the experiment carried out in this study. These inputs are detailed in Table
4.
Input Details
2 Mean rotation rate measured in revs/second
25 Variance on mean rotation rate as a percentage
250 Rotation period measured in ms
50 Variance on mean rotation rate as a percentage
Table 4: Details of user input for active grid
To initiate the active grid, the user inputs the values found in the left hand column of Table 4 and
each pair of motors is given a random rotational direction and speed which is within the variance of

19. 19
25%, as shown above. This means that the rods can rotate at any speed between 1.5 and 2.5
revolutions per second. Then, a 50% variance implies that the period can fall between 125 and 375ms
before a new rotation rate and direction are chosen for all of the motors for a particular
microcontroller, and thus the cycle repeats.
The coordinate system used within this study is highlighted in Figure 3, which also shows the general
layout of the test section.
Figure 3: Layout and coordinate frame of the wind tunnel, Nguyen (2015)
The origin of the coordinate system for this experiment is located at the centre plane of the active
grid. The time-resolved velocity measurements and the derivative of such are made in the range of
x/Mu = 35 to x/Mu = 141. The wind tunnel is 18 mesh lengths tall and 12 mesh lengths wide.

20. 20
4.3 Homogeneity
An earlier experiment in the wind tunnel laboratory validated the homogeneity of the flow in the wind
tunnel test section. The test was carried out at the end of the test section using the lowest obtainable
mean velocity, 4ms-1
. Ensuring that the flow is homogenous in this extreme condition confirms the
flow will be homogenous throughout the isotropic range. Figure 4 shows the variation of U, u2
, 𝑆 𝑢 ,
𝜕𝑢/𝜕𝑡 +, and 𝑆 𝜕𝑢/𝜕𝑥 in the transverse direction at 𝑥/𝑀5 = 142 at 4ms-1
. Over the range of
−10 ≤ 𝑦/𝑀5 ≤ 10, the mean velocity is seen to vary by less than ±1%. For −8 ≤ 𝑦/𝑀5 ≤ 10,
𝑢+/𝑈+
varies by less than ±4%; for −10 ≤ 𝑦/𝑀5 ≤ 6, 𝜕𝑢/𝜕𝑡 + varies by about ±5% and for
−10 ≤ 𝑦/𝑀5 ≤ 10, 𝑆 𝜕𝑢/𝜕𝑥 varies by less than ±5%. In summary, based on these measurements,
homogeneity is seen to occur for most quantities of interest statistics for −10 ≤ 𝑦/𝑀5 ≤ 10.

21. 21
Figure 4: (A) 𝑼, (B) 𝒖 𝟐, (C) 𝑺 𝒖 , (D) 𝝏𝒖/𝝏𝒕 𝟐 and (E) 𝑺 𝝏𝒖/𝝏𝒙 for x/Mu=142 and z/Mu=0 for
4ms-1

22. 22
4.4 Sensors
For many years, hotwire anemometry has been a valuable research tool in fluid mechanics. The term
hotwire refers to a small wire element that is exposed to a fluid medium with the intention of
measuring a property of the fluid, most commonly the velocity. The anemometer is capable of reading
instantaneous values of velocity up to very high frequencies. Therefore, its response to, and capability
of, measuring the turbulent fluctuations in the flow field has proven to be very accurate.
The wire sensors work off an electric current passing through the microscopic filament which is
exposed to cross flow. As the rate of flow varies, the heat transfer from the filament will also vary,
thus displacing the heat distribution in the wire. The wire has a very sensitive temperature coefficient
of resistance, meaning that as the temperature increases, the resistance of the wire will also increase.
These fluctuations in resistance of the wire sensors allow electronic signals to be obtained which
relate to the velocity and temperature properties of the flow.
Whilst much experimental analysis has been carried out using a single hotwire, due to its simplicity
in both operation and calibration, more recently applications have utilized a crosswire probe,
consisting of 2 wires set 90° apart from one another and angled at 45 degrees to the flow. This allows
the direct advantage of measuring multiple components of velocity and the ability to resolve high
flow angles, highlighting flow conditions that have previously gone unseen when using a single
hotwire.
Figure 5: (A) Side and (B) birds-eye view of crosswire anemometer
In this study, both a crosswire sensor and a single hotwire sensor will be used to obtain the time-
resolved velocity measurements and a single velocity component in the traverse and downstream
directions of the 6m long test section, respectively. Along with these two sensors, two other sensors,
the traverse PRT and ambient PRT, will be used to measure temperature within the test section. The
PRT measures the mean temperature in the wind tunnel throughout the course of the data-collection
(A) (B)

23. 23
period. The traverse PRT and pitot tube, which records the mean velocity inside the tunnel and is also
used in the calibration process, are mounted at the same vertical location but are displaced
horizontally, 12mm from the single hotwire and crosswire sensors.
The single hotwire is fabricated in the University of California Irvine wind tunnel laboratory using a
platinum wire with a diameter and length of 5.08um and 1mm respectively, yielding a length to
diameter ratio of approximately 200. The hot wire is operated with an overheat ratio of 1.75. Based
on a square-wave test, the hot wire is estimated to have a frequency response of 40kHz when tested
at a mean velocity of 8ms-1
.
The crosswire was purchased by Ausprex and also consists of a platinum wire with dimensions
consistent to the single hotwire; a 5.08𝜇𝑚 diameter and length of 1mm. Again, the length to diameter
ratio is found to be approximately 200. The wires have a separation distance of 1 mm. The crosswire
is operated with an overheat ratio of 1.65. Based on its square signal response test at 10 ms-1
, the
crosswire is estimated to have a frequency response of approximately 16 – 18 kHz for each wire.
Since the crosswire measures more than simply a single component of velocity, we must determine
suitable nomenclature to distinguish between these components. From this point onwards, XW-UV
will refer to the crosswire configuration which measures the u- and v-components of velocity, whilst
XW-UW will refer to the crosswire configuration used to measure the u- and w-components of
velocity.
LabView is the system design software used to store and analyse the data obtained from the sensors
in the wind tunnel, providing both the probability density functions and the power spectra of the flow.
The signals received by LabView are generated from the CTA device. Both sensor types will use an
AN-1005 constant temperature anemometer (CTA) manufactured by AA Labs Systems
(Westminster, CA).
The analog signals coming from the CTA pass through a series of analog signal conditioners before
being digitised by an analog-to-digital converter (A/D). After the CTA, the signal passes through a
low-pass filter to remove the high-frequency electronic noise before the signal is split. Both signals
then pass through an amplification/attenuation stage to ensure the fluctuating signal levels exceed at
least half of the dynamic range of the A/D but do not exceed it. The output of the upper-processing
path then passes to the input of the A/D converter. An analog differentiator is added to the second set
of processing electronics (the lower path), and the output of that path is used to determine the time-
resolved velocity derivative.

24. 24
4.5 Calibration Procedure
Most of the data obtained when using hot-wire anemometry is limited to small perturbations. There
are cases, however, where this linearization of the anemometry equation is not accurate and non-
linear effects can influence both the mean and fluctuating voltages. Since high level fluctuations can
influence the mean voltage measured across the heater wire, it is important to calibrate the probes in
flows with low levels of fluctuations, Stainback & Nagabushana (1992).
The voltage of the hotwire sensor is described by King’s Law:
𝐸+
= 𝐴ž + 𝐵ž 𝑢C
, (20)
where E is the voltage across the wire, u is the velocity of the flow normal to the wire, and Ak, Bk and
n are coefficients. However, the time-average temperature corrected voltage value, EHW
2
, is required
and so the voltage is used in the form:
𝐸+
=
Ÿ ¡
>
¢£U¢¤
, (21)
Where EHW
2
is the time-averaged voltage of the single hotwire, Tw is the temperature of the wire and
Tg is the time-averaged temperature of the gas. First, the temperature of the wire and the exponent, n,
must be calculated. The calibration constants Ak and Bk are then found by defining un
= x, and E2
=
y, and using the method of least squares which simply becomes a linear regression for y as a function
of x.
The calibration procedure is carried out with the single hotwire sensor placed inside the potential core
of an asymmetric jet. The jet is equipped with a pressurized air tank to allow precise control over the
velocity of the flow during the second part of the calibration procedure, where the mean velocity is
measured by a pitot tube. Initially, the velocity is held constant at 10ms-1
and the temperature is set
at a variety of temperatures between 30°C and 100°C, increasing in increments of 10°C. Using
Bruun’s (1995) assumption that n is equal to 0.45 and some manipulation of King’s Law, the
temperature of the wire, TW can be obtained.
Once the temperature of the hotwire has been determined, the mean temperature is held constant. The
calibration coefficients, Ak and Bk, are determined by measuring the CTA output response for a range

25. 25
of velocity values. The calibration velocities are chosen to incorporate the minimum and maximum
voltage responses that both a single hotwire and a crosswire will encounter at that particular mean
speed. For mean velocities of 4ms-1
and 6ms-1
the resulting calibration velocities are given for the
case of a single hotwire and then crosswire in Tables 5 and 6 respectively.
Mean velocity Calibration Velocities
4ms-1
2,3,4,5,6,7,8,5,4,3 ms-1
6ms-1
2,4,6,8,10,9,7,6,5,3 ms-1
Table 5: Calibration velocities for single hotwire
Mean velocity Calibration Velocities
4ms-1
2,3,4,5,6,7,8,5,4,3 ms-1
6ms-1
2,4,6,8,10,9,7,6,5,3 ms-1
Table 6: Calibration velocities for crosswire
Due to the high intensity nature of the flow an additional calibration protocol must be used to
complete the calibration of the crosswire. Alternative calibration methods may result in differences
between the statistics of experiments and so, form an important part in describing the flow
characteristics using a crosswire, Burattini and Antonia (2005).
Similar to the single hotwire, the crosswire is placed inside the potential core of an asymmetric jet
but now requires the use of an adjustable caliper fitted at the outlet to allow a variable flow angle. At
each step of the calibration, the crosswire is placed in the flow at a known and fixed angle, ∝, from
its initial position, where ∝ = 0°, by definition. The yaw angle ∝, in the plane defined by the prongs
and the wire, describes the inclination of the wire with respect to the mean speed. The measurements
should not be affected if, for ∝= 0°, the axis of the probe is not exactly aligned with the flow (Strohl
and Comte-Bellot, 1973).
The calibration pitch angle range is ± 39° as within this range the sensor is most reliable. This range
will also allow the data obtained with the crosswire to compare with, and hence validate, the data
from the single hotwire. With the start point at -39°, the angle is first increased in increments of 3°

26. 26
until -33° and then by increments of 6° until a pitch angle of +33° is reached. The range is then once
again increased by 6 ° until +39°.
At each chosen pitch angle, 7 different flow velocities were run through for 5 second sample times,
which was found to be long enough to ensure statistical stability. The calibration velocities for the
angle sweep, recorded for each mean velocity, are listed in Table 7 below.
Mean Velocity Angles Calibration Velocities
4ms-1
±33º 1, 2, 3, 4, 5, 6, 8 ms-1
6ms-1
±33º 1, 2, 4, 6, 8, 10, 14 ms-1
Table 7: Crosswire calibration angles and velocities
Once the data has been obtained, the mean speed and flow angle are both converted to give the
velocity components below:
𝑢 = 𝑈𝑐𝑜𝑠𝜃 , 𝑣 = 𝑈𝑠𝑖𝑛𝜃, (22)
Taking the values for u and v, they are plotted as functions of E1 and E2, which are the time-averaged
temperature corrected voltages. A 4th
order polynomial in u and v is used as it produces a fit closer to
that of the data obtained using the single hotwire. This leads to two equations for u and v of the form:
𝑢 𝐸h, 𝐸+ = 𝑎h 𝐸h
`
+ 𝑎+ 𝐸h
;
𝐸+ + 𝑎; 𝐸h
+
𝐸+
+
+ 𝑎` 𝐸h 𝐸+
;
+ ⋯ + 𝑎hd , (23)
𝑣 𝐸h, 𝐸+ = 𝑏h 𝐸h
`
+ 𝑏+ 𝐸h
;
𝐸+ + 𝑏; 𝐸h
+
𝐸+
+
+ 𝑏` 𝐸h 𝐸+
;
+ ⋯ + 𝑏hd, (24)
The derivative velocity signal must then be obtained by applying a time derivative to the above
equations for u(E1,E2) and v(E1,E2). The analogue differentiator produces the velocity derivative
values and is temperature corrected as before.
Calibrations are performed before and after all of the data has been obtained from the experiment,
and must consistently remain below a 2% variation between the calculated velocity and the velocity
derivative statistics to proceed to the analysis stage. The calibration results in this study yielded
similar to within 0.5% for u2
and du2
; whilst the v- and w- components of velocity exhibited a 7% and
5% variability for 4ms-1
and 6ms-1
respectively.

27. 27
5. Experimental Procedure
5.1 Single Hotwire
The single hotwire experimental procedure is carried out using 240 second samples, taken every 2
inches along the downstream direction of the test section, which corresponds to every length, x/Mu,
from 35 to 142. Day 1 of the experiment records from 35 to 90 x/Mu. The second day then records
from 85 to 142 x/Mu. Once the points have been recorded, an additional 5 hysteresis values are also
recorded on both days. These values, along with the overlapped values between 85 and 90, allows for
checks to be carried out for any deviations in data between the two days.
5.2 Crosswire
For the crosswire hotwire, it was decided that points would be taken every 1.5 Mu between 35 and
81 and every 2 Mu thereafter. The same 5 hysteresis points are again recorded for the crosswire. This
is to allow for the entire experiment to be carried out over the course of only one day.
5.3 Sampling Rates
For each mean velocity measured, the filter and sampling rates change with downstream location in
the test section. These changes are crucial to ensure that electric noise contributes to no more than
3% of the overall value of the time derivative of the downstream velocity component, (𝜕𝑢/𝜕𝑡). The
effects of electronic noise on the computational value of (𝜕𝑢/𝜕𝑡) are found by integrating the power
spectrum for (𝜕𝑢/𝜕𝑡) to the frequency at which the electronic noise is first noted and comparing it
to the value directly obtained from the time series. The corner frequency and sampling rates for
specific downstream locations in the test section are given in Table 8 for mean velocities of both 4
and 6 ms-1
.

28. 28
Table 8: Corner frequency, fc and sampling rate for specific downstream locations
The aliasing effects which cause different signals to become indistinguishable from one another
require that the sampling rate be approximately double, if not greater than, the corner frequency, fc.
6. Results
6.1 Validation
Whilst the single hotwire downstream decay experiment was carried out for mean velocities of 4, 6,
8, 10 and 12 ms-1
, the crosswire experiment only measured the decay for mean velocities of 4ms-1
and 6ms-1
and so only these velocities will be considered. To allow an accurate analysis of the high
intensity flow, the raw data collected in both the single hotwire and the crosswire experiments must
first be validated. The initial raw data obtained from the single hotwire is shown in Table 9.
U (ms-1
) x/Mu Corner frequency, fc
(Hz)
Sampling rate (Hz)
4 35 7,300 20,000
66 4,500 14,000
96 3,500 10,000
6 35 9,000 36,000
56 7,000 28,000
81 6,000 24,000
91 5,000 19,000
121 4,500 18,000
131 4,000 15,600

29. 29
Variable Mean for 4ms-1
Standard
Deviation
Mean for 6ms-1
Standard
Deviation
U 1.59719971 0.00021449 0.754565 0.000241
u 0.646992 0.000828 1.158728 0.000246
S(u) 11.11538 0.147121 1.757204 0.026376
K(u) 0.032137 0.000284 0.05289 0.000395
dU 0.244979 0.001381 1.587637 0.001728
du 0.588637 0.001645 1.329862 0.000662
S(du) 0.140628 0.00164 0.097283 0.000876
K(du) 0.313774 0.006514 0.382685 0.002662
v 2.588682 0.003959 0.238576 0.00098
dv 2.64494 0.004278 0.318379 0.001058
Ubaratron 0 0 0 0
ubaratron 0 0 0 0
T 0 0 0 0
Trms 0 0 0 0
U total 1.605209 0.000125 0.756298 0.000246
u total 0.606037 0.000457 1.146231 0.000347
𝜙 rms 1.025439 0.458339 0.458339 0.001488
Table 9: Mean and standard deviation for raw variables

30. 30
Validation of the raw data from each experiment can be obtained by comparing the mean values and
standard deviation for both the pre- and post-calibrations. For reliable data it is essential to find small
standard deviations between not only the calibrations but also the downstream and traverse
components of velocity. These values were obtained for both the single hotwire and the crosswire
experiment, as illustrated below in Table 10, indicating that the data collected is sufficient to proceed
with the analysis.
Between calibrations Between UV and UW spectrum
Mean Std. deviation Mean Std. deviation
For 4 ms 0.97453729 0.02842393 1.647593 0.0345273
For 6 ms 0.95256086 0.01503794 1.65046629 0.58955631
Table 10: Mean and standard deviations between calibrations and velocity components.
For further validation, data from a previous crosswire downstream decay experiment conducted in
the wind tunnel laboratory is used for comparison with the crosswire raw data. Though following a
similar experimental procedure, the previous crosswire angle range was ±33 and used a sampling
record of 120 seconds. In order to allow an accurate comparison, the data set obtained by this study
has been reduced to match corresponding points from the previous hotwire experiment. The
normalized velocity components and ratios between old and new crosswire data are plotted in Figures
6 and 7.

31. 31
Figure 6: (A) u2
/U2
(B) v2
/U2
(C) w2
/U2
plotted as functions of downstream position for
4ms-1
0.00E+00
5.00E-03
1.00E-02
1.50E-02
2.00E-02
2.50E-02
20 45 70 95 120 145
u2/U2
x/Mu
XW-UV
XW-UW
XW-UV reference
XW-UW reference
2.00E-03
5.00E-03
8.00E-03
1.10E-02
1.40E-02
1.70E-02
2.00E-02
20 45 70 95 120 145
v2/U2
x/Mu
XW-UV
XW-UV reference
0.00E+00
3.00E-03
6.00E-03
9.00E-03
1.20E-02
1.50E-02
1.80E-02
2.10E-02
20 45 70 95 120 145
w2/U2
x/Mu
XW-UW
XW-UW reference
(A)
(B)
(C)

32. 32
Figure 7: (A) u/w (B) u/v and (C) v/w plotted as functions of downstream position for 4ms-1
1.05E+00
1.08E+00
1.10E+00
1.13E+00
1.15E+00
1.18E+00
1.20E+00
20 45 70 95 120 145
u/w
x/Mu
XW-UW
XW-UW reference
9.50E-01
9.75E-01
1.00E+00
1.03E+00
1.05E+00
1.08E+00
1.10E+00
0 20 40 60 80 100 120 140 160
v/w
x/Mu
Components from UV &
UW
Reference components
from UV & UW
1.02E+00
1.05E+00
1.08E+00
1.11E+00
1.14E+00
1.17E+00
1.20E+00
20 45 70 95 120 145
u/v
x/Mu
XW-UV
XW-UV reference
(A)
(B)
(C)

33. 33
As can be seen from Figures 6 and 7 above, when comparing the results from the end of the test
section downstream of the active grid, though the normalized values and ratios have a small offset, it
can be seen to lay within a few percent of the previous data obtained for the hotwire.
For the current crosswire study there is also slightly less scatter shown than that of the previous
crosswire experiment. This arises from the time length of the sample record; 4 minutes compared to
2 minutes for the previous study. This is expected; increasing the sample record theoretically
increases the accuracy of the time average and thus the accuracy of the measurement taken. It should
again be noted that the number of data points have been reduced so that only points where a direct
comparison can be made are plotted.
Current Data Reference Data
4ms-1
6ms-1
4ms-1
6ms-1
u/v 1.116 1.079 1.102 1.051
u/w 1.120 1.113 1.122 1.165
v/w 1.001 1.094 0.984 1.113
Table 11: Average ratios of turbulent fluctuations
For the u/v ratio, the current data from the crosswire matches to within 1.25% for 4ms-1
and within
2.59% for 6ms-1
of the reference crosswire data. The value of 1.116 for 4ms-1
is also fairly consistent
with the value obtained by Mydlarski & Warhoft (1996), who found u/v to be 1.21 for 4ms-1
. The
trend for u/w however, is not as coherent with the reference results. Whilst the reference data shows
an increase from 1.122 to 1.165 for 4ms-1
, there is a noticeable decrease between 4ms-1
and 6ms-1
for
u/w. This would suggest that as the mean speed increases, the flow is becoming more isotropic.
However, to confirm this trend a larger sample of mean velocities would have to be measured. Again,
the traverse components of velocity match well with the reference values. The v/w ratio falls very
close to 1, suggesting that the flow is very nearly axisymmetric. This is again confirmed by the
reference data, which follows a similar trend when higher velocities are compared. As we move from
4ms-1
to 6ms-1
the ratio increases by 0.093, with the v-component of velocity almost 10% greater in
magnitude than the w-component, illustrating that the flow is becoming less axisymmetric for higher
mean velocities. The isotropic assumption aforementioned in eqn. (6) is plotted below in Figure 8 for
𝑞+
= 3𝑢+
, 𝑞+
= 𝑢+
+ 2𝑣+
, and 𝑞+
= 𝑢+
+ 2𝑤+
to illustrate the effect that the offset between ratios
has on the isotropic assumption.

34. 34
Figure 8: (A) XW-UV decay of turbulent kinetic energy and (B) XW-UW decay of turbulent kinetic
energy
1.20E-01
1.70E-01
2.20E-01
2.70E-01
3.20E-01
3.70E-01
4.20E-01
80 90 100 110 120 130 140
q
x/Mu
1.25E-01
1.75E-01
2.25E-01
2.75E-01
3.25E-01
3.75E-01
80 90 100 110 120 130 140
q
x/Mu
3𝑢2 pre-calibration u2 + 2w2 pre-calibration 3u2 post-calibration u2 + 2w2 post-calibration
(A)
(B)

35. 35
Figure 8 clearly shows that the flow is not truly isotropic; a small offset between values can be seen.
From these ratios and the above plots, we can assume that although the flow is not fully isotropic,
since they do not equal unity, the values are close enough to proceed with the assumption that the
flow is nearly isotropic. Due to this assumption, we must acknowledge that there may be some small
error resulting from no-global isotropy.
An estimation of the start of the isotropic range can also be determined from the velocity component
ratios. These start locations are given in Table 12.
Table 12: Estimation of location of the start of nearly isotropic region
The location of the start of the nearly isotropic region can quite clearly be seen to match that of the
existing reference data. To allow further validation, the single hotwire reference data will now also
be compared to the results obtained in this study.
6.2 Single Hotwire Results
A previous set of single hotwire results will now be used for reference, the data from which has been
validated and used by T. Koster (2015) in his study of The Power Decay Law in High Intensity Active
Grid Generated Turbulence. In moderately high intensity turbulent flow, crosswires are prone to
errors arising from traverse components of velocity. Since the data from the cross flow component is
irretrievable when using a single hotwire, the data obtained allows comparisons with the crosswire to
highlight when cross-flow errors become significant. The raw data for the downstream velocity
component, u, and the differential of the velocity, du, are both squared and plotted along with the
dissipation, ∈, against the corresponding data for the single hotwire.
Estimated starting point of nearly isotropy range
4ms-1
6ms-1
XW-UV 95 90
Reference XW-UV 95 90

36. 36
Figure 9: (A) u2
, (B) du2
and (C) ∈ as a function of downstream location for 4ms-1
0.03
0.09
0.15
0.21
0.27
0.33
0.39
0.45
20 45 70 95 120 145
u2
x/M
0.00E+00
2.00E+04
4.00E+04
6.00E+04
8.00E+04
1.00E+05
1.20E+05
20 45 70 95 120 145
du2
x/Mu
0.00E+00
3.00E-01
6.00E-01
9.00E-01
1.20E+00
1.50E+00
20 45 70 95 120 145
∈
x/Mu
XW-UV XW-UW Single hotwire reference
(A)
(B)
(C)

37. 37
The downstream decay of the downstream velocity variance squared, u2
, as predicted, follows a power
law decay, as do the differential downstream velocity component, du2
, and the dissipation, ∈.
Within the isotropic range (>95 for 4ms-1
) both the squared values and the dissipation remain
consistent with the single hotwire data. Though small discrepancies are still present, these can result
from any of the small differences in the mean velocity value between studies. In order to account for
the mean velocity and thus, provide a more accurate validation of the data, it is important to consider
the normalized values.
Further characterisations of the data that will also considered are the skewness and kurtosis of the
potential differential function. Whilst skewness is a measure of the symmetry, and thus, if S(u) = 0,
then the frequency distribution is normal and symmetrical, Kurtosis is a parameter that describes the
shape of a random variable’s probability distribution.
The velocity skewness, S(u), and Kurtosis, K(u), for both the velocity and the differential velocity,
S(du) and K(du), are shown in Figures 11 and 12, and compared to the single hotwire reference data.

38. 38
Figure 10: (A) u2
/U2
, (B) du2
/U2
and (C) ∈/U2
as a function of downstream location for 4ms-1
0
0.004
0.008
0.012
0.016
0.02
0.024
0.028
20 45 70 95 120 145
u2/U2
x/Mu
(A)
0.00E+00
1.00E+03
2.00E+03
3.00E+03
4.00E+03
5.00E+03
6.00E+03
7.00E+03
20 45 70 95 120 145
du2/U2
x/Mu
(B)
0.00E+00
5.00E-03
1.00E-02
1.50E-02
2.00E-02
2.50E-02
3.00E-02
20 45 70 95 120 145
ε2/U3
x/Mu
XW-UV XW-UW Single hotwire reference
(C)

39. 39
Figure 11: (A) S(u) and (B) K(u) as a function of downstream location for 4ms-1
-4.00E-02
0.00E+00
4.00E-02
8.00E-02
1.20E-01
1.60E-01
2.00E-01
20 45 70 95 120 145
S(u)
x/Mu
2.80E+00
2.85E+00
2.90E+00
2.95E+00
3.00E+00
3.05E+00
3.10E+00
3.15E+00
20 45 70 95 120 145
K(u)
x/Mu
XW-UV XW-UW Single hotwire reference
(A)
(B)

40. 40
Figure 12: (A) S(du) and (B) K(du) as a function of downstream location for 4ms-1
4.50E-01
4.80E-01
5.10E-01
5.40E-01
5.70E-01
6.00E-01
6.30E-01
6.60E-01
20 45 70 95 120 145
S(du)
x/Mu
6.00E+00
6.80E+00
7.60E+00
8.40E+00
9.20E+00
1.00E+01
1.08E+01
20 45 70 95 120 145
K(du)
x/Mu
XW-UV XW-UW Single hotwire reference
(A)
(B)

41. 41
Figure 13: (A) λ2
and (B) λ2
*U as a function of downstream location for 4ms-1
1.50E-08
3.50E-08
5.50E-08
7.50E-08
9.50E-08
1.15E-07
1.35E-07
1.55E-07
20 45 70 95 120 145
λ2*U
x/Mu
XW-UV XW-UW Single hotwire reference
6.00E-05
8.00E-05
1.00E-04
1.20E-04
1.40E-04
1.60E-04
1.80E-04
2.00E-04
20 45 70 95 120 145
λ2
x/Mu
(B)

42. 42
Figure 14: (A) λ2
and (B) λ2
*U as a function of downstream location for 6ms-1
0.0003
0.0004
0.0005
0.0006
0.0007
0.0008
0.0009
20 45 70 95 120 145
λ2*U
x/Mu
XW-UV XW-UW Single hotwire reference
0.00004
0.00006
0.00008
0.0001
0.00012
0.00014
0.00016
20 45 70 95 120 145
λ2
x/Mu
(A)
(B)

43. 43
Plots from Figures 10-13 all show consistent similarity to the reference data obtained from the
previous single hotwire study. This trend continues until the Taylor length scales are reached, Figure
13. Here, the offset continues into the nearly isotropic region (95-142) and even shows discrepancies
between the UV and UW components of velocity. Whilst this difference remains small for 4ms-1
,
these offsets grow for 6ms-1
, as shown below, highlighting that there is considerable uncertainty in
the calculation of the Taylor length scale for this study. For further confirmation, the normalised
values of Taylor length scale are provided in Figure 14.
By considering figure 11 (A), it is also clear that the first indicator of isotropy, S(u) ≈ 0, is not met by
the data measured by the crosswire, nor the single hotwire reference data when recorded close to the
active grid. However, further down the test section it is acceptable to say that the data is closest to 1
within the nearly isotropic region, with a scatter of ± 3.72e+02, again justifying the assumption of
nearly isotropic flow. Meanwhile, Figure 11 (B), where a positive kurtosis can be observed, indicates
a relatively peaked distribution of data.
6.3 Turbulent Flow Characteristics
The values of the virtual origin, V0, decay exponent, nu and the dissipation exponent, n∈, are
calculated using the method described in section X, with the addition of EXCEL tool Solver, used for
optimization of these values.
First, the estimated start range of the isotropic region must be considered, which occurs when the
solution becomes stable. Using the MACROS code found in Appendix A, the below graphs were
obtained.

44. 44
Figure 15: (A) u2
/(x/Mu-x0/Mu), (B) ∈/(x/Mu-x0/Mu) and (C) λ2
/(x/Mu-x0/Mu) as a function of
downstream location for 4ms-1
1.00E+04
1.20E+04
1.40E+04
1.60E+04
1.80E+04
2.00E+04
2.20E+04
20 45 70 95 120 145
u2/(x/Mu–x0/Mu)
x/Mu
3.00E+06
3.50E+06
4.00E+06
4.50E+06
5.00E+06
5.50E+06
20 45 70 95 120 145
∈/(x/Mu–x0/Mu)
x/Mu
0.0000009
9.15E-07
9.3E-07
9.45E-07
9.6E-07
9.75E-07
9.9E-07
20 45 70 95 120 145
λ2/(x/Mu–x0/Mu)
x/Mu
Full Data Set Estimated Isotropic Region
(A)
(B)
(C)

45. 45
The above graphs were constructed using a range of x/Mu from 92 to 141. As previously predicted,
x/Mu = 95 is shown to be the start of the stable period for 4ms-1
, confirming that it is a good estimate
for the start of the isotropic region. The results in Table 13 give the virtual origin, decay exponent
and dissipation exponent for a variety of start locations downstream of the active grid.
Virtual Origin, V0 -23.886 -24.949 -22.950 -27.226 -35.919
Start (x/Mu) 124 112 109 100 95
End (x/Mu) 141 141 141 141 141
Decay exponent, nu -1.769 -1.798 -1.798 -1.912 -2.045
Dissipation
exponent, n∈
-2.770 2.784 -2.799 -2.912 -3.047
Table 13: Flat range results for various isotropic region start points for 4ms-1
The above values deliver an important conclusion. Theoretically, the values of the virtual origin, V0,
the decay exponent, nu and the dissipation exponent, n∈, should be constant along the flat range, which
is plotted above. However, each of the values constantly alter for each alternative start point used,
offering the assumption that the solver must require more data points in the flat range for complete
optimization.
The following values were obtained using a MATLAB code, found in Appendix (B), to give the
average solution after optimization of the functions plotted above and also the maximum and
minimum of the values in that range. The results are obtained for start points of 95 Mu, which is the
first estimate of the isotropic range and then, 100 and 109, which are the next reliable start points
based on the lambda squared analysis and table 13, respectively.

46. 46
Starting Point: 95
Dataset
Mean
Velocity
Virtual Origin, V0
nu n∈
Low Average High Low Average High
Single Hotwire 4ms-1
Low -57.03 -2.31 -2.34 -2.37 -3.35 -3.37 -3.39
Average -52.33 -2.25 -2.28 -2.30 -3.26 -3.28 -3.30
High -48.11 -2.20 -2.22 -2.25 -3.18 -3.20 -3.22
XW-UV 4ms-1
Low -31.98 -2.12 -2.14 -2.15 -3.12 -3.13 -3.15
Average -30.61 -2.10 -2.11 -2.13 -3.08 -3.10 -3.11
High -29.31 -2.08 -2.09 -2.10 -3.05 -3.06 -3.08
XW-UW 4ms-1
Low -27.30 -1.99 -2.00 -2.02 -2.99 -3.00 -3.02
Average -25.89 -1.97 -1.98 -1.99 -2.95 -2.96 -2.98
High -24.55 -1.94 -1.95 -1.97 -2.92 -2.93 -2.94
Table 14: Virtual origin, V0, decay exponent, nu, and dissipation exponent, n∈, as a function of start point
x/Mu = 95
Starting Point: 100
Dataset
Mean
Velocity
Virtual Origin, V0
nu n∈
Low Average High Low Average High
Single Hotwire 4ms-1
Low -62.18 -2.30 -2.33 -2.36 -3.30 -3.38 -3.40
Average -55.47 -2.23 -2.29 -2.31 -3.20 -3.20 -3.30
High -49.22 -2.18 -2.21 -2.24 -3.17 -3.19 -3.24
XW-UV 4ms-1
Low -40.09 -2.12 -2.14 -2.15 -3.12 -3.13 -3.16
Average -35.16 -2.09 -2.10 -2.12 -3.10 -3.11 -3.15
High -31.12 -2.06 -2.09 -2.11 -3.02 -3.09 -3.10
XW-UW 4ms-1
Low -29.73 -1.99 -2.00 -2.04 -2.99 -3.00 -3.02
Average -26.98 -1.93 -1.96 -1.97 -2.95 -2.96 -2.98
High -22.59 -1.92 -1.94 -1.98 -2.90 -2.94 -2.97
Table 15: Virtual origin, V0, decay exponent, nu, and dissipation exponent, n∈, as a function of start point
x/Mu = 100

47. 47
Starting Point: 109
Dataset
Mean
Velocity
Virtual Origin, V0
nu n∈
Low Average High Low Average High
Single Hotwire 4ms-1
Low -97.64 -2.88 -2.96 -3.04 -4.00 -4.05 -4.10
Average -77.13 -2.62 -2.68 -2.76 -3.63 -3.68 -3.72
High -62.6 -2.43 -2.49 -2.56 -3.37 -3.41 -3.46
XW-UV 4ms-1
Low -35.67 -2.05 -2.36 -2.34 -3.21 -3.13 -3.27
Average -32.54 -1.98 -2.08 -2.12 -3.11 -3.10 -3.13
High -23.79 -1.67 -2.01 -2.06 -3.09 -3.06 -3.10
XW-UW 4ms-1
Low -21.98 -1.78 -2.03 -2.04 -2.92 -3.00 -3.03
Average -19.45 -1.63 -1.97 -1.98 -2.85 -2.96 -2.98
High -17.62 -1.54 -1.81 -1.87 -2.84 -2.90 -2.91
Table 16: Virtual origin, V0, decay exponent, nu, and dissipation exponent, n∈, as a function of start point
x/Mu = 95
Tables 14-16 show that for each start point there is a unique set of values for the virtual origin, V0,
decay exponent, nu, and dissipation exponent, n∈. These results are shown with only one standard
deviation accounted for; if two standard deviations were to be used, it would increase the possibility
that a common solution would be obtained for the selected start points. However, a greater concern
is the contrasting values between velocity components, XW-UV and XW-UW. This could be a result
of small differences in the mean velocity and so, normalized data must be considered.
Starting Point: 95
Dataset
Mean
Velocity
Virtual Origin, V0
nu n∈
Low Average High Low Average High
Single Hotwire 4ms-1
Low -77.83 -2.35 -2.34 -2.37 -3.39 -3.40 -3.42
Average -62.43 -2.31 -2.33 -2.35 -3.36 -3.39 -3.41
High -58.11 -2.24 -2.25 -2.27 -3.25 -3.29 -3.38
XW-UV 4ms-1
Low -41.18 -2.11 -2.16 -2.23 -3.12 -3.17 -3.20
Average -36.71 -2.10 -2.11 -2.17 -3.08 -3.10 -3.11
High -30.52 -2.09 -2.12 -2.15 -3.05 -3.06 -3.08
XW-UW 4ms-1
Low -29.48 -1.97 -2.01 -2.08 -2.99 -3.01 -3.02
Average -27.65 -1.90 -1.93 -2.02 -2.95 -2.96 -2.98
High -21.23 -1.88 -1.91 -2.00 -2.93 -2.94 -2.96
Table 17: Normalised virtual origin, V0, decay exponent, nu, and dissipation exponent, n∈, as a function of
start point x/Mu = 95

48. 48
The normalised results for a start point of 95 also exhibit a unique solution. This allows the conclusion
to be drawn that the method used is not reliable as it is not consistent with the theory that follows the
overlapping of the XW-UV and XW-UW values, but in fact illustrates that the values of the virtual
origin depends on both the mean speed and the parameters used to determine the downstream
positions where the flow is nearly isotropic. The variation of the virtual origin and the decay exponent
is consistent with George’s (1992) assertion that the value of the virtual origin and the decay exponent
depends on initial conditions.
Therefore, a new approach must be considered. An alternative to estimating the virtual origin with
variation of the Taylor length scale, the new method will utilize the dissipation decay and dissipation
exponent, an alternative to using the power decay and decay exponent.
Again, the results do not conform with the original theory and lead to new values. This method
therefore cannot be validated either. It is important to reiterate the concept suggested previously that
the number of data points measured within the flat range of the graphs must be increased to allow an
accurate optimisation of results. To validate this theory, the single hotwire data set is now reduced to
include the same number of points in the flat range as measured in the crosswire experiment. The
reduced set of results are below.
Starting Point: 95
Dataset
Mean
Velocity
Virtual Origin, V0
nu n∈
Low Average High Low Average High
Single
Hotwire
Reference
4ms-1
Low -66.30 -2.33 -2.36 -2.39 -3.41 -3.43 -3.47
Average -61.33 -2.32 -2.34 -2.34 -3.37 -3.39 -3.41
High -57.21 -2.26 -2.27 -2.28 -3.29 -3.29 -3.38
XW-UV 4ms-1
Low -63.89 -2.32 -2.34 -2.35 -3.40 -3.41 -3.45
Average -60.74 -2.30 -2.22 -2.32 -3.36 -3.36 -3.40
High -54.92 -2.26 -2.25 -2.27 -3.27 -3.28 -3.39
XW-UW 4ms-1
Low -53.42 -1.98 -2.01 -2.08 -2.99 -3.01 -3.04
Average -49.77 -1.92 -1.93 -2.02 -2.95 -2.96 -2.99
High -41.54 -1.89 -1.92 -2.01 -2.91 -2.93 -2.95
Table 18: Reduced single hotwire reference dataset comparison for virtual origin, V0, decay exponent, nu,
and dissipation exponent, n∈, as a function of start point x/Mu = 95

49. 49
As shown in Table 18, a reduced dataset from the single hotwire reference provides a much closer
common solution between the reference data and the XW-UV data. This can prove the dependability
of the results on the number of points measured in the flat range. Koster (2015) suggests that for
reliable optimization values to be obtained, each variable of interest requires at least 10 measured
points in the flat range. Hence, a total of 30 points should be required in this study to provide accurate
results. However, the single hotwire experiment carried out in this study holds enough data points to
be accurate; a point was measured every mesh length, Mu. Meanwhile, the crosswire experiment, due
to time and other contributing factors evidently obtained too few points in the flat region. This directly
results in a high uncertainty in not only the estimated value of virtual origin but also in the results of
the turbulent flow characteristics found with the XW-UV and XW-UW datasets.
Whilst the above results should theoretically be more reliable than the first method used, an unknown
uncertainty on these values means that they cannot be used to accurately describe any of the required
turbulence characteristics. Once again, the results show that the number of data points within the
isotropic region hinders the ability to find an accurate estimate of virtual origin, V0, and decay
exponent, nu.
Another contributing factor to the uncertainty in the values given above could be certain turbulence
characteristics of the flow, in particular the Taylor Reynolds number and turbulence intensity.
Max Min
Taylor Reynolds number 360 270
Intensity 9% 6%
Table 19: Taylor Reynolds number and turbulence intensity
The turbulence intensity, as shown in Table 19 is three times as much as the usual intensity found
with a passive grid. The Taylor Reynolds number also exhibits a much higher value than usual for
passive grid generated turbulence. This could explain why the value of the virtual origin, decay
exponent and dissipation exponent are inconsistent with both the previous single hotwire study and
Kolmogorov’s predicted values of virtual origin and decay exponent.
6.4 Dissipation
Using the decay constant, A, and the decay exponent, nu, calculated from the least-square fit applied
to 𝑢+
/(
?
TV
−
?u
TV
) in Figure 15 (A), the dissipation, ∈*, from eqn. (8) can be calculated. Thus, the ratio
of ∈/∈* can be calculated and those values are shown as a function of downstream position in Figure

50. 50
16. When this ratio of ∈/∈* becomes one, the flow is determined to be isotropic. It is expected that
the initial assumption of isotropy will cause an overestimation of the dissipation of ∈* and
subsequently underestimate the ratio of ∈/∈*.

51. 51
Figure 16: (A) XW-UV and (B) XW-UW for ∈/∈* as a function of downstream location for 4ms-1
7.00E-01
8.00E-01
9.00E-01
1.00E+00
1.10E+00
1.20E+00
80 90 100 110 120 130 140
∈/∈*
x/Mu
(A)
(B)

52. 52
As can be seen in figure 16, the ratio ∈/∈* is approximately a constant and there appears to be no
significant trend to the value of ∈/∈* with downstream position, despite one anomaly in the UW
configuration at a location of 139 Mu.
XW-UV XW-UW
∈/∈* 0.948 0.972
Standard deviation 0.054 0.063
Table 20: Average value and standard deviation of ∈/∈* results for isotropic region
The above table shows the mean and standard deviation for both XW-UV and XW-UW
configurations for 4ms-1
as plotted in Figure 16. The scatter is approximately ±0.05 for the UV
velocity components and ±0.06 for the UW components of velocity. This variation, whilst it is
acknowledged carries uncertainties from the raw data, remains on the order of that suggested by the
uncertainty due to lack of stationarity and electronic noise which is about ±5%. The values for ∈/∈*
once again confirm that the flow is only nearly isotropic; ∈/∈* for both configurations do not equal
unity. Whilst the data does not exhibit an overwhelming offset from 1, we must compare all
configurations to the reference data to get a clearer image of the overall uncertainty in these values.
∈/∈*
Current Data Reference Data
4ms-1
6ms-1
4ms-1
6ms-1
UV 3u2
0.948 0.891 0.951 1.090
UW 3u2
0.972 1.023 0.908 1.144
UV u2
+2v2
0.903 0.891 1.023 1.205
UW u2
+2w2
0.929 1.056 1.034 1.168
Table 21: Average of ∈/∈* for multiple velocities and turbulent kinetic energy configurations
Table 21 presents the average value of ratio ∈/∈* for 𝑞 = 3𝑢+
and 𝑞 = 𝑢+
+ 2𝑣+
for both
configurations, UV and UW, of velocity. It can be seen that there is an overestimation in the value of
∈* when assuming isotropy which results in an underestimation of the ratio ∈/∈*. For example, the
minimum underestimate occurs at 4ms-1
in the XW-UW configuration, where ∈/∈* is underestimated
by 3%. On the other hand, the maximum underestimation occurs at 6ms-1
in the XW-UV

53. 53
configuration where the ratio, ∈/∈*, is underestimated 11%. This shows that for a flow that is not
sufficiently isotropic, the range of underestimation can range anywhere between 3-11%. The
reference data produces a greater uncertainty, in the range 2-20%. This may be as a result of the
shorter sample time length, 120s, compared to 240s used in the current study. Therefore, it can be
seen that neglecting the transverse components can lead to a misinterpretation of the isotropic region.
7. Further Research
The wind tunnel laboratory used in this study is preparing for two new experiments to be carried out
with the single hotwire and crosswire in the hope of correcting and reducing the uncertainties seen in
this study. Using the same experimental procedure, the single hotwire will record data every half
mesh length, 0.5 Mu, in the flat range of the downstream velocity to ensure that the value obtained
from the single hotwire has fully converged. The second experiment, which will once again utilize
the crosswire, will take measurements every Mu; increasing the number of points recorded in
comparison to this study. The crosswire will also be carried out for a greater range of mean velocities,
Taylor Reynolds number and turbulence intensities. Once the results from the two experiments are
collected and analysed, the laboratory will make use of a scalar field in the aforementioned conditions
using a scalar passive grid inserted downstream of the active grid in the wind tunnel.
8. Conclusion
The incentive of this study was to determine whether a crosswire anemometer could achieve more
accurate results of the turbulence characteristics for the case of moderately high Taylor Reynolds
number, homogeneous-isotropic and shear-less flow, downstream of an active grid. Whilst the flow
in this study is found to be only nearly isotropic, this does not affect the capability of the power law
in accurately describing the downstream velocity variance and the dissipation rate, as clearly
illustrated in the results section.
The values of virtual origin and downstream decay exponent exceed the values published for both
passive grid and active grid studies; this study found the most reliable values of decay exponent and
virtual origin to be nu = -2.045 and V0 = 35.919 downstream of the active grid, respectively. However,
by simply increasing the number of data points recorded in the nearly isotropic region, the accuracy
of these values could be increased by allowing a greater spread over which to optimize the values.

54. 54
The measurement with the crosswire is at last as accurate as a measurement with a single hot-wire.
But, it is regrettable that the number of points recorded with the crosswire in the flat range of Figure
15, plots (A), (B) and (C), were too small to allow an accurate comparison of turbulence
characteristics such as the decay exponent and virtual origin, with the previous single hotwire
measurement. For future study it may also be worthwhile to consider the cross flow error. As indicated
by Shabbir, Beuther and George (1996), a crosswire in greater turbulence intensity flow,
approximately 40%, will incur an overestimation of about 10.4% in the variable.
Despite this, it can be concluded that the crosswire is a very useful tool in analysing decaying high
turbulence intensity flows. The crosswire allows a reliable measurement of the flow characteristics
and exhibits a lot of advantages over the single hotwire as it can be used to empirically measure parts
of the flow that the single hotwire cannot. With consideration to the above corrections, it is clear that
the crosswire can become an integral tool in measuring high intensity, turbulent flow.

55. 55
9. References
Antonia, R. a. et al., 2003. Similarity of energy structure functions in decaying homogeneous
isotropic turbulence. Journal of Fluid Mechanics, 487, pp.245–269. Available at:
http://www.journals.cambridge.org/abstract_S0022112003004713 [Accessed October 28,
2015].
Antonia, R. a. & Orlandi, P., 2004. Similarity of decaying isotropic turbulence with a passive scalar.
Journal of Fluid Mechanics, 505, pp.123–151. Available at:
http://www.journals.cambridge.org/abstract_S0022112004008456 [Accessed Septmber 29,
2015].
Van Atta, C.W. & Antonia, R. a., 1980. Reynolds number dependence of skewness and flatness
factors of turbulent velocity derivatives. Physics of Fluids, 23(2), p.252. Available at:
http://scitation.aip.org/content/aip/journal/pof1/23/2/10.1063/1.862965 [Accessed August 4,
2015].
Batchelor, G., 1947. K. 1953 The Theory of Homogeneous Turbulence. Cambridge University
Press. J. Fluid Mech. Available at:
http://scholar.google.com/scholar?hl=en&btnG=Search&q=intitle:The+Theory+of+Homogene
ous+Turbulence#2 [Accessed December 19, 2015].
Bruun, H. H., 1995, Hot-Wire Anemometry: Principles and Signal Analysis, Oxford University
Press.
Comte-Bellot, G. & Corrsin, S., 1971. Simple Eulerian time correlation of full-and narrow-band
velocity signals in grid-generated, “isotropic” turbulence. Journal of Fluid Mechanics, 48(02),
p.273. Available at: http://www.journals.cambridge.org/abstract_S0022112071001599
[Accessed September 27, 2015].
Comte-bellott, B.G. & Corrsin, S., 1966. The use of a contraction to improve the isotropy of grid-
generated turbulence. , 25.
George, W., 1992. The decay of homogeneous isotropic turbulence. Physics of Fluids A: Fluid
Dynamics. Available at: http://arxiv.org/abs/nlin/0301040 [Accessed May 22, 2013].
Heitor, M.V. and Whitelaw, J.H.,1986 Velocity, temperature and species characteristics of the flow
in gas turbine combustor, Oxford University Press.
De Karman, T. & Howarth, L., 1938. On the Statistical Theory of Isotropic Turbulence.
Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences,
164(917), pp.192–215. Available at:
http://rspa.royalsocietypublishing.org/cgi/doi/10.1098/rspa.1938.0013 [Accessed May 26,
2014].
Kang, H.S., Chester, S. & Meneveau, C., 2003. Decaying turbulence in an active-grid-generated
flow and comparisons with large-eddy simulation. Journal of Fluid Mechanics, 480(2003),
pp.129–160. Available at: http://www.journals.cambridge.org/abstract_S0022112002003579
[Accessed September 30, 2015].

56. 56
Kolmogorov, A.N., 1941. The local structure of turbulence in incompressible viscous fluid for very
large Reynolds numbers S. K. Friedlander & L. Topper, eds. Doklady Akademii Nauk Sssr,
30(1890), pp.301–305. Available at:
http://rspa.royalsocietypublishing.org/cgi/doi/10.1098/rspa.1991.0075.
Koster, T.W,2015. The Power Decay Law in High Intensity Active Grid Generated Turbulence.
Lavoie, P. et al., 2005. Effect of initial conditions on decaying grid turbulence at low R λ.
Experiments in Fluids, 39(5), pp.865–874. Available at:
http://link.springer.com/10.1007/s00348-005-0022-8 [Accessed May 28, 2014].
Loitsianskii, L., 1939. Some basic laws of isotropic turbulent flow. Available at:
http://oai.dtic.mil/oai/oai?verb=getRecord&metadataPrefix=html&identifier=ADA801020
[Accessed November 6, 2015].
Makita, H. and Sassa, K., 1991. Active Turbulence Generation in a Laboratory Wind Tunnel. ,
pp.497–505.
Mohamed, M.S. & LaRue, J.C., 1990. The decay power law in grid-generated turbulence. J. Fluid
Mech, 219, pp.195–214. Available at:
http://journals.cambridge.org/production/action/cjoGetFulltext?fulltextid=396692 [Accessed
October 22, 2014].
Mydlarski, L. & Warhaft, Z., 1996. On the onset of high-Reynolds-number grid-generated wind
tunnel turbulence. Journal of Fluid Mechanics, 320(-1), pp.331–368. Available at:
http://journals.cambridge.org/abstract_S0022112096007562.
Nguyen,B.T., 2015. Calibration of a Crosswire with Assesment of Moderatly High Reynolds
Number Flow.
Saffman, P., 1967. The large-scale structure of homogeneous turbulence. Journal of Fluid
Mechanics, 27. Available at: http://authors.library.caltech.edu/10122/ [Accessed May 22,
2013].
Shabbir, A., Beuthert, P. D., & George, W. K. (1996). X-wire response in turbulent flows of high-
intensity turbulence and low mean velocities. Experimental thermal and fluid science, 12(1),
52-56.
Stainback, P.C., and Nagabushana, K.A., 1992, "Re-investigation of Hot-WIre Anemometry
Applicable to Subsonic Compressible Flows Using Fluctuation Diagrams," JNASA-CR-4429.
Tennekes, H. & Lumley, J.L., 1972. A First Course in Turbulence. Book, p.300.
http://books.google.com/books?hl=en&lr=&id=h4coCj-lN0cC&pgis=1.

57. 57
Appendix (A)
MATLAB code for flattening the nearly isotropic range of u2
/(x/Mu-x0/Mu), ∈/(x/Mu-x0/Mu) and
λ2
/(x/Mu-x0/Mu)
Sub AdjustRange()
'
' Macro3 Macro
'
'
' Keyboard Shortcut: Ctrl+Shift+A
'
clcmd = Application.Calculation
Application.Calculation = xlAutomatic
Worksheets("4ms Analysis").ChartObjects.Delete`~~~
Range("R5").Select
StartData = ActiveCell.FormulaR1C1
Range("R6").Select
EndData = ActiveCell.FormulaR1C1
SDataS = CStr(StartData)
EDataS = CStr(EndData)
xoMT = "=4ms Analysis!$C$" + SDataS + ":$C$" + EDataS
xoMF = "=4ms Analysis!$C4:$C" + EDataS
l2FT = "=4ms Analysis!$J$" + SDataS + ":$J$" + EDataS
l2FF = "=4ms Analysis!$J4:$J" + EDataS
u2FT = "=4ms Analysis!$K$" + SDataS + ":$K$" + EDataS
u2FF = "=4ms Analysis!$K4:$K" + EDataS
epFT = "=4ms Analysis!$M$" + SDataS + ":$M$" + EDataS
epFF = "=4ms Analysis!$M4:$M" + EDataS
Names.Add Name:="xTString", RefersTo:=xoMT
Names.Add Name:="l2FTString", RefersTo:=l2FT
Names.Add Name:="u2FTString", RefersTo:=u2FT
Names.Add Name:="epFTString", RefersTo:=epFT
'Plots Lambda2 Flatten in Full and shorten range
ActiveSheet.Shapes.AddChart.Select
ActiveChart.ChartType = xlXYScatter
ActiveChart.SeriesCollection(1).Name = "=""Full Set"""
ActiveChart.SeriesCollection(1).XValues = xoMF
ActiveChart.SeriesCollection(1).Values = l2FF
ActiveChart.SeriesCollection.NewSeries
ActiveChart.SeriesCollection(2).Name = "='4ms Analysis'!$J$2"
ActiveChart.SeriesCollection(2).XValues = xoMT
ActiveChart.SeriesCollection(2).Values = l2FT
ActiveChart.SeriesCollection(2).Trendlines.Add
ActiveChart.SeriesCollection(2).Trendlines(1).Select

58. 58
'ActiveChart.Shapes("Chart 1").ScaleWidth 2, msoFalse,
msoScaleFromTopLeft
'ActiveSheet.Shapes("Chart 1").ScaleHeight 2, msoFalse,
msoScaleFromTopLeft
'Plots u2 Flatten in Full and shorten range
ActiveSheet.Shapes.AddChart.Select
ActiveChart.ChartType = xlXYScatter
ActiveChart.SeriesCollection(1).Name = "=""Full Set"""
ActiveChart.SeriesCollection(1).XValues = xoMF
ActiveChart.SeriesCollection(1).Values = u2FF
ActiveChart.SeriesCollection.NewSeries
ActiveChart.SeriesCollection(2).Name = "='4ms Analysis'!$K$2"
ActiveChart.SeriesCollection(2).XValues = xoMT
ActiveChart.SeriesCollection(2).Values = u2FT
ActiveChart.SeriesCollection(2).Trendlines.Add
ActiveChart.SeriesCollection(2).Trendlines(1).Select
'Plots Epsilon Flatten in Full and shorten range
ActiveSheet.Shapes.AddChart.Select
ActiveChart.ChartType = xlXYScatter
ActiveChart.SeriesCollection(1).Name = "=""Full Set"""
ActiveChart.SeriesCollection(1).XValues = xoMF
ActiveChart.SeriesCollection(1).Values = epFF
ActiveChart.SeriesCollection.NewSeries
ActiveChart.SeriesCollection(2).Name = "='4ms Analysis'!$M$2"
ActiveChart.SeriesCollection(2).XValues = xoMT
ActiveChart.SeriesCollection(2).Values = epFT
ActiveChart.SeriesCollection(2).Trendlines.Add
ActiveChart.SeriesCollection(2).Trendlines(1).Select
'The solver interface for finding all optimal values of x0, n and m
'Finds Optimal x0
SolverOptions Precision:=1E-12
SolverOk SetCell:="$W$5", MaxMinVal:=3, ValueOf:="0",
ByChange:="$R$4"
SolverSolve
'Finds Optimal n
SolverOk SetCell:="$W$6", MaxMinVal:=3, ValueOf:="0",
ByChange:="$R$7"
SolverAdd CellRef:="$R$7", Relation:=1, FormulaText:="-1"
SolverSolve
SolverOk SetCell:="$W$6", MaxMinVal:=3, ValueOf:="0",
ByChange:="$R$7"
SolverSolve
'Finds Optimal m
SolverOk SetCell:="$W$7", MaxMinVal:=3, ValueOf:="0",
ByChange:="$R$8"
SolverAdd CellRef:="$R$8", Relation:=1, FormulaText:="-2"
SolverSolve
SolverOk SetCell:="$W$7", MaxMinVal:=3, ValueOf:="0",
ByChange:="$R$8"
SolverSolve
End Sub

59. 59
Appendix (B)
MATLAB code for finding V0, nu and n∈
%Finds The Range of the Power Law Range from Flattening lam, u and diss
%x(:,1) = x/M
%x(:,2) = lambda^2
%x(:,3) = u^2
%x(:,4) = epsilon
clc
close all
clear all
%Finds files name
filename = uigetfile('*.xlsx');
%Tab name in Excel File
TabName = 'PLR';
%Imports file into matlabn
y = xlsread(filename , TabName);
TabConstance = 'AntoniUNorm';
%Imports file into matlabn
Con = xlsread(filename , TabConstance , 'L1:L5');
%%
%start location excel row-1
start = Con(2)-1
%Set to full data Set
endloc = Con(3)-1
x=zeros(endloc-start-1,size(y,2));
%X=zeros(N-1,3);
X=0;
%Truncates the data sets
q=0;
for k=start:endloc
q=q+1;
x(q,:)= y(k,:);
end
%%
%Shortening the Range
xT = x(:,1);
laT = x(:,2);
u2T = x(:,3);
epT = x(:,4);
%Finds x0 for the give range
[x0 x0l x0h] = FlattenLinearFit(xT,laT,-10,10,0.01)
%x0=94.65

60. 60
xMx0T = xT+x0;
[n nl nh] = FlattenPowerLaw(xMx0T, u2T, -5, -1, 0.01)
[m ml mh] = FlattenPowerLaw(xMx0T, epT, -5, -1, 0.01)
xMx0T = xT+x0l;
[nL nLl nLh] = FlattenPowerLaw(xMx0T, u2T, -5, -1, 0.01)
[mL mLl mLh] = FlattenPowerLaw(xMx0T, epT, -5, -1, 0.01)
xMx0T = xT+x0h;
[nH nHl nHh] = FlattenPowerLaw(xMx0T, u2T, -5, -1, 0.01)
[mH mHl mHh] = FlattenPowerLaw(xMx0T, epT, -5, -1, 0.01)
%%
%Creates Plots
col = 'k';
figure(5)
plot(y(:,1),y(:,2)./(y(:,1)+x0),xT,laT./xMx0T)
figure(2)
plot(y(:,1),y(:,3)./((y(:,1)+x0).^n),xT,u2T./(xMx0T.^n))
figure(3)
plot(y(:,1),y(:,4)./((y(:,1)+x0).^m),xT,epT./(xMx0T.^m))
Flatten Linear fit
%Linear fit to x,y
[m b range] = linfit(X,Y);
x0 = -round(b/m);
%Checks for Optiomal x0 by Delta
j=1;
minFlat = ones(4,17);
for i=Min:Delta:Max
j = j + 1;
xMx0 = X-x0+i;
YFlat = Y./xMx0;
[m b range] = linfit(xMx0,YFlat);
minFlat(1,j) = -x0+i;
minFlat(2,j) = m;
minFlat(3,j) = range(2,1);
minFlat(4,j) = range(2,2);
end
[M I] = min(abs(minFlat(2,:)));
x0 = minFlat(1,I)
m = minFlat(2,I);

61. 61
mError = minFlat(4,I)-m
Il = find(minFlat(2,:)-mError<0);
Ih = find(minFlat(2,:)+mError>0);
x0l = minFlat(1,Il(length(Il)));
x0h = minFlat(1,Ih(2));
Flatten Power Law
%Changes the Power Law exponent y = Ax^M to Flatten the power law
%by making y/x^M = A
function [N nl nh] = FlattenPowerLaw(X,Y,Min,Max,Delta)
minN = ones(2,15);
i=1;
for M = Min : Delta : Max
Xn = X./(X.^M);
YFlat = Y./(X.^M);
[m b range] = linfit(Xn,YFlat);
minN(2,i) = m;
minN(1,i) = M;
minN(3,i) = range(2,1);
minN(4,i) = range(2,2);
i=i+1;
end
[m2 Ie] = min(abs(minN(2,:)));
N = minN(1,Ie);
m = minN(2,Ie);
mError = minN(4,Ie)-m;
Ih = find(minN(2,:)-mError<0);
Il = find(minN(2,:)+mError>0);
nl = minN(1,Il(length(Il)));
nh = minN(1,Ih(1));
minN;
Linear Fit
function [m b range] = linfit(x,y)
%y=mx+b
X = sum(x);
Y = sum(y);
XY = sum(x.*y);
X2 = sum(x.*x);
n = length(x);

62. 62
%finding the slope
m = (n*XY-X*Y)/(n*X2-X*X);
%finding the Intercept
b = (X2*Y-X*XY)/(n*X2-X*X);
S = sqrt(sum((y-m*x-b).^2)/(n-2));
Error_m = S*sqrt(n/(n*X2-X*X));
Error_b = S*sqrt(X2/(n*X2-X*X));
%one Standard Deviation range
range = [b-Error_b, b+Error_b; m-Error_m, m+Error_m];

63. 63
Appendix (C)
6ms-1
data for V0, nu and n∈ obtained using MATLAB code given in Appendix (B)
Table 1: Virtual origin, V0, decay exponent, nu, and dissipation exponent, n∈, as a function of start point
x/Mu = 89
Table 2: Virtual origin, V0, decay exponent, nu, and dissipation exponent, n∈, as a function of start point
x/Mu = 99
Table 3: Virtual origin, V0, decay exponent, nu, and dissipation exponent, n∈, as a function of start point
x/Mu = 109

64. 64
Table 4: Normalised virtual origin, V0, decay exponent, nu, and dissipation exponent, n∈, as a function of
start point x/Mu = 89
Table 5: Reduced single hotwire reference dataset comparison for virtual origin, V0, decay exponent, nu, and
dissipation exponent, n∈, as a function of start point x/Mu = 89