Experimental Investigation into Symmetric and Asymmetric Turbulent Wakes

Chapter 1 – Analysis of Turbulent Wakes                                           
 
 
12 
 
 
Analysis of Turbulent Wakes 
 
 
1.1      Introduction to Turbulent Flows 
 
Most fluid motion presented in nature or in engineering applications, is 
dominated by turbulent motion. There are many opportunities to observe, 
turbulent flows in our everyday surrounding: a smoke coming from a chimney, a 
water in a river or in a waterfall, the buffeting of a strong wind, or a wake 
generated by any object flying in clean air [1]. Also, If we observe fluid motion 
within a straight pipe, it will be noted that the shape of the velocity curve (the 
velocity profile across any given section of the pipe) depends upon whether the 
flow is laminar or turbulent as shown in fig. n° 1.1.1. If the flow in a pipe is 
laminar, the velocity distribution at a cross section will be parabolic in shape 
with the maximum velocity at the center being about twice the average velocity 
in the pipe. In fully turbulent flow, a fairly flat velocity distribution exists across 
the section of pipe [2, 3]. 
One of the most important parameters needed to describe this behavior, is 
the Reynolds number. This parameter is given by the relation (Reynolds 1894): 
 
                                                              

D U 
 Re                                                       (1.1.1) 
where U and D are characteristic velocity and length scale of the flow, and   is 
the kinematic viscosity of the fluid considered. D could be the diameter of any
Chapter 1 – Analysis of Turbulent Wakes                                           
 
 
13 
cylindrical shape used for experiments or the chord of a wing. It is clear as D is 
directly proportional to Re, and any increase of this parameter would make Re 
larger. That is a possible way to introduce more turbulence into the flow in fact, 
as soon as we pass over a critical range of the Reynolds number, which is a range 
of values not well defined where there is a transition between laminar and 
turbulent flow, the fluid consequently changes its state from laminar to 
turbulent. This behavior is called transition state [4]. 
  
 
                      fig. n° 1.1.1: Different shape of the velocity curve [1]. 
 
In turbulent flows, the fluid velocity field varies significantly and irregularly 
in both position and time as shown in fig. n° 1.1.2 where the horizontal line 
shows the mean velocity i U and its fluctuation 
'
i
u
 
[1]. 
  
 
                   fig. n° 1.1.2: Fluctuation of the velocity around mean value [1].
Chapter 1 – Analysis of Turbulent Wakes                                           
 
 
14 
Therefore, it can be said that turbulent flows are described by these 
fundamental characteristics: 
 Irregularity: formed by random strongly unsteady motion that 
makes a deterministic approach impossible. Therefore, it is convenient 
to use a statistical method. 
 Diffusivity: which means a more rapid mixing of momentum, heat 
and mass transfer. As a consequence, for example, on an aircraft’s 
wing the shear stress (hence the drag) is much larger than it would be 
if the flow were laminar. 
 Vorticity: Turbulence is characterized by high level of vorticity 
fluctuations and random 3 – D vorticity. This characteristic plays a 
fundamental role on the turbulence’s physics. 
 Large Reynolds numbers:  e.g. in a pipe, turbulence often occurs 
when Re> 2.000; for a flat plate usually it might have turbulence for 
Re> 600; for a wake a Re> 6 – 7 might be enough. Thus, it cannot be 
defined a universal Reynolds number in which the flow become 
turbulent. This is because the Reynolds number depends upon the 
characteristic length which has been considered.   
 Dissipation: Viscous dissipation always exists in turbulent 
phenomena as also in laminar flows. The turbulence needs a constant 
“Energy supply” in order to regain the strong viscous losses. 
 Unclosed problem: One way to approach with turbulent 
phenomena is the Reynolds decomposition (see sec. 1.2). In this 
approach, the number of the unknown variables is always greater than 
the number of the equations. Thereby, a statistical method should be 
implemented. [5, 3] 
 
Another very important matter of discussion in turbulent flows, is made by 
the knowledge of Turbulent scales. This important categorization, is done by 
considering small – scale turbulence, and large – scale of motion within the flow.
Chapter 1 – Analysis of Turbulent Wakes                                           
 
 
15 
The large – scale motions are strongly correlated by the geometry of the flow, as 
the boundary conditions, and they are responsible of transport and mixing. On 
the other hand, the small – scale motions are almost completely caused by the 
rate at which they get and dissipate the energy from the large – scale, and the 
viscosity. Therefore, this last one type of scales, are almost fully independent of 
the flow geometry.  
A very important idea, introduced by Richardson (1922) is the energy 
cascade. The cleverness of this idea is to consider that Kinetic energy enters the 
turbulence at the largest scales of motion. Then, this energy is shifted by inviscid 
processes into smaller scales until, at the smallest scales, the energy is dissipated 
by viscous actions. In fact, as already discussed above, a turbulent flow can be 
considered to be composed of a large number of eddies of different size. The 
eddies with the largest range, are characterized to have the length scale 
completely comparable to the characteristic scale of the flow. Theoretical studies 
aimed at developing a tractable model, are based on the Navier – Stokes 
equations (see sec. 1.2) which describe every detail of a turbulent velocity field 
from the largest to the smallest length scales. However, as it has been seen, the 
direct approach of solving these equations is really challenging for real flows and 
different approaches have been studied during the years. One of this, described 
in sec. 1.2, is the statistical approach [1]. 
 
1.2       The Reynolds Averaged Navier – Stokes Equations 
Starting from the Navier – Stokes equations written in Eulerian convection  
form for an incompressible flow (continuity equation and momentum equation 
respectively – underlined quantities represent vectors): 
 
  0    U                                               (1.2.1) 
                                                                        (1.2.2) 
  U p U U
U
2
t
    






  


 
Chapter 1 – Analysis of Turbulent Wakes                                           
 
 
16 
It is possible to derive every quantity of the turbulent field by using the Reynolds 
decomposition, in which the flow field is divided into its mean (or average – first 
term) plus a fluctuating component. Therefore, any  instantaneous value of the 
field (as Velocity, Pressure etc...), can be written as: 
 
                                                          
'
i i i
f f f                                                         (1.2.3) 
  where 
'
i
f is the fluctuation from the average value 
i
f and also: 
 
                                                          


T
i i
dt f
T
f
0
1
                                                      (1.2.4) 
 
in which T  is an enough long time that makes 
i
f independent from the time. 
This last hypothesis is necessary if we consider that, in nature a steady turbulent 
flow won’t ever exist. Thus, it could be considered a turbulent steady flow only 
when the average of all the statistic parameters is independent from the time. 
This last assumption lead to the fact that the average of the fluctuation is always 
negligible, hence [3]: 
 
                                                            0
'
 f                                                               (1.2.5) 
 
Using the Reynolds decomposition, any instantaneous quantity in the flow field 
can be written as: 
                                                  
      
'
i
i
i
u U U                                                           (1.2.6) 
 
and the same can be done for the other quantities as Pressure and Temperature. 
By substitution into Navier – Stokes equations (1.1.2 – 3), RANSE are generated:
Chapter 1 – Analysis of Turbulent Wakes                                           
 
 
17 
  0
'
    u U                                                             (1.2.7) 
                   (1.2.8) 
 
Since it is interesting to see the average of the flow field, the mean operator is 
applied to each of the terms to yield: 
 
0    U                            (1.2.9) 
U p
x
u u
Dt
U D
j
j i
2
' '
) (
    

 
                                                                          (1.2.10) 
 
Which it can be manipulated in a more smart form considering the total mean 
stress, given by: 
 
) (
' '
j i
i
j
j
i
ij
u u
x
U
x
U
   














                                                                                (1.2.11)  
 
where the total rate of strain is made by the rate of strain for laminar flow (under 
the hypothesis of Newtonian flow), plus a term called Reynolds Stress. By this 
last assumption, equations 1.2.9 – 10 can be rewritten as [4]: 
 
0    U                         (1.2.12) 
ij
p
Dt
U D
      
                             (1.2.13)                                     
       1.2.1 – Reynolds Stresses: A Physical Overview  
  The Reynolds Stresses ) (
' '
j i ij
u u     play a crucial role in the RANS 
Equations. If 0 
ij
 , the RANS Equations, and the “Normal” Navier – Stokes 
 
' 2 2 ' ' '
'
) )( (
t t
u U p p u U u U
u U
        






    





  
Chapter 1 – Analysis of Turbulent Wakes                                           
 
 
18 
equations would have been identical [1]. Reynolds Stress, which is a second 
order tensor, has the property to be symmetric (which implies that 
ji ij
   ), 
and the diagonal components represent the normal stress – so that they are 
normal terms as pressure, unlike the off diagonal components, which represents 
Shear stresses. The first terms contribute little to the transport of mean 
momentum whereas the second ones, play the dominant role in mean 
momentum transfer [4]. 
 Some words must be dedicated to the closure of the problem. For a general 
three dimensional flow, there are four independent equations governing the 
mean velocity field (1.2.12 – 13), which contain more than four unknown 
variables. In fact, in addition to U (three components) and p (one component) 
there are also the Reynolds Stresses. Due to this fact, the RANS Equations 
cannot be solved, and the problem is unclosed unless the Reynolds Stresses are 
somehow patterned. 
 
 1.2.2 – Production Equals Dissipation 
 Further information about this subject, which is talking about turbulent 
energy and its dissipation, can be found in [6]. The purpose of this thesis is not 
to compare turbulent energy production and dissipation (even though they are 
strictly correlated to Reynolds stresses), but let just say the statement that 
turbulent flows can be considered homogeneous due to the fact that, at high 
Reynolds numbers, the flow tends to gain a state of homogeneity at the smallest 
scales characteristic of the dissipative range. This statement, as can be 
demonstrated, says that in a steady, homogeneous, pure shear flow, the rate of 
production of turbulent energy by Reynolds stresses equals the rate of viscous 
dissipation, and that this state is generally attained beyond some point 
downstream of the turbulent producing object [6].
Chapter 1 – Analysis of Turbulent Wakes                                           
 
 
19 
1.2.3 – The Turbulence Reynolds Number 
 A very important parameter needed to describe the physics of the turbulence 
is the Turbulence Reynolds number. As already described in sec. 1.1 regarding 
turbulent scales, a turbulent flow generates large and small eddies. Most of the 
transport of momentum of the flow is done by the large eddies, and between the 
two scales there are many other length of scales. As it has already discussed, 
from the largest scales of turbulence and the smallest scale there is always a 
continuous exchange of energy described by the energy cascade theory. The 
largest eddies of a turbulent flow are characterized by the parameter  , which is 
also correlated to the integral scale of turbulence. The main difference between 
the Reynolds number and the turbulence Reynolds number is that the first one 
uses the geometric characteristic
 
D as a length scale, which is the characteristic 
length scale of the object in the flow, and second one uses the length scale of the 
largest eddies so that  . Nevertheless, the two Reynolds number defined above, 
have the same order of magnitude (or nearly the same). A particular attention 
though, must be made on not confusing the latter with the “Turbulent Reynolds 
Number” which is defined by the micro – scale . The micro – scale , also 
defined as Taylor Micro – scale is one of the smallest scale in the flow but not 
the smallest at all. The smallest scale is the Kolmogorov micro – scale  [6].  
However, one possible way to estimate 

R will be shown in section 1.6 given by 
Garnet, Altman in [7].  
 
1.3      Statistical Description of Turbulent Flows 
As already discussed in sec. 1.1, in a turbulent flow the velocity field is 
random. A generic variable (which could be velocity, pressure, etc...), is random 
when it doesn’t have a unique value, but only when “this value is the same every 
time the experiment is repeated under the same set of conditions” [1]. Thereby, 
in order to give a statistical representation of turbulence, it is necessary to link
Chapter 1 – Analysis of Turbulent Wakes                                           
 
 
20 
the random nature of turbulence with its own deterministic model given by 
RANS Equations. Then, it has been noted that turbulent flows, are extremely 
sensitive to initial conditions, boundary conditions, and material properties (a 
very important issue to consider during the experiments). Therefore, it is 
important for an event to be repeated many times using the same set of initial 
conditions [1]. 
In order to give a complete representation with a statistical description on the 
subject, it could be important to recall some important statements from 
probability and statistics. In engineering applications, the probability P(A)  of 
the event A is defined as the likelihood of the occurrence of the event A. P is a 
real number, which is 1 0   P . When the event is impossible (it never occurs) 
P=0
 
, whereas when the event is sure P=1. 
Defining a Cumulative Distribution Function (CDF), it is possible to define 
the probability of any event. The CDF is given by the function: 
 
    x X P x F                                                                                                          (1.3.1) 
 
where x  is a real number, and F(x) is the likelihood that the variable X is less 
than or equal to x. Also, the Probability Density Function (PDF) of the event X 
can be defined as: 
 
 
   
x
x x X x P
dx
x dF
x p
x

   
 
  0
lim                                                                (1.3.2)
  
hence, it has been shown that the PDF (or equally the CDF), fully describes a 
random variable. It can be also state that two or more random variables having 
the same PDF are said to be statistically identical [8]. 
A very important matter, studying statistical subjects, is represented by 
statistical moments. They can be defined as:
Chapter 1 – Analysis of Turbulent Wakes                                           
 
 
21 
 


 
  dx x p x x X E
k k k
) (                                                                                         (1.3.3) 
 
which are the K power of the variable X , and where the first moment (K=1) is 
the mean value. Considering instead the differences with respect to the average 
value (as in our case), the central moments are defined: 
 
   


 
     dx x p x x x x X E
k
k k
k
) ( ) (
'
                                                                  (1.3.4) 
 
where the second moment (K=2) is called Variance and it is given by symbol 
2

while its square root is called standard deviation of the PDF. It defines the width 
of the PDF around its mean value while the third normalized moment, called 
Skewness, shows the symmetry of the PDF (fig. n° 1.3.2 a).  
 Of fundamental importance in probability theory and in turbulence is the 
normal or Gaussian distribution. If a variable x is normally distributed, its PDF 
takes the Gaussian shape as follow: 
 
 
2
2
2
2
1
) (
x
x x
x
e x p

 

                                                                                                  (1.3.5) 
 
where x is the mean of x and 
x
 the standard deviation. In fig. n° 1.3.1 it is 
possible to see a Gaussian shape and the meaning of its standard deviation.