Research Journal of Applied Sciences, Engineering and Technology 4(20): 4150-4159, 2012
ISSN: 2040-7467
© Maxwell Scientific Organization, 2012
Submitted: April 23, 2012
Accepted: May 23 , 2012
Published: October 15, 2012
Transport Equation for the Joint Distribution Function of Velocity, Temperature
and Concentration in Convective Turbulent Flow in Presence of Dust Particles
M.A.K. Azad, M.H.U. Molla and M.Z. Rahman
Department of Applied Mathematics, University of Rajshahi, Rajshahi-6205, Bangladesh
Abstract: In this study, an attempt is made to study the joint distribution functions for simultaneous
velocity, temperature, concentration fields in turbulent flow in presence of dust particles. The various
properties of the constructed joint distribution functions such as, reduction property, separation property,
coincidence and symmetric properties have been discussed. The transport equations for one and two point
joint distribution functions in presence of dust particles have been derived.
Keywords: Concentration, distribution functions, dust particles, turbulence
INTRODUCTION
In molecular kinetic theory in physics, a particle's
distribution function is a function of several variables.
Particle distribution functions are often used in plasma
physics to describe wave-particle interactions and
velocity-space instabilities. Distribution functions are
also used in fluid mechanics, statistical mechanics and
nuclear physics. A distribution function may be
specialized with respect to a particular set of
dimensions. Distribution functions may also feature
non-isotropic temperatures, in which each term in the
exponent is divided by a different temperature. The
mathematical analog of a distribution is a measure; the
time evolution of a measure on a phase space is the
topic of study in dynamical systems.
In the past, several researchers discussed the
distribution functions in the statistical theory of
turbulence. Lundgren (1967) derived the transport
equation for the distribution of velocity in turbulent
flow. Bigler (1976) gave the hypothesis that in
turbulent flames, the thermo chemical quantities can be
related locally to few scalars and considered the
probability density function of these scalars. Kishore
(1978) studied the distributions functions in the
statistical theory of MHD turbulence of an
incompressible fluid. Dixit and Upadhyay (1989)
discussed the Distribution functions in the statistical
theory of MHD turbulence of an incompressible fluid in
presence of the coriolis force. Pope (1981) derived the
transport equation for the joint probability density
function of velocity and scalars in turbulent flow.
Kollman and Janica (1982) obtained the transport
equation for the probability density function of a scalar
in turbulent shear flow. Kishore and Singh (1984)
derived the transport equation for the bivariate joint
distribution function of velocity and temperature in
turbulent flow. Also Kishore and Singh (1985) have
been derived the transport equation for the joint
distribution function of velocity, temperature and
concentration in convective turbulent flow. Sarker and
Kishore (1991) discussed the distribution functions in
the statistical theory of convective MHD turbulence of
an incompressible fluid. Also Sarker and Kishore
(1999) studied the distribution functions in the
statistical theory of convective MHD turbulence of
mixture of a miscible incompressible fluid. Azad and
Sarker (2004) discussed Statistical theory of certain
distribution functions in MHD turbulence in a rotating
system in presence of dust particles. Islam and Sarker
(2007) studied distribution functions in the statistical
theory of MHD turbulence for velocity and
concentration undergoing a first order reaction. But at
this stage, one is met with the difficulty that the N-point
distribution function depends upon the N+1-point
distribution function and thus result is an unclosed
system. This so-called closer problem is encountered in
turbulence, Kinetic theory and other non-linear system.
In this study, we have been derived the joint
distribution functions for the evolution of transport
equations and various properties of the distribution
function have been discussed for velocity, temperature,
concentration in convective turbulent flow in presence
of dust particles.
MATERIALS AND METHODS
Basic equations: The equation of motion and field
equations of temperature and concentration in presence
of dust particles are given by:
Corresponding Author: M.A.K. Azad, Department of Applied Mathematics, University of Rajshahi, Rajshahi-6205,
Bangladesh
4150
Res. J. Appl. Sci. Eng. Technol., 4(20): 4150-4159, 2012
u
u

 u   
t
x 
x 
 dx 
1  

 u  x , t 
u   x , t 

0 4 x 
x 
 x   x 
 

 

u  ƒu  v 
x  x 


 




 u
x   x 
x 
t
c
c
 
c
D
 u
x  x 
x 
t
(2)
(3)
u
v

 0
x
x
With
where,
uα (x, t)
(x, t)
c
v
ƒ
N
=
=
=
=
=
=
ρ
D
γ
cp
vα
kT
=
=
=
=
=
=
Component of turbulent velocity
Temperature fluctuation
Concentration of contaminants
Kinematics viscosity
KN/ρ = Dimension of frequency
Constant number of density of the dust
particle
Fluid density
Diffusive coefficient for contaminants
kT /ρcp = Thermal diffusivity
Specific heat at constant pressure
Dust particle velocity
Thermal conductivity
Here u and x are vector quantities in the whole
process.
Formulation of the problem: We consider the
turbulence and the concentration fields are
homogeneous, also consider a large ensemble of
mixture of miscible fluids in which each member is an
infinite incompressible heat conducting fluid in
turbulent state. The fluid velocity u, temperature θ and
concentration c are randomly distributed functions of
position and time and satisfy their field equations.
Different members of ensemble are subjected to
different initial conditions and the aim is to find out a
way by which we can determine the ensemble averages
at the initial time. The present aim is to construct a joint
distribution functions, study its properties and derive an
equation for its evolution of this joint distribution
functions in presence of dust particles.
fluid velocity u, temperature θ, concentration c at each
point of the flow field in turbulence. Lundgren (1967)
and Sarker and Islam (2002) has studied the flow field
on the basis of one variable character only (namely the
fluid velocity u) but we can study it for two or more
variable characters as well. The corresponding to each
point of the flow field, we have three measurable
characteristics. We represent the three variables by v, φ
and ψ and denote the pairs of these variables at the
points x(1), x(2),…, x(n) as (v(1), φ(1),ψ(1)), (v(2), φ(2),ψ(2)),
- - -, (v(n), φ(n),ψ(n)), at a fixed instant of time. It is
possible that the same pair may be occurring more than
once; therefore, we simplify the problem by an
assumption that the distribution is discrete (in the sense
that no pairs occur more than once). Instead of
considering discrete points in the flow field if we
consider the continuous distribution of the variables and
ψ over the entire flow field, statistically behavior of the
fluid may be described by the distribution function
F(v, φ ,ψ) which is normalized so that:
 F v,  , dv d d  1
where, the integration ranges over all the possible
values of v, φ and ψ. We shall make use of the same
normalization condition for the discrete distributions
also. The joint distribution functions of the above
quantities can be defined in terms of Dirac Deltafunctions.
The one-point joint distribution function F1(1) (v(1),
(1) (1)
φ ,ψ ) is defined in such a way that F1(1) (v(1),
φ(1),ψ(1)) dv(1) φ(1) dψ(1) is the probability that the
fluid velocity, temperature and concentration field at a
time t are in the element dv(1) about v(1), φ(1) about φ(1)
and dψ(1) about ψ(1), respectively and is given as:

   u    v   
         c       
F11 v 1 ,  1 ,
1
1 
1
1
1
1
1
(4)
where, δ is the Dirac delta-function defined as:
  u  v dv 

1 at the po int u  v
0 otherwise
Two-point joint distribution function is given by:
Joint distribution function in convective turbulence
and their properties: It may be considered that the
4151 
 
 
F21, 2    u 1  v 1   1   1  c 1   1
 u
2 
v
2 
 
2 

2 
  c
2 

2 


(5)
Res. J. Appl. Sci. Eng. Technol., 4(20): 4150-4159, 2012
And similarly:
and three point distribution function is shown by:





 
 
 
 c    u  v       
 c        u    v   
          c       
F31, 2,3   u 1  v 1   1   1
1
1
2
2
2
2
3
2
3
3
Lim
x  3   x  2  
2
(6)
3
Similarly, we can define an infinite numbers of
multi-point joint distribution functions F4(1,2,3,4),
F5(1,2,3,4,5) and so on. The joint distribution functions so
constructed have the following properties:
Reduction properties: Integration with respect to pair
of variables at one-point, lowers the order of
distribution function by one. For example:
v 2   v 1 ,  2    1
 F21, 2  dv 2  d 2  d 2   F11
And hence it follows that:
Lim
x  2   x 1   

Similarly:
 F31, 2,3 dv 3  d 3 d 3   F21, 2 
Lim
x 3   x  2  

F31, 2,3  F21, 2   v 3  v 1

  3   1    3    1  etc.
and so on.
Also the integration with respect to any one of the
variables reduces the number of Delta-functions from
the distribution function by one as:
1
 F1 dv
1
1
1
 F1 d

1

1
  
  u

1
  c
1
v
1
  c
1

1

1

 
 
 
  c   

2
Fn1, 2,r ,s ,n   Fn1, 2,s ,r ,n 
Continuity equation in terms of distribution
functions: An infinite number of continuity equations
can be derived for the convective turbulent flow and the
continuity equations can be easily expressed in terms of
distribution functions and are obtained directly by div u
= 0:

 F21, 2  dv 2     1   1  c 1   1
2 
Symmetric conditions:

and

2
0
and so on.
Separation properties: The pairs of variables at the
two points are statistically independent of each other if
these points are far apart from each other in the flow
field i.e.,
Lim
x  2   x 1

F21, 2   F11  v 2   v 1
  2    1    2    1 
 F21, 2  dv 2  d 2  d 2   F11
 
and  2    1
But also F2(1, 2) must have the property:
 F11 dv 1 d 1 d 1  1
2 
e t c.
Co-incidence property: When two points coincide in
the flow field, the components at these points should be
obviously the same that is F2 (1, 2) must be zero. Thus:
3
3
F31, 2,3   F21, 2  F13

F21, 2   F11 F12 
u1
x1
 1
u   F11 dv 1 d 1 d 1
x1



u1  F11 dv 1 d 1 d 1
x1


x1


v1 F11 dv 1 d 1 d 1
x1 

4152 
u 1 F11 dv 1 d 1 d 1
F11 1 1 1
v dv d d 1
x1
(7)
Res. J. Appl. Sci. Eng. Technol., 4(20): 4150-4159, 2012
and similarly:
0
F11 1 1 1 1
 dv d d
x1
(8)
Which are the first order continuity equations in which
only one point distribution function is involved.
For second-order continuity equations, if we multiply
the continuity equation by:

xr 
 v F
dvr  d r  d r 


xs
 v F
dvs d s d s


 



x1

1
 u  




 
 
 
u 1  u 1  v 1   1   1  c 1   1

1 1, 2  1
dv d 1 d 1
1  v F1
x 


1 F11, 2 dv 1d 1d 1
x1 
(9)

v1 FN1, 2,    N dv 1d 1d 1
x1 
 u 1 

1 

u

 x 1 
 x 1 

(10)

1 
0
(14)

 


 t         


 t  c     
  u1  v1   1   1
(11)


  u1  v1  c1  1
(12)
The continuity equations are symmetric in their
arguments i.e.,

 u1  v1
v1
1

  1   1
 1
 ut


 t

   u1  v1   1   1
1
1


1
1

   u1  v1  c1  1


1
    1   1  c1  1
and

0  1  1FN1, 2,    N dv1d 1d 1
x
F11  dv 1  d  1  d 
 1 
F1   u1  v1   1   1  c1  1
t
t

   1   1  c1  1  u1  v1
t
The Nth-order continuity equations are:
0
1 
 v
Equations for the evolution of joint distribution
functions: This, in fact is done by making use of the
definitions of the constructed distribution functions, the
transport equation for F(v, φ, ψ, x,t) is obtained from
the definition of F and from the transport Eq. (1), (2)
and (3). Differentiating Eq. (4) we get: and similarly:
0
(13)
and all the properties of the distribution function
obtained in section (4) can also be easily verified.

 u 2   v 2    2    2    c 2    2  
0
N
x1

 u 2   v 2    2    2   c 2    2  u1
x1

 s 1,2,   r    s    N 

 x 1 
and if we take the ensemble average, we obtain:

N
Since, the divergence property is an important
property and it is easily verified by the use of the
property of distribution function as:
 u 2   v 2    2    2    c 2    2  
0   u 2   v  2     2    2   c  2     2 
r  1,2,   s    r    N 


1
 ct




 c1  1
 1


Using Eq. (1), (2) and (3) in (15) we get:
4153 (15)
Res. J. Appl. Sci. Eng. Technol., 4(20): 4150-4159, 2012
 1
F1 
t


   1   1  c 1   1
 1 u 1
1   2  


2  

  u
1 
1 
 2  u
 2   u  
4 x  
x 
x 
x 
 

1
1
 1  u  v

2 
v

 1

 dx 
1
1

 x 1  x 2    x 1 x 1 u  ƒ u   v




 


 

 1


   u 1  v 1  c 1   1  u1 1   1 1  1  1   1   1
x 
x  x 
 









 
c 1


1
1
   u 1  v 1   1   1  u1 1  D 1 1 c 1 
1  c  
x 
x  x 

 





u 1 
F11
    1   1  c 1   1 u1 1 1  u 1  v 1
t
x  v







   u 1  v 1  c 1   1 u1


 1 
  1   1
x 1  1


   u 1  v 1   1   1 u1
 
   1   1  c 1   1  1
 x 




 1

 4








   1   1  c 1   1 ƒ u 1  v1


 u 1  v 1
v1

 

 
1
1
  u 1  v 1   1   1 D 1 1 c 1
1  c  
 x x 


 
 







 
1
1
 1  u  v
 v









 

 
  u 1  v 1  c 1   1   1 1  1 1   1   1
 x x 


 
 


dx 2 


2  2 
u
u


x 2  x 2 
x 1  x 2 
 
 

   1   1  c 1   1   1 1 u1  1  u 1  v 1
 v
 x x

 
 


c 1 
 c 1   1
x 1  1


0
(16)
Various terms in the above equation can be simplified as that they may be expressed in terms of one point and two
point distribution functions. The second, third and fourth term on the left hand side of the above equation are
simplified in a similar fashion and take the forms as follows:

 

u 1

 


 u 1  v 1
x 1
   1   1  c 1   1 u 1
   1   1  c 1   1 u 1

 u 1  v 1
x 1 v1




 u 1

 1  1
 v

(17)
4154 Res. J. Appl. Sci. Eng. Technol., 4(20): 4150-4159, 2012



 1 
  1   1
x 1  1




  1   1
x 1
  u 1  v 1  c 1   1 u 1
  u 1  v 1  c 1   1 u 1




(18)

 

c 1 
1
1
1
1  c  
x  

 


 c 1   1
x 1

  u 1  v 1   1   1 u 1

  u 1  v 1   1   1 u 1


(19)
Adding Eq. (17), (18) and (19) we get:
  1   1  c 1   1  u1

 u 1  v 1
x 1






  1   1
x 1






1
1
1  c  
x 

  u 1  v 1  c 1   1 u1

  u 1  v 1   1   1 u1


u 1  u 1  v 1   1   1  c 1   1
x 1

 1 1
v F1
x 1

 v1

 
Applying the properties of

distributi on function 
F11
x 1
(20)
We reduce the fifth term on left hand side of Eq. (16):


 x 1

  1   1  c 1   1  

  1

v1  4


 




2
 x x 1  x 2 


 
 1

 4

dx 2 


2  2 
u
u


x 2  x 2 
x 1  x 2 
 
1
1
 1  u  v
 v

 v 2    F 1, 2  dx 2  dv 2  d 2  d 2 
  x 2   2
 





2
We reduce the sixth term on left hand side of Eq. (16):
4155 (21)
Res. J. Appl. Sci. Eng. Technol., 4(20): 4150-4159, 2012
 
 1  
u
 u 1  v 1
 x 1 x 1   v 1

 
 
  1   1  c 1   1  




 1
1
  1  c 1   1  u 1  v 1
1
1
1 u   
v x  x 





 






v1




1
1
  1  c 1   1  u 1  v 1
1
1
1 u   
v x  x 

2 
2 
2 
2 
2 
2 
2 



 u      c    u  v
Lim
v1 x  2   x 1 x 2  x 2    1   1  c 1   1  u 1  v 1 dv 2  d 2  d 2 


 1
u   1   1  c 1   1  u 1  v 1
1
x  x 1



 




 




 2  2   v 2  F21, 2  dv 2  d 2  d 2 
1 x Lim
2

1
x
v
x  x 


(22)
We reduce the 7th term on left hand side of Eq. (16):
  1   1  c 1   1  ƒu 1  v1 

 ƒ u 1  v1



1
1
1  u  v
v



 u 1  v 1   1   1  c 1   1
v1





1
1
1
  1  c 1   1
1  u  v  
v



F11
v1
 ƒ u 1  v1
 ƒ u 1  v1





(23)
Similarly, 8th and 9th terms of left hand side of (16) can be simplified as follows:
 
  1 
1
  1

1
1  


 x x 1 


 
 


Lim  2  2    2  F21, 2  dv 2  d 2  d 2 
x  2   x 1 
x  x 
 u 1  v 1  c 1   1   


 1

 
  1 
1
1
c
1  c  
 x 1 x 1 

 
 


Lim
D 2  2   2  F21, 2  dv 2  d 2  d 2 

2

1
x x
x  x 
 u 1  v 1   1   1  D 


 1


4156 (24)

(25)
Res. J. Appl. Sci. Eng. Technol., 4(20): 4150-4159, 2012
Substituting the results (20)-(25) in Eq. (16), we get the transport equation for one point distribution function F1(1)
(v, φ ,ψ) in turbulent flow in presence of dust particles:
2
 1
 

 1, 2  2  2 2 




2
2




 v

 2   F2 dx dv d d
 2  1
2   


 4

x
x  
  x   x  




 2 2  v 2 F21, 2 dv 2 d 2 d 2
 1 Lim
2
1

v x x x  x 
1
F11

1 F1
 v
1 
t
x 
v1




1
x


1
Lim1 
2
x


 2  1, 2   2 
2
2
2
 2    F2 dv d d
x  x 
Lim D
x  2   x 1



 2  1, 2   2 
2
2
1
 ƒ u 1  v1
2
 2   F2 dv d d
1 F1  0
x  x 
v


(26)
,
Similarly, a transport equation for two-point distribution function
in turbulent flow in presence of dust
particles can be derived by differentiating Eq. (5) and using Eq. (1), (2) and (3) and simplifying in the same manner
which is:

F21, 2  1 
 2    1, 2 
v
F2
 v


 x 1
t
x2 




v2
 1

 4


 


 x 2 x 2  x 3

  
2
 1
 

 


3




 v
 
 
x3  
 1  4   x1 x 1  x3  


v 

1
,
2
,
3
3
3
3








 F3 dx dv d d 3

2




1, 2,3 3 3
3 
3  
  v 3 

F
dx
dv
d
d


   x 3  3

 


 
 


 3 3  v 3 F31, 2,3 dv 3 d 3 d 3
   1 Lim

Lim
2


3
1
3 x 2  
x

x
x
v
 v
 x  x 
 
 


 3 3   3 F31, 2,3 dv 3 d 3 d 3
   1 Lim
 2 3Lim
3
1
2 



  x x  x x  x  x
 
 


 D 1 Lim

Lim 2   3 3  3 F31, 2,3 dv 3 d 3 d 3
2 3

3

1
  x x  x x  x x 

 ƒ u 1  v1
F21,2  0
v2 


(27)
Continuing this way, we can derive the equations for evolution of F3(1, 2, 3), F4(1, 2, 3, 4) and so on. Logically, it is
possible to have an equation for every Fn (n is an integer) but the system of equations so obtained is not closed. It
seems that certain approximations will be required thus obtained.
RESULTS AND DISCUSSION
If the fluid is clean then f = 0 and the transport equation for one point join distribution function F1(1) (v, φ ,ψ ) in
turbulent flow Eq. (26) becomes:
4157 Res. J. Appl. Sci. Eng. Technol., 4(20): 4150-4159, 2012
2
 1
 

 1, 2  2  2  2 


2


2 




 v
 2   F2 dx dv d d
 2  1

2   


x
x  
 4

  x   x  




 1 Lim
 2  2   v 2  F21, 2  dv 2  d 2  d 2 

2
1
v x  x x  x 
1
F11

1 F1
 v
1 
t
x 
v1




1
Lim 
x  2   x 1 


1


 2  F21, 2  dv 2  d 2  d 2 
2 
x  x 2  
Lim D
x  2   x 1 


 2  1, 2   2 
2 
2 
0
2 
 2   F2 dv d d
x  x 
Which was obtained earlier by Kishore and Singh
(1984).
For concluding the system of equations for the
joint distribution functions, some approximations are
required. Closure scheme can be used here and closure
can be obtained by decomposing F2(1, 2) as:
F2(1,2) = (1 + ε)F1(1) F1(2)
(28)
F31,2,3  1    F11 F12 F13
(29)
2
where, ε is the correlation coefficient between the
particles. The transport equation for the joint
distribution function of velocity, temperature and
concentration field have been presented to provide the
advantageous basis for modeling the turbulent flows in
presence of dust particles.
ACKNOWLEDGMENT
The authors (M. A. K. Azad and M. H. U. Molla)
thankfully acknowledge the Ministry of Science and
Information and Communication Technology of the
Peoples Republic of Bangladesh for granting NSICT
fellowship and are
also thankful to the Department of Applied
Mathematics, University of Rajshahi for providing all
facilities during this study.
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