Research Journal of Mathematics and Statistics 4(2): 45-51, 2012
ISSN: 2040-7505
© Maxwell Scientific Organization, 2012
Submitted: April 23, 2012
Accepted: May 23, 2012
Published: June 30, 2012
Decay of Temperature Fluctuations in Homogeneous Turbulence Before the
Final Period in a Rotating System
M.H.U. Molla, M.A.K. Azad and M.Z. Rahman
Department of Applied Mathematics, University of Rajshahi, Rajshahi-6205, Bangladesh
Abstract: Using deissler’s method we have studied the decay of temperature fluctuations in homogeneous
turbulence before the final period in rotating system and have considered correlations between fluctuating
quantities at two and three point. The equations for two and three point correlation in a rotating system is
obtained and the set of equations is made to determinate by neglecting the forth order correlation in
comparison to the second and third order correlations. For solving the correlation equations are converted
to spectral form by taking their Fourier transform. Finally integrating the energy spectrum over all wave
numbers, the energy decay law of temperature fluctuations in homogeneous turbulence before the final
period in presence of coriolis force is obtained in the form 2=A (t-t0)-3/2 + B exp(-2εmij Ωm)(t – t0)-5
where, 2 is the total “energy” (the mean square of the temperature fluctuations), t is the time and A and t0
are constants determined by initial conditions. The constant B depends on both initial conditions and the
fluid Prandtl number. For large times the last term becomes negligible, leaving the -3/2 power decay law
for the final period previously found by Corrsin (1951).
Keywords: Homogeneous turbulence, rotating system, temperature fluctuations
the approaches employed in the statistical theory of
turbulence. His results pertain to the final period of
decay and for the case of appreciable convective
effects, to the “energy” spectral from in specific wavenumber ranges. Deissler (1958, 1960) introduced a
method for homogeneous turbulence which was valid
for times before the final period. Using Deissler’s
method (Loeffler and Deissler, 1961) studied the decay
of temperature fluctuations in homogeneous turbulence
before the final period. In their study they introduced a
theory, which is valid during the period for which the
quadruple correlation terms are neglected compared to
the second and third-order correlation terms. Using this
method, (Sarker and Azad, 2006; Azad and Alam
Sarker, 2008a) studied the decay of temperature
fluctuations in homogeneous turbulence before the final
period for the case of multi-point and multi-time
considering rotating system and dust particle. Azad et
al. (2006b), Azad et al. (2007-2008b) and Azad and
Sarker (2009) also studied the decay of temperature
fluctuations in dusty fluid MHD turbulence before the
final period with taking rotating system. They had
shown in each case the energy decays more rapidly than
non rotating system. In their study, they also considered
two and three point correlations and neglecting fourth
and higher order correlation terms compared to the
second and third-order correlation terms.
In the present study, we have studied the decay of
temperature fluctuations in homogeneous turbulence
INTRODUCTION
The turbulence flows in the absence of the external
agencies always decay. (Jain, 1962; Batchelor and
Townsend, 1948; Deissler, 1960) had given various
analytical theories for the decay process of turbulence
so for, Further (Monin and Yaglom, 1973) gave the
spectral approach for the decay process of turbulence.
In geophysical flows, the system is usually rotating
with a constant angular velocity. Such large-scale flows
are generally turbulent. When the motion is referred to
axes, which rotate steadily with the bulk of the fluid,
the coriolis and centrifugal force must be supposed to
act on the fluid. On a rotating earth the coriolis force
acts to change the direction of a moving body to the
right in the Northern Hemisphere and to the left in the
Southern Hemisphere. This force plays an important
role in a rotating system of turbulent flow, while
centrifugal force with the potential is incorporated in to
the pressure. (Kishore and Singh, 1984; Dixit and
Upadhyay, 1989) discussed the effect of coriolis force
on acceleration covariance in ordinary and MHD
turbulent flow. (Shimomura and Yoshizawa, 1986a;
Shimomura, 1986b) also discussed the statistical
analysis of an isotropic turbulent viscosity, turbulent
scalar flux, respectively in a rotating system by twoscale direct interaction approach.
Corrsin (1951) had made an analytical attack on
the problem of turbulent temperature fluctuations using
Corresponding Author:
M.A.K. Azad, Department of Applied Mathematics, University of Rajshahi, Rajshahi-6205,
Bangladesh
45
Res. J. Math. Stat., 4(2):45-51 , 2012
before the final period in a rotating system considering
the correlations between fluctuating quantities at two
and three point and single time. In solving the problem,
it seems logical to use the approach which has already
been employed with success for studying turbulence. In
this study, the deissler’s method is used to solving the
problem. Through the study we have obtained the
energy decay law of temperature fluctuations in
homogeneous turbulence before the final period in a
rotating system. In result it is shown that the energy
decays more rapidly than non rotating system.
 T
T     2T
 ui



 x i  r   x i  x i
 t
where, Pr =
= Instantaneous values of temperature
= Instantaneous velocity
ρ
Cp
k
xi
t
= Fluid density
= Heat capacity at constant pressure
= Thermal conductivity
= Space co-ordinate
= Time
 TT 
TT 
TT   
 ui
 ui


xi
xi  r
 t
(1)
 2 TT  2 TT  


 (5)
 xi xi xixi 
The continuity equation is:
u i u i

0
xi
xi
(6)
Substitution of Eq. (6) into (5) yields:
u TT  uiTT   2 TT  2 TT  
TT 
 


 ui i

xi
r  xi xi xixi 
xi
t
(7)
By use of a new independent variable:
ri = x′i - xi i.e.,



,
x i
ri
 T T
T
T
T
T 

 ui
 ui
 ui
 ui


t
xi
xi
xi
xi  (2)
 t
  2T
 2T 
 


 xi xi xi xi 



x i ri
TT  uiTT  uiTT  2  2 TT 



t
ri
ri
r ri ri
(8)
This equation is converted into spectral form by
use of the following three dimensional Fourier
transforms:
k
cp
From the case of homogeneity it follows that
/ xi = 0 and in addition the usual assumption is
made that
is independent of time and that i = 0;
Thus Eq. (2) simplifiesto:

    
TT rˆ    Kˆ exp i Kˆ .rˆ dKˆ (9)

46 (4)
Multiplying Eq. (3) by T′, Eq. (4) by T and taking
time average and adding the two equations gives:
Separate these instantaneous values into time
average and fluctuating components as = +T and
I= i +ui Eq. (1) may be written:
where,  
, Prandtl number, v = Kinematic
 T 
T     2T  

u

i

 
 xi  r  xixi 
 t
where,
~
T
u~i
(3)
Viscosity.
Equation (3) holds at the arbitrary point P. For the
point   the corresponding equation can be written:
Correlation and spectral equations: For an
incompressible fluid with constant properties and for
negligible frictional heating, the energy equation may
be written at the point P:
~
~
~
 T ~ T 
k  2T

u



i
 x i   c p  x i x i
 t
 

Res. J. Math. Stat., 4(2):45-51 , 2012
  
 

uiTT rˆ    i  Kˆ exp i Kˆ . rˆ dKˆ
u iT T  u iT T 
T T 
 ui

xi
xi
t
(10)

And by interchanging  and   :
uiTT rˆ  uiTT  rˆ

    
uiTT rˆ   i   Kˆ exp i Kˆ . rˆ dKˆ
    
(11)
ujTT
(11a)
uiT    i   Kˆ exp i Kˆ .rˆ dKˆ
t
ui


 
  
uiT   i Kˆ exp i Kˆ .rˆ
dKˆ
uj (14)
ujuiTT ujuiTT  2ujTT 2ujTT

 

TT
t
xixi 
xi
r  xixi
xi
The momentum equation at point  , in a rotating
system:
(11b)

u j
 

  (12)
2
  Kˆ
 iki i   Kˆ i Kˆ   k 2   Kˆ
t
r
u j
t
xi

u j ui 
xi
 2u j
1 

 2 mijmu j
xi xi
 x j

 2u j
1 

 2  mijmu j (15)
 x j
xi xi

where,
εmij: Alternating tensor
Ωm: Angular velocity of a uniform rotation
Substituted Eq. (15) into (14) the result on taking
time averages is:
Equation (12) is analogous to the two point spectral
equation governing the decay of velocity fluctuations
and therefore the quantity ττ′( ) may be interpreted as
a temperature fluctuation “energy” contribution of
thermal eddies of size 1/k. Equation (12) expresses the
time derivative of this “energy” as a function of the
convective transfer to other wave numbers and the
“dissipation” due to the action of thermal conductivity.
The second term on the left hand side of Eq. (12) is the
so called transfer to term while the term on the right
hand side is “dissipation” term.

 u u uTT  u uTT    u TT    u TT 
 u jTT

j i
i
t

2
j i
xi
xi
  1 TT    u TT   2
 u j uiTT
2
 x j
xi
2
j
r  xixi
j
xi xi
mijm
j
xixi 
u TT
j
(16)
Making use of the relations ri = x′i- xi and r′I = x′′i- x′i
allows Eq. (16) can be written as:
Three points correlation and spectral equations:
In order to obtain single time and three points
correlation and spectral equation we consider three
points P,P′ and P′′ with position vectors r̂ and rˆ are
considered.








2 ujTT 
2 ujTT
2 ujTT
 ujTT  
1r 
2r
 1r 

ri ri 
riri
riri
t
r 



  u uTT  u uTT u uTT
 ujuiTT
ri
j i
j i
j i
ri
ri
ri
   


1  TT 1  TT

2mijm ujTT
 rj
 rj
(17)
Six-dimensional Fourier transforms for quantities
this equation may be defined as:
For the two points P′ and P′′ we can write a relation
according to Eq. (7):
47  
 u j ui

t
Substitution of Eq. (9)-(11b) into (8) leads to the
spectral equation:
   
(13)



r  xix i
xixi 
Equation (13) multiplied through by uj, the j-th
velocity fluctuation component at point P. Then the
equation can be written in a rotating system at the point
P:


   2 T T   2 T T 

Res. J. Math. Stat., 4(2):45-51 , 2012

 

    
  j   exp i Kˆ .rˆ  Kˆ   rˆ dKˆ dKˆ  (18)

u jT T  

 
 
u j u iT T  
j


 
 T T  




P′′ shows that:
u j uiT  T   u j uiT T  
 
 
j


 i   exp i Kˆ .rˆ  Kˆ   rˆ  dKˆ dKˆ 
  

  j   
Using Eq. (18)-(20) into (17) then the transformed
equation can be written as:

t
t
2
r
r
r i i
r
j
(21)
r
r
2
 2r k i k i  1  r k  2

(25)
k j  j  


 
 k j  j   0 exp
 r


If the derivative with respect to xj is taken of the
momentum Eq. (16) for point P and taking time average
the resulting equation is:
x j x i
   1   k
Inner multiplication of Eq. (25) by kj and
integrating between t0 and t gives:
1
 i k j  kj   2mijm  j

΄ ΄΄ from
2
 i ki  ki j i  i ki  ki  j i
 2 u j u iT T 

 j    2  mij  m  j    0
   1  k  2 k k  1  k   
  j

Solution for times before the final period: To obtain
the equation for final period of decay the third-order
fluctuation terms are neglected compared to the secondorder terms. Analogously, it would be anticipated that
for times before but sufficiently near to the final period
the fourth-order fluctuation terms should be negligible
in comparison with the third-order terms. If this
assumption is made then Eq. (24) shows that the term
΄ ΄΄ associated with the pressure fluctuations, should
also be neglected. Thus Eq. (21) simplifies to:
     exp i Kˆ .rˆ  Kˆ   rˆ dKˆ d Kˆ  (20)
Interchanging P′

Equation (24) can be used to eliminate
Eq. (21).
 i   exp i Kˆ .rˆ  Kˆ   rˆ  d Kˆ d Kˆ  (19)
 

  k j k i  2 k j k i  k j k i
 j  i   (24)
k j k j  2 k j k j  k j k j
   1  T T 

1  r k 2  2r ki ki cos 

1   k 2  2r  
r
mij m




 t  t 
0 



 (26)
Now, letting r′ = 0 in Eq. (18) and comparing the
result with the Eq. (10) shows that:
2
 x j x j
(22)
   k     dKˆ

k i  i  Kˆ 
i
(27)
i

In terms of the displacement vectors r̂ and rˆ this
becomes:
Substituting of Eq. (26) and (27) into (12), we
obtain:

 2
2
2 
2


u j uiT  T 
(23)
 rj ri rj ri rj ri 
1  2
2
2 

 
2
T T 
  rj rj
rj rj rj rj 
 t t0 
2r
 ˆ
2
2
exp
1r  k k 2rkk cos   mijm  dK



r

Taking the Fourier transform of Eq. (23) and then
solving for ΄ ΄΄ we get:
Now, d ′ (≡ dk′1 dk′2 dk′3) can be expressed in
terms of k′ and ζ as:

48  
   

 Kˆ 2 2 ˆ
 k  K  i ki i  ki i  Kˆ  Kˆ
t
r

0
(28)

Res. J. Math. Stat., 4(2):45-51 , 2012
dKˆ   2 k  2 d cos   dk 
(29)
w 

d kˆ 
(30)
  Kˆ, Kˆ  Kˆ,Kˆ 
i


0
2
0
4
4




E 2 2

k Ew
t
r

3
5

 2
exp r mijmt t0 



k2
(34)
(35)
0
It was to be expected physically since w is a
measure of the transfer of “energy” transferred to all
wave numbers must be zero.
The necessity for Eq. (35) to hold can be shown as
and
follows if Eq. (10) is written for both Kˆ and  Kˆ and
resulting equations differentiated with respect to
ri and
added, the result is, for:


rˆ 0   
ri xi 

1 

1  k2 k2 2r k k  
 t t0   r




dcos dk
 exp

cos  2r  

r

1

mij m


 





(33a)

2
Integrating Eq. (33a) w.r.t ξ, we have:
2
49 3
0
(32)


0  r2
 w dk
(33)
w  20  k2k4 k4k2 k2k2
0

The Eq. (33) indicates that w must begin as k4 for
small k. The condition of w is fulfilled by the Eq. (34).
It can be shown, using Eq. (34) that:
where,

5 3


Multiplying both sides of Eq. (32) by
defining the spectral energy function.
E = 2 k2 ′( ) and the resulting equation is:
3 5
k12r t t0
exp

 r 1r  
 15rk4
 5r2 3 k6  r3
  
 2



 r k8

2
2
3
4 t t0 1r  1r  2t t0 1r  1r  

1  t t0  

2r

2
2
 exp
1r  k k 2rk k cos   mijm dcos dk



r
1 


(33b)
k k  k k 
22t t0 21r  2
2
2
conditions. The negative sign is placed infront of  0 in
order to make the transfer of energy from small to large
wave no. for positive value of ζ0. Substituting Eq. (31)
into (30):


t  t0  0
w 
where,  0 is a constant depending on the initial
 Kˆ 2
  2k2  Kˆ 20 k2k4 k4k2
t
r
0
0

5
    2  k k  k k  (31)

5 3
Again integrating Eq. (33b) w.r.t k′ we have:
In order to find the solutions completely and
following Loeffler and Deissler (1961), we assume that:

3 5


i ki i    Kˆ , Kˆ   i     Kˆ ,Kˆ 
k k  k k 
  t  t0  

1 r  k2  k2  2rk k  2r mijm  dk
 exp


 
  r 
0
1 

r  

2
2
  t t0  1r  k k 2rk k cos   

exp
  dcos d k

r
1 

(mijm  fs)
 



2t  t0  0


 Kˆ 2 2 ˆ
 k  K  22i ki
t
r

i

  t  t0  

1 r  k2  k2  2rk k  2r mijm 
 exp


 
  r 
Substituting Eq. (29) into (28) yields:

0


 

uiTT   iki i  Kˆ  i   Kˆ
xi

dkˆ
(35a)
Since according to the Eq. (32), (33) and (12), w ≡
2ki
Res. J. Math. Stat., 4(2):45-51 , 2012

So the Eq. (35a) can be written as -2 / xi
′ =
2
dk as d = 4 k dk for w = w (k,t) then
/2
F    e
= 0.
/
the Eq. (35) becomes
The linear Eq. (33) can be solved for w as:
k
 2 k t  t0  
 2 k t  t0 
E  exp
  w exp
 dt  J k  (36)

r
r




2
2
 2 k 2 t  t0  
k 2 exp


r


T2

2
J k  
E k , t  
N0 k

R
 2  k t  t 0  
0  
exp  

3
r
 2 2 1   72

r

exp  2 mij  m t  t 0  
  k  1  2 r t  t 0  
 exp 

r 1  r 


 3r k 4
r 7 r  6 k 6
4 3r2  2 r  3 k 8 



1 
3
5
2
2
 2 t  t 0  2
3 1  r t  t 0  2
31  r  t  t 0  2 


2
9
 8  3r  2 r  3 k F  



1
5
2


2





3
1
r
r



2



3
7
4 2 1  r  2
 exp  2 mij  m t  t 0  
  k 2 1  2 r t  t 0  
 exp 

r 1  r 






3

exp  2mijm


(43)
 r 6
21 r 1 2r 

5
2

B
20 R and
6
 9 5r 7r  6
 
16 161 2r 
35 r 3r2  2r  3
2
81 2r 

5
(44)

5
1.5422r 3r2  2r  3 1 2r 2
11
2
Corrsin (1951). T
is the total “energy” (the mean
square of the temperature fluctuations).
where,
50  t  t0 5
Equation (43) is the decay law of temperature
fluctuation in homogeneous turbulence in presence of
coriolis force before the final period. The first term of
the right side of Eq. (43) corresponds to the temperature
energy for two point correlation and the second term
represents the energy for the three point correlation.
This second term becomes negligible at large times
leaving the final period decay law previously found by
 3r k 4
r 7 r  6 k 6
4 3r2  2 r  3 k 8 



1 
3
5
2
 2 2 t  t 0  2
3 1  r t  t 0  2
31  r  t  t 0  2 


2
9
 8  3r  2 r  3 k F  



1
5
2


31  r  2 r



0 R
6
R is a function of Prandtl no.
(39)
5
8 2  t  t0 

3
2
  1 r  2
  n11 2n 1 
1 
n 
2n
 n 1 2n 1n! 2 1 r  

 0  r2
3
2
4 2  2
2
r

3
3
5
2
(42)
0
where, A  N0 r  2 ,
3
Now, substituting the values of w and J(k) as
given by the Eq. (34) and (38) into (36) gives the
equation:
2
 E k  dk
N0 r 2

(38)


 T 2 At  t0  2  Bexp  2mijm  t  t0 
where, N0 is a constant which depends on the initial
conditions. Using Eq. (37) to evaluate J(k) in Eq. (36)
yields:
N0 K
r 1  r 
Substituting Eq. (39) into (42) gives:
(37)
2
(40)
 t  t 0 
T T T2


2
2
where, J(k) is an arbitrary function of k.
For large times, Corrsin (1951) has shown the
correct form of the expression for E to be:
N0
0
(41)
Putting = 0 in Eq. (9) and we use the definition of
E given by the Eq. (39), the result is:
 2 k 2 t  t0 
exp

r


E
2

 2 e x dx
Res. J. Math. Stat., 4(2):45-51 , 2012
Azad, M.A.K. and M.S. Alam Sarker, 2008a. Decay of
temperature
fluctuations
in
homogeneous
turbulence before the final period for the case of
Multi-point and Multi- time in a rotating system in
presence of dust particle. J. Appl. Sci. Res., 4(7):
793- 802.
Azad, M.A.K., S.A. Sarker and M.A. Aziz, 20072008b. Decay of temperature fluctuations in dusty
fluid MHD turbulence before the final period.
Math. Forum, 20: 32-48.
Azad, M.A.K. and M.S. Alam Sarker, 2009. Decay of
temperature fluctuations in MHD turbulence before
the final period in a rotating system. Bangladesh J.
Sci. Ind. Res. 44(4): 407-414.
Batchelor, G.K. and A.A. Townsend, 1948. Decay of
isotropic turbulence in the final period. Proc. Roy.
Soc., London A, 194: 527-543.
Corrsin, S., 1951. On the spectrum of isotropic
temperature fluctuations in isotropic turbulence. J.
Appl. Phy., 22: 469- 473.
Deissler, R.G., 1958. On the decay of homogeneous
turbulence before the final period. Phys. Fluids, 1:
111-121.
Deissler, R.G., 1960. A theory of decaying homogenous
turbulence. Phys. Fluids, 3: 176-187.
Dixit, T. and B.N. Upadhyay, 1989. The effect of
coriolis force on acceleration covariance in MHD
turbulent dusty flow with rotational symmetry.
Astrophys. Space Sci., 153: 257-268.
Jain, P.C., 1962. Isotropic temperature fluctuations in
isotropic turbulence. Proc. Natl. Inst. Sci., 28(3):
401- 416.
Kishore, N. and S.R. Singh, 1984. Statistical theory of
decay process of homogeneous hydromagnetic
turbulence. Astrophys. Space Sci., 104: 121-125.
Loeffler, A.L. and R.G. Deissler, 1961. Decay of
temperature
fluctuations
in
homogeneous
turbulence before the final period. Int. J. Heat Mass
Trans., 1: 312-324.
Monin, A.S. and A.M. Yaglom, 1973. Statistical Fluid
Math., I., M.I.T. Press, Cambridge, MA.
Sarker, M.S.A. and M.A.K. Azad, 2006. Decay of
temperature
fluctuations
in
homogeneous
turbulence before the final period for the case of
Multi-point and Multi- time in a rotating system.
Bangladesh J. Sci. Ind. Res., 41(3-4): 147-158.
Shimomura, Y. and A. Yoshizawa, 1986a. Statistical
analysis of an isotropic turbulent viscosity in a
rotating system. J. Phys. Soc., Japan, 55:
1904-1917.
Shimomura, Y., 1986b. Two-scale direct-interaction
approach to the turbulent scalar flux in a rotating
system. J. Phys. Soc., Japan, 55: 3388-3401.
RESULTS AND DISCUSSION
Equation (43) is the decay law of temperature
fluctuation in homogeneous turbulence before the final
period in presence of corriolis force. In the absence of
the corriolis force, Ωm = 0, the Eq. (43) becomes:
3

3
N 0  r  2  2
T 2


3
2

8 2   t  t 0  2
3
 A t  t0  2  B t  t0 

5

6
t
0
R
 t0
5
which was obtained
earlier by Loeffler and Deissler (1961)
Here:
3
N  2 
A 0 r
8 2 

3
2
and
B
0 R
6
Due to the effect of coriolis force in homogeneous
turbulence, the temperature energy fluctuations decays
more rapidly then the energy for non rotating fluid
before the final period. For large times, the second term
in the Eq. (43) becomes negligible leaving the -2/3
power decay law for the final period.
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.
REFERENCES
Azad, M.A.K. and M.S. Alam Sarker, 2006a. Decay of
temperature
fluctuations
in
homogeneous
turbulence before the final period for the case of
Multi- point and Multi-time in presence of dust
particle, Rajshahi University studies. B. J. Sci., 34:
37-50.
Azad, M.A.K., M.S.A. Sarker and N.I. Mondal, 2006b.
Decay of temperature fluctuations in dusty fluid
MHD turbulence before the final period in a
rotating system. J. Eng. Appl. Sci., 1(2): 187-194.
51