Research Journal of Applied Sciences, Engineering and Technology 6(10): 1749-1756, 2013
ISSN: 2040-7459; e-ISSN: 2040-7467
© Maxwell Scientific Organization, 2013
Submitted: October 17, 2012
Accepted: December 10, 2012
Published: July 20, 2013
Homogeneous Turbulence in a First-order Reactant for the Case of Multi-point and
Multi-time Prior to the Final Period of Decay in a Rotating System
M.A. Bkar PK, M.S. Alam Sarker and M.A.K. Azad
Department of Applied Mathematics, University of Rajshahi-6205, Bangladesh
Abstract: Using Deissler’s approach, the decay for the concentration of a dilute contaminant undergoing a firstorder chemical reaction in homogeneous turbulence at times prior to the final period of decay in a rotating system
for the case of multi-point and multi-time is studied. Here two and three point correlations between fluctuating
quantities have been considered and the quadruple correlations are ignored in comparison to the second and third
order correlations. Taking Fourier transform the correlation equations are converted to spectral form. Finally,
integrating the energy spectrum over all wave numbers we obtained the decay law for the concentration fluctuations
in a homogeneous turbulence prior to the final period in a rotating system for the case of multi-point and multi-time.
Keywords: Deissler’s method, dust particle, first order reactant, Navier-stock’s equation, turbulent flow
INTRODUCTION
The mathematical models that describe chemical
reaction kinetics provide chemists and chemical
engineers with tools to better understand and describe
chemicals processes such as food decomposition,
stratospheric ozone decomposition and the complex
chemistry of biological systems. The essential
characteristic of turbulent flows is that turbulent
fluctuations are random in nature. In geophysical flows,
the system is usually rotation with a constant velocity;
such large scale flows are generally turbulent. The
Coriolis Effect is caused by the rotation of the Earth
and the inertia of the mass experiencing the effect. The
most commonly encountered rotating reference frame
in the Earth. Because the Earth completes only one
rotation per day, this force causes moving objects on
the surface of the Earth to appear to change direction to
the right in the northern hemisphere and to the left in
the southern and it has great significance in
Astrophysics, Stellar dynamics, Earth sciences.
Meteorology, physical geology and oceanography:
Following Deissler’s approaches (Deissler, 1958, 1960)
the two-point; two-time correlations are obtained by
considering the equation for the concentration of a
dilute contaminant undergoing a first order chemical
reaction. In order to solve the equations for the final
period, the triple order correlation terms are neglected
in comparison to the second-order ones. Loeffler and
Deissler (1961) used the theory, developed by Deissler
(1958, 1960) to study the temperature fluctuations in
homogeneous turbulence before the final period. In the
study of homogeneous fluid turbulence a method is
describing theoretically the concentration fluctuations
of dilute contaminant a first order reactant prior to the
ultimate phage of decay by Kumar and Patel (1974)
they studied the first order reactants in homogeneous
turbulence for the case of multi-point and single time
consideration. Kumar and Patel (1975) extended their
problem (1974) for the case of multi-point and multitime concentration correlation. In Sarker and Kishore
(1991), studied the decay of MHD turbulence at times
before the final period using Chandrasekhar’s relation
(Chandrasekhar, 1951). Islam and Sarker (2001) studied
the first-order reactant in MHD turbulence before the
final period of decay for the case of multi-point and
multi-time. PK et al. (2012) studied the first-order
reactant in homogeneous dusty fluid turbulence prior to
the ultimate phase of decay for four-point correlation in
a rotating system. Corrsin (1951) obtained the spectrum
of isotropic temperature fluctuations in isotropic
turbulence.
In our present study, we studied the fluctuation of
concentration of a dilute contaminant undergoing a
first-order chemical reaction in homogeneous
turbulence prior to the final period of decay in a
rotating system for the case of multi-point and multitime. Here, we have considered two-point and threepoint correlation equations and solved these equations
after neglecting fourth-order correlation terms. Finally
we obtained the decay law of energy fluctuations of
concentration of dilute contaminant undergoing a first
order chemical reaction in homogeneous turbulence in a
rotating system for the case of multi-point and multitime comes out to the form:
{
−3 / 2
X 2 = exp(-2RTm ) ATm
+ exp[−2ε mki Ω m ]BTm−5
Corresponding Author: M.A. Bkar PK, Department of Applied Mathematics, University of Rajshahi-6205, Bangladesh
1749
}
Res. J. Appl. Sci. Eng. Technol., 6(10): 1749-1756, 2013
where, ⟨𝑋𝑋 2 ⟩ denotes the concentration fluctuation
energy. It is seen that the demolition of the impurity is
more rapid than that in the case of pure mixing. This
result has been shown in the figure also.
Multiplying Eq. (2) by X', Eq. (3) by X and
averaging, we get:
∂ XX ′
∂t
MATERIALS AND METHODS
Basic equation: The differential equation governing
the concentration of a dilute contaminant undergoing a
first-order chemical reaction in homogeneous fluid
turbulence in a rotating system could be written as:
∂u i
∂u
∂ ui
1 ∂p
+ uk i = −
+ν
− Ru i − 2ε mki Ω m u i i (1)
∂t
∂x k
ρ ∂x k
∂x k ∂x k
2
= A random function of position and time at a
point p
𝑢𝑢𝑘𝑘 (𝑥𝑥�, 𝑡𝑡) = Turbulent velocity
R
= Constant reaction rate
D
= Diffusivity
t
= Time
= Alternating tensor
ε mki
= Constant angular velocity components
Ωm
𝑝𝑝(𝑥𝑥�, 𝑡𝑡) = Pressure fluctuation
𝜌𝜌
= Fluid density
v
= Kinematics viscosity
= Turbulent velocity component
uk
= Space-coordinate and repeated subscript in a
xk
term indicates a summation of terms, with
the subscripts successively taking on the
values 1, 2 and 3
•
•
•
∂x k
=D
∂ 2 XX ′
∂x k ∂x k
− R XX ′
(4)
(5)
where, the conditions of continuity and the fact that the
quantities at a point at a particular time are independent
of the positions at the other points have been utilized.
Using the transformations:
∂
∂
∂
∂ ∂ ∂
∂ ∂
∂
,
, =  −
,
=−
=
=
∂x k
∂rk ∂x k′ ∂rk ∂t  ∂t  ∆t ∂∆t ∂t ′ ∂∆t
Here,
𝑢𝑢𝑖𝑖 (𝑥𝑥�)
and
∂ u k XX ′
∂ XX ′
∂ uk′ XX ′
∂ 2 XX ′
+
=D
− R XX ′
∂t ′
∂xk′
∂xk′ ∂xk′
The subscripts can take on the values 1, 2 and 3.
Tow-point, two-time correlation
equations: Under the restrictions that:
+
in to Eq. (4) and (5), we obtains:
∂ XX ′
∂t
∂ u k XX ′
∂rk
∂ XX ′
∂∆t
∂X
∂X
∂2 X
+ uk
=D
− RX
∂t
∂x k
∂x k ∂x k
(2)
∂X ′
∂X ′
∂2 X ′
+ u k′
=D
− RX ′
∂t ′
∂x k′
∂x k′ ∂x k′
(3)
where, 𝑋𝑋(𝑥𝑥�, 𝑡𝑡) is a random function of position and
time. The other symbols are as usual.
+
∂ u k XX ′
(− rˆ,−∆t , t + ∆t ) −
∂rk
∂ XX ′
(6)
2
( rˆ, ∆t , t ) = 2 D
∂ u k XX ′
∂rk
∂rk ∂rk
− 2 R XX ′
(−rˆ,−∆t , t + ∆t ) = D
∂ 2 XX ′
∂rk ∂rk
− R XX ′ (7)
In order to reduce Eq. (6) and (7) to spectral form
by using three-dimensional Fourier transform:
∞
XX ′(rˆ, ∆t , t ) = ∫ θ (kˆ, ∆t , t ) exp(ikˆ.rˆ)dkˆ
spectral
The turbulence and the concentration fields are
homogeneous
The chemical reaction and the local mass transfer
have no effect on the velocity field
The reaction rate and the diffusivity are constant,
differential equation governing the concentration of
a dilute contaminant undergoing a first-order
chemical reaction we take the Navier-Stokes
equations at the point P and the concentration
equation at P′ and separated by the vector 𝑟𝑟̂ could
be written as:
+
(8)
−∞
XX ′(rˆ, ∆t , t ) =
∞
∫φ
k
(kˆ, ∆t , t ) exp(ikˆ.rˆ) dkˆ
(9)
−∞
We get:
∂θ
+ (2 Dk 2 + 2 R )θ = ik k φ k (kˆ, ∆t , t )
∂t
+ i (− k )φ (− kˆ ,− ∆t , t + ∆t )
(10)
∂θ
+ ( Dk 2 + R )θ = −ik k φ k (−kˆ,−∆t , t + ∆t )
∂∆t
(11)
k
k
k
Solution for the ultimate phase of decomposing
turbulence: For the ultimate phase of homogeneous
turbulence decompose, the third-order correlations can
be ignored in comparison to the second-order
correlations, with this approximation, the solutions of
Eq. (10) and (11) may be obtained as:
1750
θ = f 1 (kˆ, ∆t ) exp[−(2 Dk 2 + 2 R )(t − t 0 )] (12)
Res. J. Appl. Sci. Eng. Technol., 6(10): 1749-1756, 2013
θ = f 2 (kˆ, t ) exp[−( Dk 2 + R)∆t )]
∂
∂2
u i u k′ X ′X ′′ = D
u i X ′X ′′ − R u i X ′X ′′ (21)
∂x k′
∂x k′ ∂x k′
(13)
For consistent solution of Eq. (12) and (13) we must
have:
G (k ) = f (k ) exp[(−2 Dk 2 + 2 R)(t − t 0 +
∆t
)]
2
(14)
where, G (k) = 2𝜋𝜋𝑘𝑘 2 𝜃𝜃 is the concentration spectrum
function. We evaluate ƒ (k) by Corrsin (1951) i.e., ƒ (k)
= N o k2|π. where, N 0 = A constant depend on initial
condition. Thus, we obtain:
G (k ) =
N0k
2
π
exp[(−2 Dk 2 + 2 R)(t − t 0 +
∆t
)]
2
∂
u i X ′X ′′ +
∂t ′′
∂
∂2
u i u k′′X ′X ′′ = D
u i X ′X ′′ − R u i X ′X ′′ (22)
∂x k′′
∂x k′′∂x k′′
Using the transformations:
∂ ∂
∂ ∂
∂
∂
∂
∂
),
,
, t ′, t ′′
=
=
+
= −(
∂rk ∂rk′ ∂xk′ ∂rk ∂xk′′ ∂rk′ ∂t
∂xk
(15)
∂
∂
∂
∂
∂
∂
∂
,
,
=
=
−
= 
−
 ∂t  ∆t ,∆t ′ ∂∆t ∂∆t ′ ∂t ′ ∂∆t ∂t ′′ ∂∆t ′
By integrating Eq. (15) with respect to k, we obtain:
XX ′ (rˆ, t m ) =
N o D1 / 2
4 4π (t m − t o ) 3 / 2
Into Eq. (20) to (22), we get:
exp
  2C (t m − t o ) + r 2  
− 

  8 D(t m − t o )  
∂
∂
∂
+
) ui u k X ′X ′′
ui X ′X ′′ − (
∂t
∂rk ∂rk′
(16)
+
where, t m = t + Δt/2.
∂
∂
ui u k′ X ′X ′′ +
ui u k′′X ′X ′′
∂rk
∂rk′
=−
Three-point, three-time correlation and equations:
Under the same assumptions as before, we take the
Navier-Stokes equation for homogeneous fluid
turbulence in a rotating system at the point P and the
concentration equations at P′ and P′′ as:
1
(
∂
ρ ∂rk
+
∂
∂
∂ 2
) pX ′X ′′ + ν (
+
)
∂rk′
∂rk ∂rk′
ui X ′X ′′ + D (
∂u i
∂u
∂ ui
1 ∂p
+ uk i = −
+ν
− Ru i − 2ε mki Ω m u i (17)
ρ ∂x k
∂t
∂x k
∂x k ∂x k
∂2
∂2
+
) ui X ′X ′′
∂rk ∂rk ∂rk′∂rk′
− (2 R + 2ε mki Ω m ) u i X ′X ′′
(23)
∂
∂
u i X ′X ′′ +
u i u k′ X ′X ′′
∂∆t
∂rk
(24)
2
∂X ′
∂X ′
∂2 X ′
+ u k′
=D
− RX ′
∂t ′
∂x k′
∂x k′ ∂x k′
(18)
=D
∂X ′′
∂X ′′
∂ 2 X ′′
+ u k′′
=D
− RX ′′
∂t ′′
∂x k′′
∂x k′′∂x k′′
(19)
∂
∂
ui X ′X ′′ +
ui u k′′X ′X ′′
∂∆t ′
∂rk′
=D
Multiplying Eq. (17) by X′X′′, (18) by u i X'' and
(19) by u i X' and then taking space averages, we obtain:
∂
u i X ′X ′′ +
∂t
∂
u i X ′X ′′ +
∂t ′
(25)
∂
u X ′X ′′ − R X ′ui X ′′
′ rk′′ i
∂rk′∂
2
Using the six-dimensional Fourier transform of the
type:
XX ′u k′′ ( rˆ, rˆ′, ∆t , ∆t ′, t )
1 ∂
∂2
∂
u i u k X ′X ′′ = −
pX ′X ′′ + ν
u i X ′X ′′
ρ ∂x k
∂x k ∂x k
∂x k
− R u i X ′X ′′ − 2ε mki Ω m u i X ′X ′′
∂
u i X ′X ′′ − R X ′u i X ′′
∂rk′∂rk′
2
∞ ∞
=
∫ . ∫ψ
k
( kˆ, kˆ′, ∆t , ∆t ′, t ) exp(kˆ.rˆ + kˆ′.rˆ′) dkˆdkˆ′
−∞ −∞
(20)
and the assumption that the quintuple correlations
representing the transfer terms in Eq. (23) to (25) can
be neglected as they decay faster than the lower-order
1751
Res. J. Appl. Sci. Eng. Technol., 6(10): 1749-1756, 2013
∞
correlation terms. Then the Eq. (23) to (25) in Fourier
space can be written as:
W = ik (ψ ) k 2 k ′ 2 (2π ) 2 exp
∫ i i0
0
∂ψ i ˆ ˆ
(k , k ′, ∆t , ∆t ′, t ) + D[(1 + N S )k 2 + 2 N s kk ′ cos{(1 + N s )k ′ 2
∂t


∆t 
2
 + k ′ 2 (1 + N S )( t − t o )
− D (1 + N S )k  t − t o +
t
N S 
+



+ (2 R + 2ε mki Ω m ) / D}]ψ i (kˆ, kˆ ′, ∆t , ∆t ′, t ) = 0 (26)
+ [(2ε mki Ω m ) / D](t − t 0 )
∂ψ i ˆ ˆ
( k , k ′, ∆t , ∆t ′, t ) + D[
∂∆t
(k 2 + 2 R ) / D ]ψ i (kˆ, kˆ ′, ∆t , ∆t ′, t ) = 0
+ (2 R / D)
(27)
{
∆t  1

 t − t o +  ∫−1 exp[−2 N s Dkk ′(t − t 0 ) cos θ ]dk ′
2  

∂ψ i ˆ ˆ
(k , k ′, ∆t , ∆t ′, t ) + D[
∂∆t ′
(k ′ 2 + 2 R ) / D ]ψ i (kˆ, kˆ ′, ∆t , ∆t ′, t ) = 0
∞
+ ∫ [i (−k i )ψ i (−kˆ,−kˆ ′)]0 (2π ) 2 k 2 k ′ 2 exp{− D
(28)
0
where, N s = v/D, the Schmidt number
As the pressure force terms are related to higherorder correlations, therefore, these along with the
quadruple correlations are also neglected.
Integrating Eq. (26) to (28) between t o and t, we obtain:

∆t
2
(1 + N S )k  t − t o +
+
1
NS


− k ′ 2 (1 + N s )(t − t 0 + ∆t ) + [(2ε mki Ω m ) / D](t − t 0 )
ψ i = f i exp{− D[(1 + N s )k 2 + 2 N s kk ′
+
cosθ + (1 + N s )k ′ 2 + (2 R + 2ε mki Ω m ) / D](t − t 0 )}
ψ i = g i exp{[− D(k 2 + R / D)]∆t}
1
−1
where, dk' is written as 2πk ′ 2 d (cos θ )dk ′ and the
quantity (ψ i ) 0 depends on the initial conditions of the
turbulence. Now, following Deissler (1958, 1960), we
take:
For these relations to be consistent, we have:
k iψ i = k i (ψ i ) 0 exp[− D{(1 + N s )(k 2 + k ′ 2 )
(t − t 0 ) + k 2 ∆t + k ′ 2 ∆t ′ + 2 N s kk ′ cosθ (t − t 0 )
1
(2π ) 2 i[k iψ i (kˆ, kˆ ′) 0 = − δ 0 (k 2 k ′ 4 − k 4 k ′ 2 )
2
+ (2 R / D)[t − t 0 + (∆t + ∆t ′) / 2] + [(2ε mki Ω m ) / D](t − t 0 )}] (29)
where, the subscript 0 refers to the value of ψ i at t = t o ,
∆t = ∆t′ = 0 and θ is the angle between k and k'. The
relation between ϕ i and ψ i is given by:
∞
∫kψ
i
i
(kˆ, ∆t , kˆ ′,0, t )dk ′
(30)
−∞
and
1
(2π ) 2 i[−k iψ i (−kˆ,−kˆ ′) 0 = δ 0 (k 2 k ′ 4 − k 4 k ′ 2 ) (33)
2
Substituting Eq. (33) in (32) and completing the
integration, we get:
Substituting Eq. (30) and (29) into Eq. (10), we obtain:
∂G
+ (2k 2 D + 2 R )G = W
∂t
∆t
2R
(t − t 0 + )]}
D
2
× ∫ exp[−2 N s Dkk ′(t − t 0 + ∆t ) cos θ ](d cos θ )dk ′ (32)
ψ i = hi exp{[− D(k ′ 2 + R / D)]∆t ′}
k iφi ( kˆ, ∆t , t ) =



W =−
δ o N sπ 1 / 2
4 D 3 / 2 (t − to )3 / 2 (1 + N s )5 / 2
(31)

1 + 2N s
exp − k 2 D
1+ Ns

where, G = 2πk θ and
2
1752


1+ Ns
 t − t o +
∆t 
+
N
1
2
s


Res. J. Appl. Sci. Eng. Technol., 6(10): 1749-1756, 2013
∆t 

+ (2ε mki Ω m )(t − t 0 )− 2 R t − t o + 
2 

+

  N
Ns
15k 4
s
+ 5

2
2
2
 4 N s (t − t o ) D 1 + N s   1 + N s
8 D1 / 2 (3 N s2 − 2 N s + 3)k 9  (t − t o ) D 
Fk

3(1 + N s ) 5 / 2
1 + N s 

+
π Ns
δo
D 3 / 2 (1 + N s ) 7 / 2 4
2

3
 − 
2



 1 + 2N s  2 

Ns
k  t − t o +
∆t 
− D


1 + 2 N s 
 1+ Ns  
× exp 

∆t 
+ (2ε mki Ω m )(t − t 0 ) − 2 R t − t o +  
2  


3

Ns  8 

 −
k 
1 + Ns  

 
×
 N
k6
s
+ 
N s D (t − to )  1 + N s

−
δoN s π
4 D 3 / 2(t − to + ∆t ) 3 / 2 (t + N S ) 5 / 2

(7 N s − 6)k 6
3k 4
× 2
+
5/ 2
3D(1 + N s )(t − t o + ∆t ) 3 / 2
 2 D N s (t − t o + ∆t )
4 (3 N s2 − 2 N s + 3)k 8
−
3 (1 + N s ) 2 (t − t o + ∆t )1 / 2
 2 1 + 2Ns 
 
Ns
 t − to +
∆t  
− k D
1 + Ns 
1 + Ns  
× exp 

∆t 
+ (2ε mki Ω m )(t − t0 ) − 2 R t − to − 
2 




15k 4

× 2 2
2
 4 D N s (t − t o + ∆t )


  Ns
5
 N s    1 + N s
+

1+ N  
3
s 

2 −
 2
1/ 2
2
9
+ 8D (3N s − 2 N s + 3)k
(1 + N s ) 5 / 2



k6

 N s D(t − t o + ∆t )


where,
ω
 N 3
Ns  8 

s
 −
k 
+ 
 1 + N s  1 + N s  

F (ω ) = exp(−ω 2 ) exp( x 2 )dx
∫
(34)
0
This represents the transfer function arising due to
the consideration of concentration at three-point and
three-time. When ∆t = 0 and R = 0, the expression for
reduces to the case of pure mixing. It may also be noted
that (for ∆t = 0):
∞
∫0 𝑊𝑊 dk = 0
ω=k
(35)

(t - t o + ∆t )D
1+ N s
∞
XX ′
(∆t , t m ) = ∫ Gdk =
2
0
No
8D 3 / 2
+
π Ns
3/ 2
2  D (1 + N s ) 7 / 2
2
G = N 0 k exp− 2(k 2 D + 2 R) t − t + ∆t  +
o

(t − t o ) D
or k =
1+ N s
As in the previous section, by integrating Eq. (36)
with respect to k, we obtain:
This means that the conditions of continuity and
homogeneity are satisfied. Physically, it was to be
expected as W is a measure of the energy transfer and
the total energy transferred to all wave numbers must
be zero.
With the help of Eq. (31) and (34), one can get:
π
 (t − t o + ∆t ) D  (36)

Fk


1
N
+
s


∆t 

2π  T + 
2

3/ 2

∆t 

exp − 2 R T + 
2 


π
∆t
exp[−(2ε mki Ω m )] exp[−2 R(T + )]
5/ 2
D (1 + N s )(1 + N s )
2
6
× δ0

1 + 2N s 2 
(1 + N s )∆t  
δo
 
k  t − to +
− D
1 + Ns
1 + 2 N s  

4 × exp 

∆t 
+ (2ε mki Ω m )(t − t0 ) − 2 R t − t o + 
2 


4
9


1 + Ns
16T 5 / 2  T +
∆T 
1
2
+
N
s



(7 N s − 6)k 6
3k 4
+

2 N s D 2 (t − t o ) 5 / 2 3D(1 + N s )(t − t o ) 3 / 2
×
 4(3 N s2 − 2 N s + 3)k 8
−
2
1/ 2
 3(1 + N s ) (t − t o )
+
1753
5/ 2
+
9


1 + Ns
16(T + ∆T ) 5 / 2  T +
∆T 
1
2
+
N
s


5 N s (7 N s − 6)


1+ Ns
16(1 + 2 N s )T 3 / 2  T +
∆T 
1 + 2N s


7/2
5/ 2
Res. J. Appl. Sci. Eng. Technol., 6(10): 1749-1756, 2013
5 N s (7 N s − 6)
+
Therefore,


Ns
∆T 
16(1 + 2 N s )(T + ∆T ) 3 / 2  T +
1 + 2Ns


7/2
⟨𝑋𝑋 2 ⟩ = exp (-2RT m )

δα 1
No
3
+ exp[−(2ε mki Ω m )] o 6 5 

3/ 2
3/ 2
D Tm 
 4 2π D Tm
35 N s (3 N s2 − 2 N s + 3)
+


1 + Ns
8(1 + 2 N s )T 1 / 2  T +
∆T 
1 + 2N s


35 N s (3 N s2 − 2 N s + 3)
+
9/2
where,


Ns
8(1 + 2 N s )(T + ∆T )1 / 2  T +
∆T 
1 + 2N s


+ 8 N s (3 N − 2 N s + 3)(1 + 2 N s )
3.2 23 / 2 (1 + N s )11/ 2
2
s
α=
9/2
Thus, the decay law for the concentration energy
fluctuation of homogeneous dusty fluid in a rotating
system prior to the ultimate phase may be written as:
∑ n!(2n + 1)2
n =0
T
( 2 n +1) / 2
∆T 


T +
2 

( 2 n +11) / 2
+
(T + ∆T)
(2n +1)/2
∆T 


T +
2 

⟨𝑋𝑋 2 ⟩ = exp(-2RTm ){ATm−3 / 2 + exp[− F ]BTm−5 } (40)
(37)
( 2 n +1) / 2
where,
A=
where, T = t - t o . For T m = T + ΔT/2 Eq. (37) becomes:
XX ′
N0
(t m ) = exp(−2 RTm )
2
8 2π D 3 / 2T 3 / 2 m
+
δ oπ
4 D 6 (1 + N s )(1 + 2 N s )



5/ 2
9
∆T 5 / 2
∆T 5 / 2
16(Tm +
) (Tm −
)
2
2
5 N s (7 N s − 6)
1
+
∆T 7 / 2
∆T 3 / 2
16(1 + 2 N s )
(Tm +
) (Tm −
)
2
2
16(1 − 2 N s ) 
∆T
∆T  
  Tm −
 Tm +
2  
1 + 2N S




δ 0α F = 2ε Ω
,
mki
m
D6
⟨𝑋𝑋 2 ⟩ = exp(-2RTm ){ATm−3 / 2 + BTm−5 }
+....
1
3/2
,B =
It is noted that y1, y2, y3, y4, y5, y6 and y7 are
solution curves of Eq. (40) but in the absence of coriolis
force y1, y2, y3, y4, y5, y6, y7 are represented by
Eq. (41) at the different values of chemical reaction
respectively which indicated in the figures. In Eq. (40)
we obtained the concentration fluctuation energy decay
of homogeneous fluid turbulence in a rotating system.
In the absence of corriolis force i.e. if the frame is nonrotating then Eq. (40) becomes:
+
+ 5 N s (7 N s − 6)
3N 0
4 2π D 3 / 2
RESULTS AND DISCUSSION
exp[−(2ε mki Ω m )]
9
1
5/ 2
16 

∆T 
∆T
 Tm −
  Tm +
2  
1+ 2N s

π
(1 + N s )(1 + 2 N s )
9

5 N s (7 N s − 6)
+ ....
 +
+
16 16 (1 2 N s )

5/ 2
1.3.5...(2n + 9)
×
2n
(1 + N s ) n
∞
(39)
(38)
7/2
If Δt = 0, then Eq. (38) reduces to the form:
δ oπ
1
 No

 8 2π D 3 / 2 T 3/2 + 2 D 6 (1 + N )(1 + 2 N ) 
X2
m
s
s

= exp(−2 RTm ) 


2


−
(
7
6
)
N
N
9
5
s
s
exp[−(2ε Ω )]

+
mki m 

5
5

16Tm 16 (1 + 2 N s )Tm  

(41)
Which was obtained earlier by Kumar and Patel
(1975). For large times, the last term of Eq. (41)
becomes negligible and the decay law for the ultimate
period becomes exp (-2RT m ) (AT-3/2 m ) which in the
case of pure-mixing is similar to the law obtained by
Corrsin (1951).
In Fig. 1 to 4, we observe that the variation of
chemical reaction (R = 3.5, 1.75, 0.875, 0.44, 0.22, 0.11
and 0) causes significant changes in the concentration
fluctuation decay of energy of homogeneous fluid
turbulence in presence of corriolis force F = 2.75, 1.50,
0.75 and 0, respectively. It is mentioned that the energy
decay of the fluid particles decreases with the increases
of the chemical reaction R and is maximum where the
reaction rate is equal to zero. If the system is nonrotating i.e., the corriolis force is absence the energy
decay of the fluid particles more rapidly than any
1754
Res. J. Appl. Sci. Eng. Technol., 6(10): 1749-1756, 2013
3.5
y7
3
y6
Decay of energy
If in all cases we consider
F=2.75
y5
2.5
y4
2
1.5
y3
R=0
1
R=.11
R=.22
y2
R=.44
0.5
R=.875
y1
R=1.75
R=3.5
0
1
1.5
2
2.5
3
Approximation of time
3.5
4.5
4
Fig. 1: Energy decay curves of Eq. (40) for F = exp(2ε mki Ω m ) = 2.75
3.5
y7
3
y6
Decay of energy
2.5
If in all cases we consider
F=1.50
y5
y4
2
1.5
y3
R=0
1
R=.22
y2
R=.11
R=.44
0.5
R=.875
y1
R=1.75
R=3.5
0
1
1.5
2
2.5
3
Approximation of time
3.5
4
4.5
Fig. 2: Energy decay curves of Eq. (40) for F = exp(2ε mki Ω m ) = 1.5
3.5
y7
3
y6
Decay of energy
If in all cases we consider
F=.75
y5
2.5
y4
2
1.5
y3
R=0
1
R=.22
y2
R=.11
R=.44
R=.875
0.5
y1
R=1.75
R=3.5
0
1
1.5
2
2.5
3
Approximation of time
3.5
4
4.5
Fig. 3: Energy decay curves of Eq. (40) for F = exp(2ε mki Ω m ) = 0.75
rotating system which indicated in the figures below
that is increasing of Coriolis force causes the energy
decay of fluid particles and vice versa.
In Fig. 4, we notice that in the absence of Coriolis
force the energy decay of fluid particles are maximums
due to the variation of first-order chemical reaction in
1755
Res. J. Appl. Sci. Eng. Technol., 6(10): 1749-1756, 2013
3.5
y7
y6
3
If in all cases we consider
F=0
y5
Decay of energy
2.5
y4
2
y3
1.5
R=0
1
y2
R=.22
R=.11
R=.44
0.5
R=.875
y1
R=1.75
R=3.5
0
1
1.5
2
3
2.5
Approximation of time
3.5
4
4.5
Fig. 4: Energy decay curves of Eq. (41) i.e., Eq. (40) for F = exp(2ε mki Ω m ) = 0
homogeneous fluid turbulence for the case of multipoint and multi-time which mention above graphically.
CONCLUSION
This study shows that if the concentration selected
is the chemical reactant of the first order, then the effect
is that the decomposition of the concentration
fluctuation in homogeneous turbulence in the presence
of Coriolis force the energy decay of the fluid particles
is much more slowly if the system is non-rotating i.e.,
in the absence of Coriolis force. The faster rate of
decomposition is governed by exp (-2RT m ). The
decomposition of the concentration fluctuation in
homogeneous turbulence in the rotating system for the
case of multi-point and multi-time is more slowly due
to absence of Coriolis force in the first order chemical
reaction. In a normal way, it takes a lot of time to get
rid of a pollutant in the fluid. The only effective factor
in the case of chemical reactant is exp (-2RT m ) and
Coriolis force which can be taken as the correcting law
for the pure mixing case and may be applied to the data
for the case of reactant and thus, the numerical work
required for this study has been avoided. From the
above figures and discussion, we conclude that in the
rotating and non-rotating frame the decomposition of
the concentration fluctuation in homogeneous
turbulence for the case of multi-point and multi-time
are increases due to the decreases of the first order
chemical reaction and maximum at the point where the
chemical reaction is zero and more rapidly in the
absence of coriolis force.
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