Research Journal of Applied Sciences, Engineering and Technology 6(15): 2789-2798, 2013
ISSN: 2040-7459; e-ISSN: 2040-7467
© Maxwell Scientific Organization, 2013
Submitted: December 31, 2012
Accepted: January 31, 2013
Published: August 20, 2013
Decay of MHD Turbulence before the Final Period for Four-point Correlation 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: The aim of this study is to determine decay of magnetic field fluctuations in MHD turbulence for fourpoint correlation in a rotating system before the final period. Two, three and four point correlation equations have
been obtained and the set of correlation equations is made determinate by neglecting the quintuple correlations in
comparison to the third and fourth order correlation terms. The correlation equations are converted to spectral form
by taking their Fourier-transforms. Finally, integrating the energy spectrum over all wave numbers. The energy
decay of magnetic field fluctuations for four-point correlations in a rotating system is obtained and is shown
graphically in the text.
Keywords: Correlations, deissler’s method, MHD turbulence, rotating system
INTRODUCTION
Magneto-Hydrodynamics is the science which
deals with the motion of highly conducting fluids in the
presence of a magnetic field. The Coriolis force and
centrifugal force must be supposed to act on the fluid.
The coriolis and centrifugal force due to rotation plays
an important role in a rotating system of turbulent flow.
Deissler (1958, 1960) developed a theory “Decay of
homogeneous turbulence before the final period.” By
considering Deissler’s theory, Sarker and Kishore
(1991) studied the decay of MHD turbulence before the
final period. Funada et al. (1978) considered the effect
of Coriolis force on turbulent motion in presence of
strong magnetic field. Islam and Sarker (2001b) studied
the decay of dusty fluid MHD turbulence before the
final period in a rotating system. Sarker and Islam
(2001) also considered the decay of MHD turbulence
before the final period for the case of multi-point and
multi-time. Islam and Sarker (2001a) also extended
their previous problem for first order reactant. Kishore
and Golsefid (1988) discussed the effect of Coriolis
force on acceleration covariance in turbulent flow with
rotational symmetry. Loeffler and Deissler (1961)
studied the decay of temperature fluctuations in
homogeneous turbulence before the final period.
Chandrasekhar (1951) studied the invariant theory of
isotropic turbulence in magneto-hydrodynamics.
Corrsin (1951) considered the spectrum of isotropic
temperature fluctuations in isotropic turbulence. Bkar
et al. (2012) studied the decay of energy of MHD
turbulence for four-point correlation. Bkar et al. (Year)
also studied the first-order reactant in homogeneou
turbulence prior to the ultimate phase of decay for fourpoint correlation in presence of dust particles.Azad
et al. (2012) discussed the transport equatoin for the
joint distribution function of velocity, temperature and
concentration in convective tubulent flow in presence
of dust particles.
It is noted that all above cases, the researcher had
considered three-point correlations and through their
study they obtained the energy decay law prior to the
ultimate phase.
In the present study, we have studied the decay of
MHD turbulence before the final period for four-point
correlation in a rotating system. The energy decay of
MHD turbulence depends on the variation of the
magnitude of coriolis parameters in the magnetic field
and causes important role between rotating and nonrotating system. The effects of rotation in magnetic
field fluctuation of MHD turbulence are graphically
discussed. It is observed that energy decays increases
with the decreases of rotation and maximum at the
point where the rotation i.e., coriolis force is zero.
Four point correlation and equations: To find the
four point correlation equation, we take the momentum
equation of MHD turbulence at the point p and the
induction equation of magnetic field fluctuation at
p ′, p ′′and p ′′′ as:
∂u l
∂u
∂h
∂ 2ul
∂w
+ u k l − hk l = −
+ν
- 2ε pkl Ω p u l
∂t
∂x k
∂x k
∂xl
∂x k ∂x k
∂hi ′
∂hi′
∂u i′
ν ∂ 2 hi′
+ u k′
− hk′
=
- 2ε qki Ω q u i′
∂t
∂x k′
∂x k′
p M ∂x k′ ∂x k′
∂h ′j′
∂t
+ u k′′
∂h ′j′
∂x k′′
− hk′′
∂u ′′j
∂x k′′
=
ν
∂ 2 h ′j′
p M ∂x k′′∂x k′′
- 2ε rkj Ω r u ′j′
∂hm′′′
∂h ′′′
∂u ′′′
ν ∂ 2 hm′′′
+ u k′′′ m − hk′′′ m =
- 2ε skm Ω s u m′′′
∂t
∂x k′′′
∂x k′′′ p M ∂x k′′′∂x k′′′
Corresponding Author: M.A. Bkar Pk, Department of Applied Mathematics, University of Rajshahi-6205, Bangladesh
2789
(1)
(2)
(3)
(4)
Res. J. Appl. Sci. Eng. Technol., 6(15): 2789-2798, 2013
where,
ω = P/ρ+ ½|ℎ� |2 = The total MHD pressure
ρ(x,t)
= The hydrodynamic pressure
ρ
= The fluid density
= The Magnetic Prandtl number
P M = v/λ
v
= The kinematics viscosity
λ
= The magnetic diffusivity
= The magnetic field fluctuation
h i (x, t)
= The turbulent velocity
u k (x, t)
t
= The time
= The space co-ordinate and repeated
xk
subscripts are summed from 1 to 3
u l u i′hk′ (r )h′j′(r ′)hm′′′(r ′′) =
∫ ∫ ∫ φ φ ′(k )γ ′(k )γ ′′ (k ′)γ ′′′ (k ′′) exp[i(k.r + k ′.r ′ + k ′′.r ′′]d kd k ′d k ′′
∞ ∞ ∞
l i
i
j
m
− ∞− ∞− ∞
P
(9)
u l u k′′(r ′)hi′(r )h ′j′(r ′)hm′′′(r ′′) =
∞ ∞ ∞
∫∫∫
() () ( ) ( )
[
]
φl φ k′′ k γ i′ k γ ′j′ k ′ γ m′′′ k ′′ exp i (k .r + k ′.r ′ + k ′′.r ′′ d kd k ′d k ′′
− ∞− ∞− ∞
(10)
u l u ′j′(r ′)hi′(r )hk′′(r ′)hm′′′(r ′′) =
() () ( ) ( )
∞ ∞ ∞
∫∫∫
[
]
φl φ ′j ′ k γ i′ k γ ′j′ k ′ γ m′′′ k ′′ exp i(k .r + k ′.r ′ + k ′′.r ′′ d kd k ′d k ′′
− ∞− ∞− ∞
(11)
ul uk hi′(r )h′j′(r ′)hm′′′(r ′′) =
∫ ∫ ∫ φ φ γ ′(k )γ ′′(k ′)γ ′′′(k ′′) exp[i(k.r + k ′.r′ + k ′′.r′′]d kd k ′d k ′′
(12)
∞ ∞ ∞
l k i
j
m
−∞ −∞ −∞
Multiplying Eq. (1) by hi′h ′j′hm′′′ (2) by u l h′j′hm′′′ (3) by
u l hi′hm′′′ (4) by u l hi′h′j′ and adding the four equations, we
than taking the space or time averages or ensemble
average, both of the averages gives the same
representation .Space or time averages denoted by
(…
�������)
… . . and ensemble average denoted by <………>.
We get:
∂
∂
∂
(ul hi′h′j′hm′′′ ) +
(ul u k hi′h′j′hm′′′ ) −
(hk hl hi′h′j′hm′′′ ) +
∂t
∂xk
∂xk
∂
∂2
∂2
ν
( whi′h′j′hm′′′ ) +
(ul hi′h′j′hm′′′ ) +
[
(ul hi′h′j′hm′′′ ) +
∂xl
∂xk ∂xk
pM ∂xk′ ∂xk′
(
+ 2[ε pkl Ω p u l hi′h ′j′ hm′′′ + ε qki Ω q u l u i′ h ′j′ hm′′′
(
)
(
)
(5)
and interchange of point’s p/ and p etc, in the subscripts
with the facts:
u l u k′′′hi′h ′j′hm′′′ = u l u k′ hi′h′j′hm′′′; u l u k′′′hi′h′j′hm′′′ = u l u k′ hi′′h ′j′hm′′′;
and taking contraction of the indices i and j we obtain
the spectral equation is:
+ 2(ε pkl Ω p + ε qki Ω q + ε rkj Ω r + ε skm Ω s )(φl γ i′γ i′′γ m′′′) = i (k k + k k′ + k k′′)(φlφk γ i′γ i′′γ m′′′)
If we take the derivative with respect to x l of the
momentum Eq. (1) at p, we have:
−
By using:
∂
∂
∂
∂
∂
∂
, ∂ = ∂ ,
=
)
= −(
+
+
∂x k′′ ∂rk′ ∂x k′ ∂rk′ ∂x k
∂rk′ ∂rk′ ∂rk′′
(6)
into Eq. (5) and then following nine dimensional
Fourier transforms:
∫ ∫ ∫ φ l γ i′ (k )γ ′j′ (k ′)γ m′′′ (k ′′) exp[i (k .r + k ′.r ′ + k ′′.r ′′]d kd k ′d k ′′
∞ ∞ ∞
−∞ −∞ −∞
(7)
u l u k′ hi′ (r )h ′j′(r ′)hm′′′(r ′′) =
∞ ∞ ∞
∫∫∫
− ∞− ∞− ∞
() () ( ) ( )
[
m
+ i (k k + k k′ + k k′′)(φlφi′γ k′ γ i′′γ m′′′) + i (k k + k k′ + k k′′)(δγ i′γ i′′γ m′′′) .
(14)
)
′′′ hi′h′j′ ]
+ ε rkj Ω r ul u ′′j hi′hm′′′ + ε skm Ω s ul u m
u l hi′ (r )h ′j′(r ′)hm′′′(r ′′) =
j
− i (k k + k k′ + k k′′)(φl γ k γ i′γ i′′γ m′′′) − i (k k + k k′ + k k′′)(φl γ k′ γ i′γ i′′γ m′′′)
∂2
∂2
(ul hi′h′j′hm′′′ ) +
(ul hi′h′j′hm′′′ )]
∂xk′′∂xk′′
∂xk′′′∂xk′′′
)
i
−∞ −∞ −∞
∂
ν
(φlγ i′γ i′′γ m′′′) +
[(1 + pM )(k 2 + k ′2 + k ′′2 ) + 2 pM (kk ′ + k ′k ′′ + kk ′′)](φlγ i′γ i′′γ m′′′)
∂t
pM
∂
∂
∂
(ul u ′′j hi′hk′′hm′′′ ) +
(ul u k′′ hi′h′j′hm′′′ ) −
(ul u ′j′ hi′h′j′hm′′′ ) =
∂xk′′
∂xk′′′
∂xk′′′
(
∫ ∫ ∫ δγ ′(k )γ ′′(k ′)γ ′′′(k ′′) exp[i(k.r + k ′.r′ + k ′′.r′′]d kd k ′d k ′′
(13)
∞ ∞ ∞
u l u m′′′hi′h ′j′hm′′′ = u l u i′hi′hk′ h′j′hm′′′; u l u ′j′hi′hk′′hm′′′ = u l u i′hi′hk′ h′j′hm′′′;
∂
∂
∂
(ul uk′′hi′h′j′hm′′′ ) −
(ul u k hi′h′j′hm′′′ ) −
(ul ui′hk′ h′j′hm′′′ ) +
∂xk′′
∂xk′
∂xk′
−
whi′(r )h′j′(r ′)hm′′′(r ′′) =
]
∂2w
∂2
=
(u l u k − hl hk )
∂xl ∂xl ∂xl ∂xl
Multiplying Eq. (15) by
(15)
hi′h ′j′hm′′′, taking time
averages and writing the equation in terms of the
independent variables 𝑟𝑟⃗, 𝑟𝑟́⃗, r ′′ we have:
− (δγ i′γ ′j′γ m′′′ ) =
( K l K k + K l K k′ + K l K k′′ + K l′ K k + K l′ K k′ + K l′ K k′′ + K l′′K k + K l′′K k + K l′′K k′ + K l′′K k′′ )
K l K l + K l′ K l′ + K l′′K l′′ + 2 K l K l′ + 2 K l′ K l′′ + 2 K l K l′′
(φlφk γ i′γ ′j′γ′ m′′′ - γ l γ k γ i′γ ′j′γ m′′′ )
(16)
Eq. (16) can be used to eliminate (δγ i′γ ′j′γ m′′′ ) from
(8)
Eq. (14). Equation (14) and (16) are the spectral Eq.
2790
φ l φ k′ k γ i′ k γ ′j′ k ′ γ m′′′ k ′′ exp i (k .r + k ′.r ′ + k ′′.r ′′ d kd k ′d k ′′
Res. J. Appl. Sci. Eng. Technol., 6(15): 2789-2798, 2013
corresponding to the four–point correlation equation.
The spectral equations corresponding to the three-point
correlation equations by contraction of the indices i and
j are:
ν
∂
(φl β i′β i′′) +
[(1 + PM )( K 2 + K ′ 2 ) + 2 p M KK ′]
∂t
pM
i ( Kk
( K k + K k′ ) (φ l φ k′ β i′β i′′) + i( K k + K k′ ) (φl φi′β i′β i′′) +
i( k l + k l′ )γβ i′β i′′
(17)
and-(γ β i′β ′j′) = ( K l K k + K l′K k + K l k k′ + K l′K k′ )
2
2
( K l + K l′ + 2 K l K l′ )
β l β k β i′β ′j′)
(18)
()
hi hi′ r = ∫
() ( )
∞ ∞
∫∫
[
]
′ ′
′
φl β i′ k β ′j ′ k ′ exp i ( k .r + k .r d kd k
() ( )
− ∞− ∞
(19)
() () ( )
u l u k′ r hi′ r h ′j′ r ′ = ∫ ∫ φl φ k′ (k )β i′(k )β ′j′(k ′)
[
∞ ∞
]
−∞ −∞
exp i (k .r + k ′.r ′ d kd k ′
() () ( )
u l u i′ r hi′ r h ′j′ r ′ =
[
]
∫∫
() ( )
∞ ∞
∫∫
]
()
[
]
∞ ∞
∫∫
() ( )
whi′ r h ′j′ r ′ =
∫∫
φl γ i′γ ′j′γ m′′′ = φl γ i′γ ′j′γ m′′′ exp[
)
where, φl γ i′γ ′j′γ m′′′ is the value of φl γ i′γ ′j′γ m′′′ at t = t 1
1
() () ( )
(22)
that is stationary value for small values of k, k ′andk ′′
when the quintuple correlations are negligible.
Substituting of Eq. (18), (25), (30) in Eq. (17) we get:
() () ( )
β l β k′ k β i′ k β ′j ′ k ′
− ∞− ∞
(23)
[
]
(24)
A relation between φl φ k′ β i′β ′j′and φl γ i′γ ′j′γ m′′′ can be
obtained by letting 𝑟𝑟⃗`` = 0 in Eq. (7) and comparing the
result with Eq. (20):
() () ( )
]
φl γ i′ k γ ′j′ k ′ γ m′′′ k ′′ exp i (k .r + k ′.r ′ + k ′′.r ′′ d kd k ′d k ′′
ν
∂
(k kφl β i′β i′′) +
[(1 + pM )(k 2 + k ′2 ) + 2 pM kk ′](k kφl β i′β i′′)
pM
∂t
∞
γβ i′(k )β ′j ′(k ′) exp i ( k .r + k ′.r ′ d kd k ′
[
(
−ν
(1 + p M ) k 2 + k ′ 2 + k ′′ 2 + 2kk ′ + 2k ′k ′′ + 2kk ′
pM
(
− ∞− ∞
() ( ) ( )
(29)
− 2 ε pkl Ω p + ε qki Ω q + ε rkj Ω r + ε skm Ω s ](t − t1 ) (30)
φl φ k′ k β i′ k β ′j ′ k ′
∞
φl φ k′ k β i′ k β ′j ′ k ′ = ∫−∞
() ( )
(21)
exp i (k .r + k ′.r ′ d kd k ′
∞ ∞
∞
−∞
Solution neglecting quintuple correlations: As it
stands the set of linear Eq. (14), (17), (18), (19), (21),
(22) and (29) is indeterminate as it contains more
unknowns than equations. Thus neglecting all the terms
on the right side of Eq. (14), the equation can be
integrated between to t 1 and t to give:
() () ( )
exp i (k .r + k ′.r ′ d kd k ′
() ( )
(28)
−∞
α iϕ k ϕ i′ k = ∫ φl β i′ k β i′′ k ′ dk ′
φl φi′ k β k′ k β ′j ′ k ′
− ∞− ∞
u l hk hi′ r h ′j′ r ′ =
(27)
obtained by letting 𝑟𝑟́⃗ = 0 in Eq. (19) and comparing the
result with Eq. (24), then:
− ∞− ∞
u l hi′ r h ′j′ r ′ =
( )
ϕ iϕ i′ k exp i k .r dk
The relation between α iϕ k ϕ ′j (k ) and ϕ l β i′β ′j′ is
(20)
exp i (k .r + k ′.r ′ d kd k ′
[
()
∞
−∞
()
(26)
α iφ k φi′ are defined by:
1
∞ ∞
( )
and hi hk hi′ r = ∞ α iϕ k ϕ i′ (k ) exp(i k .r )dk
∫
here the spectral tensors are defined by:
u l hi′ r h ′j′ r ′ =
()
()
where, ϕ iϕ i′ and
(φl β i′β i′′) =
+ K k′ ) (φl φ k β i′β i′′) - i( K k + K k′ ) ( β l β k β i′β i′′) − i
(φlφk β i′β i′′ -
()
2ν 2
∂
k ϕ iϕ i′ k = 2ik k [ α iϕ k ϕ i′ k − α k ϕ iϕ i′ − k ]
ϕ iϕ i′ k +
pM
∂t
(25)
[a]1 ∫ exp[
−∞
(
)
exp[2(ε pkl Ω p + ε qki Ω q + ε rkj Ω r + ε skm Ω s )(t − t1 )]
∞
+ [b]1 ∫ exp[
−∞
−ν
(t − t1 ){(1 + pM ) k 2 + k ′2 + k ′′2 ) + 2 pM (kk ′ − kk ′′ }]dk ′′
pM
(
)
exp[2(ε pkl Ω p + ε qki Ω q + ε rkj Ω r + ε skm Ω s )(t − t1 )]
∞
+ [c]1 ∫ exp[
−∞
The spectral equation corresponding to the twopoint correlation equation taking contraction of the
indices is:
2791
−ν
(t − t1 ){(1 + pM ) k 2 + k ′2 + k ′′2 ) + 2 pM (kk ′ + k ′k ′′ + kk ′ }]dk ′′
pM
(
)
−ν
(t − t1 ){(1 + p M ) k 2 + k ′ 2 + k ′ 2 ) + 2 p M (kk ′ − k ′k ′ }]dk ′
pM
exp[2(ε pkl Ω p + ε qki Ω q + ε rkj Ω r + ε skm Ω s )(t − t1 )]
(31)
)
Res. J. Appl. Sci. Eng. Technol., 6(15): 2789-2798, 2013
At t 1 ,γ,s have been assumed independent of k ′′ ;
that assumption is not, made for other times. This is one
of several assumptions made concerning the initial
conditions, although continuity equation satisfied the
conditions. The complete specification of initial
turbulence is difficult; the assumptions for the initial
conditions made here in are partially on the basis of
simplicity. Substituting dk ′′ = dk1′′dk 2′′ dk 3′′ and
integrating with respect to k1′′, k 2′′ and k 3′′ , we get:
+k
∫
(k k φl β i′β i′′) =
2
ωk
2
∫ exp( x
[ν (t − t1 )(1 + p M )]
)dx}dk ′
2
∞
k2
+
)
∫
5
2π 2 i
ν
−∞
[c(k .k ′) − c(−k .− k ′)]1
exp{− (2ε pkl Ω p + 2ε qki Ω q + 2ε rkj Ω r + 2ε skm Ω s )(t − t1 )}
exp {2(ε pkl Ω p + 2ε qki Ω q + 2ε rkj Ω r + 2ε skm Ω s )(t − t1 )}
2
{- ω-1 exp[−ω 2  k 2 + 2 Pm kk ′ + (1 + 2 p M )k ′ ]
2

1 + pM
(1 + p M ) 

πp M
( )
+k`exp[-ω2 (1 + p M )(k 2 + k ′ 2 ) + 2 p M kk ′ )].
 ν (t − t1 )(1 + p M )  (1 + 2 p M ) k 2

2 p M kk ′
+
+ k ′ 2 
[b1 ] exp −

2
(
)
+
p
p
1
(
)
+
1
p
M
M
M



exp {(2ε pkl Ω p + 2ε qki Ω q + 2ε rkj Ω r + 2ε skm Ω s )(t − t1 )}
ωk ′
2
∫ exp( x
+
πp M
2
)dx}dk ′
(34)
0
[ν (t − t1 )(1 + p M )]
( )
 ν (t − t1 )(1 + p M )  2 (1 + 2 p M ) k ′ 2 2 p M kk ′  exp
[c1 ] exp −
+
k +

pM
(1 + p M )2 (1 + p M )


pkl
2
0
(
{(2ε
Ω p + 2ε qki Ω q + 2ε rkj Ω r + 2ε skm Ω s )(t − t1 )}
(32)
Integration of Eq. (32) with respect to time and in
order to simplify calculations, we will assume that [α] 1
= 0; That is we assume that a function sufficiently
general to represent the initial conditions can obtained
by considering only the terms involving [b] 1 and [c] 1
The substituting of Eq. (29) in Eq. (26) and setting:
H = 2πk 2ϕ iϕ i′ result in
∂H 2νk 2
+
H =G
∂t
pM
(33)
where H is the magnetic energy spectrum function,
which represents contributions from various wave
numbers (or eddy sizes) to the energy and G is the
energy transfer function, which is responsible for the
transfer of energy between wave numbers .In order to
make further calculations, an assumption must be made
for the forms of the bracketed quantities with the
subscripts 0 and 1 in Eq. (37) which depends on the
initial conditions:
(2π ) 2 [ k k φl β i′β i′′(k , k ′)- k k φi β i′β i′′(−k ,−k ′) ]0 =
− ξ 0 (k 2 k ′ 4 − k 4 k ′ 2 )
(35)
where, ξ 0 is a constant depending on the initial
conditions for the other bracketed quantities in Eq. (34),
we get:
7
where,
4π 2 i
ν
∞
G= k 2 2π i [ k k φl β i′β i′′(k , k ′) k k φi β i′β i′′(−k ,−k ′) ]0 .
∫
ν
pM
7
2
[b(k .k ′) − b(−k .− k ′)]1 = 4π i [c(k .k ′) −
ν
c(−k .− k ′)]1 = -2 ξ1 (k k ′ − k k ′ )
4
−∞
exp[−

(1 + p M )(k + k ′ ) + 2 p M kk ′ ]
2
[-ω
πp M
[ν (t − t1 )(1 + p M )]
1 + pM
 (1 + p M )
 ν (t − t1 )(1 + p M )  (1 + 2 p M ) k 2 + k ′ 2
2 p M kk ′ 
+
[a1 ] exp −

2
2 
p
(
)
(
+
+ p M ) 
1
p
1
M
M


+
[b(k .k ′) − b(− k .− k ′)]1
{- ω −1 exp[−ω 2  (1 + 2 p M )k + 2 Pm kk ′ + k ′ 2 ] +kexp
2


]
)
ν
exp{− (2ε pkl Ω p + 2ε qki Ω q + 2ε rkj Ω r + 2ε skm Ω s )(t − t1 )}
ν
∂
(1 + p M ) k 2 + k ′ 2 + 2 p M kk ′
(k k φl β i′β i′′) +
pM
∂t
(
2π 2 i
−∞
-2
[
5
∞
2
(t − t 0 ){(1 + p M )(k 2 + k ′ 2 ) + 2 p M kk ′}]dk ′
Remembering,
2792
6
6
4
(36)
Res. J. Appl. Sci. Eng. Technol., 6(15): 2789-2798, 2013
and
d k ′ = −2π k ′ 2 d (cos θ )dk ′ ,
kk ′ = kk ′ cos θ , θ is the angle between k and k ′ and carrying
Gγ = Gγ 1 + Gγ 2 + Gγ 3 + Gγ 4
out the integration with respect to θ, we get:
where,
∞
2
4
4
2
G = [ ξ 0 (k k ′ − k k ′ )kk ′{
∫0
ν (t − t 0 )
ν
exp[−
(t − t 0 ){(1 + p M )(k 2 + k ′ 2 ) − 2 p M kk ′}] −
pM
exp[−
ν
pM
1
Gγ 1 =
(t − t 0 ){(1 + p M )(k 2 + k ′ 2 ) + 2 p M kk ′}]} +
ξ1 (k k ′ − k k ′ )kk ′
ν (t − t 0 )
exp{(− 2ε pkl Ω p − 2ε qki Ω q − 2ε rkj Ω r − 2ε skm Ω s )(t − t1 )}
4
6
6
4
 (1 + 2 p M )k 2 2 PM kk ′

exp[−ω 2 
+ k ′ 2 ]−
2
1 + pM
 (1 + p M )


2 P kk ′ (1 + 2 p M )k ′ 2
+
exp[−ω  k 2 − M
1 + pM
(1 + p M ) 2

1 + pM

]
-
[
(1 + p M ) 2 
0
+{ k ′ exp [ − ω 2 ( (1 + p M )(k 2 + k ′ 2 ) − 2 p M kk ′ )]
- k/ exp [ − ω 2 ( (1 + p M )(k 2 + k ′ 2 ) + 2 p M kk ′ )]}
)dx

64 p 2 M
10 p 3 M
40  10
+ 2
−

k
2
3
2
− t1 ) 
ν
(
t
(
)
(
)
−
+
−
+
ν
t
t
p
ν
t
t
p
(
)
1
(
)
1
M
M
1
1

2


k 12 ]


(41)
exp
9
2
 − ν (t − t1 )(1 + p M )(1 + 2 p M − p M 2 )  2
k



p M (1 + p M )


2
2

 8+
4 pM
2 p2M
1
+
− 3
k
 2
2
2
3
3
3
ν (t − t1 ) (1 + p M ) ν (t − t1 ) (1 + p M ) ν (t − t1 ) 
8ν 2 (t − t1 ) 2 (1 + 2 p M )
-k exp [ − ω 2 ( (1 + p M )(k 2 + k ′ 2 ) + 2 p M kk ′)]} exp( x 2 )dx
∫
∫ exp( x
 − ν (t − t1 )(1 + 2 p M − p M 2 )  2.
k6
+3
k [ 4 90 p M

4


(
)
ν
(
t
−
t
)
1
+
p
p
p
(
1
)
+
1
M
M
M


1
ωk
2
exp
5
4
Gγ 2 = ξ1π 2 p M (1 + p M )
+ {k exp [ − ω 2 ( (1 + p M )(k 2 + k ′ 2 ) − 2 p M kk ′)]
ωk ′
8ν 2 (t − t1 ) 2 (1 + p M ) 5
 pM

 − 

1 + p M 

 1 + pM


ω −1 exp[−ω 2  k 2 + 2 PM kk ′ + (1 + 2 p M )k ′ 2 ]


ξ1π 2 p M 5
+ 8  p M
 (1 + 2 p M )k 2 2 PM kk ′

exp[−ω 2 
+
+ k ′ 2 ]
2
−1
p
1
+
p
+
(
1
)
M
M

+
ω
2
(40)
)] dk ′
(37)
90 p M (1 + p m )
ν 4 (t − t1 ) 4 (1 + 2 p M )
k 6+
 120 p M (1 + p m )

2 p 2 M (1 + p m ) 2
1
+
+ 3
− 3
 2
2
3
2
3 8
ν (t − t1 ) (1 + 2 p M ) ν (t − t1 ) (1 + 2 p M ) ν (t − t1 ) k




 64 p 2 M (1 + p m ) 2
10 p 3 M (1 + p M ) 3  10
40
−
+

k
2
ν (t − t1 ) ν 2 (t − t1 ) 2 (1 + 2 p M ) 3 
ν (t − t1 )(1 + 2 p M )
3
3
+  8 p M (1 + p M ) − p M (1 + p M ) k 12 ]


3
(1 + 2 p M ) 
 (1 + 2 p M )
(42)
0
where,
 ν (t − t1 )(1 + p M ) 

pM


ω = 
1
2
Gγ 3 =
1
3
2
. Integrating Eq. (37) with
exp  − ν (t − t1 )(1 + 2 p M ) 
9
ξ1π 2 p M 2


3
2
8ν (t − t1 ) (1 + p M ) 8
90 pM
[
G= G β + Gγ
 120 p M
60 p 2 M
+ 3
 2
2
−
t
t
(
)
ν
−
t
t1 ) 3 (1 + p M
(
ν
1

exp{− (2ε pkl Ω p + 2ε qki Ω q + 2ε rkj Ω r + 2ε skm Ω s )(t − t1 )}(38)
 64 p 2 M
10 p 3 M
+ 2

−
t
t
(
)
ν
−
t
t1 ) 2 (1 + p M
(
ν
1

{p
where,
Gβ = −
1
2
5
(39)
 ν (t − t 0 )(1 + 2 p M )k 2 
exp−

p M (1 + p M ) 

ν (t − t 0 ) (1 + p M )
π ξ 0 pM 2
3
2
3
2
5
2
2


pM
15 p M k 4
5 pM
3  6
+
−
. 2
k +
2
2
− t0 ) 
t
2
(
1 + pM
ν
−
+
+
−
t
t
p
p
t
t
ν
ν
4
(
)
(
1
)
(
1
)
(
)
M
M
0
0


2
M
}
−
−
30
 9
k +
ν 3 (t − t1 ) 3 
40(1 + p M ) 2  11
k +
ν (t − t1 ) 
ω1
13
0
where,
 ν (t − t1 )(1 + p M )  2
 k
ω1 = 
pM


2793
)2
)2
2
− p M (1 + p M ) k ] exp( y 2 )dy
∫
2
1
 pM 2
 
− 1k 8 

2
+
(
1
p
)
M

 
2
k7+
respect to k`we have:
ν 4 (t − t1 ) 4 (1 + pM )2
k

pM
(43)
Res. J. Appl. Sci. Eng. Technol., 6(15): 2789-2798, 2013
Gγ 4 =
1
ξ1π 2 p M
15
2
29


2 8ν (t − t1 )(1 + p M ) 2
7560(1 + p M ) 3
ν (t − t1 ) p M
4
4
2
1
pM


3
M
 20160(1 + p M ) 5 4233600(1 + p M ) 7  8
−
k6 +  3
k +
3
ν 3 (t − t1 ) 3 p M 3 
 ν (t − t1 ) p M
}
) 5 p 2 M − 128(1 + p M ) 7 k 14 + ....................]


7
8ν 2 (1 + p M ) 2

 (7 p 2 M − 6 p M )
+
3
 3ν (1 + p )(t − t ) 2
2ν 2 (t − t 0 )
M
0

 12

 8ν (3 p 2 M − 2 p M + 3)  9
+
k F (ω )]
5
1
 3(1 + p ) 2 p 2 M

M


3 pM k 4
+[
12096(1 + p M ) 5 3360(1 + p M ) 7  10  2304(1 + p M ) 5 p M 1344(1 + p M ) 9  12
− 2
−

k + 
k +
2
2
2
ν (t − t1 )
pM
ν (t − t1 ) 2 p M 2 
 ν (t − t1 )


{128(1 + p
5
H β = ξ 0π 2 p M 2 exp  − ν (t − t 0 )(1 + 2 p M ) k 2
exp  − ν (t − t1 )(1 + 2 p M ) k 2 [
5
2


p M (1 + p M )


 6  (3 p 2 M − 2 p M + 3)
k − 
1
 3(1 + p 2 M )(t − t ) 2

0


(44)
1
ω
The integral expression in Eq. (39) , the quantity
G β represents the transfer function arising owing to
consideration of magnetic field at three point correlation
equation; G γ arises from consideration of the four-point
equation. Integration of Eq. (39) over all wave numbers
shows that:
2
F( ω ) = exp(−ω ) exp( x )dx, ω =  ν (t − t 0 )  k


∫0
 p M (1 + p M ) 
∞
∫ G.d k = 0

 8
k +


(45)
2
H γ1 = −
2
ξ1π 1 / 2 p M 5
− ν (t − t1 )(1 + 2 p M − p 2 M ) 2
exp(
)k
2
5
p M (1 + p M )
8ν (1 + p M )
18 pM
15 − 6 pM + 21 p 2 M
4 pM
)k 6 + ( 3
)k 8
+
5
4ν (1 + pM ) 2 (t − t1 ) 4 ν 2 (1 + pM )(t − t1 ) 3
ν (1 + pM )(t − t1 )
[(
4
0
Indicating that the expression for G satisfies the
conditions of continuity and homogeneity, physically, it
was to be expected, since G is a measure of transfer of
energy and the numbers must be zero. From (34):
We get:
 2νk 2 (t − t 0 ) 
 2νk 2 (t − t 0 ) 
 2νk 2 (t − t 0 ) 
H = exp −

 dt + J (k ) exp −
 ∫ G exp −
p
p
pM
M
M






15 − 6 pM + 36 p 2 M − 6 p 3 M + 61 p 4 M 14 p 2 M − 40 pM − 18 10
)k
+
12ν 2 pM (1 + pM ) 3 (t − t1 ) 3
ν (1 + pM ) 2 (t − t1 ) 2
+(
14 p 4 M − 56 p 3 M − 12 p 2 M − 40 p M − 18 12
)k
p M (1 + p M ) 3 (t − t1 )
+(
J(k) = N 0 k2/ π , is a constant of integration and can be
obtained as by Corrsin (1951)
Therefore we obtained:
+(
H=
N0k 2
π
 − 2νk 2 (t − t 0 ) 
 − 2νk 2 (t − t 0 ) 
exp 
 ∫ (G β + Gγ 1 + Gγ 2 + Gγ 3 + Gγ 4 )
 + exp 
pM
pM




 − 2νk 2 (t − t0 )  dt
exp 

pM


(1 + pM ) 2 (75 − 30 pM + 180 p 2 M − 30 p 3 M + 305 p 4 M )
120νp 2 M (1 + pM ) 4 (t − t1 ) 2
+(
(1 + pM ) 2 (75 − 3 pM + 90 p 2 M − 30 p 3 M + 21 p 4 M ) 14
)k
120 p 3 M (1 + pM ) 5 (t − t1 )
ν (1 + p 2 M )(14 p 4 M − 56 p 3 M − 12 p 2 M − 40 pM − 18)k 14
+(
(46)
+
p 2 M (1 + pM ) 4 (t − t1 )
ν (1 + p M ) 3 (75 − 3 p M + 90 p 2 M − 30 p 3 M + 21 p 4 M )k 16
120 p 4 M (1 + p M ) 6
where,
ν (1 + p 2 M )tk 2
G = G β + Gγ 1 + Gγ 2 + Gγ 3 + Gγ 4
Ei (ω2 ) = ∫ exp[(
(47)
After integration Eq. (46) becomes:
H =
N0k 2
π
 2νk 2 (t − t0 )  + H + [ H + H + H + H ]
exp −
β
γ1
γ2
γ3
γ4

pM


exp{− (2ε pkl Ω p + 2ε qki Ω q + 2ε rkj Ω r + 2ε skm Ω s )(t − t1 )}
(48)
Hγ 2 = −
pM (1 + pM )
)] exp(−ω 2 ) Ei (ω 2 )
) / (t − t1 )]dt
ξ1π 1/ 2 p 5 M (1 + pM ) 4 exp
8ν 2 (1 + 2 pM ) 9 / 2
 − ν (t − t1 )(1 + 2 pM − pM 2 )  2

k


+
p
(
1
p
)
M
M


[{18 p M (1 + p M ) /ν 4 (1 + 2 p M )(t − t1 ) 5 }k 6 +
where,
2794
Res. J. Appl. Sci. Eng. Technol., 6(15): 2789-2798, 2013
{(17 + 32 pM − 2 p 2 M + 4 p 3 M + 20 p 4 M ) /(4ν 3 (1 + 2 pM ) 2 (t − t1 ) 4 )
+ {( −2115855 − 5670 pM + 12720540 p 2 M + 8436120 p 3 M − 19072032 p 4 M
− 25347840 p 6 M − 4128 p 7 M + 2304 p 8 M ) /ν (1 + pM ) 6 p 2 M (t − t1 )}k 12
+ 120 pM (1 + pM ) / 3ν 2 (1 + 2 pM )(t − t1 ) 3 }k 8
− {((−2115855 + 4226040 pM + 12731880 p 2 M − 17004960 p 3 M − 35944272 p 4 M
+ {[(17 + 49 pM + 13 p 2 M + 13 p 3 M + 98 p 4 M + 134 p 5 M + 104 p 6 M
+ 60 p 7 M ) / 4ν 3 (1 + 2 pM ) 2 (t − t1 ) 4 ] + [(52 p 4 M + 64 p 3 M − 48 p 2 M
− 40 pM ) /ν (1 + 2 pM ) (t − t1 ) ]}k
2
+ {[(1 + p M − p
2
M
+p
3
M
2
+ 12796224 p 5 M + 42264592 p 6 M + 16857280 p 7 M + 9920 p 8 M − 4864 p 9 M )k 14
10
)(17 + 49 p M + 13 p
2
M
− 13 p
3
M
+ 98 p
4
M
+ 134 p
5
+ 1344 p M )k 12 /(1 + p M ) 6 p 3 M } exp(ω 3 ) Ei (ω 3 )],
ω5 = exp{ν (1 − 2 pM )t / pM }k 2 ,
M
+ 104 p 6 M + 60 p 7 M ) / 24νp 2 M (1 + 2 p M ) 4 (t − t1 ) 2 ] + [(−40 p M − 89 p 2 M + 51 p 3 M
Ei (ω5 ) = ∫ exp[{ν (1 − 2 pM )tk 2 / pM } /(t − t1 )]dt
+ 124 p 4 M − 40 p 5 M + 36 p 6 M + 60 p 7 M ) / p M (1 + 2 p M ) 3 (t − t1 )]}k 12
+ {[(1 + p M − p 2 M + p 3 M ) 2 (17 + 49 p M + 13 p 2 M − 13 p 3 M + 98 p 4 M + 134 p 5 M
+ 104 p 6 M + 60 p 7 M ) / 24 p 2 M (1 + 2 p M ) 5 (t − t1 )]}k 14
− {[ν (1 + pM − p 2 M + p 3 M )(−40 pM − 89 p 2 M + 51 p 3 M + 124 p 4 M − 40 p 5 M + 36 p 6 M
From Eq. (48) we get:
H = H1 + H2
exp{− (2ε pkl Ω p + 2ε qki Ω q + 2ε rkj Ω r + 2ε skm Ω s )(t − t1) } (49)
+ 60 p 7 M ) / p 2 M (1 + 2 pM ) 4 ]k 14 + [ν (1 + pM − p 2 M + p 3 M ) 3 (17 + 49 pM + 13 p 2 M − 13 p 3 M
+ 98 p
4
M
+ 134 p
5
M
+ 104 p
6
M
+ 60 p
7
M
16
)k / 24 p
4
M
2
2
H 1 = N 0 k exp − 2νk (t − t0 ) 


π
pM


(1 + 2 p M ) ]. exp(ω 3 ) Ei (ω 3 )
6
(H
Ei (ω3 ) = ∫ exp[{(−ν (t − t1 )(1 + 2 pM − p 2 M )tk 2 ) /( pM (1 + 2 pM ))} /(t − t1 )]dt
and, ω3 = [{−ν (t − t1 )(1 + 2 pM − p 2 M )t} /{ pM (1 + 2 pM )}]k 2
Hγ3 = −
+ Hγ2 + Hγ3 + Hγ4
γ1
 − ν (t − t1 )(1 + 2 p M )  2 [
45 p M
k

k8 +
4
2
4
p
t
t
2
ν
(
1
+
)
(
−
)
p
M
1
M


(
)
∫ H dk =
0
)
(
2
∞
∫H
)
dk = ξ1 [ Rν −17 / 2 (t − t1 ) −15 / 2 + Sν −19 / 2 (t − t1 ) −17 / 2 ]
where, R = Q2 + Q4 + Q6 + Q7 , S = Q1 + Q3 + Q5 and Q,s
values are:
ω4 = {ν (1 − 2 pM )t / pM }k 2 and
Ei (ω4 ) = ∫ exp[{ν (1 − 2 pM )t / pM } /(t − t1 )]dt
ξ1π 1/ 2 p 9 / 2 M exp
2 ν (1 + pM )11/ 2
2
exp{− (2ε pkl Ω p + 2ε qki Ω q + 2ε rkj Ω r + 2ε skm Ω s )(t − t1 )}
+ 160 p 2 M − 60 pM − 5) / 4 p 3 M (1 + pM ) 2 )k 16 } exp(−ω4 ) Ei (ω4 )],
8
+
8 2π
ξ 0 Qν −6 (t − t 0 ) −5
− {((20 − 240 pM + 424 p 2 M − 48 p 3 M ) / pM )k 14 + ((1 − 2 pM ) 2 (20 p 4 M − 40 p 3 M
Hγ 4 = −
N 0 p 3 / 2 Mν −3 / 2 (t − t 0 ) −3 / 2
1
 (1 − 2 p M ) 20 p M − 40 p M + 160 p M − 60 p M − 5  14

k
4νp 2 M (1 + p M ) 2 (t − t1 )


3
(50)
Now, ∞
 20 p 4 M − 40 p 3 M + 160 p 2 M − 60 p M − 5 24 p 2 M − 200 p M + 20  12
+

k +
ν (t − t1 )
4ν 2 p M (1 + p M ) 2 (t − t1 ) 2


4
H2 =
)
hi hi′ ∞
= ∫ Hdk
2
0
+
) (
(
and
In Eq. (49) H 1 and H 2 magnetic energy spectrum
arising from consideration of the three and four–point
correlation equations respectively. Eq. (49) can be
integrated over all wave numbers to give the total
magnetic turbulent energy. That is:
ξ1π 1/ 2 p 4 M exp
16ν (1 + pM )15 / 2
 20 p 2 M − 70 pM − 5
60 pM  10
+ 2
k
 3
2
3
2
ν
(
1
p
)
(
t
t
)
ν
(t − t1 ) 2 
+
−
1
M

+ Hβ
Q=
 − ν (t − t1 )(1 + 2 p M )  2 [
k

pM


π.p6M
(1 + PM )(1 + 2 PM )
5
2
 9 5 PM (7 PM _ 6) 35 PM (3 p 2 M − 2 p M + 3) 8 p M (3 p 2 M − 2 p M + 3)

−
+
+ ...
 +
(1 + 2 PM )
8(1 + 2 p M ) 2
3.2 6.(1 + 2 p M ) 3
16

Q1 = −

 6+
1890 p M
k
 4
6
4 
ν (1 + p M ) (t − t1 ) 
π.p6M
5
(1 + PM ) (1 + 2 PM − p 2 M ) 7 / 2
2
15.9 15.7(15 − 6 p M + 21 p 2 M ) 15.7.3(15 − 6 p M + 36 p 2 M − 6 p 3 M + 61 p 4 M )
+
 6 +
210 (1 + 2 PM − p 2 M )
211 (1 + 2 p M − p 2 M ) 2
2
11.9.7(1 + P 2 M )(75 − 30 p + 180 p 2 M − 30 p 3 M + 305 p 4 M )
M
+

213 (1 + 2 p M − p 2 M ) 3

13.11.9.7(1 + P 2 M ) 2 (75 − 3 p + 90 p 2 M − 30 p 3 M + 15 p 4 M )
M

− ...

214 (1 + 2 p M − p 2 M ) 4
+ {( −423170 − 16938180 pM − 25381440 p 2 M − 16894080 p 3 M − 4213440 p 4 M ) /ν 3 (1 + pM ) 6 (t − t1 ) 3 }k 8
+ {(−2115855 − 4237380 pM + 4245780 p 2 M + 16927680 p 3 M + 14783328 p 4 M
+ 4218816 p 5 M + 43686 pM ) /ν 2 (1 + pM ) 6 (t − t1 ) 2 }k 10
2795
Res. J. Appl. Sci. Eng. Technol., 6(15): 2789-2798, 2013
Q2 = −
π . p 21/ 2 M
(1 + pM )
3/ 2
(1 + 2 pM − p 2 M ) 9 / 2
[ 15.7 +
26
15.9.7(14 p 2 M − 18 − 40 p M ) +
2 9 (1 + 2 p M − p 2 M )
15.11.9.7(14 p 4 M − 56 p 3 M − 12 p 2 M − 40 p M − 18) …]
−
210 (1 + 2 p M − p 2 M ) 2
Fig. 1: Energy decay curves for f = 0.75
π . p19 / 2 M (1 + pM )1/ 2
Q3 = −
(1 + 2 pM ) 2 (1 + 2 pM − p 2 M ) 7 / 2
15.7(17 + 32 p M − 2 p 2 M + 4 p 3 M + 20 p 4 M )
210 (1 + p M ) 2 (1 + 2 p M − p 2 M )
+
.[ 9.15 +
26
9.7.5(17 + 49 p M + 13 p 2 M − 13 p 3 M + 98 p 4 M + 134 p 5 M + 104 p 6 M + 60 p 7 M )
211 (1 + p M ) 3 (1 + 2 p M − p 2 M ) 2
+(
(11.9.7.5(1 + p M − p 2 M + p 3 M )(17 + 49 p M + 13 p 2 M − 13 p 3 M + 98 p 4 M + 134 p 5 M + 104 p 6 M + 60 p 7 M )
213 (1 + p M ) 4 (1 + 2 p M − p 2 M ) 3
+(
(13.11.9.7.5(1 + p M − p 2 M + p 3 M ) 2 (17 + 49 p M + 13 p 2 M − 13 p 3 M + 98 p 4 M + 134 p 5 M + 104 p 6 M + 60 p 7 M )
) − ...]
214 (1 + p M ) 5 (1 + 2 p M − p 2 M ) 4
Q4 = −
π.p
1
21
2
15.9.7(−40 p M − 48 P M + 64 p m + 52 p
2 9 (1 + p M ) 2 (1 + 2 p m − p 2 m)
2
[25.7.3 +
M
(1 + PM ) (1 + 2 PM )(1 + 2 PM − p 2 M )
2
3
4
M
9
Fig. 2: Energy decay curves for f = 0.50
2
25
)+
15.11.9.7(−40 p M − 89 p 2 M + 51 p 3 M + 124 p 4 M − 40 p 5 M + 36 p 6 M + 60 p 7 M )
...]
210 (1 + p M ) 3 (1 + 2 p M − p 2 M ) 2
19
Q5 = −
π.p 2M
(1 + PM )
19
2
(1 + 2 PM )
9
2
 45.7.53 9.7.5.3(20 p M − 70 p M − 5) 11.9.7.5.3(20 p 4 M − 40 p 3 M + 160 p 2 M − 60 p M − 5) 
+
+
 10 +
211 (1 + 2 PM )
213 (1 + 2 p M − p 2 M ) 2

 2


4
3
2
13
.
11
.
9
.
7
.
5
.
3
(
1
2
)(
20
40
160
60
5
)
p
p
p
p
p
−
−
+
−
−
M
M
M


M
M
...
−
14
3


2
(
1
2
)
p
+
M


2
Q6 = −
π.p
(1 + PM )
15
2
21
2
Fig. 3: Energy decay curves for f = 0.15
M
(1 + 2 PM )
11
2
15.9.7.5.3 11.9.7.5.3(24 p 2 M − 200 p M + 20)

+
− ...

8
11
P
2
2
(
1
+
2
)
M


Q7 = −
π . p9M
(1 + p )
(1 + 2 p )
23 / 2
7/2
M
M
7
.
5
.
3
(
4231710
16938180
25381440
+
p
+
p 2 M + 1689480 p 3 M + 4213440 p 4 M
9.7.5.3
M
[{ 11 −
}
2
213 (1 + 2 p M )
Fig. 4: Energy decay curves for f = 0
(9.7.5.3(2115855 + 4237380 p M − 4245780 p 2 M − 16927680 p 3 M − 14783328 p 4 M
−{
− 4218816 p
5
M
− 4368 p
6
M
)
214 (1 + 2 p M ) 2
}...]
Therefore from Eq. (50) we get:
hi hi′ N 0 p 3 / 2 M ν −3 / 2 (t − t 0 ) −3 / 2
=
+ ξ 0 Qν − 6 (t − t 0 ) −5 + ξ 1 [ Rν −17 / 2 (t − t1 ) −15 / 2 + Sν −19 / 2 (t − t1 ) −17 / 2 ]
2
8 2π
exp{− (2ε pkl Ω p + 2ε qki Ω q + 2ε rkj Ω r + 2ε skm Ω s )(t − t1 )} (51)
Thus, the decay of MHD turbulence for four-point
correlation in a rotating system may be written as:
Fig. 5: Energy decay curves for f = 1.5
2796
Res. J. Appl. Sci. Eng. Technol., 6(15): 2789-2798, 2013
ACKNOWLEDGMENT
This study is supported by the research grant under
the ‘National Science and Information and
Communication Technology’ (N.S.I.C.T.), Bangladesh
and the authors (Bkar, Azad) acknowledge the support.
The authors also express their gratitude to the
Department of Applied Mathematics, University of
Rajshahi for providing us all facilities regarding this
study.
Fig. 6: Decay curves of Eq. (54)
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2797
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