Decay of Energy of MHD Turbulence For Four-Point Correlation

Decay of Energy of MHD Turbulence For Four-Point Correlation

Vol. 1 Issue 9, November- 2012

1M. A. Bkar PK, 2M. A. K. Azad and 3M.S.Alam Sarker

1Assistant Professor,2Associate Professor and 3Professor, Department of Applied Mathematics, University of Rajshahi, Rajshahi-6205, Bangladesh.

Abstract

In this study we consider the decay of energy of MHD fluid turbulence for four-point correlations prior to the ultimate phase. Three and four point correlation equations are obtained. The correlation equations are converted to spectral form by their Fourier-transform

.By neglecting the quintuple correlations in comparison to the second, third and fourth order correlation terms. Finally integrating the energy spectrum over all wave

In this paper, the turbulence for three point correlations is generalized to some extent in order to analyze the four- point turbulence at higher Reynolds numbers. In this case, the quadruple correlation terms in the three- point correlation are retained and in addition, a four- point correlation equation is considered. Following Deisslers approach we studied the decay of energy of MHD turbulence for four- point correlation system. The decay law comes out to be in the form

numbers and we obtained the energy decay law of

0

MHD turbulence for magnetic field fluctuations.

p A(t

3

t0 ) 2

B(t

t ) 5

C(t

15

t1 ) 2

D(t

17

t1 ) 2 ,

where p

denotes the total energy and t is the time,

1. Introduction

   The idea of magneto hydrodynamics is that magnetic fields can induce currents in a moving conductive fluid, which create forces on the fluid. In magneto hydrodynamics we study the dynamics of electrically conducting fluids. The examples of such fluids include plasmas, liquid metals and salt water. The electrical field effects are neglected as is usually done in MHD.Taylor introduced correlation coefficients between the quantities. Chandrasekhar [1] studied the invariant theory of isotropic turbulence in magneto- hydrodynamics. S. Corrsin [2] discussed on spectrum of

   A, B, C and D are arbitrary constants determined by initial conditions.

2. Four-point correlation and spectral equations

   We take the momentum equation of MHD turbulence at the point p and the induction equation of magnetic field fluctuation four point correlation and equations at p , p and p as

   u u h 2u

   isotropic temperature fluctuations in isotropic turbulence. Deissler [3, 4] developed A theory decay of homogeneous turbulence for times before the final

   period. Using Deisslers theory Kumar and Patel [5 ]

   l uk l

   t xk

   h h

   hk l l

   xk xl xk xk

   u 2h

   (2.1)

   studied the first order reactant in homogeneous turbulence before the final period for the case of

   i + u

   t k

   i – h

   k

   xk

   i =

   xk pM

   i

   xk xk

   (2.2)

   multipoint and single time consideration. Loeffler and Deissler [6] studied the decay of temperature fluctuation in homogeneous turbulence. Patel [7]

   extended the problem [5] for the case of multipoint and multi-time concentration correlations. Islam and Sarker

   h j u h j

   t xk

   h h

   h u j

   xk k

   k

   u

   2

   h

   j (2.3)

   pM xk xk

   2 h

   m u m

   h m m

   (2.4)

   k

   [8] studied the decay of dusty fluid MHD turbulence before the final period in a rotating system. Sarker and

   t k x

   xk pM xk

   Kishore [9] studied the decay of MHD turbulence before the final periods. Azad, Aziz and Sarker [10] studied the first order reactant in magneto-

   Where

   P 1 h 2 is the total MHD pressure

   2

   hydrodynamic turbulence before the final period of decay in presence of dust particles. They considered the two and three point correlation equations and solved these equations after neglecting the fourth and higher order correlation terms.

   (x,t) is the hydrodynamic pressure, is the fluid

   P

   density, is the Magnetic Prandtl, number

   M

   is the kinematics viscosity, is the magnetic diffusivity, hi (x, t) is the magnetic field fluctuation,

   uk (x, t) is the turbulent velocity ,t is the time,

   xk is (ul u j hi hk hm )

   (ul uk hi h j hm )

   V(uol.u1 Ihsshueh9, N) ovember- 2012

   the space co-ordinate and repeated subscripts are rk rk

   summed from 1 to 3 .

   l m i j m

   k

   r

   Multiplying equation (2.1) by

   hi h j hm

   (2.2) by

   (whi h j hm )

   (whi h j hm )

   (whi h j hm )

   u h h

   (2.3) by u h h

   (2.4) by u h h

   rl rl rl

   and adding

   l j m

   l i m

   l i j

   the four equations, we than taking the space or time averages and they are denoted by …….

   (2.6)

   In order to write the equation (2.6) to spectral form, we

   or …….. .

   . We get

   can define the following nine dimensional Fourier

   transforms

   t (ul hi h j hm )

   x

   (ul uk hi h j hm )

   k

   (hk hl hi h j hm )

   x

   k

   ul hi r hj r

   l i (k)

   hm r

   j (k ) m (k ) exp[i(k.r

   k .r

   k .r )dkdk dk

   (2.7)

   x

   (ul uk hi h j hm )

   k

   (ul ui hk h j hm )

   x

   k

   (ul uk hi h j hm )

   x

   k

   ul uk hi r hj r

   hm r

   x

   (ul u j hi hk hm )

   k

   (ul uk hi h j hm )

   x

   k

   (ul u j hi h j hm )

   x

   k

   l k (k) i (k) j (k ) m (k ) exp[i(k.r

   k .r

   k .r )dkdkdk

   (2.8)

   2 2

   ul ui hi r hj r

   hm r

   x

   (whi h j hm )

   l

   xk xk

   (ul hi h j hm )

   [

   pM xk

   (ul hi h j hm )

   x

   k

   2 2

   l i (k) i (k) j (k ) m (k ) exp[i(k.r

   k .r

   k .r )dkdkdk

   (2.9)

   xk xk

   (ul hi h j hm )

   xk xk

   (ul hi h j hm )]

   (2.5)

   ul uk hi

   r hj r

   hm r

   Using the transformations

   l k (k ) i (k) j (k ) m (k ) exp[i(k.r

   k .r

   k .r )dkdk dk

   (2.10)

   ,

   xk rk xk

   r , x

   ( )

   rk rk rk

   ul u j hi

   r hk r

   hm r

   k

   k

   into equations (2.5) we get,

   2

   l j (k ) i (k) j (k ) m (k ) exp[i(k.r

   k .r

   k .r )dkdk dk

   (2.11)

   t (ul hi h j hm )

   (1 pM ) r

   r (ul hi h j hm )

   u u h

   r h

   r h r

   k k l k i j m

   2 2

   l k i (k) j (k ) m (k ) exp[i(k.r

   k .r

   k .r )dkdkdk

   (2.12)

   (1 pM )

   rk rk

   (ul hi h j hm )

   2 pM

   rk rk

   (ul hi h j hm )

   2 2

   ul ui hi r hj r

   hm r

   2 pM

   rk rk

   (ul hi h j hm )

   2 pM

   rk rk

   (ul hi h j hm )

   l i (k ) i (k) j (k ) m (k ) exp[i(k.r

   k .r

   k .r )dkdk dk

   (2.13)

   r

   (ul uk hi h j hm )

   k

   (ul uk hi h j hm )

   r

   k

   (ul uk hi h j hm )

   r

   k

   whi

   r hj r

   hm r

   (h h h h h )

   r

   (h h h h h )

   r

   (h h h h h )

   i (k) j (k ) m (k ) exp[i(k.r

   k .r

   k .r )dkdk dk

   (2.14)

   r

   l k i j m

   k

   l k i j m

   k

   l k i j m

   k

   Interchange of points

   p / and

   p , and p the

   r

   (ul uk hi h j hm )

   k

   (ul ui hk h hm )

   r

   k

   (ul uk hi h j hm )

   r

   k

   subscripts i and k; i and j results in the relations

   u u h h h

   u u h h h

   ; u u h h h

   2 2 2

   Vol. 1 I2ssue 9, November- 2012

   l k i j m

   l k i j m

   l k i j m

   2 ] (whi hj hm ) =

   u u h h h

   ; u u h h h

   u u h h h h ;

   rl rl

   rl rk

   rl rk

   rl rk

   l k i j m

   l m i j m

   l i i k j m

   ul u j hi hk hm

   ul ui hi hk hj hm ;

   2 2 2 2 2

   By use of these facts and equations (2.7) to (2.14), one can write equation (2.6) in the form

   rl rk rl rk rl rk rk rl

   2

   rk rl

   t ( l i

   j m )

   rl rk

   ]( ul uk hi hj hm

   hl hk hi hj hm )

   (2.18)

   p

   [(1

   P )K 2

   (1 p )K 2

   (1 p )K 2

   2 p KK

   2 p KK

   2 p KK ]

   ( ) (Kl Kk Kl Kk Kl Kk Kl Kk Kl Kk Kl Kk Kl Kk Kl Kk Kl Kk Kl Kk )

   M M M

   M

   M M M

   1. j m

      Kl Kl

      Kl Kl

      Kl Kl

      2Kl Kl

      2Kl Kl

      2Kl Kl

      ( l i

   2. m ) =i( K k

   K k K k ) ( l k i

   j m ) –

   ( l k i

   j m –

   l k i

   j m )

   (2.19)

   i( K k K k

   K k ) ( l k i

   j m ) –

   i( K

   K K ) (

   ) +i(

   Equation (2.19) can be used to eliminate

   i j m

   k k

   K k K k

   k

   K k ) (

   l k

   l i k

   i j m

   1. m ) +

      from equation (2.16) if we take contraction of the indices i and j in equation (2.19).

      Equations (2.16) and (2.19) are the spectral equation

      i( K k K k

      K k ) (

      i j m )

      (2.15)

      corresponding to the four point correlation equation.

      The tensor equation (2.15) can be converted to the scalar equation by contraction of the indices i and j ;

3. Three-point correlation and spectral

   t ( l

   i i m )

   equations

   M

   [(1

   pM

   P )K 2

   (1 pM

   )K 2

   (1pM

   )K 2

   2 pM KK

   2 pM KK

   2 pM KK ]

   The spectral equations corresponding to the three-point correlation equations by contraction of the indices i and j are

   ( l i i

   m ) = i( K k

   K k K k ) ( l k

   i i m ) –

   i( K k	K k	K k ) (
   i( K k	K k	K k ) (
   i( K k	K k	K k ) (
   i( K k	K k	K k ) (

   l k i i

   m ) –

   t ( l i i )

   p

   [(1 P )( K 2 K 2 ) 2 p KK ] ( )

   l k i i

   m ) +

   M M l i i

   M

   l i k

   i m ) +

   =i( K k K k ) ( l k

   i i ) –

   i i m )

   (2.16)

   ( K k K k ) ( l

   1. i i )

      i( K k K k )

      ( l k

      i i )

      i( K k K k ) ( l i

      i i )

      If we take the derivative with respect to

      xl of the

      +i( k

      k ) (3.1)

      momentum equation (2.1) at p, we have,

      2 w 2

      l

      and

   2. i i

   x x x x

   ul uk

   hl hk

   (2.17) –

   l l l l (K K K K

   K k K K )

   Multiplying equation (2.17) by

   hi h j hm , taking time

   ( ) = l k l k l k l k

   i j (K 2

   K 2 2K K )

   averages and writing the equation in terms of the independent variables r , r , r we have,

   ( l k

   i i –

   l l

   l k i j )

   l l

   (3.2)

   2

   – [

   rl rl

   2

   rl rl

   2

   2

   rl rl

   2

   rl rl

   2

   2

   rl rl

   Here the spectral tensors are defined by

   u h r h r

   The relation between

   (k)

   andVol. 1 Issue 9, Noisvember- 2012

   l i j

   i k j

   l i j

   l i (k)

   j (k )

   exp[i(k.r

   k .r

   (3.3)

   obtained by letting r

   0 in equation (3.3) and

   comparing the result with equation (4.3), Then

   ul uk (r)hi (r)hj (r )

   l k (k)

   i (k)

   j (k ) exp[i(k.r

   k .r

   (3.4)

   i k i (k)

   l i (k)

   i (k ) dk

   (4.4)

   5. Solution neglecting quintuple correlations

   ul ui (r)hi (r)hj (r )

   l i (k)

   i (k)

   j (k ) exp[i(k.r

   k .r

   (3.5)

   As it stands the set of linear equations (2.15), (3.1), (3.2), (3.5), (3.6) and (4.4) is indeterminate as it contains more unknowns than equations in equation

   ul hi (r)hj (r )

   (2.16). Neglecting all the terms on the right side of equation (2.16), the equation can be integrated between

   l i (k) j (k ) exp[i(k.r

   k .r )dkdk

   (3.6)

   t1 and t to give

   ul hk (r)hi (r)hj (r )

   l i j m =

   1. i j

      exp

   2. 1

   l k (k)

   i (k)

   j (k ) exp[i(k.r

   k .r

   M

   (3.7)

   1

   pM

   t t1

   p k 2 k 2

   k 2 2kk

   2k k

   2kk

   (5.1)

   whi (r)hj (r )

   i (k)

   j (k )

   exp[i(k.r

   k .r

   (3.8)

   where

   l i j

   is the value of

   m 1

   l i j m

   at t=

   A relation between

   l k i

   j and

   1. i j

   2. can be

   t1 that is stationary value for small values of k, k and

   k when the quintuple correlations are negligible. Equation (3.9) and (5.1) can be converted to scalar

   obtained by letting r 0 in equation (2.7) and

   comparing the result with equation (3.4)

   form by contracting the indices i and j. Equation (3.1) have been contracted already. Substituting of equation

   l k (k)

   i (k)

   j (k )

   (3.2), (3.9), (5.1) in equation (3.1), Deissler, [3, 4] We get

   l i (k)

   j (k ) m (k ) exp[i(k.r

   k .r

   k .r )dk

   (3.9)

   t (kk l

   i i )

4. Two-point correlation and spectral equations

[(1

pM

P )( K 2

K 2 )

2 pM

KK ] (kk l

i i ) =

The spectral equation corresponding to the two point correlation equation taking contraction of the indices is

[a]1

exp[

M

(t t1 ){(1 pM

p

M

)(k 2

k 2 k 2 )

2 pM

(kk k k

k k)}]dk

(k)

2 k 2

(k)

2ik [

(k)

( k) ]

(4.1)

+[b]1

exp[

p

(t t ){(1

p )(k 2 k 2

k 2 )

2 p kk

2 p k k)}]dk

t i i

i i k

M

i k i

k i i 1 M M M

p

M

where,

i i and i k

i Are defined by

+ [c]1

exp[

(t t1 ){(1

p )(k 2

M

k 2 k 2 )

2pM kk

2pM k k )}]dk

(5.2)

hi hi

and

i i (k )

exp[ i(k.r)]dk

(4.2) pM

u h h (r) =

(k)

exp[ i(k.r)dk

(4.3)

At t , ,s have been assumed independent of; that

i i i

i k i

1

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

. exp[

+

(t t0

pM

){(1 pM

)(k 2

k 2 )

V2opl. 1kIsksu}e]d9,kNovember- 2012

simplicity.

Substituting dk

dk1dk2 dk3 and integrating with

k 2 2 pM .

M

5

2

i b(k.k )

b( k.

k ) .

respect to k1

, k2

and

k3 , we get,

{- 1

1. 2 p k 2

   1

2. p kk

M

M M k 2

M

2

t (kk l

i i )

exp

2 (1

p ) 2

(1 pM )

[(1

pM

P )( K 2

K 2 )

2 pM

KK ] (kk l

i i ) =

+kexp[ (

k

2 ) (1

pM )(k

k )

2 pM kk ]

pM

(t t1 )(1

pM )

3

2 [a]1 exp

2

exp(x

0

2 )dx}dk +

2

5

(t t )(1 p ) (1 2P )(k 2 k 2 ) 2 p kk

k 2 2 pM .

i c(k.k )

c( k.

k ) 1 .

M

2

[ 1 M { M M }]+

pM (1

p )2

(1 pM )

{- 1

1 2 p k 2 2 p kk +

2

k 2 M M

pM

(t t1 )(1

pM )

3

1

2 [b] exp

exp ( 2 )

(1 pM )

(1 pM )

k exp[ 2 ( (1 pM

2 k

)(k 2 k 2 )

2 pM kk )].

[ (t

t1 )(1 pM ) {(1

2PM )(k )

2 pM kk

k 2 }]

2 2 (5.5)

M

pM (1

p )2

(1 pM )

exp(x )dx}dk

0

+ pM

(t t1 )(1

pM )

3

2

[c]1 exp[-

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

(t t )(1 p )

(1 2P

)(k 2 )

2 p kk

(5.3)

transfer of energy between wave numbers. In order to

1 M {k 2 M M }]

2

M

pM (1 p ) (1 pM )

make further calculations, an assumption must be made for the forms of the bracketed quantities with the subscripts 0 and 1 in equation (5.5) which depends on

Integration of equation (5.3) with respect to time, and in order to simplify calculations, we will assume that

the initial conditions.

k

l

a 1 0 ; That is we assume that a function

sufficiently general to represent the initial conditions can obtained by considering only the terms involving

(2 ) 2 [ k

i i (k, k )

kk l

i i (

k,

k ) ]0 =

i

[b]1 and[c]1. The substituting of equation (4.4) in

(k 2 k 4 k 4 k 2 )

(5.6)

equation (4.1) and setting H

2 k 2

0

i

, result in

H

t

where, G=

2 k 2

H G

pM

(5.4)

where 0 is a constant depending on the initial conditions. For the other bracketed quantities in equation (5.5), we get,

k 2 2

.i[ kk l

i i (k, k )

kk l

i i (

k,

k ) ]0 .

7 -k/exp

Vol. 1 Issue 9, November- 2012

2

4 pM .

i b(k.k )

b( k.

k ) 1 =

[ 2 ( (1 p

M

k

)(k 2

k 2 )

2 pM

kk )]}

4 pM .

7

2

i c(k.k )

c( k.

k ) 1

(5.7)

2

exp(x 2

0

)dx

)]

(5.8)

1

2 (k 4 k 6 k 6 k 4 )

where,

(t t1 )(1

pM

1

pM ) 2 .

Remembering that

dk

2 .k 2 d(cos )

and kk

kk cos

, is the angle between k and

Integrating equation (5.8) with respect to k .

k and carrying out the integration with respect to , we get,

G=-

We have,

G= G G

(5.9)

2

4

[ 0 (k k

k 4 k

2 )kk

{ .

where,

1 5

2 p 2

(t t )(1

2 p )k 2

(t t )

G 0 M exp

0 M

3

0 0 2 (t

3

t0 ) 2 (1

5

pM ) 2

pM (1

pM )

exp[

(t t

){(1 p

)(k 2

k 2 )

2 p kk }] .

15 p k 4

5 p 2

3 p p 2

p 0 M

M . M

M k 6

M M

1 k 8

0

M

exp[

M

(t t0

pM

){(1 pM

)(k 2

k 2 )

2 pM

kk }]}

4 2 (t

t )2 (1

pM )

(1 p )2 (t

t0 )

2 (t

t0 )

1 pM

(1 p )2

M

(5.10)

(k 4 k 6 k 6 k 4 )kk

.+ 1

and,

(t

( 1 exp[ 2

t0 )

2

(1 2 pM )k

2PM kk

k 2 ]-

G G G G G

1 2 3 4

(5.11)

2

(1 pM )

1 pM

where,

1

(1 2 p )k 2

2P kk

2 p 5

2

1 exp[

2 M

M k 2

]+ 1

G 1 M exp

(1 pM )

1 pM

1 8 2 (t t )2 (1 p )5

1

M

(t t )(1 2 p p 2 )

2P kk (1 2 p )k 2

1 M M

k 2 .

exp[

2 k 2

M M ]

2

1. pM (1 pM )

   pM (1 pM )

   90 p k 6

   [ M +3

   1 2P kk

   (1 2 p

   )k 2

   4 (t t )4 1 p

   – exp[

2. k 2

M M ] 1 M

1

2

M

p

1 pM

(1 pM )

4 pM

2 p2 M

1 k 8

+{k exp[

2 ( (1 p )(k 2

k 2 ) 2 p

M

kk )]

2 (t t )2 1 p

3 (t

t )3 1 2

3 (t

t )3

-k.exp

2 ( (1 pM

k

)(k 2

k 2 )

2 pM

kk )]}

+ 64 p 2 M

M

2

(t t1 ) 1 pM

2 (t

10 p3 M

1

1

M

p

M

t )2 1 3

40

1

(t t1 )

k10

+{ k exp2

2

exp(x )dx

M

0

[ 2 ( (1

p )(k 2

k 2 )

2 pM kk )]

2

+8 pM

pM

k 12 ]

(5.12)

128(1

p )5 p2M

128(1

p )7 k14

…..V..o..l.. .1…I.s.s..u..e ]9, November- 2012

1 pM

1

G = 2 p

1 pM

5 (1 p )4

M

M

exp

(5.15)

1 M M

1

2 9

The integral expression in equation (5.9), The quantity

8 2 (t

t )2 (1

2 pM ) 2

G represents the transfer function arising owing to

(t t )(1 p )(1 2 p p 2 )

1 M M M k 2

consideration of magnetic field at three point correlation equation; G arises from consideration of

90 pM (1

.[

pM (1

pm )

pM )

k 6 +

the four point equation. Integration of equation (5.9) over all wave number shows that

1

4 (t t )4 (1

2 pM )

G.d k 0

(5.16)

120 p (1 p )

2 p 2 M (1 p )2 1

2

M m

m 0

1

2 (t t )2 (1 2 p )

M

3 (t t )3 (1 2 p ) 3 (t t )3 k 8

1

1

M

64p2M (1 p )2 40

2

+ m

(t t1 )(1 2 pM ) (t t1 )

8 p3 (1 p )3 p (1

10p3M (1 p )3

M k10 +

1

3

M

2 (t t )2 (1 2 p )

p )

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 (5.4), we

M M M M

k12 ]

(5.13)

get,

3

(1 2 pM )

(1 2 pM )

2 k 2 (t t )

2 k 2 (t t )

2 k 2 (t t )

H exp

1 9

0 G exp

pM

0 dt pM

J (k) exp 0

pM

M

G =

3

3

8 2 (t

1. 2 pM 2

   3

   t1 ) 2 (1

   p )8

   exp

   2

   where,

   J (k )

   N 0 k

   is a constant of integration and

   (t t1 )(1

   pM

2. pM )

k 2 .[

4 (t

90 pM k 7

1

p

M

t )4 1 2

can be obtained as by Corrsin, [2]

120 p

60 p2 M

30 9 H

1

1

1

M

M k

p

2 (t

t )2

3 (t

t )3 1 2

3 (t

t )3

N0k 2

exp

2 k 2 (t

t0 )

exp

2 k 2 (t

t0 ) [G (G G

G G ) ]

p

64 p2 M

10 p3 M

40(1

p )2

k11

pM pM

1 2 3 4

M

(t t1 )

2 (t

t )2 1 2

(t t1 )

exp

2 k 2 (t

t0 )

dt (5.17)

1

M

p 2 M

pM (1

p )2

k13 ]

1

2

exp( y 2

0

)dy

(5.14)

pM

where,

M

where,

(t t1 )(1

1

pM ) 2 k

G G G G G G

1 2 3 4

(5.18) after

p

1 integration equation (5.17) becomes

1 15

N k 2

2 k 2 (t t )

1. 1 2 pM 2

   4

   exp

   29

2. 0 exp 0 +

pM

28 (t

t1 )(1

pM ) 2

H [H H H H ]

(5.19)

(t t1 )(1

pM

2 pM )

k 2 [

1 2 3 4

7560(1 p )3 20160(1 p )5 4233600(1 p )7

where,

1

1

1

M

M

M

M k 6 M M k 8

4 (t

t )4 p 2

3 (t

t )3 p

3 (t

t )3 p 3

12096(1 p )5 3360(1 p )7

2304(1 p )5 p

1344(1 p )9

M M k 10

M M

M k 12

2 (t

t )2

2 (t

t )2 p 2

(t t ) p 2

1 1 M 1 M

0 M

1 5

H = 2 p 2

exp

H Vol. 1 Issue 9, November- 2012

1

2

3 7 2 p

5 (1

p )4 exp

8 2 (1

pM ) 2

1 M M

M

9

8 2 (1 2 p ) 2

(t t0 )(1 2 pM ) k 2

(t t )(1 2 p p 2 )

p (1 p )

1 M M k 2

M M

where,

pM (1

.[ 18 pM (1

pM )

pM )

1

k 6 +

1

4 (1 2 p

M

)(t t )5

2 2 (t t ) 2

17 32 p

2 p 2 M

4 p3 M

20 p4 M

120 p (1 p )

1

1

F( )

exp(

) exp(x

)dx,

0 k

M M M k 8

0 pM (1

pM )

4 3 (1 2 p

M

+

)2 (t

t )4

3 2 (1 2 p

M

)(t t )3

and,

17 49 p 13p 2 M 13 p3 M 98 p 4 M 134 p5 M 104 p6 M 60 p7 M +

1

M

1 4 3 (1

2 5

2 pM

)2 (t

t )4

k 10

H 1 pM

exp

52 p 4 M

64 p3 M

48 p 2 M

40 p

M

1 2 5

1

8 (1

pM )

(1 2 pM

)2 (t

t )2

(t t )(1 2 p p 2 )

1 M M 2

pM (1

k

pM )

(1

.(17

pM

49 p

p2 M p3 M )2

13 p2 M 13 p3 M

98 p4 M 134 p5 M

104 p6 M

60 p7 M )

[ 18 pM k 6

2

1

M

15 6 pM 21p M

4 pM k8

M

24 p2 M (1

2 p )5 (t

t1 )

k14

M

1

1

M

4 (1 p )(t t )5

M

4 3(1 p )2 (t t )4

2 (1 p )(t t )3

(1 pM

p 2 M

p3M ).

2 3 4 2

( 40 p 89 p 2 M 51p3M 124 p 4 M 40 p5 M 36 p 6 M 60 p 7 M )

4

15 6 pM

36 p M 6 p M 61p M 14 p M

40 pM

18 10

M k 14

+

M

12 2 p (1

p )3 (t

t )3

(1 pM

)2 (t

k

1

t )2

(1 pM

p 2 M p3M )3

p 2 M (1

2 pM ) .

1

M

– .(17 49 p 13 p 2 M 13 p3M 98 p 4 M 134 p5 M 104 p 6 M 60 p 7 M )

6

M k 16

(1 pM

)2 (75

30 pM

180p2 M

30 p3 M

305p4 M )

exp( 3 )Ei( 3 )]

24 p 4 M (1

2 pM )

120 p2 M (1

+

p )4 (t t )2

k12

1

M

14 p4 M 56 p3 M 12 p2 M

40 p 18

where, Ei( 3 )

M

p (1 p )3 (t t )

(t t )(1 2 p p 2 )t

M M 1

exp

1 M M k 2

+

–

2 2 3 4

pM (1

(t

(t

2 pM )

t1 )

t )(1 2 p

dt

p 2 )t

(1 pM ) (75

3pM

90 p M

30 p M

215 p M )

k14

and, 3

1 M M k 2

120 p3 M (1

p )5 (t

t1 )

pM (1

2 pM )

2 4 3 2

M

1

14 2 p 4 M

(1 p M )(14 p M 56 p M 12 p M

40 pM

18)k

H 1 exp

(

p2M (1

pM )4

3

16 (1

15

pM ) 2

(1 p2M )3 (75

3 pM 90 p2M

30 p3M

215p4M )k16

(t t1 )(1

2 pM ) k 2

120p4M (1

pM )6

) exp(

2 )Ei( 2 )

p

M

[ 45 pM

k 8 +

1

M

where

2 4 (1 p )2 (t

t )4

1

1

M

Ei(

(1

2 ) exp(

p2M )tk 2

) /(t

t1 )dt]

20 p2 M

70 pM 5

60 pM

k10 +

pM (1

pM )

2 3 (1

p )2 (t

t )3

2 (t

t )2

20 p 4 M

40 p3 M 160 p 2 M 60 p

5 24 p2 M 200 p 20 2

M

2 2 2

M k12

exp (1

2 pM )tk

Vol. 1 Issue 9, November- 2012

4 pM (1

pM ) (t

t1 )

(t t1 )

Ei(

5 ) pM dt

M

(1 2 pM

) 20 p 4M

40 p3M

160 p 2M

60 pM

5 k14

(t t1 )

4 p 2M (1

p )2 (t

t1 )

From equation (5.19), we get,

20 240 p 424 p 2 M 48 p3 M

H H1 + H 2

(5.20)

– M k 14

pM

]

where,

(1 2 p )2 20 p 4 M 40 p 3 M 160 p 2 M 60 p 5

M M k 16 .exp( )Ei( )

M

4 p3 M (1

p )2

4 4

0

H

N k 2

1

exp

2 k 2 (t

pM

t0 )

+ H and

4 (1

2 pM )t pM

k 2 and

H H H

2

1 2

H H ;

3 4

exp

(1 2 pM )t k 2

In equation (5.20)

H1 and

H 2 magnetic energy

Ei(

4 ) =

pM dt

(t t1 )

1 9

spectrum arising from consideration of the three and four point correlation equations respectively. Equation (5.20) can be integrated over all wave numbers to give the total magnetic turbulent energy. That is

p M

H 2 2

4 1 exp

11

hi hi

Hdk

(5.21)

28 (1

pM ) 2 2 0

(t t1 )(1

pM

1890p

2 pM ) k 2

where,

3

N0 p 2 M

3

2 (t

3

t0 ) 2 + Q

6 (t

t ) 5 ,

[ M

k 6 + H1dk 0 0

1

M

4 (1 p )6 (t t )4

0 8 2

4231710 16938180p 25381440p 2 M 16894080p3 M 4213440p4 M

1

M k 8

17 15 19 17

M

3 (1 p )6 (t

t )3

H 2 dk

0

1[R

2 (t

t1 ) 2

S 2 (t

t1 )

2 ],

2115855

4237380pM

4245780p 2 M

16927680p3 M 14783328p 4 M +

4218816p5 M

4368p6 M

2 (1

M

M

p )6 p (t

1

t )2

k10

R Q2 Q4

Q6 Q7 , S

Q1 Q3 Q5

2115855

5670 pM

12720540p 2 M

8436120p3 M

19072032p 4 M

and Qs values are

25347840p 6 M

4128p 7 M

2304 p8 M

12 p 6

6 2

(1 pM ) p M (t

t1 )

k Q M

(1 pM )(1

5 . .

2 pM ) 2

9 5 p (7 p

6) 35 p (3 p2 M 2 p

M

3) 8 p (3 p2 M 2 p 3)

M

M M

M M

M M ……………

– 2115855

4226040pM

12731880p 2 M

17004960p3 M

35944272p 4 M

16 (1 2 pM )

8(1 2 p )2

3.26.(1 2 p )3

12796224p5 M

42264592p 6 M

16857280p 7 M

9920 p8 M

4864 p9 M 14

M

(1 p )6 p3 M

k 19

Q p 2 M

1344 p k 12 1

M

.exp( 5 )Ei( 5 )]

(1 pM

5

) 2 (1

2 pM

7

p 2 M ) 2

15.9 15.7(15 6 p 21p2 M )

(1 2 p )tk 2

[ M +

where, 5 exp M

pM

26 210 (1 2 p p2 M )

M

15.7.3(15 6 p

36 p2 M

6 p3 M

61p4 M )

19

Vol. 1 Issue 9, November- 2012

M p 2 M .

211 (1

2 pM

p2 M )2

(1 pM

Q

5

19

) 2 (1

9

2 pM ) 2

11.9.7(1 p 2 M )(75 30 p 180 p 2 M 30 p3 M 305p4 M )

M 45.7.5.3

9.7.5.3(20 p 2 M

70 p

5) 11.9.7.5.3(20 p 4 M

40 p 3 M

160 p 2 M

60 p 5)

213 (1

2 pM

p 2 M )3

210

211 (1

M

2 pM )

213 (1

m

2

2 pM )

13.11.9.7.5.3(1 2 p )(20 p 4 M 40 p 3 M 160 p 2 M 60 p 5)

M

M M

……………

+ 13.11.9.7(1

p 2 M )2 (75 3 p

90 p 2 M

30 p3 M

15 p4 M )

214.(1 2 p )3

M ………………. 21

214 (1 2 p p 2 M )4

M

21

p 2 M

15

<>11 .

Q .

p 2 M

2

(1 pM ) 2 (1 2 pM ) 2

Q

6

3 9

2

(1 pM ) 2 (1 2 pM p M ) 2

15.9.7.5.3 11.9.7.5.3(24 p 2 200 p 20)

M M

……………………….

[ 15.7

15.9.7(14 p2M

18 40 pM )

28 211 (1 2 p )

+

M

26 29 (1

2 pM

p2M ) Q7

15.11.9.7(14 p4 M 56 p3 M 12 p2 M 40 p 18)

M p9

.

210 (1

2 pM

p2 M )2

M

23 7

]

(1 pM ) 2 (1

2 pM ) 2

9.7.5.3

423170 16938180pM

25381440p 2 M

16894080p 3 M

4213440p 4 M

7.5.3

19

Q

p 2 M (1

3

M

(1 2 p )2 (1

1

pM ) 2

2 pM

.

7 .

p 2 M ) 2

211

…

213 2 pM )

9.15

15.7(17

32 pM

2 p2 M

4 p3 M

20 p4 M )

9.7.5.3( 2115855 4237380pM

4245780p2M

16927680p3M

214 (1 2 pM )2

14783328p4M

4218816p5M

4218816p6M

M

26 210 (1 p )2 (1

+

2 pM

p2 M )

2

11.9.7.5.3( 2115855 5670 pM 12720540p M

8436120p3 M

190720032p 4 M

9.7.5(17 49 p 13p2M 13p3M

98p4M

134p5M

104p6M 60 p7 M ) +

25347840p 6 M 4128p 7 M 2304 p8 M )

M

15 3

….

M

211(1

2

p )3 (1

2 pM

p2M )2

+

2 (1

2 pM )

11.9.7.5(1 pM p M

p3M )(17

49 pM

13p2 M

13p3M

98 p4 M

134 p5 M

104 p6 M

60 p7 M )

213 (1

M

p )4 (1

2 pM

p2 M )3

Therefore, from equation (5.21)

13.11.9.7.5(1 pM

p 2 M

p3 M )2 (17 49 p

M

13 p 2 M

13 p3 M

98 p 4 M

134 p5 M

3 3

hi hi = N0 p 2 M 2 (t

3

t0 ) 2

.Q. 6 .(t t ) 5

104 p 6 M 60 p 7 M )

M

214 (1 p )5 (1

2 pM

p 2 M )4

2 8 2

17 15

0 0

19 17

…………..]

+[ 1 R

2 (t

t1 ) 2

1 S 2 (t

t1 )

2 ] (5.22)

21

Q4 p 2 M .

5

Also, we can write equation (5.22) of the form

(1 pM

1

) 2 (1

2 pM

)(1

2 pM

9

p 2 M ) 2

p A(t

3

t0 ) 2

B(t

t0 )

C(t

15

t1 ) 2

D(t

17

t1 ) 2 ,

25.7.3

15.9.7(

40 p

48 p2 M

64 p3 M

52 p4 M )

(5.23)

This is the energy decay law of MHD turbulence for

[ M +

M

25 29 (1 p )2 (1 2 p p2 M )

M

four point correlations.

15.11.9.7( 40 p 89 p2 M 51p3 M 124p4 M 40 p5 M 36 p6 M 60 p7 M )

where,

M …..]

M

210 (1 p )3 (1 2 p p2 M )2

M

3 3

2

p h h , A= N0 p 2 M , B=2 Q 6 , C=2 ,

i i 0 1

4 2

17 19

R 2 and D=2 1 S 2 .

If R=0 and S=0 that is C=0 and D=0 in equation (5.23), than we get,

5

Decay of total energy of MHD turbulence=<p>

5

4.5

Vol. 1 Issue 9, November- 2012

p A(t

t0 )

3 / 2

B t t0

(5.24)

4

y5

3.5 y4

This is the energy decay of MHD turbulence in three- point correlations.

6. Result and discussions

3 y3

y2

2.5 y1

2

1.5

y1 at t0=.5 y2 at t0=1 y3 at t0=1.5 y4 at t0=2 y5 at t0=2.5

5

Decay of total energy of MHD turbulence=<p>

4.5

4

3.5

y5

3

1

0.5

0

2 3 4 5 6 7 8

Approximation of time=t

y1 at t0=.1 y2 at t0=.4 y3 at t0=.7 y4 at t0=1 y5 at t0=1.3

y4 y3

y2

y1

2.5

2

1.5

Fig. 6.3: Decay of energy of MHD turbulence for three- point correlation.

1

0.5

0

2 3 4 5 6 7 8

Approximation of time=t

5

Decay of total energy of MHD turbulence=<p>

4.5

4

y10

Fig. 6.1: Decay of energy of MHD turbulence for three- point correlation.

5

Decay of total energy of MHD turbulence=<p>

4.5

4

3.5 y9

3 y8

2.5 y7

2 y6

1.5

1

y6 at t0=t1=.5 y7 at t0=t1=1 y8 at t0=t1=1.5 y9 at t0=t1=2

y10 at t0=t1=2.5

3.5

3

2.5

y10

y9

y8

y6 at t0=t1=.1 y7 at t0=t1=.4 y8 at t0=t1=.7 y9 at t0=t1=1

0.5

0

2 3 4 5 6 7 8

Approximation of time=t

y7

2 y6

y10 at t0=t1=1.3

Fig. 6.4: Decay of energy of MHD turbulence for four-

point correlation.

1.5

1 5

0.5

0

2 3 4 5 6 7 8

Approximation of time=t

4.5

Decay of total energy of MHD turbulence=<p>

4

3.5

y10

y9

y8

Fig. 6.2: Decay of energy of MHD turbulence for four- point correlation.

3y6 y7

2.5 y3

y1 y2

2

y4 y5

y1 and y6 at t0=t1=.1 y2 and y7 at t0=t1=.4 y3 and y8at t0=t1=.7 y4 and y9 at t0=t1=1

y5 and y10 at t0=t1=1.3

1.5

1

0.5

0

2 3 4 5 6 7 8

Approximation of time=t

Fig. 6.5: Comparison between Figure 6.1 and Figure 6.2.

5 Vol. 1 Issue 9, November- 2012

Decay of total energy of MHD turbulence=<p>

4.5

4

y6 y7 y8 y9 y10 y1 and y6 at t0=t1=.5

y2 and y7 at t0=t1=1

3.5

3

y1 y2 y3 y4 y5

y3 and y8 at t0=t1=1.5 y4 and y9 at t0=t1=2

y5 and y10 at t0=t1=2.5

2.5

2

1.5

1

0.5

0

2 3 4 5 6 7 8

Approximation of time=t

Fig.6.6: Comparison between Figure 6.3 and Figure 6.4.

International Journal of Engineering Research & Technology (IJERT)

ISSN: 2278-0181

Vol. 1 Issue 9, November- 2012

Fig. 4.1 and Fig. 4.3 represent the energy decay of MHD turbulence for three-point correlations of equation (5.24). y1, y2, y3, y4 and y5 are solutions of equation (5.24) at t0=.1, .4, .7, 1 and 1.3 respectively; which indicated in the Figure 4.1 clearly. Similarly, in Figure 4.3; y1, y2, y3, y4 and y5 are represents solution curves of equation (5.24) at .5, 1, 1.5, 2 and 2.5 respectively, which indicated in Figure 4.2 and Figure 4.3. If the time is increases then the decay of energy is increases.

Fig. 4.2 and Fig. 4.4 represent the energy decay of MHD turbulence for four-point correlations of equation (5.23). y6, y7, y8, y9 and y10 are solutions of equation (5.23) at t0=t1=.1, .4, .7, 1 and 1.3 respectively; which indicated in the Figure 4.2 clearly. Similarly, in Figure 4.4; y6, y7, y8, y9 and y10 are represents solution curves of equation (5.23) at .5, 1, 1.5, 2 and 2.5 respectively, which indicated in Figure 4.4.

Fig. 4.5, represents the comparison between the Fig.4.1 and Fig.4.2 of three and four- point correlations of MHD turbulent flow at t0=.1, .4, .7, 1 , 1.3 and .5 , 1, 1.5, 2 , 2.5 respectively .

Fig. 4.6, represents the comparison between the Fig.4.2 and Fig. 4.4 of three and four- point correlations of MHD turbulent flow at t0=.1, .4, .7, 1 , 1.3 and .5 , 1, 1.5, 2 , 2.5 respectively .

In equation (5.23) the third and fourth term on the right hand side comes due to four point correlations. If we put C=0 and D=0 it will be in the form

From Fig. 4.5 and Fig. 4.6, we see that, in four- point correlations system energy die out faster than the three- point correlations system in MHD turbulent flow.

References

1. S. Chandrasekhar, The invariant theory of isotropic turbulence in magneto-hydrodynamics, Proc. Roy. Soc., London, A204, (1951a), 435-449.

2. S. Corrsin, On spectrum of isotropic temperature fluctuations in isotropic turbulence, J. Apll. Phys, 22(1951b), 469-473.

3. R.G.Deissler, On the decay of homogeneous turbulence before the final period, Phys .Fluids 1(1958), 111-121.

4. R.G.Deissler, A theory of Decaying Homogeneous turbulence, Phys. Fluis 3(1960), 176-187.

5. P. Kumar and S.R. Patel, First order reactant in homogeneous turbulence before the final period for the case of multi-point and single time, Phys.Fluids, 17(1974), 1362-1368.

6. A.L. Loeffler and R.G. Deissler, Decay of temperature fluctuations in homogeneous turbulence before the final period, Int. J. Heat Mass Transfer, 1(1961), 312-324.

7. S.R .Patel, First order reactant in homogeneous turbulence numerical results, Int.J.Enjng.Sci, 14(1976), 75-80.

8. M.S.Alam Sarker, and M.A. Islam, Decay of dusty

   p A(t

   3

   t0 ) 2

   B(t

   t ) 5 , which is

   fluid MHD turbulence before the final period in a rotating system, .J. Math and Math. Sci, 16(2001),

   0

   completely same with Sarker and Kshore [9] for the case of three -point correlation.

   For large times second, third and fourth terms in equation (5.23) becomes negligible leaving only

   35-48.

9. M.S .Alam Sarker and N .Kishore, Decay of MHD turbulence before the final period, Int.J. Eng. Sci, 29(1991), 1479-1485.

   A(t

   t0 )

   3

   2 power decay law.

10. M.A.K. Azad, M .A. Aziz,and M. S. Alam Sarker, First Order Reactant in Magneto-hydrodynamic Turbulence Before The Final Period of Decay in

In equation (5.23), we shows that magnetic

turbulent energy for four- point correlations systems decays more and more rapidly by exponential manner than the decays of three point correlation system.

If the quadruple and quintuple correlations were not neglected, the equation (5.23) appears that more terms in higher power of

Presence of Dust Particle in a Rotating System, Bangladesh .J. Sci. Res. 45(1) (2010),39-46.

(t t0 )and(t t1 ) would be added to the

equation (5.23).In this case, energy decays greater than the energy decays in equation (5.23) for four point correlation systems.