Group analysis of the von K�aarm�aan–Howarth equation.Part II. Physical invariant solutions

S.V. Khabirov a, G. €UUnal b,*

a Institute of Mechanics, Ufa Center of the Russian Academy of Sciences, Karl Marx Street 12, Ufa 450000, Russiab Faculty of Sciences, Istanbul Technical University, Maslak 80626 Istanbul, Turkey

Received 1 February 2002; accepted 15 February 2002

Abstract

Physically meaningful group invariant solutions of the von K�aarm�aan–Howarth equation correspondingto the optimal system of one-dimensional subalgebras have been obtained completely. These solutions maybe used to study the decay of the isotropic turbulence. � 2002 Published by Elsevier Science B.V.

PACS: 02.20.Sv; 02.20.Tw; 47.27.Gs

Keywords: Lie equivalence algebra; Optimal system; Invariant solution; Isotropic turbulence

1. Introduction

The von K�aarm�aan–Howarth (KH) equation has been obtained from the three-dimensionalNavier–Stokes equation under the assumptions of local isotropy of turbulence quantities such ascorrelation tensors, stationarity of the mean flow velocity and the incompressibility of the fluid [1].The KH equation governs the two-point double and triple longitudinal correlation functionswhich can be written as

ut ¼ 2m4

rur

�þ urr

�þ 4

rvðt; rÞ þ vrðt; rÞ; ð1Þ

where m is the kinematic viscosity, uðr; tÞ ¼ ðu0ðtÞÞ2f ðr; tÞ and vðr; tÞ ¼ ðu0ðtÞÞ3kðr; tÞ, in which u0ðtÞis the intensity of the turbulent fluctuations, f ðr; tÞ and kðr; tÞ are two-point double and triplecorrelation functions, respectively.

Communications in Nonlinear Science

and Numerical Simulation 7 (2002) 19–30

www.elsevier.com/locate/cnsns

* Corresponding author.

E-mail address: gunal@itu.edu.tr (G. €UUnal).

1007-5704/02/$ - see front matter � 2002 Published by Elsevier Science B.V.

PII: S1007-5704(02)00012-6

In order to obtain the physically relevant solutions of Eq. (1) one must impose the followingboundary conditions on the two-point double and triple correlations:

ðiÞ ujr¼0 ¼ u0ðtÞ < 1; ðiiÞ vjr¼0 ¼ 0; ðiiiÞ limr!1

u ¼ 0; ðivÞ limr!1

v ¼ 0:

To simplify the calculations in group analysis of Eq. (1) one must convert it into its canonicalform. This can be achieved by the following change of variables:

s ¼ 2mt; wðr; sÞ ¼ r2uðr; sÞ; h ¼ r2

2m4

rv

�þ vr

�: ð2Þ

The canonical form of Eq. (1) is as follows:

Ew ws wrr þ2

r2w ¼ hðr; sÞ: ð3Þ

As it can be seen from conditions (2), change of variables at r ¼ 0 and r ¼ 1 is neither localnor regular, i.e.,

u ¼ r2w; v ¼ r4 v0ðsÞ�

þ 2mZ r

0

r2hdr�:

The physical conditions on the new variables w and h must be such that

ðiÞ wjr!0 � Oðr2Þ; ðiiÞ hjr!0 � oðrÞ; ðiiiÞ wjr!1 � oðr2Þ; ðivÞ hjr!1 � oðr2Þ: ð4Þ

If the solutions of Eq. (3) satisfy the boundary conditions (4), then they are considered tobe physical solutions. In this work we only consider the physical group invariant solutions whichare obtained from the submodels of one-dimensional optimal system of subalgebras given in [2].In Section 2, we determine the boundary conditions which will be imposed on the finite trans-formations generated by the equivalence group as a consequence of conditions (4). In Section 3,we first discuss the inner automorphisms of the equivalence algebra since it is essential to de-termine the similar algebras, and then discuss the submodels of optimal system of one-dimen-sional subalgebras. Finally, in Section 4, we obtain the physical invariant solutions of thesubmodels.

2. Equivalence transformations for the solutions

The following operators act on the set of solutions of Eq. (1) [2]:

XaðsÞ ¼ a@s þr2a0@r

r2

8a00w@w þ

�� r2

8a000 þ 1

4a00�w r2

8a00

�þ a0

�h�@h;

YbðsÞ ¼ b@r r2b0w@w r

2b00

��þ 4b

r3

�wþ r

2b0h�@h;

ZcðsÞ ¼ cw@w þ ðchþ c0wÞ@h;

20 S.V. Khabirov, G. €UUnal / Communications in Nonlinear Science and Numerical Simulation 7 (2002) 19–30

Vdðr;sÞ ¼ d@w þ ds

� drr þ

2

r2d�@h;

where ‘‘0’’ stands for the derivative with respect to s, a ¼ aðsÞ, b ¼ bðsÞ, c ¼ cðsÞ and d ¼ dðs; rÞare arbitrary functions of their arguments. Finite transformations generated by these operatorscan be obtained by solving the corresponding Lie equations [2]:

Xa: �ss ¼ gð1Þð1þ gðsÞÞ; g ¼Z

dsaðsÞ ; �rr ¼ r

�aaa

!1=2

;

�aa ¼ að�ssÞ; �ww ¼ w expr2

8aða0

� �aa0Þ

�;

�hh ¼ ha�aaþ r2

8w

a00

�aa

" �aa00

aþ �aa0

2 a02

2a�aa

#exp

r2a0

8a

� �þ �aa0

4aexp

r2�aa0

8a

! a0

4�aaexp

r2a0

8a

� �;

Yb:�rr ¼ r þ bðsÞ; �ww ¼ w exp

� 1

2b0 r�

þ 1

2b��

;

�hh ¼ h

" w r

�þ 1

2b�

4b

r2ðr þ bÞ2

þ 1

2b00!#

exp

� 1

2b0 r�

þ 1

2b��

;

Zc: �ww ¼ w exp½cðsÞ ; �hh ¼ ðhþ c0wÞ exp½cðsÞ ;Vd : �ww ¼ wþ dðr; sÞ; �hh ¼ hþ Ed: ð5Þ

We now have to consider the boundary conditions on the transformed variables �ww and �hh undertransformations (5), i.e.,

Xa: ðiÞ �wwj�rr!0 � Oð�rr2Þ; ðiiÞ �hhj�rr¼0 ��aa0 a0

4�aa;

ðiiiÞ �wwj�rr!1 � oð�rr2Þ exp �rr2

8�aaða0

" �aa0Þ

#! 0; if a0 < �aa0 < 0;

ðivÞ �hhj�rr!1 ! Oð�rr2Þ;Yb: ðiÞ �wwj�rr¼bðsÞ � Oð�rr2Þ; ðiiÞ �hhj�rr¼bðsÞ � Oð1Þ;

ðiiiÞ �wwj�rr!1 � oð�rr2Þ exp� 1

2b0�rr�

! 0; if b0 > 0;

ðivÞ �hhj�rr!1 � Oð1Þ exp� 1

2b0�rr�

! 0; if b0 > 0;

Zc: ðiÞ �wwjr!0 � Oðr2Þ; ðiiÞ �hhjr!0 � oðrÞ; ðiiiÞ �wwjr!1 � oðr2Þ; ðivÞ �hhjr!1 � oðr2Þ:

Equivalence transformations may affect the behavior of solutions in physical sense. In otherwords, they may transform physical solutions into unphysical ones. But inverse equivalence

S.V. Khabirov, G. €UUnal / Communications in Nonlinear Science and Numerical Simulation 7 (2002) 19–30 21

transformations may act better in physical sense. These properties of the equivalence transfor-mations will be made use of in the sequel.

3. One-dimensional subalgebras

The operators Xa, Yb, Zc and Vd form an infinite-dimensional Lie algebra L. This algebra isdetermined by four arbitrary functions aðsÞ, bðsÞ, cðsÞ and dðr; sÞ. By using the commutator re-lations among these four operators we define the linear infinite group inner automorphism in L.These inner automorphism transformations induce the action of linear operators in the functionalspace comprising aðsÞ, bðsÞ, cðsÞ and dðr; sÞ as follows [2]:

AX : �aaðsÞ ¼ a1ðsÞaðlÞða1ðlÞÞ1; �bbðsÞ ¼ ða1ðsÞÞ1=2bðlÞða1ðlÞÞ1=2

; �ccðsÞ ¼ cðsÞ;

�ddðr; sÞ ¼ dðrða1ðlÞÞ1=2ða1ðsÞÞ1=2; lÞ exp 1

8r2ða1ðsÞÞ1ða01ðlÞ

� a01ðsÞÞ

�;

k ¼ g1ðsÞ ¼Z

ða1ðsÞÞ1ds; s ¼ gð1Þ

1 ðkÞ; l ¼ gð1Þ1 ðgðsÞ T Þ;

AY : �aa ¼ a; �bb ¼ bþ ab01 1

2b1a0; �dd ¼ dðr b1; sÞ exp

1

4b01ðb1

� 2rÞ

�;

�cc ¼ cþ 1

2ðb1b0 bb01Þ þ

1

4ab1b001

�þ a0b1b01

1

2a00b21 ab0

2

1

�;

AZ : �aa ¼ a; �bb ¼ b; �cc ¼ cþ ac01; �dd ¼ d expðc1Þ;

AV : �aa ¼ a; �bb ¼ b; �cc ¼ c; �dd ¼ d þ ad1s þ b

þ r2a0�d1r þ

r2

8a00

�þ r2b0 c

�d1;

where T is an arbitrary constant. Two subalgebras with equal dimension of L are called to besimilar if they are connected via inner automorphism [3]. Group invariant solutions of Eq. (3) forthe similar subalgebras are transformed into each other by equivalence transformations (5).

Optimal system of s-dimensional subalgebras is a collection of the representatives of thenonsimilar classes and it is denoted by Hs. Normalizer of the subalgebra L1 of algebra L is asubalgebra NLðL1Þ such that L1 is an ideal in NLðL1Þ. Operators involved in the normalizers areadmitted by the corresponding model equations (i.e., they are the symmetries of the modelequations). Optimal system H1 and its normalizers NLðL1Þ are reproduced in Table 1. As it can benoticed from this table normalizer U extends for the specific form of the function d0 for cases 1.5–1.9 for the subalgebra Vd0 .

To recapitulate, consider the following submodels (see Table 5 in [2]):

�wws ¼ �wwrr 2

r2�ww; ð6Þ

~wws ¼ ~wwrr 2

r2~ww; ð7Þ

ws ¼ wrr þ qðs; rÞw: ð8Þ

22 S.V. Khabirov, G. €UUnal / Communications in Nonlinear Science and Numerical Simulation 7 (2002) 19–30

Eqs. (6)–(8) remain invariant under the symmetry groups generated by the operators which ap-pear in the normalizers given in cases 1.1–1.3 in Table 1, respectively. Submodel Eqs. (6)–(8) haveinvariants solutions which appear as invariant solutions in the two-dimensional subalgebra.Therefore we do not consider them in detail here.

Invariants of the subalgebra Vd0 are:

s; r; h Ed0d0

w:

Notice that the last invariant is a function of the first two invariants. This enables us to obtain asubmodel of (3) in the form

ws ¼ wrr þ d10 ðd0s d0rrÞwþ qðr; sÞ; h ¼ Ed0

Pd0wþ qðr; sÞ: ð9Þ

In order to obtain a submodel equation which is similar to Eq. (3) but fully determined arbi-trary element h, we set q ¼ 0 in Eq. (8). Hence the investigation of the physical solutions of Eq. (3)reduces down to the investigation of the physical solution of Eq. (6). We shall consider cases 1.5–1.9 of optimal system H1 given in Table 1, in the subsequent sections.

4. Physical solutions constructed from the subalgebra 1

4.1. Subalgebra 1.5

In order to obtain the submodel equation corresponding to the subalgebra 1.5 given in Table 1,we substitute

d10 ¼ ekðsÞr3

into Eq. (9). Submodel Eq. (9) now takes the form:

ws ¼ wrr þ ðk0r3 9k2r4 6krÞw; h ¼ 2

r2

� 9k2r4 þ k0r3 6kr

�w: ð10Þ

Table 1

Optimal system H1 of one dimensional subalgebras and their normalizers

Case Basis of the subalgebra Normalizer

1.1 X1 fX1; Y1; Z1; VdðrÞg1.2 Y1 fY1;X1;Xs; VdðsÞg1.3 Z1 fXa; Yb; Zcg1.4 Vd0 ; d0-arbitrary function U ¼ fZ1; Vdg1.5 d1

0 ¼ ekðsÞr3 ; k00 ¼ 53k1k02 þ 36kk8=3 Xk2=3 þ Yk

1.6 d20 ¼ /ðr csÞ X1 þ Yc

1.7 d30 ¼ /ðrs1=2Þ Xs

1.8 d40 ¼ rk X1;Xs

1.9 d50 ¼ e/0 ðsÞr X1 Y2/0 Z2/02 ; Y1 þ Z/0 ;

Xs þ Y/2s/0 þ Z/0ð/2s/0Þ

S.V. Khabirov, G. €UUnal / Communications in Nonlinear Science and Numerical Simulation 7 (2002) 19–30 23

From the normalizer of this algebra it follows that Eq. (10) admit the operator Xk2=3 þ Yk, where kis a constant. We now consider the invariant solution of Eq. (10) on this one-dimensional sub-algebra. Invariants of this operator are

z ¼ rk1=3 kK; K ¼Z

kds; v ¼ wekr3 :

Therefore, a representation of the invariant solution has the form

w ¼ ekr3vðzÞ: ð11ÞSubstituting (11) into (10) we obtain the following second-order ordinary differential equation:

v00 þ ð6z2 mzþ nÞX ¼ 0; ð12Þwhere

m ¼ 12kK; n ¼ 6k2K2 1

3k5=3k0

� �kK þ kk1=3:

Notice that, since kðsÞ satisfies

k00 ¼ 5

3k1k02 þ 36kk8=3; ð13Þ

m and n become constants. Eq. (13) is integrable and some of its solutions are negative for k6 0and tend to 1 for a finite value of s. Only the solutions of Eq. (13) which lead to the physicalsolutions of Eq. (10) are given as follows:

ðiÞ k ¼ 0; k ¼ Kðs þ s0Þ3=2; K > 0 is an arbitrary constant;

ðiiÞ k < 0; k ¼ ð5ffiffiffi6

pðs0 sÞÞ6=5ðkÞ3=5

;

ðiiiÞ k < 0; k1=3 ¼ K2 r þ 1

216kK5ðs0 sÞ9ð108kÞ2

¼ 2

3

r

ðr2 þ 1Þ2þ r

r2 þ 1þ arctanr;

ðivÞ k < 0; k1=3 ¼ K2 r2 1

216kK5ðs0 sÞ9ð108kÞ2

¼ 2

3

r

ðr2 1Þ2þ r

r2 1þ 1

2ln

r 1

r þ 1

:

For all the cases (i)–(iv) solutions to Eq. (12) which satisfy the physical boundary conditions (4)have been obtained. Physical invariant solutions to (12) can only be obtained for m ¼ 12a, andn ¼ 6ða2 b2Þ, which leads to 24n < m2. Hence the physical invariant solution of Eq. (12) has theform:

v ¼ C1

Zzþ a bzþ a þ b

1=2b

dzþ C2;

where C1 and C2 are integration constants. To obtain a solution satisfying

v ¼ 0; v0 ¼ 0; at z0 ¼ a þ b;

b must be equal to 1=2. This leads to a solution of Eq. (10) in the form

24 S.V. Khabirov, G. €UUnal / Communications in Nonlinear Science and Numerical Simulation 7 (2002) 19–30

w ¼ C1ekr3 rk1=3

� kK þ a 1

2 ln rk1=3

kK þ a þ 1

2

�: ð14Þ

This solution has the property:

wj�rr!0 ¼ 0;owor �rr!0

¼ 0;

where �rr ¼ r þ bðsÞ, and bðsÞ ¼ k1=3ðkK þ a 1=2Þ: Equivalence transformation Yb in Eq. (5)allows us to introduce a new variable �rr. Therefore (14) can be transformed into �ww

�ww ¼ C1 exp kð�rr�

bÞ3 1

2b0 �rr�

1

2b��

ð�rrk1=3 ln j1þ �rrk1=3jÞ;

and h transformed into:

�hh ¼ �ww k0ð�rr�

bÞ3 6kð�rr bÞ þ 9k2ð�rr bÞ4 þ 2ð�rr bÞ2 þ �rr2�

1

2b�

4bð�rr�

bÞ2�rr2 þ 1

2b00��

:

Behavior of �hh as �rr goes to zero is as follows:

�hhjr!0 ¼ C1k2=3 exp

� kb2 þ 1

4bb0�ð1þOð�rr2ÞÞ:

It can be seen that �hh does not satisfy condition (4). We must now employ the equivalencetransformations Xa in (5) to obtain a new solution ~hh satisfying condition (ii) in (4). This leads tothe following functional differential equation for a:

a0ð�ssÞ ¼ a0ðsÞ 4C1aðsÞk2=3 exp

� kb2 þ 1

4bb0�;

Z �ss

s

dsaðsÞ ¼ 1:

As a result we obtain the following new physical invariant solution to Eq. (3):

~ww ¼ �ww exp�rr2

8aða0

" �aa0Þ

#;

~hh ¼ a�aa�ww k0ð�rr�

bÞ3 6kð�rr bÞ 9k2ð�rr bÞ4 þ 2ð�rr bÞ2 �rr�

1

2b�

4bð�rr�

bÞ2�rr2

þ 1

2b00��

þ 1

8�rr2�ww

a00

�aa

�aa00

aþ 1

2

�aa02 a0

2

a�aa

!þ 1

4

�aa0

aexp

�rr2

8

�aa0

a

! 1

4

a0

�aaexp

r2

8

a0

a

� �:

It is easy to see that when �rr ! 1 conditions (iii) and (iv) in (4) have been satisfied by ~ww and ~hh fora0 < �aa0 < 0: This solution goes to zero for finite value of s. And as �rr ! 1 it tends to zero fasterthan the solution of the linear heat equation does. This solution must correspond to a regime ofturbulence much before the final period of decay.

4.2. Subalgebra 1.6

Submodel for the subalgebra 1.6 given in Table 1 can be obtained by substituting

S.V. Khabirov, G. €UUnal / Communications in Nonlinear Science and Numerical Simulation 7 (2002) 19–30 25

d20 ¼ uðr csÞ

into Eq. (9). Submodel equation now takes the form

ws ¼ wrr u00 þ cu0

uw; h ¼ 2

r2

� u00 þ cu0

u

�w; ð15Þ

where c is a constant. Following the normalizer of the subalgebra 1.6 (see Table 1), we observethat Eq. (15) admits the operator X1 þ Yc. Therefore, invariant solution of Eq. (15) has the form:

w ¼ wðIÞ; I ¼ r Cs; ð16Þwhere C is a constant. Substituting (16) into (15) we obtain the invariant submodel:

V ðIÞ w00 þ Cw0

w¼ u00 þ Cu0

u; h ¼ 2

r2

�þ V ðIÞ

�w: ð17Þ

Introducing a new variable w ¼ f expðCI=2Þ, we obtain the normal form of Eq. (17):

f00 þ� 1

4C2 þ V ðIÞ

�f ¼ 0: ð18Þ

The physical condition on h becomes

limI!0

h ¼ limI!0

2

ðI þ CsÞ2

"þ V ðIÞ

#w ¼ oðIÞ:

This condition is satisfied for c 6¼ 0 when

limI!0

w ¼ OðI2Þ and V ðIÞ < 1:

But for this case Eq. (18) is not meaningful because the first term is finite and the second term isvery small. Therefore C must be equal to zero. This leads to the following conditions:

V ¼ 2

r2þ �ðrÞ; lim

r!0�ðrÞ ¼ oðrÞ; lim

r!1�ðrÞ ¼ oðr2Þ:

Solution f of Eq. (18) must behave as

f ¼ r2ð1þ lðrÞÞ and limr!0

lð1Þ ! 0:

Substituting this into Eq. (18) we obtain

l00 þ 4r1l0 þ ðl þ 1Þ� ¼ 0:

From this equation we can determine the behavior of �ðrÞ and wðrÞ as r ! 0. For example, if

limr!0

�ðrÞ � l0KðK þ 3Þrk2; k > 2;

then

limr!0

w � r2ð1þ l0rkÞ:

And for this case only one solution to Eq. (18) is possible to be found. Let

26 S.V. Khabirov, G. €UUnal / Communications in Nonlinear Science and Numerical Simulation 7 (2002) 19–30

�ðrÞ < 0 and limr!1

�ðrÞ ! constant6 0;

then all solutions of Eq. (18) either decay exponentially or grow no more than linear functions.Hence, both cases lead to physical solutions to (15).

Let us now consider the case 0 < r6 r0. Suppose that the function �ðrÞ has the form:

�ðrÞ ¼ h ¼ 22

3Cr4=3;

where C is a constant. It is clear that h is not a physical solution as r ! 0. This is why we excludethis point. From the third equation of (2) we obtain

v ¼ 2mr4

Z r

0

r2hdr ¼ 44

5Cmr7=3:

When r ! 0, v is not physical. For the physical case, let limr!0 v � nr, then, for

r0 ¼44Cm5n

� �3=10

;

v becomes a physical invariant solution.

4.3. Subalgebra 1.7

Submodel equation corresponding to the subalgebra 1.7 given in Table 1 can be obtained bysubstituting

d30 ¼ /ðrs1=2Þ

into Eq. (9). Submodel equation now becomes

ws ¼ wrr þ xw; x ¼ s1 /00�

þ 1

2J/0�; J ¼ rffiffiffi

sp ; h ¼ w

2

r2

�þ x

�: ð19Þ

Normalizer extends with the operator Xs in this case. Hence, invariant solution to Eq. (19) has theform:

w ¼ UðJÞ;where UðJÞ satisfies

U00 þ 12JU0

U¼

/00 þ 12J/0

/ V ðJÞ; h ¼ w

s2

J 2

�þ V ðJÞ

�: ð20Þ

Introducing a new variable U ¼ j expðJ 2=8Þ, we obtain the normal form of Eq. (20):

j00 þ� 1

4 1

16J 2 þ V ðJÞ

�j ¼ 0: ð21Þ

Physical conditions (4) imposed on h lead to the the following conditions on V, i.e.,

limr!0

V ¼ 2

J 2þ hðJÞ; h ¼ oðJÞ; lim

r!1V ¼ oðJ 2Þ:

S.V. Khabirov, G. €UUnal / Communications in Nonlinear Science and Numerical Simulation 7 (2002) 19–30 27

From these conditions it follows that all solutions to Eq. (20) decay exponentially fast. Let

j ¼ J 2ð1þ cðJÞÞ and limr!0

c ! 0:

The second-order ordinary differential equation for c has the form

c00 þ 4

Jc0 þ ð1þ cÞ hðJÞ

� 1

4 1

16J 2

�¼ 0:

As J ! 0, the dominant term c � J 2=40 for any hðJÞ ¼ oðJÞ. Thus, if

limJ!0

hðJÞ ¼ oðJÞ; limJ!1

hðJÞ ! Oð1Þ

then there exists a unique solution to Eq. (21) which has the property

limr!0

j � J 2 1

�þ 1

40J 2 þ � � �

�;

and as r ! 1 it decays exponentially fast. In this case the physical invariant solution to Eq. (3)takes the form:

w ¼ jeð1=8ÞJ2 ; h ¼ ws1hðJÞ:

4.4. Subalgebra 1.8

Submodel equation corresponding to the subalgebra 1.8 given in Table 1 can be obtained bysubstituting

d40 ¼ r4

into Eq. (9). Submodel equation now takes the form

ws ¼ wrr lðl 1Þ

r2w; h ¼ w

lþ 2 l2

r2: ð22Þ

The physical invariant solution to Eq. (22) can be found only when h ¼ 0. In fact, the physicalcondition on w at r ¼ 0 reads wð0Þ ¼ 0 and w0ð0Þ ¼ 0. Then for small r, w can be expressed as

w ¼ r2½aðsÞ þ oðs; rÞ ; a 6¼ 0:

Substituting this into Eq. (22), we have

r2ða0 þ osÞ ¼ 2ðaþ oÞ þ 4ror þ r2orr lðl 1Þðaþ oÞ:

The dominant term is lðl 1Þ ¼ 2. Hence h ¼ 0 in Eq. (22).In the classification of second-order linear partial differential equations one finds that Eq. (22)

with lðl 1Þ ¼ 2 is the one which admits more symmetries than the other equations does exceptthe heat equation [3]. Eq. (22) admits more symmetries than the ones given by the normalizers,and the Lie algebra L is spanned by the following operators:

28 S.V. Khabirov, G. €UUnal / Communications in Nonlinear Science and Numerical Simulation 7 (2002) 19–30

W1 ¼ @s; W2 ¼ s@s; W3 ¼1

2s2@s þ

1

2sr@r

1

8ðr2 þ 2sÞw@w; W4 ¼ w@w;

W1 ¼ Pðr; sÞ@w; Ps ¼ Prr lðl 1Þ

r2P:

Lie algebra L is a semidirect sum of algebra L4 and L1 ¼ fW1g, i.e., L ¼ L4 � L1. fW4g is thecenter in L4, and fW1;W2;W3g is a simple algebra. Optimal system of one-dimensional subalgebrasfor L4 is H1ðL4Þ:

W1 þ qW4; W2 þ qW4; W3 þ qW4; W1 � W3 þ qW4;

where q ¼ �1; 0. We may consider the invariant solution of all the subalgebras of this optimalsystem. To each subalgebra there is only one physical invariant solution which corresponds to thefinal period of the decay in isotropic turbulence [1].

4.5. Subalgebra 1.9

Submodel equation corresponding to the subalgebra 1.9 given in Table 1 can be found bysubstituting

d50 ¼ exp½d0ðsÞr

into Eq. (9). The submodel equation now becomes

ws ¼ wrr þ ðrd00 d02Þw; h ¼ 2

r2þ rd00 d02 :

Introducing new variables w ¼ C exp½d0ðsÞr , x ¼ r þ 2d, we obtain the linear scalar heatequation:

Cs ¼ Cxx:

This equation admits more symmetry operators than the other second-order linear partial dif-ferential equations. The optimal system of one-dimensional subalgebra and invariant solutionsare given in [4]. At r ¼ 0 and r ¼ 1 these solutions do not satisfy conditions (4). Therefore, theyare not physical invariant solutions. However, asymptotical behaviour in the small regions ofðr; sÞ plane may become physical.

5. Concluding remarks

In our previous paper [2], we have determined the two-point triple correlation function (itappears as unknown function in the KH equation) and the corresponding submodel equations bygroup theoretical means. In this paper, all the physical invariant solutions of the submodelequations obtained from the optimal system of one-dimensional subalgebras have been deter-mined. Those cases which are reducible to heat equation are relevant to final period of the decayof the isotropic turbulence.

S.V. Khabirov, G. €UUnal / Communications in Nonlinear Science and Numerical Simulation 7 (2002) 19–30 29

References

[1] Hinze JO. Turbulence. New York: McGraw-Hill; 1959.

[2] Khabirov SV, Unal G. Group analysis of the von K�aarm�aan–Howarth equation, Part I. Submodels. Commun

Nonlinear Sci Numer Simul 2002;7:3–18.

[3] Ovsiannikov LV. Group analysis of differential equations. New York: Academic Press; 1982.

[4] Ibragimov NH, editor. CRC handbook of lie group analysis of differential equations. Applications in engineering

and physical sciences, vol. 2. Boca Raton: CRC Press; 1995.

30 S.V. Khabirov, G. €UUnal / Communications in Nonlinear Science and Numerical Simulation 7 (2002) 19–30