Solution of functional equations and reduction of dimension in the local energy transfer theory
of incompressible, three-dimensional turbulence
M. Oberlack,
1
W. D. McComb,
2
and A. P. Quinn
2
1
Institut fu ¨r Wasserbau und Wasserwirtschaft, Fachgebiet Hydromechanik und Hydraulik, TU Darmstadt, Petersenstraße 13,
64287 Darmstadt, Germany
2
Department of Physics and Astronomy, The University of Edinburgh, James Clerk Maxwell Building, The King’s Buildings,
Mayfield Road, Edinburgh EH9 3JZ, Scotland, United Kingdom
Received 20 July 2000; published 26 January 2001
It is shown that the set of integrodifferential and algebraic functional equations of the local energy transfer
theory may be considerably reduced in dimension for the case of isotropic turbulence. This is achieved without
restricting the solution space. The basis for this is a complete analytical solution to the functional equations
Q( k ; t , t ' ) =H( k ; t , t ' ) Q( k ; t ' , t ' ) and H( k ; t , s ) H( k ; s , t ' ) =H( k ; t , t ' ). The solution is proved to depend only
on a single function ( k ; t ) solely determining Q and H. Hence the dimension of both the dependent and the
independent variables is reduced by one. From the latter, the corresponding two integrodifferential equations
are lowered to a single integrodifferential equation for ( k ; t ), extended by an integral side condition on the k
dependence of ( k ; t ). In the limit →0, a partial solution to the reduced set of equations is presented in the
Appendix.
DOI: 10.1103/PhysRevE.63.026308 PACS numbers: 47.27.Gs, 47.27.Eq, 02.30.Ks
I. INTRODUCTION
In the context of fluid turbulence, a certain class of theo-
ries has been developed that is formulated in wave-number
space and relies on the truncated renormalized series expan-
sion of the nonlinear convection term. The idea originated in
the pioneering work of Kraichnan 1, Edwards 2 and Her-
ring 3,4, although these early theories were incompatible
with the Kolmogorov power law 5,6. There have been nu-
merous later theories that use the Lagrangian coordinate sys-
tem but we are concerned here with the local energy transfer
LET theory that is unique in yielding Kolmogorov behav-
ior in an Eulerian framework. An overview of the topic is
given in 6.
All these theories have in common their transport equa-
tions that constitute a nonlinear integrodifferential equation
depending on the wave number k =| k| , time t, and an addi-
tional delay time t ' . It is particularly due to the latter’s de-
pendence on t ' that numerical computations may become
forbiddingly expensive. In fact, this is the primary reason,
that such theories have almost exclusively been applied to
homogeneous isotropic turbulence. The key difficulty with
respect to the t ' dependence is that at each time step a field
depending on k and t ' has to be stored, where t ' varies
between 0 and t. Hence, as time proceeds, increasingly larger
two-dimensional fields have to be kept in memory, from the
beginning up to the current time t. For this reason, usually
only very few ‘‘eddy turnover’’ times may be computed.
In the following sections it is shown that the structure of
the LET equations is such that in the case of isotropic turbu-
lence, the dimensionality of both the dependent and the in-
dependent variables may be reduced by one. This goal is
achieved without loss of information. Hence a single field
has to be stored depending solely on k and t.
II. REDUCTION OF THE LET EQUATIONS
The LET equations in their most up to date form may be
found in 7.
A. Solution of the Q and H functional equations
In the LET theory, a tensor-valued quantity H, called the
propagator, is defined; this relates the correlation tensor
Q
ij
k; t , t ' = u
i
k, t u
j
-k, t ' , 1
to itself at different instants of time ( t , t ' where t t ' ), thus
Q
ij
k; t , t ' =H
im
k; t , t ' Q
mj
k; t ' , t ' . 2
A functional equation for H is given by
H
im
k; t , s H
mj
k; s , t ' =H
ij
k; t , t ' , 3
H
ij
k; t , t =D
ij
k , 4
where D
ij
( k) is the projection operator
D
ij
k =
ij
-
k
i
k
j
| k|
2
. 5
The equations 2 and 3 imply a certain restriction on Q
and H that will be explored below for the case of isotropic
turbulence. Under this assumption, H may be written as
H
ij
k; t , t ' =D
ij
k H k ; t , t ' . 6
Substituting Eqs. 5 and 6 in Eq. 3, we find the scalar
functional equation
H k ; t , s H k ; s , t ' =H k ; t , t ' , 7
where k =| k| . A similar scalar equation may be derived from
Eq. 2 by invoking the isotropy condition
Q
ij
k; t , t ' =D
ij
k Q k ; t , t ' . 8
Using Eqs. 8, 2 and 6 we derive the scalar functional
equation
PHYSICAL REVIEW E, VOLUME 63, 026308
1063-651X/2001/632/0263085/$15.00 ©2001 The American Physical Society 63 026308-1