PHYSICAL REVIEW E 92, 067002 (2015)
Reply to “Comment on ‘Statistical symmetries of the Lundgren-Monin-Novikov hierarchy’ ”
Marta Waclawczyk
1, 2 , *
and Martin Oberlack
1, 3, 4
1
Chair of Fluid Dynamics, Department of Mechanical Engineering, TU Darmstadt, Otto-Berndt-Straße 2, 64287 Darmstadt, Germany
2
Institute of Fluid Flow Machinery, Fiszera 14, 80-231 Gdask, Poland
3
GS Computational Engineering, TU Darmstadt, Dolivostraße 15, 64293 Darmstadt, Germany
4
Center of Smart Interfaces, Alarich-Weiss-Straße 10, 64287 Darmstadt, Germany
(Received 26 August 2015; published 23 December 2015)
In this Reply we respond to the criticism of Frewer et al. presented in the Comment on “Statistical symmetries
of the Lundgren-Monin-Novikov hierarchy” by Waclawczyk et al. [Phys. Rev. E 90, 013022 (2014)]. We discuss
physical interpretation of the statistical symmetries, and respond to criticism on the violation of the causality
principle. We derive the Lundgren-Monin-Novikov equations for a flow with boundaries. Last, we stress that our
work addressed the phenomenon of “external intermittency” (separation between laminar and turbulent flow),
and not “internal intermittency” (strong fluctuations at small scales).
DOI: 10.1103/PhysRevE.92.067002 PACS number(s): 47.27.−i, 02.20.Tw, 02.50.Cw
I. INTRODUCTION
We understand that besides less important points, the key
criticism of Frewer et al. is that the statistical symmetries
derived in Waclawczyk et al. [1] are unphysical, due to the
fact that, although the probability density functions (PDFs) f
n
transform, the sample space v
(i )
remains invariant. According
to Frewer et al. the new symmetries violate the principle of
causality.
We agree with some of the minor points of the comment
by Frewer et al., and correct some of our statements. We also
explain that the issue of boundary conditions raised in [2] is
not a concern, by deriving the PDF equation in the presence
of boundaries. Still, on the whole we do not at all agree with
the conclusions of the comment. As we will show below, the
proposed statistical symmetries are physically meaningful and
the causality principle is not violated. In the following we
would like to explain our view in more detail with a point-by-
point reply to the criticism outlined in the Comment.
II. REPLY TO: VIOLATION OF THE CAUSALITY
PRINCIPLE
In Ref. [1] we derived transformations of PDFs which
follow from the scaling and translation transformations of
multipoint correlation (MPC) equations from [3,4]. For the
PDFs these transformations have the following form:
f
∗
n
= e
a
s
f
n
+ (1 − e
a
s
)δ(v
(1)
)δ(v
(1)
− v
(2)
) ...
··· δ(v
(1)
− v
(n)
) (1)
called the “intermittency symmetry” and
f
∗
n
=f
n
+ ψ (v
(1)
)δ(v
(1)
− v
(2)
) ··· δ(v
(1)
− v
(n)
), (2)
where
ψd v
(1)
= 0, called the “shape symmetry.”
The authors of the Comment on “Statistical symmetries
of the Lundgren-Monin-Novikov hierarchy” claim that the
statistical transformations derived in Ref. [1] are unphysical
as “deterministic equations due to their spatially nonlocal and
temporally chaotic behavior induce the statistical equations,
*
martaw@fdy.tu-darmstadt.de
and not vice versa.” The same criticism appears in another
contribution of Frewer et al. [5].
Before we reply to this part of the Comment we would
first like to explain the notion of “statistical symmetry.” Such
name has in fact been used in other areas of science, e.g., in
the study of dynamical systems [6], statistical physics [7], or
even sensory coding [8], to name only a few. The common
observation of these works seems to be that even if one
particular field (or image) does not verify the symmetry, it
can be observed over a large ensemble of fields (or images).
As an example, let us consider a steady laminar Poiseuille
flow in a channel which is invariant under reflection about the
center plane y = 0. This symmetry is broken when the flow
becomes unsteady, i.e., U (x,y,z) = U (x, − y,z); however, it
is once again recovered for the ensemble averaged velocity,
〈U 〉(y ) =〈U 〉(−y ). The reason is that the instantaneous
solutions U (x,y,z) and U (x, − y,z) appear in the ensemble
with equal probabilities. In Ref. [1], by “statistical symmetry”
we mean, generally, transformations of statistics (translations,
reflections, scaling, etc.) which leave the considered hierarchy
of equations invariant, which is the case for the transformations
(1) and (2).
In Ref. [1] we claimed that constant fields (or laminar fields)
can be solutions of the governing Navier-Stokes equations
and that the transformations (1) and (2) are connected with
the presence of such fields in the ensemble. Contrary to the
criticism of Frewer et al., we did not claim that the statistical
equations induce anything on deterministic equations. We
rather observed that the deterministic Navier-Stokes equations,
due to the possibly different nature of their solutions (turbulent
or nonturbulent), caused by small changes in the initial and
boundary conditions, induced transformation of the PDF
functions (1) and (2) and respective transformations of velocity
moments.
In Ref. [1] we first considered statistical turbulence motion
in free space without boundaries. If f
n
is a PDF of a turbulent
flow [cf. Fig. 1(a)], both transformations, first (1) and next (2)
with ψ = δ(v
(1)
− U
0
) − δ(v
(1)
) transform f
n
into a PDF of
an intermittent flow with a certain share of constant fields in
the ensemble
f
∗
n
= e
a
s
f
n
+ (1 − e
a
s
)δ(v
(1)
− U
0
) ··· δ(v
(n)
− U
0
) (3)
1539-3755/2015/92(6)/067002(5) 067002-1 ©2015 American Physical Society