World Journal of Mechanics, 2013, 3, 224-229
doi:10.4236/wjm.2013.34022 Published Online July 2013 (http://www.scirp.org/journal/wjm)
Turbulence Mechanics in Progress—From Classical to
Postclassical*
Jaak Heinloo
Marine Systems Institute, T al li nn University of Technology, Ta llinn, Estonia
Received May 22, 2013; revised June 22, 2013; accepted June 30, 2013
Copyright © 2013 Jaak Heinloo. This is an open access article distributed under the Creative Commons Attribution License, which
permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
ABSTRACT
This paper explains the basic steps form the classical turbulence mechanics (CTM) to the postclassical turbulence me-
chanics (PCTM). When the CTM stems from the characterization of the motion states in the infinitesimal surround ings
of the flow-field points by the flow velocity at these points then the PCTM complements this characterization by the
curvature of the velocity fluctuation streamlines passing these poin ts. The complementation is formalized by the inclu-
sion of the curvature of the velocity fluctuation streamlines to the arguments of the probability distribution of the mo-
tion states in the infinitesimal surroundings of the flow field points. The most radical physical outcome of the realized
formalism is the characterization of the turbulence viscosity properties by two types of turbulence viscosity against only
one shear viscosity within the CTM.
Keywords: Fluids Mechanics; Turbulence; Mathematical Modeling
1. Introduction
The classical turbulence mechanics (CTM) originates
from J. Boussinesq [1] and O. Reynolds [2]. It identifies
the turbulence with a chaotic form of fluids motion and
sets the turbulence description to the Reynolds-averaged
Navier-Stokes equation (RANS, called also the Reynolds
equation), gathering the turbulence effects into the sym-
metric turbulent stress tensor. The applied to this tensor
closure assumption reduces its specification to the deter-
mination of turbulent shear viscosity coefficient, which
turns the modeling (parameterization) of this coefficient
to the synonym of the CTM. Different parameterization
models of this coefficient from the semi-empirical mod-
els [3,4] to more contemporary turbulence models [5,6]
have been proposed to solve this task.
Despite the similarity of the setup of the CTM to the
setup of the classical fluid mechanics of viscous fluids
(CFM)—both are formalized within the law of momen-
tum with the symmetric stress tensor parameterized (for
incompressible fluids) by just one (shear) viscosity coef-
ficient—there is still a substantial difference b etween the
two. While the CFM grounds the symmetry of the mo-
lecular stress tensor on the constituted absence of the
energy-carrying internal rotational degrees of freedom of
the medium, then the CTM constitutes the symmetry of
the (Reynolds) stress tensor thus ruling out the energy-
carrying internal rotational degrees of freedom of turbu-
lent media. Insofar as this kind of the medium rotational
degrees of freedom in turbulent media are foreseen by
another classical conception of turbulence originating
from L. F. Richardson [7] and A. N. Kolmogorov [8]
(henceforth, the RK concep tion which stresses the turbu-
lence order reflected in its hierarchic eddy structure up-
hold by the cascading en ergy transfer through the system
of eddies of different scales with the large-scale eddies
obtaining their energy immediately from the average
flow) the grounding statements of CTM and the RK con-
ception prove contradicting.
Unlike the CTM, the po stclassical turbulence mechan-
ics (PCTM) [9,10] treats the problem of turbulence in the
context of physical doctrine of turbulence (PDT) [11].
The PDT sets the formulation of turbulence mechanics
(TM) into a systemic context [12,13], esteems the RK
conception and mandates the formulation of the TM wi-
thin the principles of statistical physics and continuum
mechanics [14,15]. The PCTM meets this mandate start-
ing from modifying the very origin of the turbulence de-
scription setup. The modification consists in distinguish-
ing the states of motion in infinitesimal surroundings of
the flow-field points by the curvature of the velocity
*The formulation of the PCTM was supported by the grants No 3157,
5009 and 9381 of the Estonian Science Foundation.

J. HEINLOO 225
fluctuation streamlines passing these points, formalized
by the inclusion of the curvature of the velocity fluctua-
tion streamlines to the set of arguments of the probability
distribution of the motion states in the infinitesimal sur-
roundings of the flow field points. Due to the modifica-
tion introduced the PCTM appears substantially different
from the CTM in its outcome. The most prominent dif-
ference is the asymmetry of the turbulent stress tensor
providing the tu rbulent media with two types of viscos ity
against just one viscosity within the CTM. The difference
ends up in not only diverging formalisms but also dem-
onstrates the insufficiency of the boundary conditions
imposed within the CTM to determine the flow situation
uniquely. Moreover, as shown in the listed in [10] appli-
cations of the theory of rotationally anisotropic turbu-
lence, the complementation of the setup of the PCTM
with the appropriate closure assumptions, the additional
introduced viscosity of turbulent media appears to be
much more essential to the description of the related ph y-
sical processes than the turbulent shear viscosity.
Being grounded on the enlarged physical background
the PCTM proves comprising the CTM as its particular
case. By comprising the CTM it comprises also the
Large-Scale-Eddy (LES) turbulence modeling [16,17],
diverging from the conventional setup of the TM by ap-
plying the averaging procedure to the small-scale turbu-
lence constituent only. The PCTM pays respect also to
several former ideas which have been remained outside
the general trends of formulation of the TM. In addition
to the RK conception it refreshes the idea of G. Mattioli
[18], who first suggested the inclusion of the equation of
moment-of-momentum to the setup of turbulence de-
scription, as well as some recent ideas like the relation of
the turbulent media to the class of micropolar fluids [19-
21] and the ideas applied in the structure-based turbu-
lence models [22,23].
The current paper is aimed to explain the PCTM in
simple terms and graphs avoiding complicated mathema-
tics and to call up all interested parties, including those
who see their mission in defending the turbulence de-
scription standards comprised in the CTM, to critically
analyze the new situation in the TM altered by the for-
mulation of the PCTM. The discussion starts (Section 2)
from decomposition of a velocity fluctuation to its con-
stituents correlating and not correlating with the curva-
ture of the velocity fluctuation streamline. Section 3 dis-
cusses the respective situation in terms of energy. Section
4 comments on some problems related to the declared in
the PCTM asymmetry of the turbu lent stress tensor. Sec-
tions 5 and 6 sum up the main points of the novelty in-
troduced by the PCTM into the turbulence description
and address the inferences drawn from available data
confirming experimentally the grounding assumptions of
the PCTM.
2. Adjusted Representation of the Turbulent
Velocity Field
The PCTM starts its formalism from the classical repre-
sentation of the turbulent flow velocity in the form v


vuv, (1)
where
uv (2)
in which the an gular brackets denote statistical averag ing
and

v denotes the fluctuating constituent of velocity.
Let now the probability density, specifying the averaging
in (2), be detailed as


,fvk , where is the curva-
ture of the velocity fluctuation streamline passing a flow
field point. By the definition
k
s
 ke in which

e

 vvand
s
is the length of the curve of

v
streamline passing the flow field point. The specification
distinguishes the flow situations in the infinitesimal sur-
roundings of the turbulent flow field points by the cur-
vature of the velocity fluctuation streamlines passing
these points (Fig ure 1).
Representing


fv,k as



 
12
fffv,kv kk, (3)
where



12
fffvkv,kk
and


2dff
kv,kv,
we have (Figure 2)


+vvv

, (4)
where (and henceforth) the over-bar denotes the averag-
ing by


1
fvk. It is evident that in (4) is statisti-
cally independent from

v

v and . Notice, that the
CTM constitutes the probability density of the averaging
procedure in (2) specified as
k
()
f
v, which declares



vv and 0



v.
dV
k
 v
Figure 1. Illustration of distinguishing the flow situations in
infinitesimal surrounding of a flow field point (dV) by the
curvature k of the velocity fluctuation streamline passing
this point: the flow situations with the same

v but oppo-
site k prove different.

J. HEINLOO 226
 v

v
 v
v
v
u
Figure 2. Decomposition of the flow velocity in turbulent
flow field accounting for the curvature of the velocity fluc-
tuation streamlines passing the flow-field points.
3. Decomposition of Turbulence Energy
3.1. Primary Decomposition
In terms of energy the decomposition of velocity fluctu a-
tion in (4) reads as
12
K
KK, (5)
where 2
1
2
K
v is the (total) turbulence energy, while
2
1
12
K
v  and 2
21
2
K
v

 are natural to interpret
as the energies of the small-scale and the large-scale tur-
bulence constitu ents.
The energy graph in Figure 3 displays the energy
situation assumed within the CTM and the situation cor-
responding to (5), for the cascading energy transfer from
the average flow with energy 2
1
2
u
K
u to the thermal
energy through the turbulent phase of motion repre-
sented by the energies 1
U
K
and 2
K
. Vertical arrows in
Figure 3 mark the levels of energy ceding and receiving,
,iji j
Qu

 denotes the work realizing the energy trans-
fer from the average flow either to
K
or to 2
K
, Q

denotes the work realizing the energy transfer from 2
K
to 1
K
and

denotes the dissipation rate of turbulence
energies
K
or 1
K
.
3.2. Secondary Decomposition
The secondary decomposition of the turbulence energy
stems from the definition of the kinematical-dynamical
pair of the Euleri an fl o w -field characteristics [10]
  vk and   MvR (6)
where 2
Rkk is the curvature radius-vector corre-
sponding to . The defined
k

(henceforth, the gyroc-
ity) has the sense of average angular velocity of rotation
of medium particles at a flow-field point in respect to the
random curvature centres of the velocity fluctuation
streamlines passing this point, and (henceforth, spin)
has the sense of average density (per unit mass) of the
moment of fluctuating constituent of momentum w
M
U
Chaos
Order
1
K
2
K
u
K
K
Q

Q

Q
Primar y decomposition (PD)
CTM
Figure 3. Primary decomposition (PD) of the turbulence
energy: the energy graphs of turbulent flow field for the
situation assumed within the CTM and for the situation
corresponding to (5).


vk and   vR  , (7)
explaining the gyrocity and the spin as the characteristics
of velocity fluctuation constituent  v only.
Using (1), (4)-(7) we have for
K
and2
K
0
K
KK

, (8)
0
22
K
KK

 (9)
and
0
21 0
K
KK, (10)
where 1
2
K

Μ, (11)

0
21
2
K

vR vk

(12)
and

01
2
K

vR vk

. (13)
Figure 4 outlines the situation corresponding to (5), (8)-
(10) as an extension of the situation shown in Figure 3
for the energy of average flow representing just one
source of turbulence energy. Notice that:
(a) the energies 1
K
and 0
2
K
characterize the mo-
tions of different scales of the same order while
K

and 0
2
K
characterize the motions of the same scale of
different order;
(b) the decomposition of into the sum of 1 and
2, realizing the energy transfer from the average flow
to the turbulence cons tituents of differen t orde r, eviden ce
about the presence of two types of turbulent viscosity
against just one viscosity within the CTM;
Q Q
Q
(c) the nature of the work 1, connecting the transla-
tory degrees of freedom of medium motion (realised in
the form of the average flow) and its rotational degrees
of freedom (realized in the form of rotation characterized
by the gyrocity and the sp in), specifies an ad ditional vis-
cosity as related to the antisymmetric constituent of the
Q
ith
R
standing for the (random) arm of the moment. Let us
note, that ss  v is statistically independent fm k
(and f oro rom a
R
), the expressions (6) can be written also as

J. HEINLOO 227
Q
0
2
K
PD
U
1
K
2
K
u
K
0
K
Q

1
Q
Q

21
QQQ
K

Q
Secondary decomposition
Figure 4. Secondary decomposition of turbulence energy:
the energy graphs corresponding to the PD and to the
secondary decomposition.
turbulent stress tensor, which abolishes the notion of
“evident symmetry” of the turb ulent stress tensor hold ing
within the CTM;
(d) for the cascading character of energy transfer the
work vanishes and Figure 4 explains the role of the
work in transforming the motion organization
quality to the form allowing its reception on the level of
small-scale turbulence constituent. (For a more detailed
discussion of the specifics of the cascading process con-
sider [24-26].)
2
Q
Q
(e) finally, all representations (4), (8)-(10) are direct
corollaries of the adopted specification of the applied
averaging and of definitio ns (6) (or (7)) while the energy
transfer directions indicated in Figure 4 correspond to a
typical but not to the only possib le situation.
4. The Setup of Description of Turbulent
Flows
The fundamental inference from definitions (6) (or (7)) is
that the turbulent media is related to the class of mi-
cropolar fluids [27-33]. The relation suggests the inter-
connection of the gyrocity and the spin




M by
2
M, (14)
where defines the characteristic average scale of mo-
tion, and delegates the description of motion to the sys-
tem of two equations—the Reynolds equation (with the
asymmetric tensor of turbulent stresses) and the equation
for the spin .

M
Referring for the details of the setup of description of
turbulent medi a with th e non-v an ish ing sp in to [9 ,10 ], we
first accent here on the inherent to this description asym-
metry of turbulent stresses—th e most conflicting point in
the relation between of the CTM and the PCTM—start-
ing from the expression for the dual vector to the anti-
symmetric constituent of turbulent stresses kkij ij e



,
where kij denote the components of the Levi-Civita
tensor, expressed as [9]
e
,kkijsij
evvR 
 
Denoting the components of the velocity fluctuation
constituent along
R
as ,
j
sjs
vvR



R from (15) we have


kkij
evv 
  Rj i
, (16)
or in the vector form
σ 



R
vv . (17)
Accounting for (4) and th e perpendicularity of

v and
we have from (17), that R
σ 

 
R
vv. (18)
Expression (18) explains the an tisymmetric constitu ent
of stresses describing the average momentum flux in
direction of
R
either accelerating of decelerating the
eddy rotation. Notice, that the velocity fluctuation con-
stituent

v, playing a crucial role in definitions of

and , does not contribute to . The second accent
concerns the work in Figure 4, represen ted in terms
Mσ
1
Q
of as σ1 Q

σω, where 1
2
ωu 1
2

ωu
is the vorticity. For the relation of closure for in the
form
σ
σ4


()

ω

[9,10], where

denotes the
coefficient of turbulence rotational viscosity, it is ev ident
that, dependent on the relative values of and ω, the
work 1 may be eith er positive or nega tive, i.e. the me-
dium rotational viscosity manages to explain the eddy-to-
mean energy transfer without introduction of notion of
“negative viscosity” [34] or without ascribing the actual
3D nature of turbulence with 2D properties. The third
accent is related to the effect of rotation of frame on the
medium turbulence, which, though not influencing ,
influences 1. The latter explains the frame rotation as a
potential cause of th e eddy-to-mean energy conversion.

Q
σ
Q
5. Discussion
The PCTM realizes a modification of the TM setup
originating from complementation of characterization of
the motion states in the infinitesimal surroundings of the
flow-field points by the curvatur e of the velocity fluctua-
tion streamlines passing these points. Th e modification is
undertaken to distinguish the flow field states in the in-
finitesimal surroundings of the flow field points depend-
ent on the curvature of the velocity fluctuation stream-
lines passing these points. The necessity for the comple-
mentation is one implication of critical analysis of the
situation in the TM from the point of view of the wid-
ened physical-historical background of the turbulence
problem specified as the PDT [11]. The analysis em-
braces the CTM together with some ideas incompatible
with the CTM. Within these ideas the leading positions
belong to the RK conception about the cascading eddy
structure of turbulence and to the idea about the turbulent
media pertaining to the class of micropolar fluids [19-21]
s
(15)

J. HEINLOO 228
(as well as to the related to it earlier idea of G. Mattioli
[18], first suggesting the inclusion of the equation of
moment-of-momentum to the setup of description of tur-
bulent flows). The analysis displays the statement of the
CTM about the evident symmetry of the turbulent stress
tensor erring against the principles of continuum me-
chanics [15], relating the solution of the symmetry prop-
erties of the stress tensor to the context of specification
of the medium internal rotational degrees of freedom.
Within the shortcomings of th e CTM note also its inabil-
ity to propose ph ysically correct explanation to the eddy-
to-mean energy conversion, explained within the PCTM
as the act of turbulence rotational viscosity neglected
within the CTM. The CTM also does not distinguish the
turbulence properties in rotating and non-rotating frames
which by now is verified observationally.
Let us underline that the PCTM does not “discover”
the additional (rotational) viscosity of turbulence—this
type of turbulence viscosity has been foreseen and de-
scribed in a sufficiently complete form by the RK con-
ception—but merely removes the obstacle from the ex-
plicit inclusion of this fundamental property of turbu-
lence into the setup of the TM. Let us highlight also, that
applying all classical motion integrals—of momentum,
of moment-of-momentum and of energy—the PCTM
formulates the turbulence description in mechanically
closed form and as such completes the formulation of the
TM.
We conclude the comments on the PCTM referring to
the paper [35], which utilized the data available within
the Global Drifter Program to estimate the gyrocity and
the spin immediately from observations. The estimated
gyrocity and spin provide the grounding declaration of
the PCTM about the non-vanishing gyrocity and spin of
the turbulent flow with the sense of experimental fact.
Finally, the PDT [11] postulates the conditions of for-
mation of probability distribution properties of momen-
tary states of motion determined as the state of motion
fixed in terms of the TM. These conditions are specified
within the CTM and within the PCTM differently. The
commented role of the TM is emphasized here to stress
the scientific merit of the TM wider than a particular
turbulence description in average terms.
6. Conclusion
The PCTM mandates a critical analysis of the results
following from the CTM. It also mandates the planning
of new tasks and research projects from the position of
the PCTM. The latter mandate is addressed to the initia-
tors of new projects rather than hug e number of scien tists
participating in turbulence-related applied projects and
following firmly established standards. This mandate is
also addressed to the people educating new generation of
specialists in the field of flu id mechanics.
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