Unformatted text preview: FUNDAMENTAL AND CONCEPTUAL
ASPECTS OF TURBULENT FLOWS
Arkady Tsinober Professor and Marie Curie Chair in Fundamental and Conceptual Aspects of Turbulent Flows
Institute for Mathematical Sciences and Department of Aeronautics, Imperial College London
Lectures series as a part of the activity within the frame
Lectures
as
of the Marie Curie Chair “Fundamental and Conceptual
Marie
Chair Fundamental
Aspects of Turbulent Flows”.
Aspects We absolutely must leave room for doubt or there is no progress and no learning.
There is no learning without posing a question. And a question requires doubt...Now
the freedom of doubt, which is absolutely essential for the development of science,
was born from a struggle with constituted authorities... FEYNMANN, 1964
FEYNMANN LECTURES IXX PHENOMENOLOGY
AND RELATED
What is phenomenology? Cascade. Is cascade a dynamical process? Is
it a kind of trivial consequence of nonlinearity and decompositions?
Is there cascade in physical space? Is "cascade" in genuine turbulence
well defined? Counterexamples. Is "cascade" of passive objects
meaningful? Nature of dissipation  is it (un)important?
I soon understood that there was little hope of developing a pure, closed theory,
and because of the absence of such a theory the investigation must be based on
hypotheses obtained in processing of experimental data... KOLMOGOROV 1985
KOLMOGOROV Correlations after experiments done is bloody bad.
Correlations
Only prediction is science, FRED HOYLE, 1957, The Black
FRED
Cloud, Harper, NY. Phenomenology  The branch of a science that
Phenomenology
classifies and describes its phenomena without any
attempt at explanation, WEBSTER'S NEW WORLD DICTIONARY, 1962
WEBSTER'S
…even wrong theories may help in designing
machines, FEYNMAN, 1996 (i.e. the right results for the wrong reasons, who cares?)
In our present state of understanding, these simple
In
models will be based, in part on good physics, in part
on bad physics, and in part on shameless
phenomenology, LUMLEY, 1992
LUMLEY,
Our present understanding of anything turbulent is
Our
at best phenomenological, SIGGIA, 1994
does such a thing exist ?
SIGGIA
does There is no definition of what is phenomenology of turbulent
There
flows. In a broad sense, it can be defined by a statement of
impotence: it is almost everything except the direct
experimental results (numerical, laboratory and field)
and/or results (a very small set indeed), which can be
obtained from the first principles, e.g. NSE. Phenomenology
involves use of dimensional analysis, variety of scaling
arguments, symmetries, invariant properties and various
assumptions, some of which are of unknown validity and
obscured physical and mathematical justification. Thus in the broad sense phenomenology includes also
Thus
most of the semiempirical approaches and turbulence
modeling. Doing all this requires insight into the basic (!)
physics of turbulence, hard experimentation and painful
efforts of interpretation. The latter may be quite
problematic, especially in models having enough free
parameters to guarantee the right results not necessarily
for the right reasons. CONVENTIONAL PHENOMENOLOGY AS IT APPEARS IN THE BOOK by ENRICO FERMI, Notes on Thermodynamics and Statistics, The University of Chicago Press, Chicago and London Midway Reprint Edition, 1988; pp. 181182. (1951 lectures) CASCADE WHAT IS IT ?
IS THERE CASCADE
IN PHYSICAL SPACE ? 66
THE FUNDAMENTAL EQUATIONS
oh. 4/8/0
On the other hand we find that convectional motions are hindered by the formation of small eddies resembling those due to
dynamical instability. Thus 0. K. M. Douglas writing of observations from aeroplanes remarks : "The upward currents of large
cumuli give rise to much turbulence within, below, and around the clouds, and the structure of the clouds is often very
complex." One gets a similar impression when making a drawing of a rising cumulus from a fixed point; the details change
before the sketch can be completed. We realize thus that: big whirls have little whirls that feed on their velocity, and little
whirls have lesser whirls and so on to viscosity— in the molecular sense…
Thus, because it is not possible to separate eddies into clearly defined classes according to the source of their energy; and as
there is no object, for present purposes, in making a distinction based on size between cumulus eddies and eddies a few metres
in diameter (since both are small compared with our coordinate chequer), therefore a single coefficient is used to represent the
effect produced by eddies of all sizes and descriptions. KOLMOGOROV 1941A
KOLMOGOROV Do you see here a cascade?
Do …. even wrong theories may help in designing machines.
(FEYNMAN, 1996) Feynmann R., 1996 Lectures on Computation, AddisonWesley.
(FEYNMAN,
Wesley. Or here?
Or …. even wrong theories may help in designing machines.
(FEYNMAN, 1996) Feynmann R., 1996 Lectures on Computation, AddisonWesley.
(FEYNMAN, One gets an impression of little, randomly structured and
One
distributed whirls in the fluid, with the cascade process
consisting of the fission of the whirls into smaller ones, after
the fashion of the Richardson poem. This picture seems to be
drastically in conflict with what can be inferred about the
qualitative structure of highReynoldsnumber turbulence from
laboratory visualization techniques and from plausible
application of Kelvin circulation theorem ... How a theoretical
attack on the inertialrange problem should proceed is far
from clear. KRAICHNAN, 1974. On Kolmogorov's inertialrange theories, J.. Fluid Mech., 62, 305On
J
330. ... the idea of conservative inertial cascade local in scale size
is consistent prima facie, provided that the actual statistics do
prima
not differ strongly from gaussian (!). It is another, and
unsettled, matter to establish that K41 or a related theory
actually describes what happens in NS flows, KRAICHNAN,
1991. Turbulent cascade and intermittency growth, Proc. R. Soc. Lond., 434, 6578
Proc.
Lond The notion that turbulent flows are hierarchical and involve
The
entities... of varying sizes is a common idea... This common notion
underlies the concept of cascade, the third key element of turbulence
theory FRISCH AND ORSZAG, 1990. All this cascade in Fourier space is a dream of linearized
physicists, BETCHOV, 1993.
BETCHOV
The three examples (jet, boundary layer, and wake)... show that
there is something wrong with this idea (the Richardson poem. In each
case turbulence begins at small scales and grows larger: not the other
way around, GIBSON, 1996.
The conceptual picture is that of a cascade organized by wall distance
The
and by eddy size, where energy is transferred to smaller scales at any
given location, and to larger ones away from the wall, JIMENEZ, 1999.
JIMENEZ,
This suggests that the Komogorov cascade process is basically incorrect,
albeit an excellent approximation, SHEN AND WARHAFT, 2000
SHEN
Much more troubling is the idea that there might not exist an energy
cascade. TENNEKES 2004 (private communication) The RichardsonKolmogorov cascade picture was formulated in physical
The
space and is used frequently without much distinction both in physical
and Fourier space, as well as some others. It was von NEUMANN 1949
NEUMANN
(see also ONSAGER 1949 who introduced the term cacasde in 1945) who stressed that
who
this process occurs not in physical space, but in Fourier space: ... the
the
system is "open" at both ends, energy is being supplied as well
dissipated. The two "ends" do not, however, lie in ordinary
space, but in its Fouriertransform...The supply of energy occurs at the
macroscopic end  it originates in the forced motions of macroscopic
(bounding) bodies, or in the forced maintenance of (again macroscopic)
pressure gradients. The dissipation, on the other hand, occurs mainly at the
microscopic end, since it is ultimately due to molecular friction, and this is
most effective in flowpatterns with high velocity gradients, that is, in small
eddies... Thus the statistical aspect of turbulence is essentially that of transport phenomenon (of energy)  transport in the
Fouriertransform space. That is, the nonlinear term in the NavierStokes equation
That
redistributes energy among the Fourier modes not ‘scales’ as
is frequently claimed, unless the ‘scale’ is defined just as an
inverse of the magnitude of the wavenumber of a Fourier
mode, which is not easy for everybody to swallow. A natural
question is then what does the nonlinear term in physical space
do? Is energy (and not only energy) transferred from large to
small scales in physical space? The answer to the last question
depends on the definition of what is a ‘scale’ in physical space. First, we recall that that there is no contribution from the nonlinear
First,
term in the total energy balance equation (and in a
homogeneous/periodic flow it's contribution is null in both the total
and the mean), since the nonlinear term in the energy equation has the
form of a spatial flux, ∂{...}/∂xj. In other words the nonlinear term
redistributes the energy in physical space, but does it do more than
that? It is straightforward to see that in a statistically homogeneous
turbulent flow the mean energy of volume of any scale (Lagrangian
and/or Eulerian) is changing due to external forcing and dissipation
only  there is no contribution in the mean of the nonlinear term,
which includes the term with the pressure. That is, if one chooses to define a ‘scale’, l, in physical space as a fluid
That
(or a fixed) volume, say, of order l³, then in a statistically
homogeneous flow there is no cascade in physical space in the sense
that, in the mean, there is no energy exchange between different
scales. This happens because the nonlinear term in the energy
equation has the form of a spatial flux, ∂{...}/∂xj, i.e. there is
conservation of energy by nonlinear terms. In other words, the
nonlinear term redistributes the energy in physical space if the flow is
statistically nonhomogeneous. So, generally, it is a misconception to
misconception
interpret this or any other process involving spatial fluxes, ∂{...}/∂xj
(e.g. momentum flux), as a ‘cascade' in physical space. FRISCH 1995
(pp. 77,78) defines a quantity which he calls (!) physicalspace energy flux At page 79 he ends up with the statement
This is correct, but this is not an energy flux relation in physical space
as just using the term physicalspace energy flux for the quantity
does not turn it into such. Neither
does help the (very useful fact ) that this quantity is related the energy
flux ΠK through the wavenumber K
appearing in the scale by scale energy budget relation in Fourier space On the other hand, in a statistically stationary state ∫F·udτ = ∫εdτ,
On
i.e. energy input, which is associated with large scales, equals
dissipation, which occurs mostly in the small scales. Is there a
contradiction? Does the equality ∫F·udτ = ∫εdτ mean that the energy
should be somehow ‘transferred' from large to small scales via some
multistep process? Not necessarily  for example, two big neighboring
eddies can dissipate energy directly through encounters with each other
at small scale  much smaller than their own scales. Such a process still
will look in Fourier space as continuous energy transfer from modes
with small to modes with large wave numbers. CASCADE VERSUS
DECOMPOSITIONS/REPRESENTATIONS Fourier transform ambiguity in turbulence… TENNEKES 1976
I think that the k space decomposition does actually obscure
the physics. MOFFATT, 1990
MOFFATT
See also LIEPMANN, 1962; LOHSE AND MÜLLERGROELING, 1996
LIEPMANN
LOHSE The resolution of the apparent contradiction lies in clarifying the meaning of the
The
term `scale', which mostly is understood as an inverse of the wavenumber
magnitude in a Fourier representation, and what is the meaning of ‘transfer of
energy (or whatever) from large to small scales’ in physical space. This issue is
directly related to the decomposition/representation of the turbulent flow field.
Indeed, the reason for the above result on the absence of energy exchange between
different scales in physical space is because no decomposition is involved in the
above definition of ‘scale'. Any decomposition (be it in physical space, Fourier or
any other) brings the ‘cascade' back to life. For example, there are various ways of
filtering the flow field widely used in large eddy simulations. However, one of the
problems with decompositions is that the nonlinear term redistributes the energy
among the components of a particular decomposition in a different way for different
decompositions, i.e. the energy exchange/transfer (and not just energy), generally is
decomposition dependent. Therefore one may ask whether quantities like energy flux are well defined. In
Therefore
other words the term ‘cascade’ corresponds to a process of interaction/exchange
of (not necessarily only) energy between components of some particular
decomposition/representation of a turbulent field associated with the nonlinearity
and the nonlocality of the turbulence phenomenon, two of the three N's:
nonlinearity, nonlocality and nonintegrability, which make the problem so
impossibly difficult . On the other hand, energy transfer, just like any physical
process, should be invariant of particular decompositions/representations of a
turbulent field. In this sense Kolmogorov's choice of dissipation (and energy
input) are well defined and decomposition independent quantities, whereas the
energy flux is (generally) not, since it is decomposition dependent. After all
Nature may and likely does not know/care about our decompositions.
our decompositions. ‘Cascade' arising from a decomposition of the flow field viewed
as a process of exchange of energy, momentum, etc. between
the components of this decomposition is a dynamical process.
This should be distinguished from ‘cascading processes’
resulting from a decomposition of some quantity, e.g.
dissipation, usually of its surrogate (∂u1/∂x1)², obtained from
experimental signals. The former is a dynamical process,
The
whereas the latter is a representation characterizing some
aspects of the spatial and/or temporal structure of some flow
characteristics. ‘In other words, ‘structure’ is not synonymous of ‘process': it
is the result of a process. Therefore, generally it is impossible
to draw conclusions about the former from the information
about the latter, though this is done quite frequently. For
example, simple chaotic systems with few degrees of freedom
only produce also ‘fine structure', possessing a continuous
spectrum with a multitude of interacting modes. Of course,
such a signal can be also cast in a multiplicative representation,
but there is no ‘cascade' whatsoever. COUNTEREXAMPLES COUNTEREXAMPLES IN
TRANSITIONAL FLOWS
Abrupt transition Pipe. Entrainment. Vortex breakdown. Turbulent spots.
Blow up instabilities. Bypass instabilities and transitions. In all laminar fluid becomes turbulent in ‘no time’ without any cascade whasoever. ABRUPT TRANSITION The transition between laminar and turbulent flows at the beginning
ing
and end of the turbulent region is abrupt relative to its duration.
and ROTTA, J. C.(1956) Experimenteller Beitrag zur Entstehung
ROTTA, turbulenter Strömung im Rohr, Ing. Arch., 24, No. 4, 258–281.
Ing.
No. ABRUPT TRANSITION
The transition between laminar and turbulent flows at the beginning
ing
and end of the turbulent region is abrupt relative to its duration.
and The transition is indeed frustratingly abrupt s
l
u
g
s Wygnanski & Champagne 1973 Durst & Unsal 2006
Durst p
u
f
f
s ABRUPT
TRANSITION A vortex ring impinging a
wall becomes turbulent in
no time as it approaches
the wall turbulent
rotational PTF  ENTRAINMENT
Mount St. Helen volcano
on 18 May 1980 The laminarturbulent
“interface” is
sharp so that
fluid particles
laminar
irrotational A turbulent jet from
testing a Lockheed
rocket engine in the
Los Angeles hills (note the Lagrangian
aspect !) “are found” abruptly
in a turbulent
environment COEXISTENCE OF
COEXISTENCE
LAMINAR AND
TURBULENT
REGIONS IN THE
SAME FLOW
SAME Vortex
breakdown Small scales are not necessarily created from large scales via a stepwise turbulent
‘cascade’: it can be bypassed, and most probably is so in turbulent flows, for
example via broadband instabilities with highest growth rate at short wavelengths
(PIERREHUMBERT AND WIDNALL, 1982) or some other approximately single step
process (BETCHOV 1976, DOUADY ET AL. 1991, OTT 1999; SHEN AND
WARHAFT 2000, VINCENT AND MENEGUZZI 1994). The problem goes back to
TOWNSEND 1951: ...the postulated process differs from the ordinary
TOWNSEND
...the
type of turbulent energy transfer being fundamentally a single
process An important example, is the complicated structure of vorticity (and passive
An
objects) with power law spectrum, (multi)fractality and significant variations down
to very small scale that can be produced by a single instability at much larger scale
without any ‘cascade’ of successive instabilities arising in a simple fluid flow via a
single instability only(!), OTT 1999. ANTICASCADE ARGUMENT BY
ANTICASCADE
CHORIN 1994 Vorticity and Turbulence, Springer, pp. 5557
CHORIN Do you see anything
wrong with Chorin’s
Chorin
argument ? So you have a kind of home work: have a
So
look and we can discuss this later.
look IS CASCADE LOCAL?
IS
# If cascade picture makes sense, one probably must have a complex
interplay between distant shells CHORIN 1994
# One of the key ideas in turbulence theory is that of locality of cascades.
The essential idea is that only modes near a given scale contribute to
transfer across that scale. The concept has a central importance in the subject
because locality is one of the conditions invoked to justify universal
statistics at small scales. If excitations are transferred by a chain of
chaotic steps from scale to scale, then the conditions at the large scales
may be forgotten and only the interactions at a sequence of adjacent scales
will determine the characteristics at small scales. EYINK 2005.
EYINK In view of the existing evidence (part of which was presented in previous lectures) the latter
(part
statement looks somewhat outdated and seems to contradict the evidence. As the
paper in question (Eyink, G.L. (2005) Locality of turbulent cascades, Physica, D207, 91116) is pretty recent
Physica
and claims mathematical rigor some comments are in place. The paper is based on
The
The regularity that we shall assume for Euler solutions in the high
The
Reynolds number limit is the Hölder type that was conjectured theoretically
[1,2] (ONSAGER 1949 AND PARISI & FRISH 1983) and is observed (in a spacemean
sense) experimentally [3] (ANSELMET ET AL 1984). In other words the whole outcome is based at best on a conjecture. More serious is
that it is not at all clear how the properties of Euler solutions can be observed
experimentally (if at all), in general, and at large Reynolds numbers, in particular.
All the experiments the author refer to (and all the other) are done at pretty low
Reynolds numbers at which there are no very long inertial ranges to allow for local
transfer to dominate (a crude estimate is that one needs about six decades for this,
i.e. Taylor microscale (!) Reynolds numbers exceeding 10⁸, which cannot be
reached in any experiment in the forseeable future). Thus the scaling exponents
obtained from experiment as in ANSELMENT ET AL 1984 and similar ones (and, of
ANSELMENT
course, numerical) later are not necessarily those which can be expected at very
large Reynolds numbers. Nobody seems to be sure at all what can be expected at
very large Reynolds numbers. For example, there is a possibility (and a number of publications clearly
For
indicating that) in the limit a pure Kolmogorov scaling will recover (it happens
pretty slowly) and that the present experimental and, of course, numerical
observations are just a finite Reynolds number (and possibly finite size) effect*.
This casts serious doubts about the multifractality hypothesis** * But this does not necessarily mean that nonlocal effects  even if they are not dominant in the sense used in the paper in
question  may not spoil the universality of small scales at any whatever large Reynolds numbers.
** In fact, the main feature of the multifractal hypothesis (assumption, see Frisch 144) is that it's authors ARE back with
universality postulating two universal exponents (i.e. a whole range) and a universal function postulated to be independent of
the mechanism of production of the flow). Unfortunately, in contrast with K41, the multifractal model (like many other
intermittency models) is an arbitrary construction in the sense that it lacks dynamical motivation (so far) in general and, with
respect to the postulated multifractal universality in particular. (Recall the question Where is the physics? by L. P. Kadanoff
1986, Physics today, 39, 6.). Since structure functions exhibit some scaling this is all necessary to make the multifractal
hypthesis "valid". indeed: Of course, having that much assumed (i.e. a whole range of exponents and a function) it is really
easy to fit to this frame almost any experimental/DNS data and whatever. In other words, though multifractality was designed
to ‘explain' the anomalous scaling, intermittency, etc., it is in fact another description and way of looking at the data as some
people do at finite Reynolds numbers. The assumption (not made in the paper) that at low Reynolds numbers the large scales
The
(whatever this means) behave the same way as at very large Reynolds numbers does not help
much: on the contrary this assumption meets lots of difficulties in view of the existing (pretty
massive) evidence that at least at all achievable (so far) Reynolds numbers the nonlocal effects
are essential.
There is a problem concerning the relation of the exponents defined in the paper (based on Lp norm on the space domain of the flow) in claiming that these are related to the usual
these
(absolute) structurefunction exponents by αp = ζp/p. The αp and ζp/p are pretty
different objects. A related (technical, but not only) question in the definitions of exponents like
αp is that first the limit ν → 0 is taken (assuming that it exists) and then r→0 whereas ζp’s
are determined not really in this way.
The above is related to the claim: It is important that we have established
It
locality for individual flow realizations without statistical averaging... This brings two questions. First, the exponents ζp are obtained essentially by statistical
This
averaging. Second, it is it is not clear in what sense the power laws exist for individual flow
realizations. The paper ends up with the statement:
The On the other hand, our estimates, like those from
closures, show that the cascade is, as Kraichnan
expressed it, only "asymptotically local" and
"diffuse". The rate of vanishing of the nonlocal
contributions with increasing scaleseparation are quite
slow powerlaw decays. Thus, very long inertial
ranges might be required for local transfer to
dominate. As mentioned very long inertial ranges (if the notion of inertial
range is meaningful at all) are not achievable in the forseeable
future. Thus even if asymptotically (as ν→0 ) the claims of the
author are correct the cascade (whatever this means) is not local
in any realistic situation. This brings us to the most popular and
difficult question about the limit ν→0. NATURE OF DISSIPATION
NATURE
IS IT (UN) IMPORTANT ? It is quite a common view that the precise nature of
dissipation is mostly unimportant in high Reynolds
number turbulence except for the smallest scales. This
This
forms, e.g. the basis for what is called inertial range. In
view if various aspects of nonlocality the natural question
is what does it mean, what kind of quantities do not
really depend on the nature of dissipation, why and and
in what sense as well as many similar closely related
questions We therefore conclude that, for the large eddies which
We
are the basis of any turbulent flow, the viscosity is
unimportant and may be equated to zero, so that the
motion of these eddies obeys Euler’s equation. ... The
viscosity of the fluid becomes important only for the
smallest eddies, whose Reynolds number is
comparable with unity. ... we may say that none of the
quantities pertaining to the eddies of sizes r >> η can
>>
depend on ν (more exactly, these quantities cannot be
changed if ν varies but other conditions of the motion
are unchanged). LANDAU AND LIFSHITZ 1954 The natural question is, therefore: Is it at all important
that this subsidiary agent be viscosity? Might other
dissipative, perturbing forces not do equally well? In
planning for a test of this question, one might first think of
investigating other forms of the law of viscosity, i.e. other
equations of flow instead of those of NavierStokes, where
viscosity might be described by a term other than ν∇2u or by
entirely different, nonlinear changes in the equations. … In
any event, it would be interesting to determine, whether such
modifications could lead to different forms of turbulence
(in the pure limiting, i.e. ν→0 form) than the NavierStokes
equations. The whole character of the KolmogoroffOnsagerWeizsa}cker theory would make one inclined to
surmise that this is not the case.
JOHN VON NEUMAN (1949) νΔu …there is nothing ``irrelevant'' in the (NSE) equation (except, may be, as ν→0, the
precise nature of dissipative term) . FRISCH (1983)…
FRISCH
… it may be that turbulence is not dependent on the NavierStokes equations. There
may be other equations, and not even necessarily diffrential equations, whose
properties have the same kind of structure as the turbulent structure of the NavierStokes equations. And these other equations may be easier to solve... one should be
prepared to consider systems of equations other than just the NavierStokes
equations. SAFFMAN (1991)
SAFFMAN
In fact, turbulence is an inertial phenomenon. That is, turbulence is statistically
indisdinguishable on energycontaining scales in gases, liquids, slurries, foams,
and many nonNewtonian media. These media have markedly different fine
structures, and their mechanisms for dissipation of energy are quire different. This
observation suggests that turbulence is an essentially inviscid, inertial
phenomenon, and is uninfluenced by the the precise nature of the viscous
mechanism.. HOLMES, BERKOOZ AND LUMLEY (1996)
HOLMES,
Causality is from large to small scale, and how the energy is dissipated in the
latter does not influence the former, as long as the amount is correct.. JIMENEZ
JIMENEZ
(2000) In any case, the dissipation processes, independently of their
nature, serve only as energy sinks, which cut off the spectrum of turbulent
fluctuations at small scales but do not affect the main turbulence scales.
BISKAMP (2003)
BISKAMP Iff the precise nature of dissipation is unimportant in high Reynolds
I
number turbulence and if the nature of dissipation is not important either,
why to work hard specifically on NSE instead of, e.g. taking some modified
version of NSE (LERAY 1934; LIONS, 1969; LADYZHENSKAYA, 1969,
1970; FRIEDLANDER AND PAVLOVIć, 2004) or lattice gas hydrodynamics
approximation (CHEN AND DOOLEN, 1998; FRISCH ET AL., 1987; HARDY
ET AL., 1976; WOLFRAM, 1986), all of which have regular solution for
any time and at any Reynolds numbers ?
The modification of NSE introduced by consists in mollifying the nonlinearity rather than changing the dissipative term as did the other authors. And why Clay Mathematics Institute (with $1M prize) insists specifically
on NSE ?
Is this really the case that the precise nature of dissipation is unimportant? It is possible that, after all, the
It
investigation in which viscosity is ignored
altogether is inappropriate to the limiting
case of a viscous fluid when the viscosity is
small... certain features of the motion
which could not enter into solutions were
the viscosity ignored from the first are
independent of the magnitude of the
viscosity .. RAYLEIGH 1892
Rayleigh, (1892) On the question of the stability of the flow of
fluids, Phil. Mag. 34, 5970
Phil. SOME PROPERTIES OF
SOME
TURBULENT FLOWS DO NOT
DEPEND OF THE NATURE OF
DISSIPATION
Turbulence is so rich that it can afford it The precise nature of dissipation is unimportant only in respect with a number of
The
manifestations of turbulence which are really weakly sensitive to the nature of
dissipation at large Re (and even more generally to specific properties of the
system as long as it is dissipative), e.g. things like 2/3, 4/5 and 4/15 laws, k5/3
spectrum and some others. The most convincing example is the 4/5 law showing
that the third order structure function is universal, i.e. it depends on the mean
energy injection rate only in the so called inertial range. However, the 4/5 law is a direct consequence of Euler equations (Duchon and
However,
Robert 2000, Eyink 2003) and it is not obvious that the same result should hold
for structure functions of higher orders and/or other objects. The argument that
this may be not the case goes as follows. ONLY SOME PROPERTIES OF
ONLY
TURBULENT FLOWS DO NOT
DEPEND OF THE NATURE OF
DISSIPATION The starting point is that viscosity/dissipation modifies substantially
The
and qualitatively the nonlinearity, i.e. it is not a passive sink of energy.
Indeed, this point is clearly seen from looking at the equations for
vorticity and enstrophy, e.g. enstrophy production is approximately
balanced by its viscous destruction 〈ωiωjsij〉 ~ 〈νωi∆ωi 〉
That is turbulence is an essential interaction between nonlinear and
linear processes and not just a simple cascade of energy or whatever
down to smaller scales. For example, in case of modified equations such
as those with hyperviscosity the enstrophy balance is quite different
from that for NSE, i.e. the nonlinearity is quite different either. Thus viscosity exerts direct influence on the vorticity field (and similarly
Thus
strain) and thereby should do the same with the velocity field either
(indeed, we remind that the velocity filed is fully defined by the field of
vorticity). Along with nonlocality (direct interaction of large and small
scales irrespective of their separation, see references in Tsinober (
2001, 2003) and broken scale invariance this means that the nature of
dissipation should be felt in large scales as well. In particilar, it is not
obvious why some statistical properties of velocity increments u(x+r) u(x) (both are functionals of vorticity and/or strain) do not depend on
viscosity. This is true even of the third order structure function: it is
interesting to explain how the viscous effects cancel out. The pure inertial range is not well defined due to nonlocality
The
of turbulence; because of scale invariance breaking,
because
the notion of inertial range is not well defined (ARNEODO ET AL., 1999). We wish to reiterate that
independence of some (statistical) parameters or properties
of viscosity at large Reynolds numbers does not mean that
viscosity is unimportant. It means only that the effect of
viscosity is Reynolds number independent. RELATED EXAMPLES SEE ALSO EXAMPLES IN THE LECTURES ON NONLOCALITY
SEE
NONLOCALITY Same dissipation – different flows: Dissipation (energy input) or drag
Same
only are not sufficient to define the properties of a turbulent flow. For
example, BEVILAQUA AND LYKOUDIS (1978) performed experiments on
flows past a sphere and a porous disc with the same drag. However, other
properties of these flows even on the level of velocity fluctuations were
quite different; see also WYGNANSKI ET AL. (1986) who performed
similar experiments with a larger variety of bodies with the same drag.
Similarly, many properties of turbulent flows with rough boundaries are
not defined uniquely by their friction factor (KROGSTAD AND ANTONIA,
1999). Turbulent LS dynamo of magnetic fields is strongly dependent (sensitive)
Turbulent
on diffusivity. POSSIBLE CONSEQUENCES
POSSIBLE
FOR THE "INVISCID" LIMIT The differences in the behavior of the systems with different dissipation at
The
finite Reynolds numbers points to a possibility that the limiting solution
will depend on the kind of dissipation we have at finite Reynolds number
(recall the above quotation by NEUMANN, 1949) even if their "inviscid
dissipation“ manifested in D(x,t) (DUCHON AND ROBERT 2000) would be
the same. Of special interest is what happens in this limit with
enstrophy/strain production and similar things. For example, assume that
for a hyperviscous case that the mean disipation ε→const as some
viscosity νh goes to zero, then velocity derivatives (both vorticity and
strain) grow as νh1/2h, which is pretty slow, if say, h= 8 as used in many
simulations. We remind again that the requirement that D(x,t) ≥ 0 (DUCHON AND
We
ROBERT 2000) is nonnegative is not sufficient to determine the "right
inviscid" solution (distribution) of Euler, and the whole issue of Redependence and finite Reeffects adds to its importance (see
KRAICHNAN, 1991). Is it possible that the Redependence is different
(which seems to be the case) but the limit is the same? More generally,
does it make sense to speak about the inviscid limit without referring to
some specific situation/properties? It is naturally to ask a more
general question: Is turbulence "slightly viscous" at whatever small
viscosity? (Isn't it similar to being slightly pregnant?) The answer
seems to be negative even for regions far from boundaries. A BIT MORE FOR
BIT
MATHEMATICALLY MINDED EULER EQUATIONS AND WEAK SOLUTIONS OF
EULER
NAVIERSTOKES EQUATIONS
NAVIER Att least since KOLMOGOROV (1941) an enormous effort is invetsed in
A
attempts to study asymptotic properties of turbulent flows at vanishingly
small viscosity. Considerable evidence shows that these flows at whatever
small viscosity possess finite dissipation, i.e. such flows are very much
unlike classical solutions of the Euler equations, which have the property
of energy conservation. One of natural conjectures in the mathematical
community was that turbulent flows may be described asymptotically
correctly by some sort of specially selected weak solutions of the Euler
equations which are called "disspative"  an approach which goes back to
ONSAGER (1949). Indeed, examples of weak (or distributional) solutions have been
Indeed,
constructed without energy conservation, see LIONS (1996), SHNIRELMAN
(2003) and references therein. It appears that there exist very different
classes of weak solutions, having little in common, and some of them are
physically meaningless (with negative dissipation, i.e. energy creation), at
least, in the context of turbulent flows. Moreover, there is no uniqueness of a
weak solutions, SHNIRELMAN (2003). In other words one needs additional
conditions to ensure physical meaning and uniqueness of solution. Simply
stated the solution has to be dissipative in the first place. DUCHON AND
ROBERT (2000) addressed this issue by considering local energy
balance for any weak solution ∂u²/2/∂t +∂/∂xk {(u²/2+p)uk}+D(x,t) = 0
where D(x;t) is shown to be a distribution defined in terms of the local
D(x;t
smoothness of velocity field u. DISSIPATIVE SOLUTIONS
DISSIPATIVE
OF EULER EQUATIONS found an explicit expression for D(x;t) which
makes the above equation identity in the sense of distributions. Thus D(x;t)
measures a possible dissipation (or production) of energy caused by a lack
of smoothness in the velocity field u in the spirit of ONSAGER (1949). For
smooth solutions D(x;t)≡0. The next step is to impose a condition that
locally D(x;t) ≥0 as physically acceptable, since for a general weak solution
of Euler equation, there is no connection of D(x;t) with viscous dissipation,
nor should D(x,t)≥0. However, as shown by DUCHON AND ROBERT
D(x,t)≥0 for a weak solution of Euler which is the strong limit of a
sequence of dissipative weak solutions of NavierStokes as viscosity goes
to zero. In this they used the fact that the condition D(x,t)≥0 is satisfied by every
In DUCHON AND ROBERT (2000) weak solution of the Navier Stokes equation obtained as a limit of a subsequence of
solutions uɛ of the regularized equation introduced by LERAY (1933, 1934). However, the condition D(x,t)≥0 does not garantee uniqueness
However,
 the phase space of Euler weak solutions seems to be too rich.
Moreover nonuniqueness is not the only problem. To quote
Moreover
SHNIRELMAN (2003): as for today, we have no weak
solution (of Euler equation) at hand which really
(of
describes a turbulent flow. In fact, having a "good candidate" it would be an extremely
difficult (if not impossible) task to decide whether it really describes
a turbulent flow. Moreover, to find such a candidate seems to
be as difficult as the "solution of the problem of turbulence"
itself. WEAK SOLUTIONS OF
WEAK
NAVIERSTOKES EQUATIONS In fact, DUCHON AND ROBERT (2000) started with weak solutions of
In
NSE and wrote a local energy balance for NavierStokes equations
∂u²/2/∂t+∂/∂xk {(u²/2+p)uk} ν∆ u²/2+ν(∂ui/∂xk)(∂ui/∂xk)+ D(x;t)=0
with D(x;t) defined in the same way as for the Euler equation. Thus  they
write  the nonconservation of energy originates from two sources:
viscous dissipation and a possible lack of smoothness in the
solution.; and stress that D(x;t) measures a possible dissipation (or
production) of energy caused by a lack of smoothness in the
velocity field u, this term is by no means related to the presence
or absence of viscosity. The latter statement being formally/mathematically nice
The is problematic from the physical point of view. As long as one is speaking about NSE
this looks definitely unphysical: so far no physical process is known that can bring an
additonal dissipation into operation which is formally described by the distribution
formally
D(x;t). The questions about the existence of a weak solution of NavierStokes
The
with D(x;t)≠0 as well as the uniqueness of such a solution with
D(x;t)
D(x;t)>0 remained unanswered. DUCHON AND ROBERT note also that
DUCHON
There is still some doubt as to whether weak
solutions of the NavierStokes equation, the
uniqueness of which is unknown, or hypothetical weak
solutions of the Euler equation, are relevant to the
description of turbulent flows at high Reynolds
numbers. One may add that, moreover, if we look at real turbulence at finite
One
Reynolds numbers (whatever large) there seems to be no need for weak
solutions at all. HOW MEANINGFUL IS "CASCADE" OF
PASSIVE OBJECTS ass described by linear equations?
a
Is there ‘enough’ analogy (more on analogies in a separate
lecture) between genuine and ‘passive’ turbulence or the
differences are essential? Nonlinear versus linear. Is
extension of Kolmogorov phenomenology justified for
systems governed by linear equations? Itt is rather common, since Obukhov (1949) and Corrsin (1951), to speak about cascade in
I
case of a passive scalar and more recently passive . The main argument is from some analogy.
Indeed, for instance in any random isotropic flow the rate of production of ‘dissipation' (i.e.
corresponding field of derivatives) of both passive scalars and passive vectors is essentially
positive, which can be interpreted as a sort of `cascade'. However, the equations describing
the behavour of passive objects are linear. Hence, there is no interaction between modes of
whatever decomposition of the field of a passive object: the princilple of superposition is valid
in case of passive objects*.
*Here by ‘mode' is meant as a solution of the appropriate equation, e.g. of the advectiondiffusion equation . Of course, there are many ways to use ‘modes’ that are not solutions of this equation, such as Fourier modes. In
this case the Fourier modes do interact, since one of the coefficients of the advectiondiffusion equation, the
velocity field, is not constant. This interaction is interpreted frequently as a 'cascade' of passive objects. But, as
mentioned, this interaction is decomposition dependent, and therefore is not appropriate for description of
physical processes, which are invariant of our decompositions. There is a point concerning the behavior of an
individual solution. Namely, the evolution of its energy spectrum is expected to exhibit positive energy transfer
to higher wave numbers as a consequence of production of the field of derivatives of the passive field. Can one
see this as a kind of ‘cascade’? Even if the answer were affirmative it is a very different kind of cascade, if at all. Therefore, it seems more appropriate to describe the process in terms of production
Therefore,
of the field of derivatives of the passive object, which is performed by the velocity
straining field, just like it is proposed above for the velocity ffield.
ield.
Hence the extension of Kolmogorov arguments and phenomenology to passive
Hence
objects seems to be much less justified. No wonder that the phenomenological
paradigms for the velocity field failed in most cases when applied to passive
objects*. We are reminded that the ‘analogy’ between the passive objects and the
active variables is, at best, very limited for several reasons, the main of which are the
linear nature of ‘passive’ turbulence, Lagrangian chaos, the irreversible effect of the
randomness of the velocity field on passive objects independently of the nature of
this randomness, e.g. even a Gaussian one, and the oneway interaction between the
velocity field and the field of a passive object .
*E.g. experiments by Villermaux et al. (2001) clearly show that this is the case. The behaviour of passive scalar in
their experiments is distinctly nonlocal in the sense that the main mechanism responsible for mixiing involves direct
interaction between large and small scales ‘bypassing' the (nonexistent) cascade. IS BELOW ANYTHING WRONG? Bogucki, D., Domaradzki, J.A. and Yeung. P.K. (1997) Direct numerical simulations of passive scalars with Pr>1 advected by
turbulent flow, J. Fluid Mech., 343, 111130. AT P. 117 THE MIXED DERIVATIVE SKEWNESS IS RELATED TO THE
NONLINEAR TRANSFER OF SCALAR VARIANCE TO SMALL SCALES AND TAKES A VALUE ZERO WHEN THERE IS NO NET
CASCADE TO HIGHER WAVENUMBERS. CONCLUDING
CONCLUDING CHOOSE WHAT YOU LIKE MORE?
Big whirls have little whirls,
Big
Which feed on their velocity.
And little whirls have lesser whirls
And so on to viscosity –
In the molecular sense
RICHARDSON (1922)
Big whirls lack smaller whirls,
Big
To feed on their velocity.
They crash and form the finest curls
Permitted by viscosity
BETCHOV (1976) The notion that turbulent flows are hierarchical, which underlies the
The
concept of cascade, though convenient, is more a reflection of the
unavoidable (due to the nonlinear nature of the problem) hierarchical
structure of models of turbulence and/or decompositions rather than
reality. This is emphasized in the case of passive objects, whose
evolution is governed by linear equations, with the velocity field
entering multiplicatively in these equations, thus making them
‘statistically nonlinear’. AND SO WE ARE BACK WITH THE QUESTIONS Is the inertial range a conceptually well defined concept or
Is
is it "mostly a pedagogical imagery"? Are its properties
really independent of the nature of dissipation? Is the
Komlogorov 4/5 law an unequivocal evidence of such a
cascade? Is "casade" Eulerian, Lagrangian or both? Are
decompositions aiding understanding or obscuring the
physics of turbulence?
AND MANY OTHER ...
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This note was uploaded on 09/16/2011 for the course ME 563 taught by Professor Staff during the Spring '11 term at Auburn University.
 Spring '11
 Staff

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