This preview shows page 1. Sign up to view the full content.
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 XIXII ANALOGIES, DIFFERENCES AND
RELATIONS BETWEEN GENUINE
TURBULENCE AND ITS ‘ANALOGS’ Our understanding of the general character of the smallscale features of
turbulent motion is very far from complete…Very few theoretical or
experimental results have been established so that for the most part we
must proceed analogy and plausible inference. BATCHELOR 1956, p 183
183 Analogies in turbulence research have a special status mainly due to
Analogies
unsatisfactory state of theory. Most analogies are aimed to look at
similarity between genuine turbulence and some "analogous" system such
as evolution of some passive object (e.g. scalar, vector, etc.) polymers,
and some other (see below) in some prescribed random (usually
Gaussian) velocity field. This led in many cases to exaggerated and
consequently misleading claims on analogous behavior between the two
and consequently to misconceptions. Hence the purpose of this lecture is
twofold. Along with critical overview of similarities the main emphasis is
given to differences rather than similarities. The primary reason for this is
that (at least some) understanding of differences is expected to aid better
understanding of both systems and avoid misconceptions associated with
extending the analogies too far. SOME EARLY ANALOGIES
# Reynolds analogy on transport of momentum and heat, REYNOLDS, O. 1874 On
REYNOLDS,
the extent and action of the heating surface of steam boilers, Proc. Lit. Phil. Soc. Manchester, 14, 712.
Proc.
# Study of fluid motion by means of `colour bands‘, REYNOLDS. O. 1894 Study of
fluid motion by means of coloured bands, Nature, 50, 161164.
Nature
# Frozennes of vorticity in the flow field in invisdid flows (and other solenoidal
fields with vanishing diffusivity, e.g. magnetic field in perfectly conducting fluids),
HELMHOLZ, H. 1858 On integrals of the hydrodynamical equations which express vortex motion.
On
Translated from German by P.G.Tait, 1867 with a letter by Lord Kelvin (W.Thomson) in London
Edinburgh Dublin Phil. Mag. J. Sci., Fourth series, 33, 485512; KELVIN, LORD (THOMSON,
KELVIN,
Phil.
W.) 1880 Vibration of columnar vortex, London Edinburgh Dublin Phil. Mag. J. Sci.,Fifth series,
Vibration
London
33, 485512.; (1910) Mathematical and physical papers, vol. 4, Cambr. Univ Press.
(1910)
# Finite diffusivity: analogy between amplification of vorticity and magnetic field
by turbulent flow BATCHELOR, G.K. (1950) On the spontaneous magnetic field in a conducting
liquid in turbulent motion, Proc. Roy. Soc. London, A201,
Proc. VORTICITY VERSUS
VORTICITY
(INFINITESIMAL) MATERIAL
LINES
Are they stretched in the same
way and for the same reason?
And what is the meaning of the
“same way” # Vorticity amplification is a result of the kinematics of
turbulence , TENNEKES, H. AND LUMLEY, J. L. (1972) First course in
TENNEKES,
turbulence, MIT Press. In the context of concern here a similar view originates
with TAYLOR 1937,1938 (see below)
# The physics of vortex stretching is well understood…, see for
see
instance, G.K. Batchelor, An introduction to Fluid Dynamics
G.K.
(Cambridge U.P., New York, 1967); and R. H. Kraichnan, J.
Fluid Mech., 64, 737 (1974) A footnote in E.. D. SIGGIA, (1977)
E `Origin of intermittency in fully developed turbulence', Phys. Rev., 15(4), 1730.
Phys.
# …amplification of the vorticity by vortex stretching, a wellunderstood mechanism in 3D Euler flow." POMEAU, Y. AND SCIAMARELLA, D. (2005) An unfinished tale of nonlinear PDEs: Do solutions of 3D incompressible Euler equations blowup in finite time? Physica, D205, 215
Physica A BIT OF HISTORY  I
BIT A small digression reminding a big mistake assumed that the
expression ∑i∑kωiωk∂ui/∂uk is zero in
the mean and that he (vK ) cannot see any
physical reason for such a correlation.
TAYLOR (1937) conjectured that there is a
TAYLOR
strong correlation between ω3² and
∂u3/∂x3 so that (the mean of) ω3² ∂u3/∂x3 is
not equal to zero (x3 is directed along vorticity)
and showed experimentally that this is really the case,
VON KARMAN (1937) TAYLOR (1938) Itt is a rather common view (and misconception) that the prevalence of vortex stretching is due to the
I
predominance of stretching of material lines. This view originates with TAYLOR 1938:
# Turbulent motion is found to be diffusive, so that particles which
were originally neighbors move apart as motion proceeds. In a
diffusive motion the average value of d²/d02 continually increases.
It will be seen therefore .., that the average value of ω²/ω₀²
continually increases.
#... the interesting physical argument that 〈ωiωjsij〉 is positive
because two particles on average move apart from each other and
therefore vortex lines are on average stretched rather than
compressed , HUNT 1973.
HUNT
# The relative diffusion of a pair of probe particles in grid turbulence at high
Reynolds numbers is treated as the most clearcut manifestaion of vortex
stretching. MORI & TAKAYOSHI, 1983.
# The latest example is found in DAVIDSON (2004, p.259. ): However, since
However, materialline stretching seems to be a norm for the broader class
of kinematically admissible fields, it should also be the norm for
the narrower class of dynamically admissible velocity fields, and
so one should not be surprised that vortexline stretching, like
material line stretching, is seen in practice. CHORIN (1994) points to the problematic aspect of such a view: Vortex
CHORIN lines are special lines, and constitute a negligible
fraction of all lines (there is one vortex direction at
each point, but an infinite number of others). All
arguments that involve averages with respect to a
probability measure may fail to hold in a negligible
fraction of cases, and thus one cannot conclude from
(5.1) (i.e d/dt<δx (t)²>0) that vortex lines stretch, even in
isotropic flow, but ends with a the statement that This conclusion
but
is, however, eminently plausible.
Indeed, it is plausible, since it is observed in the laboratory and in numerical
Indeed,
simulations. But the underlying reasons/processes are still not understood unlike in
case of passive material lines. COCKE (1969) proved the following important results. The first result is that the length of an infinitesimal material line element, l ≡ l, increases on average in
any isotropic random velocity field*. Similarly, Cocke showed that an infinitesimal
material surface element, N, identified by its vector normal, N, increases on average
in an any isotropic random velocity field as well.** Two important points have to
be stressed. First, the results of Cocke are based on statistics of ‘all’ material lines
having the property that infinitely many lines pass through each point (whereas
typically only one vorticity line is passing through a point – recall the statement by
Chorin: All arguments that involve averages with respect to a
All
probability measure (i.e. all material lines) may fail to hold in a
negligible fraction of cases (i.e. vorticity lines only).
___________________________________________________________________________________
___________________
* For references on more details and a review of other related issues see TSINOBER 2001, P. 48 AND ON.
For
**More precisely Cocke showed that ln[〈l(t)〉/l(0)] ≥ 0, and ln[〈N(t)〉/N(0)]≥ 0 for all t>0 with equality holding
**
only if there is no fluid motion at all. Arguments similar to those by Cocke (1969) show that 〈lp(t)〉 ≥ lp (0) and
〈Np(t)〉 ≥ Np(0) for any p > 0 (MONIN AND YAGLOM, 1975, PP. 579  580). Second, the results by Cocke are purely kinematic: the flow does not ‘know’
Second
about material lines and does not have to be a real one, i.e. to satisfy the
NavierStokes equations and/or to be observable in laboratory or elsewhere the only requirement is that the flow should be random and isotropic. For
example, this result is true for a Gaussian velocity field as well, which is
important for the purpose of comparison of material line elements, which are
passive, and vorticity, which is not. Namely, the enstrophy production in a
Gaussian isotropic field vanishes identically in the mean 〈ωiωksik〉 ≡ 0, but
〈ω
〈lilksik〉 >0! On the qualitative level the results by Cocke were confirmed in a number of
On
DNS experiments both for real and artificial flow fields (DRUMMOND, 1993;
GIRIMAJI AND POPE, 1990; HUANG, 1996; YEUNG, 1994) and laboratory
experiments (LÜTHI ET AL., 2005). In an inviscid flow
Dω/Dt = (ω·∇)u; Dl/Dt = (l·∇)u
D(ωl)/Dt = {(ωl)·∇}u
So ω = l att all times if initially ω – l = 0;
a
However, in a flow with ν≠ 0 whatever small
However,
ν≠ 〈ωiωksik〉 ≈ ν〈ωi∇2ωi〉, i.e. the vortex lines are not frozen into the fluid at whatever high Reynolds
number – otherwise how theFrozenness in large approximately balanced
enstrophy production can be •
by viscous terms again at any – whatever large scales number. In othetr
– Reynolds
In
words in slightly viscous flows frozennes is meaningless. Just like the claim that
turbulence is slightly viscous at whatever large Re. In this context the question:
what happens with enstrophy/strain production as ν→0 is of special interest.
ν→ The above is not entirely new (at least in part)
The
a material line which is initially coinsiding with
material
a vortex line continues to do so. It is thus
possible and convenient to regard a vortexline as
having a continuing identity and as moving with
the fluid (In a viscous fluid it is, of course,
... possible to draw the pattern of vortex lines
at any instant, but there is no way in which
particular vortexline can be identified at
different instants). BATCHELOR, 1967, p.. 274
1967 p WHAT ARE THE MAIN
PROBLEMS WITH THE ABOVE
MISCONCEPTION? A SUMMARY First, the vortex lines are not frozen into the fluid at whatever high Reynolds
First,
number  otherwise how the enstrophy production can be approximately balanced
by viscous terms.
Second, even if frozen vorticity is not a marker, it reacts back strangly: everybody
knows what is BioSavart law, or more generally ∇²u =  curl ω.
Third, even if frozen those material lines coinciding with vorticity are special and
not the other way around. Namely, the material line elements which initially and
thereby consequently coincide with vorticity are special in the sense that they are
not dynamically passive quantities anymore and react back on the flow precisely
as does vorticity. In other words, the fact that vorticity is frozen in the inviscid
flow field does not mean that vorticity behaves the same way as material lines, but
the other way around: those material lines which coincide with vorticity behave
like vorticity, because they are not passive anymore as are all the other material
lines: the rules of democarcy do not apply to science. MORE DIFFERENCES BETWEEN
MORE
VORTICITY AND MATERIAL LINES VORTICITY VERSUS MATERIAL LINES
VORTICITY
ALIGNMENT BETWEEN THE EIGENFRAME λi OF THE RATE OF
STRAIN TENSOR Sij AND
(from 3DPTV; Reλ=60, LUTHI ET AL 2005)
VORTICITY ω MATERIAL LINES l GEOMETRICAL STATISTICS ALIGNMENTS
ALIGNMENTS VORTICITY ω VERSUS MATERIAL LINES l
Dω/Dt = (ω·∇)u + ν Δω
Dl/Dt = (l·∇)u
PRODUCTION OF l²
PTV
PRODUCTION OF ω²
Again Nonlocality VORTEX STRETCHING VERSUS
VORTEX
STRETCHING OF MATERIAL LINES
A summary from Tsinober, 2001, p. 88 # The equation for a material line element l is a linear one and the vector is .
passive, i.e. the fluid flow does not `know' anything whatsoever about the vector l
(as any passive vector) does not exert any influence on the fluid flow. The material
element is stretched (compressed) locally at an exponential rate proportional to the
rate of strain along the direction of l since the strain is independent of l.
# On the contrary, the equation for vorticity is a nonlinear partial differential
On
equation and the vorticity vector is an active one  it `reacts back' on the fluid flow.
The strain does depend in a nonlocal manner on vorticity and vice versa, i.e. the rate
of vortex stretching is a nonlocal quantity, whereas the rate of stretching of material
lines is a local one. Therefore the rate of vortex stretching (compressing) is different
from the exponential one and is unknown. There are ‘fewer’ vorticity lines than the
material ones  at each point there is typically only one vortex line, but infinitely
many material lines. This leads to essential differences in the statistical properties
of the two fields. . # In the absence of viscosity vortex lines are material lines, but they are special in the sense that they are not passive as all the other passive material lines. But the
fact that vorticity is frozen in the inviscid flow field does not mean that vorticity
behaves the same way as material lines, but the other way around: those material
lines which coincide with vorticity behave like vorticity, because they are not
passive anymore as are all the other material lines. While a material element tends to be aligned with the largest (positive)
While
eigenvector of the rate of strain tensor, vorticity tends to be aligned with
the intermediate (mostly positive) eigenvector of the rate of strain
tensor: its eigenframe rotates with an angular velocity of the order of
vorticity # # For a Gaussian isotropic velocity field the mean enstrophy generation
.
vanishes identically, whereas the mean rate of stretching of material lines
is essentially positive. The same is true of the mean rate of vortex
stretching and for purely twodimensional flows. This means that in
turbulent flows the mean growth rate of material lines is larger than that
of vorticity. The nature of vortex stretching process is dynamical and not a
kinematic one as the stretching of material lines is.
# Comparing vorticity with any passive vector (also with the same diffusivity as
viscosity), the analogy is partial not just/only because the equation for
vorticity is nonlinear, but also because in the case of vorticity the process
is due to self amplification of the field of velocity derivatives, whereas in
case of a passive vector it is not. GENIUNE TURBULENCE VERSUS
GENIUNE
PASSIVE “TURBULENCE”
How analogous are the genuine and passive turbulence (if at all)?
What are the main differences? Evolution of disturbances. What
can be learned about genuine turbulence from its signature on
the evolution of passive objects? What is the importance (if any)
of statistical conservation laws in genuine turbulence (if such
exist)? # Yet statistical properties of this socalled `passive scalar'
turbulence are decoupled from those of the underlying velocity
field (are they?)... The nontrivial statistical properties of scalar turn
out to originate in the mixing process itself, rather than being
inherited from the complexity of the turbulent velocity field (but this is
just one part of the story). Study of passive scalar turbulence is therefore
decoupled from the still intractable problem of calculating the
velocity statistics, and so has yielded to mathematical analysis.
…the well established phenomenological parallels between the
statistical description of mixing and fluid turbulence itself
suggest that progress on the latter front may follow from a
better understanding of turbulent mixing (really ?). SHRAIMAN AND
SIGGIA, 2000. # Passive contaminants are transported by turbulent motions in
Passive
much the same way as momentum.... Momentum is not a passive
contaminant; "mixing" of mean momentum relates to the
dynamics of turbulence, not merely its kinematics. TENNEKES AND
TENNEKES
LUMLEY, 1972 # The advectiondiffusion equation, in conjunction with a velocity field model with turbulent characteristics (prescribed
(prescribed
a priori), serves as a simplified prototype problem for
developing theories for turbulence itself. MAJDA AND
MAJDA
KRAMER, 1999 # An important progress has been achieved in the last
decade in understanding some simpler systems
exhibiting behaviors similar to developed turbulence.
These include the socalled weak or wave turbulence ,
the advection of passive scalar and vector fields by
random velocities that mimic (do they? in what sense?) the turbulent
ones, and, to certain extent, the socalled burgulence ,
the phenomena described by the Burgers equation.
GAWEDZKI 2002
GAWEDZKI Examples of passively advected quantities are the temperature
Examples
or the impurity concentration in a fluid. Ideally one would be
interested in the statistical properties of the advected field in
the case where the underlying flow is turbulent. Significant
progress has been achieved when the velocity field is taken
random, with Gaussian statistics but decorrelated (white) in
time. One mimics the important feature of turbulent flows by
taking the velocities rough, i.e. only Hölder continuous, in
space. For such an ensemble of velocities (called the Kraichnan
model), it was possible to study the ensuing steady state of the
advected fields both analytically and numerically. It appears
to be a nonequilibrium state with nonzero flux of a conserved
quantity, again in analogy to hydrodynamical turbulence.
Moreover it exhibits intermittency in the form of anomalous
scaling of moments of scalar differences in nearby points, the
first (and so far only) nontrivial model where the anomalous
scaling has been established analytically. GAWEDZKY 2002
analytically MAJOR DIFFRENECS
MAJOR
The differences are more than essential: the evolution of passive objects
is not related to the dynamics of turbulence in the sense that the
dynamics of fluid motion does not enter in the problems in question  the
velocity field is prescribed a priory in all problems on evolution of
priory
passive objects. Consequently the problems associated with the passive
objects are linear; whereas genuine turbulence is a strongly nonlinear
problem  nonlinearity is in the heart of turbulent flows and is underlying
the main manifestations of the differences between genuine and passive
turbulence. Selfamplification of velocity derivatives.
Self Nonlinearity of genuine turbulence is the reason for the selfamplification of the field
of velocity derivatives, both vorticity and strain. In contrast there is no phenomenon
of selfamplification in the evolution of passive objects (such as material lines, gradients of passive
(such
scalar and solenoidal passive vectors with finite diffusivity). We stress that the process of selfamplification of strain is a specific feature of the dynamics of genuine turbulence
having no counterpart in the behavior of passive objects. In contrast, the process of
selfamplification of vorticity, along with essential differences (We would like to stress again
(We
that vorticity is an active vector, since it `reacts back' on the velocity (and thereby on strain) field. This is not the case
with passive objects  the process here is `one way': the velocity field does not `know' anything about the passive
object), has common features with analogous processes in passive vectors; in both the main factor is their interaction with strain, whereas the production of strain is much
more ‘self‘.
A related important difference is absence of pressure in case of
related
passive objects. Differences in structure(s) Along with some common features the
Differences mechanisms of formation of structure(s) are essentially different for the passive objects and the
dynamical variables. Among the reasons is the presence of Lagrangian chaos, which is manifested as
rather complicated structure of passive objects even in very simple regular velocity fields (On the other hand, e.g. the rampcliff structures of a passive scalar are observed in pure Gaussian `structureless' random velocity field, just like those in a
variety of real turbulent flows practically independently of the value of the Reynolds number). In other words the structure of passive objects in turbulent flows arises from two (essentially inseparable) contributions: one due to
the Lagrangian chaos and the other due to the random nature of the velocity field itself (Therefore one
cannot claim that statistical properties of this socalled `passive scalar' turbulence are decoupled from those of the underlying velocity field
Shraiman Siggia2000) , since the nontrivial statistical properties of scalar turn out to originate not only in the mixing process itself but are
inherited from the complexity of the turbulent velocity field as well. Study of passive scalar turbulence is therefore not decoupled from the still
intractable problem of calculating the velocity statistics). Among other reasons are differences in sensitivity to initial (upstream) conditions (i.e. Lagrangian `memory'), ‘symmetries', e.g. the velocity field may be
locally isotropic, whereas the passive scalar may not be and some other (see references in
TSINOBER 2001). A recent result, BAIG &CHERNYSHENKO 2005 for turbulent flow in a
TSINOBER
BAIG
plane channel is an interesting addition to the list of these differences: although the vortical
structure of the flow is the same, the scalar streak spacing varies by an order of magnitude
depending on the mean profile of the scalar concentration. Moreover, passive scalar streaks were
observed even in an artificial "structureless" flow field. Differences in structure(s)
Passive scalar dissipation
Courtesy P.K. Yeung SHE ET AL. 1991 Dissipation of energy Differences in structure(s)
Vorticity Gradient of passive scalar CHEN & CAO 1997
SHE ET AL. 1991
Isosurfaces of enstrophy Also rampcliff SAME FLOW  NOT THE SAME PATTERN
SAME
All frames (i.e. four different Lagrangian fields)
correspond to the same (Eulerian)flow. Cimbala, J.M., Nagib, H. M and Roshko, A. (1988) Large structures in the far
Cimbala
wakes of twodimensional bluff bodies, J. Fluid Mech., 190, 265298.
190 RDTlike processes/terms
dominate the flow near the wall Kolmogorov 4/5 law
Kolmogorov
versus Yaglom 4/3 law The Kolmogorov and the Yaglom laws are
The
respectively
S3(r) ≡ 〈(Δu)³〉 = (4/5) εr
〈Δu(Δθ)²〉 =  (4/3) εθr where Δu ≡ [u(x+r)u(x)]·r/r, Δθ = θ(x+r)  θ(x), ε  is
Δθ
the mean rate of dissipation of kinetic energy and εθ =
〈D ∂θ/∂xi ∂θ/∂xi〉  is the mean rate of dissipation of
∂θ
∂θ
fluctuations of a passive scalar. The analogy between
these two laws* though useful in some respects, e.g.
ANTONIA ET AL 1997, is violated for a Gaussian velocity
ANTONIA
field.
*The 4/5 Kolmogorov law follows by isotropy from the the 4/3 law for the
velocity field in the form 〈Δu(Δu)²〉 = (4/3) εr) Namely, the 4/3 law remains valid for such (as
Namely,
4/3
any other random isotropic) velocity field,
whereas the 4/5 law is not, because S3(r) ≡ 0
the
for a Gaussian velocity field. This difference is
one of the manifestations of the dynamical
nature of the Kolmogorov law as contrasted to
the kinematical nature of the Yaglom law. It
reflects the difference between genuine
genuine
turbulence as a dynamical phenomenon and
‘passive' turbulence as a kinematical process. Vorticity versus passive vectors.
Solenoidal vector fields with
nonvanishing diffusivity The usual comparison is based on looking at the equations for
vorticity ω and the (solenoidal) passive vector, B, e.g.
magnetic field in electrically conducting fluids,
BATCHELOR1950 ∂ω/∂t =∇×(u×ω) + ν∇²ω
∂B/∂t =∇×(u×B) + η∇2B Though a number of differences are known these differences
are hidden when one looks at the equations for ω and B,
which look quite `similar‘ when ν = η. What is hardest to accept in Batchelor's
discussion is the assumed simlarity between B
and ω. LUNDQUIST, 1952 However, a more ‘fair' comparison should be made between the velocity field, u, and the vector potential A, with B =∇×A,
TSINOBER & GALANTI 2003. Such a comparison allows to
see immediately one of the basic differences between the fields u
and A (apart of the first obeying nonlinear and the second linear
equation) which is not seen from the above equations. Namely,
the Euler equations conserve energy, since the scalar product of
u·(ω×u) ≡ 0. In contrast, (unless initially and thereby subsequently u ≡ A) the scalar product
A·(u×B) ≠ 0*.
It is this term A·(u×B) ≡ AiAksik+ ∂/∂xk{…}
which acts as a production term in the energy equation for A. In
other words the field A (and B), is sustained by the strain, sik of
the velocity field  in contrast to the field u. This leads, in
particular, to substantial differences in amplification of vorticity,
ω and magnetic field B, e.g. in statistically stationary velocity
field (both NSE and Gaussian) the enstrophy ω² saturates to
some constant value, whereas the energy of magnetic field B²
grows exponentially without limit (but there is much more, see below).
*the corresponding equation for the vector potential A has the form ∂A/∂t+B×u = ∇pA+η∇2A As in case of passive scalar an analogue of Kolmogorov
4/5 law* is valid for the vector A (see e.g. GOMEZ ET AL.,
1999 and references therein) 〈Δu(ΔA)²〉 =  4/3rεA where Δu≡ Δu·r/r≡ {u(x+r)u(x)}·r/r,
ΔA=A(x+r)A(x), and εA is the mean dissipation
rate of the energy of A. Again the latter relation holds
for any random isotropic velocity field including the
Gaussian one.
*It is more convenient to use the 4/3 law for the velocity field in the form 〈Δu(Δu)²〉 =(4/3) εr , which
turns into the 4/5 law by isotropy Vorticity versus passive vectors
with nonvanishing diffusivity
Evolution of disturbances Important aspects of the essential difference between the evolution of
fields ω and B arising from the nonlinearity of the equation of ω and
linearity of the equation for B are revealed when one looks at how these
fields amplify disturbances. The reason is that the equation for the
disturbance of vorticity differ strongly from that for vorticity itself due to
the nonlinearity of the equation for the undisturbed vorticity ω, whereas
the equation for the evolution of the disturbance of B is the same as that
for B itself due to the linearity of the equation for B. Consequently, the
evolution of disturbances of the fields ω and B is drastically different.
For example, in a statistically stationary velocity field the energy of the
disturbance of B grows exponentially without limit (just like the energy of
B itself), whereas the energy of vorticity disturbance grows much faster
than that of B for some initial period until it saturates at a value which is
of order of the enstrophy of the undisturbed flow. It is noteworthy that much faster growth of the energy of
disturbances of vorticityis observed during the very initial (linear in
the disturbance) regime which is due to additional terms in the
equation for the disturbance of vorticity, ω which have no
analogues in the case of passive vector B. It is important to stress
that these additional ‘linear' terms arise due to the nonlinearity of
the equations for the undisturbed vorticity. In this sense the
essential differences between the evolution of the disturbances of
vorticity ω and the evolution of the disturbance of passive vector B
with the same diffusivity can be seen as originating due to the
nonlinear effects in genuine NSE turbulence even during the linear
regime. Looking at the evolution of the disturbance Δu of some flow
realization u in a statistically steady state and similarly for
other quantities.
Active: vorticity  Δω, strain  Δs;
Active:
Passive: magnetic field  ΔB, its vector potential  ΔA,
passive scalar  Δθ and its gradient  ΔG.
For more details see
15, 35143531. , Phys. Fluids, TSINOBER AND GALANTI 2003 Note the additional linear in disturbance terms which arise due to the nonlinearity of the equations for the
undisturbed vorticity and which have no analogues in the case of passive vector B. These additional terms are
responsible for much faster growth of the energy of disturbances of vorticity during the very initial (linear in the
disturbance) regime. In this sense the essential differences between the evolution of the disturbances of vorticity and
the evolution of the disturbance of passive vector B with the same diffusivity can be seen as originating due to the
nonlinear effects in genuine NSE turbulence even during the linear regime. GROWTH OF ENERGY OF DISTURBANCES IN GENUINE
{EΔu, EΔω, EΔs} AND PASSIVE {EΔA, EΔB, EΔθ, EΔG}
TURBULENCE
EΔB
EΔω EΔu
EΔs EΔA EΔG
EΔθ Note the much faster
growth of the energy of
disturbances of active
variables such as
vorticity during the very
initial (linear in the
disturbance) regime and
decay of disturbances
associated with passive
scalar
TSINOBER & GALANTI,
2003 ADDITIONAL ISSUES
ADDITIONAL Scaling exponents and
statsitically conserved
quantities
There is a number of publications insisting in some sense on a
kind of essential linearization of genuine turbulence problem
when this concerns scaling exponents (mainly of structure
functions) and the role of statistically conserved quantities. The claims are summarized by arguing that the mechanism leading to anomalous scaling in NavierStokes equations and other nonlinear models
is identical to the one recently discovered for passively advected fields. ANGHELUTA ET AL 2006 If this is really true it means that this is just one more aspect – as in RDT  which
can be treated via a linear model which in some cases enables to handle some
aspects of turbulent flows, but not their genuine nonlinear aspects: One can thus speculate that the anomalous scaling for the genuine turbulence can also
appear as a linear phenomenon in the following sense. Let us split the total
velocity field into the two parts, the background field and the perturbation …
linearize the original stochastic equation with respect to the latter, choose an
appropriate statistics for the former … Then the smallscale perturbation
field will show anomalous scaling behavior with nontrivial exponents, which
can be calculated systematically within a kind of εexpansion. model. In
such a case the passive vector field can give the anomalous exponents for the
NS velocity field exactly. ANTONOV ET AL 2003 Similar statements are made in respect with so called statistically conserved
quantities which have been discovered for passive objects, but not really for genuine
NSE, see references in FALKOVICH AND SREENIVASAN 2006. Analogy between genuine
turbulence Lagrangian chaos This analogy is closely related to those associated with the analogies between the
genuine and passive turbulence in several respects. The main is that the former is a
dynamical phenomenon (Eturbulent) whereas the latter is a kinematic one (Lturbulent, i.e. purely Lagrangian). The flow can be purely Lturbulent (i.e. Elaminar)
at Re ~1 and Re << 1 (see exmples in TSINOBER 2001). This includes examples such as a
number of mixing issues in flows in porous media, microdevices, and kinematic
simulations of Lagrangian chaotic evolution (KS, turbulentlike motions). However if
the flow is Eturbulent (i.e. Re >> 1) it is Lturbulent as well. An important
consequence is that the structure and evolution of passive objects in genuine turbulent
flows arises from two (essentially inseparable) contributions: one due to the
Lagrangian chaos and the other due to the random nature of the (Eulerian) velocity
field itself. Hence, one can expect adequate kinematic simulation of those properties
which are insensitive (or weakly sensitive) to the differences between the genuine and
synthetic velocity fields. An important counterexample is the difference between
backwards and forwards relative dispersion (with the mean square separation
following particle pairs backwards in time being twice as large as forwards) in genuine
turbulence. ELAMINAR BUT LTURBULENT
Since the equations describing the evolution of passive objects are linear, it may seem that there is
Since
no place for chaotic behaviour of passive objects if the velocity field is not random and is regular
and fully laminar, because the chaotic behaviour appears/shows up in nonlinear systems. There
is, however, no real contradiction or paradox. This apparent contradiction is resolved via looking
at the the fluid flow in the Lagrangian setting in which the observation is made following the fluid
particles wherever they move. Here the dependent variable is the position of a fluid particle,
X(a,t), as a function of the particle label, a, (usually it's initial position, i.e. a ≡ X(0)) and time, t.
The relation between the two ways of description is given by the following equation
∂X(a,t))/∂t = u[X(a,t)]
(EL)
(E
i.e. the Lagrangian velocity field, v(a,t) = ∂X(a,t))/∂t, is related to the Eulerian velocity field,
u(x,t), as V(a,t) ≡ u[X(a,t);t]. Though the Eulerian velocity field, u(x;t) is not chaotic and is
regular and laminar, the Lagrangian velocity field v(a,t) ≡ u[X(a,t);t] is chaotic because X(a,t) is
chaotic: the equation (EL) is not integrable even for simplest laminar Euler fields with the
exception of very simple flows such as unidirectional ones. ON THE RELATION BETWEEN
EULERIAN AND LAGRANGIAN FIELDS
# Given the marker dispersion the problem is to determine the source(s) of
agitation. In general, owing to chaotic advection, this inverse problem is
impossible to solve AREF 1984
#…the possession of such relationship would imply that one had
(in some sense) solved the general turbulence problem. Thus it seems
arguable that such an aim, although natural, may be somewhat illusory
MCCOMB 1990
# What one sees is real. The problem is interpretation The relation between Eulerian and Lagrangian fields is a longstanding and most
difficult problem. The general reason is because the Lagrangian field is an extremely
complicated nonlinear functional of the Eulerian field. This issue just as the whole
theme of Lagrangian description of turbulent flows (not just kinematical chaos) will
be addressed in several lectures later. Only few general notes are given here. MIXING IN PMM, Re ~ 1 (!) KUSH & OTTINO (1992)
Re
RELEVANT TO MICROFLUIDICS with Re ~ 0 (!);
Linked twist maps (LTMs), Bernoulli mixing… The complexity and problematic aspects
of the relation between the Lagrangian
and Eulerian fields is seen in the example
of Lagrangian (kinematic) chaos or
Lagrangian turbulence (chaotic
advection) with a priori prescribed and
not random Eulerian velocity field (Elaminar). This is why Lagrangian
description  being physiclly more
transparent  is much more difficult than
the Eulerian description. In such Elaminar but Lturbulent flows the
Lagrangian statistics has no Eulerian
counterpart, as in the flow shown at the
left. Indeed, though the Eulerian velocity field, u(x;t) is not chaotic and
Indeed,
is regular and laminar, the Lagrangian velocity field v(a,t) ≡
u[X(a,t);t] is chaotic because X(a,t) is chaotic. This shows that, in
general, there does not exist a unique relation between Lagrangian
and Eulerian statistical properties in genuine turbulent flows as was
foreseen by CORRSIN 1959 : in general, there is no
in
reason to expect that Lik (the Lagrangian two point
velocity correlation tensor) and Eik (the Eulerian two point
velocity correlation tensor) will be uniquely related.
In other words it may be meaningless to look for such a relation. A list of a variety of other
attempts to analogies
Turbulence is rent by factionalism. Traditional approaches in the field are
under attack, and one hears intemperate statements against long time
averaging, Reynolds decomposition, and so forth. Some of these are
reminiscent of the Einstein–Heisenberg controversy over quantum mechanics,
and smack of a mistrust of any statistical approach. Coherent structures
people sound like The Emperor's new Clothes when they say that all
turbulent flows consist primarily of coherent structures, in the face of visual
evidence to the contrary. Dynamical systems theory people are sure that
turbulence is chaos. Simulators have convinced many that we will be able to
compute anything within a decade... The cardcarrying physicists dismiss
everything that has been done on turbulence from Osborne Reynolds until the
last decade. Cellular Automata were hailed on their appearance as the
answer to a maidens prayer, so far as turbulence was concerned .
LUMLEY 1990. BURGULENCE
In order to keep the formalism as simple as possible, we shall, work here
with the onedimensional scalar analog (!!!) to the NavierStokes euqation
proposed by Burgers³¹. In the method to be presented here, the true
poroblem is replaced by models that lead, without approximaton, to
closed equations for correlation functions and averaged Green's functions
(p. 124). The treatment of NavierStokes equation for an incompressible
fluid, which we shall discuss briefly, does not differ in essentials (p.143)
KRAICHNAN, R.H. 1961, Dynamics of nonlinear stochastic systems, J. Math Phys., 2(1), 124148)
Mathematical analysis will deal with several basic models. The simplest
one is the 1D Burgers equation with random forcing. It displays several
basic features of turbulence…3D NavierStokes systems probably need
completely new ideas. SINAI, YA.G. 1999 Mathematical Problems of Turbulence, Physica, A
263,565566 # Analogy between the Navier–Stokes equations and Maxwell’s equations: application to turbulence.
Analogy Screening.
# Beyond the Navier–Stokes equations, e,g. analogy between Boltzmann kinetic theory of fluids and
turbulence
# Modeling nearly incompressible turbulence with minimum Fisher information.
# Neural networks approach, the simulation and interpretation of free turbulence with a cognitive
neural system
# Variety of approaches from statistical physics/mechanics such as critical phenomena, Levy walks,
Gibbsian hypothesis in turbulence, Tsalis nonextensive statistics, quantum kinetic models of
turbulence
# Polymer analogies
# Stock market dynamics and turbulence: parallel analysis of fluctuation phenomena.
# Dynamical systems, e.g. low dimensional description.
There are more but all with modest succes (if at all) Perhaps the biggest fallacy about turbulence is that it can be reliably
described (statistically) by a system of equations which is far easier to solve
than the full timedependent threedimensional NavierStokes equations
BRADSHAW, 1994. CONCLUDING
CONCLUDING
The essential differences between the genuine turbulence and its
analogues (as those described above and many other not described) and
the intricacy of the relation between them (e.g. between genuine and
“passive turbulence”) require caution in promoting analogies to far
leading to grave misconceptions. On the other hand these very
differences can be effectively used to gain more insight into the dynamics
of real turbulence. The above examples also serve as a warning that flow visualizations used for studying the structure of dynamical fields (velocity,
vorticity, etc.) of turbulent flows may be quite misleading, making the question "what do we see?" extremely nontrivial. The general
reason is that the passive objects may not `want' to follow the dynamical fields (velocity, vorticity, etc.) due to the intricacy of the
relation between passive and active fields just like there is no one to one relation between the Lagrangian and Eulerian statistical
properties in turbulent flows. As mentioned one of the reasons is the presence of Lagrangian chaos, which is manifested as rather
complicated structure of passive objects even in very simple regular velocity fields. On the other hand the rampcliff structures of a
passive scalar are observed in pure Gaussian `structureless' random velocity field just like those in a variety of real turbulent flows
practically independently of the value of the Reynolds number TSINOBER 2001.
This does not mean that qualitative and even quantitative study of fluid motion by means of `color bands' (REYNOLDS1894) is
impossible or necessarily erroneous. However, watching the dynamics of material ‘colored bands' in a flow may not reveal the nature
of the underlying motion, and even in the case of right qualitative observations the right result may come not necessarily for the right
reasons. The famous verse by Richardson belongs to this kind of observation. On the other hand there are properties of passive
objects which do depend on the details of the velocity field (TSINOBER 2001, TSINOBER & GALANTI 2003). Just these very
properties can be effectively used to study the differences between the real turbulent flows and the artificial random fields, to gain
more insight into the dynamics of real turbulence. At present, however, the knowledge necessary for such a use is very far from being sufficient. With few exceptions it is
even not clear what can be learnt about the dynamics of turbulence from studies of passive objects (scalars and
vectors) in real and `synthetic' turbulence. This requires systematic comparative studies of both. An an attempt of such
a comparative study was made by TSINOBER & GALANTI 2003. This is a relatively small part of a much broader
field of comparative study of ‘passive' turbulence reflecting the kinematical aspects and genuine turbulence
representing also the dynamical processes. A BIT OF FUN
BIT Is there an analogy or
should we believe our
eyes ? ...
View Full
Document
 Spring '11
 Staff

Click to edit the document details