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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 VIIVIII NONLOCALITY
NONLOCALITY ‘KINEMATIC’ AND DYNAMIC
A SET/VARIETY OF MANIFESTATIONS/
PHENOMENA UNDER THE COMMON
NAME ‘DIRECT AND BIDIRECTIONAL
INTERACTION OF LARGE AND SMALL
SCALES’. This, however, contains two
ambiguous notions: SCALES and
SCALES
COUPLING/INTERACTION.
COUPLING/INTERACTION SCOPE. Nonlocality – kinematic and dynamic. Examples.
SCOPE
Manifestations of nonlocality. Small scale anisotropy at large
Small Reynolds numbers. Substantial lack of reflexional symmetry in small
scales in helically forced flows. Dependence of SS statistics (and
accelerations) on LS. Memory effects and predictability. Modification of
turbulence in multiphase flows. Drag reduction in turbulent flows of
flows
dilute polymer solutions and other drag reducing additives. Turbulence
induced secondary mean flows.
Related phenomena: Anistropic flows (MHD, stratified, rotating), steady
streaming in a turbulent oscillating flows, ‘negative eddy viscosity’, longrange correlations in Lagrangian setting, predictability and data
assimiliation, nature of forcing including acoustic; difference between
quasi2D and pure2D turbulent fows. From the formal mathematical point of view a process is called local if all
From
the terms in the governing equations are differential. If the governing
equations contain integral terms or other nonlocal operators then the
process is nonlocal. The NavierStokes equations are integrodifferential
for the velocity field in both physical and Fourier space (and any other).
Therefore, generally, the NavierStokes equations describe nonlocal
processes. From the physical point of view (the above meaning of)
nonlocality is associated with or long range or action at a distance. This
again is closely related to what is called direct and bidirectional coupling
of large and small scale ON SCALES AND THEIR COUPLING The ambiquity in
ON
definition of scales can be avoided by not using any
decomposition (D): the small scales (whatever this means) are
associated with the field of velocity derivatives (vorticity and
strain). Therefore, it is naturally to look at this field as the one
objectively (i.e. Dindependent) representing the small scales.
The velocity fluctuations represent the large scales.Taking the
above position one can state that in homogeneous (not
necessarily isotropic) turbulence the large and the small scales
are uncorrelated. For example, in a homogeneous turbulent flow
the mean Lamb vector 〈ω×u〉 = 0 (and 〈(u·∇)u〉 = 0 ). ON SCALES AND THEIR COUPLING. If the flow is statistically
ON
reflexionally symmetric then the mean helicity 〈ω·u〉 vanishes
too. Similarly, in a homogeneous turbulent flow velocity, ui, and
strain, sij do not correlate either, 〈uisij〉 = 0. However,
vanishing correlations do not necessarily mean absence of
dynamically important relations (vorticity and the rate of strain
tensor are precisely uncorrelated too 〈ωisij〉 ≡ 0, but their
interaction is in the heart of the physics of any turbulent flow).
Indeed, the quantities (u·∇)u ≡ ω×u+∇(u²/2) and
ω×u, are the main `guilty' for all we call turbulence. Both
contain the large scales (velocity) and small scales (velocity
derivatives). Therefore some kind of dynamically essential
coupling between the two is unavoidable. A REMINDING
REMINDING
ON WHAT CAN BE CALLED KINEMATIC NONOLCALITY The whole flow field is defined by the vorticity in the whole flow
domain with appropriate boundary conditions. This with is true
of strain too, e.g. there is a relation between the fields of velocity
and strain similar to that of BiotSavart. It is essential that the
above argument is based on vorticity and/or strain – as
invariant physical quantities – rather than any ‘surrogate’
derivatives. This can be qualified as most elementary aspect of
This
‘kinematic’ nonlocality as vortcity and starin can be seen as
(mostly) representing ‘small scales’. Vorticity and strain are not just velocity derivatives. They
Vorticity
are special for several reasons as mentioned (and will be
discussed at length below). The one to be stressed here is
that the whole flow field is determined entirely by the
field of vorticity or strain *with appropriate boundary
conditions: ∇²u = curl ω; ∇²ui=2∂sij/∂xj,
i.e. the velocity field is∇ a linear functional of vorticity u
i.e.
= F{ω}, or strain ui = G{sij},*i.e. alteration of
the field of velocity derivatives reflects on the velocity
field, vorticity and strain are not passive – they react back
and not only for the above (kinematic) reason. In the whole space the first functional is the well known BioSavart
law, and the second has a similar form (see eq. C.14’ in Tsinober 2001)
(see ui(x,t) = ∫µjjω(r)ωj(y,t)dy, µjjω (r)=  (4π)1 εijk rj/r³, ri = xiyi, ui(x,t) = ∫µjs(r)sij(y,t)dy, µjs(r)=  (2π)1 rj/r³, ri = xiyi, sij(x,t) = ∫P.V.αij(r)ωk (y,t)dy, αij (r)=  3(8π)1 {εijlrlrk+ εkjlrlri}/r5,
ri = xiyi, ∫P.V. stands for the Cauchy principal value. IN OTHER WORDS
IN Alteration of the field of velocity derivatives reflects on the
velocity field: vorticity and strain are not passive – they
velocity
react back and not only for the above (kinematic) reason
react
(likewise the field of a passive scalar is defined by its gradient G=∇θ and the field of
a vector potential A off a solenoidal passive vector B (rot A=B) iss defined by B).
A=B) i
o . The above statement is not that trivial as may seem.
The Three examples: FIRST, the purely kinematic reaction back, e.g. of vorticity in
FIRST,
the statistical sense, is seen from the following example. Taking
a Helmholz decomposition of the most significant part of the
nonlinear term in NSE, the Lamb vector ω×u = ∇α +
∇×β it can be shown that 〈(∇α)²〉 ~ 〈 ∇×β)²〉 if ω and
u are random Gaussian and not related, i.e. ω ≠ rot u.
However, if ω = rot u and u is random Gaussian, then
rot
is
〈(∇α)²〉 ~ 2〈∇×β)²〉 , i.e. ω ‘reacts back’ for purely
kinematic reasons. SECOND, kinematic coupling plus selfamplification of vorticity
and strain ‘react back' in creating the corresponding velocity field
. (since as mentioned the whole flow field  including velocity, which is mostly a large scale
object  is determined entirely by the field of vorticity or strain). In other words the small scales cannot be considered as a kind of passive objects
swept by the large scales or just ‘slaved’ to them at any (whatever
large) Reynolds numbers. This looks as a stong indication (and
warning) that the nature of dissipative processes (viscosity,
hyperviscosity, etc.) are important at any (whatever large)
Reynolds numbers (this issue will be addressed in the next leture).
It is noteworthy
(this
that due to nonlocality mostly small scale vorticity and strain are, generally, creating also some large scale velocity. (u·∇)u = ωxu + ∇(u2/2) In a plane channel flow (or any flow with d〈…〉/dy=0) d〈uv〉/dy = <ωxu>1
WEI AND WILMARTH 1989 ∇ Hence we have a clear indication of a dynamically
important statistical dependence between the large
(u) and small (ω) scales.
and Since in these turbulent
flows d〈uv〉/dy is
essentially different from
zero at any arbitrarily large
Reynolds number, one can
see from above that at least
some correlations between
velocity and vorticity are
essentially different from
zero too at any arbitrarily
large Reynolds number. The above and other aspects of nonlocality (to be addressed
The
below) contradict the idea of cascade (in physical space), which is
(in
local by definition . This means that the common view that
there exists a range of scales (the inertial range) in which
the effects of viscosity, boundary conditions, and largescale structures are unimportant is suspicous as
basically/conceptually incorrect (more in the next lecture on
(more
phenomenology and related). Nonlocality – dynamic. Long range forces due to pressure:
Nonlocality NavierStokes equations are integrodifferential. This is a property of
nonlocality of NavierStokes equations is in physical space : pressure is
a functional of the field of velocity derivatives p = P{Q}, where
where
Q=(1/4)(ω²2sijsij) = (1/4)(∂²uiuj/∂xi∂xj) is the second
Q=(1/4)(
invariant of the velocity gradient tensor ∂ui/∂xj. Again (it looks that)
(it
pressure too is completely determined by the field of velocity derivatives
(not just the field of velocity) in a nonlocal way (The pressure field can be directly expressed in terms of the field of velocity itself. However, since the latter is defined by the field of velocity derivatives,
so does the pressure). Nonlocality – dynamic. The nonlocality due to pressure is strongly
The
Nonlocality
associated with essentially non Lagrangian nature of pressure. For example,
replacing in the Euler equations the pressure Hessian ∂²p/∂xi∂xj ≡ Πij, which
is both nonlocal and non Lagrangian, by a local quantity
(1/3)δij∇²p=(1/6)ρ{ω²2sijsij} turns the problem into a local and
(1/3)
integrable one and allows to integrate the equations for the invariants of the tensor
of velocity derivatives ∂ui/∂xj in terms of a Lagrangian system of coordinates
moving with a particle. The reason for disappearance of turbulence (and formation
of singularity in finite time) in such models, called restricted Euler models, is that
the eigenframe of sij in these models is fixed in space (same is true for Burgers vortex),
(same
whereas in a real turbulent flow it is oriented randomly. This means that nonlocality
due to presure is essential for (self) sustaining turbulence: no pressure Hessian no turbulence. Accelerations. A related aspect is that the Lagrangian
Accelerations. acceleration a ≡ Du/Dt  a kind of small scale quantity – which is
dominated by the pressure gradient, ∇p. Hence the scaling
properties of the acceleration variance do not obey Kolmogorovlike
scaling (e.g. Hill, 2002) , most probably due to dominating
contribution of nonlocality. As Du/Dt = ∂u/∂t + u(∇⋅u) =
∂u/∂t + ωxu +∇(u2/2), i.e. it is a ‘mixed’ quantity due to
i.e.
presence of both velocity and velocity derivatives, this could be seen
also as a reason for the impact of nonlocality on the behaviour of
acceleration. However, the first one (i.e. ∇p) is a dynamical reason,
whereas the second one is of purely kinematic nature. ACCELERATION VARIANCE GULITSKII ET AL., 2007 Acceleration
variance does
not scale as
expected if it
would be a
small scale
quantity
The figure is based on
Fig. 8c from
GYLFASON ET AL.
2004 Loitsyanskii invariant. This is another aspect related to the
Loitsyanskii issue of the long range forces due to pressure (see ISHIDA 2006 ET AL and
(see references therein, also YAKHOT 2004).
YAKHOT assuming that that the integral above is converging (if second and third order correlations
(if
assuming
decay faster than r4, but not exponentially). Contrary to many objections based on the
importance of long range forces due to paressure at very large distances
ISHIDA 2006 ET AL gave convinsing argumnents and numerical evidence
that Loitsyanskii/Komogorov/Lanadau were right, i.e. the longrange interactions between remote eddies, as measured by the triple
correlations, are very weak. This, however, does not mean that long range forces due to pressure at NOT very large distances are unimportant. * In 1970 Loytsianskii told me about some astropysical observations which were in conformity with the invariance of I. Taking curl of the NSE and getting rid of the pressure does not remove the
Taking
nonlocality. Indeed, the equations for vorticity and enstrophy are nonlocal in
vorticity, ωi, since they contain the rate of strain tensor, sij, due to nonlocal relation
between vorticity, ω, and the rate of strain tensor, sij (‘kinematic' nonlocality)*,**
Both aspects of nonlocality are reflected in the equations for the rate of strain tensor
and total strain/dissipation, s² ≡ sijsij, and the equations for the third order
quantities ωiωjsij and sijsjkski. An important aspect is that the latter equations
contain invariant (nonlocal) quantities ωiωk ∂²p/∂xi∂xj and
sikskj∂²p/∂xi∂xj , reflecting the nonlocal dynamical effects due to pressure and
can be interpreted as interaction between vorticity and pressure and between
vorticity and and strain. These quantities are among possible candidates preventing
formation of finite time singularity in NSE.
________________________
* Nonlocality of the same kind is encountered in problems dealing with the behaviour of vortex filaments in an ideal fluid. Its importance is
manifested in the breakdown of the so called localized induction approximation as compared with the full BioSavart induction law).
** The two aspects of nonlocality are related, but are not the same. For example, in compressible flows there is no such relatively simple
relation between pressure and velocity gradient tensor as above, but the vorticitystrain relation remains the same. MANIFESTATIONS OF NONLOCALITY
MANIFESTATIONS
There exist massive evidence not only on direct interaction/coupling
between large and small scales but also that this interaction is
bidirectional such as the example of turbulent flows in a channel. We
mention also the well known effective use of fine honeycombs and
screens in reducing large scale turbulence in various experimental
facilities. One can substantially increase the dissipation and the rate of
mixing in a turbulent flow by directly exciting the small scales. The
experimentally observed phenomenon of strong drag reduction in and
change of the structure of turbulent flows of dilute polymer solutions
and other drag reducing additives is another example of such a ‘reacting
back' effect of small scales on the large scales. SS Anisotropy I. Lack of rotational symmetry.
Lack
SS One of the manifestations of direct interaction between large and small scales is the
One
anisotropy in the small scales. Though local isotropy is believed to be one of the
universal properties of high Re turbulent flows it appears that it is not so universal:
in many situations the small scales do not forget the anisotropy of the large ones.
There exist considerable evidence for this which has a long history starting
somewhere in the 50ies. Recently similar observations were made for the velocity
increments and velocity derivatives in the direction of the mean shear both
numerical and laboratory, see references in TSINOBER 2001, BIFERALE, L. AND
TSINOBER
PROCACCIA 2005, OULETTE ET AL 2006 . It was found that the stastistical
properties of velocity increments and velocity derivatives in the direction of the
mean shear do not conform with and do not confirm the hypothesis of local
isotropy. SS Anisotropy I. Lack of rotational symmetry.
Lack
SS
So it is not surprizing that in the ‘simlpe' example of turbulent channel flow the
So
flow is neither homogeneous nor isotropic even in the proximity of the midplane,
y ≈ 0 , where dU/dy ≈ 0. Indeed, though 〈u1u2〉 ≈ 0 in this region too,
dU/dy
d 〈u1u2〉 /dy is essentially nonzero and is finite independently of Reynolds
number as far as the data allow to make such a claim. This is also a clear
indication of nonlocality, since in the bulk of the flow, i.e. far from the
boundaries, dU/dy ≈ 0. Therefore the assumption that in the proximity
in
dU/dy
of the centerline of the channel flow, the local isotropy
assumption seem reasonable is incorrect. SS Anisotropy II. Lack of reflexional symmetry. Helicity.
SS The hypothesis of local isotropy (K41) includes restoring of all the
symmetries in small scales, i.e. the expectation is for restoring also of
of
reflectioninvariance at small length scales and that reflection symmetry
which is broken at large scales will tend to be restored asymptotically at
small scales (CHEN ET AL 2003, Phys. Fluids, 15, 361) . These authors claimed that this is
Phys.
15
These
the case: that there is a tendency toward equalization of energy in the + and that
components due to the nonlinear transfer between them. Hence, reflection
symmetry which is broken at large scales will tend to be restored
asymptotically at small scales. However, GALANTI AND TSINOBER 2004 showed
However,
showed that the opposite is true: to maintain finite helicity dissipation to balance the finite
to
helicity input (in a statistically stationary turbulence) the tendency to restore
reflection symmetry at small scales can not be perfect, since helicity dissipation is
associated with broken reflection symmetry at small scales, because helicity
dissipation is just proportional to the superhelicity H_{s}, showing the lack of
reflection symmetry of the small scales. Moreover, this lack of reflectional
symmetry should increase as the Reynolds number increases. SS Anisotropy II. Lack of reflexional symmetry. Helicity.
Lack
SS Helicity – H = ∫u·ωdx; superhelicity  Hs = ∫ω·curlωdx,
hyperhelicity Hh = ∫curlω·curlcurlωdx. The Equation for the helicity DH/Dt = 2νHs + FH
FH = 2∫f·ωdx is the term associated with forcing f in the right
hand side of NSE. The important point is that helicity dissipation 2νHs is
The
vanishing if reflexional symmetry holds. The Equation for the superhelicity DHs/Dt = PHs  2νHh+FHs
The term FHs = ∫curlf·curlωdx is associated with forcing f in the RHS of NSE,
and the term PHs =2∫curlω·curl(u×ω)dx is the production term of the
superhelicity Hs. There is a similar phenomenon of selfproduction of Hs and
approximate balance of PHs and 2νHh and irrelevance of forcing at this level. SS Anisotropy II. Lack of reflexional symmetry. Helicity.
SS
GALANTI AND TSINOBER 2004 Note the
tendency of
DHl²u3 to a
to
constant with
increasing Reλ
Re
[10] Chen et al 2003, Phys.
Phys.
Fluids, 15, 361 .
[16] Kurien et al. (2004) J.. Fluid
[16]
J
Mech., 515 . The ABC case. Dependence of normalized dissipation of helicity DHl²u3 and energy DElu3 on the
Taylor microscale Reynolds number Reλ. ◦  corresponds to the data from Ref. [10], □  correspond
Ref.
to the data from Ref. [16] and references therein. Along with other manifestations of direct interaction between large and small scales
Along
the deviations from local isotropy seem to occur due to various external constraints
like boundaries, initial conditions, forcing (e.g. as in DNS), mean shear/strain,
centrifugal forces (rotation), buoyancy, magnetic field, etc., which usually act as an
organizing factor, favoring the formation of coherent structures of different kinds
(quasitwodimensional, helical, hairpins, etc.). These are as a rule large scale
features which depend on the particularities of a given flow and thus are not
universal. These structures, especially their edges seem to be responsible for the
contamination of the small scales. This ‘contamination' is unavoidable even in
homogeneous and isotropic turbulence, since there are many ways to produce such
a flow, i.e. many ways to produce the large scales. It is the difference in the
mechanisms of large scales production which `contaminates' the small scales.
Hence, nonuniversality. This brings us to the next issue. Anisotropy III.
Anisotropy Statistical dependence of small and large scales.
Statistical
J. Fluid Mech., 5, 497—543 (1959). PRASKOVSKY ET AL 1993 J. Fluid Mech., 248, 493511.
248 Anisotropy III.
Anisotropy Statistical dependence of small and large scales.
Statistical Enstrophy ω2, total strain s2 and squared acceleration a2 conditioned on magnitude of the
Enstrophy
velocity fluctuation vector, Field experiment, SilsMaria, Switzerland, 2004, Reλ= 6800
(hopefully JFM, 2007) Closures and constitutive relations I.
Closures
The nonlocality due to the coupling between large and small scales is also manifested (and is a
The
concern) in problems related to various decompositions of turbulent flows and in the so called
closure problem. For example, in the Reynolds decomposition of the flow field into the mean and the
fluctuations and in similar decompositions associated with large eddy simulations (LES) the relation
between the fluctuations and the mean flow (or resolved and unresolved scales in LES, etc.) is a
nonlinear functional. That is the field of fluctuations at each time/space point depends on the mean
(resolved) field in the whole time/space domain. Vice versa the mean (resolved) flow at each
time/space point depends on the field of fluctuations (unresolved scales) in the whole time/space
domain. This means that in turbulent flows pointwise flow independent `constitutive' relations
analogous to real material constitutive relations for fluids (such as stress/strain relations) can not
exist, though the `eddy viscosity' and `eddy diffusivity' are used frequently as a crude
approximation for taking into account the reaction back of fluctuations (unresolved scales) on the
mean flows (resolved scales). The fact that the `eddy viscosity' and `eddy diffusivity' are flow (and
space/time) dependent is just another expression of the strong coupling between the large and the
small scales. Closures and constitutive relations II.
Closures
The simplest version of this approach with a scalar eddy viscosity leads always to a positive
The
subgrid dissipation (positive energy flux from the resolved to the unresolved scales), whereas a
priory tests of data from real flows (experiments and DNS) show that there exist considerable
regions in the flow with negative subgridscale dissipation (called backscatter). The exchange of
`information' between the resolved and unresolved scales is pretty reach and is not limited by
energy. For example, BOS ET AL., 2002 (see also references therein) report that the subgridBOS
scales have a variety of significant effects on the evolution of field of filtered velocity gradients. So
it is too optimistic to claim, for examle, that LES of wall bounded flows ... resolve all the
resolve
important eddies... has received increased attention, in recent years, as a tool
to study the physics of turbulence in flows at higher Reynolds number, or in
more complex geometries, than DNS (PIOMELLI AND BALARAS 2002). The qualification of large scale (resolved) eddies as the most important ones is too subjective: all
eddies are important in view of direct and bidirectional coupling of essentially all eddies. It is
doubtful that LES or any other similar approach can be used as a tool to study the physics of
turbulence, since a vitally important part of physics of turbulence resides in the unresolved scales. Memory effects. The far field statistical properties of free shear turbulent
Memory flows (mixing layers, wakes, jets) and also boundary layers (Journal of Turbulence, 5, 015) are
(Journal
known to possess strong memory (`nonlocality in time'): they are sensitive to the
conditions at their ‘start' with some properties not universal in Reynolds number,
BEVILAQUA AND LYKOUDIS 1978, DIMOTAKIS 2001, GEORGE & DAVIDSON
BEVILAQUA
2004. These flows develop in space beginning with small scales into the large ones, in apparent contradiction to the RichadrsonKolmogorov cascade ideas. It is
noteworthy that passive tracers in such flows possess even stronger memory,
CIMBALA ET AL. 1988, due to importance of Lagrangian aspects of their
CIMBALA
evolution. The problem of predictability of turbulent flows involves nonlocality in
time either: a small scale perturbation (both in time and space) perturb
substantially the whole flow including the largest scales within time of the order of
integfral time scale. In this sense instability can be seen as nonlocality in time. SAME FLOW  NOT THE SAME PATTERN
CIMBALA ET AL. 1988 Particulate flows. The basic interaction of the carrier fluid flow with
Particulate particulates occurs at the scale of the particle size, i.e. at small scales.
However, a number of essential phenomena emerge at much larger scales
in a variety of particualate flows: sedimenting suspensions, fluidized beds,
formation of bedforms and their interaction with the carrier fluid,
preferential concentration of particles/bubles (clustering) in and
modification of turbulent flows. These phenomena are treated in terms of
large scale instabilities, intrinsic convection in sedimenting suspensions,
collective phenomena, longrange multibody hydrodynamic
interactions/correlations, clusters. All these are essentially fluid mediated
phenomena/interactions as contrasted with direct particle/particle
interactions. Therefore, nonlocality is expected to be significant in these
phenomena (see TSINOBER 2003 for more and references). Particulate flows. For example, an important process in the
For
Particulate interaction of the carrier fluid flow with particles (or any other additives) is
the production (or more generally modification) of velocity derivatives, i.e.
vorticity and strain. The modified field of velocity derivatives reacts back in
changing the large scales of the flow (both velocity and pressure). It is
tempting to see this process as the one underlying the formation of the
mentioned large scale features, though the details in each case are different
and in most cases are poorly understood. These processes are modified by
specific features such as inertial bias, i.e. inertial response of particles to
fluid accelerations and preferential concentration of particles(bubbles) in
strain (vorticity) dominated regions. The latter may lead to enhanced bias
of strain dominated regions (heavy particles), i.e. regions with large
dissipation, or regions with strong enstrophy (bubbles). Clusters as manifestion of nonlocality. Both kinematic
Clusters
. (Lagrangian) aspects and dynamic and are important.
Clustering of inertial particles in kinematic simulations, Lu Chen, Ph. D. Thesis 2007 Inertial Particles and
streamfunction Inertial particles and
stagnation points The effect is considerable at
The
much larger scales than of the
order of Kolmogorov scale. For
example, see from figure 3b in
KOSTINSKI AND SHAW
KOSTINSKI
2001 in which the maximum occurrs at scales ~ 1cm, but
the effect is considerable at 1 m
and is seen even up to 10 m (see
(see also figure 5 in ROUSON AND EATON
ROUSON
2001, in which significant effect is
seen at scales of halfwidth channel). CLUSTERS –
CLUSTERS
SMALL AND
LARGE
KOSTINSKI AND SHAW 2001
figure 3b CLUSTERS –
CLUSTERS
SMALL AND
LARGE
Dependence of particle spatial
distribution for various Stokes
numbers. Each panel represents a
“slice” through the computational
domain from a direct numerical
simulation of homogeneous,
isotropic turbulence containing
particles. Figure courtesy of L. R.
Collins. CLUSTERS – SMALL AND LARGE
CLUSTERS
FALLON AND ROGERS 2002 Turbulenceinduced preferential concentration
of solid particles in microgravity conditions ALISEDA ET AL 2002 Settling velocity of heavy partricles in
a wind tunnel experiment Dilute polymer solutions. Turbulent flows can be strongly
Dilute modified by additives in even extremely small concentrations. The most
spectacular changes occur with only few parts per million of flexible
polymers added to the solvent. These changes are exhibited in a
number of flow parameters both large scales and small scales, though
the direct action of the dissolved polymers is obviously in the small
scales. Hence again nonlocality. The large scale manifestations are
represented in the first place by strong reduction of drag (up to 80%)
in turbulent shear flows. Along with this other global/ large scale
effects on turbulence structure are observed both experimentally and in
simulations. Other related issues I. The nonlocality in the sense of concern here is
Other especially strongly manifested in the atmospheric convective boundary layers in which
the common downgradient approximation is not satisfactory due to countergradient
heat fluxes, ZILITINKEVICH ET AL. 1998 and references therein. We mention also a
ZILITINKEVICH
similar phenomenon in stably stratified turbulent flows  the so called PCG, persistent
countregradient fluxes. The essence of PCG is the countergradient transport of
momentum and active scalar. It is observed at large scales when stratification is
strong, but in small scales it is present with weak stratification as well.
on
There is a class of flows with the socalled phenomenon of ‘negative eddy viscosity',
There
see references in TSINOBER 2001. It occurs in the presence of energy supply other
TSINOBER
than the mean velocity gradient. In such flows the turbulent transport of momentum
occurs against the mean velocity gradient, i.e. from regions with low momentum to
regions with high momentum (i.e. the Reynolds stresses as one of the agents of coupling the fluctuations with
(i.e.
the mean flow act in such flows in the `opposite' direction as compared to the usual turbulent shear flows). Concomitantly kinetic energy moves in the ‘opposite' direction too  from fluctuations
to the mean flow.. Other related issues II . The flows with negative eddy viscosity are akin
Other to nonturbulent but nonstationary flows in a fluid dominated by its fluctuating
components and known (since RAYLEIGH 1883) under the name (acoustic)
RAYLEIGH
steady streaming, RILEY 2001, in the sense that in these flows a mean (time
RILEY
averaged) flow is induced and driven by the fluctuations. Recently turbulent flow of
this were observed too, SCANDURA 2007.
SCANDURA
There are examples of ‘usual' turbulent flows with turbulence induced mean flows.
The best known ones are flows in pipes with noncircular crossection. In such flows
a mean secondary flow is induced which is absent in purely laminar flow. For
example, see PATTERSON REIF AND ANDERSSON 2002 for references on such
PATTERSON
for
flows in a square duct. Other related issues III . Predicatbility, data assimilation and
Predicatbility
Other
‘determining modes’. ARNOLD, 1991 SO WHAT ARE POSSIBLE
SO
CONSEQUENCES AND QUESTIONS ?
CONSEQUENCES
# NATURE OF DISSIPATION  IS IT (UN)IMPORTANT
GENERALLY (is it just stupid passive sink of energy?) # AND FOR THE SO
(is
CALLED INERTIAL RANGE (IR)? # ARE (ALL) ITS
PROPERTIES REALLY INDEPENDENT OF VISCOSITY/NATURE
OF DISSIPATION? # PROBLEMS WITH VARIOUS ASPECTS OF
PHENOMENOLOGY AND DECOMPOSTIONS SUCH AS: IS
"CASCADE" IN GENUINE TURBULENCE WELL DEFINED? * IS
THERE CASCADE IN PHYSICAL SPACE? *HOW MEANINGFUL
IS CASCADE OF PASSIVE OBJECTS? These and similar in the next lecture on
phenomenology and related at February 21
at ...
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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
 The Land

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