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 fl...
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 Fluid Dynamics, The Land, vorticity, Turbulent Flows, small scales

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