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 VVI VORTICITY AND STRAIN IN TURBULENCE
VORTICITY
Why velocity derivatives? Vorticity versus strain. Selfamplification
(SA) of both: who is responsible for enhanced disspation? Stretching
versus compressing. Geometrical statistics. Concentrated vorticity or
strain. Unrestanding the physics of SA of vorticity and strain is the heart
of approaching the turbulence problem.
Selfamplification of the strainrate field is something
I overlooked up to now ... TENNEKES 2003
TENNEKES One of the important common features of
One
processes resulting in turbulence is that all of
them tend to enhance the rotational and
dissipative properties of the flow in the process
of transition to turbulence. The first property is
associated with the production of vorticity
(turbulence is highly rotational), whereas the second
property is due to the production of strain
(turbulence is strongly dissipative). The latter is a unique
process of genuine turbulence and does not
have any analogue in ‘passive’ turbulence. VORTICITY AND STRAIN
LIVE TOGETHER CHEN, 2000 BUT HAVE A VERY
DIFFERENT LIFE DNS of forced NSE in a periodic box Reλ=220, ω = 3ωmean, s = 3smean One of the important
One
common features of
processes resulting in
turbulence is that all of
them tend to enhance the
rotational and dissipative
properties of the flow in
the process of transition to
turbulence. The first
property is associated with
the production of vorticity,
whereas the second
property is due to the
production of strain. INTEREST TO Aik= ∂ui/∂xk
VORTICITY AND RELATED
Skewsymmetric part of Aik= ∂ui/∂xk
Skew
TAYLOR 1938 ω  obsession
DISSIPATION/STRAIN
DISSIPATION/STRAIN
Symmetric part of Aik = ∂ui/∂xk, i.e. sik
Symmetric
KOLMOGOROV 1941 ε  religion Vorticity obsession since HELMHOLTZ and KELVIN
HELMHOLTZ and 1867 TAYLOR 1937, CORRSIN 1953
The turbulence syndrome includes the following symptoms:…it
is essentially nonlinear and rotational STEWART 1963
STEWART
Turbulence is rotational and three dimensional. Turbulence is
characterized by high levels of fluctuating vorticity. For this
reason, vorticity dynamics plays an essential role in the
description of turbulent flows. TENNEKES AND LUMLEY 1972
What is turbulence but a random chaotic field of vorticity,
What
whose strong nonlinear interactions makes the problem so
nonlinear
impossibly difficult? ... the concept of the coherent structure in
turbulent shear flows has led to the picture of such flows as a
superposition of organized, ‘deterministic ’ vortices whose
evolution and interaction is the turbulence. SAFFMAN 1981 WHY SMALL SCALES?
AND WHAT THEY ARE ? A great deal of interest is concentrated on the
great
Reynolds stress tensor 〈uiuj〉 (or similar quantities in
LES) and the “closure problem”, i.e. how it is related
to the mean flow (with the – not obvious  assumption that a relatively
simple relation – if at all  does exist). Such an approach is
Such
based on the view that small scales (whatever this
means) are “slaved” to the large scales and are
mostly a kind of passive sink of energy. This is the
so called “classical approach” with the view that in
order to understand turbulence one needs only to
‘resolve’ the large scales and ‘model’ the small
scales (can this be done without sufficient understanding and/or full resolution
down to all physically relevant scales?). However, today it is high time to ask how
However,
much justified is such oversimplified
treatment of small scales via methods like
eddy viscosity, and similar. The small scales
contain a great deal of essential physics of
turbulent flows, much of which is not known or
poorly understood, and which are intimately
and bidirectionally related to the large scales
(nonlocality). Apart of basic there are
numerous problems in which one has to deal
explicitly with the nature, structure and
dynamics/evolution of small scales. For instance, special information on small scale structure(s) is
For
needed in problems concerning, e.g. combustion, disperse multiphase
flow, mixing, cavitation, turbulent flows with chemical reactions,
some environmental problems, generation and propagation of sound
and light in turbulent environments, and some special problems in
blood flow related to such phenomena as hemolysis and
thrombosis. In such problems, not only special statistical properties
are of importance like those describing the behaviour of smallest
scales of turbulence, but also actual ‘nonstatistical' features like
maximal concentrations in such systems as an explosive gas which
should be held below the ignition threshold, some species in
chemical reactions, concentrations of a gas with strong dependence
of its molecular weight on concentration (such as hydrogen fluoride
used in various industries e.g. in production of unleaded petrol) and
toxic gases. THE ABOVE IS A CLEAR INDICATION
THE WHY VELOCITY DERIVATIVES
ARE SO IMPORTANT, BUT THERE IS MUCH MORE Velocity derivatives, Aij = ∂ui/∂xj, play an outstanding
Velocity
role in the dynamics of turbulence for a number of
reasons. Their importance has become especially
clear since TAYLOR, 1938 and KOLMOGOROV, 1941.
TAYLOR,
KOLMOGOROV
Taylor emphasized the role of vorticity, i.e. the
antisymmetric part of the velocity gradient tensor Aij =
∂ui/∂xj, whereas Kolmogorov stressed the importance
of dissipation, and thereby of strain, i.e. the symmetric
part of the velocity gradient tensor. It is important (see
It
(see
below) that the whole (incompressible) flow field is
fully determined by the fields of vorticity or strain
with appropriate boundary conditions. Apart from vorticity and strain/dissipation, there are many
other reasons for special interest in the characteristics of
the field of velocity derivatives, Aij=∂ui/∂xj, in turbulent
flows:.
In the Lagrangian description of fluid flow in a frame
In
following a fluid particle, each point is a critical one, i.e. the
direction of velocity is not determined. So everything happening
in its proximity is characterized by the velocity gradient
Aij=∂ui/∂xj. For instance, local geometry/topology is naturally
described in terms of critical points terminology.
The field of velocity derivatives is much more sensitive to the
The
nonGaussian nature of turbulence or more generally to its
structure, and hence reflects more of its physics.
The possibility of singularities being generated by the Euler
The
and the NavierStokes equations (NSE) and possible
breakdown of NSE are intimately related to the field of velocity
derivatives. There is a generic ambiguity in defining the meaning of the term small
There
scales (or more generally scales) and consequently the meaning of the
term cascade in turbulence research. The specific meaning of this term
and associated interscale energy exchange/`cascade' (e.g. spectral energy
transfer) is essentially decomposition/representation dependent. Perhaps,
the only common element in all decompositions/representations (D/R) is
that the small scales are associated with the field of velocity derivatives.
Therefore, it is natural to look at this field as the one objectively (i.e. D/R
independent) representing the small scales (one can – following CORRSINCORRSIN
(one
use higher order derivatives, e.g. curl ω, ∇²sij ). Indeed, the dissipation is
Indeed,
associated precisely with the strain field, sij, both in Newtonian and nonNewtonian fluids. There is a number of more specific reasons why
studying the field of velocity derivatives is so important in the dynamics of
turbulence. 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 ∇²ui = 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. WHY STRAIN TOO?
WHY
IS IT AN EQUAL PARTNER OF
VORTICITY? SStrain controls the flow in the same way as does vorticity also in the sense of possible
train
vorticity also
breakdown of smooth solutions for 3D flows, Ponce, G. (1985) Remarks on a paper J.T. Beale, T.
Ponce,
breakdown
Kato and A. Majda, Commun. Math. Phys., 98, 3453353.
Majda Commun
98 53. STRAIN – AN EQUAL PARTNER
STRAIN In some cases even more
In ‘Dolphins in phosphorescent sea’.
The inspiration for this woodcut,
created by M. C. Escher in 1923,
created
was the flowinduced
induced
bioluminescence that occurs on
dolphins when they swim through
waters that contain high levels of
waters
bioluminescent plankton. Some reasons More below 1. Most important stresses in fluids (both Newtonian and nonNewtonian) flows are defined by strain
2. Energy dissipation is directly associated with strain (both Newtonian
(both
and nonNewtonian) and not with vorticity.
3.. The energy cascade (whatever this means) and its final result 3
dissipation, are associated with predominant selfamplification of
the rate of strain and vortex compression rather than with vortex
stretching. That is another nonzero odd moment  sijsjksjk responsible for the production of strain, is not less important than
the enstrophy production ωiωjsij 4.. Vortex stretching is essentially a process of interaction of
4
vorticity and strain. “Vortices” iinteract via their strain fields.
nteract
5.. Strain dominated regions appear to be the most active/nonlinear
5
in a number of aspects.
6. Interaction of the flow field with additives (particles, polymers,
blood cells) is mostly via strain .
7. Though formally all the flow field is determined entirely by the
field of vorticity the relation between the strain and vorticity is
strongly nonlocal. In many cases, they are only weakly statistically
correlated or not correlated at all. BASIC EQUATIONS
Dω/Dt = (ω·∇)u + νΔω + ɛijk∂Fk/∂xj
(½)Dω²/Dt = ωiωjsij + νωi∆ωi + ɛijkωi ∂Fk/∂xj Dsij/Dt =  sjkski – (1/4)(ωiωj ω2δij) –
∂²p/∂xi∂xj + ν∆sij + Fij
(½)Ds²/Dt =  sijsjkski – (1/4)ωiωjsij –
sij∂²p/∂xi∂xj + νsij∆sij + sijFij In what sense vortex stretching plays a central role in the
energy cascade to small scales and dissipation? OR WHO IS THE ‘GUILTY’ ? I One of the most basic phenomena and distinctive features of threedimensional turbulence is the predominant vortex
One
stretching. This process occurs via interaction of vorticity and strain.
strain (½)Dω²/Dt = ωiωjsij + νωi∆ωi + ɛijkωi ∂Fk/∂xj
νω
It is a very common view that this process is responsible for the enhanced dissiption in 3D turbulent flows (TAYLOR 1938, ONSAGER 1949 and everybody on). It seems that the stretching of vortex filaments must be regarded as the
principal mechanical cause of the high rate of dissipation which is associated
with turbulent motion, TAYLOR 1938
Vortexline stretching plays a central role in the energy cascade to small scales and
dissipation. RHINES, 1997 (p.102)
RHINES BUT, dissipation is
BUT, 2=
2νs 2νsijsij !!
!! The true physical causal relation is between
The
dissipation and strain both in Newtonian and nonNewtonian fluids. Therefore it is a misconception to
associate dissipation directly with vorticity. However,
the rotational nature of turbulence (i.e. vorticity) is
crucial for dissipation.
As a home work show that in irrotational flow (i.e.
As
u=∇ϕ) the dissipation is either zero or volume integral
2∫s2dV over the flow domain is equal to the surface
integral ∫∇u2dV
∫∇ WHO IS THE ‘GUILTY’ ? II There exist two nonlocally connected and weakly correlated
two
processes. The ‘second’ is the selfamplification of strain.
The (½)Ds²/Dt =  sijsjkski – (1/4)ωiωjsij –
sij∂²p/∂xi∂xj + νsij∆sij + sijFij
It is this ‘second’ process which is directly responsible for the
It
directly responsible
enhanced dissipation in turbulent flows. Moreover the ‘first’ one
(i.e. the enstrophy production) is opposing production of strain.
Note that the production of strain is 1) much more ‘’self‘, 2) it is a
specific (!!) feature of the dynamics of genuine turbulence having
no counterpart in the behavior of passive objects, 3) Enstrophy
3)
production ωiωjsij has an additional role in exchanging
“energy” between enstrophy and strain. NONLOCALITY OF
NONLOCALITY OF
VORTICITY/STRAIN
RELATION AND THE
ISUE OF SURROGATES
ISUE Field experiment
2000, Israeli field
station, Reλ=104 ωiωjsij AND sjkskisij VERSUS
SURROGATE 17.5∂u1/ ∂x1
17.5 Field experiment 2004, SilsMaria, Switzerland, Reλ= 6800 THE MARIA SILS SITE, SWITZERLAND
SILS FIELD EXPERIMENT SUMMER 2004 SILS
MARIA, SWITZERLAND
Height 1850 m
Experiment was
Experiment
performed in
collaboration of Institute
of Hydromechanics and
Water Resources
Management, ETH Zurich
Management,
The
Israeli
team The calibration unit at 3 m
in the field THE PROBE
THE 3 mm hot wires cold wires Manganin is used as a
material for the sensor
prongs instead of
tungsten because the
temperature coefficient of
the electrical resistance of
manganin is 400 times
smaller than that of
tungsten. The tip of the probe with prongs made of manganin
The
manganin SELFAMPLIFICATION OF VORTICITY
AND STRAIN (½)Dω²/Dt = ωiωjsij + νωi∆ωi +
ɛijkωi ∂Fk/∂xj (½)Ds²/Dt =  sijsjkski –
(1/4)ωiωjsij – sij∂²p/∂xi∂xj +
νsij∆sij + sijFij
The property of self amplification of vorticity and strain is responsible for the fact the neither
enstrophy ω² nor the total strain s² are inviscid invariants as is the kinetic energy u² SELFSELF
AMPLIFICATION OF VORTICITY
AND STRAIN SELFRANDOMIZATION/INTRINSIC STOCHASTICITY: NO SOURCE OF
RANDOMNESS IS NEEDED, THE FORCING CAN BE CONSTANT IN TIME AT THE LEVEL OF VELOCITY
DERIVATIVES: VORTICITY AND
STRAIN (DISSIPATION)
THE EXTERNAL FORCING IS
IRRELEVANT
Three cases:
1. DNS in a periodic box, Reλ=102
2. DNS in a channel flow, Re=5600
3. Atmospheric SL, Reλ=104; Re=108 0.5
F
O
R
C
I
N
G VELOCITY DERIVATIVES 5
0 0
5
0.5
5000 10
0 5000 Assume there is no production of enstrophy in the
Assume
mean 〈ωiωjsij〉 = 0. Is there turbulence?
Is
What about a similar assumption for strain
production 〈sijsjkski〉 = 0 ?
. Enstrophy production is approximately balanced by
viscous terms at any  whatever large  Reynolds
number?*
TENNEKES AND LUMLEY (1972, P.91):
P.91 (½)Dω²/Dt = ωiωjsij + νωi∆ωi + ɛijkωi ∂Fk/∂xj
νω
Three cases:
flow, Re=5600 1. DNS in a periodic box, Reλ=102 2. DNS in a channel
3. Atmospheric surface layer, Reλ=104; Re=108 More: Spatial integrals, running averages Similar result holds for strain production
*(In this sense  but not only in this  turbulence is not slightly viscous at whatever large Reynolds number. In this context the
*(In
but
umber.
question: what happens with enstrophy/strain production as ν→0 is of special interest)
enstrophy/strain
ν→ TEMPORAL EVOLUTION OF SPATIAL INTEGRALS
IN THE ENSTROPHY BALANCE EQUATION
Reλ = 2x102 ∫(½)Dω²/Dt = ωiωjsij + νωi∆ωi + ɛijkωi ∂Fk/∂xj
νω
dEω/dt Pω Dω Fω Note i) approximate balance between Pω and Dω
and ii) irrelevance of the forcing term Fω Similar behavour is observed with hyperviscosity ARE VORTEX LINES (OR VORTICITY)
ARE
VORTICITY
APPROXIMATELY FROZEN IN FLUID FLOW
AT LARGE REYNOLDS NUMBERS?
AT
a material line which is initially coinsides
material
with 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 diffrent instants). BATCHELOR, 1967, p. 274
1967 ... STRETCHING VERSUS (AND/OR?)
STRETCHING
(AND/OR?)
COMPRESSING
In turbulent flows 〈ωiωjsij〉 > 0 due to predominant
stretching. However, there is no stretching without
compressing (div u=0), so what is the meaning of
so
predominant stretching? Also 〈 sijsjkski〉 > 0. Is it
0. Is
too due predominat stretching? To clarify this matter a
bit of geometrical statistics is needed GEOMETRY OF VORTEX STRETCHING
VORTEX STRETCHING ωiωjsij = ω2{Λk ω,λk)} cos2( BETCHOV
1956 λk – eigenvectors of the rate of strain tensor sij, Λk  eigenvectors of the rate of strain tensor sij,
Λ1 > Λ2 >Λ3; Λ1+Λ2 +Λ3 = 0 (div u=0);
Λ1 > 0, Λ3<0;
ω2{Λ1 cos2(ω,λ1)} = I
ω2{Λ2cos2(ω,λ2)} = II
II ω,λ3)} = III
III ω2{Λ3 cos2( 〈I〉: 〈II〉: 〈III〉 =
3: 1: 1
3: EIGEN CONTRIBUTIONS
EIGEN Λ α cos 2 (ω, λ α ) Λ2 cos 2 (ω, λ α )
α 1.2m 2.0m 3.0m 4.5m 7.0m 10m 1 1.44 1.60 1.36 1.31 1.53 1.04 1.37 2 0.44 0.62 0.67 0.46 0.46 0.58 0.49 0.87 1.22 1.03 0.77 0.99 1 0.53 0.33 0.29 0.52 0.46 0.49 0.49 2 0.09 0.05 0.15 0.14 0.13 0.16 0.15 0.38 0.63 0.56 0.34 0.41 0.35 0.36 1 1.77 1.56 1.63 1.91 2.08 1.55 2.19 2 0.47 0.50 0.52 0.45 0.47 0.54 0.47 3 ω2 Λ2 cos 2 (ω, λ α )
α 0.8m 3 ω2 Λ α cos 2 (ω, λ α ) α 3 Value 1.24 1.07 1.15 1.36 1.55 1 0.51 0.50 0.50 0.51 0.49 0.50 0.49 2 0.08 0.09 0.10 0.10 0.10 0.11 0.10 3 0.41 0.41 0.41 0.40 0.41 0.40 0.41 0.62 0.85 1.09 1.66 ωiωjsij = ω2{Λ1 cos2(ω,λ1)} I + ω2{Λ2 cos2(ω,λ2)} II + ω2{Λ3 cos2(ω,λ3)} III Note that the dominating term is the first one (I)
Note
associated with the first eigenvector λ1 corresponding to
the largest (purely positive) eigenvalue. This is
This comprises the meaning of ‘predominant stretching’. Yet vorticity is preferentially aligned with the second
eigenvector λ2 which is quite a bit counterintuitive. ALIGNMENT BETWEEN THE EIGENFRAME λI OF THE
RATE OF STRAIN TENSOR Sij AND VORTICITY ω Note essentially the same behavior at large and small Reλ
Field experiment; Reλ= 104 3DPTV; Reλ=60
Same in DNS PDFS OF EIGENVALUES, Λk
PDF OF EIGENVALUES
OF THE RATE OF STRAIN TENSOR sij
Field experiment
2004, SilsMaria,
Switzerland, Reλ=
6800 EIGENVALUES Λi
EIGENVALUES
OF RATE OF STRAIN, sij
Value i 0.8m 2m 10m 1 0.53 0.51 0.47 2 0.09 0.09 0.06 0.62 0.60 0.53 1/ 2 Λα / s2 3 3/2 0.40 0.40 0.41 0.04 0.04 0.06 3
Λ3 / s 2
α 1
2 Λ2 / s 2
α 0.56 0.56 0.55 1 0.48 0.54 0.60 2 0.01 0.02 0.02 0.73 0.83 1.04 3 So what about 〈 sijsjkski〉 > 0 (in homogeneous
So
flows equal to  4/3 〈ωiωjsij〉 ) is it due to
predominant stretching?
predominant
sijsjkski = Λ13 +Λ23 +Λ33 = 3Λ1Λ2 Λ3 〈Λ13〉 : 〈Λ23〉 : 〈Λ33 〉 = 1.5 : 0.05: 2.5
That is 〈 sijsjkski〉 = {〈Λ13〉 + 〈Λ23〉 +〈Λ33 〉} is
That positive due to dominant contribution from  〈Λ33 〉, i.e.
i.e.
predominant compressing rather than stretching !
compressing
There are several other (see the examples below) important processes
(see
in which predominant compressing is the main player : there is
no reason to push everywhere stretching, especially vortex
stetching. MORE EXAMPLES WITH PREDOMINANT
CONTRIBUTION OF COMPRESSING
TKE PRODUCTION IN TURBULENT SHEAR FLOWS 〈uiuk〉Sik EVOLUTION OF DISTURBANCES IN
GENUINE AND PASSIVE TURBULENCE PRODUCTION OF VORTICITY GRADIENTS IN TWO DIMENSIONAL TURBULENCE PRODUCTION OF GRADIENTS/ DISSIPATION OF
PASSIVE SCALAR TKE PRODUCTION IN SHEAR FLOWS
TKE
STRETCHING OR COMPRESSING? The turbulent energy production in a turbulent shear flow is known to
The
be represented by the term <uiuk>Sik, with ui being the components
with
of velocity fluctuations, and Sik the mean rate of strain. In turbulent
flows which are twodimensional in the mean (i.e. such that ∂〈⋯ /∂x3 =
〉
0) the production term can be represented as  <uiuk>Sik =  〈u²Λ1Scos²(u,λ1S)〉  〈u²Λ2Scos²(u,λ2S)〉 where u² = u1² + u2², ΛiS are the eigenvalues and λiS are the
corresponding eigenvectors of the mean rate of strain tensor Sik, and
Λ1S>0,Λ2S<0 (Λ3S=0). Since the term associated with the stretching
of material elements is negative,  〈u²Λ1Scos²(u,λ1S)〉 < 0, and the term
associated with the compressing of material elements is positive, 〈u²Λ2Scos²(u,λ1S)〉 > 0, the production term <uiuk>Sik can be (and
usually is) positive due to positiveness of the term associated with the
compressive (negative) eigenvalue/eigenvector {Λ2, λ2S}, of the mean
strain Sik. In this sense the turbulent energy production is due to the
predominant compressing of material elements rather than stretching. TKE PRODUCTION: STRETCHING OR
TKE COMPRESSING?
Turbulent
channel flow
(same in BL)
Phys Fluids, 16,
16
2704 (2004) The main feature  tendency of alignment of the vector u with both
λ1S and λ2 S. The difference is that the latter alignment is somewhat
stronger, which results in the positive value of the TKE production TKE PRODUCTION:
PRODUCTION: STRETCHING OR COMPRESSING?
STRETCHING Tennekes & Lumley 1972 pp 4041 TKE PRODUCTION:
TKE PRODUCTION: STRETCHING OR COMPRESSING?
STRETCHING TKE PRODUCTION: STRETCHING OR
TKE PRODUCTION: COMPRESSING? SUBGRID STRESSES TKE PRODUCTION: STRETCHING OR
TKE PRODUCTION: COMPRESSING? SUBGRID STRESSES GROWTH OF DISTURBANCES IN
GENUINE (EΔu, EΔω, EΔs) AND
GENUINE
PASSIVE (EΔA , EΔB , EΔθ , EΔG)
TURBULENCE
Looking at the evolution of the disturbance
Δu of some flow realization u in a statistically
steady state and similarly for other quantities.
For more details see Tsinober and Galanti 2003, Phys.
For
Phys.
Fluids, 15, 35143531. Again compression
Again
rather than stretching GROWTH OF DISTURBANCES IN GENUINE, EΔu, EΔω, EΔs
GROWTH
AND PASSIVE, EΔA , EΔB , EΔθ , EΔG, TURBULENCE
The process of
The
evolution and
amplification of
disturbances  both
in genuine and
`passive' turbulence
 is dominated by the
strain field of the
basic flow. In all their energy
production has
the form ΔuiΔujsij
Tsinober & Galanti, 2003
Galanti PRODUCTION OF VORTICITY GRADIENTS IN
TWODIMENSIONAL TURBULENCE
Dξ/Dt =  (ξ·∇)u + νΔξ ; ξ = ∇ω (1/2)Dξ2/Dt =  ξiξk sik + νξiΔξi
The main contribution to the production (〈ξiξksik〉 > 0)
 〈ξiξk〉sik =  ξ2Λkcos2(ξ,λk)
is associated with compressive eigenvalue Λ3: it is due to ξ2Λ3
is
cos2(ξ; λ3). Note that ‘partner’ to ξi has the same partner as does
vorticity (strain), but they (ξi and sik) are not equal partners as ξi lives
are
at much smaller scales than sij and there is no strain production (!) either.
However, production of palinstrophy, i.e. (curl ω)2 is due to predominant
contribution from the term associated with the stretching eigenvalue Λ1. PRODUCTION OF GRADIENTS/
DISSIPATION OF PASSIVE SCALAR
DISSIPATION
DΘ/Dt =  (Θ·∇)u + κΔΘ ; G = ∇Θ
(1/2)DG2/Dt =  GiGk sik + κGiΔGi
(1/2)
The main contribution to the production (〈GiGksik〉 > 0)
 GiGksik =  G2Λkcos2(G;λk) is associated with the alignment of the temperature gradient and the
eigenvector¸ λ3 corresponding to the compressive eigenvalue Λ3: it
is due to G2Λ3 cos2(G; λ3). Selected results PDF OF THE PRODUCTION GiGjsij,
PDF
The PDF of the production,
GiGjsij, obtained in our
experiments in a slightly
heated jet is very similar to
that obtained in DNS of
NSE by Tsinober and
Galanti (2003). The PDF of GiGjsij is
positively skewed
positively
and the mean
<GiGjsij > is positive.
positive 1D 2
2
G = −Gi G j sij + κGi∇ Gi
2 Dt Alignments between the temperature gradient G
and the eigenframe λi of the rate of strain tensor sij
(Λk are the corresponding eigenvalues); Λ1> Λ2> Λ3; Λ1> 0, Λ3<0; Λ1+ Λ2+ Λ3 = 0
0, GiGksik=G2[Λ1cos2(G;λ1) + Λ2cos2(G;λ2) + Λ3cos2(G;λ3)] The main effect, the alignment between the temperature gradient and the
eigenvector λ3 corresponding to the compressive eigenvalue Λ3< 0, is captured
well in the measurements and is similar to that obtained from DNS BACK TO VORTICITY
BACK
VERSUS STRAIN
STRONGER NON LINEARITY IN STRAIN
DOMINATED REGIONS
CONCENTRATED VORTICITY OR STRAIN
ARE THEY THAT IMPORTANT? JOINT PDFS OF ENSTROPY (top) AND STRAIN (bottom)
PDF
ENSTROPY AND STRAIN
PRODUCTION WITH ENSTROPHY (left) AND STRAIN (right)
PRODUCTION
FROM OUR
FROM
FIELD
EXPERIMENT
AT Reλ=104
AT
Physics of Fluids,
13, 311 (2001) Note much
larger positive
shift with strain (ω ∇ ω / ω )
2 i 2 i Rate of enstrophy
Rate enstrophy
production and its
production
viscous reduction
conditioned on
strain and vorticity
vorticity ω ωi ∇ 2 ωi / ω 2 (ω ω s
i k ik ) /ω 2 ω ωi ωk sik / ω 2 (ω ∇ ω / ω )
2 i 2 i s ωi ∇ 2 ωi / ω 2 (ω ω s
i k ik /ω 2 ωi ωk sik / ω ) NOTE THE LARGE RATE OF
OF
ENSTROPHY PRODUCTION IN
STRAIN DOMINATED REGIONS
STRAIN
(RED CURVE) AS COMPARED TO
AS
REGIONS OF LARGE ENSTROPHY
(BLUE CURVE) s 2 s / s ,ω/ ω ALL NONLINEAR
ALL
TERMS BEHAVE
THIS WAY!
THIS Eigencontributions to ωiωjsij/ω2=Λkcos2(ω,λk)
Eigen
Nonlinear terms are growing much slower in the enstrophy
dominated regions than in the strain dominated regions. Q Summary of threeSummary
dimensional,
incompressible flow
patterns/
local structure of the
MR
R
flow field in the frame
following a fluid
particle
(from Soria et al. 1994, after
Perry et al. 1990). Q = 1/4(ω²  2sijsij),
R =  1/3(sijsjkski +
+ 3/4ωiωjsij). . STRONGER NON LINEARITY IN
STRONGER
STRAIN DOMINATED REGIONS
HOW THIS LOOKS ON THE R – Q PLANE
PLANE
Q = (1/4){ω2 2s2}; R =  (1/3){sijsjkski+(3/4)ωiωjsij} –
Second and third invariants of the velocity gradient
tensor Aik= ∂ui/∂xk RQ  PLOT
Field experiment, Reλ= 6800 DNS of NSE, ReΘ= 690
PTV, Reλ= 80 velocity gradient ( ) Chacin et al 2000 12
ω − 2 sik sik
 second invariant of the velocity gradient tensor
4
1⎛
3
⎞
R = − ⎜ sik skm smi + ω iω k sik ⎟  third invariant of the velocity gradient tensor
3⎝
4
⎠ Q= The first invariant is vanishing as a consequence of incompressibility ∂ui
∂xk Q = (1/4){ω2 2s2}; R =  (1/3){sijsjkski+(3/4)ωiωjsij}
CONDITIONAL AVERAGES ON THE RQ PLANE OF REYNOLDS STRESS (LEFT) AND
REYNOLDS
TKE `generating events’ (RIGHT), CHASIN AND CANTWELL , 2001
TKE D=0
D=0 Q = (1/4){ω2 2s2}; R =  (1/3){sijsjkski+(3/4)ωiωjsij}
CONDITIONAL AVERAGES ON THE RQ PLANE OF REYNOLDS STRESS (LEFT) AND
PLANE
AND
DISSIPATION (RIGHT), CHASIN AND CANTWELL , 2001 Q
Q R R Ejections
(u+v) D=0 D=0 Sweeps (+uv)
1.3 uv/uτ2 0.3 0.13 ε/(uτ4) 0.36 One of the most interesting (from our point of view)
One
findings is that the main contribution to the shear stress,
turbulent energy production, and dissipation comes from
the regions with Q<0 with larger contribution from the
lower right quadrant, i.e. Q<0 and R>0, not only dominated
by strain, but also by production of strain, sikskmsmi, see
figure 5, also figures 8, 11 and 14 in Chacin and Cantwell
(2000). It should be emphasized that these regions are mainly not
the ones corresponding to vortices (hairpins or whatever), which are
located mostly in the regions with Q>0, or a bit more precisely in
regions with D>0, where D=((27)/4)Q³+R² is the discriminant of ∂ui/∂xj.
That is the regions of major nonlinear activity are really associated
with large strain (mainly corresponding to what Chacin and Cantwell
call `blank' spaces) rather than with regions of concentrated vorticity
with lower dissipation. In other words it seems that concentrated vorticity is not
In
that important also in turbulent shear flows and that
structure(s) associated with turbulence (not only its
energy) production are mainly due to the large strain
rather than large vorticity. Structure(s) associated with the
latter seem to be the consequence of the turbulent
dynamics rather than its dominating factor. A final remark
is that these results do not contradict the importance of
vorticity in maintaining the Reylolds stress. First, these
are relations for the mean quantities, and second, there is
no turbulent flow without vorticity. However, important
details of the relations between Reynolds stress, vorticity,
strain and their production remain not clear. The interpretation of the results by
The
Chacin and Cantwell, 2000 (and
similar) given here is not in full
agreement with their conclusions,
especially regarding the role of
vortices and concentrated vorticity
in turbulent flows (again vortex
obsession). NICE VORTICES/WORMS/FIALMENTS OR WHATEVER
vorticity velocity
velocity SHE et al. 1991 At this stage, this alternative approach (i.e. the ‘structural’) has not led to a
At
i.e.
has
generally applicable quantitative model, neither – for better or worse – has it a
for
has
major impact on the statistical approaches. Consequently the deterministic
major
viewpoint is neither emphasized nor systematically presented, POPE 2000.
POPE
nor
This does not mean that there exists “generally applicable quantitative
This
generally
model” based on statistical approaches.
model REAL SHE et al. 1991 GAUSSIAN ‘STRUCTURES’ OF INTENSE VORTICITY AND STRAIN,
Moisy & Jimenez, 2004 a b c (a)Big structure with ω>3ω; (b)Small structure with ω>3ω; (c)Big structure with ω>6ω
SIMILAR TO THOSE IN
SHE et al., 1991 and
SHE
BORATAV AND PELZ, 1997
CHEN, 2000
CHEN, a b
(a) s>2.8s; (b) s>4.2s IS THE FLOW FIELD IN ‘WORMS’
SIMPLE? IS IT QUASI2D?
JIMENEZ ET AL 1993
JIMENEZ IS THE FIELD OF VORTICITY IN ‘WORMS’ SIMPLE?
CONSEQUENCES FOR REPRESENTATION OF TURBULENT FIELDS JIMENEZ & WRAY 1996
JIMENEZ THE ‘AMOUNT’ OF COMPRESSING IN
‘WORMS’ IS THE SAME AS IN THE
WHOLE FIELD ! HENCE INADEQUATE
REPRESENTATION OF THE FLOW
FILED BY A COLLECTION OF PURELY
STRETCHED VORTICES (or other
`simple' objects), ESPECIALLY THOSE WHICH DO NOT INTERACT WITH
WITH
STRAIN.. THESE LATTER ARE
THESE DESCRIBED BY DIFFRENTIAL
EQUATIONS. THE REAL ONES ARE
DESCRIBED BY INTEGRO –
DIFFERENTIAL EQUATIONS. SOME COMMENTS ON
SOME WHAT IS A“ VORTEX” ? I DO NOT KNOW. WHO DOES? THERE ARE SOME DEFINTIONS (MOSTLY AD HOC). A COMMON
AD
DO
FEATURE IS THAT (MORE OR LESS) CONCENTRATED VORTICITY IS SURROUNDED BY PLENTY
VORTICITY
OF STRAIN , E.. G. ‘POTENTIAL’ VORTEX: VORTICITY ONLY IS NOT REALLY ‘ONLY’
STRAIN E
A popular method to look for structure(s) (and `vortices’) is to use a criterion based on one
parameter only, e.g. enstrophy ω². Though such an approach is useful and `easy', it is
inherently limited and reflects the simplest aspects of the problem. For example, even for
characterization of some aspects of the local (i.e. in a sense `point'wise) structure of the
some aspects
flow field in the frame following a fluid particle requires at least two parameters: the second
and the third invariants of the velocity gradient tensor ∂ui/∂xk: Q = 1/4(ω²  2sijsij) and R =
 1/3(sijsjkski + 3/4ωiωjsij). Therefore attempts to adequately identify/characterize finite size
3/4
structure(s) – and ‘vortices’ seem to be of this kind  by one parameter only are unlikely to
be successful, and one needs something like pattern recognition based on some
conditional sampling scheme involving definitely more (perhaps much more) than two
parameters. But then all the ‘simplicity’ (and attraction) will be gone.
So it is safer to use well defined quantities – vorticity and strain. Vorticity alone is
not ‘enough’. Concentrated vorticity (very popular) is not that important and other
regions (e.g. regions dominted by strain) play essential role in the evolution and
dynamics of turbulent flows (Tsinober, 1998, 2001) IN LIEU OF
IN
CONCLUSION Looking in more essential details (both physics and the mathematical
Looking
ones) of the processes of self amplificiation of the field of velocity
derivatives seems to be the key issue for most of the problems of
turbulence but also 3D NSE and Euler. This includes subtle geometrical
relations between vorticity and strain (which are likely to be the
main guilty of almost happening in turbulence: after all there is
no turbulence without vorticity production) and several others of
dynamical significance . Understanding of these processes and thereby
essential aspects of turbulence physics seems to form the basis for
constructive approach to a great variety of problems including effective
handling of nonlinearities which will allow to solve the standard 3DNSE
and Euler problems and not the other way round. ...
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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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