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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 XVXVI HELICITY
HELICITY
SCHRAUBENSINN− SCHRAUBENSINN WHAT IS IT AND WHAT IS IT GOOD FOR? A bit of history. Definition and elementary properties of helicity for a vector
field. Geometrical aspects, including topology of field (e.g. vortex, magnetic)
lines. Invariance in nondissipative media, for both inviscid flows and for ideally
conducting media . Effects of helicity. Kinetic helicity and turbulent dynamo.
Kinetic helicity and fluid turbulence. Applications : atmospheric physics, plasma
physics, astrophysics.
‘ BETCHOV 1961 At least intuitively it is quite clear that a vortex having a
component of velocity along its axis is characterized by
nonzero helicity, i.e. is a helical structure. There is a great
number of structures falling under this category, e.g. such
diverse structures as TaylorGortler vortices, leading edge
and trailing vortices shed from wings and slender bodies,
streamwise vortices in boundary layers and free shear flows,
Langmuir circulations in the ocean and analogous structures
in the atmosphere, tornados and rotating storms. Similar
structures are observed in space and laboratory plasmas.
Structures of this type are expected to lack reflexional
symmetry. The key quantity characterizing the reflexional
symmetry (or its lack) of a fluid flow is the so called helicity. A BIT OF HISTORY The term HELICITY in pure fluid dynamics was introduced by
BETCHOV 1961 in the context of turbulent flows A GALLERY OF HELICAL
GALLERY
STRUCTURES VORTEX BREAKDOWN Vortex breakdown visualized. (a) In a water tunnel over a slender delta wing as a result of the emission of colored
dye near the apex. (b) Around NASA’s F18 High Angle of Attack Research Vehicle (HARV) using smoke to seed the
vortex emanating from the nose of the aircraft at 201 incidence. Courtesy T. Leweke (Aachen 1989) ROTATING STIORM It has to be stressed that (quite popular) Clebsch variables
are ‘defective’ in the sense that flows which (presumably) can
be represented in these variables have oversimplified
geometry as their helicity
H = ∫ω·u dV = ∫ ω·∇ϕ dV = ∫ ∇·(ϕω)dV, i.e. it
vanishes in many cases (ω = ∇λx∇μ, i.e. ω ⊥ λ∇μ) Prototype configuration of linked field tubes with nonzero helicity, MOFFATT AND TSINOBER, 1992.. Here
the field is assumed to be identical zero except in two closed tubes with axes C1 and C2 and with vanishingly small
crosssection. The field lines are untwisted within each tube, i.e. each line is a closed curve passing once round the
tube, and unlinked with its neighbors in the same tube. In such a case HB=±2L1,2,where L1,2 is the linking (or
winding) number of the two tubes (in the figure L1,2=1), F1,2 are the fluxes (circulations κ1,2) associated with
each tube, and the + or  is chosen depending on whether the linkage is right  of lefthanded. in nondissipative media (no reconnections) In cases when the fluid
flow is influenced by the
presence of the
magnetic field, that is by
the Lorenz force {the
term curl(j ×B) added to
the LHS of NSE} the helicity
of vorticity H is not
conserved whereas the
magnetic helicity
remains conserved.
Another conserved
quantity in this latter
case is the crosshelicity
HB,ω= ∫ u·BdV. MODIFIED HELICITY
Helicity is a global quantity which in many cases is not well defined. It appears that
one can choose the gauge ϕ in such a way that the helicity density is a Lagrangian
(nondissipative) invariant, i.e. it is conserved (pointwise) along the paths of fluid
particles and therefore for any fluid volume. Such a choice is possible both for
magnetic field (ELSASSER, 1956; CHILDRESS & GILBERT, 1995) and for
nonconducting fluid flows (KUZMIN, 1983; OSELEDETS, 1989). It is possible
to do so also for a viscous flow (OSELEDETS, 1989) chosing ϕ obeying the
equation
Dφ/Dt = p − u2/2 + ν∇2φ
Then the modified helicity density hm = ω·v, with v = u+∇ϕ satisfies the
equation
Dhm/Dt = ν{∇²hm 2(∂ωi/∂xk)(∂vi/∂xk)}
i.e. is a Lagrangian invariant if ν = 0. EFFECTS OF HELICITY Note that magnetic fields are generated
Note
pretty successfully by velocity fields
without any helicity, by flows possessing
reflexional symmetry and even random
gaussian isotropic ones The reason is that this argument
is too simple as the Lamb vector
ω×u = ∇α + ∇×β has a large
potential part ∇α (such that
〈(∇α)²〉 ≥ 2〈∇×β)²〉), which
can be included into pressure just
like the Bernoulli term ∇(u2/2).
It is the solenoidal part ∇×β =
curl (ω×u) which matters, e.g.
for vorticity dynamics. Moreover,
if one cares about enstrophy (and
strain) production, i.e.
ω·∇×∇×β = uk∂ω2/∂xk –
ωiωksik, only the second tem is
of importance. 4/5 AND 2/15 KURIEN ET AL 2004 # Note that ∂jui∂jωi = 〈ω·curlω〉 is just the mean superhelicity. The helicity
dissipation h = 2ν 〈ω·curlω〉 is assumed to be finite in the mentioned above
and similar publications (see references in GALANTI AND TSINOBER 2004).
# The 2/15 law has a nonlocal character as it involves correlation of both velocity
(i.e. large scale quantities) and its differences (i.e. small scale quantities). It is
noteworthy that the Kolmogorov 4/5 law can be interpreted in the same way since
〈(ΔuL)3〉 =  3〈ΔuL [(uL(x+r) + uL(x)]2〉 Helicity dissipation seems to tend to a finite limit
Helicity
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 (helical forcing). Dependence of normalized dissipation of helicity DH l² u3 and
The
energy DE l u3 on the Taylor microscale Reynolds number Reλ. ◦  corresponds to the data from
Ref. [10], □  correspond to the data from Ref. [16] and references therein.
Ref. Cω,curlω= 0.1
0.1 Note the pretty small value of
the correlation coefficient
Cω,curlω= 0.1between ω and curlω
0.1
curl A natural question is how it is
natural
possible that both energy and
helicity dissipation (presumably)
remain finite with increasing
Reynolds number. A possible
explanation is seen from the form of
helicity dissipation as used above h
= 2ν〈ω·curlω〉: the imperfect
alignment between ω and curlω
(as shown in figure at the left) can make it
possible that the ‘singularitites’
arising from the finiteness of both
energy and helicity dissipation can
be matched also aided by the fact
that ω·curlω is not a positively
defined quantity. Helicity dissipation seems to tend to a finite limit even with
Helicity
nonhelical forcing
nonhelical GALANTI AND TSINOBER 2004 T+(T) is the total length of the time intervals
where both H and Hs are positive (negative),
etc.
The total helicity dissipation in this case is
DH = D++ D and the total helicity
production DpH = Dp++ Dp . Their
difference, DH  DpH, which is the net
dissipation of helicity, is still positive. ADDITIONAL ASPECTS OF
HELICITY ‘DISSIPATION’ from figures shown below
from SELFPRODUCTION OF HELICITY
DISSIPATION/SUPERHELICITY
The Equation for the superhelicity Hs = ∫ω·curlωdx 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.
Hyperhelicity Hh = ∫curlω·curlcurlωdx Three main results closely related to finite dissipation of helicity : i)  the selfproduction property of superhelicity
i)
production
superhelicity
H s=∫ω·curlωdx (which is proportional to helicity dissipation) in the sense that forcing does not play any role in
forcing
production of superhelicity, i.e. the dissipation of helicity; ii) finiteness of helicity dissipation is the manifestation of
i.e.
ii)
the lack of reflection symmetry of the small scales; iii)  this lack of reflexional symmetry should increase with the
iii
Reynolds number, following from our third main result that the normalized helicity dissipation tends to remain finite as
the Reynolds number increases, just like the energy dissipation, in the spirit of Kolmogorov 41. HELICAL FORCING
D
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PRODUCTION of
PRODUCTION Hs FORCING
FORCING Three main results closely related to finite dissipation of helicity : i)  the selfproduction property of superhelicity
production
superhelicity
Three
helicity i)
H s=∫ω·curlωdx (which is proportional to helicity dissipation) in the sense that forcing does not play any role in
forcing
production of superhelicity, i.e. the dissipation of helicity; ii) finiteness of helicity dissipation is the manifestation of the
ii)
lack of reflection symmetry of the small scales; iii)  this lack of reflexional symmetry should increase with the
iii)
Reynolds number, following from our third main result that the normalized helicity dissipation tends to remain finite as
the Reynolds number increases, just like the energy dissipation, in the spirit of Kolmogorov 41. NONHELICAL FORCING
NON
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PRODUCTION of Hs FORCING This
This JOINT PDFS OF HELICITY H AND SUPERHELICITY HS a) NHcase (nonhelical foring), b)ABCcase (helical forcing). In the NH case the
correlation coecient CH;Hs between H and Hs is 0:70. In case ABC it is equal to0:52
without removal of the means of H, Hs and 0.98 with the means of H, Hs removed. Itt should be stressed, however, that an important difference with
I
true dissipation is that 2Hs can have any sign, so that when H
and Hs have opposite signs the viscous term 2νHs acts as
production of helicity. It is also noteworthy that finite deviation
from reflection symmetry (i.e. finite Hs and also Hh) in small
scales should remain at whatever small viscosity (and should be
large for small viscosity), e.g. to maintain the helicity ‘cascade’
with definite sign helicity input, though this appears to be true in
some sence in case of purely nonhelical forcing as well. Three main results closely related to finite dissipation of helicity defined
Three
as the integral of the scalar product of velocity and vorticity, H=∫u·ωdx:
dx
i)  the selfproduction property of superhelicity Hs=∫ω·curlωdx
i)
(which is proportional to helicity dissipation) in the sense that forcing
does not play any role in production of superhelicity, i.e. the dissipation
of helicity; ii) finiteness of helicity dissipation is the manifestation of the
ii)
lack of reflection symmetry of the small scales; iii)  this lack of
iii
reflexional symmetry should increase with the Reynolds number,
following from our third main result that the normalized helicity
dissipation tends to remain finite as the Reynolds number increases, just
like the energy dissipation, in the spirit of Kolmogorov 41. The results
were obtained from DNS of NavierStokes equations in a simplest flow
geometry with periodical boundary conditions. COMPLEMENTARY RESULTS
LIMITATIONS OF CORRELATIONS Itt is instructive to compare the relative helicities defined as
I with the correlation coefficients HELICAL FORCING t NONHELICAL FORCING Two observations can be made here. First, the relative higher order
helicities are of the same order and not that small in both cases.
Second, the correlation coefficient Cu,curlω reflecting the coupling
between u and curlω is considerably larger than the relative helicity
Hr reflecting the coupling between u and ω . This is in spite of the
fact that the scales of u and ω are ‘closer’ than those of u and curlω .
Similarly the correlation coefficient Cω,curlcurlω, reflecting the coupling
between ω and curlcurlω is considerably larger than the relative
helicity Hsr reflecting the coupling between ω and curlω. Moreover,
in flows with reflectional symmetry both Hr and Hsr (and also
Hhr)vanish, whereas Cu,curlω and Cω,curlcurlω remain practically
unchanged. This illustrates the limited value of quantities like
correlations and correlation coefficients. This is an important
This
example of sensitivity
of passive scalar to te
nature of velocity
field. Usually it is not
so sensitive to the
nature of velocity field
so that many
properties are the
same, e.g. for genuine
turbulence and
gaussian velocity field.
We remind another example of such sensitivity: the
difference between forwards and backwards dispersion The 4/3 law is an
for genuine turbulence and artificial velocity fields.
immediate example. CONCLUDING
CONCLUDING There seems to be little doubt that helicity, i.e. H = ∫u·ωdx,
There
being a quadratic (inviscid) invariant ... has a
status comparable with that of kinetic energy¹.
Nevertheless, the issue of the role of helicity in turbulent flows is one of
the most controversial ones and is very far from a consensus. On one
hand, it is thought that helicity should be of dynamical significance in
turbulent flows, and on the other hand, helicity was shown to be
dynamically irrelevant in some cases. So it is not surprising that the
role of helicity in threedimensional turbulence is considered to be still
somewhat ‘mysterious’. However, it seems conceptually misleading to look at helicity as a causal
However,
factor as it is an integral property (as any other) of turbulent flows. An
immediate example is comprised by flows in rotating fluids. Such flows lack
reflexional symmetry and – as a consequence – possess nonvanishing
helicity. On the other hand flows of this kind are characterized by reduced
nonlinearity and consequently drag and dissipation A recent example is the
numerical study of turbulent flow in a rotating pipe by ORLANDI 1997.
He observed a clear positive correlation between increase in helicity and
decrease in dissipation. However, the cause of both is rotation, i.e. an
observed relation between two quantities/properties is not (necessarily)
synonymous to direct causal relation between them: causality is different
from relation. ...
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