Turbulence Concepts

7/29/2019 Turbulence Concepts

1/79

Lectures seriesLectures series as a part of the activity within the frameas a part of the activity within the frameof theof the Marie CurieMarie Curie ChairChair Fundamental and ConceptualFundamental and ConceptualAspects of Turbulent FlowsAspects of Turbulent Flows..

FUNDAMENTAL AND CONCEPTUALFUNDAMENTAL AND CONCEPTUAL

ASPECTSASPECTS OF TURBULENT FLOWSOF TURBULENT FLOWSArkadyArkadyTsinoberTsinober

Professor andProfessor andMarie CurieMarie CurieChair in Fundamental and Conceptual Aspects of Turbulent FlowsChair in Fundamental and Conceptual Aspects of Turbulent Flows

Institute for Mathematical SciencesInstitute for Mathematical Sciences and Department of Aeronautics, Imperial College Londonand Department of Aeronautics, Imperial College London

We absolutely must leave room for doubt or there is no progressWe absolutely must leave room for doubt or there is no progress and no learning.and no learning.

There is no learning without posing a question. And a questionThere is no learning without posing a question. And a question requires doubt...Nowrequires doubt...Now

the freedom of doubt, which is absolutely essential for the devethe freedom of doubt, which is absolutely essential for the development of science,lopment of science,was born from a struggle with constituted authorities...was born from a struggle with constituted authorities... FEYNMANNFEYNMANN, 1964, 1964


2/79

VORTICITYVORTICITYAND STRAIN IN TURBULENCEAND STRAIN IN TURBULENCE

LECTURES VLECTURES V--VIVI

Why velocity derivatives? Vorticity versus strain. Self-amplification(SA) of both: who is responsible for enhanced disspation? Stretching

versus compressing. Geometrical statistics. ConcentratedConcentrated vorticityvorticity ororstrain.strain. UnrestandingUnrestanding the physics of SA ofthe physics of SA ofvorticityvorticity and strain is the heartand strain is the heart

of approaching the turbulence problem.of approaching the turbulence problem.

SelfSelf--amplification of the strainamplification of the strain--rate field is somethingrate field is something

I overlooked up to nowI overlooked up to now ...... TENNEKES 2003TENNEKES 2003


3/79

One of the important common features ofOne of the important common features of

processes resulting in turbulence is that all ofprocesses resulting in turbulence is that all ofthem tend to enhance thethem tend to enhance the rotationalotational andanddissipativeissipative properties of the flow in the processproperties of the flow in the processof transition to turbulence. The first property isof transition to turbulence. The first property is

associated with the production ofassociated with the production ofvorticityvorticity(turbulence is highly rotational), whereas the secondwhereas the second

property is due to the production of strainproperty is due to the production of strain(turbulence is strongly dissipative). The latter is aThe latter is a uniqueuniqueprocess of genuine turbulence and does notprocess of genuine turbulence and does not

have any analogue inhave any analogue in passivepassive turbulenceturbulence ..


4/79

VORTICITY AND STRAINOne of the importantOne of the important

common features ofcommon features ofprocesses resulting inprocesses resulting in

turbulence is that all ofturbulence is that all of

them tend to enhance thethem tend to enhance therotational and dissipativerotational and dissipative

properties of the flow inproperties of the flow in

the process of transition tothe process of transition toturbulence. The firstturbulence. The first

property is associated withproperty is associated with

the production ofthe production ofvorticityvorticity,,whereas the secondwhereas the second

property is due to theproperty is due to the

production ofproduction ofstrainstrain..

CHEN, 2000CHEN, 2000

DNS of forced NSE in a periodic boxDNS of forced NSE in a periodic box

ReRe=220=220,, == 3meanmean,, ss = 3s= 3smeanmean

LIVE TOGETHERLIVE TOGETHER

BUT HAVE A VERYBUT HAVE A VERY

DIFFERENT LIFEDIFFERENT LIFE


5/79

DISSIPATION/STRAINDISSIPATION/STRAIN

Symmetric partSymmetric part ofofAAikik == uuii//xxkk, i.e., i.e. ssikikKOLMOGOROV 1941KOLMOGOROV 1941

-- religionreligion

VORTICITY AND RELATEDVORTICITY AND RELATED

SkewSkew--symmetricsymmetric part ofpart ofAAikik== uuii//xxkkTAYLOR 1938TAYLOR 1938

--

obsessionobsession

INTEREST TOINTEREST TO AAikik== uuii//xxkk


6/79

What is turbulence but a random chaotic field ofWhat is turbulence but a random chaotic field of vorticityvorticity,,

whose strongwhose strong nonlinearnonlinear interactions makes the problem sointeractions makes the problem soimpossibly difficult? ...impossibly difficult? ... the concept of the coherent structure inthe concept of the coherent structure in

turbulent shear flows has led to the picture of such flows as aturbulent shear flows has led to the picture of such flows as a

superposition of organized,superposition of organized, deterministicdeterministic vortices whosevortices whoseevolution and interaction is the turbulenceevolution and interaction is the turbulence.. SAFFMANSAFFMAN 19811981

Turbulence is rotational and three dimensional. Turbulence isTurbulence is rotational and three dimensional. Turbulence is

characterized by high levels of fluctuatingcharacterized by high levels of fluctuating vorticityvorticity. For this. For this

reason,reason, vorticityvorticity dynamics plays an essential role in thedynamics plays an essential role in the

description of turbulent flows.description of turbulent flows. TENNEKES AND LUMLEY 1972TENNEKES AND LUMLEY 1972

The turbulence syndrome includes the following symptoms:The turbulence syndrome includes the following symptoms:itit

is essentiallyis essentially nonlinear and rotationalnonlinear and rotational STEWART 1963STEWART 1963

VorticityVorticity obsessionobsession sincesince HELMHOLTZHELMHOLTZ andand KELVIN 1867KELVIN 1867TAYLORTAYLOR1937,1937, CORRSINCORRSIN 19531953


7/79

WHY SMALL SCALES?AND WHAT THEY ARE?


8/79

A great deal of interest is concentrated on theA great deal of interest is concentrated on the

Reynolds stress tensorReynolds stress tensoruuiiuujj (or similar quantities in(or similar quantities inLES) and theLES) and the closure problemclosure problem, i.e. how it is related, i.e. how it is related

to the mean flowto the mean flow ((with thewith the not obviousnot obvious -- assumption that a relativelyassumption that a relatively

simple relationsimple relation if at allif at all -- does existdoes exist).). Such an approach isSuch an approach isbased on the view that small scales (whatever thisbased on the view that small scales (whatever this

means) aremeans) are slavedslaved to the large scales and areto the large scales and aremostly a kind of passive sink of energy.mostly a kind of passive sink of energy. This is theThis is theso calledso called classical approachclassical approach with the view that inwith the view that in

order to understand turbulence one needs only toorder to understand turbulence one needs only toresolveresolve the large scales andthe large scales and modelmodel the smallthe smallscalesscales ((can this be done without sufficient understanding and/or full recan this be done without sufficient understanding and/or full resolutionsolution

downdown to all physically relevant scalesto all physically relevant scales?).?).


9/79

However, today it is high time to ask howHowever, today it is high time to ask how

much justified is such oversimplifiedmuch justified is such oversimplifiedtreatment of small scales via methods liketreatment of small scales via methods like

eddy viscosity, and similar. The small scaleseddy viscosity, and similar. The small scales

contain a great deal of essential physics ofcontain a great deal of essential physics of

turbulent flows, much of which is not known orturbulent flows, much of which is not known or

poorly understood, and which are intimatelypoorly understood, and which are intimatelyandand bidirectionallybidirectionally related to the large scalesrelated to the large scales

((nonlocalitynonlocality). Apart of basic there are). Apart of basic there arenumerous problems in which one has to dealnumerous problems in which one has to deal

explicitly with the nature, structure andexplicitly with the nature, structure and

dynamics/evolution of small scales.dynamics/evolution of small scales.


10/79

For instance, special information on small scaleFor instance, special information on small scale structure(sstructure(s) is) isneeded in problems concerning, e.g. combustion, disperse multiphneeded in problems concerning, e.g. combustion, disperse multiphaseaseflow, mixing,flow, mixing, cavitationcavitation, turbulent flows with chemical reactions,, turbulent flows with chemical reactions,some environmental problems, generation and propagation of sounsome environmental problems, generation and propagation of soundd

and light in turbulent environments, and some special problems iand light in turbulent environments, and some special problems innblood flow related to such phenomena asblood flow related to such phenomena as hemolysishemolysis andandthrombosis. In such problems, not only special statistical propethrombosis. In such problems, not only special statistical propertiesrties

are of importance like those describing theare of importance like those describing the behaviourbehaviour of smallestof smallestscales of turbulence, but also actualscales of turbulence, but also actual nonstatisticalnonstatistical' features like' features likemaximal concentrations in such systems as an explosive gas whichmaximal concentrations in such systems as an explosive gas which

should be held below the ignition threshold, some species inshould be held below the ignition threshold, some species inchemical reactions, concentrations of a gas with strong dependenchemical reactions, concentrations of a gas with strong dependenceceof its molecular weight on concentration (such as hydrogen fluorof its molecular weight on concentration (such as hydrogen fluorideide

used in various industries e.g. in production of unleaded petrolused in various industries e.g. in production of unleaded petrol) and) andtoxic gasestoxic gases..


11/79

THE ABOVE IS A CLEAR INDICATIONTHE ABOVE IS A CLEAR INDICATION

WHY VELOCITY DERIVATIVESWHY VELOCITY DERIVATIVESARE SO IMPORTANT, BUT THERE IS MUCH MOREARE SO IMPORTANT, BUT THERE IS MUCH MORE


12/79

Velocity derivatives,Velocity derivatives,AAijij == uuii//xxjj, play an outstanding, play an outstanding

role in the dynamics of turbulence for a number ofrole in the dynamics of turbulence for a number ofreasons. Their importance has become especiallyreasons. Their importance has become especially

clear sinceclear sinceTAYLOR, 1938TAYLOR, 1938

andandKOLMOGOROVKOLMOGOROV, 1941, 1941

..Taylor emphasized the role ofTaylor emphasized the role ofvorticityvorticity, i.e. the, i.e. the

antisymmetricantisymmetric part of the velocity gradient tensorpart of the velocity gradient tensorAAijij ==

uuii//xxjj, whereas, whereas KolmogorovKolmogorov stressed the importancestressed the importanceof dissipation, and thereby of strain, i.e. the symmetricof dissipation, and thereby of strain, i.e. the symmetric

part of the velocity gradient tensor.part of the velocity gradient tensor. It is importantIt is important (see(seebelow)below) that the whole (incompressible) flow field isthat the whole (incompressible) flow field is

fully determined by the fields offully determined by the fields ofvorticityvorticity or strainor strain

with appropriate boundary conditions.with appropriate boundary conditions.


13/79

Apart from vorticity and strain/dissipation, there are manyother reasons for special interest in the characteristics of

the field of velocity derivatives, Aij=u i/xj, in turbulentflows:. In theIn the LagrangianLagrangian description of fluid flow in a framedescription of fluid flow in a frame

following a fluid particle, each point is a critical one, i.e. tfollowing a fluid particle, each point is a critical one, i.e. thehedirection of velocity is not determined. So everything happeningdirection of velocity is not determined. So everything happeningin its proximity is characterized by the velocity gradientin its proximity is characterized by the velocity gradient

AijAij==

ui/ui/

xjxj. For instance, local geometry/topology is naturally. For instance, local geometry/topology is naturallydescribed in terms of critical points terminology.described in terms of critical points terminology. The field of velocity derivatives is much more sensitive to theThe field of velocity derivatives is much more sensitive to thenonnon--Gaussian nature of turbulence or more generally to itsGaussian nature of turbulence or more generally to its

structure, and hence reflects more of its physics.structure, and hence reflects more of its physics. The possibility of singularities being generated by the EulerThe possibility of singularities being generated by the Eulerand theand the NavierNavier--Stokes equations (NSE) and possibleStokes equations (NSE) and possible

breakdown of NSE are intimately related to the field of velocitybreakdown of NSE are intimately related to the field of velocityderivatives.derivatives.


14/79

There is a generic ambiguity in defining the meaning of the termThere is a generic ambiguity in defining the meaning of the term smallsmall

scales (or more generally scales) and consequently the meaning oscales (or more generally scales) and consequently the meaning of thef theterm cascade in turbulence research. The specific meaning of thiterm cascade in turbulence research. The specific meaning of this terms termand associatedand associated interscaleinterscale energy exchange/`cascade' (e.g. spectral energyenergy exchange/`cascade' (e.g. spectral energy

transfer) is essentially decomposition/representation dependent.transfer) is essentially decomposition/representation dependent. Perhaps,Perhaps,the only common element in all decompositions/representations (Dthe only common element in all decompositions/representations (D/R) is/R) isthat the small scales are associated with the field of velocitythat the small scales are associated with the field of velocity derivatives.derivatives.

Therefore, it is natural to look at this field as the one objectTherefore, it is natural to look at this field as the one objectively (i.e. D/Rively (i.e. D/Rindependent) representing the small scalesindependent) representing the small scales (one can(one can followingfollowing CORRSINCORRSIN--use higher order derivatives, e.g.use higher order derivatives, e.g. curlcurl,ssijij ).. Indeed, the dissipation isIndeed, the dissipation isassociated precisely with the strain field,associated precisely with the strain field, ssijij, both in Newtonian and non, both in Newtonian and non--Newtonian fluids. There is a number of more specific reasons wNewtonian fluids. There is a number of more specific reasons whyhy

studying the field of velocity derivatives is so important in thstudying the field of velocity derivatives is so important in the dynamics ofe dynamics ofturbulence.turbulence.


15/79

VorticityVorticity and strain are not just velocity derivativesand strain are not just velocity derivatives. They. They

are special for several reasons as mentioned (and will beare special for several reasons as mentioned (and will bediscussed at length below). The one to be stressed here isdiscussed at length below). The one to be stressed here is

thatthat the whole flow field is determined entirely by thethe whole flow field is determined entirely by thefield offield ofvorticityvorticity oror strainstrain *with appropriate boundarywith appropriate boundaryconditions:conditions: uu == --curl;; uuii==22ssijij//xxjj,,i.e. the velocity field isi.e. the velocity field is a linear functional ofa linear functional ofvorticityvorticity uu== FF{

},}, or strainor strain uuii == GG{{ssijij},},*i.e. alterationi.e. alterationof the field of velocity derivatives reflects on the velocityof the field of velocity derivatives reflects on the velocity

field,field, vorticitvorticityy andand strainstrain are not passiveare not passivethey react backthey react backand not only for the above (and not only for the above (kinematickinematic reason)reason)


16/79

uuii((xx,t,t)) == jjss

((rr))ssijij((yy,t,t))ddyy,,jj

ss((rr)=)= -- ((22))--11 rrjj//rr,, rrii == xxii--yyii,,

uuii((xx,t,t)) == jjjj((rr))jj((yy,t,t))ddyy,,

jjjj ((rr)=)= -- ((44))--11 ijkijkrrjj//rr,, rrii == xxii--yyii,,

ssijij((xx,t,t)) == P.V.P.V.ijij((rr))kk((yy,t,t))ddyy,,ijij ((rr)=)= -- 33((88))

--11 {{ijlijlrrllrrkk++ kjlkjlrrllrrii}//rr55,,

rrii == xxii--yyii,, P.VP.V.. stands for the Cauchy principal valuestands for the Cauchy principal value..

In the whole space the first functional is the well known BioIn the whole space the first functional is the well known Bio--SavartSavart

law, and the second has a similar formlaw, and the second has a similar form (see(see eqeq. C.14. C.14 inin TsinoberTsinober 2001)2001)


17/79

WHY STRAIN TOO?WHY STRAIN TOO?IS IT AN EQUAL PARTNER OFIS IT AN EQUAL PARTNER OF

VORTICITYVORTICITY??

Strain controls the flow in the same way as doesStrain controls the flow in the same way as does vorticityvorticity also in the sense of possiblealso in the sense of possible

breakdown of smooth solutions for 3D flows,breakdown of smooth solutions for 3D flows, Ponce, G. (1985) Remarks on a paper J.T. Beale, T.Ponce, G. (1985) Remarks on a paper J.T. Beale, T.

Kato and A.Kato and A. MajdaMajda,, CommunCommun. Math. Phys.. Math. Phys.,, 988, 345, 345--353.353.


18/79

STRAINSTRAIN AN EQUAL PARTNERAN EQUAL PARTNER

Dolphins in phosphorescent seaDolphins in phosphorescent sea..The inspiration for this woodcut,The inspiration for this woodcut,

created by M. C. Escher in 1923,created by M. C. Escher in 1923,

was the flowwas the flow--inducedinduced

bioluminescence that occurs onbioluminescence that occurs on

dolphins when they swim throughdolphins when they swim through

waters that contain high levels ofwaters that contain high levels of

bioluminescent planktonbioluminescent plankton.

In some cases even moreIn some cases even more

M b lM b lSSome reasons


19/79

1. Most important stresses in fluidsMost important stresses in fluids ((both Newtonian and nonboth Newtonian and non--NewtonianNewtonian))flows are defined by strainflows are defined by strain2.. Energy dissipation is directly associated with strainEnergy dissipation is directly associated with strain (both Newtonian(both Newtonianand nonand non--Newtonian)Newtonian) and not withand not with vorticityvorticity..3.. The energy cascadeThe energy cascade (whatever this meanswhatever this means) and its final resultand its final result --dissipation, are associated with predominant selfdissipation, are associated with predominant self--amplification ofamplification ofthe rate of strain and vortex compression rather than with vortethe rate of strain and vortex compression rather than with vortexx

stretching. That is another nonzero odd momentstretching. That is another nonzero odd moment -- ssijijssjkjkssjkjk --

responsible for the production of strain, is not less importantresponsible for the production of strain, is not less important thanthan

thethe enstrophyenstrophy productionproduction iijjssijij

More belowMore belowSome reasonsSome reasons


20/79

4.. Vortex stretching is essentially a process of interaction ofVortex stretching is essentially a process of interaction ofvorticityvorticity and strain.and strain. VorticesVortices interact via their strain fields.interact via their strain fields.5.. Strain dominated regions appear to be the most active/nonlineStrain dominated regions appear to be the most active/nonlineararin a number of aspects.in a number of aspects.6.. Interaction of the flow field with additives (particles, polymInteraction of the flow field with additives (particles, polymers,ers,blood cells) is mostly via strain .blood cells) is mostly via strain .7.. Though formally all the flow field is determined entirely by thThough formally all the flow field is determined entirely by theefield offield of vorticityvorticity the relation between the strain andthe relation between the strain and vorticityvorticity isisstronglystrongly nonlocalnonlocal. In many cases, they are only weakly statistically. In many cases, they are only weakly statistically

correlated or not correlated at allcorrelated or not correlated at all..


21/79

BASIC EQUATIONS

DD//DDtt = (= (

))uu ++ ++ ijkijkFFkk//xxjj

(()D)D/Dt =/Dt = iijjssijij ++ iiii ++ ijkijkii FFkk//xxjj

(()Ds)Ds//DtDt == -- ssijijssjkjksskiki(1/4)(1/4)iijjssijij

ssijijpp//

xxiixxjj ++

ssijij

ssijij ++ ssijijFFijij

DsDsijij//DtDt == -- ssjkjksskiki(1/4)((1/4)(iijj-- 22ijij))

pp//xxiixxjj ++ ssijij ++ FFijij


22/79

One of the most basic phenomena and distinctive features of threOne of the most basic phenomena and distinctive features of threee--dimensional turbulence is the predominant vortexdimensional turbulence is the predominant vortex

stretching. This process occurs viastretching. This process occurs via interactioninteraction ofofvorticityvorticity andand strainstrain..

(()D)D/Dt =/Dt = iijjssijij ++ iiii ++ ijkijkii FFkk//xxjjIt is a very common view that this process is responsible for thIt is a very common view that this process is responsible for the enhancede enhanced dissiptiondissiption in 3in 3--D turbulent flowsD turbulent flows((TAYLOR 1938,TAYLOR 1938, ONSAGERONSAGER19491949 and everybody onand everybody on).).

WHO IS THEWHO IS THEGUILTYGUILTY ?? II

In what sense vortex stretching plays a central role in theIn what sense vortex stretching plays a central role in the

energy cascade to small scales and dissipationenergy cascade to small scales and dissipation? OR? OR

It seems that the stretching of vortex filaments must be regardIt seems that the stretching of vortex filaments must be regarded as theed as theprincipal mechanical cause of the high rate of dissipation whichprincipal mechanical cause of the high rate of dissipation which is associatedis associated

with turbulent motion,with turbulent motion, TAYLOR 1938TAYLOR 1938

VortexVortex--line stretching plays a central role in the energy cascade to smline stretching plays a central role in the energy cascade to small scales andall scales anddissipationdissipation.. RHINESRHINES, 1997, 1997 (p.102)(p.102)


23/79

BUT, dissipation isBUT, dissipation is

22ss22 = 2= 2ssijijssijij !!!!

The true physical causal relation is betweenThe true physical causal relation is betweendissipation and strain both in Newtonian and nondissipation and strain both in Newtonian and non--Newtonian fluids. Therefore it is a misconception toNewtonian fluids. Therefore it is a misconception to

associate dissipationassociate dissipation directlydirectly withwith vorticityvorticity. However,. However,the rotational nature of turbulence (i.e.the rotational nature of turbulence (i.e. vorticityvorticity) is) iscrucial for dissipationcrucial for dissipation..As a home work show that inAs a home work show that in irrotationalirrotational flow (i.e.flow (i.e.uu==) the dissipation is either zero or volume integral) the dissipation is either zero or volume integral22ss22dVdV

over the flow domain is equal to the surfaceover the flow domain is equal to the surface

integralintegral uu22dVdV

WW YY


24/79

It is thisIt is this secondsecond process which isprocess which is directlydirectlyresponsible for theresponsible for theenhanced dissipation in turbulent flows.enhanced dissipation in turbulent flows. Moreover theMoreover the firstfirst oneone

((i.e. thei.e. the enstrophyenstrophy productionproduction)) is opposing production of strain.is opposing production of strain.

Note that the production of strain isNote that the production of strain is 11)) much moremuch more selfself ,, 22)) it is ait is aspecificspecific (!!)(!!) feature of the dynamics of genuine turbulence havingfeature of the dynamics of genuine turbulence having

no counterpart in the behavior of passive objectsno counterpart in the behavior of passive objects ,, 3)3) EnstrophyEnstrophy

productionproduction

ii

jjss ijij has an addit ional role in exchanginghas an additional role in exchanging

energyenergy betweenbetween enstrophyenstrophy and strain.and strain.

WHO IS THEWHO IS THEGUILTYGUILTY ?? IIII

There existThere exist twotwo nonlocallynonlocally connected and weakly correlatedconnected and weakly correlatedprocessesprocesses.. TheThe secondsecond is the selfis the self--amplification of strainamplification of strain..

(()Ds)Ds//DtDt == -- ssijijssjkjksskiki(1/4)(1/4)iijjssijijssijijpp//xxiixxjj ++ ssijijssijij ++ ssijijFFijij

YY


25/79

NONLOCALITYNONLOCALITYOFOF

VORTICITY/STRAINVORTICITY/STRAINRELATIONRELATION AND THEAND THEISUEISUEOF SURROGATESOF SURROGATES

Field experiment

2000, Israeli field

station, Re=104


26/79

iijjssijij ANDANDssjkjksskikissijijVERSUSVERSUSSURROGATESURROGATE17.517.5uu11// xx11

Field experiment 2004, Sils-Maria, Switzerland, Re= 6800

THE MARIATHE MARIA SILSSILS SITE SWITZERLANDSITE SWITZERLAND


27/79

THE MARIATHE MARIASILSSILSSITE, SWITZERLANDSITE, SWITZERLAND

FIELD EXPERIMENT SUMMER 2004 SILS


28/79

FIELD EXPERIMENT SUMMER 2004 SILSMARIA, SWITZERLAND

The

Israeli

team

The calibration unit at 3 min the field

Height 1850 mHeight 1850 mExperiment wasExperiment was

performed inperformed incollaboration ofcollaboration ofInstituteInstituteof Hydromechanics andof Hydromechanics and

Water ResourcesWater ResourcesManagement, ETH ZurichManagement, ETH Zurich


29/79

THE PROBETHE PROBE


30/79

THE PROBETHE PROBE

Manganin is used as amaterial for the sensor

prongs instead oftungsten because thetemperature coefficient of

the electrical resistance ofmanganin is 400 timessmaller than that of

tungsten.

cold wiresold wires

hot wiresot wires

The tip of the probe with prongs made ofThe tip of the probe with prongs made ofmanganinmanganin

3 mm3 mm

SELFSELF-- OF VORTICITYOF VORTICITY


31/79

(()D)D/Dt =/Dt = ii

jjss

ijij

++ iiii ++ijkijkii FFkk//xxjj

(()Ds)Ds//DtDt == -- ssijijssjkjksskiki (1/4)(1/4)

ii

jjss

ijij ss

ijijpp//xx

iixx

jj ++

ssijijssijij ++ ssijijFFijij

SELFSELF

AMPLIFICATIONAMPLIFICATION

OF VORTICITYOF VORTICITY

ANDANDSTRAINSTRAIN

The property of self amplification of vorticity and strain is responsible for the fact the neitherenstrophy nor the total strain s are inviscid invariants as is the kinetic energy u

SELFSELF-- OF VORTICITYOF VORTICITY


32/79

SELFSELF

AMPLIFICATIONAMPLIFICATION

SELFSELF--RANDOMIZATION/INTRINSIC STOCHASTICITY: NO SOURCE OFRANDOMIZATION/INTRINSIC STOCHASTICITY: NO SOURCE OF

RANDOMNESS IS NEEDED, THE FORCING CAN BE CONSTANT IN TIMERANDOMNESS IS NEEDED, THE FORCING CAN BE CONSTANT IN TIME

AT THE LEVEL OF VELOCITYAT THE LEVEL OF VELOCITYDERIVATIVES: VORTICITY ANDDERIVATIVES: VORTICITY AND

STRAIN (DISSIPATION)STRAIN (DISSIPATION)THE EXTERNAL FORCING ISTHE EXTERNAL FORCING ISIRRELEVANTIRRELEVANT

0 5000-5000

0.5

-0.5

0

5

-5

-10

0

VELOCITYVELOCITY DERIVATIVESDERIVATIVES

FF

OORRCCII

NNGGThree cases:

1. DNS in a periodic box, Re=102

2. DNS in a channel flow, Re=5600

3. Atmospheric SL, Re=104; Re=108

OF VORTICITYOF VORTICITY

ANDANDSTRAINSTRAIN


33/79

Assume there is no production ofAssume there is no production ofenstrophyenstrophy in thein themeanmean iijjssijij == 0.0. Is there turbulence?Is there turbulence?

What about a similar assumption for strainWhat about a similar assumption for strain

productionproduction ssijijssjkjksskiki = 0 ?= 0 ?..


34/79

(()D)D/Dt/Dt == ii

jjss

ijij++

ii

ii++

ijkijk

ii

FFkk//xx

jj

Enstrophynstrophy production is approximately balanced byroduction is approximately balanced byviscous terms at anyiscous terms at any - whatever largehatever large - Reynoldseynoldsnumber?umber?*

Three cases: 1. DNS in a periodic box, Re=102 2. DNS in a channel

flow, Re=5600 3.Atmospheric surface layer, Re=104; Re=108

TENNEKES AND LUMLEY (1972,TENNEKES AND LUMLEY (1972, P.91P.91):):

More: Spatial integrals, running averagesMore: Spatial integrals, running averages

*(In this sense*(In this sense

--

but not only in thisbut not only in this

--

turbulence is not slightly viscous at whatever large Reynolds nturbulence is not slightly viscous at whatever large Reynolds n

umber. In this context theumber. In this context the

question: what happens withquestion: what happens with enstrophyenstrophy/strain production as/strain production as0 is of special interest)0 is of special interest)

Similar result holds for strain production

TEMPORAL EVOLUTION OF SPATIAL INTEGRALS


35/79

(()D)D/Dt =/Dt = iijjssijij ++ iiii ++ ijkijkii FFkk//xxjjddEE

//dtdt PP

----DD

FF

IN THE ENSTROPHY BALANCE EQUATION

TEMPORAL EVOLUTION OF SPATIAL INTEGRALS

Note i)approximate balance between PP

andand ----DD

and ii)irrelevance of the forcing term FF

SimilarSimilarbehavourbehavouris observed withis observed with hyperviscosityhyperviscosity

Re

= 2x102

ARE VORTEX LINES (ORARE VORTEX LINES (OR VORTICITYVORTICITY))


36/79

. . ..a material line which is initiallya material line which is initially

coinsidescoinsides

with a vortex line continues to do so. It is thuswith a vortex line continues to do so. It is thus

possible and convenient to regard a vortexpossible and convenient to regard a vortex--

line as having a continuing identity and asline as having a continuing identity and asmoving with the fluid (moving with the fluid (In a viscous fluid it is,In a viscous fluid it is,

of course, possible to draw the pattern ofof course, possible to draw the pattern ofvortex lines at any instant, but there is no wayvortex lines at any instant, but there is no way

in which particular vortexin which particular vortex--line can be identifiedline can be identified

atat diffrentdiffrent instantsinstants).). BATCHELORBATCHELOR,, 19671967 , p. 274, p. 274

ARE VORTEX LINES (ORARE VORTEX LINES (ORVORTICITYVORTICITY))

APPROXIMATELY FROZEN IN FLUID FLOWAPPROXIMATELY FROZEN IN FLUID FLOW

AT LARGE REYNOLDS NUMBERSAT LARGE REYNOLDS NUMBERS??

STRETCHINGSTRETCHING VERSUSVERSUS (AND/OR?)(AND/OR?)


37/79

STRETCHINGSTRETCHINGVERSUSVERSUS (AND/OR?)(AND/OR?)

COMPRESSINGCOMPRESSING

In turbulent flowsIn turbulent flows iijjssijij > 0> 0 due to predominantdue to predominantstretching. However, there is no stretching withoutstretching. However, there is no stretching without

compressingcompressing ((divdivuu=0=0),), so what is the meaning ofso what is the meaning of

predominant stretchingpredominant stretching?? AlsoAlso -- ssijijssjkjk

sskiki >> 0.0. Is itIs it

too duetoo due predominatpredominat stretching? To clarify this matter astretching? To clarify this matter a

bit of geometrical statistics is neededbit of geometrical statistics is needed

GEOMETRY OFGEOMETRY OF VORTEXVORTEX STRETCHINGSTRETCHING


38/79

GEOMETRY OFGEOMETRY OFVORTEXVORTEXSTRETCHINGSTRETCHING

iijjssijij == 22{{kk coscos

22((,,kk)})}

22{{11 coscos22((,,11)})} == II22{{22coscos2

2((,,22)})} == IIII

22{{33 coscos22((,,33)} =)} = IIIIII

II:: IIII:: IIIIII ==3: 1:3: 1: --11

kkeigenvectors of the rate of strain tensoreigenvectors of the rate of strain tensor ssijij,,kk-- eigenvectors of the rate of strain tensoreigenvectors of the rate of strain tensor ssijij,,

11 >> 22 >>33;; 11++22 ++33 == 00 ((divdiv uu=0)=0);;11 > 0,> 0, 33


39/79

EIGENEIGEN CONTRIBUTIONSCONTRIBUTIONS

Value 0.8m 1.2m 2.0m 3.0m 4.5m 7.0m 10m

1 1.441.44 1.601.60 1.361.36

0.670.67-1.03

0.290.29

0.150.150.56

1.631.63

0.520.52-1.15

0.500.50

0.100.10

0.41

1.371.37

2 0.440.44 0.620.62

1.041.041.531.531.311.31

0.460.46-0.77

0.520.52

0.140.140.34

1.911.91

0.450.45-1.36

0.490.49

0.510.51

0.580.58

0.100.10

0.460.46-0.99

0.460.46

0.130.130.41

2.082.08

0.470.47-1.55

1 0.530.53 0.330.33

0.490.49

0.100.10

-0.62

0.490.49

0.160.160.35

1.551.55

0.540.54-1.09

0.500.50

0.40

0.490.49

3 0.38 0.63 0.36

0.410.110.11

0.40

1 1.771.77 1.561.56 2.192.19

2 0.470.47 0.500.50 0.470.473 -1.24 -1.07 -1.66

1 0.510.51 0.500.50 0.490.49

20.080.08

0.090.09

0.100.10

3 -0.87 -1.22 -0.85

2 0.090.09 0.050.05 0.150.15

3 0.41 0.41 0.41

( ) ,cos22

( )

,cos222

( ) ,cos2

( ) ,cos22

22{{ 22(( )} I)} I


40/79

Note that the dominating term is the first one (Note that the dominating term is the first one (II))

associated with the first eigenvectorassociated with the first eigenvector 11 corresponding tocorresponding tothe largest (purely positive)the largest (purely positive) eigenvalueeigenvalue.. This isThis is

comprises the meaning ofcomprises the meaning of predominant stretchingpredominant stretching..YetYet vorticityvorticity is preferentially aligned with the secondis preferentially aligned with the second

eigenvectoreigenvector 22 which is quite a bit counterintuitive.which is quite a bit counterintuitive.

iijjssijij == 22{{11 coscos

22((,,11)} I)} I

++ 22{{22 coscos22((,,22)} II)} II

++ 22

{{

33 coscos22

((

,,

33)} III)} III

ALIGNMENT BETWEEN THE EIGENFRAME I OF THE


41/79

ALIGNMENT BETWEEN THE EIGENFRAMEI OF THE

RATE OF STRAIN TENSORSij

AND VORTICITY

3D-PTV; Re=60Field experiment; Re= 104

Note essentially the same behavior at large and smallNote essentially the same behavior at large and small Re

Same in DNSSame in DNS

PDFPDFSS OFOFEIGENVALUESEIGENVALUES,, kk


42/79

OF THE RATE OF STRAIN TENSOROF THE RATE OF STRAIN TENSOR ssijij

Field experimentField experiment

2004,2004, SilsSils--Maria,Maria,

Switzerland, ReSwitzerland, Re=68006800

EIGENVALUESEIGENVALUES ii


43/79

EIGENVALUESU S ii

OF RATE OF STRAINOF RATE OF STRAIN,, ssijijValue i 0.8m 2m 10m

1 0.530.53 0.510.51 0.470.47

2 0.09 0.09 0.06

3 -0.62 -0.60 -0.53

1 0.400.40 0.400.40 0.410.41

2 0.04 0.04 0.06

1 0.480.48 0.540.54 0.600.60

2 0.01 0.02 0.02

3 0.56 0.56 0.55

3 -0.73 -0.83 -1.04

2/12

/ s

22 / s

2/323 / s

So what aboutSo what about -- ssijijssjkjksskiki > 0> 0 (in homogeneous(in homogeneous


44/79

So what aboutSo what about ssijijssjkjksskiki > 0 0 (in homogeneous(in homogeneous

flows equal toflows equal to -- 4/34/3 iijjssijij ) is it due tois it due topredominant stretchingpredominant stretching??ssijijssjkjksskiki ==

11

33 ++22

33 ++33

33 =33

11

22

331133 :: 223

3 :: 3333 = 1.5 : 0.05:= 1.5 : 0.05: --2.52.5

That isThat is -- ssijijssjkjksskiki == -{1133 ++ 2233 ++3333 } isispositive due to dominant contribution frompositive due to dominant contribution from -- 3333 ,, i.e.i.e.predominantpredominant compressingcompressing rather than stretching !rather than stretching !

There are several otherThere are several other (see the examples below)(see the examples below) important processesimportant processesin which predominantin which predominant compressingcompressing is the main player : there isis the main player : there isno reason to push everywhere stretching, especially vortexno reason to push everywhere stretching, especially vortex

stetchingstetching..

MORE EXAMPLES WITH PREDOMINANT


45/79

CONTRIBUTION OF COMPRESSING

TKETKEPRODUCTION IN TURBULENT SHEAR FLOWSPRODUCTION IN TURBULENT SHEAR FLOWS --uiuuiukkSSikik

EEVOLUTION OF DISTURBANCES INVOLUTION OF DISTURBANCES INGENUINE AND PASSIVE TURBULENCEGENUINE AND PASSIVE TURBULENCE

PPRODUCTION OF GRADIENTS/ DISSIPATION OFRODUCTION OF GRADIENTS/ DISSIPATION OF

PASSIVE SCALARPASSIVE SCALAR

PPRODUCTION OFRODUCTION OFVORTICITYVORTICITY GRADIENTS IN TWOGRADIENTS IN TWO--

DIMENSIONAL TURBULENCEDIMENSIONAL TURBULENCE


46/79

TKETKEPRODUCTION IN SHEAR FLOWSPRODUCTION IN SHEAR FLOWS

STRETCHING OR COMPRESSING?STRETCHING OR COMPRESSING?

The turbulent energy production in a turbulent shear flow is knoThe turbulent energy production in a turbulent shear flow is known town to


47/79

be represented by the termbe represented by the term --SSikik ,, withwith uu ii being the componentsbeing the components

of velocity f luctuations, andof velocity f luctuations, and SSikik the mean rate of strain. In turbulentthe mean rate of strain. In turbulentflows which are twoflows which are two--dimensional in the mean (i.e. such thatdimensional in the mean (i.e. such that //xx33 ==00) the production term can be represented as) the production term can be represented as

-- SSikik == -- uu11SScoscos(u,(u,11SS)) -- uu22SScoscos(u,(u,22SS))wherewhere uu = u= u11 + u+ u22,, ii

SS are theare the eigenvalueseigenvalues andand iiSS are theare the

corresponding eigenvectors of the mean rate of strain tensorcorresponding eigenvectors of the mean rate of strain tensorSSikik , and, and

11SS

>0,>0,22SS

0, the production term, the production term --SSikik can be (andcan be (andusually is) positive due tousually is) positive due to positivenesspositiveness of the term associated with theof the term associated with thecompressive (negative)compressive (negative) eigenvalueeigenvalue/eigenvector/eigenvector{{22,, 22

SS}} , of the mean, of the mean

strainstrain SSikik . In this sense the turbulent energy production is due to the. In this sense the turbulent energy production is due to thepredominant compressingpredominant compressing of material elementsof material elements rather than stretchingrather than stretching ..

TKETKEPRODUCTION: STRETCHING ORPRODUCTION: STRETCHING OR


48/79

The main featureThe main feature -- tendency of alignment of the vectortendency of alignment of the vectoruu with bothwith both

11SS andand 22

SS.. The difference is that the latter alignment is somewhatThe difference is that the latter alignment is somewhatstronger, which results in the positive value of the TKE productstronger, which results in the positive value of the TKE product ionion

TurbulentTurbulent

channel flowchannel flow

(same in BL)(same in BL)

Phys Fluids,Phys Fluids, 1616,,

2704 (2004)2704 (2004)

COMPRESSING?COMPRESSING?

Tennekes & Lumley 1972 pp 40-41TKETKEPRODUCTION:PRODUCTION:STRETCHING OR COMPRESSINGSTRETCHING OR COMPRESSING??


49/79

e e es & u ley 97 pp 0STRETCHING OR COMPRESSINGSTRETCHING OR COMPRESSING??

TKETKEPRODUCTION:PRODUCTION:


50/79

STRETCHING OR COMPRESSING?STRETCHING OR COMPRESSING?

TKETKEPRODUCTION: STRETCHING ORPRODUCTION: STRETCHING ORCOMPRESSING?COMPRESSING? SSUBGRIDUBGRID STRESSESSTRESSES


51/79

COMPRESSING?COMPRESSING?SSUBGRIDUBGRIDSTRESSESSTRESSES

TKETKEPRODUCTION: STRETCHING ORPRODUCTION: STRETCHING OR


52/79

COMPRESSING?COMPRESSING?SSUBGRIDUBGRIDSTRESSESSTRESSES

GROWTH OF DISTURBANCES INGROWTH OF DISTURBANCES IN


53/79

GROWTH OF DISTURBANCES INGROWTH OF DISTURBANCES IN

GENUINE (GENUINE (EEuu, E, E, E, Ess)) ANDANDPASSIVEPASSIVE

((EEAA , E, EBB , E, E , E, EGG))TURBULENCETURBULENCE

Looking at the evolution of the disturbanceLooking at the evolution of the disturbanceuu of some flow realizationof some flow realization uu in a statisticallyin a statisticallysteady state and similarly for other quantities.steady state and similarly for other quantities.For more details seeFor more details see TsinoberTsinober andand GalantiGalanti 2003,2003, Phys.Phys.

FluidsFluids,, 155, 3514, 3514--3531.3531.


54/79

AgainAgain compressioncompression

rather than stretchingrather than stretching

GROWTH OF DISTURBANCES IN GENUINE,GROWTH OF DISTURBANCES IN GENUINE, EEuu, E, E, E, EssAND PASSIVE,AND PASSIVE, EEAA , E, EBB , E, E , E, EGG TURBULENCETURBULENCE


55/79

The process ofThe process ofevolution andevolution andamplification ofamplification of

disturbancesdisturbances -- bothbothin genuine andin genuine and`passive' turbulence`passive' turbulence-- is dominated by theis dominated by the

strain field of thestrain field of thebasic flowbasic flow.Inn allll their energyheir energyproduction hasroduction hasthe formhe formuuiiuujjssijij

AND PASSIVE,, EAA , E, BB , E, , E, G,G,TURBULENCER C

TsinoberTsinober && GalantiGalanti, 2003, 2003

PPRODUCTION OFRODUCTION OFVORTICITYVORTICITY GRADIENTS INGRADIENTS INTWOTWO--DIMENSIONAL TURBULENCEDIMENSIONAL TURBULENCE


56/79

DD//DtDt == -- (())u +u + ;; == (1/2)D2/Dt = -

i

k

sik

+ i

i

The main contribution to the productionThe main contribution to the production ((--iikkssikik > 0> 0))

-- iikkssikik == -- 22

kkcoscos22

((,,kk))is associated withis associated with compressivecompressive eigenvalueeigenvalue 33:: it is due toit is due to --2233coscos22((;; 33).). Note thatNote that partnerpartner toto ii has the samehas the same partner aspartner as doesdoesvorticityvorticity (strain), but they ((strain), but they (ii andand ssikik)) are not equal partners asare not equal partners as ii liveslivesat much smaller scales thanat much smaller scales than ssijij andand there is no strain production (!) eitherthere is no strain production (!) either..However,However, production ofproduction ofpalinstrophypalinstrophy, i.e. (, i.e. (curlcurl))22 is due to predominantis due to predominantcontribution from the term associated with the stretchingcontribution from the term associated with the stretching eigenvalueeigenvalue 1.1.

TWOTWO-DIMENSIONAL TURBULENCEDIMENSIONAL TURBULENCE

PRODUCTION OF GRADIENTS/PRODUCTION OF GRADIENTS/

DISSIPATION OF PASSIVE SCALARDISSIPATION OF PASSIVE SCALAR


57/79

DISSIPATION OF PASSIVE SCALARDISSIPATION OF PASSIVE SCALAR

D/Dt = - ()u + ; G = (1/2)(1/2)DG2/Dt = - GiGksik+ GiGi

The main contribution to the production (-GiGksik > 0)- GiGksik= - G

2kcos2(G;k)

is associated with the alignment of the temperature gradient and the

eigenvector 3 corresponding to the compressive eigenvalue 3: itis due to -G23 cos2(G; 3).

PDF OF THE PRODUCTIONPDF OF THE PRODUCTION --GGiiGGjjssijij,,Selected resultsSelected results


58/79

The PDF of the production,-GiGjsij, obtained in ourexperiments in a slightly

heated jet is very similar tothat obtained in DNS ofNSE by Tsinober andGalanti (2003).

The PDF of -GiGjsij ispositivelypositively skewedskewedand the mean ispositivepositive.

iiijji GGsGGGDt

D 22

2

1

+=

Alignments between the temperature gradientAlignments between the temperature gradient GG

d thd th i fi f f th t f t i tf th t f t i t


59/79

The main effect, the alignment between the temperature gradient and theeigenvector3 corresponding to the compressive eigenvalue 3< 0, is captured

well in the measurements and is similar to that obtained from DNS

and theand the eigenframeeigenframe ii of the rate of strain tensorof the rate of strain tensorssijij((kk are the correspondingare the corresponding eigenvalueseigenvalues));; 11>> 22>> 33;; 11>> 0,0, 33


60/79

BACK TOBACK TOVORTICITYVORTICITY

VERSUS STRAINVERSUS STRAIN

CONCENTRATEDCONCENTRATED VORTICITYVORTICITYOR STRAINOR STRAINARE THEY THAT IMPORTANTARE THEY THAT IMPORTANT??

STRONGER NON LINEARITY IN STRAINSTRONGER NON LINEARITY IN STRAINDOMINATED REGIONSDOMINATED REGIONS

JOINTJOINTPDFPDFSS OFOF ENSTROPYENSTROPY((toptop)) ANDANDSTRAINSTRAIN ((bottombottom))PRODUCTIONPRODUCTION WITH ENSTROPHYWITH ENSTROPHY(left)(left) ANDAND STRAINSTRAIN (right)(right)


61/79

PRODUCTIONR WITH ENSTROPHYR ( f )( f ) STRAINR ( g )( g )

FROM OURFROM OUR

FIELDFIELD

EXPERIMENTEXPERIMENT

AT ReAT Re=10=1044

Physics of Fluids,Physics of Fluids,

1313,, 311 (2001)

Note muchNote much

larger positivelarger positiveshift with strainshift with strain

( )22

22

/

/ii

Rate ofRate of enstrophyenstrophyproduction and itsproduction and its


62/79

22 /ii

( )22

22

/

/

ii

sii

( )2

2

/

/

ikki

ikki

s

s

( )2

2

/

/

ikki

sikki

s

s

production and itsproduction and its

viscous reductionviscous reductionconditioned onconditioned on

strain andstrain andvorticityvorticity

/,/ss

NOTE THENOTE THE LARGE RATELARGE RATE OFOFENSTROPHY PRODUCTION INENSTROPHY PRODUCTION IN

STRAIN DOMINATED REGIONSSTRAIN DOMINATED REGIONS((RED CURVERED CURVE))AS COMPARED TOAS COMPARED TOREGIONS OF LARGE ENSTROPHYREGIONS OF LARGE ENSTROPHY((BLUE CURVEBLUE CURVE))

ALL NONLINEARALL NONLINEAR

TERMS BEHAVETERMS BEHAVE

THIS WAY!THIS WAY!

EigenEigen--contributions tocontributions to iijjssijij//22==kkcoscos22((,,kk))


63/79

j j

NonNon--linear terms are growing much slower in the enstrophylinear terms are growing much slower in the enstrophy

dominated regions than in the strain dominated regionsdominated regions than in the strain dominated regions..

Summary of threeSummary of three--

dimensionaldimensional

QQ


64/79

dimensional,dimensional,

incompressible flowincompressible flowpatterns/patterns/

local structure of thelocal structure of the

flow field in the frameflow field in the frame

following a fluidfollowing a fluid

particleparticle((fromfromSoriaSoriaet al. 1994, afteret al. 1994, after

Perry et alPerry et al.. 19901990).).

Q = 1/4Q = 1/4(( -- 2s2s ijijss ijij),),

R =R = -- 1/3(s1/3(s ijijssjkjksskiki ++

.. + 3/4+ 3/4iijjss ijij).).

RRRM


65/79

STRONGER NON LINEARITY INSTRONGER NON LINEARITY INSTRAIN DOMINATED REGIONSSTRAIN DOMINATED REGIONS

HOW THIS LOOKS ON THEHOW THIS LOOKS ON THER Q PLANEPLANEQ= (1/4){2 -2s2}; R= - (1/3){s ijsjkskiki+(3/4)ijs ij}

Second and third invariants of the velocity gradient

tensor AAikik== uuii//xxkk

RR--QQ-- PLOTPLOT


66/79

velocity gradient

Field experiment, Re= 6800 DNS of NSE, Re= 690

PTV, Re= 80

Chacin et al 2000( )

+=

=

ikkimikmik

ikik

ssssR

ssQ

4

3

3

1

24

1 2

- third invariant of the velocity gradient tensor

- second invariant of the velocity gradient tensor

The first invariant is vanishing as a consequence of incompressibility

k

i

x

u

Q= (1/4){2 -2s2}; R= - (1/3){s ijsjkskiki+(3/4)ijs ij}


67/79

D=0D=0

CONDITIONAL AVERAGES ON THE RCONDITIONAL AVERAGES ON THE R--Q PLANE OFQ PLANE OF REYNOLDS STRESS (LEFT)REYNOLDS STRESS (LEFT)ANDANDTKE `generating eventsTKE `generating events ((RIGHT),RIGHT), CHASIN AND CANTWELL , 2001, 2001

Q= (1/4){2 -2s2}; R= - (1/3){s ijsjkskiki+(3/4)ijsij}


68/79

CONDITIONAL AVERAGES ON THE RCONDITIONAL AVERAGES ON THE R--Q PLANE OFQ PLANE OF REYNOLDS STRESSREYNOLDS STRESS ((LEFTLEFT))ANDAND

DISSIPATIONDISSIPATION ((RIGHTRIGHT)),, CHASIN AND CANTWELL , 2001, 2001

uv/uuv/u22

--1.31.3 --0.30.3

Sweeps (+uSweeps (+u--vv))

EjectionsEjections

((--u+v)u+v)

QQ RR

QQ

RR

D=0D=0 D=0D=0

/(u/(u

44

))0.130.13 0.360.36

One of the most interesting (from our point of view)One of the most interesting (from our point of view)

findings is that the main contribution to the shear stressfindings is that the main contribution to the shear stress


69/79

findings is that the main contribution to the shear stress,findings is that the main contribution to the shear stress,

turbulent energy production, and dissipation comes fromturbulent energy production, and dissipation comes from

the regions with Q0, or a bit more precisely inin

regions with D>0, where D=((27)/4)Qregions with D>0, where D=((27)/4)Q+R+R is theis the discriminantdiscriminant ofof

uu ii //

xxjj ..That is the regions of major nonlinear activity are really assocThat is the regions of major nonlinear activity are really associatediated

with large strain (mainly corresponding to whatwith large strain (mainly corresponding to what ChacinChacin and Cantwelland Cantwell

call `blank' spaces) rather than with regions of concentratedcall `blank' spaces) rather than with regions of concentrated vorticityvorticity

with lower dissipation.with lower dissipation.

In other words it seems that concentratedIn other words it seems that concentrated vorticityvorticity is notis not


70/79

that important also in turbulent shear flows and thatthat important also in turbulent shear flows and thatstructure(sstructure(s) associated with turbulence (not only its) associated with turbulence (not only its

energy) production are mainly due to the large strainenergy) production are mainly due to the large strain

rather than largerather than large vorticityvorticity .. Structure(sStructure(s) associated with the) associated with thelatter seem to be the consequence of the turbulentlatter seem to be the consequence of the turbulent

dynamics rather than its dominating factor. A final remarkdynamics rather than its dominating factor. A final remark

is that these results do not contradict the importance ofis that these results do not contradict the importance ofvorticityvorticity in maintaining thein maintaining the ReyloldsReylolds stress. First, thesestress. First, these

are relations for the mean quantit ies, and second, there isare relations for the mean quantities, and second, there is

no turbulent flow withoutno turbulent flow without vorticityvorticity . However, important. However, importantdetails of the relations between Reynolds stress,details of the relations between Reynolds stress, vorticityvorticity ,,

strain and their production remain not clear.strain and their production remain not clear.


71/79

The interpretation of the results byThe interpretation of the results byChacinChacin and Cantwell, 2000 (andand Cantwell, 2000 (and

similar) given here is not in fullsimilar) given here is not in fullagreement with their conclusions,agreement with their conclusions,

especially regarding the role ofespecially regarding the role ofvortices and concentratedvortices and concentrated vorticityvorticity

in turbulent flows (again vortexin turbulent flows (again vortexobsession).obsession).

vorticityvorticity

NICE VORTICES/WORMS/FIALMENTS OR WHATEVERNICE VORTICES/WORMS/FIALMENTS OR WHATEVER


72/79

velocityvelocityvorticityvorticity

SHE et al. 1991SHE et al. 1991

At this stage, this alternative approachAt this stage, this alternative approach ((i.e. thei.e. thestructuralstructural)) has not led to ahas not led to a

generally applicable quantitative model, neithergenerally applicable quantitative model, neither for better or worsefor better or worse has it ahas it a


73/79

major impact on the statistical approaches. Consequently themajor impact on the statistical approaches. Consequently the deterministicdeterministicviewpoint is neither emphasized nor systematically presentedviewpoint is neither emphasized nor systematically presented,, POPE 2000POPE 2000..

This does not mean that there existsThis does not mean that there exists generally applicable quantitativegenerally applicable quantitative

modelmodel based on statistical approachesbased on statistical approaches..

REALREAL SHE et al. 1991SHE et al. 1991 GAUSSIANGAUSSIAN

STRUCTURESSTRUCTURES OF INTENSE VORTICITYOF INTENSE VORTICITYAND STRAINAND STRAIN,,


74/79

(a)Big structure with ||>3; (b)Small structure with ||>3; (c)Big structure with ||>6

(a) |s|>2.8s; (b) |s|>4.2s

SIMILAR TO THOSE INSIMILAR TO THOSE IN

SHE et al., 1991 andSHE et al., 1991 and

BORATAV AND PELZ, 1997BORATAV AND PELZ, 1997

CHEN, 2000CHEN, 2000

aa bb cc

aa bb

MoisyMoisy& Jimenez, 2004& Jimenez, 2004

IS THE FLOW FIELD INIS THE FLOW FIELD IN WORMSWORMS


75/79

JIMENEZ ET AL 1993JIMENEZ ET AL 1993SIMPLE?SIMPLE? IS IT QUASI-2D?

CONSEQUENCES FOR REPRESENTATION OF TURBULENT FIELDSCONSEQUENCES FOR REPRESENTATION OF TURBULENT FIELDS

IS THE FIELD OFIS THE FIELD OFVORTICITYVORTICITY ININ WORMSWORMSSIMPLE?SIMPLE?


76/79

JIMENEZ & WRAY 1996JIMENEZ & WRAY 1996

THETHE AMOUNTAMOUNT OFOF COMPRESSINGCOMPRESSING ININ

WORMSWORMS IS THE SAME AS IN THEIS THE SAME AS IN THE

WHOLE FIELD !WHOLE FIELD ! HENCE INADEQUATEHENCE INADEQUATEREPRESENTATION OF THE FLOWREPRESENTATION OF THE FLOW

FILED BY A COLLECTION OFFILED BY A COLLECTION OF PURELYPURELY

STRETCHED VORTICES (or otherSTRETCHED VORTICES (or other`simple' objects),`simple' objects), ESPECIALLY THOSEESPECIALLY THOSE

WHICH DO NOTWHICH DO NOTINTERACTINTERACTWITHWITH

STRAIN.STRAIN... THESE LATTER ARETHESE LATTER ARE

DESCRIBED BY DIFFRENTIALDESCRIBED BY DIFFRENTIAL

EQUATIONSEQUATIONS.. THE REAL ONES ARETHE REAL ONES ARE

DESCRIBED BYDESCRIBED BY INTEGROINTEGRO

DIFFERENTIAL EQUATIONSDIFFERENTIAL EQUATIONS..


77/79


78/79

IN LIEU OFIN LIEU OF

CONCLUSIONCONCLUSION

Looking in more essential details (both physics and the mathematLooking in more essential details (both physics and the mathematicalical


79/79

ones) of the processes ofones) of the processes ofselfself-- amplificiationamplificiation of the field of velocityof the field of velocityderivatives seems to be the key issue for most of the problems oderivatives seems to be the key issue for most of the problems off

turbulence but alsoturbulence but also 33--D NSED NSE andand EulerEuler. This includes subtle geometrical. This includes subtle geometrical

relations betweenrelations between vorticityorticity andand straintrain (which are likely to be the(which are likely to be themain guilty of almost happening in turbulence: after all theremain guilty of almost happening in turbulence: after all there isis

no turbulence withoutno turbulence without vorticityvorticity production)production) and several others ofand several others ofdynamical significance . Understanding of these processes and thdynamical significance . Understanding of these processes and therebyereby

essential aspects of turbulence physics seems to form the basisessential aspects of turbulence physics seems to form the basis forfor

constructive approach to a great variety of problems including econstructive approach to a great variety of problems including effectiveffectivehandling of nonlinearities which will allow to solve the standarhandling of nonlinearities which will allow to solve the standard 3Dd 3D--NSENSE

and Euler problems and not the other way round.and Euler problems and not the other way round.

Turbulence Concepts

Documents

Transcript of Turbulence Concepts