Outline I - thomas-gomez.netthomas-gomez.net/images/Lectures/Turbulence/turbulence_lectures... · Reynolds decomposition Mean NS equations Reynolds stress tensor Kinetic energy Fluctuating

Lectures in Turbulence

Thomas Gomez

LMFL

2017-2018

// [email protected] 1/374

Outline I1 The turbulence fact : Definition, observations and universal features of

turbulenceObjective of the coursePreliminary definitionsUbiquitous character of turbulence

NaturalEngineering

Two complementary approaches : Experimental and NumericalExperimentsSimulations

Essential and universal features of turbulent flowsConclusion

2 The governing equationsNavier-Stokes EquationsVorticityPressure in incompressible flowsNS equations and SymmetriesDimensionless numbers

Reynolds numberStrouhal number


Outline II

Prandtl numberNon viscous invariantsValidityCharacteristics of turbulent flowsHomogeneity and isotropyCanonical turbulent flowsExercises

Small scalesLarge scalesInterscales relationsDispersion law of a polluantBeltrami FlowsAtomic Blast

3 Statistical description of turbulenceRealization of a turbulent flowProbability density functionJoint probability density functionThe correlation functionErgodicity and statistical symmetries


Outline III

Statistical averageReynolds decompositionMean NS equationsReynolds stress tensorKinetic energyFluctuating NS equationsReynolds stress tensor equationKinetic energy of the fluctuationsExercises

Scalar dynamics

4 Turbulence modelingClosure problemModels for the closure of the systemFirst order models

zero equationOne equation modelsTwo equations modelsGeneric form for two equations models

Second order models


Outline IV

PrincipleReynolds stress model

ExerciseShear layer

5 Turbulent wall bounded flowsDescriptionWall effectsSpecific physical quantitiesMean velocity profileChannel flowsBoundary layerCoherent structures and turbulent dynamicsTurbulent drag : Generation and ControlSkin friction controlThermal boundary layerThermal boundary layer control

6 Homogeneous Isotropic TurbulenceSpectral description


Outline V

Spectral equationsSpectral phenomenological descriptionClosure spectral theory

Obukhov Model 1941Spectral Eddy Viscosity

Passive scalar dynamicsFree decaying turbulence

Kinetic energyScalar

7 Homogeneous Shear FlowsDefinition and observationsCraya’s definition of the homogeneitySlow and fast termsSimplification of the budget equationsRapid Distortion Theory

8 Results based on the equations of the dynamics in fully developedturbulence

Tensorial general expressionsvon Kármán equation


Outline VI

Kolmogorov 4/5 lawBibliographyFinal Exam


PART I

The turbulence fact :

Definition, observations and universal features

of turbulence

The Turbulence fact// [email protected] 8/374

1 The turbulence fact : Definition, observations and universal features ofturbulence

2 The governing equations

3 Statistical description of turbulence

4 Turbulence modeling

5 Turbulent wall bounded flows

6 Homogeneous Isotropic Turbulence

7 Homogeneous Shear Flows

8 Results based on the equations of the dynamics in fully developedturbulence


1 The turbulence fact : Definition, observations and universal features ofturbulence

Objective of the coursePreliminary definitionsUbiquitous character of turbulenceTwo complementary approaches : Experimental and NumericalEssential and universal features of turbulent flowsConclusion


Introduction to Turbulent Flows (20h)

Fundamental featuresCharacteristics of turbulent flowsEquations + mathematical toolsClosure problemPhysics of turbulenceModelling for numerical simulations

ObjectivesPredict : the behavior of complex turbulent flowsEstimate : lift, drag, pressure losses, acoustics, mixing, pollution,meteorological/solar forecasts. . .

Physics of turbulence =) Simulations & Experiences

The Turbulence fact/Objective of the course/ [email protected] 11/374

What is turbulence ? Preliminary definitions

Taylor and von Kármán 1937"Turbulence is an irregular motion which in general makes its appearancein fluids, gaseous or liquid, when they flow past solid surfaces or evenwhen neighboring streams of the same fluid flow past or over one another."

An attempt to give a more precise definitionA turbulent flow is a fluid flow where the different variables characterizingthe flow take random values in space and time so that statisticalvalues of these variables can be defined.

u, p, ⇢, T = random fct. of x, t

Where can we observe "irregular" flows ?

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Astrophysical flows

Collapse and fragmentation of aturbulent molecular cloud (simulation)

https://ned.ipac.caltech.edu/level5/Sept06/Loeb/Loeb5.html

Galaxy

The Turbulence fact/Ubiquitousness/Natural [email protected] 13/374

Astrophysical flows : Planetology

Atmosphere of JupiterGreat Red Spot diameter ⇠ 40000km


Astrophysical flows

The sun


Atmospheric flows

Clouds Atmospheric pollution


Atmospheric flows

Sakura-jima eruption as seen on August 18, 2013, Japan


Atmospheric flows

Combined Flights Ground Measurements, 30Mar-03Apr2011, FukushimaThe Turbulence fact/Ubiquitousness/Natural [email protected] 18/374

Oceanic flows


Atmospheric flows

Windturbine wake Wake of an island :von Karman street


Rivers

River

Leonardo Da Vinci (1452 - 1519)


Aerodynamics

Peugeot Side view mirror

The Turbulence fact/Ubiquitousness/Engineering [email protected] 22/374

Transitional flows

Transition - Wake - Recirculation region

The Turbulence fact/Ubiquitousness/Engineering [email protected] 23/374

Propellers : Aeronautic / Hydrodynamic performance

The Turbulence fact/Two approaches/ [email protected] 24/374

Jets

Jets, KwonSeo2005

With the increase of the jet velocityEarly transitionIncrease of the jet widthMore intense fluctuations

The Turbulence fact/Two approaches/Experiments [email protected] 25/374

Experiments : Flow over a bump

LFML-KF , Re ⇠ 2000

The Turbulence fact/Two approaches/Experiments [email protected] 26/374

Flow over a bump

Channel Flow,Reh = 12600, based on the half-width of the channelDNS

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Vorticity filaments

Iso-value of the vorticityDNS of Compressible flowD. H. Porter, A. Pouquet,and P. R. Woodward


Vorticity filaments


Turbulence modifies local properties

Wind tunnelFrisch 1995


Turbulence modifies global properties

Mean propertiesForces : Drag, LiftPressure lossesHeat transfer

CD = F12⇢U2S

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Drag coefficient for the flow past a sphere

CD =F

12⇢U2S

1 Laminar2 Turbulent

transition3 Drag crisis :

Turbulent BL

Sphere in rotation


Drag coefficient valueInfluence of the shape of the body

Body Drag coefficient

Aeronautics 0.005–0.010

Hydrodynamics ⇠ 0.03

Automotive record ⇠ 0.14

AX (small car in 80’s) 0.31

Clio II 0.35

Prius (2009) 0.29

CD =F

12⇢U2S

F induced force⇢ densityU velocityS frontal surface area ofthe body

Strong influence of the turbulence intensity ! ! ! i.e. ReThe Turbulence fact/Essential features/ [email protected] 33/374

Pressure losses in pipes

L : LengthD : Diameter✏ : Roughnessµ : Viscosity� pressure losscoefficientPressure Losses :

�P =1

2⇢u2�

L

D

where

Re =⇢uD

µ


Pressure losses


Major universal properties of turbulence

Non exhaustive listDisorder / Irregular flows / Complex3D / Structured by vorticity / continuous self-production of vorticityand strainInfluence on local/global properties of the flowsWide range of strongly and nonlocally interacting degrees of freedomin time and spaceTurbulent diffusivity =) Efficient mixingHighly dissipative, statistically irreversibleIntrinsic Spatio-temporal random process : Turbulence is chaos (butnot necessarily vice versa) ; its intrinsic property is self-stochastization or self-randomization.Quite unpredictable : loss of predictability, but stable statisticpropertiesStrongly nonlinear, non-integrable, nonlocal, non-GaussianMultiphysics : Scalar, MHD, Multiphase, Flotability, Stratification...


Societal concerns

IssuesDrag reductionPropulsion optimization

Reduce energy consumptionReduce polluant production

Nuclear, wind and water electrical power generationAtmospheric and Solar forecastingGlobal warmingAtmospheric CO2 balancePolluant contamination : Ocean, Atmosphere, RiverFlow control : aerodynamics, nuclear field, transportProcess engineering : mixing, plasma...Reduce noise production


Tackle the turbulence issueExperimentsReally too expensive ?...

Numerical simulations ! yes but...

Moore’s law (1965

revised in 1975) :

the number of

transistors in a

dense integrated

circuit doublesapproximatelyevery two years.


Tackle the turbulence issue

ExperimentsReally too expensive ?...

Numerical simulations ! yes but also too expensive so far...

dof ⇠ `/⌘ = Re3/4

` : Large scale⌘ : small scaleRe : Reynolds number

# of Degree ofFreedom


Energy comsumptionPower plotted against maximum speed.P / V 3 ? WHY?


Conclusion

FactsTurbulent flows are ubiquitous and complexStrong impact on : forecasting, environnemental pollution (acoustics,atmosphere, ocean), energy consumption/production, control

NecessityDevelop new tools for evaluating and anticipating the turbulent systembehaviour :

Modeling : Understand the physics of the turbulence(Physics+Mathematics).Simulate : Develop new mathematical and numerical adaptedmethods (Computing Science).Post-process : Manipulate huge data banks in order to extractpertinent and useful informations (Data Science).

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Challenge

Clay prize :Since understanding the Navier-Stokes equations is considered to be thefirst step to understanding the elusive phenomenon of turbulence, the ClayMathematics Institute in May 2000 made this problem one of its sevenMillennium Prize problems in mathematics. It offered a US$ 1, 000, 000prize to the first person providing a solution.

Problem :Prove or give a counter-example of the following statement : In threespace dimensions and time, given an initial velocity field, there exists avector velocity and a scalar pressure field, which are both smooth andglobally defined, that solve the Navier-Stokes equations.

The Turbulence fact/Conclusion/ [email protected] 42/374

Documents

Outline I - thomas-gomez.netthomas-gomez.net/images/Lectures/Turbulence/turbulence_lectures... · Reynolds decomposition Mean NS equations Reynolds stress tensor Kinetic energy Fluctuating