CFD for Turbulence lectures - thomas-gomez.net

Lectures in Turbulence

Thomas Gomez

Laboratoire de Mécanique des Fluides de Lille : Kampé de Fériet

2019-2020

// [email protected] 1/327

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

turbulenceObjective of the coursePreliminary definitionsUbiquitous character of turbulence

Natural

Engineering

Two complementary approaches : Experimental and NumericalExperiments

Simulations

Essential and universal features of turbulent flowsConclusion

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

Reynolds number

Strouhal number


Outline II

Prandtl number

NS non viscous invariantsValidityCharacteristics of turbulent flowsHomogeneity and isotropyCanonical turbulent flowsExercises

3 Statistical description of turbulenceRealization of a turbulent flowProbability density functionJoint probability density functionThe correlation functionErgodicity and statistical symmetriesStatistical averageReynolds decompositionMean NS equationsReynolds stress tensorKinetic energy


Outline III

Fluctuating NS equationsReynolds stress tensor equationKinetic energy of the fluctuationsExercises

Scalar dynamics

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

zero equation

One equation models

Two equations models

Generic form for two equations models

Second order modelsPrinciple

Reynolds stress model

ExerciseShear layer

5 Turbulent wall bounded flows


Outline IV

DescriptionWall effectsSpecific physical quantitiesMean velocity profileChannel flowsBoundary layersCoherent structures and turbulent dynamicsTurbulent drag : Generation and ControlSkin friction control

6 Homogeneous Isotropic TurbulenceSpectral descriptionSpectral equationsSpectral phenomenological descriptionClosure spectral theory

Obukhov Model 1941

Spectral Eddy Viscosity

Passive scalar dynamicsFree decaying turbulence


Outline V

Kinetic energy

Scalar

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

Tensorial general expressionsvon Kármán equationKolmogorov 4/5 lawBibliography


PART I

The turbulence fact :

Definition, observations and universal features

of turbulence

The Turbulence fact// [email protected] 7/327

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

2 The governing equations

3 Statistical description of turbulence

4 Turbulence modeling


6 Homogeneous Isotropic Turbulence




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


Introduction to Turbulent Flows

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] 10/327

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 ?

The Turbulence fact/Preliminary definitions/ [email protected] 11/327

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] 12/327

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] 17/327

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] 21/327

Transitional flows

Transition - Wake - Recirculation region

The Turbulence fact/Ubiquitousness/Engineering [email protected] 22/327

Propellers : Aeronautic / Hydrodynamic performance

The Turbulence fact/Two approaches/ [email protected] 23/327

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] 24/327

Experiments : Flow over a bump

LFML-KF , Re ⇠ 2000

The Turbulence fact/Two approaches/Experiments [email protected] 25/327

Flow over a bump

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

The Turbulence fact/Two approaches/Simulations [email protected] 26/327

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

The Turbulence fact/Essential features/ [email protected] 30/327

Drag coefficient for the flow past a sphere

CD =F

12⇢U

2S

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⇢U

2S

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] 32/327

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 freedom,"scales" in 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


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, HPC).Post-process : Manipulate huge data banks in order to extractpertinent and useful informations (Data Science, IA).

The Turbulence fact/Conclusion/ [email protected] 39/327

Why turbulence is so impossibly difficult ?

Kraichnan, 1972Turbulent flow constitutes an unusual and difficult problem of statisticalmechanics, characterized by extreme statistical disequilibrium, byanomalous transport processes, by strong dynamical nonlinearity, and byperplexing interplay of chaos and order.

Lumley, 1999The experience of 100 years should suggest, if nothing else, that turbulenceis a difficult problem, that is unlikely to suddenly succumb to our efforts.We should not await sudden breakthroughs and miraculous solutions.


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.


PART II

The governing equations

Equations// [email protected] 42/327









2 The governing equationsNavier-Stokes EquationsVorticityPressure in incompressible flowsNS equations and SymmetriesDimensionless numbersNS non viscous invariantsValidityCharacteristics of turbulent flowsHomogeneity and isotropyCanonical turbulent flowsExercises


The Navier Stokes equations

Leonardo da Vinci, Sul Volo degli Ucceli, 1505A bird flies according to mathematical principles.

Equations/NS equations/ [email protected] 45/327


Mass conservation : local and conservative form

@⇢

@t+@⇢uj

@xj

= 0 , 8 (x, t) (1)

x space variablet time⇢(x, t) densityu(x, t) velocity field



Momentum equation : HistoryLeonardo da Vinci (1452� 1519)Isaac Newton (1642� 1726) :

m⇥ � =X

F

Leonhard Euler (1707� 1783)

Equation for a fluid particle

Claude Navier in 1823

Viscous term


The Navier Stokes equations (Newtonian fluids)Momentum equation : component form (non conservative)

⇢@ui

@t+ ⇢uj

@ui

@xj

= ⇢fi +@�ij

@xj

(2)

f : mass forces density (e.g. gravity g)� : stress tensor, second-order tensor

�ij ⌘ �p�ij + 2µSij + �@u`

@x`

�ij ,

where � and µ are the Lamé coefficients.

S : deformation rate tensor, second order. Sij ⌘ 12

⇣@ui@xj

+ @uj

@xi

⌘

�ij : Kronecker symbolVolume viscosity : K ⌘ �+ 2

3µ, K = 0 for monoatomic gas.Usual approximation : � = � 2

3µ for air.Einstein summation convention



Momentum equation : component form

⇢Dui

Dt= � @p

@xi

+ µ

@2ui

@x2k

+1

3

@

@xi

@uk

@xk

�+ Fi (i = 1, 3) (3)

with the material derivative defined as

Dui

Dt⌘ @ui

@t+ uj

@ui

@xj

⇢ densityµ dynamic viscosityp pressure fieldu velocity field


The Navier Stokes equationsMomentum equation : with vectorial operators

⇢Du

Dt= ⇢f �rp +r · (2µS) +r(�r · u) (4)

with

Du

Dt=@u

@t+ u ·ru =

@u

@t+ (r⇥ u)⇥ u +ru2

2

f : mass forces density (e.g. gravity g)S : second order deformation rate tensor

Sij ⌘1

2

✓@ui

@xj

+@uj

@xi

◆

Exercice :

Show that u ·ru = (r⇥ u)⇥ u +ru2

2Equations/NS equations/ [email protected] 50/327


Thermodynamicsp = R⇢T : Ideal gas lawe = CvT : Internal energy ; Cv heat capacity at constant volumeh = CpT : Enthalpy ; Cp heat capacity at constant pressureR = Cp � Cv : Ideal gas constant� = Cp/Cv : Specific heat ratioAir : Diatomic gases around 78% nitrogen (N2) and 21% oxygen (O2).At standard conditions ⇠ ideal gas =) � ⇠ 1.4

First law for energy : E = ⇢V (e + u2)

�E = �Q� �W i.e. Heat� Work

Gibbs relation for entropy

Tds

dt=

de

dt+ P

d

dt

✓1

⇢

◆



Internal energy equation : Non-conservative form

@e

@t+ uj

@e

@xj

= �(� � 1)e@uk

@xk

+1

⇢�ijSij +

k

⇢

@2T

@x2k

(5)

e(x, t) internal energy

S : deformation rate tensor, 2nd order tensor. Sij ⌘ 12

⇣@ui@xj

+ @uj

@xi

⌘

� : viscous strain tensor, second-order

�ij = 2µSij + �@u`

@x`

�ij

where � is the second coefficient of viscosity.Stokes’ assumption for convenience : � ⌘ � 2

3µ.k thermal conductivity. Note that ⌘ k/(⇢Cp) is the thermal diffusivity.



Internal energy equation : Conservative form

@⇢e

@t+@⇢uje

@xj

= �p@uk

@xk

+ �ijSij +@

@xk

✓k@T

@xk

◆(6)

⇢uje : Internal energy flux due to advection

�p@uk

@xk

: Net work of pressure force

�ijSij : Net work of viscous force@

@xk

✓k@T

@xk

◆: Net heat transfer



3D Incompressible flowsHyp : ⇢ constant

Pb : 4 equations, 4 unknowns (ui, p), 4 independent variables (xi, t)8>><

>>:

@ui

@xi

= 0

⇢@ui

@t+ ⇢uj

@ui

@xj

= � @p

@xi

+ µ@2ui

@x2j

+ Fi (i = 1, 3)(7)

Non linear, coupled, partial differential equations !



Viscosity

µ : dynamic viscosity [M ][L]�1[T ]�1

⌫ = µ/⇢ : kinematic viscosity [L]2[T ]�1

Typical values of µ for air and water

Air at 300 K, 1 atm : µ ⇠ 18.46⇥ 10�6(Ns)m�2

Air at 400 K, 1 atm : µ ⇠ 23.01⇥ 10�6(Ns)m�2

Liquid water at 300 K, 1 atm, µ ⇠ 855⇥ 10�6(Ns)m�2

Liquid water at 400 K, 1 atm, µ ⇠ 217⇥ 10�6(Ns)m�2

Sutherland’s Formula (1893)Dynamic viscosity (Pa.s) of an ideal gas as a function of thetemperature

µ = µ0T0 + C

T + C

✓T

T0

◆3/2

C : Sutherland’s temperature (constant) for the gaseous material



Passive scalar field : Advection diffusion equation

@✓

@t+r · (✓u) = r · (r✓) + S (8)

✓(x, t) : scalar field : molecular diffusivity coefficientr✓ : molecular diffusive fluxS : source or sink

Passive=) No influence of the scalar dynamics ✓(x, t) on the velocity fielddynamics u(x, t).


Passive scalar dynamics

Advection diffusion equation

@✓

@t+@✓uj

@xj

=@

@xj

✓@✓

@xj

◆

| {z }Diffusiveflux

+S (9)

Passive scalar field : ✓(x, t)

Temperature (small fluctuations)Polluant concentration


The Navier Stokes equations + Passive Scalar

3D Incompressible flowsHyp : ⇢ constant

Pb : 5 equations, 5 unknowns (ui, p, ✓), 4 independent variables (xi, t)8>>>>>><

>>>>>>:

@ui

@xi

= 0

⇢@ui

@t+ ⇢uj

@ui

@xj

= � @p

@xi

+ µ@2ui

@x2j

+ Fi (i = 1, 3)

@✓

@t+ uj

@✓

@xj

=@

@xj

✓@✓

@xj

◆+ S

(10)


Exercice :Write the momentum equation under a conservative form


The Navier Stokes equations : Initial and Boudaryconditions

Geometry and physics of the flowPeriodic boxWall (slip / no slip)Pressure outletFlux (mass, momentum or energy)Dirichlet/Neumann. . .

=) Boundary conditions

Physics of the flow : Initial statePressure, Internal energy,Temperature, DensityVelocityPolluant concentration

=) Initial conditions


Scalar dynamics examples

Advection diffusion equation : case 1

1D flowUniform and constant velocity u1

Initial condition ✓0 at (x0, t0)

S = 0, No source ⇠ 0, No diffusivity

Advection diffusion equation : case 2

Same initial conditionNo convectionS = 0, No source 6= 0 Diffusivity


Vorticity dynamics

DefinitionVorticity :

!!! = r⇥ u

Einstein notation :

!i = "ijk

@uk

@xj

= "ijk@juk

"ijk : Levi-Civita symbol, permutation symbol, antisymmetric symbol,or alternating symbol

PropertiesDivergenceless tensor : @j!j = 0

Pseudo-tensor : don’t satisfy the parity (or mirror) symmetry

Equations/Vorticity/ [email protected] 61/327

Vorticity dynamics in turbulents flows

Boundary layer


Vorticity dynamics in turbulent flows

Compressible turbulent flow in 3D periodic cubic box

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


Vorticity dynamics

Incompressible turbulent flow in 3D cubic box


Vorticity dynamics in turbulent flows

EquationVorticity :

!!! = r⇥ u =) r⇥ (NS)

Vorticity equation :

@!i

@t+ uj

@!i

@xj

= !j

@ui

@xj

+ ⌫@2!i

@xk@xk

+r⇥ fi (i = 1, 3)

RemarksNo pressure termStretching increases vorticy intensity

ExerciseDerive the governing equation for the vorticity dynamics.Show that !j@jui = !jSij .


Pressure in incompressible flows

Non local quantity ! and non linear...

Equations/Pressure in incompressible flows/ [email protected] 66/327


Poisson problem



Filament of vorticity tracerAir bubbles =) minimum of pressure () �p� 1


NS equations and Symmetries

HypothesisUnboundedness of the space, or more strictly periodic boundary

conditions.Incompressible flow : @iui = 0.

Transformations acting on space-time functions

x �! x0 , t �! t0 , u(x, t) �! v(x0, t0)

Equations/Symmetries/ [email protected] 69/327


Symmetry group of the NS equationsSpace translation

(x, t) �! (x0 = x + r, t0 = t) , v(x0, t0) = u(x, t)

Time translation

(x, t) �! (x0 = x, t0 = t + ⌧) , v(x0, t0) = u(x, t)

Galilean transformation

(x, t) �! (x0 = x + u0t, t0 = t) , v(x0, t0) = u(x, t) + u0

ExerciseShow that NS equations are invariant under space and timetranslation.Show that NS equations are invariant under Galilean transformation.



Symmetry group of the NS equationsParity

(x, t) �! (x0 = �x, t0 = t) , v(x0, t0) = �u(x, t)

Rotation

A 2 SO(R3)(x, t) �! (x0 = Ax, t0 = �1�ht) , v(x0, t0) = Au(x, t)

ScalingCondition on h 2 R, ⌫ = 0 and h = �1, ⌫ 6= 0

(x, t) �! (x0 = �x, t0 = �1�ht) , v(x0, t0) = �hu(x, t)


Dimensionless numbers

Reynolds experiment 1883 : Laminar / Turbulent Regime

Equations/Dimensionless numbers/Reynolds number [email protected] 72/327

Dimensionless numbers : Reynolds number Re

From dimensional analysis and similarity theory

Characteristic quantitiesU : Velocity scaleL : Length scaleµ : viscosity⇢ : density

⇢u ·ru| {z }NL term

⇠ ⇢U2

Land µr2u| {z }

Viscous term

⇠ µU

L2

Reynolds number Re =⇢UL

µ⇠ Inertial effect

Viscous effect(11)


Dimensionless numbers

Reynolds number Re

Re⌧ 1 Laminar flowsRe� 1 Turbulent flowsControl parameterRec : Critical Reynolds

Re =⇢UL

µ⇠ Inertial effect

Viscous effect

(12)


The Navier Stokes equations : Non-dimensionalized form

Incompressible flowsHyp : ⇢ constant, 4 equations, 4 unknowns (ui, p), 4 variables (xi, t)

8>><

>>:

@ui

@xi

= 0

@ui

@t+ uj

@ui

@xj

= � @p

@xi

+1

Re

@2ui

@x2j

+ Fi (i = 1, 3)(13)


Re �! +1 =) NS equations �! Euler equations.Re �! +1 =) Turbulent fields !


Jets : Laminar �! Turbulent

�! �! �! �! �!

Reynolds number %


Dimensionless numbers : Strouhal number S

Characteristic quantitiesU : Velocity scaleL : Length scaleT : Time scale

@ui

@t|{z}Unsteady

⇠ U

Tand uj

@ui

@xj| {z }Advection

⇠ U2

L

Strouhal number S =L

UT=

Lf

U⇠ Unsteady

Advection(14)

Equations/Dimensionless numbers/Strouhal number [email protected] 77/327

Wakes : flows around a cylinder


Cont’d – Wakes : flow around a cylinder


Cont’d – Wakes : flow around a cylinder

Fixed cylinder


Dimensionless numbers : Prandtl number Pr

Characteristic quantities⌫ : Viscosity : Thermal diffusivityD : Mass diffusivity

Prandtl number Pr =⌫

⇠ Momentum diffusion

Heat diffusion(15)

Schmidt number Sc =⌫

D⇠ Momentum diffusion

Mass diffusion(16)

Equations/Dimensionless numbers/Prandtl number [email protected] 81/327

Dimensionless numbers : Prandtl number Pr

Prandtl number Pr =⌫

⇠ Momentum diffusion

Heat diffusion

Typical values

PrHelium, Hydrogene, Azote 0.7

carbon dioxide 0.75Air 0.7

Water vapor 1.06Liquid water 7Engine oil 10400Mercury 0.025

Liquid sodium very smallSun 10�9

Equations/Dimensionless numbers/Prandtl number [email protected] 82/327

Non viscous invariants : ⌫ = 0

Physical integrated quantitiesKinetic energy in 3D

E ⌘ 1

2

Z|u|2 dv

Helicity in 3D

H ⌘ 1

2

Z

V

u ·!!! dv

Enstrophy in 2D

⌦ ⌘ 1

2

Z

V

|!!!|2 dv

Equations/NS non viscous invariants/ [email protected] 83/327

Non viscous invariants

Conservation lawsConsidering any quantity h satisfying

@h

@t+ uj

@h

@xj

= g

The advection velocity is divergenceless

@uj

@xj

= 0

Then

8 V ,

Z

V

uj

@h

@xj

dv =

Z

S

(ujh)nj ds

⌘ 0 if u = 0 on S = @V .



Kinetic energyDefinition

E ⌘ 1

2

Z|u|2 dv

Governing equation

dE

dt=

Z

V

�r ·u

✓u2

2+

p

⇢

◆+ ⌫!!! ⇥ u

�

| {z }Flux term

dv � ⌫

Z

V

!!!2 dv

| {z }Dissipation

Then E is an invariant of the dynamics if ⌫ = 0 and u = 0 on S = @V .



Kinetic HelicityDefinition

H ⌘ 1

2

Z

V

u ·!!! dv

Governing equation when u = 0 on S = @V .

dH

dt= �2⌫

Z

V

!!! ·r⇥!!! dv

Then H is an invariant of the dynamics if ⌫ = 0.

Exercise

1 Write the flux term for the kinetic helicity equation.2 Show that the enstrophy ⌦ is a non viscous invariant of the 2D dynamics.


Non viscous 2D/3D invariants

Invariants

Cascade direction 2D 3D

NS hv2i inverse hv2i direct

h!!!2i direct hv ·!!!i direct

MHD hv2 + b2i direct hv2 + b2i direct

hv · bi direct hv · bi direct

ha2i inverse ha · bi inverse


Validity of NS equations

In the wide senseMass conservationMomentumEnergy (1st Principle)2nd Principle

Validity

No mathematical proof of existence and unicity of solutions in 3D

Validity guaranteed today if flow properties do not vary at the scale of them.f.p, i.e. molecular scale.

Knudsen number :

Kn =m.f.p.

L,

L : characteristic length scale of the flow.

Equations/Validity/ [email protected] 88/327

Validity of NS equations (cont’d)

Knudsen numberKn < 10�2 : Continuous regime.10�2 < Kn < 10�1 : Sliding regime =) local anomalies near the walls.10�1 < Kn < 10 : Transitional regime =) Boltzmann equationsneeded.10 < Kn : Free molecular regime =) Molecular collisions negligable.

On earth

m.f.p.⇠ 10�7m and L ⇠ 10�3m =) Kn ⇠ 10�4.

At 11km, m.f.p.⇠ 3.10�7m

At 20km, m.f.p.⇠ 1.5 10�6m=) NS equations valid in 99% of case.

Equations/Validity/ [email protected] 89/327

Characteristics of turbulent flows

Irregularity in space and timecf. examples in section 1


Equations/Characteristics of turbulent flows/ [email protected] 90/327

Characteristics of turbulent flowsDiffusive characterExample : Heating of a room

A room of size L.Heater in the room.

No motion in the roomHeat equation =) molecular diffusion

@✓

@t+ uj

@✓

@xj

=@

@xj

✓@✓

@xj

◆

| {z }Diffusiveflux

+ S (17)

: thermal diffusivity.Conducting characteristic time :

Tc ⇠ L2/ . (18)



Diffusive characterExample : Heating of a room

A room of size L.Heater in the room.

Turbulent motion in the roomHypothesis :

L : size of the largest turbulent structures L ⇠ 10 m.u : turbulent velocity scale u ⇠ 1 cm.s�1.Characteristic time :

Tt ⇠ L/u ⇠ 103s .

Exercise

1 Give an estimation of Tc and compare the both characteristic time Tt andTc. Conclusion ?



Turbulence develops above a threshold of Reynolds numberExamples :

Boundary layer transition.Cylinder wakes.

Boundary layer transition (NASA)



Golf ball dimples & drag (http ://www.aerospaceweb.org/)



Turbulence is 3D and rotationalExamples :

Vorticity equationAmplification of the vorticity field by strain : !jsij

DNS of Compressible flowD. H. Porter et al.



Turbulence is structured by a superposition of many ’vortices’ of differentcharacteristic scales interacting

Turbulence has a continuous spectrum of kinetic energy.Large range of active scales.



Kinetic energy spectrum



Turbulence is dissipativeExample

Friction in a boundary layer.Energy feeding of turbulence : c.f. Section on boundary layer

dynamics.



The largest scales are fixed by the characteristic size of the flowExamples

Boundary layer.Wake

Largest scale



The turbulence is non linearExample : 1D ideal flows (Burgers equation) or equivalently at largeReynolds number

@u

@t+ u

@u

@x= 0

=) Energy transfer from low to high frequencies

ExerciceConsidering the 1D ideal Burgers equation associated to an initialcondition under the form

u = u0 cos(kx) , for t = t0 ,

show that at small time an energy transfer from low to high frequenciesoccurs, i.e. from large to small scales.



The size of the smallest structures is fixed by the viscosity` : length scale of a structureu` : velocity scale of a structure

u@xu| {z }NL convective term

⇠ u2`

ànd ⌫@2

xxu| {z }

diffusion

⇠ ⌫ u`

`2

Re�1`

=⌫

u``⇠ diffusion

convection(19)

=) when ` & viscosity dominates and damps the turbulentstructures.=) small length scales () small time scales.=) statistical independance of small scales.



The size of the smallest structures is fixed by the viscosity (Kolmogorov41)

Hyp 1 : Depend only on viscosity and rate of energy transfer fromthe large scales.

Hyp 2 : Equilibrium turbulence : Transfer rate = dissipation rate (noaccumulation)Dimensional analysis : Energy 1

2u2i

and dissipation ⌫( @ui@xj

)2

=)(" (m2 · s�3) dissipation rate,⌫ (m2 · s�1) kinematic viscosity.

Kolmogorov micro-scales :

⌘ =

✓⌫3

"

◆1/4

, u⌘ = (⌫")1/4 , ⌧⌘ =⇣⌫"

⌘1/2.



The size of the smallest structures (cont’d)Non viscous estimation of "`

"` ⇠u3

`

`

Characteristic ratio :

⌘

`⇠✓

u``

⌫

◆�3/4

= R�3/4`

uL

u⌘

⇠ R1/4L

and⌧L⌧⌘⇠ R1/2

L



Turbulence is continuousSmall scales (⌘) >> mean free path (m.f.p.)Kinetic theory of gases : ⌫ ⇠ cs · m.f.p.

m.f.p.

⌘⇠ ⌫

cs

1

`

✓u`

⌫

◆3/4

=u

cs

✓u`

⌫

◆�1/4

= Mt.R�1/4t

=)m.f.p.

⌘� 1 if Mt large and Rt small ! ! !


Homogeneity and isotropy

DefinitionsSpatially homogeneous turbulence :All averaged quantities are independant of space variables.Stationnary turbulence :All averaged quantities at one point in space are independant of time.Isotropic turbulence :Averaged quantities at one point in space are independant oforientation.

ExampleGrid turbulence is a stationnary turbulence which is good approximationof homogeneous isotropic turbulence.

Equations/Homogeneity and isotropy/ [email protected] 105/327

Homogeneity and isotropy (cont’d)

Remarks"real" turbulence is much more complex than grid turbulence.

Interaction between mean and fluctuating component of motion.Steady boundary conditions for fluctuating "random" variables.Transport =) Spatial variations of fluctuating intensities.

Grid turbulence


Equations/Homogeneity and isotropy/ [email protected] 106/327

Canonical turbulent flows

ClassificationsHIT : Homogeneous Isotropic TurbulenceFree turbulence

JetsWakesMixing layer

Wall turbulenceChannel flowPipe flow

Boundary layer

Equations/Canonical turbulent flows/ [email protected] 107/327

Canonical turbulent flows (cont’d)

Similarity

Power of x for uc y1/2

Plane wake �1/2 1/2Axisym. wake �3/4 1/4Mixing layer � 1

Plane jet �1/2 1Axisym. jet �1 1Radial jet �1 1

Jet

Equations/Canonical turbulent flows/ [email protected] 108/327

Exercises

The characteristic scales of the turbulent dynamics

1. The small scalesWe assume that the small scale quantities of the turbulent flow can beexpressed only in terms of the dissipation rate by mass unit " (m2.s�3) andthe kinematic viscosity ⌫ (m2.s�1).

1 Using dimensional analysis, determine the characteristic velocity andtime scales, respectively denoted u⌘ and ⌧⌘ at the small length scale ⌘.These scales are called the micro-scales of the turbulent flows.

2 Write the Reynolds number based on these micro-scales and concludeabout the physical meaning of these scales.

Equations/Exercises/ [email protected] 109/327

Exercises


2. The large scales1 Let’s assume that the dynamics of the large scales can be characterized

by a velocity scale U and a length scale L. Using these bothcharacteristic scales, express the corresponding time ⌧L, so called theturnover time, characteristic of the kinetic energy transfer time. Thelength L is characteristic of the biggest eddies size of the flow.

2 Determine an expression for the kinetic energy transfer T through thescales from large to small scales.

3 Write the viscous characteristic time for the large scale and determinethe transfer rate T⌫ from large scales due to the viscous effects.

4 Compare the transfer terms T and T⌫ . What can we conclude whenRe � 1.


Exercises


3. Interscales relations1 Assuming the turbulence is steady, determine the transfer rate T and

T⌘ as function of ".2 Express the ratios L/⌘, ⌧L/⌧⌘ and U/u⌘ in term of the Reynolds

number Re = UL/⌫. Conclusions ?3 In order to describe the spatial scales of a turbulent 3D flow using a

numerical simulation, each space direction is discretized with a givennumber of points n. Determine the total number of points needed todescribe the 3D Flow in term of the Reynolds number.


Exercises

Dispersion law of a polluantRidchardson, Proc. Roy. Soc. London, 110, (1926).

Let’s consider two particles injected in a turbulent flow initially located ata distance d0 from each other. We assume that it exists a scale rangeL� `� ⌘, so called the inertial range, in which the mean rate of kineticenergy dissipation " is assumed to be constant.

1 Express the kinetic energy dissipation rate "` at the scale ` in terms ofthe velocity and time characteristic scales, respectively denoted u`

and t`, then in terms of u` and ` only.2 Give an expression for u` in terms of "` and `. Assuming that "` is

constant through the scales, write a differential equation satisfied by `.3 Solve this equation and give a law for the time evolution of the

distance between two particles initially at the distance d0.


Exercises

Beltrami FlowsLet’s consider a flow defined by the following velocity field :

u =

8<

:

u1 = C sin↵z + B cos↵yu2 = A sin↵x + C cos↵zu3 = B sin↵y + A cos↵x

(20)

1 Show that this flow is incompressible.2 Show that the velocity field satisfies the Beltrami property, i.e.

!!! = �u

where � 2 IR. Give � as function of ↵.3 Write the non-linear terms of the Navier-Stokes equations written for

the vorticity. What can we conclude ?


Exercises

Atomic blastLet’s assume that, in an atomic explosion, a release of a significant amount ofenergy E occurs instantaneously within a small region (one may say, at a point).In the early stage, a strong spherical shock wave develops at the point ofdetonation.

1 Let’s assume that the radius R of the shock wave front, at an interval oftime t after the explosion only depends on the quantities E, t and on theinitial air density ⇢0. Using dimensional analysis, give an expression for R interms of E, t and ⇢0.

2 Then give an expression of E in terms of R, t and ⇢0.3 From a series of high speed photograph, Taylor has measured that the front

was at R = 125m at time t = 0.025s. Give the Taylor’s estimation of theenergy of the explosion. This was considered as top secret and caused ’muchembarrassment’ in American government circles. NA : ⇢0 = 1.3kg/m3.


Exercises

Atomic blast (Cont’d)

+ =)

Figure: First atomic bomb called Gadget at the top of a tower, July 15th 1945and its Fireball 25ms after an atomic explosion on the ground.


PART III

Statistical description of turbulence

Statistical description// [email protected] 116/327









3 Statistical description of turbulenceRealization of a turbulent flowProbability density functionJoint probability density functionThe correlation functionErgodicity and statistical symmetriesStatistical averageReynolds decompositionMean NS equationsReynolds stress tensorKinetic energyFluctuating NS equationsReynolds stress tensor equationKinetic energy of the fluctuationsExercises


Statistical description of turbulenceWhy probabilistic description ?

Signal highly disorganized.Appears unpredictable in its detailed

behavior.Some statistical properties of thesignal are quite reproductible.=) probabilistic description



Realization of a turbulent flow

Set of realizationsMacroscopic conditions identical for each test.n realizations.Velocity at the position A : v1

A, v2

A, . . ., vn

A.

Statistical description/Realization of a turbulent flow/ [email protected] 120/327

Probability density function (PDF)

Random variable : v(t)

Deterministic theory ! probabilistic theory.

HistogramFrequency distributionbin i : [i�v, (i + 1)�v[ where i 2 [�1, +1].Histogram

H(i) =NX

1

n with

(n = 1 if v 2 [i�v, (i + 1)�v[

n = 0 else.

H(i) = H(i�v, (i + 1)�v, N)

= H(v, v + �v, N)

Statistical description/PDF/ [email protected] 121/327

Histogram

Construction of the histogram by binning


Histogram

Same signal : hot wire

Hot wiresampled 5000 over a time-spandof 150s.Same signal, few minutes later.S1 wind tunnel of ONERA.Gagne & Hopfinger


Probability density function (cont’d)

Normalized histogram :1

NH(i)

Probability density function of a continuous variable

P(v) = limN!+1�v!0

1

�v· 1

NH(v, v + �v, N)

Complete description of a turbulent variableAt a given location and time, given by the PDF P(v)

P(v)dv is the probability of the variable v taking a value between vand v + dv.



Some properties

P(v) � 0 , 8v 2 IR .

Z +1

�1P(v) dv = 1 .

P(v0 < v < v0 + �v) = P(v0) · �v .


PDF and moments

Moment of order n

The n-th moment vn

vn =

Z +1

�1vnP(v)dv

The mean : first moment

v =

Z +1

�1vP(v)dv

The varianceThe second moment of the perturbation quantity v0 = v � v

v02 =

Z +1

�1(v � v)2P(v)dv


PDF and moments (cont’d)

SkewnessThe third moment of v0 normalized by the variance

skewness =v03

v023/2

Kurtosis (or flatness)

The fourth moment of v0 normalized by the variance

kurtosis =v04

v022

Normal distributionskewness = 0kurtosis = 3



Classical resultCentral Limit Theorem (CLT) :

The arithmetic mean of a sufficiently large number of iterates ofindependent random variables, each with a well-defined expectedvalue and well-defined variance, will be approximately normallydistributed.Gaussian PDF (Central Limit Theorem)

P(v) =1p2⇡�

exp

✓� (v � v)2

2�

◆.

But...Turbulence is NOT Gaussian !


Probability density function (cont’d)Example of PDF for the scalar : Lee et al. 2012


Cumulative probability function F (v)

Definition and properties

P(v) =dF (v)

dv.

P(v0 < v < v0 + �v) = F (v0 + �v)� F (v0) .

P(v) = lim�v!0

F (v + �v)� F (v)

�v.

F (v) =

Zv

�1P(v0) dv0 .


Joint probability density function (JPDF)

PJ(u, v)

Turbulence involves random variables dependent on each other.Probability of finding the first random variable between u and u + du,and the second one between v and v + dv

Properties Z +1

�1PJ(u, v)dudv = 1

P (u) =

Z +1

�1PJ(u, v)dv and P (v) =

Z +1

�1PJ(u, v)du

The covarianceDefinition

C(u, v) = uv � u · v = u0v0

withuv =

Z +1

�1uv PJ(u, v)dudv

Statistical description/JPDF/ [email protected] 131/327

The correlation function

r(u, v)

Definition : The covariance normalized by the rms values

r(u, v) =u0v0pu02 · v02

Quantifies the degree of correlation between u and v.r(u, v) = 0 () uncorrelated variables.r(u, v) = ±1 () perfectly correlated functions.Statistical independance

PJ(u, v) = P (u) P (v)

P and PJ fundamental to theories of turbulence but seldom measuredor used.

Statistical description/The correlation function/ [email protected] 132/327

Ergodicity and statistical symmetries

StationarityIf all mean quantities are invariant under a translation in timeA stationary variable v is ergodic if the time average of v converges tothe mean v as the time interval extends to infinity.

1

T

Z 1

0v(t)dt = v as T �!1 .

Ergodicity : ensemble average () time average.

HomogeneityIf all mean quantities are invariant under any spatial translationSpatial average

1

L

ZL

0v(x)dx = v as L �!1 .

Ergodicity :ensemble average () spatial average.

Statistical description/Ergodicity and statistical symmetries/ [email protected] 133/327

Ergodicity and statistical symmetries (cont’d)

IsotropyIf all mean quantities are invariant under any arbitrary rotation ofcoordinates

AxisymmetryIf all mean quantities are invariant under a rotation about oneparticular axis only, e.g. stratified turbulence.

Statistical description/Ergodicity and statistical symmetries/ [email protected] 134/327

Statistical averageSampling :

N quantities obtained during p independent realizations

�i(x, t) , i = 1, N

density, velocity component, pressure, ...

Definition :Mean value

�i(x, t) ⌘ limp�!+1

1

p

0

@X

k=1,p

�(k)i

(x, t)

1

A

Variance

�0i�0

i(x, t) ⌘ lim

p�!+1

1

p

0

@X

k=1,p

(�(k)i

(x, t)� �i(x, t))(�(k)i

(x, t)� �i(x, t))

1

A

Statistical description/Statistical average/ [email protected] 135/327

Statistical average (cont’d)

DefinitionCentered value

�0i(x, t) ⌘ (�i(x, t)� �i(x, t))

2 times, 2 points correlation

�0i�0

i(x,y, t, t0) ⌘ �0

i(x, t)�0

i(y, t0) = �0

i(x, t)�0

i(x + r, t + ⌧)

= �0i�0

i(x, r, t, ⌧)

= limp�!+1

1

p

0

@X

k=1,p

(�(k)i

(x, t)� �i(x, t))(�(k)i

(y, t0)� �i(y, t0))

1

A

Statistical description/Statistical average/ [email protected] 136/327

Reynolds decomposition

Definition

�(x, t) = �(x, t) + �0(x, t)

Statistical description/Reynolds decomposition/ [email protected] 137/327

Reynolds decomposition (cont’d)

PropertiesPreservation of uniform fields : 1 = 1

Linearity : �1 + �2 = �1 + �2

Commutativity of mean/derivative operators :

@�

@xk

=@�

@xk

(k = 1, 3),@�

@t=@�

@t

Reynolds’ axioms :

�0 = 0() � = �

� = �

Statistical description/Reynolds decomposition/ [email protected] 138/327

Mean NS equations

NS equationsContinuity :

@ui

@xi

= 0

NS :

@ui

@t+@uiuj

@xj

= � @p

@xi

+ ⌫@2ui

@xk@xk

+ fi (i = 1, 3)

Statistical description/Mean NS equations/ [email protected] 139/327

Reynolds stress tensor

NS equationsSecond order moment for the fluctuating part of the velocity

uiuj(x, t) = uiuj(x, t) + u0iu0

j(x, t)

| {z }Rij(x,t)

= uiuj(x, t) + Rij(x, t)

Mean momentum equation :

@

@tui +

@

@xj

(uiuj) = � @p

@xi

+ ⌫@2ui

@xk@xk

+ fi �@

@xj

Rij (i = 1, 3)

Statistical description/Reynolds stress tensor/ [email protected] 140/327

Kinetic energy

Governing equations for the kinetic energy of the mean velocity fieldKinetic energy for the mean velocity field

K =1

2uiui

Equation for K :

@

@tK +

@

@xj

(Kuj)

| {z }I

= � @

@xi

(pui)| {z }

II

+ ⌫@2K

@xk@xk| {z }III

� ⌫ @ui

@xk

@ui

@xk| {z }IV

+ uifi|{z}V

� @

@xj

(uiRij)

| {z }V I

+ Rij

@ui

@xj| {z }V II

.

Statistical description/Kinetic energy/ [email protected] 141/327

Kinetic energy (cont’d)

Physical interpretation@

@tK +

@

@xj

(Kuj)

| {z }I

= � @

@xi

(pui)| {z }

II

+ ⌫@2K

@xk@xk| {z }III

� ⌫ @ui

@xk

@ui

@xk| {z }IV

+ uifi|{z}V

� @

@xj

(uiRij)

| {z }V I

+ Rij

@ui

@xj| {z }V II

.

I : Transport by the mean velocity field.II : Spatial diffusion due to the pressure.III : Spatial diffusion due to viscosity.IV : Dissipation by Joule effect.V : Net work of external forcing.V I : Spatial diffusion of K due to turbulence.V II : Energy transfer between mean and fluctuating part of thevelocity field.

Statistical description/Kinetic energy/ [email protected] 142/327

Fluctuating NS equations

NS equationsContinuity :

@u0i

@xi

= 0

NS :

@u0i

@t+

@

@xj

(u0iuj + uiu

0j+ u0

iu0

j�Rij) = � @p0

@xi

+ ⌫@2u0

i

@xk@xk

+ f 0i

Statistical description/Fluctuating NS equations/ [email protected] 143/327

Reynolds stress tensor equation

Reynolds stress tensor equationContinuity :

@u0i

@xi

= 0

Statistical description/Fluctuating NS equations/ [email protected] 144/327


Reynolds stress tensor equationNS with S0

ij= (@ju0

i+ @iu0

j)/2 :

@

@tRij +

@

@xk

(ukRij)| {z }

I

= �✓

Rjk

@ui

@xk

+ Rik

@uj

@xk

◆

| {z }II

� @

@xk

u0iu0

ju0

k

| {z }III

�✓

@

@xi

p0u0j+

@

@xj

p0u0i

◆

| {z }IV

+ 2p0S0ij| {z }

V

+ f 0iu0

j+ f 0

ju0

i| {z }V I

+ 2⌫

✓u0

j

@

@xk

S0ik

+ u0i

@

@xk

S0jk

◆

| {z }V II

Statistical description/Reynolds stress tensor equation/ [email protected] 145/327


Reynolds stress tensor equationNS :

@

@tRij +

@

@xk

(ukRij)

| {z }I

= � Rjk

@ui

@xk

+ Rik@uj

@xk

!

| {z }II

�@

@xk

u0iu

0ju

0k

| {z }III

�

0

@@

@xi

p0u0j +

@

@xj

p0u0i

1

A

| {z }IV

+ 2p0S0ij| {z }

V

+ f0iu

0j + f0

ju0i| {z }

V I

+ 2⌫

u0j

@

@xk

S0ik

+ u0i

@

@xk

S0jk

!

| {z }V II

I : advection by u.

II : production/destruction by interaction between u0 and u.

III : Turbulent diffusion () Closure problem ! ! !

IV : Spatial diffusion by pressure and velocity interactions.

V : Production/Destruction by p0 and S0 interactions.

VI : Net work of external forces.

VII : Dissipation by molecular viscosity.

Statistical description/Reynolds stress tensor equation/ [email protected] 146/327

Kinetic energy of the turbulent fluctuations

Governing equationsKinetic energy of the fluctuations

K =1

2u0

iu0

i=

1

2Rii

Equation for K :

@

@tK +

@

@xl

(ulK)| {z }

I

= �Ril

@ui

@xl| {z }II

� 1

2

@

@xl

u0iu0

iu0

l

| {z }III

� "|{z}IV

+ f 0iu0

i|{z}V

� @

@xl

p0u0l

| {z }V I

+ ⌫@2

@xl@xl

K| {z }

V II

Statistical description/Kinetic energy of the fluctuations/ [email protected] 147/327

Kinetic energy of the fluctuations (cont’d)

Physical interpretationEquation for K :

@@t

K +@@xl

(ulK)| {z }

I

= �Ril

@ui

@xl| {z }II

� 12

@@xl

u0iu0iu0l

| {z }III

� "|{z}IV

+ f 0iu0i|{z}

V

� @@xl

p0u0l

| {z }V I

+ ⌫@2

@xl@xl

K| {z }

V II

I : advection by u.

II : production/destruction by interaction between u0 and u.

III : Turbulent diffusion.

IV : Dissipation by molecular viscosity with " ⌘ ⌫@u0

i@xl

@u0i

@xl.

V : Net work of external forces.

VI : Spatial diffusion by pressure and velocity fluctuations interactions.

VII : Viscous diffusion.

Statistical description/Kinetic energy of the fluctuations/ [email protected] 148/327

Exercises

Scalar dynamicsLet’s consider the dynamics of a passive scalar ✓ governed by anadvection/diffusion equation :

@✓@t

+ uj

@✓@xj

= @2✓@x2

j

,

where is the scalar diffusivity and @juj = 0.1 Write the equation of the mean scalar field ✓.2 Write the equation of the scalar fluctuations ✓0.3 Write the equation for the scalar variance K✓ ⌘ ✓0✓0.4 Give a physical interpretation for each term.5 Write the equation for the scalar flux u0

i✓0.

6 Give a physical interpretation for each term.

Statistical description/Exercises/ [email protected] 149/327

Exercises

Scalar dynamics : Solution1 The scalar variance equation reads

@@t

K✓ +@

@xk

(ukK✓)| {z }

I

= � 2u0k✓0

@✓@xk| {z }

II

� @@xk

u0k✓0✓0

| {z }III

+@2

@xk@xk

K✓

| {z }IV

� "✓|{z}V

(21)


Exercises

Scalar dynamics : Solution1 The equation for the scalar flux reads

@@t

u0i✓0 +

@@xk

(uku0i✓0)

| {z }I

=

✓u0k✓0

@ui

@xk

+Rik

@✓@xk

◆

| {z }II

� @@xk

u0i✓0u0

k

| {z }III

� @@xi

p0✓0 + p0@✓0

@xi| {z }IV

+ f 0i✓0

|{z}V

+(⌫ + )@2

@xk@xk

u0i✓0

| {z }V I

� (⌫ + )@✓0

@xk

@u0i

@xk| {z }V II

�@

@xk

✓0@u0

i

@xk

� ⌫@

@xk

u0i

@✓0

@xk| {z }V III

(22)









Turbulence modeling// [email protected] 152/327

4 Turbulence modelingClosure problemModels for the closure of the systemFirst order modelsSecond order modelsExercise

Turbulence modeling// [email protected] 153/327

Closure Problem

Momentum equation

Turbulence modeling/Closure problem/ [email protected] 154/327

Closure Problem (cont’d)

Equation for Reynolds stress tensor Rij

@

@tRij +

@

@xk

(ukRij) = �✓

Rjk

@ui

@xk

+ Rik

@uj

@xk

◆� @

@xk

u0iu0

ju0

k

| {z }unknown!!!

�✓

@

@xi

p0u0j+

@

@xj

p0u0i

◆+ 2p0S0

ij

+f 0iu0

j+ f 0

ju0

i+ 2⌫

✓u0

j

@

@xk

S0ik

+ u0i

@

@xk

S0jk

◆

Turbulence modeling/Closure problem/ [email protected] 155/327

Models for the closure

Estimate the contribution of the Reynolds stress tensor to the NS equation ?First order =) Eddy viscosity model (EVM) :Turbulent diffusion =) µt = ⇢⌫t ? ! Rij ! NS equation =) u.Classified in terms of number of transport equations solved in addition tothe RANS equations :

Zero equation/algebraic model : Mixing Length, Cebeci-Smith,Baldwin-Lomax, ...One equation : Spalart-Allmaras ⌫t, K, Wolfstein, Baldwin-Barth, ...Two equations : K� ", K� !, K� �, K� L,...Three equations : K� "�A, ...Four equations : v2� f , ...

Second order model =) Solve the Reynolds stress tensor equation : Rij ?ASM : Algebraic Stress ModelRSM : Reynolds Stress Model

Turbulence modeling/Models for the closure of the system/ [email protected] 156/327

First order models

Eddy viscosity modelsTurbulent stresses act similarly to viscousstresses.Turbulent viscosity ⇠ property of the flow.Boussinesq’s Hypothesis 1877

Laminar

⌧ij = µ

✓@ui

@xj

+@uj

@xi

◆� 2

3µ�ij

@uk

@xk

Turbulent

⌧ t

ij = �⇢u0iu0j= µt

✓@ui

@xj

+@uj

@xi

◆� 2

3�ij⇢K

Turbulence modeling/First order models/ [email protected] 157/327

First order modelsEddy viscosity models

@

@tui +

@

@xj

(uiuj) = � @

@xi

(p + ⇢K) +@

@xj

�(µt + µ)Sji

�

µt : turbulent viscosity⇢K : kinetic energy of the fluctuations ⇠ PressureRemarks :

For ⌫ : characteristic spatial scale of molecular motion ⇠ m.f.p. ofmolecules ⌧ scales of macroscopic fluid motionsThis clear-cut separation does not hold between u0

i and ui velocity fields.=) The concept of turbulent viscosity becomes more accurate with theincreasing scale separation.

Problem : How to define the turbulent viscosity µt in terms of the unknowns ofthe dynamics as ui ?

Turbulence modeling/First order models/ [email protected] 158/327

Zero equation models

Mixing length modelsNo EDP for the tranport of the turbulent stress tensor = nodynamical depedence.A simple algebraic equation is used to close the systemMixing length theory ⇠ characteristic length scale of the eddiesDimensional analysis leads to

⌫t =µt

⇢⇠ ù = `m

✓`m

��du

dy

��

◆

Turbulence modeling/First order models/zero equation [email protected] 159/327

Zero equation models

Example : a very simple model for the boundary layer caseEVM :

⌫t =µt

⇢⇠ ù = `m

✓`m

��du

dy

��

◆

� : boundary layer thickness, : von Kármán constant

(`m = y pour y < �

`m = � pour y � �

Re-injected in the turbulent viscosity expression

⌫t =µt

⇢⇠ ù = `m

✓`m

��du

dy

��

◆

Re-injected in the RANS equations


Zero equation model : The boundary layer case


�u0iu0

j= ⌫t

✓@ui

@xj

+@uj

@xi

◆� 2

3�ijK

with

⌫t = `m

✓`m

��du

dy

��

◆

then

�u0v0 = `2m

��du

dy

��2


Zero equation model (cont’d)

Advantages :Simple to implementFast computing timeQuite good predictions for simple flows where experimentalcorrelations for the mixing length exist.Used in higher level models

Drawbacks :No history effect ; purely local.Flows where the turbulent length scale varies : anything withseparation or circulation.Only give mean flow properties and turbulent shear stress.Cannot switch from one type of region to another.Only used for simple external flows.Eddy viscosity is zero if the velocity gradients are zero.Not in commercial CFD code.


One equation models

EDP for kinetic energy of the fluctuationsEDP for K

K =1

2u0

iu0

i=

1

2Rii

Turbulent viscosity using K

µt = Cµ

pK`m

where Cµ is a free parameter.

Turbulence modeling/First order models/One equation models [email protected] 163/327

One equation models (cont’d)EDP for kinetic energy of the fluctuations

EDP for K from momentum conservation

@

@tK +

@

@xl

(ulK) = �Ril

@ui

@xl

� @

@xl

u0iu0

iu0

l� "

� @

@xl

p0u0l+ ⌫

@2

@xl@xl

K

Model : Transport equation for K

@

@tK +

@

@xl

(ulK) =@

@xl

✓⌫ +

⌫t

�k

◆@K@xl

�+ Pk � "

withPk ⌘ u0

iu0

l

@ui

@xl

' ⌫t

✓@ui

@xj

+@uj

@xi

◆@ui

@xl

" ⌘ ⌫ @u0i

@xl

@u0i

@xl

' K3/2

l=) " = Cd

K3/2

`m

=) 4 free adjustable parameters...Turbulence modeling/First order models/One equation models [email protected] 164/327

One equation models (cont’d)

Spalart-Allmaras model (1994)Modern one-equation models abandoned the K-equationBased on an ad-hoc Transport equation for the eddy viscosity directly

@⌫

@t+ uj

@⌫

@xj

= P⌫ � ✏⌫ +@

@xj

1

⇢

✓µ +

⌫

�⌫

◆@⌫

@xj

�

12 adjustable constants to set ! ! !Boundary/Initial conditions :

Walls : ⌫ = 0Free stream : ideally ⌫ = 0 or ⌫ ⌫

2if problem with the solver


One equation models (cont’d)

Advantages :Inclusion of the history effects.Economical and accurate for : Attached wall-bounded flows, Flowswith mild separation and recirculationDeveloped for use in unstructured codes in the aerospace industryPopular in aeronautics for computing the flow around aero planewings, etc...

Drawbacks :Weak predictions for : Massively separated flows, Free shear flows,Decaying turbulence, Jet spreading (⇠ 40% of overprediction on therate of spreading for SP model), Complex internal flows.

Characteristic length scale empirically determinedSA model : ⌫ unaffected by irrotational mean straining


Two equations models

Two unknowns K � "EDP for K

K =1

2u0

iu0

i=

1

2Rii

EDP equation for the dissipation rate "

"

=) Turbulent viscosity : using K and "

µt ⇠ ul = K1/2

✓K3/2

"

◆=) µt = Cµ

K2

"

Turbulence modeling/First order models/Two equations models [email protected] 167/327

K � " model (cont’d)Transport PDE for the dissipation "

Model : PDE for K

@

@tK +

@

@xl

(ulK) = �Ril

@ui

@xl

� @

@xl

u0iu0

iu0

l� "

� @

@xl

p0u0l+ ⌫

@2

@xl@xl

K

Model for the production term

P = �Ril

@ui

@xl

⇡ 2⌫tSilSil = Cµ

K2

"S2

ij

Model for the diffusion terms (turbulent and pressure)

� @

@xl

⇣u0

iu0

iu0

l+ p0u0

l

⌘⇡ @

@xl

✓⌫t

�K

@K@xl

◆


K � " modelModel : Two transport PDE for K and "

@K@t

+ uj

@K@xj

= Cµ

K2

"

��S��2 � "+

@

@xj

✓✓⌫t

�K+ ⌫

◆@K@xj

◆

@"

@t+ uj

@"

@xj

="

K (C"1P � C"2") +@

@xj

✓✓⌫t

�"

+ ⌫

◆@"

@xj

◆

Two supplementary scalar PDEsTwo unknowns K and " =) Boundary conditions ? Wall functions5 free parameters Cµ, �K, C"1 , C"2 , �" =) Calibration ?Standard values : Launder and Sharma (1974) Cµ = 0.09, �K = 1.0,C"1 = 1.44, C"2 = 1.92, �" = 1.3 determinded empirically.Hyp : High Reynolds numbers, isotropyModel for low or transitional Reynolds numbers : K � !, ...


K � " model (cont’d)

Advantages :Massively used, implemented in numerous CFD codes.Spatial variation of the turbulent kinetic energy.Simple to implement.Quite good predictions of the simple sheared flows.Stable calculations

Drawbacks :Not quite efficient for complex flows : recirculations, strong anisotropy,

swirling and rotating flows, flows with strong separation, axis symmetric jets,...

Ad hoc equation for ".Valid only in the fully developed turbulence zone.Wall functions implementation needed.Over-prediction of K in the strong shear regions.Over dissipating at all scales of the flows (stabilizing effect).


K � " model (cont’d)Simulations : mean velocity field



Simulations : K and "


K � " model (cont’d)ComparisonsPredicted turbulent viscosity around a transonic airfoil=) Spalart and Chien models for shear layer...

from http ://www.innovative-cfd.com



ComparisonsPredicted surface pressure coefficient and shock location2.3 degrees angle of attack and a Mach number of 0.729

The only real differences for this case lie in the predicted shock location onthe upper surface. The more sophisticated models are not always the bestones to use.


K � ! model (Wilcox 1993, Menter 1994, ...)Model : Two transport PDE for K and !

@K@t

+ uj

@K@xj

= Cµ

K!

��S��2 � "+

@

@xj

✓✓⌫t

�K+ ⌫

◆@K@xj

◆

@!

@t+ uj

@!

@xj

= CµC!1 |S|2 � C!2!2 +

@

@xj

✓✓⌫t

�!

+ ⌫

◆@!

@xj

◆

with

! = "/K and ⌫t = Cµ

K!

.

5 free parameters Cµ, �K, C!1 , C!2 , �! =) Calibration ?Developed for Boundary layer flows.Possibly with streamwise pressure gradients.


Generic formulation for two equations models

K � � models

Model : K � � with � = Kl"m

Dimensional analysis : ⌫t = CµK2+l/m��1/m

Standard formulation for �

@�

@t+ uj

@�

@xj

=�

K (C�1P � C�2") +@

@xj

✓✓⌫t

��

+ ⌫

◆@�

@xj

◆

5 free-parameters.

Turbulence modeling/First order models/Generic form [email protected] 176/327

Generic formulation for two equations modelsK � � models

� = Kl"m

Table: Examples of two-equations turbulence models for incompressible flows.� = Kl"m

Model l mChou (1945), Launder, ... K � " 0 1

Kolmogorov (1942) , Saffman, Wilcox, Menter ... K � ! -1 1Cousteix (1997), Aupoix ... K � ' -1/2 1

Rotta (1951), Smith ... K � l 3/2 -1Speziale (1990) K � ⌧ 1 -1Zeierman (1986) K �K⌧ 2 -1

Saffman (1970), Launder, Spalding, Wilcox ... K � !2 -2 2Rotta (1968), Rodi, Spalding ... K �Kl 5/2 -1

Glushko (1971) ... K � l2 3 -2

Turbulence modeling/First order models/Generic form [email protected] 177/327

Beyond first order modelsBoussinesq approximation failure

Three dimensional flowsFlows with boundary layer separationRotating ans stratified flowsFlows with sudden change in mean strain rateFlow over curved surfacesWall bounded flows with secondary motions

PrincipleUse the governing equations of the dynamics to directly determine thecomponents of the 2nd-order Reynolds stress tensor Rij , instead ofusing the Boussinesq’s hypothesis analogy.Efficient for anisotropic flows

ExamplesASM : Algebrabic Stress ModelRSM : Reynolds Stress Model

Turbulence modeling/Second order models/Principle [email protected] 178/327


Rij models

@

@tRij +

@

@xk

(ukRij)| {z }

I

= �✓

Rjk

@ui

@xk

+ Rik

@uj

@xk

◆

| {z }II

� @

@xk

u0iu0

ju0

k

| {z }III

�✓

@

@xi

p0u0j+

@

@xj

p0u0i

◆

| {z }IV

+ 2p0S0ij| {z }

V

+ f 0iu0

j+ f 0

ju0

i| {z }V I

+ 2⌫

✓u0

j

@

@xk

S0ik

+ u0i

@

@xk

S0jk

◆

| {z }V II

I, II : exact termsIII, IV , V , V II =) Model

Turbulence modeling/Second order models/Reynolds stress model [email protected] 179/327

Reynolds stress tensor equationRij models

@

@tRij +

@

@xk

(ukRij) = Pij+⇧ij + Dij + "ij

Exact :Pij = �

✓Rjk

@ui

@xk

+ Rik

@uj

@xk

◆

Approximation needed = model :

⇧ij = 2p0S0ij

where S0ij

=1

2

✓@u0

i

@xj

+@u0

j

@xi

◆

Dij = � @

@xk

hu0

iu0

ju0

k+ p0u0

i�jk + p0u0

j�iki

"ij = 2⌫@u0

i

@xk

@u0j

@xk



Rij models : Reference Linear Model@

@tRij +

@

@xk

(ukRij) = Pij + ⇧ij + Dij + "ij

Exact expression :⇧ij = 2p0S0

ij

Model of Rotta :⇧ij = ⇧(1)

ij+ ⇧(2)

ij

with⇧(1)

ij= �C1

✓Pij �

Pkk

3�ij

◆

⇧(2)ij

= �2C2

✓Rij

Rnn

� �ij3

◆


Reynolds stress tensor equationRij models

@

@tRij +

@

@xk


Exact :

"ij = 2⌫@u0

i

@xk

@u0j

@xk

Model of Hanjalic & Launder : Local isotropy

"ij =2

3"�ij

with

@"

@t+

@

@xk

(uk") = C"

@

@xj

✓K"

Rij

@"

@xi

◆+ C"1

"

KPkk

2� "2

K



Rij models@

@tRij +

@

@xk


Exact expression : Diffusion term

Dij = � @

@xk

hu0

iu0

ju0

k+ p0u0

i�jk + p0u0

j�iki

Extended gradient model :

Dij = CD

@

@xn

✓K"

Rnm

@Rij

@xm

◆


RLM model

AdvantagesBetter efficiency compared to the K � " modelBetter approximation of the mean velocity fieldBetter tendencies for the second order quantities : K, ", ...

DrawbacksIncompressible 3D : 3 (ui) + 6 (Rij) + 1 (") = 10 unknowns=) 10 scalar equations

More free parameters to calibrate...Still far from universality...


Exercise

Mixing length model for shear layer problem

Let’s consider a 2D shear layer turbulent flow with a mean velocity field as

u = (u(y, t), 0, 0).

The boundary conditions for the velocity reads u(y = ±1) = ± 12Us.

�(t) is the mixing layer width defined such as u(y = ± �

2 ) = ± 25Us.

We use the following expression for the Reynolds stress components :

u0iu0j=

2

3k�ij � ⌫T

✓@ui

@xj

+@uj

@xi

◆, (23)

where the turbulent viscosity reads

⌫T = `2m

��@u

@y

�� (Smagorinsky 1963). (24)

Turbulence modeling/Exercise/ [email protected] 185/327

ExerciseMixing length model for shear layer problem (cont’d)

1 Write the equation for u(y, t). Is that equation closed ?

2 The mixing length hypothesis for the eddy viscosity model is used considering a uniform

mixing length across the flow and proportional to its width, i.e. `m = ↵�(t), where ↵ is a

given constant. Determine the governing equation for u(y, t).

3 Show that we can obtain a self similar solution defined as u(y, t) = Usf(⇠) where

⇠(y, t) = y/�(t) and f(⇠) satisfies

� S⇠f0 = 2↵

2f0f00

, (25)

where S is a parameter to express in terms of Us and �(t).

4 Show that the equation (25) admits two solutions, denoted f1 and f2 including three

constants.

5 Write this solution in the three parts of the flow : firstly for |⇠| > ⇠?, then for |⇠| < ⇠

?, show

that

f =3

4

⇠

⇠?�

1

4

✓⇠

⇠?

◆3

,

where ⇠?

is defined by f0(±⇠

?) = 0.Hint : Show that the increasing rate S of the mixing layer width can be expressed in terms of

the mixing layer length constant ↵ and of ⇠?.

6 Give an approximation for ⇠?

by considering that, by definition of �(t), one has f( 12 ) = 2

5 .

7 Plot ⌫T as function of y.

Turbulence modeling/Exercise/ [email protected] 186/327








Turbulent wall bounded flows// [email protected] 187/327

5 Turbulent wall bounded flowsDescriptionWall effectsSpecific physical quantitiesMean velocity profileChannel flowsBoundary layersCoherent structures and turbulent dynamicsTurbulent drag : Generation and ControlSkin friction control

Turbulent wall bounded flows// [email protected] 188/327

Turbulent boundary layers

Falco 1977, Re = 4000 (momentum thickness), fog of tiny oil droplets

Turbulent wall bounded flows/Description/ [email protected] 189/327

Turbulent boundary layers (cont’d)

Velocity field in wind channel : PIV



Vorticity structures, Re = 3000, iso Q



Vorticity structures


Turbulent boundary layers (cont’d)Vorticity structuresQ-criterion colored by streamwise vorticity


Turbulent boundary layers (cont’d)

Vorticity structures : PowerFlow simulation


Turbulent boundary layers (cont’d)Pressure gradient : Favorable / Adverse (Falco 1982)


Flow over a bumpStreaks and vortices


Flow over a bumpStreaks and vortices (lower at left, upper at right and bottom)


Flow over a bump

Adverse gradient

Channel Flow,Reh = 12600, based onthe half-width of the chan-nelDNS


Wall effects

PressureKinematic effect : Maintain the no-slip condition at the wall.Dynamical effect : Echo effect due to the non locality of the pressure.

ViscosityViscous dissipation dominant at the wallInhomogeneity normal to the wallMechanisms of redistribution of the energy varying with the walldistance.

ShearDue to the no-slip conditionMean field gradientProduction of turbulent kinetic energy + Anisotropy

Turbulent wall bounded flows/Wall effects/ [email protected] 199/327

Specific physical quantities : Deficit induced by the no-slip condition

Displacement thickness

�1 =

Z�

0

✓1� u(y)

u1

◆dy

Momentum thickness

✓ =

Z�

0

u(y)

u1

✓1� u(y)

u1

◆dy

Kinetic energy thickness

�3 =

Z�

0

u(y)

u1

"1�

✓u(y)

u1

◆2#

dy

Turbulent wall bounded flows/Specific physical quantities/ [email protected] 200/327

Specific physical quantities

Wall shear stress

⌧⇤ ⌘ µdu

dy

��wall

Wall skin friction velocity

u⇤ ⌘r⌧⇤⇢

=

s

⌫du

dy

��wall

Wall skin friction length

l⇤ ⌘⌫

u⇤=

vuut⌫

du

dy

��wall

Turbulent wall bounded flows/Specific physical quantities/ [email protected] 201/327

Turbulent boundary layers structureMean velocity profile :Inner/outer region

y+ = y/l⇤

Turbulent wall bounded flows/Mean velocity profile/ [email protected] 202/327

Turbulent boundary layers structure

Channel flow : Mean velocity profile :

Pope 2000


Turbulent boundary layers structureChannel flow : Mean velocity profile :

Tennekes & Lumley


Turbulent Channel flowsFramework with @t· = 0

Mean velocity field equations

0 = �@p

@x+ ⌫

d2u

dy2� dR12

dy= �@p

@x+

d

dy

⌫

du

dy�R12

�

0 = �@p

@y� dR22

dy= � @

@y[p + R22]

0 = �dR32

dy

Turbulent wall bounded flows/Channel flows/ [email protected] 205/327

Turbulent Channel flows

Framework with @t· = 0

Pressure term

p(x, y) + R22(y) = cste

p(x, y) + R22(y) = p(x, 0) = p0(x) =) @

@xp(x, y) =

d

dxp0(x)

thend

dy

⌫

du

dy�R12

�=

(d

dxp0(x) (channel flow)

0 (boundary layer). (26)



Mean field kinetic energy equations

0 = � @

@x(pu) + ⌫

d2K

dy2� ⌫

✓du

dy

◆2

� d

dy(uR12) + R12

du

dy

= �u@p

@x+

d

dy

⌫

dK

dy� uR12

�� du

dy

⌫

du

dy�R12

�


Mean flow kinetic energySchlichting 2000



Governing equation for Reynolds stress & TKE

DDt

Production Vertical diffusion Pressure Diss.

0 = �2R12du

dy+

d

dy

✓�u0u0v0 + ⌫

d

dyR11

◆+⇧11 �"11

0 =d

dy

✓�v0(v0v0 + 2p0) + ⌫

d

dyR22

◆+⇧22 �"22

0 =d

dy

✓�v0w0w0 + ⌫

d

dyR33

◆+⇧33 �"33

0 = �R22du

dy+

d

dy

✓�u0(v0v0 + p0) + ⌫

d

dyR12

◆+⇧12 �"12

0 = �R12du

dy+

d

dy

✓�1

2v0(u0u0 + v0v0 + w0w0)� p0v0 + ⌫

d

dyK◆

�"

Main difference with homogeneous shearWall-normal diffusion term (turbulent+pressure+viscous contributions)


Reynolds stress tensor components

Channel flow, Inner layer, Schlichting 2000


TKE balance, Schlichting 2000

3 terms =) 3 mechanisms


Reynolds stress tensor components (Kim & Moser)


Streamwise RST balance - R11 (Kim & Moser)


Streamwise RST budget - R11


Wall-normal RST balance - R22 (Kim & Moser)


Wall-normal RST budget - R22


Spanwise RST balance - R33 (Kim & Moser)


Spanwise RST budget - R33


Shear stress balance - R12 (Kim & Moser)


Shear stress budget - R12


Pressure/velocity correlations (Chassaing) : Pressuredriven transfers

Anisotropy production for y+ � 12 =) u02 ! v02 and w02

�11 =2

⇢p0

du0

dx1


Boundary layer

von Kármán and Prandtl phenomenological approach (1930)2D flow : u(x) = (u(x, y), v(x, y), 0)

Almost parallel v ⌧ u

Incompressiblity : @u/@x = �@v/@y ⌧ 1

Zero pressure gradient (ZPG) :

0 =d

dy

⌫

du

dy�R12

�

⌫d

dyu(y)�R12(y) = ⌫

d

dyu(0) ⌘ ⌧⇤

⇢⌘ u2

⇤

Turbulent wall bounded flows/Boundary layers/ [email protected] 222/327

AssumingR12 : constant and negative,The friction velocity is the characteristic velocity for describing theturbulent fluctuations and therefore the Reynolds stress R12

One can define a turbulent viscosity ⌫t such that the Reynolds shear stressreads

�R12 = ⌫t

d

dyu(y) .

Dimensional analysis : (purely empirical)

⌫t = [L2][T�1] �! ⌫t(y) = VKu⇤y with VK ⇠ 0.41

The molecular viscosity is negligible compared to the turbulent viscosity

(222) =) �R12 = u2⇤ = ⌫t

d

dyu(y) (27)

Then u2⇤ = VKu⇤y

d

dyu(y) =) u

u⇤=

1

VKln (yu⇤/⌫) + B with B ⇠ 5.1


Boundary layer

Classical theory for the mean velocity profile

Layer Regions u(y)

viscous sublayer 0 y+ 3� 5 u+ = y+

buffer layer 3� 5 y+ 30� 50 empirical lawlogarithmic layer 30� 50 y+ 0.1�+ u+ = 1

VKln y+ + C1

logarithmic sublayer 30� 50⌫/u⇤ y 0.1� u = 1 VK

ln(y/h) + C2

wake y � 0.1� empirical law


Boundary layer

Simplified turbulent kinetic energy budget

Region Simplified turbulent kinetic energy budget

viscous sublayer dissipation = viscous diffusionbuffer layer production = turbulent diffusion + dissipation

logarithmic layer production = dissipation

wake turbulent diffusion = dissipation


Boundary layer

Turbulent Kinetic Energy Budget


Boundary layer

Reynolds stress component budget equations


Is asymptotic theory valid ?

Strong deviation of constant for internal flows ! ! !...

(Hoyas et al., 2006)


Is the log law observable ?

Experimental setups & computers ?LL extent : 30 < y+ < 0.1�+

=) Necessary condition : �+ > 300.1 decade if �+ > 3000

Reachable ?Larger by a factor 10 compared to existing DNS.Maximum reached in wind tunnel (LML).But lower by a factor ⇠ 10� 100 than real applications ! ! !...


Universality ?

What about the constants ?


Coherent structures and turbulent dynamics

Flow structuresVery complex instantaneous flow organization.Different flow structures.Each associated with the dynamics of BL layers.Identification and role of each structures on the global dynamics(Drag, Thermal transfer, ...).Still open issues...

Turbulent wall bounded flows/Coherent structures/ [email protected] 231/327

Turbulent boundary layersVorticity structuresQ-criterion colored by streamwise vorticity


Observations in viscous/buffer layers

Low/high speed streamwise velocity streakssinuous arrays of alternating streamwise jets superimposed on the meanshear (Kim & al., 1971)

Average streamwise length x+ = 1000

Average spanwise wavelength z+ = 50� 100 (Smith & al. 1983)Wall shear peaks where the jets associated with high speed streaks


Observations in viscous/buffer layers (cont’d)

Hairpin vortices (Eitel-Amor et al. 2014)



Quasi streamwise vorticesSlightly tilted from the wall.Typical length x+ = 200 (Jeong & al. 1997).Advected at speed c+ = 10.Several vortices associated to streaks.Some of them are connected to legs of hairpin vortices in the log layer,but most merge in un-coherent vorticity patches away from the wallWall shear increase where the jets associated with high speed streaks.


Observations in viscous/buffer layers (cont’d)Streaks and vortices (LML, lower wall at left, upper wall at right and bottom)



Quasi streamwise vorticesBy advecting the mean shear =) Streaks (Blackwelder & Eckelman1979)Are independent to the presence of the wall (Rashidi & Banerjee,1990)Strong contribution to the turbulent drag (Orlandi & Jimenez, 1994)




Boundary layer

Autonomous cycle of near wall turbulence


Boundary layer

Autonomous cycle of near wall turbulence

Stronglocalincrease

ofskinfric0on

EJECTION

genera0onof

low‐speedstreak

SWEEP

genera0onof

high‐speedstreak

Low‐

momentum

fluidpocket

u’<0

v’>0

high‐

momentum

fluidpocket

u’>0

v’<0

Streamwise

vortexStreamwise

vortex

Streamwise

vortex

‐produc)onK

‐produc)onK


Boundary layer

Thermal autonomous cycle of near wall turbulence

Stronglocalincrease

ofheattransfer

Coldwallcase@T

@y> 0

Low‐

temperature

fluidpocket

T’<0

v’>0

high‐

temperature

fluidpocket

T’>0

v’<0

Streamwise

vortexStreamwise

vortex

Streamwise

vortex

‐produc)onKT

‐produc)onKT

EJECTION SWEEP


Boundary layer

Thermal autonomous cycle of near wall turbulence

Stronglocalincrease

ofheattransfer

Hotwallcase

high‐

temperature

fluidpocket

T’>0

v’>0

low‐

temperature

fluidpocket

T’<0

v’<0

Streamwise

vortexStreamwise

vortex

Streamwise

vortex

‐produc)onKT

‐produc)onKT

@T

@y< 0

EJECTION SWEEP


Turbulent drag

Generation and Control

Cf (x, t) = u2⇤/

1

2u2

b

Turbulent wall bounded flows/Turbulent drag/ [email protected] 243/327

Turbulent drag

Many semi-empirical laws (Nagib et al. 2007)

Relation Forme originale ModificationsColes-Fernholz 1 Cf = 2[1/ VK ln(Re�⇤) + C⇤]�2 VK = 0, 384, C⇤ = 3, 354Coles-Fernholz 2 Cf = 2[1/ VK ln(Re✓) + C]�2 VK = 0, 384, C = 4, 127Karman-Schoenherr Cf = 0, 558C 0

f/[0, 558 + 2(C 0

f)�1/2]

C 0f

= [log(2Re✓)/0,242]�2 0,2385Prandtl-Schlichting Cf = 0,455(log Rex)�2,58 �A/Rex 0,3596Prandtl-Karman C�1/2

f= 4 log(Rex

pCf )� 0,4 2,12

Schultz-Grunow Cf = 0,427(log Rex � 0, 407)�2,64 0,3475Nikuradse Cf = 0, 02666Re�0,139

x-0,1502

Schlichting Cf = (2 log Rex � 0, 65)�2,3 -2,3333White Cf = 0,455[ln(0, 06Rex)]�2 0,4177Loi 1/7 Cf = 0,027Re�1/7

x 0,02358Loi 1/5 Cf = 0,058Re�1/5

x �A/Rex 0,0655George-Castillo C1/2

f= 2(55/Ci1[�+]��1 exp[A/(ln �+)↵) 56,7


Turbulent drag

Semi-empirical laws


Turbulent drag

Non local drag laws : FIK (Fukagata, Iwamoto and Kasagi

HypothesisStationaryu(x) = u(y)ex

Homogeneous in z directionStatistical symmetry plane : y = h


Turbulent drag

Non local drag laws : FIK

� @

@xp =

@

@y

R12 �

1

Reb

@

@yu

�+@

@tu +

@

@xuu +

@

@y(uv)� 1

Reb

@2

@y2u

| {z }Ix

Ix : Inhomogeneity along x direction

Triple equation of the momentum equation

�00(x, y, t) ⌘ �(x, y, t)� e�(x, t), e�(x, t) ⌘Z 1

0�(x, y, t)dy (28)

=)

Cf = 12

1

Reb�

Z 1

0

2(1� y)R12(y)dy +12

Z 1

0

(1� y2)

✓I 00x +

@p00

@x+

@@t

u

◆dy

�


Turbulent drag

FIK laws : different configurations

Configuration Relation

Plane channel Cf = 12

1

Reb

�Z 1

02(1� y)u0v0dy

�

Pipe flow Cf = 16

1

Reb

�Z 1

02ru0

ru0

zdr

�

ZPG Boundary layer Cf = 4

(1� �d)

Re�

�Z 1

0(1� y)u0v0dy

�

Fukagata et al. 2002


Turbulent drag

Compressible case M = 0.4 and M = 2 (Gomez et al. 2008)


Turbulent drag

FIK’s laws : Control Strategy (Fukagata et al. 2002)

Turbulent wall bounded flows/Skin friction control/ [email protected] 250/327

Turbulent drag

Skin friction control StrategySuction/Blowing (Fukagata et al. 2002)


Turbulent drag

Skin friction control StrategySuction/Blowing (Fukagata et al. 2002)


Turbulent drag

Skin friction control StrategyHydrophobic surface


Turbulent drag

Skin friction control Strategy : Hydrophobic surface

Min & Kim 2004


Turbulent drag

Skin friction control Strategy

Contributions to friction drag in surfactant-added channel flowN. Kasagi & K. Fukagata 2006


Exercice

Non-local expression of the skin friction - FIK formula(Fukagata et al., Phys. Fluids,

2002)

In this exercice, we want to determine the FIK non local formula.The characteristic length is taken equal to the channel half-height h. Thecharacteristic velocity is ub, which is defined as twice the bulk velocity.

1 Let the friction coefficient be Cf = u2⌧/

12 u

2b . Show that

18Cf =

ddy

✓R12(y)�

1Reb

ddy

u(y)

◆, Reb =

h2ub

⌫(29)

2 ApplyingR 1

0dy

Ry

0dy

Ry

0dy to the above formula, prove that

Cf =12Reb

+ 12

Z 1

0

2(1� y)(�R12)dy (30)

Give a physical interpretation of this formula. How can the turbulent drag bereduced ?









HIT// [email protected] 257/327

6 Homogeneous Isotropic TurbulenceSpectral descriptionSpectral equationsSpectral phenomenological descriptionClosure spectral theoryPassive scalar dynamicsFree decaying turbulence

HIT// [email protected] 258/327

Spectral description

Fourier transformDirect

f(k) =1

2⇡

Z +1

�1f(x)e�ıxk dx

Inverse

f(x) =

Z +1

�1f(k)eıxk dk

k : wavenumberi2 = �1

HIT/Spectral description/ [email protected] 259/327


Velocity correlation tensorPhysical space

Rij(x, r, t) ⌘ u0i(x, t)u0

j(x + r, t)

Rij(x, r, t) =

Z +1

�1dk1

Z +1

�1dk2

Z +1

�1dk3�ij(x,k, t)eık·r

Fourier space : Spectral correlation tensor �ij(x,k, t)

�ij(x,k, t) =1

(2⇡)3

Z +1

�1dr1

Z +1

�1dr2

Z +1

�1dr3Rij(x, r, t)e�ık·r

k : wavenumberi2 = �1



Reynolds stress tensorPhysical space

Rij(x,0, t) ⌘ u0i(x, t)u0

j(x, t)

Rij(x, t) =

Z +1

�1dk1

Z +1

�1dk2

Z +1

�1dk3�ij(x,k, t)

Fourier space :spectral density of kinetic energy ⌘ kinetic energy spectrum E(k, t).

K ⌘ 1

2u0

iu0

i(t) =

1

2Rii(t) =

Z +1

0E(k, t)dk



HypothesisHomogeneity in space :

All the statistical quantities are invariant under any arbitrary spacetranslationIn practice

Rij(x, r, t) ⌘ u0i(x, t)u0

j(x+ r, t) = Rij(r, t)

r

u(x + r)u(x)

x

x + r



HypothesisIsotropy

All the statistical quantities are invariant under any arbitrary rotationof the reference frame

Skew isotropy6= full isotropyThe mirror symmetry property is not satisfied. The mirror symmetryis defined by the invariance of all averaged quantities depending on thefluctuating fields against reflexions on arbitrary planes.Mirror symmetry means equipartitions between right and left handedhelical motions.



Turbulent kinetic energy spectrumHomogeneous turbulence

E(k, t) ⌘Z +1

�1dk1

Z +1

�1dk2

Z +1

�1dk3

1

2�ij(k, t)�(|k|� k)

�ij(k, t) assumed independent of x (HT).


Kinetic energy density spectrum

Stationnary random process, Cramer’s theorem

�ij(k) =1

(2⇡)3

Z

IR3

Rij(r)e�ık·r dr

Consider

u⇤i(k)uj(k0) =

1

(2⇡)6

ZZui(x)uj(x0)eı(k·x�k0·x0) dx dx0

=1

(2⇡)6

ZZui(x)uj(x0)eı(k·x�k0·x0) dx dx0

=1

(2⇡)6

ZZRij(r)e

�ık0·reı(k�k0)·x dx dr

=1

(2⇡)3

ZRij(r)e

�ık0·r dr · �(k� k0)

= �ij(k0) · �(k� k0) � dirac function


Kinetic energy density spectrum

PropertiesHomogeneity

Rij(r) = Rji(�r) =) �⇤ij

(k) = �ij(�k) = �ji(k) , 8 k

Incompressibility

ui(x)@uj(x + r)

@rj

= 0 =) ki�ij(k) = kj�ij(k) = 0 , 8 k

Decomposition : Symmetric (real) + Antisymmetric (pure imaginary)

�ij(k) = �(s)ij

(k) + �(a)ij

(k)

Remark :Rij(0) = ui(x)uj(x) =

Z�ij(k) dk

=) �ij(k) represents a density of contributions to ui(x)uj(x) inwavenumber space.


Energy spectrum function E(k)

Mean kinetic energy spectrum u2/2

1

2u2 =

1

2

Zu⇤

i(k)e�ık·x dk

Zui(k0)e+ık0·x dk0

=1

2

ZZu⇤

i(k)ui(k0)e�ı(k�k0)·x dk dk0

=1

2

ZZ�ii(k

0)�(k� k0)e�ı(k�k0)·x dk dk0

=1

2

Z�ii(k) dk

Energy spectrum E(k) defined by

E(k) =1

2

Z

S(k)�ii(k) dS =) 1

2u2 =

Z 1

0E(k) dk

S(k) sphere of radius k in Fourier space.


Spectral equations

Dynamics equationsMomentum equation

@

@tuj + ⌫k2uj = �ıPjlm(k)

Z

p+q=kul(p, t)um(q, t)dp

| {z }sj(k,t)

(31)

Projection operator

Pijl(k) =1

2(Pij(k)kl + Pil(k)kj) , Pij(k) =

✓�ij �

kikj

k2

◆

Perpendicular to kP (k)v ? k, 8v

HIT/Spectral equations/ [email protected] 268/327

Spectral equations

Dynamics equationsIncompressibility

r · u = 0 =) k · u(k) = 0 , 8k

Exercice :Write the momentum equation in the Fourier space as written in (31)p. 268.


Spectral equations

Triadic interactionsClassification of Waleffe 1992

Forward

Forward

Forward

Backward


Spectral equations

Lin equationEquation for the spectral density

�ij(k, t)�(k� p) = u⇤i(p, t)uj(k, t)

=) @

@tE(k, t) + 2⌫k2E(k, t) = T (k, t)

Non linear transfer term

T (k, t) = ⇡k2(u⇤i(k, t)si(k, t) + ui(k, t)s⇤

i(k, t))

Wheresj(k, t) = �ıPjlm(k)

Z

p+q=kul(p, t)um(q, t) dp


Spectral equations

Budget equationBy integration of the Lin equation

@

@t

Z +1

0E(k, t)dk

| {z }K(t)

+ 2⌫

Z +1

0k2E(k, t)dk

| {z }"(t)

=

Z +1

0T (k, t)dk

Conservative non-linear transfer termZ +1

0T (k)dk = 0


Spectral equations

Budget equationBy integration of the Lin equation

@

@t

Z +1

0E(k, t)dk

| {z }K(t)

+ 2⌫

Z +1

0k2E(k, t)dk

| {z }"(t)

=

Z +1

0T (k, t)dk

Conservative non-linear transfer termZ +1

0T (k)dk = 0


Spectral equations

Spectral flux


Spectral phenomenological description

SpectrumScenario of "cascade energy" of Ridcharson and of the viscous cut-off.Kolmogorov HypothesisReynolds numberCharacteristic scales

HIT/Spectral phenomenological description/ [email protected] 275/327


Kolmogorov spectrum

Characteristic scalesTurn-over timesIntegral, Taylor,Viscous lengthscalesKinetic energy

Reynolds numbers



characteristic scales of the spectrum

Scales Integral Taylor Kolmogorov

Space Lu =K3/2

"�g =

r10K⌫"

⌘ =

✓⌫3

"

◆1/4

Time ⌧u =K"

⌧� =

r15⌫

"⌧⌘ =

r⌫

"

Reynolds number ReL =K2

⌫"Re� =

r20

3

Kp⌫"

Re⌘ = 1



Kolmogorov theory : HypothesisH1 : At small scales l⌧ L11,1, the two-points statistical moments,separated by a distance r and at two times separated by a delay ⌧ can beexpressed by using only the quantities ", ⌫, r, ⌧ .H2 : At small scales ⌘ ⌧ l⌧ L11,1, the two-points statistical momentsseparated by a distance r and at two times separated by a delay ⌧ can beexpressed by using only the quantities ", r, ⌧ . The viscosity ⌫ is not neededanymore, this means that these scales are very weakly affected by theJoule dissipation and only experience non linear effects represented by ".

=) E(k) = C"2/3k�5/3



Inertial rangeBoundary layersGrid turbulenceChannel flowsShear layers

Kolmogorov spectrum

E(k) = C"2/3k�5/3


Spectral modeling

Budget equationSolve the Lin equation

@

@tE(k, t) + 2⌫k2E(k, t) = T (k, t)

By modeling the transfer term with an expression which preserve thekinetic energy conservation

Z +1

0T (k)dk = 0 =)

Zk

0T (k0)dk0 = �

Z +1

k

T (k0)dk0 = 0

Then we can seek for the function F such as

T (k) = � @

@kF (k)

HIT/Closure spectral theory/ [email protected] 280/327

Spectral modeling

Obukhov Model 1941Function F (k) ?

Rij

@ui

@xj

= �" = F (k)

By dimensional analysis

Rij =

Z +1

k

E(p)dp,@ui

@xj

=

Zk

0p2E(p)dp

!1/2

HIT/Closure spectral theory/Obukhov Model 1941 [email protected] 281/327

Spectral x

Heisenberg and Weizsacker 1948Function F (k) ?

F (k) = 2⌫t(k)

Zk

0p2E(p)dp

Spectral eddy viscosityHeisenberg

⌫t(k) =89K�3/2

0

Z +1

k

pp�3E(p)dp

Stewart & Townsend 1952 : c > 1

⌫t(k) =

✓Z +1

k

p�(1+1/2c)E1/2c(p)dp

◆c

HIT/Closure spectral theory/EVM [email protected] 282/327

Spectral modeling

Heisenberg and Weizsacker 1948Function F (k) ?

F (k) = 2⌫t(k)

Zk

0p2E(p)dp

Spectral eddy viscosityGeneralization of Stewart and Townsend 1951

⌫t(k) =X

i

ai

✓Z +1

k

p�(1+1/2ci)E1/2ci(p)dp

◆ci

with ai > 0 and ci > 0


Spectral model



Spectrum of free decaying turbulenceDefinition : Variance spectrum

K✓(t) =

Z +1

0E✓(⇠, t)d⇠, "✓(t) = 2

Z +1

0⇠2E✓(⇠, t)d⇠

Inertial range : Kolmogorov spectrum

E(k) = C"2/3k�5/3

Dimensional analysis

E✓(k) = c�"✓"�1/3k�5/3

HIT/Passive scalar dynamics/ [email protected] 285/327


Stan Corrsin



G. K.BatchelorPortrait by RupertShephard 1984this portrait hangs inDAMTP, CambridgeDepartment foundedunder Batchelor’s leader-ship in 1959



George BatchelorRecently elected FRSIn his office at the old CavendishLaboratoryOctober 1956



Scalar spectrumOne-dimensional power spectraVelocity (circles)Temperature (squares)Measured in a jet near the peakshear off-center positionAdapted from Corrsin andUberoi (1951)



Similitude hypothesisAdditional dimensionless parameter : The Prandtl number

Pr =⌫

General case Pr ⇠ 1

Case Pr ⌧ 1

Case Pr � 1



Characteristic scalesTheory Kolmogorov Batchelor Obukhov–Corrsin

Sim. hyp. 1 (b), 3(a), 3(b) 1(a), 1(b) 1(b), 2

Variables "T , ", ⌫ "T , ⌧⌘ , "T , ",

Length ⌘ ⌘B =p⌧⌘ = ⌘/

pPr ⌘OC = (3/")1/4 = ⌘Pr�3/4

Time ⌧⌘ ⌧B = ⌧⌘ ⌧OC =p

/" = ⌘Pr�1/2

Scalar ⌃⌘ =p"T ⌧⌘ ⌃B = ⌃⌘ ⌃OC = "T

p/" = ⌃⌘Pr�1/4



Inertio - convective rangeInertial range

E(k) = C"2/3k�5/3

Inertio - convective range

E✓(k) = c�"✓"�1/3k�5/3



When Pr 6= 1

Case Range LengthscalesPr ⌧ 1 inertio-convective L�1

T⌧ k ⌧ 1/⌘OC

inertio-diffusive 1/⌘OC ⌧ k ⌧ 1/⌘visco-diffusive 1/⌘ ⌧ k

Pr � 1 inertio-convective L�1T⌧ k ⌧ 1/⌘

visco-convective 1/⌘ ⌧ k ⌧ 1/⌘B

visco-diffusive 1/⌘B ⌧ k



Similitude hypothesis : Case Pr � 1

k 2 [1/⌘, 1/⌘✓]The passive scalar fluctuations are strongly damped by the viscouseffects, whereas the scalar fluctuations are not affected by thediffusion. The scalar fields fluctuations are driven by the velocity shearfield. This velocity shear can be evaluated by using the dimensionalanalysis as the inverse of the Kolmogorov time scale, i.e. ⌧�1

⌘=p"/⌫.

The dimensional analysis says that E✓ = E✓(k, "✓, ⌧⌘), =)

E✓(k) = c�"✓⌧⌘k�1

Proposed for the first time by Batchelor in 1959, experimentallyconfirmed in 1963.






Free decaying turbulence


10−5

100

105

10−20

10−10

100

k

E(k

)

t=0

t=103

t=107

t=1011

HIT/Free decaying turbulence/KE [email protected] 297/327


Question : Can we predict the behavior of free decaying turbulence ?

HypothesisHITDefine the relevant parameters

Reynolds numberInitial conditionsSpectrum at large scales (IR) : k�

...

Algebraic decay ?

K(t) / t�n


Kinetic energy decay exponents Meldi & Sagaut 2012


Kinetic energy decay exponents Meldi & Sagaut2012


Kinetic energy decay exponents Meldi & Sagaut2012


Kinetic energy decay exponents





Method : Comte-Bellot CorrsinHypothesis

Spectrum shape

E(k, t) =

⇢Aks kL(t) 1, 1 s 4K0"

2/3k�5/3 kL(t) � 1

Algebraic decayK(t) / t�n

Results

L(t) / (t� t0)2/(3+s)

"(t) = �dK(t)dt

=) K(t) / t"(t)

K(t) / (t� t0)�2(s+1)/(3+s)


Passive scalar decay exponents


Kinetic energy decay exponents : High Reynolds numberregime

s = 1 s = 2 s = 3 s = 4 s = +1K(t) / t�1 / t�6/5 / t�4/3 / t�10/7 / t�2

"(t) / t�2 / t�11/5 / t�7/3 / t�17/7 / t�3

L(t) / t1/2 / t2/5 / t1/3 / t2/7 CsteReL(t) Cste / t�1/5 / t�1/3 / t�3/7 / t�1



Small Reynolds number regime

K(t) ⇠Z 1/L(t)

0Aksdk

K(t) ⇠Z 1/(�

p⌫t)

0Aksdk =

A

s + 1

✓1

�p⌫

◆(s+1)/2

t�(s+1)/2

K(t) / t�(s+1)/2, "(t) / t�(s+3)/2,L(t) / t(3�s)/4, ReL(t) / t(1�s)/2



Small Reynolds number regime

K(t) ⇠Z 1/L(t)

0Aksdk

K(t) ⇠Z 1/(�

p⌫t)

0Aksdk =

A

s + 1

✓1

�p⌫

◆(s+1)/2

t�(s+1)/2

K(t) / t�(s+1)/2, "(t) / t�(s+3)/2 .

K✓(t) / t?, "✓(t) / t?,L(t) / t(3�s)/4, ReL(t) / t(1�s)/2


Scalar dynamics in free decaying turbulence

HypothesisHITAlgebraic decay

K(t) / t�n

K✓(t) / t�n✓

Previous worksReference Re PredictionsCorrsin (1951) High K / t�10/7 KT / t�6/7

Corrsin (1951) Low K / t�5/2 KT / t�3/2

Nelkin & Kerr (1981) High K / t�6/5 KT / t�6/5

Ristorcelli & Livescu (2004) High K / t�1 KT / t�1

HIT/Free decaying turbulence/Scalar [email protected] 309/327


A few experimental/DNS data



Method : Comte-Bellot Corrsin extended to the scalarHypothesis

Spectrum shape

E✓(k, t) =

⇢AT k

p kL(t) 1, 1 s 4c�"

�1/3"✓k�5/3 kL(t) � 1

Algebraic decayK(t) / t�n

Results

"✓ / (t� t0)�(s+2p+5)/(3+s)

K✓ / (t� t0)�2(p+1)/(3+s)

Rc =K"

"✓K✓

=s + 1

p + 1


Passive scalar decay exponents : High Reynolds numberregime

p s = 1 s = 2 s = 3 s = 4 s = +11 / t�1 / t�4/5 / t�2/3 / t�4/7 Cste

K✓(t) 2 / t�3/2 / t�6/5 / t�1 / t�6/7 Cste4 / t�5/2 / t�2 / t�5/3 / t�10/7 Cste1 / t�2 / t�9/5 / t�5/3 / t�11/7 / t�1

"✓(t) 2 / t�5/2 / t�11/5 / t�2 / t�13/7 / t�1

4 / t�7/2 / t�3 / t�8/3 / t�17/7 / t�1

1 = 1 = 3/2 = 2 = 5/2 ⇠ 1Rc 2 = 2/3 = 1 = 4/3 = 5/3 ⇠ 1

4 = 2/5 = 3/5 = 4/5 = 1 ⇠ 1


Decay exponents : High and low Reynolds numbers


PART VIII

NS equations based relationships in HIT

Rigorous results// [email protected] 314/327










Tensorial general expressionsvon Kármán equationKolmogorov 4/5 lawBibliography


Dynamics based results in fully developed turbulence

Tensorial general expressionsConsidering homogeneous isotropic incompressible turbulence.

Qi(r) = A(r)ri =ex.

p(x)ui(x + r)

Rij(r) = F (r)rirj + G(r)�ij =ex.

ui(x)uj(x + r)

Sij`(r) = A(r)rirjr` + B(r)(ri�j` + rj�i`) + D(r)r`�ij

=ex.

ui(x)uj(x)u`(x + r)

where A, F , G, B, D are arbitrary scalar functions of r2 ; all even functionof r (by isotropy)

Rij(r) symmetric in i, j : Rij(r) = Rji(r)

Sij`(r) symmetric in i, j : Sij`(r) = Sji`(r)

Rigorous results/Tensorial general expressions/ [email protected] 317/327

General form for HIT

Expression of Qi

Continuity condition : @iui = 0

@Qi(r)

@ri

= 3A + r@A

@r= 0 , 8r

=) A(r) = 0 assuming regularity at r = 0.First order tensor

Qi(r) ⌘ 0 , 8r.


Tensorial general form

Expression of Rij

Continuity condition :

@Rij(r)

@ri

= rj

✓4F + r

@F

@r+

1

r

@G

@r

◆= 0 , 8r

=) 4F + r@F

@r+

1

r

@G

@r= 0


Tensorial general expression

Expression of Rij

Longitunal and lateral velocity correlations

up(x)up(x + r) ⌘ u2f(r)

un(x)un(x + r) ⌘ u2g(r)

f(r), g(r) : even scalar functionsurms : u2 ⌘ u2

p = u2n = 1

3u2i


General form for HIT

Expression of Sij`

From the general form, one obtains

sij`(r) = u3

k � rk0

2r3rirjr` +

2k + rk0

4r(ri�j` + rj�i`)�

k

2rr`�ij

�,

where k(r) is the single scalar function determining the triple velocitycorrelation and

k0(r) ⌘ @k(r)

@r.

Note thatSiji(r) =

1

2u3

k0 +

4

rk

�rj ⌘

1

2K(r)rj


von Kármán equation (1938)

VKH equationWrite NS equation at points x and x0 = x + r.Summing and averaging with ui ⌘ ui(x, t) and u0

i⌘ ui(x + r, t), in

order to write@tuiu0

j= ...

Rigorous results/von Kármán equation/ [email protected] 322/327


VKH equationPrevious equation leads to

@Rij(r)

@t=

@

@r

huiuku0

j� uiu0

ku0

j

i

+

@

@ri

pu0j� @

@rj

p0ui

�

+2⌫@2Rij(r)

@r`@r`



VKH equationPutting i = j

@R(r)

@t=

1

2

✓r@

@r+ 3

◆K(r) + 2⌫

✓@2

@r2+

2

r

@

@r

◆R(r)

First integral of this equation gives

@(u2f)

@t=

✓@

@r+

4

r

◆(u3k) + 2⌫

✓@2

@r2+

4

r

@

@r

◆(u2f)


Kolmogorov 4/5 lawStructure functionsConsidering the velocity structure functions

2nd order

Bik(r) = �ui(r)�uk(r) = (ui(x + r)� ui(x))(uk(x + r)� uk(x))

3rd order

Bik`(r) = �ui(r)�uk(r)�u`(r)

= (ui(x + r)� ui(x))(uk(x + r)� uk(x))(u`(x + r)� u`(x))

And replacing in the von Kármán Howarth equation yields

�2

3"+

1

2

@Bpp

@t=

1

6r4@

@r(r4Bppp)�

⌫

r4@

@r

✓r4@

@rBpp

◆

with@E

@t=

1

2

@

@tu2

i= �" =

3

2

@

@tu2

p.

Rigorous results/Kolmogorov 4/5 law/ [email protected] 325/327

Kolmogorov 4/5 law

VKH to 4/5 lawNeglecting the time derivative term compared to " .Neglecting the dissipative term in the inertial rangeIntegrating over r

One obtains in the inertial Range

Bppp = (�up(r))3 = �4

5"r

But observed only for very large Reynolds number ! ! !

Rigorous results/Kolmogorov 4/5 law/ [email protected] 326/327

Bibliography

BooksBatchelor 1953, The Theory of Homogeneous TurbulenceFrisch 1995, TurbulencePope 2000, Turbulent FlowMonin & Yaglom, Statistical Fluids Mechanics, Mechanics ofTurbulence

Rigorous results/Bibliography/ [email protected] 327/327

Documents

CFD for Turbulence lectures - thomas-gomez.net