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Lectures in Turbulence
Thomas Gomez
Laboratoire de Mécanique des Fluides de Lille : Kampé de Fériet
2019-2020
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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
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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
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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
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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
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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
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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
5 Turbulent wall bounded flows
6 Homogeneous Isotropic Turbulence
7 Results based on the equations of the dynamics in fully developedturbulence
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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
The Turbulence fact// [email protected] 9/327
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
The Turbulence fact/Ubiquitousness/Natural [email protected] 13/327
Astrophysical flows
The sun
The Turbulence fact/Ubiquitousness/Natural [email protected] 14/327
Atmospheric flows
Clouds Atmospheric pollution
The Turbulence fact/Ubiquitousness/Natural [email protected] 15/327
Atmospheric flows
Sakura-jima eruption as seen on August 18, 2013, Japan
The Turbulence fact/Ubiquitousness/Natural [email protected] 16/327
Atmospheric flows
Combined Flights Ground Measurements, 30Mar-03Apr2011, FukushimaThe Turbulence fact/Ubiquitousness/Natural [email protected] 17/327
Oceanic flows
The Turbulence fact/Ubiquitousness/Natural [email protected] 18/327
Atmospheric flows
Windturbine wake Wake of an island :von Karman street
The Turbulence fact/Ubiquitousness/Natural [email protected] 19/327
Rivers
River
Leonardo Da Vinci (1452 - 1519)
The Turbulence fact/Ubiquitousness/Natural [email protected] 20/327
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
The Turbulence fact/Two approaches/Simulations [email protected] 27/327
Vorticity filaments
The Turbulence fact/Two approaches/Simulations [email protected] 28/327
Turbulence modifies local properties
Wind tunnelFrisch 1995
The Turbulence fact/Two approaches/Simulations [email protected] 29/327
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⇢U
2S
1 Laminar2 Turbulent
transition3 Drag crisis :
Turbulent BL
Sphere in rotation
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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
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Pressure losses
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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...
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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
The Turbulence fact/Essential features/ [email protected] 36/327
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.
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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
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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).
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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.
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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] 41/327
PART II
The governing equations
Equations// [email protected] 42/327
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 Results based on the equations of the dynamics in fully developedturbulence
Equations// [email protected] 43/327
2 The governing equationsNavier-Stokes EquationsVorticityPressure in incompressible flowsNS equations and SymmetriesDimensionless numbersNS non viscous invariantsValidityCharacteristics of turbulent flowsHomogeneity and isotropyCanonical turbulent flowsExercises
Equations// [email protected] 44/327
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
The Navier Stokes equations
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
Equations/NS equations/ [email protected] 46/327
The Navier Stokes equations
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
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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
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The Navier Stokes equations
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
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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
The Navier Stokes equations
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
⇢
◆
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The Navier Stokes equations
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.
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The Navier Stokes equations
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
Equations/NS equations/ [email protected] 53/327
The Navier Stokes equations
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 !
Equations/NS equations/ [email protected] 54/327
The Navier Stokes 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
Equations/NS equations/ [email protected] 55/327
The Navier Stokes equations
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).
Equations/NS equations/ [email protected] 56/327
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
Equations/NS equations/ [email protected] 57/327
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)
Non linear, coupled, partial differential equations !
Exercice :Write the momentum equation under a conservative form
Equations/NS equations/ [email protected] 58/327
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
Equations/NS equations/ [email protected] 59/327
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
Equations/NS equations/ [email protected] 60/327
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
Equations/Vorticity/ [email protected] 62/327
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
Equations/Vorticity/ [email protected] 63/327
Vorticity dynamics
Incompressible turbulent flow in 3D cubic box
Equations/Vorticity/ [email protected] 64/327
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 .
Equations/Vorticity/ [email protected] 65/327
Pressure in incompressible flows
Non local quantity ! and non linear...
Equations/Pressure in incompressible flows/ [email protected] 66/327
Pressure in incompressible flows
Poisson problem
Equations/Pressure in incompressible flows/ [email protected] 67/327
Pressure in incompressible flows
Filament of vorticity tracerAir bubbles =) minimum of pressure () �p� 1
Equations/Pressure in incompressible flows/ [email protected] 68/327
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
NS equations and Symmetries
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.
Equations/Symmetries/ [email protected] 70/327
NS equations and Symmetries
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)
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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)
Equations/Dimensionless numbers/Reynolds number [email protected] 73/327
Dimensionless numbers
Reynolds number Re
Re⌧ 1 Laminar flowsRe� 1 Turbulent flowsControl parameterRec : Critical Reynolds
Re =⇢UL
µ⇠ Inertial effect
Viscous effect
(12)
Equations/Dimensionless numbers/Reynolds number [email protected] 74/327
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)
Non linear, coupled, partial differential equations !
Re �! +1 =) NS equations �! Euler equations.Re �! +1 =) Turbulent fields !
Equations/Dimensionless numbers/Reynolds number [email protected] 75/327
Jets : Laminar �! Turbulent
�! �! �! �! �!
Reynolds number %
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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
Equations/Dimensionless numbers/Strouhal number [email protected] 78/327
Cont’d – Wakes : flow around a cylinder
Equations/Dimensionless numbers/Strouhal number [email protected] 79/327
Cont’d – Wakes : flow around a cylinder
Fixed cylinder
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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 .
Equations/NS non viscous invariants/ [email protected] 84/327
Non viscous invariants
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 .
Equations/NS non viscous invariants/ [email protected] 85/327
Non viscous invariants
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.
Equations/NS non viscous invariants/ [email protected] 86/327
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
Equations/NS non viscous invariants/ [email protected] 87/327
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
Wind tunnelFrisch 1995
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)
Equations/Characteristics of turbulent flows/ [email protected] 91/327
Characteristics of turbulent flows
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 ?
Equations/Characteristics of turbulent flows/ [email protected] 92/327
Characteristics of turbulent flows
Turbulence develops above a threshold of Reynolds numberExamples :
Boundary layer transition.Cylinder wakes.
Boundary layer transition (NASA)
Equations/Characteristics of turbulent flows/ [email protected] 93/327
Characteristics of turbulent flows
Golf ball dimples & drag (http ://www.aerospaceweb.org/)
Equations/Characteristics of turbulent flows/ [email protected] 94/327
Characteristics of turbulent flows
Turbulence is 3D and rotationalExamples :
Vorticity equationAmplification of the vorticity field by strain : !jsij
DNS of Compressible flowD. H. Porter et al.
Equations/Characteristics of turbulent flows/ [email protected] 95/327
Characteristics of turbulent flows
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.
Equations/Characteristics of turbulent flows/ [email protected] 96/327
Characteristics of turbulent flows
Kinetic energy spectrum
Equations/Characteristics of turbulent flows/ [email protected] 97/327
Characteristics of turbulent flows
Turbulence is dissipativeExample
Friction in a boundary layer.Energy feeding of turbulence : c.f. Section on boundary layer
dynamics.
Equations/Characteristics of turbulent flows/ [email protected] 98/327
Characteristics of turbulent flows
The largest scales are fixed by the characteristic size of the flowExamples
Boundary layer.Wake
Largest scale
Equations/Characteristics of turbulent flows/ [email protected] 99/327
Characteristics of turbulent flows
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.
Equations/Characteristics of turbulent flows/ [email protected] 100/327
Characteristics of turbulent flows
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`
`and ⌫@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.
Equations/Characteristics of turbulent flows/ [email protected] 101/327
Characteristics of turbulent flows
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.
Equations/Characteristics of turbulent flows/ [email protected] 102/327
Characteristics of turbulent flows
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
Equations/Characteristics of turbulent flows/ [email protected] 103/327
Characteristics of turbulent flows
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 ! ! !
Equations/Characteristics of turbulent flows/ [email protected] 104/327
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
Wind tunnelFrisch 1995
Equations/Homogeneity and isotropy/ [email protected] 106/327
Canonical turbulent flows
ClassificationsHIT : Homogeneous Isotropic TurbulenceFree turbulence
JetsWakesMixing layer
Wall turbulenceChannel flowPipe flow
Boundary layer
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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
The characteristic scales of the turbulent dynamics
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.
Equations/Exercises/ [email protected] 110/327
Exercises
The characteristic scales of the turbulent dynamics
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.
Equations/Exercises/ [email protected] 111/327
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.
Equations/Exercises/ [email protected] 112/327
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 ?
Equations/Exercises/ [email protected] 113/327
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.
Equations/Exercises/ [email protected] 114/327
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.
Equations/Exercises/ [email protected] 115/327
PART III
Statistical description of turbulence
Statistical description// [email protected] 116/327
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 Results based on the equations of the dynamics in fully developedturbulence
Statistical description// [email protected] 117/327
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// [email protected] 118/327
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
Wind tunnelFrisch 1995
Statistical description// [email protected] 119/327
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
Statistical description/PDF/ [email protected] 122/327
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
Statistical description/PDF/ [email protected] 123/327
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.
Statistical description/PDF/ [email protected] 124/327
Probability density function (cont’d)
Some properties
P(v) � 0 , 8v 2 IR .
Z +1
�1P(v) dv = 1 .
P(v0 < v < v0 + �v) = P(v0) · �v .
Statistical description/PDF/ [email protected] 125/327
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
Statistical description/PDF/ [email protected] 126/327
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
Statistical description/PDF/ [email protected] 127/327
Probability density function (cont’d)
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 !
Statistical description/PDF/ [email protected] 128/327
Probability density function (cont’d)Example of PDF for the scalar : Lee et al. 2012
Statistical description/PDF/ [email protected] 129/327
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 .
Statistical description/PDF/ [email protected] 130/327
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 equation
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
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)
Statistical description/Exercises/ [email protected] 150/327
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)
Statistical description/Exercises/ [email protected] 151/327
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 Results based on the equations of the dynamics in fully developedturbulence
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
⇢⇠ `u = `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
⇢⇠ `u = `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
⇢⇠ `u = `m
✓`m
����du
dy
����
◆
Re-injected in the RANS equations
Turbulence modeling/First order models/zero equation [email protected] 160/327
Zero equation model : The boundary layer case
Reynolds stress tensor
�u0iu0
j= ⌫t
✓@ui
@xj
+@uj
@xi
◆� 2
3�ijK
with
⌫t = `m
✓`m
����du
dy
����
◆
then
�u0v0 = `2m
����du
dy
����2
Turbulence modeling/First order models/zero equation [email protected] 161/327
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.
Turbulence modeling/First order models/zero equation [email protected] 162/327
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
Turbulence modeling/First order models/One equation models [email protected] 165/327
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
Turbulence modeling/First order models/One equation models [email protected] 166/327
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
◆
Turbulence modeling/First order models/Two equations models [email protected] 168/327
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 � !, ...
Turbulence modeling/First order models/Two equations models [email protected] 169/327
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).
Turbulence modeling/First order models/Two equations models [email protected] 170/327
K � " model (cont’d)Simulations : mean velocity field
Turbulence modeling/First order models/Two equations models [email protected] 171/327
K � " model (cont’d)
Simulations : K and "
Turbulence modeling/First order models/Two equations models [email protected] 172/327
K � " model (cont’d)ComparisonsPredicted turbulent viscosity around a transonic airfoil=) Spalart and Chien models for shear layer...
from http ://www.innovative-cfd.com
Turbulence modeling/First order models/Two equations models [email protected] 173/327
K � " model (cont’d)
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.
Turbulence modeling/First order models/Two equations models [email protected] 174/327
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.
Turbulence modeling/First order models/Two equations models [email protected] 175/327
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
Reynolds stress tensor equation
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
Turbulence modeling/Second order models/Reynolds stress model [email protected] 180/327
Reynolds stress tensor equation
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
◆
Turbulence modeling/Second order models/Reynolds stress model [email protected] 181/327
Reynolds stress tensor equationRij models
@
@tRij +
@
@xk
(ukRij) = Pij + ⇧ij + Dij + "ij
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
Turbulence modeling/Second order models/Reynolds stress model [email protected] 182/327
Reynolds stress tensor equation
Rij models@
@tRij +
@
@xk
(ukRij) = Pij + ⇧ij + Dij + "ij
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
◆
Turbulence modeling/Second order models/Reynolds stress model [email protected] 183/327
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...
Turbulence modeling/Second order models/Reynolds stress model [email protected] 184/327
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
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 Results based on the equations of the dynamics in fully developedturbulence
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
Turbulent wall bounded flows/Description/ [email protected] 190/327
Turbulent boundary layers
Vorticity structures, Re = 3000, iso Q
Turbulent wall bounded flows/Description/ [email protected] 191/327
Turbulent boundary layers
Vorticity structures
Turbulent wall bounded flows/Description/ [email protected] 192/327
Turbulent boundary layers (cont’d)Vorticity structuresQ-criterion colored by streamwise vorticity
Turbulent wall bounded flows/Description/ [email protected] 193/327
Turbulent boundary layers (cont’d)
Vorticity structures : PowerFlow simulation
Turbulent wall bounded flows/Description/ [email protected] 194/327
Turbulent boundary layers (cont’d)Pressure gradient : Favorable / Adverse (Falco 1982)
Turbulent wall bounded flows/Description/ [email protected] 195/327
Flow over a bumpStreaks and vortices
Turbulent wall bounded flows/Description/ [email protected] 196/327
Flow over a bumpStreaks and vortices (lower at left, upper at right and bottom)
Turbulent wall bounded flows/Description/ [email protected] 197/327
Flow over a bump
Adverse gradient
Channel Flow,Reh = 12600, based onthe half-width of the chan-nelDNS
Turbulent wall bounded flows/Description/ [email protected] 198/327
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
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Turbulent boundary layers structureMean velocity profile :Inner/outer region
y+ = y/l⇤
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Turbulent boundary layers structure
Channel flow : Mean velocity profile :
Pope 2000
Turbulent wall bounded flows/Mean velocity profile/ [email protected] 203/327
Turbulent boundary layers structureChannel flow : Mean velocity profile :
Tennekes & Lumley
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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
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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)
Turbulent wall bounded flows/Channel flows/ [email protected] 206/327
Turbulent Channel flows
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
�
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Mean flow kinetic energySchlichting 2000
Turbulent wall bounded flows/Channel flows/ [email protected] 208/327
Turbulent Channel flows
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)
Turbulent wall bounded flows/Channel flows/ [email protected] 209/327
Reynolds stress tensor components
Channel flow, Inner layer, Schlichting 2000
Turbulent wall bounded flows/Channel flows/ [email protected] 210/327
TKE balance, Schlichting 2000
3 terms =) 3 mechanisms
Turbulent wall bounded flows/Channel flows/ [email protected] 211/327
Reynolds stress tensor components (Kim & Moser)
Turbulent wall bounded flows/Channel flows/ [email protected] 212/327
Streamwise RST balance - R11 (Kim & Moser)
Turbulent wall bounded flows/Channel flows/ [email protected] 213/327
Streamwise RST budget - R11
Turbulent wall bounded flows/Channel flows/ [email protected] 214/327
Wall-normal RST balance - R22 (Kim & Moser)
Turbulent wall bounded flows/Channel flows/ [email protected] 215/327
Wall-normal RST budget - R22
Turbulent wall bounded flows/Channel flows/ [email protected] 216/327
Spanwise RST balance - R33 (Kim & Moser)
Turbulent wall bounded flows/Channel flows/ [email protected] 217/327
Spanwise RST budget - R33
Turbulent wall bounded flows/Channel flows/ [email protected] 218/327
Shear stress balance - R12 (Kim & Moser)
Turbulent wall bounded flows/Channel flows/ [email protected] 219/327
Shear stress budget - R12
Turbulent wall bounded flows/Channel flows/ [email protected] 220/327
Pressure/velocity correlations (Chassaing) : Pressuredriven transfers
Anisotropy production for y+ � 12 =) u02 ! v02 and w02
�11 =2
⇢p0
du0
dx1
Turbulent wall bounded flows/Channel flows/ [email protected] 221/327
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
Turbulent wall bounded flows/Boundary layers/ [email protected] 223/327
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
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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
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Boundary layer
Turbulent Kinetic Energy Budget
Turbulent wall bounded flows/Boundary layers/ [email protected] 226/327
Boundary layer
Reynolds stress component budget equations
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Is asymptotic theory valid ?
Strong deviation of constant for internal flows ! ! !...
(Hoyas et al., 2006)
Turbulent wall bounded flows/Boundary layers/ [email protected] 228/327
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 ! ! !...
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Universality ?
What about the constants ?
Turbulent wall bounded flows/Boundary layers/ [email protected] 230/327
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
Turbulent wall bounded flows/Coherent structures/ [email protected] 232/327
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
Turbulent wall bounded flows/Coherent structures/ [email protected] 233/327
Observations in viscous/buffer layers (cont’d)
Hairpin vortices (Eitel-Amor et al. 2014)
Turbulent wall bounded flows/Coherent structures/ [email protected] 234/327
Observations in viscous/buffer layers (cont’d)
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.
Turbulent wall bounded flows/Coherent structures/ [email protected] 235/327
Observations in viscous/buffer layers (cont’d)Streaks and vortices (LML, lower wall at left, upper wall at right and bottom)
Turbulent wall bounded flows/Coherent structures/ [email protected] 236/327
Observations in viscous/buffer layers (cont’d)
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)
Turbulent wall bounded flows/Coherent structures/ [email protected] 237/327
Observations in viscous/buffer layers (cont’d)
Turbulent wall bounded flows/Coherent structures/ [email protected] 238/327
Boundary layer
Autonomous cycle of near wall turbulence
Turbulent wall bounded flows/Coherent structures/ [email protected] 239/327
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
Turbulent wall bounded flows/Coherent structures/ [email protected] 240/327
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
Turbulent wall bounded flows/Coherent structures/ [email protected] 241/327
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 wall bounded flows/Coherent structures/ [email protected] 242/327
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
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Turbulent drag
Semi-empirical laws
Turbulent wall bounded flows/Turbulent drag/ [email protected] 245/327
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 wall bounded flows/Turbulent drag/ [email protected] 246/327
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
�
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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 wall bounded flows/Turbulent drag/ [email protected] 248/327
Turbulent drag
Compressible case M = 0.4 and M = 2 (Gomez et al. 2008)
Turbulent wall bounded flows/Turbulent drag/ [email protected] 249/327
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 wall bounded flows/Skin friction control/ [email protected] 251/327
Turbulent drag
Skin friction control StrategySuction/Blowing (Fukagata et al. 2002)
Turbulent wall bounded flows/Skin friction control/ [email protected] 252/327
Turbulent drag
Skin friction control StrategyHydrophobic surface
Turbulent wall bounded flows/Skin friction control/ [email protected] 253/327
Turbulent drag
Skin friction control Strategy : Hydrophobic surface
Min & Kim 2004
Turbulent wall bounded flows/Skin friction control/ [email protected] 254/327
Turbulent drag
Skin friction control Strategy
Contributions to friction drag in surfactant-added channel flowN. Kasagi & K. Fukagata 2006
Turbulent wall bounded flows/Skin friction control/ [email protected] 255/327
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 ?
Turbulent wall bounded flows/Skin friction control/ [email protected] 256/327
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 Results based on the equations of the dynamics in fully developedturbulence
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
Spectral description
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
HIT/Spectral description/ [email protected] 260/327
Spectral description
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
HIT/Spectral description/ [email protected] 261/327
Spectral description
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
HIT/Spectral description/ [email protected] 262/327
Spectral description
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.
HIT/Spectral description/ [email protected] 263/327
Spectral description
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).
HIT/Spectral description/ [email protected] 264/327
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
HIT/Spectral description/ [email protected] 265/327
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.
HIT/Spectral description/ [email protected] 266/327
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.
HIT/Spectral description/ [email protected] 267/327
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.
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Spectral equations
Triadic interactionsClassification of Waleffe 1992
Forward
Forward
Forward
Backward
HIT/Spectral equations/ [email protected] 270/327
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
HIT/Spectral equations/ [email protected] 271/327
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
HIT/Spectral equations/ [email protected] 272/327
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
HIT/Spectral equations/ [email protected] 273/327
Spectral equations
Spectral flux
HIT/Spectral equations/ [email protected] 274/327
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
Spectral phenomenological description
Kolmogorov spectrum
Characteristic scalesTurn-over timesIntegral, Taylor,Viscous lengthscalesKinetic energy
Reynolds numbers
HIT/Spectral phenomenological description/ [email protected] 276/327
Spectral description
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
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Spectral description
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
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Spectral phenomenological description
Inertial rangeBoundary layersGrid turbulenceChannel flowsShear layers
Kolmogorov spectrum
E(k) = C"2/3k�5/3
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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)
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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
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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
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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
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Spectral model
HIT/Closure spectral theory/EVM [email protected] 284/327
Passive scalar dynamics
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
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Passive scalar dynamics
Stan Corrsin
HIT/Passive scalar dynamics/ [email protected] 286/327
Passive scalar dynamics
G. K.BatchelorPortrait by RupertShephard 1984this portrait hangs inDAMTP, CambridgeDepartment foundedunder Batchelor’s leader-ship in 1959
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Passive scalar dynamics
George BatchelorRecently elected FRSIn his office at the old CavendishLaboratoryOctober 1956
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Passive scalar dynamics
Scalar spectrumOne-dimensional power spectraVelocity (circles)Temperature (squares)Measured in a jet near the peakshear off-center positionAdapted from Corrsin andUberoi (1951)
HIT/Passive scalar dynamics/ [email protected] 289/327
Passive scalar dynamics
Similitude hypothesisAdditional dimensionless parameter : The Prandtl number
Pr =⌫
General case Pr ⇠ 1
Case Pr ⌧ 1
Case Pr � 1
HIT/Passive scalar dynamics/ [email protected] 290/327
Passive scalar dynamics
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
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Passive scalar dynamics
Inertio - convective rangeInertial range
E(k) = C"2/3k�5/3
Inertio - convective range
E✓(k) = c�"✓"�1/3k�5/3
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Passive scalar dynamics
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
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Passive scalar dynamics
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.
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Passive scalar dynamics
HIT/Passive scalar dynamics/ [email protected] 295/327
Passive scalar dynamics
HIT/Passive scalar dynamics/ [email protected] 296/327
Free decaying turbulence
Spectral description
10−5
100
105
10−20
10−10
100
k
E(k
)
t=0
t=103
t=107
t=1011
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Free decaying turbulence
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
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Kinetic energy decay exponents Meldi & Sagaut 2012
HIT/Free decaying turbulence/KE [email protected] 299/327
Kinetic energy decay exponents Meldi & Sagaut2012
HIT/Free decaying turbulence/KE [email protected] 300/327
Kinetic energy decay exponents Meldi & Sagaut2012
HIT/Free decaying turbulence/KE [email protected] 301/327
Kinetic energy decay exponents
HIT/Free decaying turbulence/KE [email protected] 302/327
Kinetic energy decay exponents
HIT/Free decaying turbulence/KE [email protected] 303/327
Free decaying turbulence
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)
HIT/Free decaying turbulence/KE [email protected] 304/327
Passive scalar decay exponents
HIT/Free decaying turbulence/KE [email protected] 305/327
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
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Kinetic energy decay exponents
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
HIT/Free decaying turbulence/KE [email protected] 307/327
Passive scalar decay exponents
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
HIT/Free decaying turbulence/KE [email protected] 308/327
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
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Passive scalar decay exponents
A few experimental/DNS data
HIT/Free decaying turbulence/Scalar [email protected] 310/327
Passive scalar decay exponents
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
HIT/Free decaying turbulence/Scalar [email protected] 311/327
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
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Decay exponents : High and low Reynolds numbers
HIT/Free decaying turbulence/Scalar [email protected] 313/327
PART VIII
NS equations based relationships in HIT
Rigorous results// [email protected] 314/327
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 Results based on the equations of the dynamics in fully developedturbulence
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7 Results based on the equations of the dynamics in fully developedturbulence
Tensorial general expressionsvon Kármán equationKolmogorov 4/5 lawBibliography
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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)
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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.
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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
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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
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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
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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= ...
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von Kármán equation (1938)
VKH equationPrevious equation leads to
@Rij(r)
@t=
@
@r
huiuku0
j� uiu0
ku0
j
i
+
@
@ri
pu0j� @
@rj
p0ui
�
+2⌫@2Rij(r)
@r`@r`
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von Kármán equation (1938)
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)
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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.
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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
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