Spectral description -...

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

2 The governing equations

3 Statistical description of turbulence

4 Turbulence modeling

5 Turbulent wall bounded flows

6 Homogeneous Isotropic Turbulence

7 Homogeneous Shear Flows

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

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

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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

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Velocity correlation tensorPhysical space

Rij(x, r, t) ⌘ u0i(x, t)u0

j(x + r, t)

Rij(x, r, t) =

Z +1

�1dk1

Z +1

�1dk2

Z +1

�1dk3�ij(x,k, t)eık·r

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

�ij(x,k, t) =1

(2⇡)3

Z +1

�1dr1

Z +1

�1dr2

Z +1

�1dr3Rij(x, r, t)e�ık·r

k : wavenumberi2 = �1



Reynolds stress tensorPhysical space

Rij(x,0, t) ⌘ u0i(x, t)u0

j(x, t)

Rij(x, t) =

Z +1

�1dk1

Z +1

�1dk2

Z +1

�1dk3�ij(x,k, t)

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

K ⌘ 1

2u0

iu0

i(t) =

1

2Rii(t) =

Z +1

0E(k, t)dk



HypothesisHomogeneity in space :

All the statistical quantities are invariant under any arbitrary spacetranslationIn practice

Rij(x, r, t) ⌘ u0i(x, t)u0

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

r

u(x + r)u(x)

x

x + r



HypothesisIsotropy

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

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



Turbulent kinetic energy spectrumHomogeneous turbulence

E(k, t) ⌘Z +1

�1dk1

Z +1

�1dk2

Z +1

�1dk3

1

2�ij(k, t)�(|k| � k)

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


Kinetic energy density spectrum

Stationnary random process, Cramer’s theorem

�ij(k) =1

(2⇡)3

Z

IR3

Rij(r)e�ık·r dr

Consider

u⇤i(k)uj(k0) =

1

(2⇡)6

ZZui(x)uj(x0)eı(k·x�k0·x0) dx dx0

=1

(2⇡)6

ZZui(x)uj(x0)eı(k·x�k0·x0) dx dx0

=1

(2⇡)6

ZZRij(r)e

�ık0·reı(k�k0)·x dx dr

=1

(2⇡)3

ZRij(r)e

�ık0·r dr · �(k � k0)

= �ij(k0) · �(k � k0) � dirac function


Kinetic energy density spectrum

PropertiesHomogeneity

Rij(r) = Rji(�r) =) �⇤ij

(k) = �ij(�k) = �ji(k) , 8 k

Incompressibility

ui(x)@uj(x + r)

@rj

= 0 =) ki�ij(k) = kj�ij(k) = 0 , 8 k

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

�ij(k) = �(s)ij

(k) + �(a)ij

(k)

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

Z�ij(k) dk

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


Energy spectrum function E(k)

Mean kinetic energy spectrum u2/2

1

2u2 =

1

2

Zu⇤

i(k)e�ık·x dk

Zui(k0)e+ık0·x dk0

=1

2

ZZu⇤

i(k)ui(k0)e�ı(k�k0)·x dk dk0

=1

2

ZZ�ii(k

0)�(k � k0)e�ı(k�k0)·x dk dk0

=1

2

Z�ii(k) dk

Energy spectrum E(k) defined by

E(k) =1

2

Z

S(k)�ii(k) dS =) 1

2u2 =

Z 1

0E(k) dk

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


Spectral equations

Dynamics equationsMomentum equation

@

@tuj + ⌫k2uj = �ıPjlm(k)

Z

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

| {z }sj(k,t)

(42)

Projection operator

Pijl(k) =1

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

✓�ij � kikj

k2

◆

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

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Spectral equations

Dynamics equationsIncompressibility

r · u = 0 =) k · u(k) = 0 , 8k

Exercice :Write the momentum equation in the Fourier space as written in (42)p. 289.


Spectral equations

Triadic interactionsClassification of Waleffe 1992

Forward

Forward

Forward

Backward


Spectral equations

Lin equationEquation for the spectral density

�ij(k, t)�(k � p) = u⇤i(p, t)uj(k, t)

=) @

@tE(k, t) + 2⌫k2E(k, t) = T (k, t)

Non linear transfer term

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

i(k, t))

Wheresj(k, t) = �ıPjlm(k)

Z

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


Spectral equations

Budget equationBy integration of the Lin equation

@

@t

Z +1

0E(k, t)dk

| {z }K(t)

+ 2⌫

Z +1

0k2E(k, t)dk

| {z }"(t)

=

Z +1

0T (k, t)dk

Conservative non-linear transfer termZ +1

0T (k)dk = 0


Spectral equations

Budget equationBy integration of the Lin equation

@

@t

Z +1

0E(k, t)dk

| {z }K(t)

+ 2⌫

Z +1

0k2E(k, t)dk

| {z }"(t)

=

Z +1

0T (k, t)dk

Conservative non-linear transfer termZ +1

0T (k)dk = 0


Spectral equations

Spectral flux


Spectral phenomenological description

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

HIT/Spectral phenomenological description/ [email protected] 296/374


Kolmogorov spectrum

Characteristic scalesTurn-over timesIntegral, Taylor,Viscous lengthscalesKinetic energy

Reynolds numbers



characteristic scales of the spectrum

Scales Integral Taylor Kolmogorov

Space Lu =K3/2

"�g =

r10K⌫

"⌘ =

✓⌫3

"

◆1/4

Time ⌧u =K"

⌧� =

r15⌫

"⌧⌘ =

r⌫

"

Reynolds number ReL =K2

⌫"Re� =

r20

3

Kp⌫"

Re⌘ = 1



Kolmogorov theory : HypothesisHypothesis 1 : At small scales l ⌧ L11,1, the two-points statisticalmoments, separated by a distance r and at two times separated by a delay⌧ can be expressed by using only the quantities ", ⌫, r, ⌧ .Hypothesis 2 : At small scales ⌘ ⌧ l ⌧ L11,1, the two-points statisticalmoments separated by a distance r and at two times separated by a delay⌧ can be expressed by using only the quantities ", r, ⌧ . The viscosity ⌫ isnot needed anymore, this means that these scales are very weakly affectedby the Joule dissipation and only experience non linear effects representedby ".

=) E(k) = C"2/3k�5/3



Inertial rangeBoundary layersGrid turbulenceChannel flowsShear layers

Kolmogorov spectrum

E(k) = C"2/3k�5/3


Spectral modeling

Budget equationSolve the Lin equation

@

@tE(k, t) + 2⌫k2E(k, t) = T (k, t)

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

Z +1

0T (k)dk = 0 =)

Zk

0T (k0)dk0 = �

Z +1

k

T (k0)dk0 = 0

Then we can seek for the function F such as

T (k) = � @

@kF (k)

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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

HIT/Closure spectral theory/Obukhov Model 1941 [email protected] 302/374

Spectral modeling

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


Spectral model


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

HIT/Passive scalar dynamics/ [email protected] 306/374


Stan Corrsin



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



George BatchelorRecently elected FRSIn his office at the old CavendishLaboratoryOctober 1956



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



Similitude hypothesisAdditional dimensionless parameter : The Prandtl number

Pr =⌫

General case Pr ⇠ 1

Case Pr ⌧ 1

Case Pr � 1



Characteristic scalesTheory Kolmogorov Batchelor Obukhov–Corrsin

Sim. hyp. 1 (b), 3(a), 3(b) 1(a), 1(b) 1(b), 2Variables "T , ", ⌫ "T , ⌧⌘ , "T , ",

Length ⌘ ⌘B =p⌧⌘ = ⌘/

pPr ⌘OC = (3/")1/4 = ⌘Pr�3/4

Time ⌧⌘ ⌧B = ⌧⌘ ⌧OC =p

/" = ⌘Pr�1/2

Scalar ⌃⌘ =p"T ⌧⌘ ⌃B = ⌃⌘ ⌃OC = "T

p/" = ⌃⌘Pr�1/4



Inertio - convective rangeInertial range

E(k) = C"2/3k�5/3

Inertio - convective range

E✓(k) = c�"✓"�1/3k�5/3



When Pr 6= 1

Case Range LengthscalesPr ⌧ 1 inertio-convective L�1

T⌧ k ⌧ 1/⌘OC

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

Pr � 1 inertio-convective L�1T

⌧ k ⌧ 1/⌘visco-convective 1/⌘ ⌧ k ⌧ 1/⌘B

visco-diffusive 1/⌘B ⌧ k



Similitude hypothesis : Case Pr � 1

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

⌘=p

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

E✓(k) = c�"✓⌧⌘k�1

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






Free decaying turbulence


10−5

100

105

10−20

10−10

100

k

E(k

)

t=0

t=103

t=107

t=1011

HIT/Free decaying turbulence/KE [email protected] 318/374


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

HypothesisHITDefine the relevant parameters

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

...

Algebraic decay ?

K(t) / t�n


Kinetic energy decay exponents Meldi & Sagaut 2012


Kinetic energy decay exponents Meldi & Sagaut2012


Kinetic energy decay exponents Meldi & Sagaut2012


Kinetic energy decay exponents





Method : Comte-Bellot CorrsinHypothesis

Spectrum shape

E(k, t) =

⇢Aks kL(t) 1, 1 s 4K0"

2/3k�5/3 kL(t) � 1

Algebraic decayK(t) / t�n

Results

L(t) / (t � t0)2/(3+s)

"(t) = �dK(t)dt

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

K(t) / (t � t0)�2(s+1)/(3+s)


Passive scalar decay exponents


Kinetic energy decay exponents : High Reynolds numberregime

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

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

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



Small Reynolds number regime

K(t) ⇠Z 1/L(t)

0Aksdk

K(t) ⇠Z 1/(�

p⌫t)

0Aksdk =

A

s + 1

✓1

�p

⌫

◆(s+1)/2

t�(s+1)/2

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



Small Reynolds number regime

K(t) ⇠Z 1/L(t)

0Aksdk

K(t) ⇠Z 1/(�

p⌫t)

0Aksdk =

A

s + 1

✓1

�p

⌫

◆(s+1)/2

t�(s+1)/2

K(t) / t�(s+1)/2, "(t) / t�(s+3)/2 .

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


Scalar dynamics in free decaying turbulence

HypothesisHITAlgebraic decay

K(t) / t�n

K✓(t) / t�n✓

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

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

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

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

HIT/Free decaying turbulence/Scalar [email protected] 330/374


A few experimental/DNS data



Method : Comte-Bellot Corrsin extended to the scalarHypothesis

Spectrum shape

E✓(k, t) =

⇢AT k

p kL(t) 1, 1 s 4c�"

�1/3"✓k�5/3 kL(t) � 1

Algebraic decayK(t) / t�n

Results

"✓ / (t � t0)�(s+2p+5)/(3+s)

K✓ / (t � t0)�2(p+1)/(3+s)

Rc = K"

"✓K✓

= s+1p+1


Passive scalar decay exponents : High Reynolds numberregime

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

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

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

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

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

4 = 2/5 = 3/5 = 4/5 = 1 ⇠ 1


Decay exponents : High and low Reynolds numbers


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