Passive Scalar and Scalar Flux in Homogeneous...

Spectral equations HITSG HST Bibliography

Passive Scalar and Scalar Flux in HomogeneousTurbulence

Antoine BRIARDSupervisor: Thomas GOMEZ

Collaboration: Claude Cambon and Vincent Mons

Paris, ∂'Alembert Institute

GDR Grenoble, 1-3 June 2015

Antoine BRIARD Passive Scalar and Scalar Flux in Homogeneous Turbulence


1 Spectral equationsFluctuating �eldsCraya equationsLin equations

2 Homogenous Isotropic Turbulence with Scalar GradientDe�nitionsValidationDecay and growth laws

3 Homogeneous Shear TurbulenceWithout scalar gradientWith scalar gradient

4 Bibliography


Spectral equations HITSG HST Bibliography Fluctuating �elds Craya equations Lin equations

Fluctuating �elds

Velocity �eld

(∂

∂t− Alnkl

∂

∂kn+ νk2

)ˆui (k)+Mij(k)uj(k)+iPimn(k)umun(k) = 0 (1)

Mij(k) = (δin − 2αiαn)Anj , Aij mean velocity gradient matrix.

Scalar �eld(∂

∂t− Ajlkj

∂

∂kl+ ak2

)θ(k) + λj uj(k) = −ikj θuj(k) (2)

λl scalar gradient



Kinetic Craya equation

dRij

dt+ 2νk2Rij(k) + Min(k)Rnj(k) + Mjn(k)Rni (k) = TNL

ij (k) (3)

Spectral Reynolds tensor Rij(k , t)δ(k − p) =< u∗i (p, t)uj(k , t) >.

Scalar Craya equation

dET

dt+ 2ak2ET (k) + 2λlFl(k) = TT ,NL(k) (4)

Spectral scalar correlation < θ∗(p)θ(k) >= ET (k)δ(k − p) .

Scalar Flux Craya equation

dFi

dt+ (ν + a)k2Fi (k) + Mij(k)Fj(k) + λj Rij = T F ,NL

i (k) (5)

Spectral scalar �ux correlation < u∗i (p)θ(k) >= Fi (k)δ(k − p).



Spherically-averaged Lin equations(∂

∂t+ 2νk2

)E (k , t) = SL(iso)(k , t) + SNL(iso)(k, t) (6)(

∂

∂t+ 2νk2

)E (k , t)H

(dir)ij (k , t) = S

L(dir)ij (k , t) + S

NL(dir)ij (k, t) (7)(

∂

∂t+ 2νk2

)E (k , t)H

(pol)ij (k , t) = S

L(pol)ij (k , t) + S

NL(pol)ij (k , t) (8)

(∂

∂t+ 2ak2

)ET (k , t) = ST ,L(iso)(k , t) + ST ,NL(iso)(k, t) (9)(

∂

∂t+ 2ak2

)ET (k, t)H

(T )ij (k , t) = S

T ,L(dir)ij (k , t)+S

T ,NL(dir)ij (k, t) (10)(

∂

∂t+ (a + ν)k2

)EFH

(F )i (k , t) = SF ,L

i (k , t) + SF ,NLi (k, t) (11)



Spectra, energies, dissipation rates (1/2)

Kinetic energy and scalar variance spectra

E (k, t) =

∫Sk

Rii (k , t)

2d2k , ET (k , t) =

∫Sk

ET (k , t)d2k (12)

Kinetic and scalar energies and dissipation rates

K(T )(t) =

∫ ∞0

E(T )(k , t)dk, ε(T )(t) = 2ν(T )

∫ ∞0

k2 E(T )(k, t)dk

(13)

Directional anisotropy

2E (k , t)H(dir)ij (k , t) =

∫Sk

R(dir)ij (k , t)d2k (14)

Polarization anisotropy

2E (k , t)H(pol)ij (k , t) =

∫Sk

R(pol)ij (k , t)d2k (15)



Spectra, energies, dissipation rates (2/2)

Scalar directional anisotropy

2ET (k, t)H(T )ij (k , t) =

∫Sk

E (T ,dir)Pij(k , t)d2k (16)

Scalar �ux anisotropy

EFH(F )i (k, t) =

∫Sk

Fi (k , t)d2k , Fi =3

2EF0 H

(F )j Pij (17)

Cospectrum and streamwise �ux

F(k, t) = EFH(F )3 (k , t), FS(k , t) = EFH

(F )1 (k, t) (18)

Cospectrum and streamwise �ux energies and dissipation rates

K(S)F (t) =

∫ ∞0

F(S)(k , t)dk , ε(S)F (t) = (ν+a)

∫ ∞0

k2F(S)(k, t)dk

(19)


Spectral equations HITSG HST Bibliography De�nitions De�nitions Decay and growth laws

Homogeneous Isotropic Turbulence with Scalar Gradient

Cospectrum F with scalar gradient Λ

F(k, t) = EFH(F )3 (k , t), λ = (0, 0,−Λ) (20)

Spectral behavior : Lumley (1967), Bos (2005)

F(k , t) = CFΛε1/3k−7/3 (21)

Other scaling using εF

F(k , t) = CFε−1/3εFk

−5/3 (22)

εF is not conserved

Same scaling as for ET (k , t)



Spectral behavior of F(k , t)

10−2

100

102

104

10−10

100

k

E(k,t),F(k,t)

E(k, t)

F(k, t)

k−7/3

k−5/3

kL kη



Production and dissipation - DNS Overholt & Pope (1996)

εF (t) = (ν + a)

∫ ∞0

k2F(k , t)dk, PF (t) = −2

3ΛK (t) (23)

104

103

102

101

10−3

10−2

10−1

100

101

Reλ

ǫF/PF

DNS of Overholt and Pope

EDQNM

Re−0.760λ

Re−0.769λ

Re−1.072λ



Sirivat & Warhaft (1983), β = 1.78◦C .m−1, Λ = 0.152

0 50 100 150−1

−0.8

−0.6

−0.4

−0.2

0

x/M

ρwθ(t)

MandolineEDQNM

0 50 100 1500

0.2

0.4

0.6

0.8

1

1.2

x/M

LT(t)/L(t)

MandolineEDQNM

0 50 100 1500

0.5

1

1.5

2

x/M

−ΛK

F(t)/ǫT(t)

Mandoline

EDQNM

0 50 100 1500

0.5

1

1.5

2

2.5

3

x/M

RT(t)

MandolineEDQNM



Decay of the cospectrum

Power law decayKF (t) ∼ tαF

High Reynolds regime : inertial range dominant

KF (t) =

∫ ∞kL

F(k , t) ∼ k−4/3L ε1/3

αF = −σ − pF − 1

σ − p + 3, pF =

1

2(p + pT ) = 0.4075

Low Reynolds regime : Production through scalar gradient dominant

dKFdt

= PF =2

3ΛK , αF = −σ − 1

2



Cospectrum energy KF and scalar dissipation εT

10−2

100

102

104

−1.5

−1

−0.5

0

0.5

1

Reλ

αF(t)

CBC Cospectrum High Re

CBC Cospectrum Low Re

σ = 4

σ = 3

σ = 2

σ = 1



Growth of the passive scalar with the gradient Λ

Power law growth

KT (t) ∼ tαΛF

Production through the cospectrum dominant

dKT

dt∼ ΛKF (t)

High Reynolds regime

αΛT =

1

2

pT − p + 8

σ − p + 3

Agreement with Chasnov (1995) for Sa�man turbulence αΛT = 4/5.

Low Reynolds regime

αΛT = −σ − 3

2



Scalar energy KT

10−2

10−1

100

101

102

−0.5

0

0.5

1

1.5

2

Reλ

αΛ T(t)

CBC Scalar with Λ High Re

CBC Scalar with Λ Low Reσ = 1

σ = 2

σ = 3

σ = 4


Spectral equations HITSG HST Bibliography Without scalar gradient With scalar gradient

Passive scalar with an uniform shear S

Exponential decrease : Gonzalez (2000)

KT (t) = K∞T exp(γTSt), εT (t) = ε∞T exp(γTSt), γT = − εTSKT

0 10 20 30 40−0.2

0

0.2

0.4

0.6

St

bT ij,ǫ

T/(K

TS)

bT11bT22bT33bT13ǫT /(KTS)

0 10 20 30 4010

−20

10−10

100

1010

St

K(T

)(t),ǫ

T(t)

K(t)KT (t)ǫT (t)

e0.34St

e−0.52St



Rogers, Mansour & Reynolds (1989) : S = 14.142 andΛ = 2.5

Integrated quantities

Di�usivity tensor Dij(t) = − < θui > /λj

Turbulence Prandtl number PrT (t) = −R12(t)/(SDii (t))

Scalar �ux correlation ρuiθ(t) = <uiθ>√<u2i ><θ

2>= KF√

2KTRii

0 5 10 15 20−4

−2

0

2

4

6

8

St

Dij(t)/D

22(t)

Dexp11

Dexp12

Dexp21

Dexp33

D11

D12

D21

D33

0 5 10 15 20 250

0.5

1

1.5

2

St

PrT(t)

DNS 1

DNS 2

DNS 3

EDQNM 1

EDQNM 2

EDQNM 3



Tavoularis & Corrsin (1981) : S = 6.19 and Λ = 0.1823

0 5 10 15−1

−0.5

0

0.5

St

ρuθ(t),ρvθ(t)

ρuθρvθ

ρexpuθ

ρexpvθ

0 5 10 150

0.5

1

1.5

2

St

(ǫK

T)/(ǫ

TK)(t)

EDQNM

Experiment

0 5 10 150

0.5

1

1.5

2

2.5

St

Pr T(t)

EDQNM

Experiment

0 5 10 150

0.5

1

1.5

2

2.5

St

B(t)

EDQNM

Experiment



Spectral behavior of FS(k , t)

10−6

10−4

10−2

10−8

100

100

10−10

1010

k

E(k,t),F

S(k,t),F(k,t)

E(k, t), ET (k, t)

FS(k, t)

F(k, t)

k−7/3

k−5/3

k−23/9



Bibliography

Bos, Touil, Bertoglio, Physics of Fluids, Vol. 17 (2005)Bos, Bertoglio, Physics of Fluids, Vol. 19 (2007)Briard, Gomez, Sagaut, Memari, Journal of Fluid Mechanics, (2015, review)Briard, Gomez, Cambon, Journal of Fluid Mechanics, (2015, writing)Chasnov, Physics of Fluids, Vol. 7 (1995)Corrsin, Journal of Aero Science, Vol. 18 (1951)Gonzalez, Int. J. of Heat and Mass Transfer, Vol. 43 (2000)Lumley, Physics of Fluids, Vol. 10 (1967)Mons, Cambon, Sagaut, Journal of Fluid Mechanics, (2015, review)O'Gorman, Pullin, Journal of Fluid Mechanics, Vol. 532 (2005)Overholt, Pope, Physics of Fluids, Vol. 8 (1996)Rogers, Mansour, Reynolds Journal of Fluid Mechanics, Vol. 203 (1989)Sagaut, Cambon, Cambridge University Press (2008)Sirivat, Warhaft, Journal of Fluid Mechanics, Vol. 128 (1983)Tavoularis, Corrsin, Journal of Fluid Mechanics, Vol. 104 (1981)


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