Upload
others
View
4
Download
0
Embed Size (px)
Citation preview
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
Spectral equations HITSG HST Bibliography
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
Antoine BRIARD Passive Scalar and Scalar Flux in Homogeneous Turbulence
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
Antoine BRIARD Passive Scalar and Scalar Flux in Homogeneous Turbulence
Spectral equations HITSG HST Bibliography Fluctuating �elds Craya equations Lin equations
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).
Antoine BRIARD Passive Scalar and Scalar Flux in Homogeneous Turbulence
Spectral equations HITSG HST Bibliography Fluctuating �elds Craya equations Lin equations
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)
Antoine BRIARD Passive Scalar and Scalar Flux in Homogeneous Turbulence
Spectral equations HITSG HST Bibliography Fluctuating �elds Craya equations Lin equations
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)
Antoine BRIARD Passive Scalar and Scalar Flux in Homogeneous Turbulence
Spectral equations HITSG HST Bibliography Fluctuating �elds Craya equations Lin equations
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)
Antoine BRIARD Passive Scalar and Scalar Flux in Homogeneous Turbulence
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)
Antoine BRIARD Passive Scalar and Scalar Flux in Homogeneous Turbulence
Spectral equations HITSG HST Bibliography De�nitions De�nitions Decay and growth laws
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
Antoine BRIARD Passive Scalar and Scalar Flux in Homogeneous Turbulence
Spectral equations HITSG HST Bibliography De�nitions De�nitions Decay and growth laws
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
Antoine BRIARD Passive Scalar and Scalar Flux in Homogeneous Turbulence
Spectral equations HITSG HST Bibliography De�nitions De�nitions Decay and growth laws
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
Antoine BRIARD Passive Scalar and Scalar Flux in Homogeneous Turbulence
Spectral equations HITSG HST Bibliography De�nitions De�nitions Decay and growth laws
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
Antoine BRIARD Passive Scalar and Scalar Flux in Homogeneous Turbulence
Spectral equations HITSG HST Bibliography De�nitions De�nitions Decay and growth laws
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
Antoine BRIARD Passive Scalar and Scalar Flux in Homogeneous Turbulence
Spectral equations HITSG HST Bibliography De�nitions De�nitions Decay and growth laws
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
Antoine BRIARD Passive Scalar and Scalar Flux in Homogeneous Turbulence
Spectral equations HITSG HST Bibliography De�nitions De�nitions Decay and growth laws
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
Antoine BRIARD Passive Scalar and Scalar Flux in Homogeneous Turbulence
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
Antoine BRIARD Passive Scalar and Scalar Flux in Homogeneous Turbulence
Spectral equations HITSG HST Bibliography Without scalar gradient With scalar gradient
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
Antoine BRIARD Passive Scalar and Scalar Flux in Homogeneous Turbulence
Spectral equations HITSG HST Bibliography Without scalar gradient With scalar gradient
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
Antoine BRIARD Passive Scalar and Scalar Flux in Homogeneous Turbulence
Spectral equations HITSG HST Bibliography Without scalar gradient With scalar gradient
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
Antoine BRIARD Passive Scalar and Scalar Flux in Homogeneous Turbulence
Spectral equations HITSG HST Bibliography
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)
Antoine BRIARD Passive Scalar and Scalar Flux in Homogeneous Turbulence