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G. AhmadiME 639-Turbulence G. AhmadiME 639-Turbulence
OutlineOutline44Double Diffusive ConvectionDouble Diffusive Convection44Thermal ConvectionThermal Convection44Isotropic TurbulenceIsotropic Turbulence44BifurcationBifurcation44TurbulenceTurbulence
G. AhmadiME 639-Turbulence
S. A. AbuS. A. Abu--ZaidZaid and G. Ahmadi, and G. Ahmadi, ApplAppl. Math. Modeling, Vol. 13 (1989). Math. Modeling, Vol. 13 (1989)
1T
2T
1S
2S
Fluid Layer heat from below with a Fluid Layer heat from below with a salt concentration gradientsalt concentration gradient
G. AhmadiME 639-Turbulence
( ) ( ) ( ) uuuu 200t0 gpp ∇ρν+ρ−ρ+−−∇=∇⋅+∂ρ
TTT 2Tt ∇κ=∇⋅+∂ u
SSS 2St ∇κ=∇⋅+∂ u
0=⋅∇u
Governing EquationsGoverning Equations
( ) ( )( )000 SSTT1 −β+−α−ρ=ρ
2
G. AhmadiME 639-Turbulence
Stream FunctionStream Function ( )ψ∂ψ∂−= xz ,0,u
NondimensionalNondimensional Governing EquationsGoverning Equations
( )( )( )( )t,z,xz1SSS
t,z,xz1TTT
0
0
Σ+−∆=−θ+−∆=−
( )[ ] ψ∇+Σ∂−θ∂=ψ∇ψ+ψ∇∂σ− 4xSxT
22t
1 RR,J
( ) θ∇+ψ∂=θψ+θ∂ 2xt ,J
( ) Σ∇τ+ψ∂=Σψ+Σ∂ 2xt ,J
G. AhmadiME 639-Turbulence
( ) θ∂ψ∂−θ∂ψ∂=θψ xyyx,J
νκ∆α
=T
3
TTdgR
νκ∆β
=S
3
STdgR
Tκν
=σT
skκκ
=
JacobianJacobianThermal Thermal RayleighRayleigh NumberNumber SolutalSolutal RayleighRayleigh NumberNumber
PrandtlPrandtl NumberNumber Lewis NumberLewis Number
G. AhmadiME 639-Turbulence
( ) ( )*21
tXzsinxsinp22 πλπ
πλ
=ψ
( ) ( )**21
tZz2sin1tYz2sinxcosp22 π
π−π
λπ
⎟⎟⎠
⎞⎜⎜⎝
⎛=θ
( ) ( )**21
tVz2sin1tUzsinxcosp22 π
π−π
λπ
⎟⎟⎠
⎞⎜⎜⎝
⎛=Σ
G. AhmadiME 639-Turbulence
( )UrYrXX ST −+−σ=&
( )Z1XYY −+−=&
( )XYZaZ +−=&
( )V1XkUU −+−=&
( )XUkVaV +−=&
p4a
2π=
T32
2
T Rp
rλπ
=
S32
2
S Rp
rλπ
=
3
G. AhmadiME 639-Turbulence
No NoiseNo Noise With NoiseWith Noise
rrTT = 10= 10
G. AhmadiME 639-Turbulence
With Noise
rrTT = 40= 40
G. AhmadiME 639-Turbulence
With Noise
rrTT = 40= 40
G. AhmadiME 639-Turbulence
With NoiseWith Noise
rrTT = 40= 40
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G. AhmadiME 639-Turbulence
Different Different rrTT
G. AhmadiME 639-Turbulence
rrTT = 80= 80
G. AhmadiME 639-Turbulence
With NoiseWith NoiseWithout NoiseWithout Noise
rrTT = 40= 40
G. AhmadiME 639-Turbulence
(McLaughlin and (McLaughlin and OrszagOrszag, JFM, 1982), JFM, 1982)
( )θ+∇+π∇−×=∂∂ kvωvv 2Pr
t
θ∇+=θ∇⋅+∂θ∂ 2wRat
v
0=⋅∇ v
Governing EquationsGoverning Equations
5
G. AhmadiME 639-Turbulence
vω ×∇=
2v21p +=π
νκ∆β
=THgRa
3
κν
=Pr
RayleighRayleighNumberNumber
PrandtlPrandtlNumberNumber
VorticityVorticity
Pressure Pressure HeadHead
G. AhmadiME 639-Turbulence
( ) ( ) ( )∑ ∑ ∑< < =
⎟⎠⎞
⎜⎝⎛ +π
=M
21m N
21n
P
0pP
Yny
Xmxi2
z2Tet,p,n,m~t,z,y,x vv
( ) ( ) ( )∑ ∑ ∑< < =
⎟⎠⎞
⎜⎝⎛ +π
θ=θM
21m N
21n
P
0pP
Yny
Xmxi2
z2Tet,p,n,m~t,z,y,x
FourierFourier--ChebyshevChebyshev SeriesSeries
G. AhmadiME 639-Turbulence
Ra=15000 Ra=25000
G. AhmadiME 639-Turbulence
(G. Ahmadi and V.W. Goldschmidt, (G. Ahmadi and V.W. Goldschmidt, Developments in Mechanics Vol. 6, 1971)Developments in Mechanics Vol. 6, 1971)
NavierNavier--StokesStokes
f2
Lfff
f
Re1P1
tuuuu
∇+∇ρ
−=∇⋅+∂∂
0f =⋅∇ u
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G. AhmadiME 639-Turbulence
( ) ( )∑ ⋅=K
xKuxu if et,Kt,Fourier SeriesFourier Series
( ) ( )
( ) ( )
( ) ( )
( ) ( ) ⎪⎪⎪⎪
⎭
⎪⎪⎪⎪
⎬
⎫
⎪⎪⎪⎪
⎩
⎪⎪⎪⎪
⎨
⎧
−−⎥⎥⎦
⎤
⎢⎢⎣
⎡
+−
+
−−⎥⎥⎦
⎤
⎢⎢⎣
⎡
+−+
−−⎥⎥⎦
⎤
⎢⎢⎣
⎡
+−
−
+−=κκ
∂∂
∑∑
∑∑
∑∑
1̀x
1y
1̀x
1y
1̀x
1y
K K
1yy
1xx
1y
1x2
y2x
yxy
K K
1yy
1xx
1y
1x2
y2x
2x
y
K K
1yy
1xx
1y
1x2
y2x
2x
x
yxL
2y
2x
yx
t,KK,KKvt,K,KvKKKK
K
t,KK,KKvt,K,KuKK
K21K
t,KK,KKut,K,KuKK
K1K
i
t,K,KuRe
KKt,,
tu
xx--ComponentComponent
G. AhmadiME 639-Turbulence
( ) ( )
( ) ( )
( ) ( )
( ) ( )⎪⎪⎪⎪
⎭
⎪⎪⎪⎪
⎬
⎫
⎪⎪⎪⎪
⎩
⎪⎪⎪⎪
⎨
⎧
−−⎥⎥⎦
⎤
⎢⎢⎣
⎡
+−+
−−⎥⎥⎦
⎤
⎢⎢⎣
⎡
+−+
−−⎥⎥⎦
⎤
⎢⎢⎣
⎡
+−
−
⎥⎥⎦
⎤
⎢⎢⎣
⎡ +−=
∂∂
∑∑
∑∑
∑∑
1x
1y
1x
1y
1x
1y
K K
1yy
1xx
1y
1x2
y2x
2y
y
K K
1yy
1xx
1y
1x2
y2x
2y
x
K K
1yy
1xx
1y
1x2
y2x
yxx
yxL
2y
2x
yx
t,KK,KKvt,K,KvKK
K1K
t,KK,KKvt,K,KuKK
K21K
t,KK,KKut,K,KuKK
KKK
i
t,K,KvRe
KKt,K,Kv
t
( ) ( ) 0Kt,K,KvKt,K,Ku yyxxyx =⋅+⋅
yy--ComponentComponent
ContinuityContinuity
G. AhmadiME 639-Turbulence
Isotropic Isotropic TurbulenceTurbulence
G. AhmadiME 639-Turbulence
Isotropic TurbulenceIsotropic Turbulence
Longitudinal Longitudinal Lateral Lateral
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G. AhmadiME 639-Turbulence
Instantaneous Instantaneous Velocity Velocity ContoursContours
G. AhmadiME 639-Turbulence
( )Re,Nt
uu=
∂∂NavierNavier--StokesStokes
Equilibrium Equilibrium SolutionsSolutions
Time InvariantTime Invariant
TimeTime--PeriodicPeriodic
QuasiQuasi--PeriodicPeriodic
ChaoticChaotic
G. AhmadiME 639-Turbulence
BifurcationBifurcation(Supercritical)(Supercritical)
RecrRe
u Stable
Unstable
Bifurcating Bifurcating SolutionsSolutions
G. AhmadiME 639-Turbulence
HopfHopfBifurcationBifurcation
Time Time InvariantInvariant
Time Time PeriodicPeriodic
Regular Regular BifurcationBifurcation
Time Time InvariantInvariant
Time Time InvariantInvariant
Chaotic Chaotic BifurcationBifurcation
QuasiQuasi--PeriodicPeriodic
ChaoticChaotic
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G. AhmadiME 639-Turbulence
LandauLandau--HopfHopf
CountablyCountably Infinite Bifurcation Infinite Bifurcation of of NavierNavier--Stokes EquationStokes Equation
TurbulenceTurbulence
After Four Bifurcation After Four Bifurcation Solutions to Solutions to NavierNavier--Stokes Stokes Equation Become ChaoticEquation Become Chaotic
RuelleRuelle--TakensTakens
G. AhmadiME 639-Turbulence
G.I. Taylor & von G.I. Taylor & von KarmanKarman (1937)(1937)
““Turbulence is an irregular motion Turbulence is an irregular motion which in general makes its appearance which in general makes its appearance in fluids, gaseous or liquid, when they in fluids, gaseous or liquid, when they flow past solid surfaces or even when flow past solid surfaces or even when neighboring streams of the same fluid neighboring streams of the same fluid flow past or over one another.”flow past or over one another.”
G. AhmadiME 639-Turbulence
HinzeHinze (1959)(1959)
““Turbulent fluid motion is an irregular Turbulent fluid motion is an irregular condition of flow in which the various condition of flow in which the various quantities show a random variation quantities show a random variation with time and space coordinates, so with time and space coordinates, so that statistically distinct average values that statistically distinct average values can be discerned.”can be discerned.”
G. AhmadiME 639-Turbulence
Concluding RemarksConcluding Remarks44Double Diffusive ConvectionDouble Diffusive Convection44Thermal ConvectionThermal Convection44Isotropic TurbulenceIsotropic Turbulence44BifurcationBifurcation44TurbulenceTurbulence