Outline Outline u 2 x ( ) u 2 x ( ) 4 Two 4 1 Two- - - PowerPoint PPT Presentation

SLIDE 1

1

• G. Ahmadi

ME 637-Particle II

• G. Ahmadi

ME 637-Particle II

Outline Outline

4 4Two Two-

• Point Correlation Tensor

Point Correlation Tensor 4 4Longitudinal and Lateral Longitudinal and Lateral Integral Scales Integral Scales 4 4Taylor Taylor Microscales Microscales 4 4Energy Spectrum Energy Spectrum 4 4Relations between Scales Relations between Scales 4 4Order of Magnitude Analysis Order of Magnitude Analysis

) ( u2 x ′ ) ( u

1 x

′ ) ( u

1 2 x

′

) ( u

1 1 x

′

r

x

1

x

• G. Ahmadi

ME 637-Particle II

Two Two-

• Point Correlation Tensor

Point Correlation Tensor

) ( u 2 x ′ ) ( u1 x ′ ) ( u

1 2 x

′

) ( u

1 1 x

′

r

x

1

x ) ( u ) ( u ) , ( Q

j i ij 1 1

x x x x ′ ′ =

) ( Q ) , ( Q

ij ij

r x x

1 =

Homogenous Turbulence Homogenous Turbulence

2 1

x x r − =

• G. Ahmadi

ME 637-Particle II

2 1 11

Q ) r ( f u =

) ( u ) ( u Q

1 1 11 1

x x ′ ′ =

) ( u ) ( u

2 1 2 1 2 1 1

x x ′ = ′ = u

) r ( f ) r ( f − =

r 8

f

f (r)

SLIDE 2

2

• G. Ahmadi

ME 637-Particle II

2 2 22

Q ) r ( g u =

) ( u ) ( u Q

2 2 22 1

x x ′ ′ =

) r ( g ) r ( g − =

r

8

g

g (r)

• G. Ahmadi

ME 637-Particle II

) ( f 2

2 f

′ ′ − = λ ) ( g 2

2 g

′ ′ − = λ

Lateral Lateral Microscale Microscale Longitudinal Longitudinal Microscale Microscale

2 g 2 2

r 1 ) ( g r ! 2 1 1 ) r ( g λ − ≈ + ′ ′ + = L

• G. Ahmadi

ME 637-Particle II

∫

∞

= Λ

f

dr ) r ( f

∫

∞

= Λ

g

dr ) r ( g

Longitudinal Longitudinal Macroscale Macroscale Lateral Lateral Macroscale Macroscale

• G. Ahmadi

ME 637-Particle II

2 1 1 1 E

) t , ( u ) t , ( u ) ( R u τ + ′ ′ = τ x x

) ( R 2

E 2 E

′ ′ − = τ

∫

∞

τ τ =

E E

d ) ( R T

E f

UT ≈ Λ

E f

Uτ ≈ λ

) ( R ) U ( f

E τ

≈ τ

Frozen Field Approximation Frozen Field Approximation Macroscale Macroscale Microscale Microscale

x U t ∂ ∂ − = ∂ ∂

SLIDE 3

3

• G. Ahmadi

ME 637-Particle II

Lagrangian Time Lagrangian Time Macroscale Macroscale Lagrangian Lagrangian Time Microscale Time Microscale

2 L L L L

v ) t ( v ) t ( v ) ( R ′ τ + ′ ′ = τ ) ( R 2

L 2 L

′ ′ − = τ

∫

∞

τ τ =

L L

d ) ( R T

• G. Ahmadi

ME 637-Particle II

dx x k

x k⋅ − +∞ ∞ − +∞ ∞ − +∞ ∞ −

∫ ∫ ∫

π =

i ij 3 ij

e ) ( Q 8 1 ) ( E

dk k x

x k⋅ − +∞ ∞ − +∞ ∞ − +∞ ∞ −

∫ ∫ ∫

=

i ij ij

e ) ( E ) ( Q

• G. Ahmadi

ME 637-Particle II

∫

+∞ ∞ − −

π =

1 x ik 1 2 1 1 l

dx e ) x ( f ) k ( E

1 1

u

∫

+∞ ∞ −

=

1 x ik 1 1 1 2 1

dk e ) k ( E 2 1 ) x ( f u

1 1

∫

∞

π =

1 1 1 1 2 1 1 1

dx x k cos ) x ( f 2 ) k ( E u

∫

∞

=

1 1 1 1 1 1 2 1

dk x k cos ) k ( E ) x ( f u

• G. Ahmadi

ME 637-Particle II

r f (r)

f

r

e ) r ( f

Λ −

=

k1 E(k1)

2 1 2 f f 2 1 1 1

k 1 2 ) k ( E Λ + Λ π = u

SLIDE 4

4

• G. Ahmadi

ME 637-Particle II

j i j i

x u x u ∂ ′ ∂ ∂ ′ ∂ ν = ε

) ( f u 15 ) x u ( 15

2 1 2 1 1

′ ′ ′ ν − = ∂ ′ ∂ ν = ε

2 g 2 f 2 f 2 1

15 30 u 30 λ ν = λ ν = λ ′ ν = ε

2 2

u u

2

g f

λ = λ

Energy Energy Dissipation Dissipation Isotropic Isotropic Turbulence Turbulence

• G. Ahmadi

ME 637-Particle II

Macroscopic Macroscopic Estimate Estimate

2 f

30 A λ ν = Λ = ε

2 3

u u

2 / 1 f

R A 30

− Λ

= Λ λ

1

f <<

Λ λ 1 R >> ν Λ =

Λ

u

2 / 1 g

R A 15

− Λ

= Λ λ

1 g

R A 15

− λ

= Λ λ

ν λ =

λ

u R

• G. Ahmadi

ME 637-Particle II

Taylor Taylor-

• Kolmogorov

Kolmogorov

2 / 1 4 / 1 4 / 1 4 / 1 g

R 15 R ) A 225 (

λ Λ =

= η λ τ = ν ε = λ 26 . 26 .

g

u

ε ν = υ η = τ

Kolmogorov Kolmogorov-

• Time Scale

Time Scale

• G. Ahmadi

ME 637-Particle II

Velocity Gradient Velocity Gradient

2 j i j i

) ( ~ x u x u λ ν ε = ∂ ′ ∂ ∂ ′ ∂ u

2 j i j i ij ij

) ( ~ x u x u ~ d d λ ∂ ′ ∂ ∂ ′ ∂ ′ ′ u

2

~ λ ν Λ

2 3

u u

Λ λ ν

2

~ u

Deformation Rate Tenor Deformation Rate Tenor

Viscosity Viscosity

SLIDE 5

5

• G. Ahmadi

ME 637-Particle II

Taylor/Integral Taylor/Integral Kolmogorov/Integral Kolmogorov/Integral Different Scales Different Scales

1 2 / 1

R ~ R ~

− λ − Λ

Λ λ

2 / 3 4 / 3

R ~ R ~

− λ − Λ

Λ η

2 / 1 4 / 1

R ~ R ~

− λ − Λ

λ η

3 2

λ = Λ η

Kolmogorov/Taylor Kolmogorov/Taylor

• G. Ahmadi

ME 637-Particle II

Concluding Remarks Concluding Remarks

4 4Two Two-

• Point Correlation Tensor

Point Correlation Tensor 4 4Longitudinal and Lateral Scales Longitudinal and Lateral Scales 4 4Integral Scales Integral Scales 4 4Taylor Taylor Microscales Microscales 4 4Energy Spectrum Energy Spectrum 4 4Relations between Scales Relations between Scales 4 4Order of Magnitude Analysis Order of Magnitude Analysis

• G. Ahmadi

ME 637-Particle II