Outline Outline Viscous Flow Viscous Flow Turbulence - - PowerPoint PPT Presentation

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Outline Outline Viscous Flow Viscous Flow Turbulence Turbulence Mixing Length Models Mixing Length Models One One- -Equation Models Equation Models Two Two- -Equation Models Equation Models

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SLIDE 1

1

  • G. Ahmadi

ME 637-Particle II

  • G. Ahmadi

ME 637-Particle II

Outline Outline

  • Viscous Flow

Viscous Flow

  • Turbulence

Turbulence

  • Mixing Length Models

Mixing Length Models

  • One

One-

  • Equation Models

Equation Models

  • Two

Two-

  • Equation Models

Equation Models

  • Stress Transport Models

Stress Transport Models

  • Rate

Rate-

  • Dependent Models

Dependent Models

  • G. Ahmadi

ME 637-Particle II

) ( t = ρ ⋅ ∇ + ∂ ρ ∂ u

τ f u ⋅ ∇ + ρ = ρ dt d

τ τ =

T

h e ρ + ∇ + ∇ = ρ q u : τ &

T h ) T ( > ρ − ⋅ ∇ − η ρ q &

Mass Mass Momentum Momentum Energy Energy Entropy Entropy

  • G. Ahmadi

ME 637-Particle II

Viscous Viscous Fluids Fluids Material Frame Material Frame-

  • Indifference

Indifference

) u ( G p

j , i kl kl kl

+ δ − = τ

ij ij j , i

d u ω + =

) u u ( 2 1 d

k , l l , k kl

+ =

) u u ( 2 1

k , l l , k kl

− = ω

) d ( F p

ij kl kl kl

+ δ − = τ

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SLIDE 2

2

  • G. Ahmadi

ME 637-Particle II

Navier Navier-

  • Stokes

Stokes

kl kl i , i kl

d 2 ) u p ( µ + δ λ + − = τ

2 3 > µ + λ

> µ

j j i 2 i j i j i

x x u x p ) x u u t u ( ∂ ∂ ∂ µ + ∂ ∂ − = ∂ ∂ + ∂ ∂ ρ

x u

i i =

∂ ∂

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ME 637-Particle II

Reynolds Equation Reynolds Equation

i i i

u U u ′ + =

p P p ′ + =

i i

U u = ui = ′

P p = p = ′

j j i j j i 2 . i j i j i

x u u x x U x P 1 x U U t U ∂ ′ ′ ∂ − ∂ ∂ ν + ∂ ∂ ρ − = ∂ ∂ + ∂ ∂

j i T ij

u u ′ ′ ρ − = τ

Turbulent Stress Turbulent Stress

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ME 637-Particle II

Eddy Eddy Viscosity Viscosity Mixing Mixing Length Length

dy dU v u

T T 21

ρν = ′ ′ ρ − = τ

ij k k i j j i T j i T ij

u u 3 1 ) x U x U ( u u δ ′ ′ − ∂ ∂ + ∂ ∂ ν = ′ ′ − = ρ τ

y U | y U | l2

m T 21

∂ ∂ ∂ ∂ ρ = τ

| y U | l2

m T

∂ ∂ = ν

y T v T

T T

∂ ∂ σ ν = ′ ′ −

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ME 637-Particle II

Eddy Eddy Viscosity Viscosity Free Shear Flows Free Shear Flows

l u c

T ≈

ν

Scale Length = l Scale Velocity u =

λ ∝ ν c

Kinematic Kinematic Viscosity Viscosity

Path Free Mean = λ Sound

  • f

Speed c =

m

c ~ l l

y

m

κ = l

Near Wall Flows Near Wall Flows

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SLIDE 3

3

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ME 637-Particle II

Local Equilibrium Production = Dissipation Local Equilibrium Local Equilibrium Production = Dissipation Production = Dissipation Mixing length Hypothesis Mixing length Mixing length Hypothesis Hypothesis

Short Comings of Mixing Length Short Comings of Mixing Length

  • Eddy viscosity vanishes when velocity

Eddy viscosity vanishes when velocity gradient is zero gradient is zero

  • Lack of transport of turbulence scales

Lack of transport of turbulence scales

  • Estimating the mixing length

Estimating the mixing length

  • G. Ahmadi

ME 637-Particle II

Mixing Mixing Length Length

T =

ν

y U = ∂ ∂

When When max T T

| 8 . 0 ν ≈ ν

Experiment Experiment

T =

γ

U

T

ν

T T

= γ = ν

Hot

Cold

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ME 637-Particle II

Reattachment Point Reattachment Point

Mixing Mixing Length Length

T =

γ

Experiment Experiment Maximum Heat Maximum Heat Transfer Transfer

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ME 637-Particle II

Exact k Exact k-

  • equation

equation Eddy Viscosity Eddy Viscosity

l

2 / 1 T

k cµ = ν

4 43 4 42 1 4 3 4 2 1 4 3 4 2 1 4 4 4 3 4 4 4 2 1 3 2 1

Diffusion Viscous i i j j 2 n Dissipatio j i j i

  • duction

Pr j i j i Diffusion Turbulence i i k k Transport Convective i i

2 u u x x x u x u x U u u ) P 2 u u ( u x 2 u u dt d ′ ′ ∂ ∂ ν + ∂ ′ ∂ ∂ ′ ∂ ν − ∂ ∂ ′ ′ − ρ ′ + ′ ′ ′ ∂ ∂ − = ′ ′

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4

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ME 637-Particle II

Modeled k Modeled k-

  • equation

equation

3 2 1 l 4 4 4 3 4 4 4 2 1 4 43 4 42 1

n Dissipatio 2 / 3 D

  • duction

Pr j i i j j i T Diffusion j k T j

k c x U ) x U x U ( ) x k ( x dt dk − ∂ ∂ ∂ ∂ + ∂ ∂ ν + ∂ ∂ σ ν ∂ ∂ =

  • G. Ahmadi

ME 637-Particle II

K K-

  • equation

equation

3 2 1 l 3 2 1 4 4 3 4 4 2 1

n Dissipatio 2 / 3 D

  • duction

Pr Diffusion Max

k c y U ak ) Bk ( y dt dk − ∂ ∂ + ρ τ ∂ ∂ =

Short Comings of One Short Comings of One-

  • Equation Models

Equation Models

  • Lack of transport of turbulence length scale

Lack of transport of turbulence length scale

  • Estimating the length scale

Estimating the length scale

  • G. Ahmadi

ME 637-Particle II

{

Source Secondary z n Dissipatio T 2

  • duction

Pr 2 T 1 Diffusion z T

S ] k c ) y U ( k c [ z ) y z ( y dt dz + ν − ∂ ∂ ν + ∂ ∂ σ ν ∂ ∂ = 3 2 1 43 42 1 4 3 4 2 1

  • G. Ahmadi

ME 637-Particle II

Scale Time k /

2

= l Scale Frequency / k

2 =

l

Scale Vorticity / k

2 =

l

Rate n Dissipatio x u x u

j i j i

= ∂ ′ ∂ ∂ ′ ∂ ν = ε

l k z =

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5

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ME 637-Particle II

4 4 4 3 4 4 4 2 1 4 4 3 4 4 2 1 4 3 4 2 1

n distructio Viscous l k i 2 l k i 2 stretching vortex by Generation l k l i k i Diffusion j j

x x u x x u 2 x u x u x u 2 ) ' u ( x dt d ∂ ∂ ′ ∂ ∂ ∂ ′ ∂ ν − ∂ ′ ∂ ∂ ′ ∂ ∂ ′ ∂ ν − ε ′ ∂ ∂ − = ε

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ME 637-Particle II

k E(k) , , Universal Equilibrium Universal Equilibrium Inertia Inertia Subrang Subrang e e

Kolmogorov Kolmogorov

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ME 637-Particle II

ε = ν

µ 2 T

k c

ij i j j i T j i

k 3 2 ) x U x U ( u u δ − ∂ ∂ + ∂ ∂ ν = ′ ′ −

Momentum Momentum Mass Mass

j i j i i

u u x x P 1 dt dU ′ ′ ∂ ∂ − ∂ ∂ ρ − =

x U

i i =

∂ ∂

  • G. Ahmadi

ME 637-Particle II

k k-

  • equation

equation ε ε-

  • equation

equation

3 2 1 4 4 4 3 4 4 4 2 1 4 4 3 4 4 2 1

n dissipatio

  • duction

Pr j i i j j i T Diffusion j k T j

x U ) x U x U ( ) x k ( x dt dk ε − ∂ ∂ ∂ ∂ + ∂ ∂ ν + ∂ ∂ σ υ ∂ ∂ =

3 2 1 4 4 4 4 3 4 4 4 4 2 1 4 43 4 42 1

n Distructio 2 2 Generation j i i j j i T 1 Diffusion j T j

k c x U ) x U x U ( k c ) x ( x dt d ε − ∂ ∂ ∂ ∂ + ∂ ∂ ε ν + ∂ ε ∂ σ υ ∂ ∂ = ε

ε ε ε

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6

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ME 637-Particle II

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ME 637-Particle II

k k-

  • ε

ε Model Model

  • G. Ahmadi

ME 637-Particle II

k k-

  • ε

ε Model Model

  • G. Ahmadi

ME 637-Particle II

k k-

  • ε

ε Model Model

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7

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ME 637-Particle II

Algebraic Algebraic Stress Model Stress Model

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ME 637-Particle II

Algebraic Algebraic Stress Model Stress Model

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ME 637-Particle II

  • G. Ahmadi

ME 637-Particle II

  • Eddy viscosity assumption

Eddy viscosity assumption

  • Isotropic eddy viscosity

Isotropic eddy viscosity

  • Negligible convection and diffusion

Negligible convection and diffusion

  • f turbulent shear stress
  • f turbulent shear stress
  • Absence of normal stress effects

Absence of normal stress effects

k ~ u u

j i ′

′

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8

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ME 637-Particle II

k i k k i k k i k k k i 2 i k i k i

x U u ) u u ( x u u x x x u x p 1 x u U t u ∂ ∂ ′ − ′ ′ ∂ ∂ − ′ ′ ∂ ∂ + ∂ ∂ ′ ∂ ν + ∂ ′ ∂ ρ − = ∂ ′ ∂ + ∂ ′ ∂

Fluctuation Velocity Fluctuation Velocity

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ME 637-Particle II

4 4 4 4 4 4 4 4 3 4 4 4 4 4 4 4 4 2 1 4 4 3 4 4 2 1 43 42 1 4 4 4 4 3 4 4 4 4 2 1 4 4 4 3 4 4 4 2 1

Diffusion j i k ik j jk i k j i k strain essure Pr i j j i n Dissipatio k j k i

  • duction

Pr k i k j k j k i Convection j i k k

] u u x ) u u ( ' p u u u [ x ) x u x u ( ' p x u x u 2 ] x U u u x U u u [ u u ) x U t ( ′ ′ ∂ ∂ ν − δ ′ + δ ′ ρ + ′ ′ ′ ∂ ∂ − ∂ ′ ∂ + ∂ ′ ∂ ρ + ∂ ′ ∂ ∂ ′ ∂ ν − ∂ ∂ ′ ′ + ∂ ∂ ′ ′ − = ′ ′ ∂ ∂ + ∂ ∂

−

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ME 637-Particle II

Diffusion Diffusion ) x u u u u x u u u u x u u u u ( k c u u u

l j i l k l i k l j l k i l i s k j i

∂ ′ ′ ∂ ′ ′ + ∂ ′ ′ ∂ ′ ′ + ∂ ′ ′ ∂ ′ ′ ε = ′ ′ ′ − Dissipation Dissipation

ε δ = ∂ ′ ∂ ∂ ′ ∂ ν

ij k j k i

3 2 x u x u 2

  • G. Ahmadi

ME 637-Particle II

{ }

4 4 4 4 3 4 4 4 4 2 1 4 4 4 4 4 4 4 3 4 4 4 4 4 4 4 2 1

Term Rapid i j j i 1 l m 1 m l Isotropy to turn Re i j j i 1 l m m l 1 1 i j j i

) 2 ( ji ) 2 ( ij 1 ) 1 ( ij

) x u x u ( ) x u ( ) x U ( 2 ) x u x u ( ) x u x u ( ) ( G d ) x u x u ( ' p

= ϕ + ϕ = ϕ

∂ ′ ∂ + ∂ ′ ∂ ∂ ′ ∂ ∂ ∂ + ∂ ′ ∂ + ∂ ′ ∂ ∂ ′ ∂ ∂ ′ ∂ − = ∂ ′ ∂ + ∂ ′ ∂ ρ

∫

x

x x, x

Pressure Pressure-

  • Strain

Strain

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SLIDE 9

9

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ME 637-Particle II

) k 3 2 u u )( k ( c

ij j i 1 ) 1 ( ij

δ − ′ ′ ε − = ϕ

) P 3 2 P (

ij ij ) 2 ( ji ) 2 ( ij

δ − γ − = ϕ + ϕ

) x U u u x U u u ( P

k i k i k j k i ij

∂ ∂ ′ ′ + ∂ ∂ ′ ′ − =

Return to Isotropy Return to Isotropy Rapid Term Rapid Term

Production Production

  • G. Ahmadi

ME 637-Particle II

4 4 4 4 4 4 4 4 4 3 4 4 4 4 4 4 4 4 4 2 1 43 42 1 4 4 4 4 4 3 4 4 4 4 4 2 1 3 2 1 4 4 4 3 4 4 4 2 1 4 4 4 3 4 4 4 2 1

Diffusion l j i l k l i k l j l k j l i k s effects Wall ) w ( ji ) w ( ij strain essure Pr ) 2 ( ji ) 2 ( ij ij j i 1 n Dissipatio ij

  • duction

Pr k j k i k i k j Convection j i k k

]} x u u u u x u u u u x u u u u [ k { x c ) ( ) ( ) k 3 2 u u ( k c 3 2 ] x U u u x U u u [ u u ) x U t ( ∂ ′ ′ ∂ ′ ′ + ∂ ′ ′ ∂ ′ ′ + ∂ ′ ′ ∂ ′ ′ ε ∂ ∂ + ϕ + ϕ + ϕ + ϕ + δ − ′ ′ ε − ε δ − ∂ ∂ ′ ′ + ∂ ∂ ′ ′ − = ′ ′ ∂ ∂ + ∂ ∂

−

  • G. Ahmadi

ME 637-Particle II

3 2 1 4 43 4 42 1 4 4 4 3 4 4 4 2 1 4 4 3 4 4 2 1

n Destructio 2 2 Generation k i k i 1 Diffusion i i k k Convection k k

k c x U u u k c ) x u u k ( x c ) x U t ( ε − ∂ ∂ ′ ′ ε − ∂ ε ∂ ′ ′ ε ∂ ∂ = ε ∂ ∂ + ∂ ∂

ε ε ε

Dissipation Dissipation

  • G. Ahmadi

ME 637-Particle II j i j i i j j

u u x x P 1 U ) x U t ( ′ ′ ∂ ∂ − ∂ ∂ ρ − = ∂ ∂ + ∂ ∂ = ∂ ∂

i i

x U

ε ′ ′ , P , u u , U

j i i

Mass Mass Reynolds Reynolds 11 Unknowns for 11 Unknowns for 11 Equations 11 Equations

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10

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ME 637-Particle II

Gibson and Rodi (1981) Gibson and Gibson and Rodi (1981) Rodi (1981)

  • G. Ahmadi

ME 637-Particle II

Mean Velocity and Turbulence Shear Stress Mean Velocity and Turbulence Shear Stress

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ME 637-Particle II

Turbulence Intensity Turbulence Intensity

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ME 637-Particle II

Stress Transport Model Stress Transport Model k k-

  • Equation

Equation 3 2 1 4 4 4 4 4 4 3 4 4 4 4 4 4 2 1 4 4 4 3 4 4 4 2 1 4 4 4 4 3 4 4 4 4 2 1

n Dissipatio ij Strain essure Pr ij ij ij j i 1

  • duction

Pr k i k j k j k i Diffusion m m k k s j i

3 2 ) P 3 2 P ( ) k 3 2 u u ( k c x U u u x U u u ) j u i u x u u k ( x c u u dt d ε δ − δ − γ − δ − ′ ′ ε − ∂ ∂ ′ ′ − ∂ ∂ ′ ′ − ′ ′ ∂ ∂ ′ ′ ε ∂ ∂ = ′ ′

−

{

n Dissipatio

  • duction

Pr m k m k Diffusion m m k k s

x U u u ) x k u u k ( x c dt dk ε − ∂ ∂ ′ ′ − ∂ ∂ ′ ′ ε ∂ ∂ = 4 3 4 2 1 4 4 4 3 4 4 4 2 1

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11

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ME 637-Particle II

) P ( k u u ) D dt dk ( k u u D u u dt d

j i j i ij j i

ε − ′ ′ = − ′ ′ = − ′ ′

Rodi Rodi’ ’s s Assumption Assumption

) u u x u u k ( x D

j i l l k k ij

′ ′ ∂ ∂ ′ ′ ε ∂ ∂ =

k i k j k j k i ij

x U u u x U u u P ∂ ∂ ′ ′ − ∂ ∂ ′ ′ = ) x k u u k ( x D

l l k k

∂ ∂ ′ ′ ε ∂ ∂ =

l k l k

x U u u P ∂ ∂ ′ ′ =

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ME 637-Particle II

⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ − ε + δ ε − ε γ − + δ = ′ ′ ) 1 P ( c 1 1 P 3 2 P c 1 3 2 k u u

1 ij ij 1 ij j i

ε = ν

µ 2 T

k c

2 1 1 1

)] 1 P ( c 1 1 [ )] P 1 ( c 1 1 [ c ) 1 ( 3 2 c − ε + ε γ − − γ − =

µ

Simple Shear Flow Simple Shear Flow

  • G. Ahmadi

ME 637-Particle II

  • Averaged Conservation laws

Averaged Conservation laws

  • Entropy Constraints

Entropy Constraints

  • Thermodynamics of Turbulence

Thermodynamics of Turbulence

  • Constitutive Equations

Constitutive Equations

  • Rate Dependent Model

Rate Dependent Model

  • Model Predictions

Model Predictions

  • Comparison with Experimental Data

Comparison with Experimental Data

  • G. Ahmadi

ME 637-Particle II

} d d d d 3 1 d d d d Dt D ˆ d 2 { k 3 2 t

kj ik ij kl lk ij kl lk 2 ij ij T ij T ij

⎥ ⎦ ⎤ ⎢ ⎣ ⎡ − δ βτ + γτ + ατ + µ + δ ρ − =

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SLIDE 12

12

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ME 637-Particle II ki jk kj ik ij ij

d d d Dt d D ˆ ω + ω + = &

( )

i , j j , i ij

v v 2 1 d + =

) v v ( 2 1

i , j j , i ij

− = ω

ij ijd

d 2 1 = ∆

i , T i

C Q θ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ σ µ + κ =

θ

⎥ ⎦ ⎤ ⎢ ⎣ ⎡ τ τ − ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ σ µ + µ =

i , i , k T i

k k K

2 2

C θ ∂ ψ ∂ θ − =

Jaumann Jaumann Derivative Derivative Heat Flux Heat Flux

Heat Capacity Heat Capacity

Energy Flux Energy Flux

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ME 637-Particle II

v i

, i =

i j , ij kl lk 2 kj ik ij kl lk ij T ij T i , i

f ]} d d d ) d d d d 3 1 ( Dt d D ˆ [ d ) ( 2 { k 3 2 p v ρ + γτ + − δ βτ + ατ µ + µ + µ + ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ρ + − = ρ&

  • G. Ahmadi

ME 637-Particle II

r d d 2 ) C ( C

ij ij i , i , T

+ ρε + µ + ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ θ σ µ + κ = θ ρ

θ

&

ρε − ατµ + + ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ τ τ − σ µ + µ = ρ

ij ij T i , i , i , k T

d Dt d D ˆ P ) k k )( ( k &

] ) d d ( d d d d d 2 [ P

2 ji ij 2 ij kj ik ij ij T

γτ + βτ − µ =

  • G. Ahmadi

ME 637-Particle II

k C C ) k ]( k C 2 C 2 [ ] k k ][ k k [ k C P k C ) (

2 2 i , i , 2 2 2 2 T i , i , i , i , 2 k T i , i , T

2 3 1

ε ρ − ε ε ∆ ε ∆ α + α α ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ σ µ + µ + ε ε − ε ε − ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ε ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ σ µ + µ + ε + ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ε σ µ + µ = ε ρ

ε ε µ µ ε ε ε ε

&

ε ρ = µ

µ 2 T

k C

ε = τ k 93 . = α 54 . = β

005 . = γ 09 . C =

µ

45 . 1 C 1 =

ε

92 . 1 C 2 =

ε

1

k =

σ 3 . 1 = σε

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13

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ME 637-Particle II

Mean velocity Mean velocity Mean velocity Axial turbulence intensity Axial turbulence Axial turbulence intensity intensity

Comparison are with the experimental data of Kreplin and Eckelmann and DNS of Kim et al. Comparison are with the experimental data of Kreplin Comparison are with the experimental data of Kreplin and Eckelmann and DNS of Kim et al. and Eckelmann and DNS of Kim et al.

  • G. Ahmadi

ME 637-Particle II

Vertical turbulence intensity Vertical turbulence Vertical turbulence intensity intensity Lateral turbulence intensity Lateral turbulence Lateral turbulence intensity intensity

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ME 637-Particle II

Turbulence shear stress Turbulence shear stress Turbulence shear stress

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ME 637-Particle II

Mean Velocity Profiles Mean Velocity Profiles Mean Velocity Profiles

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ME 637-Particle II

Turbulence Kinetic Energy Profiles Turbulence Kinetic Energy Profiles Turbulence Kinetic Energy Profiles

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ME 637-Particle II

Axial Turbulence Intensity Profiles Axial Turbulence Intensity Profiles Axial Turbulence Intensity Profiles

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ME 637-Particle II

Vertical Turbulence Intensity Profiles Vertical Turbulence Intensity Profiles Vertical Turbulence Intensity Profiles

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ME 637-Particle II

Turbulence Shear Stress Profiles Turbulence Shear Stress Profiles Turbulence Shear Stress Profiles

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ME 637-Particle II

  • Available models can predict the mean

Available models can predict the mean flow properties with reasonable accuracy. flow properties with reasonable accuracy. ‘ ‘ First First-

  • order modeling is reasonable when
  • rder modeling is reasonable when

turbulence has a single length and velocity turbulence has a single length and velocity scale. scale. ‘ ‘ The k The k-

  • ε

ε model gives reasonable results model gives reasonable results when a scalar eddy viscosity is sufficient. when a scalar eddy viscosity is sufficient. ‘ ‘ The stress transport models have the The stress transport models have the potential to be most accurate. potential to be most accurate.

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ME 637-Particle II

‘ ‘Adjustments of coefficients are needed. Adjustments of coefficients are needed. ‘ ‘The derivation of the models are arbitrary. The derivation of the models are arbitrary. ‘ ‘There is no systematic method for There is no systematic method for improving a model when it loses its improving a model when it loses its accuracy. accuracy. ‘ ‘Models for complicated turbulent flows Models for complicated turbulent flows are not available. are not available. ‘ ‘Realizability Realizability and other fundamental and other fundamental principles are sometimes violated. principles are sometimes violated.

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ME 637-Particle II