Outline Outline 4 Definition of Turbulence 4 Definition of - - PowerPoint PPT Presentation

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• G. Ahmadi

ME 639-Turbulence

• G. Ahmadi

ME 639-Turbulence

Outline Outline 4 4Definition of Turbulence Definition of Turbulence 4 4Features of Turbulence Features of Turbulence 4 4Kolmogorov Kolmogorov Scales Scales 4 4Energy Cascade Energy Cascade

• G. Ahmadi

ME 639-Turbulence

G.I. Taylor & von G.I. Taylor & von Karman Karman (1937) (1937) “Turbulence is an irregular motion which in general makes its appearance in fluids, gaseous or liquid, when they flow past solid surfaces or even when neighboring streams of the same fluid flow past or over one another.” “ “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. Ahmadi

ME 639-Turbulence

Hinze Hinze (1959) (1959) “Turbulent fluid motion is an irregular condition of flow in which the various quantities show a random variation with time and space coordinates, so that statistically distinct average values can be discerned.” “ “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.”

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ME 639-Turbulence

Random, Irregular,Chaotic Random, Random, Irregular,Chaotic Irregular,Chaotic Statistical Approach Statistical Statistical Approach Approach Highly Diffusive Highly Diffusive Highly Diffusive Rapid Mixing Rapid Rapid Mixing Mixing Needs a Source of Energy Needs a Needs a Source of Source of Energy Energy Highly Dissipative Highly Dissipative Highly Dissipative

• G. Ahmadi

ME 639-Turbulence

High Vorticity Fluctuation High High Vorticity Vorticity Fluctuation Fluctuation Vortex Stretching Vortex Vortex Stretching Stretching Continuum Phenomena Continuum Continuum Phenomena Phenomena Kolmogrov Scale >> λ Kolmogrov Kolmogrov Scale >> Scale >> λ λ Rotational and Three-D Rotational and Rotational and Three Three-

• D

D

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ME 639-Turbulence

Manifestation of the Flow Manifestation of Manifestation of the Flow the Flow Mean Field Fluid is non-Newtonian, viscoelastic, memory-dependent, multi-temperature, nonlocal, and contains several internal variables Mean Field Fluid is non Mean Field Fluid is non-

• Newtonian,

Newtonian, viscoelastic viscoelastic, memory , memory-

• dependent,

dependent, multi multi-

• temperature,

temperature, nonlocal nonlocal, and , and contains several internal variables contains several internal variables

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ME 639-Turbulence

Instability of Shear Flows Instability of Instability of Shear Flows Shear Flows

Turbulence Turbulence Turbulence

Buoyancy Buoyancy Buoyancy Separation Separation Separation

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ME 639-Turbulence

k k E(k) E(k) , , Universal Equilibrium Universal Equilibrium Inertia Inertia Subrange Subrange Kolmogorov

Kolmogorov

• G. Ahmadi

ME 639-Turbulence

Λ = ε

3

u

Dissipation Dissipation ≅ ≅ Production Production Direct Viscous Direct Viscous Dissipation Dissipation

2 2 2

~ y U Λ υ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ υ u

Λ

= Λ Λ υ = = ε ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ υ Re 1 Rate n Dissipatio Turbulence . Dissip Viscous Direct Eddy e arg L y U

3 2 2 2

u u

ν Λ =

Λ

u Re

Reynolds Number Reynolds Number

• G. Ahmadi

ME 639-Turbulence

Large-scale turbulent motion is roughly independent of viscosity. The small-scale motion is controlled by viscosity. Small-scale motions are statistically independent of large-scale turbulent fluctuations (and/or mean motions).

• Large

Large-

• scale turbulent motion is roughly

scale turbulent motion is roughly independent of viscosity. independent of viscosity.

• The small

The small-

• scale motion is controlled by

scale motion is controlled by viscosity. viscosity.

• Small

Small-

• scale motions are statistically

scale motions are statistically independent of large independent of large-

• scale turbulent

scale turbulent fluctuations (and/or mean motions). fluctuations (and/or mean motions).

Turbulent Motions Turbulent Motions

• G. Ahmadi

ME 639-Turbulence

Small-scale turbulence is in equilibrium (independent of large-scale) and is controlled solely with dissipation rate, ε, and viscosity, ν. Small Small-

• scale turbulence is in equilibrium

scale turbulence is in equilibrium (independent of large (independent of large-

• scale) and is controlled

scale) and is controlled solely with dissipation rate, solely with dissipation rate, ε ε, and viscosity, , and viscosity, ν ν. . Kolmogorov Kolmogorov (Universal Equilibrium Theory) (Universal Equilibrium Theory)

4 1 3

⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ε ν ≡ η

2 1

⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ε ν ≡ τ

( )4

1

νε ≡ υ

Length Scale Length Scale Time Scale Time Scale Velocity Scale Velocity Scale

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ME 639-Turbulence

Length Scale Length Scale Time Scale Time Scale Velocity Scale Velocity Scale

4 3

Re ~

− Λ

Λ η

2 1

Re ~ U

− Λ

Λ τ

4 1

Re ~ U

− Λ

υ

kg W 1 = ε

For Water For Water

m 30µ ≈ η

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ME 639-Turbulence

Kolmogorov Kolmogorov -

• 5/3 Law

5/3 Law For eddies much smaller than the energy containing eddies and much larger than η, turbulence is controlled by the dissipation rate, ε, and wave number k. For eddies much smaller than the energy For eddies much smaller than the energy containing eddies and much larger than containing eddies and much larger than η η, , turbulence is controlled by the dissipation turbulence is controlled by the dissipation rate, rate, ε ε, and wave number k. , and wave number k.

3 5 3 2 2 3 1 2 k

k ~ k ) k ( ~ k ~ ) k ( E

−

ε ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ε υ

• G. Ahmadi

ME 639-Turbulence

“Big whirls have little whirls that feed on their velocity. Little whirls have lesser whirls, and so on to viscosity.” “ “Big whirls have little whirls that Big whirls have little whirls that feed on their velocity. Little whirls feed on their velocity. Little whirls have lesser whirls, and so on to have lesser whirls, and so on to viscosity.” viscosity.” Richardson Verse Richardson Verse

• G. Ahmadi

ME 639-Turbulence

8 1 e

Re d 05 .

−

= l

d U 4 Re d u 20 u ~ f

8 1 * e * e

≈ = l Energy containing Eddies Energy containing Eddies

78 .

Re d 4

−

= η

44 . 2 * 56 . k

Re / u 17 Re d U 06 . f ν = = η υ =

Kolmogorov Kolmogorov Eddies Eddies For a Pipe of Diameter d For a Pipe of Diameter d

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ME 639-Turbulence

Most dissipative Eddies (5 Most dissipative Eddies (5η η) ) Largest Eddies Largest Eddies

78 . d

Re d 20

−

= l

44 . * 56 . d

Re / u 6 Re d U 02 . f

2 ν

= =

8 1 * L

Re d U 4 . d u 2 f

−

= =

2 / d

• G. Ahmadi

ME 639-Turbulence

Table of eddy size and frequencies. Eddies Size Frequency Largest Eddies 25 mm 3.5 Hz Energy Containing Eddies 0.6 mm 140 Hz Most Dissipative Eddies 0.125 mm 450 Hz Kolmogorov Eddies 0.025 mm 1300 Hz

For a pipe with d= 5cm, V= 1.8m/s and Re=10 For a pipe with d= 5cm, V= 1.8m/s and Re=105

5

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ME 639-Turbulence

2 *

u v u ≈ ′ ′ −

y u y U

*

κ ≈ ∂ ∂

y u y U v u

• duction

Pr

3 *

κ = ∂ ∂ ′ ′ − =

y u

3 *

κ = ε

4 1 3

⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ε ν = η ν η = η+

*

u

4 1

) y (

+ +

κ = η

Kolmogorov Length Scale Kolmogorov Length Scale

• G. Ahmadi

ME 639-Turbulence

y κ = Λ

+ +

κ = ν Λ = Λ y u*

Turbulence Turbulence Macroscale Macroscale

+

y

4 1

) y (

+ + κ

= η

+ + κ

= Λ y 5 1 .2 2 1 2 1 .5 4 4 2 1 6 2 3 8 1 4 .5 4

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ME 639-Turbulence

y+

Viscous Sublayer

5 1 100 1000 10 0.1 1000

+

η

+

Λ

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ME 639-Turbulence

Concluding Remarks Concluding Remarks

4 4Definition of Turbulence Definition of Turbulence 4 4Features of Turbulence Features of Turbulence 4 4Kolmogorov Kolmogorov Scales Scales 4 4Energy Cascade Energy Cascade