Outline Outline 4 Definition of Turbulence 4 Definition of Turbulence 4 Features of Turbulence 4 Features of Turbulence 4 Kolmogorov 4 Kolmogorov Scales Scales 4 Energy Cascade 4 Energy Cascade ME 639-Turbulence ME 639-Turbulence G. Ahmadi G. Ahmadi Hinze (1959) Hinze (1959) G.I. Taylor & von Karman Karman (1937) (1937) G.I. Taylor & von “Turbulence is an irregular motion “Turbulent fluid motion is an irregular “ “Turbulence is an irregular motion Turbulence is an irregular motion “ “Turbulent fluid motion is an irregular Turbulent fluid motion is an irregular which in general makes its appearance condition of flow in which the various which in general makes its appearance condition of flow in which the various which in general makes its appearance condition of flow in which the various in fluids, gaseous or liquid, when they quantities show a random variation in fluids, gaseous or liquid, when they quantities show a random variation in fluids, gaseous or liquid, when they quantities show a random variation flow past solid surfaces or even when with time and space coordinates, so flow past solid surfaces or even when with time and space coordinates, so flow past solid surfaces or even when with time and space coordinates, so neighboring streams of the same fluid that statistically distinct average values neighboring streams of the same fluid neighboring streams of the same fluid that statistically distinct average values that statistically distinct average values flow past or over one another.” can be discerned.” flow past or over one another.” flow past or over one another.” can be discerned.” can be discerned.” ME 639-Turbulence ME 639-Turbulence G. Ahmadi G. Ahmadi 1
Random, Statistical High Vorticity Vortex Random, Random, Statistical Statistical High High Vorticity Vorticity Vortex Vortex Irregular,Chaotic Approach Fluctuation Stretching Irregular,Chaotic Irregular,Chaotic Approach Approach Fluctuation Fluctuation Stretching Stretching Continuum Rapid Kolmogrov Rapid Continuum Continuum Kolmogrov Rapid Kolmogrov Highly Diffusive Highly Diffusive Highly Diffusive Scale >> λ λ Phenomena Scale >> λ Mixing Phenomena Mixing Phenomena Mixing Scale >> Needs a Needs a Needs a Rotational and Rotational and Rotational and Source of Highly Dissipative Source of Highly Dissipative Source of Highly Dissipative Three-D Three Three- -D D Energy Energy Energy ME 639-Turbulence ME 639-Turbulence G. Ahmadi G. Ahmadi Manifestation of Instability of Manifestation of Instability of Manifestation of Instability of the Flow Shear Flows the Flow Shear Flows the Flow Shear Flows Buoyancy Turbulence Buoyancy Buoyancy Turbulence Turbulence Mean Field Fluid is non-Newtonian, Mean Field Fluid is non- -Newtonian, Newtonian, Mean Field Fluid is non viscoelastic, memory-dependent, viscoelastic, memory , memory- -dependent, dependent, viscoelastic multi-temperature, nonlocal, and multi- -temperature, temperature, nonlocal nonlocal, and , and multi Separation Separation Separation contains several internal variables contains several internal variables contains several internal variables ME 639-Turbulence ME 639-Turbulence G. Ahmadi G. Ahmadi 2
3 E(k) E(k) Dissipation ≅ ≅ Production ε = u Universal Equilibrium Universal Equilibrium Dissipation Production Λ , 2 ⎛ ∂ ⎞ Direct Viscous Direct Viscous 2 , U u υ ⎜ ⎟ υ ~ ⎜ ⎟ ∂ Λ 2 Dissipation ⎝ ⎠ Dissipation y 2 ⎛ ∂ ⎞ U 2 υ ⎜ ⎟ u ⎜ ⎟ υ Inertia Inertia ∂ ⎝ ⎠ y L arg e Eddy Direct Viscous Dissip . Λ 1 2 = = = Subrange Kolmogorov Subrange Kolmogorov ε 3 Turbulence Dissipatio n Rate Re u Λ Λ k k Λ u = Re Reynolds Number Reynolds Number Λ ν ME 639-Turbulence ME 639-Turbulence G. Ahmadi G. Ahmadi Kolmogorov (Universal Equilibrium Theory) (Universal Equilibrium Theory) Turbulent Motions Kolmogorov Turbulent Motions � Large-scale turbulent motion is roughly Small-scale turbulence is in equilibrium � Large � Small- -scale turbulence is in equilibrium scale turbulence is in equilibrium Small Large- -scale turbulent motion is roughly scale turbulent motion is roughly (independent of large-scale) and is controlled (independent of large- (independent of large -scale) and is controlled scale) and is controlled independent of viscosity. independent of viscosity. independent of viscosity. solely with dissipation rate, ε , and viscosity, ν . ε , and viscosity, ν . solely with dissipation rate, ε , and viscosity, ν . � The small-scale motion is controlled by solely with dissipation rate, � The small � The small- -scale motion is controlled by scale motion is controlled by viscosity. viscosity. viscosity. Length Scale Length Scale Time Scale Time Scale � Small-scale motions are statistically � Small � Small- -scale motions are statistically scale motions are statistically Velocity Scale Velocity Scale 1 1 ⎛ ⎞ ν independent of large-scale turbulent ν ⎛ ⎞ 3 independent of large- independent of large -scale turbulent scale turbulent 4 2 ( ) 4 1 ⎜ ⎟ η ≡ τ ≡ ⎜ ⎟ υ ≡ νε ⎜ ⎟ fluctuations (and/or mean motions). ε fluctuations (and/or mean motions). ε fluctuations (and/or mean motions). ⎝ ⎠ ⎝ ⎠ ME 639-Turbulence ME 639-Turbulence G. Ahmadi G. Ahmadi 3
Kolmogorov - -5/3 Law 5/3 Law Velocity Scale Kolmogorov Velocity Scale Length Scale Length Scale Time Scale Time Scale υ For eddies much smaller than the energy τ η 1 1 For eddies much smaller than the energy For eddies much smaller than the energy 3 − − − U containing eddies and much larger than η , 2 ~ Re 4 η , ~ Re containing eddies and much larger than η ~ Re 4 containing eddies and much larger than , Λ Λ Λ Λ Λ U turbulence is controlled by the dissipation turbulence is controlled by the dissipation turbulence is controlled by the dissipation rate, ε , and wave number k. ε , and wave number k. rate, ε rate, , and wave number k. For Water For Water 2 ⎡ ε ⎤ 1 W η ≈ 30 µ ε = m ⎢ ⎥ 1 ( ) 3 υ 2 5 2 ⎣ k ⎦ kg − ε k E ( k ) ~ ~ ~ 3 k 3 k k ME 639-Turbulence ME 639-Turbulence G. Ahmadi G. Ahmadi For a Pipe of Diameter d For a Pipe of Diameter d Richardson Verse Richardson Verse Energy containing Eddies Energy containing Eddies “Big whirls have little whirls that “Big whirls have little whirls that Big whirls have little whirls that “ 1 1 * * u u U − = ≈ = 8 feed on their velocity. Little whirls f ~ 20 Re 4 feed on their velocity. Little whirls l feed on their velocity. Little whirls 0 . 05 d Re 8 e l d d e e have lesser whirls, and so on to have lesser whirls, and so on to have lesser whirls, and so on to Kolmogorov Kolmogorov Eddies Eddies viscosity.” viscosity.” viscosity.” υ 2 ν * − η = U 17 u / 0 . 78 = = = 0 . 56 4 d Re f 0 . 06 Re k η 0 . 44 d Re ME 639-Turbulence ME 639-Turbulence G. Ahmadi G. Ahmadi 4
η ) Most dissipative Eddies (5 η For a pipe with d= 5cm, V= 1.8m/s and Re=10 5 5 ) Most dissipative Eddies (5 For a pipe with d= 5cm, V= 1.8m/s and Re=10 2 ν Table of eddy size and frequencies. − = * 0 . 78 l U 6 u / 20 d Re = = 0 . 56 Eddies Size Frequency f 0 . 02 Re d d 0 . 44 d Re Largest 25 mm 3.5 Hz Eddies Energy 0.6 mm 140 Hz Containing Largest Eddies Largest Eddies Eddies Most 0.125 mm 450 Hz * 1 − 2 u U = = Dissipative d / 2 8 f 0 . 4 Re L Eddies d d Kolmogorov 0.025 mm 1300 Hz Eddies ME 639-Turbulence ME 639-Turbulence G. Ahmadi G. Ahmadi ∂ Λ * u * U u Turbulence Turbulence Λ = κ ′ ′ 2 − ≈ ≈ + + Λ = = κ * y u v u y ∂ κ ν Macroscale Macroscale y y + κ + + Λ = 1 y + κ y 3 * + η = u ∂ 3 ( y ) 4 * U u ε = ′ ′ = − = Pr oduction u v ∂ κ κ 5 1 .2 2 y y y 1 2 1 .5 4 Kolmogorov Length Scale Kolmogorov Length Scale 4 0 2 1 6 2 0 0 3 8 0 η 1 1 * u ⎛ ν ⎞ η + = 3 η + = κ + 4 ( y ) 4 1 0 0 0 4 .5 4 0 0 η = ⎜ ⎟ ν ⎜ ⎟ ε ⎝ ⎠ ME 639-Turbulence ME 639-Turbulence G. Ahmadi G. Ahmadi 5
Λ + Concluding Remarks Concluding Remarks 1000 Viscous 4 Definition of Turbulence 4 Definition of Turbulence Sublayer 4 Features of Turbulence 4 10 Features of Turbulence + η 4 Kolmogorov 4 Kolmogorov Scales Scales 0.1 4 Energy Cascade 4 y + Energy Cascade 1 5 100 1000 ME 639-Turbulence ME 639-Turbulence G. Ahmadi G. Ahmadi 6
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