choas theory1

04/08/2304/08/23 Lecture 14 - Fundamentals of TurbulencLecture 14 - Fundamentals of Turbulence + supplemental materiale + supplemental material

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Boundary Layer ClimatologyBoundary Layer ClimatologyFundamentals of Turbulence (Arya CH 8)Fundamentals of Turbulence (Arya CH 8)

+ supplemental material+ supplemental material

Theodore von

Kármán 1881-1963

Osborne Reynolds

1842-1912

Lewis Fry Richardson1881-1953

Lecture 14

Universität Stuttgart


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Big whorls have little whorls,Big whorls have little whorls,Which feed on their velocity,Which feed on their velocity,And little whorls have lesser whorls,And little whorls have lesser whorls,And so on to viscosity.And so on to viscosity.- Lewis Fry Richardson (1922)Lewis Fry Richardson (1922)

- ‘‘energy cascade’ from higher to lower order, energy cascade’ from higher to lower order, from low to high entropy, ending in viscous from low to high entropy, ending in viscous dissipation.dissipation.

A whorl or swirl


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Turbulent development from laminar flow through a grid after Frisch (1995)


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Laminar to Turbulent in Surface Laminar to Turbulent in Surface LayerLayer

(Oke, 1987)


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The Reynolds NumberThe Reynolds Numberdimensionless ratio of the inertial force to the viscous dimensionless ratio of the inertial force to the viscous force in the Navier–Stokes equationsforce in the Navier–Stokes equations– U is velocityU is velocity– l is length (length scale)l is length (length scale)– is the kinematic viscosity of the fluidis the kinematic viscosity of the fluid

RRee is of great importance in the theory of hydrodynamic is of great importance in the theory of hydrodynamic stability and the origin of turbulence. stability and the origin of turbulence. The inertia force generates vortex stretching and The inertia force generates vortex stretching and nonlinear interactions and hence creates randomness. nonlinear interactions and hence creates randomness. Turbulence occurs when the inertial term dominates the Turbulence occurs when the inertial term dominates the viscous term, that is, when the Reynolds number is large. viscous term, that is, when the Reynolds number is large.

For many engineering flows, turbulence occurs when RFor many engineering flows, turbulence occurs when Ree > R> Recec, where the critical Reynolds number is roughly R, where the critical Reynolds number is roughly Recec = = 2100. 2100.

UlRe

Adapted from AMS glossary of meteorologyAdapted from AMS glossary of meteorology


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The “Butterfly effect”The “Butterfly effect”

sensitive dependence sensitive dependence on initial conditionson initial conditions

in 1960s, computers in 1960s, computers were slow, so MIT were slow, so MIT meteorologist Edward meteorologist Edward Lorenz made a Lorenz made a shortcut and shortcut and discovered the discovered the butterfly effectbutterfly effect


7740 years later, Hollywood discovers the butterfly effect.


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Animation: U California Irvine Source: Turner J.S., 1973, Buoyancy effects in fluids Cambridge University press. pp 368

Shear instabilityShear instability


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from (Stull, 1988)


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Reynolds DecompositionReynolds DecompositionFluctuating winds Fluctuating winds include mean (u-hat) include mean (u-hat) and fluctuating (u’) and fluctuating (u’) components.components.

'uuu

from (Stull, 1988)


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from Fritsch (1995) Turbulence, Cambridge Univ. Press


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Mean and Turbulent Kinetic EnergyMean and Turbulent Kinetic Energy

222

2

1wvuMKE

222 '''2

1wvuTKE


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Types of ‘Instabilities’Types of ‘Instabilities’1.1. Buoyant or ‘gravitational’ instabilityBuoyant or ‘gravitational’ instability

– i.e., i.e., vv decreasing with height decreasing with height

2.2. Shear instabilityShear instability3.3. Dynamic or hydrodynamic instabilityDynamic or hydrodynamic instability

– Stable if perturbations are found to decay with Stable if perturbations are found to decay with time (or distance) in the direction of the flow.time (or distance) in the direction of the flow.

– Unstable if the perturbations grow in space or Unstable if the perturbations grow in space or time.time.

– In unstable case, perturbations continually grow In unstable case, perturbations continually grow no mater how small they were initially.no mater how small they were initially.

These instabilities are These instabilities are IrreversibleIrreversible


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Turbulence generation and Turbulence generation and maintenancemaintenance

Hydrodynamic Hydrodynamic instability (H)instability (H)

Buoyancy Production Buoyancy Production (B)(B)

shear production (S)shear production (S)

Dissipation (D)Dissipation (D)

Transport (T)Transport (T)

HTDBSdtTKEd /


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Review of conceptsReview of conceptsThe butterfly effectThe butterfly effect

Types of instabilities in turbulenceTypes of instabilities in turbulence

Turbulence maintenanceTurbulence maintenance

Meaning of terms in RMeaning of terms in R ii equation equation

Reynolds DecompositionReynolds Decomposition

MKE and TKEMKE and TKE

choas theory1

Technology

reynolds number

reynolds decomposition

hydrodynamic

butterfly

turbulence

whorls

turbulence

instabilities