X.S. Bai Turbulence

Lecture 7

Turbulence

X.S. Bai Turbulence

Content

• Basic features of turbulence

• Energy cascade theory

• Turbulence scales

• Turbulence mixing

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Basic features of turbulence

What is turbulence?

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spiral galaxies NGC 2207 and IC 2163

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Turbulent jet flow

Volcano jet flow

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Clouds: turbulence in atmosphere

Large eddies and small eddies

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Water flow

Motor oil floating on

water runoff from a

street after a snowstorm

Large eddies and small eddies

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Turbulence wake flow

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Jet flow in gas turbine combustor

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Turbulent flames

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Origin of turbulence

• Shear layer instability – Kelvin – Helmholtz instability

• Transition from laminar to turbulent

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Clouds: turbulence in atmosphere

Kelvin-Helmholtz instability

Large eddies and small eddies

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A turbulent jet flow

3D eddies

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Turbulent jet flow

Direct numerical simulations

K-H instability

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Turbulent jet flow

PLIF imaging

laminar

turbulent

K-H instability

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Turbulent jet flow

Direct numerical simulations

laminar

turbulent

K-H instability

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Wall generates turbulence

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Near wall turbulence

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Basic features of turbulence

– Irregularity

• Statistical approaches, not deterministic approaches

• unsteady, random

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O. Reynolds experiment (1880s)

Osborne Reynolds

Born: 23 Aug 1842 in Belfast, Ireland

Died: 21 Feb 1912 in Watchet,

Somerset, England

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Reynolds experiment

Reynolds number

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Reynolds number and turbulence scales

Lower Re

Higher Re

Large eddy

Small eddy

Large scales: independent of Re

Small scales: dependent of Re

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The nature of turbulence: what is turbulence?

– High Reynolds numbers • Instability of laminar flows

• Non-linearity and randomness make the problem nearly intractable

– Three dimensional vorticity fluctuations • The large scale structure in tornado is not mainly

turbulent!

• Vortex stretching mechanism

– Dissipation • Energy transfer from large scales to small scale and

dissipated to heat

– Turbulence is continuum flow!

Basic features of turbulence

X.S. Bai Turbulence

What is turbulence?

• Turbulence is flow

• Turbulence is three dimensional high Reynolds number flow

• Turbulence is a random, chaotic, and irregular flow

• Turbulence is made up of 3D eddies of various size

• Turbulence is dissipative

• Turbulence is effective in transferring of mass and heat

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– Diffusivity

• Enhanced mixing & heat and momentum transfer

– Wind/ocean current momentum transfer

– Coffee/milk mixing

– Fuel/air mixing

• Prevent separation of golf ball, airfoil

Basic features of turbulence

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Other examples

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7.2 Energy cascade of turbulence

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Turbulence eddies

• Richardson’s definition of turbulence eddies (1922)

– Turbulence consists of different eddies

– An eddy is a localized flow structure

– Large eddies consists of small eddies

Lewis Fry Richardson

1881-1953

Big whirls have little whirls that feed on their velocity,

and little whirls have lesser whirls and so on to viscosity.

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Breakdown of large eddies

• Due to the non-linearity of the N-S eqaution (convective transport) large eddies tend to breakdown to smaller eddies

• If the viscous term is very large then the breakdown process is prevented: viscous terms tend to stabilize the flow (think about a very sticky fluid)

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Small scale in turbulence

• Kolmogorov eddies

– the smallest eddy must be at the length and velocity scale at

which viscous term is at least as important as the non-linear

convective term

– Physically, viscous damping should be so strong that any

velocity gradient would be quickly smooth away

Andrey Kolmogorov

1903 - 1987

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Energy Cascade

heat

inlet, other

boundaries

Energy transfer at a

‘constant’ rate e

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Energy cascade theory

• Kolmogorov’s universal equilibrium theory

– Large eddies are not affected by viscosity

– Large eddies transfer energy to small eddies

– The rate of energy transfer from large eddies must be in the

order of energy dissipation from the small eddies to heat,

since otherwise the energy at small eddies will be

accumulating or dying out!

– Since no accumulation of energy at any of the scales, one

may assume the energy transfer rate is the same at all scales

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7.2 Turbulence eddy scales

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Taylor’s estimation of the energy transfer rate

(1935)

• Large eddy size

• Large eddy speed

• one eddy turnover time

• Large eddy kinetic energy

• Large eddy loses a significant

fraction of their kinetic energy

within one eddy turnover time -

Energy transfer rate

u0

2

t0

µl

0

u0

e µu

0

2

t0

µu

0

3

l0

0l

u0

Sir Geoffrey Ingram Taylor

1886 - 1975

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Energy cascade

• Integral scale

– length: l0

– velocity: u0=IU, I=turbulent intensity, U=mean velocity

– time: t0=l0/u0

– Reynolds number: Rel=u0l0/n

• Kolmogrov scale

– length: lk

– velocity: uk

– time: tk=lk/uk

– Reynolds number: Rek=uklk/n=1 !!!

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Energy cascade

• Dissipation rate of turbulent kinetic energy e is constant at all

scales (Cascade assumption). At smallest scale, it is estimated as

2

22

32

,

,,

k

kkk

k

k

k

kk

k

k

l

ultor

l

u

u

lt

t

u

nen

ee

eµu

0

2

t0

, t0

µl0

u0

, eµu

0

3

l0

\u

0

uk

µl0

lk

æ

èçç

ö

ø÷÷

1/3

,t

0

tk

µl0

u0

uk

lk

µl0

lk

æ

èçç

ö

ø÷÷

2/3

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Scale relations

• At the viscous scale (Kolmogorov scale) convection term is in the

same order as diffusion term, i.e., Rek=1, so ...

eµ nu

k

2

lk

2µ n

uk

2

lk

2

n2

uk

2lk

2µ

n3

lk

4

or eµu

0

3

l0

µu

0

3

l0

n

u0l0

æ

èçç

ö

ø÷÷

3

u0l0

n

æ

èç

ö

ø÷

3

µn3

l0

4Re

l

3

\l0

lk

µ Rel

3/4 ,u

0

uk

µ Rel

1/4 ,t

0

tk

µ Rel

1/2

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Turbulent jet flows

Lower Re

Higher Re

Large eddy

Small eddy

3 / 40 Rel

k

l

l

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7.2 Turbulence mixing

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Turbulence mixing

u0

l0

uk, lk

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Turbulence mixing

u0

l0

uk, lk

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Turbulence mixing

u0

l0

Large eddy passing time

= l0/u0

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Turbulence mixing

u0

l0

Large eddy passing time

= l0/u0

Molecular mixing time

= l0/(D/l0)

= l02/D

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Turbulence mixing rate

u0

l0

Large eddy passing time = l0/u0

uk, lk

Kolmogorov eddy turnaround

time = lk/uk<<l0/u0 = large eddy

passing time

Molecular mixing time = lk2/D

= lk2/(uklk) = Kolmogorov eddy

turnaround time

It implies that as soon as the entire

large eddy has past the surface it

would be mixed with the local

material on the molecular scale

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Turbulence mixing

u0

l0

Mixing time of

An entire large eddy = l0/u0

uk, lk

Kolmogorov eddy turnaround

time = lk/uk<< large eddy passing

time

Molecular mixing time = lk2/D

= lk2/(uklk) = Kolmogorov eddy

turnaround time

Turbulence mixing time = l0/u0

= l02/(u0l0) = l0

2/Dt

Turbulence eddy diffusion coefficient: Dt>>D

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Estimation of turbulence mixing

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Mixing by molecule motion

• Mixing of milk and coffee

H

Molecular diffusion coefficient

nµaxµ10-6m2 / s

Coffee height H=0.06 m

Mixing time t=H2/n

= 1 hour

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Mixing by turbulent motion

• Mixing of milk and coffee

H

Molecular diffusion coefficient

ntµu

0l0µ10-3m2 / s

Coffee height H=0.06 m

Mixing time t=H2/nt

= 3.6 s

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Summary of key issues

• Concepts

– turbulent eddies

– Energy cascade

• Key parameters

– integral scales

– Kolmogrov scales

– Reynolds number

– Damköhler number

– Karlovitz number

– eddy viscosity