Scales of Turbulent Flow

TURBULENCE: THEORY AND MODELING

What is a ’scale’ ?

Turbulent scales?

• What are the ’turbulent

scales’?

• How to ’detect’?

• How big?

• How small?

• How they evolve?

Lewis Fry Richardson (1881-1953)

• Weather forcasting

• Mathematical analysis of war

• Length of coastlines and

borders

Big whirls have little whirls

Which feed on their velocity

Little whirls have lesser whirls

And so on to viscosity

– in the molecular sense

What is an eddy/whirl?

• How to visualize?

– U, p, w

l2, Q

An eddy eludes precise definition,

but is conceived to be a turbulent

motion localized within a region of

size l, that is at least moderately

coherent over this region. The

region occupied by a large eddy can

also contain smaller eddies.

S.B. Pope

What is an eddy?

Artificial velocity field: uconv + ubig_eddy + usmall_eddy

Artificial vorticityfield.

Autocorrelation functions

• “…at least moderately

coherent…”

• Assume

– Isotropic

– No mean flow

tututR jiij ,,,, rxxxr

U1

U2

U3

x X+r

k, e, Taylor-Green vortices

Longitudinal/Transversal

autocorrelation functions

longitudinal

transversal

Longitudinal autocorrelation function

Transversal autocorrelation function

U1

U2

U3

x X+r

Longitudinal autocorrelation function

• Symmetric

• f’(r)=0

• f’’(r)<0

• How big/small are the

eddies?

– Integral

– Taylor

– Kolmogorov

Integral lengthscales

L11

Taylor lengthscales

• Physical interpretation unclear

• Can be used to estimate

dissipation rate

• Useful in defining a Reynolds

number of universal character

lf

Andrey Nikolaevich Kolmogorov

(1903-1987)

• 5 years old:

• Probability theory

• Statistical theory to artillery

fire

• Turbulence

– 3 hypotheses

1=12

1+3=22

1+3+5=32

1+3+5+7=42

Kolmogorov’s hypothesis of local

isotropy (the 0th hypothesis)

• At sufficiently

high Reynolds

number, the small

scale turbulent

motions are

statistically

isotropic.

The MIT lecture movie can be found at the following link:

https://www.youtube.com/watch?v=1_oyqLOqwnI

Kolmogorov’s first similarity

hypothesis

• In every turbulent flow,

at sufficiently high

Reynolds number, the

statistics of the small

scale motions have a

universal form and are

uniquely determined by

n and e.

Kolmogorov’s second similarity

hypothesis

• In every turbulent flow, at sufficiently high Reynolds

number, there is a range of scales, much smaller than the

largest scales and much larger than the smallest scales,

where the statistics of the motions have a universal form

and are uniquely determined by e idependent of n.

Kolmogorov microscales

• Scales

– Integral

– Taylor

– Kolmogorov

• What happens in-

between?

– Any rule?

Length

Time

Velocity

Reynolds number

Fourier transform

• Determine

frequency/wavenumber

-10

-8

-6

-4

-2

0

2

4

6

8

10

0 50 100 150 200 250 300

Func

0

200

400

600

800

1000

1200

1400

0 0.02 0.04 0.06 0.08 0.1

Ampl

Energy spectrum

• Velocity spectrum tensor

• Two-point correlation

• Wave number vector

• Wave length

κ

κ

l

2

Φ𝑖𝑗 𝜅 =1

(2𝜋)3

−∞

∞

𝑅𝑖𝑗(𝒓)𝑒−𝑖𝜅𝒓𝑑𝒓

𝑅𝑖𝑗 𝒓 =

−∞

∞

Φ𝑖𝑗(𝜅)𝑒−𝑖𝜅𝒓𝑑𝜅

Energy spectrum

• Energy spectrum function

• Simpler than F

• Directional information

removed

• One dimensional spectra:

F κκ dκtκtκE ii ,,

iiii uutRκdtκE

2

1,0

2

1,

0

Turbulent kinetic

energy

Energy-containing

rangeUniversal equilibrium range

Inertial subrange Dissipation

range

Kolmogorov

Hypotheses!

Effect of Reynolds number

Fake vs. real

How do vortices evolve?

Vorticity transport equation

Vorticity: Levi-Civita epsilon:

Taking the curl of the Navier-Stokes equations one gets:

j

ij

jj

i

j

ij

i

x

u

xxxu

t

w

wn

ww 2

j

kijki

x

u

ew

3

2

2

311

x

u

x

u

x

u

j

kjk

ew

equal are indices twoif 0

npermutatio oddan isijk if 1

npermutatioeven an isijk if 1

ijke

What is

what?

Beware missing terms!

Vorticity transport equation

j

ij

jj

i

j

ij

i

x

u

xxxu

t

w

wn

ww 2

It can be shown that

𝜔𝑗Ω1𝑗 = 𝜔1Ω11 +𝜔2 Ω12 + 𝜔3Ω13

=𝜀2𝑗𝑘𝜕𝑢𝑘

𝜕𝑥𝑗Ω12 + 𝜀3𝑗𝑘

𝜕𝑢𝑘

𝜕𝑥𝑗Ω13

= 𝜀213𝜕𝑢3

𝜕𝑥1+ 𝜀231

𝜕𝑢1

𝜕𝑥3

1

2

𝜕𝑢1

𝜕𝑥2−𝜕𝑢2

𝜕𝑥1+

𝜀312𝜕𝑢2

𝜕𝑥1+ 𝜀321

𝜕𝑢1

𝜕𝑥2

1

2

𝜕𝑢1

𝜕𝑥3−𝜕𝑢3

𝜕𝑥1=0

Vorticity transport equation

j

ij

jj

i

j

ij

i

x

u

xxxu

t

w

wn

ww 2

In 2D:

3,0,0 ww j

000

02

1

02

1

2

2

1

2

2

1

1

2

2

1

1

1

x

u

x

u

x

u

x

u

x

u

x

u

Sij

0ijjSw

Vortex stretching