turbulence in geophysical flows lecture 1connected with the brownian motion of the particles, so...

Turbulence in Geophysical Flows

Lecture 1

Osborne Reynolds (1842-1912)“An experimental investigation of the

circumstances which determine whether

the motion of water shall be Direct or

Sinuous, and of the law of resistance in

parallel channels” Philosophical

Transactions, 1883

Turbulence

• “Sinuous” motion (Reynolds 1883).

• Continuous state of instability (Monin & Yaglom

1961); proposed by Lev Landau, according to

which flow undergoes an infinite sequence of

bifurcations (instabilities) before it becomes

unpredictable and chaotic.

• Irregular motion in space and time, which is not

connected with the Brownian motion of the

particles, so that average statistical values can

be discerned (Hinze 1956; Corrsin 1961).

• Irregular motion of fluid that makes appearance

during flow over boundaries or presence of

velocity shear (Karman 1937; Taylor 1937)

Statistical Averaging

Ensemble Averaging:

N

1=n N

)t,x(u~

t,xU `n

~

t,xUt,xut,x'~

~u

21N

1

221

2 t)(x,U-t)x,N

1(u~ut,xu n

Ensemble Average

Fluctuation

rms.

Conduct a large number (N) of identical experiments (realizations)

“On the Dynamical Theory of Incompressible Viscous Fluids and

Determination of the Criterion,” Phil Trans., Reynolds,1894

Stationary & Homogenous Turbulence

;),~~

~ x(U=t)x(U0=t

t),xU(

;(t)U=t)x(U0=t),x(Ux ~~j

,

)+tx(dT

lim=)x(U

T

-T~T~,

~1u

2

1x

x-~~0x

t),r+x(drx

lim=(t)U~

u

Ergodicity – all converge to ensemble averaging

at sufficiently large averaging periods

(G.D. Birkhoff, circa 1930)

Stationary homogeneous

Reynold’s (1894) Averaging Rules

gfgf

faaf

gfgf

ii x

f

x

f=

If a is a constant

'')')('(~~jijijjiiji uuUUuUuUuu +=++= 0== jiji uUuU '',

Examples:

Reynolds Averaged Equations

Continuity Equation:

0=j

j

x

u~

0==+=+

j

jjj

jj

jj

x

UuU

xx

uU)'(

)(

Momentum Equation

jj

ii

ikjijk

j

ij

i

xx

ub

x

pu

x

uu

t

u ~~~~~ ~~ 2

30

1 ++=Ω++

pppandbbb,u'Uu~~

iii +=+=+=~

( ) ( )( ) ( ) ( ) ( )jj

ii

ioikjijkjjii

j

ii

xx

uUbb

x

ppuUuUuU

xt

uU 1 2

3

++++

+=+Ω++++

+'

')(

)(( ''++=Ω++ji

uux

Uv

xb

x

pU

x

UU

t

U

j

i

jiz

ikjijk

jj

i -)1

0

+= )(''

iij

x

U

x

UKuu

ji-

Turbulent Motions

It remains to call attention to the chief outstanding difficulty of our subject…

– Sir Horace Lamb (1897, 1932)

……and this remains valid even today

uu.

We believe it means that, even after 100

years, turbulence studies are still in their

infancy.

We are naturalists, observing butterflies

in the wild. We are still discovering how

turbulence behaves, in many respects.

We do have a crude, practical, working

understanding of many turbulence

phenomena but certainly …

..nothing approaching a comprehensive

theory…

..and nothing that will provide predictions

of an accuracy demanded by designers.

Non-Linearity

0x

θθ

t

θ

(2/9)

,cos0, kxx

kxktkxtx 2sin2

cos,2

Cascade and TKE dissipation

Big whorls have little whorls, which feed on their velocity; and little whorls have lesser whorls, and so on to viscosity” –Richardson 1922

Turbulence

Viscous

22

LU~u

Inertia

LUuu

2

~

Inertia

Viscous

eRUL

(3/9)

++=+++ )(2

221-

z

u

xx

u

x

puf

z

uw

x

uu

t

ukjjk

Energy Cascade

1v

UL111

v

LU122

v

LU1~

v

LU kk

LU ,11, LU 22 , LU kk LU ,

Kolmogorov Scales

L

U 3

(4/9)

Viscosity is unimportant

Energy flux is conserved32 TLunits [ ],, ful kk =

Kolmogorov Hypothesis

413

k

v

vk ( )1 4

21

k )(Tv

(length)

(velocity)

(time)and

kE

L2

klL

k 2

Spectrum

3532 //)( kkE k=

351

321111

//)( kkF =

Spectral Analysis of Turbulent Flows

....,,n,eaxu L

nxi

n 321

2

which is the Fourier representation of the signal. Further, dxexuL

a

L

L

L

nx

n

2/

2/

21

i

Thenikx

eaxu n and ( ) ( ) dxexuk ikxL

L

2

2

/

/

=Λ

U

Now that k changes as 2

L,4

L,6

L... as n takes n 1,2,3... ... .

So that L

dk2

Now define kL

n2kULanand

(2/2)

( ) ( ) dxekxu ikxL

L

-2

22

1/

/

Λ

= U

G.I. Taylor Simplification

Then, deutu ti

ii )(ˆ)(

and

T

T

ti

ii dtetuu )(2

1)(ˆ

Hence,

detutui

ijji )()()( *

0 Ttfor

TtTfortutu ii

>=

<<= )()( -

dekFrxuxu

ikr

ijji

+

=+ )()()( *

= du iii )(2

-

A Fourier mode

0=

=+

~~

~~~

.

)(

ku

eaxu n~~x.ki

One and Three Dimensional Spectra

~~

~~~~~

~~)()()(')(' kdekrRrxuxurki

ijijji

.+

==+

321100 dkdkdkekrRrik

ijij

+

= )(),,(~

one-dimensional spectra32

~1 )()( dkdkkkF ijij

where r~

(r1, r2 ,r3) ),,( 321~

kkkkand

dekFrxuxu

ikr

ijji

+

=+ )()()( *

The wave number k

Spectra for a single wave number

The Fluid Dynamics group in the Cavendish Laboratory, April

1955. Front row: Ellison, Townsend, Taylor, Batchelor, Ursell,

van Dyke. Middle row: Barua, Thomas, Morton, Thompson,

Philips, Bartholomeusz, Thorne. Back row: Nisbet, Grant, Hawk,

Saffman, Wood, Hutson, Turner. From Huppert (2000).

Turbulent Kinetic Energy Equation

(A) rate of increase of TKE a given point

(B) gain of TKE by the advection caused by mean flow

(C) production of turbulent kinetic energy by the working of mean flow on the Reynolds stresses

(D) transport of turbulent kinetic energy by the advection by the turbulent fluctuations and turbulent pressure

> 0 represents convection and <0 stably stratified turbulence

jj

ii

j

ij

jj

iji

j

i

j

i

xx

uuub

upuu

xx

Uuu

x

u

Ut

u'

'''2/2

'

2

'2

3

0

''

2''''

22

(A) (B) (C) (D) (E) (F)

22''

ij uu

wb

TKE Dissipation

( )iiijijss= ≈2 ''

)( +=i

j

j

iij

x

u'

x

u'

2

1s'

3

-F

+=

21 F

ji

ji2

F

jj

2i2

F

jj

ii

xx

u'u'

xx2

u'

xx

u'u'

(1/2)

L

U 3

Isotropic Turbulence

A flow in which any relationship between turbulent quantities (in averaged sense) is invariant under rotation of the coordinate system, translation of the coordinate system and under the reflection with respect to the coordinate system.

iiijij ss == ''2

15u1

x1

2

For isotropic turbulence

u1

x1

2u1

2

2

Microscale Reynolds Number

Ru

(2/2)

21 /)(uLl=

WHERE IN THE SPECTRUM IS DISSIPATION?

Kolmogorov scales?

1=u

R.R. Long. 1982. A new theory of energy spectrum, B.L.M., 24, 136-160.

( )

( ) ( ) lll

ll

l

llr

r

r

r

V

l

n

V

S

u

k

d

u

<<

<<

21--

-

2

1

43

21

/ReRe

Re~

)(~

~~

~

Structure of turbulence and dissipation

She et al., Nature, Vol.

334, 226-228, 1990

Dissipation Estimation

2

2

2

x

uU

t

u

351

321111

//)( kkF =

Taylor’s hypothesis: if then

spatial autocorrelation:

1-D velocity spectrum:

Kolmogorov spectrum: for inertial subrange

1U

u

U

xt

)()()( rxuxurRxx

0

111 )( dxeRkF iwx

xx

351

321111

//)( kkF =

T-REX Central

Tower

30 m

CASES-99 main tower

October 1999, 60 m

zo=~0.16

zo=~0.02

Outdoor Three-dimensional In-situ calibrated

Hot-film anemometry System (OTIHS) – 3D probe

Poulos, Semmer, Militzer, Maclean, Horst, Oncley…..

3-d hotfilm rotates on

this axis 235o

2 kHz @ ~1 mm scale

Uses 70.15 m films.

Courtesy: Greg Poulos

40

= 0.035 mL ~14 m

fSu

= 2.8 mm500

Turbulent Kinetic Energy Equation

Typically -- homogeneous turbulence--production balances dissipation (really large scales feeding the smaller scales)

=+++=+jj

ii

jij

jj

iji

j

i

j

i

xx

uuub

upuu

xx

Uuu

x

u

Ut

u'

''')/(

''''

''''2

30

2

22

2--22

(A) (B) (C) (D) (E) (F)

-

--0 3'''' ub

x

Uuu

j

iji +≈

( ) ( ) 2-

2-

2detutuwtW

j)(~)(~, *+=

De Silva, I.P.D. and H.J.S. Fernando. 1994. Oscillating grids as a source of nearly isotropic turbulence,

Phys. Fluids, 6(7), 2455-2464.

Wigner-Ville Transforms

Turbulent Kinetic Energy Equation

No production in homogeneous turbulence --turbulence is decaying (small scales decay first and gradually seeps into larger scales)

=+++=+jj

ii

jij

jj

iji

j

i

j

i

xx

uuub

upuu

xx

Uuu

x

u

Ut

u'

''')/(

''''

''''2

30

2

22

2--22

(A) (B) (C) (D) (E) (F)

-

-2

2

≈t

u i'

Smaller scales decay fast

Energy Budgets

z

)(zU

j

jj

j

iji

j

jx

puq

u

bwx

Uuu

x

q

Ut

q 222

222

P B

Flux Richardson

Number

j

iji

f

x

Uuu

bwR Gradient Richardson

Number2

2

z

U

NRig

Diffusion Coefficients

i

hi

i

j

j

imji

x

bKbu

x

U

x

UKuu

eg.z

UKuw m

Flux versus Gradient Richardson Number

Arizona State University

Environmental Fluid Dynamics

Program

0

0.1

0.2

0.3

0.4

0.5

0.6

0.001 0.01 0.1 1 10 100

Ri g

Ri f

Strang & Fernando

MY (1982)

VTMX (2000)

TA-6 data

Townsend (1958)

Nakashini (2001)

fRi

Pardyjak, Monti and Fernando, J. Fluid Mech., 2002

Normalization of the eddy coefficients in the VTMX

Arizona State University

Environmental Fluid Dynamics

Program

K

/

/ |d

V/d

z|

g

Mw2

~

Ri

K

/

/ |d

V/d

z|H

w2~

Field Experiment

Equation (3.7)

Field experiment

Equation (3.8)

0.01 0.1 1 10 100

10

1

0.1

0.01

0.001

10

1

0.1

0.01

0.001

(

)(

)

Monti et al., J. Atmos. Sci., 59(17), 2002

Eddy Diffusivity (Semi Empirical)

Arizona State University

Environmental Fluid Dynamics

Program

34.0)34.0(/

~/

02.0

2 g

w

m RidzVd

K

5.049.0

2)08.0()08.0(

/~

/gg

w

h RiRidzVd

K

Monti et al., J. Atmos. Sci., 59(17), 2002

Eddy Diffusivity Ratio

Arizona State University

Environmental Fluid Dynamics

Program

gRi

K

/ KH

M

0.01 0.1 1 10 100

10

1

0.1

0.01

0.001

Monti et al (2002)Strang & Fernando (2001)

MRF

GS

Prandtl Number

gRi

K

/ KH

M

0.01 0.1 1 10 100

10

1

0.1

0.01

0.001

Monti et al (2002)Strang & Fernando (2001)

MRF

GS

• Lee et al. Boundary layer Meteorology, 119, 2006h

mt

K

K=Pr

Balance of a Transferable Quantity (a scalar)

˜ P

t˜ u j

˜ P

x j

D2 ˜ P

x j x j

˜ F p

pj

jjj

j Fupx

PD

xx

PU

t

P)''(or

ii

xKup- =

Fluctuations

jj

jjjj

j xx

pDuppupUuP

xt

p 2'' )(

jj

j

jj

j

j

jxx

ppDup

xx

Ppu

x

pU

t

p2

22 '2'22

(A) (B) (C) (D) (E)

(1/4)

The term (E) is of special interestjjjjjj x

p

x

pD

xx

p

Dxx

ppD 2

'''

22

2

(E1

)

(E2

)

(a) Binary distribution in space.

t0

t1 >

t0

t2 >

t1

(b) Amplitude on a cut.

t0

t1 >

t0

t2 >

t1

(c) Effect of molecular transport for t > t0.

t0

t1 >

t0

t2 >

t1

Schematic sketch of turbulent mixing of

a contaminant, both without and with

molecular diffusion. These represent a

small sample from a large statistically

homogeneous field.

(2/4)

Separation Distance

Diffusion Time D2

Separation Time u

41

vu 31

or

41

3v or

u

l

D~

2

( )

( ) 1413

31

<= Pr

~,~

/

/

requiresDl

lulif

oc

1>< Pr, orllif Koc4/1

2

4/14/13 ~,~

vDl

vuvl

B

Scalar Dissipation

41

2= vD

41

3= )(D

From Monin & Yaglom – wake past a

bullet

Incoherent and Coherent Motions

Brown & Roshko Experiment

Coherent Structures

Double Decomposition

(to explain coherent-incoherent interactions)

txftxftxf rc ,,,~~~

coherent incoherent

Stratified and Rotating Turbulent Flows

…..Discussion

Horizontal Momentum

kjjk

HH

H uRoz

uw

x

uu

t

u

Tu

L

1

x

p

u

p

H

2

0

0

2

222

z

u

L

L

xx

u

V

H

2/1Re~

HVLL

Re

1

2

2u

u~

~~ +×

2

221

z

u

xx

u

x

puf

z

uw

x

uu

t

ukjjk

ScalesVelocityuandu

ScalesTimeTandT

ScalesLengthLandL

vH

VH

VH

:

:

:

Vertical Momentum

z

p

u

p

z

ww

x

wu

t

w

Tu

L

L

L

HVH

H

H

V2

0

02

2

222

2

0 ~

z

w

xx

w

L

L

uLb

u

Lb

H

V

HHH

V

0H

V

L

L

01Re

2

22

0

1

z

w

xx

wb

z

p

z

ww

x

wu

t

w

Definitions – 3D Turbulence

Chaotic Motions, where the inertial vortex forces

dominates

viscous, Coriolis, buoyancy and all other body

forces

+

balance pressure forces

Inertial vortex forces ~ Coriolis Rotating turbulence

Inertial vortex forces ~ buoyancy forces stratified turbulence

Inertial vortex ~ Coriolis/buoyancy Geophysical Turbulence

2)z/u(~y

y

y

z

z

(Fincham et al 1996)

y

z

total

wz

bw

x

bu

NL

b

t

b

TNu

b

VbV

2

0

2

02

22

2

2

0

z

b

L

L

xx

b

v

K

NL

b

HU

HL

v

V

H

V

1

22

Nu

L

LNL

u

Growth

u

L~T

v

V

V

V

v

v

vb

2N

bLL o

EllisonV

2/12/1 Re~ ScL

L

H

V

Buoyancy Equation

bV

V LN

u~L

Enstropy Flux

332

k~)k(E

kE

kE

32

21

~ tk

Energy flux 35-

32

kkE ~)(

k

Evolution of a Turbulent Patch in a Stratified Fluid

(A) surrounding a turbulent patch due to the collapse of the patch. Note the setting up of “zero-frequency” modes.

(B) surrounding the patch. (De Silva & Fernando 1998).

(a) (b)

Formation of Intrusions

Lb

kjjk

HH

H uRz

uw

x

uu

t

u

Tu

L

1

0x

p

u

p

H

2

0

0

2

22

2

z

u

L

L

xx

u

uL V

H

HH

2

0

2

0

0 ~H

v

H U

Lb

U

p2

1

0~ vH LbU

TUL HH ~

2

0~1~HH

H

U

p

TU

L

H

H

L

UR ~1~0

d

v

H Rf

LbL

21

0~

)(~

~.

1

1

222

0

ONh

CFr

x

hN

x

C

x

puu

==

)Frm(m

Fr

H

hFr

1

2

1

2

21

150.~Nh

C

C

h

The variation of Fri (lower bound) with the

number of forward propagating wave modes

M. Three aspect ratios φ1 of the patch ( -

0.01; Δ- 0.1; and +- 0.15) are shown.

Variation of Fri

\N/f = 1.3

(Hedstrom & Armi 1988). N/f = 0.18

f/NL~f/)LbRVVD

20 f

N

L

L

V

H

Baroclinic instabilities in a two-layer fluid (Ivey 1987)

Thermohaline Circulation in Oceans

(b) An oceanographic mixing bowl. The bowl

represents a surface on which the temperature is

constant, capped by an Eckman layer in which the

wind directly drives water flow and mixing. (From

Marshall et. al. 2002)

(a) The thermohaline conveyor belt in the oceans.

Dark bands show flow of deep, cold, and salty water;

light bands show return surface flow. (From Broecker

1981).

CLASSES OF STRATIFIED TURBULENT FLOWS

Stratified Shear LayersSlow

Faster

dz

Udwu

0dz

dUwu

Theory/Laboratory Profiles

2U

bDRi

Δ=

2

2

dzUd

NRig

Stratified Shear Flow (Lab)

1gRi

(Strang & Fernando JFM, 2001)

Stratified Shear Flow #21gRi

Stratified Shear Flow #3

1>gRi

Mechanisms of Entrainment

(c) 830.gRi

(e) 211.gRi

(g) 822.gRi

(b) 360.gRi

(d) 211.gRi

(f) 781.gRi

(h) 822.gRi

(a) 120.gRi

Interfacial Measurements

Tower

Turbulence in ABL

Daytime Vertical Profile

Moum et al. 1990

DeSilva, Fernando, Hebert & Eaton, Earth Planetary

Sci. Lett. , 1996

Mechanisms of Entrainment

(c) 830.gRi

(e) 211.gRi

(g) 822.gRi

(b) 360.gRi

(d) 211.gRi

(f) 781.gRi

(h) 822.gRi

(a) 120.gRi

Stratified Wakes

Lv

Re = 3040, Fr = 1.5

Re = 3800, Fr = 3.37

Re = 200, Fr = 2.24

Lb

Nd

UFr

UdRe

N

UL v

b

Scales

b

v

L

U3

~

Ozmidov

LN

~L Rb

21

3

KbR LLL ~~

2530102N H.P. Pao/Boeing

Gibson

Fossils

Energy Spectra in a Wake

4D

15D

10D

20D

Self-Propelled Bodies

Jets in Stratified & Rotating Fluids

2N

4

2

0

2

0

udJ

00

2

04

budB

0

2

04

udQ

21

0ReJ

2

0

22

0u

LvNRi

HfL

uRo 0

xxQ 8

Uue

Turbulent Jets2

22

~U

LNRi v

x

JU

21

0~

xLv ~

41

20

N

J~hmax Limiting vertical scale

21

21

0

0

0

/

/

max Fr~Nd

u~

d

h

Transport and Dispersion(Material P)

txUtxtxU ,,'u,~

~

pj

jjj

j Fupx

PD

xx

pU

t

P)''(

txPtxtxP ,,'p,~

~

transport dispersion

source

HYSPLIT verification – Asian dust storm / TOMS data• 10-day loop, April 2001, http://www.arl.noaa.gov/ss/transport/dust/

Plumes - Convective

Plumes - Stable

Dispersion of Particles in Turbulent Fluids

(over a large number of displacements), then the mean square value is

If a fluid particle moves in successive displacements of ,iy

then the resultant displacement after N steps is .1

N

iyy

.2

1

2

11

2 yNyyyyN

i

i

N

j

j

N

i

i

If the field is homogeneous, the average of ,0, yy

(1/2)

Dispersion

,2 tvdt

dyand hence the distance .0

02

t

tdtvyty

tdttvtdtdttvtdT

tyt

002

002

2 1

(2/2)

t

r.m.s. Dispersion

( ) ( ) ( )=0 0

22

2 -2t t

LL dRdtRvty )('

at large times t > t*, RL( ) is small and negligible

L

t

L tvdRtvy 2'

2

*

0

2'

2

2 22

222

2 2 tvy '==

For short times t << t*, RL( ) ~ 1

)()()( += tvtvRL 22

Eddy Diffusivity

tvtvy LL

2'

2

2'

2

2 22

212

12

1

2'

2

2 2 tvy L

LvK 2'

22

Stratified Turbulence

( ) ( ) ( )=00

22

2 -2t

L

t

L dRdtRvty )('

Can we use this?

z

Homogeneous/Stationary (Taylor 1921)

0

22' 2 dRtTz LLw

Very small!

Dynamics of Fluid Parcel Dispersion

222

2

1~

2

1wzN

NL

Nz w

bw :~

2/1___2

Maximum Height

;02

dt

zdconst2z

zZz e

dt

dZ

dt

zd

dt

dZ

dt

dz ee

2222 ! Small!!!

No Mixing

2N

0

ez

z

0

(mix)

ez

2

N

ez

tbw

;j

jx

utdt

dz

Zw

xu

t jj ++=

dt

d

STIR MIX

tdtwZ

t

0

twt

z

Zw

2

2

1'

,w

11

Fluxes of Buoyancy

w

Phoenix Brown Cloud

mNw 50~02.0/1~/

From the Arizona Republic

mNw 50~02.0/1~/

twdt

zd

dZw '

'

2

1'

2

LwTK

22

N

2

2

<<N

2

Density changes contribute more!

i.e. Changes of mean concentrations have to be considered!

CdZdydZdyCZZ sZ

2

ZZ K

dt

d2

2

Diffusion of a Substancec

s kdt

yd2

2H

eig

ht (m

)

Concentration (scaled)

0.0 0.2 0.4 0.6 0.8 1.0-60

-40

-20

0

20

40

60

Time (days)

Se

co

nd m

om

ent (m

2)

0 50 100 150 2000

100

200

300

400

(2/2)

0dz

dUwu