Atmospheric turbulence

VII Séminaire Transalpin de Physique - Atmospheric Turbulence 1

Atmospheric turbulence

Richard PerkinsLaboratoire de Mécanique des Fluides et d’Acoustique

Université de LyonCNRS – EC Lyon – INSA Lyon – UCBL

36, avenue Guy de Collongue69134 Ecully

[email protected]

R.J. Perkins 2009


One of the great unsolved problems• From a theoretical point of view:

– Einstein/Heisenberg, Cray prize• From a practical point of view:

– Most ‘engineering’ and geophysical flows are turbulent

Impossible to define satisfactorily• But usually easy to recognise• Is it random?• Is it unpredictable?

Often described in terms of how it occurs…

What is turbulence?

Clouds over Madeira

NASA

R.J. Perkins 2009

R.J. Perkins 2009 VII Séminaire Transalpin de Physique - Atmospheric Turbulence 3

Reynolds experiment

What is turbulence?

VII Séminaire Transalpin de Physique - Atmospheric Turbulence 4R.J. Perkins 2009

Reynolds’ analysis of his pipe flow experiment

What is turbulence?

2 1

1

m sm

m s

D

U

The flow is determined by three parameters only:

: the kinematic viscosity of the fluid [: the diameter of the pipe [

: the average velocity in the pipe [ ]

Re UD

These can be rearranged into a single dimensionless number:

which determines the state of the flow.

Critical Reynolds number for transition

Re 2000 Re 2000Flow is stable (laminar) Flow becomes unstable (turbulent)


The role of Reynolds number

What is turbulence?

The wake behind a cylinder


A wide range of Length and Time Scales

What is turbulence?

Conservation of mass― for an incompressible fluid


The Governing Equations

0

0 1

0 1

12 2 2

2

2 2 2 2

2 2 2

2

Entering Leaving

z z

x x

y y

w wM w z x y M w z x

u uM u x y z

v vM v y x z M v y

yz z

x zy

M u x y zx x

y

1 0 1 0 1 0 0x x y y z zM M M M M M M t

. . 0i

i

uu v wu

x y z x

u2δx2δy

2δz

Mz0

Mz1

Mx1

Mx0

My0

My1

.

. .

.

.

.


Conservation of momentum – the Navier-Stokes equations


2 2 2

2 2 2

2 2 2

2 2 2

2 2 2

2 2 2

1

1

1

u u u u dp u u uu v w

t x y z dx x y z

v v v v dp v v vu v wt x y z dy x y z

w w w w dp w w wu v w

t x y z dz x y z

221 1.or : or :

LiNon linear near

LineaNon linear

i i

j j j

r

ij

i

u u uu dpu u p u ut t x dx x x

, , ,3N- S + continuity

4 unknowns: 4 equations:

soluble p u v w

Dimensional Analysis

The physical problem can be characterised by:• the fluid density, ρ• a characteristic length scale, L• a characteristic velocity scale, U

The dimensionless variables then become:



* * *

* * *

*

2*

2

2 2 2* * *

/ , / , // , / , /

( / )

/

1 1, ,

x x L y y L z z Lu u U v v U w w Ut t U L

p p U

Ux L x x L x t L t

Lengths: Velocities: Time :

Pressure :

Derivatives:


In Dimensionless Form:


*

*

2* * **

** * * * *

2* * **

** * * * *

0

1Re

Non linear

Non line

i

i

i

a

i ij

j i j j

i i ij

j i j j

Linear

Lir near

ux

u u udpu

t x dx UL x x

u u udput x dx x x

Continuity equation :

Navier-Stokes equation :

: or

4 variables (u1, u2, u3, p) and 4 equations1 independent parameter – the Reynolds number Re (=UL/ν) Family of solutions, as a function of Re

Very few analytical solutions available Need to solve the equations numerically


Flow between parallel plates

Laminar flow

u

xy

z

z h0u v w

0u v w

0z

z h

20 Re 1000

0

0

Steady flow

Choice of axes

Except for

U ht

yp

x x

0 0 0 0and

Continui

since at ,

ty e

everyw

quatio

r

n

he eu v w w

w z h wx y z z

2 2 2

2 2 2

1

10 is independe

Navier-Stokes equation ( compon

nt of

e

nt)w

w w w w dp w w wu v w

t x y z dz x y zdp

p zdz


Flow between parallel plates

Laminar flow

2 2 2

2 2 2

2

2

2

1

10

1 12

Navier-Stokes equation ( component)

Since is indepen

dent of we can integrate this directly:

u

u u u u dp u u uu v w

t x y z dx x y z

dp udx z

p z

dpu z

dx

0 :with boundary con ditions t a

az b

u z h

2max

01

2

Maximum speedz

dpu h

dx

32

3

12 3

1 42 3

Mass flow ratehh

h h

dp zq u dz h z

dx

dp hdx

Shear stress on the wall

hz h

u dph

z dx

2 212

dpu z h

dx


What happens at higher Reynolds numbers?

Turbulent flow

If Re 1000 the flow will start to become turbulent, and the velocities will fluctuate in space and in time.

Poiseuille flow close to the boundary, visualised with smoke

Laminar Turbulent

0

0, 0, 0

0

t

x y zw

Fransson, Talamelli, Brandt & Cossu (PRL, 2006).

Could we do the same analysis, using just the average velocities?


Reynolds Decomposition

Turbulent flow

0 10 20 30 40 50t(s)

0

5

10

15

u [m

s-1

] u

0 0 0

1 1 1 0T T T

u u u

u u dt u dt u dt u u uT T T

Reynolds decomposition:

For a steady flow we can take a time average of the velocity:

For unsteady flow we need to take an ensemble average

1

1 1

1( ) ( )

1 1( ) ( )

( ) ( )

( ) 0

n N

nn

n N n N

nn n

u t u tN

u t u tN Nu t u t

u t

h

u

u gh


Reynolds Decomposition applied to the Continuity Equation

Turbulent flow

0

0

i i ii

i i i i

i

i

u u uux x x x

ux

Taking the ensemble average:

i i iu u u

Reynolds decomposition

0i

i

ux

The Continuity Equation

0ii ii i

i i i i

uu uu ux x x x

Applying Reynolds decomposition:

Conclusions• The average velocities satisfy the continuity

equation• The fluctuating velocities satisfy the

continuity equation, at every instant


Reynolds Decomposition applied to the Navier-Stokes Equations

Turbulent flow

21Re

i ji i ij

j i j j j

u uu u p uu

t x x x x x

Taking the ensemble average:

i i iu u u

p p p

Reynolds decomposition21

Rei i i

jj i j j

u u uput x x x x

The Navier-Stokes Equations

21

Rei i j j i i i ij i j j

u u u u u u p p u ut x x x x

Applying Reynolds decomposition:

Conclusions• The average velocities do not satisfy

the Navier-Stokes equations!• Correlations between the

fluctuating velocities contribute to the mean transport of momentum.


The Reynolds stress term

Turbulent flow

2

2

12

0 0.

jj j

j j

j j

i j

uu u

x x

u u

Consider the case :

And only if

This is only true in a laminar flow.δz

0u w

0u w 0uu u zz

0uu u zz

u

z

' 0

0

uu zz

w

0

' 0

wuu zz

Reynolds stresses in the boundary layer

0u w So in the boundary layer:

• Fluctuating velocities towards the wall transport faster fluid towards the wall

• Fluctuating velocities away from the wall transport slower fluid away from the wall

• Reynolds stresses transport momentum down the momentum gradient

• The action of the Reynolds stresses is similar to the action of viscosity.

• But, the Reynolds stresses are much more effective than viscosity

They cannot be neglected


The closure problem

Turbulent flow

211 1 1

1

222 2 2

2

233 3 3

3

0

1Re

1Re

1Re

i

i

jj

j j j j

jj

j j j j

jj

j j j j

ux

u uu u p uu

t x x x x x

u uu u p uu

t x x x x x

u uu u p uu

t x x x x x

4 Equations:

Continuity:

Navier-Stokes:

1 2 3

1 1 1 2 1 3 2 2 2 3 3 3

, , ,

, , , , ,

u u u p

u u u u u u u u u u u u

10 unknowns: Need a model for the Reynolds stress terms to close the system of equations.


Closure models

Turbulent flow

2

2 2

13

3

Eddy viscosity models

jii j ij T

j i

i i

uuu u q

x x

q u u u

2

2

1 2

:

12

turbulent kinetic energy : average energy dissipation

Need Evolution equations for and

: Production

of turbulent energ

models

y

T

i ii j T

i j

k kC

kdkdtd C Cdt k k

u uu u

x

k

x

2

j

i

u

x


Numerical solutions of the Navier-Stokes equations

Turbulent flows

Direct Numerical Simulation – DNSAll the terms are computed explicitlyÞ Spatial resolution Δx, Δy ~ kη – Kolmogorov length scale

2(Re) 1mm 0.1s; for the atmosphere, ! (And !)k f k T k

Large Eddy Simulation – LES• The large scales are calculated explicitly (Δx, Δy kη)• The effect of the small scales is modelled using a sub-grid scale model

Express the derivatives as Finite Differences:

3,2 1,2 2,3 2,1

3 1 3 1

u u u uu ux x x y y y

e.g.

x

1,1u 2,1u

2x 3x

1y

1,2u

2,3u

2,2u

1,3u

3,1u

3,2u

3,3u

2y

3y

y

1x


Vertical Structure of the Atmospheric Boundary layer

Turbulence in the Atmospheric Boundary Layer


Length and Time Scales



Synoptic Scales – Radioactive plume from Chernobyl

Turbulence in the Atmospheric Boundary layer


Diurnal variations



Effect of density gradient of air

Thermal Effects in the ABL

Hydrostatic pressure:

Ideal gas :

Adiabatic movement:

Potential temperature :

1K /100mp

dp dT ggdz dz C

p RT

0pdpdq C dT

/ 2 / 71000pR C

refpT T

p p

0 :

0 :

0 :

Unstable

Neutral

Stable

ddzddzddz

0 0,p T

2 / 7

11 1 0

0

, pp T Tp


Thermal Stability


Neutral Stable Unstable


Effects on Dispersion



Inversion layers


Beirut, April 2000

29R.J. Perkins 2009 VII Séminaire Transalpin de Physique - Atmospheric Turbulence

The dispersion of hot smoke in a tunnel

The effect of stratification on turbulence


Mechanical Production of Turbulence

R.J. Perkins 2009


ˆ

Horizontal force:

Work done = Force x distance:

Power = rate of doing work:

Power/unit mass of fluid:

M

M

M

F x y

duW x y z t

dz

duP x y z

dz

P duP

x y z dz


Buoyant production/destruction of turbulence

R.J. Perkins 2009


ˆ

Buoyancy Force caused by a fluctuation in density:

Work done = Force x distance:

Power = rate of doing work:

Power/unit mass of fluid:

Averaging over all fluctuati

B

B

B

B

F V g

W V g w t

P V g w

P g w

ˆ

ons of density and velocity:

Bg

P w


Vertical Heat flux

R.J. Perkins 2009


ˆ

Density fluctuations are related to temperature fluctuations:

From which:

where is the vertical flux of sensible heat.

pB

p p

TT

g c T wg g HP T w

T T c T c

H

For an unstable (convective) boundary layer H>0: upward heat flux adds to the turbulence

For a stable boundary layer H<0:downward heat flux suppresses turbulence

Buoyant production is almost independent of height:ρ and T vary very little in the first 10m-50m

Þ At low altitudes, stability is determined principally by mechanical productionÞ At higher altitudes, stability is determined principally by buoyant production


The Richardson number

R.J. Perkins 2009


R

R

ˆˆ /

f

f

B

pM

P g HT c du dzP

Richardson defined a stability criterion as:

negative buoyant production rate mechanical production rate

- /

/

p HH c K T z

But these quantities are difficult to measure, so Richardsonassumed that the turbulent transport of Heat and Momentum could be modelled by diffusion equ

Flux Richardson

ations

r

:

Numbe

2

2

/ /

/R

/

M

Hf

M

u z K u z

T zKgT K u z

From which:

1H

M

KK

T

Often

The Temperature can be written in terms of the Potential Temperature


The Richardson number

The Effect of Stratification on Turbulence

2

/R/

R 0.25 :R 0.25 :

where: slightly turbulent flow remains turbulent

Gradient Richardson Number

turbulence is suppressed

i

i

i

g zu z


The Monin-Obukhov Length

R.J. Perkins 2009

The Effect of Stratification on Turbulence

3*

3 3* *

ˆˆ ˆ

ˆ

MO

MMO

M B

Bp

p v pMO

L

uduPdz kL

P Pg HPT c

u T c u T cL

g H k k g w

Suppose that at some height the mechanical production of turbulence

is balanced by the buoyant dissipation of turbulence:


Lagrangian dispersion

Consider the trajectories of particles passing through the source:

R.J. Perkins 2009

Turbulent dispersion coefficient

In the absence of molecular diffusion, the concentration transported by a particle remains constant.


Trajectory of a single particle

R.J. Perkins 2009


22( ) ( )( )

( ) ( )

Velocity autocorrelation

L v t v tRv t v t

( )tu

( )t t u

( 2 )t t u

( 3 )t t u

( 4 )t t u

22 220

( ) d

Lagrangian Integral timescale

L LT R


Lagrangian analysis

R.J. Perkins 2009


0

( , ) ( , ), ( ,0,0)

0,

( )

Consider a velocity field defined by:

where

Release a cloud of particles at in the volume

The position of particle at any instant is given by the n

u x t U u x t U U

N t x r

x t n t

0

1

( ) ( ( ), ) d

1( ) ( )

Lagrangian integral:

The position of the centroid of the cloud of particles at any instant is defined by:

where denotes the of t

t

n n n

n N

c n nn

n

x t u x

t

x t x t x tN

ensemble average he variable .


Diffusion by continuous movements (Taylor, 1921)

R.J. Perkins 2009


2 2 2

1

1( ) ( ) ( )

Consider dispersion in the direction; this can be characterised by the variance of the particle positions:

, the dispersion coeB fy fidefi ciennit t is the rate of change ion

n N

y nn

y

t y t y tN

22

0

0 0 0

22

( )( ) ( )2 ( ) 2 ( ) ( ) 2 ( ) ( )

:

2 ( ) ( ) 2 ( ) ( ) 2 ( ) ( )

2 ( ) ( )

of the variance:

With the change of variable

From which:

ty

T

t t t

T

LT

d y td t dy tK y t y t v t v t v s dsdt dt dt

s t

K v t v t d v t v t d v t v t d

K v t v t R

0

( )t

d


Time dependence of the dispersion coefficient

R.J. Perkins 2009


222

0

222

222 22 22

0

2 ( )

0, ( ) 1 2

, ( ) d 2

tL

T

LT

tL L L

T

K v R d

t R K v t

t R T K v T

t

.TK const

TK tKT varies with distance from the source

Atmospheric turbulence

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