Atmospheric turbulence
description
Transcript of 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
R.J. Perkins 2009
VII Séminaire Transalpin de Physique - Atmospheric Turbulence 2
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)
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The role of Reynolds number
What is turbulence?
The wake behind a cylinder
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A wide range of Length and Time Scales
What is turbulence?
Conservation of mass― for an incompressible fluid
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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
.
. .
.
.
.
R.J. Perkins 2009 VII Séminaire Transalpin de Physique - Atmospheric Turbulence 8
Conservation of momentum – the Navier-Stokes equations
The Governing 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:
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The Governing Equations
* * *
* * *
*
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:
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In Dimensionless Form:
The Governing Equations
*
*
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
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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
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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
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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?
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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
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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
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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.
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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
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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.
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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
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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
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Vertical Structure of the Atmospheric Boundary layer
Turbulence in the Atmospheric Boundary Layer
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Length and Time Scales
Turbulence in the Atmospheric Boundary Layer
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Synoptic Scales – Radioactive plume from Chernobyl
Turbulence in the Atmospheric Boundary layer
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Diurnal variations
Turbulence in the Atmospheric Boundary Layer
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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
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Thermal Stability
Thermal Effects in the ABL
Neutral Stable Unstable
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Effects on Dispersion
Thermal Effects in the ABL
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Inversion layers
Thermal Effects in the ABL
Beirut, April 2000
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The dispersion of hot smoke in a tunnel
The effect of stratification on turbulence
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Mechanical Production of Turbulence
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The effect of stratification on turbulence
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
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Buoyant production/destruction of turbulence
R.J. Perkins 2009
The effect of stratification on turbulence
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
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Vertical Heat flux
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The effect of stratification on turbulence
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
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The Richardson number
R.J. Perkins 2009
The effect of stratification on turbulence
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
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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
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The Monin-Obukhov Length
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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:
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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.
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Trajectory of a single particle
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Turbulent dispersion coefficient
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
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Lagrangian analysis
R.J. Perkins 2009
Turbulent dispersion coefficient
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 .
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Diffusion by continuous movements (Taylor, 1921)
R.J. Perkins 2009
Turbulent dispersion coefficient
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
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Time dependence of the dispersion coefficient
R.J. Perkins 2009
Turbulent dispersion coefficient
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