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