Turbulent properties: - vary chaotically in time around a mean value
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Transcript of Turbulent properties: - vary chaotically in time around a mean value
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Turbulent properties:- vary chaotically in time around a mean value- exhibit a wide, continuous range of scale variations- cascade energy from large to small spatial scales
“Big whorls have little whorlsWhich feed on their velocity;And little whorls have lesser whorls,And so on to viscosity.” (Richardson, ~1920)
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- Use these properties of turbulent flows in the Navier Stokes equations-The only terms that have products of fluctuations are the advection terms- All other terms remain the same, e.g., tUtutUtu
0
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zu
wyu
vxu
uzU
WyU
VxU
U
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dtUd
zwu
yvu
xuu
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zw
uyv
uxu
uzu
wyu
vxu
u
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zw
yv
xu
u'''
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'','','' wuvuuu are the Reynolds stressesReynolds stresses
arise from advective (non-linear or inertial) terms
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Turbulent Kinetic Energy (TKE)
An equation to describe TKE is obtained by multiplying the momentum equation for turbulent flow times the flow itself (scalar product)
Total flow = Mean plus turbulent parts = 'uU
Same for a scalar: 'tT
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Turbulent Kinetic Energy (TKE) Equation
ijijoj
ijiijijij
oji eew
gxU
uueuuuupx
udtd
22
1 2212
21
Multiplying turbulent flow times ui and dropping the primes
2
21
221
221
221
wdtd
vdtd
udtd
udtd
i
Total changes of TKE Transport of TKE Shear Production
Buoyancy Production
ViscousDissipation
i
j
j
iij x
u
xu
e21
fluctuating strain rate
Transport of TKE. Has a flux divergence form and represents spatial transport of TKE. The first two terms are transport of turbulence by turbulence itself: pressure fluctuations (waves) and turbulent transport by eddies; the third term is viscous transport
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zU
wu
yU
vu
xU
uu
xU
uuj
iji
wg
o
22
242
2i
j
j
i
i
j
j
iijij x
u
xu
x
u
xu
ee
interaction of Reynolds stresses with mean shear;
represents gain of TKE
represents gain or loss of TKE, depending on covarianceof density and w fluctuations
represents loss of TKE
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zU
uwwg
o
0
In many ocean applications, the TKE balance is approximated as:
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The largest scales of turbulent motion (energy containing scales) are set by geometry:- depth of channel- distance from boundary
The rate of energy transfer to smaller scales can be estimated from scaling:
u velocity of the eddies containing energyl is the length scale of those eddies
u2 kinetic energy of eddies
l / u turnover time
u2 / (l / u ) rate of energy transfer = u3 / l ~
At any intermediate scale l, 31l~lu
But at the smallest scales LK,
413
L Kolmogorov length scale
Typically, 356 1010 mW so that mLK
43 10610~
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Shear production from bottom stressz
u
bottom
Vertical Shears (vertical gradients)
3
2
s
m
z
Uwu
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Shear production from wind stressz
W
u
Vertical Shears (vertical gradients)
3
2
s
m
z
Uwu
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Shear production from internal stressesz
u1
Vertical Shears (vertical gradients)
u2
Flux of momentum from regions of fast flow to regions of slow flow
3
2
s
m
z
Uwu
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zU
Awu z
Parameterizations and representations of Shear Production
2
*
refB U
uC
2* refBB UCu Bottom stress:
0*
ln1
zz
uU
Near the bottom
Law of the wall
Bu *
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0
* lnz
zuu
m005.0
sm04.0
0
*
z
u
Bu *
Pa2B
Data from Ponce de Leon Inlet
FloridaIntracoastal Waterway
Florida
0033.07.0
04.022
*
refB U
uC
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Law of the wall may be widely applicable
(Monismith’s Lectures)
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Ralph
Obtained from velocity profiles and best fitting them to the values of z0 and u*
(Monismith’s Lectures)
2
*
refB U
uC
BC
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wuzz
UA
z z
wvzz
VA
z z
Shear Production from Reynolds’ stresses
Mixing of momentum
wszz
SK
z z
Mixing of property S
sm
RiK
sm
RiA
z
z
2
23
2
21
33.31
06.0
101
06.0
Munk & Anderson (1948, J. Mar. Res., 7, 276)
sm
Ri
AK
sm
RiA
zz
z
25
242
1051
1051
01.0
Pacanowski & Philander (1981, J. Phys. Oceanogr., 11, 1443)
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With ADCP:
cossin4
varvar 43 uuwu
and
cossin4varvar 21 uu
wv
θ is the angle of ADCP’s transducers -- 20ºLohrmann et al. (1990, J. Oc. Atmos. Tech., 7, 19)
zV
wvzU
wuTKE Production
wuzU
Az
wvzV
Az
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Souza et al. (2004, Geophys. Res. Lett., 31, L20309)
(2002)
wu
wv
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Day of the year (2002)
Souza et al. (2004, Geophys. Res. Lett., 31, L20309)
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Souza et al. (2004, Geophys. Res. Lett., 31, L20309)
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S1, T1
S2, T2
S2 > S1
T2 > T1
Buoyancy Production fromCooling and Double Diffusion
wg
o
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Layering Experiment
http://www.phys.ocean.dal.ca/programs/doubdiff/labdemos.html
wg
o
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From Kelley et al. (2002, The Diffusive Regime of Double-Diffusive Convection)
Data from the Arcticw
g
o
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Layers in Seno Gala
wg
o
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/s)(m seawater of viscosity kinematic the is
3...1,;2
2
2
jix
u
xu
tensorratestrain
i
j
j
i
Dissipation from strain in the flow (m2/s3)
turbulence
isotropic for
5.72
zu
(Jennifer MacKinnon’s webpage)
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From:
Rippeth et al. (2003, JPO, 1889)
Production of TKE
Dissipation of TKE
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http://praxis.pha.jhu.edu/science/emspec.html
Example of Spectrum – Electromagnetic Spectrum
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(Monismith’s Lectures)
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KSS ,
Wave number K (m-1)
S (
m3
s-2)
3
2
s
m
2
3
s
mS
m
K1
3532 KS
Other ways to determine dissipation (indirectly)
Kolmogorov’s K-5/3 law
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(Monismith’s Lectures)
3532 KS
P
equilibrium range
inertialdissipating range
Kolmogorov’s K-5/3 law
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3532 2
U
fS
325102 sm
(Monismith’s Lectures)
Kolmogorov’s K-5/3 law -- one of the most important results of turbulence theory
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Stratification kills turbulence
25.02
2
22
S
N
zv
zu
zg
Ri o
In stratified flow, buoyancy tends to:
i) inhibit range of scales in the subinertial range
ii) “kill” the turbulence
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(Monismith’s Lectures)
U3
oLU 2
325101 sm
mL
zzgN
03.0,18.0,1
10/10,1,1.0 taking;
0
2
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(Monismith’s Lectures)
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(Monismith’s Lectures)
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(Monismith’s Lectures)
(responsible for dissipation of TKE)
At intermediate scales --Inertial subrange – transfer of energy by inertial forces
nsfluctuatio of numberwave K
TKE of ndissipatio
1.5 constant
KS
3532
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(Monismith’s Lectures)
3
2
sm
Other ways to determine dissipation (indirectly)