wind and turbulence, part 1 - university of novi...
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Wind and Turbulence, Part 1
A. ProcessesA. Wind and Turbulence1. Conceptsa. Definition of Turbulence
b. Reynolds’ Numberc. Conservation equation for windd. TKE budget, conceptual
ESPM 129 Biometeorology
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Roles of Turbulence in Biometeorology
• Transfers heat, momentum and mass (water vapor, carbon dioxide, biogenic gases, pollutants) between the biosphere and the atmosphere and diffuses pollutants in the atmosphere.
• Imposes drag forces on plants, causing them to wave, bend and break
• Mixes the air and diffusing air parcels with different properties, thereby forming spatial gradients
• Gusts place loads on the surface, which can erode soils and eject dust, spores, pollen, seeds, bacteria/viruses and insect eggs into the atmosphere.
ESPM 129 Biometeorology
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Turbulence, Philosophical Side
• ‘There are two great unexplained mysteries in our understanding of the universe. One is the nature of a unified generalized theory to explain both gravity and electromagnetism. The other is an understanding of the nature of turbulence. After I die, I expect God to clarify the general field theory to me. I have no such hope for turbulence’.– Theodore Von Karman
ESPM 129 Biometeorology
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Schematic and Evolution of Boundary Layers
LaminarBoundary Layer
TurbulentBoundary Layer
ESPM 129 Biometeorology
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Transition from Laminar to Turbulent Flows
http://www.thtlab.t.u-tokyo.ac.jp/index.html
ESPM 129 Biometeorology
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Breakup of Stable Boundary Layer
ESPM 129 Biometeorology
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General Attributes of Turbulence
• Complex– Spectrum of scales– Highly Organized– Chaotic
• Mechanical turbulence– Produced by Shear
• Convective Turbulence– Produced by Buoyancy
ESPM 129 Biometeorology
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Specific Properties of Turbulence
• Turbulence is non-linear – du/dt ~ u du/dx
• acceleration is forced by advection • Turbulence is non-Gaussian
– It is skewed and kurtotic• Turbulence is three-dimensional
– Motions are rotational and anisotropic (egvortices)
• Turbulence is diffusive • Turbulence is dissipative
– Energy of motion is degraded into heat• Turbulence consists of multiple length scales
– Large scales of energy input break down into smaller and smaller scales
ESPM 129 Biometeorology
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Poetic Side of Turbulence
Great whirls have little whirlsThat feed on their velocity
And little whirls have lesser whirlsAnd so on to viscosity
Lewis Fry Richardson
ESPM 129 Biometeorology
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Wind
• Speed• direction
U u v w= + +( ) /2 2 2 1 2
u, longitudinal velocity, dx/dt
v, lateral velocity, dy/dt
w, vertical velocity, dz/dt
ESPM 129 Biometeorology
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Newton's Law of Viscosity
the shear force per unit area is proportional to the negative of the local velocity gradient, Shear
FA
P VZ
= = −µ (kg m-1 s-1)
dynamic viscosity (µ). kg m-1 s-2
ESPM 129 Biometeorology
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Kinematic viscosity
Defined as dynamic viscosity (kg m-1 s-1=Pa s) normalized by the density of the fluid:
υ µρ
= (m2 s-1)
ESPM 129 Biometeorology
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Smallest Scales of Turbulence:Kolmogorov Microscale, ~ 10-3 m
η υε
= ( ) /3
1 4
Andrei Kolmogorov
kinematic viscosity (ν) rate of dissipation (ε),scales with velocity cubedOver a length scale, u3/l.
ESPM 129 Biometeorology
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Frictional Shear Stress
τ ρν µ=∂∂
=∂∂
uz
uz
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Defining whether the flow is turbulent or laminar.
Reynolds Number
Re = ⋅d uν
Re is the ratio between inertial and viscous forces
d, physical dimensionu, fluid velocityν, kinematic viscosity
Re < 2000, laminarOsborne Reynolds
ESPM 129 Biometeorology
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Is Atmospheric Flow Laminar or Turbulent?
Re = ⋅d uν
Re .=
⋅=−
01 110
10005Leaf
Re .=
⋅=−
30 310
900005
Corn field
ESPM 129 Biometeorology
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Equation of Motion
Equation of Motion defines how wind velocity accelerates or decelerates
ρ ∆∆
∆∆
∆∆
∆∆
ut
F Ax
Px
Fx
friction= = +( / )
Changes in wind velocity with time are promoted by lateral differenceIn forces per unit area (pressure) and are retarded by differences in frictional
forces
ESPM 129 Biometeorology
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Navier-Stokes Equation
∂∂
+∂∂
+∂∂
+∂∂
≈ −∂∂
−∂∂
+∂∂
+∂∂
ut
u ux
v uy
w uz
px
ux
uy
uz
1 2
2
2
2
2
2ρµρ
[ ]
ESPM 129 Biometeorology
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Understanding non-linear terms
Total Derivative
du t x y zdt
ut
dxdt
ux
dydt
uy
dzdt
uz
( , , , )=∂∂
+∂∂
+∂∂
+∂∂
Time Rate of Change Advection
du t x y zdt
ut
u ux
v uy
w uz
( , , , )=∂∂
+∂∂
+∂∂
+∂∂
u=dx/dt v=dy/dt w=dz/dt
ESPM 129 Biometeorology
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Explaining 2nd derivative terms
∂∂
+∂∂
+∂∂
+∂∂
≈ −∂∂
−∂∂
+∂∂
+∂∂
ut
u ux
v uy
w uz
px
ux
uy
uz
1 2
2
2
2
2
2ρµρ
[ ]
Divergence of Frictional Shear Stress, τ
∂∂
=∂ ∂
∂∂
=∂∂
τ ν νz
uz
zu
z
( ) 2
2
µρ
[ ]∂∂
+∂∂
+∂∂
2
2
2
2
2
2u
xu
yu
z
ESPM 129 Biometeorology
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Mean velocity budget equation
∂∂
+∂∂
+∂∂
= −∂∂
−∂∂
ut
u ux
w uz
px
w uz
1ρ
' '
u u u= + '
Note addition of new and unknown term,The essence behind the Turbulence Closure Problem w u' '
ESPM 129 Biometeorology
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∂∂
12
2qt
'production of turbulent kinetic energy, v1.0
−∂∂
=w u uz
' ' ε
Kinetic energy, produced by shear, is in balancewith dissipation of energy into heat by viscous processes
E mv=12
2 12 2
22 2 2
q u v w' ' ' '=
+ +
ESPM 129 Biometeorology
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production of turbulent kinetic energy, v2.0
shear and buoyant production of tkemust equal the rate at which energy
is dissipated into heat by viscous processes
−∂∂
+ =w u uz
g wv
v' ' 'θ
θ ε
g, acceleration of gravityθv, virtual potential temperature
ESPM 129 Biometeorology
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Summary
• Turbulence– transfers heat, momentum and mass – imposes drag forces on plants, causing them to wave,
bend and break– mixes the air and diffusing air parcels
• Important boundary layers are:– planetary boundary layer– surface boundary layer– internal or constant flux layer– nocturnal boundary
ESPM 129 Biometeorology