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

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

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

Schematic and Evolution of Boundary Layers

LaminarBoundary Layer

TurbulentBoundary Layer

ESPM 129 Biometeorology

Transition from Laminar to Turbulent Flows

http://www.thtlab.t.u-tokyo.ac.jp/index.html

ESPM 129 Biometeorology

Breakup of Stable Boundary Layer

ESPM 129 Biometeorology

General Attributes of Turbulence

• Complex– Spectrum of scales– Highly Organized– Chaotic

• Mechanical turbulence– Produced by Shear

• Convective Turbulence– Produced by Buoyancy

ESPM 129 Biometeorology

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

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

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

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

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

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

Frictional Shear Stress

τ ρν µ=∂∂

=∂∂

uz

uz

ESPM 129 Biometeorology

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

Is Atmospheric Flow Laminar or Turbulent?

Re = ⋅d uν

Re .=

⋅=−

01 110

10005Leaf

Re .=

⋅=−

30 310

900005

Corn field

ESPM 129 Biometeorology

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

Navier-Stokes Equation

∂∂

+∂∂

+∂∂

+∂∂

≈ −∂∂

−∂∂

+∂∂

+∂∂

ut

u ux

v uy

w uz

px

ux

uy

uz

1 2

2

2

2

2

2ρµρ

[ ]

ESPM 129 Biometeorology

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

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

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

∂∂

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

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

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