Characteristics of boundary layer flow

WIND ENERGY METEOROLOGY
UNIT 2

Detlev Heinemann

ENERGY METEOROLOGY GROUP
INSTITUTE OF PHYSICS
OLDENBURG UNIVERSITY
FORWIND – CENTER FOR WIND ENERGY RESEARCH

Dienstag, 19. April 2011


CHARACTERISTICS OF BOUNDARY LAYER
FLOW
Influence of surface
‣ friction, shear, turbulence
strong vertical gradients
vertical fluxes of momentum, heat, ..

Turbulence
‣ turbulent eddies are generated mechanically by strong shear as
flow adjusts to condition at surface

‣ thermal generation of turbulence through buoyancy by
destabilized stratification (--> thermal stability)



CHARACTERISTICS OF BOUNDARY LAYER
FLOW

Governing quantities

‣ wind speed („driving“ large scale wind field)
‣ surface roughness
‣ thermal stability

3


PLANETARY BOUNDARY LAYER:
GENERAL CHARACTERISTICS

Stability
‣ Structure of PBL is influenced by underlying surface and by stability
of the PBL
‣ Surface roughness
‣ Unstably stratified PBL enhances turbulence production
-> intensified exchange
-> more uniform distribution of momentum, etc.
‣ Stably stratified PBL turbulence produced by shear is suppressed
-> weak exchange
-> weak coupling with surface
.

4



Diurnal pattern
‣ Strong mixing during daytime with upward heat flux from surface
‣ Strong turbulent mixing
-> nearly uniform vertical profiles (‘mixed layer‘)
‣ top of PBL is capped by inversion
‣ inversion height rises quickly early in the morning with max. height
of few km during daytime
‣ turbulence dies out in night time when vertical heat flux turns
-> shallow stable layer near surface
‣ nocturnal boundary layer height is typicaly 50 to 200 m only
. 5



Turbulence
‣ Diffuse processes in the PBL are dominarted by turbulence
(molecular diffusion can in general be neglected)
‣ Time scales of turbulent motion: few seconds to about half an hour
‣ Length scales: millemeters to few hundred meters
(these scales have to be parameterized in large scale models)
.

6


TURBULENCE
Karman vortex streets...

...in the laboratory, for ...in the atmosphere, for a
water flowing past a cylinder cumulus-topped boundary
layer flowing past an island

Stull (2006)

7


GENERATION OF TURBULENCE

‣ Generation can be mechanically, thermally, and inertially.
‣ Mechanical turbulence, a.k.a. forced convection, results from
shear in the mean wind. It is caused by:
‣ frictional drag (slower winds near the ground than aloft)
‣ wake turbulence (swirling winds behind obstacles)
‣ free shear (regions away from any solid surface)
‣ Thermal or convective turbulence, a.k.a. free convection,
consists of plumes or thermals of warm air that rises and cold air
that sinks due to buoyancy forces.
‣ Inertial turbulence: Generation of small eddies along the edges
of larger eddies (within the turbulent cascade). Special form of
shear turbulence (shear is generated by larger eddies)
Stull (2006)

8


TURBULENCE
Spectrum of turbulent kinetic energy (TKE)

‣ The energy spectrum indicates how much of the
total TKE is associated with each eddy scale.
‣ The total TKE is given by the area under the curve.
‣ Permanent generation of TKE from shear or
buoyancy at large scales.
‣ TKE cascades through medium-size eddies to be
dissipated by molecular viscosity at the small-
eddy scale (TKE is not conserved!).

Stull (2006)

9


SPECTRUM OF HORIZONTAL WIND VELOCITY

Source: Burton, 2001 (based on van der Hoven (1957))

10


ORIGIN OF TURBULENCE

‣ Turbulence is a natural response to instabilities in the flow and
tends to reduce the instability.
‣ Example 1: Thermal instability:
Vertical turbulent motion of cold and warm air to
reduce a large vertical temperature gradient.
‣ Example 2: Dynamic instability:
Vertical shear in the horizontal wind: Turbulence
mixes the faster and slower moving air.

‣ Persistent mechanical turbulence in the atmosphere needs a
continual destabilization by external forcings.
Stull (2006)

11


STATISTICAL DESCRIPTION OF TURBULENCE
(I)

‣ Aim: Describing net effect of many eddies, rather than the exact
behavior of any individual eddy
‣ A certain point in the atmosphere (e.g., a measurement sensor) is
affected by many eddies of all sizes -> randomness
‣ But: for a given time period, say 10 minutes, the measurement
will fluctuate around a well-defined mean value
--> turbulence is quasi-random

‣ Then, the fluctuating portion of the flow is given by subtracting
the mean from the instantaneous component:

Stull (2006)

12


(II)

‣ The intensity of turbulence (in the u direction) is then defined by
the variance:

‣ If σu2 is relatively constant with time, the flow is said to be
stationary

‣ If σu2 is relatively uniform in space, the flow is said to be
homogeneous

Stull (2006)

13


(III)

‣ Fluctuations in velocity are often accompanied by fluctuations in
scalar values
‣ E.g.: rising warm air in a field of thermals (positive potential
temperature θ, positive vertical velocity w), surrounded by
regions with sinking cold air (negative θ, negative w)
‣ A measure of the amount that θ and w vary together is the
covariance (cov):

Stull (2006)

14


(IV)

‣ If warm air parcels are rising and cold parcels are sinking, as in a
thermally direct circulation, then

‣ The variance of velocity represents the kinetic energy associated
with the motions on the scale of the turbulence.
‣ Similarly, the covariance is a measure of flux due to these
motions, such as the vertical heat flux .
‣ This ‘kinematic‘ heat flux (Kms -1) is related to the usual heat flux
QH (Wm-2) by

Stull (2006)

15


16


TURBULENT KINETIC ENERGY (I)

‣ When the kinetic energy of an air parcel is ,
the specific kinetic energy, i.e. per unit mass, associated with
turbulent fluctuations is:

‣ TKE is the turbulent kinetic energy
‣ In a laminar flow TKE = 0.
‣ Larger values of TKE indicate a greater intensity of the microscale
turbulence.

Stull (2006)

17


TURBULENT KINETIC ENERGY (II)

from a simple balance we may write a simple forecast equation for
turbulence kinetic energy:

with advection by mean wind (Ad), mechanical generation (M),
buoyant generation/consumption (B), transport by turbulence
itself (Tr), and viscous dissipation (ε) approximated by

and the dissipation length scale Lε.
The dissipation rate ε will always cause TKE to decrease.
--> Turbulence is a dissipative process.
Stull (2006)

18


TURBULENT KINETIC ENERGY (III)

statically stable atmosphere:
‣ buoyancy reduces TKE by converting it to potential energy by
moving cold air up and warm air down
‣ existence of turbulence depends on the relative strengths of
mechanical generation (M) by wind shear versus buoyant
consumption (B) by static stability
‣ ratio of these two terms defines the dimensionless Richardson
number Ri::

approximated by the vertical gradients of wind and potential temperature

Stull (2006)

19


TURBULENT KINETIC ENERGY (IV)

‣ Laminar flow becomes turbulent when Ri drops below the
critical value Ric = 0.25.
‣ Turbulent flow often stays turbulent, even for Richardson
numbers as large as 1.0, but becomes laminar at larger values
of Ri.
‣ The presence or absence of turbulence for 0.25 < Ri < 1.0
depends on the history of the flow.
‣ Flows for which Ric < 0.25 are said to be dynamically unstable.

Stull (2006)

20


TURBULENT TRANSPORT & FLUXES
Covariances can be interpreted as fluxes
Ex.: idealized eddy circulation in atmosphere with a constant
gradient of potential temperature θ
turbulent heat fluxes for small-
eddy vertical mixing

in adiabatic processes, air
parcels preserve their potential
temperature θ
left: statically unstable ( )
covariance is positive
right: statically stable ( )
covariance is negative

Stull (2006)

21


VERTICAL STRUCTURE OF THE PLANETARY
BOUNDARY LAYER

geostrophic wind, ~1-2 km

decreasing turbulent effects,
change in wind direction,
small change in wind speed

~100 m

strong vertical wind shear,
almost constant wind direction

Ekman layer and Surface layer are each characterized by different physical constraints.



EKMAN LAYER

Source: http://mathsci.ucd.ie/met/msc/fezzik/Phys-Met

23


EKMAN SPIRAL

The theoretical Ekman spiral describing the height dependence of the
departure of the wind ﬁeld in the boundary layer from geostrophic
balance.
α is the angular departure from the geostrophic wind vg.

24


LOGARITHMIC WIND PROFILE

logarithmic wind profile in
logarithmic (left) and linear
(right) height scale

u(z) = u*/k ln (z/ z0)

k ≅ 0.4: von Karman constant
u*= √τ/ρ: friction velocity
z0: roughness length

here:
z0 = 0.05 m
u* = 0.4 m/s

25


LOGARITHMIC WIND PROFILE

Validity:

‣ mean values
‣ horizontally homogeneous
‣ neutral stability

26


ROUGHNESS CLASSES AND ROUGHNESS
LENGTH TABLE
Roughnes Roughness Energy Index Landscape Type
s Class Length m (per cent)

0 0.0002 100 Water surface

0.5 0.0024 73 Completely open terrain with a smooth surface, e.g.concrete runways in airports,
mowed grass, etc.

1 0.03 52 Open agricultural area without fences and hedgerows and very scattered buildings.
Only softly rounded hills

1.5 0.055 45 Agricultural land with some houses and 8 metre tall sheltering hedgerows with a
distance of approx. 1250 metres

2 0.1 39 Agricultural land with some houses and 8 metre tall sheltering hedgerows with a
distance of approx. 500 metres

2.5 0.2 31 Agricultural land with many houses, shrubs and plants, or 8 metre tall sheltering
hedgerows with a distance of approx. 250 metres

3 0.4 24 Villages, small towns, agricultural land with many or tall sheltering hedgerows,
forests and very rough and uneven terrain
3.5 0.8 18 Larger cities with tall buildings
4 1.6 13 Very large cities with tall buildings and skycrapers

27


INFLUENCE OF ROUGHNESS ON VERTICAL
PROFILE

© EWEA 2002
The vertical wind gradient for different values of surface roughness (built up area; scrub vegetation; open flat
land or sea surface).
Note that the wind power content of the wind at the same height (30 m) varies considerably.

28


INFLUENCE OF THERMAL STABILITY ON
VERTICAL WIND PROFILE

wind speed variation with height in the surface layer for different
static stabilities, plotted on linear (left) and semi-log graph (right)
Stull (2006)

29


ATMOSPHERIC STABILITY AND ADIABATIC
MOTION

Stability regimes in dry air

Dry adiabatic lapse rate:

30


DRY ADIABATIC LAPSE RATE

Rate of temperature decrease with For an adiabatic process, the first law of
height for a parcel of dry or thermodynamics can be written as
unsaturated air rising under cp dT − αdp = 0
adiabatic conditions
For an atmosphere in hydrostatic equilibrium
„adiabatic“: no heat transfer into or
out of the parcel dp = −ρgdz
When air rises (e.g., by convection) it Combining both equations (eliminate pressure):
expands due to the lower pressure.
As the air parcel expands, it does dT g
Γd = − = = −0.98 K/100m
work at the environment, but gains dz cp
no heat.
-> It loses internal energy and its with g: standard gravity
temperature decreases. ρ: density
cp: specific heat at constant pressure
The rate of temperature decrease α: specific volume
with height is 0.98 °C per 100 m.

31


GEOSTROPHIC DRAG LAW

with

G: geostrophic wind speed,
u*: friction velocity,
k: von Karman constant,
f: Coriolis parameter,
z0: roughness length, and
A and B: dimensionless functions of stability
(for neutral conditions: A=1.8, B=4.5).

Large-scale pressure differences determine the fictitious geostrophic wind,
which is representative for the wind speed driving the boundary layer (under
certain simplifying circumstances).

The geostrophic drag law relates this wind field to the boundary layer wind.
Using information about surface roughness and stability, it is then possible to
calculate the wind speed near the surface.

32

Characteristics of boundary layer flow

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Characteristics of boundary layer flow