We've seen that wind can be generated by both global and local sources.
We will now consider how that wind is modified by interaction with
the Earth's surface.
The winds generated by global and local wind mechanisms interact with
the natural and manmade features of the Earth's surface.
At the very surface, the wind speed is zero.
At some distance from the ground,
the wind is essentially unaffected by the surface features.
At this altitude, the wind speed is determined by
the resultant of a pressure gradient force and the Coriolis force.
This wind, free of surface effects, is called the geotropic, geostropic wind.
The region between the surface and
the geostropic wind, is called the surface boundary layer.
Sheer forces at the boundary between the surface and the air generate turbulence
which which enhances mixing and dissipates the wind's energy at the surface boundary.
Intuitively it would make sense that the physical dimensions of
any surface features would have a big impact on the surface boundary layer.
In this image, the rectangular boxes represent the generic surface roughness
element, recognizing that the surface roughness elements can,
in practice, have all kinds of shapes.
Larger surface roughness elements have more surface areas to
generate sheer force which, in turn, generates more turbulence.
Once again intuitively, you know from experience with
rough surfaces like sandpaper that the friction is higher with rougher sandpaper.
Here you can think of surface roughness as indicative of surface friction or
resistance to airflow over the surface.
Roughness length is a physical dimension that is used to
characterize surface roughness.
Lettau developed an empirical relationship to calculate the roughness length for
the earth's surface boundary layer, shown here.
The parameters involved in this equation include the height of
the roughness element h, S,
which is the cross-sectional area of the roughness element facing the wind, and
a sub h, the average horizontal area available to each roughness element.
So this is in a horizontal plane not facing the wind.
In this equation, the roughness element is assumed to be si, solid.
Connecting the empirical formula with our very basic illustration of
surface roughness elements, here is a simple example.
Assume that the rectangular boxes represent houses in a suburban or
country village location.
Starting with the lot dimensions,
we have the width of the lot 30 meters in this example.
The depth of the lot 30 meters so the area for the lot is 900 meters squared.
The house dimensions, the width of the house is 20 meters,
the height is 7 meters which gives us an S value of 140 meters squared.
So substituting into the formula for roughness link we find that Z0 is
equal to 0.54 meters.
Note that in this example H, the height of the house appears twice in the numerator.
So in fact the roughness length depends on the square of the house heighth.
The various types of surface terrain have been divided into four roughness classes,
water areas, class one, open country with few surface features,
class, class one, I think I mis, missaid water areas was class zero.
Farmland with buildings and hedges class two.
Farmland with many trees, forests, villages class three.
The table provides a representative roughness length value for
each surface roughness class.
In reality there aren't just discrete values for surface roughness.
Roughness length varies over a continuum,
the table also includes a range of surface roughness values for
terrain that readily fits the description for a particular class.
You can see that the 0.54 value calculated for roughness length in the example for
suburban houses, falls within the range for class three.
So far in this discussion the surface roughness elements have been assumed to
be solid.
Some natural roughness elements, vegetation for
example are not solid to the wind, but have some porosity on the order of 0.5.
Porosity is easily accounted for by multiplying the value of roughness length
calculated using our simple empirical formula, times the porosity.
Note that for porous surface roughness elements there can be
seasonal variation in roughness length.
For example bare trees in winter have much higher porosity to wind,
than trees full of leaves in the summer.
You might gave noticed in the previous table that was no
roughness class for cities.
A very interesting situation arises when the surface roughness elements are very
closely spaced, as in cities.
When the roughness elements are so
close together, the flow is effectively lifted over them.
In this situation, the city behaves somewhat like a hill with air
flow over it.
Thus the bottom of the wind speed profile is displaced some distance upwards,
the displacement length, from the ground.
This displacement length is somewhere between half of
the maximum height of the buildings and the maximum height itself.
The exact value depending on the specific characteristics of the buildings,
their spacing and other parameters.
For a variety of reasons including this effect, cities are not a good location for
a wind turbine.
A roughness length of one meter provides a very,
very crude estimate of the surface roughness of a city.
Let's return to the wind speed profile scene in the second slide,
the slide that introduced the surface boundary layer.
It is helpful to be able to represent the wind speed profile by an equation.
Prandtl developed a logarithmic expression to represent dependence of
wind speed on height in a turbulent boundary layer, presented here.
There are three parameters in this equation.
The first is the roughness length, the length scale that characterizes surface
roughness which we've already discussed in the previous slides.
The two other parameters are V star the friction velocity and
k the Karman constant.
Approximate values of both of those are shown on the slide.
However, the constants, values of the constants V star and k
are difficult to determine accurately, so the utility of this equation is limited.
Although V star and k are difficult to evaluate.
The logarithmic form of that equation can be used to calculate wind velocity at
different heights based on a measured wind velocity at a reference height.
So that's the formula shown here.
You can see now that the only parameter remaining is the roughness length which we
can readily characterize.
So that if we have the wind speed measured at a height,
Zr, and we know the velocity of the wind at that same height, the wind speed at
that same height, we can calculate the wind speed at any other height.
Here is a frequently encountered example that uses that formula.
Wind speed measurements are tyki,
typically taken on instruments mounted on a ten meter height tower.
Yet wind turbines are mounted at much higher heights,
30 meters 50 meters, 80 meters or 90 meters.
The logarithmic equation can be to estimate wind speeds at
these heights using the wind speed measurements taken at ten meters height.
The plot shows the normalized wind speed normalized by the wind speed measured at
ten meters for four surface roughness classes and
including a very crude estimate for a city using the roughness length of one meter.
You can see that the surface roughness substantially reduces the wind speed at
lower heights.
Just for information, there is an alternative to the logarithmic wind
speed function, and that's a simple power law function shown here.
Here the exponent alpha is used to characterize the surface roughness and
fulfills this same role as the roughness length and the logarithmic equation.
I prefer the logarithmic form as it has a basis in
the fluid mechanics of boundary layer flows.
Let's now consider the practical implications of the surface
boundary layer.
This figure shows a logarithmic profile of the wind,
normalized by the geostropic, geotropic, geostropic wind.
The height at which the boundary layer affects used to be
significant varies with conditions.
Here a geostropic wind height of 500 meters was chosen for
illustration purposes.
Looking at the profile one can draw the obvious conclusions that,
higher is better for placing a wind turbine.
The further up the boundary layer the higher the wind speed available to
the turbine.
In practice there are limitations on when turbine height.
The greatest height currently in use is a 90 meter rotor hub height for
off shore wind turbines.
And 90 meter height is indicated by the dashed line.
A second conclusion is that open area is better.
So, moving to class one on the right to class three on the left
at constant hub height you can see that the wind speed available at
wind turbine hub height is reduced with increased surface roughness.
So, open area is better.
The surface wind profile also has an implication on wind turbine performance
beyond the simple effect on wind speed at rotor when turbine rotor hub height.
This plot shows how the logarithmic wind profile varies relative to
the wind speed at rotor hub height for three different roughness classes.
As wind turbines grow in size,
the total height spanned by the rotor is quite significant.
Horizontal lines on the plot show the top and bottom hubs for
an 82 meter diameter rotor, mounted at an 80 meter hub height.
The plot shows that there is signification wind speed variation from rotor bottom to
top, 15% for class 1 roughness, surface roughness,17% for
class 2, and 24% for class 3.
This wind speed variation from bottom to top of the rotor is a challenge for
wind turbine blade designers as they try to optimize performance with wind speed.
Blade performance is a topic for part B of this course.
Let's conclude by summarizing.
Surface roughness elements, both natural and
man-made, generate turbulence, which dissipates the energy of the wind.
The result of that dissipation is to reduce wind speed as
the ground is approached.
Surface roughness can be characterized by roughness length.
So, roughness length depends on the nature of the terrain and there are four
roughness classes identified for most common types of terrain.
A logarithmic profile allows wind speed at various heights to
be estimated from a measured wind speed at one height.
Finally, the surface boundary layer has some obvious implications for
the sighting of wind turbines.
First higher is better.
You want the rotor exposed to wind speeds possible.
Second, siting in an open area is better surface roughness reduces the wind
speed available to the wind turbine.
Please try the questions that follow to give you some practice with
surface boundary layer concepts.