Introduction to Wind Turbine Technology - Small Wind Turbines: Analysis, Design, and Application - page 15

Þ 2 þ 0 : 24 m ¼ 0 : 096 log 10 z ðÞþ 0 : 016 log 10 z ðÞ ð ð 1 : 16 Þ

An average value of m is about 1/6, so that, if P * U 3 , then P increases roughly
as h 1/2 . This leads to the rule-of-thumb that doubling the tower height will increase
the power output by 40%. This approximation ignores the fact that U(z) is the
average wind speed and the dependence of the average power output on average
wind speed may not be cubic. In fact, the example described in Chap. 12 shows
that the dependence is roughly linear. Furthermore, the wind speed dependence on
height is often more complex than suggested by the relations ( 1.14 ) and ( 1.15 ).
The logarithmic law, for example, is applicable only if z 0 does not vary signifi-
cantly around the site for a distance of probably 100 h, the site is flat, and there is
no heating or cooling of the air; in other words, the flow is ''neutrally buoyant''.
Thus U(z) may well depend on wind direction and time of day.

1.7 Turbulence and Wind Statistics

Rarely is the wind steady. It usually fluctuates in magnitude, as indicated in
Fig. 1.7 , and direction in an apparently random manner. The fluctuation level can
be measured by the turbulence intensity defined in terms of the root mean square
(rms) of the velocity fluctuations. To quantify this relation, assume that the wind
speed at any time t, is the sum of the mean U, and the fluctuating velocity u(t).
Note that the average value of u(t) is zero. The turbulence intensity is defined as

2 3

p

u 2

1 = 2

Z

T s

¼ 1
U

1
T s

4 5

u 2 dt

l u ¼ ð 1 : 17 Þ

U

0

where T s is the sampling time. The overbar on u 2 (in the square root) denotes a
time average—a convention that will be used throughout this topic, so that, for
example, u ¼ 0. In practice, the output of an anemometer [which measures
U ? u(t)] is usually sampled at a fixed frequency as in Fig. 1.7 and the integral in
( 1.17 ) is approximated by a summation.

Fig. 1.7 Typical wind speed
measured at 10 Hz at the
University of Newcastle

10

8

6

4

2

0

0 20 40 60 80 100 120 140

Time (seconds)