Wind turbine acoustics - Wind Power Generation and Wind Turbine Design - page 169

Figure 15: The value of the shape function in the solid and fl uid regions.

second derivative of the Lighthill stress tensor, T ij = r u i u j , which acts as a source

of quadrupole type.
Lighthill's analogy does not take into account the noise generated by solid
surfaces. To remedy this defi ciency, Ffowcs Williams and Hawkings (cited in
[24]) extended Lighthill's analogy by representing a solid object by a sur-
face S defi ned as f ( t,x,y,z ) = 0 (see Fig. 15) and extending the fl uid motion
also into the object domain but restricting the velocity to the speed of the
object itself. Using the generalized function theory the following equation has
been derived:

2

⎛ ⎞

2 2

∂ ( r uu )

∂ r ' ∂ r ' ∂ ∂ f ij

2
0

− c = − p d () f

⎜ ⎟

ij

2

∂∂ xx ∂∂ xx ∂ x ∂ x

∂ t

⎝ ⎠

i i i j i j

(11 )

⎛ ⎞

∂ ∂ f

+ rd uf ()

⎜ ⎟

0 i

∂ t ∂ x

⎝ ⎠

i

The fi rst term on the right-hand side is identical to the RHS of eqn (10) and is
the Lighthill stress tensor. The second term on the RHS is proportional to the stress
tensor p ij , refl ecting the force which the object exerts on the fl uid and it has a
dipole character. The third term on the RHS is a monopole term and is proportional
to the acceleration of the fl uid by the non-stationary solid surface.
Based on these theoretical models, researchers deduced semi-empirical models
used for the prediction of the noise generated by non-rotating and rotating air-
foils. The fi rst models were developed to predict the noise generated by airplane
wings and helicopter rotors. Later, these models were adapted for wind turbine
applications. With the increase of the available computational power and evolu-
tion of the aeroacoustic theory more and more complex models were developed