Environmental Engineering Reference
In-Depth Information
2
4
6
=+ + + +
(35)
Substituting the decomposed variables in the compressible Navier-Stokes equa-
tions and grouping terms with the same order in
e
, equation sets with different
approximations are derived. The zeroth order approximations
e
-2
describe the ther-
modynamic fi eld, the fi rst order approximations
e
-0
the hydrodynamic fi eld and the
second order approximations
e
2
describe compressibility effects. The results of
this method can be written in the form of Lighthill's acoustic analogy. For further
details the reader should consult [28].
Once the acoustic sources are determined, there are two main strategies for the
computation of the acoustic fi eld: analytical computation of the sound pressure
level at the observer position or numerical computation of the noise propagation.
When the acoustic fi eld is governed by a non-homogeneous wave equation having
elementary acoustic sources (monopoles, dipoles or quadrupoles) of the form:
TT
e
T
e
T
e
T
0
1
2
3
2
2
1
∂
pxt
'(,)
∂
pxt
'(,)
(36 )
−
=
Qxt
(,)( )
d
f
2
2
∂∂
xx
c
∂
t
i
i
and
f
is the surface over which the sources are distributed, using the free-space
Green's function
d
(
g
)/4
p r
, with
g
=
t
−
t
+
r
/
c
the solution can be written in an
integral form:
t
∞
Qy
(,)( )()
td
f
d
g
∫∫
4'(, )
p
pxt
=
dd
y
t
(37 )
r
−∞ −∞
In eqn ( 37 )
x
is the observer position vector,
y
the source position vector,
r
the
distance between the source and the observer and
t
is the source time. This equa-
tion can be integrated after variable transformations. Further details can be found
e.g. in [24] where the Ffowcs Williams
Hawkings equation is solved to determine
the noise generated by helicopter rotors. Recently, Filios
et al.
[29] combined a low
order panel method with prescribed wake-shape and the integration of the Ffowcs
Williams
−
Hawkings equations to predict the noise generated by the NREL wind
turbine. In their work only the monopole and dipole sources have been considered,
the noise generated by velocity fl uctuations (quadrupoles) being neglected.
This previously described integral approach is based on the assumption that the
observer is far enough to consider the waves propagating from individual sources
spherical. Also, propagation effects (refl ection, absorption, refraction) are neglected.
An alternative method is to solve numerically the CAA equation(s) on an acous-
tic grid (which does not need to be identical to the fl ow grid). Although this
approach is computationally more expensive, it allows the consideration of propa-
gation effects and gives a more detailed three-dimensional picture of the radiated
acoustic fi eld. Moroianu and Fuchs used Large Eddy Simulations to compute the fl ow
fi eld around a single wind turbine [30]. The acoustic sources provided by the LES
computations have been used in a separate acoustic solver to compute the corre-
sponding noise generated by the wind turbine. Figure 17 shows an instantaneous
snapshot of the radiated acoustic fi eld and the sound pressure levels around the
−
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