ERCOFTAC Bulletin - Centre Acoustique

spanwise near-field expression (9), neglecting chorwise effects
and taking spanwise effects only in the scaling term
defined in Eq. (7) is compared to the direct numerical
integration of formulation (5), without any geometrical
assumptions. The maximum difference between formulations
is up to 5 dB for z < d/20, where chordwise effects
start to appear, but improves highly the solution compared
to the far-field Amiet’s solution. If a direct numerical
integration of formulation (5) is performed with similar
assumptions than in formulation (9), identical results
are obtained between both formulations. The spanwise
effects neglected in the exponential term to obtain formulation
(9) have then a negligeable effect on the sound
results. The near-field expression (9) is attractive for
geometrical near-field computation due to its facility of
implementation and solution robustness, compared to direct
integration requiring higher computational time to
obtain similar accuracy due to the low convergence of
Monte Carlo techniques. Nevertheless the direct integration
of the Amiet’s general formulation (5) is a valuable
tool to test the different assumptions/simplifications
that could be introduced in the Amiet’s theory and to
quantify the level of accuracy relative to the different
assumptions.
3.2 Influence of acoustical far-field
assumptions
The same test case is used to evaluate the acoustical farfield
approximation introduced in formulation (5) compared
to the general formulation (4). Figure 3 (bottom)
shows the results obtained using Amiet’s theory at different
observer distances from the airfoil including geometrical
near-field terms and acoustical near-field terms, together
with combinations of the different simplifications.
A first observation is that the differences are appearing,
as expected, for a distance comparable to the magnitude
of the acoustical wavelength. The difference observed
using acoustical near-field terms on formulation (11) is
around 10 dB close to the airfoil while the difference observed
on formulation (4) is around 6-7 dB. It seems then
that the influence of acoustical near-field terms are attenuated
when geometrical near-field terms are taken into
account. Finally, we should notice that, for the particular
frequency selected (f = 2000 Hz), avoiding the use
of acoustical near-field terms in the computation can be
acceptable while for smaller frequencies, the deviations
could increase dramatically, the influence of the acoustical
near-field terms increasing with lower frequencies.
4 Concluding remarks
In this work, we have investigated the influence of the geometrical
and acoustical near-field effects that could be
important in case of observers close to the airfoil surface.
Those effects have been studied through an implementation
of a generalised formulation of Amiet’s theory using
the Monte Carlo technique to solve the five dimensional
integral involved in the formulation. This generalised
formulation revealed to be an efficient tool to verify the
assumptions and estimate the potential of simplified formulations
based on Amiet’s theory taking geometrical
near-field effects into account. Nevertheless, the Monte
Carlo technique used in the generalised formulation requires
a large computational effort to obtain accurate
results, but the implementation would be easily used in
parallel to improve its efficiency.
Acknowledgements
The work presented in this paper has been sponsored by
the European Commission under the framework of the
FP7 Collaborative Project ECOQUEST (Grant Agreement
no 233541). The authors are thankful to Prof.
Michel Roger for fruitful discussions on the near-field
generalization of Amiet’s theory.
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48 ERCOFTAC Bulletin 90