ERCOFTAC Bulletin - Centre Acoustique

Analytical and Numerical Wxtensions of Amiet’s
Incoming Turbulence Noise Theory to Account for
Near-field Effects
1 Introduction
J. Christophe 1 , K. Kucukcoskun 2,3 and C. Schram 1
1
von Karman Institute for Fluid Dynamics,
1640 Rhode-St-Genèse, Belgium
julien.christophe@vki.ac.be, christophe.schram@vki.ac.be
2
LMS International, 3001 Leuven, Belgium
3
Current address: Laboratoire de Mécanique des Fluides et d’Acoustique,
UMR CNRS 5509 & Ecole Centrale de Lyon, 69134 Ecully Cedex, France
korcan.kucukcoskun@ec-lyon.fr
Flow-generated sound is nowadays a serious concern in
many engineering applications. The noise pollution generated
is particularly problematic in transport industry.
In aeronautical applications, the noise emitted by wings
and high-lift devices [5, 20, 22, 23], is a significant component
of the overall noise radiated. In automotive industry,
the cooling fan systems [6] are examples of flowgenerated
noise that are considered as key factors in the
overall appreciation of a vehicle. In other domains of industrial
applications as in wind turbines [10], laptops or
desktops cooling fan systems [15], or Heating, Ventilating
and Air Conditioning (HVAC) [7, 16], the aerodynamic
noise is also a major concern.
These applications have in common that the generic
aerodynamic noise production mechanism is due to the
interaction between an airfoil with the turbulent flow
in which it is submerged. In some instances the turbulence
is generated upstream of the airfoil and is convected
over it with the main flow, in which case we refer
to an incoming turbulence noise production mechanism.
In other configurations the blade is placed in a
laminar incoming flow, but turbulence is still generated
in the boundary layers that develop along the chord of
the blade, and is scattered by the blade trailing edge.
This is commonly referred to as trailing-edge noise or
self-noise. Both mechanisms have been considered by
Amiet [3, 4], with an analysis based on linearized airfoil
theory to model the lift variation of the airfoil subjected
to incoming or self-generated turbulence in a first step,
and on Curle’s aeroacoustic analogy [9] to predict the
radiation of the equivalent dipoles in a second step.
A number of simplifying hypotheses are made in the
derivation of these theories. It is for example often assumed
that the listener is located in the geometrical and
acoustical far-field of the blade. While valid for certain
applications (e.g. wind-turbines), this hypothesis
is clearly violated for a fan confined within a casing, if
we want to evaluate the incident field on the casing to
compute the acoustic scattering by the rotor neighboring
surfaces. This aspect constitutes the focus of this work,
in which we investigate acoustical and geometrical nearfield
effects that are often ignored when applying Amiet’s
theory. The work presented in this paper is focused on
the incoming turbulence noise problem, but the proposed
improvements could also be applied on the trailing-edge
noise theory [4, 17, 21] as well.
2 Amiet’s theory for incoming
turbulence noise
An airfoil of chord C = 2b and span 2d is placed in a
turbulent fluid with a mean flow velocity U. The x axis
is the streamwise/chordwise direction, aligned with the
convection speed U, y is the spanwise direction and z
the crosswise direction. The origin of the coordinates
system is at the center of the airfoil. The turbulence is
assumed to be frozen and represented in terms of its spectral
wavenumber components, kx and ky, as illustrated
in Fig. 1.
Figure 1: Representation of a single skewed gust sweeping
over the airfoil.
The airfoil is assumed to be a flat plate of zero thickness,
and linearised theory is applied. For this particular
problem, an analytical solution is found for the transfer
function between the impacting gust and the airfoil
pressure jump. This is obtained by iteratively solving
scattering problems at the airfoil edges. The main
leading-edge scattering obtained by assuming the airfoil
extends toward infinity in the downstream direction, is
then corrected by a trailing-edge back-scattering contribution
which fully accounts for the finite chord length
[2, 19, 18].
The acoustic response of the airfoil subjected to incoming
turbulence involves the radiation of spanwise and
chordwise distributed dipoles obtained from the pressure
jump across the airfoil surface. The airfoil and the coordinate
system are represented on Fig. 2. The coordinates
ERCOFTAC Bulletin 90 45