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Aircraft noise generation and assessment

Installation: numerical investigation
  • M. LummerEmail author
Original Paper
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Abstract

Acoustic installation effects are considered as scattering problem. Two methods to investigate them are presented. First, a fast multipole method boundary element method (FMM-BEM) which can solve the Helmholtz wave equation for full scale aircraft configurations at frequencies of some kHz. There, low Mach number potential mean flow fields can be taken into account by a so-called Taylor transformation. Second, a discontinuous Galerkin method (DGM) which solves acoustic perturbation equations (APE) for realistic mean flow fields is presented. DGM calculations are very expensive and can be performed for full scale aircrafts only at low frequencies. Its main purpose so far is to assess the accuracy of the FMM for selected cases. The Taylor-transformed Helmholtz equation is derived and the fundamentals of the FMM are introduced. Some details of the DLR FMM code FMCAS are given. The basic DGM equations are derived for the APE and some implementation details of the DLR DGM code DISCO++ are discussed. For generic geometries results of the FMM and DGM are compared and the limits of the Taylor transformation are shown. Finally, scattering results for a 1 kHz Monopole at a full scale aircraft geometry will be presented.

Keywords

Airframe noise Acoustic shielding Acoustic scattering Discontinuous Galerkin method Fast multipole method 

List of symbols

\({c_{0}(\mathbf x )}\)

Mean flow sound speed—assumed to be constant

\({k}= \frac{\omega }{c_{0}}\)

Wave number

\({{M}}= \frac{v_{\infty }}{c_{0}}\)

Mean flow Mach number

\({R_{{\text {d}}}}\)

Radius of directivity circle

\(\mathbf{v _{0}(\mathbf x )}= \nabla \varPhi _{0}\)

Mean flow velocity

\(\mathbf{v _{\infty }}= (c_{0} M,0,0)\)

Free stream mean flow vector

\({v_{\infty }} = c_{0} M\)

Reference velocity

\({\lambda }\)

Wavelength

\({\rho _{0}(\mathbf x )}\)

Mean flow density—assumed to be constant

\({\theta }\)

Angular position on directivity circle

\({\omega }\)

Circular frequency of acoustic disturbances

FMM specific symbols

\({\mathscr {B}_{x}} = 1+\alpha \frac{\partial {}}{\partial n_{x}}\)

Burton–Miller operator

\({\mathscr {C}_{x}}= \mathscr {B}_{x}/2\)

on surface

\({\mathscr {D}_{y}}\)

Shorthand for operator \(\frac{\partial}{\partial n_{y}} - Y(\mathbf y )\)

\({F_\mathbf{a }({\hat{\mathbf{s }}})}\)

FMM far-field signature function

\({{G(\mathbf x ,\mathbf y )}}\)

Free field Green’s function

\({h_{l}^{(1)}(z)}\)

Spherical Hankel function of the first kind

\({\mathscr {K}_{\partial \varOmega }(\mathbf x )}\)

Integral operator (integration over surface \(\partial \varOmega\))

L

Cut-off index of FMM transfer function

\({M_{L,{\hat{\mathbf{s }}}_{}}(\mathbf u )}\)

FMM transfer function

\({N_\mathbf{c }({\hat{\mathbf{s }}})}\)

FMM near-field signature function

\(\mathbf{n _{y}}\)

Surface normal at point \(\mathbf y\)

\({\mathscr {O}}\)

Size of scattering surface \(\partial \varOmega\)

\({P_{l}(x)}\)

Legendre polynomials

\({p(\mathbf x )}\)

Fourier amplitude of acoustic pressure

\({{\hat{\mathbf{s }}}}\)

Surface point on unit sphere

\({\mathscr {T}(\mathbf x )} = e^{-ik{M}\varPhi _{1}{}}\)

Taylor phase factor

\(\mathbf{v (\mathbf x )}= \nabla \phi\)

Fourier amplitude of acoustic velocity

\({W_{j}^{m}}\)

Quadrature integration weight on surface element \(\varOmega _{j}\)

\(\mathbf{x _{i}}\)

Center point of surface element \(\varOmega _{i}\)

\({Y(\mathbf x )}\)

Wall admittance

\(\mathbf{y _{jm}}\)

Quadrature collocation point on surface element \(\varOmega _{j}\)

\({\alpha }\)

Burton–Miller constant

\({\varPhi _{0}(\mathbf x )} = v_{\infty }\varPhi _{1} = c_{0} M \varPhi _{1}\)

Mean flow velocity potential

\({\varPhi _{1}(\mathbf x )}= \frac{\varPhi _{0}}{v_{\infty }}\)

Mean flow velocity potential scaled by reference velocity

\({{\tilde{\phi }}(\mathbf x ,t)}\)

Acoustic velocity potential

\({\phi (\mathbf x )} = \mathscr {T}\psi\)

Fourier amplitude of acoustic velocity potential

\({\psi (\mathbf x )}= \mathscr {T}^{-1}\phi\)

Taylor-transformed Fourier amplitude of acoustic velocity potential

\({\varOmega }\)

Computational domain

\({\partial \varOmega }\)

Boundary of computational domain

\({\Box _\mathbf{c }}\)

Cube with center \(\mathbf c\)

DGM specific symbols

\({A_{\alpha \beta }^{j}}\)

Flux matrices

\({A_{\alpha \beta }^{j\mathfrak {n}}}\)

Value of flux matrix at collocation point \(\mathbf x _{\mathfrak {n}}\)

\({D_{\alpha \beta }^{p\mathfrak {n}}}= n_{j}^{p}A_{\alpha \beta }^{j\mathfrak {n}} = \mathbf D ^{p\mathfrak {n}}\)

Flux matrix on face p and collocation point \(\mathfrak {n}\)

\({F_{\alpha }^{j}}= A_{\alpha \beta }^{j} u_{\beta }\)

Flux vectors

\({F_{\alpha }^{p\mathfrak {n}}}= D_{\alpha \beta }^{p\mathfrak {n}}u_{\alpha }^{\mathfrak {n}}\)

Flux vector on face p and collocation point \(\mathfrak {n}\)

\({H_{\alpha \beta }^{p\mathfrak {n}\pm }}= \mathbf H ^{p\mathfrak {n}\pm }\)

Upwind flux matrices

\({\mathsf {M}_{}(\mathbf x )}\)

Galerkin test function

\({\mathsf {N}_{\mathfrak {l}}(\mathbf x )}\)

Lagrange interpolation polynomials on tetrahedron

\(\mathbf{R ^{p\mathfrak {n}}}\)

Matrix of right eigenvectors of \(\mathbf D ^{p\mathfrak {n}}\) on face p and collocation point \(\mathfrak {n}\)

\(\mathbf{x _{\mathfrak {n}}}\)

Collocation points on tetrahedron

\({\pmb {\varLambda }^{p\mathfrak {n}}}\)

Matrix of eigenvalues on face p and collocation point \(\mathfrak {n}\)

Super- and subscripts

\({i,j,\dots }\)

Component of space vector

\({\alpha ,\beta ,\dots }\)

Component of variable vector

\({\mathfrak {l},\mathfrak {m},\dots }\)

Index of cell collocation point

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Copyright information

© Deutsches Zentrum für Luft- und Raumfahrt e.V. 2019

Authors and Affiliations

  1. 1.Department of Technical Acoustics, Institute of Aerodynamics and Flow TechnologyGerman Aerospace Center (DLR)BrunswickGermany

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