Stochastic modelling of leading-edge noise in time-domain using vortex particles

Abstract The interaction of a wing/blade profile with turbulent inflow is one of the main sources of sound generation in turbomachines. There are already several noise prediction methodologies available which either appear not to account for the influence of geometrical and flow parameters on noise generation or have been able to account for a remarkably limited extent due to the requirement of high-performance computing for flow calculations. This leads to the motivation of the current paper, which presents a low-cost and easy-to-use noise prediction methodology based on the statistical modelling of the inflow turbulence and Lookup Table (LUT) approach for aeroacoustic design and optimisation. The development of the statistical method is divided into three parts; namely – i) calculating the background flow, ii) modelling of statistically optimised inflow disturbance, iii) computing the far-field sound pressure for individual vortex passages and superpose them linearly – this step involves repeated computation of identical vortex passages and can be therefore easily sped up using a database approach. In the framework of this work, a new approach to model the inflow turbulence using vortex particles characterised by shape functions, based on waveforms, is presented. The idea is to not conduct a time-dependent unsteady calculation of the flow field in real-time, instead to consider the mean flow around the profile in the computational domain, in which the vortex particles are convected to realise a statistical turbulent signal. The convection of these vortex particles, also, does not take place in the real-time calculation, instead, vortices of every possible size and strength are convected in a similar domain with a specific airfoil, and the acoustic radiation due to their interaction with the airfoil are computed and stored in a database. The far-field noise is predicted using Curle’s formulation. The generated database is accessed using the LUT approach to rapidly extract the acoustic signals. Through this approach, the influence of geometrical as well as flow parameters on the noise generated by airfoils can be quantified without requiring to conduct a numerical simulation every time for a new set of geometrical and flow variables. In the article, the application of the method for different blade profiles is shown, and the results obtained are compared with the standard literature.

[1]  G. Vatistas,et al.  A simpler model for concentrated vortices , 1991 .

[2]  Allan Larsen,et al.  Discrete vortex method simulations of the aerodynamic admittance in bridge aerodynamics , 2010 .

[3]  Pierre Sagaut,et al.  Large-eddy simulation for acoustics , 2007 .

[4]  P. S. Johansson,et al.  Generation of inflow data for inhomogeneous turbulence , 2004 .

[5]  Michael L. Jonson,et al.  Prediction of high frequency gust response with airfoil thickness effects , 2013 .

[6]  S. Pope.  Turbulent Flows: FUNDAMENTALS , 2000 .

[7]  M. Dieste,et al.  Random particle methods applied to broadband fan interaction noise , 2012, J. Comput. Phys..

[8]  Anurag Agarwal,et al.  Prediction method for broadband noise from unsteady flow in a slat cove , 2006 .

[9]  Cheolung Cheong,et al.  Time-domain inflow boundary condition for turbulence-airfoil interaction noise prediction using synthetic turbulence modeling , 2015 .

[10]  A. Vulpiani,et al.  Chaos: From Simple Models To Complex Systems , 2009 .

[11]  T. Kármán.  Progress in the Statistical Theory of Turbulence , 1948 .

[12]  Khaled Ibrahim Tolba,et al.  Modelling of inflow-conditions for vortex particle methods to simulate atmospheric turbulence and its induced aerodynamic admittance on line-like bluff bodies , 2018, International Journal of Computational Fluid Dynamics.

[13]  Xue-Song Bai,et al.  A fully divergence-free method for generation of inhomogeneous and anisotropic turbulence with large spatial variation , 2014, J. Comput. Phys..

[14]  D. L. Hawkings,et al.  Sound generation by turbulence and surfaces in arbitrary motion , 1969, Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences.

[15]  S. Obi,et al.  Vortex Methods for the Simulation of Turbulent Flows: Review , 2011 .

[16]  P. E. Doak,et al.  Acoustic radiation from a turbulent fluid containing foreign bodies , 1960, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences.

[17]  R. Amiet.  Noise due to turbulent flow past a trailing edge , 1976 .

[18]  Fernando Gea-Aguilera,et al.  Synthetic turbulence methods for computational aeroacoustic simulations of leading edge noise , 2017 .

[19]  J. Gershfeld,et al.  Leading edge noise from thick foils in turbulent flows , 2004 .

[20]  Ideal vortex motion in two dimensions: Symmetries and conservation laws , 1978 .

[21]  W. Desmet,et al.  Panel method for turbulence-airfoil interaction noise prediction , 2012 .

[22]  Eugene Kazantsev,et al.  Parameterizing subgrid scale eddy effects in a shallow water model , 2017 .

[23]  L. Rosenhead.  The Formation of Vortices from a Surface of Discontinuity , 1931 .

[24]  Umr Cnrs,et al.  A Stochastic Approach To Compute Subsonic-Noise Using Linearized Euler's Equations* , 1999 .

[25]  Allan Larsen,et al.  On estimating the aerodynamic admittance of bridge sections by a mesh-free vortex method , 2015 .

[26]  F. Thomsen,et al.  POTENTIAL EFFECTS OF OFFSHORE WIND FARM NOISE ON FISH , 2008 .

[27]  William J. Devenport,et al.  Sound radiation from real airfoils in turbulence , 2010 .

[28]  Andrew J. Majda,et al.  Vortex methods. I. Convergence in three dimensions , 1982 .

[29]  Lipeng Lu,et al.  Improved vortex method for large-eddy simulation inflow generation , 2018 .

[30]  Ole H. Hald,et al.  Convergence of Vortex methods for Euler's equations, III , 1987 .

[31]  Effect of Leading-Edge Thickness on High-Speed Airfoil-Turbulence Interaction Noise , 2011 .

[32]  Massimo Gennaretti,et al.  A Unified Boundary Integral Methodology for Aerodynamics and Aeroacoustics of Rotors , 1997 .

[33]  Sébastien Candel,et al.  Stochastic approach to noise modeling for free turbulent flows , 1994 .

[34]  T. Geyer,et al.  Numerical investigation of noise generation by rod-airfoil configuration using DES (SU2) and the FW-H analogy , 2019, 25th AIAA/CEAS Aeroacoustics Conference.

[35]  Thomas F. Brooks,et al.  Airfoil self-noise and prediction , 1989 .

[36]  Jae Wook Kim,et al.  An advanced synthetic eddy method for the computation of aerofoil-turbulence interaction noise , 2015, J. Comput. Phys..

[37]  Chaitanya Paruchuri,et al.  Aerofoil geometry effects on turbulence interaction noise , 2015 .

[38]  Makoto Taiji,et al.  42 TFlops hierarchical N-body simulations on GPUs with applications in both astrophysics and turbulence , 2009, Proceedings of the Conference on High Performance Computing Networking, Storage and Analysis.

[39]  S. Benhamadouche,et al.  A synthetic-eddy-method for generating inflow conditions for large-eddy simulations , 2006 .

[40]  Michaela Herr,et al.  RANS/CAA based prediction of NACA 0012 broadband trailing edge noise and experimental validation , 2009 .

[41]  J. Katz,et al.  Low-Speed Aerodynamics , 1991 .

[42]  Alexandre J. Chorin,et al.  Discretization of a vortex sheet, with an example of roll-up☆ , 1973 .

[43]  R. Benzi,et al.  The lattice Boltzmann equation: theory and applications , 1992 .

[44]  E. Sarradj,et al.  Fluctuating inflow conditions for time-domain boundary element method for airfoil-turbulence interaction noise , 2018 .

[45]  Antonello Provenzale,et al.  Elementary topology of two-dimensional turbulence from a Lagrangian viewpoint and single-particle dispersion , 1993, Journal of Fluid Mechanics.

[46]  William J. Devenport,et al.  Panel methods for airfoils in turbulent flow , 2010 .

[47]  Petros Koumoutsakos,et al.  Vortex Methods: Theory and Practice , 2000 .

[48]  Giovanni Bernardini,et al.  Novel Boundary Integral Formulation for Blade-Vortex Interaction Aerodynamics of Helicopter Rotors , 2007 .

[49]  I. J. Sharland.  Sources of noise in axial flow fans , 1964 .

[50]  William J. Devenport,et al.  Unsteady loading on an airfoil of arbitrary thickness , 2009 .

[51]  Stéphane Moreau,et al.  On the use of a uniformly valid analytical cascade response function for fan broadband noise predictions , 2010 .

[52]  Stewart A. L. Glegg,et al.  The response of a swept blade row to a three-dimensional gust , 1999 .

[53]  R. Amiet.  Acoustic radiation from an airfoil in a turbulent stream , 1975 .

[54]  Sancho,et al.  Stochastic generation of homogeneous isotropic turbulence with well-defined spectra. , 1993, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[55]  Piet Hut,et al.  A hierarchical O(N log N) force-calculation algorithm , 1986, Nature.

[56]  Ray Hixon,et al.  Toward low‐noise synthetic turbulent inflow conditions for aeroacoustic calculations , 2013 .

[57]  W. Schröder,et al.  Acoustic perturbation equations based on flow decomposition via source filtering , 2003 .

[58]  Michel Roger,et al.  Effect of Angle of Attack and Airfoil Shape on Turbulence-Interaction Noise , 2005 .

[59]  J. Gordon Leishman,et al.  Free-Vortex Filament Methods for the Analysis of Helicopter Rotor Wakes , 2002 .

[60]  N. Curle.  The influence of solid boundaries upon aerodynamic sound , 1955, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences.

[61]  G. Hancock,et al.  The unsteady motion of a two-dimensional aerofoil in incompressible inviscid flow , 1978, Journal of Fluid Mechanics.

[62]  J. Hess,et al.  Calculation of potential flow about arbitrary bodies , 1967 .

[63]  William John Macquorn Rankine,et al.  A manual of applied mechanics , 2022 .

[64]  Matthew R. Myers,et al.  Influence of camber on sound generation by airfoils interacting with high-frequency gusts , 1997 .

[65]  J. Thomas Beale,et al.  On the Accuracy of Vortex Methods at Large Times , 1988 .

[66]  Georges-Henri Cottet,et al.  Blending Finite-Difference and Vortex Methods for Incompressible Flow Computations , 2000, SIAM J. Sci. Comput..

[67]  David P. Lockard,et al.  Radiated noise from airfoils in realistic mean flows , 1998 .

[68]  E. Sarradj,et al.  Two-dimensional isotropic turbulent inflow conditions for vortex particle method , 2019, Physical Review Fluids.

[69]  T. Chong,et al.  On the leading edge noise and aerodynamics of thin aerofoil subjected to the straight and curved serrations , 2018, Journal of Sound and Vibration.

[70]  Ole Hald,et al.  Convergence of vortex methods for Euler’s equations , 1978 .

[71]  Leslie Greengard,et al.  A fast algorithm for particle simulations , 1987 .

[72]  Pierre Sagaut,et al.  Large-Eddy Simulation for Acoustics: Contributors , 2007 .

[73]  G. Hamad,et al.  Sound generated in a cascade by three-dimensional disturbances convected in a subsonic flow , 1981 .

[74]  M. Farge.  Wavelet Transforms and their Applications to Turbulence , 1992 .

[75]  A. Hirschberg.  Introduction to Aeroacoustics and Self-Sustained Oscillations of Internal Flows , 2013 .

[76]  J. G. Esler,et al.  Universal statistics of point vortex turbulence , 2015, Journal of Fluid Mechanics.

[77]  J. Gordon Leishman,et al.  Flow visualization of compressible vortex structures using density gradient techniques , 1993 .

[78]  Sheryl M. Grace.  Unsteady blade pressure - The BVI model vs. the gust model , 2001 .

[79]  M. Hirata,et al.  Wake and aerodynamics loads in multiple bodies—application to turbomachinery blade rows , 2004 .

[80]  Jean-Paul Bonnet,et al.  Generation of Three-Dimensional Turbulent Inlet Conditions for Large-Eddy Simulation , 2004 .

[81]  Gianfranco Guidati,et al.  Simulation and measurement of inflow-turbulence noise on airfoils , 1997 .

[82]  B. D. Mugridge.  Acoustic radiation from aerofoils with turbulent boundary layers , 1971 .

[83]  L. Ayton,et al.  On high-frequency sound generated by gust–aerofoil interaction in shear flow , 2015, Journal of Fluid Mechanics.

[84]  Tim Colonius,et al.  A Vortex Particle Method for Two-Dimensional Compressible Flow , 2002 .

[85]  L. Devroye.  Non-Uniform Random Variate Generation , 1986 .