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Analytical models of the wall-pressure spectrum under a turbulent boundary layer with adverse pressure gradient

Published online by Cambridge University Press:  02 September 2019

G. Grasso Open the ORCID record for G. Grasso [Opens in a new window] ,
P. Jaiswal ,
H. Wu Open the ORCID record for H. Wu [Opens in a new window] ,
S. Moreau  and
M. Roger
G. Grasso*
Affiliation:
Univ Lyon, Ecole Centrale de Lyon, INSA Lyon, Université Claude Bernard Lyon I, CNRS, Laboratoire de Mécanique des Fluides et d’Acoustique, UMR 5509, 36 Avenue Guy de Collongue, F-69134, Écully, France Département de Génie Mécanique, Université de Sherbrooke, Sherbrooke, QC, J1K 2R1, Canada
P. Jaiswal
Affiliation:
Département de Génie Mécanique, Université de Sherbrooke, Sherbrooke, QC, J1K 2R1, Canada
H. Wu
Affiliation:
Département de Génie Mécanique, Université de Sherbrooke, Sherbrooke, QC, J1K 2R1, Canada
S. Moreau
Affiliation:
Département de Génie Mécanique, Université de Sherbrooke, Sherbrooke, QC, J1K 2R1, Canada
M. Roger
Affiliation:
Univ Lyon, Ecole Centrale de Lyon, INSA Lyon, Université Claude Bernard Lyon I, CNRS, Laboratoire de Mécanique des Fluides et d’Acoustique, UMR 5509, 36 Avenue Guy de Collongue, F-69134, Écully, France
*
†Email address for correspondence: gabriele.grasso@ec-lyon.fr
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Abstract

This paper presents a comprehensive analytical approach to the modelling of wall-pressure fluctuations under a turbulent boundary layer, unifying and expanding the analytical models that have been proposed over many decades. The Poisson equation governing pressure fluctuations is Fourier transformed in the wavenumber domain to obtain a modified Helmholtz equation, which is solved with a Green’s function technique. The source term of the differential equations is composed of turbulence–mean shear and turbulence–turbulence interaction terms, which are modelled separately within the hypothesis of a joint normal probability distribution of the turbulent field. The functional expression of the turbulence statistics is shown to be the most critical point for a correct representation of the wall-pressure spectrum. The effect of various assumptions on the shape of the longitudinal correlation function of turbulence is assessed in the first place with purely analytical considerations using an idealised flow model. Then, the effect of the hypothesis on the spectral distribution of boundary-layer turbulence on the resulting wall-pressure spectrum is compared with the results of direct numerical simulation computations and pressure measurements on a controlled-diffusion aerofoil. The boundary layer developing over the suction side of this aerofoil in test conditions is characterised by an adverse pressure gradient. The final part of the paper discusses the numerical aspect of wall-pressure spectrum computation. A Monte Carlo technique is used for a fast evaluation of the multi-dimensional integral formulation developed in the theoretical part.

JFM classification

Turbulent Flows: Turbulent boundary layers Acoustics: Aeroacoustics Boundary Layers: Boundary layer structure
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 877 , 25 October 2019 , pp. 1007 - 1062
Copyright
© 2019 Cambridge University Press 

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References

Amiet, R. K. 1976 Noise due to turbulent flow past a trailing edge. J. Sound Vib. 47 (3), 387–393.Google Scholar
Bailly, C. & Comte-Bellot, G. 2015 Turbulence. Springer.10.1007/978-3-319-16160-0Google Scholar
Batchelor, G. K. 1951 Pressure fluctuations in isotropic turbulence. Math. Proc. Camb. Phil. Soc. 47 (2), 359–374.10.1017/S0305004100026712Google Scholar
Batchelor, G. K. 1953 The Theory of Homogeneous Turbulence. Cambridge University Press.Google Scholar
Bertagnolio, F., Fischer, A. & Zhu, W. J. 2014 Tuning of turbulent boundary layer anisotropy for improved surface pressure and trailing-edge noise modeling. J. Sound Vib. 333, 991–1010.Google Scholar
Blake, W. K. 1986 Mechanics of Flow-Induced Sound and Vibration, vol. I and II. Academic Press Inc.Google Scholar
Borrell, G., Sillero, J. A. & Jiménez, J. 2013 A code for direct numerical simulation of turbulent boundary layers at high Reynolds numbers in bg/p supercomputers. Comput. Fluids 80, 37–43.10.1016/j.compfluid.2012.07.004Google Scholar
Bradshaw, P. 1967 ‘Inactive’ motion and pressure fluctuations in turbulent boundary layers. J. Fluid Mech. 30 (2), 241–258.10.1017/S0022112067001417Google Scholar
Buignon, P.2013 Scikit-Monaco documentation. http://scikit-monaco.readthedocs.io/en/latest/.Google Scholar
Bull, M. K. 1996 Wall-pressure fluctuations beneath turbulent boundary layers: some reflections on forty years of research. J. Sound Vib. 190 (3), 299–315.Google Scholar
Caiazzo, A., D’Amico, R. & Desmet, W. 2016 A generalized Corcos model for modelling turbulent boundary layer wall pressure fluctuations. J. Sound Vib. 372, 192–210.Google Scholar
Carpenter, M. H., Nordström, J. & Gottlieb, D. 1999 A stable and conservative interface treatment of arbitrary spatial accuracy. J. Comput. Phys. 148, 341–365.10.1006/jcph.1998.6114Google Scholar
Catlett, M. R., Anderson, J. M., Forest, J. B. & Stewart, D. O. 2015 Empirical modeling of pressure spectra in adverse pressure gradient turbulent boundary layers. AIAA J. 54 (2), 569–587.10.2514/1.J054375Google Scholar
Chase, D. M. 1980 Modeling the wavevector-frequency spectrum of turbulent boundary layer wall pressure. J. Sound Vib. 70 (1), 29–67.Google Scholar
Christophe, J.2011 Application of hybrid methods to high frequency aeroacoustics. PhD thesis, Université Libre de Bruxelles.Google Scholar
Cipolla, K. & Keith, W. L. 2000 Effects of pressure gradients on turbulent boundary layer wave number frequency spectra. AIAA J. 38 (10), 1832–1836.10.2514/2.864Google Scholar
Cohen, E. & Gloerfelt, X. 2018 Influence of pressure gradients on wall pressure beneath a turbulent boundary layer. J. Fluid Mech. 838, 715–758.10.1017/jfm.2017.898Google Scholar
Corcos, G. M. 1964 The structure of the turbulent pressure field in boundary-layer flows. J. Fluid Mech. 18, 353–379.10.1017/S002211206400026XGoogle Scholar
Ffowcs Williams, J. E. 1982 Boundary layer pressures and Corcos model: a development to incorporate low-wavenumber constraints. J. Fluid Mech. 125, 9–25.10.1017/S0022112082003218Google Scholar
Fischer, A., Bertagnolio, F. & Madsen, H. A. 2017 Improvement of TNO type trailing edge noise models. Eur. J. Mech. (B/Fluids) (61), 255–262.10.1016/j.euromechflu.2016.09.005Google Scholar
Gerolymos, G. A., Sénéchal, D. & Vallet, I. 2013 Wall effects on pressure fluctuations in turbulent channel flow. J. Fluid Mech. 720, 15–65.10.1017/jfm.2012.633Google Scholar
Goldstein, M. E., Leib, S. J. & Afsar, M. Z. 2017 Generalized rapid-distortion theory on transversely sheared mean flows with physically realizable upstream boundary conditions: application to trailing-edge problem. J. Fluid Mech. 824, 477–512.10.1017/jfm.2017.350Google Scholar
Goody, M. 2004 Empirical spectral model of surface pressure fluctuations. AIAA J. 42 (9), 1788–1794.10.2514/1.9433Google Scholar
Gradshteyn, I. S. & Ryzhik, I. M. 2007 Table of Integrals, Series, and Products, 7th edn. Elsevier/Academic Press.Google Scholar
Grant, H. L. 1958 The large eddies of turbulent motion. J. Fluid Mech. 4 (2), 149–190.10.1017/S0022112058000379Google Scholar
Grasso, G., Jaiswal, P. & Moreau, S. 2018 Monte-Carlo Computation of Wall-Pressure Spectra Under Turbulent Boundary Layers for Trailing-Edge Noise Prediction. Katholieke Universiteit Leuven.Google Scholar
Gravante, S. P., Naguib, A. M., Wark, C. E. & Nagib, H. M. 1998 Characterization of the pressure fluctuations under a fully developed turbulent boundary layer. AIAA J. 36 (10), 1808–1816.10.2514/2.296Google Scholar
Heisenberg, W. 1948 On the theory of statistical and isotropic turbulence. Proc. R. Soc. Lond. A 195 (1042), 402–406.Google Scholar
Hodgson, T. H.1961 Pressure fluctuations in shear flow turbulence. PhD thesis, The College of Aeronautics, Cranfield.Google Scholar
Hu, N. 2018 Empirical model of wall pressure spectra in adverse pressure gradients. AIAA J. 56 (9), 3941–3506.10.2514/1.J056666Google Scholar
Hu, N., Reiche, N. & Ewert, R. 2017 Simulation of turbulent boundary layer wall pressure fluctuations via poisson equation and synthetic turbulence. J. Fluid Mech. 826, 421–454.10.1017/jfm.2017.448Google Scholar
Hunt, J. C. R. 1973 A theory of turbulent flow round two-dimensional bluff bodies. J. Fluid Mech. 61, 625–706.10.1017/S0022112073000893Google Scholar
Ince, E. L. 1956 Ordinary Differential Equations. Courier Corporation.Google Scholar
James, F. 1980 Monte Carlo theory and practice. Rep. Prog. Phys. 43, 1145–1189.10.1088/0034-4885/43/9/002Google Scholar
Jones, L. E.2008 Numerical studies of the flow around an airfoil at low Reynolds number. PhD thesis, University of Southampton.Google Scholar
Jones, L. E., Sandberg, R. D. & Sandham, N. D. 2008 Direct numerical simulations of forced and unforced separation bubbles on an airfoil at incidence. J. Fluid Mech. 602, 175–207.10.1017/S0022112008000864Google Scholar
Kamruzzaman, M., Lutz, T., Würz, W. & Krämer, E. 2011 On the length scales of turbulence for aeroacoustic applications. In Proceedings of the 17th AIAA/CEAS Aeroacoustic Conference. American Institute of Aeronautics and Astronautics.Google Scholar
Kamruzzaman, M., Lutz, T., Würz, W., Shen, W. Z., Zhu, W. J., Hansen, M. O. L., Bertagnolio, F. & Madsen, H. A. 2012 Validations and improvements of airfoil trailing-edge noise prediction models using detailed experimental data. Wind Energy 15, 45–61.10.1002/we.505Google Scholar
von Kármán, T. 1948 Progress in the statistical theory of turbulence. Proc. Natl Acad. Sci. USA 34, 530–539.10.1073/pnas.34.11.530Google Scholar
Keith, W. L., Hurdis, D. A. & Abraham, B. M. 1992 A comparison of turbulent boundary layer wall-pressure spectra. J. Fluids Engng 114, 338–347.10.1115/1.2910035Google Scholar
Kennedy, C. A., Carpenter, M. H. & Lewis, R. M. 1999 Low-storage, explicit Runge–Kutta schemes for the compressible Navier–Stokes equations. Appl. Numer. Maths 35, 177–219.10.1016/S0168-9274(99)00141-5Google Scholar
Kim, J. W. & Lee, D. J. 2003 Characteristic interface conditions for multiblock high-order computation on singular structured grid. AIAA J. 41, 2341–2348.10.2514/2.6858Google Scholar
Kraichnan, R. H. 1956a Pressure field within homogeneous anisotropic turbulence. J. Acoust. Soc. Am. 28 (1), 64–72.10.1121/1.1908224Google Scholar
Kraichnan, R. H. 1956b Pressure fluctuations in turbulent flow over a flat plate. J. Acoust. Soc. Am. 28 (3), 378–390.10.1121/1.1908336Google Scholar
Lee, S. 2018 Empirical wall-pressure spectral modeling for zero and adverse pressure gradient flows. AIAA J. 56 (5), 1818–1829.10.2514/1.J056528Google Scholar
Liepmann, H. W., Laufer, J. & Liepmann, K.1951 On the spectrum of isotropic turbulence. Tech. Rep. National Advisory Committee for Aeronautics.Google Scholar
Magnaudet, J. 2003 High-Reynolds-number turbulence in a shear-free boundary layer: revisiting the huntgraham theory. J. Fluid Mech. 484, 167–196.10.1017/S0022112003004245Google Scholar
Moreau, S., Henner, M., Iaccarino, G., Wang, M. & Roger, M. 2003 Analysis of flow conditions in freejet experiments for studying airfoil self-noise. AIAA J. 41, 1895–1905.Google Scholar
Moreau, S. & Roger, M. 2005 Effect of airfoil aerodynamic load on trailing edge noise. AIAA J. 43 (1), 41–52.10.2514/1.5578Google Scholar
Moreau, S., Sanjosé, M., Perot, F. & Kim, M.-S. 2011 Direct Self-Noise Simulation of the Installed Controlled Diffusion Airfoil. American Institute of Aeronautics and Astronautics.10.2514/6.2011-2716Google Scholar
Padois, T., Laffay, P., Idier, A. & Moreau, S. 2015 Detailed Experimental Investigation of the Aeroacoustic Field Around a Controlled-Diffusion Airfoil. American Institute of Aeronautics and Astronautics.Google Scholar
Panton, R. L. & Linebarger, J. H. 1974 Wall pressure spectra calculations for equilibrium boundary layers. J. Fluid Mech. 65 (02), 261–287.10.1017/S0022112074001388Google Scholar
Parchen, R.1998 Progress report draw: a prediction scheme for trailing-edge noise based on detailed boundary layer characteristics. Tech. Rep. TNO Institute of Applied Physics.Google Scholar
Peltier, L. J. & Hambric, S. A. 2007 Estimating turbulent-boundary-layer wall-pressure spectra from cfd rans solutions. J. Fluids Struct. 23, 920–937.10.1016/j.jfluidstructs.2007.01.003Google Scholar
Perennes, S. & Roger, M. 1998 Aerodynamic noise of a two-dimensional wing with high-lift devices. In 4th AIAA/CEAS Aeroacoustics Conference, p. 2338. American Institute of Aeronautics and Astronautics.Google Scholar
Phillips, O. M. 1956 On the aerodynamic surface sound from a plane turbulent boundary layer. Proc. R. Soc. Lond. A 234 (1198), 327–335.Google Scholar
Pope, S. B. 2000 Turbulent Flows. Cambridge University Press.10.1017/CBO9780511840531Google Scholar
Press, W. H. & Farrar, G. R. 1990 Recursive stratified sampling for multidimensional Monte Carlo integration. Comput. Phys. 4 (2), 190–195.10.1063/1.4822899Google Scholar
Prigent, S. L., Engelman, R., Salze, E. & Bailly, C. 2018 Deconvolution of the Wave Number – Frequency Spectra of Wall Pressure Fluctuations. American Institute of Aeronautics and Astronautics.Google Scholar
Proskurov, S., Darbyshire, O. R. & Karabasov, S. A. 2017 Aerofoil broadband and tonal noise modelling using stochastic sound sources and incorporated large scale fluctuations. J. Sound Vib. 411, 60–74.Google Scholar
Remmler, S., Christophe, J., Anthoine, J. & Moreau, S. 2010 Computation of wall-pressure spectra from steady flow data for noise prediction. AIAA J. 48 (9), 1997–2007.10.2514/1.J050206Google Scholar
de la Riva, D. H.2001 Turbulence interaction in a highly staggered cascade-propulsor configuration. Master’s thesis, Virginia Polytechnic Institute and State University, Blacksburg, VA.10.21236/ADA389758Google Scholar
de la Riva, D. H., Devenport, W. J., Muthanna, C. & Glegg, S. A. L. 2004 Behavior of turbulence flowing through a compressor cascade. AIAA J. 42 (7), 1302–1313.10.2514/1.3107Google Scholar
Roger, M. 2017 Microphone measurements in aeroacoustic installations. In Design and Operation of Aeroacoustic Wind Tunnel Tests for Ground and Air Transport STO-AVT-287. Von Karman Institute for Fluid Dynamics.Google Scholar
Roger, M. & Moreau, S. 2004 Broadband self noise from loaded fan blades. AIAA J. 42 (3), 536–544.10.2514/1.9108Google Scholar
Roger, M. & Moreau, S. 2005 Back-scattering correction and further extensions of Amiet’s trailing-edge noise model. Part 1: Theory. J. Sound Vib. 286 (3), 477–506.Google Scholar
Roger, M. & Moreau, S. 2012 Addendum to the back-scattering correction of AMIETS trailing-edge noise model. J. Sound Vib. 331, 5383–5385.Google Scholar
Rosenheinrich, W.2016 Tables of some indefinite integrals of Bessel functions. www.eah-jena.de/∼rsh/Forschung/Stoer/besint.pdf.Google Scholar
Rozenberg, Y., Robert, G. & Moreau, S. 2012 Wall-pressure spectral model including the adverse pressure gradient effects. AIAA J. 50 (10), 2168–2179.10.2514/1.J051500Google Scholar
Rozenberg, Y., Roger, M. & Moreau, S. 2008 Fan blade trailing-edge noise prediction using rans simulations. In Proceedings of the Acoustics ’08 Conference, Paris, June 30 – July 4.Google Scholar
Salze, E., Bailly, C., Marsden, O., Jondeau, E. & Juve, D. 2014 An experimental characterisation of wall pressure wavevector-frequency spectra in the presence of pressure gradients. American Institute of Aeronautics and Astronautics.10.2514/6.2014-2909Google Scholar
Salze, E., Bailly, C., Marsden, O., Jondeau, E. & Juve, D. 2015 An Experimental Investigation of Wall Pressure Fluctuations Beneath Pressure Gradients. American Institute of Aeronautics and Astronautics.10.2514/6.2015-3148Google Scholar
Sandberg, R. D. 2015 Compressible-flow DNS with application to airfoil noise. Flow Turbul. Combust. 95, 211–229.10.1007/s10494-015-9617-1Google Scholar
Sandberg, R. D. & Sandham, N. D. 2006 Nonreflecting zonal characteristic boundary condition for direct numerical simulation of aerodynamic sound. AIAA J. 44, 402–405.10.2514/1.19169Google Scholar
Santana, L. D., Christophe, J., Schram, C. & Desmet, W. 2016 A rapid distortion theory modified turbulence spectra for semi-analytical airfoil noise prediction. J. Sound Vib. 383, 349–363.Google Scholar
Schlinker, R. & Amiet, R. K.1981 Helicopter trailing edge noise. NASA Tech. Rep. 3470.10.2514/6.1981-2001Google Scholar
Sillero, J. A., Jiménez, J. & Moser, R. D. 2014 Two-point statistics for turbulent boundary layers and channels at Reynolds numbers up to 𝛿+ ≈ 2000. Phys. Fluids 26 (10), 105109.10.1063/1.4899259Google Scholar
Slama, M., Leblond, C. & Sagaut, P. 2018 A kriging-based elliptic extended anisotropic model for the turbulent boundary layer wall pressure spectrum. J. Fluid Mech. 840, 25–55.10.1017/jfm.2017.810Google Scholar
Smol’yakov, A. V. 2006 A new model for the cross spectrum and wavenumber-frequency spectrum of turbulent pressure fluctuations in a boundary layer. Acoust. Phys. 52 (3), 331–337.10.1134/S1063771006030146Google Scholar
Spalart, P. R. & Leonard, A. 1987 Direct numerical simulation of equilibrium turbulent boundary layers. In Turbulent Shear Flows 5 (ed. Durst, F., Launder, B. E., Lumley, J. L., Schmidt, F. W. & Whitelaw, J. H.), pp. 234–252. Springer.10.1007/978-3-642-71435-1_20Google Scholar
Stalnov, O., Paruchuri, C. & Joseph, P. 2016 Towards a non-empirical trailing edge noise prediction model. J. Sound Vib. 372, 50–68.Google Scholar
Taylor, G. I. 1938 The spectrum of turbulence. Proc. R. Soc. Lond. A 164 (919), 476–490.10.1098/rspa.1938.0032Google Scholar
Wang, M., Moreau, S., Iaccarino, G. & Roger, M. 2009 Les prediction of wall-pressure fluctuations and noise of a low-speed airfoil. Intl J. Aeroacoust. 8 (3), 177–197.10.1260/147547208786940017Google Scholar
Wilson, D. K.1997 Three-dimensional correlation and spectral functions for turbulent velocities in homogeneous and surface-blocked boundary layers. Tech. Rep. Army Research Laboratory.10.21236/ADA327709Google Scholar
Wilson, D. K.1998 Turbulence models and the synthesis of random fields for acoustic wave propagation calculations. Tech. Rep. Army Research Laboratory.10.21236/ADA352483Google Scholar
Wohlbrandt, A., Hu, N., Guérin, S. & Ewert, R. 2016 Analytical reconstruction of isotropic turbulence spectra based on the gaussian transform. Comput. Fluids 132, 46–50.10.1016/j.compfluid.2016.03.023Google Scholar
Wu, H., Laffay, P., Idier, A., Jaiswal, P., Sanjosé, M. & Moreau, S. 2016 Numerical study of the installed Controlled-Diffusion airfoil at transitional Reynolds number. In Mathematical and Computational Approaches in Advancing Modern Science and Engineering, pp. 505–515. Springer.10.1007/978-3-319-30379-6_46Google Scholar
Wu, H., Moreau, S. & Sandberg, R. D. 2019 Effects of pressure gradient on the evolution of velocity-gradient tensor invariant dynamics on a controlled-diffusion aerofoil at Re c = 150 000. J. Fluid Mech. 868, 584–610.10.1017/jfm.2019.129Google Scholar
Wu, H., Sanjosé, M., Moreau, S. & Sandberg, R. D. 2018 Direct numerical simulation of the self-noise radiated by the installed Controlled-Diffusion airfoil at transitional Reynolds number. AIAA Paper 2018-3797. American Institute of Aeronautics and Astronautics.Google Scholar