Hostname: page-component-586b7cd67f-2brh9 Total loading time: 0 Render date: 2024-11-24T18:22:32.172Z Has data issue: false hasContentIssue false
  • English
  • Français

Rapid distortion theory on transversely sheared mean flows of arbitrary cross-section

Published online by Cambridge University Press:  24 October 2019

M. E. Goldstein Open the ORCID record for M. E. Goldstein [Opens in a new window] ,
S. J. Leib  and
M. Z. Afsar Open the ORCID record for M. Z. Afsar [Opens in a new window]
M. E. Goldstein*
Affiliation:
National Aeronautics and Space Administration, Glenn Research Centre, Cleveland, OH 44135, USA
S. J. Leib
Affiliation:
Ohio Aerospace Institute, Cleveland, OH 44142, USA
M. Z. Afsar
Affiliation:
Strathclyde University, Department of Mechanical and Aerospace Engineering, 75 Montrose St., Glasgow GI 1XJ, UK
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper is concerned with rapid distortion theory on transversely sheared mean flows that (among other things) can be used to analyse the unsteady motion resulting from the interaction of a turbulent shear flow with a solid surface. It expands on a previous analysis of Goldstein et al. (J. Fluid Mech., vol. 824, 2017, pp. 477–512) that uses a pair of conservation laws to derive upstream boundary conditions for planar mean flows and extends these findings to transversely sheared flows of arbitrary cross-section. The results, which turn out to be quite general, are applied to the specific case of a round jet interacting with the trailing edge of a flat plate and are used to calculate the radiated sound field, which is then compared with experimental data taken at the NASA Glenn Research Center.

JFM classification

Acoustics: Aeroacoustics Turbulent Flows: Compressible turbulence
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 881 , 25 December 2019 , pp. 551 - 584
Check for updates
Copyright
© 2019 Cambridge University Press 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Abramowitz, M. & Stegun, I. A. 1964 Handbook of Mathematica Functions. Nat. Bureau Stand. Appl. Math. Ser., vol. 55. National Bureau of Standards US Department of Commerce.Google Scholar
Abreu, L. I., Nogueira, P. A. S., Nilton, M. M. & Cavalieri, A. V. G.2019 Resolvent analysis applied to acoustic analogies. AIAA Paper 2019-2402.10.2514/6.2019-2402Google Scholar
Baker, D. J. & Peake, N.2019 Scattering in rotational flow. AIAA Paper 2019-2554.10.2514/6.2019-2554Google Scholar
Batchelor, G. K. & Proudman, I. 1954 The effect of rapid distortion of a fluid in turbulent motion. Q. J. Mech. Appl. Maths 7 (1), 83–103.10.1093/qjmam/7.1.83Google Scholar
Bers, A. 1975 Linear waves and instabilities. In Plasma Physics (ed. Dewitt, C. & Perraud, J.), pp. 113–216. Gordon & Breach.Google Scholar
Bilka, M. J., Kerrian, P., Ross, M. H. & Morris, S. C. 2014 Radiated sound from a circular cylinder in a turbulent shear layer. Intl J. Aeroacoust. 13 (7), 511–532.10.1260/1475-472X.13.7-8.511Google Scholar
Bridges, J. & Brown, C. A.2005 Validation of the small hot jet rig for jet noise research. AIAA Paper 2005-2846.10.2514/6.2005-2846Google Scholar
Briggs, R. J. 1964 Electron Stream Interaction with Plasmas. MIT Press.10.7551/mitpress/2675.001.0001Google Scholar
Brown, C. A. 2013 Jet-surface interaction test: far-field noise results. Trans ASME J. Engng Gas Turbines Power 135 (7), 071201-071201-7.10.1115/1.4023605Google Scholar
Brown, C. A.2015a An empirical jet-surface interaction noise model with temperature and nozzle aspect ratio effects. AIAA 2015-0229.10.2514/6.2015-0229Google Scholar
Brown, C. A.2015b Empirical models for the shielding and reflection of jet mixing noise by a surface. AIAA 2015-3128.10.2514/6.2015-3128Google Scholar
Brown, C. A. & Bridges, J.2006 Small hot jet acoustic rig validation. NASA-TM-2006-214234.Google Scholar
Campbell, G. A. & Foster, R. M.1948 Fourier Integrals for Practical Applications. D. Van Nostrand Co. Inc.Google Scholar
Dennis, D. C. J. & Nickels, T. B. 2008 On the limitations of Taylor’s hypothesis in constructing long structures in turbulent boundary layers. J. Fluid Mech. 614, 197206.10.1017/S0022112008003352Google Scholar
Goldstein, M. E. 1976 Aeroacoustics. McGraw-Hill Book Company.Google Scholar
Goldstein, M. E. 1978a Unsteady vortical and entropic distortions of potential flows round arbitrary obstacles. J. Fluid Mech. 89 (3), 433–468.10.1017/S0022112078002682Google Scholar
Goldstein, M. E. 1978b Characteristics of the unsteady motion on transversely sheared mean flows. J. Fluid Mech. 84 (2), 305–329.10.1017/S002211207800018XGoogle Scholar
Goldstein, M. E. 1979a Turbulence generated by entropy fluctuation with non-uniform mean flows. J. Fluid Mech. 93 (2), 209–224.10.1017/S0022112079001853Google Scholar
Goldstein, M. E. 1979b Scattering and distortion of the unsteady motion on transversely sheared mean flows. J. Fluid Mech. 91 (4), 601–632.10.1017/S0022112079000379Google Scholar
Goldstein, M. E., Afsar, M. Z. & Leib, S. J. 2013a Non-homogeneous rapid distortion theory on transversely sheared mean flows. J. Fluid Mech. 736, 532–569.10.1017/jfm.2013.518Google Scholar
Goldstein, M. E., Afsar, M. Z. & Leib, S. J.2013b Structure of small amplitude motion on transversely sheared mean flows. NASA/TM-2013-217862.Google 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 the trailing edge problem. J. Fluid Mech. 824, 477–512.10.1017/jfm.2017.350Google Scholar
Huete Ruiz de Lira, C. 2010 Turbulence generation by a shock wave interacting with a random density inhomogeneity field. Phys. Scr. T142, 014022.10.1088/0031-8949/2010/T142/014022Google Scholar
Huete Ruiz de Lira, C., Velikovich, A. L. & Wouchuk, J. G. 2011 Analytical linear theory for the interaction of a planar shock wave with a two- or three-dimensional random isotropic density field. Phys. Rev. E 83, 056320.10.1103/PhysRevE.83.056320Google Scholar
Huete, C., Wouchuk, J. G. & Velikovich, A. L. 2012 Analytical linear theory for the interaction of a planar shock wave with a two- or three-dimensional random isotropic acoustic wave field. Phys. Rev. E 85, 026312.10.1103/PhysRevE.85.026312Google Scholar
Hunt, J. C. R. 1973 A theory of turbulent flow around two dimensional bluff bodies. J. Fluid Mech. 61, 625–706.10.1017/S0022112073000893Google Scholar
Hunt, J. C. R. & Carruthers, D. J. 1990 Rapid distortion theory and the ‘problems’ of turbulence. J. Fluid Mech. 212, 497–532.10.1017/S0022112090002075Google Scholar
Kovasznay, L. S. G. 1953 Turbulence in supersonic flow. J. Aero. Sci. 20 (10), 657–674.10.2514/8.2793Google Scholar
Leib, S. J. & Goldstein, M. E. 2011 Hybrid source model for predicting high-speed jet noise. AIAA J. 49 (7), 1324–1335.10.2514/1.J050707Google Scholar
Lighthill, M. J. 1964 Fourier Analysis and Generalized Functions. Cambridge University Press.Google Scholar
Lin, C. C. 1953 On Taylor’s hypothesis and the accelerating terms in Navier–Stokes equations. Q. Appl. Maths 10, 295–306.10.1090/qam/51649Google Scholar
Moore, F. K.1954 Unsteady oblique interaction of a shock wave with a plane disturbance. NACA Tech. Rep. 2879.Google Scholar
McKeon, B. J. & Sharma, A. S. 2010 A critical-layer framework for turbulent pipe flow. J. Fluid Mech. 658, 336–382.10.1017/S002211201000176XGoogle Scholar
Moffatt, H. K. 1967 The interaction of turbulence with strong wind shear. In Colloquium on Atmospheric Turbulence and Radio Wave Propagation (ed. Yaglom, A. M. & Tatarski, V. I.), pp. 139–156. Nauka.Google Scholar
Noble, B. 1958 Methods Based on the Wiener–Hopf Technique for the Solution of Partial Differential Equations. Pergamon.Google Scholar
Olsen, W. & Boldman, D.1979 Trailing edge noise data with comparison to theory. AIAA Paper 79-1524.10.2514/6.1979-1524Google Scholar
Orr, W. M. 1907 The stability and instability of the unsteady motions of a perfect liquid and of a viscous liquid. Proc. R. Irish Acad. A27, 69–138.Google Scholar
Pope, S. 2000 Turbulent Flows, 1st edn. Cambridge University Press.10.1017/CBO9780511840531Google Scholar
Ramakrishnan, K., Paliath, U., Pastouchencko, N., Malcevic, I., Pilon, A., Morgenstern, J., Buonanno, M., Martinez, M. & Majjigi, M.2018 Evaluation of low noise integration concepts and propulsion technologies for future supersonic civil transports. NASA CR-2018-219936.Google Scholar
Ribner, H. S.1953 Convection of a pattern of vorticity through a shock wave. NACA Tech. Rep. 1164.Google Scholar
Ross, M. H.2009 Radiated sound generated by airfoils in a single stream shear layer, Master’s thesis, University of Notre Dame, Notre Dame, IN.Google Scholar
Sagaut, P. & Cambon, C. 2018 Homogeneous Turbulence Dynamics, 2nd edn. Springer International Publishing.10.1007/978-3-319-73162-9Google Scholar
Taylor, G. I. 1938 The spectrum of turbulence. Proc. R. Soc. Lond. A 164, 476–490.Google Scholar
Townsend, A. A. 1976 The Structure of Turbulent Shear Flows, 2nd edn. Cambridge University Press.Google Scholar
Tufts, A., Wang, K. & Wang, M.2018 Computational study of sound by airfoil interaction with a turbulent shear layer. In AIAA Aerospace Sciences meeting, Kissimmee Florida, January 8–12, AIAA Paper 2018-0757.Google Scholar
Wiener, N. 1938 The use of statistical theory to study turbulence. In Proceedings of the 5th International Congress Applied Mechanics, pp. 356–360. John Wiley.Google Scholar
Wouchuk, J. G., Huete Ruiz de Lira, C. & Velikovich, A. L. 2009 Analytical linear theory for the interaction of a planar shock wave with an isotropic turbulent vorticity field. Phys. Rev. E 79, 066315.10.1103/PhysRevE.79.066315Google Scholar
Xie, Z., Karimi, M. & Girimaji, S. S. 2017 Small perturbation evolution in compressible Poiseuille flow–velocity interactions and obliqueness effects. J. Fluid Mech. 814, 249–276.10.1017/jfm.2016.795Google Scholar