Hostname: page-component-586b7cd67f-2brh9 Total loading time: 0 Render date: 2024-12-01T01:10:13.494Z Has data issue: false hasContentIssue false
  • English
  • Français

Generalized rapid-distortion theory on transversely sheared mean flows with physically realizable upstream boundary conditions: application to trailing-edge problem

Published online by Cambridge University Press:  06 July 2017

M. E. Goldstein Open the ORCID record for M. E. Goldstein [Opens in a new window] ,
S. J. Leib  and
M. Z. Afsar
M. E. Goldstein*
Affiliation:
National Aeronautics and Space Administration, Glenn Research Center, Cleveland, OH 44135, USA
S. J. Leib
Affiliation:
Ohio Aerospace Institute, Cleveland, OH 44142, USA
M. Z. Afsar
Affiliation:
Department of Mechanical and Aerospace Engineering, Strathclyde University, 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 which (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 extends previous analyses of Goldstein et al. (J. Fluid Mech., vol. 736, 2013a, pp. 532–569; NASA/TM-2013-217862, 2013b) which showed that the unsteady motion is completely determined by specifying two arbitrary convected quantities. The present paper uses a pair of previously derived conservation laws to derive upstream boundary conditions that relate these quantities to experimentally measurable flow variables. The result is dependent on the imposition of causality on an intermediate variable that appears in the conservation laws. Goldstein et al. (2013a) related the convected quantities to the physical flow variables at the location of the interaction, but the results were not generic and hard to reconcile with experiment. That problem does not occur in the present formulation, which leads to a much simpler and more natural result than the one given in Goldstein et al. (2013a). We also show that the present formalism yields better predictions of the sound radiation produced by the interaction of a two-dimensional jet with the downstream edge of a flat plate than the Goldstein et al. (2013a) result. The role of causality is also discussed.

JFM classification

Acoustics: Aeroacoustics Mathematical Foundations: Mathematical Foundations Turbulent Flows: Turbulent Flows
Type
Papers
Information
Journal of Fluid Mechanics , Volume 824 , 10 August 2017 , pp. 477 - 512
Check for updates
Copyright
© Cambridge University Press 2017. This is a work of the U.S. Government and is not subject to copyright protection in the United States. 

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

Afsar, M. Z., Leib, S. J. & Bozak, R. F. 2017 Effect of de-correlating turbulence on the low frequency decay of jet–surface interaction noise in sub-sonic unheated air jets using a CFD-based approach. J. Sound Vib. 386 (6), 177–207.Google Scholar
Afsar, M. Z., Sescu, A. & Leib, S. J.2016 Predictive capability of low frequency jet noise using an asymptotic theory for the adjoint vector Green’s function in non-parallel flow. AIAA Aero-acoustics. AIAA Paper 2016-2804.Google Scholar
Ayton, L. J., Gill, J. & Peake, N. 2016 The importance of the unsteady Kutta condition when modelling gust–aerofoil interaction. J. Sound Vib. 378, 28–37.Google 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.CrossRefGoogle Scholar
Bers, A. 1975 Linear waves and instabilities. In Plasma Physics (ed. Dewitt, C. & Perraud, J.), pp. 113–216. Gordon & Breach.Google Scholar
Bridges, J. 2014 Noise from Aft Deck Exhaust Nozzles – differences in experimental embodiments. In 52nd AIAA Aerospace Sciences Meeting – 13–17 January 2014, Nat’l Harbor, MD.Google Scholar
Bridges, J. & Brown, C. A.2005 Validation of the small hot jet rig for jet noise research. AIAA Paper 2005-2846.Google Scholar
Bridges, J., Brown, C. A. & Bozak, R.2014 Experiments on exhaust of tightly integrated propulsion systems. AIAA Paper 2014-2904.Google Scholar
Briggs, R. J. 1964 Electron Stream Interaction with Plasmas. MIT Press.CrossRefGoogle Scholar
Brinkman, K. W. & Walker, J. D. A. 2001 Instabilities in a viscous flow driven by streamwise vortices. J. Fluid Mech. 432, 127–166.Google Scholar
Brown, C.2012 Jet–surface interaction test: far-field noise results. ASME Paper GT2012-69639.Google Scholar
Brown, C. A. 2015 An empirical jet–surface interaction noise model with temperature and nozzle aspect ratio effects. In 53rd AIAA Aerospace Sciences Meeting.Google Scholar
Brown, C. A. & Bridges, J.2006 Small hot jet acoustic rig validation. NASA-TM-2006-214234.Google Scholar
Brown, S. N. & Daniels, P. G. 1975 On the viscous flow about the trailing edge of a rapidly oscillating plate. J. Fluid Mech. 67 (4), 743–761.CrossRefGoogle Scholar
Carrier, G. F., Krook, M. & Pearson, C. E. 1966 Functions of a Complex Variable. McGraw-Hill.Google Scholar
Cassel, K. W. & Conlisk, A. T. 2014 Unsteady separation in vortex induced boundary layers. Phil. Trans. R. Soc. Lond. 372 (2020), 20130348.Google Scholar
Chinaud, M., Rouchon, J. F., Duhayon, E., Scheller, J., Cazin, S., Marchal, M. & Braza, M. 2014 Trailing-edge dynamics and morphing of a deformable flat plate at high Reynolds number by time-resolved PIV. J. Fluids Struct. 47, 41–54.Google Scholar
Cowley 2001 Laminar boundary layer theory: a 20th century paradox? In Proc. ICTM 2000, Chicago, IL, August–September 2000, pp. 389–411. Kluwer.Google Scholar
Crighton, D. G. 1985 The Kutta condition in unsteady flow. Annu. Rev. Fluid Mech. 17, 411–445.Google Scholar
Dowling, A. P., Ffowcs Williams, J. E. & Goldstein, M. E. 1978 Sound propagation in a moving stream. Phil. Trans. R. Soc. Lond. A 288, 321–349; 40, 657–670.Google Scholar
Drazin, P. G. & Reid, W. H. 1981 Hydrodynamic Stability. Cambridge University Press.Google Scholar
Goldstein, M. E. 1978a Unsteady vortical and entropic distortions of potential flows round arbitrary obstacles. J. Fluid Mech. 89, 433–468.Google Scholar
Goldstein, M. E. 1978b Characteristics of the unsteady motion on transversely sheared mean flows. J. Fluid Mech. 84 (2), 305–329.Google Scholar
Goldstein, M. E. 1979a Turbulence generated by entropy fluctuation with non-uniform mean flows. J. Fluid Mech. 93 (2), 209–224.CrossRefGoogle Scholar
Goldstein, M. E. 1979b Scattering and distortion of the unsteady motion on transversely sheared mean flows. J. Fluid Mech. 91 (4), 601–632.Google Scholar
Goldstein, M. E. 2005 On identifying the true sources of aerodynamic sound. J. Fluid Mech. 526, 337–347.Google Scholar
Goldstein, M. E. 2009 A theoretical basis for identifying the sound sources in a turbulent flow. Intl J. Aeroacoust. 8 (4), 283–300.Google Scholar
Goldstein, M. E., Afsar, M. Z. & Leib 2013a Non-homogeneous rapid distortion theory on transversely sheared mean flows. J. Fluid Mech. 736, 532–569.CrossRefGoogle Scholar
Goldstein, M. E., Afsar, M. Z. & Leib2013b Structure of small amplitude motion on transversely sheared mean flows. NASA/TM-2013-217862.Google Scholar
Gradshteyn, I. S. & Ryshik, I. M. 1965 Table of Integrals, Series and Products. Academic.Google Scholar
Hunt, J. C. R. 1973 A theory of turbulent flow around two dimensional bluff bodies. J. Fluid Mech. 61, 625–706.Google Scholar
Hunt, J. C. R., Ishihara, T., Szubert, D., Asproulias, I., Hoarau, Y. & Braza, M. 2016 Turbulence near interfaces – modelling and simulations. In Advances in Fluid–Structure Interaction, pp. 283–292. Updated contributions reflecting new findings presented at the ERCOFTAC Symposium on Unsteady Separation in Fluid–Structure Interaction, 17–21 June 2013, St John Resort, Mykonos, Greece. Springer.Google Scholar
Khavaran, A., Bozak, R. F. & Brown, C. A.2016 Jet surface interaction noise in a planar exhaust. AIAA Paper 2016-2863.CrossRefGoogle Scholar
Kovasznay, L. S. G. 1953 Turbulence in supersonic flow. J. Aero Sci. 20 (10), 657–674.Google Scholar
Leib, S. J. & Goldstein, M. E. 2011 Hybrid source model for predicting high-speed jet noise. AIAA J. 49 (7), 1324–1335.Google Scholar
Lighthill, M. J. 1964 Fourier Analysis and Generalized Functions. Cambridge University Press.Google Scholar
Livescu, D. & Madnia, C. K. 2004 Small scale structure of homogenious turbulent shear flow. Phys. Fluids 16 (8), 2864–28787.Google Scholar
Mani, R. 1976 Influence of jet flow on jet noise. J. Fluid Mech. 173, 753–793.Google Scholar
Moffatt, H. K. 1967 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
Möhring, W.1976 Über Schallwellen in Scherströmungen, Fortschritte der Akustik. DAGA 76 VDI, pp. 543–546.Google Scholar
Moore, F. K.1954 Unsteady oblique interaction of a shock wave with a plane disturbance. NACS Tech. Rep. 2879.Google Scholar
Orr, W. 1907 The stability and instability of the steady motions of a perfect liquid and of a viscous liquid. Proc. R. Irish Acad. A 27, 9–68, 69–138.Google Scholar
Ribner, H. S.1953 Convection of a pattern of vorticity through a shock wave. NACA Tech. Rep. 1164.Google Scholar
Sagaut, P. & Cambon, C. 2008 Homogeneous Turbulence Dynamics. Cambridge University Press.CrossRefGoogle Scholar
Sears, W. R. 1941 Some aspects of non-stationary airfoil theory and its practical application. J. Aero. Sci. 8 (3), 104–108.Google Scholar
Taylor, G. I. 1935 Turbulence in a contracting stream. Z. Angew. Math. Mech. 15, 91–96.CrossRefGoogle Scholar
Wiener, N. 1938 The use of statistical theory to study turbulence. In Proc. 5th Int. Congress Appl. Mech., pp. 356–360. Wiley.Google Scholar
Xie, Z., Karimi, M. & Girimaji, S. S. 2017 Small perturbation evolution in compressible Poiseulle flow–velocity interactions and obliqueness effects. J. Fluid Mech. 814, 249–276.Google Scholar
Zaman, K., Brown, C. A. & Bridges, J. E.2013 Interaction of a rectangular jet with a flat-plate placed parallel to the flow. AIAA Paper 2013-2184. NASA/TM-2013-217879 (E-18684).Google Scholar