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

  • M. E. Goldstein (a1), S. J. Leib (a2) and M. Z. Afsar (a3)


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.


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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), 177207.
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.
Ayton, L. J., Gill, J. & Peake, N. 2016 The importance of the unsteady Kutta condition when modelling gust–aerofoil interaction. J. Sound Vib. 378, 2837.
Batchelor, G. K. & Proudman, I. 1954 The effect of rapid distortion of a fluid in turbulent motion. Q. J. Mech. Appl. Maths 7 (1), 83103.
Bers, A. 1975 Linear waves and instabilities. In Plasma Physics (ed. Dewitt, C. & Perraud, J.), pp. 113216. Gordon & Breach.
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.
Bridges, J. & Brown, C. A.2005 Validation of the small hot jet rig for jet noise research. AIAA Paper 2005-2846.
Bridges, J., Brown, C. A. & Bozak, R.2014 Experiments on exhaust of tightly integrated propulsion systems. AIAA Paper 2014-2904.
Briggs, R. J. 1964 Electron Stream Interaction with Plasmas. MIT Press.
Brinkman, K. W. & Walker, J. D. A. 2001 Instabilities in a viscous flow driven by streamwise vortices. J. Fluid Mech. 432, 127166.
Brown, C.2012 Jet–surface interaction test: far-field noise results. ASME Paper GT2012-69639.
Brown, C. A. 2015 An empirical jet–surface interaction noise model with temperature and nozzle aspect ratio effects. In 53rd AIAA Aerospace Sciences Meeting.
Brown, C. A. & Bridges, J.2006 Small hot jet acoustic rig validation. NASA-TM-2006-214234.
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), 743761.
Carrier, G. F., Krook, M. & Pearson, C. E. 1966 Functions of a Complex Variable. McGraw-Hill.
Cassel, K. W. & Conlisk, A. T. 2014 Unsteady separation in vortex induced boundary layers. Phil. Trans. R. Soc. Lond. 372 (2020), 20130348.
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, 4154.
Cowley 2001 Laminar boundary layer theory: a 20th century paradox? In Proc. ICTM 2000, Chicago, IL, August–September 2000, pp. 389411. Kluwer.
Crighton, D. G. 1985 The Kutta condition in unsteady flow. Annu. Rev. Fluid Mech. 17, 411445.
Dowling, A. P., Ffowcs Williams, J. E. & Goldstein, M. E. 1978 Sound propagation in a moving stream. Phil. Trans. R. Soc. Lond. A 288, 321349; 40, 657–670.
Drazin, P. G. & Reid, W. H. 1981 Hydrodynamic Stability. Cambridge University Press.
Goldstein, M. E. 1978a Unsteady vortical and entropic distortions of potential flows round arbitrary obstacles. J. Fluid Mech. 89, 433468.
Goldstein, M. E. 1978b Characteristics of the unsteady motion on transversely sheared mean flows. J. Fluid Mech. 84 (2), 305329.
Goldstein, M. E. 1979a Turbulence generated by entropy fluctuation with non-uniform mean flows. J. Fluid Mech. 93 (2), 209224.
Goldstein, M. E. 1979b Scattering and distortion of the unsteady motion on transversely sheared mean flows. J. Fluid Mech. 91 (4), 601632.
Goldstein, M. E. 2005 On identifying the true sources of aerodynamic sound. J. Fluid Mech. 526, 337347.
Goldstein, M. E. 2009 A theoretical basis for identifying the sound sources in a turbulent flow. Intl J. Aeroacoust. 8 (4), 283300.
Goldstein, M. E., Afsar, M. Z. & Leib 2013a Non-homogeneous rapid distortion theory on transversely sheared mean flows. J. Fluid Mech. 736, 532569.
Goldstein, M. E., Afsar, M. Z. & Leib2013b Structure of small amplitude motion on transversely sheared mean flows. NASA/TM-2013-217862.
Gradshteyn, I. S. & Ryshik, I. M. 1965 Table of Integrals, Series and Products. Academic.
Hunt, J. C. R. 1973 A theory of turbulent flow around two dimensional bluff bodies. J. Fluid Mech. 61, 625706.
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. 283292. 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.
Khavaran, A., Bozak, R. F. & Brown, C. A.2016 Jet surface interaction noise in a planar exhaust. AIAA Paper 2016-2863.
Kovasznay, L. S. G. 1953 Turbulence in supersonic flow. J. Aero Sci. 20 (10), 657674.
Leib, S. J. & Goldstein, M. E. 2011 Hybrid source model for predicting high-speed jet noise. AIAA J. 49 (7), 13241335.
Lighthill, M. J. 1964 Fourier Analysis and Generalized Functions. Cambridge University Press.
Livescu, D. & Madnia, C. K. 2004 Small scale structure of homogenious turbulent shear flow. Phys. Fluids 16 (8), 286428787.
Mani, R. 1976 Influence of jet flow on jet noise. J. Fluid Mech. 173, 753793.
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. 139156. Nauka.
Möhring, W.1976 Über Schallwellen in Scherströmungen, Fortschritte der Akustik. DAGA 76 VDI, pp. 543–546.
Moore, F. K.1954 Unsteady oblique interaction of a shock wave with a plane disturbance. NACS Tech. Rep. 2879.
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.
Ribner, H. S.1953 Convection of a pattern of vorticity through a shock wave. NACA Tech. Rep. 1164.
Sagaut, P. & Cambon, C. 2008 Homogeneous Turbulence Dynamics. Cambridge University Press.
Sears, W. R. 1941 Some aspects of non-stationary airfoil theory and its practical application. J. Aero. Sci. 8 (3), 104108.
Taylor, G. I. 1935 Turbulence in a contracting stream. Z. Angew. Math. Mech. 15, 9196.
Wiener, N. 1938 The use of statistical theory to study turbulence. In Proc. 5th Int. Congress Appl. Mech., pp. 356360. Wiley.
Xie, Z., Karimi, M. & Girimaji, S. S. 2017 Small perturbation evolution in compressible Poiseulle flow–velocity interactions and obliqueness effects. J. Fluid Mech. 814, 249276.
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).
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Generalized rapid-distortion theory on transversely sheared mean flows with physically realizable upstream boundary conditions: application to trailing-edge problem

  • M. E. Goldstein (a1), S. J. Leib (a2) and M. Z. Afsar (a3)


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