Hostname: page-component-7c8c6479df-7qhmt Total loading time: 0 Render date: 2024-03-29T00:13:30.017Z Has data issue: false hasContentIssue false

The trailing vorticity field behind a line source in two-dimensional incompressible linear shear flow

Published online by Cambridge University Press:  04 March 2013

Sjoerd W. Rienstra*
Affiliation:
Department of Mathematics and Computer Science, TU Eindhoven, Den Dolech 2, 5612 AZ Eindhoven, The Netherlands
Mirela Darau
Affiliation:
Department of Mathematics and Computer Science, TU Eindhoven, Den Dolech 2, 5612 AZ Eindhoven, The Netherlands Department of Mathematics and Computer Science, WU of Timişoara, Blvd. V. Parvan 4, Timisoara 300223, Romania
Edward J. Brambley
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge CB3 0EW UK
*
Email address for correspondence: s.w.rienstra@tue.nl

Abstract

The explicit exact analytic solution for harmonic perturbations from a line mass source in an incompressible inviscid two-dimensional linear shear is derived using a Fourier transform method. The two cases of an infinite shear flow and a semi-infinite shear flow over an impedance boundary are considered. For the free-field and hard-wall configurations, the pressure field is (in general) logarithmically diverging and its Fourier representation involves a diverging integral that is interpreted as an integral of generalized functions; this divergent behaviour is not present for a finite impedance boundary or if the frequency equals the mean flow shear rate. The dominant feature of the solution is the hydrodynamic wake caused by the shed vorticity of the source. For linear shear over an impedance boundary, in addition to the wake, (at most) two surface modes along the wall are excited. The implications for duct acoustics with flow over an impedance wall are discussed.

Type
Papers
Copyright
©2013 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. 1965 Handbook of Mathematical Functions. Dover.Google Scholar
Balsa, T. F. 1988 On the receptivity of free shear layers to two-dimensional external excitation. J. Fluid Mech. 187, 155177.Google Scholar
Bers, A. 1983 Space–time evolution of plasma instabilities – absolute and convective. In Handbook of Plasma Physics: Volume 1 Basic Plasma Physics (ed. Galeev, A.A. & Sudan, R.N.). pp. 451517. North Holland, chap. 3.2.Google Scholar
Brambley, E. J. 2011 Surface modes in sheared flow using the modified Myers boundary condition. AIAA Paper 2011–2736.Google Scholar
Brambley, E. J., Darau, M. & Rienstra, S. W. 2011 The critical layer in sheared flow.AIAA Paper 2011–2806.Google Scholar
Brambley, E. J., Darau, M. & Rienstra, S. W. 2012 The critical layer in linear-shear boundary layers over acoustic linings. J. Fluid Mech. 710, 545568.Google Scholar
Briggs, R. J. 1964 Electron-Stream Interaction with Plasmas, Monograph, vol. 29. MIT.Google Scholar
Crighton, D. G., Dowling, A. P., Ffowcs Williams, J. E., Heckl, M. & Leppington, F. G. 1992 Modern Methods in Analytical Acoustics. Lecture Notes, Springer.CrossRefGoogle Scholar
Criminale, W. O. & Drazin, P. G. 1990 The evolution of linearized parallel flows. Stud. Appl. Maths 83, 123157.Google Scholar
Criminale, W. O. & Drazin, P. G. 2000 The initial-value problem for a modeled boundary layer. Phys. Fluids 12 (2), 366374.Google Scholar
Drazin, P. G. & Reid, W. H. 2004 Hydrodynamic Stability, 2nd edn. Cambridge University Press.Google Scholar
Goldstein, M. E. & Leib, S. J. 2005 The role of instability waves in predicting jet noise. J. Fluid Mech. 525, 3772.Google Scholar
Jones, D. S. 1982 The Theory of Generalised Functions, 2nd edn. Cambridge University Press.Google Scholar
Kundu, P. K. & Cohen, I. M. 2002 Fluid Mechanics. Academic.Google Scholar
Lesser, M. B. & Crighton, D. G. 1975 Physical acoustics and the method of matched asymptotic expansions. In Physical Acoustics (ed. Mason, W.P. & Thurston, R.N.). vol. XI. Academic.Google Scholar
Pridmore-Brown, D. C. 1958 Sound propagation in a fluid flowing through an attenuating duct. J. Fluid Mech. 4, 393406.Google Scholar
Rayleigh, J. W. S. 1945 Theory of Sound, vol. II. Dover.Google Scholar
Rienstra, S. W. 2003 A classification of duct modes based on surface waves. Wave Motion 37 (2), 119135.Google Scholar
Rienstra, S. W. 2006 Impedance models in time domain, including the extended Helmholtz resonator model. AIAA Paper 2006–2686.Google Scholar
Suzuki, T. & Lele, S. K. 2003a Green’s functions for a source in a mixing layer: direct waves, refracted arrival waves and instability waves. J. Fluid Mech. 477, 89128.Google Scholar
Suzuki, T. & Lele, S. K. 2003b Green’s functions for a source in a boundary layer: direct waves, channelled waves and diffracted waves. J. Fluid Mech. 477, 129173.Google Scholar
Tennekes, H. & Lumley, J. L. 1972 A First Course of Turbulence. MIT.Google Scholar
Wu, X. 2002 Generation of sound and instability waves due to unsteady suction and injection. J. Fluid Mech. 453, 289313.Google Scholar
Wu, X. 2011 On generation of sound in wall-bounded shear flows: back action of sound and global acoustic coupling. J. Fluid Mech. 689, 279316.Google Scholar