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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 ,
Mirela Darau  and
Edward J. Brambley
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: [email protected]
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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.

JFM classification

Acoustics: Aeroacoustics Mathematical Foundations: General fluid mechanics Vortex Flows: Vortex shedding
Type
Papers
Information
Journal of Fluid Mechanics , Volume 720 , 10 April 2013 , pp. 618 - 636
Copyright
©2013 Cambridge University Press

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