Journal of Fluid Mechanics

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Analysis of the unstable Tollmien–Schlichting mode on bodies with a rounded leading edge using the parabolized stability equation

M. R. TURNERa1 c1 and P. W. HAMMERTONa1

a1 School of Mathematics, University of East Anglia, Norwich NR4 7TJ, UK

Abstract

The interaction between free-stream disturbances and the boundary layer on a body with a rounded leading edge is considered in this paper. A method which incorporates calculations using the parabolized stability equation in the Orr–Sommerfeld region, along with an upstream boundary condition derived from asymptotic theory in the vicinity of the leading edge, is generalized to bodies with an inviscid slip velocity which tends to a constant far downstream. We present results for the position of the lower branch neutral stability point and the magnitude of the unstable Tollmien–Schlichting (T-S) mode at this point for both a parabolic body and the Rankine body. For the Rankine body, which has an adverse pressure gradient along its surface far from the nose, we find a double maximum in the T-S wave amplitude for sufficiently large Reynolds numbers.

(Received December 21 2007)

(Revised November 05 2008)

Correspondence:

c1 Present address for correspondence: Mathematics Research Institute, School of Engineering, Computing and Mathematics, University of Exeter, Exeter EX4 4QF, UK. Email: M.R.Turner@ex.ac.uk.

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