Receptivity for a Flat Plate with a Rounded Leading-Edge

Abstract

Abstract. The receptivity due to the interaction of an acoustic wave with a body which has a rounded leading edge and a region of low wall shear is considered. The body of interest is formed by a line source in a uniform stream. The nose radius, r * n , of this body is characterised in the theory through a parameter A = 2ω r * n /3U∞ where ω is the frequency of the acoustic wave and U∞ the mean flow speed. By comparing asymptotic and numerical results, the receptivity coefficient, C1, is calculated. The receptivity coefficient is seen to decrease dramatically with a small increase in nose radius, a local rise occurs about A = 0.035 and it then gradually declines to nearly zero when A = 0.1.

Tollmien-schlichting wave amplitudes on a semi-infinite flat plate and a parabolic body: comparison of a parabolized stability equation method and direct numerical simulations

“…It has also be shown that the first of the Lam-Rott eigenmodes matches to the unstable Tollmien-Schlichting (T-S) mode of the Orr-Sommerfeld equation, which exhibits exponential streamwise growth downstream (2). A similar asymptotic structure in the leading edge region has been shown to exist on both a parabolic body (16) and more general bodies where the slip velocity tends to a constant downstream (17). An equivalent asymptotic analysis in the Orr-Sommerfeld region for bodies other than the flat plate does not yet exist due to the complex structure in this region.…”

“…This boundary condition depends upon the form of the boundary layer at ξ 0 and the interaction of the free-stream disturbance with the boundary layer upstream of this point. This paper uses the boundary condition given by (30) which is the first Lam-Rott eigenmode of the asymptotic receptivity analysis in the vicinity of the leading edge (12,17,30). This mode is chosen because it is the only discrete eigenmode which exhibits streamwise growth after the lower branch neutral stability point and hence will dominate the solution far downstream.…”

“…The constant λ = f (0) = 0.4696..., while the constant D is calculated numerically and depends upon the curvature of the body (16,17). The function E(N P ) is also calculated numerically by solving the differential equation…”

“…The Lam-Rott eigenmodes are important in receptivity analysis because they exhibit wavelength shortening as they move downstream and hence link the longwavelength free-stream disturbances to the much shorter scale instability waves in the boundary layer via a multiplicative receptivity coefficient. This receptivity coefficient has been calculated for a flat plate (15), a parabolic body (16) and a Rankine body (17) via a combination of numerical and asymptotic approaches. The solution in this leading edge region can be asymptotically matched to the large-Reynolds-number, small wavenumber form of the classical Orr-Sommerfeld equation.…”

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

“…The asymptotic Lam-Rott eigenmodes for a flat plate have been generalised for a parabolic body by Hammerton & Kerschen (1996), who also calculate the free-stream dependent receptivity coefficient as a function of the nose radius. This analysis has been generalised further by Nichols (2001) to bodies which have an inviscid free-stream velocity which tends to unity far downstream. Nichols also calculates the receptivity coefficient for the Rankine body as a function of the nose radius.…”