Asymptotic receptivity analysis and the parabolized stability equation: a combined approach to boundary layer transition

Abstract

We consider the interaction of free-stream disturbances with the leading edge of a body and its effect on the transition point. We present a method which combines an asymptotic receptivity approach, and a numerical method which marches through the Orr-Sommerfeld region. The asymptotic receptivity analysis produces a three deck eigensolution which in its far downstream limiting form, produces an upstream boundary condition for our numerical Parabolized Stability Equation (PSE). We discuss the advantages of this method against existing numerical and asymptotic analysis and present results which justifies this method for the case of a semi-infinite flat plate, where asymptotic results exist in the Orr-Sommerfeld region. We also discuss the limitations of the PSE and comment on the validity of the upstream boundary conditions. Good agreement is found between the present results and the numerical results of Haddad & Corke (1998).

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

“…For more information the reader is referred to (19) and (30). The PSE is valid within the boundary layer on a two-dimensional body with a rounded leading edge.…”

“…If T-S wave amplitude comparisons with experiments are required at the upper branch, then the nonlinear form of the PSE, which is discussed in Bertolotti et al (24), should be considered. The main assumption in the formulation of the PSE is that the streamwise variation of both α and φ P is sufficiently small (24,19). This means that ∂ 2 α/∂ξ 2 P , ∂ 2 φ P /∂ξ 2 P and the product of first derivatives ∂α/∂ξ P and ∂φ P /∂ξ P are O(R …”

“…Equations (2.5) and (2.6) are solved numerically via a spectral collocation technique using Chebyshev polynomials. This method is equivalent to that described in (24) and (19), which the reader is directed to for more details of the scheme. In this study we are only interested in the propagation of the eigenmodes, generated in the leading-edge region of the body, through the Orr-Sommerfeld region.…”

“…One method is conducting direct numerical simulations (DNS) of the T-S wave evolution using the approach examined in Haddad & Corke (18) and the other method is the parabolized stability equation (PSE) scheme laid out in Turner & Hammerton (19). The purpose of this comparison is to examine to what extent the PSE scheme describes the streamwise evolution of disturbances in the boundary layer excited by acoustic waves.…”

“…However, these approaches did not use the receptivity information at the leading edge to link the free-stream information to the amplitude of the T-S wave. This was overcome by (19) who linked the receptivity information to the T-S wave using a scheme which combined the PSE in the Orr-Sommerfeld region with an upstream boundary condition given by the Lam-Rott eigenmode from the leading edge receptivity analysis. This method allowed the streamwise amplitude of the unstable T-S wave to be calculated.…”