A Rapid Method of Calculating N-Factors for Estimating Transition Position

Abstract : Transition estimation using the E-to-N approach involves evaluating large numbers of eigenvalues of the linear perturbation equations. These eigenvalues provide the amplification rates of the modes of various frequencies and angles of orientation. The number of evaluations required to determine the N-factors can be large, especially when cross-flow modes over a leading edge are important. But since the variation in the values of the amplification rates varies relatively slowly and smoothly with the parameters involved in the problem, it seems that a fresh evaluation of the eigenvalue by solving the governing equations for every mode is somewhat of an overkill. Here we explore the idea that the behavior of the eigenvalues can be described by some compact expansion. An appropriate functional form has been found that describes the eigenvalues in terms of the controlling parameters and the necessary coefficients evaluated so that predictions of N-factors can be made for two-dimensional compressible boundary layers very rapidly indeed. The N-factor curves on a typical aerofoil boundary layer can be determined roughly 2500 times as fast as by the use of a direct solver. The accuracy is well within the requirements of the E-to-N prediction method. Extension to three-dimensional flows is currently being undertaken.