Three-dimensional linear stability approach to transition on wings and bodies of revolution at incidence

Abstract

The calculation of transition on an infinite swept wing at several angles of incidence for several sweep angles and on a body of revolution at one incidence is investigated with the en method based on the eigenvalue formulation of Cebeci and Stewartson in which the relationship between the two wave numbers a and /? are determined by making use of the saddle-point method. The method, which is based on the solution of the boundary-layer and Orr-Sommerfeld equations by a finite difference procedure, is evaluated in terms of measurements reported for the flow around a swept wing equipped with a cambered leading edge and attached to a half fuselage and for the flow around a prolate spheroid at 10-deg incidence. It is shown to be convenient to use, particularly because the neutral stability curves (zarfs) facilitate the calculation and avoid uncertainties associated with the choice of magnitude and location of the critical frequencies. In general, the calculated values of the onset of transition are in good agreement with measured values. NUMBER of correlation formulas have been developed to calculate the onset of transition and have proved very useful, although it is recognized that they lack generality and can be used outside the range of flows upon which they are based only with great care. Linear stability theory and its implementation through the en method, as proposed by Smith and Gamberoni1 and van Ingen,2 and recently reviewed by Bushnell et al.,3 offers the possibility of being able to represent the onset of transition in a wide range of flow configurations. The accuracy of the approach for incompressible flows can be assessed in terms of two-dimension al boundary layers with and without heat transfer,4 separation bubbles,5'6 rotating disks,7 and yawed cylinders.8 This paper is concerned with its application to swept wings and bodies of revolution at inci- dence. The calculation method is based on the solution of the three-dimensional laminar boundary layer and linear stability equations and is described in Sec. II for an infinite swept wing and in Sec. Ill for a body of revolution. In the former case, the boundary-layer calculations for an infinite swept wing are solved in an inverse mode with the relationship between the inviscid and viscous flows expressed through the Hilbert inte- gral. The resulting velocity profiles are used in the solution of the stability equation and, thereby, in the determination of the amplification rates in the en method to provide the location of the onset of transition. The disturbance frequency needed in the en method and the manner in which it is obtained and used are also considered in this section. Section III describes the procedure for a body of revolution where the boundary-layer equations are solved with a combination of standard and characteristic box methods for a prescribed inviscid pressure distribution and with a stability criterion to ensure numerical accuracy, as previously used by Cebeci and Su.9 The resulting profiles are then used in the solution of the stability equation described in the previous section. In Sec. IV, the method is evaluated in relation to the experiments of Arnal and Juil-