Aerotecnica Missili & Spazio, The Journal of Aerospace Science, Technology and Systems Stability Analysis of Three-Dimensional Laminar Compressible Boundary Layers based on Ray-Tracing Theory and Multiple Scale Technique ∗ R. S. Donelli a , D. de Rosa b a CIRA - Centro Italiano Ricerche Aerospaziali Programmazione Strategica e Sviluppo Business b CIRA - Centro Italiano Ricerche Aerospaziali Laboratorio di Modellistica Fluidodinamica Abstract The objective of the present work is to improve the physical understanding of specific 3D stability and transition processes in 3D mean flow by using a full linear stability theory, based on a multiple scales method and ray theory. A full 3D test case in two different flow conditions has been investigated and compared against a classical transition prediction approach. Preliminary results are presented. 1. Introduction The traditional approach to transition prediction is based on the linear stability analysis of viscous flows, generally treated as thin shear layers, consisting in de- termining the evolution, in space or in time, of small perturbations superimposed to a basic flow field [1, 2]. In the framework of the parallel flow assumption, the stability of the flow is studied with respect to each Fourier component of the disturbance, characterized by given frequency ω and wave vector  k =(α, β). Since the governing equations are homogeneous with homo- geneous boundary conditions the problem has non- trivial solution only for a particular combination of the wave parameters and the local Reynolds number, given by the so-called dispersion relation. In the present work, the linear instability of steady laminar boundary layers is approached in the frame- work of the theory of ray tracing in non-homogeneous anisotropic dispersive wave systems. The studies on the development of a ray-theory transition prediction model and its implementation for the stability analy- sis on 3D swept tapered wings has been developed in cooperation with the University of Fisciano [3–5]. The main difference between these approaches is that the ray theory approach allows to compute the ray trajec- tory along which the disturbance amplify and use this as integration path. This approach is closely a what ∗ Based on paper presented at the XXIII Congresso Nazionale AIDAA, Novembre 2015 Torino, Italia 1 c AIDAA, Associazione Italiana di Aeronautica e Astronautica really happen in the propagation of the disturbances while the traditional approach neglects the trajectory of the disturbance and just use as integration path the chordwise flow direction. The theoretical model will be presented in section 2 by emphasizing the main differences between the classical 2.5D and the present 3D approaches. Sec- tion 3 reports the results of stability analyses for a DLR wing that has been used as reference test case in two different flight conditions, Finally, section 4 will describe the most relevant conclusions. 2. Theoretical Model for Linear Stability The- ory The theoretical approach to ray-tracing theory of the propagation of disturbances in 3D incompress- ible boundary layers valid for infinite and conical swept boundary layers was presented [3]. The present method represents a continuation of the works [3, 6] in 3D compressible boundary layers. The starting equations are the classical Navier-Stokes equations for a compressible, viscous, newtonian fluid in non- dimensional form. In the hypothesis of small perturbations the generic flow quantity q can be expressed as the sum of an aver- aged quantity plus its fluctuation q =ˆ q + δq. Within this hypothesis dynamic viscosity and conductibility coefficients are considered as function of the tempera- ture and can be expressed as: 248