Quantitative results of the study of the invariant gradient criterion of stability, which was theoretically obtained by N. N. Yanenko and S. A. Gaponov, for Introduction and Formulation of the Problems. To obtain reliable data on the thermal state of supersonic and hypersonic aircraft and their drag and dynamics, it is necessary to have credible results on the Reynolds numbers of the laminar-turbulent transition and relaminarization (reverse transition) in supersonic and hypersonic near-wall boundary layers. The problems of stability and transition to turbulence in incompressible and compressible laminar near-wall flows and relarninarization of turbulent boundary layers are rated among the most complicated problems in continuum mechanics. Their importance and practical interest are emphasized [1][2][3][4][5]. Despite the great number of theoretical papers, which are reviewed, for example, in [2, [4][5][6], there is no comprehensive theory of the transition of the laminar boundary-layer flow to a turbulent state. In connection with the use of effective numerical methods, techniques based on the theory of hydrodynamic stability, both linear [2, 4, 7, 8] and nonlinear [9-13], have been developed. Taking into account the difficulties of theoretical investigation of the transition in near-wall boundary layers and its practical importance, Yanenko and Gaponov [14] noted that it is important to determine the conditions of stability and transition on the basis of local flowfield properties.It is impossible now to obtain reliable data on the transition in supersonic wind tunnels, first of all, because of the presence of an acoustic field in test sections [15, 16]. It is shown [17, 18] that the transition Reynolds numbers on the cones which were measured in flight were many-fold higher than those measured in wind tunnels. Along with the acoustic field, scaling effects make an additional contribution to the difference in ground-based and flight data on the laminar-turbulent transition for Moo >/2.0. These effects are caused by the fact that the full-scale Reynolds and Mach numbers and the temperature factor, the external flow perturbations, the effect of operating engines, and the aeroelastic properties of the bodies cannot be simultaneously simulated in supersonic and hypersonic wind tunnels. Reshotko [1] emphasized the necessity of careful preparation of flight experiments for studying the laminar-turbulent transition. The number of experiments conducted is rather small [17][18][19][20][21]. For example, a full-scale experiment was conducted at the Institute of Theoretical and Applied Mechanics (ITAM) of the Siberian Division of the Russian Academy of Sciences on the aerophysical complex "Oblako" ("Cloud") with a working engine [22]. The results obtained are in good agreement with the in-flight data of NASA [18] for free flying cones for identical Reynolds (Rel,oo) and Mach (Moo) numbers. However, the operating engines on the "Oblako" affected the transition for some Reynolds and Mach numbers.