Optimally Controlled Turbulent Boundary Layers in Supersonic Gas Flows

A variational problem for a conditional extremum of the Mayer type on the construction of the distribution law of the normal component of the velocity of injection of a cooled gas into a turbulent boundary layer under a supersonic flow regime that provides the minimum value of the convective heat flux transmitted from the hot gas to the streamlined surface is considered. The isoperimetric condition is the power of the injection control system, calculated taking into account Darcy’s filtration law. To solve the optimal problem, we use the first integral for the conjugate system with respect to Lagrange multipliers, obtained earlier by the authors using the classical theorem of E. Noether on invariant variational problems and the Lie-Ovsyannikov infinitesimal apparatus. A.A. Dorodnitsyn’s method of generalized integral relations, which has proven itself well in calculating the characteristics of boundary layers under various flow regimes, is used for the calculations. A computational experiment conducted for the case of a flow around a sphere shows the effectiveness of the optimal control law found in comparison with a uniform injection: the gain in the value of the minimized functional is 16.8%. The novelty of the study lies in the development of a method for solving the variational problem using the first integral for the conjugate system, as well as the Dorodnitsyn’s method of generalized integral relations. The scientific significance of the study lies in the development of the theory of an optimally controlled boundary layer under a turbulent flow regime in supersonic gas flows. The results obtained may be of interest in the design of systems for the active thermal protection of surfaces in high-velocity gas flows.