Occurrence of Tolman-Schlichting waves in the boundary layer under the effect of external perturbations

Abstract

Under small external perturbations, the initial stage of the laminar into turbulent flow transition process in boundary layers is the development of natural oscillations, Tolman-Schlichting waves, which are described by the linear theory of hydrodynamic stability. Subsequent nonlinear processes start to appear in a sufficiently narrow band of relative values of the perturbation amplitudes (1–2% of the external flow velocity) and progress quite stormily. Hence, the initial “linear” stage of relatively slow development of perturbations is governing, in a known sense, in the complete transition process. In particular, the location of the “transition point” depends, to a large extent, on the spectrum composition and intensity of the perturbations in the boundary layer, which start to develop according to linear theory laws, resulting in the long run in destruction of the laminar flow mode. In its turn, the initial intensity and spectrum composition of the Tolman-Schlichting waves evidently depend on the corresponding characteristics of the different external perturbations generating these waves. The significant discrepancy in the data of different authors on the transition Reynolds number in the boundary layer on a flat plate [1–4] is probably explained by the difference in the composition of the small perturbing factors (which have not, unfortunately, been fully checked out by far). Moreover, it is impossible to expect that all kinds of external perturbations will be transformed identically into the natural boundary-layer oscillations. The relative role of external perturbations of different nature is apparently not identical in the Tolman-Schlichting wave generation process. However, how the boundary layer reacts to small external perturbations, under what conditions and in what way do external perturbations excite Tolman-Schlichting waves in the boundary layer have practically not been investigated. The importance of these questions in the solution of the problem of the passage to turbulence and in practical applications has been emphasized repeatedly recently [5, 6], Only the first steps towards their solution have been taken at this time [4, 7–10], Out of all the small perturbing factors under the real conditions of the majority of experiments to investigate the flow stability and transition in the case of smooth polished walls, three are apparently most essential, viz.: the turbulence of the external flow, acoustic perturbations, and model vibrations. In principle, all possible mechanisms for converting the energy of these perturbations into Tolman-Schlichting waves can be subdivided into two classes (excluding the nonlinear interactions which are not examined here): 1) distributed wave generation in the boundary layer; and 2) localized wave generation at the leading edge of the streamlined model. Among the first class is both the possibility of the direct transformation of the external flow perturbations into Tolman-Schlichting waves through the boundary-layer boundary, and wave excitation because of the active vibrations of the model wall. Among the second class are all possible mechanisms for the conversion of acoustic or vortical perturbations, as well as the vibrations of the streamlined surface, into Tolman-Schlichting waves, which occurs in the area of the model leading edge.