Sergey A. Gaponov

Streamwise structures generated by external vortical and thermal waves in subsonic and supersonic boundary layers have been studied. Particular attention is paid to the formulation of the boundary conditions at the outer edge of the boundary layer. It has been found that the disturbances of the boundary-layer streamwise velocities can be several times larger than the amplitude of the external vortical wave. The generation efficiency decreases with increasing Mach number. It has been shown that the influence of thermal external waves on the boundary-layer flow structure is much weaker than the influence of the vortex structure of the flow.

The problem of hydrodynamic stability of a boundary layer with diffusion combustion is formulated in the Dan–Lin–Alekseev approximation and at constant Prandtl and Schmidt numbers; it is reduced to solving a system of the tenth-order ordinary differential equations with homogeneous boundary conditions. With Lewis numbers equal to unity, it may be lowered to the eighth order. In the inviscid approximation, the stability problem is reduced to the integration of a single second-order differential equation. Based on the obtained stability equations and calculations of stationary flow parameters, the stability of a supersonic boundary layer with diffusive combustion on a permeable plate with hydrogen supply through its pores is studied for the first time by direct numerical modeling. With the Mach number M = 2, the possibility of flame flow stabilization is established using calculations. It is shown that within the framework of the inviscid theory of stability, it is possible to obtain quite reliable data on the maximum degrees of the growth of disturbances.

It is found that the problem of boundary layer stability with a diffusion flame under the conditions of a constant molecular weight over the boundary layer in the inviscid approximation and in the Dunn–Lin approximation with identical Schmidt numbers can be reduced to a similar problem for a single-component gas. Approximately, this conclusion is also valid for different Schmidt numbers. Calculations of the steady-state parameters of the boundary layer show that there are two generalized inflection points testifying to a necessary condition of the “inviscid" Rayleigh instability. Simulations show that the boundary layer with a diffusion flame is most unstable to two-dimensional disturbances. As the frequency is increased, the phase velocity of the growing wave tends to the value at the generalized inflection point. Despite sufficiently large growth rates, the degree of spatial enhancement of the wave is equal with high accuracy to the ratio of the time degree of enhancement to the group velocity.

The problem of stability of a subsonic boundary layer is solved under the conditions of heat supply inside the boundary layer with injection of a homogeneous gas through a porous plate that partially simulates the problem of stability of a boundary layer with diffuse combustion. Two-dimensional waves are the most growing waves over the entire range of investigated parameters. It is found that in the case of a fixed norm of heat supply the maximum temperature in the boundary layer increases with the Reynolds number, i.e., with increase in the distance from the leading plate edge. This is in agreement with available experiments and calculations of parameters of the boundary layer with diffuse combustion. In this case the dependence of the degrees of amplification on the Reynolds number, which are maximum in the frequency, is nonmonotonic. It is shown that gas injection with heat supply destabilizes the boundary layer, as it occurs without heat supply. On the other hand, the stabilizing role of heat supply is also shown under the conditions of gas injection through a porous wall. As the frequency of growing wave increases, the phase velocity tends to the velocity at the generalized inflection point. Despite fairly large degrees of growing, the Gaster relation is valid. In accordance with this relation the spatial degree of amplification is equal to the temporal degree of amplification divided by the group velocity.

The paper presents a study of a supersonic boundary layer while heat supply to the layer’s narrow internal region. Calculations were performed for a flow with Mach number M = 2 and the dimensionless temperature (normalized to the temperature at a layer boundary edge T_w = 3.9 (630 K). It was demonstrated that heat supply causes a twofold increase in the 2D disturbance increments, while the frequency range expands by factor of three with a heat-insulated plate (zero heat input) taken as a reference. For a heated plate (with T_w = 3.9), the heat input into this layer delays the disturbance growth; that is, this facilitates the boundary layer stabilization and the disturbance increments reduce by almost four times. Along with 2D waves, the oblique waves were investigated in this paper. It was found that for low-frequency disturbance, we observe an intense growth of 3D (oblique) waves. Meanwhile, 2D disturbances enhance for a high-frequency range. These facts comply with the numerous data on boundary layer stability. However, these factors for boundary layer stability become less significant in the case of heat input to the supersonic boundary layer (compared with the zero heat input). The study also observes the longitudinal structures with a downstream decay. The heat supply makes these structures decrement lower in the supersonic boundary layer (meaning a slower decay). The results for the study of boundary layer disturbance are compared to similar data for the flow with M ≪ 1.

The interaction between vortex disturbances (Tollmien—Schlichting waves) in the supersonic boundary layer on an impermeable surface is studied within the framework of the weakly nonlinear stability theory. The first level of the nonlinear interaction, namely, that in three-wave resonance systems, is investigated. The main features of the interaction in unit triplets consisting of plane and three-dimensional components, are considered, whereupon the group mutual influence (the joint realization of several simple triplets) is studied. The streamwise dynamics of the disturbances of two types, controlled and natural, are modeled. The possibility of energy redistribution is these wave systems in the case of the nonlinear interaction between the constituting wave packets is studied. The resonance interactions are shown to be adequate to the actual nonlinear processes at the earlier transition stages.

Stability of a subsonic boundary layer with heat supply into its narrow band is studied. The maximum length of the stable boundary layer in the case of heat supply is found to be approximately three times smaller as compared to the case of a thermally insulated wall without heat supply. The frequency range of disturbances and their growth rates are approximately doubled. In contrast to the case without heat supply, three-dimensional disturbances are the most rapidly growing disturbances. In the case of a heated wall, the growth rate of disturbances decreases after heat supply into the boundary layer, which may lead to an increase in the length of the laminar region of the boundary layer.

The possibility of controlling unsteady disturbances, traveling Tollmien—Schlichting waves, and stationary streamwise structures is studied. The investigation is performed for the flat-plate boundary layer at the freestream Mach number M = 2. The possible enhancement and suppression of the growth of these waves by stationary streaky structures of the stability eigenproblem of supersonic boundary layer is studied. The problem is solved in the local-parallel approximation within the framework of the three-wave resonance interaction. The pumping wave is a stationary, near-streaky formation. It is shown that even in the stability domain Tollmien—Schlichting waves grow under the influence of the streamwise structures. It is established that under certain conditions the effect of stationary disturbances on these waves can be considerable also in the instability domain and the Reynolds number ranges in which the steady disturbances suppress traveling waves are determined.