6. Conclusions

In this work, the short review of researches on the study of BL equation singularities, which are formed when two streamline families are collided, is presented. This phenomenon can arise only in unsteady and 3D problems and has no analogue in 2D flows. A typical example of such problem is the flow around a slender cone in the vicinity of the runoff plane. In this case, solutions are found in the analytical form that allows to analyze explicitly the singularity character.

The analysis of solutions for the outer flow part revealed two singularity types. One type is in streamwise and cross-velocity viscous perturbations; it arises at values of relative cross pressure gradient k≥1 and leads to the exponential disturbance growth as the runoff plane is approached. At k ¼ 1 the singularity is logarithmic and at k . 1 it is power; its appearance is correlated with the BL separation appearance. Another singularity type at smaller values of k≥1=3 in the first-order approximation leads to the infinite growth of transverse velocity perturbations only and is not related directly with the flow separation; at k ¼ 1=3 the singularity is logarithmic, and at k . 1=3 it is power. These BL singularities correspond to some asymptotic flow structure at Re ≫ 1. This structure includes the boundary region with the dimension of the order of the BL thickness, in which the viscous transverse diffusion effect smoothes the singularity. The comparison of obtained parabolized Navier-Stokes equation solutions describing the flow in the boundary region with BL equations solutions confirms this conclusion. Second region induced by the viscous-inviscid interaction effect has the transverse dimension of the order of square root from the BL thickness and the two-layer structure. For the potential flow in the outer inviscid subregion, the integral solution representation is found on the base of the slender wing theory. The inner subregion is described by full 3D BL equations, the solution of which is obtained for the outer viscous subregion part. It was shown that the viscous-inviscid interaction does not eliminate the singularity

but drifts it in the parametric space. To eliminate the irregularity, the boundary region is needed.

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23

To find the dependence of the critical parameter of the singularity appearance kc on Mach and Prandtl numbers and the wall temperature BL equations, solutions are studied in the near-wall region beside the runoff plane. Equation subcharacteristic (streamlines) analysis showed the presence of one parameter α, the sing of which defines the qualitative change of the streamline topology and, consequently, the physical flow structure. It is shown and confirmed by comparison with all available calculations that the boundary of the solution which exists in the runoff plane corresponds to the criterion αð Þ¼ kc 0. The analysis of BL equation solutions near the runoff plane revealed the presence at α ≥0 of irregular and at α , 0 singular proper solutions. This is confirmed by numerical calculations of the flow around slender delta wing with the small aspect ratio. Singularities in the near-wall region generate the some flow structure in its vicinity, the study of which is out of this paper framework. Presented results do not depend on outer boundary conditions and are true for the full freestream velocity diapason including hypersonic flows.

Presented research allows concluding that the flow in symmetry planes, for example, on wings, has the complex structure, which is needed to take into account the numerical modeling in order to eliminate the accuracy loss. Regular flow function decompositions commonly used at solutions of BL equations are not applied near this plane, and it cannot be considered as a boundary condition plane due to a possible solution disappearance.
