Uniqueness of the self-similar solutions of three-dimensional laminar boundary-layer equations

3D Boundary Layer Theory

“…Sufficient conditions and singularity types were not studied, and real flow conditions were not considered. Singularities of numerical solutions (the nonuniqueness or the absence of a solution) were found for the laminar boundary layer in the leeward symmetry plane on a round cone at incidence [16][17][18]. Similar results were obtained inside the computation region of the 3D turbulent boundary layer on the swept wing [19].…”

“…The series decomposition of the near-wall solution in the runoff plane showed the presence of a parameter α, the linear combination of skin friction components, and the sign change of which leads to the change of the physical flow topology near this plane [24]. The analysis of BL equations in the near-wall region showed that α ¼ 0 corresponds to the critical value k c , and it was confirmed by all published numerical calculations [16][17][18]20]. In the runoff plane, the new power type singularity in solutions of full BL equations revealed that it is related with the eigensolutions appearing near this plane.…”

“…Firstly, analytical results about singularities were obtained for outer BL part for a such cone. It is understood from previous works [15][16][17][18]20], the singularity can arise when two subcharacteristic (streamlines) families collided -this is a necessary condition. Such situation arises usually in the leeward symmetry (runoff) plane over a body of revolution at an angle of attack.…”

“…Such situation arises usually in the leeward symmetry (runoff) plane over a body of revolution at an angle of attack. Unusual properties in numerical solutions of self-similar equations in this plane for a round slender cone in supersonic freestreams were studied in many works due to the practical interest of the heat exchange on flying vehicles head parts [16][17][18]20]. In this case, one parameter defines the flow.…”