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Analytical solution to the steady Navier-Stokes equation for a supersonic cone in the area of boundary layer behind the shock wave by means tunnel mathematics
  • Oleh G. Shvydkyi
Oleh G. Shvydkyi
Zaporiz'kij nacional'nij universitet

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The means of tunnel mathematics (the theory of functions of spatial complex variable) allow to find an analytical solution for problem of supersonic flow around a cone in the area of boundary layer and beyond. The peculiar feature of Navier-Stokes equations is that they allow to determine analytical velocity fields of fluids only for small number of simple problems. Of course, the problem of supersonic motion of fluid around a cone is not included in this number. Tunnel mathematics is a method for finding analytical vector velocity fields for steady flows of fluids with axial symmetry. The Navier-Stokes equations themselves are then used to determine pressure and temperature distributions. The main theorem of tunnel mathematics allows to find these distributions for planes z = const (it is similar to the constructing of slices of brain at MRI procedure). Further, collecting these “slices”, we can obtain full space distributions of pressure and temperature around a supersonic cone. At this stage of investigations, the conclusions obtained by means tunnel mathematics make it possible to qualitatively judge the thickness of boundary layer on the cone surface, the shape of the shock wave and weather the shock wave intersects the boundary layer or not. First of all, we focused on ensuring that resulting solutions corresponded to the physical pattern of phenomena. No doubt, solutions obtained by means tunnel mathematics must be confirmed experimentally.