| Popis: |
We consider the incompressible axisymmetric Navier-Stokes equations as an idealized model of tornado-like flows. Assuming that an infinite vortex line that interacts with a boundary surface resembles the tornado core, we look for stationary self-similar solutions of the axisymmetric Euler and the axisymmetric Navier-Stokes equations emphasizing the connection among them as the viscosity goes to zero. We construct a class of explicit stationary solutions for the axisymmetric Euler equations and prove that solutions of self-similar Euler equations with slip discontinuity at a finite number of points do not exist. The nonexistence result is then extended to a class of flows with mass input or loss through the vortex line, as well as to conical flows. In addition, a study of the system of self-similar axisymmetric Navier-Stokes equations provides a-priori bounds and information about the configuration of their solutions which are then verified by solving it numerically. Using techniques that are motivated by the theory of viscosity approximation for Riemann problems in conservation laws, we also prove that, under certain assumptions, solutions of the axisymmetric Navier-Stokes equations converge to solutions of axisymmetric Euler equations as the viscosity tends to zero. This also allows us to characterize the type of Euler solutions that exist. Furthermore, a new approach is proposed for solving the one-dimensional system of elastodynamics. The main idea of this approach involves the method of gradient descent to solve an implicit scheme with a constrained variational formulation and the discontinuous Galerkin finite element methods to discretize in space. The resulting optimization scheme performed well, it has an advantage on how it handles oscillations near shocks, and a disadvantage in computational cost, which can be partly alleviated by using techniques on step selection from optimization methods. |
