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arXiv · 2311.10575

Axisymmetric flows with swirl for Euler and Navier-Stokes equations

Theodoros Katsaounis·Ioanna Mousikou·Athanasios E. Tzavaras
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Abstract

We consider the incompressible axisymmetric Navier-Stokes equations with swirl as an idealized model for tornado-like flows. Assuming an infinite vortex line which interacts with a boundary surface resembles the tornado core, we look for stationary self-similar solutions of the axisymmetric Euler and axisymmetric Navier-Stokes equations. We are particularly interested in the connection of the two problems in the zero-viscosity limit. First, we construct a class of explicit stationary self-similar solutions for the axisymmetric Euler equations. Second, we consider the possibility of discontinuous solutions and prove that there do not exist self-similar stationary Euler solutions with slip discontinuity. This nonexistence result is extended to a class of flows where there is mass input or mass loss through the vortex core. Third, we consider solutions of the Euler equations as zero-viscosity limits of solutions to Navier-Stokes. Using techniques from the theory of Riemann problems for conservation laws, we prove that, under certain assumptions, stationary self-similar solutions of the axisymmetric Navier-Stokes equations converge to stationary self-similar solutions of the axisymmetric Euler equations as $ν\to0$. This allows to characterize the type of Euler solutions that arise via viscosity limits.

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Theodoros Katsaounis, Ioanna Mousikou, Athanasios E. Tzavaras. 2023-11-17. Axisymmetric flows with swirl for Euler and Navier-Stokes equations. https://arxiv.org/abs/2311.10575

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