Axisymmetric Flows with Swirl for Euler and Navier–Stokes Equations

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 an...

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Bibliographic Details
Published in:Journal of nonlinear science 2024-10, Vol.34 (5), Article 86
Main Authors: Katsaounis, Theodoros, Mousikou, Ioanna, Tzavaras, Athanasios E.
Format: Article
Language:English
Subjects:
Analysis
Axisymmetric flow
Cauchy problems
Classical Mechanics
Conservation laws
Discontinuity
Economic Theory/Quantitative Economics/Mathematical Methods
Euler-Lagrange equation
Fluid flow
Incompressible flow
Mathematical and Computational Engineering
Mathematical and Computational Physics
Mathematics
Mathematics and Statistics
Navier-Stokes equations
Self-similarity
Theoretical
Tornadoes
Viscosity
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Summary: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 → 0 . This allows to characterize the type of Euler solutions that arise via viscosity limits.
ISSN:0938-8974
1432-1467
DOI:10.1007/s00332-024-10064-0