Exact spiral solutions of the two-dimensional Euler equations

Tong Zhang, Yuxi Zheng

Research output: Contribution to journalArticle

26 Citations (Scopus)

Abstract

We construct a two-parameter family of self-similar solutions to both the compressible and incompressible two-dimensional Euler equations with axisymmetry. The equations can be reduced under the situation to two systems of ordinary differential equations. In the compressible and polytropic case, the system in autonomous form consists of four ordinary differential equations with a two-dimensional set of stationary points, one of which is degenerate up to order four. Through asymptotic analysis and computations of numerical solutions, we are fortunate to be able to recognize a one-parameter family of exact solutions in explicit form. All the solutions (exact or numerical) are globally bounded and continuous, have finite local energy and vorticity, and have well-defined initial and boundary values at time zero and spatial infinity respectively. Particle trajectories of some of these solutions are spiral-like. In the incompressible case, we also find explicit self-similar axisymmetric spiral solutions, which are, however, somewhat less physical due to unbounded pressures or infinite local energy near their swirling centers.

Original languageEnglish (US)
Pages (from-to)117-133
Number of pages17
JournalDiscrete and Continuous Dynamical Systems
Volume3
Issue number1
StatePublished - Dec 1 1997

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Euler equations
Euler Equations
Exact Solution
Axisymmetry
Numerical Solution
Ordinary differential equations
Particle Trajectory
Self-similar Solutions
Stationary point
Energy
Boundary Value
System of Ordinary Differential Equations
Asymptotic Analysis
Vorticity
Well-defined
Two Parameters
Ordinary differential equation
Asymptotic analysis
Infinity
Zero

All Science Journal Classification (ASJC) codes

  • Analysis
  • Discrete Mathematics and Combinatorics
  • Applied Mathematics

Cite this

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Exact spiral solutions of the two-dimensional Euler equations. / Zhang, Tong; Zheng, Yuxi.

In: Discrete and Continuous Dynamical Systems, Vol. 3, No. 1, 01.12.1997, p. 117-133.

Research output: Contribution to journalArticle

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