Book contents
- Frontmatter
- Contents
- Preface
- 1 An Introduction to Vortex Dynamics for Incompressible Fluid Flows
- 2 The Vorticity-Stream Formulation of the Euler and the Navier-Stokes Equations
- 3 Energy Methods for the Euler and the Navier–Stokes Equations
- 4 The Particle-Trajectory Method for Existence and Uniqueness of Solutions to the Euler Equation
- 5 The Search for Singular Solutions to the 3D Euler Equations
- 6 Computational Vortex Methods
- 7 Simplified Asymptotic Equations for Slender Vortex Filaments
- 8 Weak Solutions to the 2D Euler Equations with Initial Vorticity in L∞
- 9 Introduction to Vortex Sheets, Weak Solutions, and Approximate-Solution Sequences for the Euler Equation
- 10 Weak Solutions and Solution Sequences in Two Dimensions
- 11 The 2D Euler Equation: Concentrations and Weak Solutions with Vortex-Sheet Initial Data
- 12 Reduced Hausdorff Dimension, Oscillations, and Measure-Valued Solutions of the Euler Equations in Two and Three Dimensions
- 13 The Vlasov–Poisson Equations as an Analogy to the Euler Equations for the Study of Weak Solutions
- Index
11 - The 2D Euler Equation: Concentrations and Weak Solutions with Vortex-Sheet Initial Data
Published online by Cambridge University Press: 03 February 2010
- Frontmatter
- Contents
- Preface
- 1 An Introduction to Vortex Dynamics for Incompressible Fluid Flows
- 2 The Vorticity-Stream Formulation of the Euler and the Navier-Stokes Equations
- 3 Energy Methods for the Euler and the Navier–Stokes Equations
- 4 The Particle-Trajectory Method for Existence and Uniqueness of Solutions to the Euler Equation
- 5 The Search for Singular Solutions to the 3D Euler Equations
- 6 Computational Vortex Methods
- 7 Simplified Asymptotic Equations for Slender Vortex Filaments
- 8 Weak Solutions to the 2D Euler Equations with Initial Vorticity in L∞
- 9 Introduction to Vortex Sheets, Weak Solutions, and Approximate-Solution Sequences for the Euler Equation
- 10 Weak Solutions and Solution Sequences in Two Dimensions
- 11 The 2D Euler Equation: Concentrations and Weak Solutions with Vortex-Sheet Initial Data
- 12 Reduced Hausdorff Dimension, Oscillations, and Measure-Valued Solutions of the Euler Equations in Two and Three Dimensions
- 13 The Vlasov–Poisson Equations as an Analogy to the Euler Equations for the Study of Weak Solutions
- Index
Summary
In the first half of this book we studied smooth flows in which the velocity field is a pointwise solution to the Euler or the Navier–Stokes equations. As we saw in the introductions to Chaps. 8 and 9, some of the most interesting questions in modern hydrodynamics concern phenomena that can be characterized only by nonsmooth flows that are inherently only weak solutions to the Euler equation. In Chap. 8 we introducted the vortex patch, a 2D solution of a weak form of the Euler equation, in which the vorticity has a jump discontinuity across a boundary. Despite this apparent singularity, we showed that the problem of vortex-patch evolution is well posed and, moreover, that such a patch will retain a smooth boundary if it is initially smooth.
In Chap. 9 we introduced an even weaker class of solutions to the Euler equation, that of a vortex sheet. Vortex sheets occur when the velocity field forms a jump discontinuity across a smooth boundary. Unlike its cousin, the vortex patch, the vortex sheet is known to be so unstable that it is in fact an ill-posed problem. We saw this expicitly in the derivation of the Kelvin–Helmholtz instability for a flat sheet in Section 9.3. This instability is responsible for the complex structure observed in mixing layers, jets, and wakes. We showed that an analytic sheet solves self-deforming curve equation (9.11), called the Birkhoff–Rott equation. However, because even analytic sheets quickly develop singularities (as shown in, e.g., Fig. 9.3), analytic initial data are much too restrictive for practical application.
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- Chapter
- Information
- Vorticity and Incompressible Flow , pp. 405 - 449Publisher: Cambridge University PressPrint publication year: 2001