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The velocity field and Birkhoff-Rott integral for non-decaying, non-periodic vortex sheets

David M. Ambrose

Abstract

The Birkhoff-Rott integral expresses the fluid velocity on a vortex sheet. This integral converges if certain quantities decay at horizontal infinity, but can also be summed over periodic images in the horizontally periodic case. However, non-decaying, non-periodic cases are also of interest, such as the interaction of periodic wavetrains with non-commensurate periods (i.e. spatially quasiperiodic solutions), or non-periodic disturbances to periodic wavetrains. We therefore develop a more general single formula for the Birkhoff-Rott integral, which unifies and extends the cases of decay and periodicity. We verify that under some reasonable conditions this new version of the Birkhoff-Rott integral is the restriction to the vortex sheet of an incompressible, irrotational velocity field, with continuous normal component but with a jump in tangential velocity across the vortex sheet. We give a number of examples of non-decaying, non-periodic sheet positions and sheet strengths for which our assumptions may be verified. While we develop this in the case of two-dimensional fluids, the methodology applies equally well to three-dimensional fluids.

The velocity field and Birkhoff-Rott integral for non-decaying, non-periodic vortex sheets

Abstract

The Birkhoff-Rott integral expresses the fluid velocity on a vortex sheet. This integral converges if certain quantities decay at horizontal infinity, but can also be summed over periodic images in the horizontally periodic case. However, non-decaying, non-periodic cases are also of interest, such as the interaction of periodic wavetrains with non-commensurate periods (i.e. spatially quasiperiodic solutions), or non-periodic disturbances to periodic wavetrains. We therefore develop a more general single formula for the Birkhoff-Rott integral, which unifies and extends the cases of decay and periodicity. We verify that under some reasonable conditions this new version of the Birkhoff-Rott integral is the restriction to the vortex sheet of an incompressible, irrotational velocity field, with continuous normal component but with a jump in tangential velocity across the vortex sheet. We give a number of examples of non-decaying, non-periodic sheet positions and sheet strengths for which our assumptions may be verified. While we develop this in the case of two-dimensional fluids, the methodology applies equally well to three-dimensional fluids.
Paper Structure (15 sections, 4 theorems, 69 equations)

This paper contains 15 sections, 4 theorems, 69 equations.

Table of Contents

  1. Introduction
  2. Examples and assumptions
  3. Four illustrative examples
  4. Assumptions on $\gamma$ and $\mathcal{C}$
  5. The new expression for the velocity
  6. Rewriting the velocity
  7. The new expression for the Birkhoff-Rott integral
  8. The new velocity is well-defined
  9. Approximation by decaying vortex sheets and uniform convergence on compact sets
  10. The main result
  11. Jump conditions
  12. Limit at vertical infinity
  13. Revisiting the examples
  14. Discussion
  15. Proofs of lemmas

Key Result

Lemma 1

Let $\gamma$ satisfy Assumption assumptionA. The constant, $c,$ associated to $\gamma$ and $\Gamma$ is uniquely defined. We denote the constant as $c=c[\gamma].$

Theorems & Definitions (8)

  • Lemma 1
  • Lemma 2
  • Remark 1
  • Lemma 3
  • Theorem 1
  • proof
  • proof : Proof of Lemma \ref{['assumptionASatisfiedLemma']}
  • proof : Proof of Lemma \ref{['assumptionBLemma']}