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Singularity formation in three-dimensional vortex sheets

Hou, Thomas Y. and Hu, Gang and Zhang, Pingwen (2003) Singularity formation in three-dimensional vortex sheets. Physics of Fluids, 15 (1). pp. 147-172. ISSN 1070-6631. doi:10.1063/1.1526100.

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We study singularity formation of three-dimensional (3-D) vortex sheets without surface tension using a new approach. First, we derive a leading order approximation to the boundary integral equation governing the 3-D vortex sheet. This leading order equation captures the most singular contributions of the integral equation. By introducing an appropriate change of variables, we show that the leading order vortex sheet equation degenerates to a two-dimensional vortex sheet equation in the direction of the tangential velocity jump. This change of variables is guided by a careful analysis based on properties of certain singular integral operators, and is crucial in identifying the leading order singular behavior. Our result confirms that the tangential velocity jump is the physical driving force of the vortex sheet singularities. We also show that the singularity type of the three-dimensional problem is similar to that of the two-dimensional problem. Moreover, we introduce a model equation for 3-D vortex sheets. This model equation captures the leading order singularity structure of the full 3-D vortex sheet equation, and it can be computed efficiently using fast Fourier transform. This enables us to perform well-resolved calculations to study the generic type of 3-D vortex sheet singularities. We will provide detailed numerical results to support the analytic prediction, and to reveal the generic form of the vortex sheet singularity.

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Additional Information:©2003 American Institute of Physics. Received 14 September 2001; accepted 9 October 2002; published online 6 December 2002. The research presented here was strongly motivated by a number of fruitful discussions with Dr. Stephen Cowley during his visit to Caltech in the summer of 1998. The fact that the 3-D vortex sheet singularity is supported in a locally thin region was first observed by Dr. Cowley using a different asymptotic analysis based on a scaling argument. We have also benefited from several inspiring conversations with Dr. Dale Pullin. We are indebted to Dr. Hector Ceniceros for his valuable suggestions regarding the presentation of the original manuscript. The research was in part supported by a grant from the National Science Foundation under Contract No. DMS-0073916, and by a grant from the Army Research Office under Contract No. DAAD19-99-1-0141. P.Z. also wishes to acknowledge the support of the Special Funds for Major State Basic Research Projects No. G1999032804 from China.
Subject Keywords:vortices; boundary integral equations; fast Fourier transforms; numerical analysis; flow instability; shear turbulence
Issue or Number:1
Record Number:CaltechAUTHORS:HOUpof03
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:3127
Deposited By: Tony Diaz
Deposited On:16 May 2006
Last Modified:08 Nov 2021 19:53

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