Skip to main content Accessibility help
×
Home
Hostname: page-component-5c569c448b-qj5tk Total loading time: 0.29 Render date: 2022-07-05T12:58:36.508Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "useRatesEcommerce": false, "useNewApi": true } hasContentIssue true

Equation of motion of a diffusing vortex sheet

Published online by Cambridge University Press:  26 April 2006

M. R. Dhanak
Affiliation:
Department of Ocean Engineering, Florida Atlantic University, Boca Raton, FL 33431, USA

Abstract

Moore's (1978) equation for following the evolution of a thin layer of uniform vorticity in two dimensions is extended to the case of a non-uniform, instantaneously known, vorticity distribution, using the method of matched asymptotic expansions. In general, the vorticity distribution satisfies a boundary-layer equation. This has a similarity solution in the case of a vortex layer of small thickness in a viscous fluid. Using this solution, an equation of motion of a diffusing vortex sheet is obtained. The equation retains the simplicity of Birkhoff's integro-differential equation for a vortex sheet, while incorporating the effect of viscous diffusion approximately. The equation is used to study the growth of long waves on a Rayleigh layer.

Type
Research Article
Copyright
© 1994 Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Birkoff, G. 1962 Proc. Symp. Appl. Math. Am. Math. Soc. 13, 55.
Baker, G. R. & Shelley, M. J. 1990 J. Fluid Mech. 215, 161194.
Dhanak, M. R. 1980 The motion of vortex layers and vortex filaments. PhD dissertation. Imperial College, London.
Dhanak, M. R. 1994 Stud. Appl. Maths. In press.
Drazin, P. G. & Howard, L. N. 1962 J. Fluid Mech. 14, 257.
Goldstein, S. 1938 Modern Developments in Fluid Mechanics, vol. I. Clarendon.
Moore, D. W. 1978 Stud. Appl. Maths 58, 119140.
Moore, D. W. 1981 SIAM J. Sci. Stat. Comput. 2, 6584.Google Scholar
Rayleigh, Lord 1894 Theory of Sound, 2nd edn, chap. XXI. Macmillan.
Saffman, P. G. 1992 Vortex Dynamics. Cambridge University Press.
Tryggvason, G., Dahm, W. J. A. & Skeih, K. 1991 Trans. ASME I: J. Fluids Engng 113, 3136.
12
Cited by

Save article to Kindle

To save this article to your Kindle, first ensure coreplatform@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Equation of motion of a diffusing vortex sheet
Available formats
×

Save article to Dropbox

To save this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you used this feature, you will be asked to authorise Cambridge Core to connect with your Dropbox account. Find out more about saving content to Dropbox.

Equation of motion of a diffusing vortex sheet
Available formats
×

Save article to Google Drive

To save this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you used this feature, you will be asked to authorise Cambridge Core to connect with your Google Drive account. Find out more about saving content to Google Drive.

Equation of motion of a diffusing vortex sheet
Available formats
×
×

Reply to: Submit a response

Please enter your response.

Your details

Please enter a valid email address.

Conflicting interests

Do you have any conflicting interests? *