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Original Articles

The Navier–Stokes-α equations revisited

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Pages 303-316 | Received 29 May 2008, Accepted 16 Jan 2009, Published online: 22 Jul 2009
 

Abstract

The Navier–Stokes-α equation is a regularised form of the Euler equation that has been employed in representing the sub-grid scales in large-eddy simulations. Determined efforts have been made to place it on a secure deductive foundation. This requires two steps to be completed. The first is fundamental and consists of establishing from the equations governing the fluid flow, a relationship between two velocities called by Holm (Chaos, 2002a, 12, 518) the “filtered” and “unfiltered” velocities. The second consists of the relation between these two velocities. Until now, the preferred route to the first objective has been variational, by varying the action using Hamilton's principle. Soward and Roberts (J. Fluid Mech., 2008, 604, 297) followed that variational route and established the existence of an important but unwelcome term omitted by Holm in his derivation. It is shown here that the Soward and Roberts result may be derived from Euler's equation by a direct approach with considerably greater efficiency. Holm achieved the second objective by making a “Taylor hypothesis”, which we use here to evaluate the unwelcome term missing from his analysis of the first step. The resulting model equations differ from those of Holm's α model, and the attractive mean Kelvin's circulation theorem that follows from his α equations is no longer valid. For that reason, we call the term omitted by Holm unwelcome.

Acknowledgements

This research was almost completed in the Spring of 2008 during the Workshop on Dynamo Theory held at the Kavli Institute for Theoretical Physics at UC Santa Barbara, supported in part by the National Science Foundation under Grant No. PHY05-51164. It is preprint NSF-KITP-08-105 of the Institute. One of us (PHR) wishes to thank the National Science Foundation also for partial support through CSEDI Grant No. 0652423. We would like to thank an anonymous referee for encouraging us to seek the mean equation (Equation54), which with (Equation52c) provides an alternative to the NS-α equations.

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