Hostname: page-component-586b7cd67f-vdxz6 Total loading time: 0 Render date: 2024-11-28T14:52:18.946Z Has data issue: false hasContentIssue false
  • English
  • Français

Tri-periodic fully three-dimensional analytic solutions for the Navier–Stokes equations

Published online by Cambridge University Press:  17 March 2020

Matteo Antuono Open the ORCID record for Matteo Antuono [Opens in a new window]
Matteo Antuono*
Affiliation:
CNR-INM, Via di Vallerano 139, 00128 Rome, Italy
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper we derive unsteady tri-periodic laminar solutions of the Navier–Stokes equations. In particular, these represent fully three-dimensional (3-D) flows, since all the velocity components depend non-trivially on all three coordinate directions. We show that they belong to the class of Beltrami flows and can be gathered in two distinct solutions characterized by positive and negative helicity. These can be regarded as an extension in three dimensions of the bi-periodic vortex solution by Taylor (Phil. Mag., vol. 46, 1923, pp. 671–674). Their use as benchmarks for checking the accuracy of 3-D numerical codes and/or studying the onset of turbulence is suggested.

JFM classification

Vortex Flows: Vortex dynamics Turbulent Flows: Turbulent transition
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 890 , 10 May 2020 , A23
Copyright
© The Author(s), 2020. Published by 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

Barbato, D., Berselli, L.-C. & Grisanti, C. R. 2007 Analytical and numerical results for the rational large eddy simulation model. J. Math. Fluid Mech. 9, 44–74.CrossRefGoogle Scholar
Berselli, L. C. 2005 On the large eddy simulation of the Taylor–Green vortex. J. Math. Fluid Mech. 7, S164–S191.CrossRefGoogle Scholar
Berselli, L. C. & Cordoba, D. 2009 On the regularity of the solutions to the 3D Navier–Stokes equations: a remark on the role of the helicity. C. R. Acad. Sci. Paris Ser. I 347, 613–618.CrossRefGoogle Scholar
Brachet, M., Meiron, D. I., Orszag, S. A., Nickel, B. G., Morf, R. H. & Frisch, U. 1983 Small-scale structure of the Taylor–Green vortex. J. Fluid Mech. 130, 411–452.CrossRefGoogle Scholar
Di Mascio, A., Antuono, M., Colagrossi, A. & Marrone, S. 2017 Smoothed particle hydrodynamics method from a large eddy simulation perspective. Phys. Fluids 29, 035102.CrossRefGoogle Scholar
Di Mascio, A., Muscari, R. & Dubbioso, G. 2014 On the wake dynamics of a propeller operating in drift. J. Fluid Mech. 754, 263–307.CrossRefGoogle Scholar
Drikakis, D., Fureby, C., Grinstein, F. F. & Youngs, D. 2007 Simulation of transition and turbulence decay in the Taylor–Green vortex. J. Turbul. 8 (20), 1–12.Google Scholar
Ethier-Ross, C. & Steinman, D. A. 1994 Exact fully 3D Navier–Stokes solutions for benchmarking. Intl J. Numer. Meth. Fluids 19, 369–375.CrossRefGoogle Scholar
Goldstein, S. 1940 Three-dimensional vortex motion in a viscous fluid. Lond. Edinb. Dublin Phil. Mag. J. Sci. 30 (199), 85–102.CrossRefGoogle Scholar
Langlois, W. E. & Deville, M. O. 2014 Slow Viscous Flow, 2nd edn. Springer International Publishing.CrossRefGoogle Scholar
Muscari, R., Dubbioso, G. & Di Mascio, A. 2017 Analysis of the flow field around a rudder in the wake of a simplified marine propeller. J. Fluid Mech. 814, 547–569.CrossRefGoogle Scholar
Orszag, S. A. 1974 Numerical simulation of the Taylor–Green vortex. In Computing Methods in Applied Sciences and Engineering Part 2: International Symposium, Versailles, pp. 50–64. Springer.CrossRefGoogle Scholar
Sengupta, A., Suman, V. K., Sengupta, T. K. & Bhaumik, S. 2018a An enstrophy-based linear and nonlinear receptivity theory. Phys. Fluids 30, 054106.CrossRefGoogle Scholar
Sengupta, T. K., Sharma, N. & Sengupta, A. 2018b Non-linear instability analysis of the two-dimensional Navier–Stokes equation: the Taylor–Green vortex problem. Phys. Fluids 30, 054105.CrossRefGoogle Scholar
Sharma, N. & Sengupta, T. K. 2019 Vorticity dynamics of the three-dimensional Taylor–Green vortex problem. Phys. Fluids 31, 035106.CrossRefGoogle Scholar
Taylor, G. I. 1923 LXXV. On the decay of vortices in a viscous fluid. Phil. Mag. 46, 671–674.CrossRefGoogle Scholar
Taylor, G. I. & Green, A. E. 1937 Mechanism of the production of small eddies from large ones. Proc. R. Soc. Lond. A 158, 499–521.Google Scholar
Wang, C. Y. 1989 Exact solutions of the unsteady Navier–Stokes equations. Appl. Mech. Rev. 42, CONF-8901202–.CrossRefGoogle Scholar