Abstract
Despite the lack of a regularity proof for the 3d Navier-Stokes equations, it is an interesting and important question whether it can be shown that large fluctuations or excursions away from temporal and spatial averages can occur. If it can be demonstrated that the Navier-Stokes equations allow such fluctuations away from averages, then these must have narrow spatial and temporal bandwidths and the width of these will give information about the smallest scale in the flow. Any numerical scheme must ‘resolve’ these spikes to get an accurate representation of the flow. The question of the smallest length scale is the topic of this paper.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
C. Foins and G. Prodi. Sur le comportement global des solutions non-stationnaires des equations de navier-stokes en dimension 2. Rendiconti di Padova, 1:1, 1967.
R. Temam. Navier-Stokes Equations, Theory and Numerical Analysis. North-Holland, Amsterdam, second edition, 1979.
R. Temam. The Navier-Stokes Equations and Non-linear Functional Analysis. CBMSNSF Regional Conference Series in Applied Mathematics. SIAM, 1983.
R. Temam. Infinite Dimensional Dynamical Systems in Mechanics and Physics, volume 68 of Springer Applied Mathematics Series. Springer, Berlin, 1988.
P. Constantin, C. Foins, R. Temam, and B. Nicolaenko. Integral and Inertial Manifolds for Dissipative PDE’s, volume 70 of Springer Applied Mathematics Series. Springer, Berlin, 1988.
P. Constantin, C. Foins, O.P. Manley, and R. Temam. Determining modes and fractal dimension of turbulent flows. J. Fluid Mech., 150:427–440, 1985.
C. Foias, O.P. Manley, R. Temam, and Y.M. Treve. Asymptotic Analysis of the Navier-Stokes Equations. Physica 9D, pages 157–188, 1983.
P. Constantin, C. Foias, and R. Temam. On the Dimension of the Attractors in Two-Dimensional Turbulence. Physica 30D, pages 284–296, 1988.
P. Constantin and C. Foias. The Navier-Stokes Equations. Chicago University Press, 1989.
P. Constantin, C. Foias, and R. Temam. Attractors Representing Turbulent Flows. Memoirs of AMS, 1985.
M. Bartuccelli, P. Constantin, C.R. Doering, J.D. Gibbon, and M. Gisselfält. On the Possibility of Soft and Hard Turbulence in the Complex Ginzburg-Landau Equation. Physica D, 44:421–444, 1990.
M. Bartuccelli, C. Doering, and J.D. Gibbon. Ladder Theorems for the 2d and 3d Navier-Stokes Equations on a Finite Periodic Domain. Nonlinearity, 4:531–542, 1991.
M.V. Bartuccelli, C.R. Doering, J.D. Gibbon, and S.J.A. Maiham. Vorticity Fluctuations in the 3d Incompressible Navier-Stokes Equations. In Preparation.
W.D. Henshaw, H.O. Kreiss, and L.G. Reyna. Smallest Scale Estimates for the Incompressible Navier-Stokes Equations. Preprint.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1993 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Bartuccelli, M., Doering, C.D., Gibbon, J.D., Malham, S.J.A. (1993). Length Scales and the Navier-Stokes Equations. In: Fokas, A.S., Kaup, D.J., Newell, A.C., Zakharov, V.E. (eds) Nonlinear Processes in Physics. Springer Series in Nonlinear Dynamics . Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-77769-1_49
Download citation
DOI: https://doi.org/10.1007/978-3-642-77769-1_49
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-77771-4
Online ISBN: 978-3-642-77769-1
eBook Packages: Springer Book Archive