Abstract

A summary is given of recent work on numerical and analytical studies in nonlinear depletion in the 3D Navier–Stokes equations by Donzis et al. (2013) and Gibbon et al. (2014). These results are specifically discussed in terms of modal dependency, where the high modes are controlled by the lowest modes. The modes (frequencies) used are L2m-norms of vorticity.

Introduction

In fluid dynamics the idea that the low modes of a system might conceivably control the high modes has long been attractive, not least because it suggests a potential mechanism for the reduction in size and cost of computing high dimensional systems, in particular three dimensional turbulent flows. The idea has emerged in different guises over the last 40 years including the development of centre manifolds for ODEs (Guckenheimer & Holmes, 1997) and inertial manifolds for PDEs (Foias, Sell, & Temam, 1988, Titi, 1990, Foias & Titi, 1991, Robinson, 1996), together with the concept of determining modes and nodes (Foias & Temam, 1984). Success has been limited to the construction of an inertial manifold for the one-dimensional Kuramoto–Sivashinsky equation (Foias, Jolly, Kevrekidis, Sell, & Titi, 1988) and the achievement of an estimate for the number of determining modes and nodes for the 2D Navier–Stokes equations (Foias & Titi, 1991, Jones & Titi, 1993, Olson & Titi, 2003, 2008, Farhat, Jolly, & Titi, 2015). The 3D Navier–Stokes equations, however, have remained stubbornly resistant, the main obstacle being the issue of existence and uniqueness of solutions (Constantin & Foias, 1988, Doering & Gibbon, 1995, Foias, Manley, Rosa, & Temam, 2001, Escauriaza, Seregin, & Šverák, 2003).

Another major feature of work on the 3D Navier–Stokes equations has been the difficulty in interpreting the results of large scale numerical simulations in the context of the highly limited analytical results that are available : for early numerical results see Orszag & Patterson (1972), Rogallo (1981), Kerr (1985), Eswaran & Pope (1988), Jimenez, Wray, Saffman, & Rogallo (1993), Moin & Mahesh (1998). However, now that higher resolution is available, it is perhaps time to renew the idea of interpreting some more recent simulations to see if they can inform the analysis by suggesting new ways of considering the Navier–Stokes regularity problem: for more recent numerical work see Kurien & Taylor (2005), Ishihara, Gotoh, & Kaneda (2009), Donzis, Yeung, & Sreenivasan (2008), …