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Recent interest in the development of a unifying framework among direct numerical simulations, large-eddy simulations, and statistically averaged formulations of the Navier-Stokes equations, provides the motivation for the present paper. Toward that goal, the properties of the residual ͑subgrid-scale͒ stress of the temporally filtered Navier-Stokes equations are carefully examined. This includes the frame-invariance properties of the filtered equations and the resulting residual stress. Causal time-domain filters, parametrized by a temporal filter width 0Ͻ⌬Ͻϱ, are considered. For several reasons, the differential forms of such filters are preferred to their corresponding integral forms; among these, storage requirements for differential forms are typically much less than for integral forms and, for some filters, are independent of ⌬. The behavior of the residual stress in the limits of both vanishing and infinite filter widths is examined. It is shown analytically that, in the limit ⌬→0, the residual stress vanishes, in which case the Navier-Stokes equations are recovered from the temporally filtered equations. Alternately, in the limit ⌬→ϱ, the residual stress is equivalent to the long-time averaged stress, and the Reynolds-averaged Navier-Stokes equations are recovered from the temporally filtered equations. The predicted behavior at the asymptotic limits of filter width is further validated by numerical simulations of the temporally filtered forced, viscous Burger's equation. Finally, finite filter widths are also considered, and both a priori and a posteriori analyses of temporal similarity and temporal approximate deconvolution models of the residual stress are conducted for the model problem.

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In this paper we study the interplay between the classical and quantum anholonomy effects (Hannay's angle and Berry's phase). When a quantum system with a finite number of energy levels has a Berry phase, it also has a nonzero Hannay angle. We show how systems with infinitely many levels can evade this correspondence, and find some necessary conditions for a system with a Berry phase to have no Hannay angle.

In this paper we introduce some mathematical tools to further study the classical adiabatic holonomy effect known as Hannay's angle. In particular, we prove with purely classical methods that the area (or angle) two-form associated with this effect can be seen as a modification of the symplectic structure of the slow-variable dynamics. We also show that, as in the quantum case, degeneracies cause singularities in this two-form. We conclude with some considerations concerning the triviality or nontriviality of the phase-space bundle associated with this phenomenon.