The temporally filtered Navier–Stokes equations: Properties of the residual stress

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Abstract
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-scale) 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 0<Δ<∞, 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 Δ. 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 Δ→0, the residual stress vanishes, in which case the Navier–Stokes equations are recovered from the temporally filtered equations. Alternately, in the limit Δ→∞, 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.