Author(s): M. B. Abbott; J. Larsen

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Abstract: The question concerning the dimensionality of models is posed, both for the numerical models themselves and for the differential equations upon which they are usually based. It is observed that only in certain limits can the models considered here be regarded as strictly "two-dimensional". It is also observed that schemes of the Leendertse type, despite possessing an alternating direction algorithmic structure defined on a rectangular grid, have a property of near-isotropy in the horizontal plan when square grids are used. The limiting case of a strictly two-dimensional flow is characterised. The general notion of streamline is then considered. It is observed that classical hydraulics makes use of a hierarchy of streamlines, each level in the hierarchy being associated with a particular scale of filtering. To each such filtering there corresponds a set of defiltering terms that effectively constitutes an "antifilter". Viscosity, Reynolds' stress, bed resistance, Leonard stress and further such terms are now identified as the antifilters corresponding to the successive levels in the hierarchy of filters and it is observed that these terms may be defined, very generally, in such a way. The filter that provides the illusion of streamlines in models of the Leendertse type is described and the corresponding antifilter is derived. The action of this antifilter on the vortex sheet of a sudden expansion is described. Modelling of depth-integrated flows corresponds to a form of Large Eddy Simulation (LES). The LES process of triad interaction is described in the language of the continuum for strictly two-dimensional flows and this process is compared with the process as actually modelled in a first-order language. Finally, the Equivalence Theorem of Lax is reviewed. It is suggested that the problem of determining the detailed limiting solution of a circulating flow, as the grid step tends to zero, is an undecidable problem. The principal mathematical derivations have been relegated to four appendices to the paper. Equations taken from Part 1 of this work are prefixed by a letter A while equations in the appendices are prefixed by the number of the appendix.

DOI: https://doi.org/10.1080/00221688509499335

Year: 1985