Eddy viscosity of parity-invariant flow

A general formalism is developed to determine eddy viscosities for incompressible flow of arbitrary dimensionality subject to forcing periodic in space and time. The dynamics of weak large-scale perturbations is obtained by a multiscale analysis. The large-scale behavior is found to be formally diffusive (first order in time, second order in space) whenever the basic flow is parity invariant, that is, possesses a center of symmetry. The eddy viscosity is in general a fourth-order tensor, for which a compact representation is provided. Explicit expressions of the eddy-viscosity tensor are given (i) for basic flow with low Reynolds numbers, and (ii) when the basic flow is layered, i.e., depends only on one space coordinate and time. A special class of layered flow is two-dimensional, time-independent parallel periodic flow, an example of which is the Kolmogorov flow. Such parallel flow acquires a negative-viscosity instability to large-scale perturbations transverse to the basic flow when the molecular viscosity becomes less than the rms value of the stream function of the basic flow. For flows presenting less symmetry than the Kolmogorov flow, the first large-scale instability is usually found not to be transverse, thus breaking the spatial periodicity of the basic flow. Such nontransverse instabilities, observed in a lattice-gas simulation on the Connection Machine, are reported in the companion paper by H\'enon and Scholl [following paper, Phys. Rev. A 43, 5365 (1991)].