Passive scalar decay in chaotic flows with boundaries

Abstract

This paper considers the long-time decay rate of a passive scalar in twodimensional flow. The focus is on the effects of boundary conditions for kinematically prescribed velocity fields with random or periodic time dependence. Scalar evolution is followed numerically in a periodic geometry for families of flows that have either a slip or a no-slip boundary condition on a square or plane layer subdomain D. The boundary conditions on the passive scalar are imposed on the boundary of D by restricting to a subclass invariant under certain symmetry transformations. The scalar field obeys constant (Dirichlet) or no-flux (Neumann) conditions exactly for a flow with the slip boundary condition and approximately in the no-slip case. At late times the decay of a passive scalar is exponential in time with a decay rate γ (κ), where κ is the molecular diffusivity. Scaling laws of the form γ(κ) ζ C κ α for small κ are obtained numerically for a variety of boundary conditions on flow and scalar, and supporting theoretical arguments are presented. In particular when the scalar field satisfies a Neumann condition on all boundaries, α ζ 0 0 for a slip flow condition; for a no-slip condition we confirm results in the literature that α ζ 1/2 for a plane layer, but find α ζ 2/3 in a square subdomain D where the decay is controlled by stagnant flow in the corners. For cases where there is a Dirichlet boundary condition on one or more sides of the subdomain D, the exponent measuring the decay of the scalar field is α ζ 1/2 for a slip flow condition and α ζ 3/4 for a no-slip condition. The scaling law exponents α for chaotic time-periodic flows are compared with those for similarly constructed random flows. © 2012 The Japan Society of Fluid Mechanics and IOP Publishing Ltd.