Passive scalar decay in chaotic flows with boundaries

Abstract

This paper considers the long-time decay rate of a passive scalar in two-dimensional 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 . The boundary conditions on the passive scalar are imposed on the boundary of 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) ≃ Cκα 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 ≃ 0 for a slip flow condition; for a no-slip condition we confirm results in the literature that α ≃ 1/2 for a plane layer, but find α ≃ 2/3 in a square subdomain 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 , the exponent measuring the decay of the scalar field is α ≃ 1/2 for a slip flow condition and α ≃ 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.