Scalar transport in compressible flow

Abstract

Transport of scalar fields in compressible flow is investigated. The effective equations governing the transport at scales large compared to those of the advecting flow are derived by using multi-scale techniques. Ballistic transport generally takes place when both the solenoidal and the potential components of the velocity do not vanish, despite of the fact that it has zero average value. The calculation of the effective ballistic velocity $V_b$ is reduced to the solution of one auxiliary equation. An analytic expression for $V_b$ is derived in some special instances, i.e. flows depending on a single coordinate, random with short correlation times and slightly compressible cellular flow. The effective mean velocity $V_b$ vanishes for velocity fields which are either incompressible or potential and time-independent. For generic compressible flow, the most general conditions ensuring the absence of ballistic transport are isotropy and/or parity invariance. When $V_b$ vanishes (or in the frame of reference moving with velocity $V_b$), standard diffusive transport takes place. It is known that diffusion is always enhanced by incompressible flow. On the contrary, we show that diffusion is depleted in the presence of time-independent potential flow. Trapping effects due to potential wells are responsible for this depletion. For time-dependent potential flow or generic compressible flow, transport rates are enhanced or depleted depending on the detailed structure of the velocity field.Comment: 27 pages, submitted to Physica

Travelling fronts in stochastic Stokes’ drifts

“…The effective diffusion coefficient is determined by the traveling potential. Similar effects are known in the context of homogenization theory, see e.g., [5,6]. The stochastic Stokes' drift has the additional difficulty that it is an evolution problem in which the small parameter and the time are not independent.…”

“…Using relative entropies and homogenized logarithmic Sobolev inequalities, one can then prove (5). The function u ∞ therefore describes the asymptotic regime of u, in self-similar, traveling variables.…”

“…Questions related to the stochastic Stokes' drift have been studied before in the context of diffusive turbulent flows, see [3,4]. Also see [5] for drift velocity related issues. The effective diffusion coefficient is determined by the traveling potential.…”

Effective velocity for transport in heterogeneous compressible flows with mean drift

“…Making use of this solution, Vergassola and Avellaneda 14 showed that static potential flows cannot produce any heterogeneity induced large scale drift components, and that the large scale drift equals the local drift value. The assumption of a vanishing mean drift is crucial for this statement to hold.…”

Large-scale effects on meso-scale modeling for scalar transport

“…In this limit the goal is to derive the expression of the asymptotic diffusion coefficient renormalized by the presence of the small scale velocity field. This can be accomplished exploiting asymptotic methods (see, e.g., [6,8,9,10,11,12,13,14] among the others). However, in many physical circumstances one has that the velocity field may be though as a smallscale advecting velocity field (at scale ℓ) superimposed to a large-scale, slowly varying component (at scale L ≫ ℓ).…”