Scalar transport in compressible flow

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.