Heat and mass transport in nonhomogeneous random velocity fields

Abstract

The effective equation describing the transport of passive tracers in nonsolenoidal velocity fields is determined, assuming that the velocity field U(r,t) is a function of both position r and time t, albeit remaining locally random. Assuming a strong separation of scales and applying the method of homogenization, we find a Fickian constitutive relation for the coarse-grained particle flux, as the sum of a convective part, V(E)c, and a diffusive term, -D(s). Inverted Delta c, where V(E) is the Eulerian mean tracer velocity, c the average particle concentration, and D(s) the effective diffusivity. The latter can be written as D(s)(r,t)=D(0)I+D(r,r,t), where D0 is the molecular diffusivity, I the unit dyadic and D(r(1),r(2),t) the cross diffusion dyadic. Conversely, the Eulerian mean velocity V(E)(r,t) is the sum of the microscale mean tracer velocity V(r,t) and a particle drift velocity, V(d)(r,t)=-[(delta/delta r(2)).D(T)(r,r(2),t)](r(2)=r), which depends on the nonhomogeneity of the velocity field at the macroscale. The microscale mean particle velocity, in turn, is the sum of the mean fluid velocity and the ballistic tracer velocity, which is due to the local nonuniformity of the concentration field and is therefore structurally different from the tracer drift velocity. In the limit of large Peclet numbers, D(s) coincides with the self-diffusion dyadic, as it measures the local temporal growth of the mean square displacement of a tracer particle from its average position. In this case, the motion of a tracer particle is a random process in the manner of Stratonovich, where the smoothly varying mean tracer velocity equals the microscale mean tracer velocity and the fluctuating term is described through the cross diffusion dyadic D(r(1),r(2),t).

Effective Parameter Functions for the Richards Equation in Layered Porous Media

“…In the unsaturated zone the flow is due to gravity and capillary forces. There are therefore two time scales that could be estimated (e.g., Mauri, 2003; Lunati et al, 2002). The first is the time scale due to gravity forces on the large length scale T 1 = L / K h In the case of a small Bond number, Bo ≈ ε 0 , the typical time scale due to capillary forces on the large scale T 2 = L 2 /( K h H ) is of the same order as T 1 However, if the Bond number is large, Bo ≈ ε −1 , capillary forces and gravity forces on the large scale attain different time scales, both of which have to be taken into account, if capillary and gravity forces are also considered in the upscaled problem.…”

“…In the unsaturated zone the flow is due to gravity and capillary forces. There are therefore two time scales that could be estimated (e.g., Mauri, 2003;Lunati et al, 2002). The first is the time scale due to gravity forces on the large length scale T 1 5 L/K h .…”

“…It depends on the large‐scale boundary conditions of the problem, whether averaged higher order terms contribute to the averaged solution or not. Similarly to Mauri (1991, 2003), we assume that the averaged orders higher than the zeroth order do not contribute to the solution. The boundary conditions of the large‐scale problem should be captured by the averaged zeroth order pressure head.…”

Explicit expressions for eddy-diffusivity fields and effective large-scale advection in turbulent transport

“…Its symmetric part (which is also positive definite) contributes to the diffusion process while both the symmetric and the antisymmetric parts enter, in general, into the effective advection velocity U E which turns out to be compressible (scalar dynamics in compressible fields has been analysed, e.g. by Vergassola & Avellaneda (1997) and Mauri (2003)). The eddy diffusivity, often associated to the sole small-scale activity, here explicitly depends on the large-scale advection U.…”

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).…”