Heat and mass transport in nonhomogeneous random velocity fields

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).