The concentration distribution of massive dilute species (e.g.
aerosols,
heavy vapours,
etc.) carried in a gas stream in non-isothermal boundary layers is studied
in the
large-Schmidt-number limit, Sc[Gt ]1, including the cross-mass-transport
by thermal diffusion
(Ludwig–Soret effect). In self-similar laminar boundary layers, the
mass
fraction
distribution of the dilute species is governed by a second-order ordinary
differential
equation whose solution becomes a singular perturbation problem when
Sc[Gt ]1. Depending on the sign of the temperature gradient, the
solutions exhibit different
qualitative behaviour. First, when the thermal diffusion transport is directed
toward
the wall, the boundary layer can be divided into two separated regions:
an outer
region
characterized by the cooperation of advection and thermal diffusion and
an inner
region in the vicinity of the wall, where Brownian diffusion accommodates
the mass
fraction to the value required by the boundary condition at the wall. Secondly,
when
the thermal diffusion transport is directed away from the wall, thus competing
with the
advective transport, both effects balance each other at some intermediate
value
of the similarity variable and a thin intermediate diffusive layer separating
two outer regions
should be considered around this location. The character of the outer solutions
changes sharply across this thin layer, which corresponds to a second-order
regular
turning point of the differential mass transport equation. In the outer
zone from the
inner layer down to the wall, exponentially small terms must be considered
to account
for the diffusive leakage of the massive species. In the inner zone, the
equation
is solved in terms of the Whittaker function and the whole mass fraction
distribution is
determined by matching with the outer solutions. The distinguished limit
of Brownian
diffusion with a weak thermal diffusion is also analysed and shown to match
the two
cases mentioned above.
Principal Length Scales in Second-Order Closure Models for Canopy Turbulence
