Universitā degli studi di Roma “Tor Vergata” - ART - TORVERGATA ...

where et is the turbulent diffusion coefficient (Eddy diffusivity, of the order 10 -3 m 2 s -1 ) that is
higher the em which is of the order 10 -9 m 2 s -1 . Moreover, et depends on the flow, whereas em
depends on the fluids. et is assumed to be isotropic and homogeneous.
The equation (Eq. 2.2-11) provides an economical and a tolerable accurate model for
turbulent diffusion at the mid- and far-fields that are explained below.
The Reynolds analogy comes from the following consideration. In a turbulent flow the
turbulent velocity fluctuations transfer momentum more rapidly than in laminar flow and the
shear stresses between adjacent water layers are higher than in laminar flow. This stresses are
called turbulent or Reynolds stresses:
τ = ρ < u u
t
'
x
'
y
>
Eq. 2.2-12
τt is proportional to the correlation between the turbulent velocity fluctuations and remembering
the equation (Eq. 2.2-11) where et is proportional to the correlation between the tracer
concentration and the velocity fluctuation then because of the problem for estimating τt into the
momentum equation, Boussinesq in 1877 proposed the following relationship for turbulent
stress:
∂〈 ux 〉
τ t = ρν t
∂y
Eq. 2.2-13
Reynolds pointed out that the turbulent Eddies that transfer momentum transfer mass too
and then it is possible to write by analogy:
∂〈 c〉
= −e
∂y
J y t
As equation (Eq. 2.2-15), et and νt have the same units [m 2 /s].
Eq. 2.2-14
The equation Eq. 2.2-14 is equal to equation Eq. 2.2-11, but the first one is generally
valid for turbulent flow, while the second one stems from the analogy between the laminar and
homogeneous stationary turbulent flows where the variance of tracer cloud increases linearly
with time.
Another approach is from Reynolds analogy for tracer mass the eddy diffusivity is equal
to turbulent viscosity:
43