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

where Ix is the advective flux in x direction [g m -2 s -1 ]; ux is the average longitudinal velocity [m
s -1 ]; c is the tracer concentration [g m -3 ]. The Lagrangian system, which lies at the centre of the
tracer cloud (collection of fluid parcels whose dimension are comparable with the smallest length
scale of interest and with velocity and concentration characteristics), is use for advection.
The molecular diffusion is modelled by the Fick’s law:
∂c
J x = −em
∂x
Eq. 2.2-6
where Jx is the molecular diffusive flux in the x direction [g m -2 s -1 ]; em is the molecular diffusion
coefficient whose typical value in water changes between 0.5 – 2.0 x 10 -9 [m 2 s -1 ]; δc/δx is the
tracer concentration gradient in the x direction [g m -4 ]. The negative sign means that the
molecular diffusion develops in the opposite direction of the concentration gradient. The
molecular diffusion is defined in a fixed Eulerian coordinates.
The advection/diffusion three dimensional equation can be derived from a mass balance
on a rectangular parcel fluid moving at mean velocity is in rectangular Cartesian coordinates 2 :
∂c
+
∂t
3
∑
⎛ ∂c
⎞ ⎡
⎜
⎜u
⎟ i = em
⎢
⎝ ∂x
⎠ ⎣
∑
2 ⎛ ∂ c ⎞⎤
⎜
⎟ 2 ⎥
⎝ ∂x
⎠⎦
i= 1 i
i= 1 i
3
Eq. 2.2-7
where ui is the average velocity along the three orthogonal directions defined by the rectangular
Cartesian coordinates [m s -1 ] where 1=“x”, 2=”y” and 3=”z”.
A solution of this system for:
a. a conservative tracer;
b. known initial conditions;
c. stationary situation;
d. unbounded channel;
allows us seeing that the variance of tracer cloud increase linearly with time.
If the Reynolds number is below 500 the flow is laminar, if it is above about 2000 the
flow is turbulent which is generated by velocity shears where there are velocity gradients. In a
turbulent flow the tracer spread more rapidly than in the first one.
2 It is considered the right Cartesian coordinate system.
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