Contaminant and sediment transport by advection and diffusion 199
f
D
c
x
D
c
x
tx
t
t
= − +
∂
∂
≈ −
∂
∂
(
)
ν
(8.3)
The turbulent diffusion coefficient depends on the flow characteristics
(the mean velocities gradients and their turbulent fluctuations). In general,
determination of the function D t (x,y,z,t) is a very complex process following the difficulties involved when solving for the turbulent mean velocities.
However, the process can be simplified by assuming two separate eddy
diffusion parameters, D h and D v , where D h refers to the horizontal mixing
and D v to the vertical mixing. The difference between the two parameters
stems from the fact that in most geophysical horizontal flows, water depths
are much smaller as compared to the horizontal dimensions of the flow
domain. Therefore, the development of turbulent eddies have different geometric and energy content, and characteristics between the vertical and the
horizontal dimensions.
Development of a mass transport mathematical model for the estimation
of c(x,y,z,t) can be accomplished by using the following basic considerations:
1. The transported substance does not interfere with the pre-existing
hydrodynamic conditions. This is true for concentrations less than
0.1 ppt.
2. The transported substance may be conservative or non-conservative,
that is, its total mass may remain constant during the transport, or it
may decay due to mechanical, chemical or biological processes which
evolve with the transport. For instance, biological decomposition is
usually described by an exponential decay law (c = c o exp(–λt), where
λ is the bio-decomposition rate (with dimensions, s –1 ).
For a one-dimensional system, application of the mass conservation principle on a linear control space (Figure 8.1), and utilization of Equations 8.1
and 8.3 leads to the following mass transport equation:
∂
∂
+
∂
∂
=
∂
∂
∂
∂
c
t
uc
x
x
D
c
x
t
( )
(8.4)
f tx + df tx
f ax + df ax
f tx
f ax
dx
Figure 8.1 One-dimensional control volume for mass flux.
f
D
c
x
D
c
x
tx
t
t
= − +
∂
∂
≈ −
∂
∂
(
)
ν
(8.3)
The turbulent diffusion coefficient depends on the flow characteristics
(the mean velocities gradients and their turbulent fluctuations). In general,
determination of the function D t (x,y,z,t) is a very complex process following the difficulties involved when solving for the turbulent mean velocities.
However, the process can be simplified by assuming two separate eddy
diffusion parameters, D h and D v , where D h refers to the horizontal mixing
and D v to the vertical mixing. The difference between the two parameters
stems from the fact that in most geophysical horizontal flows, water depths
are much smaller as compared to the horizontal dimensions of the flow
domain. Therefore, the development of turbulent eddies have different geometric and energy content, and characteristics between the vertical and the
horizontal dimensions.
Development of a mass transport mathematical model for the estimation
of c(x,y,z,t) can be accomplished by using the following basic considerations:
1. The transported substance does not interfere with the pre-existing
hydrodynamic conditions. This is true for concentrations less than
0.1 ppt.
2. The transported substance may be conservative or non-conservative,
that is, its total mass may remain constant during the transport, or it
may decay due to mechanical, chemical or biological processes which
evolve with the transport. For instance, biological decomposition is
usually described by an exponential decay law (c = c o exp(–λt), where
λ is the bio-decomposition rate (with dimensions, s –1 ).
For a one-dimensional system, application of the mass conservation principle on a linear control space (Figure 8.1), and utilization of Equations 8.1
and 8.3 leads to the following mass transport equation:
∂
∂
+
∂
∂
=
∂
∂
∂
∂
c
t
uc
x
x
D
c
x
t
( )
(8.4)
f tx + df tx
f ax + df ax
f tx
f ax
dx
Figure 8.1 One-dimensional control volume for mass flux.