Transport - Environmental Modeling - page 64

Sidebar 3.1: Mass Conservation in Streams
For 1D models in streams, the mass conservation leads to a slightly different
formulation. The changing cross-section along the stream needs to be con-
sidered. Instead of (2.1) one obtains:

c ð x ; t þ D t Þ c ð x ; t Þ
D t

A D x ¼ j x ðx; tÞA x j xþ ðx; tÞA xþ

where A denotes the cross-section in the y-z - and j fluxes through cross-
sections. As the riverbed is usually changing, different upstream and down-
stream cross-sections A x and A x have to be taken into account (Fig. 3.4 ).
Division by

D x and transition to infinitesimal scale leads to the formulation

A @ c

@t ¼ @

j x ðÞ

@x

After application of Fick's Law for the cross-section the equation
becomes:

A @ c

@t ¼ @

D turb A @ c
@x

@
@x

ð v mean Ac Þ

@x

where D turb stands for mean turbulent diffusivity across a cross-section,
and v mean for mean velocity across the cross-section.
Of course, the 1D approach is a simplification of the flow regime in
a stream or channel. However, it is justified in order to capture the dominant
downstream behavior in the main flow channel. Additional features, such as
counterflow in groyne fields or along the bankline and flow in floodplains, can
be accounted for by the introduction of additional source and sink-terms in
the equation.

In case of constant D the equivalent formulation is:

2 c

y @c

@t ¼ yD @ @x 2 yv @ c

@x þ q

(3.18)

For higher dimensional problems one may use the -operator to obtain an

∇

equally short formulation for the general situation:

y @ c

@t ¼r y D rc v c

ð Þ þ q (3.19)