Diffusion

For a source contaminant concentration ${\displaystyle \ C_{0}}$ entering a flow of velocity ${\displaystyle {\vec {U}}}$ at a distance ${\displaystyle \ x}$ upstream from a point, the downstream concentration ${\displaystyle \ C}$ at that point is determined by the ratio...

${\displaystyle {\frac {C}{C_{0}}}=e^{\frac {x*{\vec {U}}}{D}}}$

Where... ${\displaystyle \ D}$ is the local dispersion coefficient determined by ${\displaystyle \ D=0.067*depth*V_{f}}$

and where friction velocity is ${\displaystyle V_{f}={\sqrt {g*depth*ChannelSlope}}}$

Velocity Distribution in an Open Channel (River)

Velocity distribution within a river follows a standard velocity profile for a confined space (pipe-flow)with the exception that the vertical distribution is truncated at the surface due to reduced friction with atmospheric gases versus the high friction against riverbed materials.

Boundary Layer thickness, also known as the displacement thickness, is defined by:

${\displaystyle \delta _{d}=\int _{0}^{H}{\frac {1}{H}}\left(1-{\frac {\vec {u}}{\vec {U}}}\right)dz}$

where ${\displaystyle {\vec {U}}}$ is the average velocity and ${\displaystyle {\vec {u}}(x)}$ is the velocity distribution in a channel of uniform depth ${\displaystyle \ H}$.

${\displaystyle {\bar {U}}={\frac {\int _{A}\left(\rho {\vec {V}}\cdot {\hat {n}}\right)dA}{\rho A}}}$

So in the case of this river, where the current is always normal to the cross-sectional area of the river (idealized)...

${\displaystyle {\bar {U}}={\frac {\int _{x}\int _{y}\left(\rho {\vec {V}}\right)dydx}{\rho A}}}$