D-R-A-F-T
Convection-Diffusion Governing Equation
The basic convection - diffusion equation is presented in Ariathurai, MacArthur, and
Krone (l977),
Equation 1
∂C
∂C
∂C ∂ ⎛
∂C ⎞ ∂ ⎛
∂C ⎞
∂y ⎟ α 1
⎟ + C +α 2
⎟ + ⎜ Dy
= ⎜ Dx
+u
+v
∂x ⎠ ∂y ⎜
∂t
∂x
∂y ∂x ⎝
⎝
⎠
where
concentration, kg/m3
C
=
t
=
time, sec
u
=
flow velocity in x - direction, m/sec
x
=
primary flow direction, m
v
=
flow velocity in y - direction, m/sec
y
=
direction perpendicular to x , m
effective diffusion coefficient in x - direction, m2/sec
Dx
=
effective diffusion coefficient in y - direction, m2/sec
=
Dy
α1
=
a coefficient for the source term, 1/sec
α2
=
the equilibrium concentration portion of the source term,
kg/m3/sec = - α1Ceq
Bed Shear Stress Calculation
The bed shear stress is needed to evaluate the bed source-sink terms in the governing
equation. Several options are available for computing bed shear stresses using
Equation 2
(u )
2
*
τb = ρ
where
ρ
=
water density
u*
=
shear velocity
Smooth-wall log velocity profile
Equation 3
⎞
⎛
*
⎜ 3.32 u D ⎟
u
= 5.75 log
ν ⎟
⎜
*
u
⎠
⎝
which is applicable to the lower 15 percent of the boundary layer when
Equation 4
8
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