CE 576 Environmental Flows
Spring 2012
Homework 2
*10. In three dimensions, the y-momentum equation, where y points up, is
∂
∂
∂
∂
∂τxy
∂τyy
∂τzy
(ρv) +
(ρuv) +
(ρvv) +
(ρwv) = −ρg +
+
+
.
∂t
∂x
∂y
∂z
∂x
∂y
∂z
Show that for all flows the y-momentum equation can be written as
ρ
∂v
∂v
∂v
∂v
+u
+v
+w
∂t
∂x
∂y
∂z
= −ρg +
∂τyy
∂τzy
∂τxy
+
+
.
∂x
∂y
∂z
11. Show that the x-momentum equation in three dimensions is
ρ
∂u
∂u
∂u
∂u
+u
+v
+w
∂t
∂x
∂y
∂z
=
∂τyx
∂τzx
∂τxx
+
+
.
∂x
∂y
∂z
*12. Edwards et al. (2005, Can. J. Fish. Aquat. Sci.) solved a one-dimensional model for
the transport of phytoplankton. The model included sinking with a velocity ws , net
growth with a rate μ, and turbulent flux qF (say). We will see later that turbulent
fluxes are often modeled with a version of Fick’s law:
qF = −K
∂F
,
∂z
where K is called the vertical eddy diffusivity. Find this paper and derive equation
(1).