Computational Project # 7
Due December 13
Consider unsteady viscous flow in an
inclined open rectangular channel.
Initially, the fluid is at rest. One may
assume that streamwise velocity at any
cross-sectional area is the same. Due to
viscous effect the fluid sticks to the walls,
i.e. velocity is zero at the walls, while the
open top boundary remains stress free all
the time. The equation describing the
evolution of the flow in the open channel
is given by:
0
u
y=
∂
∂
y
22
,
z
UUU
g
tX
ρρμ
⎛⎞
∂∂∂
=+ +
⎜⎟
∂∂
22
Y∂
⎝⎠
where U(X,Y,t) is streamwise velocity, gz
is streamwise component of acceleration
of gravity, X, Y, and Z are respectively
horizontal, vertical, and streamwise
directions,
ρ
is the fluid density, and
μ
is
viscosity. The schematic diagram of the
flow is given in the figure.
This problem can be rewritten in non-dimensional form as:
22
22
1
Fr ,
Re
uuu
xy
τ
⎛⎞
∂∂∂
=+ +
⎜⎟
∂∂∂
⎝⎠
where u is nondimensional streamwise velocity, x and y are non-dimensional coordinates,
τ
is
non-dimensional time, and Fr and Re are Froude and Reynolds numbers. The initial conditions
are
(
)
,,0 0uxy
=
,
while the boundary conditions are
()()() ()
,0, 0, , , , 0, , , 0.
u
ux t u yt ulyt xht
y
∂
=== =
∂
max
t
Δ
Discretize the problem using explicit Euler time discretization and central difference in space.
What is the accuracy of the solution in terms of Δt, Δx, Δy?
Is the scheme consistent?
Is your scheme stable?
What is the maximum time step you can take in terms of Δx and Δy?
h
l
x
0u
=
0u
=
0u=
z
g
JG
g
JG y
g
JG
z
