16.100 Homework Assignment # 6
Due: Monday, October 31 9am
Reading Assignment
Anderson, 3rd edition: Chapter 15, Sections 1-4, 7
Chapter 16, pages 745-751, 781-786
Problem 1
y
0
U
Consider an initially stationary, long flat plate. At time t
=
0, the plate is set in motion at
velocity U. As time evolves, the air above the plate will begin to move as the viscous effects
diffuse the momentum away from the plate. We will define the height of the boundary layer of
non-negligible momentum,
0
)(t
δ
, as the y-location at which the velocity is only 1% of the wall
velocity. Assume the flow is incompressible.
a) Apply the conservation of mass in differential form (i.e. not integral form) to show that
the vertical velocity, v, is zero everywhere in the flow for all time.
b) Next, show that the conservation of x-momentum can be reduced to the following form:
2
2
y
u
t
u
∂
∂
=
∂
∂
ν
(1)
c) Show that the following x-velocity is a solution to Equation (1) and that it satisfies the
initial and boundary conditions:
()
()
∫−
≡
≡
−=
η
ξξ
π
η
νη
η
0
0
2
2
4
1
deerf
ty
erf
U
u
Note that
()
η
erf is known as the error function and its values are available in Matlab and
Excel using the erf function.
d) Plot the velocity distribution as a function of
η
. Determine the boundary layer thickness
as a function of time. For air at standard day conditions, at what time does the boundary
layer thickness reach 1 inch above the plate? What time for 1 foot above the plate?
