Homework Problem set # 1
Due September 23, 2002
1) In the Hudson River off the west-side of Manhattan sea level observations reveal a tide
with a range of 1.4 meters and that the tidal wave propagates up the river at celerity c= 6
m/s. The period of the tidal motion (T) is 12.42 hours. This tidal period sea level motion
can be writes mathematically as =A sin( t + kx), where  is the frequency of the tidal
motion (2*pi/T) and k is the wave-number , k=2*pi/ where  is the wave length (
=c*T). Assuming that the channel is 10 meters deep use a form of the continuity
equation to derive an expression for the tidal current speed. What is the maximum tidal
current speed? When does this occur relative to high and low water?
2) During maximum ebb the tidal current encounters a bottom front, such as shown in
the figure below. Current measurements were made at points A and B which are 1 km
apart. At point A the along-channel current is u(z) = -0.25-0.05z , while at point B the
along channel current is u(z) =.25-.15z, where z is distance in meters from the bottom.
A) What is the vertical velocity of the sea surface at this instant in time?
B) Draw vectors to depict the vertical structure of the along-channel current.
C) Using the continuity equation calculate the vertical structure of the vertical
velocity (w(z)) between the two current observations.
B
A
H=10m
z
20
12
16
x
Schematic of bottom front located between current observations
A and B. Contours of isohalines with intervals of 2 parts per
thousand are drawn in black
3) Flow enters a pipe at a rate of 1 liter/sec. At the upstream side the pipe (figure below)
has an inside diameter of 5 cm. A smaller pipe on the down stream side has a diameter of
1 cm and is connected to the larger pipe by a reducer whose radii decreases linearly from
5 cm to 1 cm over a distance of 6 cm (i.e. D=5-2x/3, where x is distance along coupling)
A) What is the flow velocity in the 5 cm and 1 cm pipe?
B) Calculate the magnitude of the advective term at locations 1, 3 and 5 cm along the
coupling. (Hint. Calculate the velocity gradient by using the radius +/- a small
distance from each location)
C) Assuming that this advective term is balanced by a pressure gradient--what is the
pressure gradient at 1, 3 and 5 cm ?
D) With these 3 estimates of the pressure gradient estimate the pressure difference
(i.e. P2 – P1) across the coupling. Indicate which side of the coupling has high and
low pressures.
P2
P1
6 cm
1
3
5