ME 207 | Fluid Dynamics
Homework 4
Instructions

Problems marked with an asterisk should be solved during the tutorial session. Students
must submit the solutions (hardcopy) by the end of the respective tutorial session.

Other problems are part of homework assignment. Students must submit the hard copy of
homework assignment before the start of the tutorial session on February 19, 2025.
Problem 1*
Figures below are the schematics of the experimental setups of impingement of fluid jets on
different surfaces. You have performed this experiment in the laboratory considering a flat
surface and a hemispherical surface. A nozzle of diameter 6 mm directs a jet stream of water,
which impacts the center of the vane surface. Two support masses (m2 and m3) are housed on
a horizontal rod to counter the force Fi exerted by the water jet on the surface (F1 on the flat
surface and F2 on the hemispherical surface).
m1
Applying the Reynolds Transport Theorem, calculate the force exerted by the fluid jet for
each case and thus evaluate the ratio F2/F1. The larger support mass m2 is fixed at a distance
of R from the hinge that holds the horizontal rod, while the smaller support mass m3 is fixed
at a distance of xi. Use the value of Fi evaluated above in each case and apply the moment
balance at the hinge to calculate the values of xi. Evaluate the ratio x2/x1.
Experimental Data:
1. Nozzle diameter, D = 6 mm.
2. Volume flow rate, 𝑄̇ = 160 cm3/s.
3. Combined mass of the suspended mass and the vertical supporting rod, m1 = 100 g.
4. Mass of the horizontal rod (on which the two supporting masses are mounted), m4 =
700 g.
5. Diameter of the stream after impact = di
6. Mass of the larger supporting balance, m2 = 1.08 kg
7. Mass of the smaller supporting balance, m3 = 330 g.
8. Distance between the centre of mass of the larger (fixed) mass m2 and hinge, R = 15.2
cm
9. For comparison, the distance between the centre of mass of the smaller mass m3 and
hinge(x1) is experimentally obtained to be 23.9 cm.
Problem 2
Figure below shows a vane with a turning angle of 60°. The vane moves at constant speed of
U = 10 m/s and receives a jet of water that leaves a stationary nozzle with speed V = 30 m/s.
The nozzle has an exit area of 0.003 m2. Determine the force components that act on the
vane.
Problem 3
When a uniform stream flows past an immersed thick cylinder, a broad low-velocity wake is
created downstream, idealized as a V shape as shown below. Pressures p1 and p2 are
approximately equal. If the flow is two-dimensional and incompressible, with width b into
the paper, derive a formula for the drag force F on the cylinder. Rewrite your result in the
form of a dimensionless drag coefficient based on body length: 𝐶𝐷 = 𝐹 ⁄(𝜌𝑈 2 𝑏𝐿).
Problem 4
A tank of 0.1 m3 volume contains air at 800 kPa (absolute) and 15ºC. At t = 0, air begins
escaping from the tank through a valve with a flow area of 65 mm2. The air passing through
the valve has a speed of 300 m/s and a density of 6 kg/m3. Determine the instantaneous rate
of change of density in the tank at t = 0.
Problem 5
The rocket shown below has a supersonic exhaust, and the exit pressure pe is not necessarily
equal to pa. Derive an expression for the force F required to hold this rocket on the test stand.
Is this force F what we term the thrust of the rocket?
Problem 6
Water flows steadily through the 90º reducing elbow, as shown below. At the inlet to the
elbow, the absolute pressure is 200 kPa and the cross-sectional area is 0.01 m2. At the outlet,
the cross-sectional area is 0.002 m2 and the velocity is 16 m/s. The elbow discharges to the
atmosphere. Determine the force required to hold the elbow in place.
Problem 7
Gravel is dumped from a hopper, at a rate of 600 N/s, onto a moving belt, as shown below.
The gravel then passes off the end of the belt. The drive wheels are 70 cm in diameter and
rotate clockwise at 100 r/min. Neglecting system friction and air drag, estimate the power
required to drive this belt.
Problem 8
A liquid jet of velocity Vj and area Aj strikes a single 180o bucket on a turbine wheel rotating
at angular velocity Ω, as shown below. Derive an expression for the power P delivered to this
wheel at this instant as a function of the system parameters. At what angular velocity is the
maximum power delivered? How would your analysis differ if there were many, many
buckets on the wheel, so that the jet was continually striking at least one bucket?
Problem 9
Actual air flow past a parachute creates a variable distribution of velocities and directions.
Let us model this as a circular air jet, of diameter half the parachute diameter, which is turned
completely around by the parachute, as shown below. (a) Find the force F required to support
1
𝜋
the chute. (b) Express this force as a dimensionless drag coefficient, 𝐶𝑑 = 𝐹/ [2 𝜌𝑉 2 ( 4 ) 𝐷2 ].
Problem 10
Water at 20oC flows through the elbow and exits to the atmosphere, as shown below. The
pipe diameter is D1 = 15 cm while D2 = 4 cm. At a weight flow rate of 160 N/s, the pressure
p1 = 2.3 atm (gage). Neglecting the weight of water and elbow, estimate the force on the
flange bolts at section 1.