University of Florida



Fluids 1 and 2: Problem I (25 points, estimated time 30 minutes)

Given:

• The stream function for flow over a Quonset hut of radius [pic] and length [pic] is given by

[pic], where [pic]and [pic].

• The freestream velocity is [pic] and the freestream pressure is [pic] which is atmospheric.

• You may neglect end effects.

• The weight of the Quonset hut is [pic].

• Assume that the inside of the Quonset hut is at atmospheric pressure.

• Neglect end effects since [pic] and assume 2-D

Find:

a) List and explain the appropriate assumptions. (2 points)

b) The stream function is defined such that it exactly satisfies what conservation equation? (2 points)

c) The stream function provided is obtained via superposition of what two elemental stream functions? Why is superposition valid? (2 points)

d) If the flow is rotational, would the stream function provided above be valid? If not, what equations would you have to solve to obtain a solution? (Do not solve this problem.) How would the velocity boundary conditions change, if any? (3 points)

e) Determine the velocity on the surface of the hut, [pic], using [pic]. Does the result satisfy the no-slip boundary condition? Why or why not? (4 points)

f) What is the maximum and minimum velocity on surface of the hut and where do they occur? (3 points)

g) What is the gage pressure distribution on surface of the hut (i.e., [pic])? (3 points).

h) Determine [pic] that will lift the hut off of its foundation assuming it is not held fixed to the ground. (4 points) NOTE: [pic]

i) What is the drag force on the hut? Why? Is this really true in practice (2 points)

[pic]

Fluids 1 and 2: Problem II (25 points)

Given: A long cylinder of radius b moves axially at velocity U in a larger cylinder of radius a. The space between the cylinders is filled with a viscous Newtonian incompressible fluid. There is no [pic] pressure gradient driving the flow.

Schematic:

[pic]

Useful Relations:

Continuity: [pic]

[pic] momentum: [pic]

[pic] momentum: [pic]

[pic] momentum: [pic]

where: [pic]

[pic]

Note: [pic]

Please do the following (5 points each):

a) List the appropriate assumptions.

b) Simplify the continuity equation. What does this imply about [pic]?

c) Simplify the [pic] momentum equations. What do these equations and the fact that there is no [pic] pressure gradient driving the flow imply about [pic]?

d) Find the resulting velocity distribution in the fluid.

e) Simplify the solution for the case when [pic]. Physically, what is this case?

Fluids 1 and 2: Problem III (25 points, estimated time 30 minutes)

Given:

• A spherical water droplet of radius [pic] falls under the action of gravity from a rain cloud and undergoes a natural oscillation of frequency [pic] as it falls.

• To study the characteristics around the droplet, a 5:1 scale model (the model is larger than the droplet) is constructed so that velocity measurements can be made in a vertical flow wind tunnel (assume that the droplet reaches a terminal velocity, so the effects of gravity on dynamic similarity may be ignored).

• Note that surface tension [pic] is required to maintain the bubble shape.

Find:

a) Define and list the differences between geometric, kinematic, and dynamic similarity. (6 points)

b) Determine the appropriate nondimensional parameters ([pic]groups) that describe the dynamics of this problem and describe the physical meaning of each. (9 points)

c) If the model and prototype droplet are both comprised of water and the surrounding medium is air, what is the frequency ratio? (5 points)

d) Can true dynamic similarity be achieved in this case? If not, why? What must be done to the model droplet’s fluid to achieve dynamic similarity? (5 points)

Fluids 1 and 2: Problem IV (25 points, estimated time 30 minutes)

Given:

• A jet of water that is declined at an angle [pic] is filling a moving water car. The water car is moving away from the jet at a constant velocity [pic]and possesses a diameter [pic]. The liquid level in the car is [pic]. The jet has a diameter [pic] and a uniform velocity [pic]. A frictional force[pic] is acting on the car. Assume that the jet is always filling the car and ignore sloshing.

Find:

a) Draw an appropriate control volume, explicitly stating unit normal vectors and flux areas. List the appropriate assumptions (6 points).

b) What is the time rate of change of the liquid level? (7 points)

c) What is the thrust force acting on the car by the jet of water? (6 points)

d) What is the horizontal force acting on the car? (6 points)

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[pic]

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