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Transcript
```Fluid Mechanics Sample Exam 1
Please work at least three problems. If you attempt all four, your score will be based
1) Consider a large hot-air balloon rising upward on a windless day. Let the radius of the balloon
be R, the density of the warm air inside the balloon be ρi , the density of the surrounding (ambient)
air be ρo , and the total mass of the balloon plus riders be Mo ; in addition, define a z-axis as being
directed vertically upward (opposite the direction of gravity), with an origin at ground level.
a) What is the approximate buoyancy force on the balloon? State in terms of the parameters listed
above.
b) Taking into account the aerodynamic drag force, which has the generic form, D = CD ρo u2 /A,
write down the equation of motion for the system consisting of the balloon, plus riders. [Here, CD
is the drag coefficient, u is the speed of the balloon relative to the surrounding still air, and A is
the projected area of the balloon.]
c) Assuming that the balloon eventually reaches a terminal (i.e., fixed) velocity, VT , use your
equation from b) to determine VT .
d) For slow rates of ascent, we can neglect the drag force. Focusing on this limit, use the result
in b) to determine the time-dependent velocity and position of the rising balloon. In the course of
doing this calculation, introduce physically reasonable initial conditions on velocity and position.
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2) Consider steady flow past a blunt body, e.g., a car, within a wind tunnel. Assume that the
velocity distribution at the entrance to the tunnel test section is horizontal and uniform, i.e., flat,
with a magnitude Uo . Assume that the test section entrance pressure is likewise spatially uniform
and equal to Po . Using a pitot tube (or any device/devices that allows simultaneous measurement
of local velocity and static pressure), we measure both the static pressure distribution, Pe (x), and
the horizontal velocity distribution, Ue (x), over the entire test section exit plane. Our objective is
to use all of these measurements (including measured Uo and Po ) to back out the total drag force,
D, exerted by the flowing air on the car.
a) Sketch an appropriate control volume (CV) that could be used to tackle this problem. Hint:
Exclude the car from your CV.
b) In your sketch, indicate the resultant horizontal and vertical forces exerted by the excluded car
on the CV. Be sure to indicate the correct direction of the horizontal force and note that, depending
on the car and flow conditions, the vertical force can be up or down.
c) Given Pe (x), write an expression for the total pressure force that is exerted on the test section
exit plane.
d) Given Ue (x), write an expression for the total flux of horizontal momentum through the exit
plane.
e) Use your sketch from a) to create a free body diagram for your chosen CV, specifically, sketch
all of the forces acting in the horizontal direction on the CV.
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f) Finally, solve for the unknown drag force, D.
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3) Consider steady, fully-developed, pressure-driven, laminar, vertically-upward flow in a wide,
rectangular, vertical duct. Let the (upward) flow direction be x, the cross-stream direction be y,
the width of the duct in the y-direction be H, the density and viscosity of the fluid be ρ and µ,
respectively, and the magnitude of the imposed pressure gradient be G. Assume that gravity acts
in the negative x−direction.
a) Roughly speaking, and without doing any calculation, how large would you estimate the Reynolds
number to be when the flow starts to transition from laminar to turbulent flow? Hint: Consider
laminar-to-turbulent transition in circular pipes.
b) Focusing on the entry length region (over which boundary layers grow to eventually meet at the
center of the duct), use a simple scaling argument to estimate the entry length, le .
c) Simplify the Navier-Stokes equations to obtain the governing differential equations for this flow.
What boundary conditions must be imposed in order to determine the x-velocity component, u?
d) Time permitting, solve for the velocity field, u.
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4) The momentum integral equation for flow over a flat surface is given by
d
ρU
dx
2
Z
0
δ
u
u
∂u
1−
dy = µ
U
U
∂y y=0
(1)
where flow is directed in the x-direction, y begins at the plate and is directed perpendicular to the
plate (and flow), U is the (fixed) speed of the flow outside the boundary layer, u is the x-velocity
component within the boundary layer, ρ and µ are the fluid density and viscosity, respectively, and
δ = δ(x) is the position-dependent boundary layer thickness. See figure below.
a) Assuming a linear boundary layer velocity profile:
u = ay + b
(2)
determine the (ordinary) differential equation governing δ(x).
b) Solve the differential equation obtained in a) in order to determine the boundary layer thickness,
δ(x).
c) Starting with the x-component of the Navier-Stokes equation, use a scaling argument to estimate
the boundary layer thickness, δ(x), as a function of x.
Y
δ(X)
U
u
X
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```
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