Unformatted text preview: EAS 4101 – Aerodynamics – Spring 2010 02/17/10 Exam #1
Name: ________________________
I. TRUE/FALSE (10 Points total – 1 point each) 1. Extensive properties are dependent on the size of the system.
2. The Reynolds’ number, an important nondimensional parameter in fluid mechanics, is the
ratio of viscous to pressure forces.
3. Timelines, pathlines, streaklines, and streamlines are all identical in a steady flow.
4. Motion about the vertical axis of an airplane is called yaw, and is produced by deflection of
the elevators.
5. The convective derivative of a flow property is written as
·
, where is the given flow
property.
6. Flow in a boundary layer is irrotational in the presence of a favorable pressure gradient.
7. Assuming inviscid flow transforms the NavierStokes Equations into Euler’s Equations.
8. The substantial derivative is always equal to zero if the flow is steady.
9. Pathlines are a locus of particles that have passed through a given point.
10.Stokes Theorem states that the circulation of a vector around a curved surface is equal to the
flux of the curl of that vector over a surface.
II. T
F
F
F
T
F
T
F
F
T SHORT ANSWER (30 points total – 4 pts each unless otherwise marked) 11. (2 pts) The velocity boundary condition at a solid surface for an inviscidflow problem is
described by:
a. Normal velocity is equal to zero.
b. Noslip condition holds.
c. Tangential velocity is equal to the free stream velocity. 12. (2 pts) From an ideal flow solution, it is possible to integrate the pressure on the surface
of a body to determine ____Pressure________ drag but not _____Viscous______ drag. 13. In ideal flow, it is common to employ superposition solutions of various individual
solutions to Laplace’s equations in terms of the stream function and/or velocity potential.
Is it also valid to superimpose pressure fields of each individual solution? Please justify
your answer mathematically and/or physically.
No, the pressure fields are nonlinearly related to the velocity fields via Bernoulli’s
equation in ideal flow. Name: ________________________
14. Once the velocity potential is known for ideal flow over a body, how do you determine
the lift? Describe the specific steps.
1. Compute velocity field according to
2. Using Bernoulli’s equation, determine the pressure distribution.
3. Integrate the pressure distribution over the area to determine the net force.
4 .Take the component of the force in the direction opposite gravity to determine the lift. 15. The pressure distribution over the surface of a rotating cylinder of radius R is often given
. Write an integral expression for the lift per unit span
as a pressure coefficient,
acting on the cylinder in terms of
. 1
2
1
2 sin sin 16. For a lowspeed, steady flow around the thin airfoil shown below, the free stream
pressure and velocity ( and ) are known. The local static pressure at points 26 are
known from measurements. At which point(s) out of 26 may we use Bernoulli's
equation to determine the local velocity? Circle your answer(s). In a single sentence,
explain your choices. 2 3 4 5 6 Bernoulli’s equation is only valid when the flow is incompressible, irrotational, and
inviscid; flow within a boundary layer is both rotational and viscous. 17. (6 pts) Starting with the integral form of the conservation of mass equation, obtain the
substantial derivative form of continuity. (Hint: use Gauss’ theorem to convert the
surface integral into a volume integral.)
· 0
Assuming fixed control volume: · 0
Applying Gauss’ theorem:
0 · · From this it can be seen that the integrand must be equal to zero:
· 0 · · ·
· , 0 · 1 · Name: ________________________
Problem 1: (20 points total)
Given: A thin symmetric airfoil section is mounted in a lowspeed wind tunnel. A Pitot probe is
used to determine the velocity profile in the viscous region downstream of the airfoil.
This velocity distribution is
1
1
cos
, for 2
2
is a constant. The fluid is air, with a density .
where The following integrals may come in handy:
1 1
sin
2 9
16 1 1 1
cos
2 9
16 1 Find:
a) List appropriate assumptions. (3 pts)
b) Sketch the proper control volume for this problem, clearly labeling the forces acting on the
control volume and the appropriate unit normal vectors. (3 pts)
c) Determine the relationship between and . (7 pts) d) Determine the drag force per unit span acting on the airfoil. Explicitly indicate its direction with
an arrow. (7 pts) Name: ________________________
Problem 1continued: a) Required: Incompressible, steady.
Allowed: Neglect changes in height, no flow across the streamline
b) Start with the integral form of the conservation of mass:
· 0 Assuming steady, incompressible flow, and noting that there is only flux across the left and right
boundaries,
· · 0 Dividing by , plugging in the velocity, and making use of the symmetry about
2 2 1 1
cos
2
2 0, 0 Evaluating the integral,
2
and solving for 1 2 0 ,
1 c) Start with the integral form of the conservation of momentum:
· . Assuming steady, incompressible flow, plugging in velocity, and integrating via the given
integral formula with the problem,
2 2 1 1
cos
2
2 4 2 4 2
4 2 9
16 1
cos
2 1 9
16 1 1 in the negative x‐direction The force is the force required to hold the airfoil stationary and acts in the negative xdirection. Drag, by definition, is the force of the fluid acting on the airfoil, and thus is equal to
and acting in the opposite direction. in the positive x‐direction
2 1 2 or 4 or
2 2 1 9
1 16 or
1
4 8
1
. 1 Name: ________________________
Problem 2: (20 points total)
Given: Starting with the NavierStokes equation in Gibbs notation: derive Bernoulli’s equation. Hint: Use the following vector identity:
· 1
2 Find:
a) List appropriate assumptions. (5 pts)
b) Derive Bernoulli’s equation. Show and explain all steps. (15 pts) Name: ________________________
Problem 2 continued:
a) Incompressible flow / constant thermodynamic properties, Stokes hypothesis; Steady; Inviscid flow;
Irrotational flow b) Applying assumptions to the NS equation we obtain DV p f b 2V
Dt (2) V (3) V V p f b 2V t Then V V p fb Recall the vector identity V V 1 V
2 2 V V Since flow is irrotational this equation reduces to 1
V V V 2
2
Plugging back into the reduced NS equation we obtain 1 V 2 p fb
2
or 1 V 2 p fb
2 Where fb is the gravity force acting on the fluid: fb gz
Then we obtain
1 V 2 p gz
2
or
1 1 V 2 p gz 0 2 After integrating, Bernoulli’s equation is
121
V p gz const
2 Name: ________________________
Problem 3: (20 points total)
Given:
,
,
Add a source
at the origin
r r
θ Fig. a θ Fig. b Flow impinging on a flat plate (Fig. a) can be described with the stream function
1
Ψ
sin
cos
sin 2
2
where is a constant.
By adding a source at the origin , flow against a plate with a "bump" is obtained
(Fig. b).
In both figures, the streamline patterns are shown.
The fluid is air.
The following stream functions may be useful:
cos
Λ
Λ
Γ
Ψ
, Ψ
, Ψ
, Ψ
ln 2
2
2
2
The velocity field is related to the stream function by:
1Ψ
Ψ
, Find:
a) Use superposition to determine the stream function for the flow of Fig b. (2 pts)
,
,
̂
,
̂ for an arbitrary point within the flow.
b) Find the velocity
(2 pts)
c) Find the bump height in terms of and the source strength. (4 pts)
d) Identify the value of the streamline that describes the surface of the “bump.” (4 pts)
e) Find an expression for the shape of the bump,
. Write your answer in terms of
and . (4 pts)
, neglecting gravitational
f) Find the gage pressure on the surface of the bump at
effects. (4 pts) Name: ________________________
Problem 3 continued:
a)
1
2 Ψ Λ
2 sin 2 b)
1Ψ Λ
2 cos 2
Ψ sin 2 c) Recognizing that the peak of the “bump” is a stagnation point:
⁄2,
0.
At
Then evaluating, ,
, Λ
2 0 2
Λ 2
d) Plugging in the location of the stagnation point
Ψ Λ
4 e) First, use (d) to relate the two coordinates:
Λ1
sin 2
42
and making use of (c) Λ
2
2
2 2 sin 2
f) First, evaluate at : 4 2
sin 2 2 To find the gage pressure, apply Bernoulli’s equation between a point in the free stream and
the point where
1
2 1
2 Neglecting gravitational effects:
1
2
We know that: To find these values, we evaluate and
Λ , 24 , at the point
Λ ,
Λ 2 1
2 2 2 Λ 1
2 2
,
24 2 Thus
Λ 1
2
Λ 2
1
2 2
2 Λ 4
2 2 2 Plugging this back into the expression for gage pressure yields (where any substitution of the
above is sufficient):
1
2 ...
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 Spring '08
 Sheplak
 Fluid Dynamics, Velocity, pts, fb

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