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ENAE 311
Midterm I Review (2021)
Key topics/concepts:
Week1:
- What is a fluid?
- What is Aerodynamics?
- What is an airfoil?
- What are the important attributes of an airfoil (span, chord, angle of attack, leading edge, trailing
edge, NACA shape, etc.)
- What are the forces created by an airfoil and in what directions do they act?
- How do these forces arise?
- How are the values of these forces calculated?
- How is airfoil performance measured and reported?
Week 2:
- Where are airfoil forces assumed to act?
- How does one compute the net aerodynamic moment?
- How is net aerodynamic moment reported?
- What is the center of pressure?
- What is dimensional analysis?
- What is the Buckingham-Pi theorem?
- What are Pi groups?
- How does one determine the Pi groups for a particular problem?
- How does one use dimensional analysis to identify important non-dimensional parameters in a
problem?
- What is flow similarity?
- How does one determine if two flows are geometrically similar?
- How does one determine if two flows are dynamically similar?
Week 3:
- What is a vector?
- What is a tensor?
- How are vectors specified?
- How are tensors specified?
- What is a scalar field?
- What is a vector field?
- Important vector operations to know: dot product, cross product, gradient, divergence, curl.
- Line integrals along a path and along a closed contour
- Surface integrals.
- Relations between line, surface, and volume integrals
- What is the Reynolds Transport Theorem (RTT)?
- What are each of the terms in the RTT?
- What does each term in the RTT represent physically?
- What are the Eulerian and Lagrangian reference frames?
- How does one use a volume integral to represent the value of an arbitrary extensive property?
Week 4
- How does one derive conservation of mass from RTT?
- What are the integral and differential forms of mass conservation?
- What is the equation of mass conservation for steady, 1-D flow?
- What is the substantial derivative?
- What does it mean when we say that a flow is incompressible?
- What is a mass flux and how does one calculate it?
- How does one use the integral form of mass conservation to figure out how much material is
entering/leaving a control volume?
- What is the steady, 1-D version of mass conservation?
- How does one derive conservation of momentum from RTT?
- What forces acting on the fluid do we usually account for?
- What are the integral and differential forms of momentum conservation?
- What is a momentum flux and how does one calculate it?
Week 5
- How does one use the integral form of momentum conservation to find the values of forces acting
on a fluid system?
- How does one derive conservation of energy from RTT?
- What are streamlines, pathlines, and streamlines? When are they all the same?
- What is a stream tube?
- How does one derive conservation of momentum from RTT?
- What forces acting on the fluid do we usually account for?
- What are the integral and differential forms of momentum conservation?
- What is a momentum flux and how does one calculate it?
Week 6
- What is Bernoulli’s equation?
- What is a pitot tube and how is it used to determine the speed of an aircraft?
- Review of thermodynamics
o What are thermodynamic properties and how do they differ from thermodynamic states?
o What is internal energy?
o What is enthalpy?
o How are internal energy and enthalpy related to temperature for a thermally perfect gas
and a calorically perfect gas?
o How does one use specific heats to represent internal energy and enthalpy?
o How does one account for temperature variations in specific heats?
o What is the first law of thermodynamics?
o What is the second law of thermodynamics?
o What is an adiabatic process?
o What is a reversible process?
o What is an isentropic process?
o How is work defined for a reversible process?
o How does one calculate the change in entropy associated with a particular
thermodynamic process?
- How do shock and expansion waves reflect off free boundaries?
- What conditions need to be met across free boundaries?
- What is shock expansion theory?
- What is wave drag?
Week 11
‘Formulas’ to know:
𝜌𝜌𝜌𝜌𝜌𝜌
𝜇𝜇
where 𝜇𝜇 is the dynamic viscosity (N-s/m
2
𝜌𝜌
𝑎𝑎
where a is the speed of sound.
- Ideal gas law: 𝑝𝑝 = 𝜌𝜌𝑅𝑅𝜌𝜌 where 𝑅𝑅 = 𝑅𝑅
𝑢𝑢
- Definitions of lift and drag coefficients
𝐿𝐿
𝐿𝐿
1
2
𝜌𝜌
∞
𝜌𝜌
∞
2
𝐴𝐴
𝐿𝐿
𝐿𝐿
′
1
2
𝜌𝜌
∞
𝜌𝜌
∞
2
𝑐𝑐
𝜌𝜌
𝜌𝜌
1
2
𝜌𝜌
∞
𝜌𝜌
∞
2
𝐴𝐴
𝐿𝐿
𝜌𝜌
′
1
2
𝜌𝜌
∞
𝜌𝜌
∞
2
𝑐𝑐
𝑝𝑝
𝑝𝑝−𝑝𝑝
∞
1
2
𝜌𝜌
∞
𝜌𝜌
∞
2
𝐴𝐴
- Definition of dynamic pressure: 𝑞𝑞
∞
1
2
∞
∞
2
- Specifying an arbitrary extensive property B in terms of a volume integral: 𝐵𝐵
𝐶𝐶𝐶𝐶
𝐶𝐶
- Reynolds Transport Theorem
𝑠𝑠𝑠𝑠𝑠𝑠
𝐶𝐶
𝐴𝐴
- Mass flow rate: 𝑚𝑚̇ = 𝜌𝜌𝑢𝑢𝐴𝐴
- Bernoulli’s equation:
1
1
2
1
New for Midterm II:
- First law of thermodynamics: 𝜌𝜌𝑅𝑅 = 𝛿𝛿𝑞𝑞 + 𝛿𝛿𝛿𝛿. Note in this definition work is positive when done
ON the system. This also implies that 𝛿𝛿𝛿𝛿 = −𝑝𝑝𝜌𝜌𝑝𝑝 in an internally reversible process. Neither
are standard but they are what the textbook assumes. You may use the conventional definitions if
you would like ( 𝜌𝜌𝑅𝑅 = 𝛿𝛿𝑞𝑞 − 𝛿𝛿𝛿𝛿 where 𝛿𝛿𝛿𝛿 = 𝑝𝑝𝜌𝜌𝑝𝑝). Both versions of the first law will ‘work’ as
long as you use the corresponding definitions of 𝛿𝛿𝛿𝛿.
- Second law of thermodynamics: 𝜌𝜌𝑐𝑐 ≥
𝛿𝛿𝛿𝛿
𝑇𝑇
- Definition of enthalpy: ℎ = 𝑅𝑅 +
𝑝𝑝
𝜌𝜌
- Property ratios for an isentropic process:
2
1
2
1
𝛾𝛾
𝛾𝛾−
2
1
𝛾𝛾
‘Formulas’ you will be given with BRIEF explanations
Integral form:
𝜕𝜕
𝜕𝜕𝜕𝜕
𝐶𝐶
𝐴𝐴
Differential form:
𝜕𝜕𝜌𝜌
𝜕𝜕𝜕𝜕
Integral, steady 1-D: 𝑚𝑚̇ = 𝜌𝜌
1
1
1
2
2
2
- Integral form of momentum conservation
Integral form:
𝜕𝜕
𝜕𝜕𝜕𝜕
𝐶𝐶
𝐴𝐴
𝐴𝐴
𝐴𝐴
Differential form: 𝜌𝜌
𝜌𝜌𝑢𝑢��⃗
𝜌𝜌𝜕𝜕
Integral, steady, 1-D, inviscid, no body forces: 𝑚𝑚̇
[
2
1
] + (
2
2
1
1
𝑥𝑥
Integral form:
𝜕𝜕
𝜕𝜕𝜕𝜕
1
2
𝐶𝐶
1
2
𝐴𝐴
𝐴𝐴
𝐴𝐴
𝐴𝐴
𝑠𝑠ℎ𝑎𝑎𝑎𝑎𝜕𝜕
Differential form: 𝜌𝜌
𝜌𝜌𝐷𝐷
𝜌𝜌𝜕𝜕
Integral, steady, 1-D, inviscid: 𝑚𝑚̇ �(ℎ
2
1
𝑢𝑢
2
2
−𝑢𝑢
1
2
2
2
1
𝑠𝑠
- Vorticity: 2 𝜔𝜔��⃗ = 𝜉𝜉⃗ = ∇ × 𝑈𝑈
𝐶𝐶
𝑥𝑥
𝑥𝑥
𝑠𝑠
𝑠𝑠
𝑧𝑧
𝑧𝑧
𝐴𝐴⃗ × 𝐵𝐵
𝑥𝑥
𝑠𝑠
𝑧𝑧
𝑥𝑥
𝑠𝑠
𝑧𝑧
𝑠𝑠
𝑧𝑧
𝑧𝑧
𝑠𝑠
𝑧𝑧
𝑥𝑥
𝑥𝑥
𝑧𝑧
𝑥𝑥
𝑠𝑠
𝑏𝑏
𝑥𝑥
Stokes’s Theorem: ∮
𝑐𝑐
�∇ × 𝑈𝑈
𝐴𝐴
Gauss’s Theorem: ∫
𝐴𝐴
𝐶𝐶
Gradient Theorem: ∫
𝐴𝐴
𝐶𝐶
Leibnitz’s Theorem:
𝑑𝑑
𝑑𝑑𝜕𝜕
𝑅𝑅(𝜕𝜕)
𝜕𝜕𝜕𝜕
𝜕𝜕𝜕𝜕
𝑅𝑅
𝐴𝐴
- Lift and drag coefficients:
𝐿𝐿
𝑁𝑁
𝐴𝐴
𝜌𝜌
𝑁𝑁
𝐴𝐴
- Normal and Axial Forces/span
′
𝑢𝑢
′
𝑇𝑇𝑇𝑇
𝐿𝐿𝑇𝑇
𝐿𝐿
′
𝑇𝑇𝑇𝑇
𝐿𝐿𝑇𝑇
′
𝑢𝑢
′
𝑇𝑇𝑇𝑇
𝐿𝐿𝑇𝑇
𝐿𝐿
′
𝑇𝑇𝑇𝑇
𝐿𝐿𝑇𝑇
𝐿𝐿𝑇𝑇
′
′
𝑐𝑐𝑝𝑝
𝐿𝐿𝑇𝑇
′
′
𝑐𝑐/ 4
′
Cartesian coordinates: 𝜌𝜌𝑢𝑢 =
𝜕𝜕𝜓𝜓
𝜕𝜕𝑠𝑠
𝜕𝜕𝜓𝜓
𝜕𝜕𝑥𝑥
Cylindrical coordinates: 𝜌𝜌𝜌𝜌
𝑟𝑟
1
𝑟𝑟
𝜕𝜕𝜓𝜓
𝜕𝜕𝜕𝜕
𝜕𝜕
𝜕𝜕𝜓𝜓
𝜕𝜕𝑟𝑟
Incompressible flows only: 𝜓𝜓 =
𝜓𝜓
𝜌𝜌
Cartesian coordinates: 𝑢𝑢 =
𝜕𝜕𝜕𝜕
𝜕𝜕𝑥𝑥
𝜕𝜕𝜕𝜕
𝜕𝜕𝑠𝑠
𝜕𝜕𝜕𝜕
𝜕𝜕𝑧𝑧
Cylindrical coordinates: 𝜌𝜌
𝑟𝑟
𝜕𝜕𝜕𝜕
𝜕𝜕𝑟𝑟
𝜕𝜕
1
𝑟𝑟
𝜕𝜕𝜕𝜕
𝜕𝜕𝜕𝜕
𝑧𝑧
𝜕𝜕𝜕𝜕
𝜕𝜕𝑧𝑧
2
1
1
2
1
2
1
2
2
1
1
2
2
1
1
2
1
2
1
2
1
2
2
1
2
- Relationship between turning angle and velocity change through an expansion
2
- Prandtl-Meyer function and relationship to turning angle
ν
−
2
−
2
2
1
2
1
ENAE 311
Midterm I Formula Sheet
Fall 2021
𝐶𝐶
𝐴𝐴
1
1
1
2
2
2
2 𝜔𝜔��⃗ = 𝜉𝜉⃗ = ∇ × 𝑈𝑈
𝐶𝐶
𝜕𝜕
𝜕𝜕𝜕𝜕
𝐶𝐶
𝐴𝐴
𝐴𝐴
𝐴𝐴
[
2
1
] + (
2
2
1
1
𝑥𝑥
𝐶𝐶
𝐴𝐴
𝐴𝐴
𝐴𝐴
𝐴𝐴
𝑠𝑠ℎ𝑎𝑎𝑎𝑎𝜕𝜕
2
1
2
2
1
2
2
1
𝑠𝑠
𝑥𝑥
𝑥𝑥
𝑠𝑠
𝑠𝑠
𝑧𝑧
𝑧𝑧
𝐴𝐴⃗ × 𝐵𝐵
𝑥𝑥
𝑠𝑠
𝑧𝑧
𝑥𝑥
𝑠𝑠
𝑧𝑧
𝑠𝑠
𝑧𝑧
𝑧𝑧
𝑠𝑠
𝑧𝑧
𝑥𝑥
𝑥𝑥
𝑧𝑧
𝑥𝑥
𝑠𝑠
𝑏𝑏
𝑥𝑥
𝑐𝑐
�∇ × 𝑈𝑈
𝐴𝐴
𝐴𝐴
𝐶𝐶
𝐴𝐴
𝐶𝐶
𝑑𝑑
𝑑𝑑𝜕𝜕
𝑅𝑅(𝜕𝜕)
𝜕𝜕𝜕𝜕
𝜕𝜕𝜕𝜕
𝑅𝑅
𝐴𝐴
𝐿𝐿
𝑁𝑁
𝐴𝐴
𝜌𝜌
𝑁𝑁
𝐴𝐴
′
𝑢𝑢
′
𝑇𝑇𝑇𝑇
𝐿𝐿𝑇𝑇
𝐿𝐿
′
𝑇𝑇𝑇𝑇
𝐿𝐿𝑇𝑇
′
𝑢𝑢
′
𝑇𝑇𝑇𝑇
𝐿𝐿𝑇𝑇
𝐿𝐿
′
𝑇𝑇𝑇𝑇
𝐿𝐿𝑇𝑇
𝐿𝐿𝑇𝑇
′
′
𝑐𝑐𝑝𝑝
𝐿𝐿𝑇𝑇
′
′
𝐶𝐶/ 4
′
𝜕𝜕𝜓𝜓
𝜕𝜕𝑠𝑠
𝜕𝜕𝜓𝜓
𝜕𝜕𝑥𝑥
𝜓𝜓
𝜌𝜌
(inc. only)
𝜕𝜕𝜕𝜕
𝜕𝜕𝑥𝑥
𝜕𝜕𝜕𝜕
𝜕𝜕𝑠𝑠
𝜕𝜕𝜕𝜕
𝜕𝜕𝑧𝑧
𝑟𝑟
1
𝑟𝑟
𝜕𝜕𝜓𝜓
𝜕𝜕𝜕𝜕
𝜕𝜕
𝜕𝜕𝜓𝜓
𝜕𝜕𝑟𝑟
𝑟𝑟
𝜕𝜕𝜕𝜕
𝜕𝜕𝑟𝑟
𝜕𝜕
1
𝑟𝑟
𝜕𝜕𝜕𝜕
𝜕𝜕𝜕𝜕
𝑧𝑧
𝜕𝜕𝜕𝜕
𝜕𝜕𝑧𝑧
