What does lift in 3D require in terms of vortex dynamics?
Lift in 3D requires a complex vortex system, meaning the irrotational flow assumption must be partially dropped.
State Euler’s equation for an incompressible fluid of constant density.
Du/Dt = -∇p/ρ + g, with ∇·u = 0, where Du/Dt is the material derivative.
What is the evolution equation for vorticity in an inviscid incompressible flow?
Dω/Dt = ω·∇u.
What does the vorticity equation imply about irrotational flows?
If ω = 0 initially, its rate of change is zero, meaning the flow remains irrotational (principle of persistence of irrotationality).
How does the evolution of an infinitesimal vector dr relate to vorticity?
Both evolve by Ddr/Dt = dr·∇u, so if dr and ω are initially parallel, they remain parallel.
Define a vortex line.
A vortex line is a line everywhere tangent to the local vorticity vector ω.
Define a vortex tube.
A vortex tube is a surface formed by all vortex lines passing through a closed curve; it remains a vortex tube as the flow evolves.
What does ∇·ω = 0 imply about vortex tubes?
The vortex tube’s strength (∫ω·n dA) is constant along its length, analogous to continuity in a stream tube.
What is the physical interpretation of ∇·ω = 0?
It implies that vortex lines cannot start or end within the fluid—they must either close on themselves or end at a boundary.
What is the relationship between vortex tube strength and circulation?
By Stokes’ theorem, the strength of a vortex tube equals the circulation of velocity ∮u·dl along the loop.
State Kelvin’s Circulation Theorem.
For an inviscid incompressible fluid, the circulation around a closed loop remains constant as it moves with the flow.
What are the assumptions of Kelvin’s theorem?
The flow must be inviscid and incompressible, and the external body forces (like gravity) must be conservative.
How is the circulation defined?
Γ = ∮_C u·dl, where C is a closed curve moving with the flow.
What does Kelvin’s theorem imply about vortex tubes?
Their strength remains constant both along their length and over time; they are carried as material entities by the flow.
What causes the constancy of vortex tube strength at a fixed instant?
The condition ∇·ω = 0, independent of the dynamics.
What causes the constancy of vortex tube strength over time?
The dynamical laws of motion given by Euler’s equations.
How does vorticity behave when ∇u = 0 (uniform velocity field)?
dr does not change in time and ω remains constant; the fluid element preserves its shape and angular momentum.
Explain the analogy between vorticity evolution and a rotating dancer.
When a fluid element stretches or contracts (∇u ≠ 0), its rotation speed changes, like a dancer adjusting her arms.
What is the meaning of Ddr/Dt = dr·∇u physically?
It represents the deformation of an infinitesimal fluid element as it is carried by the flow.
What happens to a vortex line as it moves with the flow?
It remains a vortex line because it stays aligned with the local vorticity vector.
Why does circulation remain unchanged in Kelvin’s theorem proof?
Because the integrand’s terms (gravitational potential, pressure, and kinetic energy) take the same values at the start and end of the loop, making the integral null.
Summarize the key property of vortices in inviscid incompressible flow.
A vortex tube preserves its identity, structure, and strength as it is advected by the flow.
What is the analogy between stream tubes and vortex tubes?
In stream tubes, ∇·u = 0 ensures constant mass flux; in vortex tubes, ∇·ω = 0 ensures constant vorticity flux.
What defines a material vortex in an inviscid flow?
It is a region where the vorticity is concentrated and is carried by the flow without losing strength or structure.
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Midterm Prep - Ch 217
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Midterm Prep - Ch323
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Midterm Prep - Ch430
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Midterm Prep - Ch513
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Midterm Prep - Ch640
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Exam 2 - Chapter 724
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Exam 2 - Chapter 834
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Exam 2 - Chapter 933
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Exercises Ch88
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Exercises Ch98
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Chapter 11 - Shocks12
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Exercises Ch115
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Exercises Ch125
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Chapter 1316
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Chapter 1045
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Exercises 17
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Exercises 26
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Exercises 36
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Exercises 42
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Exercises 67
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Midterm 1 theory4
