What is the main advantage of using the complex plane in generating 2D flow fields?
Operating in the complex plane simplifies generating solutions to Laplace’s equation because analytic functions Φ(z)=φ(x, y)+iψ(x, y) have real and imaginary parts that both satisfy ∇²φ=∇²ψ=0, making them natural velocity potentials and stream functions for inviscid incompressible flows.
What defines an analytic function Φ(z)?
A function Φ(z) is analytic at a point z₀ if its derivative Φ’(z₀) exists and is independent of the direction in which the limit is taken.
Write the Cauchy-Riemann equations and explain their physical meaning.
∂φ/∂x = ∂ψ/∂y, ∂φ/∂y = -∂ψ/∂x. They express that φ and ψ are harmonic conjugates; physically, φ is the velocity potential and ψ is the stream function, with their contours orthogonal.
What is the complex velocity in terms of the potential function?
dΦ/dz = u - iv, where u and v are velocity components in x and y directions.
How can you obtain the velocity field from a complex potential Φ(z)?
Take dΦ/dz = u - iv; this gives velocity directly without computing gradients.
How can you obtain the stream function from Φ(z)?
The imaginary part of Φ(z) gives the stream function ψ(x, y).
What is the complex potential for a uniform flow in the x-direction?
Φ(z) = Uz
What is the complex potential for a uniform flow inclined at angle α?
Φ(z) = Uz e^{-iα}
What is the complex potential for a source of strength q’ at z₀?
Φ(z) = (q’/2π) ln(z - z₀)
What is the complex potential for a doublet at z₀?
Φ(z) = A/(z - z₀), where A is a complex constant defining dipole strength and orientation.
What is the complex potential for a vortex with circulation Γ?
Φ(z) = -iΓ/(2π) ln(z - z₀)
What is circulation and how is it defined?
Γ = ∮ u·dl, the line integral of velocity around a closed path. It equals the surface integral of vorticity; for a potential vortex, vorticity is infinite only at the origin.
What is the complex potential for flow around a cylinder of radius R?
Φ(z) = U(z + R²/z)
What is the potential for flow around a cylinder with circulation Γ?
Φ(z) = U(z + R²/z) - iΓ/(2π) ln z
What transformation turns a cylinder into a flat plate?
The Joukowski transformation: w = z + R²/z.
What is a conformal mapping?
A transformation w(z) that preserves angles locally; small shapes are rotated/scaled but not distorted.
Where is the Joukowski transformation singular?
At z=0 and z=±b, where dw/dz=1 - b²/z² becomes 0 or ∞.
What is the general Joukowski transformation?
w = z + b²/z, which maps circles into ellipses; for b=R, the ellipse becomes a flat plate.
What is the Kutta condition?
The flow cannot turn around a sharp trailing edge; circulation adjusts so that the rear stagnation point lies at the trailing edge.
How is circulation Γ determined from the Kutta condition?
Γ = 4πUR sinα, where α is the angle of attack.
What is the complex potential for a cylinder with α and Γ?
Φ(z) = U(ze^{-iα} + R²e^{iα}/z) + iΓ/(2π) ln z
What is the velocity distribution over a flat plate?
(u/U) = [sin(θ - α) + sinα]/sinθ
Where does the velocity become infinite for a flat plate?
At the leading edge due to the sharp nose (singularity).
What is the pressure coefficient Cp?
Cp = 1 - (u² + v²)/U²
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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
