Questions: Potential Flow Theory

Question 1

Potential flow predicts zero drag on a circular cylinder (d'Alembert's paradox). This result fails for real fluids because:

• A. Potential flow incorrectly predicts the velocity magnitude at the cylinder surface
• B. Laplace's equation cannot be solved accurately for curved geometries like cylinders
• C. The irrotational assumption breaks down in the viscous boundary layer and separated wake, where vorticity is generated
• D. The pressure distribution predicted by potential flow is incorrect everywhere around the cylinder, including far from the surface

Answer: (C)

D'Alembert's paradox arises because potential flow is fore-aft symmetric — the pressure distribution on the front half exactly mirrors the back half, producing no net pressure force. In reality, viscosity creates a boundary layer on the cylinder surface and a separated wake behind it; both regions contain strong vorticity. The real pressure distribution on the rear of the cylinder is much lower than potential flow predicts (because the separated wake maintains a low-pressure region), producing the net pressure drag that potential flow entirely misses.

Question 2

Why can complex flow patterns be constructed by adding together elementary solutions (uniform flow, source, doublet, vortex) in potential flow theory?

• A. Because the Navier-Stokes equations are linear at high Reynolds number
• B. Because irrotational flows share identical boundary conditions that allow solutions to combine
• C. Because Laplace's equation is linear, so any linear combination of solutions is also a solution
• D. Because all real flows can be decomposed into a finite number of elementary vortex structures

Answer: (C)

The superposition principle applies directly because Laplace's equation ∇²φ = 0 is linear. If φ₁ and φ₂ both satisfy ∇²φ = 0, then c₁φ₁ + c₂φ₂ does too. This is not true of the Navier-Stokes equations, which contain nonlinear inertial terms — option A is incorrect. The linearity of Laplace's equation is precisely why potential flow has a vast catalog of analytical solutions: once you know a library of elementary solutions, you can construct complex flows by superposition.

Question 3

Potential flow applies to most inviscid flows — any flow with high enough Reynolds number qualifies as potential flow, since viscosity becomes negligible.

• T. True
• F. False

Answer: False

False. Potential flow additionally requires irrotationality (∇×V = 0), which is a separate and stronger condition than inviscid flow. In many high-Reynolds-number flows, viscosity is indeed negligible in the bulk — but near solid surfaces, viscosity generates vorticity in the boundary layer, and wakes downstream contain strong vortical structures. These rotational regions violate the irrotational assumption even where viscosity is small. Potential flow applies only to the outer, irrotational region of a high-Re flow, not to boundary layers, wakes, or separated flow regions.

Question 4

In 2D potential flow, streamlines (lines of constant stream function ψ) and equipotential lines (lines of constant velocity potential φ) are always perpendicular to each other.

• T. True
• F. False

Answer: True

True. The stream function ψ and velocity potential φ satisfy the Cauchy-Riemann equations: ∂φ/∂x = ∂ψ/∂y and ∂φ/∂y = −∂ψ/∂x. This makes φ and ψ conjugate harmonic functions, and a theorem from complex analysis states that the level curves of two conjugate harmonic functions are everywhere orthogonal. Together, lines of constant φ and ψ form a flow net — an orthogonal grid that provides a powerful visual tool for analyzing 2D potential flows.

Question 5

Explain why irrotationality is both the source of potential flow theory's power and the reason for its practical limitations.

Model answer: Irrotationality allows the velocity field to be derived from a scalar potential φ satisfying Laplace's equation, which is linear. Linearity enables superposition of elementary solutions, analytical tractability, and the full machinery of complex analysis — enormous computational power. But viscosity generates vorticity wherever fluid contacts a solid surface, violating the irrotational assumption in boundary layers and wakes. Potential flow therefore correctly describes the outer flow and gives accurate pressure predictions for lift, but completely fails to predict drag, which depends on the viscous wake it ignores.

This tradeoff is the conceptual foundation of boundary layer theory: match an outer potential-flow solution (irrotational, analytically tractable) to an inner viscous boundary-layer solution (rotational, thin). The two regions capture different physics. The outer flow determines the pressure distribution that drives lift; the inner boundary layer determines skin friction and separation that drives drag. Understanding where potential flow is valid — and where it breaks down — is as important as knowing how to compute it.