Vorticity, circulation and rotational flow

Why it matters

Lift on a wing exists because of circulation around it, wing-tip vortices and wake turbulence come from shed vorticity, and the decision to use potential flow (and Bernoulli across streamlines) depends on whether the flow is irrotational. Vorticity and circulation are therefore the link between basic kinematics and aerodynamics.

Key ideas

Motion of a fluid element. A small fluid element can translate, rotate, stretch (linear strain) and shear (angular strain). Rotation is measured by the average angular velocity of two perpendicular line segments in the element.

Vorticity is the curl of velocity, ω = ∇ × V. It equals twice the local angular velocity of the fluid element. In 2-D flow in the x–y plane only the z-component exists: ω_z = ∂v/∂x − ∂u/∂y, positive for anticlockwise rotation.

Rotational and irrotational flow. A flow with ω = 0 everywhere is irrotational. Then a velocity potential φ exists with V = ∇φ, and Bernoulli's constant is the same on all streamlines (for steady, inviscid, incompressible flow). Irrotational does not mean particles move in straight lines: in a free vortex particles move in circles yet do not spin about their own axes. Conversely, a simple shear flow u = ky has straight streamlines but is rotational.

Where vorticity comes from. Viscous no-slip at walls generates vorticity, so boundary layers and wakes are rotational, while flow outside them that started from a uniform stream remains irrotational. Curved shocks and baroclinic effects (density and pressure gradients not aligned) also generate vorticity.

Circulation is the line integral of the tangential velocity around a closed curve, Γ = ∮V·dl, taken anticlockwise as positive. By Stokes' theorem it equals the flux of vorticity through any surface bounded by the curve, Γ = ∬ω·n dA. So circulation is the "total vorticity" enclosed.

Model vortices.

• Forced vortex (solid-body rotation, V_θ = Ωr): uniform vorticity 2Ω, rotational.
• Free vortex (V_θ = Γ/(2πr)): irrotational everywhere except the singular centre, where all the circulation is concentrated. Any loop enclosing the centre has circulation Γ; any loop not enclosing it has zero circulation.
• Rankine vortex: forced core plus free-vortex outer region, a realistic model of tip vortices and tornadoes.

Kelvin's circulation theorem. For an inviscid, barotropic fluid (ρ a function of p only) with conservative body forces, the circulation around a closed curve moving with the fluid stays constant. Consequence: when a wing starts from rest and develops circulation Γ around itself, an equal and opposite starting vortex is shed into the wake, so the total stays zero.

Helmholtz vortex theorems (same assumptions): vortex-line strength is constant along its length, and a vortex line cannot end in the fluid; it must close on itself or end on a boundary. This is why a wing's bound vortex turns into trailing tip vortices (horseshoe vortex model).

Kutta–Joukowski theorem. For a 2-D body in a uniform inviscid stream, the lift per unit span is L′ = ρU∞Γ, perpendicular to the free stream. The Kutta condition at a sharp trailing edge fixes Γ.

Formulas

ω = ∇ × V and in 2-D ω_z = ∂v/∂x − ∂u/∂y

• ω: vorticity (s⁻¹); u, v: velocity components (m/s). Angular velocity of element = ω/2 (rad/s).

Γ = ∮ V·dl = ∬ ω·n dA

• Γ: circulation (m²/s); dl: line element (m); dA: area element (m²).

V_θ = Ω·r (forced vortex, ω = 2Ω) and V_θ = Γ / (2π·r) (free vortex)

• Ω: angular speed (rad/s); r: radius (m).

L′ = ρ·U∞·Γ

• L′: lift per unit span (N/m); ρ: density (kg/m³); U∞: free-stream speed (m/s).

V = ∇φ exists only when ω = 0 (φ: velocity potential, m²/s).

Worked examples

Example 1 (standard): simple shear flow. Given: u = 4y, v = 0 (m/s, y in m). Find the vorticity and the circulation around the unit square 0 ≤ x ≤ 1, 0 ≤ y ≤ 1.

1. ω_z = ∂v/∂x − ∂u/∂y = 0 − 4 = −4 s⁻¹ (clockwise rotation at 2 rad/s).
2. Line integral anticlockwise: bottom (y = 0) u = 0, contributes 0; right side v = 0; top (y = 1), moving in −x: −∫u dx = −4 × 1 = −4; left side 0. Γ = −4 m²/s.
3. Stokes check: ω_z × area = −4 × 1 = −4 m²/s. Answer: ω_z = −4 s⁻¹, Γ = −4 m²/s. The streamlines are straight yet the flow is rotational.

Example 2 (standard): forced and free vortex. (a) A tank rotates as a solid body at Ω = 10 rad/s. Circulation around a circle of r = 0.2 m: Γ = 2πr·V_θ = 2π × 0.2 × (10 × 0.2) = 2.513 m²/s. Check by Stokes: ω × πr² = 20 × π × 0.04 = 2.513 m²/s. (b) A free vortex has Γ = 2π m²/s. At r = 0.5 m, V_θ = Γ/(2πr) = 2π/(2π × 0.5) = 2 m/s. The circulation around any loop that encloses the centre is 2π m²/s; around a loop that excludes it, zero. Answers: (a) 2.51 m²/s (b) 2 m/s.

Example 3 (GATE level): lift from circulation. Given: a rectangular wing of span 10 m in a stream of 60 m/s, ρ = 1.225 kg/m³, with uniform bound circulation Γ = 20 m²/s (2-D estimate).

1. L′ = ρ·U∞·Γ = 1.225 × 60 × 20 = 1470 N/m.
2. L = L′ × b = 1470 × 10 = 14 700 N. Answer: L ≈ 14.7 kN (an upper estimate; tip vortices reduce real 3-D lift).

Common mistakes

• Equating vorticity with angular velocity. Vorticity is twice the angular velocity.
• Sign errors in ω_z = ∂v/∂x − ∂u/∂y (the order matters).
• Assuming curved streamlines mean rotational flow (the free vortex is irrotational) or straight streamlines mean irrotational flow (shear flow is rotational).
• Saying circulation is always zero in irrotational flow. It is zero around loops in a simply connected irrotational region, but a loop around a free vortex or a lifting aerofoil has non-zero circulation.
• Applying Kelvin's theorem inside viscous boundary layers, where it does not hold.

For GATE AE

Expect: compute vorticity or angular velocity of a given 2-D field and decide whether it is irrotational; circulation around a rectangle or circle (directly or with Stokes' theorem); free-vortex and forced-vortex velocity and circulation; lift per unit span from L′ = ρU∞Γ; and conceptual questions on Kelvin's and Helmholtz's theorems, the starting vortex and the horseshoe vortex. Practise both the direct line integral and the Stokes shortcut.

Quick check

1. For u = 2y, v = −2x, what is ω_z?
2. What is the angular velocity of a fluid element if ω_z = 6 s⁻¹?
3. Is the free vortex V_θ = C/r rotational or irrotational away from its centre?
4. Find L′ for ρ = 1.225 kg/m³, U∞ = 50 m/s, Γ = 15 m²/s.

Answers: 1. −4 s⁻¹ 2. 3 rad/s 3. Irrotational 4. 918.75 N/m