Elementary flows: uniform flow, source, sink, doublet and vortex

Why it matters

Laplace's equation is linear, so a handful of simple solutions — uniform flow, source, sink, doublet and vortex — can be added together to build the flow past a cylinder, a Rankine body, a lifting airfoil or a whole wing (panel and vortex-lattice methods). Knowing their φ, ψ and velocity fields by heart is the toolkit for the rest of this subject.

Key ideas

All the elementary flows below are 2-D, incompressible and irrotational (except at a singular point), so each satisfies ∇²φ = 0 and ∇²ψ = 0. Polar coordinates (r, θ) are measured from the singularity, θ anticlockwise from the +x axis. The sign conventions follow Anderson's Fundamentals of Aerodynamics; some Indian texts absorb the 2π into the strength or use the opposite sign for vortex circulation — always check the convention of the question.

Uniform flow. Velocity V∞ along +x everywhere. Streamlines are horizontal lines, equipotentials vertical lines. Circulation round any closed curve is zero.

Source and sink. Purely radial flow from (source) or into (sink) a point. Strength Λ is the volume flow per unit depth (m²/s); Λ > 0 source, Λ < 0 sink. V_r = Λ/(2πr) falls off as 1/r; the point r = 0 is singular (infinite velocity), so it must lie inside a body or outside the flow region of interest. Streamlines are radial lines; equipotentials are circles.

Doublet. A source and an equal sink brought together (separation l → 0) while κ = l·Λ is held constant. Streamlines are circles passing through the doublet point, tangent to the x-axis there. On its own a doublet looks nothing like a body; added to a uniform stream it produces the flow past a circular cylinder (next topic).

Point vortex. Purely circumferential flow, V_θ = −Γ/(2πr) with Γ positive for clockwise rotation (Anderson's convention, chosen so that positive Γ gives positive lift later). Streamlines are circles, equipotentials radial lines. The flow is irrotational everywhere except at r = 0, where the vorticity is concentrated; the circulation round any curve enclosing the origin is Γ, and round any curve not enclosing it is zero. Real vortices have a viscous core of finite size (Rankine vortex model).

Superposition examples.

• Uniform flow + source → Rankine half-body: a stagnation point upstream of the source at distance Λ/(2πV∞); the dividing streamline ψ = Λ/2 is a semi-infinite body whose width tends to Λ/V∞ far downstream.
• Uniform flow + source + equal sink downstream → Rankine oval (closed body).
• Uniform flow + doublet → circular cylinder; adding a vortex → lifting cylinder.
• Source + vortex at the same point → spiral vortex (bathtub drain if a sink).

Formulas

• Uniform flow: φ = V∞·x = V∞·r·cos θ, ψ = V∞·y = V∞·r·sin θ — V∞ (m/s).
• Source/sink: φ = (Λ/2π)·ln r, ψ = (Λ/2π)·θ, V_r = Λ/(2πr), V_θ = 0 — Λ (m²/s), r (m).
• Doublet: φ = (κ/2π)·cos θ / r, ψ = −(κ/2π)·sin θ / r, V_r = −(κ/2π)·cos θ / r², V_θ = −(κ/2π)·sin θ / r² — κ (m³/s).
• Vortex: φ = −(Γ/2π)·θ, ψ = (Γ/2π)·ln r, V_r = 0, V_θ = −Γ/(2πr) — Γ (m²/s), clockwise positive.
• Rankine half-body: stagnation point at x_s = −Λ/(2πV∞); body (dividing streamline) ψ = Λ/2; asymptotic total width h = Λ/V∞.
• Bernoulli between any two points: p₁ − p₂ = ½ρ(V₂² − V₁²) and C_p = 1 − (V/V∞)².

Worked examples

Example 1 (standard). A source of strength Λ = 4π m²/s and a vortex of strength Γ = 8π m²/s (clockwise) are both at the origin in air (ρ = 1.225 kg/m³). Find the velocity at r = 2 m, its direction, and p(r = 4 m) − p(r = 2 m).

1. V_r = Λ/(2πr) = 4π/(2π × 2) = 1.0 m/s.
2. V_θ = −Γ/(2πr) = −8π/(2π × 2) = −2.0 m/s (clockwise).
3. |V| = √(1² + 2²) = 2.24 m/s, inclined at tan⁻¹(2/1) = 63.4° to the radial direction.
4. At r = 4 m: V_r = 0.5, V_θ = −1.0, so V² = 1.25 m²/s²; at r = 2 m, V² = 5 m²/s².
5. The flow is irrotational outside r = 0, so Bernoulli applies between the points: p₄ − p₂ = ½ρ(V₂² − V₄²) = 0.5 × 1.225 × (5 − 1.25) = 2.30 Pa.

Answer: |V| = 2.24 m/s at 63.4° to the radius (spiralling outward clockwise); p(4 m) − p(2 m) = 2.30 Pa.

Example 2 (GATE level). A uniform stream V∞ = 10 m/s (air, ρ = 1.225 kg/m³) is combined with a source of strength Λ = 6 m²/s at the origin. Find the stagnation point, the asymptotic width of the half-body, and C_p and the gauge pressure p − p∞ on the body directly above the source.

1. Stagnation point: V∞ + Λ/(2πx) = 0 ⇒ x_s = −Λ/(2πV∞) = −6/(2π × 10) = −0.0955 m.
2. Asymptotic width: h = Λ/V∞ = 6/10 = 0.6 m (half-width 0.3 m).
3. Body streamline ψ = Λ/2: V∞ r sin θ + Λθ/(2π) = Λ/2. At θ = 90°: 10r + 6/4 = 3 ⇒ r = 0.15 m (= Λ/(4V∞)).
4. Velocity there: u = V∞ + V_r cos 90° = 10 m/s, v = V_r = Λ/(2πr) = 6/(2π × 0.15) = 6.37 m/s (= 2V∞/π).
5. C_p = 1 − (u² + v²)/V∞² = 1 − (1 + 4/π²) = −4/π² = −0.405.
6. p − p∞ = C_p·½ρV∞² = −0.405 × 0.5 × 1.225 × 100 = −24.8 Pa.

Answer: stagnation point 0.0955 m upstream of the source; width 0.6 m; C_p = −0.405, p − p∞ = −24.8 Pa above the source.

Common mistakes

• Using a 3-D point-source formula (V = Q/4πr²) for a 2-D line source; in 2-D, V_r = Λ/(2πr) and Λ is in m²/s.
• Calling the vortex velocity "radial" — it is tangential; a source's velocity is radial.
• Mixing sign conventions: if your text takes Γ positive anticlockwise, V_θ = +Γ/(2πr) and the lift formula carries the corresponding sign.
• Believing the vortex flow is rotational. Only its centre is; everywhere else ω = 0.
• Superposing pressures rather than velocities.
• Forgetting that the singular point must lie inside the body.

For GATE AE

Expect: velocity at a point due to one or more singularities; strength of a source or vortex from a measured velocity; stagnation-point location for half-bodies and ovals; checking which combinations satisfy Laplace's equation; and circulation round a given contour. Learn the φ/ψ table both ways and state the sign convention in your working.

Quick check

1. A 2-D source gives V_r = 5 m/s at r = 2 m. Find Λ.
2. A vortex induces 4 m/s at r = 1 m. Find |Γ|.
3. What is the circulation round a circle that does not enclose a point vortex?
4. Which combination of elementary flows gives a Rankine half-body?

Answers: 1. Λ = 2πrV_r = 20π ≈ 62.8 m²/s. 2. |Γ| = 2πrV_θ = 8π ≈ 25.1 m²/s. 3. Zero. 4. Uniform flow plus a source.