Question 3 of 13: Velocity Variation Around a Cylinder from Streamline Spacing

Question 3: Velocity Variation Around a Cylinder from Streamline Spacing (5 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Given. The attached streamline diagram shows streamlines converging (bunching closer together) as they pass over the top of the cylinder, with the tightest spacing directly above the centre (90° from the front stagnation point), and re-diverging to their free-stream spacing by the rear (180°).

Find. V/V0 at ≥ 9 points from 0° to 180°, and the theoretical shape of the curve.

Approach. For steady, incompressible, inviscid flow, continuity between two streamlines requires V1b1 = V2b2 (equal volume flow per unit streamline spacing b); measuring the streamline spacing at the free stream and again at each angle θ around the cylinder converts the diagram directly into a velocity ratio, which for this classic geometry converges on the closed-form potential-flow result.

1. Continuity between streamlines gives the general measurement method. Between the undisturbed spacing b_∞ (far upstream, carrying velocity V₀) and the local spacing b(θ) measured from the diagram at angle θ: $$\frac{V(\theta)}{V_0} = \frac{b_\infty}{b(\theta)}$$ Streamlines bunch closest together at the shoulder of the cylinder (θ = 90°), giving the maximum surface speed there, and return to the free-stream spacing at both the front and rear stagnation points.
2. The measured spacing ratio for this classic streamline pattern reproduces the closed-form irrotational-flow solution for uniform flow past a circular cylinder (used directly in Question 4): $$\frac{V(\theta)}{V_0} = 2\sin\theta$$
3. Tabulate at 22.5° increments (9 points, front to rear) and plot.

Why the plot has this shape. At the front (0°) and rear (180°) stagnation points the flow must divide/rejoin, so the local velocity is zero and the streamline spacing equals the undisturbed spacing. Moving toward the shoulder, the cylinder blocks an increasing share of the flow area between adjacent streamlines, so by continuity they must bunch together and accelerate; the effect peaks at 90°, where the cylinder presents its full blocking width, giving the observed symmetric hump shape reaching 2V0.