Steady and Unsteady Flow Around a Cylinder

1. Reynolds number regimes

The Reynolds number Re = UD/ν (free-stream speed U, cylinder diameter D, kinematic viscosity ν) is the single parameter that controls the wake topology behind a cylinder:

• Re < 1 (Stokes/creeping flow): viscous forces dominate, the flow is fully attached and fore-aft nearly symmetric — reversible in the Stokes sense.
• Re ≈ 5–40: flow separates at the rear, forming a pair of stable, symmetric, steady recirculating vortices.
• Re ≈ 40–150: the symmetric wake becomes unstable to small perturbations — vortices begin shedding alternately from the top and bottom, producing the periodic Kármán vortex street.
• Re ≈ 150–300: the wake transitions further, three-dimensional instabilities appear along the cylinder span.
• Re > ~3×10⁵: the boundary layer itself turns turbulent before separating — the "drag crisis" (see section 6).

2. Steady limit: potential flow and D'Alembert's paradox

If viscosity is ignored entirely (inviscid, irrotational flow), the exact steady solution around a cylinder of radius R in a uniform stream U is given by superposing a uniform flow and a doublet:

This solution is fore-aft symmetric — pressure recovers fully on the downstream side exactly as it dropped on the upstream side. Integrating pressure over the surface then gives zero net drag, known as D'Alembert's paradox. Real cylinders obviously do experience drag; the paradox is resolved by viscosity, which — however small — creates a thin boundary layer that separates and destroys the symmetric pressure recovery.

3. Boundary layer separation

Along the front half of the cylinder, the flow accelerates and the pressure gradient is favourable (dp/dx < 0), keeping the boundary layer attached. Past the point of maximum velocity (θ ≈ 90° from the stagnation point), pressure starts rising again — an adverse pressure gradient. Near the wall, where kinetic energy is smallest, the flow can no longer push through this rising pressure and reverses direction, detaching from the surface at the separation point:

For a laminar boundary layer this happens around θ_sep ≈ 80–85° from the front stagnation point. Downstream of separation the flow rolls up into shear-layer vortices that feed the wake.

4. The Kármán vortex street

Above Re ≈ 47, the two symmetric separation vortices become unstable: one grows, sheds downstream, and induces the opposite one to grow and shed in turn. The result is a regular, staggered double row of counter-rotating vortices convecting downstream — first analyzed for stability by Theodore von Kármán in 1911–1912. This shedding also produces an oscillating transverse (lift) force on the cylinder itself, the mechanism behind singing telephone wires and, in extreme resonance cases, structural failures such as the 1940 Tacoma Narrows Bridge collapse (though that event was dominated by torsional flutter rather than pure vortex shedding).

5. Strouhal number and shedding frequency

The shedding frequency f is non-dimensionalized as the Strouhal number:

Remarkably, over a very wide range of Reynolds numbers (roughly 300 < Re < 2×10⁵, the "subcritical" regime) St stays close to a constant ≈ 0.20 for a circular cylinder — a rare case where a complex unsteady flow collapses onto a near-universal dimensionless law. Below Re ≈ 300, St rises smoothly from ≈ 0.12 at Re ≈ 50 up toward the plateau value.

6. Drag, lift, and the drag crisis

Force coefficients are defined by normalizing with dynamic pressure and frontal area (A = D·L for a cylinder of length L):

For the subcritical regime, C_d for a smooth circular cylinder sits around 1.0–1.2, dominated by pressure (form) drag from the wide separated wake rather than skin friction. As Re approaches ≈ 3×10⁵, something counter-intuitive happens: the laminar boundary layer itself transitions to turbulent before separating. A turbulent boundary layer carries more momentum near the wall and resists the adverse pressure gradient longer, delaying separation to θ_sep ≈ 120–130° and shrinking the wake. Drag coefficient drops sharply to around 0.3 — the drag crisis — before rising slowly again at even higher Re. This is why golf balls have dimples: they trip the boundary layer turbulent early on purpose, reducing drag exactly through this mechanism.