Drag of bluff and streamlined bodies

Why it matters

Drag sets the thrust, fuel and power an aircraft needs, the loads on struts, antennas and landing gear, and the way a store or droplet falls. Knowing whether a body's drag is dominated by friction or by pressure tells a designer what to change: smoother surfaces and laminar flow for streamlined bodies, shape and separation control for bluff ones.

Key ideas

Where drag comes from. The fluid acts on a body only through pressure (normal) and shear (tangential) stresses. Integrating the streamwise components gives

• skin-friction drag: from wall shear stress, dominant for streamlined bodies at small incidence;
• pressure (form) drag: from the fore–aft pressure difference caused mainly by separation and the wake, dominant for bluff bodies. Their sum is the profile drag of a 2-D section. For a finite wing there is also induced drag (from trailing vortices, linked to lift), and at transonic and supersonic speed wave drag (from shocks). Those are treated in aerodynamics.

Bluff versus streamlined. A body is bluff if its boundary layer separates over a large part of its rear surface and leaves a wide wake (cylinder, sphere, normal plate, a truck). It is streamlined if the flow stays attached almost to the trailing edge and the wake is thin (aerofoil, teardrop strut). A streamlined strut can have about one-tenth the drag of a circular cylinder of the same thickness.

Drag coefficient. C_D = D/(½ρV²A). The reference area A must be stated: frontal area for bluff bodies, planform area for wings, wetted area for skin-friction estimates. By dimensional analysis C_D = f(shape, Re, M, roughness, free-stream turbulence).

C_D against Re for a cylinder or sphere.

• Re < 1 (creeping, Stokes) flow: viscous drag dominates; for a sphere D = 3πμVd, so C_D = 24/Re.
• Moderate Re: separation appears, a steady pair of eddies forms, then (Re above about 40 for a cylinder) periodic vortex shedding in a Kármán street with Strouhal number St = fd/V ≈ 0.2.
• Subcritical range (about 10³ to 2×10⁵): laminar separation near 80°, C_D roughly constant (sphere about 0.47, long cylinder about 1.2).
• Drag crisis (about 2×10⁵ to 5×10⁵, lower with roughness or turbulence): the boundary layer turns turbulent before separating, separation moves to about 120°, the wake narrows and C_D falls sharply (sphere to about 0.1–0.2, cylinder to about 0.3). These numbers are typical values from data books; use the chart for your case.

Bodies with sharp edges (normal flat plate, disk, cube) separate at the edges at all Re above a few hundred, so their C_D is nearly independent of Re (normal 2-D plate about 2.0, disk about 1.17). Reynolds effects are weak because the separation point is fixed.

Streamlining trade-off. Lengthening a body reduces pressure drag but adds wetted area and friction. For bodies of revolution the minimum total drag for a given frontal area occurs at a fineness ratio (length/diameter) of roughly 3–4; for a given volume it is somewhat higher.

Drag reduction ideas. Fairings and trailing-edge tapering (delay separation), boat-tailing of fuselages, trips or dimples on bluff bodies near the critical Re, laminar-flow aerofoils and smooth finishes for streamlined bodies, riblets for turbulent friction.

Formulas

D = C_D·½·ρ·V²·A

• D: drag (N); C_D: drag coefficient (dimensionless); ρ (kg/m³); V (m/s); A: reference area (m²).

D = D_friction + D_pressure = ∮τ_w·sinθ dA + ∮p·cosθ dA (streamwise components)

• θ: local surface angle; τ_w: wall shear (Pa); p: surface pressure (Pa).

D = 3π·μ·V·d, C_D = 24/Re (sphere, Re = ρVd/μ < 1) V_t = (ρ_p − ρ)·g·d² / (18μ)

• V_t: Stokes terminal velocity (m/s); ρ_p: particle density (kg/m³); d: diameter (m).

St = f·d / V ≈ 0.2

• f: vortex-shedding frequency (Hz), circular cylinder, about 300 < Re < 2×10⁵.

Worked examples

Example 1 (standard): wind load and shedding on a cable. Given: cable d = 20 mm in wind V = 25 m/s; air ρ = 1.225 kg/m³, ν = 1.5×10⁻⁵ m²/s; take C_D = 1.2 (subcritical cylinder, data book).

1. Re = Vd/ν = 25 × 0.02/1.5×10⁻⁵ = 3.33×10⁴: subcritical, so C_D ≈ 1.2 applies.
2. Drag per metre: D′ = C_D·½ρV²·d = 1.2 × 0.5 × 1.225 × 625 × 0.02 = 9.19 N/m.
3. Shedding: f = St·V/d = 0.2 × 25/0.02 = 250 Hz (the source of the "singing" of wires). Answer: D′ ≈ 9.2 N/m; f ≈ 250 Hz.

Example 2 (GATE level): Stokes settling of a droplet. Given: water droplet d = 50 μm (ρ_p = 1000 kg/m³) falling in air (ρ = 1.225 kg/m³, μ = 1.8×10⁻⁵ Pa·s).

1. Weight minus buoyancy equals Stokes drag: (ρ_p − ρ)g·πd³/6 = 3πμV_t·d.
2. V_t = (ρ_p − ρ)·g·d²/(18μ) = 998.8 × 9.81 × (5×10⁻⁵)²/(18 × 1.8×10⁻⁵) = 0.0756 m/s.
3. Check: Re = ρV_t·d/μ = 1.225 × 0.0756 × 5×10⁻⁵/1.8×10⁻⁵ = 0.26 < 1, so Stokes' law is valid. Answer: V_t ≈ 7.6 cm/s.

Example 3 (GATE level): drag crisis. A 0.22 m ball at 30 m/s in the same air has Re = 30 × 0.22/1.5×10⁻⁵ = 4.4×10⁵, near the critical range. With subcritical C_D = 0.47 the drag would be 0.47 × 0.5 × 1.225 × 900 × 0.03801 = 9.85 N; if the layer is tripped and C_D falls to about 0.2, it drops to 4.2 N, less than half.

Common mistakes

• Using planform area with a C_D defined on frontal area, or the reverse.
• Believing a sharp, flat plate parallel to the flow has pressure drag. It has almost only friction drag; normal to the flow it has almost only pressure drag.
• Applying C_D = 24/Re above Re ≈ 1.
• Thinking a turbulent boundary layer always increases drag. On bluff bodies near the critical Re it lowers total drag.
• Forgetting that the drag crisis Re depends on roughness and free-stream turbulence.
• Mixing up the Strouhal number definition (use diameter and free-stream speed).

For GATE AE

Expect: drag from C_D with the right reference area, the C_D–Re curve and the drag crisis, Stokes drag and terminal velocity, vortex-shedding frequency from the Strouhal number, power needed to overcome drag (P = D·V), and conceptual MCQs on friction versus pressure drag for bluff and streamlined bodies. Practise checking the Re range before choosing a C_D.

Quick check

1. C_D = 0.2, A = 3 m², V = 40 m/s, ρ = 1.225 kg/m³. Drag?
2. Which drag component dominates on a circular cylinder at Re = 10⁵?
3. Drag on a sphere in Stokes flow?
4. If speed doubles at constant C_D, how does the power to overcome drag change?

Answers: 1. 588 N 2. Pressure (form) drag 3. 3πμVd 4. It increases eight times