Drag Coefficient Lab: Levitation Wind Tunnel, Reynolds Number and Terminal Speed | mysimulator.uk

What is modelled

A vertical wind (or water) tunnel, like the ones skydivers use. A test body sits on a grille; a fan blows fluid upward past it. When the drag force on the body exceeds its weight minus buoyancy, it lifts off; when the flow speed equals the body's terminal speed it hovers. Because drag depends on shape and Reynolds number through the drag coefficient Cd(Re), you can watch the same physics explain why a flat disk falls slowly, a streamlined body falls fast, and a golf-ball-sized sphere hits a drag crisis.

Equations

• Drag: D = ½ ρ Cd(Re) A u |u| with u = U − z′ the speed of the flow relative to the body and A the frontal area. Reynolds number Re = ρ |u| d / μ.
• Equation of motion with added mass: (ρb V + k ρ V) z″ = D − (ρb − ρ) V g. Added-mass coefficients k: sphere 0.5, cylinder 1.0, face-on disk about 4.2, cube 0.7 and a 3:1 spheroid 0.2 (ideal-fluid values, used as estimates).
• Terminal speed: solve ½ ρ Cd(Re) A v² = (ρb − ρ) V g; for Re < 1 this becomes Stokes' law v = (ρb − ρ) g d² / 18 μ for a sphere.
• Sphere: Morrison's (2013) fit to the standard drag curve, 24/Re + 2.6(Re/5)/(1+(Re/5)1.52) + …, including the drag crisis near Re = 3 x 10⁵. Cylinder across the flow: Lamb for Re < 1 and White's 1 + 10 Re−2/3 above, with a crisis, times an end-effect factor for a length of 3 d. Flat disk face-on: 1.17 + 64/(π Re). Cube: 1.05 + 25.3/Re. Streamlined 3:1 body: Hoerner's skin-friction formula Cf (3λ + 4.5/√λ + 21/λ²) plus a Stokes-like term for low Re.

Checks run in the page

• The terminal speed from root-finding matches the speed reached by integrating the equation of motion from rest, for six combinations of shape and fluid.
• A 2 mm steel bead in glycerol reproduces Stokes' law to better than 0.3 %.
• The sphere curve matches Schiller-Naumann and handbook values to within 12 % and shows the drag crisis.
• The energy theorem closes for a falling aluminium sphere (kinetic energy including added mass, work of drag, work of net weight).
• Streamlined, cylinder and disk coefficients fall inside the published ranges.

Honest simplifications

• Every correlation is an empirical fit for a body held steady in a smooth, steady stream, valid to about 10-15 %. Real falling disks, cylinders and cubes tumble and flutter; here they stay face-on and rigid in the stream.
• White's cylinder fit gives about 1.0 at Re = 10⁴, whereas tables show about 1.2; the finite-length factor for 3 d is a fit to Hoerner's values. The streamlined formula is for Re_L above about 10⁴; below that the low-Re term is a rough estimate.
• Tunnel wall blockage, turbulence level and surface roughness are ignored (the drag crisis moves with them). The fan reaches its set speed through a 0.6 s lag, and the stream is uniform across the tube.
• The blue tracers follow the potential flow around a sphere of the same frontal area, which has no wake and no separation. It is a picture of the flow ahead of and around the body, not a CFD result.
• Air is treated as incompressible (valid below about 100 m/s here).

What to try

• Press Hover for each shape and compare the fan settings: for equal weight a disk hovers in a far slower stream than a streamlined body.
• Make the foam sphere 20 cm wide and watch the terminal speed change as Cd jumps near Re = 3 x 10⁵.
• Switch the fluid to glycerol: Re drops below 1 and the drag becomes proportional to speed.