A train nearly fills a subway tunnel's cross-section. As it advances, it must push the air ahead of it somewhere — most of that air escapes backward through the thin annular gap between the train and the tunnel wall. This "tunnel piston effect" is modelled here with a simplified 1D incompressible flow (constant tunnel area A₀ ≈ 25 m², reference tunnel diameter ≈ 5.6 m):
β = A_train / A_tunnel (blockage ratio)
v_gap = v_train · β / (1 − β) (continuity, train frame)
ΔP_piston = ½ρ(v_gap² − v_train²) (Bernoulli, front → gap)
ΔP_friction = K · ½ρ·v_gap² (Darcy-Weisbach loss, K = fL/D_h)
Power = (ΔP_piston + ΔP_friction) · A_train · v_train
Re = ρ·v_gap·D_h / μ_air (μ_air = 1.81×10⁻⁵ Pa·s)
Raising the blockage ratio shrinks the gap the air must squeeze through, so v_gap — and the pressure it produces — grows nonlinearly (it diverges as β → 1, exactly the reason tunnels are bored wider than the trains that run through them). The friction loss slider adds the extra pressure drop from wall roughness and tunnel length, on top of the ideal Bernoulli term. Air density scales every pressure and power term directly, so a chilly, dense-air tunnel pushes harder than a warm one at the same speed.
- Colour of the flow field — particle speed, from calm blue to fast orange/white, brightest inside the gap.
- Pressure strip — local static pressure relative to ambient along the tunnel: a rise ahead of the train, a trough through the gap (Bernoulli), partial recovery behind.
- Regime — the gap Reynolds number classifies the bypass flow as laminar, transitional or turbulent (turbulent above ~4,000 is normal for real tunnels).