Civil, Structural & Geotechnical Engineering
Ferguson & Church (2004) Universal Sediment Settling Velocity Formula: Silt to Cobbles
Calculate terminal fall velocity w_s (m/s) of natural river/marine grains across ALL Reynolds numbers — bridging Stokes viscous silt (w ~ D^2) and turbulent gravel wake drag (w ~ SQRT(D)) without iteration — via R. I. Ferguson and M. Church (2004).
Replace iterative drag-curve lookups with Ferguson & Church’s single explicit equation!
Stokes’ creeping-flow settling law w = R * g * (D^2) / (18 * nu) fails catastrophically above grain diameter D ~= 0.1 mm (overpredicting 10 mm gravel fall speed by > 250x!), whereas turbulent Newton impact drag w = SQRT(4 * R * g * D / (3 * C_2)) fails in fine sand and silt! Combining viscous linear drag C_1 * nu / D and turbulent form drag SQRT(0.75 * C_2 * R * g * D) in quadratic drag balance, Robert I. Ferguson and Michael Church (2004, Journal of Sedimentary Research) derived one explicit closed-form equation accurate from micron silt to cobbles:
w_s = (R * g * (D^2)) / (C_1 * nu + SQRT(0.75 * C_2 * R * g * (D^3)))! Here R = (rho_s - rho_f) / rho_f (~= 1.65 for quartz in water), g = 9.80665 m/s^2, nu is kinematic viscosity (~= 1.00e-6 m^2/s at 20°C), and shape constants are (C_1 = 18, C_2 = 0.4) for Smooth Spheres vs (C_1 = 18, C_2 = 1.0) for Natural Sieved Sand/Gravel Grains (Corey Shape Factor ~0.7)!
| Quartz Grain Diameter D in 20°C Water (R = 1.65, nu = 1e-6 m^2/s) | Natural Grain Fall Velocity (C_1=18, C_2=1.0) | Smooth Sphere Fall Velocity (C_1=18, C_2=0.4) | Hydrodynamic Regime |
|---|---|---|---|
| Fine Silt (D = 0.05 mm = 0.00005 m) | 0.00222 m/s (2.22 mm/s) | 0.00224 m/s | Stokes Viscous (Re_p << 1) |
| Medium Sand (D = 0.50 mm = 0.0005 m) | 0.07103 m/s (7.10 cm/s) | 0.0888 m/s | Transitional Drag |
| Coarse Gravel (D = 10.0 mm = 0.010 m) | 0.46210 m/s (46.2 cm/s) | 0.7289 m/s | Turbulent Wake (w_s ~ D^0.5) |
Ferguson & Church (2004) Terminal Grain Settling Velocity w_s (m/s)
Evaluates (R * 9.80665 * D^2) / (C_1 * nu + SQRT(0.75 * C_2 * R * 9.80665 * D^3)) for grain diameter D_mm in B2 (converted via B2 / 1000 to meters), submerged SG R (B3), kinematic viscosity nu (B4 m^2/s), viscous C_1 (B5), and wake C_2 (B6).
=IF(OR(B2 <= 0, B3 <= 0, B4 <= 0, B5 <= 0, B6 <= 0), "Invalid Grain Diameter, Submerged R, Viscosity nu, or Drag Constants", ROUND((B3 * 9.80665 * ((B2 / 1000) ^ 2)) / (B5 * B4 + SQRT(0.75 * B6 * B3 * 9.80665 * ((B2 / 1000) ^ 3))), 5))Particle Settling Reynolds Number Re_p = w_s * D / nu
Computes dimensionless particle Reynolds number to distinguish laminar Stokes (< 1), transitional (1–1,000), and turbulent (> 1,000) settling.
=IF(OR(B2 <= 0, B3 <= 0, B4 <= 0, B5 <= 0, B6 <= 0), "Invalid Settling Inputs", ROUND((((B3 * 9.80665 * ((B2 / 1000) ^ 2)) / (B5 * B4 + SQRT(0.75 * B6 * B3 * 9.80665 * ((B2 / 1000) ^ 3)))) * (B2 / 1000)) / B4, 2))Try the example
Use the original values or change them to test your understanding.