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Civil, Structural & Geotechnical Engineering

Rubey (1933) Sediment Fall Velocity Formula: HEC-RAS Dimensionless Factor F_1 & Settling

Calculate river sand and gravel terminal settling velocity w_s = F_1 * SQRT((s - 1) * g * D) using USGS geologist William W. Rubey’s (1933) analytical viscous-plus-impact quadratic drag formula built into USACE HEC-RAS.

Understand the classic 1933 USGS settling law still powering HEC-RAS sediment routing!

More than 70 years before modern empirical fits, USGS geologist William Walden Rubey (1933, American Journal of Science, "Settling Velocities of Gravel, Sand, and Silt Particles") summed laminar Stokes viscous skin resistance (3 * pi * mu * D * w_s) and turbulent fluid-momentum impact resistance ((pi / 4) * rho_f * (D^2) * (w_s^2)) against submerged particle weight ((pi / 6) * (rho_s - rho_f) * g * (D^3))! Solving that exact quadratic for terminal fall velocity w_s yields Rubey’s famous Dimensionless Settling Factor F_1:

F_1 = SQRT((2 / 3) + (36 * (nu^2)) / ((s - 1) * g * (D^3))) - SQRT((36 * (nu^2)) / ((s - 1) * g * (D^3))), and w_s = F_1 * SQRT((s - 1) * g * D)! Notice the two asymptotic limits: for fine silt (D -> 0), binomial expansion recovers exact Stokes’ Law, whereas for coarse gravel (D >= 2 mm, where viscous term -> 0), F_1 -> SQRT(2 / 3) = 0.8165 (reduced to ~0.79 for natural angular gravel)!

Sediment Particle Size D (s = 2.65, nu = 1.00e-6 m^2/s)Rubey Dimensionless Factor F_1Terminal Fall Velocity w_s (m/s)
Very Fine Sand (D = 0.00010 m = 0.10 mm)0.20050.00806 m/s (0.81 cm/s)
Medium Sand (D = 0.00050 m = 0.50 mm)0.69390.06242 m/s (6.24 cm/s)
Fine Pea Gravel (D = 0.00400 m = 4.00 mm)0.8082 (~= SQRT(2/3))0.20623 m/s (20.62 cm/s)

Rubey (1933) Dimensionless Fall-Velocity Factor F_1 (0 < F_1 <= SQRT(2/3))

Evaluates SQRT(2/3 + K) - SQRT(K) where K = (36 * nu^2) / ((s - 1) * 9.80665 * D^3) for grain diameter D (B2 m), sediment specific gravity s > 1 (B3), and water kinematic viscosity nu (B4 m^2/s).

=IF(OR(B2 <= 0, B3 <= 1, B4 <= 0), "Requires Grain Diameter D > 0, Specific Gravity s > 1, and Viscosity nu > 0", ROUND(SQRT((2 / 3) + (36 * (B4 ^ 2)) / ((B3 - 1) * 9.80665 * (B2 ^ 3))) - SQRT((36 * (B4 ^ 2)) / ((B3 - 1) * 9.80665 * (B2 ^ 3))), 4))

Rubey Terminal Sediment Settling Velocity w_s (m/s) = F_1 * SQRT((s - 1) * g * D)

Multiplies unrounded F_1 by SQRT((s - 1) * 9.80665 * D) to return terminal fall velocity in m/s.

=IF(OR(B2 <= 0, B3 <= 1, B4 <= 0), "Requires Grain Diameter D > 0, Specific Gravity s > 1, and Viscosity nu > 0", ROUND((SQRT((2 / 3) + (36 * (B4 ^ 2)) / ((B3 - 1) * 9.80665 * (B2 ^ 3))) - SQRT((36 * (B4 ^ 2)) / ((B3 - 1) * 9.80665 * (B2 ^ 3)))) * SQRT((B3 - 1) * 9.80665 * B2), 5))

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