Civil, Structural & Geotechnical Engineering
Einstein-Brown Bedload Transport Formula: Stochastic Low-Shear & 40 * theta^3 Law
Calculate river and coastal bedload transport intensity Phi across stochastic incipient saltation (theta < 0.18) and sheet-flow high-shear transport (Phi = 40 * K_R * theta^3) using Hans Albert Einstein (1942) and Carl B. Brown (1950).
Replace hard zero-transport cutoffs with Hans Albert Einstein’s probabilistic turbulent lift law!
Unlike deterministic threshold formulas that predict strictly ZERO grain movement below theta_c = 0.047, Hans Albert Einstein (1942, 1950 — son of Albert Einstein and pioneer of stochastic sediment mechanics!) recognized that random turbulent pressure bursts occasionally lift grains even at very low mean shear stress! Synthesizing Einstein’s probabilistic step-length data in Rouse’s Engineering Hydraulics (1950), Carl B. Brown formulated the piecewise Einstein-Brown Bedload Function scaled by Rubey’s dimensionless fall-velocity factor F_1:
Across low-to-moderate Shields stress (theta = tau_0 / ((rho_s - rho_f) * g * D) < 0.18), probabilistic saltation follows Phi = 2.15 * F_1 * EXP(-0.391 / theta), whereas under vigorous flood / coastal sheet-flow shear (theta >= 0.18), bedload transitions smoothly into the celebrated Cubic Shear-Stress Law Phi = 40 * F_1 * (theta^3) (producing volumetric unit bedload q_b = Phi * SQRT((s - 1) * g * (D^3)), so q_b scales with shear velocity to the 6th power u_*^6)!
| Shields Mobility Parameter theta (with Rubey Factor F_1 = 0.75) | Active Einstein-Brown Regime | Dimensionless Intensity Phi |
|---|---|---|
| Weak Baseflow (theta = 0.05 < 0.18) | Stochastic Burst Regime: 2.15 * F_1 * EXP(-0.391 / theta) | 0.00065 (Trace Creep) |
| Moderate Flood (theta = 0.15 < 0.18) | Stochastic Saltation: 2.15 * F_1 * EXP(-0.391 / theta) | 0.1189 |
| High-Shear Sheet Flow (theta = 0.30 >= 0.18) | Cubic Regime: 40 * F_1 * theta^3 | 0.8100 |
Piecewise Einstein-Brown (1950) Dimensionless Bedload Intensity Phi
Evaluates 2.15 * F_1 * EXP(-0.391 / theta) for 0 < theta < 0.18 and 40 * F_1 * (theta^3) for theta >= 0.18 across Shields parameter theta (B2) and Rubey grain settling factor F_1 (B3 in (0, 0.85]).
=IF(OR(B2 <= 0, B3 <= 0, B3 > 0.85), "Requires Shields theta > 0 and Rubey Factor F_1 in (0, 0.85]", ROUND(IF(B2 < 0.18, 2.15 * B3 * EXP(-0.391 / B2), 40 * B3 * (B2 ^ 3)), 5))Volumetric Unit Bedload Rate q_b (m^2/s) = Phi * SQRT((s - 1) * g * D^3)
Converts Einstein-Brown Phi into solid volumetric bedload discharge per meter width across grain diameter D (B4 m) and specific gravity s > 1 (B5).
=IF(OR(B2 <= 0, B3 <= 0, B3 > 0.85, B4 <= 0, B5 <= 1), "Invalid Einstein-Brown Inputs", ROUND(IF(B2 < 0.18, 2.15 * B3 * EXP(-0.391 / B2), 40 * B3 * (B2 ^ 3)) * SQRT((B5 - 1) * 9.80665 * (B4 ^ 3)), 6))Try the example
Use the original values or change them to test your understanding.