Civil, Structural & Geotechnical Engineering
Van Rijn (1984) Sediment Transport Formula: Particle Parameter D_* & Transport Stage T
Calculate Leo C. van Rijn’s (1984, Delft Hydraulics) Dimensionless Particle Parameter D_* = d_50 * (((s - 1) * g) / nu^2)^(1/3), Excess Bed-Shear Transport Stage T, and Sand Bedload Rate q_b (m^2/s).
Eliminate iterative Shields-diagram lookups with Van Rijn’s explicit D_* and T parameters!
In his landmark three-part 1984 ASCE Journal of Hydraulic Engineering series from Delft Hydraulics, Leo C. van Rijn replaced implicit boundary Reynolds numbers on the Shields curve with an explicit, velocity-independent Dimensionless Particle Parameter D_* = d_50 * ((((s - 1) * g) / (nu^2))^(1/3))! For quartz sand (s = 2.65) in 20°C water (nu = 1.0e-6 m^2/s), 1 mm sand corresponds directly to D_* ~= 25.29 — letting engineers calculate critical Shields threshold theta_cr directly without iteration (e.g., theta_cr = 0.04 * (D_*^-0.10) for 10 < D_* <= 20, and 0.013 * (D_*^0.29) for 20 < D_* <= 150)!
Next, defining Excess Bed-Shear Transport-Stage Parameter T = MAX(0, (tau_b_prime - tau_cr) / tau_cr) (where tau_b_prime is grain-related effective bed shear stress in Pa and tau_cr is critical incipient shear stress in Pa), Van Rijn’s (1984a) Bedload Transport Equation predicts Volumetric Sand Bedload Rate (for 0.2 <= d_50 <= 2.0 mm, T < 3): q_b (m^2/s) = 0.053 * SQRT((s - 1) * g * (d_50^3)) * ((T^2.1) / (D_*^0.3))!
| Van Rijn (1984) Sand-Bed Quantity (d_50 = 0.0005 m, s = 2.65, nu = 1e-6 m^2/s) | Formula | Evaluated Value |
|---|---|---|
| Dimensionless Particle Diameter D_* | d_50 * (((s - 1) * g) / nu^2)^(1/3) | 12.647 |
| Transport-Stage Parameter T (tau_b_prime = 1.05 Pa, tau_cr = 0.25 Pa) | (tau_b_prime - tau_cr) / tau_cr | 3.200 |
| Volumetric Sand Bedload Rate q_b | 0.053 * SQRT((s-1)*g*d_50^3) * T^2.1 / D_*^0.3 | 0.000013 m^2/s (0.034 kg/m/s) |
Van Rijn Dimensionless Particle Diameter D_*
Evaluates d_50 * (((s - 1) * 9.80665 / nu^2)^(1/3)) across median sand size d_50 (B2 m), specific gravity s > 1 (B3), and kinematic viscosity nu (B4 m^2/s).
=IF(OR(B2 <= 0, B3 <= 1, B4 <= 0), "Requires d_50 > 0, Specific Gravity s > 1, and Viscosity nu > 0", ROUND(B2 * ((((B3 - 1) * 9.80665) / (B4 ^ 2)) ^ (1 / 3)), 3))Van Rijn (1984a) Volumetric Sand Bedload Rate q_b (m^2/s)
Computes T = MAX(0, (tau_b_prime - tau_cr) / tau_cr) from effective grain shear stress B5 (Pa) and critical shear stress B6 (Pa), returning 0.053 * SQRT((s - 1) * g * d_50^3) * (T^2.1) / (D_*^0.3).
=IF(OR(B2 <= 0, B3 <= 1, B4 <= 0, B5 < 0, B6 <= 0), "Invalid Van Rijn Inputs", ROUND(0.053 * SQRT((B3 - 1) * 9.80665 * (B2 ^ 3)) * ((MAX(0, (B5 - B6) / B6) ^ 2.1) / ((B2 * ((((B3 - 1) * 9.80665) / (B4 ^ 2)) ^ (1 / 3))) ^ 0.3)), 6))Try the example
Use the original values or change them to test your understanding.