Science
Shields Parameter for Incipient River Sediment Motion & Bed Shear
Calculate river bed shear stress tau_0 = rho * g * R_h * S, dimensionless Shields Stress tau_* = (rho * R_h * S) / ((rho_s - rho) * d_50), and gravel bedload mobilization status.
Predict whether flood stage scours streambed gravel or leaves riprap stable!
In open-channel hydraulics and river geomorphology, flowing water exerts boundary Bed Shear Stress tau_0 = rho * g * R_h * S (Pa), where R_h is hydraulic radius (m) and S is channel energy slope (m/m). Dividing driving shear stress by the submerged buoyant weight scale (rho_s - rho) * g * d_50 of median grain diameter d_50 cancels gravity g cleanly—producing Albert Shields's dimensionless Shields Parameter: tau_* = (rho * R_h * S) / ((rho_s - rho) * d_50)!
For rough turbulent flow over coarse sand, gravel, and cobble beds, incipient bedload motion begins once tau_* exceeds the Critical Shields Number tau_*c (typically 0.045 to 0.047 for Meyer-Peter & Mueller gravel transport, or 0.030–0.060 depending on grain packing)! Comparing tau_* against tau_*c immediately determines whether a design flood scours bridge-pier riprap or keeps the channel bed stable!
| River Stage (Quartz rho_s = 2,650 kg/m^3, Gravel d_50 = 0.030 m, S = 0.002) | Hydraulic Radius R_h (m) | Dimensionless Shields tau_* | Bed Stability (tau_*c = 0.047) |
|---|---|---|---|
| Baseflow Stage | 0.60 m | 0.0242 (< 0.047) | Stable Armored Gravel Bed |
| Threshold Bankfull Flow | 1.163 m | 0.0470 (= 0.047) | Incipient Grain Motion |
| 10-Year Flood Stage | 1.80 m | 0.0727 (> 0.047) | Active Bedload Scour & Transport! |
Dimensionless Shields Stress tau_* = (rho * R_h * S) / ((rho_s - rho) * d_50)
Evaluates dimensionless Shields parameter from water density B2, sediment density B3, hydraulic radius B4, energy slope B5, and median grain size d_50 in B6.
=IF(OR(B3 <= B2, B6 <= 0), "Invalid Density or Grain Size", ROUND((B2 * B4 * B5) / ((B3 - B2) * B6), 4))Incipient Sediment Scour vs Stable Bed Check (tau_* > tau_*c)
Flags active bedload transport whenever Shields stress exceeds critical Shields parameter B7.
=IF(((B2 * B4 * B5) / ((B3 - B2) * B6)) > B7, "Bedload Mobilized (Active Scour)", "Stable Channel Bed")Try the example
Use the original values or change them to test your understanding.