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ABSTRACT
Studying open channel flow and sediment transport in narrow flumes under non-uniform flow conditions, both sidewall and non-uniformity corrections are required for bed-shear stress. This research first reviews conventional predictive methods for bed-shear stress, including the flow-depth method, the hydraulic radius method and Einstein's sidewall correction. It then presents a novel procedure for sidewall and non-uniformity corrections based on a recent cross-sectional velocity distribution model. These methods are compared with data from the log-law under uniform and non-uniform, sub- and supercritical flow conditions, indicating that (i) the flow-depth and the hydraulic radius methods specify the upper and lower bounds for bed-shear stress; (ii) although Einstein's procedure causes a paradox for smooth flumes, it agrees with data from rough beds; and (iii) the proposed is better than Einstein's for subcritical flow, but the latter has advantage for supercritical flow. As an application, sediment inception under non-uniform flow conditions is also discussed.
Acknowledgements
This research was supported by the US FHWA Hydraulics R&D Program (Contract No. DTFH61-11D-00010) through the Genex System to the University of Nebraska-Lincoln.
Notation
| A | = | = cross-sectional area (m2) |
| Ab, Aw | = | = cross-sectional areas associated with bed and sidewalls, respectively (m2) |
| b | = | = flume width (m) |
| D50 | = | = median diameter of sediment (m) |
| dh/dx | = | = non-uniformity or pressure-gradient (–) |
| F | = | = Froude number based on h (–) |
| F | = | = functional symbol (–) |
| f | = | = friction factor (-) |
| fw | = | = sidewall friction factor (–) |
| g | = | = gravity acceleration (m s−2) |
| h(x) | = | = flow depth (m) |
| hc | = | = critical flow depth (m) |
| ho | = | = flow depth at x=0 (m) |
| ks | = | = equivalent sand-grain roughness (m) |
| M, N | = | = dummy variables (–) |
| = | = Manning's coefficients for bed and sidewalls, respectively (s m−1/3) | |
| P | = | = wetted perimeter (m) |
| Q | = | = discharge (m3s−1) |
| R | = | = Reynolds number based on 4R (–) |
| Rw | = | = sidewall Reynolds number based on 4Rw (–) |
| R | = | = hydraulic radius (m) |
| Rb, Rw | = | = hydraulic radii for bed and sidewalls, respectively (m) |
| r2 | = | = coefficient of determination (–) |
| Sf | = | = friction slope (–) |
| So | = | = bottom slope (–) |
| T | = | = temperature (°C) |
| u(y, z) | = | = cross-sectional velocity distribution (m s−1) |
| = | = shear velocity based on R (m s−1) | |
| = | = bed shear velocity based on h (m s−1) | |
| = | = average bed-shear velocity (m s−1) | |
| = | = centerline bed-shear velocity (m s−1) | |
| = | = sidewall shear velocity distribution (m s−1) | |
| = | = average sidewall shear velocity (m s−1) | |
| V | = | = cross-sectional average velocity (m s−1) |
| x, y, z | = | = coordinates (m) |
| z0 | = | = zero-velocity position from bed (m) |
| α | = | = parameter reflecting sidewall effect (–) |
| β | = | = momentum correction factor (–) |
| γ | = | = specific water weight (N m−3) |
| δ | = | = velocity-dip position from bottom (m) |
| κ | = | = von Karman constant (–) |
| λ | = | = ratio of |
| ν | = | = kinematic water viscosity (m2s−1) |
| ρ | = | = water density (kg m−3) |
| τb(y) | = | = boundary shear distribution along bed |
| τw(z) | = | = boundary shear distribution along sidewalls (Pa) |
| = | = average values of τb and τw, respectively (Pa) | |
| τbc | = | = centerline bed-shear stress (Pa) |
| τc | = | = critical bed-shear stress for sediment initiation (Pa) |
| τ0 | = | = overall boundary shear stress based on R (Pa) |
| φ(y) | = | = velocity distribution function in y-direction (–) |
| ψ(z) | = | = velocity distribution function in z-direction (–) |
| = | = average values of φ(y) and ψ(z), respectively (–) |