Estimation of the Mean-to-Surface-Velocity Ratio in Shallow Streams with Rough Beds

Estimating in a stream’s cross-section the depth-averaged velocity, V, from the free-surface velocity, vsurf, is an efficient, non-invasive hydrometric method. The ratio fv = V/vsurf is typically assumed constant at fv = 0.86 in field applications, despite observations to the contrary. Guidance is, therefore, needed in estimating actual fv-ratios when velocity profile data are absent. This work provides field-verified guidance based on the hydromechanics of the logarithmic velocity law, which shows that fv depends on the scaled resistance measure ‘friction length/depth’, yo/h, with the yo(k) function of the equivalent sand grain roughness, k. The mean-to-surface-velocity ratio in rough-bed streams is estimated from the bed roughness and stream morphology by modifying Nikuradze’s equation, yo = k/30, to yo = ck, with c(h/k) ≥ 1/30, and k ≈ D84—data fit: c ≈ 8.61(h/k)−1.821, ~5 ≤ h/k < ~30. Field-verification of the ratio’s modified hydromechanics, fv = fh/yo, with yo(h/k) evaluated from bed roughness estimated by inspection or sieve analysis shows this ratio holding within ~|10|% error for shallow streamflow over a coarse bed of gravels and rocks, giving submergences of ~5 ≤ h/D84 ≤ ~30; yo = k/30 suits large streams with smooth beds (h/k ≥ ~30, fv ≥ ~0.86). Variable roughness-estimated fv-ratios appear to be more reliable than the fixed default, fv(h/yo ≈ 1000) = 0.86. This flow-gauging concept is based on observable physical characteristics of a monitoring cross-section and facilitates the rating of hard-to-access streams draining small basins in ragged upland terrain.