Estimation of the Mean-to-Surface-Velocity Ratio in Shallow Streams with Rough Beds
Journal
Water
ISSN
2073-4441
Date Issued
April 21, 2026
Abstract
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 = f(ℎ/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.
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.
Funding
This research was partially funded by HYDRONET–Open Hydrosystem Information Network (OpenHi.net), as a part of the Hellenic Integrated Marine and Inland Water Observing, Forecasting and Offshore Technology System, HIMIOFoTS (MIS5002739), which was implemented under the Operational Programme ‘Competitiveness, Entrepreneurship and Innovation’ (NSRF 2014–2020), co-financed by Greece and the European Union.
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