ABSTRACT
Advances in sensors and drone technologies now permit remote collection of high temporal and spatial resolution data from rivers. Uses of such data can range from flow gauging (Le Coz et al. 2014) to spatial distributions of aquatic macrophytes (Biggs et al. 2018). Usual practice in remote gauging is to measure the fluctuating free-surface velocity https://www.w3.org/1998/Math/MathML"> u s https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003110958/a2ad1dd0-8968-490b-9794-8f1e87488e61/content/inline-mathch136_0250_1.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> or mean surface velocity https://www.w3.org/1998/Math/MathML"> u s ‾ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003110958/a2ad1dd0-8968-490b-9794-8f1e87488e61/content/inline-mathch136_0250_2.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> and convert this into a depth-averaged velocity https://www.w3.org/1998/Math/MathML"> U https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003110958/a2ad1dd0-8968-490b-9794-8f1e87488e61/content/inline-mathch136_0250_3.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> , using an‘ https://www.w3.org/1998/Math/MathML"> α https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003110958/a2ad1dd0-8968-490b-9794-8f1e87488e61/content/inline-mathch136_0250_4.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> ’ coefficient, where https://www.w3.org/1998/Math/MathML"> α = U / u s ‾ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003110958/a2ad1dd0-8968-490b-9794-8f1e87488e61/content/inline-mathch136_0250_5.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> . A standard https://www.w3.org/1998/Math/MathML"> α https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003110958/a2ad1dd0-8968-490b-9794-8f1e87488e61/content/inline-mathch136_0250_6.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> value assumed for field measurements is 0.85 (Rantz 1982, Johnson & Cowen 2017). However, appropriate https://www.w3.org/1998/Math/MathML"> α https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003110958/a2ad1dd0-8968-490b-9794-8f1e87488e61/content/inline-mathch136_0250_7.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> values depend on the shape of the streamwise velocity profile https://www.w3.org/1998/Math/MathML"> u ˉ z https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003110958/a2ad1dd0-8968-490b-9794-8f1e87488e61/content/inline-mathch136_0250_8.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> and, since there are significant variations in streamwise velocity profiles described in the literature, we should expect corresponding variations in the https://www.w3.org/1998/Math/MathML"> U / u s ‾ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003110958/a2ad1dd0-8968-490b-9794-8f1e87488e61/content/inline-mathch136_0250_9.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> ratio. The main culprits causing deviations from conventional velocity profiles are surface wind, streamwise coherent structures and secondary flows. To determine which https://www.w3.org/1998/Math/MathML"> U / u s ‾ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003110958/a2ad1dd0-8968-490b-9794-8f1e87488e61/content/inline-mathch136_0250_10.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> ratio is appropriate for a given site and flow, this paper investigates the effect of shear velocity, flow depth, boundary roughness and turbulence parameters. We develop a range of predictive equations to convert surface velocity to depth averaged velocity and test these with high resolution flume data. For logarithmic velocity profiles https://www.w3.org/1998/Math/MathML"> α = 1 − u ∗ / κ u s ‾ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003110958/a2ad1dd0-8968-490b-9794-8f1e87488e61/content/inline-mathch136_0250_11.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> . Assuming https://www.w3.org/1998/Math/MathML"> u ∗ = g H S 0.5 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003110958/a2ad1dd0-8968-490b-9794-8f1e87488e61/content/inline-mathch136_0250_12.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> this provides a practical method for remote sensing https://www.w3.org/1998/Math/MathML"> α https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003110958/a2ad1dd0-8968-490b-9794-8f1e87488e61/content/inline-mathch136_0250_13.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> at a range of flow conditions. For power law velocity profiles with exponent https://www.w3.org/1998/Math/MathML"> m https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003110958/a2ad1dd0-8968-490b-9794-8f1e87488e61/content/inline-mathch136_0250_14.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> , https://www.w3.org/1998/Math/MathML"> α = 1 / m + 1 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003110958/a2ad1dd0-8968-490b-9794-8f1e87488e61/content/inline-mathch136_0250_15.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> , with https://www.w3.org/1998/Math/MathML"> α = 0.857 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003110958/a2ad1dd0-8968-490b-9794-8f1e87488e61/content/inline-mathch136_0250_16.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> for Manning’s equation which is a specific case of a 1/6th power law. Measurements of turbulence at the water’s surface https://www.w3.org/1998/Math/MathML"> σ u s https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003110958/a2ad1dd0-8968-490b-9794-8f1e87488e61/content/inline-mathch136_0250_17.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> offer another approach with https://www.w3.org/1998/Math/MathML"> U ≈ u s ‾ − 3 σ u s https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003110958/a2ad1dd0-8968-490b-9794-8f1e87488e61/content/inline-mathch136_0250_18.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> and a simple “rule-of-thumb” that suggests https://www.w3.org/1998/Math/MathML"> U ≈ u s , m i n https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003110958/a2ad1dd0-8968-490b-9794-8f1e87488e61/content/inline-mathch136_0250_19.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> . Analytic considerations indicate that common https://www.w3.org/1998/Math/MathML"> U / u s ‾ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003110958/a2ad1dd0-8968-490b-9794-8f1e87488e61/content/inline-mathch136_0250_20.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> ratios may strongly overestimate depth-averaged velocity with permeable boundaries such as vegetation. Initial results are promising, however further testing of the equations with field data and refinement to account for variations in profile shape due to secondary currents or drag effects at the surface are required.
