Application of SPM for estimating the velocity index in geometric channels
Extended Abstract

Introduction
Accurate flow measurement in open channels is essential for applications such as flood risk assessment, hydropower generation, water resource management, and hydraulic modeling. Traditional in-situ discharge measurements using electromagnetic or mechanical instruments are labor-intensive, costly, and potentially hazardous under high-flow conditions. As a result, surface velocity-based methods, including radar and image velocimetry, have become increasingly popular for non-contact discharge estimation. A critical step in these methods is converting surface velocity (us) to depth-averaged velocity (Ud), typically using the velocity coefficient α. Although many studies report average α values ranging from 0.62 to 0.92 depending on bed roughness, flow depth, and channel geometry, a comprehensive understanding of α variation across geometric channels remains limited. Most existing approaches consider α as a constant, which may introduce significant error, especially in non-uniform or shallow flows. This study aims to address this gap by applying the Single Point Method (SPM), introduced by Maghrebi (2003), to quantify the variation of α across the free surface of geometric channels. Additionally, the study examines the relationship between α variations and isovel contours, and compares SPM results with previous empirical and experimental findings.

Methodology
The core concept of the velocity index α is rooted in establishing a functional relationship between the depth-averaged velocity (Ud) and a single-point measurement, typically the surface velocity (us). In this study, a power-law vertical velocity distribution is assumed to represent flow behavior, consistent with experimental observations and theoretical models. The Single Point Method (SPM) provides a semi-analytical approach to estimate point velocities within the channel cross-section based on boundary effects. Derived from analogies with the Biot–Savart law in electromagnetism, the method computes local velocities using an integral formulation that considers the geometry of the wetted perimeter, the relative roughness, and shear velocity parameters. A constant exponent m=7 is used in the velocity profile to model turbulent open-channel flow over smooth boundaries. To apply the SPM, each channel section (rectangular, trapezoidal, or triangular) is discretized into vertical strips, and point velocities at the free surface are computed at discrete locations. Under each surface point, a vertical band is analyzed to compute the depth-averaged velocity using numerical integration of the power-law profile. The local α at each surface point is then calculated as the ratio Ud/us. To obtain the overall α for the entire section, a discharge-weighted averaging approach is used. This methodology was implemented for multiple geometric configurations (varying B/H and side slope m) and validated by comparison with published data.

Results and Discussion
The SPM was applied to three types of channel cross-sections: rectangular (n=0), trapezoidal, and triangular (B=0), over a wide range of width-to-depth ratios (B/H) and side slopes (n). Isovel contours (lines of constant dimensionless velocity λ) were plotted to visualize the velocity distribution and examine the location of maximum flow velocities. In rectangular channels, increasing the B/H ratio shifts the location of maximum velocity toward the free surface, while in narrow sections (low B/H), it moves deeper into the cross-section. This pattern directly affects α: in wide channels, local α values near the centerline decrease, while in narrow channels, they increase. For trapezoidal channels, similar trends were observed. However, additional variations in α appeared near the sloped walls due to decreasing effective strip height and declining surface velocity, us. Triangular sections exhibited a strong concentration of α around the central region, with lower values at the deepest point. Average α values obtained for each cross-section type are: 0.73 to 1.04 for rectangular section (increasing in narrower channels), 0.71 to 0.88 for trapezoidal section (depending on B/H and n), and 0.78 to 0.88 for triangular section (decreasing with side slope, n). The validation of discharge results using the cross-sectional α-coefficient against the experimental discharge and the measured surface velocity indicates a relative error below 5% and demonstrates a good agreement of the results. Furthermore, these results were consistent with previous studies (e.g., Fujita 2018; Welber et al. 2016), where α values in artificial channels typically ranged from 0.8 to 0.9. The maximum relative error between SPM-based α and literature values was under 12.7%. Notably, the study confirms that α varies significantly across the free surface and is sensitive to channel geometry, highlighting the limitation of assuming a constant α. Two empirical equations were also proposed to estimate α based on geometric parameters with a maximum error of 2.18%.

Conclusion
This study demonstrated that the Single Point Method (SPM) can effectively model surface and depth-averaged velocities in geometric channels and accurately estimate both local and global α coefficients. The results emphasize that α is not a constant and exhibits systematic variation across the free surface depending on the channel geometry. The proposed method offers a cost-effective, non-contact alternative for discharge estimation and improves upon existing empirical assumptions. The findings validate the applicability of SPM for hydraulic analysis in artificial channels and support its extension to real-world flow conditions with complex geometries.