Journal of Hydraulics

Journal of Hydraulics

Investigation of the Entropy Methods for Determining Velocity Distribution in Meandering Compound Channels

Document Type : Research Article

Authors
Vahideh Mortazvai Amiri 1
Kazem Esmaili 2
1 Water Science and Engineering Department Ferdowsi University of Mashhad
2 water and science engineering Dept. Ferdowsi university of mashhad
10.30482/jhyd.2025.500845.1727
Abstract
Introduction
Hydraulic modeling in natural channels encompasses sediment transport, pollutant dispersion, channel design, and velocity distribution (Singh, 2013). In meandering channels, the distribution of velocity and its interaction with momentum are crucial for understanding flow dynamics. Various methods exist to model velocity distribution, with the concept of entropy being a key approach. The entropy principle maximizes the entropy function, minimizing errors by considering probabilistic distributions linked to the problem's nature.
The application of entropy in open-channel flow began with Chiu (1987), who established a linear relationship between mean and maximum velocities. Ksia (1997) extended this for the Mississippi River, showing its reliability even in meanders. Recent studies have focused on Shannon entropy for velocity distribution, with findings indicating its effectiveness in both laboratory and natural channels (Moramarco & Termini, 2017; Gholami et al., 2019). While entropy methods have been widely used for velocity distribution, their application in meandering floodplain channels has received limited attention.
This study aims to evaluate the effectiveness of entropy-based methods in meandering rivers with complex three-dimensional flow patterns, addressing the challenge of accurately simulating velocity distribution in such channels.
Methodology
Experimental Setup
The experimental data were collected from the Hydraulic and Physical Modeling Laboratory at Ferdowsi University of Mashhad. A physical model of a meandering floodplain channel with a rigid bed was constructed, measuring 10 meters in length and 120 cm in width. Flow conditions were steady and uniform, with a constant discharge of 22.3 L/s. Flow depth in the main channel was 25 cm, while in the floodplain, it was 10 cm. The Froude number of the flow was 0.2.
Velocity Measurement
Velocity profiles were measured using a 3D ultrasonic velocimeter (Sontek2001), capturing flow velocities along the longitudinal, lateral, and vertical axes. Measurements were taken at multiple cross-sections along the channel, and the data were processed using Vectrino and WinADV software to determine velocity distributions.
Entropy Methods
Two entropy methods were used to model the velocity distribution:
• Shannon Entropy: Maximizes entropy subject to physical constraints like mass conservation, providing a statistical model of velocity distributions.
• Tsallis Entropy: A generalized entropy approach that is more suited for complex, non-extensive flow systems, such as meandering channels.
Error Analysis
The accuracy of the entropy models was evaluated using three error metrics: Root Mean Square Error (RMSE), Mean Absolute Error (MAE), and Mean Absolute Percentage Error (MAPE), which compared the observed and predicted velocity profiles.


Results and Discussion
Longitudinal Velocity Distribution
In meandering channels, maximum flow velocity is typically observed near the inner bend. As the flow enters the bend, secondary currents and increased pressure gradients in the outer bend led to a reduction in velocity. Conversely, the inner bend experiences an increase in velocity. After exiting the bend, the maximum velocity shifts toward the channel center and eventually to the outer bend downstream (Shiono & Moto, 1998).
At CS1, CS2, and CS3, maximum velocities occur at the deepest point in the channel, which is influenced by momentum transfer at the transition between straight and meandering sections. Secondary flow effects and the pressure gradient changes near the bends likely contribute to this velocity distribution, consistent with findings by Pan et al. (2019).
Velocity Distribution Using Entropy Methods
The velocity distributions were modeled using Shannon and Tsallis entropy methods. Parameter values ϕ and M were determined for each cross-section using the respective entropy equations. The calculated values of M for CS1, CS3, and CS5 were 4.77, 5.02, and 5.92, respectively, which are within the range for rivers in the U.S. (Lü & Singh, 2011). Figure 5 compares the velocity profiles predicted by Shannon and Tsallis entropy methods with measured velocities at the three cross-sections. The Shannon entropy method provided more accurate predictions, especially toward the center of the channel, with errors increasing near the channel walls due to the unaccounted effects of proximity to the bends.
Error Analysis
Error metrics, including RMSE, MAE, and MAPE, were used to evaluate the accuracy of both entropy methods. Shannon entropy achieved the lowest RMSE (0.01) at CS3, while the Tsallis method had larger errors, particularly at the bends. The minimum MAPE for Shannon entropy was 8.59% at CS3, compared to 26.81% for Tsallis entropy.

Conclusion
The results show that both methods perform well, with Shannon entropy providing better accuracy, especially at the channel center. Despite its limitations in accounting for the influence of channel walls, entropy methods, particularly Shannon entropy, offer a simple, cost-effective, and relatively accurate approach for determining velocity distribution in natural river channels.

Keywords Shannon entropy, Tsallis, velocity distribution and meandering floodplain.
Keywords
Shannon entropy
Tsallis
velocity distribution and meandering floodplain

Subjects

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  • Receive Date 19 January 2025
  • Revise Date 12 February 2025
  • Accept Date 28 February 2025