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Comparison of Different Extremal Theories in Calculation of Stable Channel Shape
CHEN Yi-ming, YU Xiao-long, ZHOU Cai-jin, LUAN Hua-long, QU Geng, LING Jin-ping, LI Bang-hui, DAI Wen-hong
Journal of Changjiang River Scientific Research Institute ›› 2026, Vol. 43 ›› Issue (8) : 112-118.
PDF(6214 KB)
PDF(6214 KB)
Comparison of Different Extremal Theories in Calculation of Stable Channel Shape
[Objective] Extremal theory is an important approach for predicting the hydraulic geometry of stable channels. However, multiple theories coexist, and their applicability and accuracy under different bed-material conditions (sand-bed and gravel-bed rivers) remain unclear, posing challenges for model selection in engineering practice. To address this issue, this study selects the maximum flow efficiency (MFE), minimum Froude number (MFN), and maximum entropy and minimum energy dissipation rate (ME & MEDR) theories as the research objects, aiming to: (1) clarify the computational accuracy and applicability limits of different extremal theories; (2) reveal the intrinsic relationship between theory applicability and riverbed type; and (3) provide guidance for future researchers in predicting the hydraulic geometry of stable channels. [Methods] A total of 351 datasets from both sand-bed and gravel-bed rivers were used to evaluate the MFE, MFN, and ME & MEDR theories. Model performance was assessed using the mean relative error, geometric mean deviation, and correlation coefficient. [Results] (1) Among the three extremal theories, the MFE theory considered the largest number of parameters and involved the most complex calculations, whereas the MFN theory established a stable channel-width equation, requiring only discharge and median sediment size to predict the hydraulic geometry of stable channels.(2) The validation results showed that, for sandy stable channels, both the MFE and ME & MEDR theories underestimated stable channel width when the channel width was less than 2 m and overestimated stable water depth when the water depth was less than 0.1 m. For gravel-bed stable channels, the MFE theory underestimated stable channel width and overestimated stable water depth; the MFN theory underestimated stable water depth; and the ME & MEDR theory overestimated stable channel width while underestimating stable water depth.(3) Error analysis showed that the MFE theory generally exhibited larger errors than the other two theories. The MFN theory achieved the highest accuracy in predicting stable channel width, whereas the ME & MEDR theory achieved the highest accuracy in predicting stable water depth. [Conclusion] This study clarifies the optimal applicability of different extremal theories and provides a quantitative basis and recommended approaches for stable channel design under specific riverbed conditions. For predicting the hydraulic geometry of stable sandy channels, all three extremal theories provide satisfactory accuracy. For gravel channels, however, the MFN and ME & MEDR theories are recommended.
stable channel / extremal theories / formula verification / error analysis / channel morphology
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The theory of minimum rate of energy dissipation states that a system is in an equilibrium condition when its rate of energy dissipation is at its minimum value. This minimum value depends on the constraints applied to the system. When a system is not at equilibrium, it will adjust in such a manner that the rate of energy dissipation can be reduced until it reaches the minimum and regains equilibrium. A river system constantly adjusts itself in response to varying constraints in such a manner that the rate of energy dissipation approaches a minimum value and thus moves toward an equilibrium. It is shown that the values of the exponents of the hydraulic geometry relationships proposed by Leopold and Maddock for rivers can be obtained from the application of the theory of minimum rate of energy dissipation in conjunction with the Manning‐Strickler equation and the dimensionless unit stream power equation for sediment transport proposed by Yang. Theoretical analysis is limited to channels which are approximately rectangular in shape. The width and depth exponents thus derived agree very well with those measured in laboratory experiments by Barr et al. Although the theoretical width and depth exponents are within the range of variations of measured data from river gaging stations, the at‐station width adjustment of natural rivers may also depend on constraints other than water discharge and sediment load.
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Regime channels are important for stable canal design and to determine river response to environmental changes, e.g., due to the construction of a dam, land use change, and climate shifts. A plethora of methods is available describing the hydraulic geometry of alluvial rivers in the regime. However, comparison of these methods using the same set of data seems lacking. In this study, we evaluate and compare four different extremal hypothesis-based regime methods, namely minimization of Froude number (MFN), maximum entropy and minimum energy dissipation rate (ME and MEDR), maximum flow efficiency (MFE), and Millar’s method, by dividing regime channel data into sand and gravel beds. The results show that for sand bed channels MFN gives a very high accuracy of prediction for regime channel width and depth. For gravel bed channels we find that MFN and ‘ME and MEDR’ give a very high accuracy of prediction for width and depth. Therefore the notion that extremal hypotheses which do not contain bank stability criteria are inappropriate for use is shown false as both MFN and ‘ME and MEDR’ lack bank stability criteria. Also, we find that bank vegetation has significant influence in the prediction of hydraulic geometry by MFN and ‘ME and MEDR’.
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The hypotheses of minimum energy dissipation, minimum stream power, and minimum unit stream power are summarized and compared. Their derivation from analogies with laminar flow and linear thermodynamics is criticized on the grounds that these situations differ fundamentally from river flows, which are usually highly turbulent and strongly nonlinear. The authors' empirical hypothesis of maximum friction factor seems preferable to the minimization hypotheses because it is compatible with the known behavior of turbulent flows and nonlinear processes, it is applicable with a wider range of independent variables, and it is more in keeping with trends shown by experimental data under all constraints. The minimization hypotheses seem likely to give incorrect predictions when flow rate and depth are independent variables. The empirical success of the minimization hypotheses is confined to situations in which they predict similar behavior to the maximum friction factor hypothesis; they may be considered as special cases of this more general hypothesis.
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