Explicit solutions of the Manning equation for partially filled circular pipes
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Bibliographic record
Abstract
The application of the Manning equation to partly filled circular channels is considered. The four types of problem that require iterative calculations are elucidated, and explicit solutions proposed in the literature are reviewed. New and improved equations are presented for one type of problem. Explicit analytical equations are developed for the two types of problem for which no explicit solutions could be found in the literature. For each type of problem, two cases are considered: (i) constant Manning roughness coefficient, and (ii) variable Manning roughness coefficient that depends on the depth of flow. Separate equations are presented for each case. The accuracy of each equation is demonstrated by calculating and reporting its maximum deviation from the exact solution within its range of applicability. In addition to obviating the need for iterative calculations, these equations facilitate the calculation of both solutions when the problem at hand has two distinct solutions (two possible flow depths). The proposed equations are accurate enough to be used in computer calculations and sufficiently simple to be used with a hand-held calculator. Key words: explicit solution, hydraulics, Manning equation, free-surface flow, roughness coefficient, sewer design.
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Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from it