Calculation of Multiple Critical Depths in Open Channels Using an Adaptive Cubic Polynomials Algorithm
Abstract
:1. Introduction
2. The Proposed Approach
3. Adaptive Cubic Polynomials Algorithm (ACPA)
 ✓
 Check 1 (convex concave): check whether there is an inflection point inside the segment or not. In other words, if the curve turns from convex to concave or vice versa. In that case, the segment must be subdivided. The second derivative at inflection points is zero. Therefore, it is adequate to calculate H’’ at both ends of the segment and check the signs. If they differ, then at some point inside the segment H’’ becomes zero and thus the segment must be subdivided.
 ✓
 Check 2 (curvature): check whether the slope of the curve changes significantly or not. In that case the segment must be subdivided. H’ is calculated at both ends yielding ${H}_{A}^{\prime}$ and ${H}_{B}^{\prime}$. Then, the following three conditions are being checked:
 Condition 1: $\left{H}_{A}^{\prime}{H}_{B}^{\prime}\right>\delta $, where δ in this study was taken 0.5^{*}
 Condition 2: ${H}_{A}^{\prime}\cdot {H}_{B}^{\prime}<0$
 Condition 3: $\frac{MIN\left(\left{H}_{A}^{\prime}\right,\left{H}_{B}^{\prime}\right\right)}{MAX\left(\left{H}_{A}^{\prime}\right,\left{H}_{B}^{\prime}\right\right)}<rati{o}_{min}$, where ratio_{min} in this study was taken 0.4^{*}
If condition 1 is true and either condition 2 or 3 is true, the segment must be subdivided. The values used for δ and ratio_{min} were determined with experiments. Several combinations were tried and the best results occurred for these values.  ✓
 Check 3 (fit): check whether the curve is fitted well or not. In the latter case it must be subdivided. H can be easily calculated at three additional intermediate elevations y_{E}, y_{F}, y_{G} (Equations (20)–(22))$${y}_{E}={y}_{A}+\frac{{y}_{B}{y}_{A}}{6}$$$${y}_{F}={y}_{A}+\frac{{y}_{B}{y}_{A}}{2}$$$${y}_{G}={y}_{A}+5\cdot \frac{{y}_{B}{y}_{A}}{6}$$
4. Tests and Comparison
4.1. Validation of the Hydraulic Parameters’ Calculation
4.2. Determining the Critical Depths with ACPA vs. HECRAS
5. Discussion
6. Conclusions
Author Contributions
Funding
Conflicts of Interest
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Parameter  Symbol  Max. Dif.  Max. Dif. (%)  Avg. Dif.  Avg. Dif. (%) 

Total crosssectional area  A  0.08  0.0005  0.02  0.001 
Kinetic energy correction factor  a  0.034  0.626  0.00  0.002 
Specific energy  H  38.67  0.057  0.18  0.004 
Section Index  Case  Difference 

7  4  0.555 
9  5  0.4 
27  5  0.171 
39  4  0.243 
40  3  0.98 
Section Index  Case  Difference 

1  3  1.016 
5  3  0.367 
6  3  0.175 
9  3  1.366 
12  4  0.86 
14  3  1.09 
15  5  1.198 
41  3  0.101 
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Petikas, I.; Keramaris, E.; Kanakoudis, V. Calculation of Multiple Critical Depths in Open Channels Using an Adaptive Cubic Polynomials Algorithm. Water 2020, 12, 799. https://doi.org/10.3390/w12030799
Petikas I, Keramaris E, Kanakoudis V. Calculation of Multiple Critical Depths in Open Channels Using an Adaptive Cubic Polynomials Algorithm. Water. 2020; 12(3):799. https://doi.org/10.3390/w12030799
Chicago/Turabian StylePetikas, Ioannis, Evangelos Keramaris, and Vasilis Kanakoudis. 2020. "Calculation of Multiple Critical Depths in Open Channels Using an Adaptive Cubic Polynomials Algorithm" Water 12, no. 3: 799. https://doi.org/10.3390/w12030799