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Chapter 7 - Establishing an empirical formula ( B⋅ h m ). The ratios e s ⁄ B , e s ⁄ R c and e 2 s ⁄ B⋅ R c shows correlations of R 2 = 0.677 , 0.707 and 0.728 compared to the maximum scour depth. With this modification, equation 7.43 can be written as: h ---- s B e s 2 R c (7.44) Unfortunately it is impossible to get satisfying predictions with this equations or slightly modified ones, namely by testing other ratios for the macro-roughness term, by introducing adjustment constants for the whole equation and for one of the terms in brackets. The highest obtained correlation remains remained below 0.5. S e ------------- ------- θ = ⋅ ⎛ – -------⎞ B ⋅ ⎝Fr 2 Fr2⎠ d 2) Without macro-roughness Without macro-roughness, the drag force D acting on the ribs is replaced with a friction force acting on the surface of the outer side wall ( ∆φ ⋅R o ⋅ h ). The friction along the inner side wall is neglected since the water depth is small on the inner bank and the influence on the scour depth can be neglected. A dimensionless friction coefficient C F is introduced. D ∝ ρ w ⋅ V2 m ⋅C F ⋅ ∆φ ⋅ R o ⋅ h (7.45) Forming the ratios in equation 7.40 by introducing the proportionalities 7.37, 7.38 and 7.45 yields: 1 ρ w ⋅ V2 m ⋅C F ⋅ ∆φ ⋅ R o ⋅h s ρ ----------------------------------------------------------------- w ⋅V∗ 2 ⋅ ∆φ ⋅B⋅ Rc = + ----------------------------------------------------------------- ρ w ⋅g ⋅h m ⋅∆φ ⋅B ⋅R c ⋅ S e ρ w ⋅ g ⋅ h m ⋅ ∆φ ⋅ B ⋅ R c ⋅ S e (7.46) After some simplifications (like for the case with macro-roughness), we obtain: V2 1 m ⋅C F ⋅R o ⋅h s V∗2 = ------------------------------------------ g⋅ h m ⋅B ⋅R c ⋅S + ----------------------- e g⋅ h m ⋅ S e And finally the following equation for the maximum scour depth is found: h s 1 R ---- ----- c g ⋅ h ----- ------------- m S ⎛ d 90 B V2 e 1 θ ( s – 1) ------ ---- 1 ⎝ – ⋅ ⋅ ⋅ ⎞ R c S = ⋅ ⋅ ⋅ ⋅ = ---------------- ⋅ ⎛------- e – ------- θ ⎞ ⎠ m ⋅ R ⎝Fr 2 Fr2⎠ d C F R o h m S e C F o (7.47) (7.48) This equation has the same layout as the one taking into account the scour reduction due to the vertical ribs on the outer bank. Furthermore the radius of curvature appears as a parameter in this equation. The radius to channel width ratio is hidden in the ratio ⁄ ( R o = R c + B ⁄ 2 ). The last parenthesis in equations 7.43 and 7.48 can also be written as: since θ = Fr∗ 2 = ( Fr d ⋅ V∗ ⁄ V) 2 . S e θ ------- – ------- Fr 2 Fr2 d = If we compare the computed maximum scour depth (equation 7.48) to the measured values, it is difficult to obtain satisfying correlations. The best fit was obtained with: h ---- s B R c S ------- e Fr 2 – V∗ ------ ⎝ ⎛ V ⎠ ⎞2 (7.49) θ = ---------------------- ⋅ ⎛------- + 6.3 ⋅ -------⎞ , R (7.50) 0.189 ⋅ R ⎝Fr 2 Fr2⎠ 2 = 0.724 d o S e R c R o page 174 / November 9, 2002 Wall roughness effects on flow and scouring
Establishment of the scour formula 7.3.5 Genetic algorithm With the evolution of informatics, new techniques for data treatment were developed. One of these techniques are genetic algorithms allowing to search for functions fitting to a given data set. It is important to use genetic algorithms not like a magic black box, but knowing the important parameters for the process to obtain a result which has a physically correct form. To analyze the data in this research study, a genetic program called GPKernel (Genetic Programming Kernel) developed by Maarten Keijzer and Vladan Babovic at the Danish Hydraulic Institute (DHI) was used. The program looks for mathematical relations based on a set of input parameters, constants, operations, genetic parameters and on user-defined target(s) 1 . The genetic parameters like for example the size of the population, the number of generations to produce or time of run, different probabilities of mutations and genetic operations can be modified. A big advantage of the program is the possibility to perform a dimensionally aware genetic programming (KEIJZER & BABO- VIC, 1999). Additional information on GPKernel can be found in the Users Manual (RODRIGUEZ AGUILERA, 2000). The following tables (Appendix 12.1 and 12.2) give a small subset (< 0.1%) of obtained results. It can be seen that most of the previously identified main parameters appear in the equations. The following characteristics are found in the maximum scour equation: Re, Fr, Fr∗ , V∗ ⁄ V , S e and h m ⁄ B . In the relation giving the first and second scour locations, some other characteristics are dominant: σ (Peter found this parameter, too), Fr d , R c ⁄ B , h m ⁄ B and d 90 ⁄ h m . The plots in Tables 7.4 and 7.5 compare the computed and the measured relative maximum scour, and the two scour locations. All available data are included in the plots (with the tests of PETER, 1986). a) Tests without macro-roughness For the equations obtained with the genetic algorithm it is sometimes difficult to get a plausible physical explanation of the phenomena. If we analyze the first equation for the maximum scour depth (Appendix 12.1, ID 1), we obtain after regroupment of the terms and omission of terms without significant influence: h smax , R -------------- 2.4 σ 1.2 Fr Fr (7.51) h d ----- c ⎛ 1 1 ----- + ----- ⎞ d ⋅ + ⋅ ⋅ ⋅ ⋅ 90 2 ⋅ h = ⋅ ⎛------ + Fr∗ – ------------- m + θ⎞ + 1.85 m B ⎝σ 3 σ 4 ⎠ ⎝h m B ⎠ This equation has a correlation of R 2 = 0.832 which is quite high, but less that what was obtained previously (equation 7.20). Looking for a physical explanation of this equation, some parameters show the right tendency: the relative scour increases with increasing Froude numbers, Shields parameter and increasing mean flow depth to width ratio. But a smaller radius of curvature should lead to an increase of the scour depth and the equation gives an opposite tendency. Therefore this formula will not be considered for the final evaluation. 1. The program also allows multi target searches. EPFL Ph.D thesis 2632 - Daniel S. Hersberger November 9, 2002 / page 175
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WALL ROUGHNESS EFFECTS ON FLOW AND
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Abstract By applying vertical ribs
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Résumé En disposant des nervures
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Zusammenfassung • Ein markanter S
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Acknowledgments page x / November 9
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Table of contents 4. Smart & Jäggi
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Table of contents 7.6 Summary and c
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Chapter 1 - Introduction 1.1 Contex
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Index of Keywords dimensional analy
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Index of Keywords sediment sampling
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Index of Keywords parameter 77 Pete
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Publications (continued) 1997 Hersb
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