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Chapter 7 - Establishing an empirical formula 7.6 Summary and conclusions 7.6.1 Summary A dimensional analysis (§ 7.2) allowed for the identification of the main parameters influencing the scour process. These are the overall energy slope S eall , , the water depth to channel width ratio h m ⁄ B , the ratios V⋅ R h ⁄ g⋅ B 3 (replacing the Reynolds number that showed an important correlation) and R c ⋅ h m ⁄ B 2 . The frequently used ratio V∗ ⁄ V and Fr d do not seem to play a determinant role. An additional parameter was found to characterize the macro-roughness: the spacing of the ribs e s . To make this ratio dimensionless it can be divided either by the channel width B, the mean water depth h m or the radius of curvature R c . In a first step, existing scour formulae were enhanced (§ 7.3.2). Table 7.6 gives a summary of the obtained results. Since the formula of KIKKAWA ET AL. (1976) is quite frequently used in practice, an effort was made to enhance its prediction capability for mountain rivers with a coarse bed by adding additional parameters. The result is quite good but it is possible to get a better prediction with other equations. BRIDGE (1976) gives the smallest error on the predicted average scour depth. Therefore, this formula was enhanced too. By introducing two additional parameters, related to the channel geometry, the correlation for the maximum scour depth can be increased to 0.82 and the predicted cross-section shape fits quite well to the measured one. The predicted values fit well to the results obtained with scale model tests (Gurtnellen) and field data (Sacramento River) (§ 7.5). Since the observed cross-section is rather s-shaped than an exponential function (which is an assumption of most existing formulae), an attempt was made to establish a scour formula based on the shape of the cross-section (§ 7.3.3). It was assumed that the maximum bed slope in a radial direction is limited to a maximum value corresponding to the friction angle of the bed material. Different polynomial functions (Table 7.6) of the third and fifth degree were tested. A large set of boundary conditions was analyzed, by fixing the cross-section slope at the outer and / or inner bank and by admitting or omitting a symmetry point on the channel axis. The results range from poor to good. The best result was obtained with a polynomial function of the third degree (equation 7.20) obtaining a correlation of R 2 = 0.856 and fitting well to the measured cross-section. Unfortunately this equation shows some lacking (underestimation of the scour depth) when applied to additional laboratory and field data (§ 7.5 and Fig. 7.19). Hoping to increase the precision of the predicted scour depth, an approach based on the similitude and approximation theory of KLINE (1965) was explored (§ 7.3.4). Based on the most important forces acting on a control volume in the cross-section, dimensionless ratios between the different characteristics were derived. The advantage of this method is that the parameters obtained in this way are based on the physical process. Unfortunately the result is not very satisfying. The highest obtained correlations are of 0.72 and for the case with macro-roughness R 2 < 0.5 . page 188 / November 9, 2002 Wall roughness effects on flow and scouring
Summary and conclusions Finally a genetic algorithm written by KEIJZER & BABOVIC (1999) was used to search for a function fitting well with the measured data (§ 7.3.5). Without ribs, the best obtained formula had a correlation of 0.83 which is less than for the approach based on the cross-section shape. Therefore none of the equations were retained. The search for an equation predicting the scour locations was more successful. Formulae with a correlation of 0.83 respectively 0.60 are proposed to compute the first and second scour location. Finally a formula was established (eq. 7.63), allowing for the computation of the maximum scour depth in the presence of macro-roughness. The prediction capability is very satisfying ( R 2 = 0.876 ). FORMULA EQ. COM- PUTES PARAMETERS R 2 FIT TO CROSS- SECTION modified Kikkawa modified Bridge 7.5 7.6 7.7 R B h s ⁄ , h m ⁄ B , V⋅ R h ⁄ gB 3 h s ⁄ B , h m ⁄ B h s c R c ⁄ B R c 0.795 0.796 0.817 quite well quite well quite well Polynomial functions of 3rd degree, centered 3rd degree, non-centered (nc) 3rd degree, nc, vertical adjust. (va) same but dh s ⁄ dr = 0 inside 7.11 7.14 7.18 7.20 h s h s h s h s tanφ , V⋅ R h ⁄ gB 3 tanφ , S e, all , h m ⁄ B , V⋅ R h ⁄ tanφ , R c ⁄ B , h m ⁄ B , V⋅ R h ⁄ gB 3 gB 3 tanφ , h m ⁄ B , V⋅ R h ⁄ gB 3 0.72 0.82 0.82 0.856 poor less poor better good 3rd degree, nc, va, add. terms 3rd degree, nc, va, no restr.bound. 5th degree, dh s ⁄ dr = 0 outside 5th degree, dh s ⁄ dr = 0 inside 7.22 7.29 7.27 7.28 h s h s h s h s tanφ , S e, all , R c ⁄ B , h m ⁄ B tanφ , h m ⁄ B , V⋅ R h ⁄ gB 3 tanφ , S e, all , h m ⁄ B , V⋅ R h ⁄ gB 3 tanφ , S e, all , h m ⁄ B , V⋅ R h ⁄ gB 3 0.81 0.76 0.850 0.856 very poor very poor poor poor Similitude & approx. theory with macro-roughness without macro-roughness Genetic algorithm Table 7.6: 1. θ ⁄ Fr2 d can be replaced by V∗ ⁄ V 7.43 7.44 7.48 with adjustments 7.63 h s, max e s ⁄ e d , S e , θ , Fr 2 , Fr2 d h , , , , 1 s, max e 2 s ⁄ B⁄ R c S e θ Fr 2 Fr2 d h s, max R c ⁄ ( C F ⋅ R o ), S e , θ , Fr 2 , Fr2 d 1 1 --- < 0.5 0.724 most of the previously mentioned parameters , θ– θ cr , Fr, e s ⁄ R h 0.876 h s, max h s max Summary of the tested type of formulae for establishing a new scour formula - - <strong>EPFL</strong> Ph.D thesis 2632 - Daniel S. Hersberger November 9, 2002 / page 189
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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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Chapter 1 - Introduction page 4 / N
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Chapter 2 - State of the art 2.1 Fl
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Chapter 2 - State of the art 2.2.2
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Chapter 2 - State of the art Theref
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Chapter 2 - State of the art resolu
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Chapter 2 - State of the art comple
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Chapter 2 - State of the art ORLAND
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Chapter 2 - State of the art 2.6 Me
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Chapter 2 - State of the art page 2
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Chapter 3 - Theoretical considerati
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Chapter 5 - Test results 5.1 Introd
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Chapter 5 - Test results 5.2 Prelim
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Chapter 5 - Test results 5.2.2 Test
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Chapter 5 - Test results 5.5 Tests
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Page 216 and 217: Notations K S [m 1/3 /s] roughness
Page 218 and 219: Notations bed b bf c cr d e g w o o
Page 220 and 221: References de Vriend H.J. (1981). S
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Page 224 and 225: References Williams, R. (1899). Flu
Page 226 and 227: References by chapter Keulegan (193
Page 228 and 229: References by chapter page 214 / No
Page 230 and 231: Index of Authors Hoffmanns 13 Hoffm
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Page 236 and 237: Index of Figures 22] 183 Fig.7.15:
Page 238 and 239: Index of Tables Table 7.3: Table of
Page 240 and 241: Index of Keywords dimensional analy
Page 242 and 243: Index of Keywords sediment sampling
Page 244 and 245: Index of Keywords parameter 77 Pete
Page 246: Publications (continued) 1997 Hersb
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