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Chapter 3 - Theoretical considerations Measurements of this friction angle performed on a tilting table gave the following friction angles fitting well with the predicted values. The precision of the measurement is estimated to be ±1 °. GRAIN MATERIAL RANGE* AVERAGE armoring layer 36 to 40° 38° substrate (initial mixture) 36 to 40° 38° fine (point bar) material 33 to 38° 37° Table 3.2: Measured friction angle for the used sediment * The range gives the zone where the grains moved The comparison between the measured and the analytically determined friction angle show a good agreement. Therefore the method of DHAWAN can be recommended for the determination of the friction angle if no measurements are available. page 26 / November 9, 2002 Wall roughness effects on flow and scouring
3.2 Flow equations and flow resistance Flow equations and flow resistance The flow resistance is influenced by the structure of the surface layer and the presence of sediment load and suspended load transport. The roughness of the surface layer is due to the characteristics of the surface layer and the presence of bedforms. Many works and textbooks give an overview over the different flow equations and the flow resistance, like e.g. DUBOIS (1998), SINNIGER & HAGER (1989), HUNZIKER (1995), ZARN (1997). 3.2.1 Flow resistance and friction laws In his pioneer work, CHEZY (1768) related the average flow velocity to the bed slope and the water depth. V = C⋅ h ⋅ S (3.10) For the friction coefficient C , CHEZY made important simplifications. About 100 years later, GANGUILLET & KUTTER published a formula, which allowed the computation of C as a function of the bed roughness. Authors like CHÉZY, GAUKLER, FORCHHEIMER, MANNING, CHRISTEN, (see SINNIGER & HAGER, 1989, §2.2) and many others published velocity power laws. In Switzerland, the formula of STRICKLER (1923) found a wide application: V = K s ⋅ R2 3 h ⋅ ⁄ S 1⁄ 2 (3.11) The relation between STRICKLER’S roughness coefficient K s and the size of the characteristic roughness ε (usually the mean diameter of the surface layer d m ) is given by K s = Cst ------- (3.<strong>12</strong>) 6 ε STRICKLER put the value of the constant Cst to 21.1 and used as characteristic roughness ε the mean diameter. MEYER-PETER & MÜLLER (1948) used a value of Cst = 26 combined with ε = d 90 (of the substrate). The higher constant Cst is partially compensated by the bigger grain diameter. MEYER-PETER & MÜLLER justified the increased constant with a better fit to the results of NIKURADSE. JÄGGI (1984) and HUNZIKER (1995) analyzed the constant Cst for the computation of the bed friction. They concluded that the initially proposed value of STRICKLER fits better to the observed values than the one of MEYER-PETER & MÜLLER. SCHÖBERL (1981) performed a series of tests and proposed an intermediate value of 23.5. In the present study a value of 8.2 g = 25.7 is used in combination with d 90 . DARCY-WEISSBACH (VANONI, 1975) proposed the following friction law: V = 8 ⋅g ⋅R h ⋅ S ---------------------------- f (3.13) <strong>EPFL</strong> Ph.D thesis 2632 - Daniel S. Hersberger November 9, 2002 / page 27
Page 1: WALL ROUGHNESS EFFECTS ON FLOW AND
Page 4 and 5: Abstract By applying vertical ribs
Page 6 and 7: Résumé En disposant des nervures
Page 8 and 9: Zusammenfassung • Ein markanter S
Page 10 and 11: Acknowledgments page x / November 9
Page 12 and 13: Table of contents 4. Smart & Jäggi
Page 14 and 15: Table of contents 7.6 Summary and c
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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 with and w
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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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Notations K S [m 1/3 /s] roughness
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Notations bed b bf c cr d e g w o o
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References de Vriend H.J. (1981). S
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References Meyer-Peter, E., Favre,
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References Williams, R. (1899). Flu
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References by chapter Keulegan (193
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References by chapter page 214 / No
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Index of Authors Hoffmanns 13 Hoffm
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Index of Authors page 218 / Novembe
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Index of Figures Fig.4.19: Frame po
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Index of Figures 22] 183 Fig.7.15:
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Index of Tables Table 7.3: Table of
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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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