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Chapter 7 - Establishing an empirical formula h s,max h s,max /h m tan(β max ) 1 2 12 1 2 12 1 2 12 V*/V 0.043 0.058 0.055 0.207 0.087 0.210 0.296 0.078 0.153 Fr* 0.061 0.019 0.050 0.060 0.035 0.050 0.031 0.042 0.040 Fr d 0.263 0.152 0.230 0.006 0.031 0.010 0.001 0.262 0.137 Fr 0.013 0.000 0.016 0.210 0.044 0.231 0.089 0.016 0.036 Sigma 0.130 0.020 0.079 0.087 0.009 0.056 0.086 0.036 0.005 d/h m 0.028 0.055 0.028 0.251 0.053 0.275 0.205 0.026 0.064 Re* 0.775 0.734 0.822 0.348 0.140 0.338 0.153 0.163 0.165 Re 0.798 0.759 0.832 0.477 0.201 0.475 0.232 0.188 0.204 V*R h /sqrt(g*B^3) 0.812 0.751 0.839 0.475 0.210 0.477 0.229 0.204 0.217 V*R h /sqrt(g*h m^3) 0.814 0.785 0.863 0.452 0.175 0.445 0.212 0.151 0.168 S e,in 0.031 0.475 0.251 0.169 0.571 0.464 0.001 0.404 0.175 S e,mr 0.057 0.553 0.301 0.128 0.673 0.564 0.002 0.303 0.125 S e,all 0.079 0.161 0.099 0.512 0.170 0.518 0.366 0.109 0.186 S e,bend 0.004 0.027 0.005 0.268 0.071 0.285 0.186 0.012 0.041 Rc/B 0.529 0.442 0.564 0.279 0.227 0.277 0.253 0.473 0.470 h m /B 0.662 0.532 0.628 0.486 0.262 0.521 0.228 0.297 0.295 r/B 0.502 0.416 0.535 0.263 0.218 0.262 0.240 0.460 0.455 V^2/g/r 0.040 0.011 0.026 0.019 0.026 0.016 0.054 0.148 0.153 R c *h m /B^2 0.699 0.586 0.696 0.441 0.261 0.458 0.256 0.376 0.377 (R c +h m )/B 0.537 0.448 0.571 0.287 0.230 0.285 0.255 0.472 0.469 V * 0.807 0.735 0.855 0.329 0.130 0.321 0.150 0.119 0.126 V*R h * 0.798 0.759 0.832 0.477 0.201 0.475 0.232 0.188 0.204 Table 7.2: Table of correlations R 2 for the tests without macro-roughness Re*: computed with d 90 , S e,in and S e,mr are computed without Peter’s data. 1: first scour hole, 2: second scour hole, 12: maximum of the two scour holes Correlations over 0.5 are in bold characters, correlations below 0.4 on a grey background. * parameter with a dimension It can easily be seen that the frequently used parameters V∗ ⁄ V , Fr d have almost no influence on the scour process. The particle Froude number seems to have a little influence on the absolute scour depth h max and it may be useful to introduce it in combination with other parameters in a scour formula. The Reynolds number seems to be an important parameter; but since there is no plausible reason to consider the influence of the viscosity in the scour process, this phenomenon is an expression of the correlation between the velocity and the scour depth. This fact is confirmed if we consider the correlations between the velocity V , the product of the velocity with the hydraulic radius V ⋅ R h with the absolute and relative maximum scour depth. To replace the Reynolds number, a new dimensionless ratio V⋅ R h ⁄ g⋅ B 3 is proposed. It has about the same correlation as the Reynolds number. Replacing B by in this ratio would reduce the correlation. h m page 152 / November 9, 2002 Wall roughness effects on flow and scouring
Main parameters for an empirical formula The energy slope has a significant correlation with the scour depth. If we consider the maximum scour depth in the deepest of the two scour holes h 12 , the overall energy slope S eall , and the energy slope over the domain equipped with macro-roughness elements play a major role. S emr , Furthermore the geometric ratios R c ⁄ B , h m ⁄ B seem to be important to explain the maximum scour. Since the local radius r is almost constant compared to the channel width B, the ratio r⁄ B gives quite good correlations, too. But since the radius is already by R c ⁄ B , r⁄ B will not be considered as optimization parameter. The influence of the super elevation of the water surface V 2 ⁄ ( g ⋅ r) is already included in the term ⁄ B . R c Table 7.3 shows the correlations between the chosen parameters and all available tests including the ones with macro-roughness and Peter’s data. The parameters influencing the scour process without macro-roughness still play an important role except the energy slope over the domain equipped with macro-roughness. This is quite normal: because of the head losses being more important than without vertical ribs, the energy slope has to increase. Despite the increased energy slope (over the domain with macro-roughness), the scour depth is reduced. Therefore the overall energy slope will be used in the optimization process. The influence of the macro-roughness on the maximum scour seems to be dominated by the ratio of rib-spacing to average water depth e s ⁄ h m . This ratio is more important than the rib spacing compared to the channel width e s ⁄ B . The depth of the macro-roughness appears to play a subordinate role. This seems to be contradictory to the statement that the ratio of rib-depth to rib-spacing ( e d ⁄ e s ) plays a predominant role (§ 6.2.1). If we consider that the mean water depth is highly correlated with the relative scour depth, the good correlation for the ratio e s ⁄ h m gets quite obvious. Therefore the ratio rib-spacing to channel width ( e s ⁄ B ) can also be considered as a good parameter to quantify the influence of the ribs on the relative scour. Despite a somewhat smaller correlation for the rib-depth to rib-spacing ratio ( e d ⁄ e s ) an influence on the relative scour can be made out. β max As far as the transversal bed slope is concerned, the same parameters play an important role. In addition to them, the sediment Froude number has a significant correlation. If we summarize the obtained results, we see that the following parameters need to be considered in the scour process: • general parameters: V or V⋅ R h represented by V⋅ R h ⁄ g ⋅ B 3 , S eall , , R c ⁄ B , h m ⁄ B , (as well as combinations of these parameters) • macro-roughness related parameters: e s ⁄ h m , e d ⁄ h m , and eventually the same ratios relative to the channel width B. EPFL Ph.D thesis 2632 - Daniel S. Hersberger November 9, 2002 / page 153
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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 4 - Experimental setup and
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Chapter 5 - Test results 5.1 Introd
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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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