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Chapter 7 - Establishing an empirical formula 7.6.2 Conclusions Without macro-roughness, the best correlation was obtained by an approach based on the shape of the cross-section in the scour holes. A polynomial function of the third degree combines an excellent correlation ( R 2 = 0.86 ) with a good fit over the cross-section 1 . ( h s h m φ c 6.23 B r – R c – 8.8B) 3 = + tan ⋅ ⋅ ⋅ –-------------------------------------- + ( r– R (7.31) 260 ⋅ B 2 c – 8.8B) h smax , = h m + tanφ ⋅c ⋅[ 0.129 ⋅ B] (7.32) where c 290 ⎛1 3.2 h m – ⋅ -----⎞ V⋅ R h = ⋅ ⋅ ----------------- (7.33) ⎝ B ⎠ g⋅ B 3 Unfortunately, the application to results of scale model tests and field data shows that the predicted scour depth is too small compared to the measured one. Therefore the modified formula of Bridge (eq. 7.7) is proposed for the computation of the maximum scour depth without macroroughness. Despite a slightly lower correlation, this formula gives a much better prediction if applied to field data. It further takes into account the bend geometry ( R c , B ) the flow conditions ( ) and the bed characteristics ( tanφ ) h m sinβ 0.394 11 23 h m = ⋅ ⎛ – ⋅ ----- ⎞ ----- , (7.7) ⎝ B ⎠ ⋅ ⋅ tanφ ⋅ ---- R B r 2 = 0.817 In the presence of vertical ribs, equation 7.63 is recommended for the computation of the maximum scour depth ( R 2 = 0.876 compared to the whole dataset, with Peter’s tests): R c h s h -------------- smax , 7.7 e s = ⋅ ----- ⋅ Fr ⋅( 0.001 + ( θ– θ h m R cr ) 2 ) + 1.7 h 2 ) (7.63) This formula is very simple, depending only on the Froude number and the difference between the Shields stress and the critical Shields stress. The ratio rib spacing to hydraulic radius accounts for the influence of the ribs. It is important to use an optimum rib spacing to compute the maximum scour depth since the scour depth decreases linearly with the reduction of the rib spacing. If the rib spacing is reduced sufficiently (below an optimum spacing), the scour depth can become even more important than the scour depth obtained without the presence of ribs. This tendency cannot be found in equation 7.63. Figure 6.5 on page 119 gives some indications on the reduction of the scour depth as a function of the rib spacing. Furthermore it needs to be mentioned that the result h smax , ⁄ h m is close to 1.7 and that the different parameters in equation 7.63 have only a reduced impact on the result. 1. The main parameters in this equation are the friction slope , a ration combining a flow characteristic (velocity) with the channel geometry ( R h and B ) and the flow depth to width ratio. These ratios determine the scour depth compared to the mean bed level. The expression in the brackets “fits” the 3rd-degree polynome on the cross-section with a steepest slope of tanφ ⋅ c . The term r– R c – 8.8B stands for the radius corrected by the radial shift of the inflection point of the cross section (point for which the radial bed slope is maximum). 2. for θ > θ cr page 190 / November 9, 2002 Wall roughness effects on flow and scouring
Summary and conclusions The location of the scour holes (without macro-roughness) can be computed with the following equations: α 1 σ ⎛ B S 0.58 ----- e ⋅ + 12.7 ⋅ ---------------- ⎞ V = ⋅ ⎝ , (7.54) h m σ – Fr d ⎠ + 1.4 ⋅Fr d ⋅ ------ – 6.6 R 2 = 0.829 V∗ R c α 2 12.6 ⋅ Fr d 0.9 ⎛----- ⎞ 2 = – ⋅ + 91.6 ⎝ , R (7.57) B ⎠ 2 = 0.602 For the second scour, a slightly better correlation can be obtained, but with a much more complicated equation (7.56) which is quite sensitive to the choice of the used parameters. The equations for the two scour locations illustrate that the two scour holes do not depend on the same parameters. The first scour depends e.g. on σ , but not the second one. EPFL Ph.D thesis 2632 - Daniel S. Hersberger November 9, 2002 / page 191
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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 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 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 5.5 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
Page 222 and 223: References Meyer-Peter, E., Favre,
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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