History and physical significance of the roughness Froude number

References

• Aguirre-Pe, J., Olivero, M. L., & Moncada, A. T. (2003). Particle densimetric Froude number for estimating sediment transport. Journal of Hydraulic Engineering, 129(6), 428–437. doi:10.1061/(ASCE)0733-9429(2003)129:6(428)
   Web of Science ®Google Scholar
• Alexander, R. M. (1984). The gaits of bipedal and quadrupedal animals. The International Journal of Robotics Research, 3(2), 49–59. doi:10.1177/027836498400300205
   Web of Science ®Google Scholar
• Alexander, R. M. (1989). Optimization and gaits in the locomotion of vertebrates. Physiological Reviews, 69(4), 1199–1227. doi:10.1152/physrev.1989.69.4.1199
   PubMed Web of Science ®Google Scholar
• Alexander, R. M. (2005). Models and the scaling of energy costs for locomotion. Journal of Experimental Biology, 208(9), 1645–1652. doi:10.1242/jeb.01484
   PubMed Web of Science ®Google Scholar
• Alexander, R. M., & Jayes, A. S. (1983). A dynamic similarity hypothesis for the gaits of quadrupedal mammals. Journal of Zoology, 201(1), 135–152. doi:10.1111/j.1469-7998.1983.tb04266.x
   Google Scholar
• Amador, A., Sánchez-Juny, M., & Dolz, J. (2009). Developing flow region and pressure fluctuations on steeply sloping stepped spillways. Journal of Hydraulic Engineering, 135(12), 1092–1100. doi: 10.1061/(ASCE)HY.1943-7900.0000118
   Web of Science ®Google Scholar
• Bauer, W. J. (1954). Turbulent boundary layer on steep slopes. Transactions of the American Society of Civil Engineers, 119(1), 1212–1233, Paper No. 2719. doi:10.1061/TACEAT.0006971
   Google Scholar
• Bauer, W. J. (1966). Discussion of “boundary layer development and spillway energy losses”. Journal of the Hydraulics Division, 92(HY2), 370–378. doi: 10.1061/JYCEAJ.0001429
   Google Scholar
• Biewener, A. A. (1989). Scaling body support in mammals: Limb posture and muscle mechanics. Science, 245(4913), 45–48. doi:10.1126/science.2740914
   PubMed Web of Science ®Google Scholar
• Boes, R. M., & Hager, W. H. (2003a). Two-phase flow characteristics of stepped spillways. Journal of Hydraulic Engineering, 129(9), 661–670. doi:10.1061/(ASCE)0733-9429(2003)129:9(661)
   Web of Science ®Google Scholar
• Boes, R. M., & Hager, W. H. (2003b). Hydraulic design of stepped spillways. Journal of Hydraulic Engineering, 129(9), 671–679. doi:10.1061/(ASCE)0733-9429(2003)129:9(671)
   Web of Science ®Google Scholar
• Boes, R. M., & Minor, H.-E. (2000, March 22--24). Guidelines for the hydraulic design of stepped spillways. In H.-E. Minor & W. H. Hager (Eds.), Hydraulics of Stepped Spillways, Proceedings of the International Workshop on Hydraulics of Stepped Spillways, Zurich, Switzerland. London: CRC Press.
   Google Scholar
• Cain, P. (1978). Measurements within self-aerated flow on a large spillway. Ph.D. thesis. University of Canterbury, Christchurch, New Zealand. https://ir.canterbury.ac.nz/handle/10092/9283
   Google Scholar
• Cain, P., & Wood, I. R. (1981). Measurements of self-aerated flow on a spillway. Journal of the Hydraulics Division, 107(HY11), 1425–1444. doi: 10.1061/JYCEAJ.0005761. [Erratum in J. Hydraul. Eng., 1983, 109(1), 145–146.xx].
   Google Scholar
• Campbell, F. B., Cox, R. G., & Boyd, M. B. (1965). Boundary layer development and spillway energy losses. Journal of the Hydraulics Division, 91(HY3), 149–163. doi: 10.1061/JYCEAJ.0001235
   Google Scholar
• Cassidy, J. J. (1966). Discussion of “Boundary layer development and spillway energy losses”. Journal of the Hydraulics Division, 92(HY2), 370–378. doi: 10.1061/JYCEAJ.0001429
   Google Scholar
• Chanson, H. (1994a). Hydraulics of skimming flows over stepped channels and spillways. Journal of Hydraulic Research, 32(3), 445–460. doi:10.1080/00221689409498745
   Web of Science ®Google Scholar
• Chanson, H. (1994b). Hydraulic design of stepped cascades, channels, weirs and spillways. Pergamon.
   Google Scholar
• Chanson, H. (2002). The hydraulics of stepped chutes and spillways. A. A. Balkema.
   Google Scholar
• Chanson, H., & Toombes, L. (2004). Hydraulics of stepped chutes: The transition flow. Journal of Hydraulic Research, 42(1), 43–54. doi: 10.1080/00221686.2004.9641182
   Web of Science ®Google Scholar
• Cheng, X., Gulliver, J. S., & Zhu, D. (2014). Application of displacement height and surface roughness length to determination boundary layer development length over stepped spillway. Water, 6(12), 3888–3912. doi: 10.3390/w6123888
   Web of Science ®Google Scholar
• Chow, V. T. (1959). Open-channel hydraulics. McGraw-Hill.
   Google Scholar
• Di Cristo, C., Iervolino, M., Vacca, A., & Zanuttigh, B. Influence of relative roughness and Reynolds number on the roll-waves spatial evolution. Journal of Hydraulic Engineering, 136(1), 24–33. doi: 10.1061/(ASCE)HY.1943-7900.0000139
   Web of Science ®Google Scholar
• Ettema, R., Arndt, R., Roberts, P., & Wahl, T. (2000). Hydraulic modeling concepts and practice. American Society of Civil Engineers Manual No. 97. doi:10.1061/9780784404157
   Google Scholar
• Felder, S., & Chanson, H. (2009). Energy dissipation, flow resistance and gas-liquid interfacial area in skimming flows on moderate-slope stepped spillways. Environmental Fluid Mechanics, 9(4), 427–441. doi: 10.1007/s10652-009-9130-y
   Web of Science ®Google Scholar
• Felder, S. & Chanson, H. (2011). Energy dissipation down a stepped spillway with nonuniform step heights. Journal of Hydraulic Engineering, 137(11), 1543–1548. doi: 10.1061/(ASCE)HY.1943-7900.0000455
   Web of Science ®Google Scholar
• Felder, S. & Chanson, H. (2013). Aeration, flow instabilities, and residual energy on pooled stepped spillways of embankment dams. Journal of Irrigation and Drainage Engineering, 139(10), 880–887. doi: 10.1061/(ASCE)IR.1943-4774.0000627
   Web of Science ®Google Scholar
• Felder, S. & Chanson, H. (2016). Simple design criterion for residual energy on embankment dam stepped spillways. Journal of Hydraulic Engineering, 142(4), 04015062. doi: 10.1061/(ASCE)HY.1943-7900.0001107
   Web of Science ®Google Scholar
• Frizell, K. W., & Frizell, K. H. (2015, March 22–24). Guidelines for hydraulic design of stepped spillways. Hydraulic Laboratory Report HL-2015-06. U.S. Dept. of the Interior, Bureau of Reclamation, Denver, Colorado.
   Google Scholar
• Gangadharaiah, T., Lakshmana Rao, N. S. & Seetharamiah, K. (1970). Inception and entrainment in self-aerated flows. Journal of the Hydraulics Division, 96(HY7), 1549–1565. doi: 10.1061/JYCEAJ.0002566
   Google Scholar
• Gibb, A., Jayne, B., & Lauder, G. (1994). Kinematics of pectoral fin locomotion in the bluegill sunfish lepomis macrochirus. Journal of Experimental Biology, 189(1), 133–161. doi: 10.1242/jeb.189.1.133
   PubMedGoogle Scholar
• Hager, W. H., & Boes, R. M. (2000, March 22–24). Backwater and drawdown curves in stepped spillway flow. In H.-E. Minor & W. H. Hager (Eds.), Hydraulics of Stepped Spillways, Proceedings of the International Workshop on Hydraulics of Stepped Spillways, Zurich, Switzerland. London: CRC Press.
   Google Scholar
• Halbronn, G. (1952). Étude de la Mise en Régime des Écoulements Sur les Ouvrages à Forte Pente. La Houille Blanche, 38(1), 21–40. doi:10.1051/lhb/1952018
   Google Scholar
• Halbronn, G. (1954). Discussion of ‘Turbulent boundary layer on steep slopes’. Transactions of the American Society of Civil Engineers, 119(1), 1234–1239. doi: 10.1061/TACEAT.0007019
   Google Scholar
• Hale, M. E., Day, R. D., Thorsen, D. H., & Westneat, M. W. (2006). Pectoral fin coordination and gait transitions in steadily swimming juvenile reef fishes. Journal of Experimental Biology, 209(19), 3708–3718. doi:10.1242/jeb.02449
   PubMed Web of Science ®Google Scholar
• Hiller, P. H., Lia, L., & Aberle, J. (2019). Field and model tests of riprap on steep slopes exposed to overtopping, Journal of Applied Water Engineering and Research, 7(2), 103–117. doi:10.1080/23249676.2018.1449675
   Web of Science ®Google Scholar
• Hohermuth, B., Boes, R. M. & Felder, S. (2021) High-velocity air–water flow measurements in a prototype tunnel chute: scaling of void fraction and interfacial velocity. Journal of Hydraulic Engineering, 147(11), 04021044. doi: 10.1061/(ASCE)HY.1943-7900.0001936
   Web of Science ®Google Scholar
• Hunt, S. L., & Kadavy, K. C. (2013). Inception point for embankment dam stepped spillways. Journal of Hydraulic Engineering, 139(1), 60–64. doi:10.1061/(ASCE)HY.1943-7900.0000644
   Web of Science ®Google Scholar
• Hunt, S. L., Kadavy, K. C., & Hanson, G. J. (2014). Simplistic design methods for moderate-sloped stepped chutes. Journal of Hydraulic Engineering, 140(12). doi:10.1061/(ASCE)HY.1943-7900.0000938
   Web of Science ®Google Scholar
• Hunt, S. L., Kadavy, K. C., Wahl, T. L., & Moses, D. W. (2022). Physical modeling of beveled-face stepped chute. Water, 14(3), 365. doi:10.3390/w14030365
   Web of Science ®Google Scholar
• Keller, R. J. (1972). Field measurement of self-aerated high speed open channel flow. Ph.D. thesis. University of Canterbury, Christchurch, New Zealand. http://hdl.handle.net/10092/9340
   Google Scholar
• Keller, R. J., Lai, K. K. & Wood, I. R. (1974). Developing region in self-aerated flows. Journal of the Hydraulics Division, 100(HY4), 553–568. doi: 10.1061/JYCEAJ.0003932
   Google Scholar
• Keller, R. J., & Rastogi, A. K. (1975). Prediction of flow development on spillways. Journal of the Hydraulics Division, 101(HY9), 1171–1184. doi: 10.1061/JYCEAJ.0004414
   Google Scholar
• Keller, R. J., & Rastogi, A. K. (1977). Design chart for predicting critical point on spillways. Journal of the Hydraulics Division, 103(HY12), 1417–1429. doi: 10.1061/JYCEAJ.0004889
   Google Scholar
• Killen, J. M. (1968). The surface characteristics of self aerated flow in steep channels. Ph.D. thesis. Univ. Minnesota, Minneapolis, MN. https://conservancy.umn.edu/handle/11299/107504
   Google Scholar
• Kramer, M., & Chanson, H. (2018). Transition flow regime on stepped spillways: air–water flow characteristics and step-cavity fluctuations. Environmental Fluid Mechanics, 18(4), 947–965. doi: 10.1007/s10652-018-9575-y
   Web of Science ®Google Scholar
• Lai, K. K. (1971). Studies of air entrainment in steep open channels. M. Eng. thesis, University of New South Wales, Sydney, Australia. https://unsworks.unsw.edu.au/entities/publication/655e546b-829c-49fa-929d-ad620e357c1d/full
   Google Scholar
• Lane, E. W. (1939). Entrainment of air in swiftly flowing water. Civil Engineering, 9(2), 89–91. ASCE.
   Google Scholar
• Levi, E. (1967). Macroturbulence produced by vortex breakdown in high velocity flows. Proceedings of the 12th Congress of IAHR, 2, 54–60.
   Google Scholar
• Mateos, C., & Elviro, V. (1997, August 10–15). Initiation of aeration in stepped spillways. Energy and Water: Sustainable Development, Proceedings of the 27th Congress of IAHR.
   Google Scholar
• Meireles, I., & Matos, J. (2009). Skimming flow in the nonaerated region of stepped spillways over embankment dams. Journal of Hydraulic Engineering, 135(8), 685–689. doi:10.1061/(ASCE)HY.1943-7900.0000047
   Web of Science ®Google Scholar
• Ohtsu, I., Yasuda, Y. & Takahashi, M. (2001). Discussion of onset of skimming flow on stepped spillways. Journal of Hydraulic Engineering, 127(6), 519–525. doi: 10.1061/(ASCE)0733-9429(2001)127:6(519)
   Google Scholar
• Ohtsu, I., Yasuda, Y., & Takahashi, M. (2004). Flow characteristics of skimming flows in stepped channels. Journal of Hydraulic Engineering, 130(9), 860–869. doi:10.1061/(ASCE)0733-9429(2004)130:9(860)
   Web of Science ®Google Scholar
• Pfister, M., & Chanson, H. (2014). Two-phase air-water flows: Scale effects in physical modeling. Journal of Hydrodynamics, 26(2), 291–298. doi:10.1016/S1001-6058(14)60032-9
   Web of Science ®Google Scholar
• Rayner, J., Jones, G., & Thomas, A. (1986). Vortex flow visualizations reveal change in upstroke function with flight speed in bats. Nature, 321(6066), 162–164. doi:10.1038/321162a0
   Web of Science ®Google Scholar
• Straub, L., & Anderson, A. (1958). Experiments on self aerated flow in open channels. Journal of the Hydraulics Division, 84(HY7), 1–35. doi: 10.1061/JYCEAJ.0000261
   Google Scholar
• Terrier, S. (2016). Hydraulic performance of stepped spillway aerators and related downstream flow features. Thesis No. 6989. École Polytechnique Fédérale de Lausanne (EPFL). https://infoscience.epfl.ch/record/218528?ln=fr.
   Google Scholar
• Valero, D., & Bung, D. B. (2016). Development of the interfacial air layer in the non-aerated region of high-velocity spillway flows. Instabilities growth, entrapped air and influence on the self-aeration onset. International Journal of Multiphase Flow, 84, 66–74. doi:10.1016/j.ijmultiphaseflow.2016.04.012
   Web of Science ®Google Scholar
• Vaughan, C. L., & O’Malley, M. J. (2005). Froude and the contribution of naval architecture to our understanding of bipedal locomotion. Gait & Posture, 21(3), 350–362. doi:10.1016/j.gaitpost.2004.01.011
   PubMed Web of Science ®Google Scholar
• Wood, I. R., Ackers, P., & Loveless, J. (1983). General method for critical point on spillways. Journal of Hydraulic Engineering, 109(2), 308–312. doi:10.1061/(ASCE)0733-9429(1983)109:2(308)
   Web of Science ®Google Scholar