Flow resistance and form losses on stepped spillways

In a spillway chute, the water flow is driven by gravity. Although the laws of flow resistance are essentially the same in open channels and in closed pipes (ASCE 1963, Henderson 1966, Chanson 2004), the calculations of the boundary shear stress are complicated by the existence of the free surface.
A stepped spillway is basically a steep channel with a staircase invert. The construction of steps along the spillway channel may assist with the energy dissipation, thus reducing the size of the downstream energy dissipator (i.e. stilling structure). Flow resistance in skimming flows is associated with considerable form losses (i.e. form drag) and momentum transfer between the main flow and the step cavity recirculation (Rajaratnam 1990). It is much larger than on smooth-invert chutes. The Darcy-Weisbach friction factor f is equivalent to a dimensionless boundary shear stress accounting for the form drag along the stepped invert. The experimental data in terms of f are mostly within 0.1 to 0.3. That is nearly one order of magnitude larger than on smooth concrete spillways.The results are valid for fully-developed gradually-varied flows and uniform equilibrium flows on stepped spillways. For developing flows, i.e. with a developing boundary layer, a different approach is required: namely, using the momentum integral method.An in-depth scrutiny into the stepped spillway data suggests that the skimming flow friction factor data are distributed around three dominant values: f = 0.11, 0.17 and 0.30 (Chanson, Bung & Matos 2015). The flow resistance in skimming flows might not be an unique function of flow rate and stepped chute geometry. Rather, the form drag process may present several modes of excitation resulting from the vortex shedding in the shear layers downstream of each step edge. Indeed, a number of laboratory data show that the flow properties in skimming flows oscillate between adjacent step edges. The existence of such instability modes implied that the traditional concept of 'normal flow’ might not exist in skimming flows on stepped spillways.The calculations of flow resistance in stepped spillways constitute a basic application of the concept of flow resistance and total drag in form-drag dominated free-surface flows. This is directly relevant to the design of dam spillways and energy dissipators for hydraulic structures. This is further discussed in a number of relevant Youtube video movies in the same channel at: {   / @hubert_chanson  }.

Hydraulic Engineering and Applied Hydrodynamics in Hubert Chanson Youtube channel {   / @hubert_chanson  }Fundamentals of open channel hydraulics [Playlist]Advanced hydraulics of open channel flow [Playlist]Stepped spillway hydraulics [Playlist]Flow resistance in open channels (1) Basic considerations {   • Flow resistance in open channels (1) Basic...  }Stepped Spillway Prototype Operation and Air Entrainment: Hinze Dam 2013‐2021 flood events {   • Stepped Spillway Prototype Operation and A...  }Gold Creek Dam (4) Filming the stepped spillway operation in February 2022 {   • Gold Creek Dam (4) Filming the stepped spi...  }AcknowledgementsHubert Chanson thanks his students, former students and co-workers. He also thanks all the people who provided them with relevant information.

References
ASCE Task force (1963). "Task Force on Friction Factors in Open Channels: Report of the Task Force." Journal of Hydraulic Division, ASCE, Vol. 89, No. HY2, pp. 97‐143.
CHANSON, H. (1995). "Hydraulic Design of Stepped Cascades, Channels, Weirs and Spillways." Pergamon, Oxford, UK, Jan., 292 pages (ISBN 0-08-041918-6).
CHANSON, H. (2001). "The Hydraulics of Stepped Chutes and Spillways." Balkema, Lisse, The Netherlands (ISBN 90 5809 352 2).
CHANSON, H., YASUDA, Y., and OHTSU, I. (2002). "Flow Resistance in Skimming Flows and its Modelling." Canadian Journal of Civil Engineering, Vol. 29, No. 6, pp. 809-819 (DOI: 10.1139/L02-083) (ISSN 0315-1468).
CHANSON, H., BUNG, D., and MATOS, J. (2015). "Stepped spillways and cascades." in "Energy Dissipation in Hydraulic Structures." IAHR Monograph, CRC Press, Taylor & Francis Group, Leiden, The Netherlands, H. CHANSON Editor, pp. 45-64 (ISBN 978-1-138-02755-8).
HENDERSON, F.M. (1966). "Open Channel Flow." MacMillan Company, New York, USA.
NIKORA, V., STOESSER, T., CAMERON, S.M., STEWART, M., PAPADOPOULOS, K., OURO, P., McSHERRY, R., ZAMPIRON, A., MARUSIC, I., and FALCONER, R.A. (2019). "Friction factor decomposition for rough-wall flows: theoretical background and application to open-channel flows." Journal of Fluid Mechanics, Vol. 872, pp. 626-644 (DOI: 10.1017/jfm.2019.344).
RAJARATNAM, N. (1976). "Turbulent Jets." Elsevier Scientific, Development in Water Science, 5, New York, USA.
RAJARATNAM, N. (1990). "Skimming Flow in Stepped Spillways." Journal of Hydraulic Engineering, ASCE, Vol. 116, No. 4, pp. 587-591.
SCHLICHTING, H. (1979). "Boundary Layer Theory." McGraw-Hill, New York, USA, 7th edition, 817 pages.