**1. Introduction**

Flows on stepped spillways have been widely studied in various research institutions motivated by the attractive low costs related to the dam construction using roller-compacted concrete and the high energy dissipations that are produced by such structures. This is a very rich field of study for researchers of Fluid Mechanics and Hydraulics, because of the complex flow characteristics, including turbulence, gas exchange derived from the twophase flow (air/water), cavitation, among other aspects. The most common type of flow in spillways is known as skimming flow and consists of: (1) main flow (with preferential direction imposed by the slope of the channel), (2) secondary flows of large eddies formed between steps and (3) biphasic flow, due to the mixture of air and water. The details of the three mentioned standards may vary depending on the size of the steps, the geometric conditions of entry into the canal, the channel length in the steps region and the flow rates. The second type of flow that was highlighted in the literature is called nappe flow. It occurs for specific conditions such as lower flows (relative to skimming flow) and long steps in relation to their height. In the region between these two "extreme" flows, a "transition flow" between nappe and skimming flows is also defined. Depending on the details that are relevant for each study, each of the three abovementioned types of flow may be still subdivided in more sub-types, which are mentioned but not detailed in the present chapter. Figure 1 is a sketch of the general appearance of the three mentioned flow regimes.

Fig. 1. Flow patterns on stepped chutes: (a) Nappe-flow, (b) transition flow and (c) skimming flow.

Stepped Spillways: Theoretical, Experimental and Numerical Studies 239

Ohtsu et al. (2004) studied stepped spillways with inclined floors, presenting experimental results for angles of inclination of the chute between 5.7 and 55o For angles between 19 and 55o it was observed that the profile of the free surface in the region of uniform flow is independent of the ratio between the step height (s) and the critical depth (hc), that is, s/hc, and that the free surface slope practically equals the slope of the pseudo-bottom. This subsystem was named "Profile Type A". For angles between 5.7 and 19, the unobstructed flow slide is not always parallel to the pseudo-bottom, and the Profile Type A is formed only for small values of s/hc. For large values of s/hc , the authors explain that the profile of the free surface is replaced by varying depths along a step. The skimming flow becomes, in part,

Researchers like Essery & Horner (1978), Sorensen (1985), Rajaratnam (1990) performed experimental and theoretical studies and presented ways to identify nappe flows and skimming flows. Using results of recent studies, Simões (2011) presented the graph of Figure 3a, which contains curves relating the dimensionless s/hc and s/l proposed by different authors. Figure 3b represents a global view of Figure 3a, and shows that the different propositions of the literature may be grouped around two main curves (or lines), dividing the graph in four main areas (gray and white areas in Fig 3a). The boundaries between these four areas are presented as smooth transition regions (light brown in Fig 3b), corresponding to the region which covers the positions of the curves proposed by the different authors.

> Chanson (1994) Chamani and Rajaratnam (1999b) Chanson (2001) Ohtsu et al (2001) Chinnarasri and Wongwise (2004) Ohtsu et al. (2001) Chanson (2001) Boes and Hager (2003a) Chinnarasri and Wongwise (2004) Ohtsu et al. (2004)

(a) (b)

Fig. 3. Criteria for determining the types of flow: (a) curves of different authors (cited in the legend) and (b) analysis of the four main areas (white and gray) and the boundary regions

The energy dissipation of flows along stepped spillways is one of the most important characteristics of these structures. For this reason, several researchers have endeavored to provide equations and charts to allow predictions of the energy dissipation and the residual energy at the toe of stepped spillways and channels. Different studies were performed in different institutions around the world, representing the flows and the related phenomena from different points of view, for example, using the Darcy-Weisbach or the Manning equations, furnishing algebraic equations fitted to experimental data, presenting experimental points by means of graphs, or simulating results using different numerical

(light brown) between the main areas (The lines are: s/hc=2s/l; s/hc = 0.233s/l+1).

parallel to the floor, and this sub-system was named "Profile Type B".

0.0

Type B

**2.2 Skimming flow 2.2.1 Energy dissipation** 

schemes.

0.00 0.75 1.50

Skimming flow

Type A

s/l

Transition flow

1.0

2.0

s/hc

Nappe flow

The introductory considerations made in the first paragraph shows that complexities arise when quantifying such flows, and that specific or general contributions, involving different points of view, are of great importance for the advances in this field. This chapter aims to provide a brief general review of the subject and some results of experimental, numerical and theoretical studies generated at the School of Engineering of Sao Carlos - University of São Paulo, Brazil.
