**2.1 Flow regimes**

It is interesting to observe that flows along stepped chutes have also interested a relevant person in the human history like Leonardo da Vinci. Figure 2a shows a well-known da Vinci's sketch (a mirror image), in which a nappe-flow is represented, with its successive falls. We cannot affirm that the sketching of such flow had scientific or aesthetic purposes, but it is curious that it attracted da Vinci´s attention. Considering the same geometry outlined by the artist, if we increase the flow rate the "successive falls pattern" changes to a flow having a main channel in the longitudinal direction and secondary currents in the "cavities" formed by the steps, that is, the skimming flow mentioned in the introduction. Figure 2b shows a drawing from the book *Hydraulica* of Johann Bernoulli, which illustrates the formation of large eddies due to the passage of the flow along step-formed discontinuities.

Fig. 2. Historical drawings related to the fields of turbulent flows in channels and stepped spillways: (a) Sketch attributed to Leonardo da Vinci (Richter, 1883, p.236) (mirror image), (b) Sketch presented in the book of Johann Bernoulli (Bernoulli, 1743, p.368).

The studies of Horner (1969), Rajaratnam (1990), Diez-Cascon et al. (1991), among others, presented the abovementioned patterns as two flow "regimes" for stepped chutes. For specific "intermediate conditions" that do not fit these two regimes, the transition flow was then defined (Ohtsu & Yasuda, 1997). Chanson (2002) exposed an interesting sub-division of the three regimes. The nappe flow regime is divided into three sub-types, characterized by the formation or absence of hydraulic jumps on the bed of the stairs. The skimming flow regime is sub-divided considering the geometry of the steps and the flow conditions that lead to different configurations of the flow fields near the steps. Even the transition flow regime may be divided into sub-types, as can be found in the study of Carosi & Chanson (2006).

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

In this section we present some key themes, chosen accordingly to the studies described in the next sections. Additional sources, useful to complement the text, are cited along the

It is interesting to observe that flows along stepped chutes have also interested a relevant person in the human history like Leonardo da Vinci. Figure 2a shows a well-known da Vinci's sketch (a mirror image), in which a nappe-flow is represented, with its successive falls. We cannot affirm that the sketching of such flow had scientific or aesthetic purposes, but it is curious that it attracted da Vinci´s attention. Considering the same geometry outlined by the artist, if we increase the flow rate the "successive falls pattern" changes to a flow having a main channel in the longitudinal direction and secondary currents in the "cavities" formed by the steps, that is, the skimming flow mentioned in the introduction. Figure 2b shows a drawing from the book *Hydraulica* of Johann Bernoulli, which illustrates the formation of large eddies due to the passage of the flow along step-formed

Fig. 2. Historical drawings related to the fields of turbulent flows in channels and stepped spillways: (a) Sketch attributed to Leonardo da Vinci (Richter, 1883, p.236) (mirror image),

The studies of Horner (1969), Rajaratnam (1990), Diez-Cascon et al. (1991), among others, presented the abovementioned patterns as two flow "regimes" for stepped chutes. For specific "intermediate conditions" that do not fit these two regimes, the transition flow was then defined (Ohtsu & Yasuda, 1997). Chanson (2002) exposed an interesting sub-division of the three regimes. The nappe flow regime is divided into three sub-types, characterized by the formation or absence of hydraulic jumps on the bed of the stairs. The skimming flow regime is sub-divided considering the geometry of the steps and the flow conditions that lead to different configurations of the flow fields near the steps. Even the transition flow regime may

(b) Sketch presented in the book of Johann Bernoulli (Bernoulli, 1743, p.368).

be divided into sub-types, as can be found in the study of Carosi & Chanson (2006).

**2. A brief introduction and review of stepped chutes and spillways** 

São Paulo, Brazil.

explanations.

**2.1 Flow regimes** 

discontinuities.

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

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.

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