(1)
1De Marinis G., 2Fratino U. and 2Piccinni A.F.
1Dip. Meccanica, Strutture, Ambiente e Territorio, Universit¨¤ di Cassino, Cassino (FR), Italy
via G. Di Biasio, 43 - 03043 Cassino (FR) Italy
2Dipartimento di Ingegneria Civile ed Ambientale, Politecnico di Bari, Bari, Italy
Via E. Orabona, 4 - 70125 Bari -Italy
Phone +390805963321 ¨C Fax +390805963321 - E-mail u.fratino@poliba.it
Abstract: The recent improvements of the construction
techniques and the particular attention to the environment have renewed the
interest in stepped spillways. New experimental investigations have been carried out at the Technical
University of Bari, extending the available data on the hydraulic behaviour. The
classification of flow regimes has been re-analysed, taking into account the
mathematical relationships derived from the technical literature. The study
confirms the difficulties in understanding the problem and points out the need
for standard experimental procedures, for refining the hydraulic comprehension
of the problem.
Keywords: stepped spillways, flow conditions, transition flow, nappe flow, skimming flow
The flow
conditions in stepped channels are strongly connected with the step height, the
flow rate and the channel slope. For a given step height and channel slope, two
different hydrodynamic behaviours can be observed, depending on the discharge
value. In particular, for lower discharges, the water flow is composed by a
succession of free falling jets (nappe
flow or jet flow) while, for higher discharge values, a coherent stream is
obtained (skimming flow). In the
latter case, a pseudo bottom is defined by a straight line connecting the edges
of each step (Chanson, 1994).
In the past, several methods have been proposed for defining the transition between the two regimes, even if the flow process doesn¡¯t seem to give rise to well-defined limits but, as confirmed by experimental observations, it follows a transition phase in which the flow is of intermediate characteristics with respect to the above-defined regimes (De Marinis & al., 2000).
The first
criterion for defining the onset of skimming flow was proposed by Rajaratnam (Rajaratnam, 1990) who defined the onset
of skimming flow for values of the ratio between the critical depth (k) and the step height (h) greater than 0.8. This criterion,
which was based on the analysis of available data (Essery and Horner, 1971), is assumed to be valid when the ratio h/l falls between 0.4 and 0.9, where l is the step length.
Two different
ways to define the onset of skimming flow were then proposed by Chanson. In
particular, a first relation (Chanson,
1994) was obtained by means of a fitting procedure of the experimental data:
(1)
that it is valid, in a range ¡À20%, for slopes between 0.2 and 1.25. It is underlined that this procedure, referred to different data set or different construction materials, has been subsequently used by several researchers (Mondardo & Fabiani, 1995; Boes & Minor, 2000).
The second relation (Chanson, 1996) was derived by imposing that, at the onset of skimming flow, the air cavities beneath the falling nappe disappear and by using simplified jet trajectory calculations:
(2)
in which
is the Froude number at the step edge and
is the angle of the streamlines
falling from the step that, at the onset of skimming flow, should be equal to
the channel slope.
Ohtsu and Yasuda
(Ohtsu & Yasuda, 1997) proposed a
different approach to the problem by introducing a transition regime that appears to fit the available experimental
data, even if some uncertainties remain.
Recently, their
analysis has been improved, producing two equations that can be used to define
the upper limit for the nappe flow regime and the lower limit for the skimming
flow regime (Yasuda & Ohtsu, 1999).
The proposed equations are:
(3)
(4)
and they appear to be effective for chute inclination angles lower than
55¡ã.
Using an hypothesis quite similar to
Chanson¡¯s one (Chanson, 1996),
Chamani and Rajaratnam (Chamani and
Rajaratnam, 1999) proposed another criterion for defining the onset of
skimming flow. The upper limit of the nappe flow domain is defined by:
(5)
while the onset of skimming flow for steeply sloped structures (h/l ³ 1) is given by:
(6)
Equations (5) and (6) have been derived, through some empirical
relationships (Chamani & Rajaratnam,
1995; Rand, 1955). Equation (5) has been obtained by imposing that the
length of the pool under the jet equals the step length (in other words, the
inner side of the jet coincides with the tip of the step) and equation (6) has
been obtained by observing that, in incipient skimming flow, the jet becomes
parallel to the spillway slope.
However, the
authors observed that equation (5) underestimates the experimental data and
hence its applicability appears to be limited, while equation (6) provides a
rather good fitting of their data. This observation points out that the
application range of this methodology is too narrow and probably is strongly
related to their experimental data.
From the
previously recalled studies, it appears that the hydraulic behaviour of stepped
spillways is not clearly understood and that the available experimental data are
very difficult to compare with each other.
The goal of this
paper is to evaluate the various approaches, in order to verify some hypotheses
concerning the limits of validity of different flow typologies. For this
purpose, new experimental tests and some data available in the technical
literature are analysed.
The experimental tests were performed in the laboratory of Civil and Environmental Engineering Department at Technical University of Bari using 0.75 m wide stepped spillways model. The model, arranged by means of PVC plates, is able to realize three different geometric configurations having the same number of steps (14) and the same step height (24 mm) but different slopes, respectively characterized by a h/l ratio equal to 0.48, 0.24 and 0.12 [Fig. 1]. During the tests, a maximum flow rate equal to 0.05 m2/s was used in order to assure stability of the structure. The discharge measurement involved an orifice plate and a water differential manometer, even if the values have been checked using the broad-crested weir equation, with satisfactory results. Details of the performed tests are available in a previous paper (De Marinis et al., 2000).
From a general
point of view, the skimming flow conditions are defined when the air cavities
beneath the falling jet disappear in all steps of the spillway, while a
transition flow regime is assumed when some cavities along the stepped channel
are still observed.
In the present
study, the flow regimes were identified by means of a visual interpretation that
made it possible to attribute a specific character to each flow regime.
This methodology
leaves a large subjectivity to the intermediate conditions in which the observed
flows are characterised by gravitational behaviour with a partial hydraulic jump
in the initial steps and the formation of whirling zones along the last steps.
This kind of
choice makes impossible to define a precise flow rate from which the skimming
flow regime occurs or, in equivalent manner, from which a nappe flow regime
disappears. For this reason, in the Figures, our experimental data are reported
using the acronyms NFR (Nappe Flow Regime) and SFR (Skimming Flow Regime) for
indicating hydraulic behaviours certainly applying to nappe or to skimming flow
regime.
In the Figure 2, some data, derived from the technical literature and concerning only concrete stepped spillways, and criteria by Chanson (Chanson 1996) and by Rajaratnam & Chamani (Chamani & Rajaratnam, 1999) are reported.
The analysis of the figure highlights the limited reliability of both the methodologies, since the criteria appears to be able to fit the experimental data just for steeply channel slopes and not completely.
Besides, as just mentioned by other researchers (Pinhero & Fael, 2000), the curve described by (2), for any values of the Froude number at the step edge, do not seem adequate, since for values of the k/h ratio close to zero, it doesn¡¯t show an asymptotic tendency, as expected. About the curve described by (5), it doesn¡¯t match any data, so the hypothesis, advanced by the authors, to evaluate it as the limit for defining the jet flow domain appearsrather forced, not being possible any experimental check
In a second step, the analysis has been performed comparing
the experimental data with Rajaratnam, Chanson [equation (1)], Yasuda and Ohtsu
[equation (3) and (4)] and Boes¡¯ criteria (Boes & Minor, 2000) (Fig. 3). The Figure points again at the
variety of the approaches and the different ways in which the researchers
interpreted the limited amount of available data for an identification
criterion.
It should be
stressed, however, that there is an appreciable coincidence of our experimental
data and the criterion proposed by Yasuda and Ohtsu. In fact, this approach
seems to interpret the evolutionary process of the flow, as observed during our
tests, although its analytical definition requires a finer detail, being
actually strongly related to a limited data set.
The other criteria, all derived by means of fitting operation on available experimental data, present a limited applicability and, above all, they don¡¯t have any physical basis, neglecting some experimental observations regarding flow conditions in which the two regimes (nappe and skimming) clearly overlaps.
Besides, it must be pointed out as the amount of experimental data about the onset of skimming flow arouse some doubts, above all about the way in which they were defined.
It is probable, in fact, owing to the different procedures used during the experimental tests, that they don¡¯t represent equal flow conditions, but just similar ones.
This study
focused its attention on the reliability of the criteria used for defining the
flow conditions in a stepped channel. Despite the long lasting interest of the
scientific community, some uncertainties still exist on the flow regimes. It was
also found that the methodological approaches do not account for all the
relevant variables of the problem. In particular, some criteria used for
determining and classifying the flow regimes do not fully agree with the
experimental data.
As it happens
with other hydraulic phenomena characterised by two very different flow
typologies, a transition zone clearly exists in which some features of the two
hydraulic regimes overlap with each other. This flow condition, which was
observed in the experiments presented, shows that a new approach is needed to
fit the experimental data.
Moreover, the
inherent difficulty in defining the variables that determine the occurrence of
one or the other regime makes it very difficult to provide a universally
accepted criterion for the definition of the flow regime, through a combination
of geometric and hydraulic parameters. This finding, which should be taken into
account for future studies, confirms the need for a standard experimental
procedure and for a general re-examination of the present methodical approaches.
References
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