Macro-turbulent characteristcs of pressures in hydraulic jump formed downstream of a stepped spillway Características macroturbulentas das pressões em um ressalto hidráulico formado a jusante de um vertedouro em degraus

Stilling basins are structures built at the base of the spillway to dissipate energy, by means of a hydraulic jump. Hydraulic jump is a turbulent phenomenon that causes large pressure fluctuation in the stilling basin bottom, and can damage the sink structure through mechanisms such as fatigue, upflit pressure and cavitation. The use of stepped spillways allows the dissipation of a parcel of the energy while the water falls by the spillway, allowing a reduction in the stilling basin’s dimensions and cost. The present article presents the analysis of the longitudinal distribution of mean pressure, pressure fluctuations, skewness coefficient and kurtosis coefficient, derived from tests on physical hydraulic models. Pressure values measured in a stilling basin downstream of a stepped spillway (for Froude numbers between 5 and 8) were compared with data observed in a stilling basin downstream of a smooth spillway with a radius of concordance between the chute and the basin (for Froude numbers between 4.5 and 10). The results of these studies show that the mean pressures and the pressure fluctuation observed in the stilling basin downstream of stepped spillway present maximum values at the spillway’s closest point, differing, thus, from those at the smooth spillway. The longitudinal distribution of skewness and kurtosis coefficients enabled to define the positions for flow detachment start, roller ending and as well as the ending of the influence of the hydraulic jump over the flow.


INTRODUCTION
The use of a stepped spillway allows the dissipation of a significant amount of upstream energy along the chute, reducing the resulting energy at the base of the spillway if compared to the same energy in a smooth chute spillway.Therefore, one of the advantages of the stepped spillway is the reduction of the dimensions of the dissipation structure downstream of the chute.Peterka (1957) and Simões (2008) indicate that a smooth chute can dissipate up to 5% of upstream energy.Several studies carried out with stepped spillways present values for this dissipation of energy that are this extremely larger.For example, the Tozzi (1992) experiments indicated a relation between the residual energies in smooth chutes and stepped chutes ranging from 40 to 66% and, for the studies of Sanagiotto (2003), energy dissipation in stepped chutes in relation to smooth chutes reached values between 45 and 94%.
According to Peterka (1957), a stilling basin where a hydraulic jump is formed downstream of a spillway is classified as a type I basin.According to Schulz et al. (2015), the hydraulic jump can be described as the result of a shock between the subcritical and supercritical flows, resulting in a zone with circular movements (roller).This phenomenon generates great swells, fluctuations of pressure and velocity and can cause damage to the structure.The knowledge of the geometric characteristics of the hydraulic jump is fundamental for the elaboration of the design of the dissipation structures.One of the most studied geometric characteristics is the length of the hydraulic jump due to a high level of imprecision.Peterka (1957) define the end of the hydraulic jump as the section immediately downstream of the roller.Marques, Drapeau and Verrette (1997) and Teixeira (2003) devised the Equations 1 and 2 for calculating the length of the hydraulic jump, respectively.For the length of the roller, Marques, Drapeau and Verrete et al. (1997) devised the Equation 3.
Other parameters that must be known for an adequate design of the dissipation structures are the longitudinal distributions of mean pressure, pressure fluctuation, skewness and kurtosis coefficients.Marques, Drapeau and Verrette (1997) presented experimental results obtained for these parameters measured in a stilling basin downstream of smooth chute spillway with a radius of concordance between the spillway and the channel bottom.In this study the hydraulic jump was positioned so that its beginning coincided with the point of tangency between the radius of concordance of the spillway and the bottom of the channel.In this same line of positioning of the hydraulic jump, other studies were developed, such as Mees (2008), who studied the effects of the hydraulic jump in a type I stilling basin for low numbers of Froude, and Conterato (2014), who studied the effects of the phenomenon in a stilling basin with terminal sill downstream of a stepped spillway.
The present paper presents the results observed in a stilling basin downstream of a stepped spillway and compares them with those observed by Marques, Drapeau and Verrette (1997). (1) (2) ( ) (3) Where j L = hydraulic jump lenght; r y = fast conjugate height; l y = slow conjugate height; r L = lenght of the roller.

THE MODEL AND THE EQUIPMENT
Next, the characteristics of the physical model used to obtain the instantaneous pressure data in the stilling basin downstream of the stepped spillway are described, as well as the characteristics of the physical model used by Marques, Drapeau and Verrette (1997) for stilling basin measurements downstream of smooth chute.

Physical model of the stilling basin downstream of a stepped spillway
The research was developed in a reduced partial model of a stepped spillway with downstream stilling basin.The model consists of a chute with a slope of 1V: 0.75H (53.13), height of 2.45 m, width of 0.40 m and has steps with a height of 6 cm.Downstream of the chute there is a horizontal channel 1.20 m high, approximately 8 m long and 0.40 m wide.To allow visualization of the flow, 5 m of channel length have acrylic walls.Downstream of the channel, a vertical shutter-type floodgate combined with a graduated scale piezometric tube allow level control and verification.Upstream of the stepped chute, a 5 m long segment was built as well as a small reservoir to tranquilize the flow.The channel admits a specific flow of approximately 0.3 m2/s, which corresponds to a depth of water of approximately 0.27 m.A simplified schematic of the model is shown in Figure 1.
To measure the pressures, transducers of the brand Sitron, model SP96 were used, installed in 20 distinct points, distanced from the end of the stepped chute.The transducers were installed with a screwabble support placed under the bottom of the channel, thus possible distortion effects of the pressure signal due to the connection with hoses are eliminated.Figure 2 shows a schematic of the installation of the transducers.Of the 20 transducers used, 10 have a working range of -1.5 to 1.5 mH2O and 10 have a range of -1.5 to 3.0 mH2O.Transducer signals were passed to na analog digital converter of the brand National Instruments, model USB-6225, with digital resolution of 16 bits and admissible voltage range of -10 to 10 V.All the tests were carried out with a duration of 15 minutes and a frequency of 128 Hz, which generated 115200 sample points.Table 1 shows the distances, from the end of the stepped chute, and the transducer range for each tapping point.

Physical model of the stilling basin downstream of smooth chute spillway
The model used by Marques, Drapeau and Verrette (1997) consists of a Creager spillway with a height of 720 mm installed in a channel with a width of 600 mm.The authors obtained the pressure data through 22 transducers of the brand Omega, model PX800-005GV, installed externally to the channel.The taps were attached to the transducers through silicone hoses of 6.4 mm in diameter and 500 mm in length, following the recommendations of Lopardo and Henning (1986).The data was obtained with 50 Hz frequency in measurements lasting 200 s.

METHODOLOGY OF TESTS
The tests carried out in the stilling basin downstream of the stepped spillway were performed with five different flow rates: 40, 60, 80, 100 and 110 L/(s.m).The maximum flow rate was limited in order to avoid channel overflow and minimum flow, by the accuracy of the electromagnetic flow meter, which allows measuring only flows above 40 L/s.Table 2 presents the characteristics of the tests and Figure 3 shows the parameter definition.

(
) (5)  Table 1.Distances and ranges of the transducers for each pressure tapping point.Macro-turbulent characteristcs of pressures in hydraulic jump formed downstream of a stepped spillway

Tapping point
The test was performed with free hydraulic jump.Thus, the downstream level (Nj) must be equal to the slow conjugate height, so the hydraulic jump was positioned at the beginning of the straight section of the stilling basin.The values of l y were measured during the experiments, whereas the values of r y were calculated using the equation of Belanger (Equation 4).The Froude and Reynolds numbers were calculated through Equations 5 and 6, respectively.Figure 4 shows the runoff for the flow rate of 80 L/s.
With the data obtained from the tests performed it was possible to analyze the longitudinal distribution of mean pressure, standard deviation, skewness and kurtosis coefficients and compare them with the same analyzes performed by Marques, Drapeau and Verrette (1997).
Marques, Drapeau and Verrette (1997) performed their tests, also, with free hydraulic jump.Table 3 presents the characteristics of the these tests.

MEAN PRESSURE
The mean pressure data were presented as a function of the position of the hydraulic jump according to Equation 7, as suggested by Marques, Drapeau and Verrette (1997).
x r l r l r P y x f y y y y Where x P = mean pressure at the point x in meters of depth of water, x = distance from the transducer to the start point of the stilling basin.
Figure 5 presents the gross values obtained for the mean pressure in the tests performed and Figure 6 presents the obtained curves, dimensionless according to Equation 7, compared to those obtained by Marques, Drapeau and Verrette (1997).
Analyzing Figure 5 it shows that the values of gross mean pressure increase with Froude number.It is observed that the greater pressure variation occurs in the zone of impact of the jet in the basin because of the absence of curvature.
From Figure 6 it can be asserted that all the conditions tested in the present study show a similar pattern.The values obtained for the point nearest to the spillway, where the impact of the water jet in the stilling basin occurs, are higher than those observed in Marques, Drapeau and Verrette (1997) to the same point.This fact occurs due to the presence of radius of acordance between spillway and stilling basin in the study of the authors, which significantly reduces the pressure at the beginning of the basin.After this, an abrupt reduction in pressure occurs to approximately the position of 1 ( l r y y − ).From this point on the data obtained for the present study show a similar pattern to the data obtained by Marques, Drapeau and Verrette (1997), increasing with approximately the same gradient to the position of about 4 ( l r y y − ).This zone in which the values of the mean pressures increase with the same gradient is represented in Figure 6 as zone 1.Thereafter, the mean pressure continues to increase, however, more smoothly, to the approximate distance of 8.5 (zone 2).From this point on the values begin to oscillate until they become practically constant (zone 3).

PRESSURE FLUCTUATION
For the pressure fluctuation analysis, the relation presented in Equation 8was used to group the data into a single curve.Figure 7 presents the gross values obtained for the standard deviation in the tests performed and Figure 8   Novakoski et al.
dimensionless curves obtained for pressure fluctuation, according to Equation 8, compared to those obtained by Marques, Drapeau and Verrette (1997).
Where x σ = standard deviation at the point x in meters of depth of water; t H = loss of head in the hydraulic jump.Observing Figure 7 it shows that the standard deviation gross values increase with Froude number, ranging from about 0 to 0.35.
From Figure 8 it can be asserted that the pressure fluctuation values occurring in the present study are maximum at the point closest to the chute, then the flotation undergoes a strong reduction to the approximate position of 1,0 ( l r y y − ), from that point to the distance of about 4.0 ( l r y y − ) the pressure fluctuation still decreases, but more gently.After that the values tend to stabilize.Comparing with the results of Marques, Drapeau and Verrette (1997), it can be observed that, as well as the mean pressures, the pressure fluctuations obtained by the authors at points closesest to the chute are lower than those obtained in the present study.This effect occurs due to the presence of a radius of concordance between spillway and stilling basin in the study of Marques, Drapeau and Verrette (1997) which, in addition to the mean pressure values, also significantly reduces the pressure fluctuations at the beginning of the basin.After this, the curves plotted with the data of the present study tend to meet the curves obtained by the authors, presenting very similar patterns.This encounter occurs at the approximate position of 3.0 ( ) l r y y − . Figure 9 presents a detail of the same curves focusing on the pressure fluctuations up to the dimensionless value of 1.4.

SKEWNESS COEFFICIENT
The analysis of the skewness coefficient of a series of instantaneous pressures indicates how much the extreme values influence the mean.When the coefficient is positive, there are more values of instantaneous pressures highter than the mean, the function is shifted to the right and the value of the mean is elevated.Contrarily, a negative skewness coefficient indicates the greater presence of instantaneous pressure values lower than the mean, which shifts the function to the left and causes a reduction in the mean value.According to Lopardo and Henning (1986), negative skewness means areas of flow detachment from the    Macro-turbulent characteristcs of pressures in hydraulic jump formed downstream of a stepped spillway bottom of the channel.The skewness coefficient is presented in Equation 9. Figure 10 shows the values obtained for the skewness as a function of the gross distance in the tests performed and Figure 11 presents the skewness coefficient as a function of the dimensionless distance for the present study in comparison to the results of Marques, Drapeau and Verrette (1997).
Where i P = instantaneous pressure at point x; n = number of instantaneous pressures measured and x σ = standard deviation at position x.
Figure 10 indicates that skewness values range from approximately -0.7 to 0.9, reaching maximum values at positions between 0 and 100 cm and minimum values between 100 and 300 cm.
Figure 11 shows the curves obtained for the present study.It can be observed that from the vicinity of the chute until the approximate position of 4.0 ( ) l r y y − , the skewness coefficient is positive and presentes the highest value in the sample.This is due to the impact of the water jet in the basin, which causes higher instantaneous pressures.According to Marques, Drapeau and Verrette. (1997), it is also at this point that the flow begins to present a vertical componente to the velocity.From this point on, according to visual observations of the flow combined with the analysis of data acquired by the cited authors, the value of the skewness coefficient begins to reach negative values, due to the fact that the flow begins to separate (detach) from the bottom of the basin, generating lower extreme pressures.This analysis coincides with the conclusions presented by Lopardo and Henning (1986).The minimum skewness coefficient is displayed at the position of , the skewness values are higher.From that point on, the two studies have similar curves, the points at which the minimum values of the skewness coefficient occur and where this coefficient again reaches 0, are practically the same in both studies.

KURTOSIS COEFFICIENT
The kurtosis coefficient (k) is a dispersion coefficient that indicates how much the values are concentrated in relation to the mean.A distribution with values that are more concentrated in relation to the mean, which makes it more tapered, presents a kurtosis coefficient of less than three.When the coefficient is higher than three, the distribution is more flattened, presenting less concentrated values than the mean.Equation 10presents the calculation of the kurtosis coefficient.Figure 12 shows that the values of the kurtosis coefficient vary approximately from 3.0 to 5.0, and a single point with a kurtosis value close to 6.0 can be observed, which may mean an error in the acquisition of the data.
From Figure 13 it can be asserted that, from the beginning of the stilling basin to the approximate position of 1 ( ) l r y y −

Figure 1 .
Figure 1.Schematic of the stepped spillway and the stilling basin.
Where Q = flow, r v = mean velocity in the section of the fast conjugate height; r F = Froude number of the supercritical flow of the hydraulic jump; e R = Reynolds number of the supercritical flow of the hydraulic jump; g = acceleration of gravity; ν = kinematic viscosity of the fluid.

Figure 2 .
Figure 2. Schematic of transducer installation.Table1.Distances and ranges of the transducers for each pressure tapping point.

Figure 4 .
Figure 4. Runoff for the flow of 80 L/s.

Figure 6 .
Figure 6.Mean pressure as a function of the distance of the stepped spillway compared to the mean pressure obtained in the smooth chute.

Figure 7 .
Figure 7. Gross data of standard deviation.

Figure 8 .
Figure 8. Pressure fluctuation as a function of the distance of the stepped spillway compared to the mean pressure obtained in the smooth chute.
position of the end of the roller in the hydraulic jump.At this point the vertical components of the velocity are more important, and the extreme pressures lower than the mean predominate over the higher ones.From this point on the flow tends to an equilibrium, being parallel to the bottom of the basin, so the skewness coefficient tends to a constant value, close to 0.20.The coefficient reaches this value in the position of , most points tend to stabilize the skewness coefficient at values slightly above zero, no longer suffering the influence of the hydraulic jump.The data obtained byMarques, Drapeau and Verrette (1997) show a similar trend, but at the beginning of the stilling basin to the approximate position of Figure 12 presents the gross values obtained for the kurtosis coefficient in the tests performed and Figure 13 presents the results obtained for the present study compared with the values obtained by Marques, Drapeau and Verrette (1997).

Figure 9 .
Figure 9. Detail of the pressure fluctuation as a function of the distance of the stepped spillway compared to the mean pressure obtained in the smooth chute.

Figure 10 .
Figure 10.Skewness coefficient data as a function of the (gross) distance of the stepped spillway.

Figure 11 .
Figure 11.Distribution of skewness coefficients as a function of the distance (dimensionless) of the stepped spillway compared to that obtained in the smooth chute.

Figure 12 .
Figure 12.Data on kurtosis coefficients as a function of the (gross) distance of the stepped spillway.

Table 2 .
Characteristics of the test perfomed in stilling basin downstream of a stepped spillway.

Table 3 .
Characteristics of the tests in stilling basin downstream of smooth chute spillway.