HYDRAULIC JUMP AT A POSITIVE STEP

Dipartimento di Ingegneria Idraulica Ambientale e del Rilevamento, Politecnico di Milano Piazza L. Da Vinci, 32, 20133 Milano, Italy

Abstract: Designing adequate protection for stilling basins characterised by a positive step on the bottom requires the knowledge of the acting forces. Laboratory tests have been performed, sampling pressure values (with a 100 Hz frequency and a 15 min duration) on the vertical axis of the positive step positioned in a rectangular flume where supercritical flow takes place. The first four statistical moments of recorded pressure time series are analysed with the aim of characterising the pressure field as a function of flow characteristics. Some empirical relationships are proposed to describe mean and standard deviation of local pressure as a function of few significant parameters.

1 INTRODUCTION

Since long time at the Hydraulics Laboratory of the Politecnico di Milano, experimental tests are carried out in order to determine the different types of hydraulic jump that can be observed in a stilling basin.

As a first task, pressure fluctuations on a sill in the presence of a supercritical flow have been studied, analysing the main statistics and studying the structural stability with spectral analysis ([1], [3], [5], [7]). Here we present the results of a series of studies focused on the characterisation of the pressure fluctuations on a positive step in the presence of a supercritical flow (both when a backwater effect was present or not).

In order to obtain a consistent classification of the different types of hydraulic jumps that can be observed in a rectangular channel with a positive step, a dimensional analysis has been performed. The general formulation relating mean flow parameters of the phenomena is:

where s is the height of the step, h₁ is the depth of the upstream supercritical flow, h₂ is the depth of the flow in correspondence of the step, X_s is the distance between the step and the undisturbed upstream flow, V₁ is the mean velocity of the upstream supercritical flow (corresponding to depth h₁), i is the channel slope, g is gravity, r and m are density and viscosity of water, respectively.

Assuming that the channel has zero slope, neglecting the effects of the viscosity and rewriting equation (1) in a dimensionless format, the following equation can be obtained:

where L_j is the classic (undisturbed) hydraulic jump length and Fr₁ the upstream supercritical flow Froude number.

Experimental tests is allowed to establish different behaviours of the mean flow around the step, in agreement with Hager et al. [6] [10]. Different flow conditions can be identified using the three Cartesian axes F₁ = Fr₁, Y = h₂/h₁ and S = s/h₁. In particular, four different regions may be distinguished: 1) classic hydraulic jump region; 2) a* region: here the flow is subcritical close to the step and in its neighbourhood the characteristics of a forced hydraulic jump are recognisable; 3) b* region: here the flow is similar to a jet which flows to the step to become subcritical on it; 4) g* region: here the flow is similar to that in b* region, but supercritical conditions are maintained on the whole step. Hager et al. [12] [24], with a procedure based on the momentum equation, showed that transition from the different regions happens through three surfaces that have the following mathematical expression:

The equation (3) is the surface separating the a* region from the region where the classical hydraulic jump occurs. Here the hydraulic jump is termed A type, in which the main vortex ends exactly where the step begins. Equation (4) describes the boundary between regions a* and b*; in the latter the B type hydraulic jump occurs. In this case a vortex partly develops before of the step. Equation (5) is the surface separating b* and g* regions: here, flow is supercritical both upstream and downstream of the step, and the critical flow depth is reached on the step. Figure 1 summarises the above flow conditions.

Our analysis are carried out in the regions b* (a few of them in the proximity of the boundary between a* and b*) and g*, because of the limitations of the experimental set-up. In table 1 the characteristics of the flow tests are reported. In these tests the pressure fluctuations have been recorded. In many other tests, not reported, only the mean pressure values have been measured.

Test	F₁ = Fr1	S = s/h₁	Y = h₂/h₁	Flow region
A	6.41	3.00	-	g*
B	7.05	3.17	-	g*
C	10.54	4.97	-	g*
D	6.01	2.52	-	g*
E	12.28	4.26	-	g*
F	7.80	3.26	-	g*
G	7.32	2.98	4.74	b*
H	9.11	3.55	7.54	b*
I	10.88	4.46	7.89	b*
L	10.81	4.45	7.79	b*
M	9.92	4.20	7.01	b*

2 EXPERIMENTAL SET-UP AND ACQUISITION SYSTEM

Experiments were carried out in an horizontal rectangular flume (width B = 0.40 m) with a continuous positive step (height s = 0.075 m, length = 1.55 m). Figure 2 shows a sketch of the experimental set-up

Nine pressure taps (0.0015 m internal diameter) were connected to piezometers to measure pressures on the vertical axis of the positive step surface. Local pressures have been measured by means of nine piezo-resistive pressure transducers (mod. Motorola MPX10D) connected to the pressure taps by short brass pipes (0.001 m internal diameter).

The acquisition characteristics and the care for eliminating air bubbles in the connecting pipes are the same ones used for sill tests documented by [7].

The experience which has been acquired during the sill study, allowed to limit the number of preliminary tests performed to establish the sampling frequency (f = 100 Hz).

Then, the first four statistical moments of local pressure fluctuations at each tested location on the positive step have been evaluated, i.e. time-average, p_m, variance, s², skewness, S, and kurtosis, K. It was possible to verify that average and variance of pressure fluctuations tend to converge to a quite stable value when computed over blocks of more than 60000 samples, while skewness and kurtosis still have some convergence problems with the same amount of data. We then decided to adopt T = 900 s, to obtain local pressure series of N = 90000 data.

3 EXPERIMENTAL RESULTS AND STATISTICAL ANALYSIS

The main statistics of the time-recorded pressure fluctuations at the nine transducers have been analysed. In general, a significant difference between these values and the characteristics of the flow conditions experimentally obtained (b* and g*) are observed.

Mean local pressure has been analysed with the same method proposed in [11]. For tests in the region g* the following relationships are proposed:

is the mean value of the time-averaged local pressure evaluated on the vertical axis and y is the location of a given sensor from the bed of the flume.

The comparison between experimental data and equation (7) appears quite satisfactory (see Figure 3).

Operatively, once the flow characteristics (Fr₁, s/h₁) are known, it is possible to compute the p^* value from the equation (6). This, in turn, inserted into equation (7), renders the mean pressure p_m at different locations of the vertical axis of the step.

For tests belonging to b* region, the experimental data show the distribution of the mean pressure has a less accentuated S shape, with values closer to the static condition. This behaviour, different from the one observed in g* region, appears to be related to the presence of the downstream subcritical flow. The latter tends to smooth out the dynamical effect of the incoming supercritical flow. However, the link between the mean pressure at each point along the vertical axis of the step and the mean flow characteristics can still be analysed with the method used for tests in the g* region, obtaining the relations:

Figure 4 depicts the agreement between values obtained from equation (10) and the experimental data.

The standard deviation of local pressures has been analysed with the method proposed in [7], to investigate the spread of the pressure distribution around the mean. The following relationship has been obtained:

where a₀ = 0.98, a₁ = –3.87, a₂ = 19.36, a₃ = -31.07, a₄ = 21.60, a₅ = –6.28 from a multiregressive analysis and

is the weighted average of the local standard deviations along the vertical axis. The latter is in turn a function of the flow characteristics (Fr₁, s/h₁). It has not been possible, with the available experimental data, to determine a mathematical function to give the s^* value. However, the non-dimensional quantity S* = s* /(0.5 r

) has been shown to vary in the range 0.024 ÷ 0.050. Upon taking its maximum value (0.05), on the side of safety, the standard deviation can be computed by equation (11) for each location of the vertical axis of the positive step. Figure 5 shows that equation (11), which is valid for tests in the g* region and in b* region, is able to qualitatively reproduce the experimental values trend.

For statistical moments of third and fourth order, skewness and kurtosis, a clear link with the flow characteristics has not been found.

Results allow only qualitative considerations. Skewness S, which gives information on the symmetry of the probability distribution, is always positive. This means the pressure fluctuation distribution is non symmetrical and positive differences from the average value are more frequent than negative ones. Kurtosis K, which measures the spread of the probability function round the mean value when the standard deviation is the same, has always positive values. This means that positive differences from the average value are bigger than the negative ones.

Of some interest are the maximum values p_max of the pressure on the step, once transformed in non-dimensional values. The ratios between the local difference (p_max – p_m) and the corresponding standard deviation are seen to vary within the interval 4.0 ÷ 7.5.

From experiments in a stilling basin with we know that extreme pressure fluctuations on the bottom attain constant values after more than 24 hours [9]. Nevertheless, Larcan et al. [7] show that the increasing of the p_max value after few hours is of the order of 15%, in a stilling basin with a bottom sill. Therefore, the values above mentioned can be considered a significant guideline for the design.

4 CONCLUSIONS

Pressure fluctuations are measured on the face of a positive step placed in a laboratory flume. Possible flow conditions are identified around the step, as a function of mean flow parameters, in the presence of a supercritical flow impinging the step, with and without depth control at the downstream end of the flume. Some mathematical relations have been obtained which allow one to determine the pressure values to use in the determination of the design forces, once the flow characteristics near the step are known. Further insight might be obtained via a more detailed analysis of the time series in the frequency domain.

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Fig. 5 Distribution of the dimensionless time-average pressure in g* and b* region