CAVITATION BY MACROTURBULENT PRESSURE
FLUCTUATIONS IN HYDRAULIC JUMP STILLING BASINS

Claudio A. Fattor, Maria C. Lopardo, Jose M. Casado and Raul A. Lopardo

National Institute of Water and Environment (INA)

C.C. 46 (1802) Aeropuerto Ezeiza, ARGENTINA,

Tel. 54-11-4480 0869, E-mail: rlopardo@ina.gov.ar

Abstract: This paper deals with the destructive action of macroturbulent flows induced by hydraulic jumps in stilling basins with particular focus on cavitation inception and cavitation damages by severe pressure fluctuations in relatively low velocity flows. After laboratory and prototype research, the pressure fluctuation amplitude of 0.1% by more negative values is proposed as a representative amplitude in statistic data analysis for cavitation tendency. As the cavitation coefficient is not useful in this type of cavitation, physical modelling in Froudian similarity with generous scale is used for cavitation tendency verification. Methodologies based on air incorporation have been applied successfully to prevent cavitation in stilling basins with low incident Froude Number and large specific discharge.

Keywords: cavitation, pressure fluctuations, hydraulic jump, stilling basin

1    INTRODUCTION

Cavitation in hydraulic structures has a long history and there is an extensive literature on the subject. Most of them are focused on cavitation as a problem of “high velocity flows” and technical papers usually refer to the “cavitation coefficient”. This coefficient becomes useless in macroturbulent flows, such as the hydraulic jump energy dissipators, where cavitation damages are present in relatively low velocity flows and mean pressures are above vapour pressure.

The macroscopic hydraulic design of stilling basins, based on mean pressure and velocity values, was the subject of numerous research activities in different hydraulic laboratories around the world. As a result of laboratory experiments and prototype data, there are a well known “design criteria” in manuals for hydraulic engineers covering this aspect for conventional structures. Nevertheless, the problems and destruction of stilling basins are not avoided. The internal flow of hydraulic jump is essentially an unsteady flow subjected to macroturbulent random fluctuations and it was not known enough. The random nature of these fluctuating actions requires the stochastic approach to reach results for practical applications.

Hydraulic jump energy dissipation is always associated with severe pressure fluctuations acting on the floor and walls of stilling basins, downstream of spillway piers and on any appurtenances included for forced energy dissipation. Fluctuating actions may be responsible for important damages in hydraulic structures caused by the lifting of whole slabs, structural vibrations, fatigue of materials and intermittent cavitation due to instantaneous depressions.

Macroturbulence is associated with vortex motion of large size and low frequencies. Because of the random nature of the flow, the experimental study must be based on the knowledge of several statistical parameters of amplitudes and frequencies as functions of the Froude Number and the co-ordinates of the measurement point. Mathematically, turbulent fluctuations in real flows can be considered as a stochastic, steady and ergodic process. Due to the random nature of the process, the power spectra density function and the probability distribution function are the most suitable ways of presenting a quantitative description. The spectral and the probabilistic analysis provide the statistical parameters required for design purposes, such as r.m.s. amplitude, amplitudes with several percentiles of probability, the skewness of the probability density function, peak and mean frequencies, zero-crossing frequency, etc.

In order to achieve a correct similarity in macroturbulent flows, as is the case of hydraulic jump energy dissipators, Froude law scales without geometric distortion must be adopted. Using a Froudian model with length scale non lower than 1:50, incident Reynolds Number R₁ up to 100,000 and sequent depth h₁ larger than 3 cm, the amplitude and frequency simulation of pressure fluctuations in the model have excellent agreement with prototype data. These model-prototype comparisons were tested by Lopardo [1] for different appurtenances submerged in hydraulic jump stilling basins. This research included damage evaluations due to cavitation by pressure pulses in prototype with respect to instant depression measures in physical models.

2    CAVITATION BY PRESSURE PULSES

Basic research was essentially conducted to pressure fluctuations beneath free and stable hydraulic jumps, downstream sluice gates and spillway chutes, beneath submerged jumps and beneath forced hydraulic jumps. By means of these studies the influence of the incident Froude Number F₁, the gage location x/h₁ and the submergence factor on the pressure fluctuation coefficient C'_p was clearly determined. C'_p. The fluctuation coefficient is C'_p = , where  is the pressure fluctuation amplitude r.m.s. As it was shown, for a free hydraulic jump below a horizontal bed, it is possible to propose:

C'_p = C'_p (x/h₁, F₁, inflow condition).                    (1)

Experimental data of Lopardo and Henning [2] demonstrate that inflow jump conditions exert a great influence on the instantaneous pressure field, as is the case of macroturbulent flow structures in the stilling basin.

In the range of steady jumps (4.5 < F₁ < 10) downstream a spillway chute, the maximum value of C'_p occurs for locations in the range 8 < x/h₁ < 12 and the largest is for F₁ = 6.5. When the stilling basin is placed downstream a sluice gate, the maximum C'_p was obtained for F₁ = 4.5, where there is a transition between oscillating jumps and steady jumps.

The probability density function allows the calculation of several negatives and positive pressure amplitudes with different probabilities of occurrence for the instantaneous pressure recorded data. Then, from the statistical analysis it is possible to obtain the probability of occurrence of the pressure fluctuation that reaches the vapour pressure, as a tool to check the cavitation tendency.

The cavitation by pressure pulses does not require high velocities or very low mean depressions. It is not necessary to displace the cavity to higher pressure regions in order to yield the implosion, since the extremely high pressure variation takes place at the same point. It was also demonstrated that appurtenances into the macroturbulent flow (which increase the efficiency of the energy dissipation) increase the pressure fluctuation amplitudes, the spatial pressure correlation and the energy concentration around a dominant or peak frequency f_p.

Fluctuating depressions due to low pressure pulses may increase the cavitation risks in hydraulic jump stilling basins, even though mean pressure values are largely above the fluid vapour pressure. There is evidence for stilling basins with chute blocks and submerged piers that cavitation inception and severe damages may occur with mean pressure values also above the atmospheric pressure due to random pressure fluctuations, as was presented by Lopardo et al [3]. Then, the stochastic analysis of pressure fluctuation amplitudes is essential.

The use of a random parameter, such as C'_p, derived from Gaussian assumptions, is not enough to achieve a correct cavitation tendency description. Irrefutable experiments demonstrate that flow separation zones are associated with negative skewness of the pressure amplitude probability density function. Then, downstream of keen-edged structures located into the jump flow (as spillway piers, chute blocks, baffle piers and end sills), negative instantaneous fluctuations are strongly increased with regard to the Gaussian approximation.

Fig. 1    Free hydraulic jump. froude number F₁ = 4

Laboratory and prototype data showed that representative pressure fluctuation amplitude for cavitation analysis is the p'_0.1%, with 0.1% probability of being surpassed by more negative values. Figure 1 illustrates on the pressure fluctuation amplitude values (constructed with and constructed with p'_0.1%) in a free jump stilling basin for a given Froude Number.

3    CAVITATION NUMBER

The cavitation inception in hydraulic structures induced by turbulent flows can be characterised by means of the critical value s_c of the adimensional coefficient defined as cavitation number or Leroux number s. The cavitation number does not indicate by itself the possibility of cavitation inception. It is necessary to compare s with the corresponding critical value for a given boundary condition s_c, obtained from experimental laboratory tests.

The cavitation number s is not useful for macroturbulent flows, such as the internal hydraulic jump flow, because of the impossibility to obtain the critical value of s_c by means of laboratory tests. As was proposed by Hu Minglong [4], in order to use s_c obtained from conventional laboratory tests, it is necessary to use a new macroturbulent cavitation number:

s'= (p_¥–p_v) (p–p_v) p'/ (r U₁²/2)                      (2)

defined as s' = s (v'²/V²), where v' is the velocity fluctuation.

Lopardo [5] suggested for a practical analysis the use of the depression with a 0.1% probability of occurrence as the minimal statistics to estimate the cavitation inception in macroturbulent flow. By applying that hypothesis to the Hu Minglong proposal and taking into account pressure fluctuation amplitudes, for the macroturbulent flow in a free hydraulic jump stilling basin, we obtain:

s – s' = (2.23 – 1.12 S_w) C'_p                         (3)

where S_w is the skewness of the probability density function of pressure fluctuation amplitudes.

The skewness of the probability density function of pressure fluctuations is obtained from laboratory and prototype data records, by means of transducer measurements and statistical analysis. It has been demonstrated that S_w shows with excellent accuracy the boundary layer separation in hydraulic structures submerged in macroturbulent flows. According to experimental data, flow separation is always associated with negative values of S_w, stagnation point with positive values of S_w and the skewness becomes zero (Gaussian condition) for channel turbulent motion.

Then, chute blocks, sills, baffle piers and other boundary appurtenances in a hydraulic jump stilling basin can be responsible for large negative skewness and, as a consequence, large instantaneous negative pressure amplitudes, increasing seriously the risks of cavitation damages. This effect was taken into account in the proposed formula for s'.

Fig. 2    Cavitation damage due to pressure fluctuations in Salto Grande Dam chute blocks

4    CASE STUDY: ARROYITO DAM STILLING BASIN

With the purpose to give an example of the large influence that the macroturbulent fluctuations have on the cavitation coefficient to have results according with the nature, the authors present some experimental data obtained by means of a hydraulic model when prototype cavitation damages were detected. The stilling basin of Arroyito Dam on Limay River in the Argentine Patagonic Region has a line of baffle piers, as it is shown in Figure 3. The design maximum discharge is Q = 3,000 m³7s, but cavitation was detected even for one half of this discharge when abnormal spillway gate operations were accomplished.

Fig. 3  Arroyito Dam stilling basin with bafflle piers in the hydraulic jump

The critical cavitation number for these baffle piers were assumed as s_c  = 0.7 (sharp edged block) in flows with high velocity flows with normal turbulence. The experimental data using mean pressures measured in the model indicates that the local cavitation number s never becomes lower than this value (Table 1). Nevertheless, cavitation number s' for macroturbulent flows, obtained with the concepts of the present paper has values s' < s_c in all the gages were cavitation damages in prototype were detected.

Table 1    Arroyito Dam spillway. Cavitation on baffle piers in macroturbulent flows

1500             2           0.18       2.02              –0.39    1.07       0.59       cavitation

                      I.1        0.20       2.27              –0.77     1.17       0.54       cavitation

                      II.1       0.20       2.21              –0.73    0.99       0.39       cavitation

                      III.1      0.20       2.26              –0.91     1.08       0.42       cavitation

          2580      1         0.22       2.43              –1.16    1.44       0.65       cavitation

                      II.1        0.23       2.57              –0.98     1.13       0.34       cavitation

As Table 1 demonstrates, the column of s has values over than s_c  (in normal flows cavitation will not be present) but the column s' has all the values below s_c (in macroturbulent flow under the jump, cavitation will be inevitable).

5    CONCLUSION

The hydraulic structure design, based on steady state flow parameters, is sufficiently consolidated today to prevent cavitation in high velocity flows. There are design manuals based on prototype operation and laboratory research, which have established criteria for the design of conventional hydraulic structures. However, cavitation damages and problems in stilling basins and energy dissipators, are not reduced. Little contribution can be made to the design of spillways within the “steady state hydraulics”.

Stilling basins are structures submitted to macroturbulent flows, being the internal phenomena of the jump responsible for the main part of the energy dissipation process. At present, researchers from developed countries have paid little attention to this problem. This can be explained by the fact of these countries with higher technology have drafted their hydraulic structure manuals on the basis of experiments performed in the 60's. Then, they developed studies and projects on more important dams.

During the sixties, the techniques for random signal recording and digital data processing were not sufficiently spread in hydraulics laboratories and large dams had not been affected (this occurs only after important floods take place). The participation of Ina's Hydraulic Laboratory (National Institute for Water and Environment) in almost all projects in the Paraná, Uruguay and Limay River basins, has made it possible to obtain outstanding results on intermittent cavitation due to pressure fluctuations in hydraulic jump energy dissipators.

It should be pointed out that the above mentioned hydroelectric dams include specific discharges that far exceed those proposed in the manuals, stilling basins in oscillatory jump ranges, big relationships in riverbed contraction, restitution levels which are not always adapted to the jumps and generally unavoidable abnormal operating conditions.

References

[1]   Lopardo, R.A., 1988, “Stilling basin pressure fluctuations”, International Symposium on Model-Prototype Correlation of Hydraulic Structures, Colorado Springs,­ USA, 56-73.

[2]  Lopardo, R.A. and Henning, R.E., 1986, “Efectos de las condiciones de ingreso al resalto sobre el campo de presiones instantáneas”, XII Congresso Latinoamericano de Hidráulica, IAHR, Sao Paulo, Brazil, 1, 116-127.

[3]   Lopardo, R.A., De Lío, J.C., Vernet, G.F., 1982, “Physical modelling on cavitation tendency for macroturbulence of hydraulic jump”, International Conference on the Hydraulic Modelling of Civil Engineering Structures, Coventry, England, 109-121.

[4]   Hu Minglong, 1983, “Xiehongdong fanhuduan xiayou konghua texing fenxi”, Shuili Xuebao, Beijing, China, 14-21.

[5]  Lopardo, R.A., 1990, “Una aproximación al índice de cavitación en flujos con fluctuación de presiones”, IAHR, IV Latin American Congress, Montevideo, Uruguay, 1, 237-247.