## Abstract

In this research, the ejecting jet from a flip bucket downstream of a chute spillway was simulated using physical modeling. The effects of influencing parameters upon fluctuations and extreme values of dynamic pressure were investigated. The angles of 0°, 30°, 45°, and 60° were adopted for the mobile bottom wall. The discharges were set as 67, 86, 161, and 184 litre/s and the depths of water cushion on the mobile bottom wall were set as 0, 15, 30, and 45 cm. The method suggested by Castillo for computation of fluctuating coefficient of dynamic pressure (see Castillo (2007) Pressure characterization of undeveloped and developed jets in shallow and deep pool. *Proceedings of the Congress-International Association for Hydraulic Research***32** (2), 645) was validated via the laboratory data. The results showed that the increase in water cushion depth downstream has led to a decrease in mean pressure and in pressure fluctuations. The analyses showed that the fluctuating pressure coefficient was a function of water cushion depth, and its maximum value was taken when there was a water cushion on the mobile bottom wall. With an increase in discharge and mobile bottom wall angle, the maximum value of the fluctuating coefficient occurred in less water cushion depth. Moreover, with the growth of discharge, the maximum positive and negative fluctuations of the pressure increased first and then decreased.

## HIGHLIGHTS

An increase in water cushion depth in the plunge pool contributes to decreasing pressure fluctuations as well as the mean value of pressure.

The maximum value of the fluctuation pressure coefficient (C′P) occurred when there was a water cushion in the plunge pool.

In high discharges, with an increment of the mobile bottom wall angle, the maximum pressure coefficient happened at a lower water cushion depth.

As the mobile bottom wall angle was increased, the values of C′P and the maximum values of positive and negative pressure fluctuations (C + p and C − p) were reduced.

## INTRODUCTION

Scour occurring through extra flow energy is among the most important issues regarding the stability of hydraulic structures (Karami Moghadam *et al.* 2019a, 2019b). The outlet flow from the spillway may be very destructive in a function of the height of the dam. Therefore, the dissipation of energy would be a necessary subject matter (Amini *et al.* 2014; Eghbalzadeh *et al.* 2016). The outlet flow is usually discharged as a free falling jet or by means of a curved flip bucket. As an energy dissipater, the flip bucket has the capability to conduct the high velocity flow towards downstream. After passing through the flip bucket, the flow ejects into the air and, undergoing some energy dissipation, enters the plunge pool. The appropriate choice of the plunge pool dimensions would be a technical and economic decision. Cola (1965) investigated the dynamic pressure coefficient (*C _{P}*) of non-aerated rectangular jets. Their results showed that when the ratio of water cushion depth to the jet thickness (

*Y/B*) was equal to 7, the maximum value of

_{j}*C*occurred. May and Willoughby (1991) investigated the impact of pressures produced by a vertical rectangular jet that varies with velocity, water depth, and amount of air within the jet. They established a correlation between the mean dynamic pressure, the air concentration, the jet velocity, the thickness of the jet and the water depth. Heller

_{P}*et al.*(2005) showed that the parameters of jet Froude number, relative bucket curvature, and the bucket angle have a notable effect on the maximum pressure production and its extent. Castillo (2006) has distinguished the turbulence intensity of the initial jet, the break-up length, jet thickness, and the average coefficient of dynamic pressure exerted into the plunge pool as the main features of the jet. Impact pressures of turbulent high-velocity jets plunging in pools with flat bottoms were investigated by Manso

*et al.*(2007). They showed that dynamic pressures, both at the nozzle exit and at the bottom in shallow and deep pools, follow a Gaussian (Normal) distribution in the intermediate range of cumulated probability. Manso

*et al.*(2009) showed that the geometry of the plunge pool is a key element in the definition of the mean and turbulent character of dynamic impact pressures on the pool boundaries. Studying dynamical pressures caused by flip buckets, Kerman-Nejad

*et al.*(2011) have reached the conclusion that the most pressure fluctuations happened at the impact angle of 90°, and that these fluctuations reduce as the jet impact angle lessens. Yamini & Kavianpour (2011) obtained the maximum values of pressure in two simple bucket models by exploring the static and dynamic pressures on the simple flip bucket. They found that the radius of the flip bucket was an important parameter in the distribution course of dynamic pressure. Guven & Azamathulla (2012) used gene-expression programming (GEP) in prediction of scour downstream of a flip bucket spillway. The results showed that the predictions of GEP models were observed to be in good agreement with measured ones. Pfister (2012) investigated the take-off angle of the jet and explained the effect of the slope of the chute spillway by introducing an equation. Nazari

*et al.*(2013) performed experiments on the physical model of the chute spillway flip bucket with regard to five dams. They determined the pressure distribution in the central axis of the bucket and offered standards for hydraulic design of the flip bucket. Salemnia

*et al.*(2014) conducted experiments to measure the dynamic pressures arising from an impingement of non-submerged circular jet on a smooth plane. They recognized that the coefficient of dynamic pressure exerted on the pool floor was with a remarkable increase in the increment of flow discharge or decrement of the nozzle diameter. Parsaie

*et al.*(2016) undertook investigations on the cavitation phenomenon along the spillway flip bucket of the Balaroud dam using Flow 3D. Results of numerical simulation showed that occurrence of cavitation based on an index equal to 0.25 is not possible along the spillway. To scrutinize the increase in efficiency of the chute spillway, Hojjati

*et al.*(2017) applied a multi-objective evolutionary algorithm. Using the algorithm, they determined the optimum width of the chute as well as the optimal triangular bucket angle. In addition to the calculation of jet trajectory length via empirical methods, Yavuz & Yilmaz (2017) investigated the energy dissipation of the ski-jump jets as the result of air absorption on the scale of the prototype. Their results showed that energy dissipation, as a result of air absorption, was constant from a certain discharge on. Yamini

*et al.*(2018) surveyed the pressure fluctuations and the effect of the entering flow discharge on the flip bucket bed of Gotwand dam in Iran. Their results held that when the depth and discharge of the entering flow increased or when the Froude number decreased, the average pressure decreased and pressure fluctuations increased. Furthermore, the average pressure was maximum at the bucket inlet and minimum at its end.

Recognition and identification of the jet characteristics as well as the resultant dynamic pressures exerted upon the wall and floor are essential topics in designing the plunge pool and determining the least depth of water cushion. In this research, a comprehensive experimental data of simulated flow behavior in the downstream plunge pool concerning a ski-jump jet was presented. This data was used in the determination of the effects of various factors on the pressure fluctuations arising from an impingement of the jet to the plunge pool mobile bottom wall. The extracted data and the presented analyses in this research could be used in finding the relationships and more precise criteria in the design of the structure.

## MATERIALS AND METHODS

### Characteristics of jet entering into plunge pool

Figure 1(a) gives a scheme of the jet entering into the plunge pool in Karun-3 dam, Iran. Karun-3 dam is a double curvature concrete dam of 205 m height and with a design flood discharge of 21,000 m^{3}/s. It has a concrete plunge pool of dimensions 400 metre length, 70 metre width, and 50 metre height. Figure 1(b) shows the jet features when impinged on the water surface of the pool.

*h*is two times the energy head of the upstream,

_{0}*h*(Castillo 2007). At the location of the jet's impingement on the water surface of the pool, the average velocity of the jet is

*V*, and its thickness (

_{j}*B*) is represented as in Equation (1):where

_{j}*B*is the jet thickness caused by gravity, and

_{g}*ξ*is the jet lateral spread distance by the effect of turbulence. The entry of air into the jet and falling into the plunge pool results in dissipation of the jet's energy. When the jet falls from a high elevation, surface disturbances enter the jet and break it down into discrete water droplets. The distance from the fall's starting point to where the jet core disintegrates is called the jet break-up length (

*L*). From this point on, the falling jet becomes fully developed. In this case, there is no jet core, and in fact, it is deformed into a set of water droplets.

_{b}### Coefficient of dynamic pressure fluctuations

*C′*) is used to express pressure variations around the mean value. This number expresses the root mean square, RMS, of pressure fluctuations as a function of the entering kinetic energy (Bollaert & Schletss 2003a, 2003b). The coefficient is expressed as follows (Ervine

_{P}*et al.*1997):where

*V*is the jet velocity, and

_{j}*V*is the height corresponding to the velocity or kinetic energy at the impingement point. To compute the maximum value of positive (

_{j}^{2}/2 g*C*) and negative (

^{+}_{P}*C*) dynamic pressure fluctuations, Equations (3) and (4) were used, respectively (Ervine

^{−}_{p}*et al.*1997):where

*H*,

_{max}*H*, and

_{min}*H*are, respectively, the maximum, minimum, and the mean values of instantaneous pressure.

_{m}### Experimental setup

In this research, the physical simulation of the spillway was carried out in a laboratory flume in the hydraulic laboratory of Shahid Chamran University, Ahvaz, Iran. The flume was of dimensions 0.5 metre width, 9 metre length, and 2 metre height. The flow was provided by a pump from a ground reservoir. The flow discharge was regulated through a control valve and a rectangular weir at the end of the flume. After entering the flume, the flow passed through a flip bucket and entered the plunge pool, then impinged onto an embedded mobile bottom wall. Figure 2 shows the laboratory flume and a schematic drawing of models used in this research. A square Plexiglass plate of dimensions 0.5 × 0.5 metre was used for making the mobile bottom wall. To measure dynamic pressures, 37 pores with diameter of 2 mm were set at the jet impingement location for connection of the piezometer tubes (Figure 2(b)). The mobile bottom wall was able to rotate around the horizontal axis to make different angles. The water cushion depth was adjusted by the vertical movement of the mobile bottom wall. Furthermore, the flow was recirculated into the initial reservoir via a channel system.

### The experiments

The main objective of this research was to study the fluctuations of dynamic pressures caused by the jet falling from a flip bucket into the plunge pool. The variables involved in this research include discharge (*Q*), water cushion depth (*Y*), angle of mobile bottom wall (*ϴ*), jet falling height (*H*), and jet break-up length (*L _{b}*). In the experiments, the discharges of 67, 86, 161, and 184 litre/s at four water cushion depths of 0, 15, 30, and 45 cm were studied. To examine the effect of the angle of the impinging water jet on pressure fluctuations, four angles of 0°, 30ᶱ, 60°, and 90° were tested for the mobile bottom wall. In each experiment, the jet falling height was measured accordingly. Considering all the variables, a total of 64 experiments were carried out.

### Dynamic pressure fluctuations

The piezometers were used in recording fluctuations of dynamic pressures at the impact point of the jet with the mobile bottom wall. The connected piezometers were extended from the bottom of the mobile bottom wall and transformed on the dial board (Figure 2(b)). For a better observation of the fluctuations, colored fluid was employed in the piezometers' tubes. To record dynamic pressures, two piezometers with the highest pressures were connected to the electric transducer. The transducer sensor was calibrated with a graduated cylinder at the end of a piezometer. By applying various water heads to the piezometer, and reading corresponding pressure fluctuations through computers, the sensor calibration was performed. The accuracy of the transducer was found to be ±1 mm or 0.05% of the full scale output (0.2 bar). For a time of 10 minutes, 50 data of dynamic pressures per second were recorded. Then, the measured data were transformed into a computer via Data Translation Scope, DT9800. The pressures were plotted as a function of the time. By taking digital photographs from the place of jet take-off at the edge of the flip bucket as well as the point of impact with the mobile bottom wall, or water surface in the pool, the flow areas in these two sections were measured. The two areas were used in computation of the velocity of flow and jet, respectively, at take-off and impact places. In addition to velocity and dynamic pressures, the jet falling height (*H*) was also measured for each experiment.

## RESULTS AND DISCUSSION

### Water cushion depth

At discharges of 67, 86, 161 and 184 litre/s and mobile bottom wall angle of 0° with no water cushion depth, the distance of the jet peak to the impact plate was 1.03, 1.10, 1.12 and 1.14 cm, respectively. Correspondingly, for the mentioned discharges, the jet thickness (*B _{j}*) at the impingement location was 3, 3.72, 6.90 and 7.82 cm, the mean take-off velocity (

*U*) was 1.68, 1.73, 2.08 and 2.17 m/s, and the mean jet velocities at the impingement location (

_{o}*Uj*) were 4.8, 4.96, 5.13 and 5.2 m/s, respectively. With an increase in discharge, the values of these parameters increases. These results are consistent with the findings of Manso

*et al.*(2008). Figure 3 indicates an example of the recorded pressure in a discharge of 184 litre/s and at a mobile bottom wall angle of 0° over a time interval of 20 s at different depths of water cushion.

Figure 3 shows that when the water cushion depth is relatively small (0 or 15 cm), the pressure fluctuations are high, and as the former approaches 30 and 45 cm, the latter reduces conspicuously. The mean values of pressure in depths 0, 15, 30, and 45 cm are 1.18, 0.86, 0.58, and 0.48 m, respectively. This is evidence to the fact that the increase in water cushion depth in the downstream pool of the flip bucket causes decrease not only in pressure fluctuations but also in the mean value of pressure. The reason may be that at high depths of water cushion greater percentages of the energy of the impinging jet would be dissipated. Furthermore, with the increment in water cushion depth, the jet deformed into a developed jet that yields bigger vortices with lower frequencies. This conclusion is consistent with the results of Bollaert & Schleiss (2003a, 2003b).

### Coefficient of pressure fluctuations

Jet thickness at the impingement point is effective on the rate of dynamic pressures wielded on the bottom of the plunge pool (Castillo 2007). In addition, the depth of water cushion plays a key role in the reduction of fluctuations as well as dynamic pressures. Hence, an essential factor in studying dynamic pressures resulting from the impacting jet in the plunge pool is the ratio of water cushion depth to the jet thickness, *ф**=**Y/B _{j}* (Bollaert & Schletss 2003a, 2003b). Figure 4 shows the changes in the coefficient of pressure fluctuations (

*C′*) and alterations of

_{P}*ф*at various angles and corresponding discharges.

Figure 4 shows that at different discharges and a certain angle, with an increase in water cushion depth (increase in *ф*), the value of *C′ _{P}* increases first and then decreases from a certain depth on. In other words,

*C′*takes its maximum, not at the ‘no cushion’ state with direct jet impact (

_{P}*ф*

*=*

*0*), but when there is a layer of water cushion. This result has conformity with those obtained by Bollaert (2002), Jia

*et al.*(2001), Castillo (1989), and Castillo

*et al.*(1991). The reason may be tracked by recognizing the jet features. Throughout its trajectory, the jet may be cored or without a core. When the water cushion depth is sufficient, a jet without a core is created in the downstream pool, and it grows to be a developed jet. The developed jets contain low frequencies with great energy, causing high turbulence vortices. Consequently, the existence of a water cushion is the minimum requirement to make such vortices. At the zero angle, the maximum of the coefficient

*C′*falls between 0.3 and 0.35 for different discharges. Bollaert (2002) and Jia

_{P}*et al.*(2001), respectively, obtained a

*C′*value between 0.2 and 0.3, and 0.2 for a circular jet. While Castillo (1989) and Castillo

_{P}*et al.*(1991), respectively, have given the value of

*C′*as 0.25 and 0.2 for rectangular and nappe flow, which are consistent with our results. Additionally, Castillo

_{P}*et al.*(2015) proposed different expressions to calculate

*C′*in the function of the ratio

_{P}*H/L*and

_{b}*Y/B*. The maximum value of

_{j}*C′*= 0.36 to 1.00 <

_{P}*H/L*

_{b}< 1.30 and

*Y/B*= 3. Furthermore, Figure 4 shows that in different discharges, the water cushion depth corresponding to the maximal

_{j}*C′*differs at various mobile bottom wall angles. So that in higher discharges and with an increment of the mobile bottom wall angle, the maximum value of

_{P}*C′*is taken in lower water cushion depths. In low discharges, the water cushion depths pertaining to the maximal

_{P}*C′*at different angles are almost close to each other. For instance, for the discharge of 67 litre/s the maximum value of

_{P}*C′*at all the angles occurs approximately when

_{P}*ф*

*=*

*3*. It is the case, while for 184 litre/s the maximums of

*C′*at the 0° and 90° are taken when

_{P}*ф*

*=*

*3.1*and

*ф*

*=*

*1.8*, respectively

*.*The border between the cored jet and developed jet is, in fact, the depth corresponding to the maximum value of

*C′*. According to Figure 4, this border lies nearly in the range of 2 <

_{P}*ф*< 3. Castillo (1989) obtained the maximum of

*C′*at

_{P}*ф*

*=*

*4*. Bollaert (2002), Jia

*et al.*(2001), and Castillo

*et al.*(1991) found the maximum value in the range 4 <

*ф*< 6. Different discharges and jet shapes are the sources of these discrepancies. As can be seen in Figure 4, the formation of developed jets and eddies results in pressure to the vertical wall (

*θ*

*=*

*90*°), which is lower than the pressure on the pool bottom. These results are consistent with those of Manso (2006) and Manso

*et al.*(2007), who showed that the amount of pressure applied to the walls in laterally confined plunge pools is lower than that in flat bottom pools. Also, Manso

*et al.*(2009) showed that the lateral confinement limits the development of macro-turbulent rollers around the plunging jet (instead of the pool depth), the mean pressures at impact are considerably reduced, as well as

*C′*in transitional and deep pools. For shallow pools,

_{P}*C′*increases as a result of enhanced jet development.

_{P}### Empirical formula

*et al.*(2015) reported a new analysis for available data and proposed Equation (5) for calculation of the

*C′*coefficient:where a = −6.6 × 10

_{P}^{−6}, b = 0.0004, c = −0.008, d = 0.0483, and e = 0.1179. Using the obtained experimental data in this research and Equation (5), the values of

*C′*were computed and the obtained results are drawn in Figure 4 for various angles of mobile bottom wall. As shown in Figure 4, the results obtained through Equation (5), in accordance with the above constant coefficients suggested by Castillo

_{P}*et al.*(2015), are mostly consistent with the results of this research at the 0° of the mobile bottom wall (the bottom of the plunge pool). However, the same results given by Equation (5) are not so consistent with the findings of this research at the 90° angle (the vertical wall of the pool). In the present research, by doing the best fitting between the data, constant coefficients of Equation (5) were obtained at different mobile bottom wall angles and the results are presented in Table 1. The results for Equation (5), with reference to the constant coefficients of Table 1, are depicted in Figure 4. As seen, at all angles of the mobile bottom wall, particularly at 90°, the graph for Equation (5) with the constant coefficients suggested hereby in this research, are conform better to the laboratory results relative to those suggested by Castillo

*et al.*(2015).

θ
. | A . | b . | c . | d . | e . | R^{2}
. |
---|---|---|---|---|---|---|

0 | −0.0002 | 0.0052 | −0.0451 | 0.1041 | 0.2556 | 0.80 |

30 | −0.0005 | 0.0102 | −0.0736 | 0.165 | 0.1917 | 0.80 |

60 | 0.0009 | −0.0094 | 0.0126 | 0.0384 | 0.1634 | 0.84 |

90 | 0.0000 | −0.0003 | −0.0087 | 0.0395 | 0.0993 | 0.70 |

θ
. | A . | b . | c . | d . | e . | R^{2}
. |
---|---|---|---|---|---|---|

0 | −0.0002 | 0.0052 | −0.0451 | 0.1041 | 0.2556 | 0.80 |

30 | −0.0005 | 0.0102 | −0.0736 | 0.165 | 0.1917 | 0.80 |

60 | 0.0009 | −0.0094 | 0.0126 | 0.0384 | 0.1634 | 0.84 |

90 | 0.0000 | −0.0003 | −0.0087 | 0.0395 | 0.0993 | 0.70 |

### Mobile bottom wall angle

Figure 5 illustrates the changes in the fluctuation coefficient of dynamic pressure against the mobile bottom wall angle relative to the horizon in the no cushion depth condition (direct jet impact). The figure also shows that in this situation, the mobile bottom wall angle considerably affects the value of *C′ _{P}* so that with the increase of the angle, the

*C′*value decreases. Differences in values of

_{P}*C′*are highly discernible at various mobile bottom wall angles. The same decrement trend in the

_{P}*C′*values is held for different discharges as well. At low discharges (67 and 86 litre/s), the

_{P}*C′*coefficient increases parallel to the growth of discharge, whereas at high discharges (161 and 184 litre/s), the

_{P}*C′*values fall as the discharge increases. Briefly stated, the value of

_{P}*C′*for the discharge of 161 litre/s is greater than that for 184 litre/s. This is due to the effect of the ratio of the falling height to the break-up length. According to Figure 5, there is a significant difference between the values of the pressure fluctuation coefficient at 0° and in different discharges. At 90°, this difference lessens and the

_{P}*C′*values for different discharges are in the neighborhood of each other. Hence, the effect of discharge growth upon dynamic pressures is more apparent in the plunge pool bottom rather than on the vertical wall.

_{P}One of the parameters affecting the *C′ _{P}* coefficient is the break-up length (Karami Moghadam

*et al.*2019a, 2019b). In the present study, the maximum value of

*C′*occurred in the ratio of falling height to jet break-up length

_{P}*H/L*

_{b}*=*

*0.6*, which is consistent with the results of Ervine

*et al.*(1997) and Castillo (2006).

### Extreme values

In Figure 6, the diagrams relating to the changes of maximum positive and negative values of dynamic pressure fluctuations (*C ^{+}_{p}* and

*C*) against

^{−}_{p}*ф*are drawn for the corresponding discharges at different mobile bottom wall angles. Figure 6 shows that along with the increases of

*ф*, the values of

*C*and

^{+}_{p}*C*first increase and then decrease. The maximum values were obtained at nearly

^{−}_{p}*ф*

*=*

*4*. Ervine

*et al.*(1997), who studied falling circular jets, gave the maximum

*C*as almost 4 × RMS of the pressure fluctuations, and the maximum

^{+}_{p}*C*as almost 3 × RMS of the pressure fluctuations. Accordingly, the maximum positive pressure occurs at

^{−}_{p}*ф*

*=*

*10*, and the maximum negative pressure corresponds to

*ф*

*=*

*5*. When the mobile bottom wall angle increases, the values of

*C*and

^{+}_{p}*C*diminish (Figure 6). This reduction trend is more obvious in the non-cushion depth situation, that is,

^{−}_{p}*ф*

*=*

*0*. Also, the maximum values of

*C*and

^{+}_{p}*C*increase parallel to the increase in discharge. For example, when the mobile bottom wall angle is zero and the discharge increases from 67 litre/s to 184 litre/s, the maximum

^{−}_{p}*C*rises from 1.25 to 1.5, and the maximum

^{+}_{p}*C*enlarges from 0.54 to 0.66.

^{−}_{p}Figure 7 gives a comparison between the results of this research and those of other researchers. The work of Ervine *et al.* (1997) is associated with circular jets, and the rest are related to rectangular jets. The maximum values of *C ^{+}_{p}* and

*C*obtained in current research are equal to 1.45 and 0.65, respectively which take place at around

^{−}_{p}*ф*

*=*

*4*. The maximums in Ervine

*et al.*(1997) are given respectively as 0.8 and 0.6, which correspond to

*ф*

*=*

*10*and

*ф*

*=*

*5*. The findings of this research conform well to those relating to the rectangular jets.

## CONCLUSIONS

In this research, the simulation of a spillway with flip bucket was carried out in the downstream plunge pool using a physical model. The aim of the research was to investigate the effects of parameters such as water cushion depth in the plunge pool, discharge, mobile bottom wall angle, jet falling height, and jet break-up length upon fluctuations of dynamic pressures as well as their extreme values. The most vital results of this research are as follows:

An increase in water cushion depth in the plunge pool contributes to decreasing pressure fluctuations as well as the mean value of pressure. Also, the maximum value of the fluctuation pressure coefficient (

*C′*) occurred when there was at least a water cushion in the plunge pool._{P}In high discharges, with an increment of the mobile bottom wall angle, the maximum pressure coefficient happened in a lower water cushion depth. In the case of jet falling height and jet break-up length, the highest coefficient of pressure fluctuations belonged to the case of

*H/L*_{b}*=**0.6*.As the mobile bottom wall angle was increased, the values of C′

_{P}and the maximum values of positive and negative pressure fluctuations (*C*and^{+}_{p}*C*) were reduced.^{−}_{p}The maximum values of

*C*and^{+}_{p}*C*in the present research were, respectively, 1.45 and 0.65, which occurred at^{−}_{p}*ф*≈*4*.A new set of constant values was proposed, which can be used to improve the available relations to compute the pressure fluctuation coefficient.

## ACKNOWLEDGEMENT

The financial and technical support from the Shahid Chamran University, Ahvaz, Iran for this project is highly acknowledged.

## REFERENCES

*Transient water pressures in joints and formation of rock scour due to high velocity jet impact*. PhD Thesis. In: Schleiss, A. (ed.). Laboratoire de Constructions Hydrauliques. Ecole Polytechnique Fédérale de Lausanne, Communication 13

*The influence of pool geometry and induced flow currents in rock scour by high velocity plunging jets*. PhD Thesis, No. 3430, Ecole Polytechnique Fédérale de Lausanne and Communication N° 25 of the Laboratory of Hydraulic Constructions, Lausanne