Probabilistic investigation of cavitation occurrence in chute spillway based on the results of Flow-3D numerical modeling

Probabilistic designation is a powerful tool in hydraulic engineering. The uncertainty caused by random phenomenon in hydraulic design may be important. Uncertainty can be expressed in terms of probability density function, confidence interval, or statistical torques such as standard deviation or coefficient of variation of random parameters. Controlling cavitation occurrence is one of the most important factors in chute spillways designing due to the flow’s high velocity and the negative pressure (Azhdary Moghaddam & Hasanalipour Shahrabadi, 2020). By increasing dam’s height, overflow velocity increases on the weir and threats the structure and it may cause structural failure due to cavitation (Chanson, 2013). Cavitation occurs when the fluid pressure reaches its vapor pressure. Since high velocity and low pressure can cause cavitation, aeration has been recognized as one of the best ways to deal with cavitation (Pettersson, 2012). This study, considering the extracted results from the Flow-3D numerical model of the chute spillway of Darian dam, investigates the probability of cavitation occurrence and examines its reliability. Hydraulic uncertainty in the design of this hydraulic structure can be attributed to the uncertainty of the hydraulic performance analysis. Therefore, knowing about the uncertainty characteristics of hydraulic engineering systems for assessing their reliability seems necessary (Yen et al., 1993). Hence, designation and operation of hydraulic engineering systems are always subject to uncertainties and probable failures. The reliability, ps, of a hydraulic engineering system is defined as the probability of safety in which the resistance, R, of the system exceeds the load, L, as follows (Chen, 2015): p_s=P(L≤R) (1) Where P(0) is probability. The failure probability, p_f, is a reliability complement and is expressed as follows: p_f=P[(L>R)]=1- p_s (2) Reliability development based on analytical methods of engineering applications has come in many references (Tung & Mays, 1980 and Yen & Tung, 1993). Therefore, based on reliability, in a control method, the probability of cavitation occurrence in the chute spillway can be investigated. In reliability analysis, the probabilistic calculations must be expressed in terms of a limited conditional function, W(X)=W(X_L ,X_R)as follows: p_s=P[W(X_L ,X_R)≥0]= P[W(X)≥0] (3) Where X is the vector of basic random variables in load and resistance functions. In the reliability analysis, if W(X)> 0, the system will be secure and in the W(X) <0 system will fail. Accordingly, the eliability index, β, is used, which is defined as the ratio of the mean value, μ_W, to standard deviation, σ_W, the limited conditional function W(X) is defined as follows (Cornell, 1969): β=μ_W/σ_W (4) The present study was carried out using the obtained results from the model developed by 1:50 scale plexiglass at the Water Research Institute of Iran. In this laboratory model, which consists of an inlet channel and a convergent thrower chute spillway, two aerators in the form of deflector were used at the intervals of 211 and 270 at the beginning of chute, in order to cope with cavitation phenomenon during the chute. An air duct was also used for air inlet on the left and right walls of the spillway. To measure the effective parameters in cavitation, seven discharges have been passed through spillway. As the pressure and average velocity are determined, the values of the cavitation index are calculated and compared with the values of the critical cavitation index, σ_cr. At any point when σ≤σ_cr, there is a danger of corrosion in that range (Chanson, 1993). In order to obtain uncertainty and calculate the reliability index of cavitation occurrence during a chute, it is needed to extract the limited conditional function. Therefore, for a constant flow between two points of flow, there would be the Bernoulli (energy) relation as follows (Falvey, 1990): σ= ( P_atm/γ- P_V/γ+h cos⁡θ )/(〖V_0〗^2/2g) (5) Where P_atm is the atmospheric pressure, γ is the unit weight of the water volume, θ is the angle of the ramp to the horizon, r is the curvature radius of the vertical arc, and h cos⁡θ is the flow depth perpendicular to the floor. Therefore, the limited conditional function can be written as follows: W(X)=(P_atm/γ- P_V/γ+h cos⁡θ )/(〖V_0〗^2/2g) -σ_cr (6) Flow-3D is a powerful software in fluid dynamics. One of the major capabilities of this software is to model free-surface flows using finite volume method for hydraulic analysis. The spillway was modeled in three modes, without using aerator, ramp aerator, and ramp combination with aeration duct as detailed in Flow-3D software. For each of the mentioned modes, seven discharges were tested. According to Equation (6), velocity and pressure play a decisive and important role in the cavitation occurrence phenomenon. Therefore, the reliability should be evaluated with FORM (First Order Reliable Method) based on the probability distribution functions For this purpose, the most suitable probability distribution function of random variables of velocity and pressure on a laboratory model was extracted in different sections using Easy fit software. Probability distribution function is also considered normal for the other variables in the limited conditional function. These values are estimated for the constant gravity at altitudes of 500 to 7000 m above the sea level for the unit weight, and vapor pressure at 5 to 35° C. For the critical cavitation index variable, the standard deviation is considered as 0.01. According to the conducted tests, for the velocity random variable, GEV (Generalized Extreme Value) distribution function, and for the pressure random variable, Burr (4P) distribution function were presented as the best distribution function. The important point is to not follow the normal distribution above the random variables. Therefore, in order to evaluate the reliability with the FORM method, according to the above distributions, they should be converted into normal variables based on the existing methods. To this end, the non-normal distributions are transformed into the normal distribution by the method of Rackwitz and Fiiessler so that the value of the cumulative distribution function is equivalent to the original abnormal distribution at the design point of x_(i*). This point has the least distance from the origin in the standardized space of the boundary plane or the same limited conditional function. The reliability index will be equal to 0.4204 before installing the aerator. As a result, reliability, p_s, and failure probability, p_f, are 0.6629 and 0.3371, respectively. This number indicates a high percentage for cavitation occurrence. Therefore, the use of aerator is inevitable to prevent imminent damage from cavitation. To deal with cavitation as planned in the laboratory, two aerators with listed specifications are embedded in a location where the cavitation index is critical. In order to analyze the reliability of cavitation occurrence after the aerator installation, the steps of the Hasofer-Lind algorithm are repeated. The modeling of ramps was performed separately in Flow-3D software in order to compare the performance of aeration ducts as well as the probability of failure between aeration by ramp and the combination of ramps and aeration ducts. Installing an aerator in combination with a ramp and aerator duct greatly reduces the probability of cavitation occurrence. By installing aerator, the probability of cavitation occurrence will decrease in to about 4 %. However, in the case of aeration only through the ramp, the risk of failure is equal to 10%.