1. Introduction
In hydraulic jumps, the high-velocity of an incoming flow abruptly has an impact against a slower flow [
1]. The classical hydraulic jump (CHJ) occurs on the smooth bed of stilling basins. A hydraulic jump is a phenomenon with non-deterministic characteristics, and for practical purposes, can be treated with the mathematical analysis approaches. Considering that the turbulent pressure nature is highly random, the analysis is mainly based on mathematical methodologies. Therefore, the stochastic characteristics of the problem should be paid attention to [
2,
3]. This property is a function of the turbulent characteristic of the velocity and pressure field.
Knowledge of pressure fluctuations and extreme pressures allows for a better understanding of the energy dissipation process along the hydraulic jump. Notable early studies on pressure fluctuations are such as those by Bukreyev [
4], Locher [
5], Schiebe [
6], Abdul Khader and Elango [
7], Lopardo et al. [
8], Lopardo [
9], Toso and Bowers [
2], Farhoudi and Narayanan [
10], Fiorotto and Rinaldo [
11], Fiorotto and Rinaldo [
12], and Armenio et al. [
13].
According to Yan et al. [
14], the pressure fluctuations coefficient (
CʹP) and peak frequencies of the spatial hydraulic jumps are higher than the classical jumps. Onitsuka et al. [
15] found that roller oscillations affect the instantaneous flow depth and bed pressure. In addition, the instantaneous bed pressures are associated with free surface fluctuations. Lian et al. [
16] stated that the fluctuating pressure spectrum in the rolling area follows the gravity similarity law. Lopardo and Romagnoli [
17] and Lopardo [
18] used
CʹP coefficient values to estimate the turbulence intensities close to the stilling basin bed for the low incident Froude numbers. Wang et al. [
19] predicted the total pressure based upon the void fraction and velocity data, and the results were in good agreement with the experimental data. Firotto et al. [
20] studied the stability of a plunge pool lining under the fully developed jets and proposed a design approach to determine the thickness of the linings. Barjastehmaleki et al. [
21] investigated the statistical structure of fluctuating pressures within the stilling basins. Barjastehmaleki et al. [
22] evaluated an approach for the structural design of stilling basins lining in the sealed and unsealed joints. Lopardo [
9] recommended specific flow conditions to measure pressure fluctuations. According to this, the supercritical Reynolds number (Re
1) should be more than 100,000. The minimum acquisition time must be 60 seconds. The acquisition frequency can be considered between 50 and 100 Hz. The maximum length of the plastic tube between the pressure tap and transducer is equal to 55 cm with a minimum inner diameter of 5 mm.
There are some pressure estimation methodologies associated with the hydraulic jumps in the literature. Gu et al. [
23] evaluated the Smoothed Particle Hydrodynamics (SPH) model to estimate the wave profile, velocity data, and energy dissipation caused by hydraulic jumps. Güven et al. [
24] used neural networks to predict the pressure fluctuations on the bed of a sloping stilling basin under B-type hydraulic jump, was investigated in detail by Hager [
25]. Teixeira [
26] determined the extreme pressures with different non-exceedance probabilities (
P*k%) from the sample data within a stilling basin. Teixeira et al. [
27] provided the cumulative curves of
P*k% for characteristic points along the hydraulic jump. Souza et al. [
28] investigated the behavior of the hydraulic jump concerning the longitudinal distribution of pressures near the bottom of the basin in the low Froude number zone (Fr
1 ≤ 4.5). Prá et al. [
29] investigated the influence of the vertical curve between the spillway toe and the stilling basin bed. The results showed that maximum pressure fluctuations were identified at the center of the vertical curve and assume values of 1% of the flow kinetic energy at the terminal tangency point of the curve. Novakoski et al. [
30] investigated extreme pressures with different probabilities (
P*k%) on a smooth basin downstream of a stepped spillway. The results showed that the values of
P*0.1% and
P*99.9% have lower and higher values than the values observed downstream of the smooth chute, in the region near the spillway toe, respectively.
Pressure distributions along the hydraulic jumps are not described by a normal distribution [
31]. The distributions of the skewness (S) and kurtosis (
K) coefficients of the sample pressure data along the hydraulic jump differ significantly from the value 0, attributed to a normal distribution. This values is observed after the endpoint of the hydraulic jump at the dimensionless position
X* ≈ 8. According to Marques et al. [
31], the distances of pressure points can be dimensionless concerning the conjugate depths, i.e.,
X* =
X/ (
Y2 −
Y1). Analysis of
S and
K coefficients displays that there are several types of distributions along hydraulic jumps. Therefore, it is difficult to estimate the pressure data with a certain probability (
P*k%). They proposed dimensionless relationships linking pressure data of
P*k% to the mean pressure (
P*m), and the standard deviation of the sample data (
σ*X). Such relationships allow us to organize the results of different flow discharges or Froude numbers and characterize the interest points in hydraulic jumps.
Generally, mean velocity and hydrostatic pressure are considered for designing a stilling basin. However, in the turbulent flow, the characteristics of the fluctuating fields of pressure and velocity may be more important than the mean values. Accordingly, the design of the stilling basin apron requires an assessment of the pressures acting upon the bottom of the basin to optimize concrete thickness. It is essential to study the instantaneous pressures beneath the hydraulic jump. There is little information about the pressure fluctuations, because it is quite difficult to measure the pressures underneath the hydraulic jump on the bed of stilling basins in the field [
15]. Therefore, laboratory-scale experiments covering pressure fluctuations seem to be reasonable and necessary [
32]. Indeed, the United States Bureau of Reclamation (USBR) has provided the general design criteria concerning the stilling basin length, assuming that the hydraulic jump is confined within the stilling basin. However, no indications are given to the different types of hydraulic jump, pressure regime, and forces on the bed of stilling basins [
33].
Therefore, the main aim of the present study is to measure and provide useful information about the pressure fluctuations. To do this, the experimental results are compared with those obtained on the bed of smooth basins in the literature. Many laboratory-scale experiments were designed to simulate the flow patterns downstream of an Ogee spillway, cascading into a USBR type I stilling basin, and measuring the pressure fluctuations with a frequency of 20 Hz along the longitudinal axis of the basin. The focus of this study is the mathematical analysis of the extreme pressures distribution at the bottom of a smooth stilling basin for the incident Froude numbers (Fr1) ranging from 7.12 to 9.46. New relationships will be proposed for the dimensionless standard deviation (σ*X), and the statistical coefficient of the probability distribution (Nk%) to estimate the extreme pressures with different non-exceedance probabilities (P*k%).