This paper presents a brief history of the hydraulic jump and a literature review on hydraulic jumps’ experimental and numerical studies. Leonardo da Vinci noticed this phenomenon early on, but it was only later studied by Bidone in 1820. Since the beginning of the 20th century, the hydraulic jump has received a lot of attention following the development of energy dissipater designs and stilling basins. The late 1920s and early 1930s saw many experimental studies researching the surface roller profile and energy dissipation. The study of internal flow features started in the late 1950s. Starting in the 70s, it was believed that the flow of a jump must be analyzed in its actual configuration of air–water mixture, an aspect that cannot be overlooked. Several experimental studies in the late 1980s and 1990s highlighted the existence of oscillating phenomena under specific flow conditions and particularly, a cyclic variation of jump types over long-lasting experiments. The early 2000s saw many experimental studies researching the complex structure of the separated region in very large channels downstream of the lateral shockwaves. Whereas most of the experiments provide measurements at a point or on a plane, the complete flow field supplied by CFD simulations enables us to have a deeper understanding of the dynamics of coherent structures that are responsible for free-surface fluctuations and aeration in hydraulic jumps. Therefore, in recent years, the computational fluid dynamics (CFD) method, through turbulence models, has become a useful tool to study this complex environmental fluid mechanic problem.

The jump of Bidone or hydraulic jump was described by Bidone [

Although Bidone [

In 1939, Guglielmini [

The hydraulic jump occurs whenever an upstream supercritical flow is forced to become subcritical. Bidone tested many discharges and understood that the lower upstream depth _{1} was linked to the higher downstream depth _{2} with the following equation:_{1} and _{2} are the upstream and down-stream velocity, respectively.

About ten years later, Jean Baptiste Bélanger [

Harleman [

The classic hydraulic jump has been widely studied by Peterka [

Since the beginning of the 20th century, the hydraulic jump has received a lot of attention with the development of energy dissipater designs and stilling basins [

It is well known that one of the objectives of the designer is to ensure that the jump will not be swept out of the basin, and the design process would involve the determination of optimum basin floor elevation, required tailwater elevation, adequate basin length and desired blocks and end sills. A review of different types of basins can be found in Hager [

Riegel and Beebe [

In the late 1920s and early 1930s, several experimental studies looked into the surface roller profile and energy dissipation. Safranez [

In 1932–1933, the Fluid Mechanics Laboratory of Columbia University, in New York, N.Y., USA, focused its research on the longitudinal elements of the jump using the principle of dynamic similarity, and the final data were presented in generalized dimensionless form. While the dimensionless presentation in terms of dynamic similarity was a matter of course when it came to flow in closed conduits, its application to open flow was not yet widespread. Bakhmeteff and Matzke [

The late 1950s saw the start of the studies into internal flow features; the first turbulence estimations in hydraulic jumps were studied by Rouse et al. [

On this topic, the study by Rajaratnam [_{1} is the inflow Froude number, and

The dimensionless integrated shear force _{τ}_{1} is the width of the stilling basin in upstream.

Later, McCorquodale and Khalifa [

Considering the studies of Resch et al., [

Roshko [

However, the complex nature of the hydraulic jump, involving intense turbulence, velocity and pressure fluctuations and significant air entrainment, still showed that current knowledge is far from a full understanding of the phenomenon.

Most detailed air–water flow turbulent features of hydraulic jumps started to be systematically reported following [

Mossa and Tolve [

Gualtieri and Chanson [

Some experimental studies highlighted the existence of oscillating phenomena and particularly, a cyclic variation of jump types over long-lasting experiments, under specific flow conditions [

Furthermore,

Experiments by [

Experiments in [

This study showed how it is important to design basins considering the cyclic variation of jump types caused by the upstream and downstream flow conditions. The conclusions of the experiments in [

oscillations of hydraulic jump types do not depend on whether the bottom is made of erodible or non-erodible material;

a suitable time scale may be defined both for oscillations of the jump types and for fluctuations of the jump toes with a flat and outlined bottom;

analysis of the oscillating phenomena indicates a correlation among the surface profile elevations, velocity components and pressure fluctuations;

analysis of the oscillating phenomena indicates change configurations of the surface profile of a hydraulic jump, as a function of the air concentration present in the roller.

Successively, the experiments by Mossa et al. [

The experimental observations [

In fact, experiments by [

On this basis, more recently, experimental research was conducted by [

Since the fluid dynamics problems are usually too complex to be solved by analytical methods because they involve many different issues due to their nonlinearity, research has recently focused on computer power and the continuous improvement of numerical codes. In recent years, the computational fluid dynamics (CFD) method, through turbulence models, has become a useful tool to study complex environmental fluid-mechanics problems. The numerical modeling of a hydraulic jump, which involves fluctuating boundaries as well as a multiphase flow, is still challenging, considering its complexity. Furthermore, whereas most of the experiments provide measurements at a point or on a plane, the complete three-dimensional (3D) flow field supplied by a CFD simulation would enable us to have a deeper understanding of the dynamics of coherent structures that are responsible for free-surface fluctuations and aeration in hydraulic jumps.

In recent years, hydraulic jumps have been investigated using both Eulerian and Lagrangian techniques. In general, the Eulerian method discretizes the space into a mesh and defines the unknown values at the fixed points, while the Lagrangian method tracks the pathway of each moving mass point. A comprehensive review of the studies, referring to the numerical simulation of the hydraulic jump was provided by Valero et al. [

Early Eulerian numerical studies on hydraulic jumps were carried out by Longo et al. [

Long et al. [

In Chippada et al. [

In Qingchao and Drewes [

Cheng et al. [

However, these simulations of hydraulic jumps were confined to the liquid phase and ignored the effect of the entrained air; early numerical studies that considered the air entrainment in the hydraulic jump were carried out by Souders and Hirt [

Therefore, the first numerical results to be compared with experimental void fraction data were supplied by Ma et al. [

More recently, Witt et al. [

Recently, Bayon et al. [

Although less widely researched, the Lagrangian meshless method showed interesting results; in fact, meshless Lagrangian techniques appear, in general, to be more suitable for capturing the highly unsteady free surface of a hydraulic jump. Smoothed particle hydrodynamics (SPH), which is one of the most popular mesh-free methods, was introduced by [

In SPH simulations by Lopez et al. [

In SPH simulations by Federico et al. [

In Jonsson et al. [

In Chern and Syamsuri [

In De Padova et al. [

Until the end of the 90s, in spite of the many experimental studies on the hydraulic jump, the complex structure of the separated region in very large channels downstream of the lateral shockwaves was not yet understood [

The numerical results [

In De Padova et al. [

A comparison with experimental results by [

As shown in

Vortices are characterized by a clockwise rotation (positive vorticity) when the wave jump occurs (

Further details about the numerical tests can be found in [

The complicated nature of the hydraulic jump has always attracted researchers’ attention. After many years of sustained research and with many of its features now well understood, a satisfactory and full comprehension of this complex phenomenon remains a challenge.

Leonardo da Vinci first noticed this phenomenon, but it was only later in 1820 that Bidone started to study it scientifically. Since the beginning of the 20th century, the hydraulic jump has received a lot of attention, following the development of energy dissipater designs and stilling basins.

Several experimental studies in the late 1920s and early 1930s investigated the surface roller profile and energy dissipation. Then, in the late 1950s, the study of internal flow features started, and experimental research activity focused on turbulence estimations in hydraulic jumps.

In the 70s, it was believed that the flow of a jump must be analyzed in its actual configuration of air–water mixture, an aspect that cannot be overlooked. Therefore, several experimental studies were conducted to visualize the turbulent coherent structures in a hydraulic jump and to determine the location of the maximum air concentration.

The experimental results showed the existence of two regions with the greatest air concentration when the largest vortex of a hydraulic jump breaks down, causing the water to spill.

More recently, the effect of the Froude number on the basic air–water flow properties, focusing on the maximum void fraction and bubble count rate in the shear layer, was studied.

Several experimental studies in the late 1980s and 1990s highlighted the existence of oscillating phenomena and particularly, a cyclic variation of jump types over long-lasting experiments, under specific flow conditions, and investigated the surface roller profile and energy dissipation. Experimental observations led to the conclusion that the oscillating phenomena were especially significant for the analysis of turbulence characteristics.

Until the end of the 90s, in spite of the many experimental studies on hydraulic jump, the complex structure of the separated region in very large channels downstream of the lateral shockwaves was not yet understood. The first experimental research on hydraulic jumps in a very large channel suggested that the wall boundary layer is subjected to a sudden adverse pressure gradient, causing a sharp deceleration of the flow velocity near the wall; moreover, the experiments showed a recirculation of the flow immediately behind the lateral shockwaves near the wall.

While most of the experiments provide measurements at a point or on a plane, the complete three-dimensional (3D) flow field supplied by a CFD simulation enables us to have a deeper understanding of the dynamics of coherent structures that are responsible for free-surface fluctuations and aeration in hydraulic jumps.

Therefore, in recent years, the computational fluid dynamics (CFD) method, through turbulence models, has become a useful tool to study complex environmental fluid-mechanics problems.

The numerical modeling of a hydraulic jump, which involves fluctuating boundaries, as well as a multiphase flow, is still challenging, considering its complexity. Hydraulic jumps have been widely investigated using Eulerian techniques showing satisfactory results in terms of quantity and quality; Eulerian models, by solving the RANS equations, and a two-equation turbulence model have yielded accurate results for mean flow variables, including air concentrations in some cases.

In the last few decades, although less researched, the Lagrangian meshless method showed interesting results; in fact, meshless Lagrangian techniques appear in general to be more suitable for capturing the highly unsteady free surface of a hydraulic jump.

Nevertheless, it is evident that the profitable prediction of a numerical model depends on the calibration made with experimental data.

Therefore, the mean flow variables (sequent depths’ relationship, roller length, jump length, mean free surface profile, air concentrations, etc.) that have been extensively experimentally studied in the literature should constitute the minimum dataset for numerical models’ calibration.

The following are available online at

M.M. conceived and designed the experiments; M.M. performed the experiments; M.M. and D.D.P. contributed analysis tools; M.M. and D.D.P. wrote the paper. All authors have read and agreed to the published version of the manuscript.

This research received no external funding.

Not applicable.

Not applicable.

Data available presented in this study are openly available in Zenodo at

The authors declare no conflict of interest.

A hydraulic jump in the large channel of the LIC—Coastal Engineering Laboratory of the Polytechnic University of Bari, Italy. (See

A free water jet issuing from a square hole into a pool.

Hydraulic jump (by Guglielmini [

Flow conditions during experiments by Mossa et al. [

Oscillatory flow patterns between B-jumps and wave jumps (configuration B61 in Mossa et al. [

SPH simulation of hydraulic jump in a very large channel: (

SPH vorticity field; oscillatory flow patterns between A-jumps and wave jumps: (