A thorough sensitivity analysis was already performed by the authors in the application of the same SPH numerical method to hydraulic jumps and breaking wave flows ([

29,

46]): according to this analysis, the SPH simulations of the cases here studied were performed by adopting a velocity smoothing coefficient in the XSPH scheme

φ_{v} = 0.01. The ratio of the smoothing length to the initial particle spacing

Σ was maintained to a constant value of

η/Σ = 1.5 [

73] for all the simulations. A convergence analysis was carried out by choosing different initial particle spacing

Σ ranging from 0.015 to 0.005 m. The related number of SPH particles N

_{P} in the computational domain ranged from about 1000 to 9000, respectively. It can be seen that the simulation at the lowest resolution is not able to predict the oscillating characteristics and cyclic mechanisms in hydraulic jumps (

Figure 5).

The sensitivity analysis highlights that, with a value of an initial particle spacing Σ ≤ 0.010 m, SPH simulations show results in accordance with the experiments. Therefore, all the SPH simulations have been then performed with an initial particle spacing Σ = 0.010 m and η/Σ = 1.5.

Although both turbulence models yield similar results, the detailed comparison of the computed amplitude spectra with the measured ones shows that the results obtained with the mixing-length model are closer to the experimental data than the

k-ε ones (

Figure 9). In particular, the two-equation turbulence model overestimates the peak amplitude of the pressure fluctuations upstream, while predicting a lower main frequency. A possible explanation of this behaviour could reside in an underprediction of turbulent kinetic energy (or in an overprediction of it dissipation rate) by the

k-ε model, which leads to a slightly lower turbulent diffusion downstream of the jump. A possible solution to these problems could be found in a careful tuning of the model parameters, or in the application of a different two-equation model, such as the RNG k-ε model, which has been successfully applied to the Eulerian numerical simulations of hydraulic jumps [

22]. However, the obtained results showed that a good agreement with experiments could be already obtained by applying the simpler, mixing-length turbulence model and, therefore, all the remaining SPH simulations (tests T2 to T7) were performed with it.

#### Analysis of Stable vs. Oscillating Flow Behaviour

The simulated flow patterns reproduce what was observed during the experiments.

The stable states shown by tests T4 and T6 are confirmed by the numerical results, showing the formation of an A-jump for test T4 (

Figure 10) and of a B-jump for test T6 (

Figure 11), respectively: although the jump toe exhibits a certain displacement from its average position, the jump pattern is maintained during the whole simulation period.

For the flow conditions that exhibited an oscillatory pattern in the experiments, oscillatory flow patterns were also observed during numerical simulations, such as for test T2, where an A-wave pattern occurs (

Figure 12), or for the test T1 previously discussed, which shows a B-wave behaviour (

Figure 7).

The previous results show that the SPH simulations can correctly reproduce all the main characteristics of this phenomenon, which under specific flow conditions can lead to cyclic oscillations between jump types, resulting in the cyclic formation and evolution of jump vortices. As such, the complete spatial and temporal knowledge of the flow yielded by the SPH simulation can help us to improve the understanding of the phenomena by performing additional analyses of the flow field, without requiring new extensive experimental activity.

Figure 13 shows the amplitude spectra of the time series of the surface elevations, upstream and downstream of the jump for test T1. From the analysis of these spectra it is possible to observe in each of them the existence of a peak at a frequency around 0.1 Hz, which confirms the conclusions drawn by [

12], who stated the quasi-periodicity of the oscillating characteristic of wave and B jumps.

Furthermore, as the frequency of the peak in the upstream spectrum is almost equal to the downstream one, it is possible to conclude that fluctuations of the surface profile downstream of the jump also depend essentially on the alternations between B and wave jumps.

Figure 14 shows the amplitude spectrum of the time series of the surface elevations, upstream and downstream of the jump for test T2. From the analysis of the previously mentioned figure it is possible to observe the also in this case the existence of a peak in the spectrum of the time series of the surface elevations upstream of the jump, consistently with the oscillations between A and wave jumps.

However, in this case, no dominant peak frequency was noted, unlike in the T1 case with alternations between B and wave jumps. Consequently, as also suggested by [

12], the present numerical results confirm that the surface profile downstream of the roller in this case is not affected strongly by the oscillation between different jump types.

It is possible to evaluate non-linearity through the ratios between the amplitude of the two higher harmonics (A2 and A3) and that of the main component (A1), as proposed by [

74,

75]. A value of A2/A1 equal to 0.105 and 0.017 is found for surface elevation upstream of the jump in test T1 and test T2, respectively. A value of A3/A1 equal to 0.12 and 0.25 is found for surface elevation upstream of the jump in test T1 and test T2, respectively.

A value of A2/A1 equal to 0.18 and 0.35 is found for surface elevation downstream of the jump in test T1 and test T2, respectively. A value of A3/A1 equal to 0.35 and 1.056 is found for surface elevation downstream of the jump in test T1 and test T2, respectively. The tendency of these ratios to increase in the direction of the flow shows that the non-linearity of the surface waves tends to enhance while the waves propagate downstream.

The numerical results highlight also the relationship between the tailwater depth ratio

y_{t}/

y_{1} and the upstream Froude number F

_{1} as a function of the relative step height

s/

y_{1}. The results are presented in the form of diagram (

Figure 15) as function of the relative step height

s/

y_{1} equal to 0.06 to 0.1. In the diagram (

Figure 15), the basic flow pattern is indicated in the legend (A-jump, B-jump) including the oscillatory flow patterns (B-wave and A-wave).

The results show the different regions of flow conditions and the occurrence of oscillatory flow conditions between two different jump types characterised by quasi-periodic oscillations (

Figure 15).

Figure 16,

Figure 17,

Figure 18 and

Figure 19 show a part of the time series and the amplitude spectra of the pressure evaluated at different locations along the channel the bottom under the hydraulic jumps T1and T2 of

Table 1, which represent the two possible oscillating regimes (B-wave and A-wave, respectively). In particular, the pressure was evaluated at a distance of 7, 10, 20 and 100 cm from the time-averaged position of the hydraulic jump toe.

It can be seen from the pressure time history that, in the point closest to the jump toe, the bottom pressure assumes alternatively low and high pressures which can be mostly related to low and high water levels. Downstream, the cycle between low and high pressures is less regular, possibly because of the simultaneous effect of level fluctuations due to waves and of turbulent pressure fluctuations downstream of the roller. From the analysis of the pressure amplitude spectra for test T1 (

Figure 17), it is clear that even the pressure fluctuations are quasi-periodic and strongly influenced by the oscillations between the B and wave types, as they show a peak amplitude at the same frequency of the elevation spectra.

For the test T2 (

Figure 19), no dominant peak frequency was noted, unlike in the T1 case with alternations between B and wave jumps.

Figure 20,

Figure 21,

Figure 22 and

Figure 23 show a part of the time history of the horizontal u and vertical v velocity components computed at a point 0.01 m above the channel bottom under the hydraulic jumps T1 and T2 of

Table 1, at a distance of 7, 10, 20 and 100 cm from the time-averaged position of the hydraulic jump toe.

The basic characters of the oscillating flow fields depicted in

Figure 5 and

Figure 10 can be easily deduced from the velocity time histories.

In the case of test T1, the B-jump phase conserves a non-zero horizontal velocity component throughout the jump, consistent with the presence of the anti-clockwise roller on the surface; on the other hand, the wave-phase exhibits an almost vertical, downward flow at the intermediate locations downstream of the jump, which are a consequence of the strong clockwise roller, as sketched in

Figure 1b.

In the case of test T2, the oblique flow induced by the jump roller just downstream of the toe during the A-jump phase (

Figure 1a) can be deduced by the reduced values of u and by the positive values of v, while the structure of the wave-jump phase is less defined than in the previous case.

In any case, the analysis of the oscillating phenomena indicates in both oscillating flows a strong correlation among the surface profile elevations, velocity components and pressure fluctuations.

A quantitative evaluation of this correlation can be obtained by computing the correlation coefficient r:

where

x_{1} and

x_{2} are the two variables values, and the bar denotes an average of the two variables values.

For test T1, the computed values of r for (

p–

u), (

p–

v) and (

u–

v) pairs of data at a distance of 7 cm from the time-averaged position of the hydraulic jump toe, are equal to −0.72, 0.85 and −0.8, respectively (

Figure 24).

For test T2, the computed values of

r for (

p–

u), (

p–v) and (

u–v) pairs of data at a distance of 7 cm from the time-averaged position of the hydraulic jump toe, are equal to −0.98, 0.97 and −0.96, respectively (

Figure 25).

High negative values of the r coefficient for (p–u) indicate that, in general, upstream of the bottom step low levels correspond to horizontal flow (wave-jump conditions) and vice versa. This anti-correlation between pressure (or level) and u fluctuations is maintained downstream in the T2 case, while in the T1 case the two quantities are instead weakly correlated.

Farther downstream, the two velocity components and the pressures are substantially uncorrelated in the T2 case, indicating that the characteristic flow pattern of alternate near-wall and subsurface streams does not imply any preferential direction of the vertical motions, while exhibit still a non-zero degree of correlation in the T1 case, indicating that the oscillation between the B-jump and the stronger wave-jump tends to propagate its effects more than the oscillation between wave-jump and A-jump.