# Invariants of Turbulence Reynolds Stress and of Dissipation Tensors in Regular Breaking Waves

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## Abstract

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## 1. Introduction

## 2. Experimental Set-Up and Experiments

^{®}USL 80D by General Acoustics, Kiel, Germany, sensor model USS635), with an accuracy on the instantaneous water level measurements equal to $\pm 0.5$ mm.

## 3. Data Analysis and Visualization

## 4. Turbulent Stress Invariants Analysis and Relation with the Dissipation Tensor

## 5. Conclusions

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**The experimental flume adopted for the tests: (

**a**) side view of the flume; (

**b**) layout of the bar and of the volume of measurement; (

**c**) top view showing the cameras of the V3V system. The still water depth in the mid section of the bar ($X=1050$ cm) is $d=28$ cm. The dot line indicates the still water level, the dashed line is the mean water level $\overline{\eta}$ (wave set-up or set-down). Dimensions are in centimeters.

**Figure 2.**Wave profiles in the section of velocity measurements (Section D) for (

**a**) the tests with $T=1.5\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\mathrm{s}$, and (

**b**) the tests with $T=2.0\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\mathrm{s}$. The dashed lines limit the $\pm 1$ standard deviation band for the sample of 10 wave cycles during velocity acquisition.

**Figure 3.**Experiment 6b, a single snapshot in a sequence of 13 shots of the first measured wave cycle. (

**a**) Instantaneous velocity in the midplane of the flume (z = 0); (

**b**) phase-averaged velocity; (

**c**) velocity vectors difference (fluctuating velocity) between the instantaneous velocity and the phase-averaged velocity. Only velocity components in the x–y plane are shown. The inset depicts the surface elevation time series, with the symbols indicating the time of the shot. (modified from Clavero et al., 2016 [16], with permission).

**Figure 4.**The main elements of the anisotropy tensor ${a}_{ij}$ for four phases across breaking. (

**a**) Experiment 6b; (

**b**) Experiment 9b. The element $-{a}_{12}$ is shifted downward, and only one experimental data of two is shown for an easy visualization.

**Figure 5.**Experiment 6b, first four phases (crosses, circles, squares and stars), first wave cycle. (

**a**–

**c**) Lumley map; (

**b**–

**d**) barycentric map based on scalar metrics which are functions of eigenvalues of the second order stress tensor describing turbulence. The values are averaged in horizontal layers. The insets show the phase, the arrows indicate the deepest point, with data 0.4 cm apart. Only one point of three is plotted for a clear visualization.

**Figure 6.**Experiment 9b, see Figure 5 for caption.

**Figure 7.**Invariant function $F=1+9{I}_{2}+27{I}_{3}$ for the first five phases. (

**a**) Experiment 6b; (

**b**) Experiment 9b.

**Figure 8.**The main elements of the dissipation anisotropy tensor ${e}_{ij}$ for four phases across breaking. (

**a**) Experiment 6b; (

**b**) Experiment 9b. The element $-{e}_{12}$ is shifted downward, and only one experimental data of two is shown for an easy visualization.

**Figure 9.**Comparison of the invariant function of the turbulence and of the dissipation tensors for the first five phases. (

**a**) Experiment 1b; (

**b**) Experiment 2b; (

**c**) Experiment 6b; (

**d**) Experiment 9b. The dashed curve is the interpolating line. Gray symbols refer to measurements above the still water level.

**Figure 10.**Relation between ${a}_{ii}$ and ${e}_{ii}$. (

**a**–

**c**) Experiment 6b; (

**d**–

**f**) Experiment 9b. The dashed curves are the interpolating lines.

**Table 1.**Parameters of the tests. ${H}_{0}$ is the target wave height (almost coincident with the generated wave height), T is the wave period and ${H}_{0}/{L}_{0}$ is the deep-water wave steepness. ${\xi}_{0}=tan\alpha /\sqrt{{H}_{0}/{L}_{0}}$ is the Iribarren number ($\alpha $ is the bed slope), $h=d+\overline{\eta}$ is the mean water depth in the section of measurements, $\overline{\eta}$ is the wave set-up, and ${H}_{b-rms}$, ${H}_{b-1/3}$, and ${H}_{b-max}$ are the root-mean-square wave height, the mean of the highest third of the waves, and the maximum wave height, respectively, all referred to as the statistics of the breakers. ${d}_{i}$ and ${d}_{e}$ are the still-water depth at the internal and external toe of the bar, respectively, B and ${B}^{\prime}$ are the width of the crest and the total width of the bar. The still-water depth in front of the paddle is 43 cm and the breaking section is D at $X\approx 1138$ cm, with a still-water depth $d=19.2$ cm.

Exp. | ${\mathit{H}}_{0}$ (cm) | T (s) | ${\mathit{H}}_{0}/{\mathit{L}}_{0}$ | ${\mathit{\xi}}_{0}$ | h (cm) | $\overline{\mathit{\eta}}$ (cm) | ${\mathit{H}}_{\mathit{b}-\mathit{rms}}$ (cm) | ${\mathit{H}}_{\mathit{b}-1/3}$ (cm) | ${\mathit{H}}_{\mathit{b}-\mathit{max}}$ (cm) | ${\mathit{d}}_{\mathit{i}}/{\mathit{L}}_{0}$ | ${\mathit{d}}_{\mathit{e}}/{\mathit{L}}_{0}$ | $\mathit{B}/{\mathit{L}}_{0}$ | ${\mathit{B}}^{\prime}/{\mathit{L}}_{0}$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

1b | 6 | 1.5 | 0.017 | 0.765 | 19.2 | 0.0 | 5.7 | 6.2 | 6.3 | 0.060 | 0.109 | 0.199 | 0.456 |

2b | 7 | 1.5 | 0.020 | 0.708 | 19.4 | 0.2 | 7.4 | 8.1 | 8.7 | ||||

3b | 8 | 1.5 | 0.023 | 0.662 | 19.6 | 0.4 | 8.0 | 8.6 | 8.8 | ||||

4b | 9 | 1.5 | 0.026 | 0.624 | 19.7 | 0.5 | 8.7 | 9.2 | 9.9 | ||||

5b | 10 | 1.5 | 0.028 | 0.592 | 19.8 | 0.6 | 9.2 | 9.9 | 10.4 | ||||

6b | 6 | 2 | 0.010 | 1.020 | 19.4 | 0.2 | 6.1 | 6.3 | 6.5 | 0.034 | 0.061 | 0.112 | 0.256 |

7b | 7 | 2 | 0.011 | 0.944 | 19.4 | 0.2 | 7.6 | 8.3 | 8.7 | ||||

8b | 8 | 2 | 0.013 | 0.883 | 19.5 | 0.3 | 9.0 | 9.6 | 10.4 | ||||

9b | 9 | 2 | 0.014 | 0.833 | 19.6 | 0.4 | 10.3 | 11.2 | 12.8 |

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**MDPI and ACS Style**

Longo, S.; Clavero, M.; Chiapponi, L.; Losada, M.A.
Invariants of Turbulence Reynolds Stress and of Dissipation Tensors in Regular Breaking Waves. *Water* **2017**, *9*, 893.
https://doi.org/10.3390/w9110893

**AMA Style**

Longo S, Clavero M, Chiapponi L, Losada MA.
Invariants of Turbulence Reynolds Stress and of Dissipation Tensors in Regular Breaking Waves. *Water*. 2017; 9(11):893.
https://doi.org/10.3390/w9110893

**Chicago/Turabian Style**

Longo, Sandro, Maria Clavero, Luca Chiapponi, and Miguel A. Losada.
2017. "Invariants of Turbulence Reynolds Stress and of Dissipation Tensors in Regular Breaking Waves" *Water* 9, no. 11: 893.
https://doi.org/10.3390/w9110893