# Investigation of Tsunami Waves in a Wave Flume: Experiment, Theory, Numerical Modeling

## Abstract

**:**

## 1. Introduction

^{2}is the acceleration of gravity. Upon entry to the zone of shallow water, the front speed of the wave decreases sharply, and the wave height increases tens of times. Inside bays and outfalls, the amplitude of the tsunami wave increases up to 20 m or higher because of space restriction from the sides. The danger of tsunami waves is associated primarily with their unpredictable, sudden, and tremendous energy. Exploring tsunami waves in natural conditions is almost impossible. Experiments in ground facilities usually have a high cost since bringing the wave simulation parameters to natural conditions requires the creation of large-scale (up to 300 m or more) and costly facilities. Therefore, the study of tsunami waves makes extensive use of analytical methods of research, as well as numerical (computer) modeling approaches [1,2,3].

## 2. Problems of Modeling Tsunami Waves in Experimental Facilities

## 3. Mathematic Model and Numerical Method

^{4}; however, in our calculations of the turbulence model, they are omitted. The reason for this approach is the fact that the experiments in rectangular channels [13] show high enough Reynolds numbers of transition to the turbulent state $R{e}^{*}=\frac{{\rho}_{UH}}{\eta}$, where U is a fluid velocity. At that, the value Re* increases with decreasing of the distance from the channel entrance.

_{1}* = 8 × 10

^{3}and the end of the transition to Re

_{2}* = 1.8 × 10

^{4}. Furthermore, it is known that when the initial perturbation in the flow decreases, the Reynolds of the transition increases as well. In this case (at the wave length of L ≈ 3 m, H ≈ 0.1 m), the value x/H ≤ 30, and the initial perturbations before the wave are close to zero.

_{1}+ (1 − γ) ρ

_{2}, where ρ

_{1}—density of the carrier fluid, ρ

_{2}—density of air; accordingly, the viscosity: η = γη

_{1}+ (1 − γ)η

_{2}. The force caused by surface tension ${\mathit{F}}_{\sigma}=\sigma k\mathsf{\nabla}\gamma $, where σ = 72.8 N/m is the surface tension of water–air; $k=\mathsf{\nabla}\left(\frac{\mathsf{\nabla}\gamma}{\left|\nabla \gamma \right|}\right)=\mathsf{\nabla}\mathit{n}.$

_{nn}= −p

_{atm}, p

_{ns}= 0; where p

_{nn}, p

_{ns}are normal and tangential stresses, p

_{atm}is external pressure.

## 4. Experimental Equipment and Research Methods

- Initial water depth in channel H varied from 100 mm to 103 mm;
- Wave length L ≈ 3 m, averaged incident wave amplitude A in a series of experiments ranged from 0.5 mm to 15 mm.

_{r}and W

_{t}are the total energy of the incident, reflected, and transmitted waves, respectively. A

_{r}and A

_{t}are averaged amplitudes of reflected and transmitted waves.

## 5. Generation and Propagation of Waves in a Wave Flume

#### 5.1. Wave Initiation

_{0}> 90 mm. Then, the air is evacuated via tube (3) from the upper part of the generator, thereby attaining specified water level difference η

_{0}: the water level (H + η

_{0}) in the generator and the water level H in the working part of the channel. After depressurization (t = 0) of the upper part of the generator, the wave is initiated in the working part of the channel. The wave has the length L ≈ 2a and the amplitude А ≈ η

_{0}/2 (see Appendix A).

_{0}= 2A is started. Up to this point in time, the water in the generator and the working channel was at rest. At t = 0.7 s, we see that the initial level difference split into two waves: a negative wave (−A) moves inside the generator (a = 1.465 m), a wave with a height (+A) moves into the working part of the hydrodynamic channel. In each image, above the white horizontal line, we can see the profiles of the water levels in these waves, below the longitudinal velocity. Further, at t = 2 s and t = 3.3 s, we see that the wave (−A) reflected from the generator wall and formed a single wave with height (+A) and length L = 2a, which moves into the working part of the wave flume. The wave speed is $c=\sqrt{gH}=1\mathrm{m}/\mathrm{s}$. The velocity of liquid in the channel before and after the wave was zero.

_{0}= 0.015 m.

#### 5.2. Wave Propagation

_{0}= 15 mm, H = 102 mm.

#### 5.3. Transformation of Highly Nonlinear Wave, Which Interacts with Shallow Water

## 6. Interaction of Tsunami-Like Waves with Impermeable Thin Barriers

#### 6.1. Experimental and Numerical Studies

_{r}+ W

_{t}.

_{r}+ W

_{t}) < W. For example, at = 0.9H, experiments show that W

_{r}+ W

_{t}≈ 0.5W. A legitimate question arises: “Where did 50% of the incident wave energy go?” On the other hand, numerical simulation of the experimental conditions based on the full Navier–Stokes Equations (white markers in Figure 12) gives the correct result that coincides with the experiment.

#### 6.2. Theoretical Studies

_{k}+ W

_{p}= 2W

_{p}. Here W

_{k}and W

_{p}are the kinetic and potential energy of the wave. For example, the energy flux through the channel cross-section C—C (Figure 14) is:

_{t}. To do this, we use the Borda–Carnot principle [17], according to which these losses are similar to the energy losses during the inelastic impact of solid balls when one ball catches up with another, which moves at a slower speed. In this case, “the lost kinetic energy is equal to the energy of the lost velocities.” In our case, we have:

_{t}can be easily obtained from the condition of conservation of the liquid flow rate $U\left(\delta +{\delta}^{*}\right)={v}_{t}H$. In our approximation, we have $\raisebox{1ex}{${A}_{t}$}\!\left/ \!\raisebox{-1ex}{$H$}\right.\le 0.1$ and $\raisebox{1ex}{${v}_{t}^{2}$}\!\left/ \!\raisebox{-1ex}{$2$}\right.=\raisebox{1ex}{$g{A}_{t}^{2}$}\!\left/ \!\raisebox{-1ex}{$2$}\right.$. Finally, we arrive at Equations (10) and (11) in the form:

## 7. Conclusions

- Investigation of the features of modeling tsunami waves in a laboratory installation;
- Theoretical, experimental, and numerical studies of the interaction of tsunami waves with underwater obstacles;
- It is shown that at a certain optimal height of a thin impermeable barrier, its effectiveness in suppressing the energy of an incident tsunami wave is 70%, which is explained by the accumulation of energy in large-scale vortex structures near the obstacle.

- The use of precision measuring channels (sensor + equipment) for recording the water level made it possible to simulate the main dimensionless parameters of tsunami waves in a laboratory setup, equivalent to the parameters in large-scale wave flumes;
- The wave generator ensures the creation of gravity waves equivalent to theoretical ones with an instantaneous jump in water level and speed at the leading edge of the wave. In this case, the wavelength does not depend on its height and is determined only by the length of the wave generator;
- In our studies, we studied the interaction of a stationary homogeneous water flow ${u}_{x}=\frac{A}{H}\sqrt{gH}$ with underwater barriers, since the condition τ
_{s}< T is always provided, where τ_{s}is the time of the establishment of a stationary flow around.

^{4}, i.e., only laminar flows behind the wave front are considered. We do not consider the important problem of modeling the flow around underwater barriers in terms of the dimensionless number Re.

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Characteristics of Gravity Waves of the Tsunami Type, Modeled in the Hydrodynamic Channel of the IPRIM RAS

_{0}= 0.02 m corresponds to an upward shift of the beam by 491.4 mV. The sensor registered a sharp drop in the water level by −240 mV after the start of the wave generator. This level drop corresponds to a wave that propagates inside the generator and has a height of 𝐴 = − η

_{0}/2 with a measurement error of δ = 2.32%. Sensors No. 2–4 (blue and cyan) register waves in the working part of the wave tray. The distance between sensors No.2 and No.4 is 6.7 m. It can be seen from the 546 oscillogram that the time it takes the wave to cover this distance is Δt = 6.56 s. Thus, the experimentally measured wave speed is c = 1.03 m/s. The group velocity of such a single wave calculated on the basis of the linear theory of shallow water is $c=\sqrt{gH}=1.001\mathrm{m}/\mathrm{s}$. From Figure A1, it can be seen that the wave that propagates into the working channel of the flume has a time extension of about T ≈ 3 s, thus: L = 2a = cT ≈ 3 m.

_{0}in time τ [1]. In this case, if ${\tau}^{*}=\frac{\tau}{T}=\frac{2a}{\sqrt{gH}}\le 1$, then the average wave height $A={\eta}_{0}$/2, wavelength $L=a\left(2+{\tau}^{*}\right)$, and the potential energy of the wave is equal to its kinetic energy W

_{p}= W

_{k}. The total wave energy (per meter of length along the front) was calculated using Formula (7) and is equal to $W=2a\varrho g{\eta}_{0}^{2}\left(\frac{1}{2}-\frac{{\tau}^{*}}{12}\right)$, where $2a\varrho g{\eta}_{0}^{2}$ is the potential energy of the area (2a∙1 m

^{2}) of water as it rises to a height η

_{0}above the initial level of the liquid.

_{0}/2. In Figure 5, the red line shows the profile of such a model wave. The potential energy of the model wave, as expected, is equal to half of the initial potential energy of the generator: ${W}_{p}=2\rho ga{\left(\frac{{\eta}_{0}}{2}\right)}^{2}=\frac{1}{2}\rho ga{\eta}_{0}^{2}$. The calculation of the potential energy of real waves by integrating the experimental wave profile using the Formula ${W}_{p}=\frac{1}{2}\varrho g\sqrt{gH}{{\displaystyle \int}}_{0}^{T}\mathsf{\xi}\left(t\right)\mathrm{d}t$ gives the same result.

_{1}, the wave А

_{1}= −η

_{0}/2 moves inside the generator, and the wave А

_{2}= η

_{0}/2 moves into the working channel. At t = t

_{2}, wave A

_{1}is reflected from the wall and, together with wave A

_{2}, forms a single wave of length L = 2a, which moves into the working channel (see t = t

_{3}).

_{v}= c. Before the wave and behind the rear surface of the discontinuity, the water velocity is zero. The fluid velocity inside the wave is u << c; it (like the velocity c) does not depend on spatial coordinates y-z. A schematic drawing of the wave in a fixed (laboratory) frame of reference is shown in Figure A3a.

**Figure A3.**Schematic drawing of the motion of a gravitational wave in various frames of reference: (a) in a fixed (laboratory) frame of reference; (b) in a moving (with the speed of a wave) frame of reference.

**U**= ∞). However, the relationship between the wave parameters can be obtained from the conditions of mass and momentum flux conservation on both sides of the discontinuity. Since the wave propagates in one direction without changing its shape, then, passing into a moving (with speed c) frame of reference, we turn the nonstationary problem into a stationary one. In this frame of reference (Figure A3b), the conservation conditions have the form [22]:

_{1}, with an arbitrary value of the nonlinearity parameter ε = A/H, is also easy to obtain from relations Equations (A1) and (A2):

^{2}).

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**Figure 5.**Modeling the process of generating a tsunami-type wave in a wave flume by an ideal generator at H = 0.103 m, A/H = 0.1.

**Figure 6.**Comparison of experimental dependence wave height—time (green line) ξ(t) = H

_{ξ}(t) − H with the numerical calculation for ideal generator (black line) at a distance of 1.5 m from the front wall of wave generator. The red dashed line is the model wave profile.

**Figure 7.**The diagram (x-t) of gravity tsunami-like waves, which propagate in the hydrodynamic channel at the following initial parameters: η

_{0}= 15 mm, H = 102 mm, A/H = 0.074. Makers—experiments. Lines—linear theory of shallow water. The blue line is the trajectory of the incident wave; the red lines are the reflected waves. G—wave generator; W—reflecting wall; 1–4—locations of wave level sensors.

**Figure 8.**Dependence of long gravity waves velocity in the hydrodynamic channel on the nonlinearity parameter A/H: 1—linear theory of shallow water, 2—Navier–Stokes Equations, 3—nonlinear theory of shallow water (4А).

**Figure 9.**Breaking of wave interacting with shallow water. The comparison of experimental results with numerical simulation at H = 0.135 m, A/H = 0.61, H

_{sh}= 0.054 m, A/H

_{sh}= 1.52: (

**a**,

**c**)—t = 8.3 s; (

**b**,

**d**)—t = 8.4 s.

**Figure 10.**Schematic diagram of the interaction of a tsunami-type wave with impenetrable barriers No. 1 and No. 2.

**Figure 11.**Reflection coefficient as a function of dimensionless parameter h/(H + A): (1)—No. 1, A/H = 0.286 [4]; (2)—No. 1, A/H = 0.04–0.05; (3)—No. 2, A/H = 0.04–0.10; (4)—Navier–Stokes Equations, A/H = 0.07 and (5)—linear theory of long waves.

**Figure 12.**The sum of the relative energies of the waves reflected and transmitted through the barrier as a function of the generalized parameter of the barrier height: 1—experiments; 2—numerical experiment based on f Equations; 3—calculations by author’s theory at Н = 0.103 m, А = 0.007 m, and k = 0.68.

**Figure 13.**Visualization for various instants of time of velocity fields near thin barrier No. 2 (black color) in the case of transmission through it of a tsunami type wave: A/H = 0.07, parameter h/(H + A) = 0.9. The dashed line corresponds to the unperturbed water flow level.

**Figure 14.**Schematic diagram of the interaction of a tsunami-type wave with a thin impenetrable barrier. Shallow water theory is not applicable near the barrier: −L/2 < x < +L/2.

Zone | Water Depth, H (km) | Wave Height A (m) | Wave Length L (km) | Nonlinearity A/H | Dispersion H/L |
---|---|---|---|---|---|

Ocean | 4 | 1 | 400 | 0.00025 | 0.01 |

Continental shelf | 0.150 | 2.25 | 80 | 0.015 | 0.0019 |

Shallow | 0.015 | 4 | 30 | 0.27 | 0.0005 |

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**MDPI and ACS Style**

Boshenyatov, B.V. Investigation of Tsunami Waves in a Wave Flume: Experiment, Theory, Numerical Modeling. *GeoHazards* **2022**, *3*, 125-143.
https://doi.org/10.3390/geohazards3010007

**AMA Style**

Boshenyatov BV. Investigation of Tsunami Waves in a Wave Flume: Experiment, Theory, Numerical Modeling. *GeoHazards*. 2022; 3(1):125-143.
https://doi.org/10.3390/geohazards3010007

**Chicago/Turabian Style**

Boshenyatov, Boris Vladimirovich. 2022. "Investigation of Tsunami Waves in a Wave Flume: Experiment, Theory, Numerical Modeling" *GeoHazards* 3, no. 1: 125-143.
https://doi.org/10.3390/geohazards3010007