# Numerical Simulation of Propagation and Run-Up of Long Waves in U-Shaped Bays

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## Abstract

**:**

## 1. Introduction

## 2. Saint-Venant Equations and the MCS Model

#### Numerical Methods

## 3. Propagation and Run-Up in Channel ${\mathcal{B}}_{m}$

#### 3.1. Simulation of an Initial Hump

#### 3.2. The Shoreline Dynamics

## 4. Propagation and Run-Up in Channel ${\mathcal{B}}_{m}^{*}$

#### 4.1. Shoaling and Run Up of a Monochromatic Wave

#### 4.2. Run Up of a Solitary Hump

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Conflicts of Interest

## Abbreviations

MCS | Momentum Conserving Staggered-grid |

## References

- Bryant, E. Tsunami: The Underrated Hazard; Springer: Berlin/Heidelberg, Germany, 2014. [Google Scholar]
- Frederik, M.C.; Adhitama, R.; Hananto, N.D.; Sahabuddin, S.; Irfan, M.; Moefti, O.; Putra, D.B.; Riyalda, B.F. First results of a bathymetric survey of Palu Bay, Central Sulawesi, Indonesia following the Tsunamigenic Earthquake of 28 September 2018. Pure Appl. Geophys.
**2019**, 176, 3277–3290. [Google Scholar] [CrossRef] - Pelinovsky, E.; Troshina, E. Propagation of long waves in straits. Phys. Oceanogr.
**1994**, 5, 43–48. [Google Scholar] [CrossRef] - Didenkulova, I.; Pelinovsky, E. Non-dispersive traveling waves in inclined shallow water channels. Phys. Lett. A
**2009**, 373, 3883–3887. [Google Scholar] [CrossRef] - Didenkulova, I.; Pelinovsky, E. Nonlinear wave evolution and runup in an inclined channel of a parabolic cross-section. Phys. Fluids
**2011**, 23, 086602. [Google Scholar] [CrossRef] - Anderson, D.; Harris, M.; Hartle, H.; Nicolsky, D.; Pelinovsky, E.; Raz, A.; Rybkin, A. Run-up of long waves in piecewise sloping U-shaped bays. Pure Appl. Geophys.
**2017**, 174, 3185–3207. [Google Scholar] [CrossRef] - Nesje, A.; Dahl, S.O.; Valen, V.; Øvstedal, J. Quaternary erosion in the Sognefjord drainage basin, western Norway. Geomorphology
**1992**, 5, 511–520. [Google Scholar] [CrossRef] - Dartnell, P.; Normark, W.R.; Driscoll, N.W.; Babcock, J.M.; Gardner, J.V.; Kvitek, R.G.; Iampietro, P.J. Multibeam Bathymetry and Selected Perspective Views Offshore SAN Diego, California; Number 2959; US Geological Survey: Reston, VA, USA, 2007.
- Stoker, J. Water Waves; Interscience, Wiley: New York, NY, USA, 1957. [Google Scholar]
- Wu, Y.H.; Tian, J.W. Mathematical analysis of long-wave breaking on open channels with bottom friction. Ocean. Eng.
**2000**, 27, 187–201. [Google Scholar] [CrossRef] - Golinko, V.; Osipenko, N.; Pelinovsky, E.; Zahibo, N. Tsunami wave runup on coasts of narrow bays. Int. J. Fluid Mech. Res.
**2006**, 33, 106–118. [Google Scholar] - Choi, B.H.; Pelinovsky, E.; Kim, D.; Didenkulova, I.; Woo, S.B. Two-and three-dimensional computation of solitary wave runup on non-plane beach. Nonlinear Process. Geophys.
**2008**, 15, 489–502. [Google Scholar] [CrossRef] [Green Version] - Harris, M.; Nicolsky, D.; Pelinovsky, E.; Rybkin, A. Runup of nonlinear long waves in trapezoidal bays: 1-D analytical theory and 2-D numerical computations. Pure Appl. Geophys.
**2015**, 172, 885–899. [Google Scholar] [CrossRef] - Garayshin, V.; Harris, M.W.; Nicolsky, D.; Pelinovsky, E.; Rybkin, A. An analytical and numerical study of long wave run-up in U-shaped and V-shaped bays. Appl. Math. Comput.
**2016**, 279, 187–197. [Google Scholar] [CrossRef] [Green Version] - Harris, M.; Nicolsky, D.; Pelinovsky, E.; Pender, J.; Rybkin, A. Run-up of nonlinear long waves in U-shaped bays of finite length: Analytical theory and numerical computations. J. Ocean. Eng. Mar. Energy
**2016**, 2, 113–127. [Google Scholar] [CrossRef] [Green Version] - Didenkulova, I.; Pelinovsky, E. Runup of tsunami waves in U-shaped bays. Pure Appl. Geophys.
**2011**, 168, 1239–1249. [Google Scholar] [CrossRef] [Green Version] - Garayshin, V.V. Tsunami Runup in U and V Shaped Bays. Ph.D. Thesis, University of Alaska Fairbanks, Fairbanks, AK, USA, 2013. [Google Scholar]
- Pudjaprasetya, S.; Magdalena, I. Momentum conservative scheme for dam break and wave run up simulations. East Asian J. Appl. Math.
**2014**, 4, 152–165. [Google Scholar] [CrossRef] - Mungkasi, S.; Magdalena, I.; Pudjaprasetya, S.R.; Wiryanto, L.H.; Roberts, S.G. A staggered method for the shallow water equations involving varying channel width and topography. Int. J. Multiscale Comput. Eng.
**2018**, 16, 3. [Google Scholar] [CrossRef] - Swastika, P.V.; Pudjaprasetya, S.R.; Wiryanto, L.H.; Hadiarti, R.N. A Momentum-Conserving Scheme for Flow Simulation in 1D Channel with Obstacle and Contraction. Fluids
**2021**, 6, 26. [Google Scholar] [CrossRef] - Synolakis, C.E. The runup of solitary waves. J. Fluid Mech.
**1987**, 185, 523–545. [Google Scholar] [CrossRef]

**Figure 1.**(

**a**) The bathymetry of Palu Bay with a U-shaped cross-section (source: Badan Informasi Geospatial (BIG)); (

**b**) three-dimensional sketch of the channel-bays ${\mathcal{B}}_{m}$; (

**c**) the longitudinal section $d(x,0)=\alpha x$, for $x<0$; and (

**d**) the cross-sections $d(x,y)$ in the form of ${c\left|y\right|}^{m}$, for fixed m.

**Figure 3.**(

**a**) Contour plot of the normalized numerical surface $\eta (x,t)/{a}_{0}$; (

**b**) the enlarged contour plot of $\eta (x,t)/{a}_{0}$ near the shoreline; and (

**c**) the corresponding contour plot of $u(x,t)/{u}_{0}$ near the shoreline, where the red curves represent the shorelines.

**Figure 4.**Numerical versus analytical shoreline for three different U-shaped bays for: $m=3$ (

**top**); $m=1$ (

**middle**); and $m=2/3$ (

**bottom**). The analytical shorelines are digitized from [14].

**Figure 5.**Snapshots of free surface motion in a U-shaped bay of ${\mathcal{B}}_{2/3}$, our numerical result is compared with the analytical surface [14].

**Figure 6.**(

**a**) Illustration of the non-constant sloping bed ${x}^{4m/(3m+2)}$ in (16); and (

**b**–

**e**) the 3D sketches of non-reflecting bay ${\mathcal{B}}_{m}^{*}$ for $m=0.5,1,2,20$.

**Figure 7.**Free surface plot showing the shoaling process of monochromatic waves in a U-shaped bay (16), with varying m: $m=2$ (

**a**); and $m=3$ (

**b**).

**Figure 8.**(

**a**) Free surface motion in the simulation of a solitary wave run up; and (

**b**) shoreline dynamics as a function of time, resulting from the run up of a solitary wave (19) for a narrow bays with $m=2$.

**Table 1.**Run-up height of solitary wave in the parabolic bay ${\mathcal{B}}_{2}^{*}$ and plane beach, comparison between numeric and analytic.

Amplitude | Parabolic-Bay | Plane Beach | ||
---|---|---|---|---|

${\mathit{a}}_{\mathbf{0}}/{\mathit{d}}_{\mathbf{0}}$ | ${\mathit{R}}_{\mathit{num}}/{\mathit{a}}_{\mathbf{0}}$ | ${\mathit{R}}_{\mathbf{2}}/{\mathit{a}}_{\mathbf{0}}$ | ${\mathit{R}}_{\mathit{plnum}}/{\mathit{a}}_{\mathbf{0}}$ | ${\mathit{R}}_{\mathit{pl}}/{\mathit{a}}_{\mathbf{0}}$ |

0.01 | 3.20 | 3.27 | 2.71 | 2.83 |

0.02 | 4.50 | 4.62 | 3.36 | 3.37 |

0.03 | 5.40 | 5.66 | 3.78 | 3.73 |

0.04 | 6.00 | 6.53 | 4.00 | 4.00 |

0.05 | 6.48 | 7.30 | 3.93 | 4.23 |

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**MDPI and ACS Style**

Pudjaprasetya, S.R.; Risriani, V.M.; Iryanto.
Numerical Simulation of Propagation and Run-Up of Long Waves in U-Shaped Bays. *Fluids* **2021**, *6*, 146.
https://doi.org/10.3390/fluids6040146

**AMA Style**

Pudjaprasetya SR, Risriani VM, Iryanto.
Numerical Simulation of Propagation and Run-Up of Long Waves in U-Shaped Bays. *Fluids*. 2021; 6(4):146.
https://doi.org/10.3390/fluids6040146

**Chicago/Turabian Style**

Pudjaprasetya, Sri R., Vania M. Risriani, and Iryanto.
2021. "Numerical Simulation of Propagation and Run-Up of Long Waves in U-Shaped Bays" *Fluids* 6, no. 4: 146.
https://doi.org/10.3390/fluids6040146