Nonlinear wave run-up in bays of arbitrary cross-section: generalization of the Carrier–Greenspan approach

AbstractWe present an exact analytical solution of the nonlinear shallow water theory for wave run-up in inclined channels of arbitrary cross-section, which generalizes previous studies on wave run-up for a plane beach and channels of parabolic cross-section. The solution is found using a hodograph-type transform, which extends the well-known Carrier–Greenspan transform for wave run-up on a plane beach. As a result, the nonlinear shallow water equations are reduced to a single one-dimensional linear wave equation for an auxiliary function and all physical variables can be expressed in terms of this function by purely algebraic formulas. In the special case of a U-shaped channel this equation coincides with a spherically symmetric wave equation in space, whose dimension is defined by the channel cross-section and can be fractional. As an example, the run-up of a sinusoidal wave on a beach is considered for channels of several different cross-sections and the influence of the cross-section on wave run-up characteristics is studied.

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Coupled method for the numerical simulation of 1D shallow water and Exner transport equations in channels with variable cross-section

E3S Web of Conferences ◽

10.1051/e3sconf/20184005012 ◽

2018 ◽

Vol 40 ◽

pp. 05012

Author(s):

Sergio Martínez-Aranda ◽

Javier Murillo ◽

Pilar García-Navarro

Keyword(s):

Shallow Water ◽

Finite Volume ◽

Cross Section ◽

Physical Phenomenon ◽

Mathematical Expression ◽

Channel Cross Section ◽

Finite Volume Scheme ◽

Variable Cross ◽

Volume Scheme ◽

Solid Transport

This work is focused on the a numerical finite volume scheme for the resulting coupled shallow water-Exner system in 1D applications with arbitrary geometry. The mathematical expression modeling the the hydrodynamic and morphodynamic components of the physical phenomenon are treated to deal with cross-section shape variations and empirical solid discharge estimations. The resulting coupled system of equations can be rewritten as a nonconservative hyperbolic system with three moving waves and one stationary wave to account for the source terms discretization. But, even for the simplest solid transport models as the Grass law, to find a linearized Jacobian matrix of the system can be a challenge if one considers arbitrary shape channels. Moreover, the bottom channel slope variations depends on the erosion-deposition mechanism considered to update the channel cross-section profile. In this paper a numerical finite volume scheme is proposed, based on an augmented Roe solver (first order accurate in time and space) and dealing with solid transport flux variations caused by the channel geometry changes. Channel crosssection variations lead to the appearance of a new solid flux source term which should be discretized properly. Comparison of the numerical results for several analytical and experimental cases demonstrate the effectiveness, exact wellbalanceness and accuracy of the scheme.

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Analytical Solution of the Time-Dependent Microfluidic Poiseuille Flow in Rectangular Channel Cross-Sections and Its Numerical Implementation in Microsoft Excel

Biosensors ◽

10.3390/bios9020067 ◽

2019 ◽

Vol 9 (2) ◽

pp. 67

Author(s):

Patrick Risch ◽

Dorothea Helmer ◽

Frederik Kotz ◽

Bastian E. Rapp

Keyword(s):

Analytical Solution ◽

Cross Section ◽

Poiseuille Flow ◽

Exact Analytical Solution ◽

Rectangular Channel ◽

Channel Cross Section ◽

Analysis Tool ◽

Complex Flow ◽

Microsoft Excel ◽

Navier Stokes Equation

We recently demonstrated that the Navier–Stokes equation for pressure-driven laminar (Poiseuille) flow can be solved in any channel cross-section using a finite difference scheme implemented in a spreadsheet analysis tool such as Microsoft Excel. We also showed that implementing different boundary conditions (slip, no-slip) is straight-forward. The results obtained in such calculations only deviated by a few percent from the (exact) analytical solution. In this paper we demonstrate that these approaches extend to cases where time-dependency is of importance, e.g., during initiation or after removal of the driving pressure. As will be shown, the developed spread-sheet can be used conveniently for almost any cross-section for which analytical solutions are close-to-impossible to obtain. We believe that providing researchers with convenient tools to derive solutions to complex flow problems in a fast and intuitive way will significantly enhance the understanding of the flow conditions as well as mass and heat transfer kinetics in microfluidic systems.

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ShallowFlow: A Program to Model Ship Hydrodynamics in Shallow Water

Volume 1A: Offshore Technology ◽

10.1115/omae2014-23291 ◽

2014 ◽

Cited By ~ 2

Author(s):

Tim P. Gourlay

Keyword(s):

Shallow Water ◽

Cross Section ◽

Hydrodynamic Pressure ◽

Open Water ◽

Arbitrary Cross Section ◽

Far Field ◽

Shallow Water Theory ◽

Ship Hydrodynamics ◽

Cruise Ships ◽

Sinkage And Trim

In this article we present details of “ShallowFlow”, a computer program to model the hydrodynamic flow around ships in calm shallow water. The program is based on slender-body shallow-water theory. Outputs from the program include far-field hydrodynamic pressure and flow velocities; free surface drawdown; sinkage and trim. Varying transverse bathymetry including open water, dredged channels, and canals of arbitrary cross-section may be modelled. The method is best suited to displacement ships, including cargo ships, ferries, cruise ships, warships and superyachts.

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On the Godunov Scheme Applied to the Variable Cross-Section Linear Wave Equation

Finite Volumes for Complex Applications VI Problems & Perspectives - Springer Proceedings in Mathematics ◽

10.1007/978-3-642-20671-9_33 ◽

2011 ◽

pp. 313-321

Author(s):

Stéphane Dellacherie ◽

Pascal Omnes

Keyword(s):

Wave Equation ◽

Cross Section ◽

Variable Cross Section ◽

Linear Wave ◽

Variable Cross ◽

Godunov Scheme ◽

Linear Wave Equation

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Evolution of long water waves in variable channels

Journal of Fluid Mechanics ◽

10.1017/s0022112094001011 ◽

1994 ◽

Vol 266 ◽

pp. 303-317 ◽

Cited By ~ 20

Author(s):

Michelle H. Teng ◽

Theodore Y. Wu

Keyword(s):

Cross Section ◽

Wave Energy ◽

Water Waves ◽

Mass Conservation ◽

Arbitrary Cross Section ◽

Channel Cross Section ◽

Cross Sectional ◽

Reflected Waves ◽

Conservation Property ◽

Divergent Channel

This paper applies two theoretical wave models, namely the generalized channel Boussinesq (gcB) and the channel Korteweg–de Vries (cKdV) models (Teng & Wu 1992) to investigate the evolution, transmission and reflection of long water waves propagating in a convergent–divergent channel of arbitrary cross-section. A new simplified version of the gcB model is introduced based on neglecting the higher-order derivatives of channel variations. This simplification preserves the mass conservation property of the original gcB model, yet greatly facilitates applications and clarifies the effect of channel cross-section. A critical comparative study between the gcB and cKdV models is then pursued for predicting the evolution of long waves in variable channels. Regarding the integral properties, the gcB model is shown to conserve mass exactly whereas the cKdV model, being limited to unidirectional waves only, violates the mass conservation law by a significant margin and bears no waves which are reflected due to changes in channel cross-sectional area. Although theoretically both models imply adiabatic invariance for the wave energy, the gcB model exhibits numerically a greater accuracy than the cKdV model in conserving wave energy. In general, the gcB model is found to have excellent conservation properties and can be applied to predict both transmitted and reflected waves simultaneously. It also broadly agrees well with the experiments. A result of basic interest is that in spite of the weakness in conserving total mass and energy, the cKdV model is found to predict the transmitted waves in good agreement with the gcB model and with the experimental data available.

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Linear Wave Interaction with a Vertical Cylinder of Arbitrary Cross Section: An Asymptotic Approach

Journal of Waterway Port Coastal and Ocean Engineering ◽

10.1061/(asce)ww.1943-5460.0000407 ◽

2017 ◽

Vol 143 (5) ◽

pp. 04017028 ◽

Cited By ~ 4

Author(s):

N. B. Dişibüyük ◽

A. A. Korobkin ◽

O. Yilmaz

Keyword(s):

Cross Section ◽

Vertical Cylinder ◽

Wave Interaction ◽

Arbitrary Cross Section ◽

Linear Wave ◽

Asymptotic Approach

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A mathematical formulation for estimating maximum run-up height of 2018 Palu tsunami

10.5194/nhess-2020-223 ◽

2020 ◽

Author(s):

Ikha Magdalena ◽

Antonio Hugo Respati Dewabrata ◽

Alvedian Mauditra Aulia Matin ◽

Adeline Clarissa ◽

Muhammad Alif Aqsha

Keyword(s):

Shallow Water ◽

Cross Section ◽

Incident Wave ◽

Shallow Water Equation ◽

Mathematical Formulation ◽

Equation System ◽

Type Transformation ◽

Linear Equation System ◽

Nonlinear Shallow Water Equations ◽

Run Up

Abstract. Run-up is defined as sea wave up-rush on a beach. Run-up height is affected by many factors, including the shape of the bay. As an archipelagic country, Indonesia consists of thousands of islands with bays of diverse profiles, including Palu Bay, which is a well-known example of a bay with a drastically-increasing wave run-up height. In the case of the 2018 Palu tsunami, scientists found that the incident wave was amplified by the shape of the bay. The amplifying wave played a large role in the significant increase of run-up height. The run-up in question caused severe inundation, which led to a high number of casualties and damages. Therefore a mathematical model will be constructed to investigate the wave run-up. The bay's geometry will be approximated using three linearly-inclined channel types: one of parabolic cross-section, one of triangular cross-section, and a plane beach. We use the generalized nonlinear shallow water equations, which is then solved analytically using a hodograph-type transformation. As a result, the nonlinear shallow water equation system can be reduced to a one-dimensional linear equation system. Assuming the incident wave is sinusoidal, we can obtain a simple formula for calculating maximum run-up height on the shoreline.

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