A comparison of a two-dimensional depth-averaged flow model and a three-dimensional RANS model for predicting tsunami inundation and fluid forces

Qin, Xinsheng; Motley, Michael; LeVeque, Randall; Gonzalez, Frank; Mueller, Kaspar

doi:https://doi.org/10.5194/nhess-18-2489-2018

Articles | Volume 18, issue 9

https://doi.org/10.5194/nhess-18-2489-2018

Articles | Volume 18, issue 9

Research article

18 Sep 2018

Research article |

| 18 Sep 2018

A comparison of a two-dimensional depth-averaged flow model and a three-dimensional RANS model for predicting tsunami inundation and fluid forces

Xinsheng Qin, Michael Motley, Randall LeVeque, Frank Gonzalez, and Kaspar Mueller

Abstract

The numerical modeling of tsunami inundation that incorporates the built environment of coastal communities is challenging for both 2-D and 3-D depth-integrated models, not only in modeling the flow but also in predicting forces on coastal structures. For depth-integrated 2-D models, inundation and flooding in this region can be very complex with variation in the vertical direction caused by wave breaking on shore and interactions with the built environment, and the model may not be able to produce enough detail. For 3-D models, a very fine mesh is required to properly capture the physics, dramatically increasing the computational cost and rendering impractical the modeling of some problems. In this paper, comparisons are made between GeoClaw, a depth-integrated 2-D model based on the nonlinear shallow-water equations (NSWEs), and OpenFOAM, a 3-D model based on Reynolds-averaged Navier–Stokes (RANS) equation for tsunami inundation modeling. The two models were first validated against existing experimental data of a bore impinging onto a single square column. Then they were used to simulate tsunami inundation of a physical model of Seaside, Oregon. The resulting flow parameters from the models are compared and discussed, and these results are used to extrapolate tsunami-induced force predictions. It was found that the 2-D model did not accurately capture the important details of the flow near initial impact due to the transiency and large vertical variation of the flow. Tuning the drag coefficient of the 2-D model worked well to predict tsunami forces on structures in simple cases, but this approach was not always reliable in complicated cases. The 3-D model was able to capture transient characteristic of the flow, but at a much higher computational cost; it was found this cost can be alleviated by subdividing the region into reasonably sized subdomains without loss of accuracy in critical regions.

Download & links

Article (PDF, 7917 KB)

How to cite

How to cite.

Dates

Received: 19 May 2018 – Discussion started: 01 Jun 2018 – Revised: 29 Aug 2018 – Accepted: 30 Aug 2018 – Published: 18 Sep 2018

1 Introduction

1.1 Scope and motivation of the study

For many years, researchers have been working on different numerical models that can predict tsunami behavior. Tsunami prediction generally requires modeling at a wide range of spatial scales, including (from large to small scale) offshore wave propagation, beach run-up, inland inundation, and impact on individual structures.

Due to the large differences in scale for the different processes, most tsunami models solve two-dimensional depth-integrated equations, e.g., the nonlinear shallow-water equations (NSWEs) or some form of Boussinesq wave equations, to predict tsunami behavior, using computational grids that vary several orders of magnitude in spatial resolution, from several kilometers far from the shoreline to 10 m inland. The NSWEs are often used in the nearshore and inundation zone, since they can handle nonlinearities that arise in very shallow water and can be adapted to deal robustly with wetting and drying. However, it is not clear whether these equations are adequate to properly model fully three-dimensional turbulent flow, particularly at the scale necessary to determine tsunami impact and corresponding tsunami-induced forces on individual structures.

It would be preferable to solve the three-dimensional Navier–Stokes equations with a proper turbulence closure. However, this is still extremely expensive computationally relative to two-dimensional models, and only practical for detailed simulations over small spatial regions.

Although such modeling is challenging, the latest version of ASCE 7-16 (American Society of Civil Engineers) in the United States has a chapter for tsunami loads and effect for coastal structures. The provision requires site-specific inundation modeling and analysis be performed for all vertical evacuation structures. One of such examples is the design of the first vertical evacuation structure in the United States (Ash, 2015), the site-specific inundation analysis of which was conducted by González et al. (2013).

Given the challenges in tsunami inundation modeling, and the necessity of doing so for coastal structures design and tsunami hazard mitigation, the goal of this paper is to (1) study the characteristics of two different types of models when applied to tsunami inundation modeling (with an explicitly represented constructed environment); (2) show some guidance for modeling tsunamis or other flooding events in similar constructed environments; and, by showing the pros and cons of both models, (3) provide some insight to tsunami hazard researchers, coastal structure designers, and relevant government agencies.

1.2 Previous work

Before introducing the two numerical models used in the current study, a brief review of previous research involving different types of models is given below.

The two-dimensional depth-integrated equations are the most widely used tsunami models for their simplicity and computational efficiency. Popinet (2012) simulated the 2011 Tōhoku tsunami by solving the 2-D NSWE with dynamically adapted spatial resolution that varied from 250 m in flooded areas nearshore up to 250 km offshore. The model accurately predicted long-distance wave and coarse-scale flooding; the initial surface elevation was determined from a source model based on seismic inversion (as opposed to inversion of DART buoys and tidal gauge time series). This also showed that an accurate and consistent model of tsunami wave propagation can sometimes be constructed using only seismic wave inversion. Wei et al. (2013) used the Method of Splitting Tsunamis (MOST) model to simulate the same tsunami event. The MOST model solves the shallow-water equations in spherical coordinates with numerical dispersion. Their results demonstrated that it may be possible to forecast near-field tsunami inundation in real time. Hu et al. (2000) presented an NSWE model that can simulate storm waves propagating in the coastal surf zone and overtopping a sea wall. They found that waves overtopping a vertical wall may be approximately modeled by representing the wall as a steep slope and that the overtopping rate is sensitive to the bottom friction and the minimum friction depth. The two-dimensional NSWE model of wave run-up and overtopping by Hubbard and Dodd (2002) features an adaptive mesh refinement (AMR) algorithm. Their model can accurately reproduce 1-D and 2-D wave transformation, run-up, and overtopping in physical experiments. Their modeling of seawall overtopping by off-normal incident waves showed that there can be more flooding in such a situation than at normal incidence. Lynett (2007) simulated long-wave run-up obstructed by an obstacle and concluded that the obstacle can help reduce run-up and maximum overland velocity if the wave is highly nonlinear (with a ratio of wave height to shelf water depth ≥0.5). The sensitivity study also showed that in cases of breaking waves the Boussinesq model was more accurate than the nonlinear shallow-water equations in terms of wave run-up (maximum differences up to 10 %). For nonbreaking long waves, differences between the two were negligible. Shi et al. (2012) developed a high-order adaptive time-stepping total variation diminishing (TVD) solver for a fully nonlinear Boussinesq model and validated it against a series of laboratory experiments for wave shoaling and breaking and a suite of benchmark tests for wave run-up. The results showed that the model was able to accurately model wave shoaling, wave breaking, and wave-induced nearshore circulation. With a Boussinesq model, Lynett et al. (2010) simulated overtopping of levees of the Mississippi River–Gulf Outlet (MRGO) during Hurricane Katrina at four characteristic transects along the 20 km long stretch of the levees. The predicted overtopping rates agreed well with the observed data.

As computing power increases, it becomes possible to model the tsunami run-up process by solving three-dimensional Navier–Stokes equations with a proper turbulence closure. Choi et al. (2007) solved three-dimensional Reynolds-averaged Navier–Stokes (RANS) equations to simulate wave run-up on an conical island and compared different turbulence closure models including k-ϵ, RNG (renormalization group methods; Yakhot et al., 1992), and LES (large-eddy simulation). Their results showed that LES and RNG k-ϵ are similar and more accurate than k-ϵ is worse than those two. Williams and Fuhrman (2016) solved incompressible RANS equations with a transitional variant of the standard two-equation k-ω turbulence closure to study boundary layer flow induced by tsunami-scale waves. Their results indicated that the boundary layer generated by a tsunami is both current-like due to the long duration and wave-like due to its unsteadiness. The study also indicated that an existing expression for maximum bed shear stress under wind wave scale can be reasonably extrapolated to full tsunami scale. Mayer and Madsen (2000) investigated wave breaking in the surf zone by solving the RANS equations with a k-ω turbulence model. They found that the volume-of-fluid method could be used successfully to simulate wave breaking and that, although some instabilities occurred in applying the RANS equations, they can be eliminated by an ad hoc modification of the turbulence model. Other relevant studies include Biscarini (2010), Montagna et al. (2011), and Larsen et al. (2017), which model landslide generated tsunamis and tsunami-induced scours around coastal structures.

The prediction of tsunami impact on individual structures is also important because it provides guidance on designing coastal structures in tsunami inundation zones. The two-dimensional depth-integrated model may not work properly for these scenarios since the problems are more three-dimensional with large variation in the vertical direction and with transient and turbulent flow impacting the structure. In these cases, a three-dimensional model that solves the Navier–Stokes equation may give much better results. Researchers at the University of Washington (UW) modeled a series of “dam break” experiments by solving the 3-D RANS equations for bore-type impact of a wave on a series of 1∕20-scale model girder bridges to assess the 3-D effects on bridge skew (Motley et al., 2015; Wong, 2015).

The scale of modeling tsunami inundation inland with an explicitly represented constructed environment lies between that of modeling the large-scale tsunami wave propagation offshore and the small-scale tsunami impact on individual structures. This process is actually even more challenging to model since, for two-dimensional depth-integrated models, inclusion of the constructed environment increases the complexity of the topography, and the flow begins to have more variation in the vertical direction; while for the three-dimensional model that solves the Navier–Stokes equations, a fine mesh needs to be generated around each individual structure, which dramatically increases the number of cells in the computational domain.

Some researchers have tried to model this process with two-dimensional models. Ozer Sozdinler et al. (2015) used the numerical code NAMI DANCE to investigate hydrodynamic parameters in tsunami inundation zones with idealized structures – three rows of 20 blocks representing three-story concrete buildings. The code solved the NSWE using a finite-difference technique in a staggered leapfrog scheme. The effect of wave period, wave shape, protection structures, building layout, and Manning's coefficient are discussed. Some major conclusions included that the coastal protection structures like seawalls and breakwaters have very limited effect if the waves are able to overtop them and that it is preferable to use different Manning's coefficients for the sea, land, and buildings if more accurate values of hydrodynamic parameters are needed, albeit at the expense of more computational time. Similar conclusions on the Manning's coefficient were presented by Park et al. (2013). They simulated tsunami inundation in part of Seaside, Oregon, and compared flow parameters with their physical experiment. The comparison showed that the flow parameters were sensitive to the friction coefficient, especially for the momentum flux, which is proportional to tsunami loads on structures. For instance, decreasing the friction coefficient by a factor of 10 increased the predicted momentum flux by 208 %. Muhari et al. (2011) compared three different tsunami inundation models to evaluate tsunami impact on coastal communities: (1) the Constant Roughness Model (CRM) which uses a constant friction coefficient, does not include the constructed environment, and assumes that all buildings are not able to withstand the tsunami; (2) the Topographic Model (TM), which includes the constructed environment by incorporating building shape and height information into the topography; and (3) The Equivalent Roughness Model (ERM), which represents the building by using a different equivalent friction coefficient at the site of a building on the original topography (with only terrain information but not building height). Both the TM model and the ERM model gave more reliable prediction than the CRM model did, which confirmed the importance of taking the constructed environment into consideration.

However, few researchers have tried to use a three-dimensional model for inundation in a complex built environment. Shin et al. (2012) applied a 3-D LES model with two-phase flow to simulate inland tsunami inundation in a coastal city with hundreds of buildings and compared the prediction with experimental measurements. However, a fairly coarse mesh was used on land, and each building had only three to five mesh cells along its edge in the along-shore or cross-shore direction, so that the resulting agreement in flooding depth can only be considered qualitative. Qin et al. (2018) used 3-D RANS equations to predict tsunami inundation process and loads on individual buildings in part of Seaside, Oregon, and demonstrated that the whole part can be modeled using subsections with proper width without loss of accuracy in areas of interest.

In this paper, the results from a two-dimensional NSWE model and a 3-D Navier–Stokes model are presented for the test case of flow through a scale model of a portion of Seaside, Oregon. Two open-source models are used: the 2-D GeoClaw software from Clawpack (Clawpack Development Team, 2015), which is widely used for modeling tsunamis (both global propagation and local inundation), and the 3-D OpenFOAM software (The OpenFOAM Foundation, 2014). The two models are first compared and validated against an experiment in which a simple bore impinges on a single column and then compared for the Seaside model.

2 Simulation methodology

2.1 Two-dimensional model

The nonlinear shallow-water equations can be written as

\begin{array}{l} (1) & h_{t} + (u h)_{x} + (v h)_{y} = 0, \\ (2) & (h u)_{t} + (h u v)_{y} + {(h u^{2} + \frac{1}{2} g h^{2})}_{x} = - g h B_{x} - D u, \\ (3) & (h v)_{t} + (h u v)_{x} + {(h v^{2} + \frac{1}{2} g h^{2})}_{y} = - g h B_{y} - D v, \end{array}

where $u (x, y, t)$ and $v (x, y, t)$ are the depth-averaged velocities in the two horizontal directions, h is the water depth, g is gravitational acceleration, B(x,y) is elevation of the seabed, and $D = D (h, u, v)$ is the drag coefficient. The drag coefficient D could have many forms; in this study it is represented by

\begin{matrix} (4) & D = \frac{g M^{2} \sqrt{(u^{2} + v^{2})}}{h^{1 / 3}}, \end{matrix}

where M is the Manning's coefficient and is set to 0.025 s m $^{- 1 / 3}$ for all two-dimensional simulations in this study. The subscripts in these equations represent first-order partial derivatives.

The GeoClaw model (LeVeque et al., 2011; Berger et al., 2011) features AMR and is released as a submodule of the Clawpack software (Clawpack Development Team, 2015), an open-source package for solving hyperbolic systems of partial differential equations (PDEs) of one, two, and three dimensions, through finite-volume implementation of high-resolution Godunov-type “wave-propagation algorithms”. Cell averages of the solution variables q are computed over the volume of each cell and updated with waves propagating into the cell from all surrounding cell edges. The wave at each edge is computed by solving a “Riemann problem” with initial piecewise constant data determined by cell averages on each side of the edge. This method is especially good at solving problems with discontinuous solutions like shock waves, which usually arise in the solution of nonlinear hyperbolic equations (e.g., bores in the case of NSWEs).

Specifically, GeoClaw uses a variant of the f-wave formulation of the wave-propagation algorithms that allow incorporation of the topography source terms on the right-hand side of Eqs. (2) and (3) into the Riemann problem directly. The augmented Riemann solver in GeoClaw combines the desirable qualities of the Roe solver (Roe, 1981), HLLE-type (Harten, Lax, van Leer and Einfeldt) solvers (Einfeldt, 1988; Einfeldt et al., 1991), and the f-wave approach (Bale et al., 2003). The Roe solver provides an exact solution for the single-shock Riemann problem. It is also depth-positive semidefinite like the HLLE solvers, has a natural entropy fix by providing more than two waves, and yields a better approximation for problems with large rarefactions. A large class of steady states is also preserved, even for non-stationary steady states with nonzero fluid velocity. In addition, it is able to handle the presence of dry states in the “Riemann problem”, in which one state is wet (h>0) while another is dry (h=0), or both states are dry. It also works robustly in situations where the topography changes abruptly from one cell to another by an arbitrarily large value. For more details on the augmented Riemann solver in GeoClaw, see George (2008).

A typical characteristic of tsunami inundation models, especially those that incorporate the built environment, is that the spatial scale of regions of interest may vary from kilometers to meters. For regions several kilometers offshore, grid cells can be as large as thousands of meters on a side, while for regions near the shoreline or in a built environment onshore, grid cells must be refined to several meters or less, since the size of a building may be only several meters and an adequate number of grid cells are required to achieve acceptable accuracy. In GeoClaw, a patch-based AMR technique can efficiently handle these situations (LeVeque et al., 2011; Berger and Leveque, 1998).

2.2 Three-dimensional model

For the three-dimensional model, version 2.3.1 of the open-source computational fluid dynamics (CFD) package OpenFOAM was used (The OpenFOAM Foundation, 2014). The package comes with different solvers for different types of flow. For tsunami inundation, in which there are two immiscible fluids (air and water) with a free interface, the interFoam solver can be chosen which uses the PISO algorithm to solve the RANS equations with a volume-of-fluid (VOF) approach to model the free surface. For details on these numerical methods, readers can refer to Hirt and Nichols (1981) and Versteeg and Malalasekera (2011). The VOF approach defines a scalar field α_water, which represents fractional volume of water in each cell. A cell full of water (ρ=1000 kg m⁻³, $ν = 1.0 \times 10^{- 6}$ m² s⁻¹) has α_water=1.0, while a cell full of air (ρ=1.22 kg m⁻³, $ν = 1.48 \times 10^{- 5}$ m² s⁻¹) has α_water=0.0. Here ρ is the mass density of the fluid and ν is the kinematic viscosity. A cell with α_water between 0 and 1 contains the interface. A special transport equation is solved to advance the α_water field. To close the RANS equations, Menter's k-ω shear stress transport (SST) model (Menter and Esch, 2001) was applied.

There are many other turbulence closure models, among which the k-ϵ model is also very popular. It is suitable for fully turbulent and non-separated flows and has the shortcoming of numerical stiffness in the viscous sublayer, which can result in stability issues (Menter, 1993). It was also applied to model the inundation process in this study but became unstable during the simulation. The k-ω SST is generally more stable and behaves better in modeling partially separated flows, which is the case in the current study (flow becomes separated after passing around the built environment).

With the assumption of an incompressible fluid, the RANS equations are listed below:

\begin{array}{l} (5) & \frac{\partial {\overline{u}}_{i}}{\partial x_{i}} = 0, \\ (6) & ρ \frac{\partial {\overline{u}}_{i}}{\partial t} + ρ {\overline{u}}_{j} \frac{\partial {\overline{u}}_{i}}{\partial x_{j}} = - \frac{\partial \overline{p}}{\partial x_{i}} + μ \frac{\partial^{2} {\overline{u}}_{i}}{\partial x_{j} \partial x_{j}} - \frac{\partial ρ \overline{u_{i}^{'} u_{j}^{'}}}{\partial x_{j}}, \end{array}

where ${\overline{u}}_{i}$ is the mean velocity in the i direction, ${u_{i}}^{'}$ is the fluctuating component of velocity in the i direction, and $\overline{p}$ is the mean pressure. If u_i is the velocity component in the i direction, then $u_{i} = {\overline{u}}_{i} + {u_{i}}^{'}$ . The Reynolds stress term in Eq. (6) is

\begin{matrix} (7) & - ρ \overline{u_{i}^{'} u_{j}^{'}} = ν_{t} ρ [\frac{\partial {\overline{u}}_{i}}{\partial x_{j}} + \frac{\partial {\overline{u}}_{j}}{\partial x_{i}}] - \frac{2}{3} k ρ δ_{i j}, \end{matrix}

where k is the turbulence kinetic energy and ν_t is the turbulence eddy viscosity. The equations above need to be closed with some closure model. Here Menter's k-ω SST model (Menter and Esch, 2001) was applied:

\begin{array}{l} (8) & \frac{\partial k}{\partial t} + \nabla \cdot (U k) = \tilde{G} - β^{*} k ω + \nabla \cdot [(ν + α_{k} ν_{t}) \nabla k], \\ \frac{\partial ω}{\partial t} + \nabla \cdot (U ω) = γ S^{2} - β ω^{2} + \nabla \cdot [(ν + α_{ω} ν_{t}) \nabla ω] \\ (9) & + (1 - F_{1}) {CD}_{k ω}, \end{array}

where ν is the kinematic viscosity of fluid and $\tilde{G}$ is defined as $\tilde{G} = min \{G, c_{1} β^{*} k ω\}$ , where G is the production term and defined as

\begin{matrix} (10) & G = ν_{t} S^{2}; \end{matrix}

S is the invariant measure of the strain rate, defined by

\begin{matrix} (11) & S = \sqrt{2 S_{i j} S_{i j}}; \end{matrix}

and S_ij is the strain rate tensor, defined by $S_{i j} = \frac{1}{2} (\nabla U + U^{T})$ . F₁ is a blending function defined by

\begin{matrix} (12) & F_{1} = \tanh \{{\{min [max (\frac{\sqrt{k}}{β^{*} ω y}, \frac{500 ν}{y^{2} ω}), \frac{4 α_{ω 2} k}{{CD}_{k ω}^{*} y^{2}}]\}}^{4}\}, \end{matrix}

where ${CD}_{k ω}^{*}$ is defined by

\begin{matrix} (13) & {CD}_{k ω}^{*} = max ({CD}_{k ω}, 10^{- 10}) \end{matrix}

and CD_kω is defined by

\begin{matrix} (14) & {CD}_{k ω} = 2 σ_{ω 2} \nabla k \cdot \frac{\nabla ω}{ω} . \end{matrix}

After solving Eqs. (8) and (9), ν_t can be calculated by

\begin{matrix} (15) & ν_{t} = \frac{a_{1} k}{max (a_{1} ω, S F_{2})}, \end{matrix}

where F₂ is a second blending function, defined as

\begin{matrix} (16) & F_{2} = \tanh \{{[max (\frac{2 \sqrt{k}}{β^{*} ω y}, \frac{500 ν}{y^{2} ω})]}^{2}\} . \end{matrix}

All other constants are computed using a blend from the corresponding constants associated with the k-ϵ and k-ω models via blending functions like $ϕ = ϕ_{1} F_{1} + ϕ_{2} (1 - F_{1})$ . Values for these constants are α_k1=0.85013, α_k2=1.0, α_ω1=0.5, α_ω2=0.85616, β₁=0.075, β₂=0.0828, γ₁=0.5532, γ₂=0.4403, $β^{*} = 0.09$ , a₁=0.31, and c₁=10.0 (Menter et al., 2003).

A force vector, F, on a structure is computed by summing forces from pressure, F_p, and from viscous stress, F_v.

\begin{matrix} (17) & F = F_{p} + F_{v} \end{matrix}

F_p and F_v are calculated respectively by

\begin{array}{l} (18) & F_{p} = \sum_{i} (- p_{i} A_{i} {(α_{water})}_{i} n_{i}), \\ (19) & F_{v} = \sum_{i} \{(τ_{i} \cdot n_{i}) A_{i} {(α_{water})}_{i}\}, \end{array}

where i is the index of cell faces on the building on which forces need to be evaluated; p_i is the total pressure on face i; A_i is area of face i; ${(α_{water})}_{i}$ is volume fraction of water in the adjacent cell of face i; n_i is the unit normal vector of face i pointing into the computational domain; and τ_i is the viscous stress tensor at face i, which can be expressed by $τ_{i} = \{ρ (ν + ν_{t}) [\nabla U + \nabla U^{T}]\}$ on face i.

3 Initial comparison of the 2-D and 3-D numerical models

An initial comparison of the two numerical models was conducted by modeling the interaction between a bore and a free-standing coastal structure, with experimental results from Árnason (2005). The experiment was performed at the Charles W. Harris Hydraulics Laboratory at UW, Seattle. In the experiment, a square column was placed in a 16.6 m long, 0.6 m wide, and 0.45 m deep wave tank, and aligned in parallel to the tank side walls (Fig. 1).

A thin gate separated water in the tank into two parts with different depths: 0.02 m deep on the square column side and 0.25 m deep on the other side. When the gate was lifted to the top of the tank in 0.2 s by a 6.4 cm diameter pneumatic piston, a bore formed and propagated toward the square column downstream. The square column with a 12×12 cm square-shaped cross section was placed 5.2 m downstream from the gate. To measure hydrodynamic forces, the column was supported from above and connected with a force sensor.

Both the three-dimensional and two-dimensional models were developed at model scale to simulate the physical experiment. The three-dimensional OpenFOAM model incorporated the column into the computational domain by simply cutting off a block of mesh of the same shape from the computational domain. The mesh was coarse far from the column (1 cm by 1 cm by 0.5 cm in the x, y, and z directions, where z is the vertical direction) and was refined gradually to 0.125 cm by 0.125 cm by 0.0625 cm in the x, y, and z directions near the column surface. These distances are evaluated to be 70, 70, and 35, respectively, in terms of dimensionless wall distance defined as $y^{+} = \frac{y u^{*}}{ν}$ , where y is the distance, u^∗ is the friction velocity, and ν is the kinematic fluid viscosity. The mesh was finer in the z directions to better capture the water surface. Forces on the column were obtained by integrating pressure and shear forces from fluid on the surface of the column.

https://www.nat-hazards-earth-syst-sci.net/18/2489/2018/nhess-18-2489-2018-f01

Figure 1Schematic of the experimental setup for the interaction between bore and square column. The top figure shows a plan view, and the bottom figure shows a cross section through the center of the column, also illustrating the bore. (Reprinted with permission from Motley et al., 2015. Copyright by ASCE.)

A comparison of a two-dimensional depth-averaged flow model and a three-dimensional RANS model for predicting tsunami inundation and fluid forces

1.1 Scope and motivation of the study

1.2 Previous work

2.1 Two-dimensional model

2.2 Three-dimensional model

4.1 The physical experiment

4.2 Setup of numerical models

4.2.1 OpenFOAM model

4.2.2 GeoClaw model

4.3 Comparison of flow parameters

4.3.1 Onshore time series near initial impact

4.3.2 Onshore time series in post-impact region