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 , the site-specific inundation analysis of which was conducted by .

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.

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. 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. 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. 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 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. 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. 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, 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. 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; ), 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. 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. 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 , , and , 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 .

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. 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 . 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%. 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. 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. 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 , which is widely used for modeling tsunamis (both global propagation and local inundation), and the 3-D OpenFOAM software . 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.

The nonlinear shallow-water equations can be written as ht+(uh)x+(vh)y=0,(hu)t+(huv)y+hu2+12gh2x=-ghBx-Du,(hv)t+(huv)x+hv2+12gh2y=-ghBy-Dv, 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 D=gM2(u2+v2)h1/3, 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 features AMR and is released as a submodule of the Clawpack software , 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. () and () into the Riemann problem directly. The augmented Riemann solver in GeoClaw combines the desirable qualities of the Roe solver , HLLE-type (Harten, Lax, van Leer and Einfeldt) solvers , and the f-wave approach . 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 .

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 .

For the three-dimensional model, version 2.3.1 of the open-source computational fluid dynamics (CFD) package OpenFOAM was used . 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 and . The VOF approach defines a scalar field αwater, which represents fractional volume of water in each cell. A cell full of water (ρ=1000 kg m-3, ν=1.0×10-6 m2 s-1) has αwater=1.0, while a cell full of air (ρ=1.22 kg m-3, ν=1.48×10-5 m2 s-1) has αwater=0.0. Here ρ is the mass density of the fluid and ν is the kinematic viscosity. A cell with αwater between 0 and 1 contains the interface. A special transport equation is solved to advance the αwater field. To close the RANS equations, Menter's k-ω shear stress transport (SST) model 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 . 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: ∂u‾i∂xi=0,ρ∂u‾i∂t+ρu‾j∂u‾i∂xj=-∂p‾∂xi+μ∂2u‾i∂xj∂xj-∂ρui′uj′‾∂xj, where u‾i is the mean velocity in the i direction, ui′ is the fluctuating component of velocity in the i direction, and p‾ is the mean pressure. If ui is the velocity component in the i direction, then ui=u‾i+ui′. The Reynolds stress term in Eq. () is -ρui′uj′‾=νtρ∂u‾i∂xj+∂u‾j∂xi-23kρδij, where k is the turbulence kinetic energy and νt is the turbulence eddy viscosity. The equations above need to be closed with some closure model. Here Menter's k-ω SST model was applied: ∂k∂t+∇⋅(Uk)=G̃-β∗kω+∇⋅ν+αkνt∇k,∂ω∂t+∇⋅(Uω)=γS2-βω2+∇⋅ν+αωνt∇ω+1-F1CDkω, where ν is the kinematic viscosity of fluid and G̃ is defined as G̃=min⁡G,c1β∗kω, where G is the production term and defined as G=νtS2; S is the invariant measure of the strain rate, defined by S=2SijSij; and Sij is the strain rate tensor, defined by Sij=12∇U+UT. F1 is a blending function defined by F1=tanh⁡min⁡max⁡kβ∗ωy,500νy2ω,4αω2kCDkω∗y24, where CDkω∗ is defined by CDkω∗=max⁡CDkω,10-10 and CDkω is defined by CDkω=2σω2∇k⋅∇ωω. After solving Eqs. () and (), νt can be calculated by νt=a1kmax⁡a1ω,SF2, where F2 is a second blending function, defined as F2=tanh⁡max⁡2kβ∗ωy,500νy2ω2.

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=ϕ1F1+ϕ21-F1. Values for these constants are αk1=0.85013, αk2=1.0, αω1=0.5, αω2=0.85616, β1=0.075, β2=0.0828, γ1=0.5532, γ2=0.4403, β∗=0.09, a1=0.31, and c1=10.0 .

A force vector, F, on a structure is computed by summing forces from pressure, Fp, and from viscous stress, Fv. F=Fp+Fv Fp and Fv are calculated respectively by Fp=∑i-piAiαwaterini,Fv=∑iτi⋅niAiαwateri, where i is the index of cell faces on the building on which forces need to be evaluated; pi is the total pressure on face i; Ai is area of face i; αwateri is volume fraction of water in the adjacent cell of face i; ni is the unit normal vector of face i pointing into the computational domain; and τi is the viscous stress tensor at face i, which can be expressed by τi=ρν+νt∇U+∇UT on face i.

A 1 : 50 scale physical model of part of Seaside, Oregon, adjacent to the Cascadia Subduction Zone (CSZ), was constructed in the Tsunami Wave Basin at the O.H. Hinsdale Wave Research Laboratory at Oregon State University, and a series of experiments were conducted to measure flow velocities and water levels at 31 locations within the model-scale community. For full details on the experiment, one can refer to .

The rectangular basin for the experiment is 48.8 m long, 26.5 m wide, and 2.1 m deep. Figure  shows the top and side view of the basin. The still water depth at the wave maker is 0.97 m and decreases as it approaches the shoreline. A 0.04 m height (model-scale) seawall was also constructed between all idealized buildings and the shoreline and was parallel to the wave maker. Figures  and show the locations of the 31 gauges where water level and flow velocity were measured at a sampling frequency of 50 Hz. The gauges are grouped into four groups – A, B, C, and D (from bottom to top) – and marked by different symbols. Buildings in blue are large commercial buildings like hotels and hospitals. All red buildings are of the same size and represent small commercial buildings. Buildings in yellow are residential structures and are also all the same size.

In the experiment, the piston-type wave maker was designed to generate an initial wave with a wave height of approximately 0.2 m (model scale) at the lower horizontal section of the basin; this is equivalent to 10 m at full scale, which corresponds to a 500-year CSZ tsunami for this region . Note that this is not a solitary wave but a long single-peak wave. Experimental measurement of the wave maker speed was fit with a Gaussian function of the form s(t)=Aeβ(t-t0)2, with β=0.25, t0=14.75, and amplitude A=0.51. Equation () was used as input to generate numerical wave in current simulation. The experiment was repeated many times with identical initial conditions. Data from multiple trials were averaged to obtain the results presented here to smooth out stochastic features of the experiment, more details of which were presented in .

The predicted free-surface elevation, cross-shore velocity, and corresponding momentum flux from the two numerical models will be compared and discussed in this section. All experimental data in this study were provided by the NTHMP Mapping and Modeling Benchmarking Workshop: Tsunami Currents , and descriptions of the physical experiments to gather the data are provided by and .

Gauges were positioned as shown in Figs. –. Ultra-sonic surface wave gauges were used to measure the free surface. The bore front propagation speed was obtained by analysis of imagery gathered by two high-resolution video cameras located above the wave basin . Fluid velocity measurements were acquired by acoustic Doppler velocimeter (ADV) only after peaks; air entrainment in the bore at and shortly after the initial impact rendered the ADV measurements inconsistent in repeated trials . then assumed that the propagation speed and fluid velocity at the bore front are equal and fit a second-order polynomial to that value and ensemble-averaged ADV measurements in this region.

Time histories of the free-surface elevation, cross-shore velocity, and corresponding momentum flux at selected gauges are shown in Figs. –. After the peak (initial impact), there appears to be a significant drop in discrepancies between modeled and measured water level and fluid velocity; therefore, the discussion that follows will separately compare the results before and after the peak.