Development of Connecting Method between two-dimensional and three-dimensional Tsunami Numerical Model

Kazuhiko Honda

Tsunami Group, Marine Information and Tsunami Division, Port and Airport Research Institute Yokosuka, Kanagawa, Japan

ABSTRACT In this study, the connecting method between horizontally 2D and 3D tsunami numerical models is developed; the vertical distribution of horizontal flow velocity as the boundary condition of 3D domain is estimated by not only the conservation of flux from 2D domain but also referring the vertical distribution of horizontal flow velocity of the neighbor calculation cell in 3D domain. The present method is applied to tsunami propagation in the model basin and in the actual port. The present method shows good performance in comparison with the typical connecting method without considering the vertical distribution of horizontal flow velocity. KEY WORDS: Tsunami; numerical simulation; 2D model; 3D model; two-way. INTRODUCTION In order to estimate damage due to possible tsunamis, such as tsunami inundation area, inundation depth and tsunami debris, we usually use numerical models to calculate tsunami propagation from tsunami source to coastal area and tsunami inundation on land. Generally, tsunami propagating offshore can be estimated by horizontally two-dimensional numerical models (2D models) with the hydrostatic pressure assumption, because those tsunamis are extreme long waves and the vertical distribution of horizontal flow velocity of those tsunamis is uniform. On the other hand, tsunami around complicated topography and structures near coast has three-dimensional flow structure with non-hydrostatic feature. Therefore, we have to use three-dimensional non-hydrostatic models (3D models) for estimating tsunami flow around structures. However, it is necessary to use smaller grids in the simulation by those 3D models and then the calculation cost is very high. In order to reduce the calculation cost of simulation from tsunami propagation offshore to tsunami inundation on land, nesting system is effective: connecting method between calculation domains which have different grid size. When we conduct the nesting system between the domain with horizontally 2D model and the domain with 3D model, there is a problem about connecting of horizontal flow velocity at the

boundary between those domains because of the difference of vertical distribution of horizontal flow velocity between 2D model and 3D model: vertically uniform flow in 2D model and vertically non-uniform flow in 3D model. In this study, the connecting method between 2D and 3D tsunami numerical models is developed; the vertical distribution of horizontal flow velocity as the boundary condition of 3D domain is estimated by not only the conservation of flux from 2D domain but also referring the vertical distribution of horizontal flow velocity of the neighbor calculation cell in 3D domain, and the horizontal flow velocity as the boundary condition of 2D domain, which is vertically uniform, is estimated by only the conservation of flux from 3D domain. The present proposed connecting method is applied to tsunami propagation in the model basin and in the actual port. Then, the present proposed method shows good performance in comparison with the typical connecting method, which is without referring the vertical distribution of horizontal flow velocity of the neighbor cell. NUMERICAL MODEL Tsunami Numerical Model In order to calculate tsunami propagation in the ocean and tsunami runup on land interacting with coastal structures and buildings, Tomita et al. (2006) developed the numerical simulator which is named STOC (Storm surge and Tsunami simulator in Oceans and Coastal areas). STOC consists of two sub-models; STOC-IC and STOC-ML. Fig.1 illustrates the example of combination of STOC-IC and STOC-ML. STOC was applied to physical experiments and actual tsunamis, and the calculation results by STOC were in good agreement with the experimental results and the observation data (Tomita et al., 2006; Tomita and Honda, 2007; Honda and Tomita, 2008). STOC-IC is a three-dimensional Reynolds Averaged Navier-Stokes (RANS) model which is applied to estimate tsunamis in the coastal area, in order to estimate three-dimensional flow structure of tsunamis. To reduce computational costs for tsunami simulation in coastal areas, the free water surface is detected by the vertically integrated continuity equation.

The governing equations of STOC-ML, which is a multi-layer model, are same as those of STOC-IC except hydraulic pressure: assuming of hydrostatic pressure in STOC-ML. STOC-ML can be applied to simulate the tsunami propagation in the ocean, because this assumption provides more reduction of computational costs. In order to verify the connecting method proposed in this study, STOC is applied for numerical simulation of tsunami propagation and tsunami runup: STOC-IC as 3D model and STOC-ML with single-layer as 2D model.

Figure 1. Example of combination of STOC-IC and STOC-ML

Figure 2. Boundary condition at interface of 2D and 3D domains with no vertical distribution of horizontal flow velocity

Figure 3. Boundary condition at interface of 2D and 3D domains with considering vertical distribution of horizontal flow velocity

Connecting Method Generally, horizontal flow is assumed to be vertically uniform in horizontally 2D model. On the other hand, vertical distribution of horizontal flow is non-uniform. This difference of vertical distribution of horizontal flow between 2D and 3D models causes a problem for 3D model at interface of 2D and 3D domains. Two-way coupling method is adopted in STOC. For the boundary condition of 2D model at the interface of 2D and 3D domains, the depth average velocity in STOC-ML (2D model) at the interface is evaluated from the vertical distribution of horizontal velocity in STOC-IC (3D model). For the boundary condition of 3D model at the interface, there is no considering of vertical distribution of horizontal flow velocity in STOC; the horizontal flow velocity means both of orthogonal and parallel components to the interface. Fig. 2 shows the boundary condition of the horizontal flow component orthogonal to the interface in STOC-IC: similar assumption for the parallel component. When the flow is from 2D domain to 3D domain, the decrease of accuracy of numerical simulation may not be so much. However, when the flow direction is reverse, the accuracy is much decreased in numerical simulation, because of the discontinuity of vertical distribution of horizontal flow. Fujima et al. (2002) proposed the connecting method between 2D and 3D models, which is to solve the problem due to difference of vertical distribution of horizontal flow. They assumed that the information of discharge rate (flux) at the interface is shared between 2D and 3D models, and the following equations of deviation from the depth averaged velocity as boundary condition of 3D model:

(1)

0 (2)

where is the orthogonal direction to the interface of 2D and 3D domains, means horizontal velocity in -direction; is the depth-averaged velocity of , is the deviation from . In addition, they assumed the similar manner is used for out of 3D domain as boundary condition of 3D model (Eqs. 3 and 4), in order to evaluate the advection term of momentum equation in -direction, where means parallel component to the interface of horizontal velocity.

(3)

0 (4)

However, it is necessary to estimate out of 3D domain as boundary condition of 3D model not only to the advection terms of momentum equations but also to the viscosity terms of momentum equations. Therefore, in this study, the same method as that of Fujima et al. (2002) is applied not only to the advection terms but also to the viscosity terms. Fig. 3 illustrates the modified boundary condition of the horizontal flow component orthogonal to the interface in STOC-IC (3D model) in this study: similar assumption for the parallel component. The finite difference of Eqs. 1 and 2 for the orthogonal component are as followed.

, , (5)

∑ ,⁄ (6)

∑ , , ∑ , (7) Eq. 5 is the finite difference formula of deviation from the depth-averaged velocity, Eqs. 6 and 7 mean the depth-averaged velocity; the index is number of layers in vertical direction, the index means the value of parameters in neighbor cell, the index and mean the value of parameters in STOC-ML (2D model) and STOC-IC (3D) model respectively, is the thickness of layers, means the discharge rate (flux) at the interface in STOC-ML. PERFORMANCE OF THE PRESENT MODEL In order to confirm the performance of the present proposed method, numerical simulation for three kinds of model tests were carried out: comparison the results with the typical boundary condition between 2D and 3D models and with the present proposed boundary condition. STOC-ML and STOC-IC were used as 2D model and 3D model respectively. For the comparing the different effects of boundary condition at the interface of 2D and 3D models, three kind of boundary condition are applied to each model test: the typical boundary condition without considering vertical distribution of horizontal flow velocity (Case-1), the boundary condition with vertical distribution of the only orthogonal component for horizontal flow (Case-2), the present proposed boundary condition with considering vertical distribution of both of orthogonal and parallel components for horizontal flow velocity (Case-3).

Figure 4. Plain layout of bathymetry and topography and initial water surface: yellow rectangle is the interface of STOC-ML and STOC-IC domains

Figure 5. Initial water surface elevation

Figure 6. Snapshots of water surface of numerical simulation in each case

Obliquely Incident Tsunami The model test was carried out, where the incident tsunami propagate obliquely to the interface of 2D and 3D domains. Fig. 4 illustrates the bathymetry and topography in this model test, in which green and blue mean above and below the mean water surface respectively: flat sea bottom with 50m depth connects to uniform slope, of which the angle is 1/30. The slope has the submerged breakwater, of which the crown height is -5m, and around which the depth is 14m. Initial water surface elevation as tsunami source is shown in Fig. 5. In this model test, the horizontal resolution of the calculation grids is 15m in STOC-ML domain and 5m in STOC-IC domain. The thickness of layers is non-uniform in STOC-IC domain, which is from 2m to 3m: single-layer in STOC-ML domain. Fig. 6 illustrates the snapshots of tsunami propagation for each case of boundary condition. When tsunami propagates from 2D domain to 3D domain at the interface offshore (40s), there is no difference among each case. The reason is that there is no vertical distribution of horizontal flow, because long wave theory can be assumed at the interface. When tsunami propagates to shallow water area (60s, 80s, and 100s), scattering waves are generated along the interface near coast in Case-1 and Case-2. On the other hand, there is no scattering wave along the interface in Case-3. At the interface near coast, the effect of the vertical distribution of horizontal flow velocities, which are orthogonal and parallel to the interface, cannot be neglected, because the flow structure behind submerged breakwater has vertical change. Therefore, the present proposed model shows the good performance to this model test, according to water surface change due to tsunami propagation. Tsunami around Open Mouth of Breakwater The next model test is for tsunami propagation around breakwaters in port which has open mouth with submerged breakwater (Fig. 7); the crown height of breakwaters is 20m and the crown height of submerged breakwater is -20m. In this model test, the shape of initial water surface elevation as tsunami source is the same as a sinusoidal wave: 13.3km wave length and 6m wave height. Fig.8 illustrates the layout of numerical domains in this model test. STOC-ML with single-layer is applied from Domain-1 to Domain-4, and STOC-IC is applied to Domain-5. The horizontal resolution of the calculation grids of Domain-1 to Domain-5 is 150m, 50m, 25m, 5m, and 5m respectively. The thickness of layers is non-uniform in STOC-IC domain, which is from 2m to 5m. Fig. 9 shows the snapshots of tsunami propagation for Case-3. Tsunami propagates from offshore to port, the tsunami enters to port from the open mouth of breakwaters, and then tsunami is reflected at coast and propagates to offshore. The snapshots of tsunami propagation near the open mouth of breakwaters for each case of boundary condition are shown in Fig. 10. Until tsunami propagates at the front of breakwaters, there is no difference among each case at the interface of 2D and 3D domains (270s and 300s). However, scattering waves are generated along the interface behind breakwaters for case-1 and case-2 and the simulation was stopped with an error because of numerical instability (305s). On the other hand, there is no scattering wave along the interface and the tsunami propagates for all time in case-3 (Fig. 9). Therefore, the present proposed model shows the good performance.

Figure 7. Layout of bathymetry and topography and initial water surface

Figure 8. Layout of numerical domains

Figure 9. Snapshots of water surface of numerical simulation in Case-3

Figure 10. Snapshots of water surface of numerical simulation in each case

Figure 11. Layout of domains with STOC-ML and STOC-IC and maximum horizontal velocity of numerical simulation in Case-1 and Case-3

Actual Topography and Bathymetry In the previous two model test, the present proposed model shows good performance according to water surface change due to tsunami propagation. In order to verify the performance to estimate flow velocity of the present model, the results of flow velocity in numerical simulation for tsunami propagation in actual topography and bathymetry with the typical boundary condition (Case-1) and with the present proposed boundary condition (Case-3) were compared. The layout of STOC-ML domain and STOC-IC domain, and the numerical results of maximum horizontal velocity in time-series of Case-1 and Case-3 are shown in Fig.11. The upper is the port side, and the lower is the offshore in the figure. The maximum horizontal velocity in STOC-IC domain in Fig. 11 means the maximum value of horizontal velocity not only in time-direction but also in vertical direction, because STOC-IC is 3D model. In Case-1, there is unnatural flow in STOC-ML domain along the western interface of 2D and 3D domains. On the other hand, there is no unnatural flow along all interfaces in both of STOC-ML and STOC-IC domains. In order to verify the time series of flow velocity, the results of horizontal velocity at three points in Case-1 and Case-3 are compared. Fig. 12 shows the three output points near the interface, and the time history of flow velocity at these points are illustrated in Fig. 13; and mean horizontal flow velocity in the -direction and y -direction respectively. In STOC-IC, and are the depth-averaged velocity.

Figure 12. Output point of time-series of horizontal flow velocities

Figure 13. Time-series of horizontal flow velocities at each output point in Case-1 and Case-3

When tsunami propagates from offshore to port, there is no noise of the velocity in each case. However, after the flow direction is changed due to tsunami reflection at coast, noise of velocity is generated at the interface in Case-1. On the other hand, there is no noise in Case-3. Therefore, the present proposed model shows the good performance to flow velocity due to tsunami propagation. CONCLUSIONS In this paper, the connecting method between 2D and 3D tsunami numerical models is developed; the vertical distribution of horizontal flow velocity as the boundary condition of 3D domain is estimated by not only the conservation of flux from 2D domain but also referring the vertical distribution of horizontal flow velocity of the neighbor calculation cell in 3D domain. The present proposed connecting method is applied to tsunami propagation in the model basin and in the actual port, and the numerical results by using the present method are compared with those of the typical connecting method without considering the vertical distribution of horizontal flow velocity. The present proposed method shows good performance in comparison with the typical connecting method, according to water surface elevation change and horizontal flow velocity due to tsunami propagation. DISCUSSION In this study, the vertical distribution of horizontal flow is considered as

boundary condition at the interface of 2D and 3D domains. However, the total momentum transport across the interface from 3D domain to 2D domain, which is generated by the gap of horizontal velocity from the depth-averaged velocity, is not considered. The verification of effect due to momentum transport across the interface is future work. ACKNOWLEDGEMENTS This research was carried out as part of the "Enhancement of Technology to Develop Tsunami-Resilient Community" research project supported by JST/JICA, SATREPS (Science and Technology Research Partnership for Sustainable Development), Japan. REFERENCES Fujima, K, Masamura, K, and Goto, C (2002). “Development of the

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Honda, K and Tomita, T (2008). “Tsunami Inundation Simulation by Three-Dimensional Model,” Proc 31st Int Conf Coastal Eng, ASCE, 1433-1445.

Tomita, T, Honda, K, and Kakinuma, T (2006). “Application of three-dimensional tsunami simulator to estimation of tsunami behavior around structures,” Proc 30th Int Conf Coastal Eng, ASCE, 1677-1688.

Tomita, T, and Honda, K (2007). “Tsunami estimation including effect of coastal structures and buildings by 3d-model,” Proc 5th Int Conf Coastal Structures, ASCE, 681-692.