Three-dimensional numerical analysis to predict behavior of driftage carried by tsunami
© The Society of Geomagnetism and Earth, Planetary and Space Sciences (SGEPSS); The Seismological Society of Japan; The Volcanological Society of Japan; The Geodetic Society of Japan; The Japanese Society for Planetary Sciences; TERRAPUB. 2012
Received: 7 December 2010
Accepted: 30 November 2011
Published: 24 October 2012
This study aims to develop a three-dimensional (3D) numerical analysis code for the prediction of driftage behavior during a tsunami. The main features of this code are as follows: (1) it can simulate the six degree-of-freedom motion of driftage in a 3D flow field; (2) it can consider the interaction between fluid flow and driftage motion; and (3) it can compute the impact of the collision with a wall based on the Lagrangian equation of impulsive motion. In this code, we assume that the fluid pressure and viscosity cause driftage motion and that driftage motion affects fluid flow through deformation of the boundary between the fluid and itself. The code was applied to a hydraulic experiment carried out by subjecting a wooden body to an abrupt flow of water. The obtained numerical solution of driftage motion agreed well with the experimental result. It is concluded that our code can be used to successfully predict the behavior of driftage carried by a tsunami.
Key wordsThree-dimensional numerical analysis tsunami driftage six degree-of-freedom motion
During the Indian Ocean Tsunami in 2004, rubble, cars, etc., drifted toward coastal areas with the receding waves and destroyed buildings and structures. Rubble from destroyed structures was also adrift, causing increased damage. To reduce such damage, it is essential to predict the behavior and collision force of tsunami driftage. Ushijima et al. (2006) and Kawasaki et al. (2006) proposed methods for the three-dimensional (3D) simulation of tsunami driftage. These methods can simulate driftage behavior accurately by treating the driftage as a fluid; however, they also have to simulate the air flow. In this study, we have developed a numerical method that does not require the simulation of the air flow to predict the driftage behavior across a wide coastal area. Yoneyama et al. (2002) performed a numerical analysis of the locally high run-up caused by the 1993 Hokkaido-Nansei-Oki seismic tsunami. The wave height calculated by them agreed well with the actual wave height. We believe that the driftage behavior can be simulated with a high degree of accuracy by appending a driftage simulation function to their fluid analysis code. We had developed a vertical two-dimensional (2D) analysis code based on this method, and had verified its validity (Nagashima et al., 2008). In this study, we developed a new 3D numerical analysis code that can simulate the six degree-of-freedom motion of driftage. We verified the validity of this code by comparing the obtained results with a numerical result and with the results of hydraulics model test.
2. Numerical Analysis Method
The movement of driftage causes a change in the porosity ratio and in the aperture ratio of the computational cell. This change affects the fluid flow; this effect can be expressed by the following continuity equation.
2.1 Basic equations for fluid flow
The basic equations of fluid flow are shown below.
Our method is based on the SIMPLE method (Patankar and Spalding, 1972); we use discretized equations (Eqs. (1), (2), and (3)) on the Cartesian coordinate system to represent fluid flow. The definition points of the flow velocity and the others were at the center of the boundary phase between the cells and at the centers of the cells, respectively. The discretizations of time, advective term, and others yielded the forward difference, third-order upwind difference, and centered difference, respectively. Moreover, we discretized Eq. (3) using the volume of fluid (VOF) method (Hirt et al., 1981). We also devised a few countermeasures to conserve fluid volume (Yoneyama, 1998).
2.2 Basic equations of driftage motion
where m is the mass of the driftage, v g is the driftage centroid velocity vector, and and are the vectors of the fluid pressure and viscous force, respectively, acting on the segment surface. ω is an angular velocity vector on the inertia principal-axis coordinate system. is an inertia tensor that consists of the inertia moment of driftage. is the position vector of the centroid of a segment surface on the inertia principal-axis coordinate system.
2.3 Algorithm of fluid force that acts on driftage
2.3.1 Pressure Fpr
2.3.2 Viscous force Fvis
2.4 Calculation related to collision
Calculate the velocity of the driftage centroid, v g , and the angular velocity of the inertia principal-axis, ω, using the basic equations of driftage motion.
Calculate the time of the collision occurrence (t + Δtcol) and the position vector of the collision point on the inertia principal-axis rcol using v g and ω.
- iii)Calculate the velocity of the driftage centroid and the angular velocity of the inertia principal-axis ω′ immediately after collision using the following equations.ncol is the normal vector of the collision surface (wall, bed etc); and J, the impulse force, given by the following expression.
2.5 Calculation procedure
Read the input data.
Calculate the flow velocity at time t + Δt using the discretized form of Eq. (2).
Calculate the error in the continuity equation D using the discretized form of Eq. (1). If D exceeds the limit Dmax, then correct the pressure pn+1 on the basis of the solution of the pressure error equation and return to step iv). If not, then proceed to step viii).
Calculate the fluid-filling ratio Fn+1 at time t + Δt.
If it is time to stop, then stop the calculation. If not, increase the time and return to step ii).
3. Application and Discussion
3.1 Case 1: driftage in sea
The experimental conditions for calculations using our method were as follows: Water levels were H1 = 40 cm, H2 = 5 cm. The driftage was a cylinder with a diameter of 8 cm and a height of 20 cm. The initial position of the centroid of this object was Y = 8.95 m. The computation conditions are as follows: The grid spacing is 6.5 cm along the direction across the flow, 6 cm along the flow direction, and 3 cm in the vertical direction. The computational time interval Δt is 1.0 × 10−3 s, The maximum permissible error of the continuity equation Dmax is 1.0 × 10−5, fluid density ρ is 1.0 × 103 kg/m3, kinematic viscosity ν is 1.0 × 10−6 m2/s, density of the driftage ρ d is 0.5 × 103 kg/m3, and the reflection coefficient between the driftage and vertical wall e is 0.5. For computational purposes, the cylindrical driftage was modeled as an octagonal pillar with cross-sectional area and volume equal to that of the actual cylinder (driftage).
As shown in Fig. 8, the computed driftage trajectory, time variation of the position, and point of collision are in good agreement with the corresponding experimental results. Therefore, we concluded that our code can successfully predict the behavior of driftage in the sea.
3.2 Case 2: driftage on land
As shown in Fig. 11, the computed trajectory elevation of the driftage between the initial position and the vertical wall was higher than the experimental result. Therefore, the collisions that occurred at low elevations in the experiment did not show in the computation results. This difference might be caused by the initial gap between the driftage and the ground surface. In future works, it is necessary to understand the mechanism of initial movement and to find a suitable initial condition.
However, in general, the computed drifting behavior agrees well with the experimental results despite the initial movement problem. Therefore, we concluded that our code can successfully predict the behavior of driftage on land.
A numerical analysis code has been developed to predict the behavior of driftage carried by a tsunami. The main features of the code are as follows.
It can simulate driftage motion with six degrees-of-freedom in a 3D flow field.
It can consider the interaction between a fluid flow and a driftage motion.
It can determine the impact of the collision of driftage with a wall on the basis of the La-grangian equation of impulsive motion.
To verify the validity of the code, the obtained computational results were compared with the results of two hydraulic experiments. The behavior of driftage and the time variation of its position as calculated using our code were in good agreement with the experimental results.
It can be concluded that our code can be used to successfully predict the behavior of driftage carried by a tsunami.
In the future, the code will be applied to various drifting motions and it will be improved such that it can be used to simultaneously predict the behavior of several drifting objects. Furthermore, a method will be developed to determine the collision force; this was not discussed in this paper.
The authors are indebted to Dr. Masaaki Ikeno, Central Research Institute of Electric Power Industry, Japan, for kindly supplying his experiment data.
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