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THE FLOW PARAMETERS ESTIMATION DURING THE LONG WAVES RUN-UP MODELING

THE FLOW PARAMETERS ESTIMATION DURING THE LONG WAVES RUN-UP MODELING. Andrei G.Marchuk and Alexandr A.Anisimov. Institute of Computational Mathematics and Mathematical Geophysics, Siberian Division of the Russian Academy of Sciences, Novosibirsk, RUSSIA. mag@omzg.sscc.ru.

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THE FLOW PARAMETERS ESTIMATION DURING THE LONG WAVES RUN-UP MODELING

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  1. THE FLOW PARAMETERS ESTIMATION DURING THE LONG WAVES RUN-UP MODELING Andrei G.Marchuk and Alexandr A.Anisimov Institute of Computational Mathematics and Mathematical Geophysics, Siberian Division of the Russian Academy of Sciences, Novosibirsk, RUSSIA. mag@omzg.sscc.ru

  2. STATEMENT OF THE PROBLEM Numerical modeling of this process was carried out based on the one-dimensional nonlinear shallow-water model  , , where u - velocity,  - surface elevation, H - the value of depth. . The incident tsunami wave is generated by water motion and moving of a free surface on the left boundary of computational area , , . When the time value t will become equal to 2/b, the “free” boundary conditions will be assigned to this boundary.

  3. Figure 1. The statement of the runup problem.

  4. NUMERICAL ALGORITHM The explicit finite difference equations with central differences are used for computations in the inner points of the area. On the right boundary (the coast) where the depth value is equal to zero or is very close to this value (~0.001m) the “free” boundary conditions are used until the moment, when the surface disturbance will reach this boundary grid-point. From this moment the special algorithm for computation of the flow parameters is used.

  5. Figure 2. Definition of the water edge location.

  6. New values of velocity and elevationin a midpoint between grid-points with indexes M-1 and M we use the following difference equations: ) And in a midpoint between points M and B (values and

  7. Then with the help of a linear interpolation values of velocity and elevation on the new time level are define in the grid-point with number M Here the value R represents distance along the horizontal axis from the most right computational grid-point up to a current position of the water edge point, and the gridded variables with index M+1- are values of flow parameters in the mobile point of a water edge. The water-edge velocity is determined from the mass and energy conservation laws. where u0 is the preliminary water edge velocity, estimated from the one time-step increment of the triangle MAB square value (fig. 2) and tg() is the shore inclination between M and M+1 grid-points.

  8. Beach slope angle 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Run-up height 19.5 19.3 18.9 18.5 18.1 17.8 17.5 17.3 17.1 16.9 RESULTS OF NUMERICAL EXPERIMENTS The first series of test experiments was carried out for the uniformly sloping shore with the following parameters: Spatial step of the grid: dx = 10 meters. Initial number of computational grid-points: M = 400 Time step : dt = 0.2 seconds Length of an incident wave (expression 3) B = 0.05 Initial wave height h = 2 meters Table 1. Calculated wave run-up height according to the beach slope angle.

  9. Figure 3. The moment of highest run-up of a wave to the uniform slope.

  10. Animation will be here

  11. Figure 4. Wave run-up on a shore with the curvature parameters C = +4 and C = -6.

  12. FLOW PARAMETERS BEHAVIOR DURING WAVE RUN-UP AND DRAWN-DOWN Uniform bottom and shore slope (tg = 0.1)

  13. CONCLUSIONS The new algorithm of tsunami wave run-up computation a on a shore of an arbitrary profile is developed. The results of the wave run-up on a sloping shore are in good agreement with the results obtained earlier with the help of analytical and numerical methods. The computational experiments of a tsunami impact on a shore of an arbitrary profile have revealed such type of shore relief, which gives the highest climb of the water on the shores of various profiles with identical initial wave. The destructive tsunami effect due to the water-flow force is bigger when the affected structure is closer to the initial water edge. The wash-away force is much greater than water flow force during run-up.

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