Tsunami benchmark cases: Tsunami benchmark cases: benchmark # 3benchmark # 3

The third International The third International workshop on long-wave workshop on long-wave

runup models, June 2004runup models, June 2004

Stéphan Grilli, Enet FrançoisDepartment of Ocean Engineering,University of Rhode Island

ForewordForeword

Due to lack of time, only Due to lack of time, only case Bcase B was was solved as this is the most solved as this is the most demanding modeling case and is demanding modeling case and is more likely to exhibit more likely to exhibit nonlinearitiesnonlinearities that the LSWE can't model but that that the LSWE can't model but that the numerical FNPF solution the numerical FNPF solution accurately models.accurately models.

IntroductionIntroduction

• Benchmark parametersBenchmark parameters

• Numerical modelNumerical model

• ResultsResults

• ConclusionsConclusions

Benchmark parameters:Benchmark parameters:

• Slide shape as function of time: Slide shape as function of time:

22

0

0

tan2exp),(

)tan()(

:

),(),(),(

tgx

txh

xxH

where

txhtxHtxh

Numerical ModelNumerical Model

• Fully nonlinear potential flow higher-Fully nonlinear potential flow higher-order 2D-BEM model order 2D-BEM model – Grilli and Subramanya (1996)Grilli and Subramanya (1996)– Grilli and Horrillo (1997)Grilli and Horrillo (1997)– Grilli and Watts (1999)Grilli and Watts (1999)

Boundary conditions and Boundary conditions and geometry:geometry:

• 1/101/10thth slope slope

• Constant depth region offshoreConstant depth region offshore

• Absorbing piston offshoreAbsorbing piston offshore

• Slide truncated at the bottom of the Slide truncated at the bottom of the slopeslope

• 2 domains2 domains1.1. L=40m, dL=40m, dmaxmax=3.5m=3.5m

2.2. L=80m, dL=80m, dmaxmax=7.5m=7.5m

Boundary conditionsBoundary conditions• The deforming slide is modeled The deforming slide is modeled

analytically and truncated either at 1% of analytically and truncated either at 1% of maximum thickness delta (1 cm), or at the maximum thickness delta (1 cm), or at the maximum depth of the discretization.maximum depth of the discretization.

• Kinematics on the moving boundary Kinematics on the moving boundary calculated analytically.calculated analytically.

• Both Both ΦΦtt and and ΦΦtntn are needed as BC for the are needed as BC for the two BEM problems needed for the second two BEM problems needed for the second order time stepping:order time stepping:

Kinematics:Kinematics:

0tan hxh

2

tan2

0 exp

t

gx

h

21

1cos

xh

23

21

1

x

xx

h

h

R

costn h

nn

ssn

nsttttn RRhh

sincos

RemarksRemarks

• Care was taken to have enough adaptive Care was taken to have enough adaptive integration subdivision in the runup region integration subdivision in the runup region which becomes very shallow.which becomes very shallow.

• the runup point is forced to follow the slide the runup point is forced to follow the slide shape by keeping x(t) as obtained from shape by keeping x(t) as obtained from the Taylor series expansion providing the the Taylor series expansion providing the time stepping and calculating the time stepping and calculating the corresponding elevation z analytically corresponding elevation z analytically using the slide shapeusing the slide shape

Discretization:Discretization:

• Total of 470 or 472 nodes and 383 or Total of 470 or 472 nodes and 383 or 384 elements384 elements– mid-interval elements (potential)mid-interval elements (potential)– cubic splines (geometry)cubic splines (geometry)

• Free surface:200 cubic boundary Free surface:200 cubic boundary elements (dx= 0.2 or 0.4 m)elements (dx= 0.2 or 0.4 m)

• Slope: dx = 0.14 or 0.28 mSlope: dx = 0.14 or 0.28 m

Time stepping:Time stepping:

• Based on a mesh Courant condition and Based on a mesh Courant condition and varies as (Lagrangian) nodes movevaries as (Lagrangian) nodes move

• average time step is about 0.015 or 0.02 saverage time step is about 0.015 or 0.02 s

• 900 time steps to compute up to t' = 5s900 time steps to compute up to t' = 5s

• CPU time on a Mac G4 1.33 GHz laptop is CPU time on a Mac G4 1.33 GHz laptop is 2-2.5 sec per time step (40 min)2-2.5 sec per time step (40 min)

Accuracy:Accuracy:

• Relative accuracy on Boundary fluxes Relative accuracy on Boundary fluxes is better than 5 10is better than 5 10-8-8

• Volume conservation better than Volume conservation better than 5.105.10-6-6

Results: Results:

Results:Results:

Results:Results:

Results:Results:

Results: Tsunami exits the Results: Tsunami exits the domaindomain

ConclusionsConclusions

• The analytical LSWE solution provides The analytical LSWE solution provides a good prediction of tsunami shape a good prediction of tsunami shape given by the full FNPF solution only given by the full FNPF solution only up to t'=1 for the results provided. up to t'=1 for the results provided.

• Larger differences with the FNPF Larger differences with the FNPF solution occur at later time due to the solution occur at later time due to the depth limitation and to the proximity depth limitation and to the proximity of the open BCof the open BC

ReferencesReferences

• Grilli, S.T. and Subramanya, R. 1996. Numerical Modeling of Grilli, S.T. and Subramanya, R. 1996. Numerical Modeling of Wave Breaking Induced by Fixed or Moving Boundaries. Wave Breaking Induced by Fixed or Moving Boundaries. Computational Mechanics, 17(6), 374-391.Computational Mechanics, 17(6), 374-391.

• Grilli, S.T. and Horrillo, J. 1997 Numerical Generation and Grilli, S.T. and Horrillo, J. 1997 Numerical Generation and Absorption of Fully Nonlinear Periodic Waves. Journal of Absorption of Fully Nonlinear Periodic Waves. Journal of Engineering Mechanics, 123 (10), 1060-1069. Engineering Mechanics, 123 (10), 1060-1069.

• Grilli, S.T. and Watts, P. 1999 Modeling of waves generated Grilli, S.T. and Watts, P. 1999 Modeling of waves generated by a moving submerged body. Applications to underwater by a moving submerged body. Applications to underwater landslides. Engng. Analysis Boundary Elemt., 23, 645-656.landslides. Engng. Analysis Boundary Elemt., 23, 645-656.