Modelling Tsunami Waves using Smoothed Particle Hydrodynamics (SPH) R.A. DALRYMPLE and B.D. ROGERS Department of Civil Engineering, Johns Hopkins University

Introduction Motivation  multiply-connected free-surface flows Mathematical formulation of Smooth Particle Hydrodynamics (SPH) Inherent Drawbacks of SPH Modifications - Slip Boundary Conditions - Sub-Particle-Scale (SPS) Model

Numerical Basis of SPH SPH describes a fluid by replacing its continuum properties with locally (smoothed) quantities at discrete Lagrangian locations  meshless SPH is based on integral interpolants (Lucy 1977, Gingold & Monaghan 1977, Liu 2003) (W is the smoothing kernel) These can be approximated discretely by a summation interpolant

The Kernel (or Weighting Function) Quadratic Kernel

SPH Gradients Spatial gradients are approximated using a summation containing the gradient of the chosen kernel function Advantages are: –spatial gradients of the data are calculated analytically –the characteristics of the method can be changed by using a different kernel

Equations of Motion Navier-Stokes equations: Recast in particle form as (XSPH)

Closure Submodels Equation of state (Batchelor 1974): accounts for incompressible flows by setting B such that speed of sound is Viscosity generally accounted for by an artificial empirical term (Monaghan 1992): Compressibility O(M 2 )

Dissipation and the need for a Sub-Particle-Scale (SPS) Model Description of shear and vorticity in conventional SPH is empirical  is needed for stability for free-surface flows, but is too dissipative, e.g. vorticity behind foil

Sub-Particle Scale (SPS) Turbulence Model Spatial-filter over the governing equations: (Favre-averaging) = SPS stress tensor with elements: Eddy viscosity: Smagorinsky constant: C s  0.12 (not dynamic!) S ij = strain tensor

Boundary conditions are problematic in SPH due to: –the boundary is not well defined –kernel sum deficiencies at boundaries, e.g. density Ghost (or virtual) particles (Takeda et al. 1994) Leonard-Jones forces (Monaghan 1994) Boundary particles with repulsive forces (Monaghan 1999) Rows of fixed particles that masquerade as interior flow particles (Dalrymple & Knio 2001) (Can use kernel normalisation techniques to reduce interpolation errors at the boundaries, Bonet and Lok 2001) Boundary Conditions (slip BC)

Determination of the free-surface Caveats: SPH is inherently a multiply-connected Each particle represents an interpolation location of the governing equations Far from perfect!!

JHU-SPH - Test Case 3 R.A. DALRYMPLE and B.D. ROGERS Department of Civil Engineering, Johns Hopkins University

SPH: Test 3 - case A -  = 0.01 Geometry aspect-ratio proved to be very heavy computationally to the point where meaningful resolution could not be obtained without high-performance computing ===> real disadvantage of SPH hence, work at JHU is focusing on coupling a depth- averaged model with SPH e.g. Boussinesq FUNWAVE scheme Have not investigated using  z <<  x,  y for particles

SPH: Test 3 - case B  = 0.1 Modelled the landslide by moving the SPH bed particles (similar to a wavemaker) Involves run-time calculation of boundary normal vectors and velocities, etc. Water particles are initially arranged in a grid-pattern …

Test 3 - case B  = 0.1 SPH settings:  x = 0.196m,  t = s, Cs = particles Machine Info: –Machine: 2.5GHz –RAM: 512 MB –Compiler: g77 –cpu time: 71750s ~ 20 hrs

Test 3B  = 0.1 animation

Test 3 comparisons with analytical solution

Free-surface fairly constant with different resolutions

Points to note: Separation of the bottom particles from the bed near the shoreline Magnitude of SPH shoreline from SWL depended on resolution Influence of scheme’s viscosity

JHU-SPH - Test Case 4 R.A. DALRYMPLE and B.D. ROGERS Department of Civil Engineering, Johns Hopkins University

JHU-SPH: Test 4 Modelled the landslide by moving a wedge of rigid particles over a fixed slope according to the prescribed motion of the wedge Downstream wall in the simulations 2-D: SPS with repulsive force Monaghan BC 3-D: artificial viscosity Double layer Particle BC did not do a comparison with run-up data

2-D, run 30, coarse animation 8600 particles,  y = 0.12m, cpu time ~ 3hrs

2-D, run 30, wave gage 1 data Huge drawdown little change with higher resolution  lack of 3-D effects

2-D, run 32, coarse animation particles,  y = 0.08m, cpu time ~ 4hrs breaking is reduced at higher resolution

2-D, run 32, wave gage 1 data Huge drawdown & phase difference Magnitude of max free-surface displacements is reduced lack of 3-D effects

3-D, run 30, animation Ps,  x = 0.1m (desktop) cpu time ~ 20hrs

Conclusions and Further Work Many of these benchmark problems are inappropriate for the application of SPH as the scales are too large Described some inherent problems & limitations of SPH Develop hybrid Boussinesq-SPH code, so that SPH is used solely where detailed flow is needed