Modelling Tsunami Waves using Smoothed Particle Hydrodynamics SPH

1
Modelling Tsunami Waves using Smoothed Particle
Hydrodynamics (SPH)

• R.A. DALRYMPLE and B.D. ROGERS
• Department of Civil Engineering, Johns Hopkins
  University

2
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

3
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

4
The Kernel (or Weighting Function)

• Quadratic Kernel

5
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

6
Equations of Motion

• Navier-Stokes equations
• Recast in particle form as

(XSPH)
7
Closure Submodels

• Equation of state (Batchelor 1974)
• accounts for incompressible flows by setting
  B such that speed of sound is

Compressibility O(M2)

• Viscosity generally accounted for by an
  artificial empirical term (Monaghan 1992)

8
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

9
Sub-Particle Scale (SPS) Turbulence Model

• Spatial-filter over the governing equations
• (Favre-averaging)

SPS stress tensor with elements
Sij strain tensor

• Eddy viscosity
  

• Smagorinsky constant Cs ? 0.12 (not dynamic!)

10
Boundary Conditions

• 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)

(slip BC)
11
Determination of the free-surface
Far from perfect!!

• Caveats
• SPH is inherently a multiply-connected
• Each particle represents an interpolation
  location of the governing equations

12
JHU-SPH - Test Case 3

• R.A. DALRYMPLE and B.D. ROGERS
• Department of Civil Engineering, Johns Hopkins
  University

13
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
• gt 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 ltlt ?x, ?y for
  particles

14
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

15
Test 3 - case B ? 0.1

• SPH settings
• ?x 0.196m, ?t 0.0001s, Cs 0.12
• 34465 particles
• Machine Info
• Machine 2.5GHz
• RAM 512 MB
• Compiler g77
• cpu time 71750s 20 hrs

16
Test 3B ? 0.1 animation
17
Test 3 comparisons with analytical solution
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Test 3 comparisons with analytical solution
Free-surface fairly constant with different
resolutions
19
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 schemes viscosity

20
JHU-SPH - Test Case 4

• R.A. DALRYMPLE and B.D. ROGERS
• Department of Civil Engineering, Johns Hopkins
  University

21
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

22
2-D, run 30, coarse animation
8600 particles, ?y 0.12m, cpu time 3hrs
23
2-D, run 30, wave gage 1 data

• Huge drawdown
• little change with higher resolution
• ?lack of 3-D effects

24
2-D, run 32, coarse animation
10691 particles, ?y 0.08m, cpu time 4hrs
breaking is reduced at higher resolution
25
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

26
3-D, run 30, animation
38175 Ps, ?x 0.1m (desktop) cpu time 20hrs
27
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