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Modelling Tsunami Waves using Smoothed Particle Hydrodynamics (SPH)

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

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Title: 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
18
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
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