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

'd,' ' rrrrr hWAA

j

jN

jjj

mhWAA

1

, rrrr

The Kernel (or Weighting Function)

• Quadratic Kernel

1

4

1

2

3, 2

2qq

hhrW

W(r-r’,h)

Compact supportof kernel

WaterParticles

2h

Radius ofinfluence

r

| | , barh

rq rr

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

ijijj j

ji WA

mA

ijij

jijii Wm . . uuu

Equations of Motion• Navier-Stokes equations:

• Recast in particle form as

ijj ij

jiji

i Wmt

vv

r

d

d

ijj

ijiji Wm

t vvd

d

iijj

iijj

j

i

ij

i Wpp

mt

Fv

22d

d

v.d

d

t

iopt

Fuv

21

d

d

0

d

d

t

mi

(XSPH)

Closure Submodels • Equation of state (Batchelor 1974):

accounts for incompressible flows by setting B such that speed of sound is

max10d

dv

pc

• Viscosity generally accounted for by an artificial empirical

term (Monaghan 1992):

1

o

Bp

0.

0.

0

ijij

ijij

ij

ijij

ij

c

rv

rv

22

.

ij

ijijij r

h rv

Compressibility O(M2)

Dissipation and the need for a Sub-Particle-Scale (SPS) Model

• Description of shear and vorticity in conventional SPH is empirical

22 01.0

.

hr

hcΠ

ij

jiji

ij

ijij

rruu

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)

u~.D

D

t

τugu

.1~1~

2

oP

Dt

D

= SPS stress tensor with elements:τ

ijijkkijtij kSS 32

32 ~~

2

• Eddy viscosity: SlCst2

• Smagorinsky constant: Cs 0.12 (not dynamic!)

2/12 ijij SSS

Sij = strain tensor

ff ~

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

b

a f = n R(y) P(x)

y(slip BC)

Determination of the free-surface

Caveats:• SPH is inherently a multiply-connected

• Each particle represents an interpolation location of the governing equations

g

Free-surface

2h

Free-surface defined by

water 21x

where

j j

jjj

jjjj

mWVW

xxxxx

x

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 …

t1 t2

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

Test 3B = 0.1 animation

Test 3 comparisons

with analytical solution

tND = 0.5

-1

-0.5

0

0.5

1

1.5

2

0 20 40 60 80 100 120

x (m)

free

-su

rfac

e (m

)

SPH

Analytical

tND = 1.0

-1

-0.5

0

0.5

1

1.5

2

0 20 40 60 80 100 120

x (m)

free

-su

rfac

e (m

)SPH

Analytical

Test 3 comparisons

with analytical solution

tND = 2.5

-1

-0.5

0

0.5

1

1.5

2

0 20 40 60 80 100 120

x (m)

free

-su

rfac

e (m

)

SPH

Analytical

tND = 4.5

-1

-0.5

0

0.5

1

1.5

2

0 20 40 60 80 100 120

x (m)

free

-su

rfac

e (m

)SPH

Analytical

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

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0 0.5 1 1.5 2 2.5 3

time (s)

fre

e-s

urf

ac

e (

m)

experimental data

SPH

2-D, run 32, coarse animation

10691 particles, y = 0.08m, cpu time ~ 4hrs

breaking is reduced at higher resolution

2-D, run 32, wave gage 1 data

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0 0.5 1 1.5 2 2.5 3

time (s)

free

-su

rfac

e (m

)

experimental data

SPH

• Huge drawdown & phase difference

• Magnitude of max free-surface displacements is reduced

• lack of 3-D effects

3-D, run 30, animation

38175 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