modelling tsunami waves using smoothed particle hydrodynamics (sph) r.a. dalrymple and b.d. rogers...

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