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