High-Fidelity Numerical Simulation of Shallow Water Waves
Amir Zainali
Dissertation submitted to the Faculty of the
Virginia Polytechnic Institute and State University
in partial fulfillment of the requirements for the degree of
Doctor of Philosophy
in
Geosciences
Robert Weiss, Chair
Scott D. King
Jennifer L. Irish
Heng Xiao
Nina Stark
December 2, 2016
Blacksburg, Virginia
Keywords: tsunami, dispersive waves, coastal vegetation
Copyright 2016, Amir Zainali
High-Fidelity Numerical Simulation of Shallow Water Waves
Amir Zainali
ABSTRACT
Tsunamis impose significant threat to human life and coastal infrastructure. The goal of
my dissertation is to develop a robust, accurate, and computationally efficient numerical
model for quantitative hazard assessment of tsunamis. The length scale of the physical
domain of interest ranges from hundreds of kilometers, in the case of landslide-generated
tsunamis, to thousands of kilometers, in the case of far-field tsunamis, while the water
depth varies from couple of kilometers, in deep ocean, to few centimeters, in the vicinity of
shoreline. The large multi-scale computational domain leads to challenging and expensive
numerical simulations. I present and compare the numerical results for different important
problems — such as tsunami hazard mitigation due to presence of coastal vegetation, boulder
dislodgement and displacement by long waves, and tsunamis generated by an asteroid impact
— in risk assessment of tsunamis. I employ depth-integrated shallow water equations and
Serre-Green-Naghdi equations for solving the problems and compare them to available threedimensional results obtained by mesh-free smoothed particle hydrodynamics and volume
of fluid methods. My results suggest that depth-integrated equations, given the current
hardware computational capacities and the large scales of the problems in hand, can produce
results as accurate as three-dimensional schemes while being computationally more efficient
by at least an order of a magnitude.
High-Fidelity Numerical Simulation of Shallow Water Waves
Amir Zainali
GENERAL AUDIENCE ABSTRACT
A tsunami is a series of long waves that can travel for hundreds of kilometers. They can be
initiated by an earthquake, a landslide, a volcanic eruption, a meteorological source, or even
an asteroid impact. They impose significant threat to human life and coastal infrastructure.
This dissertation presents numerical simulations of tsunamis. The length scale of the physical
domain of interest ranges from hundreds of kilometers, in the case of landslide-generated
tsunamis, to thousands of kilometers, in the case of far-field tsunamis, while the water
depth varies from couple of kilometers, in deep ocean, to few centimeters, in the vicinity of
shoreline. The large multi-scale computational domain leads to challenging and expensive
numerical simulations. I present and compare the numerical results for different important
problems — such as tsunami hazard mitigation due to presence of coastal vegetation, boulder
dislodgement and displacement by long waves, and tsunamis generated by an asteroid impact
— in risk assessment of tsunamis. I employ two-dimensional governing equations for solving
the problems and compare them to available three-dimensional results obtained by mesh-free
smoothed particle hydrodynamics and volume of fluid methods. My results suggest that twodimensional equations, given the current hardware computational capacities and the large
scales of the problems in hand, can produce results as accurate as three-dimensional schemes
while being computationally more efficient by at least an order of a magnitude.
Acknowledgments
I would like to express my great appreciation to Dr. Robert Weiss, as my adviser, for his
guidance, encouragement, and patience during the course of this work. I would also like to
thank my dissertation committee members, Drs. Scott King, Jennifer L. Irish, Nina Stark,
and Heng Xiao for their helpful comments on the draft of this thesis. I would also like to
thank Kannikha Kolandaivelu and Roberto Marivela for reviewing the earlier versions of this
dissertation.
The work presented in here is based upon work partially supported by the National Science
Foundation under Grants No. NSF-CMMI-1208147 and NSF-CMMI-1206271.
iv
Contents
Acknowledgement
iv
List of Figures
ix
List of Tables
xvii
Nomenclature
xix
1 Introduction
1
1.1
Theoretical Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6
1.2
Contributions and Outline of the Dissertation . . . . . . . . . . . . . . . . . . . 10
2 Boulder Dislodgement and Transport by Solitary Waves: Insights from
Three-Dimensional Numerical Simulations†
2.1
14
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
v
vi
Contents
2.2
2.3
2.4
Theoretical Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.2.1
Governing Equations and Numerical Method . . . . . . . . . . . . . . . . 17
2.2.2
Model Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.2.3
Non-Dimensional Parameters . . . . . . . . . . . . . . . . . . . . . . . . . 20
Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.3.1
Validation of Numerical Results . . . . . . . . . . . . . . . . . . . . . . . 23
2.3.2
Boulder Transport by Solitary Waves . . . . . . . . . . . . . . . . . . . . 25
Discussion and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3 High-Fidelity Depth-Integrated Numerical Simulations in Comparison to
Three-Dimensional Simulations
3.1
3.2
32
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.1.1
Three-Dimensional Methods . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.1.2
Depth-Integrated Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.2.1
Non-Breaking Solitary Wave Interaction with a Group of Cylinders . . 39
vii
Contents
3.2.2
Breaking Solitary Wave Run-Up on a Sloping Beach . . . . . . . . . . . 42
3.2.3
Breaking Solitary Type Wave Run-Up on a Sloping Beach . . . . . . . . 42
4 Numerical Simulation of Nonlinear Long Waves in the Presence of Discontinuous Coastal Vegetation
†
45
4.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
4.2
Theoretical Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
4.3
4.2.1
Initial and Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . 50
4.2.2
Wave Breaking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
4.3.1
Validation of Numerical Results . . . . . . . . . . . . . . . . . . . . . . . 52
4.3.2
Breaking Solitary-Type Transient Wave Run-Up in the Presence of
Macro-Roughness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
4.4
4.3.3
Effects of Macro-Roughness on the Local Maximum Local Water Depth 56
4.3.4
Effects of Macro-Roughness on the Local Maximum Momentum Flux . 57
4.3.5
Maximum Run-Up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
Discussion and Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
viii
Contents
5 Some Examples for Which Dispersive Effects Can Change the Results
Significantly†
5.1
64
Numerical Simulation of Hazard Assessment Generated by Asteroid Impacts
on Earth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
5.1.1
Numerical Simulation of Tsunami Waves Generated by an Asteroid
Explosion near the Ocean Surface . . . . . . . . . . . . . . . . . . . . . . 67
5.1.2
Numerical Simulation of Tsunami Waves Generated by an Asteroid
Impact into the Ocean . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
5.2
Non-Breaking Cnoidal Wave Interaction with Offshore Cylinders . . . . . . . . 73
5.3
Hazard Assessment Along the Coastline from the Gaza Strip to the Caesarea,
Israel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
6 Future Work
79
Bibliography
81
List of Figures
1.1
Location of tsunamis that have happened since 1900 to present day; ( ):
caused by a volcanic activity; ( ) caused by a landslide; ( ) caused by an
unknown source; ( ) caused by an earthquake. Color codes are as following:
dark-red represents the tsunamis that caused more than 1000 casualties or
more than 1 billion dollars damage in total; red represents the tsunamis that
caused more than 100 casualties or more than 100 million dollars damage in
total; orange represents the tsunamis that caused more than 10 casualties or
more than 10 million dollars damage in total; yellow represents the tsunamis
that caused less than 10 casualties or less than 10 million dollars damage in
total . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
ix
2
x
List of Figures
1.2
Histograms of the number of earthquakes versus the death and damage toll.
I: less than 10 casualties; II casualties between 10 and 100; III casualties
between 100 and 1000; IV casualties between 1000 and 10000; casualties more
than 10000. A: damage less than 10 million dollars; B: damage between 10 and
100 million dollars; C: damage between 0.1 and 1 billion dollars; D: damage
between 1 and 10 billion dollars; E: damage between 10 and 100 billion dollars;
F: damage more than 100 billion dollars; The y-axis is in logarithmic scale.
1.3
.
5
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6
Casualties and economic loss due to I: unknown source; II: earthquake; III:
landslide; IV: volcano; V: meteorological source; The y-axis is in logarithmic
scale.
1.4
Schematic of a long wave.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.5
Tsunami wave propagation caused by a landslide. The initial ratio between the
7
wavelength and water depth is more than 10. The figure shows the comparison
of the three different simulations. (blue) SGN high resolution; (red) SGN low
resolution; (green) SWE high resolution.
2.1
. . . . . . . . . . . . . . . . . . . . . . 11
Sketch of dam-break scenario at t = 0. Water behind the gate starts to flow
after sudden gate removal at t = 0. To compare the numerical results with
experiments presented by Imamura et al. (2008), we applied: lx = 10 m, ly =
0.45 m, lz = 0.3 m, lx h = 5.5 m, lw = 3 m and hw = 0.15, 0.20, 0.25, 0.3 m.
. . . . 19
xi
List of Figures
2.2
Comparison of numerical and experimental data presented. Experimental
data are from Imamura et al. (2008). The case ρb = 1550 kg m−3 is colored in
black and case with ρb = 2710 kg m−3 is colored in blue; (
results and (
2.3
) experimental
) SPH simulations. . . . . . . . . . . . . . . . . . . . . . . . . . . 22
Time snapshots for scenarios CA, CB and CC with α = 0, (left column) and
scenarios CA, CB and CC with α = 1, (right column) with wave height of
0.15 m. Please note that different cases are superimposed on the same domain
just for illustration purposes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.4
Contour plots of boulder maximum displacement, db , as a function of nondimensional parameters. a: aspect ratio in logarithmic scale versus Froude
number; (a-1) β = −8/30 (a-2) β = 0 and (a-3) β = 4/30. b: submergence
factor versus Froude number; (b-1) α = −1, (b-2) α = 0 and (b-3) α = 1. c:
submergence factor versus aspect ratio in logarithmic scale; (c-1) F r = 1.08,
(c-2) F r = 1.15 and (c-3) F r = 1.22. White regions indicate no significant
boulder movement, i.e. db lb−1
z < 0.1. For gray regions, the distance boulder
travels is in between 0.1 < db lb−1
z < 2.0. Color contours indicate boulders moved
significantly, i.e. db lb−1
z > 2.0 (colormap is in logarithmic scale) . . . . . . . . . 30
xii
List of Figures
3.1
A schematic sketch of the solitary wave passing through and around cylinders.
The coordinate of the domain is located at the center of the cylinder on the
right hand side of the domain. Wave propagates along the x-axis. The red dots
represent the location of the center of the cylinders. The blue dots represent
the location of the wave gauges. Gauge 1: (-3.04 m, -0.14 m); Gauge 2: (-1.82
m, -0.14 m); Gauge 3: (-0.83 m, 0.00 m); Gauge 4: ( 0.00 m, -0.88 m); Gauge
5: ( 0.85 m, 0.00 m); Gauge 6: ( 1.82 m, 0.00 m); . . . . . . . . . . . . . . . . . 34
3.2
Comparison of numerical and experimental data presented. Experimental and
√
three-dimensional simulations are from Mo (2010); t∗ = t/ h/g, and ζ ∗ = ζ/H;
(
) present simulation (2D-SGN), ( ) experimental results and (
) three-
dimensional simulation (3D-VOF). . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.3
Comparison of numerical and experimental data presented. Experimental and
√
three-dimensional simulations are from Mo (2010); t∗ = t/ h/g, and ζ ∗ = ζ/H;
(
) present simulation (2D-SGN), ( ) experimental results and (
) three-
dimensional simulation (3D-VOF). . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.4
Schematic of the solitary wave run-up on a sloping beach with the slope of
1:19.85.
3.5
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
Breaking solitary wave run-up compared to experimental results of Synolakis
(1987) and three-dimensional GPUSPH simulation of Marivela et al. (Under
review). (
(
) present simulation (2D-SGN), ( ) experimental results and
) GPUSPH. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
xiii
List of Figures
3.6
Free-surface elevation at (a) gauges 1-2, and (b) gauges 3-4. Experimental
results are summarized in Yang et al. (2016); (
SGN), ( ) experimental results and (
3.7
) GPUSPH.
. . . . . . . . . . . . . . . 43
Local water depth at gauges 5-16. Experimental results are summarized in
Yang et al. (2016); (
sults and (
4.1
) present simulation (2D-
) present simulation (2D-SGN), ( ) experimental re-
) GPUSPH. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
Free surface elevation of the solitary-type transient wave over constant water
depth at (a) t = 5 s, (b) t = 15 s, and (c) t = 25 s; (
(
) 1D-NSW. The following parameters are used: h0 = 0.73 m, H = 0.50 m,
k0 = 0.54 m−1 .
4.2
) 1D-SGN, and
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
Schematic of the breaking process. The vertical dashed lines indicate the
boundary of subdomains. The governing equations in the left subdomain are
SGN and NSW equations elsewhere. The boundary follows the leading wave;
(I) t < t1 ∶ SGN in the whole domain, (II) t1 < t < t2 ∶ SGN in the left
subdomain and SW in the right subdomain, and (III) t > t2 ∶ NSW in the
whole domain. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
4.3
Free-surface elevation at (a) gauges 1-2, and (b) gauges 3-4. Experimental
and COULWAVE results are summarized in Yang et al. (2016); (
simulation (2D-SGN), ( ) experimental results and (
) present
) COULWAVE. . . . . 52
xiv
List of Figures
4.4
Local water depth at gauges 5-16. Experimental and COULWAVE results
are summarized in Yang et al. (2016); (
experimental results and (
) present simulation (2D-SGN), ( )
) COULWAVE.
. . . . . . . . . . . . . . . . . . . 53
4.5
Sketch of the macro-roughness patches. . . . . . . . . . . . . . . . . . . . . . . . 54
4.6
Local water depth at gauges 5-16 for Scenario 3. Experimental results are
summarized in Irish et al. (2014); (
experimental results.
4.7
) present simulation (2D-SGN), and ( )
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
Maximum local water depth h∗max for (a) Scenario 1, (b) Scenario 2 (c) Scenario
3, and (d) Scenario 3 in which all the patches are removed except the first
patch. The maximum water depth for each scenario is normalized with the
reference values in the absence of macro-roughness patches. . . . . . . . . . . . 58
4.8
∗
Maximum momentum flux Fmax
for (a) Scenario 1, (b) Scenario 2, and (c)
Scenario 3. The momentum flux for each scenario is normalized with the
reference values in the absence of macro-roughness patches. . . . . . . . . . . . 59
4.9
Propagation of bore-lines in the presence of macro-roughness patches (Scenario 2). Irish et al. (2014); (a) experimental results, and (b) present simulation (2D-SGN).
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
4.10 Propagation of bore-lines for (a) Scenario with no macro-roughness patches
(b) Scenario 1, (c) Scenario 2 (d) Scenario 3.
. . . . . . . . . . . . . . . . . . . 61
xv
List of Figures
5.1
Comparison of water surface elevation between our SWE model and GeoClaw
results. Waves are generated by an asteroid with a diameter of 140 m exploding at the altitude of 10 km. Wave gauges are located at (
0.2LD, (
5.2
) 0.5LD, and (
) 0.05LD, (
) 0.8LD, where LD = 111 km. . . . . . . . . . . 66
Comparison of water surface elevation obtained using SWE equations (
and SGN equations (
)
),
) at (a) t = 400 s, (b) t = 800 s, and (c) t = 1200 s.
Waves are generated by an asteroid with a diameter of 140 m exploding at
the altitude of 10 km.
5.3
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
Water surface elevation obtained using SWE equations (
tions (
), and SGN equa-
) at (a) t = 1000 s, (b) t = 2000 s, (c) t = 3000 s, and (d) t = 4000 s.
Left y-axis shows the topographical variation in logarithmic scale. . . . . . . . 68
5.4
Water surface elevation of a tsunami wave generated by an impact into water
obtained using SGN equations at (a) t = 1000 s, (b) t = 5000 s, (c) t = 10000
s, and (d) t = 15000 s.
5.5
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
Maximum wave height as a function of distance from the impact center. (
SWE (
) SGN.
)
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
5.6
Schematic of a cnoidal wave interacting with a cylinder. . . . . . . . . . . . . . 73
5.7
Contour plots of the maximum water elevation for (a) Case1, (b) Case2, (c)
Case3. Blue lines denote the vertical cross sections at 1: (x − x0 )/h0 = 9.83,
2: (x − x0 )/h0 = 17.65, 3: (x − x0 )/h0 = 41.25.
. . . . . . . . . . . . . . . . . . . 74
List of Figures
5.8
xvi
Maximum momentum flux along the vertical cross sections at top: (x −
x0 )/h0 = 9.83, middle: (x − x0 )/h0 = 17.65, bottom: (x − x0 )/h0 = 41.25. . . . . 75
5.9
Tsunami run-up along the coastline from the Gaza Strip to the Caesarea,
Israel. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
List of Tables
2.1
Physical and numerical simulation parameters (hw = 0.3 m. hb represents the
depth of the water at the location of boulder. c is the solitary wave celerity).
√
Submergence factor: β = hhwb (y axis); Froude number: F r = c( ghw )−1 ; boulder aspect ratio in logarithmic scale: α = log2 (lby lb−1
x ) [x − y plane; boulder
−3
normalized width: Wb = lby h−1
w ; boulder normalized volume: Vb = lbx lby lbz hw ;
density ratio: ρb ρ−1
w ; friction coefficient: µ;
3.1
. . . . . . . . . . . . . . . . . . . . . 25
Running time for three-dimensional SPH simulation, and one-dimensional
SGN simulation of solitary wave run-up on a sloping beach.
. . . . . . . . . . 40
4.1
Wave gauge coordinates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
4.2
Geometrical parameters of macro-roughness patches. dr ∶ distance between
two horizontally (or vertically) aligned cylinders inside a patch; Ncp ∶ total
number of cylinders inside a patch; dp ∶ distance between two horizontally (or
vertically) aligned patches; Cf p ∶ coordinate of the center of the first patch;
Dc ∶ diameter of the cylinders; Dp ∶ diameter of the patches. . . . . . . . . . . . 55
xvii
List of Tables
5.1
xviii
Simulation parameters of cnoidal waves interacting with a cylinder. . . . . . . 73
Nomenclature
Abbreviations
SGN
Serre-Green-Naghdi equations
SPH
Smoothed particle hydrodynamics
SWE
Nonlinear shallow water equations
VOF
Volume of fluid
Latin symbols
u
Velocity vector
ζ
Free surface elevation
b
Topographical variation
g
Gravitational acceleration
h
Local water depth
p
Pressure
t
Time
xix
Nomenclature
Non-dimensional numbers
α
Aspect ratio
β
Submergence factor
µ
Shallowness parameter
Re
Reynolds number
ε
Nonlinearity parameter
Fr
Froude Number
xx
Chapter 1
Introduction
Majority of the world’s mega-cities are located near oceans. More than 50% of the world’s
population lives within the 60 km of ocean1 . In the United States, according to NOAA2 ,
more than 39% of the population lived in the counties next to an ocean. This number is
expected to increase to 47% by 2020. Tsunami waves are one of the most common forms
of natural disaster that affects the life of human beings living close to coastal areas. They
impose significant threat to human life and coastal infrastructure.
A tsunami is a series of long waves that can travel for hundreds of kilometers through the
ocean. Figure 1.1 shows the locations that are impacted by tsunamis since 1900. As we can
see from Figure 1.1, tsunamis are usually initiated by a sudden displacement of large amount
of water in the ocean or large lakes due to one or combination of the following reasons:
(a) Earthquakes are responsible for more than 80% of the tsunamis. Two of the most recent
deadliest tsunamis, i.e. 2004 Indian Ocean tsunami and 2011 Tohoku tsunami, were
1
2
http://www.unep.org/urban environment/issues/coastal zones.asp
National Ocean and Atmospheric Administration; http://oceanservice.noaa.gov/facts/population.html
1
Introduction
2
Figure 1.1: Location of tsunamis that have happened since 1900 to present day; ( ):
caused by a volcanic activity; ( ) caused by a landslide; ( ) caused by an unknown source;
( ) caused by an earthquake. Color codes are as following: dark-red represents the tsunamis
that caused more than 1000 casualties or more than 1 billion dollars damage in total; red
represents the tsunamis that caused more than 100 casualties or more than 100 million
dollars damage in total; orange represents the tsunamis that caused more than 10 casualties
or more than 10 million dollars damage in total; yellow represents the tsunamis that caused
less than 10 casualties or less than 10 million dollars damage in total
Introduction
3
caused by earthquakes. The 2011 Tohuku tsunami occurred on March 11, 2011 caused
by a magnitude 9.0 earthquake. It resulted in more than 15, 000 deaths and a total
economic loss of over 220 billion dollars. Indian Ocean tsunami occurred on December
26, 2004 and killed more than 230, 000 people. The estimated total economic losses due
to this tsunami is about 15 billion dollars. Maximum recorded wave height for Indian
Ocean and Tohoku tsunamis is more than 50 and 40 meters, respectively.
(b) Landslides are the second most probable cause of tsunamis. Tsunamis caused by landslides are usually generated by a solo landslide or a landslide following an earthquake or
a landslide initiated from a structural failure of a volcano. The wavelength of a tsunami
due to a landslide is usually shorter (it will only affect the areas at the proximity of the
origin). However the amplitude of these tsunamis can be much larger than the amplitude
of the tsunamis caused by earthquakes. In fact the maximum recorded wave-height in
history is more than 500 meters and belongs to 1958 Lituya Bay tsunami which was
caused by a landslide following an earthquake. Another notable tsunami that was due
to a landslide initiated by an earthquake is the 1964 Alsaka tsunami which killed more
than 100 people.
(c) Volcanic eruptions also generate tsunamis with shorter wave lengths. Thus they will
affect only the local areas. One of the most notable tsunamis caused by a volcanic
eruption is the 1741’s tsunami in the Oshima-Oshima region in Japan. This tsunami
caused lots of damages to coastal infrastructure and the total number of casualties, for
this event, is estimated to be about 2000 (Satake, 2007).
(d) A tsunami can also be generated by a meteorological source or an asteroid impact. Another historically important tsunami in the South Pacific ocean was generated by Eltanin
Introduction
4
impact which occurred about 2.5 million years ago. Even though the estimated initial
wave amplitude, according to the numerical simulations, was about 1 km, due to the
dispersive nature of the generated tsunami it is thought to have dissipated very fast with
an estimated wave height, close to Chilean coast, of around 10 meters. This makes the
tsunami as dangerous as Indian Ocean tsunami or Sumutra tsunami (Weiss et al., 2015;
Wünnemann and Weiss, 2015). The most notable recorded historical tsunami caused
by a meteorological source is the Hooghly River disaster that happened on October 11,
1737. This natural disaster is one of the deadliest recorded hazards in the history. The
cyclone caused a tsunami with a wave height in the range of 10 to 15 meters. The estimated casualties due to the tsunami and cyclone caused by this event is estimated to
be around 300, 000 people. Some researchers (e.g., Dominey-Howes et al., 2007) argued
only 10% of the overall deaths were due to the tsunami and the remaining 90% were
caused by the initial storm surge and cyclone.
Overall more than 2500 tsunamis have been recorded from 2000 B.C. through present day,
according to the Global Historical Tsunami Database provided by NGDC3 /WDS4 . More
than 900,000 people have died because of these natural disasters. The total economic loss
due to these events is estimated to be around 550 billion dollars. Figure 1.2 shows the
histograms of the number of earthquakes versus the death and damage toll caused by them.
Less than 10% of the tsunamis have caused casualties more than 10 or total economical loss
of more than 10 million dollars. Only two recorded event caused more than 100, 000 deaths
or more than 100 billion dollars damage. We note here that there is no direct correlation
3
4
National Geophysical Data Center
World Data Service
5
Introduction
between the number of deaths and total economical loss. In other words, tsunamis with the
largest amount of casualties do not necessarily correspond to the largest amount of property
damage. Figure 1.3 shows the plot bar representing the casualties and deaths caused by
tsunamis. Earthquakes are responsible for more than 65% of casualties and more than 85%
of the economical loss. About 30% of deaths are caused by tsunamis with meteorological
sources and landslides, and volcanic activities cause the remaining losses/damage due to
Number of Earthquakes
Number of Earthquakes
tsunamis.
104
103
102
101
100
10
I
II
III
IV
V
VI
A
B
C
D
F
G
4
103
102
101
100
Figure 1.2: Histograms of the number of earthquakes versus the death and damage toll.
I: less than 10 casualties; II casualties between 10 and 100; III casualties between 100 and
1000; IV casualties between 1000 and 10000; casualties more than 10000. A: damage less
than 10 million dollars; B: damage between 10 and 100 million dollars; C: damage between
0.1 and 1 billion dollars; D: damage between 1 and 10 billion dollars; E: damage between 10
and 100 billion dollars; F: damage more than 100 billion dollars; The y-axis is in logarithmic
scale.
6
Introduction
Total number of casualties
Damage in million dollars
106
105
104
103
102
101
100
10−1
I
II
III
IV
V
Figure 1.3: Casualties and economic loss due to I: unknown source; II: earthquake; III:
landslide; IV: volcano; V: meteorological source; The y-axis is in logarithmic scale.
1.1
Theoretical Background
Figure 1.4 shows the schematic of a typical long wave. Assuming a flat bathymetry, two
non-dimensional parameters that represent the problem are the nonlinearity parameter
ε=
a
,
h0
(1.1)
µ=
h20
.
λ2
(1.2)
and the shallowness parameter
Here a is the nominal wave amplitude, h0 is the water depth when at rest, and λ is the wave
length.
7
Introduction
Nonlinearity: ε =
a
h0
Shallowness: µ =
h02
λ2
λ
a
h0
Figure 1.4: Schematic of a long wave.
ε is typically in the order of 10−4 for far field tsunamis. For example, the 2004 Indian
Ocean tsunami’s wave height, while traveling in deep ocean (h0 ∼ 5000 m) was about 1 m.
However as the tsunami approaches the coast the wave height increases and the nonlinearity
parameter can rise up to ε ∼ 0.5. The nonlinearity parameter for storm waves will be in the
order of 10−2 to 10−1 .
A typical shallowness parameter for far field tsunamis will range from 10−5 < µ < 10−4 . However, as the tsunami wave approaches the shore and the wave height increases, the wave speed
and consequently the wavelength decreases. This results in an increased nonlinearity parameter in the proximity of the coast. For coastal waves, the wavelength will be significantly
shorter which usually results in shallowness parameter of µ > 10−2 .
Figure 1.5 shows a tsunami wave propagation and run-up. The initial wavelength is about
8
Introduction
6 km and the initial wave amplitude is 4.5 m. The water depth at the origin of the wave
is about 400 m. Thus the initial nonlinearity and shallowness parameters are ε ∼ 10−2 , and
µ ∼ 10−2 . We solved this problem using the nonlinear shallow water equations and weaklydispersive fully-nonlinear Boussinesq-type equations. As we can see in Figure 1.5 the results
obtained from different equations significantly differs from each other. We will explain the
importance of choosing an appropriate model for the specific problem we want to solve in
the following paragraphs.
The nonlinear shallow water equations (SWE) are the most common form of equations used
in the numerical simulation of long waves:
∂t h + ∇ ⋅ (hu) = 0,
(1.3)
1
∂t (hu) + ∇ ⋅ (hu ⊗ u + gh2 I) = −gh∇b + O(µ),
2
(1.4)
where u is the depth-integrated velocity vector, h is the water depth, b represents the bottom
variations, and I is the identity tensor. To derive the nonlinear shallow water equations from
the Euler equations, one has to assume hydrostatic pressure distribution in the vertical
direction. This assumption implies that there is no acceleration in the vertical direction,
i.e. velocity is constant along the z−axis. SWE are hyperbolic partial differential equations
and thus they have shock capturing capability in their conservative form. Furthermore,
there exists many mature numerical schemes for solving these equations which makes them
computationally a good and efficient choice.
However, while SWE can produce satisfactory results for far field tsunamis (µ ∼ 10−4 ), they
become very inaccurate for waves with shorter wavelengths (such as landslide generated
tsunamis, or storm waves). To overcome this drawback, different classes of higher-order
9
Introduction
depth-integrated equations, based on Boussinesq wave theory, have been derived and presented in the literature on simulating shallow water flows: Wei et al. (1995); Liu (1994);
Lannes and Bonneton (2009). In this study, we employ fully nonlinear weakly dispersive
Boussinesq equations, also known as Serre-Green-Naghdi equations (SGN, Lannes and Bonneton, 2009):
∂t h + ∇ ⋅ (hu) = 0,
(1.5)
1
∂t (hu) + ∇ ⋅ (hu ⊗ u + gh2 I) = −gh∇b + D + O(µ2 ),
2
(1.6)
and
⎛
⎞
−1
⎜g
⎟
1
1
1
⎟
D = h⎜
⎜ α ∇ζ − h (I + αhT h ) ( α gh∇ζ + hQ(u))⎟ ,
⎜
⎟
´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¸¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¶⎠
⎝
B
(1.7)
or
(I + αT ) B =
1
g∇ζ + Q(u),
α
(1.8)
where
ζ = h + b.
(1.9)
Here ζ denotes the free-surface elevation. T is an operator acting on a scalar field given by
T [h, b](s) = R1 [h, b] (∇ ⋅ s) + R2 [h, b] (∇b ⋅ s) ,
(1.10)
Q is an operator acting on a vector field given by
Q[h, b](s) = −2R1 [h, b] (∂1 s ⋅ ∂2 s⊥ + (∇ ⋅ s)2 ) + R2 [h, b] (s ⋅ (s ⋅ ∇) ∇b) ,
(1.11)
where s = (−sx, sy)T . R1 and R2 are differential operators acting on scalar fields:
h
1
R1 [h, b]w = − ∇ (h3 w) − w∇b,
3
2
(1.12)
10
Introduction
and
1
R2 [h, b]w = ∇ (h2 w) − w∇b.
2
(1.13)
These equations have two advantages compared to others: (a) They only have spatial derivatives up to second order while other equations usually include spatial derivatives of third
order, and (b) they do not have any additional source term in the conservation of mass.
These properties improve the computational stability and robustness of the model. While
using the SGN equations we can simulate waves with shorter wavelengths, solving these
equations is associated with finding a solution to a linear system with 4 × nx × ny unknowns
which makes them computationally more expensive in comparison to SWE. This computational cost can become significant with further refining the grid in order to properly resolve
the effects of shorter waves on longer waves.
1.2
Contributions and Outline of the Dissertation
The remaining of this dissertation is organized as follows:
• In chapter two, a study on the motion and dislodgement of boulders by solitary waves
is presented. The results are obtained using a three-dimensional SPH method (a meshfree method). This is the first time that three-dimensional numerical simulations of
boulder transport were made. The results show that nonlinear processes are important.
It is important to note that these nonlinear processes can only be captured with numerical models. Proposed analytical approaches simplify the boulder transport problem.
11
Elevation (m) Elevation (m) Elevation (m) Elevation (m) Elevation (m) Elevation (m) Elevation (m)
Introduction
20
0
20
0
200
a-I
400
20
0
0
200
b-I
400
20
0
0
200
c-I
400
20
0
0
200
d-I
400
20
0
0
200
e-I
400
20
0
0
200
f-I
400
20
0
0
g-I
10
200
0
10
20
Distance (km)
30
40
400
0
a-II
20
0
b-II
20
0
c-II
20
0
d-II
20
0
e-II
20
0
f-II
20
0
g-II
Figure 1.5: Tsunami wave propagation caused by a landslide. The initial ratio between
the wavelength and water depth is more than 10. The figure shows the comparison of the
three different simulations. (blue) SGN high resolution; (red) SGN low resolution; (green)
SWE high resolution.
Introduction
12
From a physical point of view, waves transform into a three-dimensional flow field in
the vicinity of the shoreline by breaking or by the interaction with boulders. The main
conclusion of our study is that, there is no linear trend in the parameter space other
than the boulders move larger distances for larger values of the parameters. Furthermore, this project showed that while it is important to simulate the problem using a
three-dimensional scheme, computational cost of a three-dimensional simulation make
it impractical for a large-scale real-world problem.
• In chapter three, we present an extensive comparison of the results obtained using
depth-integrated equations to the existing three-dimensional and experimental data.
Three-dimensional simulation relies on less number of assumptions. In addition, they
can fully resolve the coherent turbulent structures of the flow. However, given the
domain size of the numerical geophysical problems we are interested in solving, i.e.
lx × ly ∼ 1000 km × 1000 km, in addition to the limits imposed on grid resolution by the
current computational facilities, we argue that it is not feasible to simulate long waves,
such as tsunamis and storms, in three dimensions. We demonstrate that using depthintegrated equations, we can simulate the problems using finer grids which results in
more accurate results.
• In chapter four, we present numerical simulation of nonlinear long waves interacting
with arrays of emergent cylinders. Our model employs the fully nonlinear and weakly
dispersive Serre-Green-Naghdi equations (SGN) until the breaking process starts, while
we changed the governing equations to nonlinear shallow water equations (NSW) at the
vicinity of the breaking-wave peak and during the run-up stage. Using this approach,
we avoid the numerical oscillations that can be caused by dispersive terms in SGN in the
Introduction
13
vicinity of the shoreline and the coastal areas. In the literature, coastal vegetation (here
cylinders) is usually approximated as macro-roughness friction. We model the cylinders
as physical boundaries. This eliminates the approximation associated with defining the
friction coefficient. In addition, the presented numerical results are vigorously validated
against existing experimental and numerical data. Given the summary above, we think
our findings merit a publication in the Marine Geology.
• In chapter five, we present the risk assessment on the coast along the Gaza strip
imposed by possible earthquakes, landslides, and volcanic eruptions. In addition we will
present some preliminary results on numerical simulation of tsunamis by meteorological
impacts.
Chapter 2
Boulder Dislodgement and Transport
by Solitary Waves: Insights from
Three-Dimensional Numerical
Simulations†
† Citation:
A. Zainali, and R. Weiss (2015), Boulder dislodgement andtranspor t by solitary
waves: Insights from three-dimensional numerical simulations, Geophys. Res. Lett., 42,
44904497, doi:10.1002/2015GL063712.
14
3D Simulation of Boulder Dislodgement
15
Abstract
The analysis of boulder motion and dislodgement provides important insights into the physics
of the causative processes, i.e., whether or not a boulder was moved during a storm or tsunami
and the magnitude of the respective event. Previous studies were mainly based on simplified
models and threshold considerations. We employ three dimensional numerical simulation of
the hydrodynamics coupled with rigid-body dynamics to study boulder dislodgement and
transport by solitary waves. We explore the effects of three important non-dimensional
parameters on the boulder transport problem, the Froude number F r, the aspect ratio in
logarithmic scale α and the submergence factor, β. Our results indicate that boulder motion
and dislodgement is complex, and small changes in one of the non-dimensional parameters
result in significantly different behavior during the transport process and the final resting
place of boulders. More studies are need to determine the role of boulders in tsunamis and
storms hazard assessments.
2.1
Introduction
Boulders can be found along the coastlines of our planet. In general it is assumed only
high-energy events, such as storm and tsunami waves, can move these boulders because of
their large mass that can exceed 50 tons. Other more common marine or coastal processes
are unable to move these boulders (Paris et al., 2010; Barbano et al., 2010; Goff et al., 2006;
Scheffers and Scheffers, 2006).
Boulders are, therefore, important event deposits that record the frequency and characteris-
3D Simulation of Boulder Dislodgement
16
tics of the storm or tsunami waves moved them. The analysis of boulder movement can be
an important component of robust and defensible hazard assessments for future storms and
tsunamis. One major problem, however, is that it is very difficult to distinguish boulders
moved during storms from those moved during tsunamis, especially in areas where storms
and tsunamis are competing processes for boulder movement (Barbano et al., 2010; Noormets
et al., 2004). The reason is that boulder displacement depends on many environmental and
wave parameters. Among them are wave height and period, shape of the boulder, their initial location, as well as other parameters such as presence of macro-roughness elements Goto
et al. (2010b,a). If coastal boulders are included in the statistics describing past tsunamis
and storms, to predict future ones, we must be able to determine which event type moved
which boulder.
Simplified theories and models has been developed in the past to model boulder transport,
i.e. Nott (2003); Weiss (2012); Imamura et al. (2008). From a physical point of view,
boulder motion is complex due to the boulder’s complicated geometry, potentially heterogeneous density distribution, and the non-linear interactions between the turbulent flow field
and the boulder itself. Numerical models that are capable of handling this complex transport situation have to contain state-of-the art representations of turbulent flow, incorporate
cutting-edge numerical solution techniques, and require significant computational resources.
Therefore, there is a significant benefit in simplifying boulder motion to develop simple theories. However, for these theories to be useful, simplifications made of the transport problem
must be physically robust and lead to reliable and consistent results. This can be done
through rigorous testing and evaluation of these models by more rigorous three-dimensional
models.
3D Simulation of Boulder Dislodgement
17
Previous studies commonly assumed that the main cause of boulder movements are large
tsunamis (Imamura et al., 2008). In this study, we design a broader framework to investigate boulder dislodgement and transport by solitary waves with a large amplitude range.
Different representations of long waves have been discussed (Madsen et al., 2008; Madsen
and Schaeffer, 2010; Tadepalli and Synolakis, 1994). However, we note that solitary wave
theory still remains the most common (Tadepalli and Synolakis, 1996; Liu et al., 1995). The
interaction between boulders and solitary waves causes boulder dislodgement and movement,
and is classified in three different categories: no motion, motion but no dislodgement, and
dislodgement (for more details see, Weiss and Diplas, 2015). To solve our problem in a threedimensional domain, we employ GPUSPH (Hérault et al., 2010; Herault et al., 2006-2014).
GPUSPH is a numerical model that uses Smoothed Particle Hydrodynamics to solve the
governing equations of fluid motion.
2.2
2.2.1
Theoretical Background
Governing Equations and Numerical Method
We employ Smoothed Particle Hydrodynamics (SPH) to simulate the flow of water around
boulders. The respective conservation of mass and momentum in their weakly compressible
form can be written as (Yoshizawa, 1986)
Dρ̄
∂ ũi
+ ρ̄
=0
Dt
∂xi
(2.1)
Dũi
1 ∂ p̄
∂ 2 ũi
1 ∂τij∗
=−
− gδi3 + ν
+
,
Dt
ρ̄ ∂xi
∂xj ∂xj ρ̄ ∂xj
(2.2)
18
3D Simulation of Boulder Dislodgement
where “∼” refers to the Favre-filtering operator φ̃ = ρφ(ρ)−1 . The Large Eddy Simulation
approximation (LES) is employed to consider sub-particle scales. Dalrymple and Rogers
(2006) suggested the following representation of the LES approximation in SPH for the
sub-particle stress tensor:
2
2
τij∗ = ρ̄ (2νt S̃ij − S̃kk δij ) − ρ̄CI ∆2 δij
3
3
∂ ũi
in which S̃ij = − 12 ( ∂x
+
j
∂ ũj
∂xi )
(2.3)
and CI = 1/1500. The turbulent viscosity is calculated using the
standard Smagorinsky model: νt = (Cs ∆) ∣S̄∣ (Smagorinsky, 1963, Smagorinsky constant is
2
Cs = 0.12). The parameter ∆ is the initial spacing of the SPH particles. u denotes the
velocity vector, p represents pressure, ρ is the density, ν is the kinematic viscosity, and g
refers to the gravitational acceleration. δ denotes the Kronecker delta.
As mentioned earlier, we employ SPH to solve the aforementioned equations of fluid motion.
Because of its Lagrangian nature, the SPH method is an appropriate method for simulating
complicated flow situations. Refer to Dalrymple and Rogers (2006); Weiss et al. (2011);
Shadloo et al. (2015) for more information about water waves modeling using SPH and to
Monaghan (1992, 2005a, 2012) for a more complete description of the SPH method and a
comprehensive list of its applications.
To employ the SPH method for our flow situation, we employ GPUSPH. GPUSPH solves
the governing equations with the aid of graphical processing units (Hérault et al., 2010;
Herault et al., 2006-2014). To enforce incompressibility of the flow, GPUSPH employs an
artificial speed of sound of c = 10umax such that the density variations in the fluid are limited
to about 1%. For all boundaries between fluid and solid, we use the Lennard-Jones type
repulsive force (Monaghan, 1994a). GPUSPH is coupled with the Open Dynamics Engine
19
3D Simulation of Boulder Dislodgement
(ODE; Smith, 2006) to simulate the interaction between a rigid body, such as a boulder,
and the fluid. Aside from computing the motion of boulders, the ODE library allows the
inclusion of, for example, Coulomb’s friction law between the boulders and the underlying
boundary surface and solves an equation of motion for rigid body.
2.2.2
Model Setup
The geometrical setup of the computational domain consists of a box with length of lx = 10m,
height of ly = 0.45 m and width of lz = 0.3 m (Figure 2.1). We included a sloping beach
(S=1:10) whose toe is located at lxh = 5.5 m. The water depth is denoted with hw . We
consider two different scenarios: A dam-break problem for validation, and solitary wave
runup for long waves. The gate that is lifted to create the dam break is located at lw .
y
Gate
lz
ly
hw
.
z
Boulder
x
lw
lxh
lx
Figure 2.1: Sketch of dam-break scenario at t = 0. Water behind the gate starts to flow
after sudden gate removal at t = 0. To compare the numerical results with experiments
presented by Imamura et al. (2008), we applied: lx = 10 m, ly = 0.45 m, lz = 0.3 m, lx h = 5.5 m,
lw = 3 m and hw = 0.15, 0.20, 0.25, 0.3 m.
To create solitary waves, we employ the wavemaker function as defined by Goring (1978),
√
√
for which the wavenumber is κ = (3η)(4h3 )−1 and the phase speed is c = g(h + η) where
20
3D Simulation of Boulder Dislodgement
η is the wave amplitude.
2.2.3
Non-Dimensional Parameters
The one-dimensional shallow water equations in the absence of topographical variations
are known to predict the wave runup accurately (Li and Raichlen, 2002; Synolakis, 1987).
However, in the presence of topographical variations, flow field in its vicinity becomes threedimensional violating the one-dimensional flow assumption.
For a brief theoretical analysis, we will continue with the one-dimensional shallow water
equations to drive the important non-dimensional parameters controlling the main wave
characteristics. Later on, we will introduce other important parameters that can characterize the three-dimensional effects due to the presence of topographical variations, namely a
boulder in our study. One-dimensional shallow water equations can be written in the form
of:
(h)t + (hu)x = 0,
(hu)t + (hu2 )x +
1
(gh2 )x = −ghbx .
2
(2.4)
in which h is the water depth, u represents the velocity, and the bottom topography is
denoted by b. The variables and parameters x, t, h, u and b are normalized with xref , tref ,
href , uref and bref . With the help of these normalizations, Eq. (2.4) can be rewritten to:
St (h∗ )t∗ + (h∗ u∗ )x∗ = 0,
St (h∗ u∗ )t∗ + (h∗ u∗ 2 )x∗ +
1
β
(h∗ 2 )x∗ = − 2 h∗ b∗x∗ .
2
2F r
Fr
(2.5)
where
St =
uref
bref
xref
, Fr = √
, β=
tref uref
href
ghref
(2.6)
3D Simulation of Boulder Dislodgement
21
are the Strouhal number, Froude number, and submergence factor which is the ratio between
reference height and reference topographical variations. Superscript ∗ refers to normalized
parameters. Here we choose the following reference values: xref = κ−1 , tref = (cκ)−1 , href =
hw , uref = c and bref = lby , where lby denotes boulder height and c and κ denote phase speed
and wave number.
For simplicity, we will only consider boulders with constant width, i.e., lbz = constant.
Keeping the volume of the boulder constant, we define a parameter α = log2 (lby lb−1
x ), the
aspect ratio in logarithmic scale of the boulder length in the x-y plane, as another important
non-dimensional parameter. Boulder density can alter dislodgement process significantly.
As a result, the ratio between the boulder density and water density, ρb ρ−1
w , will play an
important role in this problem as well.
Due to our simplifying assumptions compared to the real flow situation, it is impossible to
match every possible non-dimensional parameter with real-world data. For our analysis, we
will continue with the Froude number, submergence factor and aspect ratio in logarithmic
scale. We note that the Froude number is an estimate of the wave energy. The potential
energy of a wave is given by Ep = 0.5ρw gη 2 where η is the wave surface elevation and overbar
denotes the averaged value over the wave period. Assuming that the wave motion is under
quasi equilibrium condition, using the equipartition theory we can assume that kinetic energy
Ek = Ep . Total energy becomes equal to Et = 2Ep . From the Froude number definition and
solitary wave theory we can approximate Et = ζρw g(F r2 − 1)λ where ζ and λ are constants.
The α and β parameters provide us with necessary information regarding the location of the
boulder with respect to incoming water wave. Due to simplicity of the geometrical shape we
22
3D Simulation of Boulder Dislodgement
have chosen here, α parameter is intended to capture three-dimensional effects. However,
for more complicated geometrical shapes, other non-dimensional parameters may be needed
to fully characterize shape-dependent processes. One example would be the orientation of
the boulder with respect to incident wave direction.
3
db(max)
2
1
0
0.0
0.1
0.2
0.3
0.4
h
Figure 2.2: Comparison of numerical and experimental data presented. Experimental data
are from Imamura et al. (2008). The case ρb = 1550 kg m−3 is colored in black and case with
ρb = 2710 kg m−3 is colored in blue; ( ) experimental results and ( ) SPH simulations.
3D Simulation of Boulder Dislodgement
2.3
2.3.1
23
Results
Validation of Numerical Results
We validate the employed numerical model with experiments carried out by Imamura et al.
(2008). These experiments are based on a dam-break scenario. The dam break is located
at lw = 3 m, and the water depth varies from hw = 0.15 m to hw = 0.30 m in steps of five
centimeters. The boulders have a constant size with 0.032×0.032×0.032m3 , but two different
densities, ρ1b = 1550 kg m−3 and ρ2b = 2710 kg m−3 , are applied.
Figure 2.2 shows comparisons for the distance, db , the boulder traveled from its original
location between experimental and simulated data. For a density of ρ1b , the comparison is
excellent (black color). However, for the density of ρ2b , there is a noticeable difference between
experimental and numerical results. This difference is due to the constant friction coefficient
in the numerical model, while in experiments, different scenarios come with different surface
roughness values associated with different materials, however, the exact value of friction
coefficient is not reported for the scenarios. Despite this difference, we argue that GPUSPH
coupled with the ODE library capture all processes important for boulder transport.
24
3D Simulation of Boulder Dislodgement
(a) t = 0 (s)
Case CA(α=0)
Case CB(α=0)
Case CC(α=0)
Case CA(α=1)
(c) t = 4.7 (s) Case CB(α=1)
Case CC(α=1)
(b) t = 0 (s)
(d) t = 4.7 (s)
(e) t = 4.9 (s)
(f) t = 4.9 (s)
(g) t = 5.1 (s)
(h) t = 5.1 (s)
Velocity magnetiude [ms−1 ]
Figure 2.3: Time snapshots for scenarios CA, CB and CC with α = 0, (left column) and
scenarios CA, CB and CC with α = 1, (right column) with wave height of 0.15 m. Please
note that different cases are superimposed on the same domain just for illustration purposes.
25
3D Simulation of Boulder Dislodgement
2.3.2
Boulder Transport by Solitary Waves
For our numerical experiments, we assume a water depth of hw = 0.3m. Although the selected
waves are not a rigorous representation for the entire wave spectrum, in terms of wave shape
and period, they provide an appropriate wave energy range to study the primary factors of
the dislodgement process. We first created a series of subsequent solitary waves in order to
see if a boulder can be moved by two or three subsequent waves for which the amplitude
of each individual wave is too small to cause boulder transport. It turns out, however, as
long as the first wave does not cause any movement, the effects of subsequent waves can
be ignored. Therefore, boulder dislodgement can be compared on a wave by wave basis.
For more detailed analysis, we varied the aspect ratio in logarithmic scale of the boulder,
submergence of the boulder and the Froude number, but all boulders have the same mass
(constant density, ρ = 2000 kg m−3 .) Table 2.1 summarizes the different parameters and
defines the different scenarios that are considered in our study.
Table 2.1: Physical and numerical simulation parameters (hw = 0.3 m. hb represents the
depth of the water at the location of boulder. c is√the solitary wave celerity). Submergence
factor: β = hhwb (y axis); Froude number: F r = c( ghw )−1 ; boulder aspect ratio in logarith−1
mic scale: α = log2 (lby lb−1
x ) [x − y plane; boulder normalized width: Wb = lby hw ; boulder
−1
normalized volume: Vb = lbx lby lbz h−3
w ; density ratio: ρb ρw ; friction coefficient: µ;
Scenario:
CA
Submergence factor:
−8/30
Froude number:
1.08, 1.15, 1.22
boulder aspect ratio:
−1, 0, 1
boulder normalized width:
4/30
boulder normalized volume:
43 /303
density ratio:
2
friction coefficient:
0.6
CB
CC
−4/30
0
1.08, 1.15, 1.22 1.08, 1.15, 1.22
−1, 0, 1
−1, 0, 1
4/30
4/30
3
3
4 /30
43 /303
2
2
0.6
0.6
CD
4/30
1.08, 1.15, 1.22
−1, 0, 1
4/30
43 /303
2
0.6
3D Simulation of Boulder Dislodgement
26
For the numerical experiments, we define four different scenarios that vary the initial location
of the boulder with regard to the still water shoreline. We employ the submergence factor to
define the location from the shoreline. For scenario CA, the boulder is farthest in the water
and will remain fully submerged for all aspect ratios in logarithmic scale. For scenario CB,
the boulder is submerged for an aspect ratio in logarithmic scale of -1. However, for α = 1,
the boulder is only partially submerged in water. The boulder in scenario CC is located
directly at the shoreline, and CD defines a scenario in which boulder is initially located on
dry land. In our numerical experiments, we apply three aspect ratios in logarithmic scale,
−1, 0, and 1, as well as three different Froude numbers, 1.08, 1.15, and 1.22, to each scenario.
The Froude numbers presented here are consistent with field observations (Etienne et al.,
2011; Jaffe et al., 2011; Fritz et al., 2006; Spiske et al., 2010). In terms of scaling up to the
prototype scale, assuming a coastal water level of 30 m, our experimental scale represents
approaching wave heights between 3.3 and 10 m resulting in a boulder with the volume of
18 m3 and a boulder mass of approximately 36 × 103 kg.
Figure 2.3a shows the initial position of the boulders for an aspect ratio in logarithmic scale
of α = 0, and Figure 2.3b for an aspect ratio in logarithmic scale of α = 1. Different cases
are plotted together just for better illustration. The subsequent subplots show the impact
of a wave on the different boulders (Figure 2.3c,e,g for α = 0 and Figure 2.3d,f,h for α = 1).
We see that for α = 0 only the boulder CC moves visibly. Boulders CA and CB did not.
For the aspect ratio in logarithmic scale α = 1, all boulders, initially, flip over due to their
unstable position; but again, only boulder CC moves shoreward from its original location. It
should be noted that for an aspect ratio in logarithmic scale of α = −1, none of the boulders
moved. Boulders can saltate, rotate, slide, or a combination of all three (Nandasena et al.,
3D Simulation of Boulder Dislodgement
27
2011b,a; Imamura et al., 2008). Nandasena and Tanaka (2013), however, noted that the
dominant transport mode is sliding. Our numerical modeling confirms this observation. We
also find that rotation can play an important role in boulder transport in the very early
stage of the transport process. It should be noted that if rotation occurs even in the early
stages of boulder transport, transport paths of the boulders are very different especially for
the cases with α >> 0, as highlighted by red and green circles. Boulders highlighted by the
red circles start to move but do not dislodge, while boulders highlighted by green circles
dislodge completely.
Boulder movement can be characterized in three different distinct modes: no motion, motion
but no dislodgement, and dislodgement. Figure 2.4a-c contains the information on these
modes for F r, α and β plotted against each other (Figure 2.4a: F r vs α, b: F r vs β, and c:
α vs β). The white areas in Figure 2.4a-c mark no motion in which the normalized distance
that boulder travels is db lbz < 0.1, the gray color indicates motion but no dislodgement
(0.1 < db lbz < 2), and the contours represent dislodgement. For the latter, it should be noted
that the darker the color, the larger the distance a boulder travels from its original position.
Boulder motion and dislodgement is achieved easily if all three non-dimensional parameters
are large.
With increasing F r number, the total energy the wave carries increases. As a result, the
distances boulders travel increase with increasing F r number. In our numerical experiments,
the waves break in the vicinity of shoreline. After breaking, the flow in the wavefront
becomes supercritical with significantly larger local Froude numbers, compared to average
Froude numbers of the wave before breaking (1.0 < F r < 3.0; Matsutomi et al., 2001). After
3D Simulation of Boulder Dislodgement
28
wave breaking and as the broken wave propagates shore-ward, the flow becomes shallower
and starts to accelerate. Due to this acceleration, boulders located at β = 4/30 travel larger
distances compared to boulders located at β = 0, and boulders located at β = 0 travel larger
distances compared to boulders located at β = −4/30. However, with increasing β, gravity
and friction forces in the opposite direction of flow will start to dominate the flow after a
critical value for β is exceeded. Thus the flow will start to decelerate, and as a result, the
distance boulders travel will decrease. Finally, as we mentioned earlier, rotational movement
becomes very important for boulders with α >> 0 which, generally, results in larger travel
distance.
2.4
Discussion and Conclusions
In this contribution, we studied the motion and dislodgement of boulders by solitary waves
with the state-of-the-art hydrodynamic model GPUSPH. This is the first time that three
dimensional numerical simulations of boulder transport were made. The results show that
nonlinear processes are important. It is important to note that these nonlinear processes
can only be captured with numerical models. Proposed analytical approaches simplify the
boulder transport problem. From a physical point of view, waves transform into a threedimensional flow field in the vicinity of the shoreline by breaking or by the interaction with
boulders.
The solitary waves in our numerical experiment can be interpreted as storm and tsunami
waves, based on the total energy they carry. Without loss of generality, we can assume
that small solitary waves represent storm waves, and large solitary waves represent tsunami
3D Simulation of Boulder Dislodgement
29
waves. Because the flow around boulders is three dimensional, we focus on the local energy
and not, for example, solely on the wave speed or amplitude. We employed the Equipartition
Theory to argue for a spectrum of energy that storm and tsunami waves can have and not
discrete sets of wave amplitudes and periods.
We considered boulders with very simple geometries and a homogeneous density distribution.
With the help of dimensional analysis, we identified the aspect ratio in logarithmic scale,
the submergence factor and the Froude number as crucial parameters of boulder transport.
Figure 2.4a-c depicts the result of this parameter study. From these subfigures, we can
see that there is no consistent trend other than boulders are transported larger distances
for larger values of the different parameters. However, there is no systematic trend in the
way contours exist in this parameter space. We note here that considering more realistic
scenarios, predicting the distance a boulder can travel is even more difficult. A typical
example of such a scenario would be a boulder with heterogeneous rather than homogeneous
density distribution or a scenario in which boulder is oriented at an angle to the incident
wave direction. Including these effects will result in the identification of more critical nondimensional parameters coming from more complete theoretical analysis. More research is
needed to identify the structure of these additional parameters.
30
3D Simulation of Boulder Dislodgement
mo
vin
g
(a)
0.8
a-I
a-II
a-III
0.4
α 0.0
−0.4
−0.8
1.09
1.13
1.17
1.21 1.09
1.13
Fr
(b)
0.1
1.17
1.21 1.09
1.13
Fr
b-I
1.17
1.21
1.17
1.21
Fr
b-II
b-III
0.0
β
−0.1
−0.2
1.09
1.13
1.17
1.21 1.09
1.13
Fr
(c)
0.1
1.17
1.21 1.09
1.13
Fr
c-I
Fr
c-II
c-III
0.0
β
−0.1
−0.2
−0.8 −0.4
0.0
0.4
0.8
−0.8 −0.4
0.0
0.4
0.8
α
α
Normalized maximum boulder displacement
2
4
6
8
10
16
−0.8 −0.4
22
0.0
0.4
0.8
α
28
34
Figure 2.4: Contour plots of boulder maximum displacement, db , as a function of nondimensional parameters. a: aspect ratio in logarithmic scale versus Froude number; (a-1)
β = −8/30 (a-2) β = 0 and (a-3) β = 4/30. b: submergence factor versus Froude number; (b-1)
α = −1, (b-2) α = 0 and (b-3) α = 1. c: submergence factor versus aspect ratio in logarithmic
scale; (c-1) F r = 1.08, (c-2) F r = 1.15 and (c-3) F r = 1.22. White regions indicate no
significant boulder movement, i.e. db lb−1
z < 0.1. For gray regions, the distance boulder travels
is in between 0.1 < db lb−1
<
2.0.
Color
contours indicate boulders moved significantly, i.e.
z
db lb−1
>
2.0
(colormap
is
in
logarithmic
scale)
z
More field studies are needed to see how the submergence, for example, can be better estimated. Therefore, three-dimensional simulations, such as ours, help to identify important
gaps in understanding boulder transport and eventually will help to narrow the gap between field and theory based research. The main conclusions of our study are that firstly,
3D Simulation of Boulder Dislodgement
31
the flow around a boulder needs to be considered in a three-dimensional fashion for which
numerical simulations are needed. Second, there is no linear trend in the parameter space
other than the boulders move larger distances for larger values of the parameters. The contours in the parameter space have complex and nonlinear shapes. These two conclusions
reveal shortcomings of the presently used standard methods and show that more work with
three-dimensional simulations is needed to successfully employ boulders as event deposits
in storm and tsunami hazard assessments. Perhaps the inevitable conclusion is that such
three-dimensional simulations, including variables such as angle of incident wave or heterogeneous mass distribution within the boulder, are the only credible and reliable approach to
improve our understanding of boulder dislodgement. Given the model complexities as we
observed earlier, it remains to be seen what role boulders can play in storms and tsunamis
hazard assessment.
Chapter 3
High-Fidelity Depth-Integrated
Numerical Simulations in Comparison
to Three-Dimensional Simulations
32
2D vs 3D
33
Abstract
In this study, we present the numerical simulation of the breaking solitary wave run-up and
non-breaking wave interaction with offshore cylinders using Serre-Green-Naghdi equations.
We compared the numerical results of our model to those obtained by a three-dimensional
volume of fluid method (VOF), and a mesh-free three-dimensional smoothed particle hydrodynamics (SPH) method and existing experimental data. Our results suggest that depthintegrated equations can produce results as accurate as three-dimensional schemes while
being computationally more efficient.
3.1
Introduction
Multiphase flow, where two or more fluid have inter-facial surfaces, is one of the challenging
and difficult areas in the field of Computational Fluid Dynamics (CFD), which plays an
important role in many industrial and natural systems. A free-surface flow can be considered
as a subclass of multiphase flows for which we can neglect the stresses at the interface of
the two interacting-fluid flows. The density and viscosity ratio between water and air are
large and the surface tension coefficient is small between the two fluids. In addition, the
Reynolds number of the oceanic waves are large. Thus, free-surface flow assumption is a
good representation of flow behavior in tsunami and storm waves.
In the following paragraphs, we will describe the different numerical schemes used for numerical simulation of free-surface flows.
6
34
2D vs 3D
R=0.61
4
2
(0.0,0.0)
Gauge 1
0
2R
Wave Direction
Gauge 6
Gauge 2
Gauge 4
−2
Gauge 3
Gauge 5
2.43 m
−4
−6
−10 sketch of the solitary
−5 wave passing through
0 and around cylinders.
5
Figure 3.1: A schematic
The coordinate of the domain is located at the center of the cylinder on the right hand side
of the domain. Wave propagates along the x-axis. The red dots represent the location of
the center of the cylinders. The blue dots represent the location of the wave gauges. Gauge
1: (-3.04 m, -0.14 m); Gauge 2: (-1.82 m, -0.14 m); Gauge 3: (-0.83 m, 0.00 m); Gauge 4: (
0.00 m, -0.88 m); Gauge 5: ( 0.85 m, 0.00 m); Gauge 6: ( 1.82 m, 0.00 m);
3.1.1
Three-Dimensional Methods
Mesh-Dependent Methods
Multiphase flow problems, so far, have been studied widely using mesh dependent techniques. Nevertheless, because of the complexity of these problems mainly associated with
the necessity of tracking interface evolution, most of the current works have not gone beyond
the simple problems. As can be inferred, the interface evolution is crucial to modeling of
multiphase flows and thus, needs to be modeled correctly and carefully in order to obtain
reliable simulation results. In mesh-dependent methods, an additional set of equations has
to be solved to track inter-facial surfaces, and furthermore, depending on the problem in
2D vs 3D
35
hand (i.e., if the the topology of the flows deforms significantly), mesh-refinement might be
required. In mesh-dependent methods, different techniques have been proposed and studied
in literature to capture the interface evolution on a regular grid. Among them, Volume of
fluid (VOF), (e.g., Hirt and Nichols, 1981), and Level set methods (LS), (e.g., Sethian and
Smereka, 2003) are Eulerian methods widely used in computational fluid dynamics. These
methods are generally associated with the difficulties in handling large topological deformation. Another category belongs to Lagrangian-Eulerian hybrid methods, (e.g., Tryggvason
et al., 2001; Unverdi and Tryggvason, 1992) in which external elements or markers are used to
track the interface explicitly. The main advantage of these methods over the Eulerian counterparts is their ability of tracking interfaces in a sharper manner. However, front tracking
methods are in general computationally more expensive than the Eulerian methods.
Mesh-Free Methods
An alternative to the above-mentioned methods can be purely Lagrangian methods. The
Lagrangian nature makes them potentially better candidates for tracking the interfaces with
large deformations. Being a well-advanced member of Lagrangian methods, the SPH technique due to its Lagrangian nature in particular is an excellent candidate to model complex
multiphase flow problems. It offers a simplified approach for tracking the interface evolution as well as incorporating the surface tension force into the linear momentum balance
equations.
36
2D vs 3D
1.5
3D-VOF
2D-SGN
Experiment
(a) Gauge 1
ζ∗
(d) Gauge 4
0.5
−0.5
1.5
ζ∗
(b) Gauge 2
(e) Gauge 5
(c) Gauge 3
(f) Gauge 6
0.5
−0.5
1.5
ζ∗
0.5
−0.5
−5.0
2.5
t∗
10.0 −5.0
2.5
10.0
t∗
Figure 3.2: Comparison of numerical and experimental √
data presented. Experimental and
∗
three-dimensional simulations are from Mo (2010); t = t/ h/g, and ζ ∗ = ζ/H; (
) present
simulation (2D-SGN), ( ) experimental results and (
) three-dimensional simulation (3DVOF).
Smoothed Particle Hydrodynamics (SPH) is one of the members of meshless Lagrangian particle methods used to solve partial differential equations widely encountered in scientific and
engineering problems (e.g., Fang et al., 2006; Melean et al., 2004; Tartakovsky and Meakin,
2005; Cleary et al., 2002). Unlike Eulerian (mesh-dependent) computational techniques such
as finite difference, finite volume and finite element methods, the SPH method does not re-
2D vs 3D
37
quire a grid, as field derivatives are approximated analytically using a kernel function. In
this technique, the continuum or the global computational domain is represented by a set
of discrete particles. Here, it should be noted that the term particle refers to a macroscopic
part (geometrical position) in the continuum. Each particle carries mass, momentum, energy
and other relevant hydrodynamic properties. These sets of particles are able to describe the
physical behavior of the continuum, and also have the ability to move under the influence
of the internal/external forces applied due to the Lagrangian nature of SPH. Although originally proposed to handle cosmological simulations (Gingold and Monaghan, 1977; Lucy,
1977), SPH has become increasingly generalized to handle many types of fluid and solid
mechanics problems (e.g., Monaghan, 2005b; Sigalotti et al., 2003; Rafiee and Thiagarajan,
2009; Liu et al., 2003; Hosseini et al., 2007; Rook et al., 2007; Tartakovsky et al., 2007). The
SPH method has recently received a great deal of attention for modeling multiphase flow
problems (e.g., Monaghan, 1994b; Monaghan and Kocharyan, 1995; Monaghan et al., 1999;
Landrini et al., 2007; Ferrari et al., 2010; Tartakovsky et al., 2009; Fang et al., 2006, 2009;
Zhang, 2010; Colagrossi and Landrini, 2003) owing to its obvious advantages such that it
notably facilitates the tracking of multiphase interfaces and the incorporation of inter-facial
forces into governing equations, allows for modeling large topological deformations in flow,
and does not require connected grid points for calculating partial differential terms in governing equations. Out of the above cited excellent multiphase SPH studies, works on 2D
simulations of splashing processes and near-shore bore propagation (Landrini et al., 2007),
and a 3D dam breaking simulations (Ferrari et al., 2010) deserve particular mention in that
they have attempted to illustrated the true power and the reliability of the multiphase SPH
method as a viable CFD approach by validating their results with experiments.
38
2D vs 3D
1.5
3D-VOF
2D-SGN
Experiment
(a) Gauge 1
(d) Gauge 4
ζ ∗ 0.5
−0.5
1.5
(b) Gauge 2
(e) Gauge 5
(c) Gauge 3
(f) Gauge 6
ζ ∗ 0.5
−0.5
1.5
ζ ∗ 0.5
−0.5
−5.0
2.5
t
10.0
∗
−5.0
2.5
t
10.0
∗
Figure 3.3: Comparison of numerical and experimental √
data presented. Experimental and
∗
three-dimensional simulations are from Mo (2010); t = t/ h/g, and ζ ∗ = ζ/H; (
) present
simulation (2D-SGN), ( ) experimental results and (
) three-dimensional simulation (3DVOF).
3.1.2
Depth-Integrated Methods
Three-dimensional schemes can represent the flow behavior very accurately. However, the
scale of the geophysical problems we are interested to solve is very large. In addition,
tracking the interface accurately is a challenging problem and requires a very fine resolution,
39
2D vs 3D
especially around the interface, in order to make sure that the conservation of mass is properly
preserved. Depth-integrated equations are a very popular alternative to solving the Euler
equations since these equations increase the computational efficiency significantly. These
class of equations reduce the dimension of the governing equations from three to two by
integrating the velocity along the vertical axis and assuming that the flow acceleration is
zero or negligible along z-axis.
The nonlinear shallow water equations are the most common form of the depth-integrated
equations used in the literature for simulating the long waves such as tsunamis and storms.
These equations assume that the flow velocity is constant along the vertical direction. This
assumption makes them only suitable for waves with very long wavelengths (i.e. λ/h0 >>
1). Different classes of higher-order depth-integrated equations, based on Boussinesq wave
theory, have been derived and presented in literature on simulating shallow water flows:
Wei et al. (1995); Liu (1994); Lannes and Bonneton (2009); Dutykh and Dias (2007); Kirby
(2003); Yamazaki et al. (2009); Ma et al. (2012). These equations take into account the
small variations in velocity along the vertical axis.
3.2
3.2.1
Results
Non-Breaking Solitary Wave Interaction with a Group of
Cylinders
Solitary wave interaction with a group of three vertical cylinders is investigated by Mo and
Liu (2009) in which they performed three-dimensional numerical simulations using volume of
40
2D vs 3D
fluid method (VOF) and compared their numerical results to experiments. We compare our
results from two-dimensional SGN with the experiments and the three-dimensional VOF
√
simulations by Mo and Liu (2009). A solitary wave with H = 0.3h0 , k0 = 3H/4h30 , and
x0 = 15 m inside a rectangular domain of (48.8 m, 26.5 m) interacts with some cylinders.
The diameter of cylinders is 1.22 m and they are located at (27.7 m, 0 m), (30.1 m, 1.2 m)
and (30.1 m, −1.2 m). The still water depth is h0 = 0.75 m. Uniform rectangular grid with
∆x/L0 = 1024 is used where L0 = 48.8 m is the length of the computational domain.
z
x
Figure 3.4: Schematic of the solitary wave run-up on a sloping beach with the slope of
1:19.85.
Table 3.1: Running time for three-dimensional SPH simulation, and one-dimensional SGN
simulation of solitary wave run-up on a sloping beach.
Processing time:
Hardware cost:
3D-SPH
1D-SGN
2D-SGN
≈ 3 day
1 Tesla GPU(≈5000$)
≈ 10 seconds
1 Xeon CPU(≈500$)
≈ 10 minutes
1 Xeon CPU(≈500$)
Figures 3.2, and 3.3 depict the comparison of the wave elevation at different wave gauges
obtained using SGN with those of Mo (2010) for one (only kept the cylinder in (0.0, 0.0) in
Figure 3.1) and three cylinders, respectively. The results are in excellent agreement except
for the wave gauges immediately in front and back of the first cylinder (see Figures 3.2(c-e)
41
2D vs 3D
and 3.3(c-e)). Our model overestimates the leading wave height in front and underpredicts
the wave height in back of the first cylinder. Similar errors are reported between experiments
and Boussinesq models in Zhong and Wang (2009). These differences are due to the fact
that the three-dimensional effects of the flow become more relevant in close proximity to the
cylinders. Using SGN, secondary wave elevations are closer to the experimental data than
the three dimensional simulation. This demonstrates the capability of SGN to simulate the
effects of shorter waves that can become very important in the near shore regions.
0.4
t∗ = 10
t∗ = 15
t∗ = 20
t∗ = 25
t∗ = 35
t∗ = 50
ζ ∗0.2
0.0
0.4
ζ ∗0.2
0.0
0.4
ζ ∗0.2
0.0
−30
−20
−10
x∗
0
10 −30
−20
−10
x∗
0
10
Figure 3.5: Breaking solitary wave run-up compared to experimental results of Synolakis
(1987) and three-dimensional GPUSPH simulation of Marivela et al. (Under review). (
)
present simulation (2D-SGN), ( ) experimental results and (
) GPUSPH.
42
2D vs 3D
3.2.2
Breaking Solitary Wave Run-Up on a Sloping Beach
In this section, we compared the run-up of a breaking solitary wave on a sloping beach to
the experimental results of Synolakis (1987), and to the three-dimensional SPH simulation
presented in Marivela et al. (Under review).
The schematic of the problem is shown in Figure 3.4. The sloping beach’s toe, with the
√
slope of 1:19.85, is located at 3.64 m. A solitary wave with H = 0.28h0 , k0 = 3H/4h30 starts
propagating toward the beach at t = 0 s.
Figure 3.5 compares the obtained results using SGN equations to the experimental data
and the three-dimensional SPH simulations. We can see that our results are closer to the
experimental data when compared to the SPH simulations. Table 3.1 summarizes the computational time obtained using one-dimensional SGN equations to three-dimensional SPH
simulation. We can see that the one-dimensional simulation is, approximately, four orders
of magnitude faster. We note here that two-dimensional SGN simulation will be faster by
only two orders of magnitude.
3.2.3
Breaking Solitary Type Wave Run-Up on a Sloping Beach
We investigated solitary-type transient wave run-up on a steep sloping beach (1:10) following
the experimental setup in Irish et al. (2014). We discretized the computational domain (52
m, 4.4 m) using a rectangular uniform grid with ∆x = ∆y = 52/1024 m. The toe of the
sloping beach is located at x = 32 m. The still water depth is h0 = 0.73 m. The transient
solitary wave is generated by setting H = 0.50 m, k0 = 0.54 m−1 , and x0 = 10 m at t = 0 s. A
43
2D vs 3D
symmetry boundary condition is imposed on bottom and top boundaries.
(a) Gauges 1-2
0.6
(b) Gauges 3-4
0.4
ζ
0.2
0.0
2
6
10
t
14
2
6
10
14
t
Figure 3.6: Free-surface elevation at (a) gauges 1-2, and (b) gauges 3-4. Experimen) present simulation (2D-SGN), ( )
tal results are summarized in Yang et al. (2016); (
experimental results and (
) GPUSPH.
Yang et al. (2016) studied this problem experimentally. We compared the free surface elevation (wave gauges 1-4) and the local water depth (wave gauges 5-16) with the results
presented in Yang et al. (2016) as well as to the three-dimensional simulations obtained
using GPUSPH code. The wave gauge coordinates are summarized in Table 4.1. Figure 4.3
shows the free-surface elevation ζ at gauges 1-4. While both methods capture the evolution
of the free surface accurately, three-dimensional simulations are closer to the experimental
data for the offshore wave gauges.
Figure 4.4 depicts the local water depth at onshore gauges 5-16. Our results are systematically closer to the experimental data. While GPU simulations were closer to the experimental
data, the three-dimensional simulations fail to provide more accurate predictions during the
run-up stage. This is mainly due to the insufficient resolution at the run-up stage imposed
44
2D vs 3D
by the hardware and computational time restrictions.
0.150
(a) Gauge 5
(b) Gauge 6
(c) Gauge 7
(d) Gauge 8
(e) Gauge 9
(f) Gauge 10
(g) Gauge 11
(h) Gauge 12
(i) Gauge 13
(j) Gauge 14
(k) Gauge 15
(l) Gauge 16
11 13 15 17 19
t
11 13 15 17 19
t
h 0.075
0.000
0.150
h 0.075
0.000
0.150
h 0.075
0.000
11 13 15 17 19
t
11 13 15 17 19
t
Figure 3.7: Local water depth at gauges 5-16. Experimental results are summarized in
Yang et al. (2016); (
) present simulation (2D-SGN), ( ) experimental results and (
)
GPUSPH.
Chapter 4
Numerical Simulation of Nonlinear
Long Waves in the Presence of
Discontinuous Coastal Vegetation †
† Citation:
A. Zainali, R. Marivela, R. Weiss, J. L. Irish, Y. Yang, Numerical simulation
of nonlinear long waves interacting with arrays of emergent cylinders; arXiv:1610.00687
[physics.geo-ph].
45
Macro-Roughness Vegetation
46
Abstract
We presented numerical simulation of long waves, interacting with arrays of emergent cylinders inside regularly spaced patches, representing discontinues patchy coastal vegetation. We
employed the fully nonlinear and weakly dispersive Serre-Green-Naghdi equations (SGN)
until the breaking process starts, while we changed the governing equations to nonlinear
shallow water equations (NSW) at the vicinity of the breaking-wave peak and during the
runup stage. We modeled the cylinders as physical boundaries rather than approximating
them as macro-roughness friction. We showed that the cylinders provide protection for the
areas behind them. However they might also cause amplification in local water depth in
those areas. The presented results are extensively validated against the existing numerical
and experimental data. Our results demonstrate the capability and reliability of our model
in simulating wave interaction with emergent cylinders.
4.1
Introduction
Tsunamis, caused by different events such as earthquakes and landslides, and storms pose
significant threat to human life and offshore and coastal infrastructure. The effects of coastal
vegetation on long waves have been investigated extensively in the past (e.g. Mei et al., 2011;
Anderson and Smith, 2014). It has been considered that continuous vegetation provides protection for the areas behind them (e.g. Irtem et al., 2009; Tanaka et al., 2007). However,
limited studies focused on the propagation and run-up of long waves in the presence of discontinuous vegetation (see Irish et al., 2014). The following question arises: Do discontinuous
Macro-Roughness Vegetation
47
arrays of cylinders, representing coastal vegetation such as mangroves, coastal forests, and
man-made infrastructure, act as barriers or as amplifiers for energy coming ashore during
coastal flooding events?
Three dimensional numerical simulation is the superior choice when computational accuracy
is concerned. However, three dimensional models are computationally very expensive, and
the large scale of real-world problems limits their applications in practice. Furthermore, the
vertical component of the flow acceleration is small compared to the horizontal components.
Thus, high fidelity depth integrated formulations, such as nonlinear shallow water equations
(NSW), can be an attractive alternative for practical problems.
NSW have been extensively used for simulating long waves. Due to their conservative and
shock-capturing properties, they represent a suitable approximation of the wave breaking as
well as inundation. However, these equations are only valid for very long waves, and they
cannot properly resolve dispersive effects before wave breaking. Thus, they become inaccurate in predicting the effects of shorter wavelength which is important for simulating the
nearshore wave characteristics. On the other hand Boussinesq-type equations take dispersive effects into account and can be used to simulate the nearshore wave propagation until
the breaking point more accurately. The importance of dispersive effects, until the breaking
process starts, is demonstrated in Figure 4.1. Using SGN the transient leading wave almost
maintains its shape, during propagation over constant water depth. However, if we ignore
the effects of dispersive terms (i.e. by using NSW), after traveling a sufficiently large distance
(for this scenario 80 m) the wave height decreases by the factor of 3.
Different classes of higher-order depth-integrated equations, based on Boussinesq wave the-
48
Macro-Roughness Vegetation
(a) t = 5 s
0.6
ζ
(b) t = 15 s
(c) t = 25 s
0.4
0.2
0.0
0
5 10 15 20 25 30 35 40
x
30 35 40 45 50 55 60 65 70
60 65 70 75 80 85 90 95 100
x
x
Figure 4.1: Free surface elevation of the solitary-type transient wave over constant water
depth at (a) t = 5 s, (b) t = 15 s, and (c) t = 25 s; (
) 1D-SGN, and (
) 1D-NSW. The
following parameters are used: h0 = 0.73 m, H = 0.50 m, k0 = 0.54 m−1 .
ory, have been derived and presented in literature on simulating shallow water flows: Wei
et al. (1995); Liu (1994); Lannes and Bonneton (2009). In this study, we employ fully nonlinear weakly dispersive Boussinesq equations, also known as Serre-Green-Naghdi equations
(SGN, Lannes and Bonneton, 2009). These equations have two advantages compared to others: (a) They only have spatial derivatives up to second order while other equations usually
include spatial derivatives of third order, and (b) they do not have any additional source
therm in the conservation of mass. These properties improve the computational stability
and robustness of the model. Furthermore, unlike earlier studies in which vegetation effects
are approximated by an ad-hoc bottom friction coefficient (e.g. Yang et al., 2016; Mei et al.,
2011), we model the cylinders as physical boundaries. This enables us to simulate the wave
propagation through discontinuous arrays of cylinders without making any assumptions. We
demonstrate the capability of the SGN equations in simulating of the waves interacting with
both offshore and coastal structures while sustaining the computational performance.
Macro-Roughness Vegetation
4.2
49
Theoretical Background
We are interested in simulating long waves; thus µ = (h0 /λ)2 << 1, where λ denotes the
wavelength, and h0 is the still water depth. Assuming an inviscid and irrotationl flow (in
the vertical direction), expanding the Euler equations into an asymptotic series and keeping
terms up to O(µ2 ), SGN equations can be written in the form of
∂t h + ∇ ⋅ (hu) = 0,
1
{ ∂t (hu) + ∇ ⋅ (hu ⊗ u + gh2 I) = −gh∇b + D,
2
(4.1a)
(4.1b)
where D is a nonlinear function of the free-surface elevation ζ, the depth-integrated velocity
vector u, and their spacial derivatives; h is the water depth; b represents the bottom variations; and I is the identity tensor. See Bonneton et al. (2011) and Lannes and Bonneton
(2009) for the complete form of the equations and their derivation. Note that by setting
D = 0, Eqs. (4.1a)-(4.1b) will reduce to the NSW equations (accurate to O(µ)).
We employ the Basilisk code to solve the governing equations (Basilisk, URL: basilisk.fr,
Popinet, 2015). The computational domain is discretized into a rectangular grid and solved
using the second order accurate scheme in time and space. We refer to Popinet (2015) for
more information about the numerical scheme. Friction effects become important as the
wave approaches the coast. Following Bonneton et al. (2011), we added a quadratic friction
term f = cf h1 ∣u∣u to Eq. (4.1b) where the friction coefficient is cf = 0.0034.
50
Macro-Roughness Vegetation
(III)
t2
t1
(II)
t
(I)
t0
z
t1 t2
x
Figure 4.2: Schematic of the breaking process. The vertical dashed lines indicate the
boundary of subdomains. The governing equations in the left subdomain are SGN and NSW
equations elsewhere. The boundary follows the leading wave; (I) t < t1 ∶ SGN in the whole
domain, (II) t1 < t < t2 ∶ SGN in the left subdomain and SW in the right subdomain, and
(III) t > t2 ∶ NSW in the whole domain.
4.2.1
Initial and Boundary Conditions
We imposed the initial conditions
ζ (x) ∣t=0 = H sech2 [k0 (x − x0 )] , u(x)∣t=0 =
cζ
,
ζ + h0
(4.2)
to generate solitary-type transient long waves (Madsen et al., 2008; Madsen and Schaeffer,
√
2010), in which c = g(h0 + H) is the phase speed. H represents the initial wave height, c
denotes the phase speed, and k0 is the characteristic wave number. This wave type removes
the constraint that exists between the wave height and the wave length and will result in
a more realistic representation of tsunami waves (e.g. Baldock et al., 2009; Rueben et al.,
√
2011; Irish et al., 2014) By choosing k0 = 3H/4h30 , Eq. (4.2) will reduce to the free-surface
elevation of a solitary wave with a permanent shape.
51
Macro-Roughness Vegetation
Table 4.1: Wave gauge coordinates.
Gauge
Gauge
Gauge
Gauge
Gauge
Gauge
4.2.2
Location (m)
1:
(18.24, 0.00)
4:
(36.42, 0.00)
7:
(44.34, 0.83)
10: (49.52, 1.10)
13: (42.90, 0.00)
16: (46.20, 1.10)
Location (m)
2:
(24.43, 0.00)
5:
(46.75, 0.55)
8:
(48.40, 2.18)
11: (47.30, 2.18)
14: (45.10, 0.00)
3:
6:
9:
12:
15:
Location (m)
(29.28, 0.00)
(45.10, 2.18)
(45.65, 0.60)
(47.30, 0.00)
(47.85, 1.65)
Wave Breaking
We utilize the conditions suggested in Tissier et al. (2012) to estimate the start time of the
breaking process. As noted earlier, NSW are employed as governing equations in the vicinity
of the breaking-wave peak while modeling the rest of the domain using SGN. Here, we divide
the computational domain into two sub-domains. The boundary that separates these two
sub-domains follows the leading breaking wave (see Figure 4.2). We solve the domain behind
the wave peak using SGN and the rest of the domain using NSW (t1 < t < t2 in Figure 4.2).
The distance between the wave peak and the boundary is given by xb − xw = 2h∣xw where xb
is the coordinate of the boundary and xw is the coordinate of the wave peak.
Note that in the very shallow areas, i.e., in the vicinity of the shoreline and the coastal areas,
we always model the flow using NSW equations to avoid the numerical dispersions that can
be caused by dispersive terms in SGN. In this study, we considered the run-up of a single
wave. After the wave breaks completely (t > t2 in Fig. 4.2), to increase the computational
efficiency, we model the entire domain using NSW.
52
Macro-Roughness Vegetation
(a) Gauges 1-2
0.6
(b) Gauges 3-4
0.4
ζ
0.2
0.0
2
6
10
14
2
t
6
10
14
t
Figure 4.3: Free-surface elevation at (a) gauges 1-2, and (b) gauges 3-4. Experimental
) present simulation
and COULWAVE results are summarized in Yang et al. (2016); (
(2D-SGN), ( ) experimental results and (
) COULWAVE.
4.3
4.3.1
Results
Validation of Numerical Results
Breaking Solitary-Type Transient Wave Run-Up
We investigated solitary-type transient wave run-up on a steep sloping beach (1:10) following
the experimental setup in Irish et al. (2014). We discretized the computational domain (52
m, 4.4 m) using a rectangular uniform grid with ∆x = ∆y = 52/1024 m. The toe of the
sloping beach is located at x = 32 m. The still water depth is h0 = 0.73 m. The transient
solitary wave is generated by setting H = 0.50 m, k0 = 0.54 m−1 , and x0 = 10 m at t = 0 s. A
symmetry boundary condition is imposed on bottom and top boundaries.
53
Macro-Roughness Vegetation
0.150
(a) Gauge 5
(b) Gauge 6
(c) Gauge 7
(d) Gauge 8
(e) Gauge 9
(f) Gauge 10
(g) Gauge 11
(h) Gauge 12
(i) Gauge 13
(j) Gauge 14
(k) Gauge 15
(l) Gauge 16
11 13 15 17 19
t
11 13 15 17 19
t
h 0.075
0.000
0.150
h 0.075
0.000
0.150
h 0.075
0.000
11 13 15 17 19
t
11 13 15 17 19
t
Figure 4.4: Local water depth at gauges 5-16. Experimental and COULWAVE results
are summarized in Yang et al. (2016); (
) present simulation (2D-SGN), ( ) experimental
) COULWAVE.
results and (
Yang et al. (2016) studied this problem experimentally as well as numerically using COULWAVE (Cornell University Long and Intermediate Wave Modeling Package, Lynett et al.,
2002) which solves the Boussinesq equations described in Liu (1994) and Lynett et al. (2002).
We compared the free surface elevation (wave gauges 1-4) and the local water depth (wave
gauges 5-16) with the results presented in Yang et al. (2016). The wave gauge coordinates
are summarized in Table 4.1. Figure 4.3 shows the free-surface elevation ζ at gauges 1-4.
While both methods capture the evolution of the free surface accurately, we observe that
COULWAVE captures the breaking process slightly better. This can be due to the differ-
54
Macro-Roughness Vegetation
ences in the how wave breaking is handled. COULWAVE employs an ad-hoc viscosity model
in the breaking process. In contrast, we solve NSW in the breaking zone and incorporate a
shock-capturing scheme in the breaking process. Consequently, our model develops a steeper
wave profile.
Dp
dp
y
dp
dc
dc
x
Figure 4.5: Sketch of the macro-roughness patches.
Figure 4.4 depicts the local water depth at onshore gauges 5-16. Our results are systematically closer to the experimental data. (Park et al., 2013) also reported about overpredicting
the experimental results using COULWAVE. Yang et al. (2016) argued that the difference
between the simulation and the experiment can be due to the existence of an slight leakage from the sloping beach among some other possible reasons. This can explain the slight
overpredictions we observed using our model.
4.3.2
Breaking Solitary-Type Transient Wave Run-Up in the Presence of Macro-Roughness
A schematic sketch of the problem is shown in Figure 4.5. The computational domain,
boundary conditions, and the initial conditions are the same as the ones in section 4.3.1.
Three different scenarios are considered here. Each macro-roughness patch consists of regu-
55
Macro-Roughness Vegetation
Table 4.2: Geometrical parameters of macro-roughness patches. dr ∶ distance between two
horizontally (or vertically) aligned cylinders inside a patch; Ncp ∶ total number of cylinders
inside a patch; dp ∶ distance between two horizontally (or vertically) aligned patches; Cf p ∶
coordinate of the center of the first patch; Dc ∶ diameter of the cylinders; Dp ∶ diameter of
the patches.
Scenario 1
Scenario 2
Scenario 3
dr (m)
0.1885
0.0943
0.1885
Ncp
21
69
129
dp (m) Cf p (m) Dc (m) Dp (m)
2.2
(34, 0) 0.01333
0.6
2.2
(34, 0) 0.01333
0.6
2.2
(34, 0) 0.01333
0.6
larly spaced vertical cylinders. The geometrical parameters of the patches for each scenario
are summarized in Table 4.2. We used a nested mesh with
∆x = ∆y = 52/1024 m
∆x = ∆y = 52/8192 m
x < 41 m,
x > 41 m.
Local water depth at gauges 5-16 is shown in Figure 4.6. The simulations are in good
agreement with the experimental data. Some differences in transient wave peaks can be due
to the ensemble-averaged experimental data. Ensemble averaging can smooth out some of
the sharp transitions as suggested by Yang et al. (2016) and Baldock et al. (2009)
56
Macro-Roughness Vegetation
0.150
(a) Gauge 5
(b) Gauge 6
(c) Gauge 7
(d) Gauge 8
(e) Gauge 9
(f) Gauge 10
(g) Gauge 11
(h) Gauge 12
(i) Gauge 13
(j) Gauge 14
(k) Gauge 15
(l) Gauge 16
11 13 15 17 19
t
11 13 15 17 19
t
h 0.075
0.000
0.150
h 0.075
0.000
0.150
h 0.075
0.000
11 13 15 17 19
t
11 13 15 17 19
t
Figure 4.6: Local water depth at gauges 5-16 for Scenario 3. Experimental results are
summarized in Irish et al. (2014); (
) present simulation (2D-SGN), and ( ) experimental
results.
4.3.3
Effects of Macro-Roughness on the Local Maximum Local
Water Depth
The maximum local water depth is given by
hmax =
max(h) − max(h)∣ref
,
max(h)∣ref
(4.3)
where max(h)∣ref is the maximum water depth in the absence of the macro-roughness
patches. Unlike momentum flux we do observe water depth amplification up to 1.7 times
57
Macro-Roughness Vegetation
behind the first patch in the presence of the macro-roughness patches. When the flow reaches
the first patch the flow refracts away from the center of the patch toward the other patches.
However, the patches located in the second row refract and reflect the water toward the centerline. This process causes the water to amplify behind the first patch (red area in Figure
4.7). As we can see in Figure 4.7(d), no amplification in local water depth occurs behind the
patch in the case were all other patches are removed.
4.3.4
Effects of Macro-Roughness on the Local Maximum Momentum Flux
The momentum flux represents the destructive forces of the incident wave. To study the effects of the macro-roughness patches on momentum flux we defined the maximum normalized
momentum flux as
Fmax =
max(h∣u∣2 ) − max(h∣u∣2 )∣ref
,
max(h∣u∣2 )∣ref
(4.4)
where max(h∣u∣2 )∣ref is the maximum momentum flux in the absence of the macro-roughness
patches. Figure 4.8 shows the Fmax for Scenarios 1-3. We can see that the patches provide
protection for the areas behind them for Scenario 1 (Figure 4.8(a)). With increasing density
of the cylinders in patches, Fmax is decreased. Thus the level of protection against incident
waves increases. However we did not observe any significant changes with further increasing
the density of the cylinders (see Figure 4.8).
Macro-Roughness Vegetation
58
Figure 4.7: Maximum local water depth h∗max for (a) Scenario 1, (b) Scenario 2 (c)
Scenario 3, and (d) Scenario 3 in which all the patches are removed except the first patch.
The maximum water depth for each scenario is normalized with the reference values in the
absence of macro-roughness patches.
4.3.5
Maximum Run-Up
Maximum run-up is another important criteria in determining the effectiveness of the macroroughness patches in mitigating tsunami hazard risks. Figure 4.9 shows the comparison of
the simulation to the experimental results of the bore line propagation for the Scenario 2.
The results are in good agreement. The maximum deviation from the experimental results
Macro-Roughness Vegetation
59
is less than 4%.
∗
Figure 4.8: Maximum momentum flux Fmax
for (a) Scenario 1, (b) Scenario 2, and (c)
Scenario 3. The momentum flux for each scenario is normalized with the reference values in
the absence of macro-roughness patches.
Figure 4.10 shows the bore-line propagation for different Scenarios. The maximum run-up
decreases with increasing the vegetation density inside the macro-roughness patches. For
the very low density vegetations (Fig. 4.10(b)) this reduction is more or less uniform along
the shore. However with increasing the vegetation density (Figure 4.10(c)) we observe more
reduction behind the patches. With further increasing the vegetation density (Figure 4.10(d)
in comparison to Figure 4.10(c)) while the maximum run-up behind the patches continues
to decrease, it increases within the channel between the patches slightly. The reason for this
increase is the level by which the flow is channelized between the patches.
60
Macro-Roughness Vegetation
y
(a)
2.0
1.5
1.0
0.5
0.0
(b)
43
44
45
46
47
48
49
50
x
Figure 4.9: Propagation of bore-lines in the presence of macro-roughness patches (Scenario
2). Irish et al. (2014); (a) experimental results, and (b) present simulation (2D-SGN).
61
Macro-Roughness Vegetation
2.0
(a)
y
1.5
1.0
0.5
0.0
2.0
(b)
y
1.5
1.0
0.5
0.0
2.0
(c)
y
1.5
1.0
0.5
0.0
2.0
(d)
y
1.5
1.0
0.5
0.0
43
44
45
46
47
48
49
50
x
Figure 4.10: Propagation of bore-lines for (a) Scenario with no macro-roughness patches
(b) Scenario 1, (c) Scenario 2 (d) Scenario 3.
4.4
Discussion and Conclusion
We presented numerical simulations of long water waves interacting with emergent cylinders.
We demonstrated that higher order depth integrated equations, such as SGN, are a suitable
Macro-Roughness Vegetation
62
tool to simulate the wave interaction with emergent cylinders accurately except in very close
vicinity of the cylinders. Three-dimensional effects cannot be ignored at the proximity of the
cylinders and we argue that they are the main reason for the existing differences between
results from our model and the experimental data.
Cylinders, representing coastal vegetation, are usually approximated as macro-roughness
friction (e.g. Yang et al., 2016; Mei et al., 2011). However there is no analytical solution
for a correlation between macro-roughness patterns and the implemented friction coefficient.
Thus these models need to be calibrated against available experimental data. However for
scenarios with no experimental prototype these models can become inaccurate. Here, with
further refining the grid in the coastal areas, we modeled these macro-roughness patches as
physical boundaries.
We observed that discontinuous coastal vegetation can provide protection for the areas behind them. Friction forces become dominant when the flow becomes shallower. In addition,
for the areas located on the onshore slope we also have the gravitational force acting in the
negative flow direction. Macro-roughness patches elongate the path of the incident wave,
causing a longer local inundation period. Thus the flow will be subjected to the negative
gravitational and friction force for a longer time and the maximum recorded velocity will
decline in comparison to the scenario where no macro-roughness patches exist. This will
lead to a reduced local momentum flux and increased protection against the destructive
wave force. However, in terms of local maximum water depth this conclusion does not apply.
Even though the decreased local velocity will result in more protection against the wave, we
observed for the studied macro-roughness the maximum local water depth actually increases
Macro-Roughness Vegetation
63
behind the patches. We observed amplifications up to 1.6 times in local water depth. This
will increase the chance of that area being flooded. We note here that the main reason the
Fukushima Daiichi nuclear disaster happened was the fact that the water overtopped the
protecting wall and reached the electric generators (Synolakis and Kânoğlu, 2015).
We demonstrated the capability of the model in analyzing the wave interaction with coastal
structures. However, our study was limited to the emergent cylinders and simple macroroughness patch patterns. Studying submerged coastal vegetations and different patch patterns will be the direction of our future studies.
Chapter 5
Some Examples for Which Dispersive
Effects Can Change the Results
Significantly†
Contribution
The pressure boundary condition for the airburst obtained through personal communication
with Micheal J. Aftosmis, Advanced Supercomputing Division, NASA.
† Citation:
A part of this chapter is in consideration for publication in PNAS:
N. Hoffman, et. al., Novel Integrative Approach Reveals past Mediterranean tsunami events
affecting Gaza to Tel Aviv.
64
Some Applications
5.1
65
Numerical Simulation of Hazard Assessment Generated by Asteroid Impacts on Earth
Probably the collision of Comet Shoemaker-Levy 9, the first direct observation of a comet
collision, with the Jupiter which happened in July 1994, was one of the most important
events that motivated scientists to try and understand the physics behind the collision of
asteroids with the Earth and to find the possible ways that can be used to mitigate the
hazards associated with them. NASA’s Near Earth Object Program1 monitors the asteroids
that orbit close to earth and publishes the probability of them hitting the Earth at SENTERY
web-page ( An Automatic Near-Earth Asteroid Collision Monitoring System)2 .
Assuming an unbiased asteroid impact, given the fact that 70% of the Earth is covered with
water, the probability of an asteroid impact in the ocean is 70%. Thus it is important to
study the propagation of waves generated by such an impact. Most of the asteroids will
explode before reaching the Earth’s surface. However some of the larger ones can have a
physical impact on the Earth’s surface. Smaller asteroids will explode before reaching the
Earth’s surface, while larger asteroids with a diameter approximately equal to 500 m, or
larger, will hit the surface.
For simplicity, in this section we will solve the SGN equations (Eqs. 1.5-1.6) in cylindrical
form which are given by
1
∂t h + ∇ ⋅ (rhu) = 0,
r
1
1
∂t (hu) + ∇ ⋅ (rhu ⊗ u) + ∇ ( gh2 ) = −gh∇b + D + Fp + O(µ2 ),
r
2
1
2
http://neo.jpl.nasa.gov/
http://neo.jpl.nasa.gov/risk/
(5.1)
(5.2)
66
Some Applications
6
Geoclaw
NSW
4
2
ζ [m]
0
−2
−4
−6
−8
0
100
200
300
400
500
600
700
t [sec]
Figure 5.1: Comparison of water surface elevation between our SWE model and GeoClaw
results. Waves are generated by an asteroid with a diameter of 140 m exploding at the
altitude of 10 km. Wave gauges are located at (
) 0.05LD, (
) 0.2LD, (
) 0.5LD,
) 0.8LD, where LD = 111 km.
and (
where the Fp is given by
Fp =
h
∇p.
ρ
(5.3)
Here p represents the pressure distribution at the surface of the water. Assuming the wave
propagation is axisymmetric, Eqs. 1.5-1.6 can be further simplified to:
∂h ∂ (hu)
+
= −hu,
∂t
∂r
∂hu ∂ (hu2 ) 1 ∂ (gh2 )
∂b
+
+
= −gh − hu2 + D + O(µ2 ),
∂t
∂r
2 ∂r
∂r
(5.4)
(5.5)
67
Some Applications
5.1.1
Numerical Simulation of Tsunami Waves Generated by an
Asteroid Explosion near the Ocean Surface
An asteroid with a diameter of 140 m, and density of 2000 kg/m3 , approaching the Earth
with a velocity of 18 km/sec, which explodes at the altitude of 10 km can be approximated
with an air-burst with a total energy equal to 100 Mt. Assuming that about 10% of the
energy goes into radiation this will generate a pressure distribution on the water surface
given by
P (r) = G (1 − 6
R −r
R −r
)exp(−3.8
),
2W
2W
(5.6)
where
G = 96 exp (−4 (
R 2
R 2
) ) + 96 exp (−2 (
))
C
1.8C
(5.7)
R 8
R
+ 5 exp (−2 (
) ) + 34exp(−2 (
)),
5C
12C
R = 0.3915 t, C = 12.74 and W = 30.
1
(a)
(b)
(c)
ζ [m]
0
−1
−2
0
200
Distance [km]
400
0
200
Distance [km]
400
0
200
Distance [km]
400
Figure 5.2: Comparison of water surface elevation obtained using SWE equations (
),
and SGN equations (
) at (a) t = 400 s, (b) t = 800 s, and (c) t = 1200 s. Waves are
generated by an asteroid with a diameter of 140 m exploding at the altitude of 10 km.
68
ζ [m]
Some Applications
0.5
−10 0
0.5
−10 1
1.5
−10 2
ζ [m]
2.5
ζ [m]
−10 3
0.5
−10 0
0.5
−10 1
1.5
−10 2
2.5
(b)
−10 3
0.5
−10 0
0.5
−10 1
1.5
−10 2
2.5
ζ [m]
(a)
(c)
−10 3
0.5
−10 0
0.5
−10 1
1.5
−10 2
2.5
50
(d)
−10 3
100
150
200
Distance [km]
250
300
Figure 5.3: Water surface elevation obtained using SWE equations (
), and SGN equations (
) at (a) t = 1000 s, (b) t = 2000 s, (c) t = 3000 s, and (d) t = 4000 s. Left y-axis
shows the topographical variation in logarithmic scale.
Some Applications
69
Figure 5.1 shows the water surface elevation at different wave gauges caused by the air-burst
in comparison to the results obtained using GeoClaw 3 showing an excellent agreement. Figure 5.2 compares the numerical results obtained using SGN and SWE. There is a significant
difference between the results obtained by these two sets of equations. While one might
argue that SWE produce more conservative results by generating waves with larger amplitude. However, as shown in Figure 5.3, while the wave profile obtained by SWE decays as it
approaches the coast, we observe formation of an undular bore using SGN. In other words,
the wave amplitude obtained by SGN becomes comparable to the wave amplitude obtained
by SWE. We note here that using SWE, we usually observe one leading wave followed by
smaller waves with negligible wave amplitudes. However using SGN, the predicted wave pack
consists of 3-4 leading waves with amplitude of the same order of magnitude as the leading
wave of SWE. This can lead to a significantly larger wave run-up.
5.1.2
Numerical Simulation of Tsunami Waves Generated by an
Asteroid Impact into the Ocean
To model a tsunami wave generated by an asteroid impact into the ocean, we calculate the
water surface elevation at a specific distance from impact center (in this study 20.0 km) using
three-dimensional hydro-code, iSALE (impact-SALE)4 . To generate a wave, we impose the
obtained time series on the left boundary and simulate the wave propagation using SGN
equations. Note that here, we set Fp = 0
3
4
http://www.clawpack.org/geoclaw
Simplified Arbitrary Lagrangian Eulerian: http://www.isale-code.de/redmine/projects/isale/wiki/ISALE/
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Some Applications
200
(a)
ζ [m]
100
0
−100
−200
20
(b)
ζ [m]
10
0
−10
−20
20
(c)
ζ [m]
10
0
−10
−20
20
(d)
ζ [m]
10
0
−10
−20
0
1000
2000
Distance [km]
3000
4000
Figure 5.4: Water surface elevation of a tsunami wave generated by an impact into water
obtained using SGN equations at (a) t = 1000 s, (b) t = 5000 s, (c) t = 10000 s, and (d)
t = 15000 s.
We modeled the Eltanin impact which occurred approximately 2.15 million years ago. The
Eltanin impact occurred due to an impact of an asteroid with a diameter approximately
equal to 750 m (Weiss et al., 2015; Wünnemann and Lange, 2002; Mader, 1998; Shuvalov
71
Some Applications
and Gersonde, 2014). Following Weiss et al. (2015), we assumed the rock is made of basalt
ANEOS with the density of 2700 kg/m3 and hits the water surface with the velocity of u = 12
km/s. Here we assumed a perpendicular impact for simplicity. We validated our results by
comparing the wave elevation at wave gauges located at 25, 30, 35, 40 km from the center of
the impact to the results obtained using iSALE. The depth of the ocean is assumed as 5000
m at the location of impact and the bottom of the ocean is assumed to be covered with 250
m thick sediment layer made of basalt ANEOS.
Maximum Elevation [m]
103
ζ∝
102
1 1.0
x
101
ζ∝
100 2
10
1 1.3
x
103
Distance [km]
Figure 5.5: Maximum wave height as a function of distance from the impact center. (
SWE (
) SGN.
)
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Some Applications
The leading wave of an earthquake generated tsunami usually has larger amplitude. This is
because according to shallow water theory, the phase speed of a long wave is given by c =
√
g(H + a) which means that the waves with larger amplitude will travel among the leading
waves in the wave packet. Figure 5.4 shows the propagation of the tsunami wave generated
by the Eltanin impact. However, we see that, unlike the earthquake generated tsunamis, the
waves with larger wave amplitudes travel slower compared to waves with smaller amplitudes.
This indicates that the wavelength associated with the asteroid generated waves, as expected,
are shorter than the earthquake generated tsunamis.
Figure 5.5 shows the decaying of the maximum wave height versus distance from the center
of impact. Neglecting friction forces for shallow water waves, the total wave energy will
decay proportional to geometrical spreading, i.e. E ∝ f ( 1r ). Since the wave height is related
to wave energy by ζ ∝
√
E. Thus the wave height decay for shallow water waves are given
by
1
ζ ∝ f (√ )
r
(5.8)
However, in reality, earthquake generated waves decay at faster rates. The decay rate for
asteroid generated tsunamis while using SWE as governing equations is given by
1
ζ ∝f( )
r
(5.9)
while, as expected, it decays faster while replacing SWE with SGN, i.e.
ζ ∝f(
1
r1.3
)
(5.10)
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Some Applications
Table 5.1: Simulation parameters of cnoidal waves interacting with a cylinder.
Case1
Case2
Case3
Case4
a
m
λ/h0
0.9
6.
0.99
12.
0.999999 650.
1.0
∞
r/h0
2.67
2.67
2.67
2.67
z
λ
x
h0
x
y
Figure 5.6: Schematic of a cnoidal wave interacting with a cylinder.
5.2
Non-Breaking Cnoidal Wave Interaction with Offshore Cylinders
It is widely believed by researchers that the offshore and coastal structures dissipate wave
energy and act as a barrier against the incident wave. However, recent studies suggest
that these structures, sometimes, can resonate the wave passing through them rather than
providing a protection zone behind them. Hu and Chan (2005) studied the influence of an
array of vertical bottom-mounted cylinders on the propagation of long water waves. They
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Some Applications
observed focusing of water waves in a regions behind the cylinders with a maximum intensity
of 3.8.
Figure 5.7: Contour plots of the maximum water elevation for (a) Case1, (b) Case2, (c)
Case3. Blue lines denote the vertical cross sections at 1: (x − x0 )/h0 = 9.83, 2: (x − x0 )/h0 =
17.65, 3: (x − x0 )/h0 = 41.25.
To generate a cnoidal wave train, we impose the following conditions on the left boundary
(Wiegel, 1960):
√
3a
ζ (t − T ) ∣left = ζ2 + a cn2 [
c(t − T ); m] ,
4mh30
(5.11)
and
u(t − T )∣left =
cζ
,
ζ + h0
(5.12)
where
ζ2 =
a
E(m)
(1 − m −
),
m
K(m)
(5.13)
and
c=
√
gh [1 +
a
1
3 E(m)
(1 − m −
)] .
mh0
2
2 K(m)
(5.14)
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Some Applications
The phase lag T is calculated iteratively such that ζ(0 − T ) = 0. Here K(m) denotes the
complete elliptic integral of the first kind and E(m) represents the complete elliptic integral
of the second kind.
The schematic of the problem is shown in Figure 5.6. The computational domain of 160 m ×
160 m is discretized into a 2048 × 2048 uniform grid. The cylinder of 2 m diameter is located
at the center of the domain. The water depth at rest is h0 = 0.75 m. Simulation parameters
for the cases considered here are summarized in Table 5.1.
1.75
(hu2 )∗
1.25
0.75
0.25
1.75
(hu2 )∗
1.25
0.75
0.25
(hu2 )∗
1.75
1.25
0.75
0.25
−60
0
y∗
60
−60
0
y∗
60
−60
0
y∗
60
−60
0
60
y∗
Figure 5.8: Maximum momentum flux along the vertical cross sections at top: (x−x0 )/h0 =
9.83, middle: (x − x0 )/h0 = 17.65, bottom: (x − x0 )/h0 = 41.25.
Some Applications
76
Figure 5.7 shows the contour plots of the maximum water elevation. We can see that as soon
as the leading wave reaches the cylinder, it generates secondary reflected waves spreading
in a radial direction. This generates elliptically shaped maximum wave height contour plots
spreading out from the cylinder (see the contour pattern outside the region bounded by
the white boundary in Figure 5.7(a-c)). However, inside the region bounded by the white
curves, we observe very complicated patterns. We can distinguish three main mechanisms
that causes the propagation of the wave in this region. The leading wave that tries to retrieve
the original travel path in x−axis direction. The tailing waves with smaller amplitudes and
smaller phase velocities mainly traveling in a radial direction. We also observe the formation
of vorticities behind the cylinder. This causes a very complicated maximum wave height
contours behind the cylinder as we can see in Figure 5.8, where we observe both reduction
and increase in the maximum momentum flux behind the cylinder.
5.3
Hazard Assessment Along the Coastline from the
Gaza Strip to the Caesarea, Israel
In this section, we assess the risk associated with the tsunamis at the Gaza Strip. The
possible triggering sources of the tsunami can be by an earthquake, volcanic eruption or
a landslide in the Mediterranean Sea. Here, we assess the risk associated with all three
possible scenarios. For an earthquake generated tsunami, we assume an earthquake with
an epicenter located at the south-east portion of the Hellenic arc (35.58N, 28.14E) with the
following parameters used in the Okada model:
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Some Applications
Depth = 45 km
Strike = 225o
Dip = 30o
Rake = 90o
Length = 140 km
Width = 60 km
For the volcanic eruption, we model the tsunami caused by Thera eruption (Novikova et al.,
2011; McCoy and Heiken, 2000). Here, we simply approximate the Thera eruption by a
cylindrical dam break centered at (36.5N, 26.E) and
h = 0 if (x − x0 )2 + (y − y0 )2 < 0.022 .
Here, x and y represent the longitude and latitude in degrees. And finally we approximate
a landslide with the total volume of 3 km3 with
ζ = 10.8 sin (
2π
√
).
55 gh0
Figure 5.9 summarizes the tsunami run-up along the Gaza Strip. As we can see the tsunamis
generated by an earthquake impose the most significant threat at the coast followed by
tsunamis generated by a landslide. However, because of the large distance between the
possible origin of the volcanic eruption close to Gaza Strip, the eruption-generated tsunami
imposes no significant threat on the coast along the Gaza Strip. Here we note while we
observed smaller run-up for the tsunami generated by landslide, the observed maximum
wave height was larger for it compared to the tsunami generated by the earthquake. This
can potentially make this tsunami more hazardous for the fishing boats and ships near the
Strip.
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Some Applications
Earthquake
A
Ashkelon
Ashdod
10
Run up [m]
8
Landslide
Volcanic Eruption
Tel Aviv
A0
Caesaria
Gaza City
6
4
2
0
Gaza
0
50
100
150
Distance along coast [km]
Figure 5.9: Tsunami run-up along the coastline from the Gaza Strip to the Caesarea, Israel.
Chapter 6
Future Work
Computational power is increasing very rapidly, thanks to the fast evolving computing hardwares. However, harnessing the underlying hardware potential is becoming difficult as well.
Codes, commonly used in industry, need to be altered significantly in order to make it compatible with the newer computing devices such as Intel MIC due to their design patterns.
While GPU computing can potentially speed-up the simulation by an order of magnitude,
the current limitations with the GPU memories and the necessity to communicate between
the CPU and GPU is the current bottleneck to reach the optimum speed-up. Higher order
numerical schemes can achieve the same numerical accuracy on the coarser meshes. This
makes them a suitable candidate for numerical codes targeting GPUs.
We are developing a software program based on higher order finite difference WENO schemes
(Xing and Shu, 2005; Vukovic and Sopta, 2002) for solving common problems in water waves
and sediment transport that can be easily modified to harness the power of any new hardware
that emerges in the market, while, at the same time, keeping the program easy to maintain
79
Future Work
80
and develop. Python is chosen to create a user-friendly interface. On the other hand,
computationally expensive parts of the algorithm will be translated into low level native
codes targeting clusters of CPUs, NVIDIA GPUs, and Intel MIC. This is accomplished by
leveraging a domain specific language (DSL) derived from the Mako templating engine which
is also based on Python. Using DSL language decouples the code development part from
the hardware specific parts of it, enabling us to easily modify the code according to other
possible hardwares that might emerge in the future.
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