sph numerical modelling of impulse water waves generated ...sph numerical modelling of impulse water...

SPH numerical modelling of impulse water

waves generated by landslides

Thesis submitted for the degree of

Doctor of Philosophy

by

Tatiana Capone

Hydraulic, Transportation and Roads Department

Sapienza University of Rome

Tutor: Prof. Ing. Alberto Noli Co-tutor: Ing. Andrea Panizzo

Ai miei genitori

e alle mie sorelle

ii

Summary

In this thesis landslides generated waves are studied. This kind of ”tsunami” waves

can be very dangerous for the human communities. Starting from results presented

by past studies, that found in SPH (Smoothed Particles Hydrodynamics) numerical

model a good tool to study this kind of free surface problems, this work is intended to

provide a more detailed testing and implementation of the above mentioned numerical

model.

Simply two and three dimensional tests was used in order to test the capability of

our SPH code to represent correctly first and second derivative of a field variable. Af-

ter that two dimensional flows were used to test different kind of viscosity for different

formulations founded in literature.

A detailed study was carry on in order to found a rheological model capable to

represent the interaction between a submarine deformable landslide and the water.

The model was implemented in SPH in a very simple and versatile way, and was tested

through a simple two dimensional test.

Finally the interaction between a mud flow and the water was studied with great

results.

iii

Publications

During the authors candidature, several articles relating to the subject matter of this

thesis were published or submitted for publication:

• Panizzo, A., Capone, T., Darlymple R. A., ”Accuracy of numerical schemes in

SPH”, Journal of Computational Physics, under review.

• Panizzo, A., Capone, T., Longo, D., Del Guzzo, A., ”SPH simulation of bench-

mark case 1”, 1st Spheric Workshop, 10-12 May 2006, University of Rome ”La

Sapienza”.

• Capone, T., Panizzo, A., Cecioni, C., Darlymple, R. A., ”Accuracy of numer-

ical schemes in SPH”, 2nd Spheric Workshop, 23-25 May 2007, Universidad

Politecnica de Madrid, Spain.

• Capone, T., Panizzo, A., ”SPH simulation of non-newtonian mud flows”, 3rd

Spheric Workshop, 4-6 June 2008, Ecole Polytechnique Federale de Lausanne,

Switzerland.

• Capone, T., Kajtar, J., Monaghan, J., ”SPH molecules - A model of granular

materials”, 3rd Spheric Workshop, 4-6 June 2008, Ecole Polytechnique Federale

de Lausanne, Switzerland.

• Panizzo, A., Capone, T., Marrone, S., ”Experiences of SPH with the lid driven

cavity problem”, 3rd Spheric Workshop, 4-6 June 2008, Ecole Polytechnique

Federale de Lausanne, Switzerland.

• Capone, T., Panizzo, A., Monaghan, J., J., ”SPH modelling of water waves gener-

ated by Submarine Deformable Landslides”, ASCE, 31st International Conference

on Coastal Engineering, Hamburg, Germany.

• De Girolamo, P., Cecioni, C., Montagna, F., Capone, T., Briganti, R., Panizzo,

A., Chen, Q., Bellotti, G., Di Risio, M., ”Numerical Modeling of Landslide Gen-

erated Tsunamis Around a Conical Island”, ASCE, 31st International Conference

on Coastal Engineering, Hamburg, Germany.

iv

• Capone, T., Panizzo, A., Monaghan, J., J., ”SPH modelling of water waves

generated by submarine landslides”, Journal of Hydraulic Research, SPH special

session, under publication.

v

Acknowledgements

I would like to thank Prof. Monaghan for all the help and support, as a professor

and as a friend. Thanks to Jenny for the friendship so important in a lot of difficult

moments. Thanks to Jules, without him my English would be even worse than now.

I want to thank also my tutor Professor Noli, I’m very proud to be one of his stu-

dents. Thanks to Professor De Girolamo for all the important suggestions given in

these university years.

A particular thank to Andrea, for the great opportunity to use his SPH code and to

discover the world of particles. Thanks to Sauro for all the happy and funny moments

of this experience.

A particular thank to my family, my parents and my sisters, without all of them

my life simply didn’t have any sense. They remember me every day how lucky I am.

Contents

1 Introduction 1

1.1 Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Aims of the present work . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.3 Thesis Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 Landslides generated waves 5

2.1 Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.2 Submarine Landslides . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.3 Numerical Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3 Smoothed Particles Hydrodynamics 15

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.2 Basic formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3.2.1 Integral Interpolation method . . . . . . . . . . . . . . . . . . . 16

3.2.2 First Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.3 Kernel Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3.3.1 Gradient Kernel Correction . . . . . . . . . . . . . . . . . . . . 19

3.4 Fluid Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.4.1 Continuity Equation . . . . . . . . . . . . . . . . . . . . . . . . 21

3.4.2 Momentum Equation . . . . . . . . . . . . . . . . . . . . . . . . 22

3.4.3 Equation of State . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.4.4 Variable smoothing length . . . . . . . . . . . . . . . . . . . . . 23

3.5 Viscosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.6 Timestepping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

viii CONTENTS

3.6.1 Verlet Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.6.2 Velocity Verlet Algorithm . . . . . . . . . . . . . . . . . . . . . 27

3.6.3 Beeman Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.6.4 Courant Condition . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.7 Numerical recipes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.7.1 Interpolation Schemes . . . . . . . . . . . . . . . . . . . . . . . 30

3.7.2 XSPH Correction . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.7.3 Correction of the tensile instability . . . . . . . . . . . . . . . . 31

3.8 Boundaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

3.8.1 Repulsive Forces . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.8.2 Ghost Particles . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.9 Computational Strategies . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.10 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

4 Test Cases 39

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

4.2 SPH accuracy from interpolation and time integration schemes . . . . . 40

4.2.1 Two dimensional test . . . . . . . . . . . . . . . . . . . . . . . . 40

4.2.2 Three dimensional test . . . . . . . . . . . . . . . . . . . . . . . 45

4.3 SPH evaluation of second derivative of a field variable . . . . . . . . . . 49

4.3.1 Periodic Shear Flow . . . . . . . . . . . . . . . . . . . . . . . . 50

4.3.2 Couette Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

4.3.3 Poiseuille Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

4.4 SPH Benchmark Test Cases . . . . . . . . . . . . . . . . . . . . . . . . 57

4.4.1 Wave Impact on a Tall Structure . . . . . . . . . . . . . . . . . 57

4.4.2 Landslides Generated Waves . . . . . . . . . . . . . . . . . . . . 65

4.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

5 Non Newtonian Mud Flows with SPH 73

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

5.2 Non-Newtonian fluids . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

5.3 Rheological Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

5.4 Rheological Model in SPH . . . . . . . . . . . . . . . . . . . . . . . . . 80

5.5 Governing Equations of Our SPH Model . . . . . . . . . . . . . . . . . 80

CONTENTS ix

5.6 Numerical Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

5.6.1 Annular Viscometer . . . . . . . . . . . . . . . . . . . . . . . . . 82

5.6.2 Dam Break . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

5.7 Numerical Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

5.7.1 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

5.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

6 Conclusions and Future Developments 105

A Annular Viscometer 109

B Invariants 111

C Deformation Rate 115

x CONTENTS

List of Figures

2.1 Principal phases in the phenomenon of subaerial landslide generated

waves. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.2 Different types of impulse waves defined as function of the landslide vol-

ume, represented by the dimensionless parameter λ/d, and the landslide

velocity, represented by the Froud number Fr (picture taken from Noda,

1970). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.3 Three different main cases defined as a function of the ratio between the

volume of the landslide and that of the water body. . . . . . . . . . . . 8

2.4 Generation mechanism of submarine landslides generated waves. . . . . 8

2.5 Classification of submarine mass movements adapted from sub-aerial

classification proposed by the ISSMGE Technical Committee on Land-

slides (TC-11) (Locat and Lee, 2000). . . . . . . . . . . . . . . . . . . . 9

3.1 Kernel functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.2 Ghost Particles as boundaries. . . . . . . . . . . . . . . . . . . . . . . 34

3.3 Particle interaction in the SPH model: neighbors particles of a given

particle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.4 Particle grid cells in the fluid domain (on the left) and linked list through

neighbors grid cells (on the right). . . . . . . . . . . . . . . . . . . . . . 35

3.5 Reduced linked list algorithm over five grid cells. . . . . . . . . . . . . . 36

3.6 Oversized neighbors lists. . . . . . . . . . . . . . . . . . . . . . . . . . . 37

xii LIST OF FIGURES

4.1 Two dimensional tests. Dots refer to the analytical solution, the con-

tinuous line reports the SPH results. Left panel: best obtained result,

using a QU kernel with mixed gradient and kernel correction, and the

Beeman algorithm (cumulated error = 0.5159 m). Right panel: worst

obtained result, using a non corrected GJ kernel and the Euler algorithm

(cumulated error = 1295.20 m). . . . . . . . . . . . . . . . . . . . . . . 41

4.2 Two dimensional test results showing errors and computational time

associated with each integration algorithm and kernel combination. The

kernels are GJ: Gordon Johnson; CU: Cubic spline; QU: Quintic spline;

WE: Wendland; GA: truncated Gaussian. . . . . . . . . . . . . . . . . . 43

4.3 Two dimensional test results showing errors and computational time

associated with each kernel and integration algorithm combination. The

integration algorithms are Bee: Beeman; MBee: modified Beeman; Eul:

Euler; Ver: Verlet; VVer: velocity Verlet; TVVer: two step velocity Verlet. 44

4.4 Three dimensional test Noeud de trefle. The continuous line refers to

the analytical solution, dots report the SPH results. Left panel: best ob-

tained result, using a QU kernel with mixed gradient and kernel correc-

tion, and the Beeman algorithm (cumulated error = 15.0204 m). Right

panel: worst obtained result, using a non corrected CU kernel and the

Euler algorithm (cumulated error = 279.99 m). . . . . . . . . . . . . . 46

4.5 Three dimensional test results. . . . . . . . . . . . . . . . . . . . . . . . 47

4.6 Three dimensional test results. . . . . . . . . . . . . . . . . . . . . . . . 48

4.7 Periodic flow. Particles positions (center panel), density (left panel) and

velocity field (right panel), as a function of y. t=2.0 s. . . . . . . . . . 52

4.8 Periodic flow modelled using the artificial viscosity. Second derivative

of the velocity field as a function of y at t=2.0 s. Dots report the

SPH solution, the continuous line reports the analytical one. Left panel:

Watkins method. Center panel: Morris method. Right panel: Takeda

method. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

4.9 Periodic flow modeled using NSE. Second derivative of the velocity field

as a function of y at t=2.0 s. Left panel: Watkins method. Center panel:

Morris method. Right panel: Takeda method. . . . . . . . . . . . . . . 53

LIST OF FIGURES xiii

4.10 Couette flow. Comparison of SPH (dots) and analytical (line) solutions

using the Morris (left panel), Takeda (right panel) and Monaghan (center

panel) methods. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

4.11 Comparison of SPH (dots) and analytical (line) solutions for Poiseuille

flow using Morris (left panel), Takeda (right panel) and Monaghan (cen-

ter panel) methods. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

4.12 Sketch of the experimental setup. . . . . . . . . . . . . . . . . . . . . . 58

4.13 Position Particles at the beginning of the simulation. Top panel for a

lateral view, bottom panel for the perspective one. . . . . . . . . . . . . 59

4.14 Particles positions at the beginning of the simulation. Ghost (red cir-

cles), Repulsive (red dots) and Fluid Particles (blue dots). . . . . . . . 60

4.15 Experimental results in term of Force exerted on the structure (top

panel) and velocity in front of it (bottom panel). . . . . . . . . . . . . 61

4.16 Particles positions at four different instants of the simulation. . . . . . 62

4.17 Comparison with the experimental results. Top plot for the force on the

structure and bottom one for the velocity. . . . . . . . . . . . . . . . . 63

4.18 Schematic view of the physical model. . . . . . . . . . . . . . . . . . . 65

4.19 Picture of the experimental model. . . . . . . . . . . . . . . . . . . . . 66

4.20 Picture and dimension of the physical model of the landslide. . . . . . 67

4.21 Sketch of the water waves field generated by a landslide. . . . . . . . . 67

4.22 Picture during a laboratory experiments of the water wave field gener-

ated by a landslide. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

4.23 Particles positions at four different instants of the simulation. . . . . . 69

4.24 Comparison between experimental (blue line) and numerical (red line)

water surface elevation. . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

4.25 Comparison analysis of the generated edge waves. . . . . . . . . . . . . 70

xiv LIST OF FIGURES

5.1 Typical behaviour of a non-Newtonian liquid showing the interrelation

between the different parameters. The same experimental data are used

in each curve. (a) Viscosity versus shear stress. Notice how fast the

viscosity changes with shear stress in the middle of the graph; (b) Shear

stress versus shear rate. Notice that, in the middle of the graph, the

stress changes very slowly with increasing shear rate. The dotted line

represents ideal yield-stress (or Bingham plastic) behaviour; (c) Viscos-

ity versus shear rate. Notice the wide range of shear rates needed to

traverse the entire flow curve. . . . . . . . . . . . . . . . . . . . . . . . 75

5.2 Constitutive Laws (from Vola et al. (2004)). . . . . . . . . . . . . . . . 78

5.3 Flow curves for a synthetic latex (taken from Barnes and Walters (1985)):

(a and b) Bingham plots over two different ranges of shear rate, showing

two different intercepts; (c) Semi-logarithmic plot of data obtained at

much lower shear rates, showing yet another intercept; (d) Logarithmic

plot of data at the lowest obtainable shear rates, showing no yield-stress

behavior; (e) The whole of the experimental data plotted as viscosity

versus shear rate on logarithmic scales. . . . . . . . . . . . . . . . . . 79

5.4 Geometrical domain for the annular viscometer test. . . . . . . . . . . 83

5.5 Analytical solutions. Line: classical Bingham model; dots: our Bing-

ham model with alpha=50; stars: our Bingham model with alpha=100;

circles: our Bingham model with alpha=1000. . . . . . . . . . . . . . . 84

5.6 Particles positions (top plot) and tangential velocity (bottom plot) for

the simulation with the parameter α equal to 50. . . . . . . . . . . . . 86


the simulation with the parameter α equal to 100. . . . . . . . . . . . . 87


the simulation with the parameter α equal to 1000. . . . . . . . . . . . 88


the simulation considering a newtonian fluid with the variable h scheme. 89

5.10 Comparison between tangential velocity for the simulations considering

a newtonian fluid with the variable h scheme with 4074 particles (top

plot) or without with 7068 (bottom plot). . . . . . . . . . . . . . . . . 90

LIST OF FIGURES xv


the simulation with the parameter α equal to 50 and the variable h scheme. 91

5.12 Geometrical domain for the dambreak test. . . . . . . . . . . . . . . . 92





5.15 Comparison with experimental results. Red Circles: experimental re-

sults. Blue Dots: SPH solution. . . . . . . . . . . . . . . . . . . . . . . 95

5.16 Geometrical domain for the experiment by Rzadkiewicz et al. (1997). . 96

5.17 Particles Positions at the beginning of the simulation. . . . . . . . . . . 97

5.18 Particles Positions at time t=0.4s. . . . . . . . . . . . . . . . . . . . . . 98

5.19 Particles Positions at time t=0.8s. . . . . . . . . . . . . . . . . . . . . . 99

5.20 Comparison between experimental (red circles) and numerical (blue lines)

solutions in terms of elevation of the free surface. . . . . . . . . . . . . 100

5.21 Comparison between slide profile calculated with SPH (brown dots) and

calculated with Rzadkiewicz et al. (1997) model (black circles), at time

t=0.4 s. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

5.22 Comparison between slide profile calculated with SPH (brown dots) and

calculated with Rzadkiewicz et al. (1997) model (black circles), at time

t=0.8 s. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

A.1 Sketch of a coaxial cylinder viscometer. . . . . . . . . . . . . . . . . . 109

List of Notation

Symbols

v Velocity vector

x Position vector

a Acceleration vector

p Pressure

c Speed of sound

ρ Density

m Mass

V Volume

γ Equation of state constant

Re Reynolds number

µ Dynamic viscosity

ν Kinematic viscosity

D Rate of deformation tensor

|D| Second invariant of the rate of deformation tensor

Chapter 1

Introduction

1.1 Preface

The term Tsunami comes from a Japanese word that means ”harbor” (tsu) and ”wave”

(nami). The translation of the word tsunami used to be tidal waves. However in the

international scientific community this word refers to the impulse waves generated by

sudden movements of the ocean floor. Usually this kind of movements are triggered by

earthquakes, volcanic eruptions or underwater explosions. Tsunamis can be considered

as transition phenomena because of their impulsive origins, they are characterized by

a long wave length (hundreds of kilometers) and period (from minutes to hours). They

can travel for thousand of kilometers across open ocean at speeds of 600–800 km per

hour, and the effects can be seen hours later on shores. As a tsunami approaches to

the coast, it reduces the wave celerity and increases the wave height, reaching values

of tens of meters, thus having a very high power of destruction.

There are many events in the past that occurred in the Pacific Ocean such as the

one on December 26th in 2004 that caused hundreds of thousands of deaths. However

the highest tsunami wave, ever registered, occurred in Alaska in 1958, in Lituya Bay,

and it was not caused by an earthquake itself or a volcanic eruption but by the falling

of a great landslides generated by the earthquake. The giant wave was of hundreds of

meters and this kind of tsunami are known as megatsunamis. The generation of these

great events it’s always closer to the shore and it’s related to little basin or reservoirs.

2 Introduction

Waves generated by landslides are a particular type of tsunamis waves. They can

occur in artificial reservoirs (Vajont valley, Italy, 1963) or natural lakes, or they are

triggered by landslides on sea shorelines (Lituya Bay, Alaska, 1958; Stromboli volcano,

Italy, 2002). All these events were interested by subaerial landslides and in this case,

differently from tsunamis waves generated by underwater mass movements, splashes

and complex three dimensional water flows are present close to the impact area. Then,

the perturbation travels for long distances if in oceanic areas, causing disasters far

away from the generation area, producing high wave runups on shorelines. In the case

of artificial reservoirs the most dramatic consequences are the dam overtopping and

the seiching waves of the basin. Sometimes landslide events in artificial reservoirs may

be triggered by the filling up of the artificial reservoir itself, by virtue of the induced

change in watershed. At present, in artificial basins where landslide risk exists, the wa-

ter is kept well below the maximum possible level to minimize risk. As a consequence

the dam and the basin potentialities are under–utilized, causing great economical losses.

The understanding and the forecasting of landslide generated waves is important

both for the safety of people and properties which are close to the reservoir or along sea

shorelines, and for artificial reservoir and dam management. Past studies carried out on

this topic presented mathematical theories (Stoker (1957); Prins (1958); Kranzer and

Keller (1959); Pezzoli (1965, 1972); Le Mehaute and Wang (1996)), physical model

experiments (Wiegel (1955); Wiegel et al. (1970); Kamphuis and Bowering (1972);

Huber and Hager (1997); Watts et al. (2003); Panizzo et al. (2005)) and numerical

simulations (Heinrich (1992); Watts (1997); Monaghan and Kos (2000); Monaghan

et al. (2003)). However at present time no general method is available to accurately

predict landslides generated waves. The aim of the present work is to give new insights

in the characterization of landslide generated waves, using new numerical tools.

1.2 Aims of the present work

The present work deals with the complex problem of numerical modelling landslides

generated waves. Despite the huge amount of work on the subject, there are many

aspects that are not yet completely solved, for example the development of an early

warning system capable to alert people in time. The complexity of the problem lies

1.3 Thesis Structure 3

on the mechanism and the physics of the phenomenon that is fully three dimensional

in the near field. Moreover the dynamical description of the landslide is very compli-

cated to simulate as well as the interaction with the water. The aim of this work is to

implement and apply a numerical model able to provide detailed simulations of water

waves generated by landslides. The capability of this model to represent the dynamic

of the landslide introducing a rheological model is also investigated.

Thus only the first simulations are conducted considering the landslide as a rigid body

and using Smoothed Particle Hydrodynamics (SPH) numerical model, after that SPH

is used to simulate the landslide as a deformable body, testing and applying a partic-

ular rheological model. Finally 2D simulation of deformable landslides falling into the

water are performed.

The choice of this particular fully lagrangian model is due to the possibility of simu-

lating generation, propagation and runup of the impulse waves, without any particular

constraint on the free surface.

1.3 Thesis Structure

The following PhD thesis is structured as follows. In Chapter 2 an overview on the

landslides generated waves is given, underlining in particular the physics of the phe-

nomenon and the actual experimental and numerical tools present in literature.

Chapter 3 is an introduction on the Smoothed Particles Hydrodynamics numerical

model and gives us both the basics of the code and the particular numerical tools used

in order to improve computational strategies.

In Chapter 4 a series of tests are shown, they are both taken from literature and im-

plemented for the first time and from 2D to 3D code.

Chapter 5 is a brief resume of the rheological models present in literature with special

regards to the ones that are more widespread in the debris flows simulations. There

is also the introduction of a rheological model in SPH tested with a 2D annular vis-

cometer test and consequently an application on the waves generated by a submarine

landslide following in particular experimental results given by Rzadkiewicz et al. (1997).

4 Introduction

Chapter 2

Landslides generated waves

2.1 Physics

The generation of impulse waves due to the impact of a subaerial landslide with water

is a complex phenomenon, involving several physical aspects. Huber and Hager (1997)

proposed a simplification of the considered phenomenon, individuating four distinct

phases. With reference to Figure 2.1, at the beginning (circle 1 in the Figure) the

landslide starts moving, accelerates and then falls into water. The characterization

and the forecasting of a landslide mechanism is a very complex task, involving lots of

different aspects and physical variables. The study of this part of the impulse waves

generation process lies in the scientific field of geology and soil mechanics, and is not

going to be treated in the present work. Step 2 sketches the impact of the landslide

with water. This part of the process is at the bases of waves generation, due to the en-

ergy exchange mechanism between the landslide and the water. The principal landslide

parameters, such as its volume, impact velocity, density, porosity, shape of the front,

slope inclination angle, influence the features of the subsequent water wave motion.

In phase 3 impulse waves propagate in the reservoir or in the open sea, presenting

a wave energy dispersion which is both longitudinal and directional. Waves features

change as a function of the water depth, and refraction, diffraction and shoaling may

occur. Finally, phase 4 is related to the impulse wave interaction with shorelines or

dams. The impulse wave runup may cause the flood of coastal areas, and, in the case

of an artificial reservoir, the dam can be overtopped thus flooding downstream areas.

Usually, seiches waves of the artificial water basin are also triggered.

6 Landslides generated waves

Figure 2.1: Principal phases in the phenomenon of subaerial landslide generated waves.

As far as the generated water waves are concerned, they may present very different

shapes and dispersive features. Prins (1958), Wiegel et al. (1970) and Noda (1970)

performed several physical experiments generating impulse waves by the falling of a

solid block in a two dimensional wave flume. They concluded that, depending on the

local water depth, the energy exchange between the landslide and the water, and the

landslide volume, impact waves present different characteristics. Their experimental

observation can be summarized by Figure 2.2, which presents a map of different wave

types observed during impulse waves generation due to the vertical fall of a box (λ

is the box width, d is the local water depth, Fr = v/√

gd is the dimensionless box

falling velocity). The typical time series of water surface elevation at a given point are

represented on the right part of Figure 2.2. Basically four types of impact waves were

observed by these authors: (A) leading wave with oscillatory wave characteristics, (B–

C) leading wave with solitary wave characteristics, followed by a trough connecting it

with the dispersive wave pattern, (D–E) leading wave being a single wave with solitary

wave characteristics, separated by the dispersive wave pattern, (F) solitary wave with

complex form (bore in the first stage). As a general rule the generated waves type vary

from (A) to (F) gradually as the values of λ/d and Fr increase. When the dimensions

2.1 Physics 7

of the falling body are large in comparison to water depth, solitary waves are to be

expected, vice versa a train of dispersive waves is likely to be generated in relatively

deep water.

Figure 2.2: Different types of impulse waves defined as function of the landslide volume,

represented by the dimensionless parameter λ/d, and the landslide velocity, represented

by the Froud number Fr (picture taken from Noda, 1970).

In view of the above observations, the characterization of subaerial landslide gen-

erated water waves has to be carried out making simplifying assumptions about the

landslide principal parameters. Referring to Watts’ works (Watts, 1997, 1998, 2000;

Watts et al., 2001, 2003), the water wave field can be recognized to change from the near

field, where the water motion is complex and three dimensional, presenting splashes

and bores, to the far field, where the water motion is dominated by propagating water

waves, which can be classified within one of the four introduced types. Some further

considerations can be made on the expected general aspects of the phenomenon con-

sidering the ratio between the volume of the landslide and that of the water body

where the impulse flow takes place. Basically, three different cases may be recognized

to exist. The first case (see panel A of Figure 2.3) is the case of a large landslide

falling into a small reservoir: in this limit case the far-field may not exist. The second,

intermediate case (panel B of Figure 2.3) consists of near and far-field of compara-

ble extents, i.e. their dimensions having the same order of magnitude. In the last

case (panel C on Figure 2.3) the far-field extension is several times the extension of the


near-field. This is the case of small landslides falling into large reservoirs or into the sea.

A B C

Figure 2.3: Three different main cases defined as a function of the ratio between the

volume of the landslide and that of the water body.

Another important aspect to be considered is the dynamic of the landslide itself,

and not only the interaction between the landslide and the water. As one can imagine

the phenomenon is very complex and its complexity lies also in the heterogeneity of

the events that can occur.

Figure 2.4: Generation mechanism of submarine landslides generated waves.

2.2 Submarine Landslides 9

2.2 Submarine Landslides

Up till now we only talk about subaerial landslide, but, because of their tsunamigenic

potential, submarine landslides such as ocean-island flank collapses and rockslides in

fjords have been identified as the most dangerous of all landslide related hazards (Mas-

son et al., 2006). Landslide, in this case, is used as a generic term encompassing all

forms of slope failure, irrespective of process. Other terms can be used, including slide,

debris flow, debris avalanche and turbidity current, each imply a particular process as

defined below:

- Slide: movement of a coherent mass of sediment bounded by distinct failure planes.

- Debris flow: laminar, cohesive flow of clasts in a fine-grained matrix (e.g. wet con-

crete).

- Debris avalanche: rapid flow of cohesionless rock fragments with energy dissipation

by grain contact.

- Turbidity current: gravity flow in which sediment grains are maintained in suspension

by fluid turbulence.

Figure 2.5: Classification of submarine mass movements adapted from sub-aerial classi-

fication proposed by the ISSMGE Technical Committee on Landslides (TC-11) (Locat

and Lee, 2000).


In terms of volume of gravity-driven sediment transport in the ocean, only slides,

debris flows and turbidity currents make a significant contribution. Debris avalanches

are less significant in terms of total transport, but they pose a particular threat to

human populations. There is a generally accepted paradigm that landslides in cohe-

sive sediments evolve down slope from slide to debris flow to turbidity current through

gradually increasing disintegration and entrainment of water. However, this is proba-

bly an oversimplification in that some landslides travel many hundreds of km without

appreciable transformation into turbidity currents, while others transform entirely into

turbidity currents very close to source. In truth, the formation of large turbidity cur-

rents, in which a few hundred km3 of (usually cohesive) sediment are rapidly mixed

with much larger volumes of seawater, is a very poorly understood process (Talling,

2002). Large landslides in continental margin sedimentary sequences are often complex

events, and elements of slide, debris flow and turbidity current may all be evident in

the aftermath of a single landslide.

Despite the variability of submarine landslides that might cause tsunamis, many such

tsunamis show similar general characteristics. In particular, these tsunamis often have

very large run-ups close to the landslide site but appear to propagate much less effi-

ciently than earthquake tsunami, so have limited far-field effects, as already underlined

in previous paragraphs.

Submarine landslides may have huge dimensions and long run-out distances. Modelling

the entire three-dimension problem is a huge computational task, and usually has to

be reduced to a two-dimension problem through depth averaging or through restric-

tion to cross sections to save computation resources. The loss of information about

the vertical profiles of the velocity and density is usually insignificant, in particular for

large-scale events. However, even a two-dimension simulation is a non-trivial task for

large landslides. If it is known that the lateral spreading of the flowing mass is weak

or limited, the flow evolution in the transversal direction may be neglected and one-

dimension models may be applied. Models should preferably describe the multi-layer

structure of a submarine landslide with a dense debris flow at the bottom and a dilute

turbidity current (suspension flow) above it. Hence, the vertical density variations and

the associated variation of the mechanical properties should be taken into account.

The dense debris flow is often considered either as a saturated mass of cohesionless

material, or as a visco-plastic material where no deformation takes place until a spec-

2.3 Numerical Modelling 11

ified stress is applied to the material, after which deformation is driven by the excess

of the stress beyond the yield stress. An example is the frequently used Bingham fluid

model (Bingham, 1922), describing a viscous Newtonian fluid combined with a yield

stress. Such a fluid moves as a dense plug flow riding on top of a shear flow. Material

properties, including clay rheology, are of great importance to the flow dynamics and

travel distance for the majority of events.

The complete modelling of the mass gravity flow and the generation and propagation

of the tsunami may require an optimal application and combination of a diversity of

hydrodynamic models. Both the combination of computational tools and the need to

parallelize the heavy computations point to a domain decomposition strategy. Such

techniques, together with related techniques on nesting have been applied for a while,

but more development and testing are needed, in particular for the combination of

genuinely different models.

2.3 Numerical Modelling

As already mentioned in the Introduction, this work deals with the possibility of sim-

ulating landslides generated waves with the help of Smoothed Particle Hydrodynamics

(SPH) numerical model. In the past two dimensional numerical simulations, based on

linear wave theory, have been proposed by Iwasaki (1987) and Harbitz (1992). Hein-

rich (1992) and Rzadkiewicz et al. (1997) used a volume of fluid (VOF) solution of

Navier Stokes equations in their two-dimensional (2D) simulations, and Verriere and

Lenoir (1992) used linearized potential flow equations. Nonlinear shallow water wave

equations (NSW) were used by Jiang and Leblond (1992, 1993, 1994), Imamura and

Gica (1996) and Fine et al. (1998). The wavemaker formalism developed by Watts

(1998) was applied to a 2D fully nonlinear potential flow (FNPF) model by Grilli and

Watts (1999) and validated with experimental results, e.g., in Watts (2000). The same

formalism was implemented in a 3D FNPF model by Grilli et al. (2002). This three-

dimensional (3D) Numerical Wave Tank (NWT) has been validated both numerically

and experimentally by Grilli et al. (2001), for solitary wave shoaling and breaking over

slopes.

Other authors tried to use SPH in order to represent landslides generated waves, in the


following chapters details about the SPH numerical simulations present in literature

are given for each particular subject.

2.4 Summary 13

2.4 Summary

In this chapter we gave a very brief review about the landslides generated waves. At the

beginning, section 2.1, we started talking about the physics of the phenomenon, under-

lying the different phase in which it is divided, generation, propagation and runup. We

gave an idea on the several kind of impulse waves that can be generated as a function

of the dimension of the landslide.

Then we analyzed the water field that can be divided into the near field, where the

water motion is more complex, and the far field in which the phenomenon is charac-

terized by propagating waves.

After that we just resume the different mechanism of the dynamic of the landslides

itself.

In section 2.3, finally, we listed a series of works that represent the state of the art

about the numerical modelling of landslides generated waves, except for SPH simula-

tions that are extensively presented in the following chapters.

Chapter 3

Smoothed Particles Hydrodynamics

3.1 Introduction

The way to numerically solve differential equations of fluid dynamics is of increasing

interest in the last decades. In particular the standard approach to face with this prob-

lem is to define field variables on a fixed grid or a fixed volume. In problems involving

great changes of the free surface or particular interactions between fluid and structures

the use of a meshfree method is preferable. In these models the fluid is represented as

a series of points each one carrying out a particular mass, so we can refer to that as

”particles”.

Derivatives are calculated by interpolation between neighboring particles.

Smoothed Particles Hydrodynamics (SPH) is a lagrangian numerical model introduced

in Astrophysics by Gingold and Monaghan (1977) and Lucy (1977), because of its

capability to reproduce complex and asymmetric problems in a relatively easy way.

Beginning from the first 90’s SPH was utilized to model free surface flows (Monaghan,

1992) and then applied to a wide range of fluid dynamical problems. The main advan-

tages of SPH compared to finite difference code can be resumed as follows: firstly it is

possible to simulate very complex phenomena in a relatively easy way, then there aren’t

problems at the interface modelling different materials, in contrast with the finite dif-

ference schemes and finally it is possible to represent open boundaries in an extremely

easy way, this is important in particular in astrophysical applications. There are also

some disadvantages in the use of SPH, such as the need of high computational time,

or the particular attention needed to treat with complex boundary, all this aspects are

16 Smoothed Particles Hydrodynamics

to be carefully taken in consideration especially in engineering applications, and this

is, for certain way, our challenging intention.

SPH was already used to model landslide generated waves (Panizzo and Darlymple,

2006; Panizzo et al., 2006). The choice of SPH upon the others numerical methods

came from the fact that SPH is capable to reproduce by itself generation, propagation

and run-up of the impulse wave without any particular restriction and any condition

on the shoreline or the free surface.

The following chapter is organized as follows: in the first part there is a briefly resume

of all the principal characteristics of the code, focusing in particular to the aspects that

are relevant in our context, then a detailed description of all the kernel functions, the

time integration algorithms and the numerical recipes used in our SPH code.

3.2 Basic formalism

3.2.1 Integral Interpolation method

The basic idea of SPH, and in general of the lagrangian methods, is to represent a

generic field variable A as an interpolation, through ”interpolation integrals”, of all the

known values of the same variable in the domain of interest, as in the following identity

A(r) =∫

A(r′)W (|r− r′| , h)dr′ (3.1)

where r is the position at which A is valuated by interpolating the known values at

points r′. In 3.1 W is the so-called kernel function, it represents a weight function of

support h that satisfies the following conditions:

- The limit for h to 0 of W gives a Dirac delta function

limh→0

W (r− r′, h) = δ(r− r′) (3.2)

- The integral of W in space is equal to one

∫W (r− r′, h)dr′ = 1 (3.3)

- Kernel function gradient is symmetric, so W is an even function

3.2 Basic formalism 17

∇W (r− r′, h) = −∇′W (r− r′, h) (3.4)

In Lagrangian numerical model, and in particular in SPH, the fluid is represented as

a finite number of particles, so it is possible to rewrite equation 3.1 in a discrete form,

replacing the integral by a summation and the mass element ρdV with the particle

mass m:

A(r) =∫ A(r′)

ρ(r′)W( |r−r′| , h)ρ(r′)dr′ + O(h2)

≈N∑

b=1

mbAb

ρb

W (|r− rb| , h) (3.5)

Hereinafter b index is referred to the neighbor particle b of known characteristic.

3.2.2 First Derivatives

In SPH the approximation to evaluate first derivative is extremely simple. Beginning

from the following equation:

∇A(r) =N∑

b=1

mbAb

ρb

∇W (|r− rb| , h) (3.6)

or in a compact form:

∇Aa =N∑

b=1

mbAb

ρb

∇Wab (3.7)

hereinafter Wab denotes W (|r− rb| , h).

It is clear that this derivative doesn’t vanish if A is constant, to do so it is possible

to introduce a differentiable function Φ in order to have:

∇Aa =1

Φa

N∑

b=1

mbΦb

ρb

(Ab − Aa)∇Wab (3.8)

if Φ is set equal to one, equation 3.8 becomes:

∇Aa =N∑

b=1

mb

ρb

(Ab − Aa)∇Wab (3.9)

that represents a usual form for first derivatives in SPH.


3.3 Kernel Functions

The simplest class of kernel is given by Gaussian functions such as:

W =σ

hνe−(s/h

)2

if 0 ≤ s ≤ 3 (3.10)

with normalization factor σ = [1/π, 1/π√

π], in 2D and 3D simulations respectively.

This function has the advantage to be infinitely differentiable and therefore has good

stability properties. The original kernel definitions does not present a compact sup-

port: several authors use this kernel function employing a cut off limit at r = 3h.

The functions introduced by Johnson et al. (1996), have the general formulation

written as follows:

W =σ

hν

[3

8s2 − 3

2s +

3

2

]if 0 ≤ s ≤ 2 (3.11)

with ν equal to the number of dimensions and the normalization factor σ =

[1/π, 15/64π], for 2D and 3D simulations respectively.

Hereinafter we refer to it as Gordon Johnson kernel (GJ).

One of the most used kernel is that based on Cubic Spline, or CS (Monaghan and

Lattanzio, 1985):

W =

σhν (2− s)3 if 1 ≤ s ≤ 2σhν (4− 6s2 + 3s3) if s < 1

0 otherwise

(3.12)

with normalization factor σ = [10/7π, 1/π] for 2D and 3D dimensions respectively.

This kernel has a compact support of size 2h.

By increasing the smoothing length to 3h it is possible to define another kernel that

is the Quintic Spline (QS):

3.3 Kernel Functions 19

W =

σhν (3− s)5 if 2 ≤ s ≤ 3σhν

[(3− s)5 − 6 (2− s)5

]if 1 ≤ s ≤ 2

σhν

[(3− s)5 − 6 (2− s)5 + 15 (1− s)5

]if s < 1

0 otherwise

(3.13)

with normalization factor σ = [7/478π, 1/120π], in 2D and 3D dimensions respec-

tively (Morris, 1996).

Another class of positive definite and compactly supported radial functions was

introduced by Wendland (1995), that are defined as:

W =σ

hν

[(1− s)ν+1 ((ν + 1) (s + 1))

]if 0 ≤ s ≤ 3 (3.14)

with s = r/3h and normalization factor σ = [5/9π, 7/18π], in 2D and 3D dimensions

respectively. The Wendland radial basis functions in 2D and 3D are quartic and quintic

spline weight functions respectively.

3.3.1 Gradient Kernel Correction

In order to have angular momentum conservation, both gradient and mixed gradient

and kernel correction were implemented following the work by Bonet and Lok (1999).

This is particularly important every time we have rotational motion.

The gradient kernel correction is obtained by the multiplication of the gradient of kernel

itself with the inverse of a correction matrix

∇Wb (xa) = La∇Wb (xa) (3.15)

that is given by the kronecker product of the gradient of the kernel function with

the position vector:

La =

(N∑

b=1

Vb∇Wb (xa)⊗ (xb − xa)

)−1

(3.16)

Mixed gradient and kernel correction add the normalization of the kernel function

itself:


0 0.5 1 1.5 2 2.5 3−3

−2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

r/h0 0.5 1 1.5 2 2.5 3

−3

−2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

r/h0 0.5 1 1.5 2 2.5 3

−3

−2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

r/h

0 0.5 1 1.5 2 2.5 3−3

−2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

r/h0 0.5 1 1.5 2 2.5 3

−3

−2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

r/h

Figure 3.1: Kernel functions.

3.4 Fluid Equations 21

∇Wb (xa) = La∇Wb (xa) (3.17)

where

Wb (xa) =Wb (xa)

N∑b=1

VbWb (xa)(3.18)

3.4 Fluid Equations

Considering the momentum and continuity equations used in hydraulics to simulate

the flow of a fluid with constant viscosity µ, we have:

ρdv

dt= ρf −∇P + µ∇2v +

1

3µ∇ (∇ · v) (3.19)

dρ

dt+ ρ∇ · v = 0 (3.20)

Equations 3.19 and 3.20 are the well known Navier–Stokes Equations (NSE) and

the continuity equation for a compressible liquid. Here, v is the fluid velocity vector,

ρ the fluid density, f a body force, P the fluid pressure, and µ the dynamic viscosity.

3.4.1 Continuity Equation

Using SPH approximation it is possible to write the evolution of density in a simple

discrete form:

ρa =N∑

b=1

mbWab (3.21)

while ∇ · v is evaluated as:

∇ · va =1

ρa

N∑

b=1

mb (vb − va) · ∇aWab (3.22)

In this way it is possible to write the continuity equation 3.20 in the following form:

dρa

dt=

∑

b

mb (va − vb) · ∇aWab (3.23)


3.4.2 Momentum Equation

The momentum equation 3.19 written for a non-viscous fluid in absence of gravity is

the well known Euler equation:

dv

dt= −1

ρ∇P (3.24)

The pressure gradient, present in 3.19, can be rewritten as:

∇P

ρ= ∇

(P

ρ

)+

P

ρ2∇ρ (3.25)

that in SPH form becomes

∇a

(Pa

ρa

)=

N∑

b=1

mb

ρb

Pb

ρb

∇aWab =

=N∑

b=1

mbPb

ρ2b

∇aWab (3.26)

for the first term, and for the second

∇aρa =N∑

b=1

mb∇aWab (3.27)

Combining terms we have that

∇aP

ρa

=N∑

b=1

mbPb

ρ2b

∇aWab +Pa

ρ2a

N∑

b=1

mb∇aWab

=N∑

b=1

mb

(Pb

ρ2b

+Pa

ρ2a

)∇aWab (3.28)

Thus the Euler equation can be rewritten in SPH formalism as follows:

dva

dt= −

N∑

b=1

mb

(Pb

ρ2b

+Pa

ρ2a

)∇aWab (3.29)

3.4.3 Equation of State

Fluids simulated by using the SPH are compressible, as a stiff equation of state is used.

This is to avoid solving the Poisson equation for incompressible fluids at each time step.

The equation of state used in the present work is taken from Batchelor (1967), and is:

3.4 Fluid Equations 23

p = B

[(ρ

ρ0

)γ

− 1

](3.30)

with γ = 7, ρ0 = 1000kg/m3 is the initial density of the fluid, B is a constant

chosen to make sure the Mach number M = uc

is small enough to avoid fast sound

waves which would require small ∆t . B is defined considering that the Mach number

has to be M < 0.2. In simulations that will be presented in the present work, the Mach

number was set M = 0.1, and so:

M2 =u2

c2= 0.01 (3.31)

According to the definition of sound speed under constant entropy:

c2 =∂p

∂ρ(3.32)

which results in:

c2 =∂p

∂ρ=

γB

ρ0

(ρ

ρ0

)γ−1

(3.33)

the constant B is defined assuming a value of the velocity u which should be char-

acteristic of the problem at hand. In free surface studies, u is taken equal to√

2gH,

being H a representative depth Monaghan (1994). In this approach the sound speed c

is much slower than the actual speed of sound in the water.

3.4.4 Variable smoothing length

At the beginning of its use SPH simulations used a fixed smoothing length. The

smoothing length h determines the radius of interaction for each SPH particle. How-

ever allowing each particle to have its own associated smoothing length which varies

according to local conditions increases the spatial resolution substantially (Hernquist

and Katz (1989) and Benz et al. (1990)). Usually:

ha =

(1

ρa

)(1/ν)

(3.34)

where ν is the number of spatial dimensions.

Equation 3.1 represents an implicit problem, because ρa depends from ha itself. Many


different approaches have been used over the years to perform this calculation. A

simple one is to use the time derivative of 3.1:

dha

dt= − ha

νρa

dρ

dt(3.35)

which can then be evolved alongside the other particle quantities. This rule works

well for most practical purposes. However, it has been known for some time that, in

order to be fully selfconsistent, extra terms involving the derivative of h should be

included in the momentum and energy equations. By expressing the smoothing length

as a function of r we can specify h as a function of the particle co-ordinates Monaghan

(2002). That is we have h = h(ρ) where ρ is given by:

ρa =∑

b

mbW (rab, ha) (3.36)

Considering the time derivative we have:

dρa

dt=

1

Ωa

∑

b

mbvab∇aWab (ha) (3.37)

where

Ωa =

[1− ∂ha

∂ρa

∑

k

mk∂Wab (ha)

∂ha

](3.38)

The equation of motion can then be found (Monaghan (2002)):

dva

dt= −∑

b

mb

[Pa

Ωaρ2a

∇aWab (ha) +Pb

Ωbρ2b

∇aWab (hb)

](3.39)

Calculation of the quantities Ω involve a summation over the particles and can

be computed alongside the density summation 3.36. To be fully self-consistent we

solve 3.36 iteratively to determine both h and ρ self-consistently. We do this using the

predicted smoothing length from 3.35, the density is initially calculated by a summation

over the particles. A new value of smoothing length hnew is then computed from this

density using 3.34. Convergence is determined according to the criterion:

|hnew − h|h

< 1.0× 10−2 (3.40)

For particles which are not converged, the density of (only) those particles are re-

calculated (using hnew). This process is then repeated until all particles are converged.

3.5 Viscosity 25

Note that a particles smoothing length is only set equal to hnew if the density is to

be recalculated (this is to ensure that the same smoothing length that was used to

calculate the density is used to compute the terms in the other SPH equations). Also,

the density only needs to be recalculated on those particles which have not converged,

since each particles density is independent of the smoothing length of neighboring par-

ticles. This requires a small adjustment to the density calculation routine (such that

the density can be calculated only for a selected list of particles, rather than for all),

but is relatively simple to implement and means that the additional computational cost

involved is negligible (at least for the problems considered in this thesis). Note that in

principle the calculated gradient terms 3.38 may also be used to implement an iteration

scheme such as the Newton-Raphson method which converges faster than our simple

fixed point iteration. Where the variable smoothing length terms are not explicitly

calculated, we use a simple averaging of the kernels and kernel gradients to maintain

the symmetry in the momentum equations (Hernquist and Katz (1989); Monaghan

(1992)), that is to say:

Wab =1

2[Wab (ha) + Wab (hb)] (3.41)

and

∇aWab =1

2[∇aWab (ha) +∇aWab (hb)] (3.42)

This simple solution to the problem was adopted in this thesis, we’ll not talk about

all the test done in this sense.

3.5 Viscosity

Different formulations for viscosity have been used over the years in SPH. The intro-

duction of a viscous term in the momentum equation is necessary not only to consider

viscid fluids and no slip boundary conditions, but also to prevent inter-particle pene-

tration and to improve code stability. The first formulation was introduced by Lucy

(1977), and then improved by Monaghan and Gingold (1983) in order to conserve linear

and angular momentum.


This kind of viscosity is one of the most commonly used, the viscous term Πab is added

to pressure terms in the momentum equation

dva

dt= −∑

b

mb

(Pa

ρ2a

+Pb

ρ2b

+ Πab

)∇aWab (3.43)

where

Πab = −ν

(vab · rab

r2ab + εh2

ab

)(3.44)

where ε ∼ 0.01 is a corrective term introduced to prevent singularity when rab = 0

and ν is given below:

ν =αhabcab

ρab

(3.45)

When this formulation was introduced, Monaghan and Gingold (1983), it was found

to be good for shocks of moderate strength, thus when Mach number increases it is

necessary to introduce an extra term to ν which take the following form introduced by

Monaghan (1992)

ν =hab

ρab

(αcab − β

habvab · rab

r2ab + εh2

ab

)(3.46)

α and β are constants that depend upon the kind of problem to simulate. Good

results have been obtained with 1 and 2 values respectively.

In Monaghan (1997) the artificial viscosity is used in a different form, that is analogous

to that one used for Riemann solutions. In this form

Πab = −Kvsig (vab · rab)

ρab |rab| (3.47)

K is a constant typically equal to 0.5 for shocks, in Monaghan (2006a) the kinematic

shear viscosity for the cubic spline kernel is given as an expression of this parameter K

ν =15

112Kvsigh (3.48)

using the integral expressions given by Monaghan (2005).

3.6 Timestepping 27

3.6 Timestepping

The so called timestepping stands for the integration scheme algorithm that can be

used in order to update particles positions in time. Different time integration schemes

have been employed in SPH and in other Lagrangian numerical models. Some of these

schemes are the Euler, the Verlet, the Velocity Verlet, the Two Step Velocity Verlet

(Monaghan, 2006b), Runge-Kutta, and the Beeman (Beeman, 1976) algorithms. In the

following we will focus on the mathematics of the Verlet, Velocity Verlet and Beeman

algorithms only.

3.6.1 Verlet Algorithm

The Verlet algorithm is probably the most commonly used time integration scheme in

molecular dynamics. The updated position in time is:

x(t + ∆t) = 2x(t)− x(t−∆t) + a(t)∆t2 + O(∆t4) (3.49)

One could compute the velocities from the positions by using:

v(t) =x(t + ∆t)− x(t−∆t)

2∆t(3.50)

The error associated to this scheme is of order ∆t2 for velocity and ∆t4 for position.

3.6.2 Velocity Verlet Algorithm

The Velocity Verlet algorithm provides both positions and velocities at the same time,

and for this reason may be regarded as the most complete form of the Verlet algorithm.

The basic equations are as follows:

x(t + ∆t) = x(t) + v(t)∆t + (1/2)a(t)∆t2 (3.51)

v(t + ∆t/2) = v(t) + (1/2) a(t)∆t (3.52)

The value of a(t+∆t) is then calculated from the governing equations of the modeled

phenomenon, and finally a(t + ∆t) is used to calculate:


v(t + ∆t) = v(t + ∆t/2) + (1/2) a(t + ∆t)∆t (3.53)

In this form a half-step is taken via Eq. 3.52, using the acceleration and velocity

from time step t. This is sufficient to calculate particle’s position at time step t + ∆t

using Eq. 3.51. Finally, using acceleration calculated from the new particle position

x(t + ∆t), the half step velocity is updated to the full step velocity v(t + ∆t), using

Eq. 3.53.

3.6.3 Beeman Algorithm

The Beeman algorithm is a numerical integration scheme similar to the Verlet scheme,

accurate to ∆t4 for position and to ∆t3 for velocity. It starts by considering values of

x (t), a (t), a (t−∆t) and v (t) to update x:

x (t + ∆t) = x (t) + v (t) ∆t +2

3a (t) ∆t2 − 1

6a (t−∆t) ∆t2 + O

(∆t4

)(3.54)

The value of position x at t+∆t is used to compute the acceleration at time t+∆t,

and these are used to update v:

v (t + ∆t) = v (t) +1

3a (t + ∆t) ∆t +

5

6a (t) ∆t− 1

6a (t−∆t) ∆t

+O(∆t3

)(3.55)

In SPH, like in many other models, the acceleration is evaluated as a function of

both velocity and position, thus a predictor-corrector form of the Beeman algorithm

has to be considered. It reads as:

v (t + ∆t) (predicted) = v (t) +3

2a (t) ∆t− 1

2a (t−∆t) ∆t (3.56)

v (t + ∆t) (corrected) = v (t) +1

3a (t + ∆t) ∆t +

5

6a (t) ∆t

−1

6a (t−∆t) ∆t (3.57)

3.6 Timestepping 29

In this case the numerical scheme is accurate to ∆t4 for both position and velocity.

A variant of the original Beeman algorithm is here proposed. A predictor corrector

scheme for position is also used, such as:

r (t + ∆t) (predicted) = r (t) + v(t)∆t +2

3a (t) ∆t2 − 1

6a (t−∆t) ∆t2 (3.58)

v (t + ∆t) (predicted) = v (t) +3

2a (t) ∆t− 1

2a (t−∆t) ∆t (3.59)

r (t + ∆t) (corrected) = r (t) + v (t) ∆t +1

6a (t + ∆t) ∆t2

+1

3a (t) ∆t2 (3.60)

v (t + ∆t) (corrected) = v (t) +5

12a (t + ∆t) ∆t +

8

12a (t) ∆t

− 1

12a (t−∆t) ∆t (3.61)

The scheme is still accurate to ∆t4 for both position and velocity.

3.6.4 Courant Condition

In order to determine the timestep, the Courant-Friedrichs-Levy stability condition has

to be satisfied:

dtc = Ccour min

(h

vsig

)(3.62)

where 0 < Ccour < 1 is the Courant number and vsig is the maximum signal velocity

between pairs of particles. Although this condition is sufficient for all of the simulations

described here, in general it is necessary to pose the additional constraint from the

forces:

dtf = Cf min

(h

|aa|

)1/2

(3.63)

where aa is the acceleration on particle a and typically Cf is equal to 0.25.


3.7 Numerical recipes

3.7.1 Interpolation Schemes

As we already said at the beginning of this chapter, the main characteristic of SPH is to

interpolate a generic field variable using interpolation integrals that became summation

because of the fact that it’s a lagrangian model. This approximation find its consistency

in the use of the interpolation function that we called kernel. In general in SPH the

integral of the kernel function is not equal to one, as it has to be, because of the domain

discretization.

In this sense and in general to reproduce correctly uniform function (i.e. to have a

zero integral of kernel gradient) a first variant of the interpolation technique is the

one proposed by Shepard (1968) that consists in a normalization of the value of kernel

through its integral:

W Sb (xa) =

Wb (xa)∑k

Wk (xa) dVk

(3.64)

It is possible also to have higher order interpolation, one it’s the so called MLS

interpolation technique Belytschko et al. (1998).

It consists in the introduction of a linear operator β:

WMLSb (xa) = [β0 (xa) + β1 (xa) (xb − xa) + β2 (xa) (xb − xa)] Wb (xa) (3.65)

In order to have uniform and linear functions correctly evaluated, β comes out from

the solution of the following algebraic formulation:

Aab =∑

b

1 xb − xa yb − ya

xb − xa (xb − xa)2 (xb − xa) (yb − ya)

yb − ya (xb − xa) (yb − ya) (yb − ya)2

Wb (xa) dVb

Aab

β0 (xa)

β1 (xa)

β2 (xa)

=

1

0

0

(3.66)

with this correction it is possible to evaluate a WMLSb but also a ∇WMLS

b in the

following way:

3.7 Numerical recipes 31

∂W MLSb

∂x(xa) = [β3 (xa) + β4 (xb − xa) + β5 (yb − ya)] Wb (xa)

∂W MLSb

∂y(xa) = [β6 (xa) + β7 (xb − xa) + β8 (yb − ya)] Wb (xa)

Aab

β3 (xa)

β4 (xa)

β5 (xa)

=

0

1

0

(3.67)

Aab

β6 (xa)

β7 (xa)

β8 (xa)

=

0

0

1

3.7.2 XSPH Correction

For free surface problems, a numerical filter was proposed by Monaghan (1994). The

correction modifies the update of the particles motion by correcting the velocity ac-

cording to the following expression:

dva

dt= va + ε∆va (3.68)

where

∆va =N∑

b=1

mb (vb − va)

ρab

Wab (3.69)

Indeed, the XSPH correction represents a smoothing of the velocity field considering

the neighboring particles velocities. It is particularly efficient when fast flow fields or

impacts are considered. The coefficient ε has to be chosen in the range [0; 1].

3.7.3 Correction of the tensile instability

In order to prevent particles clustering, Monaghan and Kos (2000) proposed a mod-

ification of the cubic spline based kernel, by introducing a suitable function which

increases as the separation between particles decreases:

fab =W (rab)

W (∆p)(3.70)


where ∆p is the initial particle spacing. The correction of ”tensile instability” is

introduced by replacing:

Pa

ρ2a

+Pb

ρ2b

+ Πab (3.71)

with

Pa

ρ2a

+Pb

ρ2b

+ Πab + Rfnab (3.72)

Monaghan and Kos (2000) suggests to assume the following values for the variables:

n = 4, h = 1.3∆p, while the factor R could be determined as a function of the pressure.

So that:

R = Ra + Rb (3.73)

and factor Ra is determined by the rule (if Pa < 0.0)

Ra =ε |Pa|ρ2

a

(3.74)

while a typical value of ε is 0.2. Moreover, Monaghan suggests to employ the

correction also for positive values of the field pressure, due to the formation of local

linear structures in case of liquids simulations. So, if Pa > 0.0 and Pb > 0.0,

R = 0.01

(Pa

ρ2a

+Pb

ρ2b

)(3.75)

It is to be noticed that the problem of particle clustering can be also avoided em-

ploying the kernel function proposed by Gordon R. Johnson or by Wendland, in an

easier and straightforward way.

3.8 Boundaries

SPH was firstly applied to astrophysical problems, in a way that at the beginning es-

pecially periodic boundary conditions were used. When it became to be applied in

hydraulics the problems of dealing with the boundaries became important.

3.9 Computational Strategies 33

3.8.1 Repulsive Forces

Firstly Monaghan (1994) proposed fixed particles at the boundaries that exerted ”Lennard-

Jones” forces on the centers of particles:

f (r) = D((

r0

r

)p1

−(

r0

r

)p2)

r

r2(3.76)

In Monaghan and Kos (1999) a smoother form of repulsive force is given as follows:

f (∆t, ∆n) = n ·R (∆n) · P (∆t) (3.77)

where f is the force per unit mass on a fluid particle, and it is evaluated considering

two different contributions.

3.8.2 Ghost Particles

The use of Ghost Particles as boundaries means the reproduction of a portion of the

fluid domain, close to the boundary at hand, outside of it. The main characteristic of

this particles, such as pressure, velocity, density but also position, are deduced by the

mirrored fluid particles.

The mirroring rules are the followings:

xgh = 2xb − xi

ungh = 2Unb − unp

pgh = pp

uygh = utp

(3.78)

where un and ut are normal and tangential velocity respectively with Unb for the

local displacement of the rigid boundary, x is the position vector and in this case we

are considering free slip conditions.

3.9 Computational Strategies

Because of the neighbors particles searching at every time step, SPH numerical model

needs great computational time. In order to reduce this computational time, several


Figure 3.2: Ghost Particles as boundaries.

numerical techniques have been proposed to optimize the definition of the interacting

neighbors of each particle of the fluid domain: Figure 3.3 shows the neighbors (black

particles) of the particle at hand. In the followings two of the most successful SPH

optimizing techniques will be introduced.

Figure 3.3: Particle interaction in the SPH model: neighbors particles of a given

particle.


Linked list

The principal idea of linked list, or Verlet list, methods is to reduce the neighbor search

algorithm by setting the particles in an ordered grid, characterized by a grid cell size

equal to 2h, as sketched in Figure 3.3. In this way, it is clear that each particle interacts

at least with particles of the same grid cell, and at most with particles placed in the

eight adjacent grid cells, as it is shown in the left panel of Figure 3.4. So that, a linked

list algorithm consists of searching neighbors particles looking at the eight grid cells

around the cell at hand: the right panel of Figure 3.4 shows one of the possible way to

look at the neighbors cells. An optimization of this method is obtained by employing

the action-reaction principle, according to which each interaction value A between a

couple of particles i and j could be defined as Aij = −Aji, as the value of the kernel W

is a function only of the distance between particles. So, the basic idea is to compute

only once Wij between two particles. This can be attained taking into account grid

cells by a main direction over the grid. In this optic it is sufficient to look only at five

of the grid cells of the previously defined linked list, as it is shown in Figure 3.5, while

all interaction values have to be stored in dedicated vectors, with a small wasting of

memory. In this example, the five neighbors cells are placed around the right upper

corner of the cell at hand, so the main direction according to which grid cells have to

be considered starts from the left bottom corner of the computational domain, going

to the right upper corner.

Figure 3.4: Particle grid cells in the fluid domain (on the left) and linked list through

neighbors grid cells (on the right).


Figure 3.5: Reduced linked list algorithm over five grid cells.

Oversized neighbors list

The main idea of this technique, which is introduced here for the first time, is to reduce

the number of time the neighbors particles list is computed, keeping the same particle

list for a certain number of time steps. The method is introduced making reference to

Figure 3.6, which shows the circle (r= 2h) of neighbors particles (blacks) of the particle

at hand. By considering an oversized radius r = 2h+ δ in the formation of the particle

neighbors list, some non interacting particles are taken into account (grey particles).

Even if unnecessary particles are considered in the list, it is possible to keep the same

list for lots of numerical time steps of the model. Considering at each time step (ti) the

maximum particle velocity (vmaxi ) in the flow field, the maximum particle movement

is computed as (dsmaxi = vmax

i · dt). The neighbors list is computed every time the

summation of dsi over successive time steps reaches the value of δ, that’s to say:

ds =NNL∑

i=1

dsmaxi =

NNL∑

i=1

vmaxi · dti ≥ δ (3.79)

being NNL the number of time steps necessary before the neighbors list updating.

Making reference to Figure 3.6, it is clear that smaller values of δ result in smaller

number of the useless particles interactions enclosed in the neighbors list, but the list

is frequently updated (left plot). On the contrary, greater values of δ result in greater

number of useless particles enclosed in the neighbors list, but the list is infrequently

updated (left plot).

In the present work, both the linked list (Verlet list) and the oversized neighbors

list methods have been implemented and tested. The best performances have been


Figure 3.6: Oversized neighbors lists.

obtained using the linked list with the action-reaction principle, which is computed

each time step. The tested oversized neighbor list was computed each n time steps

using the linked list algorithm, being n a parameter calculated as a function of the

velocity of the flow field, with typical values in the range [10; 200]. It is to be stressed

that, considering both the proposed method, if n=1 the optimization is reduced to the

linked list method. The parameter δ was considered in the range [0.001; 0.1]: the best

value to adopt is a function of the problem at hand.

Moreover, a parallel version of the original SPH algorithm was used using the OpenMP

parallel libraries. In particular, using both the linked list and the oversized neighbors

list computational strategies made it possible to employ a technique which assigns, at

any time step, the computation of variables of any given particle to the first available

CPU. This can be obtained assigning a parallel do loop cycle when updating the prop-

erties of particles in the fluid domain. In particular, with this technique, the work of

each CPU is equally balanced with that of other CPUs, and the parallel architecture

results well optimized. The reduction of computational time using an IBM parallel

machine with 8 CUPs resulted to be less than 40% the time required by a simple se-

quential computation.


3.10 Summary

In this chapter all the details about our SPH numerical model were given in details.

Firstly a brief resume about the general formalism of SPH was given, 3.2. After that

in 3.3 the different kernel functions were introduced. Because of the range of interest

of our applications, in 3.4 the formalism of fluid equations in SPH was described. A

review on the two most used boundary conditions is given in 3.8, while in 3.5 the for-

malisms for viscous term in SPH is shown. The integration algorithm are presented in

3.6, and in 5.7 and 3.9 all the numerical recipes used.

Chapter 4

Test Cases

4.1 Introduction

In the following chapter we will test different kernel functions that are used in SPH for

interpolation and different algorithms that can be used to integrate the SPH equations

in time. Analytical studies on the performance of different kernel functions have been

presented in the past by Price (2004) and Morris (1996). Our target is to numeri-

cally test kernels and numerical integration algorithms using simple SPH applications.

Two techniques to correct the gradient of the kernel function (Bonet and Lok, 1999),

introduced in the previous chapter, are also considered.

The capability of the interpolation technique used in SPH to evaluate second deriva-

tives of a given function is also tested. In particular, some different ways to implement

second derivatives (Watkins et al., 1996; Morris et al., 1997; Takeda et al., 1994) are

tested. At this aim, a periodic flow field with a given velocity distribution is repro-

duced, second derivatives of the velocity field are calculated and results from the three

different considered formulations are compared.

Finally, an accurate SPH numerical model is applied to simulate viscous flow fields

at low Reynolds numbers, such as the Couette and the Poiseulle flows.

40 Test Cases

4.2 SPH accuracy from interpolation and time in-

tegration schemes

Since the accuracy given by each of the algorithms used for the numerical integra-

tion, see previous chapter, is known, it would seem that the highest accuracy in SPH

would be gained by employing an accurate interpolation kernel, the gradient kernel

correction, and the time integration scheme with highest accuracy. On the other hand,

given the typical SPH time steps (small) and number of interpolation points, the role

played by the accuracy in the interpolation and time integration schemes has not been

investigated. It could happen, in fact, that a SPH model could have low accuracy even

though a high order time integration scheme is employed, if an inaccurate interpolation

is provided by the chosen kernel function and its evaluation of field variables gradients.

In addition, the SPH model has a high computational cost due to the large number of

particles and small step sizes. Thus, the choices of a kernel and of a time integration

scheme are often also related to their computational cost.

We would like to investigate the role of accuracy on SPH using different kernels and

different time integration schemes.

Two and three dimensional tests are carried out considering simple SPH reproduction

of a single particle’s periodic movement in time, which is known analytically. The ac-

curacy is calculated referring to the error on the particle position at the end of the test.

To update the particle position in time, a time step typical of SPH has been used, of

the order of ∆t = 10−5 s. Otherwise, the number of interpolation points is a function

of the width of the kernel support, and thus, is a function of h. In all the two and

three dimensional tests that are presented here we assumed a value of h = 1.3 · inh,

inh being the particle diameter.

4.2.1 Two dimensional test

The two dimensional test consists in reproducing the motion of a single particle with

mean position (xo, yo) and a trajectory described by the following equations:

x = x0 + sin(ωt) y = y0 + 0.5ω sin(2ωt) (4.1)

The angular frequency of the motion is ω, defined as ω = 2π/T , T being the period

4.2 SPH accuracy from interpolation and time integration schemes 41

−1.5 −1 −0.5 0 0.5 1 1.5−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

x

y

−1.5 −1 −0.5 0 0.5 1 1.5−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

x

y

Figure 4.1: Two dimensional tests. Dots refer to the analytical solution, the continuous

line reports the SPH results. Left panel: best obtained result, using a QU kernel with

mixed gradient and kernel correction, and the Beeman algorithm (cumulated error =

0.5159 m). Right panel: worst obtained result, using a non corrected GJ kernel and

the Euler algorithm (cumulated error = 1295.20 m).

of the motion, in this case set as T = 1.0s.

This figure-eight motion can be obtained by prescribing an analytical pressure field

in the momentum equation 3.19, neglecting terms related to viscosity (µ = 0). Thus,

from Eq. 4.1 it is possible to define the following:

P (x, y, t) = ρω2 (x sin (ωt) + 2y sin (2ωt)) (4.2)

To move the particle due to pressure gradients obtained by interpolating on points,

we here consider that a finite number of neighbor particles are placed on a rigid grid,

which move according to the motion of the considered particle. Pressure values are

prescribed to these neighbors as a function of time and position x and y, according to

Eq. 4.2.

All the above kernels, the different kernel corrections, and the various time inte-

gration schemes, have been used to simulate the motion of the particle in this test,

for a total of 90 runs. In the performed numerical simulation, the density has been

set to ρ = 1000.0 Kg/m3. To evaluate the test performance, we refer to the observed

error between the calculated particle position, using SPH, and the position given by

the analytical expressions in Eq. 4.1, after 10 periods (t = 20π/ω). The error we will

refer to is that summed up step by step during the numerical test. Figure 4.1 shows

42 Test Cases

the comparison between the computed particle trajectory and the analytical solution,

after 10 s of simulation, which corresponds to 10 periodic cycles around the reference

position [x0 = 0.0; y0 = 0.0]. In particular, the left panel of Figure 4.1 shows the best

obtained result, while the right panel refers to the worst obtained result.

Test results are summarized and reported in Figure 4.2, where the observed errors

are represented using bars, and their values are scaled as a function of the highest

observed error, which in this case is equal to 1295.20 m. In particular, each panel

in the Figure refers to a time integration scheme. In each panel, five groups of three

bars refer to the five considered kernel functions, indicated using two letters. Black

bars report errors of simulations without any kernel correction; grey bars refer to the

gradient kernel correction; white bars refer to the mixed kernel and gradient correction.

Dots connected by lines refer to the computational time of each simulation normalized

to the higher computational time. Different colors refer to different kernel correction

techniques, as before.

Obtained results show that the observed error is two orders of magnitude smaller

considering a corrected kernel function rather than a kernel function with no correc-

tion. The use of an integration scheme or another is relevant in simulations where

corrected kernel functions have been considered. In particular, the best results are

obtained using the Beeman’s and the two step Velocity Verlet algorithms. Otherwise,

as expected, the worst results are given by the Euler direct integration method.

It is interesting to notice that the interpolation technique plays the major role in

the differences on observed errors. In fact, while black bars in Figure 4.2 and Figure

4.3 show no appreciable error differences as a function of the time integration scheme,

formulations to correct the kernel interpolation (Bonet and Lok, 1999) provide a great

gain in accuracy.

It is also to be noted that no relevant differences exist between errors from the

gradient and the mixed kernel and gradient correction. These two techniques, instead,

present different computational cost, the mixed kernel and gradient correction resulting

about 10% slower. About computational cost, it is evident that 3h supported kernels


GJ CU QU WE GA10

−5

10−4

10−3

10−2

10−1

100 Beeman algorithm

0

0.2

0.4

0.6

0.8

1

GJ CU QU WE GA10

−5

10−4

10−3

10−2

10−1

100 Euler algorithm

0

0.2

0.4

0.6

0.8

1

GJ CU QU WE GA10

−5

10−4

10−3

10−2

10−1

100 Verlet algorithm

0

0.2

0.4

0.6

0.8

1

GJ CU QU WE GA10

−5

10−4

10−3

10−2

10−1

100 Velocity Verlet algorithm

0

0.2

0.4

0.6

0.8

1

10−5

10−4

10−3

10−2

10−1

100 Modified Beeman algorithm

GJ CU QU WE GA GJ CU0

0.2

0.4

0.6

0.8

1

10−5

10−4

10−3

10−2

10−1

100 Two Step Velocity Verlet algorithm


0.2

0.4

0.6

0.8

1

Figure 4.2: Two dimensional test results showing errors and computational time asso-

ciated with each integration algorithm and kernel combination. The kernels are GJ:

Gordon Johnson; CU: Cubic spline; QU: Quintic spline; WE: Wendland; GA: truncated

Gaussian.

44 Test Cases

10−5

10−4

10−3

10−2

10−1

100 Gordon Johnson Kernel

Bee MBee Eul Ver VVer TVVer0

0.2

0.4

0.6

0.8

1

10−5

10−4

10−3

10−2

10−1

100 Cubic Spline Kernel


0.2

0.4

0.6

0.8

1

10−5

10−4

10−3

10−2

10−1

100 Quintic Kernel


0.2

0.4

0.6

0.8

1

10−5

10−4

10−3

10−2

10−1

100 Wendland Kernel


0.2

0.4

0.6

0.8

1

10−5

10−4

10−3

10−2

10−1

100 Gaussian Kernel


0.2

0.4

0.6

0.8

1

Figure 4.3: Two dimensional test results showing errors and computational time as-

sociated with each kernel and integration algorithm combination. The integration

algorithms are Bee: Beeman; MBee: modified Beeman; Eul: Euler; Ver: Verlet; VVer:

velocity Verlet; TVVer: two step velocity Verlet.


perform slower than 2h supported ones, since there are more particles to take into

account. Furthermore, it is to be said that in SPH models with disordered particles,

Cubic and Quintic spline kernels present a major computational time due to the fact

that they are defined on more than one interval of the variable s. This aspect affects

the particle’s neighbors search, which has not been considered in numerical tests per-

formed in the present work.

On the basis of these results it is possible to conclude that Wendland kernel, with

applied gradient kernel correction, and the use of the Beeman’s algorithms, present the

best compromise between accuracy and time computational cost. As a general rule to

keep in mind, if typical values of the time step used in SPH are considered, here set

as dt = 10−5 s, different time integration schemes improve the code accuracy only if

corrected kernel functions are implemented. The interpolation technique, which in this

case means corrected kernel functions, plays a major role.

4.2.2 Three dimensional test

The particle motion in the three dimensional test is described by the figure called

“Noeud de trefle”, which is defined by the following equations:

x = x0 + cos(ωt) + 2 cos(2ωt)

y = y0 + sin(ωt)− 2 sin(2ωt)

z = z0 + 2 sin(3ωt)

(4.3)

As before, the particle motion is obtained from the momentum equation 3.19 pre-

scribing an appropriate pressure field, which in this case is:

P (x, y, z, t) =

ρω2 (x (cos(ωt) + 8 cos(2ωt)) + y (sin(ωt)− 8 sin(2ωt)) + 18 sin(3ωt)) (4.4)

Results are presented using the same notation and colors used to represent two

dimensions test results. Figure 4.4 shows the particle trajectory for the best (left

panel) and the worst (right panel) simulations. Figure 4.5 and Figure 4.6 present all

the obtained results.

46 Test Cases

−3−2

−10

12

3

−3

−2

−1

0

1

2

3−2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

−3−2

−10

12

3

−3

−2

−1

0

1

2

3−2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

Figure 4.4: Three dimensional test Noeud de trefle. The continuous line refers to the

analytical solution, dots report the SPH results. Left panel: best obtained result, using

a QU kernel with mixed gradient and kernel correction, and the Beeman algorithm

(cumulated error = 15.0204 m). Right panel: worst obtained result, using a non

corrected CU kernel and the Euler algorithm (cumulated error = 279.99 m).

As in the two dimensional test, it is evident that the time integration algorithm

selected does not improve the SPH accuracy, if the correction of the kernel function is

not implemented. However, the choice of kernel plays an important role in the accuracy.

Another important fact to emphasize is that if the kernel function is corrected as in

section 3.3.1, there are no appreciable differences between one kernel and another.

If we consider test run with corrected kernel functions, the use of an integration

scheme or another one becomes relevant. The smallest cumulated error has been

observed using a Quintic kernel with mixed gradient and kernel correction, and the

Beeman algorithm. Three dimensional tests confirm that the use of kernel correction

techniques improves the SPH accuracy more than the use of high order time integration

schemes, at typical time steps and dimensions of SPH models.


10−2

10−1

100 Beeman algorithm


0.2

0.4

0.6

0.8

1

10−2

10−1

100 Modified Beeman algorithm


0.2

0.4

0.6

0.8

1

10−2

10−1

100 Euler algorithm


0.2

0.4

0.6

0.8

1

10−2

10−1

100 Verlet algorithm


0.2

0.4

0.6

0.8

1

10−2

10−1

100 Velocity Verlet algorithm


0.2

0.4

0.6

0.8

1

10−2

10−1

100 Two Step Velocity Verlet algorithm


0.2

0.4

0.6

0.8

1

Figure 4.5: Three dimensional test results.

48 Test Cases

10−2

10−1

100 Gordon Johnson Kernel


0.2

0.4

0.6

0.8

1

10−2

10−1

100 Cubic Spline Kernel


0.2

0.4

0.6

0.8

1

10−2

10−1

100 Quintic Kernel


0.2

0.4

0.6

0.8

1

10−2

10−1

100 Wendland Kernel


0.2

0.4

0.6

0.8

1

10−2

10−1

100 Gaussian Kernel


0.2

0.4

0.6

0.8

1

Figure 4.6: Three dimensional test results.

4.3 SPH evaluation of second derivative of a field variable 49

4.3 SPH evaluation of second derivative of a field

variable

The evaluation of the second derivative of a field variable is a problem of increas-

ing interest in SPH simulations. In fact in order to correctly evaluate viscous terms

in NSE is important to have a correct formulation of the Laplacian of the velocity field.

Several approaches have been introduced in the literature to calculate second deriva-

tives and to model the complete set of terms in the momentum equation (see 3.19). In

literature it is possible to find the following approaches:

- The first formulation is that proposed by Takeda et al. (1994), who considers

a kernel function differentiable more than once, such as the Gaussian. Under this

hypothesis it is possible to write the laplacian of velocity as:

(∇2v

)a

=∑

b

mb

ρb

vb

∑

αβγ

[1

rab

∂Wab

∂ra

+xαβγ

ab

rab

∂

∂ra

(xαβγ

ab

rab

∂Wab

∂ra

)] (4.5)

Takeda et al. (1994) proposed to model the viscous terms of the NSE as follows:

µ

ρa

(∇2v +

1

3∇ (∇ · v)

)

a= −µ

mamb

ρaρb

[vab

(n +

1

3

)1

rab

∂W

∂rab

+(vabxab

3xab + vabx

2ab

)1

rab

∂

∂rab

(1

rab

∂W

∂rab

)](4.6)

In the following we refer to this formulation as the Takeda method.

- The second formulation was introduced by Brookshaw (1985) and Morris et al.

(1997) and consists in solving the Laplacian of the velocity combining a standard SPH

gradient estimation with a finite difference approximation of a first derivative, such as:

(∇2v

)a

= 2∑

b

mb

ρb

(1

rab

∂Wab

∂ra

)vab (4.7)

It follows that:

(1

ρ∇ · µ∇

)v

a

=∑

b

mb (µa + µb) rab · ∇aWab

ρabρab (r2ab + 0.01h2)

vab (4.8)

We will refer to this formulation as the Morris method.

50 Test Cases

- The third approach, introduced by Watkins et al. (1996), avoids direct calcula-

tion of second derivatives by using only first derivatives approximation. The laplacian

of the velocity field is written as follow:

∇2v = ∇ (∇ · v)−∇× (∇× v) (4.9)

to evaluate the second term of this expression it is possible to firstly calculate the

term ∇× v using the SPH approximation:

(∇× v)a =∑

b

mb

ρb

(va − vb)×∇aWab (4.10)

and then it is possible to calculate ∇×∇× v

(∇×∇× v)a =∑

b

mb

ρb

[(∇× v)a − (∇× v)b]×∇aWab (4.11)

This formulation will be called the Watkins method.

4.3.1 Periodic Shear Flow

In order to evaluate the second derivative of given velocity field, a simple periodic

shear flow test is considered here. The flow is assumed to be periodic in both x and

y directions, and confined in a 2D box of dimensions 2x1m; 1250 particles are set in a

lattice with an initial spacing of 0.04m. An initial velocity vx = sin(2πy) m/s is given

to the particles, with the component vy = 0, resulting in a Reynolds number Re = 103.

The initial value of the density is set to ρ = 1.02 Kg/m3, and not 1.0, in order to give

an initial value for the pressure different from zero (Monaghan, 2006a). Viscosity is

modelled by adding an artificial pressure term in the momentum equation, which is

rewritten in the SPH formalism as (Monaghan (1997) and Monaghan (2005)):

dva

dt= −∑

b

mb

(Pa

ρ2a

+Pb

ρ2b

+ Πab

)∇aWab (4.12)

Πab being defined as:

Πab = −kvsigvab · rab

ρab |rab| with k = 0.01 (4.13)

where vsig = ca + cb is the signal velocity, ca is the speed of sound of the particle

a, and K is a parameter. This kind of viscosity is not only applied to approaching


particles but also to receding ones, thus giving the possibility to model a laminar

viscosity, comparable to the other formulations given in details above.

The K parameter in the following applications is calculated in order to have a certain

value of the Reynolds number (Re=1000), but it’s possible to relate this parameter to

the kinematic viscosity ν of the considered fluid with the relation that follows:

ν =15

112Kvsigh (4.14)

depending upon the signal velocity vsig and the particles spacing.

For this test, it is possible to calculate the following:

∇ (∇ · v) = 0 (4.15)

∇×∇× v =(4π2 sin (2πy) , 0, 0

)(4.16)

Figure 4.7 presents results on the SPH particle position at t=2.0 s (on the center

higher panel), and the density and velocity field (on the bottom left and bottom right

panels respectively) as a function of y, always at t=2.0 s. There are no fluctuations in

density and the velocity field is in perfect agreement with the analytical solution.

In Figure 4.8 results for the second derivative of the velocity field calculated following

Watkins, Morris and Takeda (on the left, the center and the right panels respectively)

formulations are presented. It is visible that the best result is given by Watkins method,

which presents a slight underestimation of peak values.

A further test has been carried out removing the artificial viscosity Πab, and instead

modeling the viscous terms of NSE using the Watkins, Morris and Takeda techniques.

Figure 4.9 presents results on second derivatives of the velocity field using the same

plot layout of Figure 4.8. Results are similar to the previous case, the Watkins method

performing better than the other two.

4.3.2 Couette Flow

A Couette flow at low Re is modelled using the complete set of NSE (Eq. 3.19)

considering the three formulations (Watkins, Takeda and Morris) to model the viscous

terms. The Couette flow is a two dimensional flow evolving in the x direction, and

52 Test Cases

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

0

0.2

0.4

0.6

0.8

1

x

y

Time:2s

0 0.2 0.4 0.6 0.8 10.5

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

y

ρ

Time:2s

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

y

Vx

Time:2s

Figure 4.7: Periodic flow. Particles positions (center panel), density (left panel) and

velocity field (right panel), as a function of y. t=2.0 s.

0 0.2 0.4 0.6 0.8 1−50

−40

−30

−20

−10

0

10

20

30

40

50

y

∂ 2 v

/∂ x

2

Time:2s

0 0.2 0.4 0.6 0.8 1−50

−40

−30

−20

−10

0

10

20

30

40

50

y

∂ 2 v

/∂ x

2

Time:2s

0 0.2 0.4 0.6 0.8 1−50

−40

−30

−20

−10

0

10

20

30

40

50

y

∂ 2 v

/∂ x

2

Time:2s

Figure 4.8: Periodic flow modelled using the artificial viscosity. Second derivative of

the velocity field as a function of y at t=2.0 s. Dots report the SPH solution, the

continuous line reports the analytical one. Left panel: Watkins method. Center panel:

Morris method. Right panel: Takeda method.


0 0.2 0.4 0.6 0.8 1−50

−40

−30

−20

−10

0

10

20

30

40

50

y

∂ 2 v

/∂ x

2

Time:2s

0 0.2 0.4 0.6 0.8 1−50

−40

−30

−20

−10

0

10

20

30

40

50

y

∂ 2 v

/∂ x

2

Time:2s

0 0.2 0.4 0.6 0.8 1−50

−40

−30

−20

−10

0

10

20

30

40

50

y

∂ 2 v

/∂ x

2

Time:2s

Figure 4.9: Periodic flow modeled using NSE. Second derivative of the velocity field as

a function of y at t=2.0 s. Left panel: Watkins method. Center panel: Morris method.

Right panel: Takeda method.

consists in a fluid between two infinite plates, initially at rest. The fluid starts moving

due to the motion imposed on the upper plate, which moves with constant velocity V0

in the x direction. After a while, the fluid reaches a steady state with linear velocity

profile.

The analytical solution for the fluid velocity profile in x direction is given by Batchelor

(1967):

vx (y, t) =V0

Ly +

∞∑

n=1

2V0

nπ(−1)n sin

(nπ

Ly)

exp

(−ν

n2π2

L2t

)(4.17)

V0 being the velocity imposed to the upper plate, L the distance between the plates,

ν the fluid kinematic viscosity. We here present the test modeled using SPH, with 2450

particles initially disposed in a lattice with a particle spacing equal to 0.05 m, with

periodic boundary conditions in the x direction. The length of the box is L = 10−3 m,

the velocity of upper plate is set as V0 = 1.25× 10−5 m/s. The considered fluid has a

kinematic viscosity ν = 10−6 m2/s and the initial value for density is ρ = 103 kg/m3.

The flow presents a Reynolds number Re = 1.25× 10−2.

It has been possible to carry out this test employing only Takeda, Morris and Monaghan

formulations, since the Watkins formulation requires the prescription of the vorticity

values at the boundaries.

Figure 4.10 presents the velocity field as a function of y at subsequent time steps,

considering Morris (left panel), Takeda (right panel) and Monaghan (bottom panel)

formulations to model viscous terms. It is evident the perfect agreement between the

analytical and the SPH solutions using the Morris method that represents the best

54 Test Cases

0 0.2 0.4 0.6 0.8 1

x 10−3

0

0.2

0.4

0.6

0.8

1

1.2

x 10−5

t=0.

0200

s

y

vx

Velocity field

t=0.

0400

st=

0.11

00s

t=0.

2200

st=

∞

0 0.2 0.4 0.6 0.8 1

x 10−3

0

0.2

0.4

0.6

0.8

1

1.2

x 10−5

t=0.

0200

s

y

vx

Velocity field

t=0.

0400

st=0.

1100

st=

0.22

00s

t=∞

0 0.2 0.4 0.6 0.8 1

x 10−3

0

0.2

0.4

0.6

0.8

1

1.2

x 10−5

t=0.

0200

s

y

vx

Velocity field

t=0.

0400

st=

0.11

00s

t=0.

2200

st=

∞

Figure 4.10: Couette flow. Comparison of SPH (dots) and analytical (line) solutions

using the Morris (left panel), Takeda (right panel) and Monaghan (center panel) meth-

ods.


result. Otherwise, results of the model which employs the method of Takeda present

an underestimation of velocity close to the upper boundary, thus being the worst.

4.3.3 Poiseuille Flow

The SPH simulation of the two dimensional Poiseuille flow at low Re is here presented.

The fluid is initially at rest, and it is bounded by two horizontal walls, which remain

fixed during the flow evolution. The vertical distance between them is equal to L. At

t=0.0 s, the fluid starts moving due to a constant body force F in the x direction.

The analytical solution for velocity in the x direction is given by:

vx (y, t) =F

2νy (y − L) +

∞∑

n=0

4FL2

νπ3 (2n + 1)3

sin(

πy

L(2n + 1)

)exp

(−(2n + 1)2 π2ν

L2t

)(4.18)

The SPH simulation of this particular flow has been carried out using 2450 parti-

cles, which are initially set in a lattice with particle spacing 0.05 m. Periodic boundary

conditions in the x direction are used. The length of the box is L = 10−3 m, the force

F imposed to the fluid in the momentum equation is set to F = 10−4 m/s2. The fluid

considered here has a kinematic viscosity ν = 10−6 m2/s and an initial value of density

ρ = 103 Kg/m3. The fluid reaches a maximum velocity equal to V0 = 1.25× 10−5 m/s.

The flow presents a Reynolds number Re = 1.25× 10−2.

As in the Couette flow test, in this test only the Takeda, Morris and Monaghan

formulations have been used, for the same reasons mentioned above.

Figure 4.11 presents results by comparing the SPH particles velocity as a function

of y, and the analytical solution. Left and right panels report results of the simula-

tion carried out using the Morris and the Takeda methods, respectively, and bottom

panel using Monaghan formulation . It is evident the perfect agreement between the

analytical solution and the SPH simulation using the Morris method. SPH with the

use of the Takeda formulation presents a general overestimation of the velocity field,

in particular at t = ∞.

56 Test Cases

0 0.2 0.4 0.6 0.8 1

x 10−3

0

0.2

0.4

0.6

0.8

1

1.2

x 10−5

t=0.0200s

y

Vx

Velocity field

t=0.0400s

t=0.1100s

t=0.2200s

t=∞

0 0.2 0.4 0.6 0.8 1

x 10−3

0

0.2

0.4

0.6

0.8

1

1.2

x 10−5

t=0.0200s

y

Vx

Velocity field

t=0.0400s

t=0.1100s

t=0.2200s

t=∞

0 0.2 0.4 0.6 0.8 1

x 10−3

0

0.2

0.4

0.6

0.8

1

1.2

x 10−5

t=0.0200s

y

Vx

Velocity field

t=0.0400s

t=0.1100s

t=0.2200s

t=∞

Figure 4.11: Comparison of SPH (dots) and analytical (line) solutions for Poiseuille flow

using Morris (left panel), Takeda (right panel) and Monaghan (center panel) methods.

4.4 SPH Benchmark Test Cases 57

4.4 SPH Benchmark Test Cases

With the 3D SPH code written in Fortran by Panizzo (2004), two different benchmark

test cases were implemented and tested; the first one involving the collapse of a dam

and the successive impact with a tall structure, comparing the results with the exper-

imental ones given by Gomez-Gesteira and Dalrymple (2004) in term of force on the

structure and velocity in the close proximity of it.

The second set of simulations represent a series of experiments carried out with an

ellipsoidal rigid landslide falling into the water, they were conducted at the LIAM lab-

oratory at L’Aquila University, Italy.

4.4.1 Wave Impact on a Tall Structure

Interaction between waves and structures is a topic of increasing interest. In the fol-

lowing we tried to reproduce with our 3D SPH code the benchmark test case 1 given

by Gomez-Gesteira and Dalrymple (2004). In Figure 4.12 a sketch of the experimental

setup is shown. In a box of dimension 1.6 × 0.61 × 0.5 a dam, of height equal to 0.3,

collapses on a tall structure. The bottom of the box is covered by a thin layer of water.

In Figure 4.13 two different views of the initial particles positions are shown.

The main difficulties were found in order to reproduce the wet bottom correctly, such

a thin layer of particles can be affected by problems of stability, in fact, in our case,

just one layer of particles was used. Then the problems of boundaries, a problem like

this is very sensitive to the choice of boundaries. We decided to use Monaghan repul-

sive force for the structure and ghost particles for the rest of boundaries, see Figure 4.14.

The experimental results are given in terms of force exerted on the structure and

on velocity calculated in proximity of it, as already mentioned at the beginning of this

section. In Figure 4.15 the experimental results are shown.

58 Test Cases

Figure 4.12: Sketch of the experimental setup.

The simulations were carried on with different initial particle spacing (i.e. number

of particles), different kind of viscosity. At the end the best results were found with

an initial particles spacing of 0.0225m, ∼ 17000 particles, with the artificial viscosity

α parameter equal to 0.03 and with the XSPH correction.

Results are shown on the following plots.


Figure 4.13: Position Particles at the beginning of the simulation. Top panel for a

lateral view, bottom panel for the perspective one.

60 Test Cases

Figure 4.14: Particles positions at the beginning of the simulation. Ghost (red circles),

Repulsive (red dots) and Fluid Particles (blue dots).


Figure 4.15: Experimental results in term of Force exerted on the structure (top panel)

and velocity in front of it (bottom panel).

62 Test Cases

Figure 4.16: Particles positions at four different instants of the simulation.


Figure 4.17: Comparison with the experimental results. Top plot for the force on the

structure and bottom one for the velocity.

64 Test Cases

In Figure 4.17 comparisons with experimental results are shown in terms of force

on the structure and velocity of the fluid, measured in a point of known coordinates.

From these results it is possible to see a good agreement between experimental and

numerical solution. The oscillations of pressure are due to the weakly-compressible na-

ture of SPH and to the difficulties in the representation of solid boundaries, moreover

velocity profile is slightly underestimated even if with a good agreement with the profile.


4.4.2 Landslides Generated Waves

The first simulations of the landslide generated waves were carried out with the 3D code

following a series of experiments conducted at Maritime and Environmental Hydraulics

(LIAM) of the Engineering faculty at l’Aquila University. These experiments were

carried out in a tank 10.8m long, 5.4m wide, with a maximum depth of 0.8m. Figure

4.18 presents a sketch of the physical model. PVC sheets sustained by steel bars

represent an impermeable beach with slope of 1:3 (see Figure 4.19), this slope was

chosen referring to the slope of ”Sciara del Fuoco” flank (Stromboli Island), because

the research work was funded by the Italian Civil Protection Agency about the event

occurred at Stromboli island in 2002.

Figure 4.18: Schematic view of the physical model.

15 resistance wave gauges were used to measure the instantaneous surface elevation

in the tank. 5 of these are located on a section normal to the undisturbed shoreline

at a distance of 1 m from the landslide in order to reconstruct the modal shape of the

transient edge waves. 10 run-up gauges measure the movements of the instantaneous

shoreline and a high resolution video camera frames the area where the landslide falls

into the water. The landslide (see Figure 4.20) is represented by a rigid fibreglass body

66 Test Cases

Figure 4.19: Picture of the experimental model.

of elliptical shape (of equation x2/a2 + y2/b2 + z2/c2 = 1, with a = 0.2m, b = 0.4m,

c = 0.05m for a total volume of V = 2/3πabc = 0.0084m3 ), with a length of 0.8 m,

a width of 0.4 m and the maximum height of 0.05 m . The slide is equipped with an

accelerometer that is used to reconstruct its position in time. The landslide model was

reproduced similarly to Watts and Grilli (2005).

In Figure 4.21 a plot representing the water waves field generated by a landslide is

shown. It can be seen that two kind of waves are generated: the edge waves, which

propagate along the shore, and a principal train of waves which is free to propagate

in the offshore direction. In Figure 4.22 is represented a picture made during the

laboratory experiment, where it is possible to note the physical aspects introduced

in the Figure 4.21. Both the radiating and the edge waves were simulated with the

numerical model.

On the following plots (see Figure 4.23) particles positions at four different instants

is shown. The simulation was carried on with 25844 particles, α parameter for viscosity


Figure 4.20: Picture and dimension of the physical model of the landslide.

Figure 4.21: Sketch of the water waves field generated by a landslide.

equal to 0.015 and XSPH correction.

The wave trains radiating in the offshore direction, were observed in the numerical

simulations, in order to check if they were well simulated. A comparison between

the water surface elevation in the experimental and the numerical simulations was

performed. This comparison was performed looking at the water surface in a point

which was 2.96m distant from the landslide impact point. That point corresponds, in

68 Test Cases

Figure 4.22: Picture during a laboratory experiments of the water wave field generated

by a landslide.

the physical model, to the place were a gauge is located. By most of the comparisons,

the numerical simulations overestimated the water waves, especially in the wave crest.

The numerical simulation which better compares the water wave features in that point

is the one represented in Figure 4.24.

Considering the edge waves, Figure 4.25 is a superimposition of three frames of the

experiment with three frames of the numerical simulation, at the same instants.


Figure 4.23: Particles positions at four different instants of the simulation.

70 Test Cases

Figure 4.24: Comparison between experimental (blue line) and numerical (red line)

water surface elevation.

Figure 4.25: Comparison analysis of the generated edge waves.

4.5 Summary 71

4.5 Summary

In this chapter a series of test cases are implemented and tested. The first series in-

volving two dimensional and three dimensional tests are shown in 4.2, they are used to

understand the capability of SPH to reproduce first derivative of a given field variable.

In 4.3 three different formulations for the reproduction with SPH of second deriva-

tive field variable are given. These formulations are tested using a periodic shear flow

test, and then implemented in the viscous term of NSE and tested with simple two

dimensional flows. Finally in 4.4 two tests of our 3D SPH code are given, the first

one involving the wave impact on a tall structure and the second one a rigid landslide

falling into the water.

72 Test Cases

Chapter 5

Non Newtonian Mud Flows with

SPH

5.1 Introduction

In the present thesis the capability of SPH to model non-newtonian mud flows will be

analyzed and tested. This kind of fluids consist in solid grains immersed into a water

matrix; it is possible to say that from a certain concentration of solid grains into the

mixture they behave like non Newtonian fluids Komatina and Jovanovic (1997). For

these fluids the Bingham model (Bingham, 1922) seems to be one of the best to rep-

resent the viscoplastic behaviour Masson et al. (2006) in which no deformation takes

place until a certain stress is applied, as extensively reported in literature Barnes et al.

(1989). In the present work a particular kind of Bingham viscosity model is used.

In order to avoid the discontinuity between the yielded and the unyielded regions, a

smooth switch is introduced, giving a higher viscosity for low shear stresses values. In

literature these kind of constitutive equations are known as biviscosity models Beverly

and Tanner (1992). A two dimensional annular viscometer test is used in order to

carry on the steady flow analysis and to set the parameters necessary to give a cor-

rect evaluation of the viscosity field, following the analytical solutions calculated for

this particular model implemented and comparing it with other non-newtonian and

newtonian models. Then an unsteady flow analysis is presented consisting in a dam

break flow of a mixture of kaoline and water, with different percentage of solid grains

following the experimental results of Komatina and Jovanovic (1997).

74 Non Newtonian Mud Flows with SPH

5.2 Non-Newtonian fluids

In order to talk about the non-Newtonian behavior of a fluid we have firstly to under-

stand what Newtonian behavior is, in the context of the shear viscosity. Newtonian

behavior in experiments conducted at constant temperature and pressure has the fol-

lowing characteristics:

- The only stress generated in simple shear flow is the shear stress τ , the two normal

stress differences being zero.

- The shear viscosity does not vary with shear rate.

- The viscosity is constant with respect to the time of shearing and the stress in the

liquid falls to zero immediately the shearing is stopped. In any subsequent shearing,

however long the period of resting between measurements, the viscosity is as previously

measured.

- The viscosities measured in different types of deformation are always in simple pro-

portion to one another, so, for example, the viscosity measured in a uniaxial extensional

flow is always three times the value measured in simple shear flow.

A liquid showing any deviation from the above behavior is non-Newtonian.

As soon as viscometers became available to investigate the influence of shear rate on

viscosity, workers found departure from Newtonian behaviour for many materials, such

as dispersions, emulsions and polymer solutions. In the vast majority of cases, the

viscosity was found to decrease with increase in shear rate, giving rise to what is now

generally called ”shear-thinning” behavior although the terms temporary viscosity loss

and ”pseudoplasticity” have also been employed. We shall see that there are cases,

although very few, where the viscosity increases with shear rate. Such behavior is gen-

erally called ”shear-thickening” although the term ”dilatancy” has also been used. For

shear-thinning materials, the general shape of the curve representing the variation of

viscosity with shear stress is shown in Figure 5.1. The corresponding graphs of shear

stress against shear rate and viscosity against shear rate are also given. The curves

indicate that in the limit of very low shear rates (or stresses) the viscosity is constant,

while in the limit of high shear rates (or stresses) the viscosity is again constant, but

at a lower level. These two extremes are sometimes known as the lower and upper

Newtonian regions, respectively, the lower and upper referring to the shear rate and

not the viscosity.

5.2 Non-Newtonian fluids 75

Figure 5.1: Typical behaviour of a non-Newtonian liquid showing the interrelation

between the different parameters. The same experimental data are used in each curve.

(a) Viscosity versus shear stress. Notice how fast the viscosity changes with shear

stress in the middle of the graph; (b) Shear stress versus shear rate. Notice that, in

the middle of the graph, the stress changes very slowly with increasing shear rate. The

dotted line represents ideal yield-stress (or Bingham plastic) behaviour; (c) Viscosity

versus shear rate. Notice the wide range of shear rates needed to traverse the entire

flow curve.


The terms ”first Newtonian region” and ”second Newtonian region” have also been

used to describe the two regions where the viscosity reaches constant values. The

higher constant value is called the ”zero-shear viscosity”. Note that the liquid of Fig-

ure 5.1 does not show ”yield stress” behavior although if the experimental range had

been 104s−1 to 10−1s−1 (which is quite a wide range). In Figure 5.1(b) it is possible to

see the so-called ”Bingham” plastic behaviour for comparison purposes. By definition,

Bingham plastics will not flow until a critical yield stress σy, is exceeded. Also, by

implication, the viscosity is infinite at zero shear rate and there is no question of a first

Newtonian region in this case. There is no doubt that the concept of yield stress can

be helpful in some practical situations, but the question of whether or not a yield stress

exists or whether all non-Newtonian materials will exhibit a finite zero-shear viscosity

is not so easy to ask for.

It’s just possible to say that for dilute solutions and suspensions, there is no doubt that

flow occurs at the smallest stresses: the liquid surface levels out under gravity and there

is no yield stress. For more concentrated systems, particularly for such materials as

gels, lubricating greases, ice cream, margarine and stiff pastes, there is understandable

doubt as to whether or not a yield stress exists. It is easy to accept that a lump of

one of these materials will never level out under its own weight. Nevertheless there is

a growing part of researchers that suggest that even concentrated systems flow in the

limit of very low stresses. These materials appear not to flow because the zero shear

viscosity is so high.

Equations that predict the general shape of the viscosity for this kind of fluids are a

lot and reconstructed in laboratories for different materials. On the following section

we will introduce some of them.

5.3 Rheological Model

Rheological models consist in equations that predict the so called ”general flow curve”

that represents the shape of the viscosity versus the shear rate. One such is the ”Cross”

model given by Cross (1965):

5.3 Rheological Model 77

η − η∞η0 − η∞

=1

(1 + (Kγ)m)(5.1)

or, that is the same,

η0 − η

η − η∞= (Kγ)m (5.2)

this kind of equations usually need at least four parameters, in this case, η0 and η∞

represent the asymptotic values for viscosity at very low and very high shear stress,

respectively, K is a constant parameter with the dimension of time and m is a dimen-

sionless constant.

It is possible to do some approximations to the Cross model, and in such a way, it is

possible to reduce to other rheological model, very popular in literature. For example,

if we consider η << η0 and η >> η∞ the Cross model become:

η =η0

(Kγ)m (5.3)

that can be written as:

η = K2γn−1 (5.4)

This is the well known ”power-law” model and n is called the power-law index. K,

is called the ”consistency” (with the unit of Pa.sn).

Further if η << η0 we have:

η = η∞ +η0

(Kγ)m (5.5)

which can be rewritten as:

η = η∞ + K2γn−1 (5.6)

Equation 5.6 represent the so called ”Sisko Model” (Sisko (1958)).

If n parameter is equal to zero in the Sisko model, we obtain:


η = η∞ +K2

γ(5.7)

which, with a simple redefinition of parameters can be written as

τ = τy + ηP γ (5.8)

where τy is the yield stress and ηP , the plastic viscosity (both constant). This is the

”Bingham model” equation. The Bingham equation describes the shear stress/shear

rate behavior of many shear-thinning materials at low shear rates, but only over a

one-decade range (approximately) of shear rate (see Figure 5.3). In some literature

Bingham behavior and Power law model are considered as limit cases of the Herschel-

Bulkley model, see table 5.2 for comparison of this three constitutive laws.

Figure 5.2: Constitutive Laws (from Vola et al. (2004)).

5.3 Rheological Model 79

Figure 5.3: Flow curves for a synthetic latex (taken from Barnes and Walters (1985)):

(a and b) Bingham plots over two different ranges of shear rate, showing two differ-

ent intercepts; (c) Semi-logarithmic plot of data obtained at much lower shear rates,

showing yet another intercept; (d) Logarithmic plot of data at the lowest obtainable

shear rates, showing no yield-stress behavior; (e) The whole of the experimental data

plotted as viscosity versus shear rate on logarithmic scales.


5.4 Rheological Model in SPH

In the past few years many authors have studied the possibility to represent, with SPH,

the behavior of a non-Newtonian fluid. Ellero et al. (2002) used SPH to reproduce a

viscoelastic material trough a Maxwell model. Rodriguez-Paz and Bonet (2003) pre-

sented the development and application of a SPH model to the simulation of debris flow

and avalanches, also with the use of a frictional approach for the boundary conditions.

Shao and Lo (2003), instead, used SPH for the simulation of a non-Newtonian free

surface model through the use of the Cross model, already seen in the previous section.

The same Cross rheological model was used by Ataie-Ashtiani and Shobeyri (2008) in

order to represent landslides generated waves. Recently Hosseini et al. (2007) proposed

and tested a new method to deal with the non-Newtonian fluid, we started from that

trying to give a final solution for the treatment of granular material immersed in a

water matrix.

An SPH model for granular landslides impacting with the water was applied by Falappi

and Gallati (2007), in that work a different value of effective viscosity was given to ev-

ery particle (i.e. every grain), depending upon the value of internal friction angle and

the second invariant of the rate of deformation tensor, thus considering single phase

material to model aerial landslides.

5.5 Governing Equations of Our SPH Model

Generally speaking, in order to model a Bingham fluid flow we have to refer to the

constitutive equation of a generalized newtonian fluid, it differs from a newtonian one

in the way that the shear stress τ depends upon the shear rate tensor (i.e. the gradient

of the velocity field) at a particular time:

D =1

2

(∇u+∇tu

)(5.9)

where D has the form shown below:

D =

∂u∂x

12

(∂u∂y

+ ∂v∂x

)

12

(∂u∂y

+ ∂v∂x

)∂v∂y

(5.10)

5.5 Governing Equations of Our SPH Model 81

Usually this tensor is used in the formulation of the viscosity field in the form of

its second invariant:

|D| =√

D : D =√∑

ij

DijDij (5.11)

The constitutive law for the generalized newtonian fluids is thus of the following

form:

τ = µ (|D|)D (5.12)

in which the relation between viscosity and the second invariant of the shear rate

tensor is specified .

In the presence of classical newtonian fluids this relation becomes:

τ = 2µD (5.13)

The Bingham plastic behavior is represented by the presence of a critical yield stress

under which no deformation takes place. Above this level viscosity depends upon the

second invariant of the shear rate tensor, as usual for the generalized newtonian fluids,

and the yield stress. The constitutive equations are the following:

|τ | ≤ τy ⇔ D = 0 (5.14)

|τ | > τy ⇔ τ =

(τy

|D| + 2µ

)D (5.15)

In this thesis we propose a slightly different Bingham model that is present in

literature with the name of ”bi-viscosity model” Beverly and Tanner (1992). The main

difference between this kind of model and the classical Bingham one is that in the un-

yielded region a greater viscosity is given to the fluid, instead of having no deformation

at all, thus having a smooth switch between the two zones.

|D| ≤ τy

2αµ⇔ τ = 2αµD (5.16)

|D| > τy

2αµ⇔ τ =

(τy

|D| + 2µ

)D (5.17)


In (5.16) and (5.17) are reported the expression of the stress tensor in the un-

yielded and in the yielded regions, where µ is the classical dynamic viscosity and α is a

coefficient that gives the order of magnitude of the increased viscosity in the un-yielded

region.

In our SPH code Bingham rheology is modelled in the way to have an effective viscosity

in the momentum equation (see Monaghan (2005))

dva

dt= −∑

b

mb

(Pa

ρ2a

+Pb

ρ2b

+ Πeffab

)∇aWab (5.18)

where

Πeffab = −Keffvsigvab · rab

ρab |rab| (5.19)

The difference between this non-newtonian viscosity and the usual shear viscosity

for a newtonian fluid is in the evaluation of K that is calculated using the expression of

the kinematic shear viscosity for the cubic spline kernel ν = 15112

Kvsigh (see Monaghan

(2005)). In our model this coefficient Keff depends upon the value of the second

invariant of the shear rate tensor following (5.16) and (5.17).

5.6 Numerical Tests

The viscoplastic behavior of the non-newtonian viscosity was tested with the help of

two different numerical simulations, one for the steady flow analysis, reproducing the

flow in an annular viscometer and another one for the unsteady analysis, using the

experimental results of a dam break simulation of mixture of kaolinite and water.

5.6.1 Annular Viscometer

The first test proposed in the present work is a two dimensional annular viscometer.

It consists in an axisymmetrical problem of a flow between two coaxial cylinder that

can be treated as a fully bi-dimensional test. The outer cylinder is kept fixed while

the inner one has an angular velocity of ω = 1 rad/s. This test give us the possibility

to carry on a steady state analysis of our Bingham model and to set the rheological

parameters in the correct way.

5.6 Numerical Tests 83

The geometrical domain of the viscometer is shown in Fig. 5.4. The inner cylinder

has a radius of 0.5 cm while the outer fixed one has a radius of 1.0 cm.

Fixed fluid particles are used in order to model boundaries.

Figure 5.4: Geometrical domain for the annular viscometer test.

The steady state solution for the annular viscometer is expressed in term of tangen-

tial velocity and is present in literature for newtonian and non-newtonian fluids Bird

et al. (1987). The analytical solutions for our Bingham model are shown below (5.20

and 5.21). They differ from the expressions typical of the classical Bingham model for

the presence of the factor α.

r < rs ⇔ vt = ωr+

+τyr√2µ

ln

(r

r1

)+

(r2s

2

) (1

r2− 1

r21

) (1 +

1

α

)(5.20)

r > rs ⇔ vt =τyr

2sr

2√

2αµ

(1

r2− 1

r22

) (1 +

1

α

)(5.21)


Figure 5.5: Analytical solutions. Line: classical Bingham model; dots: our Bingham

model with alpha=50; stars: our Bingham model with alpha=100; circles: our Bingham

model with alpha=1000.

In Fig. 5.5 the plots of the analytical tangential velocity versus radius are shown

for different values of α. The density is 1000 kg/m3, viscosity µ = 1.0 Pa · s and

yield stress τy = 10 N/m2, following the parameters used in Vola et al. (2004). It is

possible to recognize the shear zone (i.e. the yielded region) and the plug zone (i.e.

the un-yielded region), divided by a value of r, rs that hereinafter we called switch

radius. In the case of the classical Bingham model the plug zone is characterized by a

zero velocity region (in the graph straight line), and it is possible to have it with our

Bingham model giving a value of α →∞.


Results

The simulations were conducted following the viscosity parameters given by Vola et al.

(2004) with different values of α and with different initial particles spacing.

In Figures 5.6, 5.7 and 5.8 the plots corresponding to three different values of α are

shown, respectively α = 50, α = 100 and α = 1000. The plots on the top represent

particles positions and on the bottom the comparison with the analytical results. In

literature the value of α = 1000 is the most used (see Beverly and Tanner (1992)

and Hosseini et al. (2007)). It is important to notice that this value depends upon

the characteristic of the particular non-newtonian fluid simulated, because it gives the

difference between the values of viscosity at low and high shear rates, but in our sim-

ulation (as clear by fig. 5.6, 5.7 and 5.8) the best results are for low value of α.

It is true that for α = 1000 (fig. 5.8) the analytical solution is closer to the classical

Bingham one, because of the low value of velocity in the plug zone, but the shear zone

is not well evaluated, and consequently the switch radius is not well evaluated.

Then we ran a simulation with a variable-h code in order to have a good resolution

for the correct evaluation of the switch radius. After some tests we divided the region

in three zones one having 1.5 times greater h than the other. In order to test the

variable resolution scheme for this particular test case we ran a simulation considering

a newtonian fluid. In Figure 5.9 the results of this simulation are shown (on the top

particles positions on the bottom comparison with the analytical result for a newtonian

fluid). There is a perfect agreement between the analytical and numerical results, and

from the plot 5.10 it is possible to see that we obtained the same results even if with

a less number of particles.

In Fig. 5.11 the results of the simulation with the non-newtonian fluid are shown, in

terms always of particles position and comparison of tangential velocity, after more

than 3 seconds of simulation. It is possible to distinguish the three different h zone,

and the shear zone (with red dots). Analytically the switch radius is 0.68 cm and in

our simulations is 0.7 cm.


Figure 5.6: Particles positions (top plot) and tangential velocity (bottom plot) for the

simulation with the parameter α equal to 50.



simulation considering a newtonian fluid with the variable h scheme.


Figure 5.10: Comparison between tangential velocity for the simulations considering a

newtonian fluid with the variable h scheme with 4074 particles (top plot) or without

with 7068 (bottom plot).



simulation with the parameter α equal to 50 and the variable h scheme.


5.6.2 Dam Break

In order to carry on an unsteady analysis a dam break simulation was implemented

and tested, following the experimental results of Komatina and Jovanovic (1997).

They consider different water-kaolinite clay mixture and different dam break geometry.

In our simulations we considered the concentration of 27.4% of kaolinite in the water,

with a density of 1452 kg/m3, µ = 0.07 Pa · s and τy = 25 N/m2. In Fig. 5.12 a sketch

of the dam break simulations is shown. Also in this case we used a variable-h code

with a greater resolution at the front. Ghost particles are used to model boundaries,

giving them zero tangential velocity to reproduce no–slip condition.

Figure 5.12: Geometrical domain for the dambreak test.

Results

In Fig. 5.13 particles positions at different times are shown, while in Fig. 5.14 a zoom

on the front is shown at the same times.

In Fig. 5.15 the comparison between experimental and numerical results are shown, in

term of non dimensional front position versus non dimensional time.


Figure 5.15: Comparison with experimental results. Red Circles: experimental results.

Blue Dots: SPH solution.


It is possible to say that there is a good agreement between experimental and nu-

merical solutions, expect for a general greater velocity of the front, this can be due

also to the experimental setup, of which the geometry is not perfectly clear. The main

difficulties were found on the treatment of the boundaries, because no-slip boundary

conditions in SPH represent a challenging problem yet. Anyway with the variable res-

olution scheme we had the possibility to increase the resolution of the front and to

reduce the problem in this sense. The final shape of the front and of the shear zone

are finally well represented.

5.7 Numerical Applications

In order to understand the capability of our SPH code to model submarine landslides,

the experiments of Rzadkiewicz et al. (1997) were tested and performed, consisting in

a mass of sand allowed to slide from a slope of 45. A sketch of the geometrical domain

of the experiment is shown in Figure 5.16. The bulk of the landslide is 0.65 m high

and long, lying on the incline surface.

Figure 5.16: Geometrical domain for the experiment by Rzadkiewicz et al. (1997).

This physical model experiment was used to test the capability of SPH to simulate

deformable landslides and their interactions with the water, in the work of Ataie-

Ashtiani and Shobeyri (2008). In that work the authors used the Cross rheological

model Cross (1965), with viscosity simulated as a function of the shear rate using four

5.7 Numerical Applications 97

different parameters (see also Shao and Lo (2003)). The Cross rheological model is

widely used to simulate the behaviour of polymeric materials, but it is difficult to un-

derstand the values of the parameters to be used. In the work of Ataie-Ashtiani and

Shobeyri (2008), no testing of the proposed rheological model was presented.

In this work, to model the deformable landslide behaviour, the rheological model pre-

sented and tested in the previous sections is considered. The simulations were per-

formed with 16819 particles, corresponding to an initial spacing of 0.015m. Ghost

particles represent our boundaries, being a portion of fluid with mirrored character-

istics on the opposite side of the boundary. In Figure 5.17 particles positions at the

beginning of the simulation is shown.

Figure 5.17: Particles Positions at the beginning of the simulation.

The fluid density is ρ = 1000kg/m3 and the viscosity is set as µ = 0.001Pas. The

sand density is ρ = 1950kg/m3, this is the value defined in the experiments. The

rheological parameters were not measured in the experiments but chosen by trials and

errors by Rzadkiewicz et al. (1997) for the numerical simulations. Always with this


approach we set this parameters as µ = 1.0Pas for the viscosity and ρ = 1000N/m2,

they differ from the values used in the numerical simulations of the experiment by

Rzadkiewicz et al. (1997) only for the viscosity and not for the yield stress that is

the same. This is due to the necessity for our particular rheological model to give a

non-zero value to the viscosity, that is indeed the approach used by Rzadkiewicz to

have an immediate liquefaction of the material.

5.7.1 Results

Experimental results are given in terms of elevation of the free surface at t = 0.4s and

t = 0.8s. In Figures 5.18 and 5.19 at the same times, particles positions are plotted, it

is possible to distinguish in the mass of sand (brown dots) with green dots the shear

zone that is typical of a Bingham fluid flow.

Figure 5.18: Particles Positions at time t=0.4s.

Comparison between experimental and computed waves are shown in Figures 5.20:


Figure 5.19: Particles Positions at time t=0.8s.


Figure 5.20: Comparison between experimental (red circles) and numerical (blue lines)

solutions in terms of elevation of the free surface.


−1.6 −1.4 −1.2 −1 −0.8 −0.60.6

0.8

1

1.2

1.4

1.6

1.8Time:0.4sec

x

y

Figure 5.21: Comparison between slide profile calculated with SPH (brown dots) and

calculated with Rzadkiewicz et al. (1997) model (black circles), at time t=0.4 s.


−1.6 −1.4 −1.2 −1 −0.8 −0.60.6

0.8

1

1.2

1.4

1.6

1.8Time:0.8sec

x

y

Figure 5.22: Comparison between slide profile calculated with SPH (brown dots) and

calculated with Rzadkiewicz et al. (1997) model (black circles), at time t=0.8 s.


In 5.21 and 5.22 comparisons between the slide profile at the two different instant,

calculated with SPH and by Rzadkiewicz et al. (1997) model, are shown. The dif-

ferences, especially at t=0.8 sec can be due to the different boundary conditions and

viscosity values.

In conclusion reproducing a mass of sand falling into the water it is possible to

obtain excellent agreement with experimental results, in terms of interaction with the

water and shape of the landslide, with only a slight underestimation of the wave ampli-

tude especially at the beginning of the slide motion. This can be due to the resolution

of the problem and to the boundary conditions, in any case it is important to underline

the great results achieved with not so many particles and with a simple and versatile

rheological model.


5.8 Summary

In this chapter a rheological model for a non-newtonian mud flows is shown with tests

and applications. In 5.2 a resume of the characteristics of the non-newtonian fluids is

given. In 5.3 a brief review of the rheological model present in literature is shown, while

in 5.4 our particular rheological model is presented. In 5.6 the annular viscometer test

is used for the validation of the model and finally in 5.7 the numerical simulation of

a landslide falling into the water is shown, following the experimental results given by

Rzadkiewicz et al. (1997).

Chapter 6

Conclusions and Future

Developments

This work has studied the numerical modelling of landslides generated waves with SPH

(Smoothed Particles Hydrodynamics) numerical model.

The first part of the thesis is dedicated to the study and testing of the accuracy

of the numerical model, trough simple two and three dimensional tests. Moreover in

order to understand the correct evaluation of second derivatives three different formu-

lations founded in literature were implemented and tested, it has been possible after

that to have a better view on the evaluation of the viscous term in Navier Stokes

equation with SPH and to analize the behavior of this viscosity with two dimensional

flow tests. This work is actually under review on the Journal of Computational Physics.

With the 3D SPH code written in Fortran by Panizzo (2004), two different bench-

mark test cases were implemented and tested; the first one involving the collapse of a

dam and the successive impact with a tall structure, comparing the results with the

experimental ones given by Gomez-Gesteira and Dalrymple (2004) in term of force on

the structure and velocity in the close proximity of it. The second simulation involved

a series of experiment about an ellipsoidal rigid landslide falling into the water, con-

ducted at the LIAM laboratory at L’Aquila University, Italy. Good agreement between

experimental and numerical solutions were achieved.

106 Conclusions and Future Developments

The possibility to introduce a deformable landslide in the simulations involving

landslides generated waves was analyzed and tested, through the SPH numerical model.

The best solution, in terms of accuracy and capability to reproduce a non-newtonian

mud flow of a mixture of granular particles immersed in a water matrix, was achieved

with a ”bi-viscosity” model, i.e. a Bingham model with a smooth switch between the

yielded and the unyielded region. This constitutive equations were tested with the nu-

merical simulation of a two dimensional annular viscometer test. Very good agreement

between the numerical and the calculated analytical solutions were found, especially

with the introduction in our SPH code of a variable h scheme. Using this rheological

model a test of a dambreak of a mixture of kaolinite and water was implemented and

tested. Also in this case good agreement was achieved with a variable h scheme, thus

having problem in the implementation of no-slip boundary conditions in SPH, through

ghost particles with zero tangential velocity.

At the end of our work all the information and indications of the testing phase

were used to simulate a deformable landslide falling into this water. With this purpose

the experiments of Rzadkiewicz et al. (1997) were implemented. Very good agreement

was achieved for the comparison with the experimental results in terms of elevation

of the free surface in two different instant of the simulation. Moreover the same good

agreement was achieved in terms of shape of the landslide and reproduction of the

shear zone. Also in this case no-slip boundary conditions with the ghost particles were

used.

Other constitutive laws were implemented and tested during this research work, the

results are not included in this thesis but in the future they could be part of the devel-

opment of this study, also for the extreme versatility of the presented SPH rheological

model.

A great effort will be done in order to bring in a 3D code the model, and to test

it with the cylindrical viscometer and with a 3D dam break of a non-newtonian fluid,

considering the 3D spreading of the fluid on an inclined plate.

Appendix A

Annular Viscometer

Annular viscometer consists in two mechanical components, separated by a fluid, that

rotates one by each other around a vertical axes. Measuring torque M necessary to

take one of the component in rotation with constant angular velocity w is thus possible

to obtain the rheological parametes of the fluid. Let us consider two coaxial cylinder

of radius R1 and R2 (with R2¿R1) and height H, and let us suppose that the inner

cylinder rotates with a constant angular velocity w (Figure A.1).

Figure A.1: Sketch of a coaxial cylinder viscometer.

108 Annular Viscometer

If the two cylinder are sufficiently long thus being possible to neglect the effect of

the two limit layer (of Eckman) generated in proximity of the extremes, and if the

rotation velocity of the inner cylinder is sufficiently small, the fluid movement is only

circular. In this case is easy to determine teorically the characteristic of the movement,

that is referred to a cylindric coordinate system (r,q,z) with z axes coincident with the

cylinder’s axes.

Newtonian Fluid

Let us indicate with v = (vr, vΘ, vz)T the three component of the velocity vector. In

the hypothesis of only a circular motion, it is possible to say:

v = (0, vΘ(r), 0)T (A.1)

The circular velocity satisfies identically the continuity equation, while the circular

component of the Navier-Stokes equations impose:

(d2

dr2 + 1r

ddr− 1

r2

)vθ = 0

vθ|r=R1= ΩR1

vθ|r=R2= 0

(A.2)

Equations A.2 admit the following solution:

vθ =c1

r+ c2r = c1r

(1

r2− 1

R22

)= Ω

R21

r

(R22 − r2)

(R22 −R2

1)(A.3)

In this way it’s easy to determine the tangential velocity uniformly distributed on

the inner cylinder:

vt = Trθ|r=R1

= µ

(dvθ

dr− vθ

r

)∣∣∣∣∣r=R1

= µΩ

[−2− 2R2

1

(R22 −R2

1)

]= −2µΩ

[R2

2

(R22 −R2

1)

]

(A.4)

and eventually torque on the inner cylinder:

M = 4πHµΩ

[R2

1R22

(R22 −R2

1)

](A.5)

Once known geometric characteristics of the device, eq. A.5 let us determine dy-

namic viscosity of the fluid, measuring torque and angular velocity of the inner cylinder.

Appendix B

Invariants

If we can calculate principal stresses, it means that we can represent the stress state

of matter by three independent values, and every different stress state is equivalent if

principal stresses are the same for all of them. This means, in particular, that various

physical phenomena taking place under stress (for example, probability for a quasi-

liquid to stick to a vertical wall and stay motionless in spite of action of gravitational

force; rupture of solid bodies; slow movement and transition to spurt for snow, sand

or mud on slopes, etc.) can be considered as a consequence of an action of principal

stresses only. Principal stresses represent stress state in a body (at a given point),

regardless of any possible set of normal and shear stresses at any arbitrary orientation

of this point. In other words, they are invariant in respect to the choice of orientation.

Then the question is: how to calculate principal stresses if all components of the stress

tensor are known for some arbitrary coordinate system? Theory of tensors gives an

answer to this question in the form of a cubic algebraic equation:

σ3 − I1σ2 + I2σ − I3 = 0 (B.1)

and principal stresses are three roots of this equation designated as σ1, σ2, σ3. It is

clear that the roots are expressed through coefficients of Eq B.1, I1, I2, and I3. These

coefficients are constructed by means of all components of stress tensor for arbitrary

orthogonal orientations in space as:

I1 = σ11 − σ22 + σ33 (B.2)

110 Invariants

I2 = σ11σ22 − σ11σ33 + σ22σ33 −(σ2

12 + σ223 + σ2

13

)(B.3)

I2 = σ11σ22σ33 − 2σ12σ23σ13 −(σ11σ

223 + σ22σ

213 + σ33σ

212

)(B.4)

As the roots of Eq B.1, the σ1, σ2 and σ2 do not depend on orientation of axes of

a unit cube (at a point) in space, on one hand, and they are expressed through values

of I1, I2, and I3, on the other. This leads to the conclusion that I1, I2, and I3 are also

invariant in respect to the choice of directions of orientation and that is why they are

usually called invariants of a stress tensor at a point. According to its structure (the

power of components), I1 is a linear, I2 is a quadratic, and I3 is a cubic invariant.

Certainly any combination of the invariants, I1, I2, and I3, is also invariant in respect

to the orientation of axes in space. Various elegant or cumbersome structures of in-

variants are possible to be built but it is important to know that three and only three

independent values of such kind do exist.

Invariants are characteristics of the physical state of a matter. It means that neither

any stress by itself nor its arbitrary combination but only invariants determine a pos-

sibility of occurrence of various physical effects such as, for example, phase transitions,

storage of elastic energy or dissipation (transition to heat) of work of external forces.

There is a fundamental principle saying that physical effects must be independent of

choice of a coordinate system and that is why invariants, which are values indepen-

dent of a coordinate system, govern physical phenomena occurring due to application

stresses.

In many practical problems, we deal a with two-dimensional or so-named plane stress

state. The very typical example of it are thin-walled items with stress-free outer sur-

faces. Thin means that the size in direction normal to the surface is much smaller than

other two dimensions. Stresses in planes parallel to the free surfaces can be assumed to

be absent in comparison with stresses at two other oriented planes. This can be proven

by a pure geometrical argument: because stresses are zero on both faces, they might

not vary appreciably over the small distance (thickness of the item). In the plane stress

state, all the components containing the index 3” vanish and the full stress tensor looks

like this:

111

σij =

∣∣∣∣∣∣∣∣∣

σ11 σ12 0

σ21 σ22 0

0 0 0

∣∣∣∣∣∣∣∣∣(B.5)

In this case, one principal stress, σ3, is zero and two others, σ1 and σ2, are the roots

of a quadratic (but not cubic) algebraic equation:

σ1,2 =σ11 + σ22

2±

(σ11 − σ22

2

)2

+ σ212

1/2(B.6)

Typical examples of thin (or two-dimensional) elements are various balloons, and

many others, we can use the analysis of the two-dimensional (or plane) stress state.

There are two particularly simple cases of the plane stress state, which are simple (or

unidimensional) tension and simple shear.

112 Invariants

Appendix C

Deformation Rate

If velocity (as a vector), at any point of a body, is the same, it means that a body

moves as a whole and no deformation takes place. The deformation appears only as

a consequence of velocity gradient at a point, which means that two neighboring sites

(the distance between them being infinitesimally small) move with different velocities.

If velocity is v (a vector value), its gradient is calculated as:

aij = dv/dr (C.1)

where space coordinates are described by radius-vector, r. Thus aij is a tensor

determined by two vectors (v and r). The velocity is the rate of displacement, i.e.,

v = du/dt. The relationship between gradient of velocity and gradient of displacement

can be found from the equation:

aij =dv

dr=

d

dr

(du

dt

)=

d2u

drdt=

d

dt

(du

dr

)=

dgij

dt(C.2)

The whole gradient of displacement is not controlling deformation, only its sym-

metric part. The same is true for the deformation rate. Differentiation in respect to

scalar - time, d/dt, - adds nothing new to the result. By decomposing tensor aij into

symmetrical and antisymmetrical components,

aij =1

2

(∂vi

∂xj

+∂vj

∂xi

)+

1

2

(∂vi

∂xj

− ∂vj

∂xi

)(C.3)

one obtain:

114 Deformation Rate

aij = Dij + wij (C.4)

where Dij is the rate of deformation tensor, and wij is the so-called vorticity

tensor. The rate of deformation tensor characterizes local changes of shape. The

deformation is related to the first term in Eq C.3, while the vorticity tensor describes the

rate of rotation of local elements of a body without their deformation. The difference

between tensors aij and Dij can be easily illustrated by a simple example. Let us

analyze the rotation of a solid (non-deformable) body around some axes. The velocity,

v, at a point located at the distance, r, from the axes is equal to ωr, where ω is the

constant angular velocity. Thus v = ωr, and the gradient of velocity, gradv = dv/dr,

is evidently equal to ω. It means that in rotation of a solid body the gradient of

velocity does exist but there is no deformation (because the body was assumed to be

non-deformable). The principal deformations (strains) are calculated in the same way

as principal stress.

Bibliography

Ataie-Ashtiani, B. and G. Shobeyri (2008). Numerical simulation of landslide impulsive

waves by incompressible smoothed particle hydrodynamic. International Journal for

Numerical Methods in Fluids 56, 209–232.

Barnes, H. and K. Walters (1985). The yield stress myth? Rheol. Acta 24, 323–326.

Barnes, H. A., J. Hutton, and K. Walters (1989). An introduction to rheology. Elsevier,

Amsterdam.

Batchelor, G. B. (1967). An introduction to Fluid Dynamics. Cambridge Univ. Press.

Cambridge.

Beeman, D. (1976). Some multistep methods for use in molecular dynamics calcula-

tions. J. Comp. Phys. 20, 130–139.

Belytschko, T., Y. Krongauz, J. Dolbow, and C. Gerlach (1998). On the completeness

of meshfree particle methods. Int. J. Numer. Mech. Engng. 43, 785–819.

Benz, W., R. Bowers, A. Cameron, and W. Press (1990). Dynamic mass exchange in

doubly degenerate binaries. i. 0.9 adn 1.2 m¯. Astrophysical Journal 348, 647–667.

Beverly, C. and R. Tanner (1992). Numerical analysis of three-dimensional bingham

plastic flow. Journal of Non-Newtonian Fluid Mechanics 42.

Bingham, E. (1922). Fluidity and Plasticity. McGraw-Hill Book Co., New York.

Bird, R., R. Armstrong, and O. Hassager (1987). Dynamics of Polymeric Liquids,

Volume 1. Wiley-Interscience, New York.

116 BIBLIOGRAPHY

Bonet, J. and T.-S. L. Lok (1999). Variational and momentum preservation aspects of

smooth particle hydrodynamic formulations. Computer Methods in applied mechan-

ics and engineering 180, 97–115.

Brookshaw, L. (1985). A method of calculating radiative heat diffusion in particle

simulations. Proc. Astron. Soc. 207.

Cross, M. (1965). Rheology of non-newtonian fluids: a new flow equation for pseudo-

plastic systems. J. Colloid Sci. 20, 417–437.

Ellero, M., M. Kroger, and S. Hess (2002). Viscoelastic flows studied by smoothed

particle dynamics. J. Non-Newt. Fluid Mech. 105, 35.

Falappi, S. and M. Gallati (2007). Sph simulation of water waves generated by granular

landslides. In IAHR 2007 proceedings, Venice.

Fine, I. V., A. B. Rabinovich, E. A. Kulikov, R. E. Thompson, and B. D. Bornhold

(1998). Numerical modeling of landslide-generated tsunamis with application to the

skagway harbor tsunami of november 3, 1994. In Proc Tsunami Symp, Paris.

Gingold, R. A. and J. J. Monaghan (1977). Smoothed particle hydrodynamic: theory

and application to non-spherical stars. Monthly Notices of the Royal Astronomical

Society 181, 375–389.

Gomez-Gesteira, M. and R. Dalrymple (2004). Using a three-dimensional smoothed

particle hydrodynamics method for wave impact on a tall structure. Journal of

Waterway, Port, Coastal and Ocean Engineering 130, 63–69.

Grilli, S. T., P. Guyenne, and F. Dias (2001). A fully non linear model for three

dimensional overturning waves over arbitrary bottom. Int J of Numer Methods

Fluids 35 (7), 829–867.

Grilli, S. T., S. Vogelmann, and P. Watts (2002). Development of a 3d numerical wave

tank for modeling tsunami generation by underwater landslides. Engng Analysis with

boundary Elements 26, 301–313.

Grilli, S. T. and P. Watts (1999). Modeling of waves generated by a moving sub-

merged body: applications to underwater landslides. Engng Analysis with boundary

Elements 23, 645–656.

BIBLIOGRAPHY 117

Heinrich, P. (1992). Nonlinear water waves generated by submarine and aerial land-

slides. ASCE J. of Waterways, Port, Coastal and Oc. Eng. 118 (3), 249–266.

Hernquist, L. and N. Katz (1989). Treesph: A unification of sph with the hierarchical

tree method. The Astrophysical Journal Supplement Series 70, 419–446.

Hosseini, S., M. Manzari, and S. Hannani (2007). A fully explicit three-step sph

algorithm for simulation of non-newtonian fluid flow. Int. Journal of Num. Meth.

for Heat and Fluid Flow 17 (7), 715–735.

Huber, A. and W. K. Hager (1997). Forecasting impulse waves in reservoirs. In Commis-

sion internationale des grands barrages, Dix-neuvieme Congres des Grands Barrages,

Florence, pp. 993–1005.

Imamura, F. and E. C. Gica (1996). Numerical model for tsunami generation due to

subaqueous landslide along a coast. Sci Tsunami hazards 14, 13–28.

Iwasaki, S. (1987). On the estimation of a tsunami generated by underwater landslide.

Proc., Int. Tsunami Symp., Vancouver, B. C., 134–138.

Jiang, L. and P. H. Leblond (1992). The coupling of submarine slide and the surface

waves which it generates. J Geophys Res 97 (C8), 12731–12744.

Jiang, L. and P. H. Leblond (1993). Numerical modeling of an underwater bingham

plastic which it generates. J Geophys Res 98 (C6), 10303–10317.

Jiang, L. and P. H. Leblond (1994). Three dimensional modeling of tsunami generation

due to submarine mudslide. J Phys Ocean 24, 559–573.

Johnson, G. R., R. A. Stryk, and S. R. Beissel (1996). SPH for high velocity impact

computations. Computer Methods in applied mechanics and engineering 139, 347–

373.

Kamphuis, J. W. and R. J. Bowering (1972). Impulse waves generated by landslides.

In Proc. of 12th ICCE, ASCE, pp. 575–588.

Komatina, D. and M. Jovanovic (1997). Experimental study of steady and unsteady

free surface flows with water-clay mixtures. Journal of Hydraulic Research 35(5).

118 BIBLIOGRAPHY

Kranzer, H. C. and J. B. Keller (1959). Water waves produced by explosions. Journal

of applied physics 30 (21-202), 398–407.

Le Mehaute, B. and S. Wang (1996). Water waves generated by underwater explosion.

World Scientific.

Locat, J. and H. Lee (2000). Submarine landslides: Advances and challenges. In Proc.

of the 8th International Symposium on Landslides, Cardiff, U.K., June 2000.

Lucy, L. B. (1977). A numerical approach to testing the fission hypothesis. The

Astronomical Journal 82 (12), 1013–1924.

Masson, D. G., C. B. Harbitz, R. B. Wynn, G. Pedersen, and F. Løvholt (2006).

Submarine landslides: processes, triggers and hazard prediction. Phil. Trans. R.

Soc. A 364, 2009–2039.

Monaghan, J. J. (1992). Smoothed particle hydrodynamics. Annual review of Astron-

omy and Astrophysics 30, 543–574.

Monaghan, J. J. (1994). Simulating free surface flows with SPH. J. Comp. Phys. 110,

399–406.

Monaghan, J. J. (1997). SPH and Riemann Solvers. J. Comp. Phys. 136, 298–307.

Monaghan, J. J. (2002). SPH compressible turbulence. Advanced Methods for Compu-

tational Fluid Dynamics 335, 843–852.

Monaghan, J. J. (2005). Smoothed particle hydrodynamics. Reports on Progress in

Physics 68, 1703–1759.

Monaghan, J. J. (2006a). Smoothed particle hydrodynamic simulations of shear flow.

Monthly Notices of the Royal Astronomical Society 365, 199–213.

Monaghan, J. J. (2006b). Time stepping algorithms for sph. Internal Report, Monash

University .

Monaghan, J. J. and R. A. Gingold (1983). Shock simulation by the particle method

sph. Journal of Computational Physics 52, 374–389.

BIBLIOGRAPHY 119

Monaghan, J. J. and A. Kos (1999). Solitary waves on a Cretan beach. ASCE J. of

Waterways, Port, Coastal and Oc. Eng. 125 (3), 145–154.

Monaghan, J. J. and A. Kos (2000). Scott Russell’s wave generator. Physics of Flu-

ids 12, 622–630.

Monaghan, J. J., A. Kos, and N. Issa (2003). Fluid motion generated by impact.

Journal of waterway, port, coastal and ocean engineering 129, 250–259.

Monaghan, J. J. and J. Lattanzio (1985). A refined method for astrophysical problems.

135-143 149.

Morris, J. P. (1996). Analysis of Smoothed Particle Hydrodynamics with Applications.

Ph. D. thesis, Monash University.

Morris, J. P., P. J. Fox, and Y. Zhu (1997). Modeling low reynolds number incom-

pressible flows using sph. Journal of Computational Physics 136, 214–226.

Noda, E. (1970). Water waves generated by landslides. ASCE J.Waterways, Harbours,

and Coastal Engineering Division, 835–855.

Panizzo, A., G. Cuomo, and R. A. Darlymple (2006). 3d-sph simulation of landslide

generated waves. In Proceedings of ICCE 2006.

Panizzo, A. and R. A. Darlymple (2006). Sph modelling of underwater landslide gen-

erated waves. In Proceedings of ICCE 2006.

Panizzo, A., P. De Girolamo, and A. Petaccia (2005). Forecasting impulse

waves generated by subaerial landslides. J. Geophys. Res. 110, C12025,

doi:10.1029/2004JC002778.

Pezzoli, G. (1965). Sulla teoria delle onde d’emersione e di impulso. una soluzione

rigorosa ad energia finita del problema di cauchy e poisson per moto piano (in italian).

Lincei - Rend. Sc. fis. mat. e nat. XXXVIII, 660–669.

Pezzoli, G. (1972). Perturbazioni impulsive in bacini di profondita limitata (in italian).

Lincei - Rend. Sc. fis. mat. e nat. LII, 903–911.

Price, D. J. (2004). Magnetic fields in Astrophysics. Ph. D. thesis, Monash University.

120 BIBLIOGRAPHY

Prins, J. E. (1958). Characteristics of waves generated by a local disturbance. Trans-

actions, American Geophysical Union 39 (5), 865–874.

Rodriguez-Paz, M. X. and J. Bonet (2003). A corrected smooth particle hydrodynamics

method for the simulation of debris flows. Wiley Periodicals, Inc., 140–163.

Rzadkiewicz, S., C. Mariotti, and P. Heinrich (1997). Numerical simulation of subma-

rine landslides and their hydraulic effects. Journal of Waterway, Port, Coastal and

Ocean Eng. 123 (4), 149–157.

Shao, S. and E. Y. M. Lo (2003). Incompressible SPH method for simulating Newtonian

and non-Newtonian flows with a free surface. Advances in Water Resources 26, 787–

800.

Shepard, D. (1968). A two dimensional function for irregularly spaced data. In ACM

National Conference.

Sisko, A. (1958). The flow of lubricating greases. Znd. Eng. Chem. 50, 1789–1792.

Stoker, J. J. (1957). Water waves. Interscience: New York.

Takeda, H., S. M. Miyama, and M. Sekiya (1994). Numerical simulation of viscous

flow by smoothed particle hydrodynamics. Progress of Theoretical Physics 92 (5),

939–960.

Talling, P. e. a. (2002). Experimental constraints on shear mixing rates and pro-

cesses: implications for the dilution of submarine debris flows. in glacier-influenced

sedimentation on high-latitude continental margins. Geological Society Special Pub-

lication 203, 89104.

Verriere, M. and M. Lenoir (1992). Computation of waves generated by submarine

landslides. Int. J. Num. Methods Fluids 14, 403–421.

Vola, D., F. Babik, and J. Latch (2004). On a numerical strategy to compute gravity

currents of non-newtonian fluids. Journal of Computational Physics 201, 397–420.

Watkins, S. J., A. S. Bhattal, N. Francis, J. A. Turner, and A. P. Whitworth (1996). A

new prescription for viscosity in smoothed particle hydrodynamics. Astronomy and

Astrophysics Supplement Series 119, 177–187.

BIBLIOGRAPHY 121

Watts, P. (1997). Water waves generated by underwater landslides. Ph. D. thesis,

California Inst. of Tech., Pasadena, CA.

Watts, P. (1998). Wavemaker curves for tsunamis generated by underwater landslides.

ASCE J. of Waterways, Port, Coastal and Oc. Eng. 124, 127–137.

Watts, P. (2000). Tsunami features of solid block underwater landslides. ASCE J. of

Waterways, Port, Coastal and Oc. Eng. 126, 144–152.

Watts, P. and S. T. Grilli (2005). Tsunami generation by submarine mass failure.

ASCE J. Waterway, Port, Coastal and Ocean Engng. 11-12, 283–310.

Watts, P., S. T. Grilli, J. T. Kirby, G. J. Fryer, and D. R. Tappin (2003). Land-

slide tsunami case studies using a boussinesq model and a fully nonlinear tsunami

generation model. Nat. Hazards Earth Syst. Sci. 3, 391–402.

Watts, P., F. Imamura, A. Bengston, and S. Grilli (2001). Benchmark cases for

tsunamis generated by underwater landslides. In Proc. Ocean Wave Measurements

and Analysis - ASCE, Volume 2, pp. 1505–1514.

Wendland, H. (1995). Piecewise polynomial, positive definite and compactly supported

radial functions of minimal degree. Advances in Computational Mathematics 4 (1),

389–396.

Wiegel, R. L. (1955). Laboratory studies of gravity waves generated by the movement

of a submarine body. tagu 36 (5), 759–774.

Wiegel, R. L., E. K. Noda, E. M. Kuba, D. M. Gee, and G. Tornberg (1970). Water

waves generated by landslides in reservoirs. ASCE J. of Waterways, Port, Coastal

and Oc. Eng. ww2 (96), 307–333.

sph numerical modelling of impulse water waves generated ...sph numerical modelling of impulse water...

Documents