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A corrected smooth particle hydrodynamics formulationof the shallow-water equations
Miguel Rodriguez-Paz, Javier Bonet *
Civil and Computational Engineering Centre, School of Engineering University of Wales Swansea, Swansea SA2 8PP, UK
Accepted 30 November 2004
Abstract
A shallow-waters formulation based on a variable smoothing length SPH method is presented. This new formulation
of the SPH equations treats the continuum as a Hamiltonian system of particles where the constitutive relationships for
the materials are introduced via an internal energy term. Some of the advantages of the new SPH formulation are evi-
dent in the solution of the shallow-water equations for expanding flows. The shallow-waters approach incorporates the
terrain into the equation of motion through terrain properties evaluated using SPH methodology. Several examples are
presented on the simulation of breaking dams on different geometries. A comparison with the analytical solutions is also
included.
2005 Elsevier Ltd. All rights reserved.
Keywords: Shallow-waters; SPH; Breaking dam; Free surface flows
1. Introduction
During the last decade, a number of changes in glo-
bal climate and other man-induced changes in nature
such as deforestation and pollution, have triggered envi-
ronmental problems, in particular water related issues:
floods and mudslides. On the other hand, the manage-
ment of water resources is also a main area of research,
which includes the prediction of storm surges, hazardprediction of dam breaks, sediment transport and coast-
al tides. Flow models that realistically represent the
physical properties of the flow and the complex topo-
graphic that are found in regions where debris ava-
lanches occurrence is high, can help in the hazard
prediction of such phenomena and help to mitigate their
destructive power. Numerical techniques are a viable
alternative when the phenomenon is difficult to repro-
duce in the laboratory due to many factors, such as,
its scale and magnitude.
Many of the hydrodynamic models for reservoirs and
tidal predictions are based on the solution of the depthaveraged shallow-water equations using finite differences
or finite element procedures. Other numerical methods
used for the solution of the shallow-water equations
for bore wave propagation include the finite element
method and more recently finite volume methods [14].
Most of these methods are based on elements or cells,
with the dependence on the grid and mesh refinement
to resolve the complex topography and evolving flow
features. However, most of the techniques that have
0045-7949/$ - see front matter 2005 Elsevier Ltd. All rights reserved.
doi:10.1016/j.compstruc.2004.11.025
* Corresponding author. Tel.: +44 1797 295 689; fax: +44
1792 295 676.
E-mail addresses: [email protected] (M. Rodri-
guez-Paz),[email protected](J. Bonet).
Computers and Structures 83 (2005) 13961410
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been developed based on the shallow-water equations,
assume a small gradient of the terrain and do not con-
sider a vertical component of the velocity.
Meshless methods, and in particular Smooth Particle
Hydrodynamic (SPH) methods have already been used
to simulate free surface flows [58]. SPH is a robust
numerical technique introduced by Lucy and Gingold[9,10]. Since its introduction in astrophysics the method
has been applied to simulate problems with complicated
physics such as multiphase flows [11] and high strain
dynamics. SPH has also been applied to simulate geo-
physical flows like debris flows [12]and ice fields[13].
In this paper, a novel variational formulation of the
Lagrangian shallow-water equations is presented. This
formulation uses the variable smoothing length ap-
proach for SPH developed by Bonet[14]and which will
be also presented in the following sections. This new
methodology is intended to deal with problems of flows
over a steep and non-uniform general terrain, like in thecase of avalanches and debris flows.
As the algorithm presented is explicit and solves only
two components of the 3-D space, the storage require-
ments are minimum and can be implemented on per-
sonal computers or small workstations. This fact
presents some importance to practising engineers and
geophysicists who may wish to experiment with the
code.
2. General assumptions
The shallow-water assumption is based on a 2-D plan
view projection of the problem domain. In this way, for
the case of a SPH discretisation of the resulting 2-D do-
main, each particle represents a column of fluid of a cer-
tain height. These particles move according to the
topography of the terrain but always in a direction tan-
gent to the terrain.
Consider a plan view of a terrain as shown inFig. 1.
The terrain is represented by a general function H(x,y),
which gives the height at each point. The continuum is
discretised with a system of Lagrangian particles, in
which each particle represents a column of water of
height ht with constant mass m, which moves over theterrain. The basic assumption is that the velocity
through all the height of the vertical column is uniform
and parallel to the terrain. This implies that the instan-
taneous spatial variation of ht is small. The motion of
the Lagrangian particles is then followed in time.
The motion of the columns of water is constrained to
follow the terrain. This implies that the globalz position
of the bottom of each column is given by (see Fig. 2)
z Hx;y 1
Differentiating with respect to time, the vertical compo-
nent of the velocity can be evaluated as
vz rH v 2
where $His the gradient of the terrain at the current po-
sition occupied by the column and v = (vx, vy) is the 2-D
velocity vector containing the x and y velocity compo-
nents of the column. The unknowns of the problem
are therefore the x and y co-ordinates of every particle
at each time and the height of the water column ht.
Alternatively, instead of the column heightsht, it is also
possible to use a related variable given by the 2-D pro-jected density of the fluid q, that is the amount of mass
per unit of area. Given that the fluid in motion will be
assumed to be incompressible, this density and the col-
umn height are simply related by
q htqw 3
whereqw denotes the constant 3-D density of the fluid.
Using this variable instead of the column height renders
the problem formally analogous to a standard 2-D SPH
formulation of a compressible fluid.
In order to derive the governing equations, a
variational approach is used [14]. This is based on the
Fig. 1. Discretisation of the fluid with a system of Lagrangian
particles that move on top of a general terrain.
Fig. 2. Velocity components of each column of water.
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expression of the total energy of the system as a function
of the particle positions and will therefore require the
evaluation of the projected density in terms of such par-
ticle positions.
3. Density evaluation
3.1. Standard SPH
In a standard SPH formulation with constant
smoothing length, the density at a particle is smoothed
over a sub-domain that is defined by a circle of radius
2h, where h is the smoothing length of a kernel function
W(seeFig. 3) [6].
If the particles have a mass mI, the discrete system of
particles has a density function of the form
^qx XI mIdxxI 4where d is the Dirac delta. In order to reconstruct the
smooth continuum, the kernel-smoothing concept can
be applied to the discrete density(4)to give a continuum
density approximation over a domain D as
qx
RD
qx0Wxx0dx0RDWxx0dx0
5
whereWx 0 x; h Wx; x0; hrepresents a suitable kernel
function[16]. Substituting Eq.(4)into the above expres-
sion and noting the Diracs delta properties, gives
qI
PJmJWIxJ; hIR
DWIxJ;hIdV
6
Typically, the kernel function is scaled so that the inte-
gral in the denominator becomes one. However, in order
to deal with rigid boundaries, Bonet et al. [14,15] have
introduced a function gamma as
cIxI; hI
ZD
WIx; hI dV 7
and substituting it into Eq. (6)gives
qI
PJmJWIxJ; hI
cIxI; hI 8
In the standard SPH formulation and in the absence of
boundaries the gamma function is cI(xI, hI) = 1. This is
no longer valid for particles that are within a distanceto a rigid boundary that is less than 2h. The evaluation
of the integral defined by Eq.(7)can be complicated for
the case in which a particle is within certain distance
from a rigid boundary. Kulasegaram et al. [15]propose
a numerical method to approximate this integral. In the
following sections, however, we assume there are no ri-
gid boundaries, and therefore c = 1. This is appropriate
for many shallow-water applications.
3.2. Variable-h SPH
The evaluation of the density using a constantsmoothing length is normal practice in SPH. In the case
of fluids with certain compressibility, however, a simple
case of uniform expansion or contraction can only be ex-
actly represented if the smoothing length is allowed to
vary. In the shallow-water approximation, the fluid will
follow the terrain and its projected 2-D density will ex-
pand or contract according to the height of the water
column as shown by Eq. (3). A variable smoothing
length is therefore needed in order to maintain accuracy
of the solution. In general, h must change according to
[16]
qhdm constant q0hdm0 9
where dm is the number of space dimensions, q0, h0 are
the initial density and smoothing length, respectively,
and q and h are the current values of density and
smoothing length for any particle. This gives an equa-
tion for the instantaneous smoothing length h for a par-
ticle Ias
hIh0q0qI
1=dm10
It is important to note that the above equation for the
density is implicit as the density qI is itself a functionof hI. In particular, the density is evaluated according
to Eq. (6), which for the case where there are no rigid
boundaries (i.e., c = 1) gives
qIXJ
mJWIxJ; hI 11
Note that this is now a non-linear equation for qI due
to the above dependency of hI on qI. Fortunately, the
equation for each particle are not coupled and can there-
fore be solved independently. A simple NewtonRaph-
son iteration to achieve this is described in Section 5.1
below.Fig. 3. Particle interpolation and kernel function.
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4. Equations of motion: non-dissipative case
Consider now the fluid described by a system of SPH
particles that are located in a 2-D Cartesian space (x,y),
the position of each particle defined by the vector xIand
the horizontal component of the velocity of the particle
by vI. Each particle I will represent a column of waterwith a total mass mI, which will remain constant during
the motion and hence, conservation of mass will be
ensured.
The EulerLagrange equations of motion of the sys-
tem of particles are, in the absent of dissipative forces,
e.g., bottom friction and viscosity effects [17]:
d
dt
oL
ovI
oL
oxI0 12
where the Lagrangian functional L is defined in terms of
the kinetic energy Kand potential energy p, as
L K p 13
and noting that p is only a function of the positions of
particles xI, Eq.(12)can be written as
d
dt
oK
ovI
oK
oxI
op
oxI14
The expressions for the kinetic and potential energies are
presented in the following sections.
4.1. Kinetic energy
The kinetic energy of the system of particles can be
approximated as the sum of the kinetic energy of eachparticle, which with the help of Eq. (2), gives
K1
2
XI
mIvI vI v2z; vz vI rHI 15
It is important to notice the termv2z vI rHI2
, which
takes into account the vertical component of the veloc-
ity. This term is often neglected in other shallow-water
approaches, since it is considered to be too small for
small terrain gradients. In order to be able to deal with
large gradients in the terrain, it is important to retain it
throughout the derivation of the equations of motion.
4.2. Potential energy
The potential energy of each particle is calculated at
the centre of gravity of each water column, i.e.,H 12ht.
Hence, the total potential energy of the system of parti-
cles can be expressed as the sum of the potential energy
of each particle
pXI
mIgHI1
2
X1
mIght;I 16
where g denotes the gravity acceleration. The first
term represents the external energy term, whilst the sec-
ond term can be interpreted as the pseudo-internal
energy,
pext XI
mIgHI 17
pint XI mI1
2ght;I 18The internal energy component can be expressed in
terms of an internal energy per unit mass w(q) as
pint XI
mIwqI 19
where
wqI 1
2g
qIqw
20
This expression matches that presented in [14] for gen-
eral fluid applications. The corresponding pressure term
defined asp q2dwdq
becomes the well-known height inte-
grated hydrostatic pressure, as usual in the case of shal-
low-water applications, and is given by
p q2dw
dq
1
2gqwh
2t 21
4.3. Evaluation of inertial forces
Eq.(14)can be re-written in the more usual format of
Newtons second law as
d
dt
oK
ovI
oK
oxI|fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl}inertial forces
FI TI; FI
opext
oxI ; TI
opint
oxI
22
where the left-hand side represents the inertial forces of
the system,FIand TIare the external and internal forces
of the system, respectively, which will be discussed in de-
tail in the section below.
The equation of motion can also be expressed as
IIFI TI 23
where the inertial forces IIcan be evaluated with the help
equation(15)as
II d
dt
oK
ovI
oK
oxI24
and substituting the expression for Kgives after simple
algebra:
II d
dtmIvI mIvI rHIrHI mIvI rHIkIvI
25
wherekIis the curvature tensor of the surface H(x,y) at
the position occupied by particle Iand is given by
k rrH 26
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Note that the last term in Eq. (25)could be interpreted
as the centripetal acceleration of the column of water.
4.4. External forces
The external forces can be evaluated with the help of
Eqs.(22) and (17) to give
FI opext
oxI mIgrHI 27
4.5. Internal forces
The internal constitutive forces are calculated using
the procedure described in [13] and are given by the
expression
TIopint
oxI28
The evaluation of this term, in general, requires the con-
stitutive definition of the material. For the case of fluids
with no viscosity, the internal forces are calculated using
Eqs.(19) and (11) as
TIopint
oxI
XJ
mImJpJaJq
2J
rWJxI; hJ pIaIq
2I
rWIxJ;hI
29
where a is a correction factor that emerges in the vari-
able-h formulation (for details see [14]):
aI XJ
mJrIJdWI
drIJ30
andp denotes the hydrostatic pressure given by Eq.(21).
4.6. Acceleration
An expression for the acceleration of the particles can
be found by substituting Eqs.(25), (27) and (29)into Eq.
(23)and solving for aIgives, after some simple algebra:
aI
gvI kIvI tI rHI1 rHI rHI rH
I tI 31
wheretI= TI/mI. Eq.(31)includes a term for the curva-
ture of the terrain as well as one term for the gradient of
the terrain. It can be noted that for the case of a flat ter-
rain ($H= 0; k = 0), Eq.(31)becomes aI=tI. In gen-eral, Eq. (31) can be easily implemented in an explicit
time integration scheme. For the examples included in
this paper, the values of$Hand kIcan be easily evalu-
ated finding the derivatives of the equation that defines
the surface for the terrain. For more general applica-
tions, an interpolation of the terrain from given grid
points will be necessary.
5. Numerical implementation
5.1. Numerical evaluation of density
As previously mentioned, Eq.(11)is non-linear inqI.
In order to solve it, a NewtonRaphson solution proce-
dure can be used. Let us define a residualRqkI for eachiterative value kof the density as
RqkI qkI
XJ
mJWIxJ; hkI 32
where the smoothing lengthh is a function of the density
through Eq.(10)as
hkI h
0I
q0I
qkI
!1=dm33
and h0I and q0I are the initial values of hI and qI,
respectively.
A simple possible iterative solution for qIis given by
qk1I
XJ
mJWIxJ;hIqkI 34
However, a much better option is to use a Newton
Raphson solution by the iteration of
qk1I q
kI
RkI
dRdq
kI
35
Substituting Eq.(32)gives the iterative solution as
qk1I q
kI
qkI
PJmJWIxJ; h
kI
dRdq
kI
36
where the derivative of the residual is calculated as
dR
dq 1
XJ
mJdWI
dhI
dhI
dqI
1XJ
mJ
qIdmWIdm rIJ
dWI
drIJ
37
After simple algebra, this last equation becomes
dRdq
1 1qI
XJ
mJWIxJ; hI aIdmqI
38
Finally, substituting Eq.(38)into Eq.(35), gives
qk1I q
kI 1
RkI dm
Rk
I dmakI
" # 39
which is the NewtonRaphson approximate solution for
the density equation at iteration k+ 1.
In order to start up the iteration process an initial
guess is required. For this purpose, the equation for
the rate of change of density [18] can be integrated in
time to give
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_qIqIdm
aI
XJ
mJrWIxJ; hI vJ vI 40
which enables a simple guess of the density at step n + 1
to be evaluated in terms of the density at step n as
q0
I;n1 q
I;nekn 41
where
kdmDt
aI
XJ
mJvJ vI rWIxJ;hI
" # 42
5.1.1. Convergence
In order to stop the NewtonRaphson iterative pro-
cedure, a tolerance that is within the machine precision
must be defined. Convergence is achieved when
jRk1I j
qkI6 e 43
Typical values in a double precision machine are
e= 1015, which is usually achieved within a few itera-
tions. A cheaper alternative is to evaluate the density
using Eq. (41) and the value of the smoothing length
from the previous time step.
Once the new values for the density have been evalu-
ated, new heights of the water columns can be obtained
using Eq.(3) or
ht;IqIqw
44
It is important to mention that by evaluating the density,new values for the smoothing length and the correction
factor a are also calculated in each iteration. The up-
dated values of the smoothing lengths are then used in
all the subsequent SPH interpolations within the time
step.
5.2. Time integration scheme
The equation of motion is assembled using Eqs.(23)
(31)and in order to update the position of particles, an
explicit time integration scheme is used, namely the leap-
frog method, defined as
vn1=2I v
n1=2I Dta
nI 45
xn1I xnI Dt
n1vn1=2I 46
where
Dt1
2Dtn Dtn1 47
Due to the explicit nature of the scheme, the Courant
FriedrichsLewy (CFL) stability criteria must be
satisfied. This implies that the time step size must be less
than
Dt CFL hmin
maxcI kvIk; 0 6 CFL 6 1.0 48
where c is the wave speed of propagation or speed, de-
fined as[19]
cI ffiffiffiffiffiffiffiffiffight;Iq 49andhminis the minimum smoothing length of the system
of particles. The magnitude of the velocity is also consid-
ered. Although this equation should provide time steps
that would satisfy the stability condition, in the numer-
ical examples presented in the following section, the
CFL factor considered was
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Fig. 4. Dimensions of the channel and the dam.
Fig. 5. SW-SPH results for t= 0.01 s and t = 0.10 s. Colours indicate depths. (For interpretation of colour in this figure legend the
reader is referred to the web version of this article.)
0.00
0.10
0.20
0.30
0.40
0.50
0.60
0.70
0.80
0.90
1.00
0.05 0.15 0.25 0.35 0.45 0.55 0.65
depth
h
h @ x=2.0m
Analytical Solution
Depth at gate
t (s)
Fig. 6. SW-SPH results vs. analytical solution for depth at x = 2.0 m.
0.00
0.10
0.20
0.30
0.40
0.50
0.60
0.70
0.80
0.901.00
0.05 0.15 0.25 0.35 0.45 0.65
(g*h)^0.5
Vel @ x=2.0m
Analytical Solution
Velocity at gate
t (s)
0.55
Fig. 7. SW-SPH results vs. analytical solution for velocity atx= 2.0 m.
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t= 0.64 s. The SW-SPH results are shown inFig. 7, com-
pared to the value predicted by the analytical solution. In
the same manner, the velocity of the fluid at the point of
the gate (x= 0.0) should remain constant and equal to 2/3
c0, where c0 ffiffiffiffiffiffiffigh0
p 3.1314 m s1. The results are
shown inFig. 7, normalized with respect to c0.
0.00
0.20
0.40
0.60
0.80
1.00
1.20
0.00 1.00 2.00 3.00 4.00 5.00 6.00
Analytical Solution
SW-SPH
t=0.40 s
x pos
depth(m)
0.00
0.20
0.40
0.60
0.80
1.00
1.20
0.00 1.00 2.00 3.00 4.00 5.00 6.00
Analytical Solution
depth(m)
x pos
SW-SPH
t=0.60 s
Fig. 10. Comparison of the profile for t = 0.40 s and t = 0.60 s. The dots represent the SW-SPH solution and the continuous line the
analytical solution.
Fig. 11. Lateral view of the collapse of a cylindrical column of water. From left to right:t = 0.0 s, t = 0.10 s, t = 0.20 s and t = 0.30 s.
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Using the formula given in Ref.[20], the position of a
particle at the front is given by
x 3ffiffiffiffiffigh
p v02
ffiffiffiffiffiffiffigh0
p t 50
Ifh is known for a particle located towards the front, the
position of the particle given by the program is com-
pared with the one provided by Eq. (50). Fig. 8 shows
the results for the front of the flow. The numerical re-
sults match very well those given by the analytical solu-
tion (Figs. 9 and 10).
6.4. Cylindrical dam on a horizontal plane
In this example a cylindrical column of water col-
lapses on a horizontal plane. The column has 1 m in
diameter and 1 m in height. In this case the properties
of the terrain are k = 0, $H= 0 (flat terrain). The results
are shown inFigs. 11 and 12. It can be seen that the ini-tial circular configuration is perfectly kept throughout
time, which would not be the case for a constant
smoothing length approach, as it was pointed out in
the previous chapter. A total of 5133 particles were used.
Fig. 12. Top view of the breaking dam.
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The results are displayed as a surface, using standard
matlab graphics.
6.4.1. Constant-h results comparison
The same example was solved using a constant-hcor-
rected SPH formulation of the shallow-water equations.
As shown in Fig. 13, the results of the constant-h ap-
proach are not able to simulate the problem, as the par-
ticles near the expanding boundary of the circle have less
neighbour particles as the simulation continues. On the
other hand, the results for the SW-SPH approach pre-
sented in this paper show a uniform distribution of the
particles, without losing the circular shape. The same
distribution of particles was used in both cases and no
bottom friction was included.
In this example the terrain was considered as per-
fectly smooth, i.e., no bottom friction.
6.5. Dam in triangular channel
In this example a channel with triangular cross-sec-
tion is used to represent the terrain on which the flow
will move. The geometric properties and dimensions
for the initial set-up are shown in Fig. 14. The problem
consists of the instantaneous breaking of a dam curtain,which makes the whole water volume move down the
steep hill. The sequence is indicated in Fig. 17.
For this case the gradient and the curvature of the
terrain are defined as
rH0.4
0.4 signy
; k 0
The other parameters employed for this simulation are-
g= 9.806 m s2;q0= 1000 kg m3; no viscosity and bot-
tom friction were considered.
A total of 3793 particles were used in the simulation.
The results are shown in Fig. 15. The initial configura-
tion is a dam that contains the water in a triangular
channel. At time t = 0.0 s the curtain is removed instan-
taneously and the water starts moving down the chan-
nel. A top view is presented for better clarity and the
colours of the graphs represent the depth of the fluid
over the channel.
Once again, it is important to mention that the mesh-
less nature of the methodology allows for large changes
in the geometry of the domain of the fluid to take
place without the need of re-meshing. This is also possi-
ble due to the variational consistent equations for the
density, which in a way work as an adaptive procedure
by changing the smoothing length for each particle as
changes in particle distribution occur throughout the
simulation.
6.6. Steep parabolic channel
A more complex geometry is used in this case to
model the surface on which the fluid moves. It consists
of a surface defined by
z y2
2Rmx
whereRis a radius for the parable and mis a slope in the
x-direction. In this simulation R= 1.1 m and m= 40.
The geometric properties, gradient and curvature of
the terrain are therefore
Fig. 13. (a) Standard SPH for a collapsing column of water, (b) variational SPH.
Fig. 14. Cross-section for the channel.
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rH0.839
0.909y ; k
0.0 0.0
0.0 0.909 And the material parameters are g= 9.806 m s2; q0=
1000 kg m3. In Fig. 16, the results of the simulation
are shown in top views, i.e., XYplane.
Fig. 15. Breaking dam in a triangular channel, the colours indicate the depth of the fluid over the channel. (For interpretation of
colour in this figure legend the reader is referred to the web version of this article.)
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The terrain is represented as contour curves. The
cylindrical column of water is released at time t= 0.0 s.
This column of water has a diameter of 60 cm and an
initial constant coordinate for the free surface of
Z= 0.10 m. A total of 4595 particles were used in this
simulation. The colours of the graphs represent in this
case the depth of the flow over the channel.
The results are for times t = 0.0 s, t = 0.50 s, t = 2.0 s
and t = 4.0 s, respectively. It is important to notice that
there are some parts of the fluid which have almost nildepth, indicate by the darker blue area, representing
wetted regions of the channel. The simulation was car-
ried out on a PC with 256 MB of physical RAM with 2
Pentium III processors with a clock speed of 550 MHz.
The total time for the simulation up to t= 4 s was
achieved in just a few hours.
6.6.1. Eccentric case
Considering now the same geometric properties of
the terrain used in the last example but now the column
of water is initially placed in an eccentric manner on the
channel. The purpose of this is to test the ability of the
method to deal with the curvature of the terrain. The
column of water was supposed to be a cylinder of
0.50 m of diameter, 0.25 m of height with centre at
x= 0.25, y = 0.50.
Fig. 17shows the results of the simulation. It is clear
how the fluid moves along the channel in a direction dri-
ven by its geometry. A sudden break initiates the flow
and then the fluid finds the centre of the channel and
starts moving down the slope along it. A total of 4100
particles were used to represent the fluid. The simulationwas run up to a time t = 8.0 s.
Taking this example as reference, if the same spacing
used for the initial distribution of particles used for the
fluid were to be used to mesh the entire channel, the
number of particles would be approximately over
187,000. The approach followed by other numerical
techniques is based on gridding the whole terrain and
then track the flow, with wet and dry cells. This SW-
SPH approach, however, discretises only the fluid and
the interaction of the terrain friction and any other dis-
sipative effect are included as forces affecting the motion
of each particle, acting at the position of that particle.
Fig. 16. Breaking dam in a parabolic channel. Dam is symmetrically located along the centre of the channel. The colours show the
depths of the flow for different times. (For interpretation of colour in this figure legend the reader is referred to the web version of thisarticle.)
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7. Concluding remarks
A new methodology for the numerical simulation of
shallow-water-like fluids over a general terrain was pre-
sented. The new set of equations is based on the varia-
tional SPH formulation presented by Bonet et al. [14].
The formulation has been denominated: variational
SPH formulation of Lagrangian shallow-water equa-
tions. Although the new formulation includes some of
the standard shallow-water equations assumptions, it
incorporates a new treatment of the terrain, as it allows
more general terrains to be considered. The resulting
technique shows a lot of potentiality for problems deal-
ing with breaking dams, flooding, debris flows [18,21],
avalanches and tidal waves, among others. Numerical
results show good agreement with analytical solutions.
Fig. 17. Breaking dam eccentrically in a parabolic channel. A total of 4100 particles were used in the simulation. The graphs show the
depth of the flow at different times. Initial depth 0.25 m.
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The results show that the technique is robust and stable,
which is in agreement with the previous work showing
that this type of particle methods are stable in the pres-
ence of compressive pressure values[22]. Given that the
pressure in the shallow-water model is proportional to
the height squared, its compressive nature is always
physically and numerically assured. Another advantageof the method over traditional numerical techniques that
use grids is that there is no need for a mesh for the entire
terrain; only the fluid is discretised with particles. This
enables the method to be implemented in serial machines
such as personal computers, since the memory require-
ments are vastly reduced. At present the cost of the
implementation appears high. This is very possibly due
to the type of searching technique used to determine par-
ticle neighbours, which is based on the Alternating Dig-
ital Tree method [23]. It is likely that other types of
searching methodologies may be more efficient for this
type of application and should be explored in the future.The method can also incorporate more sophisticated
constitutive models for different materials and bottom
friction forces. The method shows the potential to be ap-
plied in the simulation of geophysical flows and can help
in the hazard prevention for natural disasters.
Acknowledgment
Financial support from EPSRC through grant GR/
R72013 is gratefully acknowledged.
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