Modified Incompressible SPH method for simulating free...
Transcript of Modified Incompressible SPH method for simulating free...
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Modified Incompressible SPH method for simulating free surface problems
B. Ataie-Ashtiani *,a & G. Shobeyri b, L. Farhadi
Department of Civil Engineering, Sharif University of Technology, Tehran, Iran
Submitted to Fluid Dynamics Research On 19 March 2006 * Corresponding author E-mail addresses: [email protected], [email protected]
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Abstract
A modified Smooth Particle Hydrodynamics (SPH) formulation is presented to simulate
free surface incompressible fluid problems. The governing equations are mass and
momentum conservation equations that are solved in a Lagrangian form using a two-step
fractional method. In the first step, velocity field is computed without enforcing
incompressibility. In the second step a Poisson equation of pressure is used to satisfy
incompressibility condition. Laplacian, gradient and divergence operators are transformed
to interaction among moving particles using SPH formulation. The source term (the
variation of the particle density) in the Poisson equation for the pressure is approximated,
based on the SPH continuity equation, by an interpolation summation involving the relative
velocities between this particle and its neighboring particles. A new form of source term
for the Poisson equation is proposed and a modified Poisson equation of pressure is used to
satisfy incompressibility condition of free surface particles. By employing these corrections,
the stability and accuracy of SPH method are improved. In order to show the ability of SPH
method to simulate fluid mechanical problems, this method is used to simulate five test
problems such as 2-D dam-break and wave propagation. The test simulations indicate the
modification provides enhanced stability and accuracy of SPH applied to free surface
problems.
Keywords: Incompressible flow; Free surface flow; Numerical method; Lagrangian method; Smoothed Particle Hydrodynamics, Dam break
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1. Introduction
Free surface hydrodynamic flows are of significant industrial and environmental
importance. These problems are difficult to simulate due to the existence of the arbitrarily
moving surface boundary conditions. The marker and cell (MAC) and volume of fluid
(VOF) methods are two of the most flexible and robust approaches for treating such flows
in which the Navier-Stokes equations are solved on a fixed Eulerian grid. The former uses
marker particles to define the free surface while the latter solves a transport equation for the
volume fraction of the fluid. They have been successfully applied to a wide variety of flow
problems involving free surfaces. In spite of recent advances in numerical modeling of free
surface flows, still there are difficulties to analyze problems in which the shape of the
interface changes continuously or fluid structure interactions where large deformation
should be considered (Harlow and Welch, 1965; Hirt and Nichols, 1981).
Recently particle methods have been used in which each particle is followed in a
Lagrangian manner. Moving interfaces and boundaries can be analyzed by mesh-less
methods much easier. Furthermore, in Lagrangian formulations, the convection terms are
calculated without any numerical diffusion. Thus the numerical diffusion error that appears
due to advection term of Navier-Stokes equations in the grid methods does not arise in
Particle methods (Ataie-Ashtiani and Farhadi, 2006; Farhadi and Ataie-Ashtiani, 2004)
Different particle methods have been proposed and developed over the recent years.
The first idea was proposed by Monaghan for the treatment of astrophysical hydrodynamic
problems with the method called Smooth Particle Hydrodynamics (SPH) in which kernel
approximations are used to interpolate the unknowns (Gingold and Monaghan, 1977;
Monaghan, 1994). This method was later generalized to fluid mechanic problems.
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The main advantages of SPH arise directly from its Lagrangian nature. SPH does not
use a grid and a kernel function is used for smoothing discritized values at particles
positions. An interpolation function is used in order to transform governing equation to a
discrete form including particles as interpolating points. The interpolation is based on the
theory of integral interpolants using a differentiable kernel function (Monaghan, 1994). In
other words, the dependent field variables are expressed by the summation of interpolants
over neighboring particles.
SPH method can also successfully simulate incompressible flows. Two different
approaches can be used to extend SPH method to incompressible or nearly incompressible
flows. In the first approach, real fluids are treated as compressible fluids with a sound
speed that is much greater than the speed of bulk flow (Monaghan, 1994). In other words
the real fluid is approximated by an artificial fluid that is more compressible. This artificial
compressibility can cause problems with sound wave reflection at boundaries and high
sound speed leads to a stringent CFL time step constraint (Shao and Lo, 2003). On the
other hand, because of explicit computation to estimate pressure of particles by a stiff
equation of state, this approach leads to a lower computational costs, and it has proved to
be an effective method in tracking free surface problems (Monaghan, 1994, 1996, 1999).
The second approach works directly with the constraint of constant density. It employs a
strict incompressible formulation that is similar to the SPH projection method (Cummins
and Rudman, 1999). Unlike compressible SPH, in incompressible SPH method the pressure
is directly obtained by solving a Poisson equation of pressure that satisfies
incompressibility. The advantage of this method lies in its ease and efficiency of free
surface tracking using Lagrangian particles and the straightforward treatment of wall
boundaries (Lo and Shao, 2002) .
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SPH method has been shown to be applicable to a wide range of problems such as wave
propagation (Monaghan, 1999; Shao and Gotoh, 2004), study of gravity currents
(Monaghan, 1996), free surface Newtonian and Non-Newtonian flows ( Shao and Lo,
2003) and wave impact on tall structures (Gomez-Gesteria et. al., 2004).
A similar approach is the Moving Particle Semi-implicit (MPS) method proposed by
Koshizuka and Oka (1996). In the MPS method, motion of each particle is calculated
through interactions with neighboring particles through an approximate kernel (weight)
function and Laplacian, gradient and divergence operators are transformed to interaction
among moving particles. This method has been applied in the hydrodynamics and nuclear
mechanics such as the dam-breaking (Ataie-Ashtiani and Farhadi, 2006; Koshizuka and
Oka, 1996), Solitary wave breaking on a mild slope (Shao and Gotoh, 2004). In spite of the
extensive applications of MPS method, still there are limitations for getting a stable
solution by this method. Various kernel functions and different methods of solving the
Poisson equation of pressure were considered and applied to improve the stability and
accuracy of MPS method (Ataie-Ashtiani and Farhadi, 2006).
In this paper, some modifications for the conventional SPH method applied for
incompressible flows are presented. A new form of source term for the Poisson equation of
pressure and a modified Poisson equation of pressure, to enforce incompressible condition
to free surface particles, is proposed. These modifications considerably improve the
stability and accuracy of the incompressible SPH method.
The Modified Incompressible SPH method is used to simulate dam-break problem,
solitary wave moving over a uniform depth, evolution of an elliptical water bubble, solitary
wave breaking on a mild slope, and the dam break flow with a downstream slope.
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2. SPH formulation
2.1. Interpolation
The SPH formulation is obtained as a result of interpolation between a set of disordered
points known as particles. The interpolation is based on the theory of integral interpolants
that uses kernel function to approximate delta function. Each particle carries mass [M], a
velocity [LT-1], and all the properties of fluid with it. The key idea in this method is to
consider that a function A(r) can be approximated by:
)|,(| )( hrrWAmrA bab b
bba
rrr−=∑ ρ
(1)
Where, a is the reference particle and b is its neighboring particle. mb [M] and ρb [ML-3]
are mass and density respectively, W is interpolation kernel, h [L] is smoothing distance
which determines width of kernel and ultimately the resolution of the method [7].
Thus by summing over the particles the fluid density at particle a, ρa [ML-3], is evaluated
according to “Eq. (2)”.
)|,(| hrrWm bab
barr
−= ∑ρ (2)
Based on “Eq. (2)” we can deduce that the density of particle a increases when particle b
is getting closer it.
2.2. Kernel (Weight) function
Kernel (Weight) functions should have specific properties such as positivity, compact
support, normalization, monotonically decreasing and delta function behavior [16]. Many
different kernel functions satisfying the required conditions have been proposed by
researchers. Monaghan (1992) introduced a kernel function which has a spline form
described as:
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2q 0),(
2q 1 )2(28
10),(
1q )43
231(
710),(
32
322
>=
<<−=
<+−=
hrW
qh
hrW
qqh
hrW
π
π
(3)
Where h [L] is the smoothing distance, r [L] is distance between particles and q=r/h.. Since
the size of the area around particle a, which is covered by the weight function is limited,
the particle interacts with a finite number of neighboring particles. If the weight function is
not limited, the operation count is the scale of N2 where N is the total number of particles
(Koshizuka and Oka, 1996).
2.3. Gradient Model
The gradient term in the Navier-Stokes equation can have different forms in SPH
formulation. A model of gradient that conserves linear and angular momentum is
(Monaghan, 1994):
(4) )()1( 22 WPPmP ab
b
a
a
bba ∇+=∇ ∑ ρρρ
2. 4. Laplacian Model
Laplacian will lead to the second derivative of the kernel function that is very sensitive
to particle disorder (Shao and Lo, 2003). In Laplacian of pressure this can cause pressure
instability. Thus developing a model of Laplacian, which prevents this instability, is very
important. Lo and Shao (2002) introduced a model of Laplacian that has this specific
characteristic and is stable.
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(5) 2
8122 η|r|
W.r P)ρρ(
mP)ρ
.(ab
abaabab
babba +
∇+
=∇∇ ∑ r
r
Where 0.1h][ and r-r ][r ,][ baab21 ==−=−− LLPPTMLP baab η
rrr.
The corresponding coefficient matrix of the linear equations (“Eq. (5)”) is scalar,
symmetric and positive definite and can be more efficiently solved by an iterative scheme
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2. 5. Viscosity term
Viscosity term is formulated as a hybrid of standard SPH first derivative in which the
first derivative is approximated by a finite difference approximation.
)()|(|)(
.)(4)( 222
2ba
b abba
abaabbaba uu
rWrm
u rrr
r−
++∇+
=∇ ∑ ηρρµµ
ρµ (6)
Where µ [ML-1T-2] is the viscosity coefficient (Shao and Lo, 2003)
3. Mathematical and numerical formulation
The governing equation of viscous fluid flows which are mass and momentum
conservation equations are presented in the following equations respectively.
0 .1=∇+ u
DtD rρ
ρ (7)
ugPDt
uD rrr
21∇++∇−=
ρµ
ρ (8)
Where ρ [ML-3] is density, u [LT-1] is velocity vector, P [ML-1T-2] is pressure and g [LT-
2] is gravitational acceleration.
The computation of the Incompressible SPH method is composed of two basic steps.
The first step is the prediction step in which the velocity field is computed without
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enforcing incompressibility. In The second step, which is called the correction step,
incompressibility is enforced in the calculations through Poisson equation of pressure.
Incompressible SPH method can be summarized in a simple algorithm combined of 5
steps (Koshizuka and Oka, 1996).
1. Initialize fluid: u0, r0
For each time step:
2. Compute forces by considering only gravitational and viscosity terms. Apply them to
particles and find temporary particle positions and velocities: ,*ur *rr
tugu ∆∇+=∆ )( 2*
rrr
ρµ (9)
** uuu trrr
∆+= (10)
turr t ∆+= **rrr (11)
Where =− }[ ],[ 1 LrLTu ttrr particle velocity and position at time t;
=− ][ ],[ *1
* LrLTu rr temporary particle velocity and position respectively;
=∆ − ][ 1* LTur change in the particle velocity during the prediction step.
Incompressibility is not satisfied in this step and the fluid density ρ* that is calculated
based on the temporary particle positions “Eq. (2)”, deviates from the constant density (ρ0).
3. The Correction step; in this step the pressure term, obtained from the mass
conservation (Eq.7), is used to enforce incompressibility in to the calculation (Shao and Lo,
2003)
0).(1**
*0
0
=∆∇+∆− utρρ
ρ (12)
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tPu t ∆∇−
=∆ +1*
**1
ρr (13)
***1 uuutrrr
∆+=+ (14)
By combining “Eq. (12)” and “Eq. (13)”:
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*01
*
1t
)Pρ
.( t ∆−
=∇∇ + ρρρ
(15)
After employing the relevant SPH formulation “Eq. (5)” for the Laplacian operator, a
linear equation is obtained and solved efficiently by available solvers. The source term of
Poisson equation of pressure is the variation of particle densities and can be expressed as:
∑=b
abb
a
dtWd
mdt
d )(ρ (16)
ababaabababababab VW
dtdy
yW
dtdx
xW
dtyxdW rr
.)..(),(
∇=∂∂
+∂
∂= (17)
dtdtd
dtt 002
0
*0 11ρ
ρρρ
ρρ==
∆−
ababab
b WVVm ∇−∑rrr
)( (18)
By combining “Eq. (15)” and “Eq. (18)”, the Poisson equation of pressure can be
described as:
dtdtd
dtPt
001*
11)1.(ρ
ρρρ
==∇∇ + ababab
b WVVm ∇−∑rrr
)( (19)
In this form of Poisson equation of pressure, the numerical errors generated in the
pervious time steps do not affect the current results. This may considerably increase the
stability and accuracy of SPH method, and to the authors’ knowledge this form of equation
has not been used before.
4. New Particle velocities are computed by using “Eq. (10)”, “Eq. (13)” and “Eq. (14)”
5. Finally, the new position of particles is centered in time.
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tuu
rr tttt ∆
++= +
+ 21
1
rrrr (20)
Where ][ ,1 Lrr ttrr
+ =position of particle at time t and t+1.
4. Boundary condition
4.1. Wall boundaries
Solid boundaries are represented by one line of particles. The Poisson equation of
pressure is solved on these particles. This balances the pressure of inner fluid particles and
prevents them from accumulating in the vicinity of solid boundary. In addition, in order to
ensure that particle density number is computed accurately and wall particles are not
considered as free surface particles (Koshizuka and Oka, 1996), several lines of dummy
particles should be placed outside the wall boundary.
There are at least two methods to place the dummy particles. In the first method, the
dummy particles are fixed in space. In the second method, image particles that mirror the
physical properties of inner fluid particles are used (Lo and Shao, 2002).
In this study, we used the first method for placing dummy particles and employed a
smoothing length of h=1.2*l0 ,(l0 = Initial spacing between particles ). Thus two layers of
dummy particles were placed outside the solid boundary. The pressure of a dummy particle
is set to that of a wall particle in the normal direction of the solid walls.
4.2. Free surface
Since there are no particles in the outer region of free surface, the particle density
decreases on this boundary. A particle that satisfies “Eq. (21)” is considered to be on free
surface. In this equation β is the free surface parameter.
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0* βρρ < (21)
Most SPH formulations are presented in symmetric form. The symmetric particle
configuration is violated on the free surface and density falls discontinuously. This leads to
a spurious pressure gradient (Shao and Lo, 2003). To avoid this problem, special treatments
should be considered when computing gradient operator for free surface particles. Let us
assume that s is a surface particle with zero pressure and i is an inner fluid particle with
pressure Pi. In order to calculate the pressure gradient between these two particles, a mirror
particle, (m) with pressure –Pi should be placed in the direct reflection position of inner
particle i through the surface particle s. In this way, the zero pressure condition on the free
surface is satisfied.
The gradient of the pressure between the free surface particle (s), mirror particle (m) and
inner particle (i) is expressed as:
WPPmWPPmP am
m
s
sa
i
i
s
ss ∇++∇+=∇ )()()1( 2222 ρρρρρ
(22)
WsiWsm ss
s
im
0P -PP
−∇=∇==
(23)
Combining “Eq. (22)” and “Eq. (23)” gives:
WPmP ai
is ∇=∇ )(2)1( 2ρρ
(24)
Therefore the computed amount of gradient for free surface particles is double. In
addition by employing “Eq. (24)”, it is deduced that the Laplacian between free surface
particle(s) and inner fluid particle (i) is expressed as:
(2)1.( =∇∇ siPρ 2
822 η|r|
W.r P)ρρ(
msi
sissisi
iss +
∇+
r
r
) (25)
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This new form of equation satisfies the incompressibility condition of free surface
particles. The modification in the Poisson equation of pressure “Eq. (25)” which enforces
incompressibility to free surface particles and the new form of source term presented in
“Eq. (18)” and “Eq. (19)”, improves the stability and accuracy of Incompressible SPH
method.
5. Numerical convergence and particle link list
5.1 Convergence analysis
Since individual fluid particles are discrete points and cannot deform as the real fluid
does, the number of particles employed in the computation must be large enough to give
numerical convergence and a realistic flow simulation. Convergence is achieved by
increasing the number of fluid particles until the numerical solutions are essentially
unchanged (Shao and Lo, 2003).
The computation time must satisfy the following Courant condition
max
01.0V
lt ≤∆ (26)
Where l0 [L] is initial particle spacing and Vmax [LT-1] is maximum particle velocity in
the computation (Shao and Lo, 2003).
5.2 Algorithm for list generation
In SPH method, each fluid particle needs a list of neighboring particles within a distance
of kernel range (2h in this study). The whole list, which should be updated in each time
step, requires the scale of N2 operations for the calculation of distances between all pairs of
particles, where N is the number of particles. This list generation can dominate the
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computation time in large problems involving many particles. In this study in order to
decrease the number of operations, an algorithm proposed by Koshizuka (Koshizuka et. al.,
1998)was used.
6. Model application
6.1. Breaking dam analysis
An idealized two-dimensional dam-break problem is simulated in the section. The
instantaneous removal of a barrier holding a body of water at rest commences a free surface
flow. The schematic of the problem is shown in Fig. 2. The water column is represented by
648 particles which are located like a square grid. The distance between two neighboring
particles (l0) is 0.008m. The left, right and bottom walls are represented by 474 particles.
Their coordinates are fixed, and velocities are zero. In the computations, time step and
smoothing length are 0.0008s and 0.0096m respectively. In order to show the ability of
improvement of Modified Incompressible SPH method, Dam break problem is solved
using Incompressible SPH method (Fig.3) and Modified Incompressible SPH method
(Fig.4).
As seen in the figures, the Incompressible SPH method successfully simulates the
collapse of water column till 1s, but the shape of the free surface is not consistent with the
experimental results of Koshizuka and Oka (1996) after 0.3s. Particles are dispersed after
water impinges the right vertical wall at 0.3s and a large number of particles abnormally
satisfy the free surface condition.
The smooth shape of the free surface and the well agreement with the experimental
results of Koshizuka and Oka (1996) in simulating the water column collapse using
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Modified Incompressible SPH method (Fig. 4) proves the efficiency improvement of the
modified form of Incompressible SPH method.
In dam break problem simulation, the collapsing water runs on the bottom wall at 0.2s.
Accelerated water impinges the right vertical wall and rises up at 0.3s. At 0.4s, the water
goes up losing its momentum and at t=0.5s it begins to come down. A mushroom shape is
clear at t=0.7s and the waves falls down in the remaining water at t=0.8s. Around t=1s the
main water reaches he left wall again. The computed motion of leading edge is compared
with experimental data of Koshizuka and Oka (1996)are shown in Fig. 5. From this figure
it can be clearly observed that the speeds of the leading edge obtained from experiments are
slower than those of the calculations. This might be due to the friction between the fluid
and the bottom wall that is neglected in the calculations.
Parameters used in the current model are investigated with test calculations of the
collapse of a water column. β Is the free surface parameter that is used to judge whether
the particle is on the free surface or not. Fig. 6 shows the number of particles considered as
the free surface using different free surface parameter (β ). The trajectories are almost the
same from β= 0.8 to 0.99, although they are shifted lower in parallel when the parameter is
smaller. In this range of β values no instability in computation is observed. We can
conclude that the free surface parameter is not effective to the calculation result if the
calculation proceeds stably. In this paper β=0.95 was selected.
Similar analyses have been performed in order to obtain the smoothing length or kernel
range (h) (Fig. 7). The results of this analysis clearly show that by using free surface
parameter β=0.95 and the smoothening length or kernel range of h= 1.2l0, the number of
free surface particles at the start of simulation is appropriate and will change smoothly over
time. Thus, h= 1.2l0 has been selected as the kernel range in this study.
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Satisfying incompressibility condition provides an appropiate self-check on the accuracy of
incompressible numerical models. For I- SPH and M-I-SPH models proposed in the paper,
a quantitative measurement of the conservation of mass is provided by computing the
difference of time-dependant particle densities ρ(t) and ρ0.
( ) ∑=
−=N
itdensity abs
NtE
100 )/)(1 ρρρ (27)
Where N is the number of fluid particles.
In Fig. 8 the normalized time-dependent particle density error ( )tEdensity for both I-SPH and
M-I-SPH are shown. It is shown that the accuracy of both methods is satisfactory especially
before the water hits the right wall (t<0.3s) but after this time the accuracy of M-I-SPH is
much better than that of I-SPH which demonstrates the ability of new source term (right
hand side of Eq. “19”) in avoiding affecting previous error result of the current time.
6.2. Evolution of an elliptical water bubble
Simulating the evolution of an elliptic water bubble in 2 dimensions is another simple
test for verifying the presented Modified Incompressible SPH formulations. The velocity
field is linear in the coordinates and is expressed by “Eq. (28)”,
V = (-100x; 100y) (28)
This problem is studied on two axis (a, b) and the initial configuration of particles is a unit
circle. The numerical results of simulating the evolution of water bubble over time are
shown in Fig. 9.
The evolution of an elliptic water bubble can also be solved in an analytical way
( Monaghan, 1994).
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The theoretical values of Semi-Major Axis (b) of the drop at different times and the values
computed by Modified Incompressible SPH method and the Incompressible SPH method
are shown in Table 1. The computation errors of Modified incompressible SPH method are
less than 2.5% while they are less than 4% when Incompressible SPH method is used. The
decrease in computation errors using Modified Incompressible SPH method, again, proves
the improvement in the resulted obtained by the modified Incompressible SPH method.
The vertical velocity of particles along the major axis of the drop at time t=0.008s using
Modified Incompressible SPH method, are compared with the related analytical solution
values in Fig. 10. As clearly observed in the figure, there is an excellent agreement between
numerical and analytical results.
6.3. Solitary wave in a simple tank
Another example which was considered in this study to verify the presented SPH code
is the simulation of a solitary wave moving over a uniform depth.
Analytical solution for the wave profile is derived from the Boussinesq equation.
Η(x, t) = )](43[sec 3
2 ctxdaha − (29)
Where η [L] is the water surface elevation, a [L] = wave amplitude, d [L] = water depth
and ][)( 1−+= LTadgc is the solitary wave celerity. The horizontal velocity
underneath the wave profile is presented in “Eq. (30)”. (Lo and Shao, 2002)
(30) dgu η=
Two solitary waves with wave amplitudes of a/d=0.3 and a/d=0.5 were considered in
this study. The water depth (d) is 0.1m. Particles were placed on a regular grid with square
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cells, and particles that were above the specified profile were eliminated. This leads to a
slightly jagged initial profile but the fluid rapidly adjusts itself with the solitary wave
profile (Lo and Shao, 2002). The crest of the wave is at x=0.7m at T=0s.
Simulated wave profiles are compared with the analytical solutions for wave amplitudes
of a/d=0.3 and a/d=0.5 in Fig. 11. The comparison between simulated and analytical
horizontal velocity “Eq. (30)” of free surface particles is presented in Fig. 12. The
agreement between numerical simulations and analytical results is very good.
6.4. Solitary wave breaking on a mild slope
The laboratory breaking solitary wave experiment of Synolakis is used as another
convincing test to show the capability of the modified Incompressible SPH method. In the
experiment the still water depth was d=0.21m, the slope of the beach was s0=1:20 and the
incident wave height was a/d=0.28. The initial particle spacing is∆ x=0.0191m and totally
about 2700 particles are used in the simulations. The computational domain started from
the front of the foot of the slope and extended to the location beyond the maximum run-up
point. The initial solitary wave profile was produced according to Monaghan and Kos
(1999).The computed wave profiles by the present method are shown in Fig. 13. The good
agreement between the computed and experimental wave profiles demonstrates the
capability of the modified Incompressible SPH method again. The maximum run-up height
computed by the present method is about 0.52d which is close to 0.48d reported by Lin et
al. (1999). This upper-prediction of the maximum run-up might be due to the neglecting
theof the viscosity term in the momentum equation. The same problem solved by Lo and
Shao (2002) by I-SPH method. They used about 1000 particles in their computations. In
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spite of fewer particles used in the present method (2700 particles), the results are as good
as theirs demonstrating the accuracy of the present method.
6.5. Dam Break flow with a slope downstream
In this example a simplified breaking dam problem with a slope downstream is simulated.
The water column is 0.1m wide and 0.1m high. The same problem was simulated by
Mosqueira et al. (2002). The results are shown in Fig.14. Agreement with the numerical
results of Mosqueira et al. (2002) who used a corrected SPH method to solve this example
is excellent.
5. Conclusion
A modified formulation of Incompressible SPH method is introduced and applied to
simulate incompressible flows with free surface. In this method grids are not necessary and
particles are used to simulate the flow. Thus, because of the lagrangian nature of this
method numerical diffusion error that is due to advection term of Navier-Stokes equations
in grid methods does not arise. Using a new form of source term for the Poisson equation
of pressure and enforcing incompressibility to free surface particles, stability and accuracy
of the conventional SPH method are improved. The Modified Incompressible SPH method
was applied to model the Breaking dam, Solitary wave in a simple tank, Evolution of an
elliptic water bubble solitary wave breaking on a mild slope and dam break problem with a
downstream slope. The ability of this method to successfully simulate these problems
proves the ability of the presented Modified Incompressible SPH method to simulate a
wide range of fluid mechanics problems such as the breaking wave, fluid-fluid impacts and
fluid-solid impacts.
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23
List of figures:
Fig. 1- Free surface boundary treatment-relationship between inner, mirror and free surface
particle.
Fig. 2-Geometry of the collapse of water column.
Fig. 3- Numerical Simulation of collapse of water column at different times using the
Incompressible SPH method.
Fig. 4- Numerical Simulation of collapse of water column at different times using the
Modified Incompressible SPH method.
Fig. 5- Comparison between calculated motion of leading edge and experimental data.
Fig. 6- Effect of free surface parameter (β) on the number of particles on the free
surface(h=1.2l0).
Fig. 7- Effect of smoothing length (h) on the number of particles on the free surface
(β=0.95).
Fig. 8 Time-dependent density errors by I-SPH and M-I-SPH computaions.
Fig. 9- Particle positions for the evolution of an elliptical drop.
Fig. 10- Comparison between simulated and analytical solutions of vertical velocity of the
drop.
Fig. 11- Particle configurations and comparison between simulated and analytical wave
profiles.
Fig. 12- Comparison between simulated and analytical horizontal velocity of free surface
particles.
Fig. 13- Particle configurations and comparison of computed and experimental surface profiles of solitary wave . Fig. 14-. Numerical simulation of a breaking dam with a slope downstream.
24
s
i
m
0=Ps
PiPm −=
Pi
Fig. 1. Free surface boundary treatment-relationship between inner, mirror and free surface particle.
surfacefree
25
Fig. 2. Geometry of the collapse of water column.
26
Fig. 3. Numerical Simulation of collapse of water column at different times using Incompressible SPH method.
27
Fig. 4. Numerical Simulation of collapse of water column at different times using Modified Incompressible SPH method.
28
Fig. 5. Comparison between calculated motion of leading edge and experimental data.
29
Fig. 6- Effect of free surface parameter (β) on the number of particles on the free surface (h=1.2l0).
30
Fig. 7- Effect of smoothing length (h) on the number of particles on the free surface (β=0.95).
31
Fig. 8 Time-dependent density errors by I-SPH and M-I-SPH computaions.
32
Fig. 9- Particle positions for the evolution of an elliptical drop using Modified Incompressible SPH method.
33
Fig. 10- Comparison between simulated and analytical solutions of vertical velocity of the drop.
34
Fig. 11- Particle configurations and comparison between simulated andanalytical wave profiles.
35
Fig. 12- Comparison between simulated and analytical horizontal velocity of free surface particles.
36
Fig. 13. Particle configurations and comparison of computed and
experimental surface profiles of solitary wave .
37
Fig. 14-. Numerical simulation of a breaking dam with a slope downstream.
38
Table1. Comparing theoretical values of b (Semi –Major Axis) of the elliptical drop with
computed values of Modified Incompressible SPH method and Incompressible SPH method .
Time (sec)
Theoretical value
Modified Incompressible SPH method Incompressible SPH method
( Computed value) (% error) ( Computed value) (% error) 0.005 1.595 1.591 0.251 1.59 0.313 0.007 1.863 1.854 0.483 1.845 0.966 0.01 2.277 2.25 1.186 2.235 1.845
0.012 2.56 2.517 1.680 2.496 2.500 0.015 2.977 2.913 2.150 2.886 3.057 0.018 3.4 3.32 2.353 3.28 3.529