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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: ataie@sharif.edu, shobeyri@mehr.sharif.edu

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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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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