comput. methods appl. mech....

Comput. Methods Appl. Mech. Engrg. 200 (2011) 1008–1020

Contents lists available at ScienceDirect

Comput. Methods Appl. Mech. Engrg.

journal homepage: www.elsevier .com/locate /cma

Improved Incompressible Smoothed Particle Hydrodynamics methodfor simulating flow around bluff bodies

Mostafa Safdari Shadloo, Amir Zainali, Samir H. Sadek, Mehmet Yildiz ⇑Faculty of Engineering and Natural Sciences, Advanced Composites and Polymer Processing Laboratory, Sabanci University, 34956 Tuzla, Istanbul, Turkey

a r t i c l e i n f o a b s t r a c t

Article history:Received 19 March 2010Received in revised form 14 September 2010Accepted 1 December 2010Available online 7 December 2010

Keywords:Meshless methodsIncompressible Smoothed ParticleHydrodynamics (ISPH)Airfoil problemSquare obstacle problemBluff bodySolid boundary treatment

0045-7825/$ - see front matter � 2010 Elsevier B.V. Adoi:10.1016/j.cma.2010.12.002

⇑ Corresponding author.E-mail address: [email protected] (M. Yil

In this article, we present numerical solutions for flow over an airfoil and a square obstacle using Incom-pressible Smoothed Particle Hydrodynamics (ISPH) method with an improved solid boundary treatmentapproach, referred to as the Multiple Boundary Tangents (MBT) method. It was shown that the MBTboundary treatment technique is very effective for tackling boundaries of complex shapes. Also, we haveproposed the usage of the repulsive component of the Lennard-Jones Potential (LJP) in the advectionequation to repair particle fractures occurring in the SPH method due to the tendency of SPH particlesto follow the stream line trajectory. This approach is named as the artificial particle displacement method.Numerical results suggest that the improved ISPH method which is consisting of the MBT method, arti-ficial particle displacement and the corrective SPH discretization scheme enables one to obtain very sta-ble and robust SPH simulations. The square obstacle and NACA airfoil geometry with the angle of attacksbetween 0� and 15� were simulated in a laminar flow field with relatively high Reynolds numbers. Weillustrated that the improved ISPH method is able to capture the complex physics of bluff-body flows nat-urally such as the flow separation, wake formation at the trailing edge, and the vortex shedding. The SPHresults are validated with a mesh-dependent Finite Element Method (FEM) and excellent agreementsamong the results were observed.

� 2010 Elsevier B.V. All rights reserved.

1. Introduction

Smoothed Particle Hydrodynamics (SPH) is one of the membersof meshless Lagrangian particle methods used to solve partialdifferential equations widely encountered in scientific and engi-neering problems [1–4]. Unlike Eulerian (mesh-dependent) com-putational techniques such as finite difference, finite volume andFinite Element Methods, the SPH method does not require a grid,as field derivatives are approximated analytically using a kernelfunction. In this technique, the continuum or the global computa-tional domain is represented by a set of discrete particles. Here, itshould be noted that the term particle refers to a macroscopic part(geometrical position) in the continuum. Each particle carriesmass, momentum, energy and other relevant hydrodynamic prop-erties. These sets of particles are able to describe the physicalbehavior of the continuum, and also have the ability to move underthe influence of the internal/external forces applied due to theLagrangian nature of SPH. Although originally proposed to handlecosmological simulations [5,6], the SPH method has becomeincreasingly generalized to handle many types of fluid and solidmechanics problems [7–11].

ll rights reserved.

diz).

Due to being a relatively new computational method for engi-neering applications (roughly two decades old), there are still afew significant issues with SPH that need to be scrutinized. It is stilla challenge to model physical boundaries correctly and effectively.In addition, there are various ways to construct SPH equations (dis-cretization), and a consistent approach has not gained consensus.Highly irregular particle distributions which occur as the solutionprogresses may cause numerical algorithms to break down, there-by making robustness another significant issue to be addressed.Namely, it is well-known by SPH developers that when passingfrom one test case to another, new problems are faced. For exam-ple, instabilities due to clamping of SPH particles which is not ingeneral present in modeling a dam-breaking problem show them-selves in the simulation of flow over bluff bodies, especially at theleading and trailing edges. These shortcomings are not insur-mountable. The underlying factors causing these shortcomingscan be understood through extensive research on the verificationof SPH against a wide variety of possible applications as being donein the SPH literature.

In most engineering problems, the domain of interest has, ingeneral, solid boundaries. The SPH formulations being valid forall interior particles are not necessarily accurate for particles closeto the domain boundary since the kernel function is truncated bythe boundary. Therefore, the application of boundary conditions is

http://dx.doi.org/10.1016/j.cma.2010.12.002

mailto:[email protected]

http://dx.doi.org/10.1016/j.cma.2010.12.002

http://www.sciencedirect.com/science/journal/00457825

http://www.elsevier.com/locate/cma

M.S. Shadloo et al. / Comput. Methods Appl. Mech. Engrg. 200 (2011) 1008–1020 1009

problematic in the SPH technique. Consequently, the proper andcorrect boundary treatments have been an ongoing concern foran accurate and successful implementation of the SPH approach[12,13] as well as other meshless methods [14,15] in the solutionof engineering problems with solid walls. Improper boundarytreatment has two important consequences. The first originatesfrom the penetration of fluid particles into boundary walls, whichthen leave the computational domain. The second is that kerneltruncation at the boundary produces errors in the solution. Hence,over the years, several different approaches have been used forthe boundary treatment such as specular reflections, or bounce-back of fluid particles with the boundary walls [16], Lennard-Jones Potential (LJP) type force as a repulsive force [17,18], ghostparticles [19–21], and Multiple Boundary Tangent (MBT) method[22].

Additionally, the homogeneity of the particle distribution isquite important for the accuracy and the robustness of SPH models.The formation of ill particle distributions during the simulationmay result in the numerical solution to fail. For instance, if thepressure field is solved correctly thereby imposing the incompress-ibility condition as accurately as possible, the particle motion clo-sely follows the trajectory of streamline, hence resulting in a linearclustering and in turn fracture in particle distribution. In these re-gions due to the lack of sufficient number of particles, or inhomo-geneous particle distribution, the gradients of field variables cannot be computed reliably. Such a situation leads to spurious fields,especially erroneous pressure values in the ISPH approach. As thecomputation progresses, errors in computed field variables accu-mulate whereby blowing-up the simulation.

This paper suggests an improved ISPH algorithm that includesthe implementation of (i) the multiple boundary tangent methodto treat geometrically complex solid boundaries in a flow field,(ii) the artificial particle displacement (particle fracture repair)procedure for eliminating particle clustering induced instabilities,and (iii) the corrective SPH discretization scheme to improve theaccuracy of the computation.

2. Smoothed Particle Hydrodynamics

For clarity of the presentation, it is worthy of introducing nota-tional conventions to be used throughout this article. All vector andtensorial quantities are written either using suffix notation withLatin indices denoting the components or direct notation withboldface letters. These components will be written either as sub-scripts (when particle identifiers are not used) or superscripts(when particle identifiers are used). As well, throughout this articlethe Einstein summation convention is employed, where anyrepeated component index is summed over the range of the index.These superscripts do not represent any covariant or contravariantnature. Latin boldface indices (i, j) will be used as particle identifi-ers to denote particles and will always be placed as subscripts thatare not summed, unless indicated with a summation symbol. Forexample, the position vector for particle i is ~ri ¼ xk

i~ek where xk

i

denotes the components of the position vector and ~ek is a basevector. The distance vector between a pair of particles is indicatedby ~rij ¼~ri �~rj ¼ xk

i � xkj

� �~ek ¼ rk

ij~ek, and the magnitude of the

distance vector k~rijk is denoted by rij.The three-dimensional Dirac-delta function d3(rij), also referred

to as a unit pulse function, is the starting point for the SPH tech-nique. This function satisfies the identity

f ð~riÞ ¼Z

Xf ð~rjÞd3ðrijÞd3~rj; ð1Þ

where d3~rj is a differential volume element and X represents the to-tal bounded volume of the domain.

The SPH approach assumes that the fields of the particle ofinterest are affected by that of all other particles within the globaldomain. The interactions among the particles within the global do-main are achieved through a compactly supported, normalized andeven weighting function (smoothing kernel function) W(rij,h) witha smoothing radius jh (cut off distance, localized domain) beyondwhich the function is zero. Hence, in computations, a given particleinteracts with only its nearest neighbors contained in this localizeddomain. Here, the length h defines the support domain of the par-ticle of interest, j is a coefficient associated with the particular ker-nel function, and rij is the magnitude of the distance between theparticle of interest i and its neighboring particles j. If the Dirac del-ta function in Eq. (1) is replaced by the kernel function W(rij,h), theintegral estimate or the kernel approximation to an arbitrary func-tion f ð~riÞ can be introduced as

f ð~riÞ ffi hf ð~riÞi �Z

Xf ð~rjÞWðrij;hÞd3~rj; ð2Þ

where the angle bracket hi denotes the kernel approximation, and~ri

is the position vector defining the center point of the kernelfunction.

Approximation to the Dirac-delta function by a smoothing ker-nel function is the origin of the Smoothed Particle Hydrodynamics.The Dirac-delta function can be replaced by a smoothing kernelfunction provided that the smoothing kernel satisfies the followingseveral conditions; namely, (i) normalization condition: the area un-der the smoothing function must be unity over its support domain,R

X Wðrij;hÞd3~rj ¼ 1, (ii) the Dirac-delta function property: as thesmoothing length approaches to zero, the Dirac-delta functionshould be recovered, limh?0 W(rij,h) = d3(rij), (iii) compactness prop-erty: which necessitates that the kernel function be zero beyond itscompact support domain, W(rij,h) = 0 when rij > jh, and (iv) thekernel function should be spherically symmetric even function,W(rij,h) = W(�rij,h), and be positive within the support domainW(rij,h) > 0 when rij < jh. Finally, the value of the smoothing func-tion should decay with increasing distance away from the centerparticle.

The smoothing function can be represented in a general form asW(rij,h) = (1/hd)f(rij/h) where d is the dimension of the problem,and f is a function. In literature, it is possible to find a wide varietyof kernel functions which satisfy above-listed conditions, such asGaussian, cubic or quintic kernel functions [18,23]. The smoothingkernels can be considered as discretization schemes in meshdependent techniques such as finite difference and volume. Thestability, the accuracy and the speed of an SPH simulation heavilydepend on the choice of the smoothing kernel function as well asthe smoothing length. Considering the stability and the accuracyof the simulations, throughout the present work, the compactlysupported two-dimensional quintic spline kernel is used at the ex-pense of higher computational cost. For example, the utilization ofthe higher order quintic spline in simulations is at least two timescomputationally more expensive than that of the cubic spline

Wðrij;hÞ ¼7

478ph2

ð3� sijÞ5 � 6ð2� sijÞ5 þ 15ð1� sijÞ5 if 0 6 sij < 1;

ð3� sijÞ5 � 6ð2� sijÞ5 if 1 6 sij < 2;

ð3� sijÞ5 if 2 6 sij 6 3;0 if sij P 3;

8>>>><>>>>:

ð3Þ

here sij = rij/h. The spatial resolution of SPH is affected by thesmoothing length. Hence, depending on the problem solved, eachparticle can be assigned to a different value of smoothing length.However, for a variable smoothing length, it is probable to violateNewton’s third law. For example, it might be possible for a particlej to exert a force on particle i, and not to experience an equal andopposite reaction force from particle i. To ensure that Newton’s

1010 M.S. Shadloo et al. / Comput. Methods Appl. Mech. Engrg. 200 (2011) 1008–1020

third law is not violated and the pair wise interaction among parti-cles moving close to each other is achieved, the smoothing length issubstituted by its average, defined as hij = 0.5(hi + hj). The averagedsmoothing length ensures that particle i is within the influence do-main of particle j and vice versa.

2.1. Spatial derivatives and particle approximation in SPH

The SPH approximation for the gradient of an arbitrary function(i.e., scalar, vectorial, or tensorial) can be simply written throughthe substitution f ð~rjÞ ! @f ð~rjÞ=@xk

j in Eq. (2) to produce

@f ð~riÞ@xk

i

ffi @f ð~riÞ@xk

i

* +�Z

X

@f ð~rjÞ@xk

j

Wðrij;hÞd3~rj: ð4Þ

Upon integrating Eq. (4) by parts and using the compactness prop-erty of the kernel function as well as noting that @Wðrij;hÞ=@xk

i ¼�@Wðrij;hÞ=@xk

j for a constant smoothing length h, it can be shownthat

@f ð~riÞ@xk

i

ffi @f ð~riÞ@xk

i

* +�Z

Xf ð~rjÞ

@Wðrij;hÞ@xk

i

d3~rj: ð5Þ

Using a Taylor series expansion and the properties of a second-rankisotropic tensor, as derived in Appendix A, a more accurate SPHapproximation for the gradient of an arbitrary function can be intro-duced as

@f pð~riÞ@xk

i

aksij ¼

XN

j¼1

mj

qjðf pð~rjÞ � f pð~riÞÞ

@Wðrij; hÞ@xs

i

; ð6Þ

where aksij ¼

PNj¼1ðmj=qjÞrk

jið@Wðrij;hÞ=@xsiÞ is a corrective second-

rank tensor. This form is referred to as the corrective SPH gradientformulation that can be used to eliminate particle inconsistencies. Itshould be noted that the corrective term aks

ij is ideally equal toKronecker delta dks for a continuous function. On using Kroneckerdelta in Eq. (6), one can obtain the SPH gradient formulation of avector-valued function commonly seen in the SPH literature. Itshould be noted that the corrective SPH formulation requires theinversion of aks

ij , which might be singular for irregular and ill particledistribution. We have not experienced any difficulties related to aks

ij

corrective tensor being singular for a wide variety of benchmarkproblems solved (i.e., lid driven cavity, flow over backward facingstep, Taylor vortices, and bubble deformation). However, if such asingularity problem exists for a given particle which can easily bedetermined through monitoring whether the determinant of theaks

ij is zero or not, the corrective SPH formulation on that particleshould not be imposed through setting aks

ij ¼ dks, thereby leadingto the utilization of the standard SPH discretization scheme.

Additionally, there are two main forms of the SPH approxima-tion for the Laplacian of a vector-valued function [19,24]. For thesake of completeness, we have in Appendix A shown that thesetwo forms can be derived from the SPH representation of the sec-ond order spatial derivative of a vector field @2f pð~riÞ=@xk

i @xli as

@2f pð~riÞ@xk

i @xki

apmij ¼ 8

XN

j¼1

mj

qjðf pð~riÞ � f pð~rjÞÞ

rpij

r2ij

@Wðrij;hÞ@xm

i

; ð7Þ

@2f pð~riÞ@xk

i @xki

2þ allij

� �¼ 8

XN

j¼1

mj


rsij

r2ij

@Wðrij;hÞ@xs

i

: ð8Þ

Likewise, upon replacing the corrective second rank tensor in Eq. (7)and its trace in Eq. (8) by Kronecker delta and its trace, respectively,one can obtain the SPH Laplacian formulations of a vector-valuedfunction suggested by Clearly and Monaghan [24], and Morriset al. [19], correspondingly

@2f mð~riÞ@xk

i @xki

¼ 8XN

j¼1

mj


rpij

r2ij

@Wðrij; hÞ@xm

i

; ð9Þ

@2f pð~riÞ@xk

i @xki

¼ 2XN

j¼1

mj


rsij

r2ij

@Wðrij;hÞ@xs

i

: ð10Þ

Unlike Eq. (8), Eq. (7) can only be used for divergence free vector-valued functions as shown in Appendix A.

Throughout this work, all modeling result are obtained with theusage of corrective SPH discretization schemes since correctiveSPH formulations produce more accurate and stable results thanthe standard SPH equations. We have also observed that Eq. (7)performs better for the discretization of the Laplacian of velocityfields than Eq. (8). Therefore, in all presented results, Eq. (7) is usedfor the Laplacian of velocity, while Eq. (8) is used for the Laplacianof pressure in the Poisson pressure equation.

2.2. Instabilities and their possible remedies in the SPH method

To prevent the particle clustering, the trajectory of particles canbe disturbed by adding relatively small artificial displacement drk

i

to the advection of particles computed by the solution of theequation of motion. Recall the form of the Lennard-Jones Potential(LJP)-type force used in the SPH literature as a repulsive force forthe solid boundary treatment,

Fki;LJP ¼

XN

j¼1

ðro=rijÞn1 � ðro=rijÞn2� �brk

ijv2max

r2ij

; ð11Þ

where Fki;LJP is the force per unit mass on fluid particle i due to the

neighbor particles j, n1 and n2 are constants, b is a problem-dependent parameter, ro is the cutoff distance, and vmax is thelargest particle velocity in the system. In this work, vmax is set tobe equal to the absolute value of the maximum vx velocity. If thesecond term (attractive interaction) on the right-hand side of LJPforce is neglected, and n1 = 2, and the force Fk

i;LJP, and vmax arereplaced by drk

i =ðDtÞ2 and rij/Dt, one can write the relationship

drki ¼ b

XN

j¼1

rkij

r3ij

r2ovmaxDt; ð12Þ

where drki is an artificial particle displacement vector.

Here, the cut-off distance can be approximated as ro ¼PN

j¼1rij=N.Given that rk

ij=r3ij is an odd function with vanishing integral, one can

writePN

j¼1rkij=r3

ij ¼ 0 for a spherically symmetric particle distribu-tion. However, if the particle distribution is asymmetric, and clus-tered, the term

PNj¼1rk

ij=r3ij – 0 is no longer equal to zero, whereby

implying the region with clustered particle distribution. The artifi-cial particle displacement is only influential in the clustered regionand negligibly small in the rest of the computational domain due toPN

j¼1rkij=r3

ij ffi 0 provided that the particle distribution is closely uni-form. The offset vector dr̂k

ii between the particle i and the center ofmass of its influence domain can be presented as

dr̂kii ¼ xk

i � x̂ki ¼

XN

j¼1

rkij=N; ð13Þ

where xki and x̂k

i are the coordinates of the particle i and the centerof mass for the influence domain of the particle i.

The comparison of the artificial particle displacement and offsetvectors implies that upon using the particle displacement vector inthe advection equation, the particle i moves towards the dilutedparticle region (the region which is away from the center of mass).Hence, the fractures in the simulation domain are repaired. Sincethe near boundary fluid particles have influence domains truncatedby boundaries, with the usage of the artificial particle displace-ment vector in the computation, these fluid particles will tend to


move towards the boundary and stick to it. Even though this situ-ation may appear as a problem and the deficiency of the approach,it might be used in advantageous way such that fluid particlesclose to boundaries are then artificially forced to move in confor-mation with boundaries. Hence pressure forces on the boundariescan be computed much more accurately in that particle deficiencyis no longer problem therein. In our simulations, since the ghostparticles are used for near boundary fluid particles, the influencedomain of the kernel function is fully populated, and thereforesuch a problem is not an issue. The artificial particle displacementvector is added to the particle advection equation in both predic-tion and correction steps. This approach repairs all the clusteringand fracture in the domain gradually without inducing significanterrors in the computation, and enabling a quite robust SPH ap-proach. It is due to this approach that it becomes possible to runsimulations with higher Reynolds numbers, which are otherwiseimpossible to achieve.

2.3. SPH solution algorithms

The governing equations used to solve the fluid problems in thisarticle are the mass and linear momentum balance equationswhich are expressed in the Lagrangian form and given in directnotation as

Dq=Dt ¼ �qr �~v; ð14Þ

qD~v=Dt ¼ r � rþ q~fB: ð15Þ

In the present simulations, the fluid is assumed to be incompress-ible and Newtonian. Hence, the incompressibility condition requiresthat the divergence of the fluid velocity r �~v ¼ 0 be zero. Here, q isthe fluid density,~v is the divergence-free fluid velocity, r is the totalstress tensor, and~fB is the body force term, respectively. The totalstress is defined as r = �pI + T, where p is the absolute pressure, Iis the identity tensor, T ¼ lðr~v þ ðr~vÞTÞ is the viscous part of thetotal stress tensor, where l is the viscosity. Finally, D/Dt is thematerial time derivative operator defined as D/Dt = @/@t + vl@/@xl.

There are two common approaches utilized in the SPH literaturefor solving the balance of the linear momentum equation. The firstone is widely referred to as the Weakly Compressible SPH (WCSPH)where the pressure term in the momentum equation is determinedthrough an artificial equation of state. In the second approachknown as the Incompressible SPH (ISPH), the pressure is computedby means of solving a pressure Poisson equation.

The ISPH approach is based on the projection method [25–27],which uses the principle of Hodge decomposition. Upon using theHodge decomposition, any vector field can be broken into a diver-gence-free part plus the gradient of an appropriate scalar potential.The Hodge decomposition can be written for a velocity field as

~v� ¼ ~v þ ðDt=qÞrp; ð16Þ

where ~v� is the intermediate velocity, and ~v is the divergence-freepart of the velocity field. The projection method begins by ignoringthe pressure gradient in the momentum balance equation. Thesolution of Eq. (15) without the pressure gradient produces theintermediate velocity ~v�, which does not, in general, satisfy massconservation. However, this incorrect velocity field can be projectedonto a divergence-free space after solving a pressure Poisson equa-tion, from which the divergence-free part of the velocity field ~v canbe extracted. The pressure Poisson equation is obtained by takingthe divergence of Eq. (16) as

r �~v�=Dt ¼ r � ðrp=qÞ: ð17Þ

Eq. (17) is subjected to Neumann boundary conditions that can beobtained by using the divergence theorem on the pressure Poissonequation as

ðq=DtÞ~v� �~n ¼ rp �~n; ð18Þ

where ~n is the unit normal vector. Having solved the pressure Pois-son equation to obtain the pressure field and then computed thepressure gradient, we can use the Hodge decomposition equationto determine the correct, incompressible velocity field ~v. Often,the boundary conditions for~vðnþ1Þ are used for ~v�. This reduces Neu-mann boundary condition to rp �~n ¼ 0 for no slip boundary condi-tion. Note that in order for differentiating between spatial andtemporal indices, the time index n is put within brackets.

One of the main advantages of using ISPH is the elimination ofthe speed of sound parameter in the time-step conditions. Muchlarger time steps can be used in this approach, at the computa-tional expense of having to solve the pressure Poisson equationat each time step. The algorithm stability is controlled by the Cou-rant–Friedrichs–Lewy (CFL) condition, where the recommendedtime-step is Dt 6 CCFLhij,min/vmax where hij = 0.5(hi + hj), hij,min isthe minimum smoothing length among all i–j particle pairs, andCCFL is a constant satisfying 0 < CCFL 6 1 (in this work, CCFL = 0.125).

For time marching of the ISPH approach, we have used a first-order Euler time step scheme. This technique is an explicit timeintegration scheme, and is relatively simple to implement. Particlepositions, and velocities are computed respectively as D~ri=Dt ¼~vi; D~vi=Dt ¼~fi. The general form of the ISPH algorithm is as fol-low. The predictor step starts with determining an estimate~r�i forall particle locations, given the previous particle positions~rðnÞi andthe previous correct velocity field ~vðnÞi as ~r�i ¼~r

ðnÞi þ~v

ðnÞi Dt where

~r�i is the intermediate particle position. The intermediate velocityfield ~v�i is computed on the temporary particle locations throughthe solution of the momentum balance equations with forwardtime integration by omitting the pressure gradient term as~v�i ¼ ~v

ðnÞi þ~f

ðnÞi Dt. The pressure Poisson equation is solved to obtain

the pressure pðnþ1Þi which is required to enforce the incompressibil-

ity condition. The pressure Poisson equation is solved using abiconjugate gradient stabilized method or a direct solver basedon the Gauss elimination method. The Hodge decomposition prin-ciple is employed to solve for the actual velocity field ~vðnþ1Þ

i byusing the computed pressure pðnþ1Þ

i . Finally, with the correct veloc-ity field for time-step n + 1, all fluid particles are advected to theirnew positions ~rðnþ1Þ

i using an average of the previous and currentparticle velocities as~rðnþ1Þ

i ¼~rðnÞi þ 0:5 ~vðnÞi þ~vðnþ1Þi

� �Dt.

2.4. Mesh generation and MBT boundary treatment for airfoilgeometry

This section presents the geometry and mesh generation, ghostparticle creations and the implementation of the MBT boundarytreatment for the airfoil geometry. Initially, a Cartesian mesh iscreated over a rectangular domain with the length and height ofL and H, respectively. Then, an airfoil geometry is created usingEqs. (20) and (21). Since the leading edge of the airfoil has a curvewith a steeper slope, the chord is split into two parts to be able tolocate more boundary particles towards the leading edge. Discretepoints on the chord are created with the formula

xc ¼ ½ði� 1Þ=ðilen� 1Þ�n � idis ð19Þ

where xc is coordinates along the chord of the airfoil, i is a nodal in-dex, ilen is the number of nodes along the chord, idis is the length ofthe chord, and n is the geometrical progression coefficient whichcontrols the distance between points on the chord. Given the chordlength of 1, 6 inequidistant nodal points created through the geo-metrical progression coefficient of 2 are located along the 5% ofthe chord length starting from the leading edge. The remaining sec-tion of the chord has 50 equidistant nodal points. The mean camberline coordinates are computed using Eq. (20)


yc ¼ m 2pxc � x2c

� �=p2

yc ¼ mð2pðxc � 1Þ þ 1� x2c Þ=ð1� p2Þ

0 6 xc 6 p;p < xc 6 1;

�ð20Þ

where m is the maximum camber in percentage of the chord, whichis taken to be 5%, p is the position of the maximum camber in per-centage of the chord that is set to be 50%, and t is the maximumthickness of the airfoil in percentage of the chord, which is 15%.The thickness distribution above and below the mean camber lineis calculated as

yt ¼ 5t 0:2969x0:5c � 0:126xc � 0:3516x2

c þ 0:284x3c � 0:1015x4

c

� �:

ð21Þ

The final coordinates of the airfoil for the upper surface (xU,yU) andthe lower surface (xL,yL) are determined using the followingrelations: xU = xc � ytsinu, yU = yc + ytcosu, xL = xc + ytsinu, yL = yc �ytcosu, and u = arctan(dyc/dx). Having obtained all coordinates ofthe airfoil geometry, the upper and lower surface lines are curve fit-ted using the least square method of order six. In so doing, it be-comes possible to compute boundary unit normals, tangents andslopes for each boundary particles. All the initial particles fallingbetween the upper and lower camber fitted curves are removedfrom the rectangular computational domain, then remaining fluidparticles are combined with the boundary particles to form a parti-cle array of the computational domain.

The various steps of implementing the MBT boundary treat-ment technique to the airfoil geometry, as depicted in Figs. 1 and2, are as follows:

(a) At each or prescribed time steps, all near boundary fluid par-ticles (particles with boundary truncations) as well asboundary particles are associated with their neighbor bound-ary particles using the cell array data structure (the Fortran90 derived data type) as also described in our earlier work[22], see Fig. 1a. When dealing with a thin solid objectenclosed by flow such as the trailing edge of the airfoil, forinstance, near boundary fluid particles or boundary particlespositioned above/on the upper camber take contributionfrom fluid and boundary particles located below the uppercamber since the weighting function W(rij,h) has an influ-ence domain that spans over the smoothing radius jh. Phys-ically, particles flanking a solid wall should not affect eachother. Consequently, the neighbor list computed throughthe standard box-sorting algorithm has to be modified, and

Fig. 1. Boundary treatment for a s

then updated at each time step. The neighbor boundary par-ticles of a given near boundary fluid particle are sorted inaccordance with the distance between boundary particlesand the fluid particle in ascending order. Then, the fluid par-ticle in question is given the unit normal vector of its nearestboundary neighbor particle. For instance, fluid particle i = 25in Fig. 1a is given the unit normal of the boundary particlei = 7. The neighbor lists of all particles are updated by com-puting the angles ð~ni �~njÞ among unit normal vectors of par-ticles and their neighbors. Here, ~ni and ~nj are the unitnormal of a given particle and its neighbors. Only the parti-cles with angles smaller than the preset value (130� used inthis study) are regarded as neighbors to each other, wherebyforming the updated neighbor list. To be more specific, ascan be seen from Fig. 1b, the updated neighbor list of theparticle i = 25 includes only those particles enclosed by asquare frame since other neighbor particles do not satisfythe preset angle condition even though they are in the influ-ence domain of the particle i = 25.

(b) As in the case of step-a, all near boundary fluid and bound-ary particles are associated with their updated neighborboundary particles. Associating near boundary fluid particlesand boundary particles with their neighbor boundary parti-cles, and sorting and then storing these neighbor boundaryparticles in accordance with the shortest distance amongthem allows for (i) the computation of the overlapping con-tributions of mirrored particles from each boundary particle,(ii) the confinement of the mirrored particles into the soliddomain, (iii) defining solid boundaries by the envelope ofboundary tangent lines, as well as (iv) associating mirroredparticles with near boundary fluid particles.

(c) Given that each boundary particle has fluid particles in itsinfluence domain as neighbors, these fluid particles are mir-rored with respect to the tangent line of the correspondingboundary particle as indicated in Fig. 2. The fluid particlesshould satisfy the condition~rbf �~nb P 0, where~rbf is a posi-tion vector between the boundary particle and its neighborfluid particles directing towards the fluid particles and ~nb

is the unit normal of the boundary particle. This conditionensures that only fluid particles above the associated bound-ary particle tangent line are mirrored. The second condition~rnbg �~nnb 6 0 to be fulfilled is that mirrored particles shouldbe confined into the solid region, meaning that mirrored

ubmerged thin object: step-a.

Fig. 2. Boundary treatment for a submerged thin object; (a) step-c and (b) step-d.


particles associated with a boundary particle have to beinside of the all tangent lines of the neighbor boundary par-ticles of the boundary particle in question, where~rnbg is theposition vector between the ghost particles and the neighborboundary particles of the given boundary particle and, ~nnb isthe unit normal vector of the neighbor boundary particlesfor the boundary particle in question. Using the cell arraystructure, every boundary particle is associated with its cor-responding mirrored particles. Spatial coordinates and parti-cle identification numbers of mirrored particles are stored ina cell array. To be more precise, mirrored particles are asso-ciated with the particle identification number of the fluidparticle from which they are originated (referred to as the‘mother’ fluid particle). For example, for a fluid particleindexed with i = 25, the ghost particle mirrored about aboundary particle tangent line (for example, boundary parti-cle 7) will also be associated with i = 25 as shown in Fig. 2a.Note that fluid and boundary particles have numerical iden-tifications that are permanent, whereas mirrored particleshave varying (dummy) indices throughout the simulation.A ghost particle is given the same mass, density and trans-port parameters, such as viscosity, as the corresponding fluidparticle. As for the field values (i.e. velocities) of a ghost par-ticle, they are obtained depending on the type of boundarycondition implemented. For instance, for no-slip boundaryconditions, the following relation is implemented: ~vg ¼2~vb �~vf where ~vg ; ~vb and ~vf are the velocities of the ghost,boundary, and fluid particles, respectively. As for the imple-mentation of a zero-gradient at the boundary, a ghost parti-cle is given the same field values as the corresponding fluidparticle; ~vg ¼ ~vf , and pg = pf, where pg and pf are the pres-sures of the ghost and fluid particles. If the boundaries arestationary walls, the ghost particles will have the velocity~vg ¼ �~vf for no-slip boundary conditions.

(d) In a loop over all particles, if a fluid particle has a boundaryparticle or multiple boundary particles as neighbor(s), thenthe fluid particle will become a neighbor of all mirrored par-ticles associated with the corresponding boundary particleson the condition that (i) the mirrored particles are in theinfluence domain of the fluid particle in question, and (ii)for a mirrored particle, its mother particle has to be withinthe influence domain of the fluid particle in question. During

the creation of ghost particles, there is an over-creation ofghost particles due to the fact that the influence domain ofneighboring boundary particles overlaps. The overlappingcontributions of mirrored particles can be eliminated bydetermining the number of times a given fluid particle ismirrored into the influence domain of the associated fluidparticle with respect to a boundary particle’s tangent line.For computational efficiency, the fluid particle might onlybe associated with the mirror particles of its several nearestboundary particle rather than all neighbor boundary parti-cles as explained in Fig. 2b. Boundary particle i = 7 has theshortest distance to i = 25 compared to other boundary par-ticles neighbor to the fluid particle i = 25. Hence, mirror par-ticles of i = 7 also become the neighbor of i = 25 providedthat above two conditions (i and ii) are satisfied. Nearboundary fluid particles hold the information of spatial coor-dinates and fluid particle identity numbers, boundary parti-cle identity numbers (i.e. the particle number for a boundaryparticle to which mirrored particles are associated initially),and over-creation number for mirrored particles in the cellarray format. During the SPH summation over ghost parti-cles for a fluid particle with a boundary truncation, the massof the ghost particles are divided by the number of corre-sponding over-creations.

3. Flow around a square obstacle and an airfoil

In this work, to be able to test the effectiveness of the improvedSPH algorithm proposed in Section 2 (involving the utility of theMBT method together with the artificial particle displacementand the corrective SPH discretization scheme) for modeling fluidflow over complex geometries, we have solved two benchmarkflow problems; namely, two-dimensional simulations of a flowaround a square obstacle and a NACA airfoil. Mass and linearmomentum balance equations are solved for both test cases on arectangular domain with the length of L = 15 m and the height ofH = 6 m.

A square obstacle with a side dimension of 0.7 m is positioned inthe computational domain with its center coordinates at x = L/3 andy = L/2. Initially, a 349 � 145 array (in x- and y-directions, respec-tively) of particles is created in the rectangular domain, and thenparticles within the square obstacle are removed from the particle


array. This set of initial particles are created regularly through theformulations x = (i � 1) � L/(ilen � 1), and y = (j � 1) � H/(jlen � 1)where x and y are coordinates of particles, i and j are nodal indicesin the x- and y-directions, respectively, and ilen and jlen are thenumber of nodes along the x- and y-directions in the given order.The boundary particles on the square obstacle are created such thattheir particle spacing is almost the same as the initial particle spac-ing of the fluid particles. The implementation of the MBT methodfor the square obstacle follows the same steps described previouslyfor the airfoil geometry. The simulation parameters, fluid density,dynamic viscosity and body force are taken as respectivelyq = 1000 kg/m3, l = 1 kg/ms, and Fx

B ¼ 3:0� 10�3 N=kg. The massof each particle is constant and found through the relationmi = qi/ni where ni ¼

Pj¼1Wðrij;hÞ is the number density of the par-

ticle i. The smoothing length for all particles is set equal to 1.6 timesthe initial particle spacing.

The slightly modified periodic boundary condition is imple-mented for inlet and outlet particles in the direction of the flow.Particles crossing the outflow boundary are reinserted into theflow domain at the inlet from the same y-coordinate positions withthe velocity of inlet fluid region with its coordinates of x = 0, andy = 3 so that the inlet velocity profile is not poisoned by the outletvelocity profile. The no-slip boundary condition is implemented for

Fig. 3. A close-up view for particle positions around the square obstacle forReynolds number of 300.

Fig. 4. A comparison of ISPH (left) and FEM (right) velocity contours for two different Revelocities are given as a velocity magnitude.

the square obstacle. For upper and lower walls bounding the sim-ulation domain, the symmetry boundary condition for the velocityis applied such that vy = 0, and @vx/@y = 0 which is discretized byusing Eq. (6). As for the pressure, a zero-pressure gradient condi-tion rp �~n ¼ 0 is enforced on all solid boundary particles.

The channel geometry and the boundary conditions for the sec-ond benchmark problem are identical to the first one except thatthe square obstacle geometry is replaced by the NACA airfoil witha chord length of 2 m. The leading edge of the airfoil is located atCartesian coordinates (L/5,H/2). Initially, a 300 � 125 array (inx- and y-directions, respectively) of particles is created in the rect-angular domain, and then, particles within the airfoil are removedfrom the particle array. Subsequently, boundary particles are cre-ated and then distributed on solid boundaries. The smoothinglength for all particles is set equal to 1.6 times the initial particlespacing. To show convergence, we run a test case with three differ-ent particle arrays, namely, 150 � 62, and 300 � 125 and400 � 167 were used. It was observed that 300 � 125 array of par-ticles is sufficient for particle number independent solutions.

The flow around the airfoil and square obstacle positioned in-side the channel were simulated for a range of Reynolds numbersRe = qlcvb/l, which is defined by the characteristic length, lc (setequal to the side length for the square obstacle, and the chordlength for the airfoil geometry), the density, the bulk flow velocityvb and the dynamic viscosity l. Both test cases are validatedthrough comparing SPH results with those obtained by a FiniteElement Method (FEM) based solver of a Comsol multiphsicssoftware tool. The ISPH and FEM results are compared in termsof velocity contours for both test cases, and the pressure envelopefor the airfoil.

Fig. 3 shows the close-up view of particle positions around thesquare obstacle for Reynolds number of 300. One can conclude thatthe implementation of the MBT method together with the artificialparticle displacement and the corrective SPH discretization schemecan successfully hinder the formation of chaotic particle distribu-tion and fracture (unlike those observed in Ref. [22]) in the entireflow domain (particularly in the vicinity of the boundary) andeffectively prevent the particle penetration through the boundary(see Fig. 3), whereby producing simulation results in excellentagreement with FEM method as shown in Fig. 4. It is noted that

ynolds number values, 100 and 200. It should be noted that in this presentation, all


for all of the test cases, the artificial particle movement coefficientis kept constant and equal to 0.01. It is also noted that this coeffi-cient should be selected carefully such that it should be small en-ough not to affect the physics of the flow, but also large enough toprevent the occurrence of clustering and particle deficiency. Fig. 4compares ISPH and FEM velocity contours for two different Rey-nolds number values, 100 and 200.

It is well-known from both earlier experiments and numericalstudies that vortex shedding is observed at the rear edge of thesquare obstacle at higher Reynolds numbers [28]. In light of this,to be able address whether the SPH method can capture vortexshedding as accurately as mesh dependent solvers, we here presentsimulation results for Reynolds number of 300, where Fig. 5 (leftfigures) shows the simulation results for the vortex tail downand up. On comparing ISPH and FEM results, one can notice that re-sults are satisfactorily in agreement with each other regarding themagnitude of velocities as well as the position and the number ofvortices. However, there is a slight discrepancy between ISPH andFEM results in terms of the separation point of the vortices fromthe rear edge of the obstacle. For the sake of brevity, without pre-senting further results, we can safely assert that the SPH method ishighly successful in predicting changes of the topology of the vor-tex shedding behind a square obstacle with the Reynolds number.

To have a closer look at the vortex shedding behavior capturedat the trailing edge of the square obstacle by the ISPH method, inFig. 6 are presented the streamlines of vortex shedding for a fullperiod for the Reynolds number of 300. One can immediately no-tice the development of a small vortex at the rear top edge of thesquare obstacle, which is growing in size during its motion towardsthe rear lower edge, and then separating from therein. The nextvortex starts at the lower rear edge, and grows in size while mov-ing towards the top rear edge, and finally leaves the top rear edge.

The accuracy of the ISPH method can be further validated byconsidering the Strouhal number, which is defined as St = xlc/vb,where x is the frequency of vortex shedding. The computed valueof the Strouhal number for the ISPH method for the Reynolds num-ber of 300 is 0.142, which is also consistent with the experimentalresult reported in the literature [29].

Having showed the capability and effectiveness of the improvedISPH algorithm on a geometry with sharp corners, we furthertested our algorithm on a more general and complex geometrywith curved boundaries and a thin body section. Fig. 7 presents aclose-up view of particle positions around airfoils with the angles

Fig. 5. The comparison of vortex shedding contours obtained with ISPH (left) and FEM (rithe Reynolds number of 300.

of attack of 5� and 15� for a simulation time of 100 s, correspondingto the Reynolds number of 570.

The proposed algorithm is also very successful in simulating theflow around the airfoil geometry with any geometrical orientationsacross the flow field, whereby producing simulation results in verygood agreement with the FEM method as shown in Figs. 8 and 9.Fig. 8 compares SPH and FEM velocity contours for the angle of at-tack of 15� with the Reynolds number value of 570.

The pressure envelope results obtained by ISPH and FEM forsmaller angles of attack are also in very good agreement with eachother with some local deviations as shown in Fig. 9 (left). As for theangle of attack of 15�, there is a slight departure in pressure valuesfrom FEM results for the upper camber in the vicinity of leadingedge and the stagnation point. These discrepancies in pressure val-ues might be attributed to the dynamic nature of the SPH methodsince fluid particles are in continuous motion. Hence, there mightbe a local temporary depletion of particles nearby the solid bound-aries as shown in Fig. 7, which deteriorates the accuracy of thecomputed pressure because the SPH gradient discretizationscheme is rather sensitive to the particle deficiencies within theinfluence domain of the smoothing kernel function. Except the lo-cal deviations in pressure values on the airfoil, SPH results compareexcellently well with FEM solutions. Additionally, the pressure dif-ferences between upper and lower camber which correlate withthe lift force are in match with FEM results for a given positionon the boundary.

In order to investigate the sensitivity of the numerical solutionsto particle numbers in the ISPH method and the underlying meshin FEM, the velocity field over the airfoil at the same values ofthe parameters have been computed on three different sets of par-ticles (i.e., 150 � 62 (coarse), 300 � 125 (intermediate), and400 � 167 (fine)) and two sets of meshes (i.e., 11,953 number oftriangular elements with 54,688 degrees of freedom (DOF)(coarse), and 191,248 number of triangular elements with864,206 DOF (fine)). Fig. 10 shows velocity contours in terms ofvelocity magnitudes for the airfoil with the angle of attack of 15�at Reynolds number of 420. The comparison of results on thecoarse, medium and the fine particle numbers demonstrates evi-dently that the intermediate particle number provides solutionswith sufficient accuracy considering the trade-off between compu-tational costs and capturing the features being studied. Since finermeshes are computationally expensive, the intermediate particlenumber is chosen for the numerical simulations presented in this

ght) methods for the cases of vortex tail down (up figures) and up (down figures) for

Fig. 6. The streamlines of vortex shedding for a full period at the Reynolds number of 300 plotted on particle distribution where colors denote the velocity magnitude. Oneperiod T is 9.85 s. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)


article. The simulations are performed on a workstation withthe configuration of Intel (R) Core (TM) i7 (@9500 @3.07 GHz)processor under Windows XP (64 bit) operating system. The

computational cost in terms of CPU time for the coarse, mediumand fine particle numbers are 1.36, 3.24 and 14.05 s per each timestep, respectively.

Fig. 7. Close-up view of particle positions around airfoils with the angles of attack of 5� and 15� for a simulation time of 100 s, corresponding to the Reynolds number of 570.For the angle of attack of 15�, there is slight particle depletion in the close vicinity of the stagnation point, which is not observed for lower values of the angle of attack.

Fig. 8. A comparison of ISPH (left) and FEM (right) velocity contours for the angle of attack of 15� with the Reynolds number value of 570.

Fig. 9. The comparison of pressure envelopes for the angles of attack of 5� (left) and 15� (right) for the Reynolds numbers of 420 (up) and 570 (down).


Fig. 10. The velocity fields in terms of velocity magnitudes over the airfoil (with the angle of attack of 15� at Reynolds number of 420) computed on three different sets ofparticles for the ISPH method and on two sets of meshes for FEM.

Fig. 11. The comparison of vortex shedding contours produced by ISPH (left) and FEM (right) methods for the angle of attack of 10� and the Reynolds number of 1600.


Fig. 11 compares ISPH and FEM results in terms of the vortexshedding contours for the angle of attack of 10� with the Reynoldsnumber of 1600. As in the case of the presented square obstacle re-sults, SPH results are also satisfactorily in agreement with FEMregarding the magnitude of velocities as well as the position andthe number of vortices for the airfoil geometry.

The vortex shedding behavior computed by the SPH method ispresented in Fig. 12 as a close-up view magnifying the region aboutone airfoil length behind the trailing edge in terms of the stream-lines of vortex shedding for a full period for the Reynolds numberof 1400 and the angle of attack of 10�. There are two vortices nearthe trailing edge of the airfoil, one being on the upper camber, andthe second is at the end of the trailing edge. The first vortex ispushing the second vortex away from the trailing edge during itsdownward motion towards the trailing edge. When the secondvortex disappears, and the first vortex reaches to the trailing edge,

a new vortex appears on the upper camber again, thereby complet-ing a full period.

4. Conclusion

In this work, we have presented solutions for flow over an air-foil and a square obstacle using an improved ISPH algorithm thatcan handle complex geometries with the usage of the MBT method,and eliminate particle clustering induced instabilities with theimplementation of artificial particle displacement (particle frac-ture repair) procedure as well as the corrective SPH discretizationscheme. The close-up views for the distribution of fluid particlesaround bluff bodies illustrate that the proposed boundary treat-ment prevents both particle deficiency and particle penetrationacross the solid boundary without disturbing flow structure. Wehave shown that the improved ISPH method can be effectively

Fig. 12. The streamlines of vortex shedding for a full period at the Reynolds number of 1400 plotted on particle distribution where colors denote the velocity magnitude. Oneperiod T is 3.2 s. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)


used for flow simulations over bluff-bodies with Reynolds num-bers as high as 1600, which is not achievable with standard ISPHformulations. It should be noted that the improved ISPH algorithmis not limited to simulations for Reynolds numbers of 1600 in lam-inar flow regime. It can run to higher Reynolds number valueswithout any interruption. Our simulation results are validated withan FEM method, and excellent agreements among the results wereobserved. We illustrated that the improved ISPH method is able tocapture the complex physics of bluff-body flows naturally such asflow separation, wake formation at the trailing edge, and vortexshedding without any extra effort to increase the particle resolu-tion in some specific areas of interest. As well, our future work in-cludes the implementation of artificial particle displacementalgorithm for the flow problems with free surfaces as well as theutilization of the MBT boundary treatment method in multiphaseflow problems.

Acknowledgement

Funding provided by the European Commission Research Direc-torate General under Marie Curie International Reintegration Grantprogram with the Grant Agreement number of 231048 (PIRG03-GA-2008-231048) is gratefully acknowledged.

Appendix A

The following section provides derivations for the SPH approx-imation to first- and second-order derivatives of a vector-valuedfunction. The derivations are carried out in Cartesian coordinates.The SPH approximation for the gradient of a vectorial functionstarts with a Taylor series expansion of f pð~rjÞ so that

f pð~rjÞ ¼ f pð~riÞ þ rlji@f pð~riÞ@xl

i

~rj¼~ri

þ 12

rljir

kji@f pð~riÞ@xl

i@xki

~rj¼~ri

: ðA:1Þ

Upon multiplying Eq. (A.1) by the term, @Wðrij;hÞ=@xsj , and then

integrating over the whole space d3~rj, one can write,

ZXðf pð~rjÞ � f pð~riÞÞ

@Wðrij;hÞ@xs

j

d3~rj

¼ @f pð~riÞ@xl

i

ZX

rlji@Wðrij; hÞ

@xsj

d3~rj|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}Ils

þ12@f pð~riÞ@xl

i@xki

�Z

Xrl

jirkji@Wðrij;hÞ

@xsj

d3~rj|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}Ilks¼0

: ðA:2Þ

Note that the first and the second integrals on the right hand side ofEq. (A.2) are, respectively, second- and third-rank tensors. Thethird-rank tensor Ilks can be integrated by parts, which, upon usingthe Green–Gauss theorem produces Eq. (A.3) since the kernelW(rij,h) vanishes beyond its support domain

Ilks ¼ �Z

XWðrij; hÞ

@

@rsj

rljir

kji

� �d3~rj

¼ �Z

XWðrij; hÞ rl

jidsk þ rk

jidls

� �d3~rj ðA:3Þ

Recalling that the kernel function is spherically symmetric evenfunction and the multiplication of an even function by an odd func-tion produces an odd function. Integration of an odd function over asymmetric domain leads to zero

Ilks ¼ �dskZ

Xrl

jiWðrij; hÞd3~rj|fflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}¼0

�dlsZ

Xrk

jiWðrij;hÞd3~rj|fflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}¼0

¼ 0: ðA:4Þ

Following the above described procedure identically, the secondrank tensor Ils can be written as

Ils ¼ �dlsZ

XWðrij;hÞd3~rj|fflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflffl}

¼1

¼ �dls: ðA:5Þ

On combining Eq. (A.2) with Eqs. (A.4) and (A.5), one can write,

@f pð~riÞ@xs

i

¼Z

Xðf pð~rjÞ � f pð~riÞÞ

@Wðrij; hÞ@xs

i

d3~rj: ðA:6Þ


Note that in Eq. (A.6), the relationship @Wðrij;hÞ=@xsj ¼

�@Wðrij;hÞ=@xsi has been used. Replacing the integration in Eq.

(A.6) with the SPH summation over particle ‘‘j’’ and settingd3~rj ¼ mj=qj, we can obtain the gradient of a vector-valued functionin the form of the SPH interpolation as

@f pð~riÞ@xs

i

¼XN

j¼1

mj


@Wðrij; hÞ@xs

i

: ðA:7Þ

It is important to note that the second rank tensor Ils, shown to beequal to Kronecker delta for a continuous function, may not beequal to Kronecker delta for discrete particles. Hence, for the accu-racy of the computations, this term should be included in the SPHgradient interpolation of a function. From Eq. (A.2), we can write

XN

j¼1

mj


@Wðrij;hÞ@xs

i

¼ @f pð~riÞ@xl

i

XN

j¼1

mj

qjrl

ji@Wðrij;hÞ

@xsi

: ðA:8Þ

Eq. (A.8) can be written in matrix form as

PNj¼1

f ð1Þji að1Þj

PNj¼1

f ð1Þji að2Þj

266664

377775 ¼

PNj¼1

rð1Þji að1Þj

PNj¼1

rð2Þji að1Þj

PNj¼1

rð1Þji að2Þj

PNj¼1

rð2Þji að2Þj

266664

377775

@f ð1Þi

@xð1Þi

@f ð1Þi

@xð2Þi

2664

3775; ðA:9Þ

where asj ¼ ðmj=qjÞ @Wðrij;hÞ=@xs

i

� �.

Starting with the relation for the SPH second-order derivativeapproximation [22] of a vector valued-function f pð~riÞ given in Eq.(A.10)

2Z

Xðf pð~riÞ � f pð~rjÞÞ

rsij

r2ij

@Wðrij;hÞ@xm

i

d3~rj

¼ 2n@2f pð~riÞ@xs

i@xmi

þ 1n@2f pð~riÞ@xk

i @xki

dsm; ðA:10Þ

which, upon contracting on indices p and s, one can obtain

2Z


rpij

r2ij

@Wðrij;hÞ@xm

i

d3~rj ¼1n@2f pð~riÞ@xk

i @xki

dpm: ðA:11Þ

Note that the first term on the right hand side of Eq. (A.10) becomes@2f pð~riÞ=@xp

i @xmi and consequently drops off if the vector-valued

function f pð~riÞ is assumed to be a divergence-free velocity field.Here, the coefficient n takes the value of 4 and 5 in two and threedimensions, respectively. We have shown in Eqs. (A.2) and (A.5)that Kronecker delta can be written as,

dpm ¼Z

Xrp

ji

@Wðrij;hÞ@xm

i

d3~rj: ðA:12Þ

Casting Eq. (A.12) into Eq. (A.11) leads to

2Z


rpij

r2ij

@Wðrij;hÞ@xm

i

d3~rj

¼ 1n@2f pð~riÞ@xk

i @xki

ZX

rpji

@Wðrij;hÞ@xm

i

d3~rj ðA:13Þ

Eq. (A.13) can be written in matrix form as

Xj¼1

f ð1Þij rð1Þij þ f ð2Þij rð2Þij

� � 8r2

ij

að1Þj

að2Þj

24

35 ¼

Pj¼1

rð1Þji að1Þj

Pj¼1

rð2Þji að1ÞjPj¼1

rð1Þji að2Þj

Pj¼1

rð2Þji að2Þj

2664

3775

@2f ð1Þi

@xki@xk

i

@2f ð2Þi

@xki@xk

i

2664

3775

ðA:14Þ

Upon contracting on indices s and m of Eq. (A.10), an alternativeform of the Laplacian for a vector field can be obtained as

8XN

j¼1

mj


rsij

r2ij

@Wðrij; hÞ@xs

i

¼ ð2þ dssÞ @2f pð~riÞ@xk

i @xki

: ðA:15Þ

If the trace of the Kronecker delta in Eq. (A.15) is replaced by thetrace of Eq. (A.12), one can obtain an alternative form of the correc-tive SPH interpolation for the Laplacian.

References

[1] A. Rafiee, M.T. Manzari, S.M. Hosseini, An incompressible SPH method forsimulation of unsteady viscoelastic free-surface flows, Int. J. Non-Linear Mech.42 (10) (2007) 1210–1223.

[2] Y. Mele’an, L.D.G. Sigalotti, A. Hasmy, On the SPH tensile instability in formingviscous liquid drops, Comput. Phys. Commun. 157 (2004) 191–200.

[3] A.M. Tartakovsky, P. Meakin, A smoothed particle hydrodynamics model formiscible flow in three-dimensional fractures and the two-dimensionalRayleigh Taylor instability, J. Comput. Phys. 207 (2) (2005) 610–624.

[4] P.W. Cleary, J. Ha, V. Aluine, T. Nguyen, Flow modelling in casting processes,Appl. Math. Mod. 26 (2002) 171–190.

[5] L.B. Lucy, A numerical approach to the testing of the fission hypothesis, Astron.J. 82 (1977) 1013.

[6] R.A. Gingold, J.J. Monaghan, Smooth particle hydrodynamics: theory andapplication to non-spherical stars, Mon. Not. R. Astron. Soc. 181 (1977) 375.

[7] L.D.G. Sigalotti, J. Klapp, E. Sira, Y. Mele’an, A. Hasmy, SPH simulations of time-dependent Poiseuille flow at low Reynolds numbers, J. Comput. Phys. 191(2003) 622–638.

[8] A. Rafiee, K.P. Thiagarajan, An SPH projection method for simulating fluid-hypoelastic structure interaction, Comput. Methods Appl. Mech. Engrg. 198(2009) 2785–2795.

[9] M.B. Liu, G.R. Liu, K.Y. Lam, A one-dimensional meshfree particle formulationfor simulating shock waves, Shock Waves 13 (2003) 201–211.

[10] S.M. Hosseini, M.T. Manzari, S.K. Hannani, A fully explicit three-step SPHalgorithm for simulation of non-Newtonian fluid flow, J. Numer. Methods HeatFluid Flow 17 (7) (2007) 715–735.

[11] R.A. Rook, M. Yildiz, S. Dost, Modelling 2D transient heat transfer using SPHand implicit time integration, J. Numer. Heat Transfer B 51 (2007) 1–23.

[12] S. Kulasegaram, J. Bonet, R.W. Lewis, M. Profit, A variational formulation basedcontact algorithm for rigid boundaries in two-dimensional SPH applications,Comput. Mech. 33 (2004) 316–325.

[13] J. Feldman, J. Bonet, Dynamic refinement and boundary contact forces in SPHwith applications in fluid flow problems, Int. J. Numer. Methods Engrg. 72(2007) 295–324.

[14] Y. Krongauz, T. Belytschko, Enforcement of essential boundary conditions inmeshless approximations using finite elements, Comput. Methods Appl. Mech.Engrg. 131 (1996) 133–145.

[15] I. Alfaro, F. Yvonnet, R. Chinesta, E. Cueto, A study on the performance ofnatural neighbour-based Galerkin methods, Int. J. Numer. Methods Engrg. 71(2007) 1436–1465.

[16] J.C. Simpson, M. Wood, Classical kinetic theory simulations using smoothedparticle hydrodynamics, Phys. Rev. E 54 (2) (1996) 2077–2083.

[17] J.J. Monaghan, A. Kos, Solitary waves on a cretan beach, J. Waterway PortCoastal Ocean Engrg. – ASCE 125 (1999) 145–154.

[18] J.J. Monaghan, Smoothed particle hydrodynamics, Rep. Prog. Phys. 68 (2005)1703.

[19] J.P. Morris, P.J. Fox, Y. Zhu, Modeling low Reynolds number incompressibleflows using SPH, J. Comput. Phys. 136 (1997) 214–226.

[20] A. Colagrossi, M. Landrini, Numerical simulation of interfacial flows bysmoothed particle hydrodynamics, J. Comput. Phys. 191 (1) (2003) 448–475.

[21] H. Takeda, S.M. Miyama, M. Sekiya, Numerical simulation of viscous flow bysmoothed particle hydrodynamics, Prog. Theor. Phys. 92 (5) (1994) 939–960.

[22] M. Yildiz, R.A. Rook, A. Suleman, SPH with the multiple boundary tangentmethod, Int. J. Numer. Methods Engrg. 77 (10) (2009) 1416–1438.

[23] M.B. Liu, G.R. Liu, Smoothed particle hydrodynamics (SPH): an overview andrecent developments, Arch. Comput. Methods Engrg. 17 (2010) 25–76.

[24] P.W. Cleary, J.J. Monaghan, Conduction modelling using smoothed particlehydrodynamics, J. Comput. Phys. 148 (1999) 227–264.

[25] A.J. Chorin, Numerical solutions of the Navier–Stokes equations, Math.Comput. 22 (1968) 745–762.

[26] A.J. Chorin, On the convergence of discrete approximations to the Navier–Stokes equations, Math. Comput. 23 (1969) 341–353.

[27] S.J. Cummins, M. Rudman, An SPH projection method, J. Comput. Phys. 152(1999) 584–607.

[28] J. Bernsdorf, Th. Zeiser, G. Brenner, F. Durst, Simulation of a 2D channel flowaround a square obstacle with Lattice–Boltzmann (BGK) automata, Int. J. Mod.Phys. C 9 (8) (1998) 1129–1141.

[29] A. Okjima, Strouhal numbers of rectangular cylinders, J. Fluid Mech. 123(1982) 379–398.

comput. methods appl. mech....

Documents