a detailed study of lid-driven cavity flow at moderate ... detailed study of lid-driven cavity flow...

A detailed study of lid-driven cavity flow at moderate Reynoldsnumbers using Incompressible SPH

Shahab Khorasanizade and Joao M. M. Sousa*,†

Instituto Superior Técnico, Mechanical Engng. Dept., Av. Rovisco Pais, 1049-001 Lisboa, Portugal

SUMMARY

Lid-driven cavity flow at moderate Reynolds numbers is studied here, employing a mesh-free methodknown as Smoothed Particle Hydrodynamics (SPH). In a detailed study of this benchmark, the incompress-ible SPH approach is applied together with a particle shifting algorithm. Additionally, a new treatment forno-slip boundary conditions is developed and tested. The use of the aforementioned numerical treatment forsolid walls leads to significant improvements in the results with respect to other SPH simulations carried outwith similar spatial resolution. However, the effect of spatial resolution is not considered in the presentstudy as the number of particles used in each case was kept constant, approximately reproducing the sameresolutions employed in reference studies available in the literature as well. Altogether, the detailed compar-isons of field variables at discreet points demonstrate the accuracy and robustness of the new SPH method.Copyright © 2014 John Wiley & Sons, Ltd.

Received 14 February 2014; Revised 11 June 2014; Accepted 26 July 2014

KEY WORDS: incompressible smoothed particle hydrodynamics; mesh-free method; no-slip boundarycondition; lid-driven cavity

1. INTRODUCTION

Traditional mesh-based numerical methods have been widely applied in different fields of scienceand engineering, but the growing interest in complex geometry simulations limits the applicationof this kind of methods. One of the ways to overcome this drawback is to use mesh-less methods in-stead. Smoothed Particle Hydrodynamics (SPH) is among this category. By nature, it is a Lagrangianmethod, which discretizes the domain by introducing particles as computational points that move inthe flow according to the equation of motion. It was first introduced by Lucy [1] and Gingold andMonaghan [2] for compressible astrophysical flows. The application of SPH to incompressible flowsstarted with the work of Morris et al. [3] and continued to this day. Although most SPH studies havebeen restricted to free surface flows, thanks to its unique abilities to simulate such flows, several effortshave been made to improve the method for confined flows as well [4–9].In order to deal with incompressible flows, researchers resort to two different types of

approaches. The first scheme is referred to as Weakly Compressible SPH (WCSPH) [4]. It assumesthat the fluid has some compressibility, all the flow equations are solved explicitly and an equationof state is used to relate density, and pressure. Compared with the second scheme, which is based ona projection method and often called Incompressible SPH (ISPH), classical WCSPH exhibits a fewdrawbacks such as reduced accuracy in the calculation of pressure [4, 10]. Moreover, the stabilityrequirement to use considerably smaller increments in time advancement is another limitation facedby WCSPH, although ISPH incurs a supplementary computational cost in order to solve the

*Correspondence to: Joao M. M. Sousa, Instituto Superior Técnico, Mechanical Engng. Dept., Av. Rovisco Pais, 1049-001Lisboa, Portugal.

†E-mail: [email protected]

Copyright © 2014 John Wiley & Sons, Ltd.

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDSInt. J. Numer. Meth. Fluids 2014; 76:653–668Published online 28 August 2014 in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/fld.3949

pressure Poisson equation implicitly. Recently, researchers dedicated their efforts to improve theaccuracy of SPH for both algorithms, in a number of studies [5–7,11–13] among others. WithinWCSPH, the SPH-ALE algorithm, first introduced by Vila [14], shows great capabilities as pointedout by Marongiu et al. [13]. In addition, δ-SPH also allows major improvements over classicalWCSPH via the addition of a diffusive term to the equations [11].The treatment of solid walls in SPH simulations, however, requires great care. A variety of

methods has been proposed, such as the use of image particles [10], dummy particles [4], themultiple boundary tangent method [15], and semi-analytical approaches [16], to name a few. Allof these methods, except the latter, which is based on a variational formulation, can be used to placethe particles inside the solid wall, while the choice of the velocity boundary condition (BC) is lessstraightforward and has a large effect on the results as investigated by Basa et al. [17]. Some ofthese features have also been studied by González et al. [18] for various implementations, thusdemonstrating that an inconsistent shear force calculation at initial steps can later be balanced byanother boundary force appearing when slip velocities are present at the boundary. Anyway, theforegoing issues become of great concern when dealing with confined flow problems such as thelid-driven cavity.The lid-driven cavity problem involves viscous flow inside a square cavity in which the top wall

moves with a constant velocity, whereas the other walls remain stationary [19]. It is tackled fre-quently as a benchmark case in the framework of SPH [20]. Numerical schemes are tested in thatcontext with respect to robustness, accuracy, and efficiency. No analytical solution exists for thisflow problem, but it has been extensively studied, employing panoply of methods [19, 21–24].Erturk [21] divides numerical studies of lid-driven cavity flow into three categories: steadysolutions, hydrodynamic stability analyses, and direct numerical simulations of the transition fromsteady regime to unsteady flow. A number of studies carried out for moderate Reynolds numbersusing SPH can also be found in the literature (e.g., [4, 5, 17]).Within the first of the aforementioned categories, there are controversial issues such as the limit

of steadiness. Some researchers, such as Ghia et al. [19], showed steady solutions up to a Reynoldsnumber Re = 10000, while others [25] claimed that the flow is steady up to a value as high asRe = 21000. According to Erturk [21], one should always remember that steady, two-dimensional lid-driven cavity flow does not occur at high Reynolds numbers in reality andsuch flows are therefore fictitious. Moreover, he pointed out that the second and third categories ofstudies for this problem contradict the first category in the sense of steadiness [21].In the present work, we have developed a new solid wall BC treatment via changing the velocity

profile for the particles placed inside the wall, with the aim of calculating wall viscous forces moreaccurately. The implementation is tested on lid-driven cavity flow at four Reynolds numbers,namely Re = 100, 400, 1000, and 3200, and the results are compared with available data in literature[5, 17, 19, 22, 23]. Additionally, a much more detailed flow analysis that those performed in previ-ous SPH studies of this benchmark is provided. In Section 2, we review SPH formulations,governing equations and the ISPH algorithm used in present investigation. Section 3 is devotedto the proposed BC, and its effect on the velocity distribution inside the wall is discussed. InSection 4, the complete SPH method is tested on lid-driven cavity flow at moderate Reynoldsnumbers, and the results are extensively scrutinized. Major improvements with respect to other SPHsimulations previously published were found upon the use of the methodology described herein.

2. SPH METHODOLOGY

According to Monaghan [26], SPH is based on the filtering (smoothing) of any generic field f with aconvolution integral extended over the fluid domain Ω

fh i rð Þ ¼ ∫Ω f r′ð Þw r′� r; hð ÞdV ′; (1)

where w is a weighting function, which has a compact support of size h bounded to Ω, and r representsthe spatial coordinates of a generic point. The weighting function is known in the context of SPH as akernel function, and it is defined as positive and symmetric. Apart from the foregoing properties, this

654 S. KHORASANIZADE AND J. M. M. SOUSA

Copyright © 2014 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Fluids 2014; 76:653–668DOI: 10.1002/fld

function is also expected to exhibit a few other specific characteristics, as described in detail byMonaghan [26] and Liu et al. [27]. Equation (1) can be applied to the gradient of a function as well.After mathematical manipulation, it reads as

∇fh i rð Þ ¼ ∫Ω f r′ð Þ∇w r′� r; hð ÞdV ′þ ∫∂Ω f r′ð Þw r′� rð Þn′dS′ ; (2)

where ∇w stands for the derivative with respect to r, ∂Ω is the boundary of Ω, and n′ indicates thenormal unit vector of the boundary. It can be shown that if the particle’s compact support is withinthe domain of integration, the surface integral in Equation (2) vanishes, whereas in the case of near-boundary particles, kernel truncation occurs. However, the latter case can be remedied by the usageof boundary particles [27]. Based on these equations, the SPH scheme is obtained by substitutingcontinuous fields with discrete Lagrangian particles carrying field values (e.g., velocity and pressure).In order to interpolate a field f at a calculation point i, one uses the following convolution summation,which is the particle representation of Equation (1):

f i ¼ ∑j

mj

ρjf jwij; (3)

where i and j denote the target and its neighboring particle, respectively, m and ρ stand for the particlemass and density, respectively, and wij is the kernel function centered on i as calculated at j. Here, thegradient of a field is given by the formula used in [5], which reads as follows:

∇f i ¼ ∑j

mj

ρjf j � f i

� �∇wij: (4)

2.1. Governing equations

Two different approaches can be used to simulate incompressible flows in SPH. The first one iscalled WCSPH, which solves the Navier–Stokes (NS) equations in Lagrangian form fully explicitlywith the help of an equation of state that relates pressure and density. In this method, fluid isconsidered to have a small compressibility. The second method, which is the focus of this work,is truly ISPH, thereby solving the Lagrangian NS equations by means of a projection method.Various projection methods exist [5], among which we choose the divergence free method. Withrespect to WCSPH, this methodology has the advantage of allowing larger time steps, thanks toits implicit nature. Nevertheless, solving the ensuing system of equations may be very costly as wellin some situations. The Lagrangian NS equations to be used for ISPH are

∇ � u ¼ 0 ;

dudt

¼ �∇pρ

þ ν∇2uþ F ;

drdt

¼ u ;

8>>>><>>>>:

(5)

where u is the velocity vector, p is the pressure, ρ and ν stand for the density and viscosity of the fluid,respectively, F represents the external forces (e.g., gravity or other driving force), and r is the positionvector. In order to simulate fluid flows using the system of Equations (5), it is convenient to split it intoa predictor step and a corrector step. Intermediate values (within a time step Δt) are initially calculatedat the predictor step and used later, at the corrector step, in the solution of a Poisson equation for thefield pressure. In simple versions of the so-called projection method, sometimes referred to as the‘homogeneous scheme’ [7], the pressure gradient is omitted in the calculation of the intermediatevelocity field. In this case, the predictor step relations are simply as follows:

r�i ¼ rni þ Δt uni ; (6)

u�i ¼ uni þ ν∇2uni þ Fni

� �Δt: (7)

655A DETAILED STUDY OF LID-DRIVEN CAVITY FLOW USING INCOMPRESSIBLE SPH


According to Equations (6) and (7), an intermediate position (ri*) and velocity (ui*) for particle i iscalculated based on the values of position, velocity, and external force from the previouslycompleted iteration (superscript n). Subsequently, the pressure is used to project the velocity intoa divergence free space, thus allowing the calculation of the pressure field in the corrector step,as well as the final velocity field to move the particles to their final position at the end of this timeiteration. The corrector step relations are

∇ � 1ρ∇pnþ1

� �¼ 1

Δt∇ � u�i ; (8)

unþ1i ¼ u�i �

Δtρ∇pnþ1

i ; (9)

rnþ1i ¼ rni þ Δt

unþ1i þ uni

2

� �: (10)

In Equations (8)–(10), corrected values of the pressure field obtained via the correspondingPoisson equation, velocity, and position for the next time iteration (superscript n+ 1) are for particlei. An alternative formulation is followed by Lee et al. [4], calculating the intermediate velocity atthe position given by the previous iteration instead. The consequences of this option have alreadybeen investigated [5], and the results were found to be identical to those obtained by Cumminsand Rudman [10].In the present study, the so-called negative form of the relations for both the pressure gradient and

velocity divergence is used, despite the more widespread use of the ‘positive form’ in the context ofISPH [4, 10]. Following Basa et al. [17], the discreet form of these equations is

∇pi ¼ ∑j

mj

ρjpj � pi� �

∇wij; (11)

∇ � ui ¼ ∑j

mj

ρjuj � ui� � � ∇wij; (12)

where the kernel gradient ∇wij is calculated at the position of the jth particle with respect to ith spatialcoordinates.Various forms of SPH discretization exist for the viscous term in the system of Equations (5).

This subject has been extensively studied by Basa et al. [17] in the framework of WCSPH. Inaddition, in a recent investigation carried out by Shahriari [6], the results of two differentapproaches were compared in the context of WCSPH as well. There, it is shown that the formulaproposed by Morris et al. [3] produces better results in a wider range of Reynolds numbers. Basedon these indications, together with those obtained from a few more studies [5, 7], it was decided touse the following relation:

ν∇2ui ¼ ∑j

2mj ν rij :∇wij

ρjr2ij

uij; (13)

where rij= ri� rj and rij= jrij j.The solution of Equation (8) requires the assembly of a coefficient matrix and the corresponding

right-hand side. The latter is plainly obtained by applying Equation (12), while at least two different pro-cedures [10] may be considered when constructing the matrix. Here, the following expression is used:

∇ � 1ρ∇pi ¼ ∑j

2mj

ρ2pijrij � ∇wij

r2ij; (14)

where pij= pi� pj. It must be noted that the symbol ρ in Equation (14) does not carry anymore a subscriptaddressing any particle. This is a consequence of fluid incompressibility, and hence, the fluid density ρremains a constant. Naturally, this notion should be extended to all equations used in the present study.



2.2. Kernel corrections

Simply based on a Taylor series expansion, the interpolation among neighboring particles exhibitssecond-order accuracy [28]. However, as particles move and change their spatial position,additional numerical errors are introduced. Aiming to improve this problem, a number of remedieshave been proposed and analyzed [28–31]. Among these methods, and also following [5, 7], theprocedure proposed by Oger et al. [28] is implemented here:

L rð Þi ¼∑jV j xj � xi

� �∂wij

∂x∑jV j xj � xi

� �∂wij

∂y

∑jV j yj � yi� �∂wij

∂x∑jV j yj � yi

� �∂wij

∂y

0BB@

1CCA

�1

; (15)

∇ewij ¼ L rð Þi∇wij; (16)

where L is the kernel gradient correction operator, x and y denote the horizontal and verticalcoordinates of a particle, and V represents the volume of the particle, which is directly calculated herefrom m/ρ. The corrected values resulting from Equation (16) replace the original kernel gradients in theflow equations.

2.3. Particle shifting algorithm

As mentioned earlier in Section 2.2, the numerical accuracy of SPH is compromised once theparticles start moving. Monaghan [26] proposed the so-called XSPH variant, which modifies theparticle movement based on the local average velocity to attain a more isotropic particle distribu-tion. However, as pointed out by Shahriari et al. [6] and Khorasanizade et al. [8, 32], the use ofXSPH does not produce significant improvements, and in some cases, it leads to non-physicalresults. Another procedure to overcome the aforementioned problem is to use the particle shiftingalgorithm developed by Xu et al. [5]. This method has been used in other studies as well [7, 33],but without the correction of the variables proposed in the original work [5]. The complete set ofrelations to be applied for this purpose in the present ISPH method is

δri ¼ CαRi ;

Ri ¼ ∑Nij¼1

rijr3ij

r2i ;

ri ¼ 1Ni∑Ni

j¼1rij ;

8>>>><>>>>:

(17)

where δr stands for the shifting vector, C is a constant in the range of 0.01–0.1 [5] and usually taken tobe 0.04, α=umax Δt, where umax represents the maximum velocity, and Ni is the number of neighboringparticles around particle i. As noted before in several studies [8, 32, 33], the use of shifting is mandatory inISPH to avoid major particle clumping and voids.After the particles have been shifted to their new positions in accordance to the set of Equations (17),

the field variables must be corrected. Here, the following relation is applied:

ϕi′ ¼ ϕi þ δrii′: ∇ϕð Þi; (18)

where i and i′ denote the old and new positions of a particle, respectively, and ϕ can be replaced bypressure or any of the velocity components.

3. BOUNDARY CONDITIONS

The widely used technique of Image Particles [10] is employed to model the presence of solidwalls. Throughout the present study, no-slip BC for velocity and a homogenous Neumann condi-tion for pressure are used. The implementation of no-slip BC can be carried out in different ways(see, e.g., [3, 10]). As noted by Basa et al. [17], the no-slip condition obtained by the simple use



of image particles [10] generally results in the underestimation of wall viscous forces. A correctevaluation of these forces is of great interest, so the foregoing authors suggested the use of a linearextrapolation applied to the tangential velocity component. However, a new proposal stillallowing major improvements in the calculation of the wall viscous forces is made here. It is builton the same idea as the concept of Morris et al. [3] but with a slightly different formulation.Namely, the present methodology is applied to both velocity components.Similar to [3], a parameter β is defined, but the following relation is used instead:

β ¼ max βmax; 1þdBCdf

� �; (19)

where βmax is an empirical value to be defined for the case of interest. The second term inside theparenthesis corresponds to a linear extrapolation of velocity inside the wall, where dBC and df denotenormal distances to the boundary of the image particle and the fluid particle, respectively. The velocitydistribution of image particles is later shaped according to

uf � uBC ¼ β uf � uwall� �

; (20)

where the subscripts f, BC, and wall represent fluid, image particle and wall values, in that order.In order to better understand how this implementation of the BC operates, a sketch of the

calculated velocities inside the wall and the velocity differences appearing on the left-hand sideof Equation (20) is portrayed in Figure 1. For illustration purposes, a typical velocity distributionnear the wall (the vertical dash line) is assumed for the fluid particles, and the resulting calculatedvelocities inside the wall are shown for different values of βmax. It can be seen that with βmax= 2, thevelocity distribution inside the wall up to the distance of a fluid particle’s image exhibits a constantratio to the target fluid particle, but beyond that location, a linear extrapolation is applied instead. Asβmax increases, the region of constant velocity inside the wall is extended further, as expected.Another observation drawn from Figure 1 is the drastic increase obtained in the velocity differenceterm (directly used in the SPH governing equations) as βmax is raised, thus contributing to bring theusually underestimated wall viscous forces closer to their physically correct value.Hence, the design idea behind this BC treatment is mainly to improve the calculation of the wall

viscous force at the position of the target fluid particle, but still ensuring wall impermeability.Consequently, as the solid wall is approached, higher values of β must be applied, in accordanceto Equation (19). However, the present simulations show that there is an upper limit to the valueof βmax, which depends on the Reynolds number. Three additional notes should also be made at thisstage. First, Basa et al. [17] pointed out that the use of (wall) edge particles does not have a signif-icant impact on the results, and therefore, such particles are not used in the present implementation.Second, the first fluid particle from the wall is placed at half the initial particle spacing, a scenarioalso reproduced in Figure 1. Third, the value of βmax is chosen in all calculations considered in thepresent study by taking the highest value for which the numerical simulation remains stable at eachparticular Reynolds number.

Figure 1. Calculated velocity distributions near the wall (top) for different values of βmax and resultingvelocity differences (bottom), based on a typical velocity distribution assumed for fluid particles (right).

Dash lines indicate the target fluid particle for which each calculation is made.



4. TEST CASES

Despite the fact that several SPH studies have considered before the lid-driven cavity problem,detailed flow information is usually not provided and mere comparisons of velocity and/or pressureprofiles with reference results are made. In the present work, extensive numerical data are collectedfrom ISPH simulations at Reynolds numbers of 100, 400, 1000, and 3200, employing a correspond-ing spatial resolution similar to the lower of the set used by Ghia et al. [19]. A single exception wasmade for the last case, for which a higher resolution had to be used as discussed later in this section.The Reynolds number is based on the lid velocity, size of the cavity, and fluid viscosity. Fromsimulation to simulation, the size of cavity (L) and the kinematic viscosity of the fluid remainunchanged, set to 1m and 10�6m2/s, respectively, whereas the lid velocity (Ulid) varied accordingto the Reynolds number. A schematic of the flow problem is shown in Figure 2 along with thedefinition of the various vortices expected to be formed inside the cavity.In all the cases studied, the results are compared in detail with those obtained by Ghia et al. [19]

and, when available, also with the corresponding ISPH results of Xu et al. [5] for their L/160resolution. Contours of normalized stream function (ψ) and normalized vorticity (ω) are alwayspresented, in the following sections, for the levels listed in Table I, which reproduces exactly thoseused in [19] to facilitate a visual comparison.

4.1. Study at Re = 100

The first ISPH simulation was made here at Re = 100 for which a value of βmax= 8 was used. Thespatial resolution was chosen to be L/130 in order to approximate that used in [19], as mentionedbefore, despite that the two computational methodologies differ enormously (a finite differencemethod applied to a stream function – vorticity formulation of the flow equations in the latter case).The filtering in Equation (1) was numerically performed with the ‘quintic’ kernel [3]. A value of 1.3was specified for the ratio of the smoothing length to the initial particle spacing and kept constantthroughout this work.Profiles of v- and u-velocity along horizontal and vertical geometric centerlines, respectively, are

shown in Figure 3. These profiles are in very good agreement with the reference results also pre-sented. Although the present ISPH simulation employed about 1/3 less particles than in the workof Xu et al. [5], the implemented BC treatment allowed us to obtain virtually the same results, as

Figure 2. Schematic of the flow problem and definition of vortices.



in this case the remainder of both SPH algorithms coincide. A minor deviation is found, however, inthe vertical velocity component next to the right side wall, presumably due to the high velocitygradients above.Basa et al. [17] have computed the value of nondimensional kinetic energy (k) for this flow

employing a finite difference method. As shown in Table II, these authors have obtained a valueof 0.034, whereas in the current ISPH simulation, it is calculated to be approximately 0.033. As acuriosity, it should be noted that when the traditional image BC treatment was used instead, thisquantity fell sharply to 0.027. Hence, as expected for this flow problem, the new BC adds energyto the system. This mechanism should also have a beneficial impact on the formation of the flowvortices inside the cavity, as it will be analyzed next. Again, if the traditional image BC is used,the method is not able of capturing any of the bottom vortices (BL and BR, see Figure 2). Thechange to the new BC results in the appearance of the BR vortex, but the BL vortex is still notclearly observed at this Reynolds number. The values of various flow characteristics are shownin Table II, namely for the primary (P) and BR vortices, together with the maximum and the minimaof the velocity along the centerlines. Except for the stream function value at the center of vortex BR,the calculated errors are always below 6%.Contours of vorticity and stream function are depicted in Figure 4. The contour levels are from

Table I, as mentioned earlier. Because these two flow quantities are not primitive variables in theISPH method, post-processing calculations must be performed to compute them from the velocityfield, thus introducing additional errors. Because of the nature of the method and the flow problem,

Table I. Contour levels used in the graphical representation of stream function and vorticity.

Stream function Vorticity

Value of ψ Contour label Value of ψ Contour label Value of ω Contour label

�0.1175 1 �1.0 × 10�10 13 �3.0 1�0.1150 2 1.0 × 10�08 14 �2.0 2�0.1100 3 1.0 × 10�07 15 �1.0 3�0.1000 4 1.0 × 10�06 16 �0.5 4�0.0900 5 1.0 × 10�05 17 0.0 5�0.0700 6 5.0 × 10�05 18 0.5 6�0.0500 7 1.0 × 10�04 19 1.0 7�0.0300 8 2.5 × 10�04 20 2.0 8�0.0100 9 5.0 × 10�04 21 3.0 9�1.0 × 10�04 10 1.0 × 10�03 22 4.0 10�1.0 × 10�05 11 1.5 × 10�03 23 5.0 11�1.0 × 10�07 12 3.0 × 10�03 24

Figure 3. Velocity profiles along vertical (left) and horizontal (right) centerlines for Re= 100 compared withthe results of Xu et al. [5] and Ghia et al. [19].



these mainly affect the region adjacent to the walls, where the maximum errors were found.However, despite the inability of these ISPH simulations to clearly capture the BL vortex, a directcomparison of the two flow maps with the corresponding ones published by Ghia et al. [19] furthersubstantiates the quality of the present results, namely bearing in mind that both methods wereapplied to this flow problem employing basically the same spatial resolution.

4.2. Study at Re = 400

The flow at Re = 400 was studied by Ghia et al. [19] using two different spatial resolutions and alsoreporting a negligible difference between the corresponding results. Here, a resolution L/130 is usedonce again in the ISPH method, now with a value of βmax= 6.3. Besides the primary (central) vortexand the two secondary vortices (BR and BL) previously mentioned, minute additional vortices havealso been observed for this regime [19], but these have not been captured by the ISPH simulation.However, this cannot be judged as surprising because the size of such additional vortices wasexpected to be comparable with the particle spacing itself.The comparison of centerline velocities with reference data is shown for this case in Figure 5. It is

important to notice that published ISPH results [5] employing a spatial resolution L/160 displaylarger discrepancies with respect to the results of Ghia et al. [19] than the present ISPH simulationusing a resolution L/130. The overall agreement obtained at this regime for the flow velocitiesseems to be slightly worse than for the latter one, but the results are still very good.

Table II. Detailed comparison of flow characteristics and relative errors at Re= 100.

Present Ghia et al. [19] Error (%)

k 0.0333 0.034† 2.10xP 0.6178 0.6172 0.10yP 0.7401 0.7344 0.77ψP �0.0977 �0.1034 5.83ωP 3.1346 3.1665 1.02xBR 0.9287 0.9453 1.79yBR 0.0591 0.0625 5.75ψBR 8.618 × 10�06 1.254 × 10�05 45.5ωBR �0.03225 �0.03308 2.57umin

‡ �0.2024 �0.2109 4.20vmin

‡ �0.2368 �0.2453 3.59vmax

‡ 0.1669 0.1753 5.03

†Data from Basa et al. [17].‡Minima and maximum along geometric centerlines.

Figure 4. Field contours of vorticity (left) and stream function (right) for Re = 100. Contour levels in eachfigure are from Table I.



Aiming to investigate the effect of the new BC on the pressure field as well, a profile obtainedalong the diagonal line connecting the lower left corner and the upper right is also compared withdata from Xu et al. [5], which further includes results from a commercial code (Star CD). The pres-sure profile in Figure 6 shows that improvements were obtained in the region of minimum pressure.However, in the vicinity of the upper right corner of the cavity, larger discrepancies can be seen.This may be due to the numerical treatment of the foregoing corner in this flow problem, whichwas found mandatory here to prevent fluid particles from escaping the domain. A downward cornerpoint velocity equal to half of the lid velocity was imposed in the present ISPH simulations tosmooth out the discontinuity between the high-speed lid and the stationary side wall. Nothing ismentioned about a special corner treatment in the work of Xu et al. [5].Contours of stream function and vorticity are now depicted in Figure 7. As in the previous case,

additional errors were introduced in the near-wall region (mostly visible in the flow streamlines)because of the need for post-processing the raw SPH data. However, good agreement is stillobserved with the corresponding figures published by Ghia et al. [19]. Detailed flow characteristics

Figure 5. Velocity profiles along vertical (left) and horizontal (right) centerlines for Re= 400 compared withthe results of Xu et al. [5] and Ghia et al. [19].

Figure 6. Pressure profile along the diagonal line connecting lower left corner and the upper right corner forRe = 400 compared with the results of Xu et al. [5].



are provided for this regime in Table III along with reference data, but only the three vortices thatcould be clearly captured in the ISPH simulations have been considered. It can be seen that,excluding the ‘weaker’ BL vortex, the computed errors remained relatively low, even for derivedquantities such as vorticity and stream function.

4.3. Study at Re = 1000

Lid-driven cavity flow at Re = 1000 has been extensively studied in many publications. In one ofthese studies, Erturk [21] has questioned whether the flow at this Reynolds number remains steadyand two-dimensional. The present two-dimensional ISPH simulation with a spatial resolution L/130and βmax= 5 produced a solution free of any flow oscillations.Figure 8 shows the stream function and vorticity contours at this regime. Again, only three

vortices were clearly captured by the ISPH method. An additional vortex developing on the bottomright corner could not be adequately resolved. As for the previous cases, the flow streamlinesexhibit irregularities in the close vicinity of the walls that may be attributed to post-processing.However, in this case, some noise with unknown origin is also detected in vorticity at the centralregion (level 8).


Table III. Detailed comparison of flow characteristics and relative errors at Re=400.


k 0.0397 – –xP 0.5568 0.5547 0.38yP 0.6066 0.6055 0.18ψP �0.1088 �0.1139 4.69ωP 2.2793 2.2947 0.68xBL 0.0692 0.0508 26.6yBL 0.0422 0.0469 11.1ψBL 9.423 × 10�06 1.420 × 10�05 50.7ωBL �0.05514 �0.05697 3.32xBR 0.8815 0.8906 1.03yBR 0.1262 0.1250 0.95ψBR 5.890 × 10�04 6.424 × 10�04 9.07ωBR �0.4232 �0.4335 2.43umin

‡ �0.3158 �0.3273 3.64vmin

‡ �0.4363 �0.4499 3.12vmax

‡ 0.2898 0.3020 4.21

‡Minima and maximum along geometric centerlines.



Xu et al. [5] also studied this flow regime using a similar ISPH algorithm and three differentspatial resolutions. In Figure 9, velocity profiles along cavity centerlines generated by the present ISPHmethod are compared with those obtained by the aforementioned authors employing a resolutionL/160, together with data fromGhia et al. [19]. It can be seen that the newBC yields a drastic reductionof the errors in velocity fields obtained with ISPH [4, 5]. Hence, very good agreement with referenceresults [19] could be obtained also at Re=1000, using a similar spatial resolution in ISPH.Once more, it is important to scrutinize the effect of the new BC in the pressure field as well.

Pressure profiles along the centerlines of the cavity have been reported in the literature by Botellaand Peyret [23] for this regime. A direct comparison between the foregoing spectral solution and thepresent ISPH results is depicted in Figure 10. The general agreement is good, but noticeable discrep-ancies can still be observed near the four cavity walls, presumably due to the finer mesh spacing usedin this region by the spectral method as a consequence of a Chebyshev–Gauss–Lobatto collocation.A detailed analysis of flow characteristics and a comparison with reference data [4, 5] were also

carried out for Re = 1000. In this case, flow parameters have been tabulated in Table IV both for thetraditional image BC and the new BC. In addition, the effect of varying βmax up to the maximumacceptable value for this regime can also be appreciated from the results listed in this table. Thecontinuous increase of βmax from zero (linear extrapolation of velocity inside the wall) up to thesolution convergence limit strongly improves the results in every aspect. It is therefore reasonableto assume that the previous findings concerning the velocity profiles obtained from the ISPH sim-ulations can be generalized to the more global analysis supported by the parameters in Table IV.


Figure 9. Velocity profiles along vertical (left) and horizontal (right) centerlines for Re = 1000 comparedwith the results of Xu et al. [5] and Ghia et al. [19].



Figure 10. Pressure profile along vertical (left) and horizontal (right) centerlines for Re = 1000 comparedwith the results of Botella and Peyret [23].

Table IV. Detailed comparison of flow characteristics at Re = 1000 for different BC and various valuesof βmax.

Present BC

Image BC βmax= 0 βmax= 2 βmax= 3 βmax= 4 βmax= 5 Ghia et al. [19]

k 0.0306 0.0376 0.0390 0.0413 0.0426 0.0433 0.0436†

xP 0.5396 0.5325 0.5316 0.5309 0.5309 0.5307 0.5313yP 0.5742 0.5669 0.5676 0.5653 0.5656 0.5646 0.5625ψP �0.0949 �0.1079 �0.1096 �0.1124 �0.1138 �0.1143 �0.1179ωP 1.7185 1.9178 1.9287 1.9234 1.9685 2.0569 2.0497xBL 0.0801 0.0999 0.1001 0.0994 0.0976 0.0968 0.0859yBL 0.0729 0.0634 0.0649 0.0696 0.0724 0.0764 0.0781ψBL 1.090 × 10�04 1.029 × 10�04 1.135 × 10�04 1.387 × 10�04 1.590 × 10�04 1.777 × 10�04 2.311 × 10�04

ωBL �0.1902 �0.2841 �0.3082 �0.3335 �0.3432 �0.3453 �0.3618xBR 0.8516 0.8614 0.8602 0.8599 0.8604 0.8606 0.8594yBR 0.1116 0.1148 0.1155 0.1164 0.1167 0.1172 0.1094ψBR 1.261 × 10�03 1.362 × 10�03 1.416 × 10�03 1.462 × 10�03 1.514 × 10�03 1.538 × 10�03 1.751 × 10�03

ωBR �0.7441 �0.9741 �0.9897 �1.0430 �1.0870 �1.0860 �1.1547umin

‡�0.3073 �0.3512 �0.3575 �0.3687 �0.3741 �0.3768 �0.3829vmin

‡ �0.4147 �0.4751 �0.4836 �0.4994 �0.5066 �0.5114 �0.5155vmax

‡ 0.2903 0.3368 0.3435 0.3553 0.3612 0.3640 0.3710

†Data from Bruneau and Saad [22].‡Minima and maximum along geometric centerlines.

Figure 11. Velocity profiles along vertical (left) and horizontal (right) centerlines for Re = 3200 comparedwith the results of Ghia et al. [19].



Unfortunately, to the authors’ best knowledge, the values of these parameters have never beenreported before in the context of SPH simulations.

4.4. Study at Re= 3200

This section is closed with the presentation of the results from the present ISPH method for the lid-driven cavity at Re = 3200 with βmax= 2.2. In this case, a comparison is made with the reference dataof Ghia et al. [19] only. Le Touzé et al. [34] also carried out SPH simulations for this flow regimeemploying a spatial resolution of L/200. Unfortunately, because of the low quality of the figuresin [34], data for a direct comparison could not be obtained. However, those authors concludedthat their SPH method, using the resolution above, was not able of adequately reproducing theresults of Ghia et al. [19]. This task was eventually achieved in their study, but it required theuse of a Finite Volume Particle Method instead.

Figure 12. Field contours of vorticity (left) and stream function (right) for Re= 3200. Contour levels in eachfigure are from Table I.

Table V. Detailed comparison of flow characteristics and relative errors at Re=3200.


k 0.0425 – –xP 0.5157 0.5165 0.16yP 0.5416 0.5469 0.98ψP �0.1160 �0.1204 3.79ωP 1.8416 1.9886 7.98xBL 0.0851 0.0859 0.94yBL 0.1129 0.1094 3.10ψBL 1.095 × 10�03 9.782 × 10�04 10.7ωBL �1.2018 �1.0630 11.6xBR 0.8162 0.8125 0.45yBR 0.0868 0.0859 1.04ψBR 2.856 × 10�03 3.140 × 10�03 9.94ωBR �2.1357 �2.2737 6.46xTL 0.0560 0.0547 2.32yTL 0.8970 0.8984 0.16ψTL 6.762 × 10�04 7.277 × 10�04 7.62ωTL �1.5977 �1.7116 7.13umin

‡ �0.4176 �0.4193 0.41vmin

‡ �0.5437 �0.5405 0.59vmax

‡ 0.4159 0.4277 2.84

‡Minima and maximum along geometric centerlines.



The velocity profiles along cavity centerlines generated by the present ISPH simulation,employing the same spatial resolution suggested by Le Touzé et al. [34], are shown in Figure 11.It can be seen that these profiles are in very good agreement with the corresponding results of Ghiaet al. [19]. A more global view of the flow field is provided in Figure 12, where stream function andvorticity contours are depicted. The formation of a new secondary vortex (TL) on the top of the leftwall of the cavity is now evident, as anticipated. The characteristics of this flow structure, alongwith the primary and other secondary vortices within the cavity, are, once again, compared in detailwith the reference data of Ghia et al. [19] in Table V. It must be emphasized that despite the rather highvalue of the Reynolds number for this regime, the relative errors affecting the parameters listed in thetable are always small, although an expected increase has been found for non-primitive variables.

5. CONCLUSIONS

The lid-driven cavity problem has been studied for four different Reynolds numbers up to 3200employing an ISPH method. A new BC treatment for solid walls was also tested in this problem.It was designed to improve the calculation of viscous forces acting on the particles in the vicinityof the walls, still ensuring impermeability conditions. The implementation requires setting a singleparameter, βmax, which controls the wall velocity difference magnitude used in the SPH governingequations. In the present cases, it was found that there is an optimum value of βmax for eachReynolds number, limited by stability constraints of the simulations.To the authors’ best knowledge, the solutions obtained for this benchmark problem have been

analyzed with unprecedented detail in the context of SPH simulations. Close agreement with refer-ence data has been obtained, hence demonstrating the potential capabilities and competitiveness ofISPH methods with respect to more traditional methods in terms of numerical accuracy. Moreover,the application of the new BC treatment yielded significant improvements over other SPH results ofthe same problem available in the literature.

ACKNOWLEDGEMENTS

This work has been partially supported by Fundação para a Ciência e a Tecnologia (FCT) via grantPTDC/EME-MFE/103640/2008. The financial support for the PhD studies of Sh. Khorasanizadevia FCT scholarship SFRH/BD/75057/2010 is also acknowledged. The authors want also to thankDiogo Chambel Lopes for his contribution regarding computing facilities.

REFERENCES

1. Lucy LB. A numerical approach to the testing of the fission hypothesis. The Astronomical Journal 1977; 82:1013–1024.doi:10.1086/112164.

2. Gingold RA, Monaghan JJ. Smoothed particle hydrodynamics – theory and application to non-spherical stars.Monthly Notices of the Royal Astronomical Society 1977; 181:375–89.

3. Morris JP, Fox PJ, Zhu Y.Modeling low Reynolds number incompressible flows using SPH. Journal of ComputationalPhysics 1997; 136:214–26. doi:10.1006/jcph.1997.5776.

4. Lee ES, Moulinec C, Xu R, Violeau D, Laurence D, Stansby P. Comparisons of weakly compressible and truly incom-pressible algorithms for the SPH mesh free particle method. Journal of Computational Physics 2008; 227:8417–36.doi:10.1016/j.jcp.2008.06.005.

5. Xu R, Stansby P, Laurence D. Accuracy and stability in incompressible SPH (ISPH) based on the projection methodand a new approach. Journal of Computational Physics 2009; 228:6703–25. doi:10.1016/j.jcp.2009.05.032.

6. Shahriari S, Hassan IG, Kadem L. Modeling unsteady flow characteristics using smoothed particle hydrodynamics.Applied Mathematical Modelling 2013; 37:1431–50. doi:10.1016/j.apm.2012.04.017.

7. Hosseini SM, Feng JJ. Pressure boundary conditions for computing incompressible flows with SPH. Journal ofComputational Physics 2011; 230:7473–87. doi:10.1016/j.jcp.2011.06.013.

8. Khorasanizade S, Sousa JMM, Pinto JF. On the use of a time-dependent driving force in SPH simulations. Proceedingsof 7th International SPHERIC Workshop. Prato, Italy: 2012.

9. Shadloo MS, Zainali A, Yildiz M, Suleman A. A robust weakly compressible SPH method and its comparison withan incompressible SPH. International Journal for Numerical Methods in Engineering 2012; 89:939–56.doi:10.1002/nme.3267.

10. Cummins SJ, Rudman M. An SPH Projection method. Journal of Computational Physics 1999; 152:584–607.doi:10.1006/jcph.1999.6246.



11. Antuono M, Colagrossi A, Marrone S, Molteni D. Free-surface flows solved by means of SPH schemes with numer-ical diffusive terms. Computer Physics Communications 2010; 181:532–49. doi:10.1016/j.cpc.2009.11.002.

12. Colagrossi A, Bouscasse B, Antuono M, Marrone S. Particle packing algorithm for SPH schemes. ComputerPhysics Communications 2012; 183:1641–53. doi:10.1016/j.cpc.2012.02.032.

13. Marongiu J-C, Leboeuf F, Caro J, Parkinson E. Free surface flows simulations in Pelton turbines using an hybridSPH-ALE method. Journal of Hydraulic Research 2010; 48:40–9. doi:10.1080/00221686.2010.9641244.

14. Vila JP. On particle weighted methods and smooth particle hydrodynamics. Mathematical Models and Methods inApplied Sciences 1999; 09:161–209. doi:10.1142/S0218202599000117.

15. Yildiz M, Rook RA, Suleman A. SPH with the multiple boundary tangent method. International Journal for NumericalMethods in Engineering 2009; 77:1416–38. doi:10.1002/nme.2458.

16. Ferrand M, Laurence DR, Rogers BD, Violeau D, Kassiotis C. Unified semi-analytical wall boundary conditions forinviscid, laminar or turbulent flows in the meshless SPH method. International Journal for Numerical Methods inFluids 2013; 71:446–72. doi:10.1002/fld.3666.

17. Basa M, Quinlan N, Lastiwka M. Robustness and accuracy of SPH formulations for viscous flow. InternationalJournal for Numerical Methods in Fluids 2009; 60:1127–48. doi:10.1002/fld.

18. González L, Souto-Iglesias A, Macià F, Colagrossi A, Antuono M. On the non-slip boundary condition enforcementin SPH methods. Proceedings of 6th International SPHERIC Workshop. 2011. 283–290.

19. Ghia U, Ghia K, Shin C. High-Re solutions for incompressible flow using the Navier-Stokes equations and amultigrid method. Journal of Computational Physics 1982; 48:387–411.

20. Rogers B. Test3 – SPHERIC benchmark test 3. 2011. https://wiki.manchester.ac.uk/spheric/index.php/Test3(accessed November 2013).

21. Erturk E. Discussions on driven cavity flow. International Journal for Numerical Methods in Fluids 2009; 60:275–94.doi:10.1002/fld.

22. Bruneau C-H, Saad M. The 2D lid-driven cavity problem revisited. Computers & Fluids 2006; 35:326–48.doi:10.1016/j.compfluid.2004.12.004.

23. Botella O, Peyret R. Benchmark spectral results on the lid-driven cavity flow. Computers & Fluids 1998; 27:421–33.doi:10.1016/S0045-7930(98)00002-4.

24. Magalhães JPP, Albuquerque DMS, Pereira JMC, Pereira JCF. Adaptive mesh finite-volume calculation of 2D lid-cavity corner vortices. Journal of Computational Physics 2013; 243:365–81. doi:10.1016/j.jcp.2013.02.042.

25. Erturk E, Corke TC, Gökçöl C. Numerical solutions of 2-D steady incompressible driven cavity flow at highReynolds numbers. International Journal for Numerical Methods in Fluids 2005; 48:747–74. doi:10.1002/fld.953.

26. Monaghan JJ. Smoothed particle hydrodynamics. Annual Review of Astronomy and Astrophysics 1992; 30:543–74.doi:10.1146/annurev.aa.30.090192.002551.

27. Liu GR, Liu MB. Smoothed particle hydrodynamics: a meshfree particle method. World Scientific Publishing Co.Pte. Ltd. 2003.

28. Oger G, Doring M, Alessandrini B, Ferrant P. An improved SPH method: towards higher order convergence.Journal of Computational Physics 2007; 225:1472–92. doi:10.1016/j.jcp.2007.01.039.

29. Zhang GM, Batra RC. Modified smoothed particle hydrodynamics method and its application to transient problems.Computational Mechanics 2004; 34:137–46. doi:10.1007/s00466-004-0561-5.

30. Zheng X, Duan WY, Ma QW. Comparison of improved meshless interpolation schemes for SPH method andaccuracy analysis. Journal of Marine Science and Application 2010; 9:223–30. doi:10.1007/s11804-010-1000-y.

31. Jiang T, Ouyang J, Ren JL, Yang BX, Xu XY. A mixed corrected symmetric SPH (MC-SSPH) method for compu-tational dynamic problems. Computer Physics Communications 2012; 183:50–62. doi:10.1016/j.cpc.2011.08.016.

32. Khorasanizade S, Pinto JF, Sousa JMM. On the use of inflow/outflow boundary conditions in incompressible internalflow problems using smoothed particle hydrodynamics. Proceedings of ECCOMAS 2012. Vienna, Austria: 2012.

33. Shadloo MS, Zainali A, Sadek SH, Yildiz M. Improved Incompressible smoothed particle hydrodynamics methodfor simulating flow around bluff bodies. Computer Methods in Applied Mechanics and Engineering 2011;200:1008–20. doi:10.1016/j.cma.2010.12.002.

34. Le Touzé D, Barcarolo DA, Kerhuel M, Oger G, Grenier N, Quinlan N, Lobovsky L, Basa M, Leboeuf F, Caro J,Colagrossi A, Marrone S, de Leffe M, Guilcher PM, Marongiu JC. SPH benchmarking: a comparison of SPH var-iants on selected test cases within the NextMuSE initiative. Proceedings of 7th International SPHERIC Workshop.2012. 409–16.



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