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Arabian Journal for Science andEngineering ISSN 1319-8025Volume 37Number 6 Arab J Sci Eng (2012) 37:1723-1735DOI 10.1007/s13369-012-0286-2

Mesoscale Numerical Prediction of FluidFlow in a Shear Driven Cavity

Nor Azwadi Che Sidik & Fudhail AbdulMunir

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Arab J Sci Eng (2012) 37:1723–1735DOI 10.1007/s13369-012-0286-2

RESEARCH ARTICLE - MECHANICAL ENGINEERING

Nor Azwadi Che Sidik · Fudhail Abdul Munir

Mesoscale Numerical Prediction of Fluid Flow in a ShearDriven Cavity

Received: 8 February 2010 / Accepted: 25 December 2010 / Published online: 17 April 2012© King Fahd University of Petroleum and Minerals 2012

Abstract In this paper, a detailed analysis of the lattice Boltzmann method is presented to simulate an incom-pressible fluid flow problem. Thorough derivation of macroscopic hydrodynamics equations from the contin-uous Boltzmann equation is performed. After showing how the formulation of the mesocale particle dynamicsfits into the framework of lattice Boltzmann simulations, numerical results of lid-driven flow inside square andtriangular cavities are presented to highlight the applicability of the approach. The objective of the paper is togain better understanding of this relatively new approach for applied engineering problems in fluid transportphenomena.

Keywords Lattice boltzmann method · Mesoscale · Multiscale · Lid-driven flow

1 Introduction

Computational fluid dynamics (CFD) has emerged as a powerful tool for the analysis of system involving fluidflow, heat transfer and associated phenomena such as chemical reactions, evaporation, condensation, etc. From1960s onwards, the aerospace industry has integrated CFD techniques into the design, research and develop-ment and manufacturing of aircraft and jet engines. More recently, CFD has been applied to the design ofinternal combustion engines, combustion chambers of gas turbines and furnaces. Furthermore, motor vehiclemanufacturers now routinely predict drag forces, under-bonnet airflows and the in-car environment with CFD[1–4].

The fundamental law of any fluid flow problems is the Navier–Stokes equations, which define any single-phase fluid flow. These equations can be simplified by removing terms describing viscosity to yield the Euler

N. A. C. Sidik (B) · F. A. MunirDepartment of Thermofluid, Faculty of Mechanical Engineering, Universiti Teknologi Malaysia,UTM Skudai, 81310 Johor, MalaysiaE-mail: azwadi@fkm.utm.my

F. A. MunirE-mail: fudhail@utem.edu.my

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equations. Further simplification, by removing terms describing vorticity yields the full potential equations.Finally, these equations can be linearized to yield the linearized potential equations.

There are few numerical methods that simulate the evolution of fluid flow at particle level. Among themare direct simulation Monte Carlo (DSMC) [5] and molecular dynamic (MD) [6] methods. In these methods,the trajectories of every particle together with their position in the system are predicted using the second lawof Newton. From the knowledge of the forces on each atom, it is possible to determine the acceleration of eachatom in the system. These methods are deterministic; once the positions and velocities of each atom are known,the state of the system can be predicted at any time in the future or the past. But remember, a cup of watercontains 1023 number of molecules. Even when a gas is being considered where there are fewer molecules anda larger time-step can be used, because of the longer mean free path of the molecules, the number of moleculesthat can be considered is still limited. However, the question is, do we really need to know the behavior of eachmolecule or atom? The answer is no. It is not important to know the behavior of each particle; it is importantto know the function that can represent the behavior of many particles (mesoscale). As a result, in 1988, thelattice Boltzmann method (LBM) [7], a mesoscale numerical method based on statistical distribution functionhas been introduced to replace MD and DSMC methods.

Historically, LBM was derived from lattice gas automata (LGA) [8]. Consequently, LBM inherits somefeatures from it precursor, the LGA method. The first LBM model was a floating-point version of its LGAcounterpart. Each particle in LGA model (represented by single bit Boolean integer) was replaced by a single-particle distribution function represented by a floating-point number. The lattice structure and the evolutionrule remain the same. One important improvement made to enhance the computational efficiency of the LBMwas that the linearization of collision operator [9]. The uniform lattice structure was remaining unchanged.

The LBM has a number of advantages over other conventional CFD approaches. The algorithm is simpleand can be implemented with a kernel of just a few hundred lines [10]. The algorithm can also be easilymodified to allow for the application of other, more complex simulation components. For example, the LBMcan be extended to describe the evolution of binary mixtures, or extended to allow for more complex boundaryconditions [11–13]. Thus the LBM is an ideal tool in fluid simulation.

The objective of present paper is to introduce and discuss the formulation of LBM in simulating fluidflow problem. The derivation of macroscopic continuity and momentum equations from mesoscale Boltzmannequation is discussed in detail. After showing how the formulation of LBM fits in to the framework of mac-roscale flow, numerical results of lid-driven flow in various types of cavity are presented to highlight theapplicability of the approach.

This paper is arranged as follows. The next section begins with the derivation of lattice Boltzmann BGKequation from the continuous classical Boltzmann equation. The complicated collision integral is replaced withthe equilibrium function to simplify the collision process. The capability of the present approach in formulatingmacroscopic dynamics equation of fluid flow using multiscale time approach (Chapman–Enkogs expansion)is then demonstrated. The simulation results of lid-driven flow in cavity are shown and discussed in sectionthree. The fluid flow behavior is expressed in terms of streamline, size, and number of vortices, which areformed in the cavity. The final section concludes current study.

2 Mesoscopic Lattice Boltzmann Model

Ludwig Boltzmann (1844–1906) introduced a transport equation based on statistical mechanics describing theevolution of gas particle in a system as

∂ f

∂ f+ c

∂ f

∂x= � (1)

where f, c, and � stand for density distribution function, mesoscopic speed, and collision function, respec-tively. Any solution of the Boltzmann equation, Eq. (1), requires an expression for the collision operator �. Ifthe collision is to conserve mass, momentum, and energy, it is required that

∫ ⎡⎣ 1

cc

⎤⎦�dc = 0 (2)

However, the expression for � is too complex to be solved. Even if we only consider two-body collision, thecollision integral term needs to consider the scattering angle of the binary collision, the speed and direction

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Fig. 1 D2Q9 lattice model

before and after the collision, etc. Any replacement of collision must satisfy the conservation law as expressedin Eq. (3). The idea behind this replacement is that large amount of detail of two-body interaction is not likelyto influence significantly the values of many experimental measured quantities [14].

There are a few versions of collision operator published in the literature. However, the most well-acceptedversion due to its simplicity and efficiency is the Bhatnagar–Gross–Krook collision model [9] with a singlerelaxation time. The equation that represents this model is given by

�( f ) = − f − f eq

τ(3)

where f eq is the equilibrium distribution function and τ is the time to reach equilibrium condition duringcollision process, and is often called the relaxation time. Eq. (3) also describes that 1/τ of non-equilibriumdistribution relaxes to equilibrium state within time τ on every collision process. Substituting Eq. (3) intoEq. (1) gives

∂ f

∂t+ c

∂ f

∂x= − f − f eq

τ(4)

which is known as the Boltzmann BGK equation.In lattice Boltzmann formulation, magnitude of c is set up so that in each time step �t, every distribution

function propagates in a distance of lattice node spacing �x . This will ensure that distribution function arrivesexactly at the lattice nodes after �t and collides simultaneously.

In order to apply Eq. (4) into the digital computer, the mesoscopic velocity space has to be discretized. Thiscan be done by discretizing the physical space into uniform lattice nodes. Every node in the network is thenconnected with its neighbors through a number of lattice velocities to be determined through the model chosen.The general form of the lattice velocity model is expressed as DnQm where D represents spatial dimensionand Q is the number of connection (lattice velocity) at every node [15]. There are many lattice velocity modelspublished in the literature; however, the most well used due to its simplicity is D2Q9 and shown in Fig. 1.

After discretization in velocity space, Eq. (4) can be rewritten in the following form:

∂ fiσ

∂t+ ciσ

∂ fiσ

∂x= − fiσ − f eq

iσ

τ(5)

where i refers to the number of discrete velocity. σ refers to the direction of mesocopic velocity where σ = 0when i = 0, σ = 1 when i = 1, 3, 5, 7, and σ when i = 2, 4, 6, 8.

The Eulerian expression of the left-hand side of Eq. (5) can be transformed into the Lagrangian form. Todo this, take Euler time step in conjunction with an upwind spatial discretization and then set the grid spacingdivided by the time step equal to the velocity. This leads to the well-known lattice Boltzmann BGK equation

fiσ (x + ciσ ε, t + ε) − fiσ (x, t) = − fiσ − f eqiσ

τ(6)

where ε is a small lattice time unit in physical unit.

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2.1 Derivation of Macroscopic Equations

The equilibrium distribution function f eq is chosen so that it can reconstruct the hydrodynamics of fluid flow.The general form of f eq can be written as

f eqiσ = Aσ + Bσ (ciσ · u) + Cσ (ciσ · u)2 + Dσ u2 (7)

Here Aσ , Bσ , Cσ , and Dσ are the coefficients to be determined based of Chapmann–Enskog procedure. Forthe rest particle, Eq. (7) becomes

f eqi0 = A0 + D0u2 (8)

and

f eqi1 = A1 + B1(ci1 · u) + C1(ci1 · u)2 + D1u2 (9)

f eqi2 = A2 + B2(ci2 · u) + C2(ci2 · u)2 + D2u2 (10)

for other particles. This gives

B0 = C0 = 0 (11)

The symmetric properties of the tensor∑

i ciσαciσβ . . . are needed in the derivation and given as follows:

• The odd orders of tensor are equal to zero.• The second-order tensor satisfies

∑i ciσαciσβ = 2c2

σ δαβ where δαβ is the Kronecker delta and c1 = 1 andc2 = √

2.

for σ = 1 and∑

i ciσαciσβciσγ ciσθ = 4�αβγ θ − 8δαβγ θ for σ=2 where �αβγ θ = δαβδγ θ +δαγ δβθ +δαθ δβγ .Considering conservation laws of

∑σ

∑i f eq

σ i = ρ and∑

σ

∑i f eq

σ i cσ i = ρu results in∑

σ

∑i

f eqσ i = A0 + 4A1 + 4A2 + (2C1 + 4C2 + D0 + 4D1 + 4D2)u

2 = ρ (12)

∑σ

∑i

f eqσ i cσ i =(2B + 4B2)u = ρu (13)

and Dσ as

A0 + 4A1 + 4A2 = ρ (14)

2C1 + 4C2 + D0 + 4D1 + 4D2 = 0 (15)

and

2B1 + 4B2 = ρ (16)

To satisfy Eq. (14) we chose A0 = 49ρ, A1 = 1

9ρ and A2 = 136ρ.

Next, the timescale is decomposed into slow and fast timescale. This is to represent two different phenomenaoccurring at different timescales such as advection and diffusion.

∂

∂t= ∂

∂t0+ ε

∂

∂t1(17)

where ε plays the role of Knudsen number [16]. We also expand f about f eq,

fσ i = f eqσ i + ε f 1

σ i + ε2 f 2σ i + O(ε3) (18)

Here∑

σ

∑i f (n)

σ i = 0 and∑

σ

∑i f (n)

σ i cσ i = 0 to imply that the non-equilibrium distributions do not con-tribute to the local values of density and momentum.

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Taylor expanding of Eq. (5) and retaining terms up to O(ε2) results in

ε

[∂

∂t+ (cσ i · ∇)

]fσ i + ε2

2

[∂

∂t+ (cσ i · ∇)

]2

fσ i + O(ε3) = − fσ i − f eqσ i

τ(19)

Substituting Eqs. (17) and (18) into Eq. (19), the equation to order of ε and ε2 are(

∂

∂t0+ (cσ i · ∇)

)f eqσ i = − f 1

σ i

τ(20)

and(

∂

∂t0+ (cσ i · ∇)

)f 1σ i +

(∂

∂t0+ (cσ i · ∇)

)f eqσ i + ∂ f eq

σ i

∂t1= − f 2

σ i

τ, (21)

respectively. Eq. (21) can be further simplified using Eq. (20) which gives

∂ f eqσ i

∂t1+

(∂

∂t0+ (cσ i · ∇)

)(1 − 1

2τ

)f 1σ i = − f 2

σ i

τ(22)

The first-order continuity equation can be obtained by taking summation of Eq. (20) with respect to σ and i

∂ρ

∂t0+ ∇ · (ρu) = 0 (23)

Taking the same summation of Eq. (22) gives

∂ρ

∂t1= 0 (24)

Combining Eqs. (23) and (24) gives the correct form of the continuity equation:

∂ρ

∂t+ ∇ · (ρu) = 0 (25)

Equation (20) is next multiplied with cσ i and taking the summation as above gives

∂

∂t0(ρu + ∇ · �eq) = 0 (26)

EMBED Eq. (3).∏eq = ∑

σ

∑i cσ i cσ i f eq

σ i is the momentum flux tensor. Substituting the expression of theequilibrium can be written as

�eqαβ = [

2A1 + 4A2 + (4C2 + 2D1 + 4D2)u2] δαβ + 8C2uαuβ + (2C1 − 8C2)uαuβδαβ (27)

which are the pressure term and two nonlinear terms. This gives

8C2 = ρ (28)

and

2A1 + 4A2 = c2s ρ (29)

where c2s = 1/3 is speed of sound. In order to obtain a velocity-independent pressure and Galilean invariance,

it is chosen that

4C2 + 2D1 + 4D2 = 0 (30)

and

2C1 − 8C2 = 0 (31)

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This gives the final expression for �eq as

�eqαβ = c2

s ρδαβ + ρuαuβ (32)

Substituting Eq. (32) into Eq. (26) results in Euler equation as

∂

∂t0(ρu) + ∇ · (ρuu) = −∇(c2

s ρ) (33)

and the pressure is given by p = c2s ρ.

Next, multiply Eq. (22) with cσ i and taking the summation with respect to σ and i

∂

∂t1(ρu) + ∇ ·

(1 − 1

2τ

)�1 = 0 (34)

Substituting the expression of the non-equilibrium distribution and using Eqs. (22) and (29) leads to

�1αβ = −τ

{−c2

s δαβ∂γ (ρuγ ) + ∂

∂t0(ρuαuβ) + ∂α(2B1 − 8B2)uβδαβ + 4∂γ (B2uγ )δαβ

+4∂α

(B2uβ

) + 4∂β (B2uα)

}(35)

To maintain isotropy, it is set that

2B1 − 8B2 = 0 (36)

Recalling Eq. (16), gives the expression for B1 and B2

B1 = ρ

3, B2 = ρ

12(37)

Using Eq. (30) in the form

∂

∂t0(ρuαuβ) = −uα∂β(c2

s ρ) − uβ∂α(c2s ρ) − ∂γ (ρuαuβuγ ) (38)

Eq. (32) can be written as

�1αβ = −τ

{(1

3− c2

s

)∂γ (ρuγ )δαβ + 1

3∂α(ρuβ) + 1

3∂β(ρuα) − uα∂β(c2

s ρ) − uβ∂α(c2s ρ) − ∂γ (ρuαuβuγ )

}

(39)

Combining Eqs. (30), (31) and (35) gives the momentum equation in two dimensions

∂

∂t(ρuα) + ∂β(ρuαuβ) = −∂α p + ∂β

{μ

(∂αuβ + ∂βuα − 1

2uγ γ δαβ

)}(40)

where

μ = ρ2τ − 1

6(41)

For an incompressible fluid, the momentum equation becomes

∂

∂t(ρuα) + ∂β(ρuαuβ) = −∂α p + ∂β

{μ(∂αuβ + ∂βuα)

}(42)

The remaining coefficients are determined as follows:

A0 = 4

9ρ, B0 = 0, C0 = 0, D0 = −2

3ρ

A1 = 1

9ρ, B1 = 1

3ρ, C1 = 1

2ρ, D1 = −1

6ρ (43)

A2 = 1

36ρ, B2 = 1

12ρ, C2 = 1

8ρ, D2 = − 1

24ρ

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Finally, the equilibrium distribution functions can be written as follows:

f eq0i = 4

9ρ

[1 − 3

2u2

]for i = 0. (44)

f eq1i = 1

9ρ

[1 + 3(c1i · u) + 9

2(c1i · u)2 − 3

2u2

]. for i = 1, 3, 5, 7 (45)

and

f eq2i = 1

36ρ

[1 + 3(c1i · u) + 9

2(c1i · u)2 − 3

2u2

]for i = 2, 4, 6, 8 (46)

From the above derivation, it can be seen that the mesoscale Boltzmann equation can lead to the macroscopicNavier–Stokes equation by the Chapman Enskog expansion.

Fig. 2 Streamline plots for different Reynolds numbers

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Table 1 Location of the center of the main vortex in square cavity

Re Ghia et al. [17] Vanka et al. [18] LBM

100 (0.6172, 0.7344) (0.6188, 0.7375) (0.6200, 0.7400)400 (0.5547, 0.6055) (0.5563, 0.6000) (0.5600, 0.6000)1,000 (0.5313, 0.5625) (0.5438, 0.5625) (0.5300, 0.5650)3,200 (0.5165, 0.5469) − (0.5200, 0.5400)5,000 (0.5117, 0.5352) (0.5125, 0.5313) (0.5150, 0.5350)7,500 (0.5117, 0.5322) − (0.5150, 0.5325)10,000 (0.5117, 0.5333) − (0.5133, 0.5283)

3 Numerical Investigation of Lid-Driven Cavity Flow

Lid-driven cavity flow is a phenomenon where the fluid is set into motion by a part of the containing boundary.This type of flow is crucial for analyzing fundamental aspects of recirculating fluids: in spite of the apparentlysimple geometry, lid-driven cavity flows may involve high degree of complexity.

In this paper, the numerical results obtained by mesoscale lattice Boltzmann computation is first demon-strated for lid-driven flow in a square cavity at various values of Reynolds numbers. The benchmark numericalsolutions based on macroscale Navier–Stokes formulation are brought for the sake of validation. The mesoscalecomputation for lid-driven flow is then extended to isosceles triangular cavities. The computational results willbe demonstrated based on the plot of streamlines to predict the behavior of vortices inside the respective cavity.

3.1 Mesoscale Code Validations

Simulation results obtained by Ghia et al. [17] and Vanka [18] are brought for the sake of comparisons with thesimulated results in present study. Ghia et al. solved the stream function-vorticity using finite different approachwith multigrid method to accurately reproduce the structure of main and secondary vortices in the cavity. Onthe other hand, Vanka maintained the formulation of primitive variables and applied the finite different schemeusing the staggered location of the flow variables.

Figure 2a–g shows plots of streamline for the Reynolds numbers considered. It is apparent that the flowstructures are in good agreement with the results published in the literature by Ghia et al. [17]. For low Reynoldsnumber (Re = 100), the center of the vortex is located at about one-third of the cavity depth from the top. Asthe Reynolds numbers increase, the primary vortex moves towards the cavity center and increasing circular.In addition to the primary, a pair of counter rotating eddies develop at the lower corners of the cavity. AtRe = 3,200, a third secondary vortex is evolved in the upper left corner of the cavity. As the value of Reynoldsnumber further increases, the size of the secondary vortices become larger.

The location of the center of the main vortex is then computed for every Reynolds number and comparedwith the results in the literature [17,18]. The results are shown in Table 1.

From the results presented in Table 1, it can be seen that the mesoscale LBM is able to produce an excellentagreement with the results predicted by conventional macroscale numerical methods.

The velocity components along the vertical and horizontal lines through the cavity center together with thebenchmark solutions are shown in Fig. 3a–g. Good agreement between the mesoscale LBM and the benchmarksolutions is observed. It is noted that the mesoscale LBM is able to capture the critical points in the testedproblem.

3.2 Lid-Driven Flow Triangular Cavity

In this sub section, the flow behaviors at different Reynolds numbers in a triangular cavity are demonstratedusing plots of streamline in the system. The isosceles triangle with 90◦ at bottom corner angle is first consid-ered. These plots revealed two significant features. First, as Reynolds numbers is increased, the primary vortexmoves downstream to the right. For example, at low Reynolds number (Re = 100), the location of primaryvortex is located about one-third from the top horizontal wall as shown in Fig. 4. Apart from the primary vortex,no other vortex is visible for this value of Reynolds number. However, for Re = 400, the secondary vortex canbe seen and located near the stagnant corner of the triangle. The shape of this secondary vortex becomes largeras Reynolds numbers is further increased up to Re = 1,000 as shown in Fig. 4d. The second significant featureis the number of vortices in the cavity which is increased as the Reynolds number is increased. Starting from

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Fig. 3 Velocity components along the vertical and horizontal lines through the cavity center (lines LBM, symbol Ghia et al. [17])

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Fig. 4 Streamline plots for isosceles right triangle with 90◦ being at the corner angle

one main vortex for low Reynolds number, Fig. 4 shows that the secondary vortex appears at Re = 400 and itssize becomes larger and larger as the Reynolds number increases. The tertiary vortex appears at the bottom tipof the cavity at the simulation of Reynolds number 3,000. As the Reynolds number further increases, a fewvortices appear at the left and right diagonal boundaries of the cavity.

Table 2 Location of the center of the main vortex for isosceles triangle with 90◦ at corner angle

Re Location of primary vortex

x y

100 0.5450 0.7600400 0.6100 0.7500700 0.5950 0.71501,000 0.5875 0.71503,000 0.7400 0.82005,000 0.7350 0.81007,000 0.7583 0.620010,000 0.4117 0.5375

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Fig. 5 Streamline plots for 90◦ right corner triangular cavity

Table 2 shows the computed center of primary vortex for every simulated Reynolds numbers. To the bestof authors’ knowledge, such information on the coordinate of primary vortex at steady state has not beenreported. Therefore, no comparison can be made to verify the computed results.

We then considered two types of an isosceles right triangle. They are 90◦ at right corner and 90◦ at leftcorner of the triangular cavities. The flow in these considered triangles is predicted up to Re = 2,500. Figures 5and 6 show the streamline plot for every simulated case. In Tables 3 and 4, the location of the center of primaryeddy is computed and compared with Erturk and Gokcol [19], who solved the stream function-vorticity for-mulation in curvilinear coordinate, and found very good agreement. From the results predicted above, it can

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Fig. 6 Streamline plots for 90◦ left corner triangular cavity

be said that the flow behaves very differently as we increase the Reynolds number and is greatly affected bythe shape of the geometry. In addition to that, these also demonstrate the capability of the current mesoscalemethod in predicting complex fluid flow behavior in this complex system.

4 Summary and Concluding Remarks

In this paper, a mesoscale numerical method, the lattice Boltzmann scheme was used to investigate the steadyfluid flow behavior in a lid-driven cavity. The capability of lattice Boltzmann scheme to derive the macroscopicdynamics of fluid flow equations via the multiscale expansion has been shown. This led the current scheme

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Table 3 Location of the center of the main vortex

Re Ref. [19] LBM

100 (0.7090, 0.8320) (0.7100, 0.8300)500 (0.7070, 0.7676) (0.7100, 0.7650)1,000 (0.6992, 0.7559) (0.7000, 0.7550)1,500 − (0.7000, 0.7467)2,000 − (0.7000, 0.7467)2,500 (0.6973, 0.7441) (0.7000, 0.7433)

Table 4 Location of the center of the main vortex

Re Ref. [19] LBM

100 (0.4473, 0.8516) (0.4450, 0.8500)500 (0.5469, 0.8496) (0.5550, 0.8500)1,000 (0.6094, 0.8691) (0.6050, 0.8650)1,500 (0.6582, 0.8848) (0.6567, 0.8833)2,000 (0.6953, 0.8965) (0.6900, 0.8933)2,500 (0.7227, 0.9043) (0.7167, 0.9033)

to produce results with second-order accuracy in space and time. The flow structure in square and triangularcavities has been successfully reproduced and compared with the benchmark solution available in the literature.Excellent agreement has been found. This gives us confidence to apply the current method to investigate othervarious types of fluid flow problems.

Acknowledgments The authors would like to thank Universiti Teknologi Malaysia and Malaysia government for supportingthis research activity.

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