modelling and simulationsazwadi/pdf_publication/scopus27.pdf · simulation of lid-driven cavity for...

International Review on Modelling and Simulations

(IREMOS)

Contents:

(continued from Part B)

Empirical Models for the Correlation of Global Solar Radiation Under Malaysia Environment by H. A. Rahman, K. M. Nor, M. Y. Hassan, M. S. Majid

1864

Multi-Objective Single Facility Location Problem: a Review by Vaishali Wadhwa, Deepak Garg

1871

Free Vibration Functionally Graded Material Circular Cylindrical Shell with Various Volume Fraction Laws Under Symmetrical Boundary Conditions by M. Setareh, M. R. Isvandzibaei

1876

On New Solutions for Non-Newtonian Visco-Elastic Fluid in Pipe by N. Moallemi, I. Shafieenejad, H. Davari, A. Fata

1883

Integration of a Recent Profile Reduction Method in Finite Element Program for Movement Simulation with Moving Band Method by R. Saraoui, Y. Boutora, N. Benamrouche, M. Ounnadi

1889

Thermal-Hydraulic Modeling of a Radiant Steam Generator Using Relap5/Mod3.2 Code by A. L. Deghal Cheridi, A. Chaker

1896

Response to Customer Reliability Requirements with Reserve Market Management by M. Najafi, M. Simab, M. Hoseinpour, R. Ebrahimi

1900

Lattice Boltzmann Simulation of Cavity Flows at Various Reynolds Numbers by M. A. Mussa, S. Abdullah, C. S. Nor Azwadi, R. Zulkifli

1909

Transient Analysis Vibration of Two Type FGM Circular Cylindrical Shell Based on Third Order Theory Using Hamilton's Principle with Simply Support-Simply Support Boundary Conditions by M. R. Isvandzibaei

1920

Effect of Shear Theory on Analysis Free Vibration of Two Kinds Functionally Graded Material Hollow Circular Cylindrical Shell According to a 3D Higher-Order Deformation Theory with Free-Simply Support Boundary Conditions by M. R. Isvandzibaei

1929

(continued on inside back cover)

ISSN 1974-9821Vol. 4 N. 4

August 2011

PART

C

International Review on Modelling and Simulations (IREMOS)

Editor-in-Chief: Santolo Meo Department of Electrical Engineering FEDERICO II University 21 Claudio - I80125 Naples, Italy [email protected]

Editorial Board: Marios Angelides (U.K.) Brunel University M. El Hachemi Benbouzid (France) Univ. of Western Brittany- Electrical Engineering Department Debes Bhattacharyya (New Zealand) Univ. of Auckland – Department of Mechanical Engineering Stjepan Bogdan (Croatia) Univ. of Zagreb - Faculty of Electrical Engineering and Computing Cecati Carlo (Italy) Univ. of L'Aquila - Department of Electrical and Information Eng. Ibrahim Dincer (Canada) Univ. of Ontario Institute of Technology Giuseppe Gentile (Italy) FEDERICO II Univ., Naples - Dept. of Electrical Engineering Wilhelm Hasselbring (Germany) Univ. of Kiel Ivan Ivanov (Bulgaria) Technical Univ. of Sofia - Electrical Power Department Jiin-Yuh Jang (Taiwan) National Cheng-Kung Univ. - Department of Mechanical Engineering Heuy-Dong Kim (Korea) Andong National Univ. - School of Mechanical Engineering Marta Kurutz (Hungary) Technical Univ. of Budapest Baoding Liu (China) Tsinghua Univ. - Department of Mathematical Sciences Pascal Lorenz (France) Univ. de Haute Alsace IUT de Colmar Santolo Meo (Italy) FEDERICO II Univ., Naples - Dept. of Electrical Engineering Josua P. Meyer (South Africa) Univ. of Pretoria - Dept.of Mechanical & Aeronautical Engineering Bijan Mohammadi (France) Institut de Mathématiques et de Modélisation de Montpellier Pradipta Kumar Panigrahi (India) Indian Institute of Technology, Kanpur - Mechanical Engineering Adrian Traian Pleşca (Romania) "Gh. Asachi" Technical University of Iasi Ľubomír Šooš (Slovak Republic) Slovak Univ. of Technology - Faculty of Mechanical Engineering Lazarus Tenek (Greece) Aristotle Univ. of Thessaloniki Lixin Tian (China) Jiangsu Univ. - Department of Mathematics Yoshihiro Tomita (Japan) Kobe Univ. - Division of Mechanical Engineering George Tsatsaronis (Germany) Technische Univ. Berlin - Institute for Energy Engineering Ahmed F. Zobaa (U.K.) Univ. of Exeter - Camborne School of Mines

The International Review on Modelling and Simulations (IREMOS) is a publication of the Praise Worthy Prize S.r.l.. The Review is published bimonthly, appearing on the last day of February, April, June, August, October, December. Published and Printed in Italy by Praise Worthy Prize S.r.l., Naples, August 31, 2011. Copyright © 2011 Praise Worthy Prize S.r.l. - All rights reserved. This journal and the individual contributions contained in it are protected under copyright by Praise Worthy Prize S.r.l. and the following terms and conditions apply to their use: Single photocopies of single articles may be made for personal use as allowed by national copyright laws. Permission of the Publisher and payment of a fee is required for all other photocopying, including multiple or systematic copying, copying for advertising or promotional purposes, resale and all forms of document delivery. Permission may be sought directly from Praise Worthy Prize S.r.l. at the e-mail address: [email protected] Permission of the Publisher is required to store or use electronically any material contained in this journal, including any article or part of an article. Except as outlined above, no part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without prior written permission of the Publisher. E-mail address permission request: [email protected] Responsibility for the contents rests upon the authors and not upon the Praise Worthy Prize S.r.l.. Statement and opinions expressed in the articles and communications are those of the individual contributors and not the statements and opinions of Praise Worthy Prize S.r.l.. Praise Worthy Prize S.r.l. assumes no responsibility or liability for any damage or injury to persons or property arising out of the use of any materials, instructions, methods or ideas contained herein. Praise Worthy Prize S.r.l. expressly disclaims any implied warranties of merchantability or fitness for a particular purpose. If expert assistance is required, the service of a competent professional person should be sought.

International Review on Modelling and Simulations (I.RE.MO.S.), Vol. 4, N. 4

August 2011

Manuscript received and revised July 2011, accepted August 2011 Copyright © 2011 Praise Worthy Prize S.r.l. - All rights reserved

1909

Lattice Boltzmann Simulation of Cavity Flows at Various Reynolds Numbers

M. A. Mussa1, S. Abdullah1, C. S. Nor Azwadi2, R. Zulkifli1 Abstract – The lattice Boltzmann method (LBM) is a numerical method evolved from the statistical approach that has been well-accepted as an alternative numerical scheme for computational fluid dynamics (CFD). In comparison to other numerical schemes, the lattice Boltzmann method (LBM) can be regarded as a “bottom-up” approach that derives the Navier-Stokes equation through statistical behavior of particle dynamics. Hence, this paper presents the simulation of lid-driven cavity for deep and shallow flow using the lattice Boltzmann method where the effect of the Reynolds number on the flow pattern at aspect ratios of 0.25, 0.5, 1.5 and 4.0 was studied. These types of flow exhibit a number of interesting physical features but are scarcely simulated using the LBM scheme. The source code was established based on the BGK model on rectangular lattice geometry. The comparison of the results was in excellent agreement with those gathered from the literature even with relatively coarse grids applied to the numerical calculation. Copyright © 2011 Praise Worthy Prize S.r.l. - All rights reserved. Keywords: Lattice Boltzmann Method, Distribution Function, Microscopic Velocity, Lid-Driven

Cavity Flow, BGK Model

Nomenclature c Microscopic velocity f Distribution function feq Equilibrium distribution function H Characteristic height L Mean free path of particle P Pressure distribution t Time u Velocity U Centerline velocity at exit W Height of microchannel x, y Co-ordinates

Greek symbols ν Viscosity of fluid ρ Density τ Time relaxation Ω Collision operator

I. Introduction The lattice Boltzmann method can be considered as a

numerical method to solve the Boltzmann equation in discrete phase space and discrete time. In the Boltzmann equation, the velocity space of the particle is continuous, while in the lattice Boltzmann method the velocity space of the particle is discrete. In the lattice Boltzmann method, as in the Boltzmann equation, the particle is in one of the two processes: the first, moving with discrete

velocity to the neighboring node and the second is the collision process. The imaginary “propagation” and “collision” actions of fluid particles are reformulated during the development of the LBM code. These processes are represented by the evolution of particle distribution function, f(x,t) which describes the statistical population of particles at location x and time t.

The most important features of LBM are the simplicity of formulation, suitability to work on parallel computing and ease in dealing with complex boundary conditions, and in addition, and because it is built on the basis of kinetic theory, LBM is more effective in the handling and analysis of complex systems such as fluid flow multi-component, multiphase flow [1]-[3], flow in porous media [4], the flow of suspensions [5], and turbulent flow [6]. The advantages of LBM include simple calculation procedures, suitability for parallel computations, ease and robust handling of multiphase flow, complex geometries, interfacial dynamics and others [7]. A few standard benchmark problems have been simulated using LBM and the results are found to agree well with the corresponding Navier-Stokes solutions [8]. The lid-driven cavity flow is one of the most important benchmarks for new numerical method to be developed. It represents the flow of a rectangular or square geometry where the flow is driven by a tangential motion with constant velocity of a single lid, representing the Dirichlet boundary conditions. Moreover, the driven-cavity flow exhibits a number of interesting physical features [9], [10]. However, on the

M. A. Mussa, S. Abdullah, C. S. Nor Azwadi, R. Zulkifli

Copyright © 2011 Praise Worthy Prize S.r.l. - All rights reserved International Review on Modelling and Simulations, Vol. 4, N. 4

1910

other hand, the simulation of lid-driven flow inside cavity is scarcely performed using LBM.

Cavity flow is so important to researchers due to its relation to a lot of industrial applications and its similarity to many of common flow phenomena such as corner vortices, longitudinal vortices and turbulence [9], [11]. The numerical study which was done by Burggraf [12] was among the earlier research of steady state flow in a cavity, where he numerically solved the full Navier-Stokes equations for the Reynolds number up to 400. Furthermore, he showed that at this Reynolds number there will be a two secondary vortices present in the bottom corners of the cavity in addition to the primary large vortex in the flow core.

Schreibera and Keller [13] supported this finding at high Reynolds number where there were vortices formed near the bottom corners. Mehta and Lavan [14] numerically studied the lid driven cavity flow for the aspect ratios of 0.5, 1 and 2 with the Reynolds numbers of 1, 10, 100, and they concluded that the strength of the large vortex increases with the increase in the Reynolds number with no vortices at the bottom corners due to low number of nodes and low Reynolds number. Cheng and Hung [15] also confirmed Burggraf results for the vortex structure of a two-dimensional viscous flow in a lid-driven cavity of rectangular section, where they found that the major feature of flow in a deep rectangular cavity was that the domain was filled with counter-rotating large vortices and their size, the central position and the number of the vortices depend on both the Reynolds number and the aspect ratio.

Up to date, only a few studies have dealt with the influence of the height of cavity on the flow inside. In comparison with the flow inside a square cavity, one new parameter has to be taken into account, the aspect ratio of the cavity, K = W/H, W being the cavity width and H the depth. McNamara and Zanetti [16] were the first group of researchers which used the lattice Boltzmann equation as a numerical model, where they suggested it as an alternative technique to the lattice-gas automata for the study of hydrodynamic properties. This approach completely eliminates the statistical noise that plagues the usual lattice-gas simulations and therefore permits simulations that demand much less computer time, which was thought to be more efficient than the lattice-gas automata for intermediate to low Reynolds number. Bhatnagar et al. [17] simplified the collision operator by introducing a new model which applied a single relaxation time approximation, and this model is referred to as the Lattice Boltzmann Bhatnagar-Gross-Crook (BGK) model. As shown by Hou et al. [9] and Shen and Floryan [18], the physics of driven flow in a cavity indicated the existence of critical aspect ratio at which the corner eddies merged and formed a primary eddy. The dependence of the vortex structure to the aspect ratio at different Reynolds numbers were investigated by Pan and Acrivos [19], where a flow visualization experiment was performed to study effects of inertia force in the

flow structure in the Reynolds number range of 20 ≤ Re ≤ 4000. Due to experimental limitation, the shallow cavity flow experiment usually demonstrated a single inviscid core of uniform vorticity, while the vortex structure below the primary vortex in the deep cavity could not be captured.

Based on the review, it is necessary for the formation of eddies and vortices to be investigated further for a certain type of flows which has been used as benchmark for validation of CFD codes so that the outcome of the simulation will be still the same, regardless of the method used. Therefore, the objectives of the present article are to utilize the LBM simulations for cavity flows and to investigate and interpret the results with a particular focus on the structure of primary and secondary eddies.

II. Numerical Methododology Historically, LBM is the logical development of lattice

gas automata (LGA) method [20]. Like in LGA, the physical space is discretized into uniform lattice nodes. Every node in the network is then connected with its neighbours through a number of lattice velocities to be determined through the model chosen.

The lattice Boltzmann equation is given by:

( )i ii , i

f fc f

t xαα

∂ ∂+ = Ω

∂ ∂ (1)

where if is the distribution function for particles with velocity ci,α at position xα and time t. Equation (1) consists of two parts; propagation (left-hand side) which refers to the propagation of distribution function to the next node in the direction of its probable velocity, and collision ( )fΩ (right-hand side) which represent the collision of the particle distribution function. In LBM, the magnitude of ci,α is set so that in each time step ∆t, every particle denoted by appropriate distribution function propagates in a distance of lattice nodes spacing ∆x. This will ensure that the particles under consideration arrive exactly at the lattice nodes after ∆t and simultaneously collide with other adjacent particles.

There are a few versions of collision operator ( )fΩ available in literature. However, the most well accepted version was the Bhatnagar Gross Krook collision model (BGK) [17] due to its simplicity and efficiency [21], [22]. The equation that represents this model is given by:

( ) ( ) ( )eqf , ,t f , ,tf

τ−

Ω = −x c x c

(2)

where eqf is the equilibrium distribution function and τ is the time to reach equilibrium condition during collision process and is called relaxation time. Equation



1911

(2) of the BGK collision model describes that 1/τ of non-equilibrium distribution relaxes to equilibrium state within time τ on every collision.

By replacing the BGK collision model into the Boltzmann equation, the BGK Boltzmann Equation is obtained:

eqf f f fc

t xαα τ

∂ ∂ −+ = −

∂ ∂ (3)

The general form of the lattice velocity model is

expressed as DnQm where D represents spatial dimension and Q is the number of connection (lattice velocity) at every node. In this paper, the 9-microscopic velocity or 9-bit model (D2Q9) is used. The lattice geometry is shown in Fig. 1 and the equilibrium distribution function of the 9-bit model is [23]:

( ) ( )2 29 31 32 2

eqi i i if ρω ⎡ ⎤= + ⋅ + ⋅ −⎢ ⎥⎣ ⎦

c u c u u (4)

where the weights are: 1 2 5 6 9

4 1 19 9 36~ ~, ,ω ω ω= = = (5)

and the microscopic velocity components are:

0 1 0 1 0 1 1 1 10 0 1 0 1 1 1 1 1

− − −⎡ ⎤= ⎢ ⎥− − −⎣ ⎦

c (6)

The macroscopic quantities can be calculated from:

i

ifρ = ∑ (7)

i i

i if f= ∑ ∑u c (8)

5c

1c 2c

3c

4c

6c7c

8c 9c5c

1c 2c

3c

4c

6c7c

8c 9c

Fig. 1. Lattice geometry (D2Q9)

II.1. Derivation of the LBM-form of Flow Equations

In this subsection, the derivation of continuity and Navier-Stokes equations using the Chapman-Enskog expansion [24] is discussed.

If a two-dimension nine-bit model is used, then the time evolution lattice Boltzmann equation can be expressed as:

( ) ( )

( )1i i , i

eqi i

f x c t ,t t f x ,t

f f

α α α

τ

+ ∆ + ∆ − =

= − − (9)

Using the Taylor series expansion and retaining up to

the second order terms, the left hand side of (9) can be rewritten as follows:

( )

( )21 22

t i , i

t t i , i , i , i

c f

c : c c f

α

α α α

∂ + ∇ ⋅ +

+ ∂ + ∂ ∇ ⋅ + ∇∇ (10)

In order to relate the lattice Boltzmann equation with a

macroscopic equation, it is necessary to isolate different time scales. This is to indicate different scales for different physical phenomena which are treated separately in the final macroscopic equation. Hence, the space and time derivatives are expanded in terms of the Knudsen number ε [25] as follows:

( )2 3

1 2t t t Oε ε ε∂ = ∂ + ∂ + (11)

( )2Oε ε∇ = ∇ + (12)

Expanding the distribution function fi about eq

if gives: ( )1 2 2 3eq

i i i if f f f Oε ε ε= + + + (13)

where: 0 for 1n n

i i i,αi i

f f c n= = ≥∑ ∑ (14)

Equation (13) implies that the non-equilibrium

distribution functions ( )nif do not contribute to the local

values of density and momentum. Substituting Equations (11), (12) and (13) into Equation (10) and regrouping the equation up to the first order of ε yields:

( ) 11

1eqt i , i ic f fα τ

∂ + ⋅∇ = − (15)

Equation (15) is then simplified to the ε2 order using

Equation (13), which gives:

( ) 1 21 1

1 112

eqt i t i , i i ,f c f fα ατ τ

⎛ ⎞∂ + ∂ + ⋅∇ − = −⎜ ⎟⎝ ⎠

(16)



1912

A summation of Equation (15) for all values of i and α is then performed to produce the first order continuity equation, i.e.: ( )1 0t ρ ρ∂ + ∇ ⋅ =u (17)

Multiplying Equation (15) with i ,c α and carrying out summation the same way Equation (17) is produced leads to: ( )1 0eq

t ρ∂ + ∇ ⋅Π =u (18) where: ( )eq

i , i , ii

c c fα αΠ = ∑ (19)

is the momentum flux tensor. After some simple

mathematical manipulation to satisfy Galilean invariance and the property of an isotropic tensor, the final expression for eqΠ is: 2eq

sc u uβχ β χρδ ρΠ = + (20)

Replacing Equation (20) into Equation (18) leads to:

( ) ( ) ( )21t scρ ρ ρ∂ + ∇ ⋅ = −∇u uu (21)

Equation (21) is actually the Euler equation, where the

pressure can be obtained from the right-hand side term, i.e.: 2

sp c ρ≡ (22) where sc is the speed of sound. Similarly, the equation for ρ and u can be obtained from Equation (18) for ε2. Taking summation for all i and α gives: 2 0t ρ∂ = (23)

Then, multiplying Equation (16) with i ,c α and taking the summation as above yields:

( ) 12

11 02t ρτ

⎛ ⎞∂ + ∇ ⋅ − Π =⎜ ⎟⎝ ⎠

u (24)

where:

( ) ( ) ( )( ) ( ) ( )

( )

1 21 13 3

2 213

s

s s

c u u

u u c u c

u u u

γ γ βχ β χ

χ β β χ χ β

γ β χ γ

τ δ ρ δ ρ

ρ ρ ρ

ρ

Π = − − + ∂ +

+ ∂ − ∂ − ∂ +

−∂

(25)

Combining these equations for ( )O ε and ( )2O ε

gives the correct form of the continuity and momentum equations for incompressible flow, which are respectively as follows: 0∇ ⋅ =u (26)

( ) ( )

( )2 2 126

t

s

u u u

c S

β χ β χ

β χ βχ

ρ ρ

τρ ρ

∂ − ∂ =

−⎛ ⎞= −∂ + ∂ ⎜ ⎟⎝ ⎠

(27)

where ( )12S u uβχ β χ χ β= ∂ + ∂ , 2

sp c ρ= and sc is given by the following relation:

2 13sc = (28)

Finally, the kinematic viscosity of a fluid ν is related

to the time relaxation τ in mesoscopic scale as follows:

132

τ ν= + (29)

From the above derivations, we can see that the

evolution of Equation (9) can lead to the incompressible Navier-Stokes equation through the Chapman-Enskog expansion. Equation (29) describes that the value of τ must be kept higher than 0.5 in order to avoid negative value of kinematic viscosity. This limits our simulation to low Reynolds number. However, this limitation can partly be solved by using high number of nodes but would lead to longer computational time. This aspect will be addressed in our future works.

III. Results and Discussion This section demonstrates the application of the LBM

scheme discussed in the preceding section to simulate the lid-driven cavity flow at different Reynolds numbers. The simulations were considered to have reached a steady state condition when the r.m.s. change in horizontal and vertical velocity decreased to a magnitude of 10-6 or less.

III.1. Validation of the LBM Scheme

For the purpose of code validation, we carried out the simulation of lid-driven flow in a square cavity, K = 1 and compared with ‘benchmark’ results produced by Ghia et al. [26] which is theoretical simulation based on Navier-Stokes equations and experiments research conducted by Tsorng et al. [27]. Lid-driven flow in a square cavity is well-known as a standard test case for the numerical schemes of fluid flows. Supposed that the lid is located at the top boundary y = H, and moves with a constant speed U from left to right. The Reynolds number for the system is given by:



1913

ULReν

= (30)

-1.0

-0.5

0.0

0.5

1.0

-1.0 -0.5 0.0 0.5 1.0

Ghia et alX VelocityY Velocity

Fig. 2. Comparison of Velocity profiles at mid-height (y-velocity) and mid-width (x-velocity) of cavity between LBM and Ghia et al. [26]

results for Re = 100

-1.0

-0.5

0.0

0.5

1.0

-1.0 -0.5 0.0 0.5 1.0

Ghia et al

X Velocity

Y Velocity

Fig. 3. Comparison of Velocity profiles at mid-height (y-velocity) and mid-width (x-velocity) of cavity between LBM and Ghia et al. [26]

results for Re = 1000

(a)

(b)

Figs. 4. Comparison of Velocity profiles at mid-height (y-velocity)

and mid-width (x-velocity) of cavity between (a) LBM and (b) Tsorng et al. [27] results for Re = 400

(a)

(b)

Figs. 5. Comparison between (a) LBM streamlines and (b) Tsorng et al.

[27] experimental study streamlines for Re = 100

(a)

(b)





1914

(a)

(b)



Simulations were performed at Re = 100, 400 and 1,000 using a grid size of 200 × 200. Figs. 2 and 3 show the comparison of the velocity profile for the Reynolds numbers considered between LBM and Ghia et al. [26] simulation while Figs. 4 show the same comparison but with Tsorng et al. [27] results for Re = 400. Plot of stream function comparison have been showed in Figs. 5,6 and 7 . It is apparent that the flow structure and velocity distribution are in good agreement with the previous work of Ghia et al. [26] and Tsorng et al. [27].

III.2. Deep Cavity Flow

The LBM numerical simulation was performed to analyse fluid flow in a deep cavity with aspect ratios K = 0.25 and 0.5. The D2Q9 lattice geometry with the BGK collision model was used. Figs. 8 show the streamlines of a cavity with aspect ratio 0.5. These three figures correspond to Re = 100, 400 and 1,000. This simulation employed a 100 × 200 grid system.

It can be seen from Figs. 8 that for low Reynolds number simulation Re = 100, a counter-rotating vortex is formed below the moving lid. As the Reynolds number increases (Re = 400), the center of the primary eddy begins to move downwards with respect to the top lid.

For the case of high Reynolds number (Re = 1,000), the primary eddy is formed at the center of the geometry. The secondary vortex can also be clearly seen at low Reynolds number and initially formed at the lower right of the geometry. As the Reynolds number increases, this vortex shift to the center left of the test case geometry.

This is due to the effect of viscous effect produced by the primary vortex. It is also noticeable that the vortices strength increased as Reynolds number increased. This is in agreement with the results of Cheng and Hung [15].

(a) Re = 100 (b) Re = 400

(c) Re = 1,000

Figs. 8. Streamlines at steady state condition

for K = 0.5

III.3. Shallow Cavity Flow

In this section, the numerical solution of the LBM scheme for rectangular shallow cavities is presented. Two sets of geometries were chosen which represent aspect ratios of K = 1.5 and 4.0, respectively.

For the first case where the aspect ratio is K = 1.5, a 150 × 100 grid system was employed. Figs. 9 show that for, the primary vortex center descends to the center of cavity with increased strength as the Reynolds number increases.



1915

(a) Re = 100

(b) Re = 400

(c) Re = 1,000

Figs. 9. Steamlines at steady state condition

for K = 1.5

From the figure, two secondary vortices of different sizes are visible at the lower corners of the cavity with the left vortex being bigger than the right vortex due to a flow direction from the right corner to the left corner. The difference in size becomes obvious with increasing Reynolds numb. .Most of these results are in excellent agreement with the results reported by Burggraf [12] and Schreibera and Keller [13].

For an aspect ratio of K = 4 which represent a more shallow cavity, the simulation employed a 400 × 100 grid system. As Reynolds number increases, Figs. 10 show that the primary vortex began to split into two vortices. However, there is no secondary vortices observed in corners even when the Reynolds number increases. Hence, it can be said that the enlarged cavity in the flow direction will suppress the generation of vortices at the corners.

III.4. Verification of the LBM Scheme with the Results Obtained from FLUENT

This section discusses the verification and comparison between the results obtained from the LBM schemes

with those obtained from a commercial CFD software, namely FLUENT.

(a) Re = 100

(b) Re = 400

(c) Re = 1,000

Figs. 10. Steamlines at steady state condition

for K = 4.0 The main difference is that the LBM scheme utilizes

the concept of statistical molecular dynamics in simulating fluid flow, while common CFD software employs a finite volume method to simulate advancement of fluid mass in the computational domain.

For the purpose of comparison, the aspect ratios of K = 0.25, 0.5, 1.5 and 4 are selected and the flow velocities are varied to produce Re = 100, 400 and 1000. The comparison were shown for the x-velocity component at the centerline of the cavity where the results obtained from the LBM scheme and from FLUENT were plotted in the same graph for all cases. For K = 0.25, the comparison between both results is illustrated in Fig. 11. From the figure, it is obvious that the flow at the location along the centerline from y = 0.7 and above is disturbed by the lid velocity, thus producing a strong primary vortex. The higher the Reynolds number, the stronger the vortex.

For a deep cavity with higher aspect ratio, namely K = 0.5, the x-velocity profiles are depicted in Figs. 12 for all the three Reynolds numbers. From the figures, it can be seen that stronger vortices were formed from y = 0.5 and above.

Below y = 0.5, a weaker secondary vortex is generated in an opposite direction, which is identical to the previous case of K = 0.25. These vortices can clearly be seen in Figs. 8.

Both cases represent deep cavities (K = 0.25 and 0.5) and it was observed that the present LBM results are in excellent agreement with the results obtained using FLUENT and the primary vortex grew bigger towards the bottom of the cavity as the secondary vortex got smaller.



1916

This secondary vortex started to break up into two vortices at the bottom left and right corners as the aspect ratio approached 1.0 where the results for streamlines and velocity profiles can be referred to Figs. 2-4.

0

0.2

0.4

0.6

0.8

1

-0.5 0 0.5 1

LBMFLUENT

(a) Re = 100

0

0.2

0.4

0.6

0.8

1

-0.5 0 0.5 1

LBMFLUENT

(b) Re = 400

0

0.2

0.4

0.6

0.8

1

-0.5 0 0.5 1

LBMFLUENT

(c) Re = 1,000

Figs. 11. The x-velocity at the centerline for

the LBM scheme and FLUENT for K = 0.25

On the other hand, the simulation for the shallow cavity produced different phenomena. The cases for shallow cavities are represented by geometries having aspect ratios of K = 1.5 and 4.0. For K = 1.5, it is illustrated in Figs. 13.

0

0.2

0.4

0.6

0.8

1

-0.5 0 0.5 1

LBMFLUENT

(a) Re = 100

0

0.2

0.4

0.6

0.8

1

-0.5 0 0.5 1

LBMFLUENT

(b) Re = 400

0

0.2

0.4

0.6

0.8

1

-0.5 0 0.5 1

LBMFLUENT

(c) Re = 1,000





1917

0

0.2

0.4

0.6

0.8

1

-0.5 0 0.5 1

LBMFLUENT

(a) Re = 100

0

0.2

0.4

0.6

0.8

1

-0.5 0 0.5 1

LBMFLUENT

(b) Re = 400

0

0.2

0.4

0.6

0.8

1

-0.5 0 0.5 1

LBMFLUENT

(c) Re = 1,000



From the figures, the resulting velocity profiles

indicated a primary vortex dominated the central region of the cavity, while two smaller secondary vortices were shifted to the bottom corners as can be seen in Figs. 9 for Re = 400 and 1,000.

For Re = 100 however, the vortices formed were too weak to be visible as streamline.

0

0.2

0.4

0.6

0.8

1

-0.5 0 0.5 1

LBMFLUENT

(a) Re = 100

0

0.2

0.4

0.6

0.8

1

-0.5 0 0.5 1

LBMFLUENT

(b) Re = 400

0

0.2

0.4

0.6

0.8

1

-0.5 0 0.5 1

LBMFLUENT

(c) Re = 1,000



When the cavity became shallower, as can be seen in

Figs. 14 for K = 4.0, the weaker vortices diminished while the dominant primary vortex developed eccentricity within itself especially for higher Reynolds number. This is due to the dominant inertia force in a higher Reynolds number flow as can be seen in Figs. 7(b) and 7(c). Nevertheless, the comparison with the results from FLUENT produced an excellent agreement, which is similar to the cases for deep cavities.



1918

IV. Conclusion The flow structure of a two-dimensional viscous flow

in lid-driven deep cavities has been numerically studied and analysed using the lattice Boltzmann scheme. The dynamic of the primary and secondary vortices are well captured by these simulations. For cavity flow there are a primary vorticity formed in the core centre of the cavity, when the cavity is deep there will be a two secondary vortices in the lower cavity corners, these two secondary vortices become larger when the Reynolds number increase. For shallow cavity the secondary vortices disappear as the aspect ratio increase.

In addition, the present results are found to be in good agreement with the results obtained by a CFD software, namely FLUENT version 6.1. It is also in good agreement with the previous studies by Ghia et al. [26] and Tsorng et al. [27], which proves that the lattice Boltzmann scheme is a powerfull alternative tool for solving complex fluid flow problems with high accuracy. However, the effects of boundary conditions in lateral directions are not yet considered since this paper only presents two-dimensional case studies and three-dimensional cases will be considered in our future study.

Acknowledgements The authors would like to acknowledge the Ministry

of Science, Technology and Innovation, Malaysia for sponsoring this work.

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Scheme for Simulating Two-Phase Flows, JSEM International Journal, Series B, Vol. 43(2): 305-313, 2000.

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[16] G. R. McNamara and G. Zanetti, Use of the Boltzmann Equation to Simulate Lattice-Gas Automata, Physical Review Letters, Vol. 61(20): 2332–2335, 1988.

[17] P. L. Bhatnagar, E. P. Gross, and M. Krook, A Model for Collision Processes in Gases. I. Small Amplitude Processes in Charged and Neutral One-Component Systems, Physical Review, Vol. 94(3): 511-525, 1954.

[18] C. Shen and J. M. Floryan, Low Reynolds number flow over cavities, Physics of Fluids, Vol. 28(11): 3191-3202, 1985.

[19] F. Pan and A. Acrivos, Steady flows in rectangular cavities, Journal of Fluid Mechanics, Vol. 28(04): 643-655, 1967.

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Authors’ information 1Department of Mechanical and Materials Engineering, Universiti Kebangsaan Malaysia, 43600 UKM Bangi, Selangor. 2Faculty of Mechanical Engineering, Universiti Teknologi Malaysia, 81310 Skudai, Johor.

M. Mussa was born in 1975, in Baghdad, Iraq. He received his M.Sc. in Mechanical Engineering from the University of Baghdad. Also, he received his B.Sc. in Mechanical Engineering from the University of Baghdad. He has worked on different topics such as computational analysis of heat transfer, fluid mechanics and microfluidic, he has some papers

were published in different journals and conferences..



1919

M. A. Mussa is member of Iraqi Engineers Association and member of the Association of University Lecturers, Iraq.

S. Shahrir was born in 1969, in Terengganu, Malaysia. He received his PhD in Mechanical Engineering from University of Wales Swansea, United Kingdom in 1997, his M.Sc. in Design and Economic Manufacture from University of Wales Swansea, United Kingdom in 1994, and his B.Sc. in Mechanical Engineering from Universiti Kebangsaan Malaysia in 1992. He is

currently a Professor at Universiti Kebangsaan Malaysia. He has authored several technical papers in the field of combustion and fuel engineering, powertrain engineering, machine design, computation theory and mathematics and Numerical analysis. His current research focuses on applied sciences and technologies, Nanotechnology, Micro electro-mechanical system (MEMS). S. Abdullah is registered engineer of Board of Engineers Malaysia, graduate member of Institute of Engineers Malaysia, member of Society of Automotive Engineers (SAE) and corporate member of Institute of Engineers Malaysia.

C. S. Nor Azwadi was born in 1977, in Kelantan, Malaysia. He received his PhD in Mechanical Engineering from Keio University, Japan in 2007, his M.Sc. in Thermal Power and Fluid Engineering from University of Manchester Institute Science and Technology, United Kingdom in 2003, and his B.Sc. in Mechanical Engineering and Material Science

from Kumamoto university, Japan in 2001. He is currently an Asst. Professor at Universiti teknologi Malaysia. He has authored several technical papers in the field of Multiphase flow, Microfluidic, Convective Heat Transfer, Fluid-Structure Interaction, Computational Methods, Lattice Boltzmann Method and Physics of Fluid. C.S. Nor Azwadi is registered engineer of Board of Engineers Malaysia.

R. Zulkifli was born in 1971, in Selangor, Malaysia. He received his PhD in Mechanical Engineering from Universiti Kebangsaan Malaysia, Malaysia in 2010, his MSc in Advanced Engineering Materials, University of Liverpool, United Kingdom in 1996, and B.Eng(Hons) in Mechanical Engineering, University of Liverpool, United Kingdom in

1994. He is currently a senior lecturer at the Department of Mechanical and Materials Engineering, Universiti Kebangsaan Malaysia. He has authored many technical papers in the field of Engineering Sciences, Mechanical Engineering, Material Sciences, and Advanced Composite Materials.

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