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Numerical Heat Transfer, Part B: FundamentalsAn International Journal of Computation and Methodology

ISSN: 1040-7790 (Print) 1521-0626 (Online) Journal homepage: http://www.tandfonline.com/loi/unhb20

Novel hybrid lattice Boltzmann technique withTVD characteristics for simulation of heat transferand entropy generations of MHD and naturalconvection in a cavity

Amir Javad Ahrar & Mohammad Hassan Djavareshkian

To cite this article: Amir Javad Ahrar & Mohammad Hassan Djavareshkian (2017) Novelhybrid lattice Boltzmann technique with TVD characteristics for simulation of heat transfer andentropy generations of MHD and natural convection in a cavity, Numerical Heat Transfer, Part B:Fundamentals, 72:6, 431-449, DOI: 10.1080/10407790.2017.1409528

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NUMERICAL HEAT TRANSFER, PART B 2017, VOL. 72, NO. 6, 431–449 https://doi.org/10.1080/10407790.2017.1409528

Novel hybrid lattice Boltzmann technique with TVD characteristics for simulation of heat transfer and entropy generations of MHD and natural convection in a cavity Amir Javad Ahrar and Mohammad Hassan Djavareshkian

Faculty of Engineering, Mechanical Engineering Department, Ferdowsi University of Mashhad, Mashhad, Iran

ABSTRACT In the present study, a novel FD_LBM hybrid numerical method with total variation diminishing (TVD) characteristics is proposed to solve the famous problems of MagnetoHydroDynamics (MHD) and natural convection in a closed cavity. Here, the momentum and energy equations are carried out via lattice Boltzmann method (LBM) and FD techniques, respectively. To enhance the stability and performance of the finite difference scheme for higher Ra numbers, two renowned Superbee and Minmod flux limiter functions were used. The results for heat transfer and entropy generation features for a wide range of Rayleigh and Hartman numbers of 103 � Ra � 108 and 0 � Ha � 100 are presented. In addition for comparison purposes, two multiple relaxation time (MRT) and single relaxation time (SRT) algorithms presented in the open literature, were applied in the same case definitions. Not only the tests revealed an excellent agreement between the TVD method results and the published data, but they also proved that this new technique is numerically much more efficient and stable than the SRT and MRT methods, and hence one can assume the present method as a fantastic tool in the numerical solution of laminar convection problems.

ARTICLE HISTORY Received 10 August 2017 Accepted 14 November 2017

1. Introduction

Lots of important phenomena in nature can be modeled by a set of differential equations. Examples of these phenomena can be found in incompressible or compressible fluids flow (Navier–Stokes), energy and particle concentration transfer, turbulence’s energy and dissipation, etc. [1]. However, as we know, a precise and straightforward analytical solution is not always available for every ordinary dif-ferential equation, and leaves alone a set of coupled partial differential equations (PDE). In those cases, scientists had to either examine individually each case in the laboratory or find an alternative way. Considering the expense of necessary equipment and the difficulties of experimental trial and error procedures, numerical methods were born and quickly gained a foothold in the fluid dynamics. This branch of fluid mechanics was named as computational fluid dynamics and it was pioneered by Richardson [2].

Because of their understandable definition, finite difference and finite volume are two of the earliest and most prevalent computational fluid dynamics (CFD) techniques that are developed until now. Finite difference method divides the whole geometry into rectangular meshes and its primitive problem rises while encountering curved boundaries. To deal with such boundary conditions, one should use conformal mapping [3]. On the other hand, finite volume methods do not suffer from such difficulties since the mesh can be non-orthogonal in FV methods, yet still simulating complex geometries with

none defined

CONTACT Mohammad Hassan Djavareshkian [email protected] Faculty of Engineering, Mechanical Engineering Department, Ferdowsi University of Mashhad, Mashhad 9177948974, Iran. Color versions of one or more of the figures in the article can be found online at www.tandfonline.com/unhb. © 2017 Taylor & Francis

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https://doi.org/10.1080/10407790.2017.1409528http://orcid.org/0000-0002-5148-5648https://crossmark.crossref.org/dialog/?doi=10.1080/10407790.2017.1409528&domain=pdf&date_stamp=2017-12-22mailto:[email protected]://www.tandfonline.com/unhb

multiple obstacles are not that simple even with finite volume technique. Also, finite element method has not gained as much popularity in fluid mechanics as it has in structural mechanics [4].

In 1986, Frisch et al. [5] proposed a hexagonal lattice1 for the lattice gas automata (LGA) which was originated by Hardy et al. [6]2 to obtain the correct Navier–Stokes equations. However, the method was still suffering from serious numerical instabilities. To eliminate these statistical noises, McNamara and Zanetti [7] introduced an average distribution function for particle transport in the LGA and by doing so not only they reached better stability in solving flow equations, but also they could save much more computational resources. This was the birth of a new CFD technique: lattice Boltzmann method (LBM).

Afterward, the renown Bhatnagar_Gross_Krook (BGK) linearized collision model [8] was applied to the lattice Boltzmann equation by Koelman [9]3 and to relate the Macroscopic values of density and velocity components to the Mesoscopic value of particles’ distribution function, he offered a local equilibrium [10] distribution function. At this point, the LBM was almost fully functioning for the low Mach flow regimes. The method had almost impressive advantages regarding the classical CFD methods such as finite volume method (FVM) or finite difference method (FDM). First of all, thanks to the famous Bounce-Back boundary condition [1], it is much simpler to add cylinders and obstacles in the flow and generally dealing with complex geometries is much more straightfor-ward. Second, the nature of LBM provides us much better parallel data analysis environment. And above all, LBM coding is usually much easier than an FV or FD code for the same set of equations and geometry. These are some and not all benefits that LBM can offer to its users, that is why many researchers nowadays are attempting to simulate their problems using it and many others are trying to expand its boundaries.

Just like any other CFD technique, LBM has its own limitations. Like FD techniques, LBM is applied on a rectangular mesh, and so it has almost the same problem with the curved boundaries. Filippova and Hänel [11] were the first ones who tried to propose the accurate curved boundary. They used a fictitious equilibrium distribution function to model the post-collision distribution function on the solid wall. Later, Bouzidi et al. [12] suggested an interpolation scheme based on the bounce-back BC to simulate the exact position of the solid wall. Independently, other researchers proposed

Nomenclature

B magnetic field c lattice speed ci discrete particle speeds Cp specific heat at constant pressure F external forces f density distribution functions feq equilibrium density distribution functions g internal energy distribution functions geq equilibrium internal energy distribution

functions gy gravity Ha Hartman number M lattice number in y-direction Ma Mach number Nu Nusselt number Pr Prandtl number Ra Rayleigh number Sx entropy generation rate due x T temperature TVD_S hybrid TVD scheme with Superbee limiter

TVD_M hybrid TVD scheme with Minmod limiter u, v velocity components in x- and y-directions x, y Cartesian coordinates

Greek letters α thermal diffusivity β thermal expansion coefficient c magnetic field inclination angle μ dynamic viscosity ρ density σ electrical conductivity τα relaxation time for temperature τυ relaxation time for flow υ kinematic viscosity ωi weighted factor in i-direction wðrÞ limiter function

Subscripts c cold f fluid h hot

1FHP. 2HPP. 3The model was also independently proposed by other authors.

432 A. J. AHRAR AND M. H. DJAVARESHKIAN

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different ways to resolve the difficulty [13, 14]. However, the boundary condition is not the biggest dilemma in working with LBM.

Guo et al. in their outstanding book [15] express that due to the definition of collision model the Mach number cannot exceed 0.3 and if it does, compressibility errors and instabilities are influencing the solution process. This phenomenon which is acknowledged and discussed by several other authors [1, 16] can be defined differently from the computational point of view. It is known that there are two opposing features when solving PDE. The convective fluxes in the flow produce dispersion which can lead to oscillating results and consequently numerical instabilities. On the other hand, the diffusion in the system would damp these oscillations by its dissipation effects [17–19]. Raising the fluid flow velocity with respect to the characteristics velocity (which is usually the speed of sound in LBM) would increase the Mach number and as a result, these oscillating instabilities would gain power in the system.

Several authors attended to this issue and tried to use different schemes to overcome the stability problem. These publications include multiple relaxation time (MRT) [20, 21], entropic lattice Boltzmann [22, 23], and entropic filtering [24] schemes. Almost all these methods bring stability raise in the system by adding extra dissipations to the system. These dissipations would damp the dispersion effects alright, but they can augment the solution procedure elapsed time, drastically. In numerical point of view, these methods lower the transferring data from one node to its neighbors so the solution would stay stable, and by and by the whole geometry can feel the portions of the data coming from the boundaries [25].

In recent years, publications on the LBM stability tend to involve and combine with different classical CFD techniques. Matin et al. presented a finite element lattice Boltzmann scheme for binary fluids flow [26] in 2017. They declared that unlike others who integrated intermolecular forcing terms in the advection term, the FE-LBM scheme applies collision and forcing terms locally for a much simpler formulation. Moreover, they tested their results against some published benchmarks and excellent accuracy was observed.

Motivated by these works, in the present paper, it is attempted to suggest a new hybrid FD-SRT LBM with total variation diminishing (TVD) characteristics. The method applies some classical limiter functions (namely Superbee and Minmod) to give rise to the stability of the system when facing high Ra problems in a square cavity. The results of fluid flow, heat transfer, and entropy generation rates under the influence of an external magnetic field for a wide range of Ra (103–108) and Ha numbers (0–100) are presented and the results are validated against published data. Moreover, two widely used benchmark test cases are studied and the results are presented to show the accuracy and veracity of the present scheme.

2. Governing the equations

The dimensional form of equations of conversation of mass, momentum, and energy including the external force terms related to Buoyancy force with Boussinesq assumption and the renown Lorentz force due to the external magnetic field, are as follows:

quqxþqvqy¼ 0 ð1Þ

q quð Þqtþq qu2ð Þqx

þq quvð Þqy

¼ � rP þ mr2uþ rB20 v sin c cos c � u sin2 c

� �ð2Þ

q qvð Þqtþq quvð Þqx

þq qv2ð Þqy

¼ � rP þ mr2vþ qbg T � Tcð Þ

þ rB20 u sin c cos c � v cos2 c

� �ð3Þ

NUMERICAL HEAT TRANSFER, PART B 433

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q qcTð Þqt

þq qcuTð Þqx

þq qcvTð Þqy

¼ r � KrTð Þ ð4Þ

The above equations are nondimensionalized with the aid of these scales and dimensionless numbers:

X;Y ¼xH

oryH; h ¼

T � TcTh � Tc

; Pr ¼tf

af; U ¼

uaf�

H; V ¼

vaf=H

;

Ra ¼gbf Th � Tcð ÞH3

tfaf; s ¼

tH2�af; P� ¼

Pqfa

2f

� ��H2

; Ha ¼ B0Hffiffiffiffiffirf

mf

r

:

It is worth mentioning that these scales and parameters may differ for different cases especially when the velocity of the flow is nondimensionalized with tf . Above-mentioned parameters lead to the following nondimensionalized equations:

qUqXþqVqY¼ 0 ð5Þ

qUqsþq U2ð ÞqX

þq UVð ÞqY

¼ �qPqXþ Prr2U þ Pr Ha2 V sin c cos c � U sin2 c

� �ð6Þ

qVqsþq UVð ÞqX

þq V2ð ÞqY

¼ �qPqYþ Prr2V þ Ra Pr hð Þ

þ Pr Ha2 U sin c cos c � V cos2 c� �

ð7Þ

qh

qsþq Uhð ÞqX

þq Vhð ÞqY

¼ r2h ð8Þ

So, these nondimensionalized equations are to be solved to obtain the fluid convection results. But there is a fine and rather important issue that must be dealt with before progressing. As was previously discussed, when the convective fluxes on the left-hand side of momentum and energy equations gain power in the system (usually by increasing the system’s velocity magnitude), dispersion effects with oscillating manner raise in the solutions obtained from numerical methods which can consequently lead to unwanted noise and even worth divergence of the iteration process. Here, it is tried to resolve the mentioned issue with the use of an explicit hybrid LBM technique enhanced by TVD schemes. Also, to show the method’s accuracy and efficiency, the standard D2Q9 SRT and MRT codes simulated the same geometry and boundary conditions and the results regarding the convergence behaviors and CPU usage of each code are presented. Here, we present a brief introduction to the applied SRT and MRT codes as well as a more detailed discussion about the novel hybrid LBM method with TVD characteristics.

2.1. Single relaxation time LBM

The LBM method with standard two-dimensional, nine velocities (D2Q9) system for both temperature and flow field is applied in this study. The discretized LBM equations with external force in nine directions can be written as:

fi x þ ciDt; t þ Dtð Þ � fi x; tð Þ ¼ �1st

fi x; tð Þ � feqi x; tð Þ½ � þ Dtwi

ci!: F!

c2s; ð9Þ

gi xþ ciDt; t þ Dtð Þ � gi x; tð Þ ¼ �1sa

gi x; tð Þ � geqi x; tð Þ½ �: ð10Þ


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Equations (9) and (10) are used to model the flow and temperature fields, respectively. Equation (9) recovers the continuity and momentum Eqs. (5)–(7), where the total body forces were considered by the external force (F) of Eq. (9). Equation (10) describes the evaluation of the internal energy and leads to Eq. (8). In the above equations, f eqi and g

eqi are the equilibrium distribution

functions for flow and temperature field, respectively, and can be calculated as follows:

f eqi ¼ xiq 1þci � u

c2sþ

12

ci � uð Þ2

c4s�

12

u � uc2s

" #

ð11Þ

geqi ¼ xiT 1þci � u

c2sþ

12

ci � uð Þ2

c4s�

12

u � uc2s

" #

ð12Þ

where cs is the lattice speed of sound which is equal to cs ¼ c� ffiffiffi

3p

and the discrete velocities, ci for D2Q9 are also defined as:

ci ¼0 for i ¼ 0c cos ip2 �

p2

� �; sin ip2 �

p2

� �� for i ¼ 1 � 4

cffiffiffi2p

cos ip2 �9p4

� �; sin ip2 �

9p4

� �� for i ¼ 5 � 8

8<

:ð13Þ

In the above equations, c is equal to Dx=Dt, with Δx and Δt being the lattice space and lattice time step, respectively. The weighting factors for the D2Q9 model are obtained as:

xi ¼

4�

9 for i ¼ 01�

9 for i ¼ 1 � 41�

36 for i ¼ 5 � 8

8<

:ð14Þ

The kinematic viscosity tf and the thermal diffusivity af are then related to the relaxation times by these correlations [1]:

tf ¼ st �12

� �

c2s Dt ð15Þ

af ¼ sa �12

� �

c2s Dt ð16Þ

2.2. Multi-relaxation time LBM

In the SRT scheme, all modes relax to their equilibria with the same rate. However, from a physical point of view, these rates should be different during the collision process. To overcome this limitation, d’Humieres proposed a collision matrix with different eigenvalues (relaxation times). The lattice Boltzmann equation (LBE) with an MRT collision operator is defined as [20]:

f x þ ciDt; t þ Dtð Þ � f x; tð Þ ¼ � K f � f eqð Þ i ¼ 0 � b � 1ð Þ ð17Þ

In the above equation, b is the number of lattice directions and λ is the collision matrix. The above equation is in the velocity space, but one can express it in the moment space, too. The relation between the moments and distribution functions can be suggested as:

m ¼ Mf ¼ m0;m1; . . . ;mb� 1ð ÞT ð18Þ

where M is the invertible transformation matrix and is composed of individual vectors in all lattice directions. So one can rewrite Eq. (17) as:

m x þ ciDt; t þ Dtð Þ � m x; tð Þ ¼ � S m � meqð Þ ð19Þ


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Here, S ¼ MKM� 1 and usually is a diagonal matrix. Also, meq ¼Mfeq is the equilibrium state in the moment space. Practically, when applying MRT method usually the collision step is carried out in the moment space and the streaming process is still performed in the velocity space. Hence, we can write:

f x þ ciDt; t þ Dtð Þ � f x; tð Þ ¼ � M� 1S m � meqð Þ ð20Þ

In Eq. (19), the S matrix can be tuned freely to enhance the performance of the algorithm. The corresponding moments for the D2Q9 model are:

m ¼ q; e; e; jx; qx; jy; qy; pxx; pxy� �T

; ð21Þ

While the equilibria in the moment space are:

meq ¼ q 1; � 2þ 3u2; aþ bu2; ux; � ux; uy; � uy; u2x � u2y; uxuy

� �T; ð22Þ

Here, a and b are free parameters, and when we choose a ¼ 1 and b ¼ � 3 the equilibrium is the same as D2Q9 BGK model. The transformation matrix is applied as:

M ¼

1 1 1 1 1 1 1 1 1� 4 � 1 � 1 � 1 � 1 2 2 2 2

4 � 2 � 2 � 2 � 2 1 1 1 10 1 0 � 1 0 1 � 1 � 1 10 � 2 0 2 0 1 � 1 � 1 10 0 1 0 � 1 1 1 � 1 � 10 0 � 2 0 2 1 1 � 1 � 10 1 � 1 1 � 1 0 0 0 00 0 0 0 0 1 � 1 1 � 1

2

6666666666664

3

7777777777775

ð23Þ

Also, the relaxation rates corresponding to the moments are:

S ¼ diag 1:0; 1:4; 1:4; s3; 1:2; s5; 1:2; s7; s8ð ÞT; ð24Þ

Again in the above equation, s3and s5 are arbitrary parameters which are set to zero. Finally, s7 and s8 are defined as:

s7 ¼ s8 ¼2

1þ 6t

� �

ð25Þ

For more information on MRT LBM, one can refer to [15].

2.3. The hybrid LBM with TVD scheme

In this scheme, the normal D2Q9 LBM is opted to solve the flow equations. Followed by Frisch et al. [27] and Mohamad and Kuzmin [28] and based on LGA, a distribution function with appropriate source terms for simulating the fluid flow is applied as:

fi x þ ciDt; t þ Dtð Þ � fi x; tð Þ ¼ �1st

fi x; tð Þ � feqi x; tð Þ½ � þ Dtwi

ci!: F!

c2s; ð26Þ

The equilibrium distribution function, Lattice discrete velocities, and weight factors, as well as the momentum relaxation time, are completely similar to those of the single relaxation time method [Eqs. (11–15)] and therefore, they have not been repeated here again. The major modification which is used here is related to the energy equation modeling:

qh

qsþq Uhð ÞqX

þq Vhð ÞqY

¼ r2h ð27Þ


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It is known that there are three different ways of treatment for solving scalar equations coupled with the Navier–Stokes equations when the Boltzmann equation is solved instead of the flow equations, namely multispeed lattice, double distribution function, and hybrid methods. According to Guo and Shu [15], the most stable treatment is using an extra distribution function for the extra conservation equation. Also, it is a common belief that the hybrid method usually suffers numerical instabilities and is not suitable for low dissipation flow regimes, but we can see in the next section that against the common belief, not only the present TVD Hybrid schemes do not suffer the mentioned problem but they can even improve the stability, precision, and convergence rate of the numerical solution. In the present research, we attempted to use the hybrid finite difference method for solving the energy equation. The main discretized equation for temperature field is as follows:

hnewP � hP

Dsþ

Uhð Þe� Uhð ÞwDX

þUhð Þn� Uhð Þs

DY¼

hE � 2hP þ hWDX2

þhN � 2hP þ hS

DY2

� �

ð28Þ

To solve the mentioned equation with a relatively low dissipation (high Ra cases), a TVD scheme is applied to the convective values of h (i.e., he, hw, …). High-resolution techniques are precise numerical solution schemes in FD and FV methods which are developed and applied in several numerical studies during the last decades. Hirsch [29] presents a detailed study of the novel TVD Limiter Functions w ¼ wðrÞ with r being r ¼ ðhP � hW=hE � hPÞ to calculate the convective terms in the flows with sever parameter gradients. As an example, he can be obtained from:

he ¼ hP þ12w rð Þ hE � hPð Þ; ð29Þ

wðrÞ may change and produce any TVD or regular estimation for the convective fluxes. Each one of these schemes has a specific accuracy and stability and for more information on TVD schemes, one can refer to [30] . Not only because of its TVD characteristics but also because of its fine accuracy, the Superbee limiter function is very popular and is considered as one of the best available Limiter functions. On the other hand, Minmod is considered less accurate due to the additional dissipations, but compared to Superbee it possesses better stability and faster convergence. Thereby, these two Limiter Functions proposed by Roe [31] are used to calculate the convective fluxes:

TVD S scheme : wSuperbee ¼ max 0;min 2r; 1ð Þ;min r; 2ð Þ½ � ð30Þ

TVD M scheme : wMinmod ¼min r; 1ð Þ if r > 00 if r � 0

�

ð31Þ

Also to keep Mach number constant for all three models, the kinematic viscosity is calculated from the following equation:

tf ¼ M �Ma � cs

ffiffiffiffiffiffiPrRa

r

ð32Þ

In which Ma ¼ 0.1 and Pr ¼ 0.71 are fixed parameters and Ra is defined per case. Also, the thermal diffusivity can be defined from the Prandtl number definition as:

af ¼tf

Prð33Þ

So we can assume that the computational conditions for all three models are consistent. Moreover, the flowchart of the hybrid algorithm is presented in Figure 1.

2.4. Entropy generation in the system

Three types of entropy generation sources are taken into account in this study: Entropy generation due to the friction of the flow, heat transfer, and magneto-hydrodynamics. According to the local


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thermodynamic equilibrium of the linear transport theory [32] and with regard to dimensionless parameters for a 2D-Cartesian coordinate, the entropy generation rates are as follows:

Sfriction ¼ /ir 2qUqX

� �2

þqVqY

� �2" #

þqUqYþqVqX

� �2( )

ð34Þ

SHeat ¼qh

qX

� �2

þqh

qY

� �2

ð35Þ

SMHD ¼ /ir �Ha2 U sin c � V cos c½ �2

ð36Þ

Figure 1. The flowchart algorithm of the proposed hybrid FD_LB method with TVD characteristics.


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In the above equations, /ir is the irreversibility distribution ratio /ir ¼ mT0 a�

LDT� �2�k, which is

assumed to be constant and equal to 10� 4 in the present study. Also, total entropy generation can be computed from:

Stot ¼Z

V

Sfriction þ SHeat þ SMHDð Þ dV ð37Þ

The local Bejan number indicates the strength of the entropy generation due to heat transfer irreversibility with respect to the total entropy generation of the system:

Be ¼SHeat þ SMHD

Stotð38Þ

This parameter can define the relative importance of heat transfer and MHD irreversibility to the friction irreversibility. To show the overall dominant irreversibility mechanism, the average Bejan number is calculated as [32]:

Beave ¼R

A BeðX;YÞ dARA dA

ð39Þ

3. Results and discussion

The novel hybrid TVD codes, as well as the SRT and MRT codes, are tested in two different case studies. At first, the well-known natural convection of air in a closed cavity with two isothermal walls and two adiabatic walls is simulated, and the results for fluid flow, heat transfer, and entropy generation are presented for various Ra numbers. Then, the same case is considered with the influence of an external magnetic field and the results for the above-mentioned features are illustrated. Moreover, the convergence behavior, CPU usage, and accuracy of the new TVD codes are compared to the classical SRT and MRT models.

3.1. Natural convection benchmark (Ha ¼ 0)

Figure 2a presents the streamlines, U and V velocity components as well as the isothermal lines of the flow for different Ra numbers. We should keep in mind that in these cases no external magnetic field source is available and the flow is only driven by the buoyancy force. By the first look, one can see two important flow patterns: the main circulating eddy in the center of the cavity and the boundary layer regimes near the vertical walls. In lower Ra numbers like 103, one strong vortex is developed in the center of the cavity and the circulating pattern is dominant. But when Ra is increased (i.e., higher than Ra ¼ 105), the core eddy breaks into two weaker counter-rotating eddies. For this reason after Ra ¼ 106, mostly the boundary layer pattern is prominent. U and V velocity component contours reveal that for lower Ra numbers there are two eddies which develop throughout the entire geometry. But as Ra increases, these two eddies move toward the horizontal walls. This is because of the thinner boundary layer in these cases. Same feature is obvious in isothermal contour, so for lower Ra numbers the isotherms are almost vertical which is due to the diffusion dominant behaviors, but as Ra increases especially in higher Ra numbers like Ra ¼ 107 or 108 we can see that these isotherms develop a very thin thermal boundary layer and are mostly horizontal in the entire cavity, which is due to the change in the dominant regime from diffusion to convection. It is worth mentioning that all of the flow and heat transfer patterns are in excellent accordance with the open literature data [33].

The rate of entropy generation of the system along with its two main components: fluid friction and heat transfer entropy generation rates, as well as the Be number contours, are illustrated in Figure 2b. As can be seen, in lower Ra numbers the entropy generation rate of the system is strongly under the influence of the heat transfer component. It goes without saying that in these Ra numbers, fluid velocity is relatively low and because of that the frictional entropy generation mechanism is weak


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in these Ra regimes. If we look to the Be contour, we can see that in more than 80% of the geometry, the heat transfer mechanism is the dominant form of entropy generation. But as the Ra grows, it is obvious in the Be contours that by and by the irreversibility type is changed, and the regions with Be >0.5 (heat transfer dominancy) are almost moving aside for the frictional entropy generation mechanism to come in. This phenomenon is so strong that for high Ra numbers like Ra ¼ 107 or 108, one can hardly find a heat transfer dominant region. Also, it is worth mentioning that with the rise of Ra number, not only the frictional irreversibility but also the heat transfer and total entropy generation rates of the system augments but the heat transfer rate increment is in a lower order. Again, the entropy generation and the Bejan number contours are in a very fine agreement with the previously published data [34].

Absolute velocity and T residuals of three different codes are presented in Figure 3, namely MRT code, SRT code, and the present TVD algorithm with Superbee limiter function (TVD_S). The results

Figure 2. (a) Streamlines, horizontal (U), and vertical (V) velocity components, and temperature contours for various Ra numbers. (b) Entropy generation rate components, total entropy generation rate, and the Bejan number contours for various Ra numbers.


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of convergence of TVD scheme with minmod limiter function (TVD_M) were quite the same as TVD_S and therefore they are not presented here. The Ra number which is chosen for this case is Ra ¼ 106 and the rest of the fixed parameters are as previously defined. As can be seen, the present code shows much faster and smoother convergence behavior for both residual histories. As is expected, MRT code has a better stability and convergence behavior than SRT code, yet still, there is a significant difference between the MRT code convergence rates and those of the TVD codes. This faster convergence is owing to the flux limiter functions of the high-resolution schemes. Since on these TVD schemes, the interpolation is carried out over at least two and when necessary three upstream and downstream points, the unwanted numerical oscillations are omitted faster from the computational calculations. Moreover, the additional dissipation is only added wherever they are essential and hence the solution will not suffer the error ensued from additional dissipations.

Figure 4 demonstrates the rate of CPU elapsed time per iterations for all four schemes versus different mesh numbers. It is well predicted that the MRT code consumes the highest CPU time

Figure 2. Continued.


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and numerically is almost 30% less efficient of any other scheme. But the interesting aspect of this figure lays in this expression that although the TVD schemes are expected to be more time- consuming than the SRT code (because of their additional calculus) but yet a fine comparison reveals that the new hybrid codes are not relatively less efficient than the SRT code and to be precise, they are even less CPU consuming than the D2Q9 SRT code. This advantage can be related to the artificial intelligence of the TVD schemes which unlike the classic models do not force a blind dissipation into the whole geometry and they add dissipation wherever system needs it.

Table 1. Comparison of the presented numerical schemes with coarse meshes at Ra ¼ 106. Model Ref. [33] 50 � 50 75 � 75 100 � 100 SRT 8.976 N–C N–C 8.8166

Normal error N–C N–C � 0.01776 MRT 8.976 8.7554 8.8019 8.815641

Normal error � 0.024 � 0.0194 � 0.01787 TVD_M 8.976 8.7812 8.8662 8.925

Normal error � 0.0217 � 0.0122 � 0.00568 TVD_S 8.976 8.923 8.9498 8.973736

Normal error � 0.0059 � 0.0029 � 0.00025

MRT, multiple relaxation time.

Figure 3. The absolute velocity residual (left) and the temperature residual (right) profiles for SRT, MRT, and present TVD code with Superbee limiter function at Ra ¼ 106. Note: MRT, multiple relaxation time.

Figure 4. The rate of CPU elapsed time per iteration for all four numerical schemes versus different mesh sizes.


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In Table 1, the accuracy test for all schemes is presented at Ra ¼ 106 in coarser meshes. In this table, white rows represent the exact value of Nusselt, while the gray rows show their deviation from the suggested value of a published data. As can be seen, all schemes converge to the reference value quite fast, but the TVD_S scheme can achieve this rather faster than the others. Also, it is worth mentioning that due to the numerical oscillations, the computational error for SRT and MRT codes does not decrease as fast as the TVD schemes. Moreover, it is seen that the results of MRT and SRT codes in 100 � 100 mesh combination are less accurate than a coarser mesh of 75 � 75 in TVD schemes, so we can assume that for the same problem not only the TVD schemes perform faster, but they also need less resolution than classical SRT and MRT codes.

Table 2 compares the maximum, minimum, and average values of Nu number for present TVD_S and MRT codes against four published data. Also, the values in parenthesis indicate the location of minimum and maximum points on Nu graph. As is expected, the increment of Ra number causes an augmentation in the average Nu value which means an increase in the heat transfer rate from the vertical wall. The comparison of presented predictions with the references reveals that extremum and average values of Nu graph, as well as their locations, are in very good accordance.

3.2. MHD convection benchmark

In addition to the previous case study, a fine and precise benchmark is presented for MHD flows. Figure 5a presents the stream functions, U and V velocity components and isothermal lines at Ha ¼ 100 versus different Ra numbers obtained from TVD_S scheme. As can be seen, in lower Ra numbers the core eddy breaks into half and therefore, loses strength by two weaker counter-rotating vortices. As a result, the isothermal lines are developed similar to the no-convection case. This is due to the braking effects of external magnetic field. At Ra ¼ 106, the buoyant convective forces are comparable to the MHD force which is opposing the fluid flow. This leads to a combination of the effects and new patterns are obtained for stream function and velocity components. From this Ra on, the convective fluxes overcome the MHD forces; hence, convective heat transfer rate, as well as the Nu number, is enhanced.

Table 2. Validation of the present TVD code with Superbee limiter and MRT code against four different published data.

Ra Nu Ref. [31] Ref. [36] Ref. [35] Ref. [33] Present study

Superbee Present study

MRT

103 Max. 1.50 (0.092) 1.47 (0.109) 1.506 (0.089) 1.501 (0.08) 1.5143 (0.08) 1.5216 (0.09) Min. 0.692 (1) 0.623 (1) 0.6913 (1) 0.691 (1) 0.6921 (1) 0.7019 (1) Av. 1.12 1.074 – 1.117 1.121 1.128

104 Max. 3.53 (0.143) 3.47 (0.125) 3.530 (0.142) 3.579 (0.13) 3.5576 (0.14) 3.590 (0.14) Min. 0.586 (1) 0.497 (1) 0.5850 (1) 0.577 (1) 0.5963 (1) 0.6117 (1) Av. 2.243 2.084 – 2.254 2.260 2.285

105 Max. 7.71 (0.08) 7.71 (0.08) 7.708 (0.083) 7.945 (0.08) 7.811 (0.08) 7.9668 (0.07) Min. 0.729 (1) 0.614 (1) 0.728 (1) 0.698 (1) 0.7624 (1) 0.795 (1) Av. 4.52 4.3 – 4.598 4.564 4.649

106 Max. 17.92 (0.038) 17.46 (0.039) 17.530 (0.037) 17.86 (0.03) 17.9973 (0.03) 17.8843 (0.03) Min. 0.989 (1.0) 0.716 (1.0) 0.984 (1.0) 0.9132 (1.0) 1.1026 (1.0) 1.091 (1.0) Av. 8.8 8.743 – 8.976 8.9737 8.8156

107 Max. – 30.46 (0.024) 41.024 (0.038) 38.6 (0.015) 41.7687 (0.01) 41.3073 (0.01) Min. – 0.787 (1.0) 1.3799 (1.0) 1.298 (1.0) 1.7428 (1.0) 1.427 (1.0) Av. – 13.99 – 16.656 16.7148 16.3850

2 � 107 Max. – – – 48.84 (0.015) 54.279 (0.01) 51.9445 (0.01) Min. – – – 1.437 (1.0) 2.094 (1.0) 1.0872 (1.0) Av. – – – 19.97 20.131 19.588

4 � 107 Max. – – – 61.69 (0.015) 70.0692 (0.01) 63.3986 (0.01) Min. – – – 1.59 (1.0) 2.5197 (1.0) 0.5946 (1.0) Av. – 23.64 – 23.96 24.2640 23.3034

108 Max. – – 91.209 (0.067) 91.16 (0.010) 93.5960 (0.01) 77.3936 (0.01) Min. – – 2.044 (1.0) 1.766 (1.0) 2.9283 (1.0) 4.5641 (1.0) Av. – – – 31.486 31.0623 29.2656

MRT, multiple relaxation time.


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Figure 5b illustrates the total entropy generation rate of the system and its three main components for various Ra numbers. Contours of frictional, heat transfer, and MHD entropy generations and their dominance are compared with the total entropy generation rate. For Ra ¼ 103 and lower than that, it is obvious that heat transfer irreversibility is dominant throughout the entire system. However, as Ra is increased other two parties are empowered in the cavity. Stot contour at Ra ¼ 104 indicates that these components’ strength is analogous to the former one. From this Ra on, the heat transfer entropy generation rate tends to lose its dominance, and therefore at higher Ra numbers, the whole system’s entropy generation is under the influence of MHD and frictional irreversibility components.

Table 3 presents the average Nu number and the maximum, minimum points of Nu graph along with their corresponding locations on the walls for Ra ¼ 103–108 and Ha ¼ 0–100. It is seen that with Ra increment, the average heat transfer rate along with its representative Nu number increases gradually. However, Ha augmentation acts completely different. The external magnetic field tends

Figure 5. (a) Streamlines, horizontal (U), and vertical (V) velocity components, and temperature contours for various Ra numbers at Ha ¼ 100. (b) Entropy generation rate components, total entropy generation rate, and the Bejan number contours for various Ra numbers at Ha ¼ 100.


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to oppose the conductive fluid crossing its fluxes; hence, Ha increment would decrease the stream function and eventually the heat transfer rate. Also, it is worth mentioning that usually, the maximum heat transfer rate of the hot wall occurs in the left-bottom corner and the minimum Nu is achieved near to the left-top corner of the cavity. This feature is on the contrary for the cold wall which shows the maximum heat transfer rate at the right-top corner and the minima is observed near the right-bottom corner of the cavity.

Figure 6 demonstrates the vertical center-line temperature distribution in the cavity for different Ra and Ha numbers. The left picture represents the evolution of temperature profile for Ra ¼ 103–108

at Ha ¼ 0. As can be seen, the profile is almost linear for Ra numbers up to 103 and as the Ra increases it tends to the isothermal walls. This feature, which occurs due to higher stream functions, states that the thermal boundary layer is getting thinner and thinner as the Ra augments (Figure 2a). Exactly the same feature but for a different reason takes place in the right graph. In the right plot, air flows at Ra ¼ 106 and for a range of Ha numbers between 0

to a linear temperature profile which means lower temperature gradient on the walls and eventually lower heat transfer rate and Nu number on the walls.

Table 4 presents the total entropy generation rate of the system and average values of its related parameters: Sfriction, SHeat, SMHD, and Be number. This table implies that there are three different entropy regimes available in the pre-mentioned geometry and boundary conditions, which in terms of Ra and Ha numbers, one of them may become prominent in the system. It goes without saying that in lower Ra numbers, or in weaker flow stream functions the frictional entropy generation is negligible in the system. Also, MHD forces and entropy generation rate are directly proportional to the fluid velocity so one can conclude that at these flow rates, the MHD entropy generation rate is also unimportant. The main entropy generation in these cases is due to heat transfer rate. On the other hand, in high Ra numbers like Ra ¼ 107 or 108, there is a large stream function available in the system which would suggest higher frictional irreversibility and as was discussed above increment of flow rate causes higher MHD irreversibility, too. At these Ras, the dominant entropy generation type of the system is mostly the frictional irreversibility. There is also a third regime which usually comes into light in moderate Ra and high Ha numbers. Although the MHD entropy generation rate is generated by the fluid flow, theoretically it has a ratio coefficient of square Ha number (Ha2) which may raise its effect on the system, drastically. As an example, in Ra ¼ 105, the magnetic irreversibility

Figure 6. The vertical midsection temperature profile for: (a) Ra progress at Ha ¼ 0 and (b) Ha progress at Ra ¼ 106.

Table 3. The maximum, minimum, and average Nu number evolution per Ha and Ra increment. Ra Nu Ha ¼ 0 Ha ¼ 25 Ha ¼ 50 Ha ¼ 75 Ha ¼ 100

103 Max. 1.5143 (0.08) 1.0933 (0.01) 1.0297 (0.01) 1.0147 (0.01) 1.008 (0.01) Min. 0.6921 (1.0) 0.9199 (1.0) 0.9713 (1.0) 0.9855 (1.0) 0.9912 (1.0) Av. 1.121 1.006 1.000 1.000 1.000

104 Max. 3.5576 (0.14) 2.11 (0.05) 1.34145 (0.01) 1.1592 (0.01) 1.0924 (0.01) Min. 0.5963 (1.0) 0.5564 (1.0) 0.7639 (1.0) 0.8689 (1.0) 0.918 (1.0) Av. 2.260 1.297 1.038 1.009 1.003

105 Max. 7.811 (0.08) 7.033 (0.06) 4.9523 (0.03) 3.2081 (0.02) 2.2783 (0.01) Min. 0.7624 (1.0) 0.4811 (1.0) 0.4258 (1.0) 0.4605 (1.0) 0.5422 (1.0) Av. 4.564 3.4833 2.2125 1.5278 1.2435

106 Max. 17.9973 (0.03) 17.947 (0.03) 16.4305 (0.03) 14.2327 (0.02) 11.9327 (0.02) Min. 1.1026 (1.0) 0.7813 (1.0) 0.4785 (1.0) 0.3933 (1.0) 0.371 (1.0) Av. 8.9737 8.1645 6.3951 4.9097 3.8701

107 Max. 41.7687 (0.01) 42.344 (0.01) 42.528 (0.01) 41.666 (0.01) 39.7889 (0.01) Min. 1.7428 (1.0) 1.528 (1.0) 1.0377 (1.0) 0.715 (1.0) 0.5501 (1.0) Av. 16.7148 16.240 14.8818 13.1883 11.4251

108 Max. 93.5960 (0.01) 95.7359 (0.01) 96.757 (0.01) 97.867 (0.01) 98.1579 (0.01) Min. 2.9283 (1.0) 2.8779 (0.98) 2.584 (0.98) 2.194 (0.98) 1.779 (0.99) Av. 31.0623 31.215 30.1522 28.822 27.1997


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is responsible for about 50% (Ha ¼ 25), 70% (Ha ¼ 50), 75% (Ha ¼ 75), and 76% (Ha ¼ 100) of the total entropy generation rate in the system. Thereby, in high Ha and moderate Ra numbers, the MHD irreversibility may play an important role in the system.

4. Conclusion

In the present study, a hybrid TVD lattice Boltzmann code with two different limiter functions is used to revisit the famous natural and MHD convection problems in a closed cavity. The TVD code’s accuracy and performance were tested against simple D2Q9 SRT and MRT codes and the results were validated against several published data. The main findings of this research are as follows: .� A new hybrid FD-LBM scheme with TVD characteristics is proposed, in which the fluid flow

parameters are obtained in LBM and any scalar equation can be resolved in a high-resolution finite difference scheme with two different flux limiter functions.

.� The well-known Superbee and Minmod functions were opted as flux limiters to be applied in two benchmark case studies and excellent correspondence with open literature results was achieved.

.� The novel numerical technique was also tested for its computational expenses and it was seen that unlike the common belief the hybrid method is much more numerically efficient than the MRT code for a laminar flow inside the cavity, and in fact, the CPU time rate is usually even less than D2Q9 SRT code.

.� Moreover, a mesh refinement study for coarser mesh combinations revealed that an accurate result can be obtained with the TVD schemes sooner than the MRT or SRT codes, which means that for an identical case definition we can use the proposed methods with a mesh of about 25% coarser than SRT or MRT meshes to achieve the same accuracy for the results.

Table 4. Average values of entropy generation parameters versus Ra and Ha numbers progress. Ra Entropy gen. Ha ¼ 0 Ha ¼ 100 Ha ¼ 75 Ha ¼ 50 Ha ¼ 25

103 Sfriction (ave) 0.03692 0.001985 0.00036 0.00013 0.000061 SHeat (ave) 1.1668 1.04514 1.041 1.0409 1.04084 SMHD (ave) 0 0.005388 0.002 0.00101 0.000614 S (total) 1.20373 1.05252 1.0435 1.04205 1.041518 Be (ave) 0.968 0.998 0.999 0.9996 0.9998







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.� Finally, the most important advantage of these new techniques was observed in the convergence rate of the solution. For similar case definitions, the TVD schemes present much faster and smoother convergence rate which means less CPU elapsed time and less computational costs.

ORCID

Amir Javad Ahrar http://orcid.org/0000-0002-5148-5648

References

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http://orcid.org/0000-0002-5148-5648

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1. Introduction2. Governing the equations2.1. Single relaxation time LBM2.2. Multi-relaxation time LBM2.3. The hybrid LBM with TVD scheme2.4. Entropy generation in the system

3. Results and discussion3.1. Natural convection benchmark (Ha = 0)3.2. MHD convection benchmark

4. ConclusionORCIDReferences

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