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Numerical Heat Transfer, Part B:Fundamentals: An International Journalof Computation and MethodologyPublication details, including instructions for authors andsubscription information:http://www.tandfonline.com/loi/unhb20

A Hybrid Finite-Element/Finite-Difference Scheme for Solving the3-D Energy Equation in TransientNonisothermal Fluid Flow over aStaggered Tube BankSeyyed Mahmood Aboulhasan Alavia, Mohammad Reza Safaeib, OmidMahianb, Marjan Goodarzib, Hooman Yarmandc, Mahidzal Daharic &Somchai Wongwisesd

a Department of Mechanical Engineering, Mashhad Branch, IslamicAzad University, Mashhad, Iranb Department of Mechanical Engineering, Ferdowsi University ofMashhad, Mashhad, Iranc Department of Mechanical Engineering, Faculty of Engineering,University of Malaya, Kuala Lumpur, Malaysiad Fluid Mechanics, Thermal Engineering and Multiphase FlowResearch Laboratory (FUTURE), Department of MechanicalEngineering, Faculty of Engineering, King Mongkut’s University ofTechnology Thonburi, Bangmod, Bangkok, ThailandPublished online: 29 May 2015.

To cite this article: Seyyed Mahmood Aboulhasan Alavi, Mohammad Reza Safaei, Omid Mahian, MarjanGoodarzi, Hooman Yarmand, Mahidzal Dahari & Somchai Wongwises (2015) A Hybrid Finite-Element/Finite-Difference Scheme for Solving the 3-D Energy Equation in Transient Nonisothermal Fluid Flowover a Staggered Tube Bank, Numerical Heat Transfer, Part B: Fundamentals: An International Journalof Computation and Methodology, 68:2, 169-183, DOI: 10.1080/10407790.2015.1012440

To link to this article: http://dx.doi.org/10.1080/10407790.2015.1012440

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Numerical Heat Transfer, Part B, 68: 169–183, 2015Copyright # Taylor & Francis Group, LLCISSN: 1040-7790 print/1521-0626 onlineDOI: 10.1080/10407790.2015.1012440

A HYBRID FINITE-ELEMENT=FINITE-DIFFERENCESCHEME FOR SOLVING THE 3-D ENERGY EQUATIONIN TRANSIENT NONISOTHERMAL FLUID FLOW OVERA STAGGERED TUBE BANK

Seyyed Mahmood Aboulhasan Alavi1, Mohammad RezaSafaei2, Omid Mahian2, Marjan Goodarzi2, HoomanYarmand3, Mahidzal Dahari3, and Somchai Wongwises41Department of Mechanical Engineering, Mashhad Branch,Islamic Azad University, Mashhad, Iran2Department of Mechanical Engineering, Ferdowsi University of Mashhad,Mashhad, Iran3Department of Mechanical Engineering, Faculty of Engineering, Universityof Malaya, Kuala Lumpur, Malaysia4Fluid Mechanics, Thermal Engineering and Multiphase Flow ResearchLaboratory (FUTURE), Department of Mechanical Engineering, Faculty ofEngineering, King Mongkut’s University of Technology Thonburi, Bangmod,Bangkok, Thailand

This article presents a hybrid finite-element=finite-difference approach. The approachsolves the 3-D unsteady energy equation in nonisothermal fluid flow over a staggeredtube bank with five tubes in the flow direction. The investigation used Reynolds numbersof 100 and 300, Prandtl number of 0.7, and pitch-to-diameter ratio of 1.5. An equilateraltriangle (ET) tube pattern is considered for the staggered tube bank. The proposedhybrid method employs a 2-D Taylor-Galerkin finite-element method, and the energyequation perpendicular to the tube axis is discretized. On the other hand, the finite-difference technique discretizes the derivatives toward the tube axis. Weighting the 3-D,transient, convection-diffusion equation for a cube verifies the numerical results. The L2

norm of the error between numerical and exact solutions is also presented for threedifferent hybrid meshes. A grid independence study for the energy equation preceded thefinal mesh. The outcome is found to be in acceptable concurrence with those from theprevious studies. After the temperature field is attained, the local Nusselt number iscomputed for the tubes in the bundle at different times. The isotherms are also obtainedat different times until a steady-state solution is reached. The numerical results convergeto the exact results through refining the mesh. The implemented hybrid scheme requiresless computation time compared with the conventional 3-D finite-element method,requiring less program coding.

Received 2 October 2014; accepted 6 December 2014.Address correspondence to Mohammad Reza Safaei, Department of Mechanical Engineering,

Ferdowsi University of Mashhad, Mashhad, Iran. E-mail: cfd_safaei@yahoo.com

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1. INTRODUCTION

Heat transfer and fluid flow analysis over tube banks is of great importance invarious design applications such as cooling towers [1], boilers [2], various heatexchangers [3–5], chemical filtration [6], and nuclear reactors [7]. Heat exchangerdesign depends on such parameters as the number of pipes of various lengths anddiameters, geometric dimensions, arrangement of tubes, and fluids that carry heat[8, 9]. There have been numerous experimental and 2-D numerical studies on theflow over tube banks, but Le Feuvre [10] and later Launder and Massey [11] werethe ones who developed numerical applications for fluid flow and heat transfer intube banks. Alavi and Goshayeshi [12] and Alavi [13] used a finite-element (FE)method to assess the laminar forced convection of air past an in-line and staggeredtube bank. Recently, Wu and Che [14] employed a finite-volume (FV) method toexamine turbulent forced-convection heat transfer of vapor=air mixture flow pasta staggered tube bank. They studied a wide range of tube rows (from 2 to 7), tubediameters, vapor concentration, wall temperature, and gas velocity for an equilateraltriangle of tube banks.

However, 2-D analysis is not always sufficient, and a full 3-D analysis isrequired in some circumstances for more accuracy [15], especially of transfer in heatexchangers [16–18].

There have been a number of 3-D numerical studies on the heat transfer andfluid flow over tube banks. Fan, Ding, He, and Tao [19] developed a code accordingto nonorthogonal curvilinear coordinates, using the finite-volume method to solve3-D fluid flow and heat transfer equations around a dimpled fin-and-tube structure.Yusuf, Halvorsen, and Melaaen [20] have studied 3-D turbulent forced-convectionheat transfer in a cylindrical pin-fin channel with a staggered and aligned arrange-ment, using a finite-element commercial code, ANSYS CFX 11. Afgan [15] employedthe commercial CFD code STAR-CD to examine the turbulent forced convection ofan incompressible Newtonian fluid in a square in-line tube bank. His results showedthe 2-D simulations could not capture the complete flow physics. More recently,Iacovides, Launder, Laurence, and West [21] and Iacovides, Launder, and West

NOMENCLATURE

K matrixM mass matrixP dimensionless pressure�P Pressure, PaPe Peclet numberPr Prandtl numberRe Reynolds numbert dimensionless time�t time, sT temperature, KU velocity, m=s�V velocity vectorV¼ (u, v, w) dimensionless velocity vectorX dimensionless coordinate vector

�X coordinate vectorh dimensionless temperaturem dynamic viscosity Pa s

Subscripts and Superscriptsa advectionbd balancing diffusiond diffusion(e) elementnp total number of nodes in each

perpendicular planew wall1 free-stream

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[22, 23] examined the turbulent forced-convection heat transfer of a Newtonian fluidaround in-line tube banks, using the large-eddy simulation (LES) and unsteadyReynolds-averaged Navier-Stokes (URANS) models in the finite-volume code,Code Saturne.

Recently, the hybrid scheme has helped overcome the difficulties in generating3-D meshes. Dai and Nassar [24] studied the heat conduction equation in a double-layered 3-D thin film using a hybrid finite-difference=finite-element method. Theirmodel predictions were in good agreement with earlier finite-difference numericalstudies. Chaabane, Askri, and Ben Nasrallah [25] employed the lattice Boltzmannmethod (LBM) to solve the transient conduction–radiation heat transfer equationin a cylindrical enclosure. In that work, the radiation was solved through a hybridfinite-element=control-volume (CVFEM) approach. Wang, Han, and Sun [26] useda hybrid finite-difference=finite-element scheme to model 1-D and 2-D nonclassical-conduction heat transfer and its related thermal stresses in various configurations.Arefmanesh and Alavi [27] solved the unsteady energy equation in 3-D heat transferand fluid flow crossing over a circular tube using a hybrid finite-element=finite-difference method.

The aforementioned simulation approaches have proven applicable to 3-Dproblems. However, complex cases requiring both high computational time andverification of results require a new method. For this article, the hybrid finite-element=finite-difference method, using an in-house FORTRAN code, has numericallyanalyzed unsteady 3-D, nonisothermal heat transfer and fluid flow past a staggeredtube bank. The flow regime’s simulation results contrasted with the exact solutionresults in the literature. Attention focused on computation time and program codingcomplexity of the proposed method, compared to the 3-D finite-element method.

2. METHODOLOGY

This study solved a 3-D transient energy equation for a nonisothermal fluidflow passing over a staggered tube bank inserted in a channel. To do this, weintroduced a hybrid finite-element=finite-difference technique. In this scheme, thefinite-difference method discretizes the energy equation along with the tube axis,whereas the finite-element method handles the perpendicular direction. Appointingequally spaced grid points (mþ 1) along the tube length (H) divides it into identicalparts (m). A normal plane with respect to the tube axis is plotted at each point, withDz the distance between any two sequential planes, as shown in Figure 1 (perpendi-cular planes). Eventually, symmetry creates an identical 2-D finite-element meshincluding triangular elements of three nodes on each of the planes. Thereafter,aligned lines, in parallel with the tube axis, connect the nodes of homologicalelements in adjacent planes. This creates a hybrid 3-D mesh, as shown in Figure 1.

The following steps create the hybrid finite-element=finite-difference scheme.First, a Navier-Stokes equation solver based on the finite-element method helpscalculate the temperature distribution of the flow field. After the velocity distributionis obtained, a 2-D Taylor-Galerkin finite-element scheme helps discretize the energyequation over each perpendicular plane. Finally, finite-difference equations replaceresulting derivatives in the obtained semidiscretized equations, which are in thedirection of the tube axis (z coordinate).

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Dimensionless transient equations of continuity, momentum, and energy for3-D, incompressible flow of a Newtonian fluid in laminar regime are presented asfollows:

r � V ¼ 0 ð1ÞDVDt

¼ �rPþ 1Re

r2V ð2Þ

DhDt

¼ 1Pe

r2h ð3Þ

where the dimensionless variables can be expressed as

Pe ¼ qCpU1Dk

Re ¼ qU1Dm

P ¼�P

qU21

X ¼�XD

t ¼ �tU1D

h ¼ T � T1Tw � T1

V ¼�VU1

ð4Þ

A physical model of 2-D flow around a staggered tube bank with five tubes inthe flow direction is shown in Figure 2a. The axis of the first tube is positioned in theupstream at a distance from the input plane equivalent to 3D (Lus). The computa-tional domain should be long enough on the x axis to fully develop boundaryconditions at the outlet plane. Therefore, this length was selected at 12D, which isup to the fifth tube axis in the downstream (Lds). All the above lengths were chosenaccording to Tezduyar and Shih [28] and Arefmanesh and Alavi [29]. Variousarrangements can be chosen for the tubes in a staggered tube bank. The equilateraltriangle (ET) [30, 31] is a tube arrangement used in this work, and its graphicaldefinition is given in Figure 2b.

Figure 3 shows the computational domain of the studied problem and itsrelated boundary conditions. The tubes are between two upper and bottom walls

Figure 1. The hybrid 3-D mesh used in the present study.

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at distance H. The side walls consist of symmetric planes located between the tubes.No-slip condition is assumed in this simulation, and the tube surface and upper andbottom walls are considered of equal temperature (T¼Tw). The x derivatives of tem-perature and velocity at the outlet plane are set as zero, because this plane is far fromdownstream and therefore the flow is considered fully developed. For the symmetrycondition, y derivatives of all the dependent variables together with the velocity vec-tor in the y direction are considered zero along the side walls. At the outlet plane, azero value is assigned for the pressure. The initial conditions are [29]

VðX; 0Þ ¼ 0 ð5ÞhðX; 0Þ ¼ 0 ð6Þ

The problem is mathematically modeled by Eqs. (1)–(3) in a coupled systemalong with the initial conditions in Eqs. (5) and (6) and the boundary conditionsin Figure 3. Reynolds numbers of 100 and 300, Prandtl number of 0.7, and apitch-to-diameter ratio (PDR) of 1.5 are chosen for the investigation.

Figure 2. Physical model of the analyzed configuration.

Figure 3. Domain and boundary conditions for the analyzed configuration.

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3. NUMERICAL MODELING

The flow field needs to be generated before solving the energy equation. ANavier-Stokes equation solver based on the finite-element technique does this. A lin-ear shape function helps solve the velocity field within tetrahedral elements with fournodes, and the pressure is assumed unchanged in each segment within the elements(piecewise constant pressure) [32]. These approximations are substituted in thePetrov-Galerkin weighted-residual weak form of the Navier-Stokes equations [33]and the time derivatives are discretized. The result is nonlinear algebraic equationsin a coupled system for the unknown pressure and velocity components in the nodes[28]. Solution of this system of equations at each time step gives the distributions ofvelocity and pressure in the computational domain.

The proposed hybrid method is applied to discretize the energy equation[Eq. (3)]. The equation is discretized in the normal planes with respect to thetube axis (x–y planes), as shown in Figure 1. This is done with a 2-D finite-elementscheme and 3-noded triangular elements. Thereafter, the finite-difference methoddiscretizes the obtained semidiscretized ODEs in the direction of the tube axis(z direction).

We used the Taylor-Galerkin technique, a proper choice in transient cases, topresent the dimensionless form of temperature with respect to time in a second-order-accurate truncated Taylor series in the following equation:

hnþ1 ¼ hn þ Dtqhn

qtþ Dt2

2q2hn

qt2ð7Þ

where n and nþ 1 denote sequential time steps. Using Eq. (3), the tem-peraturederivatives of first and second orders with respect to time are given as follows:

qhn

qt¼ �V � rhn þ 1

Per2hn

q2hn

qt2¼ � qVn

qt� rhn � ðVn � rÞ qh

n

qtþ 1Pe

r2 qhn

qt

� �ð8Þ

Substituting the expressions of Eq. (8) into Eq. (7), and replacing qhn=qt byhnþ1 � hn=Dt in the ensuing equation, we get a time-discretized form of the energyequation:

1Dt

� 12 Pe

r2

� �hnþ1 ¼ 1

Dtþ 12 Pe

r2 � Vnþ1=2 � r þ Dt2ðVn � rÞðVn � rÞ

� �hn ð9Þ

The Galerkin finite-element method discretizes Eq. (9) is in the x–y planes.To do so, the following linear interpolation is used to estimate the dimensionlesstemperature in a typical 3-noded triangular element, Xe, within the 2-D mesh(Figure 1):

�hðeÞðx; y; z; tÞ ¼

X3i¼1

NðeÞi ðx; yÞhðeÞi ðz; tÞ ð10Þ

where �hðeÞ

is the linear approximation of the temperature in the dimensionless form,

NðeÞi is the usual linear shape functions, and hðeÞi is the value of the dimensionless

temperature at the nodes, both for i¼ 1–3.

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The Galerkin weighted residual formulation of the problem can be achieved bymultiplying Eq. (9) by the shape functions and the integral of the obtained formulaover the elements and setting the formula to zero:

ZXe

1Dt � 1

2 Per2� �

hnþ1�1Dt þ 1

2 Per2 � Vnþ1=2 � rþ Dt

2 ðVn � rÞðVn � rÞ� �

hn

8><>:

9>=>;Nj dx dy ¼ 0 ð11Þ

Using Gauss’s theorem and substituting the linear approximations for hn andhnþ 1 from Eq. (10) results in the following system of ordinary differential equationsODEs for the typical element:

MðeÞ

DtþKðeÞ

d

2 Pe

!hðeÞn onþ1

�MðeÞ

2 Pe

d2 hðeÞn onþ1

dz2¼ MðeÞ

Dt�KðeÞ

d

2 Pe� ðKðeÞ

a þKðeÞbd Þ

!hðeÞn on

þ 12 Pe

þ Dt2w2

� �MðeÞ

d2 hðeÞn on

dz2� wnþ1=2MðeÞ � wDtKðeÞ

a

� d hðeÞn on

dzð12Þ

where {h(e)}, for i¼ 1–3, is the vector of nodal unknowns. Local node numbers areused to express the elements of the matrices in the following form:

MðeÞij ¼

ZXe

NðeÞi NðeÞ

j dx dy

KðeÞdij

¼ZXe

ðNðeÞi;x N

ðeÞj;x þ NðeÞ

i;y NðeÞj;y Þ dx dy

KðeÞaij ¼

ZXe

ðunþ1=2NðeÞj;x N

ðeÞi þ vnþ1=2NðeÞ

j;y NðeÞi Þ dx dy

KðeÞbdij

¼ Dt2

ZXe

ðu2NðeÞi;x N

ðeÞj;x þ v2NðeÞ

i;y NðeÞj;y þ uvNðeÞ

i;x NðeÞj;y

þ uvNðeÞi;y N

ðeÞj;x Þ dx dy

for i; j ¼ 1� 3 ð13Þ

Discretization of the system of ODEs [Eq. (12)] along with the z axis is thenext step of solving the energy equation, done with the finite-difference method.The vector of nodal unknowns in each element is presented according to the globalnode number:

hðeÞn o‘

¼hIhJhK

8<:

9=;

‘

for ‘ ¼ n; nþ 1 ð14Þ

where I, J, and K denote the global node numbers in each element (Figure 1). Theseglobal node numbers can be used to rewrite the difference quotients for the first andsecond derivatives with regard to the z axis. Therefore, the dimensionless form of thetemperature in Eq. (12) can be given as

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dh‘idz

¼ h‘iþnp � h‘i�np

2Dzi ¼ I; J;K ð15Þ

d2h‘idz2

¼ h‘i�np � 2h‘i þ h‘iþnp

Dz2‘ ¼ n; nþ 1 ð16Þ

where z is the distance between successive planes (Figure 1). Replacing Eq. (12) withEqs. (15) and (16) yields a fully discretized form of the energy equation for the element:

MðeÞ

DtþKðeÞ

d

2 Pe

!hif gnþ1�MðeÞ

2 Pe

hi�np � 2hi þ hiþnp �nþ1

Dz2

¼ MðeÞ

Dt�KðeÞ

d

2 Pe� ðKðeÞ

a þKðeÞbd Þ

!hif gn

þ 12 Pe

þ Dtw2

2

� �MðeÞ hi�np � 2hi þ hiþnp

�nDz2

� wnþ1=2MðeÞ � wDtKðeÞa

� hiþnp � hi�np �n

2Dzð17Þ

Eventually, Eq. (17) is the result for the elements in the hybrid mesh. Applying thenecessary boundary conditions for the temperature, the obtained algebraic equationsare solved and, for each time step, the temperature values are obtained at all nodes.

4. RESULTS AND DISCUSSION

This work considered transient, 3-D, and nonisothermal fluid flow passing over astaggered tube bank. It solved the relevant energy equation using a proposed hybridfinite-difference=finite-element technique. In this technique, a 2-D finite-elementmethod discretizes in the x–y planes, and the finite-difference method approximatesthe derivatives in the direction of the z axis. The Taylor-Galerkin method ensured astable discretization at high Peclet numbers. As a test case, D¼ 0.01m, Tw¼ 400K,and T1¼ 300K are considered, leading to the discussion that follows.

4.1. Validity of the Method

The performance of the introduced scheme was evaluated by implementing themethod on a 3-D transient, convection–diffusion flow in a unit cube:

qTqt

þ aqTqx

þ bqTqy

þ cqTqz

¼ mq2Tqx2

þ q2Tqy2

þ q2Tqz2

� �ð18Þ

The exact, steady-state solution of Eq. (18) for the following boundaryconditions,

Tð0; y; z; tÞ ¼ 1� exp½ðy� 1Þb=m�1� expð�b=mÞ

� 1� exp½ðz� 1Þc=m�1� expð�c=mÞ

� Tð1; y; z; tÞ ¼ 0 ð19aÞ

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Tðx; 0; z; tÞ ¼ 1� exp½ðx� 1Þa=m�1� expð�a=mÞ

� 1� exp½ðz� 1Þc=m�1� expð�c=mÞ

� Tðx; 1; z; tÞ ¼ 0 ð19bÞ

Tðx; y; 0; tÞ ¼ 1� exp½ðx� 1Þa=m�1� expð�a=mÞ

� 1� exp½ðy� 1Þb=m�1� expð�b=mÞ

� Tðx; y; 1; tÞ ¼ 0 ð19cÞ

and the initial condition

Tðx; y; z; 0Þ ¼ 0 ð20Þ

are defined by [27]

Taðx; y; z; tÞ ¼ 1� exp½ðx� 1Þa=m�1� expð�a=mÞ

� 1� exp½ðy� 1Þb=m�1� expð�b=mÞ

� 1� exp½ðz� 1Þc=m�1� expð�c=mÞ

� ð21Þ

In the present study, the coefficients a, b, c, and m in Eq. (18) were consideredequal to unity and the equation was solved for the said boundary and initialconditions. The solution was based on the discussed hybrid scheme, employing three

different meshes. The L2 norm of the error ( ek k0 ¼RX ðTa � TÞ2dX

h i0:5, ¼ {0� x, y,

z� 1}) between the exact solution and numerical simulation of Eq. (18) wereobtained in the steady-state condition and are demonstrated in Figure 4, consideringthe number of nodes.

It is apparent from the figure that by refining the mesh, the result of thenumerical simulation converges to the exact solution. This shows the strength ofthe proposed scheme in solving 3-D problems.

We used an Intel Pentium 4A to compare the time of calculation of the pro-posed scheme to that of the 3-D finite-element method. The time recorded for thefinest applied hybrid mesh, having 1,331 nodes, was 2,800 s for 100 time steps. How-ever, the time for the same number of time steps and nodes was 8,400 s for the 3-Dfinite-element scheme with tetrahedral elements of four nodes.

Lower connectivity between the nodes in the hybrid scheme is the main reasonwhy its time of computation is lower. However, the codes the authors used for bothcases were in-house research versions rather than optimized ones. Nevertheless,based on previous research, optimization of the methods is believed to decreasecalculation time [34, 35].

4.2. Mesh-Independent Study

Mesh-independent tests were carried out before adopting the final mesh. There-fore, three grid distributions were tested to assure that the computational resultswere grid-independent. These were a coarse mesh with the number 6,869, a mediumone with the number 12,684, and a fine mesh with the number 21,679. These types ofmesh were used in the computations consisting of 2-D elements in the planesperpendicular to the tube axis until the mesh was optimized and results remainedthe same even with numbers greater than these. In the third dimension, the

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derivatives parallel to the tube axis were set at 10 parts in all tests [29]. The grid ismore refined in the vicinity of tube walls, where significant temperature and velocitygradients occur.

Figure 5 illustrates the local Nusselt number at different angles for differentgrids. Based on the presented results, a 12,684� 10 grid was selected for all modelingcases.

Figure 4. Alteration of the L2 norm of the error for Equation (18).

Figure 5. Grid independence tests for the present study by comparison of local Nusselt numbers in variousmesh concentration.

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4.3. Isotherms

The isotherms for flow over a staggered tube bank with five tubes in the flowdirection with Reynolds numbers 100 and 300, at different times until steady stateare shown in Figure 6. It is possible to predict the amount of heat at different timesand at the various points of the tube bank.

It is obvious that in places where the isotherms are closer to each other, thetemperature gradient is greater and heat transfer is higher. In Figure 6a, the iso-therms at �t ¼ 0:1 are drawn symmetrically over all the tubes. This symmetry tendsto vanish over time due to the recirculation happening between the tubes(Fig. 6b). Figure 6c demonstrates that, in the steady state, the isotherms are swarmedon the front half of the first tube and with less intensity over the front half of thesecond tube, which indicates a high radial heat flux.

Figure 6. Isotherms in the staggered tube bank in mid-plane.

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4.4. Nusselt number

Figure 7 shows the distribution of the local Nusselt number around the tubes atthe steady-state condition, in comparison with the result of Wang, Penner, and Ormis-ton [36]. The local Nusselt number and the bulk temperature are defined as [36, 37]

Nu ¼ � DðTw � TbÞ

qTqn

����w

� �¼ � 1

1� hb

qhq�n

����w

� �ð22Þ

Tb ¼R SD=2D=2 quT dy

qðSD=2ÞU1ð23Þ

Figure 7 shows that the fluid has the same local Nusselt number over all the tubesin the initial phase. The intense temperature gradient around these tubes increases itand normalizes it. As time passes, it decreases and differs among tubes. The first tubein the steady state has the highest Nusselt number, almost the same, at angles between0� and 80�. For the third tube, the maximumNusselt number is at the front stagnationpoint, and with the growth of the boundary-layer flow, decreases.

For the first and third tubes, the minimum Nusselt number occurs in the wakeflow behind the tube.

5. CONCLUSION

This work studied a nonisothermal fluid flow passing over a staggered tubebank in a channel. To solve the related 3-D transient energy equation, we proposedand implemented a hybrid finite-difference=finite-element method. This method sawdiscretization performed with a 2-D finite-element scheme for the perpendicular

Figure 7. The local Nusselt number at the steady state in the mid-plane.

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planes and the finite-difference technique for the derivatives in the direction of thetube axis. To achieve stable solutions at high Peclet numbers, the Taylor-Galerkinmethod was used for the finite-element scheme.

The time of computation for the introduced hybrid schemewas evaluated by com-parison with the calculation time for a 3-D finite-element method. Simulations for thetest case were performed by the two methods with the same computer and for the sametime steps and nodes. The results found the computational time of the finite-elementmethod to be about three times greater than that for the hybrid scheme, due to thehybrid scheme’s lower nodal connectivity. This is the advantage of this scheme inmassive simulations of 3-D problems, where calculation time is a determining factor.Then, the present method was thoroughly benchmarked against an exact solution inthe literature and was found to be in very good agreement, especially in fine meshes.

After that, the method was applied for the simulation of 3-D convection heattransfer in a nonisothermal fluid flow passing over a staggered tube bank in achannel at the conditions of Re¼ 100 and 300 and Pr¼ 0.7. The related mesh-independent evaluations were also performed.

The results showed that the isotherms were symmetrically crowded over all of thetubes at the first stage. However, the recirculation between the tubes destroys this sym-metry over time. At the steady state, the isotherms huddle over the front half of thefirst tube and with less intensity over the front half of the second tube, indicating a highradial heat flux. At steady state, the maximumNusselt number for the first tube occursbetween the angles of 0� and 80�. However, Nusselt number for the third tube is high-est at the front stagnation point, and it decreases as the flow boundary layer grows.The Nusselt number has the lowest value at the wake flow region behind the tube.

FUNDING

The authors gratefully acknowledge High Impact Research Grant UM.C=HIR=MOHE=ENG=23, UMRG Grant RP012C-13AET, and the Faculty ofEngineering, University of Malaya, Malaysia, for support in conducting thisresearch work. The seventh author would like to thank the “Research Chair Grant”National Science and Technology Development Agency (NSTDA), the ThailandResearch Fund (IRG5780005), and the National Research University Project fortheir support.

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