International Journal of Numerical Methods for Heat & Fluid FlowA second order finite volume technique for simulating transport in anisotropic mediaJayantha Pasdunkorale A. Ian W. Turner

Article information:To cite this document:Jayantha Pasdunkorale A. Ian W. Turner, (2003),"A second order finite volume technique for simulatingtransport in anisotropic media", International Journal of Numerical Methods for Heat & Fluid Flow, Vol. 13Iss 1 pp. 31 - 56Permanent link to this document:http://dx.doi.org/10.1108/09615530310456750

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A second order finite volumetechnique for simulating

transport in anisotropic mediaJayantha Pasdunkorale A. and Ian W. Turner

Centre in Statistical Science and Industrial Mathematics, School ofMathematical Sciences, Gardens Point Campus, Queensland University

of Technology, Brisbane, Australia

Keywords Flux, Finite volume

Abstract An existing two-dimensional finite volume technique is modified by introducing acorrection term to increase the accuracy of the method to second order. It is well known that theaccuracy of the finite volume method strongly depends on the order of the approximation of the fluxterm at the control volume (CV) faces. For highly orthotropic and anisotropic media, first orderapproximations produce inaccurate simulation results, which motivates the need for betterestimates of the flux expression. In this article, a new approach to approximate the flux term at theCV face is presented. The discretisation involves a decomposition of the flux and an improved leastsquares approximation technique to calculate the derivatives of the dependent function on the CVfaces for estimating both the cross diffusion term and a correction for the primary flux term. Theadvantage of this method is that any arbitrary unstructured mesh can be used to implement thetechnique without considering the shapes of the mesh elements. It was found that the numericalresults well matched the available exact solution for a representative transport equation in highlyorthotropic media and the benchmark solutions obtained on a fine mesh for anisotropic media.Previously proposed CV techniques are compared with the new method to highlight its accuracy fordifferent unstructured meshes.

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NomenclatureAk ¼ length of a control volume face [m]Cp ¼ specific heat [J/kg/K]Fk ¼ midpoint of a control volume facehb ¼ heat transfer coefficient [W/m2/K]K ¼ conductivity tensor [W/m/K]L ¼ dimension of the domain in the x

direction [m]M ¼ dimension of the domain in the y

direction [m]m ¼ order of the Taylor expansionN ¼ a representative node around a

control volume

NP ¼ number of nodes around a controlvolume

n ¼ unit outward normal vector at acontrol volume face

P ¼ node surrounded by a control volumeR ¼ remainder of the Taylor expansionSb ¼ set of boundary points of the solution

domaint ¼ time [s]t ¼ unit vector perpendicular to vu ¼ unit vector along a control volume

face

Both authors would like to thank the anonymous reviewer for the many fruitful comments thatimproved the presentation of the manuscript. The first author wishes to acknowledge thefinancial support provided by the Queensland University of Technology under the IPRSscholarship program and the leave granted for the studies from the University of Ruhuna,Sri Lanka, where he is affiliated as a Lecturer.

A second orderfinite volume

technique

31

Received September2001

Revised July 2002Accepted August 2002

International Journal of NumericalMethods for Heat & Fluid Flow

Vol. 13 No. 1, 2003pp. 31-56

q MCB UP Limited0961-5539

DOI 10.1108/09615530310456750

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IntroductionThe finite volume technique is used extensively for solving a range of differentproblems arising in many fields of science, technology and engineering (Baileyet al., 1999; Davidson and Stolcis, 1995; Demirdzic and Muzaferija, 1995; Helfand Kuster, 1994; Moraga and Medina, 2000; Perre and Turner, 1999; Vijayanand Kallinders, 1994). Typically, due to the conservative nature of the schemeand its ability to be implemented on either structured or unstructured meshes,it is the preferred method implemented in many industrial computational fluiddynamics codes. The accuracy of the scheme depends to a large extent on theapproximation of the flux at the midpoint of the control volume (CV) face(Turkel, 1985). For many problems this approximation is crucial and theclassical linear models are often inadequate, especially for anisotropic mediawhen the anisotropy ratio is large ( Jayantha and Turner, 2001b).

The literature highlights only a limited number of computational schemesthat accurately model transport in highly anisotropic media on completelyunstructured meshes using a generalised finite volume formulation (Chow et al.,1996; Croft, 1998; Demirdzic et al., 2000; Hermeline, 2000; Murthy and Mathur,1998; Turner and Ferguson, 1995b). The accuracy of the existing schemes isquestionable under extreme anisotropic ratios such as the one that exists inwood for permeability (of the order of 1 : 1,000 in the radial versus longitudinaldirections).

Typically, wood drying simulations require the computation of the moisture,temperature and pressure fields in a radial-longitudinal cross-section of theboard for the purposes of determining the stress distribution (Ferguson andTurner, 1996). Under extreme drying conditions, the non-linear variation in therelative permeability tensors causes the transport problem to become highlyorthotropic in the longitudinal direction. Most of the existing codes that arebased on finite volume discretisations implemented on unstructured meshes inorder to capture local heterogeneities in the wood structure, either diverge, orproduce misleading results under these circumstances. This lacking of a robust

v ¼ vector through a CV face connectingadjacent nodes [m]

w ¼ the vector representing K Tnx ¼ coordinate length [m]y ¼ coordinate length [m]

Greek symbols

dt ¼ discrete time step size [s]dVP ¼ area of the control volume [m2]dx ¼ vectors emanating from the midpoint

of the face [m]1 ¼ correction term in the approximation

of primary flux term [K]

l ¼ a parameter in equation (10)r ¼ density [kg/m3]f ¼ transported quantity – Temperature

[K]f0 ¼ initial temperature [K]fs ¼ surrounding temperature [K]

Superscripts and subscripts

b ¼ boundary pointk ¼ index for the control

volume facesn ¼ represents nth time

step

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numerical method for such important industrial problems provides themotivation for this research.

One can note that the use of the midpoint rule in the finite volume technique,which is described later in the text, can maintain second order accuracy only ifthe flux approximation at the CV face is at least second order accurate ( Murthyand Mathur, 1998). The aim of this work is to propose a high order fluxapproximation technique that achieves this objective.

The numerical treatment of transport equations for orthotropic andanisotropic media using finite volume methods is a non-trivial task. In order toexplain the underlying problems that occur when approximating a term of theform ðK7fÞ·n or 7f·w ¼ 7f:ðK TnÞ; the following generalised form of theflux approximation at a CV face is considered (Figure 1(c)):

Figure 1.Control volume

schematics used for thedevelopment of the

proposed fluxapproximation

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ð7fÞ·w ¼ C1 7f·v þ C2 ð7f·uÞ

Primary term Secondary term

. C1 ðfN 2 fPÞzfflfflfflfflfflfflffl}|fflfflfflfflfflfflffl{

þ C2 ð7f·uÞzfflfflffl}|fflfflffl{ ð1Þ

where the constants C1 and C2 depend on the tensor K and the geometry of theCV mesh – see Croft (1998), Demirdzic and Muzaferija (1995), Jasak andGosman (2000) and Mathur and Murthy (1997), for similar types ofdecompositions. The primary term is the first order Taylor approximation of7f·v and the gradient required for the secondary or the cross-diffusion termcan be computed using either the weighted least squares methods or the Green-Gauss gradient reconstruction (Barth, 1994; Croft, 1998; Jayantha and Turner,2001a). One notes, however, that even if the secondary term is approximatedwith a high order accuracy, the error in the flux estimation is still dominated bythe lower order associated with the primary term.

The above techniques have achieved considerable success for isotropicproblems. However, for anisotropic problems it has been shown that suchstrategies can be inaccurate when the anisotropy ratio is large (Jayantha andTurner, 2001b). This inaccuracy arises because the implicitly treated primaryflux approximation term can be less important than the secondary term, whichis usually treated explicitly in the underlying matrix system (Chow et al., 1996;Jayantha and Turner, 2001b; Turner and Ferguson, 1995a), if the preferentialdirection of transport (the direction given by the vector w in Figure 1(c)) is farfrom the direction v defined by the nodes adjoining the CV face. Hence thesystem matrix may not contain sufficient information of the flow. Therefore,most, if not all, of the important flux information can be lost from the matrixleading to inaccurate and misleading results. Note, however, that the problemof non-alignment of the flux vector and the vector defined by the nodesadjoining the cell face is not particularly of anisotropic media.

The difficulty that arises concerns an accurate decomposition of the flux atthe CV faces. In order to develop a new decomposition technique, an extensionof the weighted least squares approximation to reconstruct the derivatives ofthe function at the CV face for determining the cross-diffusion term and animportant correction to increase the accuracy of the primary term of the fluxapproximation will be analysed here.

One way of including the information from all the neighbours of the CV intothe resulting matrix system is to employ the control-volume finite-elementmethod as discussed in Perre and Turner (1999) or Jayantha and Turner (2001a,b). However, even this method becomes inaccurate for highly anisotropictransport problems, because of the linear approximation used for estimatingthe flux.

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Bearing all of these facts in mind, higher order methods that exploitquadratic or cubic Taylor series expansions to approximate the flux at the CVface are proposed.

The flux decomposition and subsequent approximation at the CV facepresented here is novel and in most of the circumstances that were tested in thisresearch, produces excellent results for this difficult problem. The strategy isdifferent from previous work, (Chow et al., 1996; Croft, 1998; Jasak and Gosman,2000; Murthy and Mathur, 1998; Turner and Ferguson, 1995a) because thistechnique introduces a correction for the primary term of the equation (1).Furthermore, the improved least squares technique discussed here producesnot only derivatives of the function, but also the function value at the requiredpoint on the CV face.

The overall methodology is summarised by the following fluxdecomposition formula:

ð7fÞ·w ¼ C1 7f·v þ C2 ð7f·uÞ

Primary term Secondary term

. C1 ðfN 2 fPÞzfflfflfflfflfflfflffl}|fflfflfflfflfflfflffl{

þ C2 ð7f·uÞzfflfflffl}|fflfflffl{ ð2Þ

The estimation of the parameter 1 and the gradient, 7f, at the CV faces, usingan improved least squares function reconstruction provides an excellentapproximation that increases the overall order of the estimation of the flux to ashigh as cubic. This strategy increases the accuracy of the primary term as wellas the secondary or cross diffusion term; therefore, the finite volume schememaintains second order spatial accuracy. However, there still remains someproblems for which this strategy does not provide good results. For example,under extreme cases of anisotropy ratios like 1 : 10,000, because of the explicittreatment of the secondary term and the correction in the CV solution process.The authors are presently working on a new technique that will address thisproblem.

To assess the accuracy and efficiency of the proposed schemes the followingtwo-dimensional anisotropic transport problem for a finite rectangular domainis considered:

7·ðK7fÞ ¼ rCp›f

›t; 0 # x # L; 0 # y # M ; t . 0 ð3Þ

where

K ¼kxx kxy

kyx kyy

!·ðK7fÞ·nb ¼ hbðfs 2 fÞ; at boundary Sb; t . 0

where nb is outward unit normal vector at boundary b, and initially

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fðx; y; 0Þ ¼ f0; 0 , x , L; 0 , y , M ;

and note that the coefficients kxy and kyx are zero for the orthotropic case.Exact solutions for the orthotropic case, and the numerical results computed

on very fine meshes for the anisotropic cases where analytical solutions areunavailable, will be used to assess the accuracy of the new scheme.

This paper is organised as follows. In the next section, the theoreticalsolution used to benchmark the new numerical scheme is presented. Thereafter,the complete analysis of the finite volume technique, together with the fluxapproximation are given. Then there follows a discussion of the numericalsimulation results offered by the new finite volume methodology. It can beobserved that the new method produces an excellent fit with analyticalsolutions obtained by using the methods in Ozisik (1980) for all orthotropiccases tested. The conclusions of the work are summarised in the final section ofthe paper.

Theoretical solutionFor the purposes of assessing the accuracy of the proposed finite volumescheme, case studies will compare the simulation results with analyticalsolutions of equation (3), for the orthotropic case (i.e. when kxy ¼ 0 and kyx ¼ 0).The system (equation (3)) is transformed into dimensionless form (Ozisik, 1980)by using the following parameters:

X ¼x

L; Y ¼

y

M; t ¼

kxxt

rCpL2; Bi1 ¼

h1M

kyy; Bi2 ¼

h2L

kxx; Bi3 ¼

h3M

kyy;

Bi4 ¼h4L

kxx; QðX ;Y ; tÞ ¼

ðf2 fsÞ

ðf0 2 fsÞ; 0 # Q # 1

to obtain

›2Q

›X 2þ

kyyL2

kxxM 2

›2Q

›Y 2¼

›Q

›t; 0 # X # 1; 0 # Y # 1; t . 0 ð4Þ

with boundary conditions

2›Q

›Xþ Bi4Q ¼ 0; X ¼ 0;

›Q

›Xþ Bi2Q ¼ 0; X ¼ 1;2

›Q

›Yþ Bi1Q ¼ 0;

Y ¼ 0 and›Q

›Yþ Bi3Q ¼ 0; Y ¼ 1; for t . 0

and initial condition QðX ;Y ; 0Þ ¼ 1; for 0 , X , 1; 0 , Y , 1: Assuming aseparation in the form QðX ;Y ; tÞ ¼ Q1ðX ; tÞQ2ðY ; tÞ it can be shown that

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Q1ðX ; tÞ ¼X1m¼1

cðbm;XÞ

NxðbmÞsinbm 2

Bi4

bmcosbm þ

Bi4

bm

e2b2

mt

where

cðbm;XÞ ¼ bm cosðbmXÞ þ Bi4 sinðbmXÞ for 0 , X , 1;

ðb2m 2 Bi2Bi4Þ tanbm ¼ ðBi2 þ Bi4Þbm;

NxðbmÞ ¼1

2ðb2

m þ Bi24Þ 1 þBi2

b2m þ Bi22

!þ Bi4

" #

and

Q2ðY ; tÞ ¼X1n¼1

xðmn;Y Þ

NyðmnÞsinmn 2

Bi1

mncosmn þ

Bi1

mn

e2m2

nr 2t

where

r 2 ¼kyy

kxx

L2

M 2; and

xðmn;Y Þ ¼ mn cosðmnY Þ þ Bi1 sinðmnY Þ for 0 , Y , 1;

ðm2n 2 Bi1Bi3Þtanmn ¼ ðBi1 þ Bi3Þmn;

NyðmnÞ ¼1

2ðm2

n þ Bi21Þ 1 þBi3

m2n þ B2

3

!þ Bi1

" #:

ð5Þ

Using the above equations, the analytical solution of equation (3) for theorthotropic case can be obtained as

fðx; y; tÞ ¼ fs þQ1ðX ; tÞQ2ðY ; tÞðf0 2 fsÞ for 0 , x , L;

0 , y , M ; t . 0:

Finite volume techniqueThe finite volume technique (Patankar, 1980; Turkel, 1985) concerns the directdiscretisation in physical space of the integral formulation of a givenconservation law. The method enables an arbitrary mesh to be employed forthe computations and a variety of options exist for the definition of the CVs

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around which the conservation law is expressed. It should be noted, however,that the sum of all CVs must cover the entire computational domain and theCVs cannot overlap without having common surfaces.

The possibilities of modifying the shape and location of the CVs associatedwith a given mesh point, together with the complete flexibility in evaluating thefluxes through the CV faces make the method a popular choice for use in avariety of applications in science, engineering and technology. The mostimportant property of the method is the conservative nature of the scheme.Basic quantities such as mass, momentum and energy remain conserved at thediscrete level.

Figure 2 exhibits four background finite element triangular meshesgenerated using “EasyMesh” (Version 1.4) developed by Niceno, B. (It is afreely available mesh generator on the Web site: http://~-dinma.univ.trieste.it/~nirftc/research/easymesh/ ). The CVs are constructed around the vertices ofthe triangles by joining the centroids of adjacent elements. These particularmeshes have been chosen in order to demonstrate the complete flexibility of thenew flux approximation scheme proposed in this work.

Figure 1 exhibits typical CVs for a two-dimensional framework within anunstructured mesh of the vertex-centred CV approach. The CV with thecentroid point, P, has neighbouring nodes, Nk, k ¼ 1; 2; . . .; p:

The discretised form of the differential equation (3) is derived by integratingthe equation over the CV. The use of the divergence theorem in the plane leadsto

rCpd �f

dt2

1

dVP

IGP

ðK7fÞ·ndG . 0 ð6Þ

where

�f ¼1

dV P

ZdVP

fdV ; ð7Þ

is the average of f in a CV. As there is no approximation made to this point, theequation (6) together with equation (7) is exact (Turkel, 1985). Discretisingequation (6) one can obtain

rCpd �f

dt2XNP

k¼1

{ðK7fÞ·n}FkAk . 0; ð8Þ

which is second order in space if the term (K7f)·n is accurately evaluated atthe midpoint of the CV face.

Assuming that fP represents the averaged value of f over dVP andconsidering the time integral from ndt to (n+1)dt, the discrete form of equation(8) can be written as

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Figure 2.(a) A fine mesh (809

nodes, 1,504 elements),(b) a coarse mesh (131

nodes, 218 elements), (c)a very coarse mesh (68

nodes, 106 elements), and(d) a distorted mesh (150

nodes, 237 elements)

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rCpdVP

dtðf

ðnþ1ÞP 2 f

ðnÞP Þ

2 lXNP

k¼1

{ðK7fÞ·n}ðnþ1ÞFk

Ak þ ð1 2 lÞXNP

k¼1

{ðK7fÞ·n}ðnÞFkAk

( )

. 0: ð9Þ

The term (K7f)·n can be expressed as 7f·(K Tn) and hence, the aboveequation can be rewritten as

rCpdV P

dtðf

ðnþ1ÞP 2 f

ðnÞP Þ

2 lXNP

k¼1

{ð7fÞ·w}ðnþ1ÞFk

Ak þ ð1 2 lÞXNP

k¼1

{ð7fÞ·w}ðnÞFkAk

( )

. 0 ð10Þ

where w ¼ K Tn: The parameter, l ¼ 1 gives a fully implicit scheme, l ¼ 0leads to a fully explicit scheme and l ¼1=2 provides a second order scheme intime. The parameter l is set to 1 for this study.

The accuracy of the scheme hinges around the approximation of the fluxterm at the CV face. The discrete form (equation (10)) allows a first orderaccuracy in time and a second order approximation in space when the fluxapproximation at the midpoint of the CV faces is at least second order accurate(Murthy and Mathur, 1998).

Flux approximation using decomposed vectors (FADV)Figure 1(b) shows a CV face in a two-dimensional framework. The unit vectorsnk and uk are perpendicular to each other, and the vector joining the points Pand Nk, vk, is perpendicular to tk. The vector wk ¼ K Tnk can be decomposed asfollows. Note that subscript k is suppressed in the following equations.

The two vector equations v ¼ ðv·uÞu þ ðv·nÞn and w ¼ ðw·uÞu þ ðw·nÞngive the following formulae:

n ¼v 2 ðv·uÞu

v·n

and

w ¼ ðw·uÞu þ ðw·nÞv 2 ðv·uÞu

v·n¼

w·n

v·nv þ w·u 2 w·n

v·u

v·n

� �u: ð11Þ

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Therefore, an expression for 7f·w ¼ 7f·ðK TnÞ can be written as

7f·w ¼w·n

v·n7f·v þ w·u 2 w·n

v·u

v·n

� �7f·u: ð12Þ

which is valid for the two-dimensional case. In order to estimate the primaryterm 7f·v, consider the following analysis.

Write the vectors emanating from the point F, the midpoint of the CV face, tothe points P and N as (Figure 3(a)) dx2 ¼ ðdx21 ; dx22 Þ and dxþ ¼ ðdxþ1 ; dxþ2 Þ,respectively. Consider Taylor expansions of the function f:

fðxF þ dxþÞ ¼Xm

k¼0

1

k!ðdxþ·7ÞkfðxFÞ þ Rþ ð13Þ

and

fðxF þ dx2Þ ¼Xm

k¼0

1

k!ðdx2·7ÞkfðxFÞ þ R2 ð14Þ

where the remainder R has the Lagrange form and for example,

Rþ ¼1

ðm þ 1Þ!ðdxþ·7Þðmþ1ÞfðxF þ udxþÞ; 0 # u # 1:

Subtracting equation (14) from equation (13) and assuming that Rþ 2 R2 . 0(i.e. negligible) the following expression for ð7fÞF ·v can be obtained;

ð7fÞF ·v . ðfN 2 fPÞ2 1np ð15Þ

where

1np .Xm

k¼2

1

k!{ðdxþ·7Þk 2 ðdx2·7Þk}fðxFÞ: ð16Þ

It is possible to assume that 1np . 0 for fine meshes, however, it may not besuitable to use fine meshes because this requires more storage and memoryrequirements for the numerical simulations. Substitution of equation (15) intoequation (12) gives the following expression for the flux at the CV face:

{ðK7fÞ·n}ðnþ1ÞFk

.w·n

v·nðf

ðnþ1ÞNk

2 fðnþ1ÞP Þ

þ w·u 2 w·nv·u

v·n

� �ð7f·uÞðnþ1Þ

Fk2

w·n

v·nð1npÞ

ðnþ1ÞFk

: ð17Þ

To complete the flux approximation, 7f·u and 1np must be evaluated at the

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CV faces k. The following section describes approximation techniques toevaluate the required terms using the function values at the nth time step sincethe function values are not readily available at time step n þ 1:

Improved least squares gradient reconstruction (ILSGR)The Taylor series expansion has been used to estimate the functions of interestby Jasak and Gosman (2000) in order to analyse errors in regular finite volumetechniques. In this article, the Taylor series expansion of the function isconsidered to estimate the derivatives of the function and to use thosederivatives to estimate the correction term 1np which contributes the finitevolume discretisation see equations (2), (15) and (16) for the required correctionterm. Consider the truncated Taylor expansion of the function f:

Figure 3.(a) Representativevectors used in Taylorseries for determiningthe correction 1np, (b) atypical boundary controlvolume, (c) a possibledistribution of nodesconsidered for the fluxapproximation on a CVface

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fðxF þ dxdÞ .Xm

k¼0

1

k!ðdxd ·7ÞkfðxFÞ ð18Þ

Writing equation (18) for each node, d ¼ 1; 2; . . .r; see Figure 3(c) for thenodes indicated by small circles, connected to the point F, the followingover-determined system of equations is obtained (for m ¼ 3):

1 Dx1 Dy1 · · ·Dx3

1

6

Dy31

6

Dx21Dy1

2

Dx1Dy21

2

1 Dx2 Dy2 · · ·Dx3

2

6

Dy32

6

Dx22Dy2

2

Dx2Dy22

2

..

. ... ..

. ... ..

. ... ..

.

1 Dxr Dyr · · ·Dx3

r

6Dy3

r

6Dx2

rDyr

2DxrDy2

r

2

0BBBBBBBB@

1CCCCCCCCA

fF

ð›f›xÞF

ð›f›yÞF

..

.

ð ›3f

›x›y 2ÞF

0BBBBBBBBBB@

1CCCCCCCCCCA

¼

f1

f2

..

.

fr

0BBBBBB@

1CCCCCCA ð19Þ

which can be written as

AX ¼ B:

At a boundary control face, see Figure 3(b), an equation related to the boundaryconditions is added to the above system. For example, for the face F shown inFigure 3(b), the equation,

kxx

� ›f›x

�Fþ kxy

� ›f›y

�F¼ hwðfF 2 fwÞ; or

hwfF 2 kxx

� ›f›x

�F2 kxy

� ›f›y

�F¼ hwfw;

is also inserted to the system of equations given by equation (19).The components that minimise kAX 2 Bk

2in the least squares sense with

respect to a weighted inner product on R r can be determined by multiplyingthe above system by Wr£r ¼ DiagðwkÞ and A T, to arrive at the normalequations

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ðATWAÞX ¼ ðATWBÞ: ð20Þ

Note that the weight coefficients, wk’s, are chosen so that more importance isgiven to the directions that are the closest neighbours of the point F as opposedto the nodes that are further away from the point F, and wk ¼ kvkk

2c; c ¼

0; 1; 2 is used for the numerical simulations. Solving the above system, it ispossible to find the function value and the first, second and third derivativesaccurately at the point F on the CV face and, therefore, the terms 7f·u and1np required for equation (17) can be estimated.

It should be noted that the technique discussed here estimates the functionvalue on the CV face also with a high accuracy. The typical least squaresgradient reconstruction technique found in Barth (1994), Ilinca et al. (2000),Jayantha and Turner (2001a) or Ollivier-Gooch (1996) only approximates thederivatives of the function.

The special cases of the above technique as shown in Table I(b) areconsidered to approximate the terms 7f·u and 1np. For all of these leastsquares methods the closest nodes to the midpoint F (Figure 3(c)) on each CVface are used. Although it was found that the closest node points for each face issufficient for most unstructured meshes, more node points were used for thehighly distorted mesh to cover every gradient direction around each CV face.Those additional node points are chosen to ensure that every node connected tothe node points P, N, A and B were included for each face.

Numerical simulationsTo test the flux approximation techniques discussed in the earlier sections, awood like material is used with the physical properties: L ¼ 0:1 m; M ¼0:04 m; hb ¼ 10 W=m2=K; r ¼ 600 kg=m3 and Cp ¼ 1:6886 £ 103 J=kg=K; withf0 ¼ 308C and fs ¼ 1408C: The values given in Table I(a) are used for thetensor K to obtain the results presented here. The time step, dt, used for thesimulations is 1 s and the results shown are obtained after 1,000 s. The meshesshown in Figure 2 are used to implement the flux decomposition techniques.

(a) Case 1 Case 2 Case 3 Case 4 Case 5 Case 6 Case 7 Case 8

kxx 154 0.154 6.54 0.154 77.4 0.462 77.08 100.85kxy 0 0 215.83 0 0 0.308 76.92 273.16kyx 0 0 214.31 0 0 0.308 76.92 273.16kyy 0.154 154 41.42 30.8 0.154 0.462 77.08 53.31

(b) ILSGR1 ILSGR2 ILSGR3 ILSGR4m 1 2 3 3r 5 9 15 151np 0 Use equation (16) Use equation (16) 0

Table I.(a) The values usedin the tensor K, (b)acronyms and theirmeanings in termsof the parametersused for differentflux approximationtechniques

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Throughout the tests, c ¼ 2 is used for the weights wk ¼ kvkk2c: The results

are not accurate for c ¼ 0 and c ¼ 1; when the distorted mesh shown inFigure 2(d) is used. However, c ¼ 0; c ¼ 1 and c ¼ 2 provide accurate results onother regular unstructured finite-volume meshes.

At the initial stage of the numerical procedure, the matrix A, the diagonalentries of W, the decomposed form the matrix A TWA in equation (20) and thematrix A TW are stored for each CV face because those matrices depend onlyon geometrical terms and constant parameters. During the processing one canform A TWB and process forward and backward substitution to obtain theleast squares solution vector X at each time step. The vector X provides thefunction value and its derivatives at each CV face for each time step duringthe CV solution process. This strategy is used for every method discussedabove.

Each method produces the gradient 7f·u on the midpoint of each CV faceand enables 1np to be evaluated. Therefore, the flux through the CV faces can becalculated using equation (17) and hence equation (10), with the choice l ¼ 1;produces a system of linear equations when every node point on a mesh isvisited. The resulting system is solved using the BiCGSTAB iterationtechnique (van der Vorst, 1992; Turner and Perre, 1996) for each time level toobtain the final result.

The terms 7f·u and 1np are treated explicitly in the finite volumeformulation, however, for some meshes and very large anisotropy ratios thisapproach may not be sufficient. The authors are presently working on theimplicit treatment of the complete flux decomposition.

The total number of iterations (TI), maximum error (ME), and root meansquare error (RMSE) are tabulated in Table II for each mesh and each methodfor the cases 1 and 2 where the exact solutions are found using the methoddescribed earlier in this paper. The RMSE is calculated using the formula

RMSE ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPMn

k¼1 ðfk;exact 2 fkÞ2PMn

k¼1 ðfk;exactÞ2

vuut ð21Þ

where fk is the value of the variable at node point k, fk,exact is the exact solutionat the same point, and Mn is the number of node points on a mesh.

A comparison of the contour plots for cases 1 and 2 obtained on the distortedmesh, (Figure 2(d)), is shown in Figure 4. Figure 5 exhibits the results forthe cases 4 and 5 on the coarse mesh, (Figure 2(c)). Figures 6 and 7 depictcomparisons of the results for the same mesh for cases 1 and 2, respectively.Figure 8 shows the numerical results for the anisotropic cases 3 and 6, whileFigures 9-11 show the results for the anisotropic cases 7 and 8.

Note that the attraction for using the orthotropic model, in cases 1, 2, 4 and 5,is that exact solutions are available for an accuracy check on the proposed fluxapproximation scheme (see Table II).

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An important observation from Table II, and Figures 4-7 indicates that theimproved least squares gradient reconstruction (ILSGR)3 technique, whichuses the new scheme with a high order (cubic) least squares approximation,produces accurate results for each mesh, including the distorted mesh as shownin Figure 2(d), for all of the orthotropic cases tested. One can see from Table II(RMSE and ME columns) that the different anisotropy ratios can influence theaccuracy of the ILSGR2 technique, whereas the accuracy of the ILSGR3technique remains unaffected.

Results have also been computed for the orthotropic cases where kxx :kyy ¼ 1 : 10; 1 : 100, 10 : 1, 100 : 1 using the ILSGR3 technique and excellentagreement with the exact solutions was found in all the meshes, and thereforethose results were not exhibited here. Note that these case studies are typicalscenarios that arise in wood drying simulation and represent real two-dimensional tests for the numerical scheme, since the strongly orthotropic caseis nearly one-dimensional.

Figure 4 shows a close agreement between the exact solutions and theresults provided by ILSGR3 on the distorted mesh (Figure 2(d)). The true

(a) (b) (c)RMSE ME TI RMSE ME TI RMSE ME TI

Case 1 Kxx : kyy¼1,000:1ILSGR1 0.138 16.19 22,821 0.156 19.13 16,350 0.151 17.65 40,863ILSGR2 0.012 1.60 23,731 0.010 1.30 16,448 0.027 3.34 45,733ILSGR3 0.012 1.72 23,879 0.010 1.36 16,396 0.020 2.56 43,383ILSGR4 0.10 12.83 23,664 0.073 8.56 16,293 * * 43,210Hybrid 0.066 8.19 28,095 0.081 9.62 18,829 0.125 14.68 53,065

Case 2 Kxx : kyy¼1:1,000ILSGR1 0.036 5.707 24,112 0.050 8.46 17,297 0.069 9.04 31,443ILSGR2 0.064 8.30 24,234 0.058 7.69 17,583 0.062 7.92 31,338ILSGR3 0.007 1.22 24,322 0.007 1.46 17,602 0.007 1.26 31,906ILSGR4 0.033 5.71 24,576 0.52 9.90 17,647 0.139 17.21 31,813Hybrid 0.028 3.73 28,492 0.037 6.91 21,043 0.024 4.33 48,616

Case 4 Kxx : kyy¼1:200ILSGR1 0.024 3.52 11,680 0.033 4.62 9,042 0.042 5.35 21,694ILSGR2 0.037 4.65 11,746 0.050 6.46 9,008 0.030 4.00 21,890ILSGR3 0.005 0.75 11,755 0.005 0.63 9,062 0.005 0.86 21,910ILSGR4 0.015 1.68 11,783 0.026 3.97 9,019 0.016 2.25 21,646Hybrid 0.018 2.51 13,973 0.027 3.87 10,472 0.011 2.21 30,229

Case 5 Kxx : kyy¼500:1ILSGR1 0.126 14.96 17,474 0.145 17.66 12,536 0.148 17.52 35,261ILSGR2 0.008 1.10 17,756 0.008 1.03 12,678 0.017 1.84 37,708ILSGR3 0.009 1.26 17,911 0.008 1.11 12,531 0.014 1.99 34,716ILSGR4 0.075 9.62 17,790 0.047 6.00 12,784 * * 36,607Hybrid 0.046 5.58 19,946 0.056 7.08 13,935 0.105 12.82 39,195

Notes: * Numerical scheme diverged.

Table II.Summary of resultson (a) the coarsemesh, (b) the verycoarse mesh and (c)the distorted mesh.RMSE – root meansquare error. ME –maximum error.TI – total numberof BiCGSTABiteration

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symmetry of the solutions are slightly concealed in these figures due to themesh irregularity and limitations of the plotting software used. In fact, thisbehaviour is evident also in the exact solutions (Figure 4(a) and (c)), again dueto the interpolation method used in the plotting software.

Figure 5 exhibits the results obtained for the cases 4 and 5, where the exactsolutions are compared to the results obtained using ILSGR3 and ILSGR4.These cases use lower anisotropy ratios to emphasise the two-dimensionaltransport process. One can see that when the primary correction term isincluded the ILSGR3 scheme is able to capture the exact results, and that

Figure 4.Results on the distorted

mesh (Figure 2(d)) forcases 1 and 2:

(a) kxx : kyy ¼ 1; 000 : 1 –exact, (b) kxx : kyy ¼

1; 000 : 1 – ILSGR3,(c) kxx : kyy ¼ 1 : 1; 000 –

exact, (d) kxx : kyy ¼1 : 1; 000 – ILSGR3

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ILSGR4 fails even though the cross diffusion term is estimated using a cubicapproximation. Further evidence of these facts can be seen in Table II, wherethe ILSGR2-3 schemes generate the lowest RMSE. It also can be seen from thetable that the ILSGR4 scheme did not converge for cases 1 and 5, whereas it didconverge for cases 2 and 4.

It is possible to see from Figure 6 that ILSGR1, ILSGR4 and the hybridtechnique produce inaccurate results for case 1, whereas ILSGR2 and ILSGR3produce results that match well with the exact solution. However, according toFigure 7, while the hybrid technique provides a reasonable result for case 2,ILSGR1, ILSGR2 and ILSGR4 fail to produce accurate results and againILSGR3 produces good agreement with the exact solution. In fact, Figures 6and 7 identify the differences, in terms of the accuracy of the differenttechniques, between the cases 1 and 2, which can be considered as nearly one-dimensional. It can be concluded from these results that the combination of thegeometry of the solution domain and the direction of anisotropy can affect theaccuracy of the numerical solution technique.

The results shown in Figures 6 and 7 highlight that ILSGR1, ILSGR2 andthe hybrid method provide poor accuracy for the strongly orthotropic cases. Itis worthwhile to note that a significant error arises in ILSGR4 when thecorrection 1np, which plays a major role in increasing the order of the primary

Figure 5.Results for the cases (a)kxx : kyy ¼ 1 : 200 and(b) kxx : kyy ¼ 500 : 1 onthe very coarse mesh(Figure 2(c)) for cases 4and 5. Top : exact,middle: ILSGR3, bottom:ILSGR4

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term in the flux approximation, is neglected from ILSGR3. These findings arean important conclusion to be drawn from this research, especially for thedevelopment of accurate wood drying simulators.

Since the ILSGR3 technique provides accurate results on all of theunstructured meshes investigated here for the orthotropic cases, this methodwas used on the fine mesh shown in Figure 2(a) to produce the benchmarkingresults for the anisotropic problems discussed next in cases 3 and 6-8 forassessing the results computed using the very coarse mesh.

Figure 8 provides the results for cases 3 and 6, which are anisotropicproblems. Figure 8(a) and (c) depicts the results using ILSGR3 on a very fineunstructured mesh. Figure 8(b) and (d) again shows that the ILSGR3 method

Figure 6.Comparison of results on

the very coarse mesh(Figure 2(c)) for the case1; kxx : kyy ¼ 1; 000 : 1

(a) exact, (b) ILSGR1,(c) ILSGR2, (d) ILSGR3,

(e) ILSGR4, (f) Hybrid

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used on a very coarse mesh provides good agreement with the benchmarksolutions, due to the accuracy of the gradient approximation used.

Figures 9, 10 and 11 exhibit the results for cases 7 and 8, which are stronglyanisotropic problems. The results for case 7, shown here as contour plots inFigure 9 and as a cross-sectional view in Figure 10, again show that theILSGR2-3 techniques match the benchmark solution well. Clearly, omitting thecorrection term (ILSGR4) produces an inconsistent behaviour in the solutionand the results of the first order methods (ILSGR1 and hybrid method) are farfrom the benchmark solution. Similar conclusions can be drawn from theresults presented for case 8 in Figure 11.

Some final reflections from Table II concerning the computationalperformance of each scheme indicate that the hybrid technique has usedmore solver iterations than ILSGR3 to produce the results and ILSGR3 alwaysproduces more accurate results than the hybrid technique. It can be seen thatall the ILSGR techniques use approximately the same number of solveriterations, however, the hybrid technique always uses more. While the hybridtechnique, which is a fully implicit scheme, fails to produce accurate results,ILSGR3, which is not a fully implicit scheme, always produces excellent resultsdue to its higher order accuracy of the flux approximation.

Figure 7.Results on the verycoarse mesh (Figure 2(c))for case 2; kxx : kyy ¼ 1 :1; 000

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Figure 8.Results using ILSGR3:

(a) and (b) for case 3, (c)and (d) for case 6. Left

column: on the very finemesh. Right column: on

the very coarse mesh

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All the results shown in this section support the conclusion that ILSGR3 isthe best technique amongst the methods considered here for solvingtransport in both anisotropic media and strongly orthotropic media,including transport problems that involve large anisotropy ratios rangingfrom 1 : 1,000 or 1,000 : 1. Note however, that ILSGR2, although not alwaysas accurate as ILSGR3, can be used to solve such problems and given that itrequires less computational overhead than ILSGR3, it may well be thefavoured solution strategy.

Figure 9.Results for case 7 (a)ILSGR3 on the fine mesh,(b) ILSGR1, (c) ILSGR2,(d) ILSGR3, (e) ILSGR4and (f) Hybrid on thevery coarse mesh

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ConclusionsIn this work a highly accurate flux approximation technique appropriate foruse in simulating transport in strongly orthotropic and anisotropic media intwo dimensions using a finite volume method implemented on unstructuredmeshes was presented in detail. The results offered by this new methodhighlight the possibilities of the proposed scheme. The computational cost ofthe scheme is minimal when one considers the excellent comparison with theanalytical results.

In summary, the finite volume technique discussed here has combined a newflux decomposition with the use of a high order least squares approximationtechnique, which estimates both the cross-diffusion term and the correction 1np

for the primary term, to increase the overall accuracy of the flux approximationat the CV face. Such a strategy ensures that the finite volume method achievessecond order spatial accuracy.

It is felt that the method outlined in this research represents an original andimportant contribution to the field of finite volume techniques for solvingtransport problems in highly orthotropic and anisotropic media. The next stageof the research is to analyse its performance when implemented in a dryingsimulator and thereafter, to extend this technique to a three-dimensionalframework.

Figure 10.Results for case 7: (a)

ILSGR3 on the fine mesh,(b) ILSGR2, (c) ILSGR3and (d) ILSGR4 on the

very coarse mesh

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Barth, T.J. (1994), “Aspects of unstructured grids and finite-volume solvers for the Euler andNavier-Stokes equations”, Lecture Notes Presented at the VKI Lecture Series, 05 February1994.

Chow, P., Cross, M. and Pericleous, K. (1996), “A natural extension of the conventional finitevolume method into polygonal unstructured meshes for CFD application”, AppliedMathematical Modelling, Vol. 20, pp. 170-83.

Croft, N. Unstructured mesh - finite volume algorithms for swirling, turbulent, reacting flowsPhD. thesis University of Greenwich, London, UK.

Davidson, L. and Stolcis, L. (1995), “An efficient and stable solution procedure of turbulent flowon general unstructured meshes using transport turbulence models”, AIAA - 33rdAerospace Sciences Meeting and Exhibit, Reno, NV, US, AIAA-95-0342.

Demirdzic, I. and Muzaferija, S. (1995), “Numerical method for coupled fluid flow, heat transferand stress analysis using unstructured moving meshes with cells of arbitrary topology”,Computer Methods in Applied Mechanics and Engineering, Vol. 125, pp. 235-55.

Figure 11.Results for case 8 (a)ILSGR3 on the fine mesh,(b) ILSGR1, (c) ILSGR2,(d) ILSGR3, (e) ILSGR4and (f) hybrid on the verycoarse mesh

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Demirdzic, Horman, I. and Martinovic, D. (2000), “Finite volume analysis of stress anddeformation in hygro-thermo-elastic orthotropic body”, Computer Methods in AppliedMechanics and Engineering, Vol. 190, pp. 1221-32.

Ferguson, W.J. and Turner, I.W. (1996), “A control volume finite element numerical simulation ofthe drying of spruce”, Journal of Computational Physics, Vol. 125, p. 59.

Helf, C. and Kuster, U. (1994), “A finite volume method with arbitrary polygonal control volumesand high order reconstruction for the eulet equations”, CFD Conference, EuropeanCommunity on Computational methods in Applied Sciences - 94, John Wiley and Sons, Ltd.

Hermeline, F. (2000), “A finite volume method for the approximation of diffusion operators ondistorted meshes”, Journal of Computational Physics, Vol. 160, pp. 481-99.

Ilinca, C., Zhang, X.D., Trepanier, J.-Y. and Camarero, R. (2000), “A comparison of three errorestimation techniques for finite-volume solutions of compressible flows”, ComputerMethods in Applied Mechanics and Engineering, Vol. 189, pp. 1277-94.

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Jayantha, P.A. and Turner, I.W. (2001a), “A comparison of gradient approximations for use infinite volume computational models for two-dimensional diffusion equations”, NumericalHeat Transfer, Part B: Fundamentals, Vol. 40 No. 5, pp. 367-90.

Jayantha, P.A. and Turner, I.W. (2001b), “Generalised finite volume strategies for simulatingtransport in strongly orthotropic porous media”, Proceedings of the 10th BiennialComputational Techniques and Applications Conference, University of Queensland,Australian and New Zealand Industrial and Applied Mathematics Journal, (in press).

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Murthy, J.Y and Mathur, S.R (1998), “Computation of anisotropic conduction using unstructuredmeshes”, Journal of Heat Transfer, Vol. 120, pp. 583-91.

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Perre, P. and Turner, I. (1999), “TransPore: a generic heat and mass transfer computationalmodel for understanding and visualising the drying of porous media”, Drying TechnologyJournal, Vol. 17 No. 7, pp. 1273-89, (Invited paper).

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Further reading

Jayantha, P.A. and Turner, I.W. “Generalised finite volume strategies for simulating transport instrongly orthotropic porous media”, Proceedings CTAC2001, Australian and New ZealandIndustrial and Applied Mathematics Journal (in press).

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