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Solution of steady-state,convective transportequation using an upwindfinite element schemeP. S. HuyakornDepartment of Civil Engineering, Princeton University, Princeton, NJ 08540, USA(Received IS August 1976)

A new finite element scheme is presented for overcoming theproblem of numerical oscillation associated with the steady stateconvectiondiffusion equation when its convective terms areimportant. This scheme differs from the standard Galerkin scheme inthat discretization is performed using a general weighted residualprocedure and weighting functions of asymmetric forms which aredependent on the direction of flow velocity along each side of afinite element. These functions are referred to as upwind weightingfunctions and are developed for linear quadrilateral and triangularelements. Numerical results have been obtained for three testexamples and compared with the results obtained using theconventional Galerkin procedure. It is shown that in all cases theproposed upwind scheme produces solutions which are oscillation-free and considerably more accurate than the Galerkin solutionsprovided the recommended procedure for minimizing false or artificialnumerical dispersion is used.

IntroductionNumerical models are being used extensively tosimulate various transport phenomena governed by thegeneral convection-diffusion equation. Several studieshave shown that the numerical solution of thisequation presents serious difficulties when convectiveterms are important. These difficulties concern theproblem of oscillation in the computed solution, andstem from the combination of the diffusive andconvective terms.With the finite difference method, numericalinstability of this type has long been recognized. It hasbeen shown to occur in central differenceapproximation when the mesh size is such that thelocal Peclet number exceeds a certain critical value.The numerical difficulties have been overcome by usingbackward (upwind) differences to approximate theconvective terms. Recent investigations have shownthat this modification leads to a considerable reductionin accuracy of the solution obtained. As a consequence,various formulae which combine central and backwarddifferences have been proposed by several workers,2,3.The same problem of numerical instability ariseswith the finite element method. As the standard

Galerkin formulation invariably leads to a set ofalgebraic equations similar to that obtained usingcentral differences. Although the problem has beenrecognized for some time, until very recently it was nottreated. Indeed, this has been one of the most seriousshortcomings which has limited the advantages of thefinite element method as compared to the finitedifference approach in fluid mechanics applications.

In the context of a simple situation involvingsteady-state, one-dimensional transport, Christie, et aL4showed that the numerical difficulties could beovercome by formulating the finite element method viaa general weighted .residual procedure and usingweighting functions of non-symmetric forms. Theydeveloped such functions for linear iso-parametric lineelements and showed that stability of the numericalsolution could be established at any value of Pecletnumber.This paper aims to generalize the approach outlinedby Christie et al4 to deal with the two-dimensionalsituation. Weighting functions are given for linear,triangular and quadrilateral elements. It is shown thatby an appropriate selection of values of their free

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Steady-state, convective transport equation using an upwin d finite element scheme: P. S. Huyakornparameters, referred to as damping factor, accurateand stable solutions can be obtained for moderate andhigh values of Peclet number using relativelycoarse meshes. The performance of triangular andquadrilateral elements is also assessed.

Typical case of severe numerical oscillation

0.75

tW 0.50

0.25

To illustrate the essential nature of the numericalinstability associated with the standard Galerkin finiteelement scheme, a typical problem of thermal transportin entry flow through a rectangular duct is considered.The problem is governed by the Navier-Stokesequations and the heat transport equation. which areuncoupled. Details of the standard velocity-pressure-temperature formulation have been presented byseveral workers, and will not be repeated here. Aschematic sketch which depicts the boundaryconditions under consideration is given in FigLive f.

Using the mesh shown in Figure 2h and a mixedinterpolation scheme which consisted of quadraticshape functions for velocities and temperature and

u= Iv=oT=0

Figure 1L6.

Thermal transport in entry flow through a rectangularduct. U. Y, components of velocities In the x, and y directionsrespectively; P, pressure; T, temperature. Dimensionlessparameters; Reynolds No (Re = Zudlv where u = entry velocity,d = half duct width, v = kinematic viscosity of fluld); PrandtlNO. (Pr = V/K). Re = 75; Pa = Re x Pr; cT= u= 0 (x direction);$=g= 0 (y direction).

dy ?y

a 0.50-

\ 025.

o- ---- - -0 I 2 3 4 5 6

x

b o-so-\ 0.25

O-0 I 2 3 4 5 6I 1 I I

xFigure 2 (a), Mesh for analysis of entry flow and thermaltransport; (b), mesh for analysis of thermal transport

Figure 3 Temperature profiles obtained using standard Galerkinscheme. Mesh 1 (--O_); Mesh 2 (-- x --_); Pe = 75 _.

linear shape functions for pressure, complete numericalresults were obtained for a case in which Re = Pe= 75. To serve as a comparison, a solution of the heattransport equation was performed separately using asecond mesh which consisted of linear rectangularelements, (Figure 2~). Computed values of thetemperature at the line of symmetry (4 = 0) are plottedin Figure 3. It can be seen that severe numericaloscillation occurs throughout the entire length of theduct and that no substantial improvement was gainedby using higher-order elements. It is particularlyinteresting that temperature values at the corner nodesof the parabolic elements in mesh 1 appear to be ingood agreement with corresponding values at theconsecutive nodes of the linear elements in mesh 2.Although one can obtain a smooth temperature profileusing these values, it will be shown below that thisdoes not provide a satisfactory approximation to thetrue answer.

Thus. if the standard Galerkin finite element schemeis used, one has no alternative but to refine the mesh,in order to avoid this type of numerical problem. Sinceoscillation occurs in a global manner, the entire flowregion has to be rediscretized so that each nodalspacing in the .u-direction remains small below acertain critical value. The heat transport equation hasto be resolved at an increased expense which dependsdirectly on the Peclet number.Treatment by upwind finite element schemeGeneml weighted residual , f i)rmulcrt iorzTo overcome the problem of numerical oscillation, theheat transport equation has to be approximated by useof a general weighted residual procedure in which.unlike the Galerkin procedure, the weighting functionsused can be different from the basis functions.

For two-dimensional, steady, incompressible flow.the heat transport equation can be written as:(1)

Let trial solutions be expressed in the form:? = N,( .Y, ) 7; (I = 1,2,...n) (2)

where N,(s, y) are the basis functions which are chosento be the standard shape functions; T, are nodalparameters. and summations over repeated subscriptsare employed.

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Steady-state, convective transport equation using an upwin d finite element scheme: P. S. HuyakornTo simplify the mathematical notation, the

following differential operator is introduced:,,,,+$+$) GE-v% (3)

Let W,(x,y) be a set of piecewise weighting functions Substitution of equations (9) and (10) into (7a) and (7b)with Z = 1.2,. . n. One requires: leads to:

i W&(?)dR = 0 (I = 1,2....n) (4)R (11)where R is the flow region.After substituting for L(T) and using Greenstheorem to avoid second-order derivatives, equation (4)becomes

W,(x) = N,(x) + 3,Xf - 3,;X2W,(x) = N2(x) - 3q + 3~; (12)

where the shape functions N,(x) and N,(x) are givenby_

dN_l dN,uFax + WF T,dR+ (5)

Bwhere B is the flow boundary; and n , and n, arecomponents of the outward unit normal vector.

The major task which remains is to find appropriateexpressions for the weighting functions. This will becarried out for both one-and two-dimensional linearelements.Weighting jimctions,fbr one-dimensional elementsFor a one-dimensional transport problem, wheretemperature values are prescribed at the two extremepoints of the flow domain, equation (5) reduces to:

Kz d$dR + sdN,uW,--- x dR1 TJ = 0 (6)Re

Lwhere R is an element subregion and xdenotessummation over all elements.Consider a linear line element shown in Figure 4. Itis assumed that the flow velocity is in the positivedirection from node 1 to 2. Let the weighting functionsfor nodes 1 and 2 be expressed in the form:

W,(x) = N,(x) - F(x) O 0 and vice versa). Thus, for any node Z in thenetwork, greater weighting must always be given to theupstream element which contributes to that node, ifoscillation is to be dampened. In view of this, theweighting functions so determined are termed upwindweighting functions. To further demonstrate the roleof 3 in dampening the numerical oscillation, equations(1 lHl4) are substituted into equation (6) to result inthe following algebraic equation for node Z in thenetwork:

+ ,+&+I) T,_,=o1 (16)

Figure 4 Weighting functions for one-dimensional lmearelements. ( -- -- )W( ?=l ). (- ~-_) N,

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Steady-state, convective transpo rt equation using an upwind finite element schem e: P. S. Huyako rnEquation (16) can also be written in finite differenceform by introducing the following notation:

6T, =(T,+1 - T,_,)1212 (17a)VT, = (T - T,_,)/h (17b)6=T = (T,,, - 2T, + T,_,)/h2 (17c)

where 6 and, V denote central and backward differenceoperatqrs respectively. Substitution of (17a), (17b) and(17~) into (16) leads to:

I-;6 /h-lcl 0

Figure 5 One-dimensional concentration transport problem.Pe = uh/Dabsolute error at x = 5 versus CI. It can be seen that foreach value of Pe, there is an optimum value of a wherecomplete accuracy can be obtained. For higher valuesof Pe, the error curves have identical slopes.

?(I - 1)6T, - $aVT, + h26=T, = 0 (18)The one-dimensional heat transport equation is nowwritten as:

udT =-_- +dTzoK dx dx= (19)It can be seen that equation (18) is simply a finitedifference approximation of equation (19) with acentral difference approximation to d2T/dx2 and alinear combination of central and backward differencesto dT/dx. These approximations are given by:

ST=~ (20a)(1 - (x)6T, + CrvT = g GOb)

Clearly, the proposed numerical scheme, which leads to(16), should become more stable as r increases. Itshould be unconditionally stable for r 2 1. This factcan be substantiated by writing the theoretical solutionof equation (16) in the form:

T,=A+B 1 + (U + l)uh/(2K) 1 + (x - l)uh/(2K) 1 WC)where A and B are coefficients which depend on theboundary conditions. From equation (21), it can bededuced that the numerical solution is free ofoscillation if;

X21uh

(2Od)or (1 - CY)~ < 2 when M < 1 (2Oe)since x directly controls oscillation, it will be termed adamping factor.Optimum value I$ a ,ftir one-dimensional elementsTo establish numerical stability with minimum loss ofaccuracy, the value of u must be carefully chosen. Theeffect of 2 on accuracy of numerical solution obtainedis illustrated for a simple case of steady, one-dimensional concentration transport as shown inFigure 5. It is assumed that there is a uniform flowvelocity in the positive x-direction and that thedispersion coefficient is constant. The exact solution ofthis problem is given by:

c = [exp(ux/D) - l]/[exp(lu/D) - I] (21)Numerical results were obtained for selected values ofthe local Peclet number (Pe = uh/D) and comparedwith the exact solution. Figure 6 shows a plot of

The expression for optimum cxwas obtainedtheoretically by Christie et d4 It can be written in theform

kpr = c&hwhere u is the uniform flow velocity; h is the meshspacing, and D is the dispersion coefficient.

(22)

Weighting ,jirnctions jbr quadrilateral elementsConsider a quadrilateral element shown in Figure 7.Let (t, q) represent a local iso-parametric coordinatesystem. Since the element belongs to the Lagrangianfamily, its weighting function can be obtained bytaking appropriate products of functions in eachcoordinate. In forming such products, the fact that thedamping factor s1 can vary from one element side toanother is also taken into account. Thus, in Figure 7,thevaluescc=a1,cc=a2,a=B1 andn=fi,areassigned to sides 12, 43, 23 and 14 respectively. Thesense of direction of flow velocity which corresponds topositive CI s also indicated for each side.By way of an example, the weighting functions fornodes 1 and 3 are obtained as:

w,((, I) = G2(53 a2W2(rl, /31)

where the expressions for G1. G2, H, and H, can be

(23)(24)

derived directly from the one-dimensional weightingfunctions in equations (11) and (12). Thus. G1 is

0 0.2 0.4 06 0.8 I.2Damping factor,a

Figure 6 Effect of dampmg factor r on accuracy of numericalsolution. (----), Oscillation; (-- ) stable solution

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Steady-state, con vectiv e transpo rt equation usin g an upw ind finite element scheme: P. S. Huyakor nF,(L, = 0) = 0 (29b)

3

t 8: =i PI

III =I 2t

(Fi(Lj = O)l 3ajLk Li ( i # j # 4 (294The first condition ensures that the sum of theweighting functions is unity. The second conditionensures the value of the weighting function for aparticular node is zero along the side opposite to thatnode. The third condition ensures that if quadrilateraland triangular elements are used to form the mesh,there is compatibility along each common side of bothtypes of elements.

Figure 7 Weighting functions for linear isoparametricquadrilateral elements

obtained by writing equation (11) in terms of 5 asfollows:

(25)x/h = (5 + 1)/2 (26)

Substituting equation (26) into (25), one obtains:G,(LaJ = ;[(I + 5)(3~15 - 3~1 - 4 + 41 (27

In a similar manner, Gz, HI and H2 may be obtainedas:

H&,/L) = ;[(I + 1)(3Pzr - 382 - 2) + 41 (27b)

Gdtr~(d = $1 + 5)(- 3@25+ 3% + 2) (27~)

ff,(rl,Pd = $1 + 9)(-3P1r + 381 + 2) P7d)Final expressions of all weighting functions and theirderivatives are given in Appendix 1. The function W, issketched in Figure 7 for two sets of values a2 and fi2,namely (a2, p2) = (1,O) and (a2, b2) = (1,1) respectively.

Weighti ng functi ons and el ement mat ri ces for tr iangul arelementsConsider a typical triangular element (Figure 8). Let(L,, L2, L3) represent a local area coordinate system,and let a1,a2, and czs be values of the damping factorassociated with the element sides opposite to nodes 1,2, and 3 respectively. Using the approach describedabove for one-dimensional elements, the weightingfunctions for linear triangles are written in the form:

~ = Ni + Fi = Li + Fi (i = 1,2,3) (28)where Fi denote the parabolic functions which arechosen to satisfy the following conditions:

It can be shown that the following expressions forFi satisfy equations (29a) to (29~)

F, = 3(Ct2L3L1 M IL DL Y) (304F2 = 3(QL 1L2 - ctIL JL2) (3Ob)F3 = 3(CQL zL3 c(~L~ L~ ) (304

Final expressions of W,, W,, W, and their derivativesare given in Appendix 2. For (a1 = a2 = aj = l), WIand W, are sketched in Figure 8. Since all weightingfunctions are expressed in terms of area coordinates,the element matrices can be generated in a simplemanner by performing exact integration using thefollowing formula:

LTL;L$ dA = 2A(m!n!p!)(m+n+p+2)!A

(31)

where A is the area of the triangleFor steady-state, two-dimensional transport

problems, the complete element matrix may be written as :[HI = LB1+ [PI

where

(Z,J = 1,2,3)p =13

(Z,J = 1,2,3)

(324

(324

(32~)

I 2

Figure 8 Weighting functions for linear triangular elementsa, coordinates: 1, (l,O, 0); 2, (0, 1,O); 3, (O,O, 1).b, W,(a, = a2 = c(~= 1); c, W3(ct, = a2 = c(~= 1)

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Steady-state, convective transport equation using an upwind finite element scheme: P. S. Huyakorn

Figure 9 Subdtvision of quadrilateral subdomain into triangularelements. Pattern 1, (right-hand side); Pattern 2, (left hand side)

isI .3+ and PI /J9 are influence coefficients whichare given byir = (1 - X3 + C1JA; ?,, = -u3A; ,I3 = a2A24 = c(~A; /I5 = (1 -a, + u3)A; /I6 = --alA2, = -ccZA; 2, = cc1A; & = (1 + !_X* a&I

;

The remaining coefficients h, and cJ are given inAppendix 2.Sign and optimum t:nlue cf x , fbr uadrilateral undtriangular elementsFor a general two-dimensional flow fluid, the velocityvector may vary in magnitude and direction frompoint to point. Thus, it is necessary to adopt a suitablecriterion for determining the sign of the dampingfactor for each side of a two-dimensional element. Thecriterion used may be described as follows:Let QJ denote the direction vector of a particularside IJ, and let C; and V, denote velocity vectors atnodes I and J respectively. The component of anaverage velocity along this side is given by taking thescalar product:

r =$I +v,) .f,, (34)The sign of c( is determined in accordance with:

x > 0 if r > 0 (35a)cc

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Steady-state, convective transport equation using an upwind finite element scheme: P. S. Huyakorn

b

Yli xI -Q =3OOOm /dayc =I C

7=Oc =. 2000m1

Figure 7 1 (a) Pollutant transport caused by an injection well;(b) Mesh 1; (c), Mesh 2; (d) Concentration distribution along xaxis (mesh 1, standard Galerkin vs. upwind scheme with z = 1);(e) concentration distribution along x axis showing effect of r onaccuracy of upwind scheme; (f) concentration distribution alongmain diagonal showing effect of CIon accuracy of upwindscheme. (-) Central scheme; (---) Upwind (c( = 1); A,Pe = 100; B, Pe = 40; C, Pe = 10: (e) (-) Central scheme(Mesh 2); 0, a=l.O (Mesh 1); x; ~~0.5 (Mesh 1)

In addition to the above criterion, the followingformula is proposed for calculating an optimumabsolute value of c( for side ZJ.

opt = coth [Ih _2k2k vhwhere h is the length of side IJ.It will be demonstrated that the use of equation (36)overcomes the problem of accuracy loss inherent in theupwinding process. Indeed, accurate solutions can beobtained at high values of Peclet number usingrelatively coarse meshes.Results and discussionThe two-dimensional weighting functions and theproposed numerical scheme have been incorporatedinto a general computer program. Results wereobtained for two test examples using iso-parametricquadrilateral elements and quadrilateral subdomainswhich were subdivided into triangular elements. Twopatterns of subdivision were used and these are givenas shown in Figure 9. In pattern 1, each quadrilateralsubdomain consists of four superimposed triangleswhile in pattern 2, it consists of four triangles having acommon internal node. The results are compared withcorresponding results obtained from the standardGalerkin procedure:

0.25 -

Oe I 14 x

0.25 -

I.00bo-75 -u

050 -0.25 -

800 dlfiExample I: In this example, the problem of thermaltransport specified in Figure 1 is reconsidered.Solutions are presented for two cases in which Pe = 75and 25. For each case, values of c( calculated fromequation (36) were used. Typical plots of calculatedtemperature values at y = 0 and y = 0.42 are shown inFigures IOa and lob. Temperature profiles at variouscross-sections (x = 0.27, 1.81 and 6) are also plotted(Figure 10~). It can be seen that using the proposedupwinding scheme, numerical oscillation wascompletely eliminated. Excellent agreement betweenthe sets of results obtained by use of rectangularelements and various combinations of triangularelements may also be observed. To serve as acomparison, corresponding results given by thestandard Galerkin procedure, referred to as centralscheme, are illustrated in Figures 1Od and 1Oe. Ofparticular interest is the fact that oscillation tends tobe slightly more severe with the rectangular elementsthan with the triangular elements. It should also benoted that at Pe = 75, the smooth temperature profileswhich can be obtained by joining consecutive nodes ofrectangular elements are substantially different fromthe corresponding profiles given by the upwind scheme.To check the accuracy of the standard Galerkinscheme versus the upwind scheme, separate sets ofresults were obtained using another mesh much morerefined than that in Figure 2b such that oscillation did

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Steady-state, convective transport equation using an upwin d finite element scheme: P. S. Huyakornnot occur when the Galerkin scheme was employed. Itwas found that these results compared much moresatisfactorily with the results of the upwind scheme,(Figure IOa and ZOb).Example 2: In this example, a common practicalproblem of pollutant transport in groundwater flow isconsidered. Figure Ila shows a plan view of aquadrant of a confined aquifer with an injection welllocated at the origin of the coordinate axes. Theparameters depicted are described as follows:Coefficient of hydraulic conductivity in the x and ydirections K, and K, lOm/day; thickness of aquifer,d = 40 m ; well injection rate, Q = 3000 m3/day ;Dispersion coefficient of aquifer = D m/day; Pecletnumber, Pe = Q/(&f). The governing equations for thisproblem are given by;

where h is the hydraulic head; c is concentration ofpollutant; Qi is the nodal flux at point i whichcorresponds to a point sink or source, 6 is the diracdelta function; u and u are seepage velocities which aregiven by:u_-&!!

xax

U=-KhyayBoundary conditions considered are given in Figure1 la, where zero normal hydraulic head andconcentration gradients are implied along the x and yaxes. The problem was studied on two meshes (Figurel l b and c). The second mesh produced no oscillationwith the standard Galerkin scheme (central scheme)and therefore, served as a calibration mesh. It was usedto check the accuracy of the upwind scheme applied tothe first mesh.(K.&z KyQ?$i Q&x - Xi>_V - YJ (37)i=l

-u$ck=OaY (38)

I.00 a0.75 -

c,0.50 -

Figure 12 Concentration distributions obtained using calculatedoptimal a. (a), along x axis; (b). along main diagonal; (c) along xaxis. (- ) Central scheme (Mesh 2); 0 Upwind scheme aopl(Mesh 1). X, Triangles (Pattern 2) 0, Triangles (Pattern I);

Pe=lOO;A,y=O;B,y=x

(394

To study the effect of the damping factor M:onaccuracy of the numerical solution, results wereobtained for three cases in which Pe = 10, 40 and 100using preset values of CIand optimum M calculatedfrom equation (36) Figures I c, Il d and Ile showconcentration profiles obtained using iso-parametricrectangular elements with preset values of CI= 1 and c(= 0.5 in the radial flow direction. In Figure Ilc, it isevident that at lower values of Peclet number wherethe central scheme is oscillation free, there is stillmarked difference between its numerical results and theresults given by the upwind scheme with CI= 1. Acomparison with the more accurate solutions obtainedfrom mesh 2 shows that the use of c1= 1 leads toconsiderable errors caused by numerical dispersioninherent in the upwind scheme. As a is reduced from 1to 0.5, a marked improvement of performance isobtained. Further improvement in the accuracy of thenumerical solution can be achieved by use of optimumcalculated values of a as illustrated in Figur e 12a and12b. Excellent agreement between the two sets ofresults given by meshes 1 and 2 may be observed.

To assess the performance of triangular elements,results obtained using the two triangular combinationsand optimum c1 are also given as shown in Figure 12cfor Pe = 100. It can be seen that pattern 2 givesslightly better agreement with the more accurateresults from mesh 2 than pattern 1.

ConclusionsA generalized upwind finite element scheme has beendeveloped for steady-state, convective transportequation. Using this scheme, the problem of numericaloscillation commonly encountered in the standardGalerkin finite element and central difference solutionsof this equation has been overcome.The proposed formulation is such that the loss ofaccuracy due to false numerical dispersion inherent inthe upwinding process can be minimized by varyingthe damping factor from element side to element side.Indeed, accurate and stable solutions have beenobtained for moderate and high values of Pecletnumber by use of relatively coarse meshes, whichconsist of iso-parametric linear quadrilateral or lineartriangular elements.

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Steady-state, convective transport equation using an upw ind fi nit e element scheme: P. S. Huy akornIt should be emphasized that although the proposed

finite element scheme leads to a three-point combinedcentral/backward difference formula in one dimension,it bears no relationship to such a combined five-pointdifference scheme in two dimensions.Although the present paper concerns the steadystate solution of the equation, extension of theprocedure to transient situation has been achieved.This will be dealt with in a future paper.References

910

1112.

Spalding, D. B. Int. .I. Num. Math. Engrg. 1972 4, 551Runchal, A. K. Int . J. M ath. Engrg. 1972 4, 541Barrett, K. E. J. M ech. and Appl. M ath. 1974 27, (I), 57Christie, I., Griffiths, A. R., Mitchell, A. R. and Zienkiewicz, 0.C. Int . J. Num. Meth. Engrg. In pressTaylor, C. and Hood, P. Comp. Fluids 1973, 1, 73Hood, P. PhD D issertat ion Univ. Wales, Swansea, (1974)Barakat, H. Z. and Clark, J. A. Proc. 3rd I nt. Heat TransferConJ: (1966) 2, 152Gosman. A. D., Punn, W. M., Runchall, A. K., Spalding, D. B.and Wolfstein, M. Heat and mass transfers in recirculatingflows, Academic Press, London, 1969Huyakorn, P. and Taylor, C. Int. Cot$ Water Resources,Princeton University, 1976Jacobs, D. A. H. In Numerical methods in fluid mechanics,Brebbia, C. A. and Connors, J. J. (Eds.), Pentech Press,London, 1973. p 433Piva, R. and DiCarlo, A. 2nd Conf: Mathematics of finiteelements and applications, Brunei University, 1975Roache, P. J. Computational fluid dynamics, HermosaPublishers, Albuquerque. (1972)

Appendix 1Using the notation shown in Figure 7, the weightingfunctions and derivatives for quadrilateral elements aregiven by:

aw,__ = $[(l + yl)(3BZV - 3BZ - 2) + 41(3a,5 - 1)a