~\...:aj~~t&..~~j~~.J:I~\j.\w~4Wu...~\~~~).&..~\.J~'i~J

S.M.F. Hasani* and Sleiman A. Abdallah

Department ofMechanical Engineering The University ofAkron Akron, OR 44325-3903, U.S.A.

: L.::,")l;L.1

~\...:aj~~t&..~~j~~.J:I~\j.\w~4Wu...~\~~~).&..~\.J~'i~J

I. -_~...r-

ui\

_{' "}"~I

'J (0

¹⁵ ~ ~ ^J

. _t\)

~~\ ~

r . b..:,..

:i.•.

.Ilw

~

-\:ill\ .J .

W:i.:i.\ -~ ~ 'J

.tJ:. ".'

..!.,)-YI u..a ~\ ~ o~ loT_

t\

~\'J.J~'i\~t....~~~~~iJ..\~u.-wfo~\'J'~'J~u.-Js

~ w~\

;',bW

~~ ~fI

e:- 40

X

20 u.- wfo

~JA ~ ~~ u..~\

.'i.JI..;.=J\

(a) u\~t,j..,.;..~\ ~\jl\ ~\ u'i~~~~\ ~'J~ ~ o)..;.=J\ 4.J~ill:..

LJj.,,!'J .\~\~6. 'i\i:ij\ a=100~'Ji~6. 'i\i:ij\ a=10~~wj ,~~'i\4~~

~\

u'iW\ u.-

~W\ ~~\ ~4

(pe' =

00) .J~'i\

r.la.1

~\ ~\ ~\

- fI

-

)~'i\) ~

..

bt....~)~'i\Ji\ ~pe'

=

500'Jpe'

= 10

~Ufo\ ~'J .~J.)~\

u~ ~wo~~·Lt6"A'~\ ~~ i~~ ,·\·-'i\~W~\~l.ti( . .J..

~

...,- .J

~ "J W~ • - ~

,--'\

~~ wlJ i.b.:iJ\ ~ 'J ~\'J ~.ll\ .J~ 'i4 u~i \~!J .20 x 10 'J 80 x 40 ~ u.­

.~J.)~\ ~~ u.- o~~.;si ~.,;al\ ~~'i\ ~ u\~ QUICK

*To whom correspondence should be addressed.

October 1997 The Arabian Journal for Science and Engineering, Volume 22, Number 2B. 205

ABSTRACT

A comparative study is done on six convective schemes used to model flows with steep gradients. First-order and second-order upwinding, Fromm's, Johnson & Mackinnon's, QUICK, and QUICK with transverse curvature term are considered. The results of the study are tested by the bench-mark problem first introduced by Smith and Hutton, which contains many of the essential features common to practical convection-diffusion problems. A 40 x 20 staggered grid mesh is used with prescribed stream function values at the boundaries. The temperature at the inlet boundary is given in terms of a hyper~olic

tangent function with a parameter, a, to control the steepness of this function. Values of 10 and 100 for a are used representing sharp and very sharp transition, respectively. The exact analytical solution for the case of no diffusion Pe' =00 is compared with numerical solutions from the six schemes studied. Two other values of Pe'

=

^{10 and Pe'}

=

^{500 are}

investigated in order to demonstrate the effect of diffusion on the two-dimensional steady state convection-diffusion equation. A grid-refinement analysis is done by considering two other mesh sizes, 20 x 10 and 80 x 40. Based on a computational cost, accuracy, and error analysis, QUICK with the transverse curvature term is found to be the most efficient scheme for convection dominant problems.

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The Arabian Journalfor Science and Engineering, Volume 22, Number 2B. October 1997

COMPARISON OF NUMERICAL UPWINDING SCHEMES FOR FLOWS INVOLVING STEEP GRADIENTS

NOMENCLATURE x

y u

v v h At T Pe' U

r

a 1V

V

CXW

CYS FLUXW FLUXS CF CURVNW CURVNS CURVIW CURVTS SGN(X)

Subscripts w s j

b NW SW

SE

x-coordinate y-coordinate

velocity in x-direction velocity in y-direction velocity vector grid spacing time step

convective quantity (temperature) Peelet Number (= Ul/r)

reference velocity reference length diffusion parameter steepness parameter stream function 2-D Laplacian operator West face Courant Number South face Courant Number West face flux

South face flux Curvature Factor

West face normal curvature South face normal curvature West face transverse curvature South face transverse curvature

built-in change of sign FORTRAN function (+1 for X>O and -1 for X<O)

West face South face

Node points in x-direction Node points in y-direction Outlet boundary

North West South West South East INTRODUCTION

The literature on numerical fluid mechanics abounds with finite-difference discretization schemes all intended, in principle, for the solution of transient and steady-state convection/diffusion problems. The range of complexity of these schemes, both in space and time, is considerable. This spatial discretization is achieved by methods as simple as first-order upwind approximation for convection [1] and as complex as higher-order spline-on-spline and Hermitian formulations [2]. It is fair to say that of the multitude of schemes available, only thoS'e utilizing simple low-order spatial discretization for both

October 1997 The Arabian Journalfor Science and Engineering. Volume 22, Number 2B.

207

diffusion and convection have found widespread application in practice [3-5]. Perhaps this is because the overshoot (or undershoot) problems associated with higher-order methods can lead to obviously unphysical results, such as locally negative densities or turbulence kinetic energy [6]. But, the low order-methods are also usually highly unphysical-although not always obviously so. These low order-methods introduce a potentially significant truncation error, termed artificial or false diffusion [7].

Some earlier investigators tested some of these schemes or their variants for a variety of classical test problems [8-12].

From these studies, it has been established that curvature strongly influences the physics of the problem and failure to account for curvature effects will lead to significant errors. In order to identify and correct these effects, it is necessary to first minimize artificial diffusion. One way of doing this in low-order algorithms is by refining the grid, which can be expensive and restricted by computer-time and storage limitations. The alternative is to use more complex and sophisticated discretization methods for the convection terms. In 1982, a test problem that involves near-discontinuities and strong streamline curvature was introduced by Smith and Hutton [13]. Researchers have solved this bench-mark problem using different methods (finite element, finite- difference, coordinate transformation, and approximate techniques). Three finite- difference schemes were used: Hybrid upwind differencing [14]; Patankar's power law [5]; and QUICK [6,15]. Leonard and Mokhtari [16] utilized the Smith-Hutton problem to compare Hybrid and other exponential-based schemes with QUICK. In this study an extensive investigation of the performance of six other upwinding schemes is done. The schemes chosen for this study are frrst-order upwinding [1], second-order upwinding [7], Fromm's scheme [17], Johnson and Mackinnon's scheme [18] hereafter referred to as (J & M) , QUICK, and QUICK with the transverse curvature term [6,15].

THE SMITH-HUTTON TEST PROBLEM

This is a two-dimensional, steady state test problem frrst introduced by Smith and Hutton to investigate the behavior of different numerical schemes under stringent conditions of strongly curved streamlines. The scaler field used in this study is temperature, T, with prescribed velocity field v(x,y) and known constant diffusivity,

r.

The non-dimensional governing equation is:

1 2

v.VT - - V T (1)

Pe'

where Pe' =

ulIr

is the Peclet number. U and l are the respective reference velocity and reference length used to non­

dimensionalize Equation (1).

The flow domain considered is a rectangle: -1 :sx:s 1, O:sy:s 1, and the velocity field is specified as:

2 all'

u - 2y (1- x ) - - (2)

ay

v = -2x(1 (3)

The corresponding stream function is:

(4)

The inlet and outlet temperature boundary conditions are respectively defined as:

Tint (X) = 1+ tanh[a(1 + 2x)] for y 0 and - 1 :s X :s 0 (5)

- = aT 0 _{for y} = 0 and 0 < x < 1. (6)

ay

The exact analytical solution for pure convection case is easily obtained as:

T(x,y) = 1 + tanh

[U(1- ^2~1

⁺ ll'(X,y) ]. (7)

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The Arabian Journal/or Science and Engineering, Volume 22, Number 2B. October 1997

COMPUTATIONAL FORMULATION

The Smith-Hutton problem is discretized using three different mesh sizes. Temperature results presented in this paper are those for a 40x20 uniform mesh (41 x21 staggered grid nodes with ~x

=

^~y

=

^h). Two other mesh sizes (20x 10 and 80x40) are investigated for computational cost, accuracy, and error analysis. The'lP nodes are placed at the boundaries.

A typical staggered grid arrangement with temperature control volume is shown in Figure 1.

o

0

tVi-l,j tVi,j

0 0 0

tVi-l,j-l tVi,j-l

0

Figure 1. Staggered Mesh. 0 Temperature nodes; • Stream function nodes.

o

The average cell convecting velocities can be obtained from:

'*'

^{NW -}

'*'

^sw

=-...;..:.--...;...;:...- '*'

^{i-I, j -}

'*'

^{i-I. j-I} ₍₈₎

U

=

w h h

'Psw - 'PSE _ 'Pi_I,j_1 - 'P_{i •}j _1

V - ---'---";;';;';" (9)

S h h

and the corresponding cell-face Courant numbers are given by:

U w ~t

(CXW)j,j= h ⁽¹⁰⁾

Vs ~t

(CYS);,j = (11)

h

The outlet numerical boundary condition is treated by assuming that T(y) follows a parabola near the outlet for yC!:O,

T(y) = Tb + ay +

bi

⁽¹²⁾

with the conditions

October 1997 The Arabian Journalfor Science and Engineering, Volume 22, Number 2B.

209

(13)

(14)

iJT) _ 0

( ⁽¹⁵⁾

iJy y-o

substitution of Equations (13), (14), and (15) into Equation (12) yields:

(16) The cell-face fluxes are calculated from:

(FLUXW)j,j - (CXW)j,j

*

Tw -

r *

(1i,j -1i-I,j) ⁽¹⁷⁾

(FLUXS) . . - (CYS) . . I., I"

* 1'. - r * (Ti,· -Ti"-l)'

, . ⁽¹⁸⁾

The cell-face temperatures for higher-order schemes can be represented in terms of a curvature factor (CF). The west face temperature, T_{w '} can be written as:

(T . + T 1 .)

T _ ^{I.} 1- , } _} CF

*

CURVNW ₍₁₉₎

w 2

defining the upwind-weighted "normal curvature" at the west face as:

CURVNW = CRVAVW _ SGN(CXW)

*

THIRDW ₍₂₀₎

2

where the average second-difference and third-difference across the west-face respectively are:

CRVAVW

=

^{Ti+I,j - Ti.j - Ti-l,j + Ti-2,j} ₍₂₁₎

2

THIRDW - T;+l.j - 31i. j + 31i_I.j -1i-2.j . ⁽²²⁾

Similarly, for the south face:

(T . + T .

I)

T _ I,} I , } - _ CF

*

CURVNS ₍₂₃₎

s 2

CURVNS = CRVAVS _ SGN(CYS)

*

THIRDS ₍₂₄₎

2

2 (25)

THIRDS - 1i.i+1 - 3T;.j + 31i,i-l -1i,j-2' ⁽²⁶⁾

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The Arabian Journalfor Science and Engineering. Volume 22. Number 2B. October 1997

CRVAVS

=

The curvature factor (CF) takes the values of 112 for second-order upwinding, 114 for Fromm's scheme, 116 for J & M, and 118 for QUICK. If the transverse curvature effects are also considered, then for the QUICK scheme an additional term appears in the formulation. The west face temperature is modified as:

T - (Ti,j + Ti-),j) CURVNW CURV1W

w - 2 + (27)

8 24

where

CURV1W 7;-1, j+l - 27;-1. j + 7;-1. j-I for CXW> 0

(28)

= 7;,j+1 - 27;,j + 7;,j-1 for CXW < O.

Similarly, the south face temperature takes the form:

T - (Ti,j + Ti,j-) CURVNS CURVfS

+ (29)

s - 2 8 24

CURVTS ... 7;+l,j-1 - 27;,j-l + 7;-I,j-1 for CYS> 0 for CYS < O. (30)

The final temperature update equation can be written as:

T·n)~w _I,

=

^T~I~ _I,) ⁺ (FLUXW) .. - (FLUXW). _{I,) 1+ ,]} I . + (FLUXS) . . - (FLUXS)· . _{I.] 1,]+} I (31)

DISCUSSION OF RESULTS 40x20 Uniform Grid

The inlet and outlet temperature profiles for Peclet numbers of infinity, 500 and 10 with a

=

¹⁰ are plotted in Figures 2a to 2c respectively. It can be seen from Figure 2a that for values of very high Pe' (pure convection), all the schemes show overshoots and undershoots except for the first order, which is highly diffusive. Compared with the exact analytical solution, QUICK with the transverse curvature term has a better agreement than the other schemes. For Pe'=500 (representing a case of convection as well as diffusion) as shown in Figure 2b, all schemes essentially have close results with slight overshoots and undershoots except for the first order which is still very diffusive. For Pe'

=

¹⁰ (representing a case of high diffusion) as shown in Figure 2c, all schemes including f1£st order give very similar results which is expected for low advection flows.

Figures 3a-3c are plotted for the same three values of Pe' but for a=100 (steeper function). Same trends can be observed as in Figures 2a through 2c except for the fact that the overshoots and undershoots are larger, thus resulting in a larger error which shows the strong influence that the streamline curvature has on all these upwinding schemes. Figures 4a to 4g are three-dimensional representation of temperature profiles for a=100 and Pe'=oo. From these figures, these overshoots and undershoots can be observed in the whole temperature field rather than just at the inlet and outlet boundaries. The total error, L), for Figures 4a to 4g is computed, based on the following definition:

L) -

~ ~ I~xact

- 7;;omputedl h2. ₍₃₂₎

i j

The L) error results for a values of 10 and 100 are summarized in Table 1. Since the test problem is a steady-state problem, the selection of time-step Atis important only to determine the stability of the scheme and does not have any direct influence on the actual time to reach steady state. The computations for all the schemes under study are performed for a fixed number of iterations (5000 iterations are used since more iterations did not produce any significant improvement in error). To select the proper time step, At is varied from 0.01 to 0.001 where the L) error and CPU time are recorded for each scheme. It is observed that changes in error with varying At are minimal compared to the changes in CPU time. Therefore, for each

October 1997 The Arabian Journal/or Science and Engineering, Volume 22, Number 2B.

211

scheme Ilt that corresponds to the smallest CPU time is chosen. The CPU times for 5000 iterations on an IBM 3090 mainframe computer are summarized in Table 2.

Grid Retinement and Optimal Cost Effectiveness

A grid refinement study is also carried out where two other grid sizes, 20x10 and 80x40, are considered for the case of infinite Pe' and a=100. Error and CPU time for the six schemes under study are plotted versus the number of grid points in Figures 5a and 5b respectively. It can be observed from these figures that although finer grids cause the error to decrease, more CPU time is required to achieve the desired accuracy. To address the question whether it is better to use a low-order scheme on a very fine grid or a higher-order scheme on a coarser grid, the optimal cost effectiveness of all the six schemes is calculated in a manner similar to the one outlined in reference [16], by preassigning the same desired level of accuracy to each scheme. If the schemes under comparison are solved on successively finer and fmer grids and the CPU time (representing cost) for each of them is recorded at successive grid refinements, a global accuracy can be assigned to each scheme for a prescribed cost, i.e. preassign an available computational time to Figure 5b and crossplot it onto Figure 5a, the result is then plotted in Figure 5c. An alternate way is to preassign an acceptable error and compute the CPU time for each scheme as shown in Figure 5d. From Figures 5c and 5d it can be seen that the frrst order upwinding is highly inefficient, because, although the cost per grid point is low, the accuracy within a prescribed computational budget is low and the number of grid points necessary to achieve a desired accuracy makes it very expensive. Among the higher-order schemes, QUICK with the transverse curvature term is found to be the most cost-effective scheme both for a prescribed accuracy or for a prescribed budget.

Table 1. Comparison of the Total Error for 40 x 20 mesh.

Total Error

Scheme Pe' =00, a=10 Pe' =00, a=l00

1 st Ord. Upwinding 0.2026641 0.2733017

2nd Ord. Upwinding 0.0702167 0.1493020

Fromm (CF 114) 0.0510866 0.1293970

J & M (CF= 1/6) 0.0426829 0.1206871

QUICK (CF = 1/8) 0.0380465 0.1171665

QUICK (wITCF = 1124) 0.0304717 0.1047864

Table 2. Comparison of CPU Time for 40x 20 mesh.

CPU Time in Seconds Scheme

1st Ord. Upwinding 2nd Ord. Upwinding Fromm (CF

=

114) J & M (CF= 116) QUICK(CF 118) QUICK (wITCF = 1124)

Pe' =00, a=10 54.39 81.31 80.53 80,48 82.07 99.63

Pe' = 00, a =100 54.23 82.79 82.16 82.50 82.71 99.52

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The Arabian Journal/or Science and Engineering. Volume 22. Number 2B. October 1997

- - EXACT

Pel=oo a=10

---+- Ist-ORDER

2.0 - 6 - 2nd-ORDER

---0--- FROMM

- 6 - J&M

1.5 _{- + -} QUICK

---<>- QUICK W/TCF

T

^1.0

0.5

0.0

1---­

~.5~~~--~~~~~~~~~~~~--~~~--~~~--~~

-1.0 ~.5 0.0 0.5 1.0

x

Figure 2a. Inlet and Outlet Temperature Profiles.

- - - 1st-ORDER

- 6 - 2nd-ORDER

Pe'=SOO a=10

2.0 ---<>- FROMM

- 6 - J&M

- + - QUICK

- - 0 - QUICK WI TCF 1.5

T

^1.0

0.5

0.01---~

~.5~~~~--~~~--~~~~--~~~----~~~--~~~

-1.0 ~.5 0.0 0.5 1.0

x

Figure 2b. Inlet and Outlet Temperature Profiles.

October 1997 The Arabian journal/or Science and Engineering, Volume 22, Number 2B. 213

- + - lst-ORDER

Pe'=10 a=10

-..- 2nd-ORDER

2.0

_{- 0 - FROMM}

- A - J&M - . - QUICK

- 0 - QUICK W/TCF

1.5

T

^1.0

0.5

0.0 ...- - - ­

-0.5

'----'----'---'-_'---...I.-....I---'----'_"'__-'----'----I---II....-""'---l----'---"_"'__....L---J

-1.0 -0.5 0.0

0.5

1.0

x

Figure 2c.lnlet and Outlet Temperature Profiles.

- - EXACT

Pe'=oo a=100

- + - lst-ORDER

2.0

-..- 2nd-ORDER

- 0 - FROMM

- A - J&M - . - QUICK

- 0 - QUICK W/TCF

1.5

T

^1.0

0.5

0.0 ...- - - '

-O.5'----'--~~--"'--~-L~--"'---'--~~--~-'----'--~---'--~~---I

-to

-0.5

0.0 0.5

1.0

x

Figure 3a. Inlet and Outlet Temperature Profiles.

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The Arabian Journal/or Science and Engineering. Volume 22, Number 2B. October 1997

- + - lst-ORDER

Pe'=SOO a=100

--...- 2nd-ORDER

2.0

1.5

T

^1.0

0.5

0.0

I---~

- 0 - FROMM

- 6 - J&M

- + - QUICK

- 0 - QUICK W/TCF

~.5~~--~--~--~--~~--~--~~~--~~~~--~~~

-1.0 -a,5 0.0

0.5

1.0

x

Figure 3b.lnlet and Outlet Temperature Profiles.

Pe'=10 a=100

- + - 1st-ORDER

- . t r - 2nd-ORDER

2.0

1.5

T

^1.0

0.5

0.0 I - - - J

October 1997

- 0 - FROMM

- 6 - J&M -+- QUICK

- 0 - QUICK W/TCF

x

Figure 3c.lnlet and Outlet Temperature Profiles.

The Arabian Journalfor Science and Engineering, Volume 22, Number 2B.

215

0.1

0.5­

^'1.0

'0.6 '0.8 2.0­

1.5­

T

_1.0­

0.5­

0.0 ­

-1.0­

-0.5 ­

+ ^0.0­

1.0 - -

'0.2 '0.4 '0.0

y

Figure 4g. 3-D View of T(x,y) for QUICK WITCF. Pe'.oo, a=100.

Pe -.-- 1st-ORDER

l

.... ^co a=1 00 _{- A -} 2nd-ORDER

- 0 - FROMM

- 6 - J&M

- + - QUICK

--<>- QUICK W/TCF

100 1000 10000

N

Figure 5a. Error versus Grid Refinement.

October 1997 The Arabian journalfor Science and Engineering. Volume 22. Number 2B. 219

Pel=OD a=100

.9

^u

f-4 100

~

U

10L---~--~--~~~~~~----~--~--~~~~~

100 1000 10000

N

Figure 5b. Cost versus Grid Refinement.

- + - 1st-ORDER .--.- 2nd-ORDER

- 0 - FROMM - - J & M - + - QUICK

- 0 - QUICK W/TCF

1 ~---, lst-ORDER

2nd-ORDER FROMM

J&M

QUICK QUICK WI TCF

... g

^0.1

~

0.01 ~---~~~~~~L-~~~~~~L-~~~~~~L-_____~

o

^{1 2 3 4 5} 6 7

Scheme

Figure 5c. Error for a Prescribed Budget.

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The Arabian Journalfor Science and Engineering, Volume 22, Number 2B. Ocrober 1997

1st-ORDER

1000---~---~

FROMM

100 J&M

QUICK

QUICKW/TCF

10~---==¥~--~~~~~~~~~~--~~~~~~~--~

o 1

2 3 4

5 6

7

Scheme

Figure 5d. Cost for a Prescribed Accuracy.

SUMMARY AND CONCLUSION

In this comparative study the six upwinding schemes have been tested for the streamline curvature effect of highly diffusive, convective-diffusive, and highly convective flows. The accuracy of the results depend upon the steepness of the gradient as well as the mesh size used. Steeper gradients cause larger total errors and the accuracy of the results can be improved by mesh refinement. Based on the optimal cost effectiveness of each scheme, it can be concluded:

• For highly diffusive flows ( low Pet ), all schemes behave essentially alike in terms of accuracy, therefore, first-order upwinding has advantage over the other schemes because of its low cost per grid point.

• For convective-diffusive flows ( moderate Pet ), all higher-order schemes show similar results in accuracy while the first-order upwind scheme is lacking in accuracy because of its inherent artificial diffusion. Fromm's, J & M, and QUICK schemes require nearly the same CPU time, therefore, they can be used interchangeably.

• For highly convective flows ( infinite Pet ), the detailed optimal cost analysis performed on these schemes show that QUICK with the transverse curvature term is the most cost effective scheme to solve the convective problems with steep gradients.

Finally, this study provides a comparison of accuracy and cost-effectiveness of some familiar upwinding schemes in their original form as they appear in the literature. Further improvement in accuracy of these higher-order schemes can be achieved by the use of limiters such as those employed by Leonard [19]. These limiters will eliminate the overshoots and undershoots encountered in these higher-order schemes. This is suggested as a topic for future work.

October 1997 The Arabian Journalfor Science and Engineering. Volume 22. Number 2B. 221

REFERENCES

[1] R. Courant, E. Isaacson, and M. Rees, "On the Solution of Non-Linear Hyperbolic Differential Equations by Finite-Differences", Comms. Pure Appl. Math., 5 (1952), p. 243.

[2] S.O. Rubin, P.K. Khosla, and S. Rivera, "Turbulent Boundary Layer Studies Using Polynomial Interpolation", Proc. ofa Symposium on Turbulent Shear Flows, Penn. State Univ., PA, USA, 1977.

[3] O.D. Raithby and K.E. Torrance, "Upstream-Weighted Differencing Schemes and their Applications to Elliptic Problems Involving Fluid Flow", Computers and Fluids, 2 (1974), p. 191.

[4] E.O. Macagno and T.K. Hung, "Computational and Experimental Study of a Captive Annular Eddy", 1. of Fluid Mechanics, 28 (1967), p. 43.

[5] S. V. Patankar, Numerical Heat Transfer and Fluid Flow. New York: Hemisphere, 1980.

[6] B.P. Leonard, "Elliptic Systems: Finite Difference Method IV", in Handbook of Numerical Heat Transfer. ed. W.J. Minkowycz eOllNew York: Wiley, 1988, pp. 347-378.

[7] C.A.l Fletcher, Computational Techniquesfor Fluid Dynamics. Vol. I, 2nd ed. Springer-Verlag, 1991.

[8] A. AI-Rabeh, "On the Computational Efficiency of Certain Upwinding Schemes", Computer Meth. in Appl. Mech. Eng., 109 (1993), pp. 131-141.

[9] M. Sharif and A. Busnaina, "Assessment of Finite Difference Approximation for the Advective Terms in the Simulation of Practical Flow Problems", 1. ofComputational Physics, 74 (1988), pp. 143-176.

[10] W. Shyy, "A Study of Finite Difference Approximation to Steady State Convection Dominated Flow Problems", 1. ofComputational Physics, 57 (1985), pp. 415-438.

[11] P.O. Huang, B.E. Lauder, and M.A. Leschziner, "Discretization of Non-Linear Convection Process: A Broad Range Comparison of Four Schemes", Computer Meth. in Appl. Mech. Eng., 48 (1985), pp. 1-24.

[12] M.A. Leschziner, "Practical Evaluation of Three Finite Difference Schemes for the Computation of Steady State Recirculating Flows", Computer Meth. in Appl. Mech. Eng., 23 (1980), pp. 293-312.

(13] R.M. Smith and A.O. Hutton, "The Numerical Treatment of Advection: A Performance Comparison of Current Methods", Num.

Heat Transfer, 5 (1982), pp. 439-461.

[14] D.B. Spalding, "A Novel Finite Difference Formulation for Differential Expressions Involving both First and Second Derivatives", Int. 1. ofNum. Meth. Eng., 4 (1972), pp. 551-559.

[15] B.P. Leonard, "A Stable and Accurate Convective Modelling Procedure Based on Quadratic Upstream Interpolation", Computer Meth. in Appl. Mech. Eng., 19 (1979), pp. 59-98.

[16] B.P. Leonard and S. Mokhtari, "Ultra-Sharp Solution of the Smith-Hutton Problem", Int. 1. of Num. Heat Transfer & Fluid Flow, 2 (1992), pp. 407-427.

[17] lE. Fromm, "A Method for Reducing Dispersion in Convective Difference Schemes", 1. ofComp. Physics. 3 (1968), p. 176.

[18] R. W. Johnson and R.J. Mackinnon, "Equivalent Versions of the QUICK Scheme for Finite Difference and Finite Volume Numerical Methods", Comm. Appl. Num. Methods, 8 (1992), pp. 841-847.

[19] B. P. Leonard, "The ULTIMATE Conservative Difference Scheme Applied to Unsteady One Dimensional Advection", Compo Meth.

in Appl. Mech. Eng .. 88 (1991), pp. 17-74.

Paper Received 23 August 1995; Revised 22 September 1996; Accepted 23 October 1996.

222 The Arabian Journal/or Science and Engineering, Volume 22, Number 2B. October 1997