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Numerical Heat Transfer, Part B: Fundamentals: AnInternational Journal of Computation and MethodologyPublication details, including instructions for authors and subscription information:http://www.tandfonline.com/loi/unhb20

EVALUATION OF A HIGHER-ORDER BOUNDEDCONVECTION SCHEME: THREE-DIMENSIONAL NUMERICALEXPERIMENTSSeok Ki Choi a , Ho Yun Nam a & Mann Cho aa Liquid Metal Reactor Coolant Department, Korea Atomic Energy Research Institute, P.O. Box105, Yousung, Daejon, 305-600, Korea

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To cite this article: Seok Ki Choi, Ho Yun Nam & Mann Cho (1995): EVALUATION OF A HIGHER-ORDER BOUNDED CONVECTIONSCHEME: THREE-DIMENSIONAL NUMERICAL EXPERIMENTS, Numerical Heat Transfer, Part B: Fundamentals: An InternationalJournal of Computation and Methodology, 28:1, 23-38

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EVALUATION OF A HIGHER-ORDER BOUNDEDCONVECTION SCHEME: THREE-DIMENSIONALNUMERICAL EXPERIMENTS

Seok Ki Choi, Ho Yun Nam, and Mann ChoLiquid Metal ReactorCoolant Department, Korea Atomic EnergyResearchInstitute, P.O. Box 105, Yousung, Daejon, 305-600, Korea

This article presents a detailed three-dimensional evaluation oj a higher-order boundedconvection scheme. A high-resolution and bounded discretization scheme named COPLA,which is composed oj piecewise linear functions in the normalized variable diogram, isproposed. The scheme is applied to the three-dimensional test problems 10 assess itscapabilities. The results oj numerical experiments show that the tested bounded schemeresolves the boundedness problem while retaining the accuracy oja higher-order scheme.

INTRODUCTION

Most flows of engineering interest occur in domains of three-dimensionalgeometries. Numerical solutions of such flows with available computer resourcesdemand an efficient and accurate algorithm for the solution of a complete set ofNavier-Stokes equations in the three-dimensional form. Development of an effi­cient convection scheme that would simultaneously possess accuracy, stability,boundedness, and algorithm simplicity may be one of the major tasks to meet suchdemands.

Several different treatments of convection terms have been proposed in thepast, but there exists a conflicting issue of accuracy and boundedness among theschemes. The classical upwind scheme, the hybrid central/upwind (HYBRID)scheme, and the power-law scheme [I], are unconditionally bounded, and highlystable but highly diffusive when the flow direction is skewed relative to the gridlines. On the other hand, the higher-order schemes such as the QUICK (quadraticupstream interpolation for convective kinematics) scheme [2], the second-orderupwind scheme [3], and the skew-upwind scheme [4] have been successful inincreasing the accuracy of the solution, but all suffer from the boundednessproblem; that is, the solutions display unphysical undershoots and overshoots inregions of steep gradient, which can lead to numerical instability.

A variety of procedures have been proposed to resolve the boundednessproblem, such as the flux-corrected transport method of Zalesak [5] and the localoscillation damping algorithm of Zhu and Leschziner [6]. In 1988 Leonard [7]

Received 18 May 1994; accepted 27 February 1995.Address correspondence to Dr. Seok K. Choi, Liquid Metal Reactor Coolant Department, Korea

Atomic Energy Research Institute, P.O. Box 105, Yousung, Daejon, 305-606, Korea

Numerical Heat Transfer, Part B, 28:23-38, 1995Copyright © 1995 Taylor & Francis

1040-7790/95 $10.00 + .00 23

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24 S. K. CHOI ET AL.

NOM ENCLATUR E

A coefficient in the discretization

'"transport variable

equation J, normalized transport variableb! geometric coefficients

JC" C2 , C3 geometric interpolation factor Superscripts

defined in Eq. (IS)D! geometric coefficients defined in n-l previousiteration levelJ

Eq. (3) + pertaining to positive flowF total flux or functional direction

relationship pertainingto negative flowJ Jacobian of the inverse coord i- direction

nate transformation

s~ source term of variable '" Subscriptss· nonorthogonal source term of~

variable '"u; Cartesian velocity components e, w, n, east, west, north, south, top, and

U; curvilinear contravariant velocity S, t, b bottom faces of a control volumecomponents E,W,N,S, east, west, north, south, top, and

Xi transformed coordinate system T,B bottom neighbors of the gridy' Cartesian coordinate system point PAV volume P pertaining to the grid point Pr diffusivity WW pertaining to the grid point WWp density

'"pertaining to the transport

+, + indicators of local flow direction variable '"(TW' Uw

developed a normalized variable formulation and proposed a high-resolution,bounded scheme named SHARP (simple high-accuracy resolution program).Based on the normalized variable analysis, Gaskell and Lau [8] formulated aconvection boundedness criterion for implicit steady-state flow calculation anddeveloped a higher-order bounded scheme named SMART (sharp and monotonicaIgorithm for realistic transport). However, the numerical experiments conductedby Zhu [9] have shown that both the SMART and SHARP schemes need anunderrelaxation treatment at each of the control-volume cell faces in order toovercome the oscillatory convergence behavior. This deficiency leads to an increaseof the computer storage requirement, which may pose a practical constraint totheir use in complex three-dimensional turbulent flow calculations.

In the present study, the SMART scheme is slightly modified to overcome theoscillatory convergence behavior without deteriorating the accuracy of the originalscheme. The proposed scheme is formulated on a nonorthogonal, nonuniform gridso that it can be applicable to the simulation of practical engineering problemsemploying a general coordinate system. The scheme is tested through applicationsto two well-documented three-dimensional flow problems to assess its capabilitiesfor the simulation of complex three-dimensional flows. The computed results arecompared with previous calculations and available experimental measurements.For a better comparison with existing popular schemes, the computed results bythe second-order bounded SOUCUP (second-order-upwind-central-differencing­first-order upwind) scheme [10] is included together with those by the HYBRIDand QUICK schemes. It is noted that the SOUCUP scheme considered in the

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THREE-DIMENSIONAL NUMERICAL EXPERIMENTS 2S

present study is based on the normalized variable formulation and the convectionboundedness criterion, and is essentially the same as the MINMOD (minimummodulus) scheme of Roe [11], widely used in compressible flow calculations.

MATHEMATICAL FORMULATION

Governing Equations

The conservative form of transport equation for a general dependent variablecp in a generalized coordinate system xi can be written as [12],

a ( [</>. acjJ)-. pUcjJ - -Dl - = JSaxl 1 J m ax m </>

where the contravariant velocity components ~ and the geometric coefficients D~

are defined as

(2)

D~ = bib;:'

and the geometric coefficients bi represent the cofactor of ayj / ss! in theJacobian matrix of the coordinate transformation y' = y'(x/) and J is the determi­nant of the Jacobian matrix. In these equations, p is the density of the fluid, [</> isthe diffusion coefficient of variable cjJ, U j are the Cartesian velocity components inthe i directions, and S</> denotes the source term of variable cp.

Discretization of Transport Equations

The computational domain is divided into hexahedral control volumes, andthe discretization of transport equation is performed in the physical solutiondomain following the finite-volume approach. The general transport equation, Eq.(1), is integrated over a control volume as shown in Figure 1. The resultingequation can be written as

(4)

where F represents the total flux of cjJ across the cell face and S~ is the sum of thenonorthogonal diffusion terms. The total flux at the west face, for example, can bewritten as follows, with the diffusion term approximated by the central differencingscheme:

(5)

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26 S. K. cum ET AL.

••I

Figure I. A typical control volume.

\

The evaluation of tPw plays a key role in determining the accuracy and the stabilityof numerical solutions. For example, tPw is evaluated as follows when one uses thefirst-order upwind scheme:

(6)

where u:); and u:;; are the indicators of the local velocity direction such that

u:;; = 1 - u:);

Incorporation of Eq. (6) and Eqs. (4)-(5) and similar expressions for the other cellfaces leads to the following general difference equation:

The details of implementation of the higher-order bounded schemes will beoutlined in the following section.

HIGHER-ORDER BOUNDED SCHEMES

The current higher-order bounded schemes are based on the variable normal­ization by Leonard [7] and the convection boundedness criterion by Gaskell andLau [8]. Consider, without loss of generality, the west face of the control volume.We introduce a normalized variable such that

A tP - tPutP=--­

tPD - tPu(9)

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THREE·DIMENSIONAL NUMERICAL EXPERIMENTS 27

where the subscripts U and D denote the upstream and the downstream locations.Equation (9) can be rewritten in terms of nodal point values:

J. ~ - ~ww + ~ - ~E _'+' = a: + (Tw

~p - ~ww w ~w - ~E(10)

Using the above upwind biased normalized variable, four schemes can be written asfollows.Central difference scheme:

First-order upwind scheme:

(12)

Second-order upwind scheme:

(13)

QUICK scheme:

where

(14)

sx;C1 = - - - - - ­

s x ; + tiXww

sx;C2 = - - - - ­

tiXw + tiXp

are the geometric interpolation factors defined in terms of the size of the controlvolume cell. For example, tiXp is the size of the control volume around thecalculation point P and is defined as (see Figure 1)

(16)

The normalized diagrams of these well-known schemes for a uniform grid situation(UW > 0) are shown in Figure 2.

Gaskell and Lau [8] formulated the following convection bounded ness crite­rion. Define a continuous increasing function or union of piecewise continuousincreasing function F relating the modeled normalized face value ~w to the

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28

$,

S. K. CHOI ET AL.

Lcaend(l) «ntr.l .1lff.'Un~11IC

Figure 2. Normalized variable diagramfor various well-known schemes.

normalized upstream nodal value ~w (Uw > 0), that is, ~w = F(~w). Then a finitedifference approximation to ~w is bounded if:

1. For 0 ;;; ~w ;;; 1, F is bounded below by the function ~w = ~w and aboveby unity and passes through the points CO,O) and (1,1);

2. For ~w < 0, ~w > 1, F is equal to ~w.

The convection boundedness criterion is a necessary and sufficient condition forachieving computed boundedness if only three neighboring nodal values are usedto approximate the face values. The diagrammatic representation of the convectionboundedness criterion is shown in Figure 3.

According to Leonard [7), for any (in general nonlinear) characteristics in thenormalized variable diagram (Figure 2):

1. Passing through Q is necessary and sufficient for second-order accuracy;2. Passing through Q with a slope of 0.75 (for a uniform grid) is necessary

and sufficient for third-order accuracy.

----,l~---.L.----~w

Figure 3. Diagrammatic representationof the convection boundedness criterion.

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THREE·DIMENSIONAL NUMERICAL EXPERIMENTS 29

The horizontal and vertical coordinates of point Q in the normalized variablediagram and the slope of the characteristics at the point Q for preserving thethird-order accuracy for a nonuniform grid can be obtained by simple algebra usingEqs. (11)-(14):

(17)

For a uniform grid, XQ = 0.5, YQ = 0.75, and SQ = 0.75.Following the above criteria by Gaskell and Lau [8] and by Leonard [7], one

may choose several bounded characteristics in the normalized variable diagramwhose order of accuracy is determined by the shape of the characteristics. Forexample, the SOUcUP scheme [10] employs union of piecewise linear characteris­tics passing through the points, 0, Q, and P in the normalized variable diagram asshown in Figure 4. The SOUCUP scheme is a composite of the second-orderupwind, central differencing, and first-order upwind schemes. We see that theSOUcUP scheme is second-order-accurate according to the criterion of Leonard[7].

The COPLA Scheme

The order of accuracy of the scheme may be increased to the third order ifone introduces a characteristic curve in the normalized variable diagram whoseslope at the intersection point Q is the same as that of the third-order-accurate

Figure 4. Normalized variable diagramfor saucup and CaPLA boundedschemes.

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30 S. K. CHOI ET AI..

QUICK scheme. Such a scheme is proposed in the present study, employing thefollowing characteristics in the normalized variable diagram:

o~ Jc ~ O.5XQ

O.5XQ ~ 4>c ~ 1.5XQ

1.5XQ s 4>c ~ 1

(18)

where

and

otherwise

aw = 0

b = 2YQ - SQXQw X

Q

Cw = YQ - SQXQ

a; = SQ

3XQ - 2YQ - SQXQe = --=------==------'=----=-w 3X

Q- 2

2YQ + SQXQ - 2f w = -=----=----=---

3XQ - 2

(19)

(20)

The present CaPLA (combination of piecewise linear approximation) schemeemploys a composite of piecewise linear characteristics in which the QUICKscheme is employed in a range of O.5XQ ~ Jc ~ 1.5XQ. The scheme is similar tothe SMART scheme [8], but is free of convergence oscillation. The normalizedvariable diagram of the CaPLA scheme is shown in Figure 4 together with that ofthe saucup scheme.

The implementation of the CaPLA scheme is quite simple. A part of Eq. (18)can be expressed in terms of the unnormalized variable:

~w = {~w + (~p - ~ww)[a: + (b: -0(:: ~::: )]}u:+ {~p + (~w - ~E)[a: + (b; - 0(:: ~ ::) ]}u; (21)

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THREE·DlMENSIONAL NUMERICAL EXPERIMENTS

Given the switch factors,

31

For o; > 0,

For u; <0, a~ = 1

a~ = 0

if I<pp - 2<pw + <Pwwl < I<pp - <Pwwl

otherwise

otherwise

(22)

(23)

the unnormalized form of Eq. (21) can be rewritten as

where

(24)

After the evaluation of the additional term (il<pw), the implementation of thisscheme is the same as that of the first-order upwind scheme. In the present studythe additional terms are treated in a deferred correction way proposed by Khoslaand Rubin [13]. The present CaPLA scheme looks a little complicated, however,the computer code implementation of this scheme requires an addition of only afew lines to the subroutine based on the SOUCUP scheme. This additional effort isnot SO grave when one considers the high accuracy achieved by the present scheme.

APPLICATIONS TO TEST PROBLEMS

The higher-order bounded schemes described in the previous section areimplemented in a general-purpose computer code designed to solve fluid flow andheat transfer in complex geometries. The computer code uses a nonorthogonal,nonstaggered grid arrangement and the SIMPLE [1] algorithm for pressure-veloc­ity coupling. The momentum interpolation practice by Rhie and Chow [14] isemployed for calculating the cell-face mass fluxes to avoid the pressure oscillation.

The test problems include: (1) pure convection of a box-shaped scalar step bya uniform velocity field; (2) laminar flow in a lid-driven cubic cavity; (3) laminarflow in a square duct of 900 bend. The computed results are compared with theanalytic solution, the available experimental data, and other computed resultsreported in the literature.

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32 S. K. cum ET AI..

Pure Convection of a Scalar Variable

Before performing detailed three-dimensional calculations, the results of alinear problem involving purely convective transport of scalar tracers containingdiscontinuities by the prescribed velocity field are presented to check the bounded­ness of each scheme. We consider a box-shaped profile as shown in Figure 5, whichis generated by imposing a step profile along the bottom and left-hand walls of thesquare solution domain. Calculations are performed with two different meshes,22 X 22 and 42 X 42. The predicted profiles along the vertical centerline (x = 0.5,o ;;; y ;;; 1) are shown in Figure 6. We can observe that the solutions by theHYBRID scheme are very diffusive: Even the grids are increased by a factor of 2.The QUICK scheme results in severe overshoots when the grid is coarse (22 X 22),but shows relatively low undershoots. We also observe that the overshoots andundershoots are not diminished with grid refinement. Both accuracy and bounded­ness are achieved by the bounded schemes. The bounded schemes resolve fairlywell the steep gradient on either side of the peaked profile, without introducingsupurious overshoots and undershoots. The SOUCUP scheme is more diffusivethan the COPLA scheme, but is much better than the HYBRID scheme.

Calculations are also performed for the same configuration, but with adifferent inlet profile of semielliptical shape. Figure 7 shows the predicted profilesalong the vertical centerline for this case. We can see that the QUICK scheme bestresolves this narrow extreme of a semielliptical shape, especially when the numeri­cal grid is relatively coarse (22 X 22). However, slight overshoots are also observedwhen the grid is refined (42 x 42). The difference between the QUICK solutionand the COPLA solution is a little pronounced when the grid is coarse (22 X 22).However, the accuracy of the COPLA solution is substantially increased with aslight grid refinement (42 X 42). The COPLA scheme resolves fairly well thesmooth narrow extreme, strictly preserving the boundedness of the solution. Ingeneral, the performance of each scheme is not much affected by the differentinlet profiles.

Figure S. Pure convection of a box­shaped scalar step by a uniform velocityfield.

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THREE·DIMENSIONAL NUMERICAL EXPERIMENTS 33

++I.(;OENI)

- £:KACT

30.30 HVBRID

$OOOO~ + QUICK

0 0 0 soucuP

• COPI. ...

• 0°0 eo 0o 0

0ij

0$ 0

.&2~0 0

0

·~o:,o

'.MY

(0)

Figure 6. q, profiles along the centerline(step profile): (a) 22 X 22 grid; (b) 42 X42 grid.(b)

'.MY

I..E:G,LND

.++"0++ - eXACT

.,,~o* 0 ,",YBRln

0 0 + QUICK

0°000000 SOUCuP

o 00 • COP'LA

~

J ~O:0

0 00

ID 0

_no':D~ .~~........

Laminar Flow in a Lid-Driven Cubic Cavity

Laminar flow in a lid-driven cubic cavity, shown schematically in Figure 8, issolved as an example of three-dimensional recirculating-type flows. Calculationsare performed for Reynolds number 1,000 using 43 X 43 X 43 uniform grids. Thecomputed results are compared with the solution of Ku et al. [15], which wasobtained by the pseudo-spectral method using 31 X 31 X 16 grids.

The computed U-velocity profiles along the centerline are given in Figure 9.We can observe that the QUICK scheme results in the most accurate solution. Thesolution by the CaPLA scheme is comparable to that of the QUICK scheme and isslightly more accurate than that of the saucup scheme. The HYBRID solution isdiffusive, questioning that this scheme can be confidently applicable to the complexthree-dimensional flow calculations.

The predicted secondary motions at the X-Z center planes are given inFigure 10. All the schemes show nearly the same features, except that theHYBRID scheme exhibits a slightly different vortex formation. These figures alsoshow that the HYBRID scheme underpredicts the strength of various vortices.

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34 S. K. CHOI ET AL.

l..£C;£ND

- ClC"'CT

• IolY8A'ID66

QUICC+

0°°0 0 SOUCUp

~6 cOPI. ...

9poooo 90 • < 0

e i g•••.ag~ \ .~~~.

..~y

(a)

I..CQCND

Cll"'CT

o H"aRID

+ QUIClC

o £QUCUP

d COPL""

e•~y

(bl

Figure 7. q, profiles along the centerline(sernielliptical profile): (a) 22 X 22 grid;(b) 42 X 42 grid.

II

.)-,/

,/,...- Figure 8. Laminar flow in a lid-drivencubic cavity.

Figure 9. Centerline u-velocity profile.

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THREE·DIMENSIONAL NUMERICAL EXPERIMENTS 35

(,,)

(e)

(b)

(,[)

Figure 10. Predicted secondary flow inX-Z centerplane: (a) HYBRID; (b)QUICK; (e) SOUCUP; (d) COPLA.

Laminar Flow in a Square Duct of 90° Bend

Laminar flow in a square duct of 90° bend, shown schematically in Figure 11,is considered in the present study as a typical three-dimensional flow involving astrong secondary motion caused by the centrifugal force and pressure gradient.This particular problem was studied experimentally by Humphrey et al. [16). TheReynolds number based on the hydraulic diameter and the bulk velocity is 790.Only a symmetric half of the solution domain is solved, employing 47 X 32 X 17nonuniform grids.

Figure 12 shows the predicted streamwise velocity profiles at the 90° and 60°planes, together with the measured data. We note that the HYBRID schemeconsistently results in diffusive solutions. In particular, it underpredicts the peakedvelocity profiles. The QUICK and COPLA schemes result in nearly the same

._"mmT~ __b • 40m~ "'-__::.1.----

Figure 11. Laminar flow in a square duct of 90' bend.

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36 S. K. cum ET AI..

.. ·.1I1D

__ soveuP

.... '.41 '.M ....("-"l)~(.. o-rl)

«(/l

. ..(,.-,., >/Cro-rl)

(b)

OUICK

:,.:., ..-~~

o ~.""",,,,,,,.

.... .... .... ....( ..-rl)~(..g-rl)

(c)

. SOUCUP

co~...

"" ..<,.-rot )/("'0-"')

(dl

Figure 12. Predicted streamwise velocity profiles: (0) 60' plane,z/b = 0.5; (b) 60' plane, z/b = 0.25; (c) 90' plane, z/b = 0.5; (d)90' plane, z/b = 0.25.

solutions, whereas the SOUCUP scheme produces a slightly more diffusive solu­tion than the QUICK or COPLA scheme.

Figure 13 shows the vector plots of the predicted secondary flow at the 90°plane. The QUICK, COPLA, and SOUCUP schemes result in nearly identicalsecondary flow features. The HYBRID scheme produces a slightly different result,especially the secondary motion near the symmetry line and the strength and thelocation of the primary vortex.

In order to investigate the effect of grid refinement on the solution, numeri­cal experiments have been performed employing three different numerical grids(37 X 22 X 12,47 X 32 x 17,62 X 42 x 22). Figure 14 shows their results at the90° symmetry plane. The HYBRID solution is not grid-independent and changesgradually as the grid is refined. The solutions by the higher-order schemes nearlyreach the grid-independent solution. We can see that the higher-order schemespredict the peak velocity profiles well. It is of interest to see that the coarsestQUICK solution is better than the finest HYBRID solution.

CONCLUSION

A high-resolution bounded convection scheme that employs a combination ofpiecewise linear characteristics in the normalized variable diagram is proposed and

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THREE·DIMENSIONAL NUMERICAL EXPERIMENTS 37

(a) (b)

(e) td)

Figure 13. Predicted secondary flow in the 90° plane: (a) HYBRID;(b) QUICK; (c) SOUCUP; (d) COPrA.

tested through applications to one linear pure-convection problem and two three­dimensional-flow problems. The results of numerical experiments show that theproposed scheme resolves the boundedness problem while retaining the accuracyof the higher-order scheme. The scheme is simple to implement, stable, and free ofconvergence oscillation. All these desired features make the present scheme a good

.... .... ....(,.-r' ''''(''0-''')

.... .... ....<...-,.., )'("0-"')

6Z·"Z.Zo! 62.<12.22.7.:lIZ_\ ,

37.22.12 37.22.\2

0 t._,., ...."" 0 E_c.."''''.''''

(al (b)

.... ..... ....(r-'" )/<"0-'" )

.... .... '.M(,.-,., ) .. ( r-0·... ' >

(e) (d)

Figure 14. Effect of grid refinement in the 90° symmetry plane: (a)HYBRID; (b) QUICK; (c) SOU CUP; (d) COPrA.

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38 S. K. eHOI ET AL.

alternative to many existing schemes for the calculation of complex three-dimen­sional-flow problems.

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Quadratic Interpolation, Camp. Meth. Appl. Mech. Eng., vol. 19, pp. 59-98, 1979.3. W. Shyy, A Study of Finite Difference Approximations to Steady State Convection

Dominated Flows, J. Camp. Phys., vol. 57, pp. 415-438, 1985.4. G. D. Raithby, Skew Upstream Differencing Schemes for Problem Involving Fluid Flow,

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A New Boundedness Preserving Transport Algorithm, Int. J. Numer. Meth. Fluids, vol.8, pp. 617-641, 1988.

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