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					4.6 Solution Procedure


       To be able to make use of the description of a single phaseflow field describedby

the abovediscussionit is necessary prescribea method of solution for equations(4.12)
                                  to

and (4.14) together with the correct boundary and initial conditions. Though analytical

solutionwould bepreferable, to the highly coupled and non-linear natureof (4.12) and
                           due
(4.14) this approach is not possible except in the simplest, trivial cases.This lack of
                                      the
analytical solution method necessitates use of numerical methods.
       The numerical solution method chosen for this work draws heavily on that of
Patankar Spalding (1970), (1972a), (1972b), (1974a), (1974b), (1978) and is in fact a
       and
direct implementation the SIMPLER algorithm outlined in the book of Patankar(1980)
                    of
"NumericalHeat andMassTransfer".This is a semi-implicit pressurecorrection codebased

on a staggerdgrid implementation.Though the method developedfor the solution of the
requiredconservationequationsis relatively simple in derivation it hasbeenchosenfor its
robustnessof fon-nand easeof application to the Eulerian conservationequationused to
        the secondphase,seethe following chapters.
represent
       Only two-dimensionalsteady situations are consideredin this work, though the
.   lementation     valid for both three-dimensional unsteadyflows. Consequently
              remains                              and
only brief mention is made of the treatment of time dependentcases.'I'lie equations
considered in the following sections are presented in Cartesian form, in one or two
          thoughthe algorithm also remainsvalid for all curvi-linear coordinatesystems
dimensions,

with only the needfor geometricalconsiderations.
       The methodis designed solve the generalised
                            to                    differential equation(4.9) with the

specificationof the diffusion coefficient and sourceterm dependingon the variableunder
consideration.As briefly mentioned                                         in
                                   previously,all of the equationsdiscussed this chapter
can be castinto this generalisedform. Ile correct specificationof the diffusion coefficient

andsourceterm for the continuity and momentumequations,togetherwith thoserequired
for the k-e turbulent closuremodel can be found in table 4.1. Specialconsiderationneeds
to be givento the solution of the momentumequationsdue to the presenceof the pressure
gradient term. This leads both to the need for a staggeredgrid and the use of pressure

                                           100
correction. Both of thesespecialproblems are discussedbelow.



            Equation              Diff-usionCoefficient, r,,           SourceTerms, S,,


           Continuity                            0                                0




           Momentum                          PT                               ap spu
                                                                              axi .



       Kinetic Energy                         Ily                 lir ! U1.aUj aU,
                                             P ak                  p   clxj axi axj



                                                                        (             )
                                                 pr                          U,           ýU
                                                                                               ,   C2. C2
           Dissipation                                            kp                                   k
                                             p                              clxj CIX,     axi

Table 4.1 : Diffusion Coefficients and SourceTerms


       A more detailed description of the model can be found in the book of Patankar
(1980).


4.6.1 Numerical Solution

       The numerical solution of these continuous differential equationsrequires their

reduction into a set of linear algebraic equationswhose solution at a finite number of

calculation points can be taken to representthe continuum solution for the flow. This
introduces the idea of a numerical solution grid consisting of these specific solution
locations. Another set of assumptionsneedsto be madeto representthe behaviourof the
variablesbetweenthe grid points.
          Variousnrthods exist to obtain thesealgebraicequationsand the profiles between
the grid points. Someof the most common methodsare briefly discussedbelow.

                                            101
4.6.2 Discretisation


                         discretisation
       One of the simplest                     usedfor derivingthe required algebraic
                                      procedures
equationsis that of a truncatedTaylor seriesexpansion.This allows approximation of the
derivatives found in the differential equations.

                                                 case,seefigure 4.3, and truncating
       Considering straightforward one-dimensional
                  a
the sequenceafter the quadratic term
                                                            )2                     (ý&ý
                                                                                          2
                                       AX                                   X)2
                                                                  2
                                                                                                   2--...




and
                                              (             )2                                t)
                                  4)2-, & x        dý             1. (, &
                        4ý3                                                 X)2
                              -                    dx            12
                                                                                     dc
                                                                                                   2




We get after adding and subtractingthe two aboveequations
                                                                      ý3-    101
                                                  2"C

                                                             2         2Ax


and

                                                                  4yiO3-2 102
                                                                       (, &X)2
                                            dL_
                                            _xt
                                                        2




                                                          into equation (4.9) leadsto the
Substitution of theseequations,for the differential ten-ns,

required finite-difference equation.
       Though this method is straightforward it containsan intrinsic assumptionthat the

profile of the dependentvariable has the form of a polynomial. This can lead to problems

when consideringvariableswith an exponentialdistribution.

                                                                 102

				
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