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)
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
(4.14) this approach is not possible except in the simplest, trivial cases.This lack of
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
direct implementation the SIMPLER algorithm outlined in the book of Patankar(1980)
"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.
Only two-dimensionalsteady situations are consideredin this work, though the
. lementation valid for both three-dimensional unsteadyflows. Consequently
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
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
correction. Both of thesespecialproblems are discussedbelow.
Equation Diff-usionCoefficient, r,, SourceTerms, S,,
Continuity 0 0
Momentum PT ap spu
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
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.
One of the simplest usedfor derivingthe required algebraic
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
the sequenceafter the quadratic term
( )2 t)
4)2-, & x dý 1. (, &
- dx 12
We get after adding and subtractingthe two aboveequations
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.