Elsden_Thesis_199500035

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					        1.2zlO*l
                                          E&T FhddPointData
                     -    -------         Convection




        8.0XIO,



        6.OA104



        4. OX104




        2.OxIO'4



         o. 0110,                                                                                        J


                    0.0               0.1              0.2         03                    OA              0.5
                                                                              t (secs)




Figure 3.11 : Comparisonof Vector Model - With Convection, with E&T
Fluid Point




  1.2x104
              I                E&T Fluid Point Data
                               Vector Model With Convection
  I.OX10*1 ................... Integralof Autocoffelation


 8.OX104




 6.OxIO"




 4.OxIO4




 2.OxIO'4




  O. OxIop

             0.0                    0.1                0.2              0.3                   0.4                 0.5
                             twýJ           toý,
                                                   )                                                Time (secs)



 Figure3.12: Comparison Integralof AutocorrelationFunction with E&T
                        of
 Fluid Point, Showing CorrespondingCorrelationTimes.

                                                              82
                                                                  CHAPTER 4
                                                        Single Phase Calculation


4.1 Introduction


       In order to modelthe behaviourof a dilutely dispersedsecondphase,in a turbulent
fluid, it is necessary be able to specify the behaviourof the carrier phase.Experimental
                     to
investigation provide the someof the required data but tends to be expensive.Also the
             can
memurement  of some.quantitiesof interestcan be extremelydifficult. 'niough experiments
remain of great importancein the investigationof flow phenomenamuch use is made of
analytical and numerical methods.
        Equationsto representfluid flow havebeenknown since the turn of the century, the

well known to Navier-Stokesequations,but due to their highly coupled, non-linear nature

         solutionsare
analytical                         in                        Though numerical methods
                      only possible the most trivial of cases.
have becomee
           widely availablewith the adventof affordableand powerful computersonly
recentlyhasit beenpossibleto solve theseequationsat sufficiently small scalesto simulate
turbulence,and this using the most powerful computersand considering only simple, low
Reynoldsnumberflows.
       Becauseof the problemsencounteredwhen attempting to solve the equationsfor
fluid flow exactly, various approximation techniqueshave been developed. Foremost

amongsttheseis that proposedby OsbornReynolds.
     This chapter provides a basic overview of the derivation of the equationsof fluid
flow together with their transformation into mean equations following the method of
Reynolds. 7le turbulence closure problem is then discussedtogether with the required
boundaryconditionsfor the solutionof the resultantequations.A simple numerical solution

scheme is then presented based heavily on the work of Patankar (1980) Finally this

numerical algorithm is applied to a simplepipe flow and comparedto the data of Laufer
(1954).
                        in
       7le work presented this chapteris by no means comprehensive
                                                   a             discussi6nof the


                                          83
                     fluid dynamicsand as such containslittle or no referenceto many of
field of computational
the currentdiscretisation solution methodsfound in the literature. No mention is made
                         and
of direct numerical simulation (DNS) or large eddy simulation (LES) or even higher order
                 One of the main goals in the use of the solution procedurediscussed
quadratureschemes.
below is that it can be run, in an acceptableamount of time, on a desktop personal

computer. Another major aim was to develop a solution algorithm which was directly

applicable to the solution of the secondphase conservation equations presentedin the
following chapter.The straightforward pressure-correctioncode chosenfulfils both of the
above goals as well as being well testedand acceptedwithin the literature.


4.2 Conservation Equations


       To be able to describe the flow situations required it is necessaryto develop

conservation equations for the important variables. Derived here are the conservation
equations for mass, often called the continuity equation, and momentum. Using these

equationsasa baseit is possible developa general
                               to               form of differential equationapplicable

to mostvariablesof interest.Ibis generalform has the addedadvantage enablingdirect
                                                                      of
application of many of the results developed in this chapter to the calculation of the
Euleriandescription the second
                   of              This second
                              phase.          phasecalculationis discussed detail
                                                                         in
in the following chapter.
       Firstly the massconservationequationwill be developed.7bis is then extendedto

the required momentumequationsand thoseof Navier and Stokes.


4.2.1 Mass Conservation


       Consider the fluid volume, V with surfacearea S, given in figure 4.1. Also let p

representthe mass unit volumeof the flukl. The massflux through a small areadSj with
                   per
a normal  in the xj coordinate direction is hencegiven as pudSj. 7lie total masstransfer

through the surfaceof the volume is then given by



                                           84

				
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