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					                                          Xi

Inclusion of this term leadsto the classicalclosureproblem. 'Mis problem is discussedin

the next section


4.4 Turbulent Closure


       The presence the Reynoldsstressterms in (4.13) imply that the set of equations
                   of
defined by (4.12) and (4.13) contains more unknowns than equations. To calculate the

required Reynolds stressterms it would be necessaryto averagefor the next order in

equation(4.13).This would in itself introduce a triple correlation term leading to a similar

problem. Therefore a solution is not possiblewithout further approximation. In order to
addressthis problem various forms of closurehypothesishave beenproposed.
       Two types of model exist. Firstly those which use a so-called turbulent viscosity

concept to approximatethe additional stressterms of (4.13) and secondly those models
which develop equationsto describethe transport of the stresses
                                                              themselves.
       Thesetwo typesof model are briefly discussedbelow and a preferenceis declared
for a model basedon the idea of an eddy viscosity. A comprehensivereview of closure

models can be found in Nallasamy(1987).



4.4.1 Turbulent Viscosity Model


       This form of modellingtreatsthe Reynoldsstresstenns found in (4.13) asadditional

viscous stresses the fonn
               of

                          Reynolds Stress - Iir    aU, au1)
                                                   a Xi .
                                                          axi

where                  viscositydue to turbulence.Substituting for the Reynoldi"stresses
      PTis an additional
in (4.13) we get

                                            91
                  aui   a (uduý          ( (P ) auf)
                 p- +pi                      - g,.                               (4.14)
                   at  ax      . -ap .aaxj
                                  axj              a           x,
Again assunfmg gradient of turbulent viscosity tenn is zero. This results in an equation
             the
                           form as the original momentumequation (4.8) with the simple
which is of exactlythe same,
addition of the extra viscosity term.
       17he
          variationsbetween modelsof this type arise from the method usedto obtain
                           the
the turbulent Yiscosity.
                  form of this type of modelarises
       M-c simplest                               from nixing lengtharguments,Prandd

                   analysisrevealsthat PTdivided by the density, p hasthe dimensions
(1926).A dimensional

of length multiplied by velocity. This implies that PT is a function of the density, a
             lengthscaleandthe local flow velocity.Usingthe expressionfor the turbulent
representative
shearstressas a model it can be postulated

                                          12
                                               a u,
                                                  .au,
                                                       )
                                               axj axi

where q is a modelconstantand 1the representativelength scale.Both of thesequantities

needed to be specified.This is accomplishedby consideration of experimentaldata for
simple flows. This type of model is called a zero-equation model since no transport

equationsneedto be solved to obtain the value Of PT*
       Ilie next typeof modelis the one-equation
                                               Model.In this form of model an equation
for the turbulent Idnetic energy,k, is derived basedon the generalisedform (4.9), Abbott

and Basco (1989).
                                           )
            ak
                  U,
                       ak    a(    Pf akpr( au, aui) IU,                         (4.15)
            at         ax         aXj
                            aXi Pok   , pax. , ax, ax( -E
                        i
This transportequationfor k is then solved, the value of e being specifiedempirically.The
turbulent viscosity is obtainedfrom

                                        pr - PC
                                                   il
                                                 k -I
                                                11



                                               92
Again this type of modelrequiresthe specification of the mixing length, I togetherwith the

various constantsfound in (4.15)
       The final modelof this type introducesa secondtransportequationwhosedependent

variableis a functionof the representativelength scale.The most commonly usedequation
is that of the turbulent dissipation,e, which has the form, again from Abbott and Basco
(1989).


       ae U,aE                                     !
                                              Ipy -U,,  auj) nuf 2c !ý
          + a -. a            Po.          'IC.
                                              kp(a
                                                                -c  k                (4.16)
       at    X, X,                  aX,              xi
                                                        ax, axj
                     in
Again the coefficients both equation(4.15) and (4.16) needto be specified.The turbulent

viscosity is given by

                                             CP



                   two-equationmodel,first derivedby LaunderandSpalding (1974), and
71iisis the standard
is usedextensively throughout the literature, and is usedin this work.
        The main benefit of these types of models is their simple form and ease of

computation.Conversely,the requirementto specify either a turbulent length scale,or the

modelconstants the two-equationmodel, implies the dependence fitting experimental
              in                                                of
data and a possiblelack of universality.
          Oneof the maindrawbacks thesemodelsis that the use of an eddy viscosity leads
                                of
to the assumption of isotropic turbulence, as a result of the scalar variables used in its
definition. Consequentlythis type of model is generally inappropriate to situations where the

flow is strongly anisotropic, though extensions to anisotropy can be found in the literature

through the specification of a multi-dimensional turbulent viscosity. Only the isotropic form

of equations (4.15) and (4.16) is considered in this work.



4.4.2 Stress Models


          17he
             secondmaintype of turbulentmodelis concernedwith modelling the Reynolds

                                             93

				
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