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
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
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
Reynolds Stress - Iir aU, au1)
a Xi .
where viscositydue to turbulence.Substituting for the Reynoldi"stresses
PTis an additional
in (4.13) we get
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
form as the original momentumequation (4.8) with the simple
which is of exactlythe same,
addition of the extra viscosity term.
variationsbetween modelsof this type arise from the method usedto obtain
the turbulent Yiscosity.
form of this type of modelarises
M-c simplest from nixing lengtharguments,Prandd
analysisrevealsthat PTdivided by the density, p hasthe dimensions
of length multiplied by velocity. This implies that PT is a function of the density, a
lengthscaleandthe local flow velocity.Usingthe expressionfor the turbulent
shearstressas a model it can be postulated
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 a( Pf akpr( au, aui) IU, (4.15)
at ax aXj
aXi Pok , pax. , ax, ax( -E
This transportequationfor k is then solved, the value of e being specifiedempirically.The
turbulent viscosity is obtainedfrom
pr - PC
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
ae U,aE !
Ipy -U,, auj) nuf 2c !ý
+ a -. a Po. 'IC.
-c k (4.16)
at X, X, aX, xi
Again the coefficients both equation(4.15) and (4.16) needto be specified.The turbulent
viscosity is given by
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
data and a possiblelack of universality.
Oneof the maindrawbacks thesemodelsis that the use of an eddy viscosity leads
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
secondmaintype of turbulentmodelis concernedwith modelling the Reynolds