VII.5 Moment (and related) methods

In a previous lecture we discussed turbulence modelling, in which the aim was to remove the ‘small scale’ detail of turbulence and replace it with a (cheaper) turbulence model. This is done by an averaging process. A similar approach is used in combustion modelling to eliminate the small scale turbulent effects and to derive models for the combustion processes. Since the density of the fluid is variable we introduce a density-weighted average called a Favre average. As usual we split the flow up into mean and fluctuating '' components

ux(x, t) = ux + u''x

(VII.5)

For a normal averaging procedure we would assume that u_x' = 0. Favre averaging is constructed by requiring that the density-weighted fluctuating component

---- rux = 0

(VII.6)

This is equivalent to

---- rux- ux = r-

(VII.7)

Of course, Favre averaging can be applied to any flow variable, not just the velocity component u_x. This density-weighted average is more difficult to evaluate, requiring simultaneous measurement of

and u_x. However when it comes to manipulating the fluid flow equations (NSE, plus the transport equations for

_i), Favre averaging is much easier to use. Frequently we find ourselves manipulating terms such as

------ ruxuy, and looking for the average ruxuy

Using a conventional average (non-density weighted) we end up with 4 terms

------ ------ ------- ------- ------- ------- ruxuy = ruxuy + r u'xu'y + r'u'xuy + r'u'yux + r'u'xu'y

none of which can be removed. Using Favre averages produces a much simpler expression

ru-u--= ru u + ru''u'' x y x y x y

Favre averaging can be used in a similar manner to that described in the previous lecture, to derive mean flow equations and equations for k and

for compressible flow. Thus a version of the k -

model for turbulent compressible flows can be derived.

We can also apply Favre averaging to equation (VII.4). After some manipulation we arrive at the equation

@ryi -- -------- -- -- -----+ \~/ .ryiu-= \~/ .rDi\ ~/ yi - \~/ .ru''y'i'+ rSyi @t

(VII.8)

All the terms on the l.h.s. contain variables already known (from the solution of the mean flow equations). Those on the r.h.s. however are unknown as yet, and typically require modelling. In high Re flows, the molecular diffusivity D is small, so the term $\~/$ .

D_i $\~/$

_i can safely be ignored. This leaves two terms

-	$\~/$ .		representing turbulent transport, and
			the mean chemical source term

Both terms cause problems for modelling. Various approaches have been applied to modelling these various terms, leading to a wide range of combustion models ranging from simple, cheap and inaccurate to sophisticated, accurate and computationally very expensive. Approaches include

Eddy Break Up model.

Devised by Spalding (1971), this is based on the assumption that the turbulent cascade mixing process controls the chemical reactions - i.e. that mixing of the reactants is the rate-determining step in the process. This can be used to derive a simple model for the chemical source term. However this model requires detailed tuning of coefficients, an unsatisfactory arrangement, and is not physically very accurate. It has been formulated primarily for premixed combustion, although it is possible to develop formulations for non-premixed combustion.

PDF transport model

Turbulence can be discussed statistically, as a quasi-random flow with particular statistical properties. These properties can be described in terms of correlations between flows at different points, or the same point at different times. The basic quantity involved is the PDF, or Probability Density Function. Simple engineering models such as the k - model ignore this, providing information only about the mean levels of turbulence. However it is possible to model turbulence in a statistical manner, providing information about these turbulent correlations as a PDF transport equation. Such models are computationally costly but can be very accurate.

Extending this to cover combustion involves providing a transport equation for the so-called joint pdf of velocity and reactive scalars. This is denoted

P (u,y;x, t)

and has the interpretation that

P (u,y; x,t) dudy-

is the probability that the flow at point

at time t has velocity

and is in the reactive state described by

. This can provide the information necessary to specify the additional terms in (VII.8). However, although this method is potentially more accurate than other approaches, and can be applied to premixed, non-premixed and partially premixed cases, it is also significantly more expensive, and so is more used for theoretical investigation of combustion than for solving engineering problems.

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