Geoscience Reference
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In the same way, averaging Eq. (1.29) for a conserved scalar and neglecting its
averaged molecular term produces
∂c
∂t +
∂c u i
∂x i
∂f i
∂x i ,
=−
(1.41)
where f i =
c u i is the turbulent flux of the scalar.
In Chapter 4 we'll derive conservation equations for these turbulent fluxes. They
involve further unknowns, and so in practice the turbulent fluxes (or their conserva-
tion equations) are modeled - approximated in some way. The two broad categories
of turbulent flow models correspond to the type of averaging that produced them.
Although the type of averaging used is often not indicated explicitly, you can often
infer it from careful reading of the model description.
In ensemble averaging the velocity and scalar fields are broken into ensemble-
mean and fluctuating (turbulent) parts, and the flux produced is due to all the
turbulence. In space averaging the filter separates the fields into resolvable and
unresolvable (also called subgrid-scale or subfilter-scale) parts. Here the flux is due
to the unresolvable turbulence.
cu i
1.7.1 Ensemble-averaged turbulence models
The earliest approximation for turbulent fluxes was a gradient-diffusion model like
that used for the molecular fluxes, but with a much larger, “eddy” diffusivity. The
“mixing-length” model used by Prandtl and Taylor is the simplest example, but it
is now seldom used in computations. We discuss it briefly in Chapter 4 to motivate
your physical understanding of turbulent fluxes.
A more recent approach, which we discuss in Chapter 5 , is to compute the
turbulent fluxes through models of their evolution equations. This second-order-
closure modeling, as it is often called, has been computationally feasible since the
late 1960s.
“Pdf modeling” is a type of ensemble-average modeling. While the technique for
deriving evolution equations for turbulent fluxes has been known for many decades,
that for deriving the evolution equation for the probability density function, or pdf,
of a turbulent flowvariable ismuch newer, being presented first by Lundgren ( 1967 ).
We shall discuss the pdf and its evolution equation in Part III .
1.7.2 Space-averaged turbulence models
The space averaging introduced by Reynolds in 1895 has been generalized recently
to spatial filtering of the governing fluid equations. Averaging implies a uniform
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