Geoscience Reference
In-Depth Information
5
Conservation equations for covariances
5.1 Introduction and background
We saw in
Chapter 1
that direct numerical solutions of the turbulence equations
can be done only at relatively small values of the turbulence Reynolds number
R
t
.
Calculations of the vastly larger
R
t
flows in engineering and geophysical appli-
cations use averaged forms of these equations. The averaging produces important
turbulent fluxes
that must be specified before the equations can be solved numer-
ically. That specification allows today's calculations of averaged turbulence fields
in applications ranging from flow in the atmospheric boundary layer to the general
circulation of the earth's atmosphere and convection on the sun.
In this chapter we derive, interpret, and scale the conservation equations for sev-
eral covariances, including turbulent fluxes, that arise from ensemble averaging.
At very coarse resolution ergodicity (the property that any unbiased average con-
verges to the ensemble average,
Chapter 2
) blurs the distinction between ensemble
and space averaging, so these covariance equations are used also in traditional
mesoscale-modeling and weather-forecasting applications.
Monin and Yaglom
(
1971
) credit a 1924 paper by Keller and Friedmann as the
first to present a method of deriving turbulence moment equations, of which the
turbulent-flux budgets are an example. Because they contain unknown terms, the tur-
bulence moment equations were little used until the late 1960s, when computers
were large enough to solve approximate versions of them.
The literature on models of these flux budgets can be bewildering. The rationale
for their closure approximations is not always discussed, and the models go under
several names - e.g., “second-order closure,” “Reynolds-averaged Navier-Stokes
(RANS),” “single-point closure,” “higher-order closure,” and “invariant model-
ing.” Occasionally the model developers' names are used, as with “Mellor-Yamada
level 2.5.”
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