Large-eddy dynamics, the energy cascade, and large-eddy simulation - Turbulence in the Atmosphere - page 116

engineering fluids community, fromwhich came its present and remarkably precise
name “large-eddy simulation” or LES. † It is the principal type of fine-resolution,
space-averaged modeling used today.
In this chapter we shall derive and discuss the space-averaged equations of motion
and explore their dynamics in the LES limit. We'll also do a “thought problem”
in equilibrium homogeneous turbulence that sheds light on the energy cascade in
turbulence.

6.2 More on space averaging
6.2.1 A one-dimensional, homogeneous example
Perhaps the simplest space-averaging operator is the local average defined in Eq.
(3.29) . In its one-dimensional form it averages a function f(x,t) over an x -interval
of length to produce a smoothed version:

/ 2

1

f r (x,t,) x ,t)dx . = f(x + (6.1)

− / 2

This local average is sometimes called a running mean. We use the notation
introduced in Chapter 3 , denoting the smoothed variable by a superscript r, for
resolvable .
In Chapter 2 we represented a real, homogeneous function f(x) as a complex,
infinite Fourier series. Here we'll use the finite form

N

f(κ n )e iκ n x .

f(x) = (6.2)

n =− N

In the averaging defined by Eq. (6.1) the contributions to the sum in Eq. (6.2) from
Fourier components having wavelength small compared to (those with κ n 1)

are strongly attenuated, since they have many cycles over the averaging interval.
Those of wavelength large compared to (those with κ n 1) are minimally

affected, since they are nearly constant over the averaging interval.
We can quantify these smoothing properties of the running-mean operator, Eq.
(6.1) , by using the Fourier representation (6.2) in the averaging expression (6.1)
and integrating:

x + / 2

N

1

f(κ n )e iκ n x dx

f r (x) =

x − / 2

n =− N

† According to Parviz Moin (personal communication), the late Bill Reynolds of Stanford University coined the
name large-eddy simulation .