In this chapter we introduce the concept of an ensemble average, which allows one to form averages for time-dependent processes. One such ensemble average statistical measure is the autocovariance (or autocorrelation) function. It gives information about the average time dependence of a process. The Fourier transform of the autocovariance, in turn, describes the frequency contents of the process. For two random functions of time one can define cross covariance between values of the two functions at different times. The Fourier transform of the cross covariance with respect to delay time gives the cross-spectral density. When these measures are independent of the choice of time origin, the processes are stationary. We shall look at examples of how one can derive a propagation speed from cross covariances or cross spectra and also see how one can find decay times and other properties of a random process. A useful application of spectra and covariance functions is to the relationship between input and output statistical measures for a linear system. From observations of the excitation and the response one is able to draw conclusions about the dynamics of a system. If one knows the system dynamics and some of the statistical properties of the input, one can find the statistical properties of the output, and vice versa.
Many fluid flows can be approximated by linear systems of equations. This means that, in turn, some flows may react to excitation in ways that we can analyze, especially if the excitation is weak.
An example of a linear response of a flow field to turbulence is the emission of acoustic waves from a turbulent jet, as first analyzed by Lighthill (1952) and discussed in Chapter 10. Flows that respond to excitation by divergent oscillation are unstable; such flows are discussed in Chapter 7.
Correlations and spectra depend upon the second moments of a joint probability density. In order to relate correlations and probability distributions, this chapter also outlines some of the elements of probability theory, including the central limit theorem and the normal distribution.
As an illustration of a non-normal distribution the log-normal distribution is presented.
Ensemble averages
A random function is a function that cannot be predicted from its past. An example of a random function of space and time is the velocity field in a turbulent jet.