Correlation functions and spectra

To describe the evolution of a fluctuating function , we need to know the manner in which the values of at different time instants are related. For this purpose a correlation function between the values of at different times may be considered. This is called the autocorrelation. Since we are working with stationary randoms (i.e. mean values are not functions of time). The autocorrelation gives no information about the origin of time. It depends only on the difference of time, .

The autocorrelation is a symmetric function of t .

Schwarz's inequality leads to

(22.1)

For stationary variables

Thus it is usual to define an autocorrelation coefficient, ρ(t), by

(22.2)

From (22.1) and (22.2), we obtain

(22.3)

We can define an integral scale by

(22.4)

In turbulence it is always assumed that the integral scale is finite. The value of is a rough measure of the interval over which is correlated with itself. Fig. 22.1 is a sketch of ρ ( ) .