Averaging used in the analysis of turbulent flows

The complete sample space and its associated joint probability density function are theoretical constructions. When faced with measured or numerically simulated data practical methods of averaging have to be used.

A general definition of the time average is

This is the convenient form of average often used in practical measurements of statistically stationary flows, i.e., flows in which averaged quantities are independent of time. In statistically time-dependent flows, T needs to be chosen to be much greater than all the time scales of the fluctuations and much less than the time scale on which U varies with t , if possible.

The spatial average is defined in a similar way to the time average. For example,

This expression will be most convenient if is expected to be independent of x₃=z . Spatial averages, which exploit spatial symmetries, are particularly useful in numerical simulations.

Higher-order statistics, such as variances and co variances, can also be calculated from time averages and spatial averages. For example,

In principle, statistics based on time averages implicitly provide an approximation to the probability density function of statistically stationary turbulent flows. Similarly statistics based on spatial averages provide an approximation to the probability density function of statistically homogeneous turbulent flows.

The ensemble average

Consider an ensemble, { u⁽ⁿ⁾ ; n = 1, . . . ,N} , obtained by carrying out the same experiments in N identical apparatuses or more realistically by repeating the same experiment N times. N should be large. The ensemble average is defined by,