2: The conditional averaging

2 The conditional averaging

In this section we define averaging techniques for describing flow properties inside the two-phase surface layer . Let ? ? denote an averaging process such that if is a generic flow variable then is the corresponding average. Here x is a position vector, and t is time. The most appropriate averaging process is the ensemble average in which the average is evaluated at each space point as the arithmetic mean of all the single realizations over a set of possible equivalent realizations i = 1, N

Many of the flows studied in the laboratory are statistically stationary with respect to time. This means that in a suitable coordinate system velocity components at a fixed point are stationary random functions of time. If this is so, the ergodic hypothesis asserts that the mean value with respect to time

is equivalent to the ensemble average i.e. .

The averaging process, whatever its nature, defines and satisfies:

where the first two equations are called Reynolds' rules, the third is known as Leibnitz' rule and the fourth is Gauss' rule.

In order to obtain conditioned equations for each phase (i.e. air and water) we introduce a phase function or intermittency function such that for a fixed point x

We deal with I( x, t) as a generalized function. Hence a derivative of I( x, t) can be defined as a generalized function in terms of...