Time averages in two-phase flows

The following manner of defining the time average of a hydrodynamic parameter in turbulent two-phase flow is studied: The time history function y(x,t) of one of the parameters of interest is expanded into a Fourier series. The signal, that is, the part of the expansion corresponding to pulsations below an arbitrary cut-off, serves as the basis for a definition of the average. The rest of the expansions constitutes the noise. After integrating over an interval it is hoped that the time average operator acts like a low-pass filter such that y(x,t) is nearly equal to the signal, and that the first derivative of y with respect to t is conserved. By analyzing the behavior of one harmonic, the author shows that the first condition is fulfilled but the second is not. However, applying the time average operator twice is shown to satisfy both conditions.