Fourth-order velocity statistics

Abstract

An investigation of several fourth-order velocity statistics is described. Whereas local isotropy and local scaling are applicable to structure functions expressible as averages of differences of velocity, local isotropy and local scaling are inapplicable to the structure functions that we study. Data from wind tunnel grid turbulence show the behavior of the fourth-order statistics for nearly isotropic turbulence. The scaling relations predicted by the joint Gaussian assumption (JGA) are considered, as are those from the statistical independence assumption (SIA). The basis of the JGA is that velocities at several points are joint Gaussian random variables, whereas the basis of the SIA is that locally averaged velocity is statistically independent of velocity difference. The JGA and SIA relate fourth-order statistics to the second-order velocity structure function, as well as to the velocity covariance. For various fourth-order statistics, the predictions of the JGA and SIA are compared with data. These comparisons quantify how accurately our fourth-order statistics follow the scaling dependence on second-order velocity structure functions and on velocity covariance as predicted by the JGA and SIA. Our measured structure functions are in agreement with the scaling predicted by the SIA with no exceptions and with that predicted by the JGA with two exceptions. As is known, one exception is the structure function that obeys local isotropy [e.g., ⟨(ui−ui′)4⟩] (which the SIA does not predict). The other exceptions are called anomalous components [e.g., ⟨(ui2−ui′2)(uj2−uj′2)⟩]. These anomalous components are shown to be sensitive indicators of intermittency for locally isotropic turbulence, whereas they are indicators of anisotropy for the case of anisotropic turbulence. The anomalous components are in reasonable agreement with the scaling predicted by the SIA but not with the JGA.