SELF-SIMILAR PROPERTIES OF THE REYNOLDS SHEAR STRESS PROBABILITY DISTRIBUTION IN WALL-BOUNDED FLOWS

Probability distributions of the normalized Reynolds shear stress (RSS) and joint cumulants of the normalized streamwise and wall-normal velocity components up to sixth order are presented spanning over three decades of friction Reynolds numbers Re_τ in several wall-bounded flows. It is shown that the fourth-order joint cumulants are non-zero and essentially invariant with Reynolds number for Re_τ & 2000 despite changes in the correlation coefficient between the two velocity components. Several sixth order joint cumulants are also non-zero and appear to become invariant with Reynolds number, albeit more slowly than the fourth-order ones. This indicates that the interactions between the streamwise and wall-normal velocities do not become reducible to those implied by their second-order moments (i.e. their covariance matrix) with increasing Reynolds number, but also that the Reynolds number dependence of the RSS distribution (at least out to a substantial number of standard deviations) is attributable almost entirely to that of the second-order moments. The fidelity of a model distribution given by Antonia & Atkinson (J. Fluid Mech., vol. 58, 1973, pp. 581–593) that includes the effects of cumulants up to 4th order is also evaluated against the experimental data. It is shown that this distribution is accurate to roughly 25 times the mean RSS in the negative tail and 15 times the mean RSS in the positive tail. Beyond this, the model distribution predicts negative probabilities in the negative tail when the present experimental cumulant values are inputted, and thus a higher order expansion is necessary if one aims to obtain a general model distribution equation for the RSS.