Invariant properties of the Reynolds shear stress probability distribution in turbulent wall-bounded flows

Understanding the relationship between high- and low-order statistical moments of the fluctuating velocity is at the heart of the turbulence closure problem. Here, we examine properties of the Reynolds shear stress (RSS) probability distribution through the lens of the joint moments of the streamwise and wall-normal velocities in turbulent wall flows. A scaling for the fluctuating RSS is proposed based on properties of the mean dynamical equation. We also present a corrected version of an existing model equation for the RSS probability distribution, and evaluate this model for its fidelity to DNS data via the Kullback-Leibler divergence. It is shown that the fourth-order joint cumulants that comprise most of the model coefficients are essentially invariant with Reynolds number in the log layer (once one is established), despite changes in the correlation coefficient between the two velocity components. The ramifications of this finding for both the shape of the probability distribution as well as the scaling relationships for the fluctuating RSS in the log layer are discussed.