Dynamics of three-dimensional turbulence from Navier-Stokes equations

In statistically homogeneous and isotropic turbulence, the average value of the velocity increment , where and are two positions in the flow and is the velocity in the direction of the separation distance , is identically zero, and so to characterize the dynamics one often uses the Reynolds number based on , which acts as the coupling constant for scale-to-scale interactions. This description can be generalized by introducing structure functions of order , , which allow one to probe velocity increments including rare and extreme events, by considering for large and small . If , the theory for the anomalous exponents in the entire allowable interval is one of the long-standing challenges in turbulence (one takes absolute values of for negative ), usually attacked by various qualitative cascade models. We accomplish two major tasks here. First, we show that the turbulent microscale Reynolds number (based on a suitably defined turbulent viscosity) is 8.8 in the inertial range with anomalous scaling, when the standard microscale Reynolds number (defined using normal viscosity) exceeds that same number (which in practice could be by a large factor). When the normal and turbulent microscale Reynolds numbers become equal to or fall below 8.8, the anomaly disappears in favor of Gaussian statistics. Conversely, if one starts with a Gaussian state and increases beyond 8.8, one ushers in the anomalous scaling; the inference is that and remains constant at that value with further increase in . Second, we derive expressions for the anomalous scaling exponents of structure functions and moments of spatial derivatives, by analyzing the Navier-Stokes equations in the form developed by Hopf. We present a novel procedure to close the Hopf equation, resulting in expressions for in the entire range of allowable moment order , and demonstrate that accounting for the temporal dynamics changes the scaling from normal to anomalous. For large , the theory predicts the saturation of with , leading to several inferences, two among which are (a) the smallest length scale , where Re is the large-scale Reynolds number, and (b) that velocity excursions across even the smallest length scales can sometimes be as large as the large-scale velocity itself. Theoretical predictions for each of these aspects are shown to be in excellent quantitative agreement with available experimental and numerical data.