The Universal Scaling Exponents of Anisotropy in Turbulence and their Measurement

The scaling properties of correlation functions of non-scalar fields (constructed from velocity derivatives) in isotropic hydrodynamic turbulence are characterized by a set of universal exponents. It is explained that these exponents also characterize the rate of decay of the effects of anisotropic forcing in developed turbulence. This set has never been measured in either numerical or laboratory experiments. These exponents are important for the general theory of turbulence, but also for modeling anisotropic flows. We propose in this letter how to measure these exponents using existing data bases of direct numerical simulations and by designing new laboratory experiments. Fundamental studies of turbulence tend to stress the model of isotropic, homogeneous turbulence, and most theories and experiments since Kolmogorov’s seminal work of 1941 [1] considered the universal (anomalous) exponents that characterize the isotropic characteristics of turbulent flows (see for example [2, 3] for recent reviews). In fact, most turbulent flows are not forced isotropically, and moreover even in isotropic flows there are important fields that are constructed from velocity derivatives that transform under rotation as vectors or tensors rather than scalars. It has been known for quite a while that the secondorder structure function (that depends on one separation vector) becomes more and more istotropic as the the separation scale goes down (see below). Moreover, the rate of this isotropization process is governed by a universal exponent [4, 5, 6, 7]. In recent papers [8, 9] it was pointed out that this exponent is one of an infinite family of universal anomalous scaling exponents that were never considered in experiments and numerical simulations. Moreover, it was shown that in the context of passive scalar convection the anomalous scaling exponents that characterize the scaling properties of anisotropic fields also govern the rate of isotropization of the properties of the flow in the cascade process down to smaller and smaller scales [8]. In this Letter we show that the same ∗Department of Chemical Physics, The Weizmann Inst. of Science, Rehovot 76100, Israel