Logarithmic distribution of mean velocity and turbulent kinetic energy in a pipe flow

A Lie-group based similarity theory is developed for both momentum and energy distributions in a turbulent pipe flow, leading to asymptotic logarithmic profiles of mean velocity and turbulent kinetic energy. Both channel and pipe data over a wide range of Re yield 0.45 to be the universal Karman constant. A new spatial invariant characterizing outer dynamics is discovered and validated by reliable experimental data. The theory predicts the mean velocity profile (MVP) with 99% accuracy for high Re experimental data (up to 40 millions), and offers a quantitative explanation for recent observation of logarithmic kinetic energy distribution by Hullmak et al. (Phys. Rev. Lett. 108, 094501). Turbulent flows over objects form thin vorticity layers called boundary layers. As it is widely accepted that near-wall flow physics is autonomous and independent of the flow being external or internal, pipe flow forms an experimentally and numerically expedient canonical flow for the study of wall turbulence. Despite extensive efforts, the prediction of the mean velocity still relies on empirical functions [1] having limited accuracy and limited range of Reynolds numbers (Re). Hence, the problem continues to receive vivid attention with great experimental [2, 3] and theoretical [4] efforts. From a statistical physics point of view, turbulent pipe flows are at a far-from-equilibrium state encompassing not only a cross-scale energy flux (cascade) but also momentum and energy fluxes in space. Understanding physical principles governing the non-homogeneous transport and non-uniform distribution of the mean momentum and kineticenergy is a log-standing goal of the research. Nearly eighty years ago, Prandtl [5] and von Karman [6], independently proposed the concept of mixing length with a linear dependence on the distance from the wall, predicting a logarithmic MVP and hence friction coefficient. However, this empirical model has led to controversies: Barenblatt et al [7] have claimed that power-law is a better description; Goldenfeld [8] has proposed a model for friction coefficient using a power-law description. A recent model of L’vov et al. [4] is particularly noteworthy, as its log-law description yields predictions of reasonable accuracy over a range of finite Re (see Fig.2). Recently, a logarithmic scaling for the streamwise mean kinetic energy profile (MKP) is reported [9], with no explanation. Clearly, a deductive theory for joint MVP and MKP is still missing. Here, we present a Lie-group based similarity theory for turbulent channel and pipe flows. The original idea was presented in [10] and formulated rigorously in [11]. The goal of the theory is to find invariant solutions of the averaged flow equations based on a symmetry analysis of a set of new quantities, called order functions, which are introduced in close analogy to order parameter [12] in the study of critical phenomena. Adding the order function to dependent variables in the equations and then performing a dilation-group transformation yields a set of new, candidate invariant solutions, which defines a new method to measure the Karman constant. In this Letter, we compare the prediction to measured MVP in a turbulent pipe [13]. In addition, with a system similarity argument, we find a new spatial invariant which predicts a logarithmic MKP at high Re. Theory for mean velocity In a pipe, the mean momentum equation (MME) is [4]: