On new scaling laws in a temporally evolving turbulent plane jet using Lie symmetry analysis and direct numerical simulation

Abstract

A temporally evolving turbulent plane jet is studied both by direct numerical simulation (DNS) and Lie symmetry analysis. The DNS is based on a high-order scheme to solve the Navier–Stokes equations for an incompressible fluid. Computations were conducted at Reynolds number $\mathit{Re}_{0}=8000$, where $\mathit{Re}_{0}$ is defined based on the initial jet thickness, $\unicode[STIX]{x1D6FF}_{0.5}(0)$, and the initial centreline velocity, $\overline{U}_{1}(0)$. A symmetry approach, known as the Lie group, is used to find symmetry transformations, and, in turn, group invariant solutions, which are also denoted as scaling laws in turbulence. This approach, which has been extensively developed to create analytical solutions of differential equations, is presently applied to the mean momentum and two-point correlation equations in a temporally evolving turbulent plane jet. The symmetry analysis of these equations allows us to derive new invariant (self-similar) solutions for the mean flow and higher moments of the velocities in the jet flow. The current DNS validates the consequence of Lie symmetry analysis and therefore confirms the establishment of novel scaling laws in turbulence. It is shown that the classical scaling law for the mean velocity is a specific form of the current scaling (which has a more general form); however, the scaling for the second and higher moments (such as Reynolds stresses) has a completely different structure compared to the classical scaling. While the failure of the classical scaling for the second moments of the fluctuating velocities has been noted from the jet data for many years, the DNS results nicely match with the present self-similar relations derived from Lie symmetry analysis. Key ingredients for the present results, in particular for the scaling laws of the higher moments, are symmetries, which are of a purely statistical nature. i.e. these symmetries are admitted by the moment equations, however, they are not observed by the original Navier–Stokes equations.