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.

Direct numerical simulations of turbulent viscoelastic jets

“…We are well aware (and well familiar) with the present ongoing discussion on the validity of the classical theory of jets (and of other flows) as described in several recent references e.g. Sadeghi et al (2018), Layek & Sunita (2018) and Cafiero & Vassilicos (2019). However, it seems that, in the present case, the Newtonian jet recovers the classical scaling laws i.e.…”

“…A problem of considerable interest in the theory of canonical turbulent free shear flows is the description of the far field of turbulent planar jets. For a Newtonian fluid, a vast amount of knowledge obtained from carefully conducted experiments and numerical simulations is available and a theory of the turbulent plane jet has been established (Tennekes & Lumley 1972;Townsend 1976;Pope 2000), even if recently, this classical theory has been challenged by a number of experimental and numerical works (Layek & Sunita 2018;Sadeghi, Oberlack & Gauding 2018;Cafiero & Vassilicos 2019). † Email address for correspondence: carlos.silva@tecnico.ulisboa.pt 899 A11-2 M. C. Guimarães, N. Pimentel, F. T. Pinho and C. B.…”

New scaling laws of passive scalar with a constant mean gradient in decaying isotropic turbulence

“…As noted earlier, this is particularly important in fluid mechanics and turbulence studies, as we can construct invariant solutions from first principles through symmetry. It should be noted that the current definition of a group invariant solution has been previously provided in Sadeghi et al (2018Sadeghi et al ( , 2020 and here it is only briefly reviewed due to its necessity for the present work. Similar to Sadeghi et al (2018Sadeghi et al ( , 2020 we define y = ζ (x) as an invariant solution of a differential equation if:…”

“…The application of Lie symmetry theory in turbulence has been developed by Oberlack and co-workers in several papers (see e.g. Oberlack 2001;Oberlack & Rosteck 2010;Oberlack et al 2015;Sadeghi, Oberlack & Gauding 2018, 2020. Their recent developments have been to derive a variety of classical and new scaling laws in turbulence through applying Lie symmetry theory to the infinite series of two-point correlation (TPC) and multi-point correlation (MPC) equations.…”

Scaling of heated plane jets with moderate radiative heat transfer in coupled DNS

“…Proceeding with these studies, Klein et al [12] numerically investigated the influence of the Reynolds number and the inflow conditions in a constant-density plane jet using DNS. A recent study of Sadeghi et al [13] proposes new scaling laws for the higher moments in constant-density temporally evolving plane jet which were derived using Lie symmetry analysis.…”