title: Turbulent Round Jet Flows: Direct Numerical Simulations and a Symmetry Analysis
Creators: name: Nguyen, Cat Tuong
Publisher: name: UNSPECIFIED
Year: 2024

description: A spatially evolving turbulent round jet flow is studied using Lie symmetry analysis and validated against data from direct numerical simulations (DNS) of the Navier-Stokes equations (NSEs). The, in the literature, unprecedented simulations were performed at two Reynolds numbers of Re=3500 and Re=7000 based on the orifice diameter D and the bulk velocity at the orifice Ub with a passive scalar Θ at a Prandtl number of Pr=0.71 in a long numerical box of z/D=75.
To achieve self-similarity in the round jet DNS, turbulent pipe flows at the corresponding Reynolds numbers and a length of z/D=5 were used as upstream inflow boundary conditions. A fast convergence of self-similarity was achieved by this approach. High quality statistics were generated, e.g., by ensemble averaging over 200 washouts of a particle for the simulation at Re=3500. Analysis of mean velocities, Reynolds stresses, and turbulent kinetic energy budgets revealed nearly perfect classical scaling based on a similarity coordinate η=r/z with the radius r. In addition, statistical analysis of probability density functions (PDFs) for the axial velocity Uz showed Gaussian behavior along the jet axis, with a transition to heavier tails and skewness away from the axis.
Using the new DNS data of the high-velocity moments, the statistical behavior of the turbulent round jet is further analyzed by applying Lie symmetry analysis for pure hydrodynamics. The turbulent velocity scaling laws derived by Lie symmetry analysis reveal a possible variation of the turbulent decay behavior due to the statistical symmetry. Interestingly, the statistical symmetry is found only in the multi-point moment equations and not in the NSEs. This variation is possible for all moment orders n except for the second order moment. However, the present simulations break this symmetry, leading to a classical scaling behavior characterized by η and a scaling of the velocity moments with z⁻ⁿ, which has been validated up to moment order 10. The prefactors of the scaling laws are exponential in n. Notably, Gaussian behavior is observed in the Uz-moments, although the Gaussian exponent shows non-linear behavior in n, which implies significant intermittency in the mean axial velocity moments. The statistical symmetry, which gives a measure of intermittency, does not play a role in the base scaling laws, but is the basis for high-moment scaling for η in turbulent jet flows.
Additionally, moments and PDF statistics of Θ are analyzed with the two simulations. Furthermore, Lie symmetries applied to multi-point velocity-scalar correlation equations lead to a generalization of the passive scalar and velocity scaling laws, where the scaling prefactors are exponential for varying velocity and scalar moment order n and m, respectively. Further, inflow variation is controlled solely by the velocity inlet. Gaussian distribution of instantaneous Θ-moments and mixed Uz-Θ-moments with η is observed. Similar to the Uz-moments, non-linear coefficients in the Gaussian exponent are traced back to intermittency. Unlike the velocity PDF statistics, scalar PDF statistics deviate from the Gaussian distribution on the jet axis, but also become skewed and heavy-tailed with increasing η.