Energy dispersion in turbulent jets. Part 1. Direct simulation of steady and unsteady jets

Abstract

AbstractWe study the physics of unsteady turbulent jets using direct numerical simulation (DNS) by introducing an instantaneous step change (both up and down) in the source momentum flux. Our focus is on the propagation speed and rate of spread of the resulting front. We show that accurate prediction of the propagation speed requires information about the energy flux in addition to the momentum flux in the jet. Our observations suggest that the evolution of a front in a jet is a self-similar process that accords with the classical dispersive scaling$z\sim \sqrt{t}$. In the analysis of the problem we demonstrate that the use of a momentum–energy framework of the kind used by Priestley & Ball (Q. J. R. Meteorol. Soc., vol. 81, 1955, pp. 144–157) has several advantages over the classical mass–momentum formulation. In this regard we generalise the approach of Kaminskiet al. (J. Fluid Mech., vol. 526, 2005, pp. 361–376) to unsteady problems, neglecting only viscous effects and relatively small boundary terms in the governing equations. Our results show that dispersion originating from the radial dependence of longitudinal velocity plays a fundamental role in longitudinal transport. Indeed, one is able to find dispersion in the steady state, although it has received little attention because its effects can then be absorbed into the entrainment coefficient. Specifically, we identify two types of dispersion. Type I dispersion exists in a steady state and determines the rate at which energy is transported relative to the rate at which momentum is transported. In unsteady jets type I dispersion is responsible for the separation of characteristic curves and thus the hyperbolic, rather than parabolic, nature of the governing equations, in the absence of longitudinal mixing. Type II dispersion is equivalent to Taylor dispersion and results in the longitudinal mixing of the front. This mixing is achieved by a deformation of the self-similar profiles that one finds in steady jets. Using a comparison with the local eddy viscosity, and by examining dimensionless fluxes in the vicinity of the front, we show that type II dispersion provides a dominant source of longitudinal mixing.