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

Abstract

We 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 ∼ √ 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 Kaminski et 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.

Generalised unsteady plume theory

“…For further details pertaining to the derivation of (2.5)-(2.8) the reader is referred to Craske & van Reeuwijk (2015a). Having been obtained via invertible manipulations, the momentum-energy system (2.6) and (2.8), is equivalent to the volume-momentum system, (2.5) and (2.6).…”

“…In unsteady jets and plumes, unlike their statistically steady counterparts, the energy flux plays an independent role in the governing equations and therefore, the assumption of a particular velocity profile does entail a loss of generality. A useful result in this regard is that for a Gaussian mean velocity profile w(r), illustrated in figure 1(c), the dimensionless energy flux γ m can be determined exactly (Craske & van Reeuwijk 2015a) as…”

“…For top-hat profiles γ g = β g = 1, the entrainment coefficient is insensitive to the integral acceleration of the plume and the second and third terms on the right-hand side of (2.21) are equal to zero. For further details regarding the physical interpretation of the first three terms on the right-hand side of (2.21), the reader is referred to Craske & van Reeuwijk (2015a), as they also appear in the equations describing an unsteady jet. The final contribution in (2.21) is only found in plumes and is proportional to the flux-balance parameter Γ of Morton (1959).…”

“…For a detailed description of the finite volume method used to discretise the governing equations, the reader is referred to Craske & van Reeuwijk (2015a conditions that allow fluid to enter and leave the finite computational domain in a manner that is consistent with a semi-infinite unbounded domain. The formulation, testing and implementation of these boundary conditions is described in Craske & van Reeuwijk (2013).…”

“…In referring to this disturbance as a wave, we follow Whitham (1974) and regard it as a recognisable signal propagating with a certain velocity. Here the notion of a wave is perhaps more appropriate than the notion of a front, which was employed in Craske & van Reeuwijk (2015a), as it encompasses a wider variety of possible disturbances, which can be comprised of several fronts. More precisely, in § 5.1 we will see that the wave is comprised of characteristic surfaces along which wave fronts travel.…”