Analysis of small-scale scalar mixing processes in highly under-expanded jets

Abstract

In the present study, we numerically investigate turbulent scalar mixing taking place downstream of highly under-expanded jets. The focus is placed on two inter-related issues: (i) the closure of the mean scalar dissipation rate (SDR) and (ii) the turbulence–scalar interaction (TSI) term. It is indeed commonly admitted that the former, i.e., the SDR, which is defined as the product of the scalar diffusivity with the squared scalar gradient, provides a good measure of the mixing efficiency. In turbulent flows, the mean (turbulent) SDR requires a specific closure to be settled. It is generally obtained within the approximation of a linear relaxation of scalar fluctuations or linear relaxation model. We will first evaluate herein the performance of this widely used closure. The analysis is further developed by means of the consideration of the mean SDR transport equation which shows that, in gaseous conditions, the SDR is mainly driven by two terms: (i) a dissipation contribution and (ii) the third-order correlation between the velocity gradient tensor and small-scale scalar anisotropy tensor. The scalar mixing efficiency thus appears to be controlled by the latter quantity, which is often denoted as the TSI term. It can be shown that only the symmetric part (rate of strain) of the velocity gradient tensor contributes to this term; the anti-symmetric part modifies, indeed, the orientation of the scalar gradient but not its magnitude. The classical approach is to analyze this contribution in the eigenframe of the rate of strain tensor. Such analyses show that, in homogeneous isotropic turbulence, the scalar gradient tends to align with the most compressive direction, thus leading to SDR production. However, the present conditions, which are far from homogeneity and involve strong density variations, may modify this classical picture. The present study analyzes this possible influence.