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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.

The primary object of this study is to derive new scaling laws for passive scalar statistical quantities in temporally and spatially evolving plane turbulent jet flows. We apply Lie symmetry analysis to the equations governing the evolution of the first three statistical moments for the passive scalar quantities. The analysis is based on novel forms of the two-point velocity–scalar and scalar–scalar correlation equations, which are naturally based on the statistical moments derived from the instantaneous velocities and not on those of the fluctuation velocities from the Reynolds decomposition. The newly derived invariant solutions recover the gaps in the classical self-similarity analysis from three major perspectives. First, the scaling laws are constructed as the direct consequence of the symmetry approach, while an a priori set of similarity scales is not required. Second, unlike in the classical laws, which show self-similarity primarily for the first moments, here self-similarity is theoretically shown up to the third moments. Third, there is a symmetry breaking induced by the integral invariant of the mean temperature that connects the scaling symmetries of momentum and passive scalar equations, which results in a close coupling of the scaling exponents of the first two moments and, further, determines the scaling exponents of all higher moments. To verify the new theoretical findings, we employ data from two direct numerical simulations of the mixing of a passive scalar driven by a temporally evolving turbulent jet. The direct numerical simulation data very clearly validate the new scaling laws up to the third moments.

An analytical framework is proposed to explore the structure and kinematics of iso-scalar fields. It is based on a two-point statistical analysis of the phase indicator field which is used to track a given iso-scalar volume. The displacement speed of the iso-surface, i.e. the interface velocity relative to the fluid velocity, is explicitly accounted for, thereby generalizing previous two-point equations dedicated to the phase indicator in two-phase flows. Although this framework applies to many transported quantities, we here focus on passive scalar mixing. Particular attention is paid to the effect of Reynolds (the Taylor based Reynolds number is varied from 88 to 530) and Schmidt numbers (in the range 0.1 to 1), together with the influence of flow and scalar forcing. It is first found that diffusion in the iso-surface tangential direction is predominant, emphasizing the primordial influence of curvature on the displacement speed. Second, the appropriate normalizing scales for the two-point statistics at either large, intermediate and small scales are revealed and appear to be related to the radius of gyration, the surface density and the standard deviation of mean curvature, respectively. Third, the onset of an intermediate ‘scaling range’ for the two-point statistics of the phase indicator at sufficiently large Reynolds numbers is observed. The scaling exponent complies with a fractal dimension of 8/3. A scaling range is also observed for the transfer of iso-scalar fields in scale space whose exponent can be estimated by simple scaling arguments and a recent closure of the Corrsin equation. Fourth, the effects of Reynolds and Schmidt numbers together with flow or scalar forcing on the different terms of the two-point budget are highlighted.

The combined effect of internal and external intermittency on the statistical properties of small-scale turbulence is investigated in temporally evolving, planar turbulent jet flows at different Reynolds numbers using highly resolved direct numerical simulations. In turbulent jet flows, the phenomenon of external intermittency originates from a sharp layer, known as the turbulent/non-turbulent interface, that separates the turbulent core from the surrounding irrotational fluid. First, it is shown that low-order and higher-order structure functions in both the core and the shear layer of the jet satisfy complete self-preservation, which means that structure functions are invariant with time and collapse over the entire range of scales, regardless of the set of length and velocity scales used for normalization. Next, the impact of external intermittency on small-scale turbulence is studied along the cross-wise direction by the self-similarity of structure functions. It is shown that structure functions exhibit from the centre toward the edge of the flow a growing departure from self-similarity and the prediction of classical scaling theories. By analysing statistics conditioned on the turbulent portion of the jet, it is demonstrated that this departure is primarily due to external intermittency and the associated similarity-breaking effect.

The effect of differential diffusion of two passive scalars having Schmidt numbers of unity and 0.25, respectively, is investigated using direct numerical simulation of a temporally evolving jet. The objective of the research is twofold: (i) to compare the turbulent/non-turbulent (T/NT) interface position using the scalar criterion between the unity- and low-Schmidt-number scalar; and (ii) to determine the impact of the T/NT interface on differential diffusion. For the latter, the T/NT interface is detected using the vorticity criterion. To quantify the effect of differential diffusion, a normalised differential diffusion parameter is analysed, clearly showing the dominance of differential diffusion at the T/NT interface. A transport equation for the scalar differences is then evaluated, which shows that differential diffusion originates at the interface. Further, the separation between the passive scalars, arising due to differential diffusion, is studied using conventional and conditional statistics with respect to the interface distance. Since differential diffusion is known to be present at large and small scales, the connection between them is analysed using the scalar dissipation rate. Moreover, the physical mechanism responsible for the departure of the two scalars is analysed using the scalar gradient alignment, the ratio of the diffusive fluxes and by a transport equation for the scalar gradients.

The dissipation element analysis is applied to the mixture fraction fields of a series of datasets from direct numerical simulations of non-premixed temporally evolving jet flames with jet Reynolds numbers ranging from 4500 to 10 000 and varying stoichiometric mixture fractions. Dissipation elements are space-filling regions where a scalar field behaves monotonically and allow for the analysis of scalar fields in homogeneous isotopic turbulence as well as in complex, highly inhomogeneous and anisotropic flows such as turbulent flames. Statistics of the dissipation element parameters of non-premixed flames are compared to those obtained from non-reacting jets. It is found that the universality of the normalized length distribution of the dissipation elements observed in non-reacting cases also holds true for the reacting flows. The characteristic scaling with the Kolmogorov micro-scale $\\eta$ is obtained as well. The effects of combustion on the scalar difference in the dissipation elements are shown and are found to diminish as the Reynolds number and the fuel dilution is increased. The dissipation elements provide the means for a local comparison of the turbulent and characteristic flame scales. A new regime diagram for non-premixed combustion is introduced using coherent structures in the scalar fields, the dissipation element parameters for a local classification of the turbulent flame surface into flamelet-like zones and fine-scale mixing zones in addition to the burning and non-burning zones. The soundness of the regime diagram and the potential consequences for combustion modelling in the individual regimes is demonstrated by the investigation of the correlation between the chemical field and the dissipation element parameters in the individual regimes.

Subject of this work is the recently introduced extended Representative Interactive Flamelet (RIF) model for multiple injections. First, the two-dimensional laminar flamelet equations, which can describe the transfer of heat and mass between two-interacting mixture fields, are presented. This is followed by a description of the various mixture fraction and mixture fraction variance equations that are required for the RIF model extension accounting for multiple injection events. Finally, the modeling strategy for multiple injection events is described: Different phases of combustion and interaction between the mixture fields resulting from different injections are identified. Based on this, the extension of the RIF model to describe any number of injections is explained. Simulation results using the extended RIF model are compared against experimental data for a Common-Rail DI Diesel engine that was operated with three injection pulses. Simulated pressure curves, heat release rates, and pollutant emissions are found to be in good agreement with corresponding experimental data. For the pilot injection and the main or post injection, respectively, different ignition phenomena are pointed out and the influence of the scalar dissipation rate on these ignition phenomena is detailly investigated.

What elements of contemporary American life contribute to the United States having the greatest number and highest share of public mass shootings around the globe?The editors and contributors to  All-American Massacre  seek to answer this question by exploring how masculinity, racism, politics, media, fame, education, gun culture, and mental.