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The equation describing the evolution of the probability density function of the temperature field in a turbulent axisymmetric heated jet is presented. A closure problem is present and some possible ways of attacking it are suggested. A closure at the first-order level is then tentatively tried and similarity arguments are exploited. A hyperbolic first-order variable-coefficient quasi-self-preserving partial differential equation is obtained. The constants appearing in those coefficients are evaluated from available experiments. Some questions are raised on the uncertainty of the computed constants due to the experimental scattering of the velocity-temperature correlation at the centerline. The probability density function is obtained analytically as a function of downstream location along the centerline if it is prescribed at a reference centerline position. In particular, the probability density function is taken as Gaussian at ten diameters downstream. The computed mean and variance are compared with existing experiments and display a reasonably good agreement. Values for the skewness and flatness factor tend to indicate that deviations from Gaussianity along the centerline are very small.

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A three-dimensional direct numerical simulation of a propagating turbulent premixed flame is performed using one-step Arrhenius model chemistry. The interaction of the flame thermochemical processes with the local geometries of the scalar field and flow topologies is studied. Four regions (“fresh reactants,” “preheating,” “burning,” and “hot products”), characterized by their reaction rate and mass fraction values, are examined. Thermochemical processes in the “preheating” and “burning” regions smooth out highly contorted iso-scalar surfaces, present in the “fresh reactants,” and annihilate large curvatures. Positive volumetric dilatation rates, −P = ∇ · u, display maxima for elliptic concave and minima for convex scalar micro-structures. Constant average tangential strain rates, aT, with large fluctuations, occur throughout the flow domain, whereas normal strain rates, aN, follow the trends of volumetric dilatation rates. Focal topologies, present in the “fresh reactants,” tend to disappear in favor of nodal structures as moving towards the “hot products.” The vorticity vector is predominantly tangential to the iso-scalar surfaces. The Unstable Node/Saddle/Saddle and Stable Focus/Stretching topologies, present in the “fresh reactants,” correlate with large values of aN and aT providing hints on the flow topologies fostering scalar mixing.