Probability density function approach for a turbulent axisymmetric heated jet. Centerline evolution

Abstract

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.