Exact solutions for the probability density function of turbulent scalar fields

The basic property of a turbulent scalar field is its probability density function p(θ x, t). Here, for the first time, some exact solutions for p(θ x, t) are derived and discussed. These apply to the case of a finite mass of passive scalar — called a cloud for short — dispersing in simple, but conceptually important, turbulent flows, namely those associated with constant rates of strain. Extensions of the solutions to cases where the cloud is meandering, and where there are several clouds, are obtained. Applications of the results are discussed, with particular emphasis on their potential value for testing and validating approximate closure schemes applied to the evolution equation for p(θ x, t).