Lagrangian-Eulerian correlation functions

The accurate description of dispersion of particles in turbulent flow is still an unsolved problem. Usually, in CFD software packages the mean particle concentration is described by a convection-diffusion equation and an eddy-diffusivity assumption is used for the unknown correlation of particle concentration and velocity. It is known, however, that such an approach does not have a sound physical basis and coeffecient in the equation have to be fitted for each situation. An alternative approach is based on a Langevin equation for the particle velocity. This is a stochastic differential equation in which the particle acceleration contains a deterministic part and a random part involving Gaussian white noise. The coefficients of the random part follow from Kolmogorov`s theory. The coefficients in the deterministic term can be determined for homogeneous flow, but for inhomogeneous flow they can only be assessed from Lagrangian velocity correlation functions, which are very difficult to measure and calculate. Recently, a relationship between Lagrangian and Eularian statistics has been established, which makes it possible to determine the coefficients in the Langevin equation from Eularian space-time correlation functions in a frame which moves with the local mean velocity. In our group a research project has been started to validate and study this theory by means of experiments and numerical simulations of fully-developed turbulent pipe flow. Concerning the numerical part direct numerical simulation is used. To this end a spectral numerical method has been developed and tested at relatively low Reynolds number. Mean flow quantities, such as rms of velocity fluctuations and terms in the kinetic energy budget, agree well with other DNS results and measurements. In order to validate the relationship between Lagrangian statistics are needed. Preliminary results already show a reasonable agreement between Lagrangian and Eularian quantities, but for a better study long-time Langrangian averages are necessary. Furthermore, in order to study the dependence on Reynolds number simulations at a higher Reynolds number are required. Both these goals involve a large amount of calculation time.