A Probability Distribution Function for Turbulent Flows

Abstract

There is current interest in being able to account for the presence of fluctuations when calculating velocity, temperature, and composition distributions in turbulent flow fields with the effect of energetic chemical reactions included. Examples of this type of flow occur in combustors and afterburning rocket plumes. There are two major approaches to solving this problem. The first is evaluation of the higher moments of the equations of motion, the energy equation, and the species continuity equations.¹ This approach requires information which is currently not available experimentally. Methods using the concept of probability density functions (PDF’s) for the velocity and the scalar properties of the flow field have been described in Refs. 2, 3, and 4. In a turbulent flow each property varies with time at any given point in the flow field, and this variation can be represented using a PDF. The integral of a PDF between two values of a parameter in the field represents the fraction of time the parameter has a value between these two limits. Some attempt has been made to calculate PDF’s a priori^{2, 3} however, these methods require some rather drastic assumptions and are limited to rather simple cases. An alternate approach is to use PDF’s which are derived from available experimental data.⁴

The research reported in this paper was conducted at the Arnold Engineering Development Center, Air Force Systems Command, Arnold Air Force Station, Tennessee. Research results were obtained by ARO, Inc., contract operator at AEDC. Major financial support was provided by the Air Force Office of Scientific Research under Program Element 61102F, Project 9711; Dr. B. T. Wolfson was project monitor. Further reproduction is authorized to satisfy the needs of the U. S. Government.