The methods presented in Chapter 5 attempt to close the chemical source term by making a priori assumptions concerning the form of the joint composition PDF. In contrast, the methods discussed in this chapter involve solving a transport equation for the joint PDF in which the chemical source term appears in closed form. In the literature, this type of approach is referred to as transported PDF or full PDF methods. In this chapter, we begin by deriving the fundamental transport equation for the one-point joint velocity, composition PDF. We then look at modeling issues that arise from this equation, and introduce the Lagrangian PDF formulation as a natural starting point for developing transported PDF models. The simulation methods that are used to ‘solve’ for the joint PDF are presented in Chapter 7.

Introduction

As we saw in Chapter 1, the one-point joint velocity, composition PDF contains random variables representing the three velocity components and all chemical species at a particular spatial location. The restriction to a one-point description implies the following.

• The joint PDF contains no information concerning local velocity and/or scalar gradients. A two-point description would be required to describe the gradients.

• All non-linear terms involving spatial gradients require transported PDF closures. Examples of such terms are viscous dissipation, pressure fluctuations, and scalar dissipation.

The one-point joint composition PDF contains random variables representing all chemical species at a particular spatial location. It can be found from the joint velocity, composition PDF by integrating over the entire phase space of the velocity components. The loss of instantaneous velocity information implies the following.