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3. Pioneering efforts to compute the PDFs

Dopazo and O'Brien (1974) drew attention to the fact that differential transport equations could be formulated for the PDF itself. However, they did not provide numerical solutions.

When Pope (1980) did address the numerical-solution problem, he concluded that it was too difficult to solve with available methods and computers. He therefore introduced a Monte-Carlo approach to PDF transport, which has been adopted by the majority of later workers.

This approach undoubtedly reveals much about the phenomena to which it has been applied. Unfortunately its computational expense is still (rightly?) judged by practising engineers to be too great.

It appears also that Monte Carlo techniques cannot easily be used with some of the physical hypotheses which intuition suggests as being worthy of study in the present context.

Nevertheless, the pioneering work of these and other authors has influenced the recent developments of the multi-fluid approach.