1. The task to be performed

Computer simulation of many turbulent-flow phenomena, especially those influenced by body forces or chemical reactions, requires:

1. quantitative description of the fluid state in terms of probability-density functions (PDFs), because knowledge of mixture-averages and of root-mean-square fluctuations is not enough;
2. quantitative physical hypotheses for the heat-and-mass-transfer, micro-mixing, mechanical, chemical and other processes which tend to change the PDFs;
3. incorporation of these hypotheses, together with the conservation laws of physics, into mathematical equations which are capable of being solved numerically;
4. computation of the solutions to these equations, which then yield the required PDFs.

The following picture shows what is meant by a probability-density function.

The PDF is the curve on the left of the picture. The task is to calculate its shape.

On the right is a graphical reminder of the fact that a turbulent fluid consists of a random- seeming assembly of fluid fragments in various states.

A one-dimensional probability-density function, with probability- density plotted vertically and temperature horizontally.

The "spikes" on the left and right show that there are large amounts of extremely-cold and extremely-hot fluid.

The curve between them shows how much of the fluid is in the intermediate temperature ranges.

This information is needed for predicting, say:

• How much fluid will tend to rise, and how much to fall, in a gravitational field.
• What will be the total amount of radiation (proportional to the fourth power of temperature);
• At what rate any possible (strongly temperature-dependent) chemical reaction will proceed.

Knowledge of the average temperature is of little or no use for these purposes.