1. The task to be performed

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

- 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; -
quantitative physical hypotheses for the heat-and-mass-transfer,
micro-mixing, mechanical, chemical and other
**processes which tend to change the PDFs**; -
incorporation of these hypotheses, together with the
conservation laws of physics, into
**mathematical equations**which are capable of being solved numerically; **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.