Statistical constraints on scalar variables in turbulent flows

We consider the statistical behavior of scalar variables in turbulent flows. The presence of chemical reactions requires statistical models for third and higher moments in order to close the rate equations. A realistic analysis should not be restricted to small fluctuations. The results presented here are completely free from such restrictions. The essential points to be demonstrated are: (1) Given A-bar and A-squared-bar, we can obtain stringent statistical bounds on a-squared B1)-bar and related moments. These bounds are found to be of interest in discussing recent experiments. (2)Maximum and minimum third (and higher) moments can be reached only with Dirac functions (i.e., discrete distributions). (3)We can always realize a statistically acceptable choice of A-bar and A-squared bar with a few Dirac functions (the minimum number is two distinct ones). The statements given above hold true when several moments (rather than only A-bar and A1-squared-bar) are given as well as when the means of several variables are given. As one consequence, we shall see that we can find, for the purposes of modeling, a discrete distribution that represents the desired set of moments and that is statistically legitimate for all allowed values of the derived moments (box model).