Benjamin Franklin and Computational Fluid Dynamics: The Population Approach to Turbulence

Abstract

Turbulent fluids can be usefully considered as populations, the members of which are distinguished by their positions in nongeometric spaces of which the dimensions may be arbitrarily chosen. These dimensions can be discretised in the same manner as is customary for distance and time. In a distancetime “cell,â€‌ variations in mass per unit volume are computed from balances of mass, energy, etc., in which the influences of neighbouring cells are represented in terms of convection and diffusion. Just so can distributions in population space be deduced from balances of mass migration effected by other physical or chemical processes. One such process is movement in population space, as when exothermic chemical reaction moves fluid from a lower to a higher temperature interval; it is akin to convection in geometric space. Another is coalescence, by way of which population members having differing attributes merge and are replaced by elements of intermediate attribute. A third is differential convection, whereby the differing body forces experienced by population elements of differing densities, by reason of the pressure gradients which they share, cause “siftingâ€‌ and “filteringâ€‌ to change the population composition. Just as turbulent diffusion is expressed by mancreated hypotheses, so hypotheses must be invented for movement in population space, coalescence, and differential convection. The above ideas are explained in the present paper and represented mainly by reference to the age distribution in human populations, to swirling flows and to the distributions of mixture ratio and temperature within turbulent combusting gases.