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By numerically integrating the Fourier transformed exact conservation equations of reactants over time, using a numerical simulation of a homogeneous isotropic turbulent velocity field for the convection velocity, O'Brien's and Toor's statistical independence hypotheses were tested using numerical experiments in which reaction and mixing condi­ tions were carefully controlled. Two simulated turbulent velocity fields that obey continuity were used for the tests. The first has a constant specified energy spectrum (and hence constant time and distance scales) but lacks interaction between Fourier modes. The second velocity field, which is generated using a modified form of the Navier-Stokes equation, has interaction and conserves turbu­ lent energy but the energy spectrum is tremsient. For single component second-order reactions, O'Brien's statistical independence hypothesis successfully predicted that the decay of concentration fluctuation intensity for turbulent mixing with reaction can be estimated by the product of the intensity decay for pure mixing without reac­ tion and intensity decay for pure reaction without mixing. The results suggest that counteracting or canceling of two opposing physical tendencies may be responsible for the success of O'Brien's hypothesis. However, O'Brien's hypoth­ esis of Fourier mode independence exhibited significant

Turbulent mixing with chemical reaction and Collection of inertialess particles on cylinders and spheroids with electrical forces and gravitation