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This thesis describes a new method for simulating high Reynolds number turbulence which requires much less computing power. This involved both theoretical work - to understand and model the important processes - and computational work, to implement the model efficiently. There are 'many different techniques for modelling particle dispersion in turbulent flow (e.g. K-theory and Random Flight) but they make assumptions about the fluid-particle interaction and require empirical coefficients. Theoretical work on the motion of bubbles and varticles in idealised flows has shown that the instantaneous structure of the velocity field is important in determining particle trajectories, and that particle motion cannot currently be modelled reliably in terms of time- or ensemble-averaged fluid velocities. Therefore the solution of many practical problems requires the simulation of the instantaneous structure of a turbulent velocity field. This can now be provided with the very large computers and large amounts of computer time; even then, only low Reynolds number turbulence can be simulated. In the method developed here, the velocity field of homogeneous isotropic turbulence is simulated by a large number of random Fourier modes varying in space and time. They are chosen so that the flow field has certain properties, namely (i) it satisfies continuity, (ii) the two point Eulerian spatial spectra have known form (e.g. the Kolmogorov inertial subrange), (iii) the time dependence is modelled by dividing the turbulence into large- and small-scales eddies, and by assuming that the large eddies advect the small eddies which also decorrelate as they are advected, (iv) the large- and small-scale Fourier modes are each statistically independent and Gaussian. Computations of the streamlines in a sequence of realisations of the flow show that they have a similar structure to that obtained from direct numerical simulations. New results for the statistics of high Reynolds number turbulent flows are obtained, for the velocity and pressure fields . Particle statistics are obtained by computing the trajectories of many particles and taking the ensemble average. Particle dispersion has been computed for a range of particle parameters and the results agree well with experimental measurements such as those of Snyder and Lumley; this enables us to compute empirical coefficients (e.g. Lagrangian timescales) for use in simpler models such as Random Flight, and for modelling other processes such as combustion and mixing. Rapid Distortion Theory is used to investigate the effects of high shear rate on the structure of homogeneous turbulence in chapter 4. The results show that an important effect of the shear acting on initially isotropic turbulence is the selective amplification of structures having large length scale in the mean flow direction.