This review attempts to summarize the current state of fluid flow simulations with particular regard to numerical discretization techniques, treatment of large Reynolds numbers, and past and future accomplishments of the simulation methodology. Three alternative mathematical representations of the flow are presented and compared for an idealized two-dimensional flow, namely, use of the velocity and pressure as dependent variables (primitive equation), use of vorticity and stream function as dependent variables, and expansion of either of these in a truncated set of orthogonal eigenfunctions of the boundary-value problem (whose coefficients become the dependent variables). Various classes of finite-difference algorithms are discussed with regard to their conditional stability as well as their ability to preserve invariants of the continuous equations. Methods of removing aliasing of high-frequency solutions are discussed. Parameterization and other techniques for dealing with the high-Reynolds-number regime in which solutions exhibit a wide range of scales are considered in depth. We conclude that certain types of information can be better obtained by numerical simulation than by other techniques.

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