to boundaries via new algorithms derived from the renormalization group method. Such procedures have the potential to eliminate much uncertainty in the small-scale modeling required for LESs. On the next generation of computers, LESs should be applied to study flows in complex geometries and with complex physics, such as combustion and multiphase flow effects. Other extensions of LESs include such problems as cloud-top entrainment instability mechanisms. Here, fine resolution is essential and, at present, unavailable.

Numerical simulation of thermal convection (and stably stratified flows) has proceeded to resolution sufficiently high (192³) that fully turbulent, high aspect ratio convection may be simulated (at least at moderate Prandtl numbers). Of interest here is not only verification (or repudiation) of traditional scaling laws but also the statistics (distribution function level) of various flow quantities such as vorticity and temperature differences. Here recent theoretical/numerical developments by Sinai and Yakhot are particularly significant. We should note that recent high-precision experiments have shown a sequence of transitions toward progressively "stronger" turbulence through which the flow proceeds as the basic buoyancy driving force is increased. Recent numerical simulations have been able to track these transitions. With increased computing power, it should be possible to explore the asymptotics of this important class of flows.

In summary, John von Neumann's 1949 prediction that computers would prove particularly useful to the study of turbulent flows has come true. Because of still limited theoretical understanding of nonlinear phenomena, engineers, atmospheric scientists, astrophysicists, fluid dynamicists, and others are in great need of computer simulations. To continue recent advances in many aspects of turbulence study, the power and memory of new supercomputers and massively parallel computers are required. In basic research the need for such capabilities will continue until we can achieve resolutions in which asymptotic regimes are manifest, probably not less than 10243 resolution and often much more, depending on the problem.