using Euler or Navier-Stokes models to simulate flow around a full aircraft was impossible, and efforts concentrated on a compressible variant of potential flow embodied in the full potential models of gas dynamics.

During the 1970s upwinded finite difference, finite volume, and finite element methods were developed, leading to the creation of several industrial codes. In addition to upwinding, these codes use multigrid and other fast direct or iterative solvers. By the 1980s transonic flow past a complete aircraft was simulated and used as a production tool in computer design subsonic airplanes with supercritical wings.

During the 1980s the development of compressible Euler solvers, again based on efficient combinations of upwind or artificial viscosity methods, finite element or finite volume approximations, flux-limiting concepts, and multilevel techniques, was an essential step in the simulation of more complicated flows. With these solvers some simulations of three-dimensional inviscid flow around complete aircraft or space vehicles, at reasonably large incidence angles and Mach numbers (including the hypersonic range), have been made. At a recent European/U.S. meeting on CFD, excellent agreement of the results of compressible Euler test cases was reported despite the fact that they were three-dimensional hypersonic flow simulations and that participants were using a large variety of numerical methods. On the other hand, discrepancy among the results of the viscous flow simulations remains large, and three-dimensional unsteady simulations remain out of reach of contemporary CFD capabilities. Additionally, difficulties in modeling pure convection phenomena still exist and researchers point to open questions that remain in developing reliable Euler codes for unsteady three-dimensional flows.

The analysis of propulsion systems, such as the space shuttle main engine and cooling systems for nuclear reactors, has encouraged much research on new efficient incompressible viscous flow simulators. Today codes exist that can simulate three-dimensional flow at Reynolds numbers of the order of several thousand, and several of these codes have turbulent flow simulation options, usually based on k-Є turbulence models. As yet, there are no viscous flow simulators able to simulate accurately and routinely three-dimensional incompressible and compressible viscous flow at Reynolds numbers greater than 10⁵ in complicated geometries. The study of computer simulation of air-and water-borne acoustical phenomena has come to the forefront of computational mechanics research in recent years. When fluid-structure interactions are considered, such as the interaction of submerged elastic struc-