Elsewhere in this report several important areas of computational fluid dynamics (CFD) are discussed in some detail. In particular, aside from general methods and approaches applicable to all areas of computational mechanics (such as adaptive methods, parallel computing, artificial intelligence, nonlinear equation solvers, and stochastic processes), the importance of future research efforts on computer simulation of pollution, chemically reacting flows, combustion, and turbulence is addressed in separate appendices. In this appendix, additional topics on CFD research are discussed that are expected to be crucial to further developments in several key technological areas.

CFD has always been driven primarily by applications. Aerodynamics, numerical weather prediction, acoustics and fluid-structure interaction, propulsion systems, and nuclear reactor design are among the major applications that have encouraged CFD research.

In the aerodynamics field, particularly aircraft design, CFD was originally regarded merely as a complement to wind tunnel experiments, essentially limited to the simulation of incompressible potential flow. For many years, aircraft-design-oriented CFD was mostly concerned with the solution of potential flow equations by panel methods.

Panel methods were widely used to simulate external flow around aircraft. To evaluate viscous effects, the flow field predicted by those methods was also used as a boundary condition for boundary layer calculations based on some form of the Prandtl equations. This methodology clearly had difficulties with separation and shock phenomena. To simulate such phenomena, it was necessary to use more complicated mathematical models than potential flow, the obvious candidates being the Euler and Navier-Stokes equations. However, with the computers available in the late 1960s and early 1970s,

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12
GENERAL COMPUTATIONAL FLUID DYNAMICS
Elsewhere in this report several important areas of computational fluid dynamics (CFD) are discussed in some detail. In particular, aside from general methods and approaches applicable to all areas of computational mechanics (such as adaptive methods, parallel computing, artificial intelligence, nonlinear equation solvers, and stochastic processes), the importance of future research efforts on computer simulation of pollution, chemically reacting flows, combustion, and turbulence is addressed in separate appendices. In this appendix, additional topics on CFD research are discussed that are expected to be crucial to further developments in several key technological areas.
HISTORICAL COMMENTS AND CURRENT STATUS
CFD has always been driven primarily by applications. Aerodynamics, numerical weather prediction, acoustics and fluid-structure interaction, propulsion systems, and nuclear reactor design are among the major applications that have encouraged CFD research.
In the aerodynamics field, particularly aircraft design, CFD was originally regarded merely as a complement to wind tunnel experiments, essentially limited to the simulation of incompressible potential flow. For many years, aircraft-design-oriented CFD was mostly concerned with the solution of potential flow equations by panel methods.
Panel methods were widely used to simulate external flow around aircraft. To evaluate viscous effects, the flow field predicted by those methods was also used as a boundary condition for boundary layer calculations based on some form of the Prandtl equations. This methodology clearly had difficulties with separation and shock phenomena. To simulate such phenomena, it was necessary to use more complicated mathematical models than potential flow, the obvious candidates being the Euler and Navier-Stokes equations. However, with the computers available in the late 1960s and early 1970s,

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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-

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tures with acoustical fluids as encountered in modern submarine design, the deficiencies of contemporary methods and mechanics are evident. Significant advances in acoustical simulation techniques are needed to resolve pressing problems in acoustical-structural interactions. To achieve these types of flow simulations within the next decade, research on parallel computing and smart algorithms must accelerate.
The impact of numerical weather prediction on CFD clearly provides a strong motivation to develop better turbulence models and, certainly during the past 15 years, was a practical motivation behind the strong development of spectral and pseudo-spectral methods. More detailed comments are given below on specific areas in CFD that require research advances during the next decade.
COMPUTATIONAL AERODYNAMICS
Modern aircraft, missiles, and reentry vehicles operate under a wide spectrum of conditions. These conditions range from very low speed incompressible flows where small general aviation aircraft operate to the very high Mach number flight regimes of the National Aerospace Plane (NASP) machines such as the aero-assisted orbital transfer vehicles (AOTV). Delineation of the different flight regimes usually proceeds with a comparison between the mean free molecular path and the characteristic length of the flow field. This ratio is the Knudsen number. When the mean free path is much smaller than the characteristic length of flow, the Navier-Stokes equations are considered to be applicable and the fluid is considered to be a continuum. In the short history of computational aerodynamics, the largest research effort has been expended in the continuum regime.
When the Knudsen number is of order one, the flow is said to fall into the slip flow regime. Here the Navier-Stokes equations may not be applicable, although some success in predicting gas flow in this regime has been achieved by solving the Navier-Stokes equations with modified boundary conditions.
When the mean free path is large compared to the characteristic body length, the flow regime is said to be "free molecule." This is an environment, experienced by orbiting vehicles, where molecules are observed as discrete particles. The study of this flow regime is sometimes referred to as superaerodynamics, a name coined in the early literature on free molecule flow.

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A major area of concentration in computational aerodynamics research has been the development of better ways to numerically model convection phenomena characterized by the so-called convective terms in the Navier-Stokes equations. Methods have been derived that provide an accurate description of the flow physics where a series of Riemann problems are solved to obtain changes in flow variables in each cell. Central to this approach is the problem of establishing the correct flux terms at cell boundaries. In computing these fluxes, either flux splitting or flux difference splitting schemes are used with modern upwind methods. A number of deficiencies in these ideas remain and need to be investigated further.
The Riemann problem is defined for one-dimensional flow. As employed in present methods, the fluxes and the solution for the dependent variables are determined by the ensuing wave field produced when two gases at different states are allowed to interact. In using Riemann solvers for one-dimensional problems, solutions can be computed that can include shock waves with as few as one transition zone. However, the extension to two and three dimensions is presently accomplished by assuming a series of one-dimensional waves, and a truly satisfactory three-dimensional Riemann solver presently does not exist. Basic considerations attest to the importance of such solvers. For example, vorticity is nonexistent in one dimension where the classic Riemann methods are derived. Yet when multidimensional applications are made, shear waves naturally appear. The implication is that the multidimensional solutions using such one-dimensional modeling ideas are inappropriate.
Along this line, the development of effective three-dimensional solvers requires that substantial information be available about any shock present in the flow. It is necessary to deduce both wave orientation and propagation information from the given solution. This is a result of the nonuniqueness of the local solution to the Riemann problem in several space dimensions. These issues lead to questions regarding the comparison of classical shock fitting and solutions with three-dimensional Riemann solvers. Both approaches need to be pursued. In addition, the flux limitation necessary to produce monotone shock transition needs to be studied in detail. This issue becomes especially important when time asymptotic solutions are computed. Typical limiting problems are evidenced by convergence rates that reach a plateau and level out at a reasonable level. The convergence rate and level depend on both the form of limiter and the particular variable

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limited in the solver. Further research is required to provide insight into this behavior.
The complex modeling requirements and delineation of the various flight regimes lead one to question several current approaches used to solve the equations governing fluid flow, particularly in low-density hypersonics. A more satisfying approach may be to attempt to model these flows with more general flow theories. In this light it is worthwhile to expend effort in direct attacks on solutions of the Boltzmann equation. Perhaps some simplification can be achieved by using model distribution functions, which retain the essential features in the flow regime of interest.
ROTORCRAFT FLOWS
Vortex-dominated flows represent another area of major concern. Correct representation of fluid physics is critical in such applications as high-angle-of-attack aerodynamics, helicopter rotor flows, and turbomachinery aerodynamics. To date, conventional numerical schemes have been used to compute flows in the category with limited success. In turbomachinery flows some additional modeling has been incorporated to make three-dimensional calculations feasible. However, in helicopter rotor flows, improved methods are required for solutions to both the hover and forward flight cases. In no other problem is the correct vorticity transport as critical. Methods that use vorticity conservation as an auxiliary constraint would be of great value. Improved induced velocity fields and wake characterization would provide better information for rotor analysis and design than is presently available.
Numerical methods applied to high-angle-of-attack aerodynamics problems lead to difficulties in lee-side flow where flow separation occurs. Present methods exhibit deficiencies that need to be addressed. In inviscid calculations a surprising amount of misunderstanding exists regarding calculations for problems such as vortex roll up over delta wing configurations. Emphasis in this research area will illustrate the need to pose properly such problems and interpret the results as those due to an inviscid solution. Separation at high angle of attack also remains as a critically difficult problem area. Flows over intakes at high angle of attack separate and cause significant losses in thrust and propulsion system efficiency. Turbulence models and the turbulence closure problem are major stumbling blocks in computer modeling in many of these cases.

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FULL-SCALE AIRCRAFT SIMULATION
Large-scale simulation of flow over entire aircraft is necessary to maximize the impact of computational aerodynamics on the design process. To date, very few entire aircraft flow fields have been computed. There needs to be a continued emphasis on applying current methods to compute the flow over full configurations. Here mesh generation is a major issue and remains a major deterrent to full-scale computer simulation. While this will become easier to resolve as faster machines with more available memory are introduced, more emphasis should be given to mesh-generation technology.
MONTE CARLO AND BOLTZMANN EQUATIONS METHOD
In the superaerodynamics area, Monte Carlo methods have been used as the major simulation tool. In both slip flow and free molecular regime, new ideas and approaches are needed. Present emphasis on hypersonic military and commercial vehicles will necessitate development of better techniques for flow calculations in these regimes. A better understanding of nonequilibrium chemical effects is also important. With the emphasis on planetary probes, exotic gases usually not considered must be used in planetary entry calculations. Radiation heat transfer is also a major factor in these cases. While the numerical modeling of chemistry problems is covered elsewhere, their importance in the hypersonic regime cannot be overemphasized. For some flow problems involving complicated physics and/or rarefied gas, simulation methods based on Boltzmann equations and Monte Carlo particle methods seem quite appropriate. Indeed, impressive results have been obtained by Japanese investigators using a Boltzmann-equation-based method to simulate multidimensional hypersonic flow. With the anticipated advances in super and parallel computers, these approaches may be attractive methods in the future.
UNSTEADY FLOWS
Virtually all flow in nature is unsteady. However, to date, simulations have largely been concentrated on computing solutions to steady flow problems. Rocket launch booster separation, store separation from aircraft, and forward flight of a helicopter rotor are all examples where the unsteady effects

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cannot be neglected. Little work has been done in this area partly due to inadequacies in models and partly to lack of computer memory and processing power. Continuing research is needed, particularly to compute solutions using the unsteady Navier-Stokes equations to better understand transient effects on aerodynamics of vehicle components.
ACOUSTICS AND FLUID-STRUCTURE INTERACTION
Simulation of air-borne and water-borne acoustical phenomena by computational methods has become a subject of considerable interest in recent years. With increased performance demands for modern submarines, simulation of acoustical scattering and radiation in complex submerged deformable bodies has emerged as a crucial research area. Today, realistic computer simulations of acoustical phenomena connected with moving submarines is impossible, and several breakthroughs must be made before these very large scale problems can be treated with confidence.
Research is needed on boundary element methods, which seem to be naturally well suited for the exterior problems of acoustics, parallel computing, modeling techniques for the effects of turbulent boundary layers, and new algorithms for treating resonance and problems with multiple scales. Many research areas discussed elsewhere in this report impact on acoustical simulations and must be pursued if there is to be any progress.
WEATHER PREDICTION
A long-time goal of CFD has been reliable weather prediction—a goal that has not yet been met satisfactorily. Much of the research on turbulence, if successful, could have a positive impact on our ability to predict weather.