P. L. RoeCollege of Aeronautics, Cranfield Institute of Technology,Cranfield MK43 0AL, EnglandIntroductionComputer simulations of fluid flow provide today the sort of detailedinformation concerning special cases that could previously only beobtained from experime.nts. The computer is attractive as a replacement forexperiments that are difficult, dangerous, or expensive, and as an alternativeto experiments that are impossible. Nevertheless, a computer simulationdoes not have quite the same status as a physical experiment because atpresent there usually remains some doubt about its accuracy. Even thoughthe computer code may be free of error to the extent that it operates exactlyas its author intended, it is seldom possible to give a rigorous proof thatthese intentions were in all respects correct. Most of the practical codeswritten to solve complicated problems contain empirical features, some-times in the form of "adjustable constants" whose values must be "tuned"by appeal to the experiments that the simulations are intended to displace.A computer code is described as being "robust" if it has the virtue ofgiving reliable answers to a wide range of problems without needing to beretuned. The ideal code would be one that fully met some declaredspecification of accuracy and problem range, and whose every line was anecessary contribution to that aim. Few codes yet approach that ideal; amajor impediment is that we presently have little idea what properties canbe specified without contradiction.In recent years, however, our understanding of computer codes for aparticular class of problems has advanced some way toward completeness.The problems are sufficiently complex that naive numerical techniques canproduce disaster, yet sufficiently simple that well-understood physics can3370066-4189/86/0115-0337502.00www.annualreviews.org/aronline Annual Reviews

Characteristic-based schemes for the euler equations

The development of upwind relaxation algorithms for obtaining efficient steady-state solutions to the compressible Navier-Stokes equations is described. The method is second-order accurate spatially and naturally disipative, using third-order flux splitting of the pressure and convective terms and second-order central differencing for shear and heat flux terms. A line Gauss-Seidel relaxation approach, shown to be unconditionally stable for model convection and diffusion equations, is used. The algorithm is demonstrated for several flows using the thin-layer form of the equations, including the problem of shock-induced separation over a flat plate.

Upwind relaxation algorithms for the Navier-Stokes equations

A systematic procedure for constructing semi-discrete families of 2m - 1 order accurate, 2m order dissipa-tive, variation diminishing, 2m + 1 point band width, conservation form approximations to scalar conservation laws is presented. Here m is an integer between 2 and 8. Simple first order forward time discretization, used together with any of these approximations to the space derivatives, also results in a fully discrete, variation diminishing algorithm. These schemes all use simple flux limiters, without which each of these fully discrete algorithms is even linearly unstable. Extensions to systems, using a nonlinear field-by-field decomposition are presented, and shown to have many of the same properties as in the scalar case. For linear systems, these nonlinear approximations are variation diminishing, and hence convergent. A new and general criterion for approximations to be variation diminishing is also given. Finally, numerical experiments using some of these algorithms are presented.

Very High Order Accurate TVD Schemes

Fluid flows within turbomachinery tend to be extremely complex in nature Understanding such flows is crucial to improving current designs of turbomachinery The computational approach can be used to great advantage in understanding flows in turbomachinery A finite difference, unsteady, thin layer, Navier-Stokes approach to calculating the flow within an axial turbine stage is presented The relative motion between the stator and rotor airfoils is made possible with the use of patched grids that move relative to each other The calculation includes endwall and tip leakage effects An introduction to the rotor-stator problem and sample results in the form of time averaged surface pressures are presented The numerical data are compared with experimental data and the agreement between the two is found to be good

Three-dimensional Navier-Stokes simulations of turbine rotor-stator interaction

Fluid flows within turbomachinery tend to be extremely complex. Understanding such flows is crucial to efforts to improve current turbomachinery designs, and the computational approach can be used to great advantage in this regard. This study presents a finite-difference, unsteady, thin-layer Navier-Stokes approach to calculating the flow within an axial turbine stage. The relative motion between the stator and rotor airfoils is made possible with the use of patched grids that move relative to each other. The calculation includes end-wall and tip-leakage effects. The numerical methodology is presented in detail in the present paper (Part I). The computed results and comparisons of these results with experimental data are presented in a companion paper (Part II).

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S. Osher

Papers

Application of a new class of high accuracy TVD schemes to the Navier-Stokes equations