par Degrez, Gérard
Référence Computational Fluid Dynamics, Springer Berlin Heidelberg, page (183-234)
Publication Publié, 2009
Partie d'ouvrage collectif
Résumé : The compressible Euler and Navier-Stokes equations represent the most sophisticatedmodels of single-phase flows of single-component Newtonian fluids. As such,they allow the analysis of complex inviscid and viscous flow phenomena includingrotational flows caused by curved shock waves or viscous/inviscid interactions leadingto flow separation. As a counterpart, the numerical techniques required to solvethese equations are also the most sophisticated and the numerical effort needed toobtain them is also the greatest. This is schematically represented in Fig. 9.1 takenfrom Green's [18] review of the state- of-the-art in numerical methods in aeronauticalfluid dynamics.The difficulties of solving complex steady compressible flows were alreadypointed out in the first part of this volume, in which the blunt body problem wastaken as an illustrative example. It was shown that the crux of the difficulty lies inthe mixed character of the flow, involving regions governed by 'elliptic' equationsand others governed by 'hyperbolic' equations. Finally, the solution to the problemwas found by solving the time dependent equations using a time marching method,taking advantage of the uniform nature of the unsteady equations with respect totime, independently of the subsonic or supersonic character of the flow.1Following that breakthrough, many methods were developed to integrate the unsteadyEuler or Navier-Stokes equations. These methods can be classified in twomain categories: explicit and implicit methods (Part I, Sect. 5.3).Historically, explicit methods were developed earlier, because of their greatersimplicity. Several examples were given in Part I, Chap. 7. The major limitation ofthese methods is their stability characteristics which impose an upper bound on theusable integration time step. In recent years, implicit methods have been developedto overcome this limitation and have proved more efficient than the former explicitmethods, which justifies their study.In Sect. 9.2, we shall examine solution techniques for simpler flows and explainwhy these techniques fail for the solution of the steady compressible Euler/Navier-Stokes equations. In Sect. 9.3, stability properties of numerical integrationtechniques will be studied in detail first for ordinary differential equations, then forpartial differential equations. In Sect. 9.4, it will be shown how an implicit methodcan be constructed to solve partial differential equations such as the Euler or Navier-Stokes equations. It will be seen that this can be subdivided into three steps, thechoice of an explicit discretization scheme, the choice of an implicit operator andfinally the choice of a solution strategy, which will be discussed in turn. For the firststep, the issue of numerical dissipation will turn out to be crucial, and this conceptwill be discussed in detail. As in Part I, only the finite difference method is consideredas the space discretization technique, but, as will be mentioned in the lecture,most of the concepts discussed and of the basic methods described apply equallyto finite volume discretizations (especially on structured meshes) and some to finiteelement discretizations.The content of these notes will remain rather basic except in a few instances, inaccordance with the objectives of this book. In particular, no individual scheme willbe examined in great detail. For additional information, the reader is referred to thevery comprehensive survey of CFD methods by C. Hirsch [22, 23] and, finally, tothe original literature. © Springer-Verlag Berlin Heidelberg 2009.