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This volume contains the proceedings of the Ninth GAMM-Conference on Numerical Methods in Fluid Mechanics, held at the Ecole Polytechnique Federale de Lausanne, Switzerland, on September 25-27, 1991. This conference, as weil as the preceding eight ones, was organized by the GAMM Committee on Numerical Methods in Fluid Dynamics. 1t was probably also the last one in this successful series of conferences, since in the future, a bi-annual European meeting on Computational Fluid Dynamics will be organized by ECCOMAS, a new organization, representing all existing European professional societies working in this field. The conference was attended by about 100 registered participants coming from all corners of the world. These proceedings contain the written version of the 56 papers presented during the meeting. In order to eliminate all kinds of errors, omissions, spelling mistakes, etc. these papers have been reviewed by an ad hoc scientific committee. As a result of this work a good deal of the papers were sent back to the authors for correction. This procedure has somewhat delayed the publication of this volume. We feel, however, that this has been a worth while effort. The subjects treated during the meeting represent weil current interests in CFD. For instance, multigrid and multiblock techniques for viscous as weil as inviscid 3D flows were presented. Similarly, finite rate chemistry hypersonic flows still attracts many scientists due to the continuation of the Hermes Project.

This volume contains the Proceedings of the Eighth GAMM Conference on Numerical Methods in Fluid Mechanics, held in Delft, The Netherlands, September 27-29, 1989, at Delft University of Technology. This international conference is organized bi-annually at various locations in Europe by the Committee on Numerical Methods in Fluid Dynamics of the Gesellschaft fiir Angewandte Mathematik und Mechanik, chaired at present by W. Kordulla. At the moment, the organizing committee draws its-members from ten European countries. The conference attracted a worldwide attendance of about 140 registered partici­ pants. The book contains the written versions of the 61 papers that were presented and a report on the GAMM-Workshop "Numerical Simulation of Oscillatory Convec­ tion in Low-Pr Fluids" held in Marseille, October 12-14, 1988. The organizing committee had a difficult task in selecting the papers from 145 submitted abstracts. In view of the high quality of the abstracts it was decided to allow more papers than in previous conferences in this series. The contents of this volume reflect current trends in computational fluid dynamics. The renaissance of hypersonic aerodynamics manifests itself clearly, strongly stimulated in Europe by the Hermes project and plans for a later future. Notable advances are reported in adaptive and multigrid computing methods, visual­ ization and computing in three space dimensions. The local organization was taken care of by the Congress Office of Delft Universi­ ty of Technology, under the able directorship of Mrs. M.H.P. Komen-Zimmerman and Mrs. A. de Bruin.

The GAMM-Commi ttee for Numerical Methods in Fluid Mechanics (GAMM-Fachausschuss für Numerische Methoden in der Strömungsmechanik) has sponsored the organization of a GAMM Workshop dedicated to the numerical simulation of three­ dimensional incompressible unsteady viscous laminar flows to test Navier-Stokes solvers. The Workshop was held in Paris from June 12th to June 14th, 1991 at the Ecole Nationale Superieure des Arts et Metiers. Two test problems were set up. The first one is the flow in a driven-lid parallelepipedic cavity at Re = 3200 . The second problem is a flow around a prolate spheroid at incidence. These problems are challenging as fully transient solutions are expected to show up. The difficulties for meaningful calculations come from both space and temporal discretizations which have to be sufficiently accurate to resol ve detailed structures like Taylor-Görtler-like vortices and the appropriate time development. Several research teams from academia and industry tackled the tests using different formulations (veloci ty-pressure, vortici ty­ velocity), different numerical methods (finite differences, finite volumes, finite elements), various solution algorithms (splitting, coupled, ... ), various solvers (direct, iterative, semi-iterative) with preconditioners or other numerical speed-up procedures. The results show some scatter and achieve different levels of efficiency. The Workshop was attended by about 25 scientists and drove much interaction between the participants. The contributions in these proceedings are presented in alphabetical order according to the first author, first for the cavi ty problem and then for the prolate spheroid problem. No definite conclusions about benchmark solutions can be drawn.

Der Band enthält den Abschlußbericht des DFG-Schwerpunktprogramms "Flußsimulation mit Höchstleistungsrechnern". Es führt die Arbeiten fort, die schon als Band 38 in der Reihe "Notes on Numerical Fluid Mechanics" erschienen sind.Work is reported, which was sponsored by the Deutsche Forschungsgemeinschaft from 1993 to 1995. Scientists from numerical mathematics, fluid mechanics, aerodynamics, and turbomachinery present their work on flow simulation with massively parallel systems, on the direct and large-eddy simulation of turbulence, and on mathematical foundations, general solution techniques and applications. Results are reported from benchmark computations of laminar flow around a cylinder, in which seventeen groups participated.

This volume entitled "European Computational Aerodynamics Research Project (ECARP)" contains the contributions of partners presented in two work­ shops focused on the following areas: Task 3 on Optimum Design and Task 4.2 on Navier Stokes Flow algorithms on Massively Parallel Processors. ECARP has been supported by the European Union (EU) through the Indus­ trial and Materials Technology Programme, Area 3 Aeronautics, with the Third Research Framework Programme (1990-1994). Part A of this volume is focused on computational constrained optimization as a follow up of the EU research project" Optimum Design in Aerodynamics" , (AERO-S9-0026) dealing with more viscous flow based real applications. It pro­ vides the reader with a set of optimization tools and referenced data useful in modern aerodynamic design. Task 3 of the project entitled "Optimum Design" brought together 13 Euro­ pean partners from the academic and industrial aeronautic oriented community showing state of the art expertise in traditional automated optimization software on current computer technology to improve the capability to optimize aircraft shapes.

Contains selected contributions to a workshop on "Numerical Methods for the Navier-Stokes Equations" held at the IBM Scientific Center, Heidelberg during October 25-28, 1993. The topics treated are advanced computational methods for the Navier-Stokes equations in fluid mechanics.

While the first two of us have worked mainly on algorithmic problems, the third has been concerned off and on with modelling and application problems of Euler methods. The problem which we soon faced, was that the field still is moving at a fast pace, for instance because hyper sonic computation problems became more and more important.

Computational Fluid Dynamics (CFD) has made remarkable progress in the last two decades and is becoming an important, if not inevitable, analytical tool for both fundamental and practical fluid dynamics research. The analysis of flow fields is important in the sense that it improves the researcher's understanding of the flow features. CFD analysis also indirectly helps the design of new aircraft and/or spacecraft. However, design methodologies are the real need for the development of aircraft or spacecraft. They directly contribute to the design process and can significantly shorten the design cycle. Although quite a few publications have been written on this subject, most of the methods proposed were not used in practice in the past due to an immature research level and restrictions due to the inadequate computing capabilities. With the progress of high-speed computers, the time has come for such methods to be used practically. There is strong evidence of a growing interest in the development and use of aerodynamic inverse design and optimization techniques. This is true, not only for aerospace industries, but also for any industries requiring fluid dynamic design. This clearly shows the matured engineering need for optimum aerodynamic shape design methodologies. Therefore, it seems timely to publish a book in which eminent researchers in this area can elaborate on their research efforts and discuss it in conjunction with other efforts.

This volume contains 37 contributions in which the research work is summarized which has been carried out between 1984 and 1990 in the Priority Research Program "Physik abgeloster Stromungen" of the Deutsche Forschungsgemeinschaft (DFG, German Research Society).

In full multigrid methods for elliptic difference equations one works on a sequence of meshes where a number of pre- and/or postsmoothing steps are performed on each level. As is well known these methods can converge very fast on problems with a smooth solution and a regular mesh, but the rate of convergence can be severely degraded for problems with unisotropy or discontinuous coefficients unless some form of robust smoother is used. Also problems can arise with the increasingly coarser meshes because for some types of discretization methods, coercivity may be lost on coarse meshes and on massively parallel computers the computation cost of transporting information between computer processors devoted to work on various levels of the mesh can dominate the whole computing time. For discussions about some of these problems, see (11). Here we propose a method that uses only two levels of meshes, the fine and the coarse level, respec­ tively, and where the corrector on the coarse level is equal to a new type of preconditioner which uses an algebraic substructuring of the stiffness matrix. It is based on the block matrix tridiagonal structure one gets when the domain is subdivided into strips. This block-tridiagonal form is used to compute an approximate factorization whereby the Schur complements which arise in the recursive factorization are approximated in an indirect way, i. e.

The Dutch Association for Numerical Fluid Mechanics (Kontaktgroep Numerieke Stromingsleer, KNSL) was founded in The Netherlands in November 1974. Since then, the Association has organized meetings twice a year. The present volume contains the proceedings of the 25th meeting, held on October 20, 1986, at Delft University of Technology. The purpose of the KNSL is to provide an opportunity for researchers in numerical fluid mechanics to meet regularly and to inform each other about their research in an informal atmosphere. Presentations preferably describe work in progress, and discussion of unsolved problems and unresolved difficulties is encouraged. The working language is Dutch. Nevertheless, science and technology are worldwide activities, and therefore it was decided to publish the proceedings of the 25th meeting in English. The nine contributions to the 25th meeting were selected by profs. A.I. van de Vooren, C.B. Vreugdenhil and the editor. These works are far from covering completely all activity in this field in this country, but they are typical of what is going on. A wide range of subjects is discussed, including fundamental aspects of spectral methods, solution methods for the Euler equations and aeronautical applications, viscous ship hydrodynamics, shallow water equations, viscous flows with capillary and non-Newtonian effects, and turbulent heat transfer with industrial applications. The 25th meeting of the KNSL was supported financially by ECN (Netherlands Energy Research Foundation), MARIN (Maritime Research Institute Netherlands), NLR (National Aerospace Laboratory), WL (Delft Hydraulics Laboratory), VEG Gasinstituut, Delft University of Technology and University of Twente.

This volume contains 23 contributions to the 10th GAMM-Seminar, which was held in Kiel on the 14-16 of January 1994. The central topics are advanced numerical techniques for solving flow problems. Five papers are devoted to parallel algorithms, a further one to domain decomposition techniques.

This volume contains 18 papers presented at the Kiel Seminar in 1993. The Kiel Seminars are well-known, and cover special areas in numerical methods for partial differential equations, numerical linear algebra, numerical methods for integral equations and related areas.

This volume contains contributions to the BRI-TE/EURAM, Area 5, pilot phase programme on the validation of computational fluid dynamics (CFD) codes, EUROVAL. Two-and three-dimensional flows are investigated with particular emphasis on precise comparison against measurements.

This volume contains 22 papers presented at the Kiel Seminar in 1992. The Kiel Seminars are well known, and cover special areas in numerical methods for partial differential equations, numerical linear algebra, numerical methods for integral equations and related areas.

11 The GAMM Committee for Efficient Numerical Methods for Partial 11 Differential Equations organises workshops on subjects concerning the algorithmic treatment of partial differential equations. The topics are discretisation methods like the finite element and the boundary element method for various types of applications in structural and fluid mechanics. Particular attention is devoted to the advanced solution methods. The series of such workshops was continued in 1991, January 25- 27, with the 7th Kiel-Seminar on the special topic 11 11 Numerical techniques for boundary element methods at the Christian-Albrechts-University of Kiel. The seminar was attended by 57 scientists from 8 countries. The list of topics contained applications of the boundary element method (BEM) to various problems of practical interest, algo­ rithmic aspects of the BEM (coupling with finite element method, parallelisation), convergence analysis, and in particular the treatment of the numerical integration. In six contributions the quadrature of weakly singular, Cauchy singular, and hypersingular integrals is analysed. 11 11 The editor thanks the DFG-Schwerpunkt Randelementmethoden for its support. He also likes to express his gratitude to all persons involved in the organisation of the seminar.

The most frequently used method for the numerical integration of parabolic differential equa­ tions is the method of lines, where one first uses a discretization of space derivatives by finite differences or finite elements and then uses some time-stepping method for the the solution of resulting system of ordinary differential equations. Such methods are, at least conceptually, easy to perform. However, they can be expensive if steep gradients occur in the solution, stability must be controlled, and the global error control can be troublesome. This paper considers a simultaneaus discretization of space and time variables for a one-dimensional parabolic equation on a relatively long time interval, called 'time-slab'. The discretization is repeated or adjusted for following 'time-slabs' using continuous finite element approximations. In such a method we utilize the efficiency of finite elements by choosing a finite element mesh in the time-space domain where the finite element mesh has been adjusted to steep gradients of the solution both with respect to the space and the time variables. In this way we solve all the difficulties with the classical approach since stability, discretization error estimates and global error control are automatically satisfied. Such a method has been discussed previously in [3] and [4]. The related boundary value techniques or global time integration for systems of ordinary differential equations have been discussed in several papers, see [12] and the references quoted therein.

This volume contains the papers of the 11th Symposium of the AG STAB (German Aerospace Aerodynamics Association). In this association those scientists and engineers from universities, research-establishments and industry are involved, who are doing research and project work in numerical and experimental fluid mechanics and aerodynamics for aerospace and other applications. Many of the contributions are giving results from the "Luftfahrtforschungsprogramm der Bundesregierung (German Aeronautical Research Programme). Some of the papers report on work sponsored by the Deutsche Forschungsgemeinschaft, DFG, which also was presented at the symposium. The volume gives a broad overview over the ongoing work in this field in Germany.

This volume contains papers presented to a EUROMECH-Colloquium held in Munich, September 30 to October 2, 1985. The Colloquium is number 199 in a series of colloquia inaugurated by the European Mechanics Committee. The meeting was jointly organized by the 'Lehrstuhl fur Stromungsmechanik' at the 'Technische Universitat Munchen' and the 'Institut fur Physik der Atmosphare' of the 'Deutsche Forschungs- und Versuchsanstalt fur Luft- und Raumfahrt' (DFVLR) in Oberpfaffenhofen. 'Direct' and 'large eddy simulation' are terms which denote two closely con­ nected methods of turbulence research. In a 'direct simulation' (DS), turbu­ lent motion is simulated by numerically integrating the Navier-Stokes equations in three-dimensional space and as a function of time. Besides ini­ tial and boundary conditions no physical simplifications are involved. Com­ puter resources limit the resolution in time and space, though simulations with an order of one million discrete points in space are feasible. The simu­ lated flow fields can be considered as true realizations of turbulent flow fields and analysed to answer questions on the basic behaviour of turbulence. Direct simulations are valid as long as all the excited scales remain within the band of resolved scales. This means that viscosity must be strong enough to damp out the not resolved scales or the simulation is restricted to a lim­ ited integration-time interval only. In summary, DS provides a tool to investigate turbulent motions from first principles at least for a finite band of scales.

Contains 14 papers, presented at the Symposium on High Performance Scientific Computing organized by FORTWIHR and held on June 17-18, 1993 in Munich, Germany. The papers give a survey on recent research efforts and industrial applications of numerical simulation in various areas.

It has been recognized by the European Union (EU) that the survival of the Aeronautical Industries of Europe in the highly competitive World Aviation Market is strongly dependent on such factors as time-to-market of a new or derivative aircraft and on its manufacturing costs but also on the achievement of a competitive technological advantage by which an increased market share can be gained. Cooperative research is therefore continuously encouraged and co-financed by the European Union in order to strengthen the scientific and technological base of the Aeronautical Industries thus facilitating the future design and manufacture of civil air­ craft products and providing the technological edge needed for survival. Targets of research within Area 3, Aeronautics, of the Industrial and Materials Technology Programme (1991 - 1993) have been identified to be aircraft efficiency, cost effectiveness and environmental im­ pact. Concerning aircraft efficiency - relevant to the present research - a reduction in aircraft drag of 10010, a reduction in aircraft fuel consumption of 30010 and a reduction in airframe, en­ gine and system weight of 20% are envisaged. Meeting these objectives has, of course, also a strong positive impact on the environment. In order to further technology, the philosophy is to avoid spreading the available resources too thinly and rather concentrate on the feasibility demonstration of a limited number of tech­ nologies of high economic and industrial impact.

The computation of complex turbulent flows by statistical modelling has already a long history. The most popular two-equation models today were introduced in the early sev­ enties. However these models have been generally tested in rather academic cases. The develope­ ment of computers has led to more and more acurate numerical methods. The interactions betwe~n numerical and modelling techniques are generally not well mastered. Moreover, computation of real life cases, including 3D effects, complex geometries and pressure gra­ dients based on two-equation models with low-Reynolds treatment at the proximity of walls are not really of common use. A large number of models has been proposed; this is perhaps the sign that none of them is really satisfactory, and then the assessment of their generality is not an easy task: it requires a lot of understanding of the physics and a lot of work for testing the large number of relevant cases in order to assess their limits of validity which is a condition for an improved confidence in engineering applications. This is probably why workshops and working groups are frequent and the ETMA consor­ tium has choosen to build a state of the art in theoretical and numerical statistical turbu­ lence modelling for real life computations by taking some marks with respect to previous workshops such as the Stanford meetings (1980,1981); some problems are kept or updated by new experiments, some problems are discarded, some new problems are introduced; the focus is kept on flows with 2D geometries.

This volume contains the papers of the 10th AG STAB (German Aerospace Aerodynamics Association). In this association all those scientists and engineers from universities, research-establishments and industry are involved, who are doing research and project work in numerical and experimental fluid mechanics and aerodynamics for aerospace and other applications. Many of the contributions are giving first results from the "Luftfahrtforschungsprogramm der Bundesregierung (German Aeronautical Research Program) 1995-1998". Some of the papers report on work sponsored by the Deutsche Forschungsgemeinschaft, DFG, which also was presented at the symposium. The volume gives a broad overview over the ongoing work in this field in Germany.

This volume contains 20 contributions to the 1st GAMM-Seminar at ICA Stuttgart, which was held in Stuttgart, October 12 - 13, 1995. In the field of environmental sciences, numerical procedures for the simulation of ecological problems are growing increasingly topical. The solution of typical problems in environmental research is closely connected with numerical supercomputing. The main subject of the seminar was the modeling and numerical simulation of ground water and soil water. Further topics were multi-scale modeling, special discretization schemes, adaptivity, multi-grid methods, heterogenity, parameter identification, homogenization, density driven groundwater flow, and coupling of transport and chemistry.

The GAMM Committee for "Efficient Numerical Methods for Partial Differential Equations" organizes seminars and workshops on subjects concerning the algorithmic treatment of partial differential equations. The topics are discretisation methods like the finite element and the boundary element method for various type of applications in structural and fluid mechanics. Particular attention is devoted to the advanced solution methods. The series of such seminars was continued in 1995, January 20-22, with the 11th Kiel-Seminar on the special topic Numerical Treatment of Coupled Systems at the Christian-Albrechts-University of Kiel. The seminar was attended by 100 scientist from 9 countries. 23 lectures were given, including two survey lectures. Different kinds of couplings are considered in this volume. The coupling of different components may occur in the physical model. On the other hand, a coupling of subsystems can be generated by the numerical solution technique. General examples of the latter kind are the domain decomposition (see p. 128) or subspace decomposition (p. 117). The local defect correction method couples different discretizations of the same problem in order to improve the results, although the basic linear system to be solved remains unchanged (p. 47). In general, the aim of the numerical coupling is to make use of (efficient) subsystem solvers (p. 1). The combination of different discretization techniques is mentioned on page 59.

This book contains the main results of the German project POPINDA. It surveys the state of the art of industrial aerodynamic design simulations on parallel systems. POPINDA is an acronym for Portable Parallelization of Industrial Aerodynamic Applications. This project started in late 1993. The research and development work invested in POPINDA corresponds to about 12 scientists working full-time for the three and a half years of the project. POPINDA was funded by the German Federal Ministry for Education, Science, Research and Technology (BMBF). The central goals of POPINDA were to unify and parallelize the block-structured aerodynamic flow codes of the German aircraft industry and to develop new algorithmic approaches to improve the efficiency and robustness of these programs. The philosophy behind these goals is that challenging and important numerical appli­ cations such as the prediction of the 3D viscous flow around full aircraft in aerodynamic design can only be carried out successfully if the benefits of modern fast numerical solvers and parallel high performance computers are combined. This combination is a "conditio sine qua non" if more complex applications such as aerodynamic design optimization or fluid structure interaction problems have to be solved. When being solved in a standard industrial aerodynamic design process, such more complex applications even require a substantial further reduction of computing times. Parallel and vector computers on the one side and innovative numerical algorithms such as multigrid on the other have enabled impressive improvements in scientific computing in the last 15 years.

This volume entitled "EUROPT - A European Initiative on Optimum Design in Aerodynamics" contains the results of the contributors during a workshop which took place in Barcelona in June 1992. This workshop was organized in the framp­ work of the Brite/Euram Aeronautics project "Optimum Design in Aerodynamics" (AERO-89-0026). The project brought together nine European partners from Acade­ my and Industry with big experience in numerical optimization techniques applif'd to automated optimum design. The manuscript is directed to the optimization field and its goal is to provide the reader with useful numerical optimization techniques for optimum design in aerody­ namIcs. The field of numerical optimization techniques has been growing since the early 50's. But only recently, with the recent advent of powerful computers. it has yielded real-life applications that demonstrate industrial potential for design purposes. Nowadays, optimization is at the stage where those scientists and engineers who have worked in the field are to be paid to apply numerical optimization to automated design of real problems, namely aerodynamic shapes in Aerospace Engineering. For the above reasons, it was evident that a Workshop on numerical optimization techniques for the validation of optimum design methods would be of interest to most of the partners involved in the AERO-S9-0026 project in order to compare in terms of accuracy and efficiency several optimization soft wares performed on the same selected flow problems.