CONCHA: COmplex flow simulatioN Codes based on High-order and - - PDF document

CONCHA

CONCHA:

COmplex flow simulatioN Codes based on High-order and Adaptive methods

Research unit: INRIA Futurs Theme: Num Localization: Laboratoire de Math´ ematiques Appliqu´ ees (LMA), Universit´ e de Pau et des Pays de l’Adour (UPPA), UMR1 5142 CNRS2 – UPPA Keywords: CFD, complex fluids, turbulent flows, combustion, numerical analysis, high-order methods, discontinuous and stabilized finite element methods, adaptivity, multigrid, DWR-method, numerical sensitivity analysis.

Contents

1 Introduction 4 2 Members 4 3 Overall Objectives 5 4 Scientific Foundation 7 4.1 Goals: accuracy and efficiency . . . . . . . . . . . . . . . . . . . . . . 7 4.2 Difficulties related to numerical simulations of reacting flows . . . . . . 7 4.2.1 Physical coupling . . . . . . . . . . . . . . . . . . . . . . . . . 7 4.2.2 Reaction mechanisms . . . . . . . . . . . . . . . . . . . . . . . 8 4.2.3 All-Mach regimes . . . . . . . . . . . . . . . . . . . . . . . . 8 4.2.4 Turbulence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 4.3 Numerical tools: High-order discretization methods . . . . . . . . . . . 9 4.3.1 Motivation for discontinuous finite elements . . . . . . . . . . . 9 4.3.2 Overview on discontinuous Galerkin and other stabilized finite elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 4.3.3 Challenges related to DGFEM . . . . . . . . . . . . . . . . . . 12 4.3.4 Approximation of solutions with shocks . . . . . . . . . . . . . 13 4.3.5 All-Mach approach . . . . . . . . . . . . . . . . . . . . . . . . 14 4.4 Numerical tools: Adaptivity . . . . . . . . . . . . . . . . . . . . . . . 15

1Unit´

e mixte de recherche

2Centre national de recherche scientifique

1

CONCHA 4.4.1 Mesh and order adaptation . . . . . . . . . . . . . . . . . . . . 15 4.4.2 Automatic model selection . . . . . . . . . . . . . . . . . . . . 15 4.4.3 DWR-method (dual-weighted residual) . . . . . . . . . . . . . 16 4.4.4 Parameter identification and numerical sensitivities . . . . . . . 16 4.5 Further topics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 5 Application Domains 17 5.1 Targeted issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 5.1.1 Working principle of combustion chambers . . . . . . . . . . . 18 5.1.2 Safety issue: accidental boring of a combustion chamber . . . . 20 5.1.3 Engine efficiency: improved cooling of a combustion chamber wall . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 5.1.4 Generating compact propulsive systems: mixing processes and combustion in micro-devices . . . . . . . . . . . . . . . . . . . 23 5.2 Test problems related to the targeted issues . . . . . . . . . . . . . . . . 24 5.2.1 Highly underexpanded supersonic jets . . . . . . . . . . . . . . 24 5.2.2 Subsonic jet in cross flow . . . . . . . . . . . . . . . . . . . . 24 5.2.3 DNS of mixing in micro-channels . . . . . . . . . . . . . . . . 26 6 Software 27 6.1 Aims and Scope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 6.2 Development strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 6.2.1 Validation and comparison with other software . . . . . . . . . 28 6.2.2 Comparison with experiments . . . . . . . . . . . . . . . . . . 28 6.2.3 Distributed hierarchical software development . . . . . . . . . 28 7 Expected Results 28 7.1 Development and analysis of algorithms . . . . . . . . . . . . . . . . . 28 7.2 Software development . . . . . . . . . . . . . . . . . . . . . . . . . . 29 7.3 Validation of algorithms and numerical approaches . . . . . . . . . . . 29 8 Positioning with respect to other research projects 29 8.1 Positioning within INRIA . . . . . . . . . . . . . . . . . . . . . . . . . 29 8.2 Positioning on the national and international level . . . . . . . . . . . . 30 8.2.1 Reactive flow simulations . . . . . . . . . . . . . . . . . . . . 30 8.2.2 External collaborators . . . . . . . . . . . . . . . . . . . . . . 31 9 Industrial partners 31 9.1 Turbomeca . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 9.2 Airbus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 10 Dissemination 32 2

CONCHA 10.1 Education . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 10.2 Scientific community . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 10.3 Participation in conferences, workshops . . . . . . . . . . . . . . . . . 33 11 Agenda and internal organistaion 33 11.1 Short term (2-years) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 11.2 Long term (5-years) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 11.3 Organisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 A Appendix: Short curriculae of members 35 A.1 Roland Becker . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 A.2 Pascal Bruel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 A.3 Daniela Capatina . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 A.4 Robert Luce . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 A.5 Eric Schall . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 A.6 David Trujillo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 3

CONCHA 1 Introduction

1 Introduction

CONCHA is concerned with the numerical simulation of complex flow problems, with special emphasis on aeronautics. Our particular interest is in problems related to propul- sion engines, which lead to the study of physical phenomena such as high-speed flow, chemical reactions, combustion, and turbulence. The objective of this project is to develop simulation codes which are able to handle the difficulties implied by the physics of our applicational domain: strong nonlinearities, large spectrum of time and space scales, stiff couplings, model uncertainties, and the im- portant size of the discrete systems to be solved; see Section 4.2 for details. Therefore,

• ur special interest lies in robustness with respect to the different physical parameters

and good efficiency, defined to be the computational work for a given accuracy. Our tools are high-order and adaptive methods, which are being developed by the math- ematical community since the last two decades; see Section 4.3 for details. Of special interest is the confrontation with specific problems (see Section 5.2 for the definition

• f test problems) from the considered domain of applications. The medium-term goal
• f the project is to evaluate the potential of these modern techniques in view of their

possible integration into industrial codes. The long-term objective of this project is to provide reliable software for optimization problems related to complex flow problems. Here, our special focus is on optimal de- sign and tools for the coordinating simulation and experimentation, including parameter estimation and the computation of numerical sensitivities, see Appendix 4.4.4. The project relies on an interdisciplinary collaboration between numerical analysts and specialists in fluid dynamics. A particular advantage of the composition of the members is the confrontation of computational results with experiments, made possible by the presence of experimental facilities common to UPPA and Turbomeca.

2 Members

Team members from UPPA

• BECKER Roland, Professeur, LMA (Team Leader)
• BRUEL Pascal, CR1 CNRS, HDR, LMA
• CAPATINA Daniela, Maˆ

ıtre de Conf´ erences, LMA

• LUCE Robert, Maˆ

ıtre de Conf´ erences, LMA 4

CONCHA 3 Overall Objectives

• SCHALL Eric, Maˆ

ıtre de Conf´ erences, HDR, LaTEP3

• TRUJILLO David, Maˆ

ıtre de Conf´ erences, LMA Phd - students (supervisor, sponsor)

• MOGUEN Yann (Amara-Schall, CR Aquitaine)
• MOST Aurelien (Bruel, Cifre Turbomeca)
• TAAKILI Abdelaziz (Becker-Gagneux, MRT)

Postdocs (supervisor, sponsor)

• NGUYEN Phuc Danh (Becker-Bruel, CR Aquitaine)

The members of the team have different scientific background: Pascal Bruel [21, 24, 25, 27] and Eric Schall [26, 28, 33, 32, 34] work on modeling and experiments in fluid mechanics ( CNU4-section 60,62), while Daniela Capatina [2, 3, 1, 23], Robert Luce [5, 6, 31, 30], and David Trujillo [2, 3, 4, 35, 36] work on numerical methods for partial differential equations (CNU-section 26). For more details see Section A. The project relies on the combination of the competence of the physicists on combustion and tur- bulence with the skill of the mathematicians in numerical analysis and software devel-

• pment. In order to bring together the two groups with different scientific background,

we organize a weekly internal seminar as well as smaller interdisciplinary projects con- cerning concrete questions on modeling, programming, and experiments.

3 Overall Objectives

The main objective of this project is the development of innovative algorithms and effi- cient software tools for the simulation of complex flow problems. On the one hand, our contributions concern modern discretization methods (high-order and adaptivity) and goal-oriented simulation tools (prediction of physical quantities, numerical sensitivities, and inverse problems) on the other. Concrete applications originate from simulations of flows related to propulsion devices featuring important and challenging problems for numerical simulations in this field; see Section 5.1. Reactive flow problems lead to very complex coupled systems of equations which are

• ften not solvable in routine way with industrial software. This is due to the physical

complexity of the system of equations to be solved and the mathematical difficulties

3Laboratoire de Thermique, Energ´

etique et Proc´ ed´ es

4Conseil National des Universit´

es

5

CONCHA 3 Overall Objectives they imply: the extremely stiff reaction terms in combustion, the different flow regimes ranging from low Mach numbers to hypersonic flow, the presence of turbulence, and two-phase flows. Our medium-term goal is to develop flow solvers based on modern numerical meth-

• ds such as high-order discretization in space and time and self-adaptive algorithms.

Adaptivity based on a posteriori error estimators has become a new paradigm in scien- tific computing, first because of the objective to give rigorous error bounds, and second because of the possible speed-up of simulation tools. A systematic approach to these questions requires an appropriate variational framework and the development of corre- sponding software tools. It is our goal to study at hand of concrete applications the possible benefits and difficul- ties related to these numerical approaches in the context of complex fluid mechanics. Therefore, prototypical applications are chosen in order to represent important chal- lenges in the our field of application. The main ingredients of our numerical approach are adaptive finite element discretiza- tions combined with multilevel solvers and hierarchical modeling. In view of our ap- plications described above in Section 5, it is natural to consider discontinuous and sta- bilized finite element methods, for example the so-called discontinuous Galerkin ap- proach (DGFEM), since it generalizes classical finite volume methods and offers a uni- fied framework for the development of adaptive higher order methods. The enhance- ments of these methods and their application to challenging physical problems induce numerous mathematical investigations. Our long-term goals are described as follows. Having appropriate software tools at our disposal, we may attack questions going beyond sole numerical simulations: parameter identification, optimization, and validation of different numerical techniques (including validation of software relying on the particular technique). The disposal of such tools is also a prerequisite for testing of physical models concerning for example turbulence, chemical kinetics, and their interaction. Nowadays it seems clear that many engineering problems in the field of complex flow problems can neither be solved by experiments nor by simulations alone. In order to improve the experiment, the software has to be able to provide information beyond the results of simple simulation. Here, information on sensitivities with respect to selected measurements and parameters is required. The parameters could in practice be as dif- ferent in nature as a diffusion coefficient and a velocity boundary condition. CONCHA has as long-term objective the development of the necessary computational framework and to contribute to the rational interaction between simulation and experiment. 6

CONCHA 4 Scientific Foundation

4 Scientific Foundation

We first give a short overview on typical systems of equations arising in the considered domain of applications. Then we describe some typical difficulties in this field which require the improvement of established and the development of new methods. Next we describe the research directions underlying our project for the development of new software tools. They are summarized under the two headlines ’high-order methods’ and ’adaptivity’. Our approach for the discretization of the Euler and related equations is based on the discontinuous Galerkin finite element method (DGFEM), which offers a flexible variational framework in order to develop high-order methods. Finally some perspectives are outlined.

4.1 Goals: accuracy and efficiency

Accurate predictions of physical quantities are of great interest in fluid mechanics, for example in order to analyze instabilities, especially in reacting and/or turbulent flows. Due to the complex and highly nonlinear equations to be solved, it is difficult to predict how fine the spatial or temporal discretization should be and how detailed a given phys- ical model has to be represented. We propose to develop a systematic approach to these questions based on high-order and auto-adaptive methods. We note that most of the physical problems under consideration have a three-dimensional character and involve the coupling of models. This makes the development of fast nu- merical methods a question of feasibility.

4.2 Difficulties related to numerical simulations of reacting flows

For the modeling of reactive flows see [51, 116]. Numerical simulations of reactive flows have rapidly gained interest [63, 94, 105]. For an overview on state-of-the-art modeling of combustion and turbulent reactive flows see [106, 114]. 4.2.1 Physical coupling The coupling between the variables describing the flow field and those describing the chemistry is in general stiff. Our efforts will therefore be concentrated on coupled implicit solvers based on Newton-type algorithms. A good speed-up of the algorithms requires a clever combination of iteration and splitting techniques based on the structure

• f the concrete problem under consideration.

7

CONCHA 4.2 Difficulties related to numerical simulations of reacting flows 4.2.2 Reaction mechanisms The modeling of chemistry in reactive flow is still a challenging question. On the one hand, even if complex models are used, estimated physical constants are frequently involved, which requires an algorithm for their calibration. On the other hand, mod- els with detailed chemistry are often prohibitive, and there exists a zoo of simplified equations, starting with flame-sheet-type models. The question of model reduction is of great interest for reacting flows, and different approaches have been developed [95, 108]. Although first attempts exist for generalization of a posteriori error estimators to model adaptation [3, 48, 20] and [104], it remains a challenging question to develop numerical approaches using a hierarchy of models in a automatic way, especially combined with mesh adaptation. 4.2.3 All-Mach regimes The development of solvers able to deal with different Mach regimes simultaneously is a challenging subject. For a long time, the CFD community has been divided into two communities: one oriented towards the Euler equations and the other towards incom- pressible flow equations. The main reason for this is that both fields have to deal with major difficulties which have few in common: The first is the approximation of - and in addition physically correct - shock positions, and the second difficulty is the stable approximation of the pressure and the bad conditioning of the systems. The methods that have been developed are consequently of quite different nature: finite volumes with sophisticated numerical fluxes based on approximate Riemann solvers combined with fast explicit time stepping schemes on the one hand, and special finite elements with implicit time discretization and multigrid solvers on the other. For the first family we refer to the text books [75, 93, 97, 111] whereas for the second one we cite exemplarily [68, 73, 76]. The recent interest in questions related to low-Mach number flows is for example documented in the proceedings [38, 46, 70]. 4.2.4 Turbulence The flows under consideration are in general turbulent. This is a major difficulty from the computational point of view, since the resolution of the finest scales still requires a prohibitive number of unknowns in the flow field alone. We note that special difficulties are due to coupling of the flow with chemistry. Recently, it has been observed that certain turbulence models have similarities to finite element stabilization techniques for 8

CONCHA 4.3 Numerical tools: High-order discretization methods the Navier-Stokes equations, and the corresponding idea that rational turbulence models might be based on adaptive techniques has gained much attention, see [72, 82] and related work.

4.3 Numerical tools: High-order discretization methods

4.3.1 Motivation for discontinuous finite elements The discontinuous Galerkin finite element method (DGFEM) offers interesting perspec- tives, since it offers a framework for the combination of techniques developed in the in- compressible finite-element (well-founded treatment of incompressibility constraints, pressure approximation, and stabilization for high-Reynolds-number flows) and the compressible finite-volume community (entropy solutions, Riemann solvers and flux limiters). In addition, the order limit of finite volume discretizations is broken by the variational formulation underlying DGFEM, which makes it possible to develop discretization schemes with local mesh refinement and local variation of the polynomial degree (hp- methods). At the same time, the well-established finite-element knowledge for saddle- point problems can be set on work. Noting that different approaches based on discontinuous Galerkin methods have been used in recent years for the solution of challenging flow problems, DGFEM seems to be a natural framework for the present project. Since the project team members have experience with (these and other) stabilized finite element methods, a combination of the different techniques is expected to be beneficial in order to gain efficiency. 4.3.2 Overview on discontinuous Galerkin and other stabilized finite elements Overview articles concerning DGFEM and stabilized finite element methods are avail- able, see for example [59, 58]. The literature on these methods has an important growth rate since the last decade of the last century. Here, we only give a very short introduction concerning the most important aspects in light of the present project. Hyperbolic equations. The first discontinuous Galerkin method for hyperbolic equa- tions is generally attributed to Reed and Hill [107], where the authors propose a scheme for the neutron transport equation. The first analysis of the method has been presented by Lesaint and Raviart [96]. Their result has been further improved and the analysis sub- stantially broadened by Johnson, Pitk¨ aranta and N¨ avert [89, 90]. We illustrate DGFEM 9

CONCHA 4.3 Numerical tools: High-order discretization methods for the linear first-order scalar equations β · ∇u = f in Ω with boundary conditions u = g on the inflow part of the boundary ∂Ω− (defined by the condition β · n ≤ 0 with n the unit outer normal field at ∂Ω). We seek an approximation uh which is allowed to be discontinuous across all interior edges of a given mesh h. On a given cell K of the mesh, the restriction uK of uh to K is a polynomial. It is determined by the equation to be satisfied by all local test functions vK: −

• K

uKβ · ∇vK dx +

• ∂K

F(uh, nK)vK ds =

• K

fvK dx, (1) where F(uh, n) is a numerical flux function which is used to distinguish between in- and outflow values. Since we are only allowed to prescribe boundary data on the inflow (think of K = Ω), F(uh, n) equals for example (β · n)g on ∂K ∩ ∂Ω− and (β · n)uK

• n ∂K+.

DGFEM became only popular ten years later when it was applied to nonlinear hyper- bolic equations within the work of Cockburn and Shu [60, 62] who combine a spatial DGFEM discretization with an explicit Runge-Kutta method in time. The breakthrough

• f DGFEM for the Euler equations was achieved by Bassi and Reabay [45]. Since then,

there is a growing research activity concerning the application of DGFEM to complex flow problems. It even found its way into lecture notes, see for example [67]. Elliptic equations. Also in the 1970s, DGFEM for elliptic and parabolic equations were proposed. The different variants were generally called interior penalty (IP) meth-

• ds and their development remained independent of the development of the DGFEM

methods for hyperbolic equations. The starting point of these methods is the celebrated paper of Nitsche [103] where a variational scheme for the so-called weak implementa- tion of Dirichlet boundary conditions was developed and analyzed. In order to solve the Dirichlet problem −∆u = 0 in a bounded polygonal domain Ω with boundary condition u = g on ∂Ω, Nitsche’s method determines a finite element approximation uh to u in a finite element space Vh as the solution of the variational equation to be satisfied for all test functions vh ∈ Vh:

• Ω

∇uh · ∇vh dx −

• ∂Ω

Fγ(uh)vh ds −

• ∂Ω

uhFγ(vh) ds = −

• ∂Ω

g ˜ Fγ(vh) ds, (2) where Fγ(wh) =

∂wh ∂n − γwh and ˜

Fγ is defined by consistency. The beauty of this formulation lies in its symmetry and obvious consistency properties. The next step made by Arnold in his 1979 thesis (summarized in [39]) was based on the fundamental idea to enforce continuity of finite element approximations on the interior edges of a given mesh in a similar way as the Nitsche method does for the boundary con- ditions; see also Wheeler [115] for a collocation finite element method and the methods 10

CONCHA 4.3 Numerical tools: High-order discretization methods

• f Douglas and Dupont [64] and Baker [42] where similar ideas have been used in order

to enforce the continuity of the derivatives of finite element functions across interior

• edges. It seems that these early methods could not gain much attention at that time and

have therefore been fallen into sleep for twenty years. The renewed interest in discontinuous finite element methods for elliptic equation seems to be related to the success of the DGFEM methods for hyperbolic equations. Being known to lead to accurate and robust discretization of the transport equation, the next step is to consider the convection-diffusion, see for example [10]. In order to deal with the Navier-Stokes equations, new ideas for the discretization of the diffusive terms have been developed in Bassi and Rebay [44] and Cockburn and Shu [61]. A unified framework of DGFEM methods for elliptic equations has been developed in [41]. The DGFEM framework offers many interesting possibilities. We mention here a robust scheme for the elasticity system [79], the discretization of Maxwell’s equation [69, 83], its use for the discretization on non-matching meshes without Mortar spaces [11], and hp-methods [84, 109]. It is generally accepted that an important advantage of DGFEM beside its flexibility is the fact that it is locally conservative. At the same time, its drawback is its relative high numerical cost. For example, compared to continuous P 1 finite elements on a triangular mesh, the number of unknowns are increased by a factor of 6 (and a factor of 2 with respect to the Crouzeix-Raviart space); considering the system matrix even leads to a more disadvantageous count. Concerning higher-order spaces, standard DGFEM has a negligible overhead for polynomial orders starting from p = 5, which is probably not the most employed in practice. The question of how to increase efficiency of DGFEM is an important topic of recent research. Our approach in this field is based on comparison with stabilized FE methods. New generation stabilized FE methods. Standard finite element schemes do not lead to satisfactory schemes for convection dominated problems. Modifications of standard Galerkin methods by introducing additional so-called stabilization terms in order to in- crease robustness have been introduced by Hughes and co-workers for the Stokes equa- tions and convection-dominated problems [50, 85, 86]. In case of the linear transport equation β · ∇u = f in Ω with boundary conditions u = 0 on the inflow part of the boundary ∂Ω− a typical variational formulation reads: Find uh ∈ Vh such that for all test functions vh ∈ Vh there holds

• Ω

β·∇uh vh dx+

• ∂Ω− |β·n|uhvh ds+
• Ω

δ(β·∇uh−f)(β·∇vh) dx =

• Ω

fvh dx. (3) The third term on the left of (3) is the stabilization term which allows to control the streamline derivative of the discrete solution. The resulting so-called streamline diffu- 11

CONCHA 4.3 Numerical tools: High-order discretization methods sion method SDFEM (or sometimes Galerkin least squares or SUPG) has been analyzed in detail by Johnson and co-workers [91, 92]. The method has been extended in many variants to the incompressible Navier-Stokes equations, see for example [71, 110]. One difficulty in the application of these methods to complex flow equations is the coupling of various terms and unknowns which can be anticipated from (3) if f = f(u) depends on u. These difficulties are avoided by a new family of methods which replace the stabilization term in (3) by terms of the form

• Ω δ(β · ∇uh −

β · ∇uh)(β · ∇vh− β · ∇vh) dx where β · ∇uh is a different approximation of β·∇u, constructed for example with the help of an additional coarser mesh, see Guermond’s two-level scheme for the transport equation [77] and the local projection stabilization [7, 18]. Another approach in this family is the stabilization based on the jumps of the derivatives of finite element functions which goes back to [64] and has recently been developed in a systematic way for convection-diffusion and incompressible flow problems by Burman and Hansbo [53, 55]. Relation between DGFEM and other stabilized FE methods. There are many simi- larities between DGFEM and SDFEM based on piecewise linears for the transport equa- tion from a theoretical point of view, see for example the classical text book [88]. Re- cently, attempts have been made to shed brighter light on the relations between these methods [49][52, 57]. There is also a renewed interest in stabilizing non-conforming finite elements for the Navier-Stokes equations [98], [54, 56]. A better understanding of the relations between these methods will contribute to the development of more efficient schemes with desired properties. As outlined before, the goal is to cut down the computational overhead of standard DGFEM, while retaining its robustness and conservation properties. 4.3.3 Challenges related to DGFEM Formulation of discretization schemes based on discontinuous finite element spaces is nowadays standard. However, some important questions remain to be solved:

• How to combine possibly higher-order spaces with special numerical integration

in order to obtain fast computation of residuals and matrices ? How to stabilize such higher-order DGFEM ?

• Treatment of quadrilateral and hexahedral meshes:

Hexahedral meshes are economical for simple geometries. However, arbitrary hexahedra (the image of the unit cube under a trilinear transformation) lead to challenging questions of discretization. For example, some standard methods 12

CONCHA 4.3 Numerical tools: High-order discretization methods such as mixed finite elements surprisingly lead to bad convergence behavior [40]. Standard conforming Qp finite elements are costly to compute and have disadvan- tages on anisotropic meshes.

• Time-discretization:

The choice of dg time-discretization is natural in view of its good stability and conservation properties. However, the higher-order members of this family lead to coupled systems which have to be solved in each time-step. In order to be fully conservative, the time discretization has to be implicit: for example for the transport equation it seems reasonable not to distinguish between time and space variables and it is therefore natural to discretize both time and space with discontinuous finite elements. Fortunately, taking into account the tensor-product structure of the space-time mesh, it is still possible to organize a code in a time-marching procedure. The dg time-discretization leads to implicit Runge-Kutta schemes [66], which are quite costly, since one has to solve for several levels of unknowns simultaneously. Therefore, a competitive realization has to take advantage of the special structure of these systems.

• Solution of the discrete systems:

The computing time largely depends on the way the discrete nonlinear and linear systems are solved. Concerning the solution of the nonlinear systems, we note that the strong stiffness induced by combustion requires special solvers, based on homotopy methods, time-stepping and specially tuned Newton algorithms. Although direct solvers are often competitive in two-dimensional computations, the complexity of three-dimensional problems makes iterative solvers unavoid- able; here, multilevel solvers are able to ensure optimal complexity for stationary

• problems. It should however be denoted, that time-dependent problems lead to

different situations, and the most efficient solution largely depends on the con- crete application. The question of parallelization is intrinsically related and is of particular importance for the solution of the systems resulting from higher-order space- or even time- discretization. 4.3.4 Approximation of solutions with shocks The shock capturing term for DGFEM plays a similar role as flux limiters in classical finite volume schemes. The choice of an effective shock capturing term is important for the success of the discretization. Authors seem to agree about this point, but the precise form differs significantly. It can be noticed that there is a lack of theory. This also concerns the treatment of the resulting additional nonlinearity. Within the framework of discontinuous Galerkin methods, it is possible to formulate 13

CONCHA 4.3 Numerical tools: High-order discretization methods methods which allow to locally adapt the approximation without necessarily adapting the mesh. Such a procedure has been proposed in [78] and applied to fracture problems in solid mechanics in [81]. We propose to develop a similar approach for the tracking

• f shock waves and combustion fronts.

4.3.5 All-Mach approach There are many attempts to improve codes for compressible flows in order to deal with small Mach numbers, see for example [34], as well as attempts to generalize codes for incompressible flows to deal with high Reynolds numbers and varying densities. It is important to notice that there is a physical difficulty in finding an appropriate model and a numerical difficulty in finding a stable discretization. Concerning the numerical difficulty, we know from the theory of numerics for incompressible flows that the pres- sure gradient has to be discretized in a stable manner. However, this causes important difficulties in a solver for the compressible flow equations since here, the pressure is not a direct variable but is instead determined by the state equation. There is a mod- ern tendency to use a stable discretization for the incompressible flow equations also in the compressible case. This can either be done by respecting certain relations between pressure and velocity approximation [73] or by introduction of stabilization terms [85] (or for more recent techniques [7, 18, 55]). As a general technique which allows to obtain a discretization which is stable for all Mach number regimes we consider the discontinuous Galerkin method (DGFEM). On the one hand, it is relatively easy to generalize standard finite element schemes to the discontinuous case. On the other hand, discontinuous finite elements have a long tra- dition [89, 96] and have recently regained attention [59, 43, 44]. We remark here, that the discretization of second-order terms as the diffusion terms has a sound theoretical background [9, 11], [39, 74]. Despite its potential for robust and accurate discretization, further research has to be carried out in order to make the DGFEM approach competitive. Among the different questions arising in this context we cite:

• Which variables to use ?

Whereas conservation properties are expressed in conservative variables and ac- curate shock approximation necessitates its use, diffusion and boundary condi- tions are expressed in primitive variables.

• How to solve the coupled system ?

Decoupling algorithms as the pressure correction schemes for transient incom- pressible flows are well understood. However, the generalization to higher Mach 14

CONCHA 4.4 Numerical tools: Adaptivity numbers, in particular for combustion, does not seem to be straightforward due to the different role of pressure. Concerning the physical difficulties, one has to be aware that the full system of flow equations comprises different physical phenomena with different time-scales. One stan- dard technique consists in developing the pressure with respect to the Mach number [102, 99]. One is then able to construct numerical schemes for the resulting equations in the small Mach number limit [117, 112]. However, in that case fast scales as acous- tic waves need to be captured, and consequently, it is important to develop separate equations since an efficient numerical treatment has to take these different aspects into account.

4.4 Numerical tools: Adaptivity

4.4.1 Mesh and order adaptation The possible benefits of local mesh refinement for fluid dynamical problems is nowa- days uncontested; the obvious arguments are the presence of singularities, shocks, and combustion fronts. The use of variable polynomial approximation is more controversial in CFD, since the literature does not deliver a clear answer concerning its efficiency. At least at view of some model problems, the potential gain obtained by the flexibility to locally adapt the order of approximation is evident. It remains to investigate if this estimation stays true for the applications to be considered in the project. The design and analysis of auto-adaptive methods as described above is a recent research topic, and only very limited theoretical results are known. Concerning the convergence

• f adaptive methods for mesh refinement, only recently there has been made signifi-

cant progress in the context of the Poisson problem [47, 101][12], based on two-sided a posteriori error estimators [113]. The situation is completely open for p-adaptivity, model-adaptivity or the DWR method5. In addition, not much seems to be known for nonlinear equations. It can be hoped that theoretical insight will contribute to the devel-

• pment of better adaptive algorithms.

4.4.2 Automatic model selection An important parameter of adaptation is the level of modeling. Here, the ideal situation is to dispose of a hierarchy of models which can be locally adapted in oder to yield a desired accuracy at lowest possible cost.

5Dual-weighted-residual method [15], see below.

15

CONCHA 4.4 Numerical tools: Adaptivity A general approach towards the systematic a posteriori error estimation of modeling er- rors has been developed in [3, 48, 20] and [104]. The crucial point for practical purposes seems to be the parameterization of different models. In the context of reacting flows, model reduction techniques in order to reduce the great number of unknowns are used. It is of interest to understand, how the errors produced by the model reduction can be controlled in a systematic way. 4.4.3 DWR-method (dual-weighted residual) The three outlined fields of adaptation have in common the following questions:

• What is the criterion ?
• Does, and if so, how fast does the algorithm converge ?
• How to practically adapt the parameters ?
• How to realize an efficient adaptive algorithm ?

Concerning the first point, we note that most of the mathematical literature deals with a posteriori error estimators in the energy norm related to linear symmetric problems. A more praxis-oriented approach to error estimation is the DWR method, developed in [14]; see also the overview paper [15], application to laminar reacting flows in [8], and application to the Euler equations in [80]. The idea of the DWR method is to consider a given, user-defined physical quantity as a functional acting on the solution space. This allows the derivation of a posteriori error estimates which directly control the error in the approximation of the functional value. This approach has been applied to local mesh- refinement for a wide range of model problems [15]. Recently, it has been extended to the control of modeling errors [48]. The estimator of the DWR method requires the computation of an auxiliary linear partial differential equation. So far, relatively few research has been done in order to use possibly incomplete information from, e.g., coarse discretization of this equation. 4.4.4 Parameter identification and numerical sensitivities Numerical simulations generally involve parameters of different nature. Some parame- ters reflect physical properties of the materials under consideration, or describe the way they interact. In addition to these parameters the values of which are often determined by experiments and sometimes only known with accuracy under certain conditions, the development of a computational model involves additional quantities, which could for example be related to boundary and initial conditions. 16

CONCHA 4.5 Further topics The generalization of the DWR method to parameter identification problems has been developed in [16], and [13] for time-dependent equations. The case of finite-dimensional parameters, which is theoretically less challenging than the infinite-dimensional case and has therefore been less treated in the literature, is of particular interest in view of the presented applications (for example the estimation of a set of diffusion velocities). The goal of numerical simulations are in general the computation of given output values I which are obtained from the approximated physical fields by additional computa- tions, often termed post-processing. The DWR method places these output values in the center of interest and aims at providing reliable and efficient computations of these quantities. In the context of calibration of parameter values with experiments, it seems to be natural to go one step beyond the sole computation of I. Indeed, the computation of numerical sensitivities or condition numbers ∂I/∂qi where qi denotes a single parameter can be expected to be of practical and theoretical interest, either in order to improve the design

• f experiments, or in order to help to analyze the outcome of an experiment.

It turns out, that similar techniques as those employed for parameter identification can be used in order to obtain information on parameter sensitivities and corresponding a posteriori error analysis [17].

4.5 Further topics

Here, we list some possible research topics which could be of interest for the long-term perspective of the project.

• thermal, acoustic couplings
• hierarchical turbulence modeling
• other complex flows such as polymers
• direct computation of periodic solutions
• stability prediction

5 Application Domains

The targeted applications belong to the field of fluid mechanics with special emphasis

• n numerical simulation of complex flows. Of special interest are generic flow config-

17

CONCHA 5.1 Targeted issues urations related to the development of combustion chambers for jet or helicopter en- gines. In the following, we first give an introduction to the physics of combustion chambers. Then we present concrete example problems illustrating the challenging numerical dif- ficulties described above. These examples serve as prototypical test problems.

5.1 Targeted issues related to combustion chamber development and engine certification

5.1.1 Working principle of combustion chambers As an introduction, we briefly remind hereafter the very basic aspects of airplane propul-

• sion. The engine provides a certain amount of energy thanks to the heat released in the

combustion chamber by the oxidation of kerosene. The usable part of this energy is used to increase the kinetic energy of the mass flow rate of air channeled by the air in-

• lets. The combustion chamber is the place where the initial amount of energy is being

brought to the system before being transformed into mechanical energy and losses. The pressure drops through the different turbine stages are used to entrain the compressor stages and the reminder serves to produce thrust through the diverging exhaust (turbojet)

• r to entrain a propeller (turboshaft engine) or a rotor (helicopter). A schematic view of

a diluted jet engine is given in Figure 1. The air mass flow streamed by the air intakes is then divided into two streams, the first one passing through the combustion chamber and the second one whose energy increases when passing through the fan. The dilution is the ratio between these two air fluxes. The higher this ratio, the better the propulsive efficiency since the ejection velocity decreases. Figure 1 presents an overview of a re- cent jet engine (by Rolls-Royce) where it appears clearly that the combustion chamber, which can be considered as the primary source of energy for the system, has a rather modest volume compared to that of the complete engine! Thermodynamically speaking, the efficiency of the relevant cycle is directly controlled by the temperature level reached at the exit of the combustion chamber. So, the ten- dency through decades of engines development is to increase as much as possible the temperature and pressure levels in the chamber. However, there is a limit due to the resistance of the various parts of the chamber and turbine (spatial homogeneity of the temperature profile in front of the turbine) and due to the respect of objectives in term

• f pollutants emission.

18

CONCHA 5.1 Targeted issues Figure 1: Functioning diagram of a diluted jet engine (left) and recent commercial plane engine (right, Rolls-Royce Trent 500). Development of new combustion chambers In the following, we describe three major issues in the development of new combustion chambers: the environmental norms for pollution, the safety constraints, and improved wall cooling efficiency. Pollution The combustion chamber being at the source of most of the engine pollu- tants emissions, it is naturally the subject of continuous efforts aimed at reducing as much as possible the level of its pollutants emissions in order to satisfy to the constantly more stringent international certification procedures. Among the different undesirable Figure 2: Example of sheet established during the certification process (CFM- International family). 19

CONCHA 5.1 Targeted issues chemical species, one can cite the various carbon oxides (which leads to ozone forma- tion, greenhouse effect), the nitrogen oxides (provoking ozone formation locally and in the high atmosphere) and sulfur (which causes atmosphere acidity), unburned hydro- carbons (provoking ozone formation locally, greenhouse effect) and soot (dangerous for human health). The International Civil Aviation Organization (ICAO) is in charge of setting up the various procedures required to measure the pollutant emission levels that any new engine must undergo before being marketable. A standard sequence of take-

• ff/flight/landing is thus defined in Annex 16 - Volume II [87]. In this framework, the

monitored species are unburned hydrocarbons, soot, carbon monoxide, nitrogen oxides. Figure 2 presents a typical result produced during such tests of a new engine (source www.qinetiq.com). Bearing in mind the various mechanisms that produce such oxides, new combustion systems are actively developed and tested especially in the framework

• f European research programs such as LOPOCOTEP and MOLECULES which group

academic and industrial partners around this common objective. Many of the present de- vice improvements rely on the recourse to Lean Premixed Prevaporized (LPP) combus- tion based systems characterized by a tight control of the stoichiometry field properties within the combustion chamber. Such innovative concepts prove to be very promising but they still suffer from intrinsic instabilities over the range of operational conditions and therefore additional studies are still required to minimize, if not suppress, the neg- ative impact of these phenomena. With this respect, the development of reliable and efficient CFD tools incorporating up-to-date and extensively validated physical model- ing is of paramount importance in order to supplement experiments. 5.1.2 Safety issue: accidental boring of a combustion chamber The development of new combustion chambers is not only concerned by pollution or efficiency matters but also by safety problems. Indeed, if the engine manufacturer has to demonstrate the various capabilities of its engine regarding reliability, easy servicing

• r low operating costs, the airplane manufacturer (Airbus or Boeing for instance) who

powered its plane with such engines has also to face some certification rules regarding safety issues. For instance, it is of its responsibility to demonstrate that the plane and its passengers will not be endangered in the case of an accidental boring of one of the engines combustion chambers (this happens from time to time, often near the kerosene injection ports). In that respect, the FAR 25903 (d)(1) stipulates the type of testing that can ensure that such conditions are satisfied. During such a test, the pylon truss must stand a three-minute impact of an highly underexpanded supersonic jet whose stagnation conditions are strictly imposed (see Figure 3). This three-minute period of time is thought of as being sufficient for the pilot to take any necessary action to detect and stop the faulty engine. For instance, Airbus France 20

CONCHA 5.1 Targeted issues Figure 3: Domain of possible interaction between the jet created by the boring of a combustion chamber and the engine pylon (from Airbus). recently developed a partnership with the Moscow Aviation Institute to carry out this type of test. Considering the wide family of airplanes that a single manufacturer can

• ffer, with different engines, pylon positioning and so forth as well as the cost and the

difficulty of conducting the experiments required by the certification authorities, it is clear again that accurate numerical simulations conducted a priori or jointly with the experiments are of great help in the course of the certification procedure. Now, what is the type of flow, simulations would have to deal with in such a case? If one considers a reservoir discharging into a quiescent atmosphere, it is the nozzle pressure ratio (NPR) between the static pressure at the nozzle exit and that of the ambient atmosphere that can be used to discriminate between the various possible flow patterns. For weak or moderate NPR’s (typically for 1 < NPR < 2), these jets are characterized by three main zones: i) the near field, whose “diamond-like” shock structure is a rep- etition of incident and reflected oblique shock waves; ii) the transition region, wherein these structures are swallowed by the development of the shear layer; and finally, iii) the far field, with a more classical self-similar jet. In these cases, the jet core remains mainly supersonic so that the global features of the flow structure are reasonably pre- dicted just by taking into account the compressible effects on turbulence. This is not the case for situations involving much higher NPR’s. In that case, the expansion fan 21

CONCHA 5.1 Targeted issues gives rise to a highly curved incident “barrel” shock which irregularly reflects through a strong, curved ”Mach Disk” which strongly interacts with the jet shear layer, while a large subsonic zone develops within the jet core. This flow configuration represents a great challenge as far as the modeling of the prominent phenomena is concerned, e.g. compressible turbulence, interaction between turbulence and shock, strong compression

• r expansion zones, curvature and non-equilibrium effects on the shear layer growth. A

comprehensive survey of the sensitivity of the flow structure to various parameters is given in [29]. When such a flow is impacting a solid surface, the resulting flow struc- ture is even more complex and extremely sensitive to the impact angle, the distance between the nozzle and the surface and the NPR. In the case of a normal impact, a double shock structure (Mach disk and the so-called plate shock) may appear and the interaction with the boundary layer developing on the impacted surface can lead to the appearance of recirculation bubbles prone to low frequency unsteadiness. In the case

• f a non-normal impact, the near surface flow morphology changes again (additional

reflected shocks and triple points) and can yield a pressure field that can reach peak values on the impacted surface three times as much as those observed when the impact is normal. Considering the complexity of such a type of flow, it appears that the pre- liminary necessary step paving the way towards accurate simulations of such type of fluid-surface interaction, is the development of numerical tools able to simulate accu- rately the structure of free highly underexpanded supersonic jets involved in such an

• interaction. In addition, such jets are well suited for testing newly developed numerical

tools since they can be simulated by considering increasingly complex physical model- ing, starting with the Euler equations (2D, 2D-axysemmetric, and 3D), then considering the Navier-Stokes equations before going to turbulence modeling. 5.1.3 Engine efficiency: improved cooling of a combustion chamber wall The film cooling technique is now widely used in combustion chambers. The contin- uous injection of air through numerous holes drilled through the combustion chamber wall permits the formation of a film which protects the wall from the high temperature resulting from the kerosene+air combustion process. If this type of wall flows has been extensively studied in order to determine the layout of the holes that maximizes the cooling efficiency, the knowledge of the influence of the hole shape on the near wall flow structure is not extensively documented. This issue has to be addressed, since it is well known that the drilling techniques usually employed in industry (laser or electron beams) produce holes of different shapes. A partnership between LMA and Turbomeca in order to address this topic exists. This collaboration has lead to the installation of a test facility at UPPA, see Figure 4. Its goal is to study experimentally confined wall flows representative of those present in a real 22

CONCHA 5.1 Targeted issues Figure 4: Experimental bank at UPPA. combustion chamber. This experimental rig is intended to be used in order to compare with computational results. 5.1.4 Generating compact propulsive systems: mixing processes and combustion in micro-devices If jet or helicopter engine combustion chambers are the primary systems in sight, it is worth noticing that recently, micro combustion chambers have received a renewed in- terest either as a propulsive tool for drones or to produce electricity for various portable

• devices. In France, ONERA, who is the leader in this domain, has recently developed

a test facility in Palaiseau to study experimentally the various aspects and problems re- lated to these devices. A doctoral thesis on a related question with a UPPA-ONERA co-supervison is planned. Since the codes developed in the present project should have the capability of dealing with laminar and turbulent reacting flows, the numerical sim- ulation of the flow in this type of micro-combustion chamber is an interesting step to assess the capability of the developed tools to deal with reacting flows. 23

CONCHA 5.2 Test problems related to the targeted issues

5.2 Test problems related to the targeted issues

5.2.1 Highly underexpanded supersonic jets Figure 5: Underexpanded jet: main parameters of the flow. This kind of flow corresponds to the jets experimentally studied by Y¨ uce¨ ıl and ¨ Ot¨ ugen [118] and ¨ Ot¨ ugen et al. [37]. Figure 5 presents a schematic view of the flow config- uration along with the main flow parameters to be considered. The main objective of this test configuration is to correctly predict the Mach disk location as well as its radial

• extension. The complex shock pattern including the Mach disk is presented in Figure 6

(taken from [29]). The underlying model are the standard Euler equations. Highly underexpanded supersonic jets have been the subject of the doctoral thesis of G. Lenasch (supervisor: P. Bruel). The computational and experimental results obtained in this thesis showed the necessity to increase the accuracy of computations for this type of problems. They form an ideal basis for comparison with the software to be developed. The objective of this test problem is to evaluate the potential gain in efficiency of high-

• rder and adaptive methods based on the discontinuous Galerkin finite element methods

applied to the Euler equations. A natural extension of this test problem is to include viscous terms. 5.2.2 Subsonic jet in cross flow This kind of flow corresponds to the experimental configuration investigated by Miron et al. [100]. Figure 7 presents a schematic view of the flow configuration along with 24

CONCHA 5.2 Test problems related to the targeted issues Figure 6: Schematic plot of the structure of an highly underexpanded supersonic jet : 2. incident shock, 3.isobaric line of the shear layer 4. Reflected shock, 5. Jet shear layer,

• 6. Mach disk, 8. Mach disk shear layer.

the main flow parameters to be considered. The first objective is to simulate accurately the flow within the hole and the near field of the jet to cross-flow interaction zone, in

• rder to be able to assess the discharge coefficient sensitivity to the detailed shape of

the hole which is in general non-circular. The flows are turbulent, thus this geometry will be specifically used for testing the capability of the numerical tools developed in the project, especially combined with turbulence modeling. The discharge coefficient is the mean of the normal velocity component over a whole, and can therefore be considered as a functional of the velocity. Our particular interest is in deriving self-adaptive methods for the efficient computation of this functional. We therefore intend to employ the DWR-method [15] in order to perform automatic mesh selection. As a mid-term goal we wish to develop a similar approach for self- adaptive turbulence modeling. A first effort in this direction is the post-doctoral work

• f Phuc Nguyen Dahn (supervised by P. Bruel and R. Becker), which is financed by the

CDAPP6.

6Communaut´

e d’agglom´ eration de Pau Pyr´ en´ ees

25

CONCHA 5.2 Test problems related to the targeted issues Figure 7: Subsonic jet in a cross-flow: main parameters of the flow configuration. 5.2.3 DNS of mixing in micro-channels This kind of flow has been experimentally studied in the doctoral thesis of Dumand [65] at ONERA (supervised by V. Sabel’nikov) . The flow geometry is sketched in Figure 8. Figure 8: Micro combustion chamber: overview of the mixing channel geometry. It represents a view of the mixing channel, e.g., the channel used to mix the oxidizer stream of air and the fuel stream which is simulated by an injection of nitrogen and

• acetone. The objective of this device is to mix the two streams as quickly as possible in
• rder to feed the combustion chamber with a mixture as homogeneous as possible. Due

to the very small dimensions of the channel, the Reynolds number of the flow is below 500, and the Mach number is very small. In order compute the flow of this test problem, we intend to develop an incompressible flow solver. In addition to the pressure and velocity field, the concentration of each considered species has to be determined. We intend to extend the solver to the case of compressible low-Mach number flows. 26

CONCHA 6 Software

6 Software

6.1 Aims and Scope

Goals of the project are to evaluate the potential of recent numerical methods and to de- velop new approaches in the context of industrial CFD problems. It is therefore impor- tant to possess flexible and extendable software which is able to integrate the methods under consideration such as local adaptive mesh refinement, anisotropic meshes, hierar- chical meshes, h-p methods, and DGFEM. At the same time the codes have to be able to deal with the physics of complex reactive flow problems. The software architecture is designed in such a way that a group of core developers can contribute in an efficient manner, and that independent development of different physical applications is possible. Further, in order to accelerate the integration of new members and in order to provide a basis for our educational purposes (see Section 10.1), the software proposes different entrance levels.

6.2 Development strategy

In the beginning of the project we concentrate on the design of an efficient and suffi- ciently broad software structure. The principal intended users of our software are the members of the team and their collaborators and phd-students. Therefore the goal is to develop tools which are allowed to require a certain mathematical and programming knowledge, and offer in return high flexibility with respect to questions related to discretization, such as adaptivity, higher-

• rder methods, finite element spaces, and different variational formulations.

In order to use existing software, it is planned to follow a modular approach. This is particularly important for mesh generation and linear solvers. The main focus of

• ur software development is on discretization algorithms, their implications on iterative

solution strategies, as well as the design of adaptive strategies. Our development is based on the experience of the team members, especially concerning the development of the toolkits Gascoigne and Salome by R. Becker and D. Trujillo, respectively, see Annexe A for details. It is intended to publish, at least sufficiently general parts of the software under the GNU public license (or similar). This is important for the life-cycle of the software, and also for the visibility of the project. 27

CONCHA 7 Expected Results LMA offers its computational facilities. A small-size cluster (32 processors Opteron 64KB dual-core), which is sufficient for the development in the beginning, has been partially financed with foundings from “r´ egion Aquitaine” and Turbomeca. 6.2.1 Validation and comparison with other software We intend to validate our software with respect to other commercial and research tools in the domain, such as A´ ero3d (INRIA-Smash), AVBP (CERFACS), Cedre (ONERA), Fluent (ANSYS), FluidBox (INRIA-Scallaplix), OpenFoam (OpenCfd). 6.2.2 Comparison with experiments Experience shows that the development of CFD software benefits in an important mea- sure from in-house experiments. With this respect, we emphasize that there exists a test facility of confined inert flows developed in another research program at UPPA. Its flow geometry and the metrology are adequate for the purpose of comparison with our

• simulations. It is planned to create a open data basis which could serve for comparison

with other simulation software and experiments. 6.2.3 Distributed hierarchical software development The idea is to use different levels of libraries : the lower the level, the more general and the more validated the tools should be. A library on a given level allows for bifurcation in different directions, such that simultaneous development is possible, based on com- mon lower level tools. In order to coordinate the practical work, we make use of the Gforge offered by INRIA.

7 Expected Results

7.1 Development and analysis of algorithms

As outlined above, the project is based and generates different research topics. Espe- cially, the interaction between fluid mechanics and numerical analysis as well as the interaction between software development and experiments is crucial for this project. The composition of the project team consists of mathematicians and physicists, and the support from in-house experiments is available. 28

CONCHA 7.2 Software development

7.2 Software development

In the first phase we intend to develop high-order adaptive flow solvers. The availability

• f such tools is crucial for DNS-based turbulence and combustion model development.

The software to be developed in the first phase of the project is a necessary condition for the realization of the more advanced algorithms related to model prediction, inverse problems, and numerical sensitivity analysis.

7.3 Validation of algorithms and numerical approaches

Testing of new numerical methods from the mathematical community at hand of con- crete problems is a necessary research task. What is the practical gain of discontinuous Galerkin schemes for a realistic flow simulation ? What can be expected from adap- tive discretization algorithms ? Is it possible to guide hierarchical modeling and model reduction based on a posteriori error analysis ? What might be the gain in computing numerical sensitivities ?

8 Positioning with respect to other research projects

8.1 Positioning within INRIA

This project has a clear orientation towards reactive flow simulation and is therefore unique within the family of INRIA projects. On the other hand a natural point of col- laboration with other projects is general CFD (comparison of different approaches, ex- change of experience and know-how). Here we mention the projects MC2, MOISE, NACHOS, REO, SIMPAF, SMASH. Concrete collaboration is planned with SCALAPPLIX with respect to different sub-

• jects. First, the CFD-part of SCALAPPLIX (R. Abgrall) is concerned with similar

physical objects: viscous and inviscid, compressible and incompressible flows. Since the employed methods are different (basically continuous versus discontinuous approx- imations), fruitful collaboration concerning comparison of test computation is expected. A further common research topic is local mesh refinement, especially the DWR-method. Second, Scallaplix has important experience in large-scale parallel direct solvers, which defines a natural point of collaboration, since the developed technology has already been employed in CFD. In addition, collaboration concerning coupling of parallel simulation and parallel visualization (O. Coulaud) is planned. 29

CONCHA 8.2 Positioning on the national and international level Further collaboration is planned with GAMMA on automatic mesh adaptation, espe- cially with anisotropic meshes. The use of strongly anisotropic meshes is not standard in reactive flow computation. A necessary prerequisite seems to be discretizations which are robust and accurate on quasi-degenerate meshes. In addition, collaboration with MAGIQUE3D concerning appropriate treatment of bound- ary conditions and discontinuous Galerkin methods is planned. Collaboration in the long-term might concern acoustic waves approximation in turboshaft engines. Finally, the long-term goal being to treat optimization problems related to CFD, natural interactions will appear with the projects concerned with numerical optimization; here we mention CORIDA, OPALE, and TROPICS.

8.2 Positioning on the national and international level

Numerical methods for complex fluid mechanical applications is a very huge subject. Here we focus on the most significant aspects related to the project. 8.2.1 Reactive flow simulations

• Research network: Pˆ
• le de comp´

etivit´ e AESE7 The themes of this project enter the ’pˆ

• le de comp´

etitivit´ e’ AESE (a´ eronautique, espace et syst` eme embarqu´ es) which brings together research activities in this domain and is located in the south-west of France.

• Research network: INCA8

Concerning the field of combustion, ONERA, CNRS, and the group SAFRAN (containing TURBOMECA), have created in 2002 the common project INCA (Advanced Combustion Initiative) which aims to extract added value from French combustion research, and to position Snecma among the world leaders in this

• technology. As a consequence, INCA provides a natural framework for technol-
• gy transfer in the field of combustion.

We intend to collaborate in the framework of these research networks. The participation is important since they provide a forum for discussion of industry relevant research

• topics. We intend to participate by means of doctoral thesis.

7http://www.aerospace-valley.com 8http://www.cerfacs.fr/inca

30

CONCHA 9 Industrial partners 8.2.2 External collaborators

• BRAACK Malte, Professor, University Kiel, Germany

common research (see publications), software development, organization of con- ferences.

• BURMAN Erik, Ass. Professor, EPFL, Switzerland

common research (invited professor at LMA in 12/2007).

• HANSBO Peter, Professor, Chalmers, Sweden

common research (see publications, invited professor at LMA in 4/2005 and 2/2007).

• SABELNIKOV Vladimir, DR, Onera, France

common research (see publications), supervision of phd-thesis (see CV P. Bruel).

9 Industrial partners

Potential industrial partners of this project are Airbus and TURBOMECA. Interaction with industry should profit from the different research networks listed above. The pur- poses of this project being to develop, analyze, and test new algorithms, collaboration with industry through the presented test problems will be natural. We intend to evolve these test problems by feed-back with our industrial partners. Technology transfer in form of integration of new methods into existing industrial codes is intended and could be the goal of phd-theses.

9.1 Turbomeca

Turbomeca (Safran Group)9 is a leading constructor of turbines for helicopters and tur- bojet engines for aircraft and missiles. It is located in Bordes, close to Pau. The interest

• f collaboration between UPPA and Turbomeca on different levels has been manifested

and lead to the establishment of a ’contrat cadre’10. CFD plays a crucial role in the development and certification of turbomachines and is used at Turbomeca in different contexts : heat transfer, cooling, compressors, and com-

• bustion. Reactive flow problems and combustion are an important field of collaboration

between UPPA and the manufacturer of turbines Turbomeca (SAFRAN group). Al- though numerical approaches have been part of it, the collaboration is at the moment

9http://www.turbomeca.com/ 10contract coordinating common research

31

CONCHA 9.2 Airbus mostly centered on experiments. Due to its experimental banks, the TURBOMECA site of Pau-Bordes is a reference center in the field of turbo machinery on the European level. It is to be noted that the need to concentrate applied research in combustion from an industrial point of view had lead to the foundation of INCA.

9.2 Airbus

The second test problem presented above (accidental boring) has been the subject of common research activities between Airbus and members of the project. P. Bruel has supervised the doctoral thesis of G. Lehnasch partially supported by Airbus through a research contract. The subject could give raise to future collaboration.

10 Dissemination

10.1 Education

The LMA has proposed a new Master program starting in 2007, which is called MMS (Math´ ematiques, Mod´ elisation et Simulation) and has a focus on analysis, modeling, and numerical computations in PDEs. The core of this education is formed by lec- tures in four fields : PDE-theory, mechanics, numerical analysis, and simulation tools. We plan that our software has a special part devoted to educational purposes (library ’ConchaBase’11). It is intended to have a simple transparent structure in order to allow teaching of the low-level details of modern finite element programming. The purpose is to provide a basis for teaching and to gradually introduce the student to the use and development of more elaborated tools. The second year of this master program includes lectures on physical applications, one

• f the three proposed fields is CFD; lectures are provided by the members of the project.

The second semester of the second year is devoted to internships in industry, which de- fines a practical means of collaboration with our industrial partners such as CERFACS, ONERA, TOTAL, and Turbomeca.

11http://web.univ-pau.fr/ becker/ConchaBase/ConchaBase.html

32

CONCHA 10.2 Scientific community

10.2 Scientific community

The participants have activities as referees in international journals such as Int. J. for

• Numer. Methods in Fluids, Combustion Science and Technology, Int. Symposium on

Combustion, J. Thermophysics, J. Comp. Phys., Numer. Math., SIAM J. on Opt. and Control, SIAM J. Numer. Anal., SIAM J. Sci. Comp., J. of Comp. Mech., Comput- ing.

10.3 Participation in conferences, workshops

The participants of the project participate in international scientific conferences such as Mafelap06, Finite Element Fair 2006, Oberwolfach 2007, ECM 2007, ICDERS 2007, CFM 2007, 43rd Joint Propulsion Conference and Exhibit 2007.

11 Agenda and internal organistaion

The first period of the project is devoted to the development of fundamental tools. The aim is to have at our disposal state-of-the-art solvers for the basic equations (two- dimensional Euler equations, two- and three-dimensional incompressible flows with varying density). The first stage of the project, devoted to the 2D Euler equations, is used to gain experience with the numerical methods, develop the competence of the members and the computational platform, and finally test different approaches. As a benchmark problem of particular interest is the computation of the highly underexpanded super- sonic jet described in Section 5.2.1. This application can serve as a benchmark prob- lem at different levels: 2D-axisymmetric Euler, Navier-Stokes, 3D, and time-dependent

• problems. At the end of this first step we plan to make some strategical decisions con-

cerning the further development based on the aforementioned benchmark computations and comparison with standard software. The second period is devoted to the advanced modeling topics: turbulence, all-Mach number flows, and combustion. The development at this stage includes numerical tools which will be necessary to tackle optimization problems in a long term perspec- tive. 33

CONCHA 11.1 Short term (2-years)

11.1 Short term (2-years)

The main goal is to develop adaptive high-precision solvers which are based on common software libraries. On the one hand we aim at an hp implementation of DGFEM for the two-dimensional Euler equations, and on the other hand we develop two- and three- dimensional solvers for the incompressible Navier-Stokes equations based on different stabilized finite element methods. The development includes the following topics which are fundamental for the long-term perspective of the project: i) Development of a flexible library structure. ii) Embedding of the parallel direct solvers developed by Scallaplix. iii) Embedding of other INRIA tools concerning mesh generation and refinement. iv) Benchmarking. We wish to validate the considered approaches, compare with

• ther software, and provide a data basis for test computations.

11.2 Long term (5-years)

The second period is devoted to the treatment of advanced physical models as mentioned

• before. It is intended to use the experience of the first period in order to focus our
• efforts. At this point we also develop the tools required for computation of numerical

sensitivities and parameter estimation.

11.3 Organisation

The structure of the project, which is determined by the goal to attribute to each member an individual task and to organize the interaction with the local collaborators, is as

• follows. First, there are three subgroups:
• DGFEM for the Euler equations (Roland Becker, Eric Dubach, Robert Luce, Eric

Schall, Tarik Kousksou),

• Turbulence and mixing (Roland Becker, Pascal Bruel, Daniela Capatina, Didier

Graebling, David Trujillo),

• Experimantal devices (Pascal Bruel, Erisc Schall, Tarik Kousksou).

Second, there are technical services, which coordinate the work with respect to the most urgent needs:

• Benchmarking (Daniela Capatina),

34

CONCHA A Appendix: Short curriculae of members

• Experiments (Pascal Bruel),
• Gforge INRIA and collaborations (Roland Becker),
• Library CONCHA (Robert Luce),
• Mesh tools (Eric Schall),
• Parallelization and visualization (David Trujillo)

Finally, we organize an open seminar with a regular schedule, completed by talks given by experts and industrial collaborators from outside. The purpose is to complete the knowledge in the fields of application and to keep track on the growing literature con- cerning DGFEM and related methods. The schedule of the seminar is accessible on the internet12.

A Appendix: Short curriculae of members

A.1 Roland Becker

Personal data date of birth 18.3.1965 place of birth Bergisch Gladbach, Germany address 126 avenue de Montardon, F-64000 Pau Education 2001 Habilitation, Universit¨ at Heidelberg 1996 Doctoral thesis, Universit¨ at Heidelberg 1991 DEA d’Analyse Num´ erique , Paris 6 1990 Maˆ ıtrise en Math´ ematiques, Marseille Positions depuis 2003 Professor, Universit´ e de Pau et des Pays de l’Adour 2002 Associate professor, Universit¨ at Magdeburg 1999-2001 Wissenschaftlicher Assistent, Universit¨ at Heidelberg 1998–99 Post-Doc, INRIA Sophia-Antipolis 1994–97 Wissenschaftlicher Mitarbeiter Institut f¨ ur Angewandte Mathematik, Universit¨ at Heidelberg

12http://web.univ-pau.fr/ becker/ConchaHtml/agenda.html

35

CONCHA A.1 Roland Becker Administrative duties

• Reponsable Master MMS (mat´

ematiques, mod´ elisation et simulation) 2007-

• Reponsable Master MCS (mod´

elisation et calcul scientifique) 2004-2007

• Responsable ´

equipe analyse num´ erique (LMA), 2004-

• Member ’commission de sp´

ecialistes section 26 (suppl´ eant)’ 2004-

• Director of project A2 ’Navier-Stokes-Gleichungen und chemische Reaktionen’

in SFB 359, 2000-2003 Industrial contracts

• BMBF project ’Simulation of a Wankel engine’, in collaboration with Wankel-

Rotary GmbH, 1996-98 Main publications

• R. Becker and B. Vexler, “Mesh refinement and numerical sensitivity analysis for

parameter calibration of partial differential equations,” J. Comp. Phs., vol. 206,

• no. 1, pp. 95–110, 2005.
• R.Becker, M.Braack, and B.Vexler. Parameter identification for chemical models

in combustion problems. Appl. Numer. Math., 54(3-4):519536, 2005.

• R.Becker, P.Hansbo, and R.Stenberg. A finite element method for domain decom-

position with non-matching grids. , 37(2), 2003.

• R. Becker and R. Rannacher, “An optimal control approach to a-posteriori er-

ror estimation,” in Acta Numerica 2001 (A. Iserles, ed.), pp. 1–102, Cambridege University Press, 2001.

• R. Becker and M. Braack, “A finite element pressure gradient stabilization for the

Stokes equations based on local projections,” Calcolo, vol. 38, no. 4, pp. 173–199, 2001.

• R.Becker and P.Hansbo. Discontinuous Galerkin methods for convection-diffusion

problems with arbitrary Peclet number. In W.Scientific, editor, Numerical Mathe- matics and Advanced Applications: Proceedings of the 3rd European Conference, pages 100109, 2000.

• R. Becker, M. Braack, and R. Rannacher, “Numerical simulation of laminar flames

at low Mach number with adaptive finite elements,” Combust. Theory Modelling,

• vol. 3, pp. 503–534, 1999.

36

CONCHA A.2 Pascal Bruel

• R.Becker and R.Rannacher. A feed-back approach to error control in finite ele-

ment methods: Basic analysis and examples. East-West J. Numer. Math., 4:237264, 1996.

• R. Becker, An Adaptive Finite Element Method for the Incompressible Navier-

Stokes Equations on Time-Dependent Domains. PhD thesis, Universit¨ at Heidel- berg, 1995. Software development

• RoDoBo13 (with D. Meidner and B. Vexler), Optimization of stationary and non-

stationary PDEs

• Gascoigne14 (with M. Braack), Adaptive finite element library in 2D and 3D
• VisuSimple15 (with Th. Dunne), Scientific visualization based on VTK

Phd-students (name, subject, year of defence, actual position)

• Dominik Meidner (with Rolf Rannacher and Boris Vexler), Adaptive finite ele-

ments for time-dependent optimial control, 2007, Post-doc

• Thomas Dunne (with Rolf Rannacher), Goal-oriented error estimation for fluid-

structure interaction, 2007, Post-doc

• Boris Vexler (with Rolf Rannacher), Adaptive finite elements for parameter esti-

mation, 2004, Assistant RICAM Linz, Austria

• Hartmut Kapp (with Rolf Rannacher), Adaptive finite elements for optimal con-

trol, 2001, Bank employee, Luxemburg

• Malte Braack (with Rolf Rannacher), Goal-oriented error estimation for laminar

flows with complex chemistry, 1998, Professor Universit¨ at Kiel, Germany

A.2 Pascal Bruel

Personal data

13http://www.rodobo.uni-hd.de 14http://www.gascoigne.de 15http://visusimple.uni-hd.de

37

CONCHA A.2 Pascal Bruel date of birth 19.1.1960 place of birth Cambrai (59) address 5 Chemin Tuquet, F-64290 - Estialescq Education

• Habilitation `

a Diriger les Recherches, Universit´ e de Poitiers, France, 1999.

• Doctoral thesis, Universit´

e de Poitiers, France, 1988.

• Engineer from ´

Ecole Nationale Sup´ erieure de l’A´ eronautique et de l’Espace (Sup’A´ ero), Toulouse, France, 1983.

• Maˆ

ıtrise de Physique , Universit´ e de Poitiers, France, 1981. Positions Since 1/1/2005 Laboratoire de Math´ ematiques Appliqu´ ees (LMA), Pau, France. 1989-2004 Laboratoire de Combustion et de D´ etonique (LCD), Poitiers, France. Since 1989 Charg´ e de recherche au Centre national de la recherche scientifique, France. 1988-1989 Research assistant, Cambridge University, UK. Administrative duties

• Deputy-Dean of the Centre d’´

Etudes A´ erodynamiques et Thermiques (CEAT) de l’Universit´ e de Poitiers, France, 2001-2003.

• Member of the Commission de sp´

ecialistes (section 62), Universit´ e de Poitiers, France, since 2001.

• 1999-2003, Head of ”Combustion et Turbulence ” research team of Laboratoire

de Combustion et de D´ etonique, Poitiers, France.

• 2003-2006, Treasury of Groupement Franc

¸ais de Combustion (French Section of the Combustion Institute). Industrial contracts

• Bruel, P., contract Turbomeca-CNRS-UPPA, 2005-2007.
• Bruel, P., Lehnasch, G., ”´

etude du comportement d’une flamme et mod´ elisation : mod´ elisation de la flamme torche”, contract EADS 780-729, Delivery 3/3, August 2002. 38

CONCHA A.2 Pascal Bruel

• Bruel, P., Lehnasch, G. et Champion, J.L., ”´

etude du comportement d’une flamme et mod´ elisation : synth` ese bibliographique ”, contract with EADS 780-729, de- livery 2/3, January 2002.

• Champion, J.L. et Bruel, P., ”Modification du banc d’essai THALIE, r´

ealisation d’un ´ ecoulement d’entrefer ”, contract with SNECMA EGC 780-609, final report, December 2001.

• Bruel, P. et Champion, J.L., ”´

etude du comportement d’une flamme et mod´ elisation : assistance aux essais ”, contract with EADS 780-729, delivery 1/3, October 2001.

• Bruel, P., Champion, M., et Deshaies, B., ”´

etude qualitative et grandeurs dimen- sionnantes des conditions de la stabilisation de la combustion dans une zone de recirculation ”, Contract with A´ erospatiale 03315379, 1991. Main publications

• Lehnasch, G. et Bruel, P., Aspects ph´

enom´ enologiques des jets supersoniques libres ou impactants et ´ el´ ements relatifs ` a leur simulation num´ erique, Journ´ ee Th´ ematique Transferts thermiques par impact de jets , Soci´ et´ e Franc ¸aise de Ther- mique, Paris, France, Mars 2006.

• Corvellec, C., Bruel, P., and Sabel’nikov, V.A., Turbulent premixed flames in

flamelet regime: burning velocity spectral properties in presence of counter-gradient diffusion , Combustion and Flame, Vol. 120, No 4, p.585-588, 2000.

• Besson, M., Bruel, P., Champion, J.L., and Deshaies, B., Experimental analysis
• f combusting flows developing over a plane symmetric expansion, Journal of

Thermophysics and Heat Transfer, Vol. 14, No 1, p. 59-67, 2000.

• Corvellec, C., Bruel, P., and Sabel’nikov, V.A., A time-accurate scheme for the

calculations of unsteady reacting flows at zero Mach number , Int. Journal for Numerical Methods in Fluids, Vol. 29, p. 207-227, 1999.

• Karmed, D., Champion, M. and Bruel, P., Two-dimensional numerical modeling
• f a turbulent premixed flame stabilized in a stagnation flow , Combustion and

Flame, Vol. 119, No 3, p.335-345, 1999.

• Dourado, W.M.C., Bruel, P., and Azevedo, J.L.F., A steady pseudo-compressibility

approach based on unstructured hybrid finite volume techniques applied to turbu- lent premixed flame propagation , Engenharia T´ ermica, No 4, p. 41-48, 2003.

• Dourado, M.C.W, Bruel, P., and Azevedo, J.L.F., A time-accurate pseudo-compressibility

approach based on unstructured hybrid finite volume techniques applied to un- 39

CONCHA A.2 Pascal Bruel steady turbulent premixed flame propagation , Int. Journal for Numerical Methods in Fluids, Vol. 14, No 10, p. 1063-1091, 2004.

• Lehnasch, G. and Bruel, P., Some specific aspects of the simulation of highly

underexpanded supersonic jets , 5th Asian-Pacific International Symposium on Combustion and Energy Utilization, ISBN 962-367-451-1, Hong-Kong, China, December 2004.

• Bruel, P., Dourado, M.C.W, Azevedo, J.L.F., “The artificial compressibility method:

an alternative for the simulation of zero Mach number inert and reactive flows”, 9th International Conference Zaragoza-Pau on Applied Mathematics and Statis- tics, Jaca, Spain, September 2005. Phd-students (name, subject, year of defence, actual position)

• St´

ephane DUPLANTIER Defense on december 8, 1995 (Co-supervision with B. Deshaies, actual position: unknown)

• Bernard PONS Defense on december 13, 1996 (Co-supervision with B. Deshaies,

actual position: Turbomeca, Bordes, France).

• St´

ephane SANQUER Defense on november 4, 1998 (Co-supervision with B. De- shaies, actual position: CSTB, Nantes, France)

• Catherine CORVELLEC Defense on december 18, 1998 (Co-supervision with B.

Deshaies, actual position: Le Moteur Moderne, Paris, France)

• Magali BESSON Defense on september 11, 2001 (Co-supervision with B. De-

shaies, actual position: Renault, Lardy, France)

• Aur´

elie FAIX Defense on december 18, 2001 (Co-supervision with B. Deshaies, actual position: Dassault Systemes, Paris, France)

• Wladymir DOURADO Defense on november 25, 2003 (Co-supervision with L.F.

Azevedo, Brazil, actual position: Centro Tecnico Aeroespacial, Sao Jos´ e dos Campos, Brazil )

• Phuc Danh NGUYEN Defense on december 2, 2003 (Actual position: post doc-

toral position, LMA, Pau , France)

• Guillaume LEHNASCH Defense on june 29, 2005 (Actual position: post-doctoral

position, Laboratoire d’´ etudes a´ erodynamiques, Poitiers, France)

• Bernardo MARTINEZ Defense on december 22, 2005, (Actual position: post-

doctoral position, Limoges, France) 40

CONCHA A.3 Daniela Capatina

• S´

ebastien REICHSTADT, in progress (co-supervision with A. Ristori, ONERA)

• Said LAHRAICHI, in progress (co-supervision avec V. Sabel’nikov, ONERA)
• Raphaelle LECOT, in progress (co-supervision avec D. Gaffi´

e, ONERA)

• Aur´

elien MOST, with Turbomeca, in progress.

A.3 Daniela Capatina

Personal data date of birth 1.5.1971 place of birth Iasi, Roumanie address 4 rue Ren´ e Coty, F-64000 Pau Education 1997

• PhD. Thesis, (supervisor: Jean-Marie Thomas), Universit´

e de Pau 1994 DEA de Math´ ematiques Appliqu´ ees, Universit´ e de Pau 1993 Maˆ ıtrise de la Facult´ e de Math´ ematiques, Universit´ e A.I. Cuza, Iasi, Roumanie Positions depuis 1998 Maˆ ıtre de Conf´ erences, Universit´ e de Pau 1997-1998 A.T.E.R., Universit´ e de Pau Administrative duties

• Responsable of the Numerical Analysis Seminar at LMA (2000-2002 )
• Responsable of the ERASMUS Programme of the Department of Mathematics,

UPPA (2000-2004 )

• Member of the Commission des Sp´

ecialistes 25-26, UPPA (from 2000) Industrial contracts

• IFREMER : European Project LITHEAU (DG 14), april 2000-april 2002. Partic-

ipation on the theme Hydrodynamique de l’Adour. 41

CONCHA A.3 Daniela Capatina

• TOTAL : Grant Mod´

elisation 2D des temp´ eratures aux puits (PhD. Thesis of B. Denel, nov. 2001-nov.2004, amount 135 000 euros including the salary)

• TOTAL : Grant MOTHER (PhD Thesis of L. Lizaik, nov. 2005-nov.2008, amount

36 000 euros excluding the salary) Main publications

• M. Amara, D. Capatina-Papaghiuc, A. Chatti, “Bending Moment Mixed Method

for the Kirchhoff-Love Plate Model”, SIAM J. Num. Anal., vol. 40, n. 5, p. 1632-1649, 2002.

• M. Amara, D. Capatina-Papaghiuc, A. Chatti,“ New Locking-Free Method for the

Reissner-Mindlin Plate Model“, SIAM J. Num. Anal., vol. 40, n. 4, p. 1561-1582, 2002.

• M. Amara, D. Capatina-Papaghiuc, D. Trujillo, “Hydrodynamical modelling and

multidimensional approximation of estuarian river flows“, Computing and Visu- alization in Science, vol. 6, n. 2-3, p. 39-46, 2004.

• M. Amara, D. Capatina-Papaghiuc, B. Denel, P. Terpolilli, “Mixed Finite Element

Approximation for a Coupled Petroleum Reservoir Model“, M2AN, vol. 39, n. 2,

• p. 349-376, 2005.
• M. Amara, D. Capatina-Papaghiuc, B. Denel, P. Terpolilli, “Numerical modelling
• f flow with heat transfer in petroleum reservoir“, Int. J. Numer. Method. Fluids,
• vol. 47, n. 8, p. 955-962, 2005.
• M. Amara, D. Capatina-Papaghiuc, D. Trujillo, “Stabilized finite element method

for Navier-Stokes equations with physical boundary conditions“, Math. Comp.,

• vol. 76, p. 1195-1217, 2007.

Software development

• Separate and Coupled Reservoir and Wellbore Thermomechanical Models ( PhD.

Thesis of B. Denel and L. Lizaik). Phd-students (name, subject, year of defence, actual position)

• Amna Chatti (co-supervised by M. Amara) Analyse des m´

ethodes d’´ el´ ements finis pour l’hydrodynamique, 1999, Assistant Professor, ISSAT of Sousse (Tunisie). 42

CONCHA A.4 Robert Luce

• Bertrand Denel (co-supervised by M.Amara) Simulation num´

erique et couplage de mod` eles thermo-m´ ecaniques puits-milieux poreux, 2004, Research and Devel-

• pment Engineer, TOTAL (CDI).
• Layal Lizaik (co-supervised by M. Amara) Couplage de mod`

eles puits et r´ eservoir. Mod` ele multi-phasique de r´ eservoir avec prise en compte de la thermom´ etrie, started in oct. 2005, PhD student at LMA and TOTAL (grant CIFRE).

A.4 Robert Luce

Personal data date of birth 14.4.1962 place of birth Guise (02) address 11 imp Georges Bizet, 64140 Lons Education

• Doctorat de l’Universit´

e de Technologie de Compi` egne, sp´ ecialit´ e contrˆ

• le des

syst` emes Directeur de recherche: J.P. Kernevez Titre du m´ emoire: Probl` emes

• Inverses. Contrˆ
• labilit´

e Exacte de Syst` emes R´ egis par des Equations aux D´ eriv´ ees

• Partielles. Soutenue le 11 f´

evrier 1991 sous la pr´ esidence de J.L. Lions Positions

• Maˆ

ıtre de Conf´ erences en 26` eme sectionen 1992 au Laboratoire de Math´ ematiques Appliqu´ ees UMR5142 (Universit´ e de Pau et des Pays de l’Adour) . Administrative duties

• Master MMS ”Math´

ematiques, Mod´ elisations et Simulation”, 2007-

• Master MCS ”Mod´

elisations et Calcul Scientifique”, 2004-2007 Main publications

• S. N. Antontsev, G. Gagneux , R. Luce, G. Vallet. A Non Standard Free Boundary

Problem Arising From Stratigraphy. Analysis and Applications, 4(3):209–236, 2006. 43

CONCHA A.4 Robert Luce

• R. Luce, B. Wolhmuth. A local a posteriori estimator based on equilibrated
• fluxes. Vol 42 N◦4,PP 1394-1414. Dec 2004. SIAM J. Numer. Anal.
• R. Luce, S. Perez. A numerical upscaling method for an elliptic equation with

heterogenous tensorial coefficients. International Journal for Numerical Methods in Engineering (2002) 54: P537-556

• R. Luce, S. Perez. A finite volume scheme for an elliptic equation with heteroge-

neous coefficients. Application to an homogenization problem. Applied Numeri- cal Mathematics 38 (2001) P.427-444.

• C. Amrouche, R. Luce, S. Perez. Identification of the Thickness of a Thin Layer

by Boundary Measuments. Inverse Problems. Vol 17, 1703-1716 (2001) Phd-students (name, subject, year of defence, actual position)

• Sylvie Perez. Directeurs de th`

ese R. Luce (85% ), J.M. Thomas (15 % ) . Titre ”Identification et homog´ en´ eisation de param` etres dans des ´ equations aux d´ eriv´ ees partielles” D´ ebut Th` ese: septembre 95 Fin th` ese: 23 septembre 1999. Financement: Bourse du Minist` ere de la Recherche. Situation du Doctorant: Enseignante ` a l’IUFM de Pau

• C´

ecile Poutous. Directeurs de th` ese : J.M Thomas (20%), R. Luce (80%) Titre : Mod´ elisation asymptotique et analyse num´ erique d’un probl` eme de cou- plage fluide-structure D´ ebut Th` ese : Dec 2002, Fin Th` ese : Oct 2006, Fimancement : Bourse du Conseil R´ egional d’Aquitaine Situation du Doctorant: ATER ` a l’UPPPA

• Fabien Dahoumane. Directeurs de th`

ese: Guy Vallet 50%, Robert Luce 50%. Titre: Etude math´ ematique de mod` ele d’´ ecoulement sous l’hypoyh` ese d’approximation hydrostatique. D´ ebut de la th` ese : septembre 2006. Fin : d´ ecembre 2009 Financement : Al- locataire boursier du Minist` ere de la Recherche et Moniteur de l’enseignement sup´ erieur.

• Amar Mockrani. Directeur de th`

ese: G´ erard Gagneux 80%, R. Luce 20 % Titre: Probl` emes pseudo-paraboliques ` a vitesse asservie. Application en prospec- tion p´ etroli` ere. 44

CONCHA A.5 Eric Schall D´ ebut de la th` ese : septembre 2005. Fin : d´ ecembre 2008. Financement : Bourse du laboratoire LMA.

A.5 Eric Schall

Personal data date of birth 11/06/1965 place of birth Grenoble, France address 13 rue des chˆ enes, 64111 Maucor Education 1995 Th` ese de Doctorat en M´ ecanique-Energ´ etique (f´ elicitations du jury) Institut Universitaire des Syst` emes Thermiques, Universit´ e de Provence 1992 DEA de M´ ecanique Energ´ etique , Marseille 1992–90 Diplˆ

• me d’Ing´

enieur Thermicien Institut Universitaire des Syst` emes Thermiques, Universit´ e de Provence 1989 Maˆ ıtrise d’A´ eronautique ` a l’Institut de M´ ecanique des Fluides , Marseille II Positions since 1999 Maˆ ıtre de Conf´ erences (section 60), IUT G´ enie Thermique et Ernergie Universit´ e de Pau et des Pays de l’Adour 1999–98 Ing´ enieur Expert de l’ INRIA Projet Sinus , Sophia Antipolis 1998–97 Chercheur de l’Ecole Nationale des Ponts et Chauss´ es Projet Caiman, CERMICS-INRIA Sophia Antipolis 1997–95 Chercheur Contractuel, Universit´ e de Provence Industrial contracts

• Rapport d’activit´

e contractuel r´ egion Aquitaine/FEDER/UPPA ”Projet Dirigeable Gros Porteur”, Activit´ e Octobre 2002, Septembre 2003, 150 pages. Contacts et/ou partenaires industriels: PME/PMI d’Aquitaine En collaboration avec M. Amara. 45

CONCHA A.5 Eric Schall

• Rapport interm´

ediaire EFAISTOS 1998: ”A Mixed Finite Volume/Finite Element Method and an adaptative Finite Element Method Applied to Combustion in Mul- tiphase Medium”, CERMICS-ENPC. Aot 1998: 26 pages.

• M. Braack, E. Schall, N. Glinsky-Olivier.
• Rapport annuel d’activit´

e du contrat pass´ e par la Direction des Recherches, Etudes et Techniques (D´ el´ egation G´ en´ erale pour l’Armement (DGA)): ”Aide ` a l’Optimisation d’un Laser de Puissance HF/DF Puls´ e”, Convention N. 96.34.059.00.470.75.65. Juin 1997, 47 pages. Contacts et/ou partenaires industriels: Soci´ et´ e LASERDOT, DGA

• E. Schall,, O. Chaoul et D. Zeitoun.
• Rapport semestriel d’activit´

e du contrat pass´ e par la Direction des Recherches, Etudes et Techniques (D´ el´ egation G´ en´ erale pour l’Armement): ”Aide ` a l’Optimisation d’un Laser de Puissance HF/DF Puls´ e”, Convention N. 96.34.059.00.470.75.65. Janvier 1997, 33 pages. Contacts et/ou partenaires industriels: Soci´ et´ e LASERDOT, DGA

• E. Schall, O. Chaoul et D. Zeitoun.

Main publications

• S. Soubacq, P. Pignolet, E. Schall, J. Batina, “Investigation of a gas breakdown

process in a laser-plasma experiment,” Journal of Physics D: Applied Physics,

• vol. 37, pp. 2686–2702, 2004.
• E. Schall, D. Leservoisier, B. Koobus, A. Dervieux, “Mesh Adaptation as a tool

for certified Computation Aerodynamics,” International Journal for Numerical Method in Fluid, vol. 45, pp. 179–196, 2004.

• E. Schall, C. Viozat, B. Koobus, A. Dervieux, “Computation of low Mach thermi-

cal flows with implicit upwind methods,” International Journal of Heat and Mass Transfer, vol. 46, pp. 3909–3926, 2003.

• E. Schall, R. Lardat, B. Koobus, A. Dervieux, C. Farhat, “Aeroelastique Cou-

pling between a Thin Divergent and High Pressure Jets,” Revue Europ´ eenne des El´ ements Finis, Fluid-Structure Interaction, vol. 9, pp. 835–851, 2000.

• S. Seror, J.Ph. Brazier, D. Zeitoun, E. Schall, “Asymptotic Defect Boundary

Layer, Theory applied to Thermochemical Nonequilibrium Hypersonic Flows,” Journal of Fluid Mechanics, Fluid-Structure Interaction, vol. 339, pp. 213–238, 1997.

• D. Zeitoun, E. Schall, Y. Burtschell, M.C Druguet, “Vibration-Dissociation Cou-

pling in Nonequilibrium Hypersonic Viscous Flows,” AIAA Journal, vol. 33, n.1, 46

CONCHA A.6 David Trujillo

• pp. 79–85, 1995.

Phd-students (name, subject, year of defence, actual position)

• S. Soubacq,“Etude de la d´

etente dynamique d’un plasma laser. Influence du champ effectif laser,” D´ ecembre 2003, Responables: P. Pignolet, E Schall

• Y. Bentaleb,“Mod´

elisation et simulation num´ erique de la turbulence compressible par des mod` eles non-lin´ eaires ` a bas Reynolds,” D´ ecembre 2007, Responables: J.P. Dumas, E. Schall

• Y. Moguen,“Mod´

elisation et simulation num´ erique ` a bas nombre de Mach,” D´ ecembre 2008, Responables: M. Amara, E. Schall

• J. Alessi,“M´

ethode de mapping en advection chaotique,” D´ ecembre 2008, Re- sponables: S. Gibout, E. Schall

• T. El Rhafiki,“Changement de phase et d´

es´ equilibre thermodynamique d’un fluide frigoporteur en mouvement,” D´ ecembre 2009, Responables: E. Schall, Y. Zeraouli

A.6 David Trujillo

Personal data date of birth 3.09.1967 place of birth Pau, France address 5 rue de Gavarnie, F-64000 Pau Education 1994

• PhD. Thesis, (supervisor: Jean-Marie Thomas), Universit´

e de Pau 1991 DEA de Math´ ematiques Appliqu´ ees, Universit´ e de Pau 1990 Maˆ ıtrise d’Ingnierie Math´ ematique, Universit´ e de Pau Positions Since 1995 Maˆ ıtre de Conf´ erences, Universit´ e de Pau Administrative duties

• Responsable of the second year undergraduate program

47

CONCHA A.6 David Trujillo Industrial contracts

• Contract RNTL (n03-2-93-0547) : Amount: 47000 euro, Period: 2003-2006 Ob-

ject: Project ”Salome2”, development of a C++ library for mixed finite element and finite volume methods in 2D and 3D. Contract with IFP (Institut Franais de Ptrole) (n 24749): Montant: 80KF Period : 10-1999 10-2002, Object: Modelling and simulation of sedimentary basinsł, Phd: L. Nadau.

• European Contract with IFREMER : Amount : 295KF Period: from 01-2001 to

12-2001, Object: Fluvial hydrodynamics

• Contract with Institution Adour: Montant : 100KF Period: from 11-2000 to 05-

2001, Object: prevention of the risks of flood Main publications

• M. Amara, D. Capatina-Papaghiuc, D. Trujillo, “Hydrodynamical modelling and

multidimensional approximation of estuarian river flows“, Computing and Visu- alization in Science, vol. 6, n. 2-3, p. 39-46, 2004.

• B. Lacabanne, G. Gagneux, D. Trujillo : ”Multicomponent Flow in a Porous
• Medium. Adsorption and Soret Effect Phenomena Local Study and Upscaling

Process”, M2AN, Vol. 35, N 3, pp. 481-512 , 2001,

• J.-M. Thomas, D. Trujillo : ”Mixed Finite Volumes Methods”, Internat. J. for

Numerical Methods in Engineering ,1999, 46, 1351–1366.

• J.-M. Thomas, D. Trujillo : ”Finite Volume Methods for Elliptic Problems; Con-

vergence on Unstructured Meshes”, Numerical Methods in Mechanics, Pitman

• Res. Notes in Mathematics Series, C. Conca & G. Gatica, eds. Addison Wesley

Longmann,1997, pp124–136.

• J.-M. Thomas, D. Trujillo : ” Finite Volume Variational Formulations. Applica-

tion to Domain Decomposition Method”, A. Quarteroni, J. Periaux, Y. A. Kus- netsov, O. B. Widlund, coll. Domain Decomposition Methods in Sc. & Engineer- ing, #157, 1994, pp 127–132 Software development

• Erreka: Fluvial hydrodynamics (Contract with Ifremer)
• Project ”Salome2”, development of a C++ library for mixed finite element and

finite volume methods in 2D and 3D. 48

PUBLICATIONS OF THE PROJECT MEMBERS Phd-students (name, subject, year of defence, actual position)

• Bruno Lacabanne: ”Thermodiffusion in porous media”, june 2001, actual posi-

tion: engineer at Schlumberger.

• Lionel Nadau: ”Modelling and simulation of sedimentary basins”, september

2003, actual position: engineer at GDF (Gaz de France)

• Agnes Petrau : ”Coupling of 2D and 3D hydrodynamical models”, since septem-

ber 2006, in collaboration with IFREMER.

Publications of the project members

[1] M. Amara, D. Capatina, B. Denel, and P. Terpolilli. Numerical modelling of flow with heat transfer in petroleum reservoirs. Int. J. Numer. Methods Fluids, 47(8- 9):955–962, 2005. [2] M. Amara, D. Capatina-Papaghiuc, and D. Trujillo. Hydrodynamical modelling and multidimensional approximation of estuarian river flows. Comput. Vis. Sci., 6(2-3):39–46, 2004. [3] M. Amara, D. Capatina-Papaghiuc, and D. Trujillo. Variational approach for the multiscale modeling of a river flow. part 1 : Derivation of hydrodynamical models. Technical report, LMA, UPPA, 2006. [4] M. Amara, E.Chac´

• n Vera, and D. Trujillo. Vorticity-velocity-pressure formula-

tion for Stokes problem. Math. Comput., 73(248):1673–1697, 2004. [5] Cherif Amrouche, Robert Luce, and Sylvie Perez. Identification of the thickness

• f a thin layer from boundary measurements. Inverse Probl., 17(6):1703–1716,

2001. [6] Stanislav N. Antontsev, G´ erard Gagneux, Robert Luce, and Guy Vallet. New uni- lateral problems in stratigraphy. ESAIM, Math. Model. Numer. Anal., 40(4):765– 784, 2006. [7] R. Becker and M. Braack. A finite element pressure gradient stabilization for the Stokes equations based on local projections. Calcolo, 38(4):173–199, 2001. [8] R. Becker, M. Braack, and R. Rannacher. Numerical simulation of laminar flames at low Mach number with adaptive finite elements. Combust. Theory Modelling, 3:503–534, 1999. 49

PUBLICATIONS OF THE PROJECT MEMBERS [9] R. Becker and P. Hansbo. A finite element method for domain decomposition with non-matching grids. Technical report, INRIA, http://www.inria.fr, 1999. [10] R. Becker and P. Hansbo. Discontinuous Galerkin methods for convection- diffusion problems with arbitrary P´ eclet number. In World Scientific, editor, Nu- merical Mathematics and Advanced Applications: Proceedings of the 3rd Euro- pean Conference, pages 100–109, 2000. [11] R. Becker, P. Hansbo, and R. Stenberg. A finite element method for domain de- composition with non-matching grids. M 2AN, 37(2), 2003. [12] R. Becker, S. Mao, and Z.-C. Shi. A convergent adaptive finite element method with optimal complexity. Technical report, Pau, 2007. [13] R. Becker, D. Meidner, and B. Vexler. Efficient numerical solution of parabolic

• ptimization problems by finite element methods. Optimisation Methods and Soft-

ware, 2007. [14] R. Becker and R. Rannacher. A feed-back approach to error control in finite ele- ment methods: Basic analysis and examples. East-West J. Numer. Math., 4:237– 264, 1996. [15] R. Becker and R. Rannacher. An optimal control approach to a-posteriori error

• estimation. In A. Iserles, editor, Acta Numerica 2001, pages 1–102. Cambridge

University Press, 2001. [16] R. Becker and B. Vexler. A posteriori error estimation for finite element discretiza- tions of parameter identification problems. Numer. Math., 96(3):435–459, 2004. [17] R. Becker and B. Vexler. Mesh refinement and numerical sensitivity analysis for parameter calibration of partial differential equations. J. Comp. Phs., 206(1):95– 110, 2005. [18] M. Braack and E. Burman. Local projection stabilization for the Oseen problem and its interpretation as a variational multiscale method. SIAM J. Numer. Anal., 43(6):2544–2566, 2006. [19] M. Braack and A. Ern. A posteriori control of modeling errors and discretization

• errors. Multiscale Model. Simul., 1(2):221–238, 2003.

[20] M. Braack and A. Ern. Coupling multimodelling with local mesh refinement for the numerical computation of laminar flames. Combust. Theory Model., 8(4):771– 788, 2004. 50

PUBLICATIONS OF THE PROJECT MEMBERS [21] P. Bruel, D. Karmed, and M. Champion. A pseudo-compressibility method for reactive flows at zero Mach number. Int. J. Comput. Fluid Dyn., 7(4):291–310, 1996. [22] E. Burman and P. Hansbo. Edge stabilization for Galerkin approximations of the generalized Stokes’ problem. Comput. Methods Appl. Mech. Engrg., 195, 2006. [23] Daniela Capatina-Papaghiuc and Jean-Marie Thomas. Nonconforming finite ele- ment methods without numerical locking. Numer. Math., 81(2):163–186, 1998. [24] C. Corvellec, P. Bruel, and V.A. Sabel’nikov. A time-accurate scheme for the cal- culations of unsteady reactive flows at low Mach number. Int. J. Numer. Methods Fluids, 29(2):207–227, 1999. [25] Wladimyr M.C. Dourado, Pascal Bruel, and Jo˜ ao L.F. Azevedo. A time-accurate pseudo-compressibility approach based on an unstructured hybrid finite volume technique applied to unsteady turbulent premixed flame propagation. Int. J. Numer. Methods Fluids, 44(10):1063–1091, 2004. [26] S. Gibout, Y. Le Guer, and E. Schall. Coupling of a mapping method and a ge- netic algorithm to optimize mixing efficiency in periodic chaotic flows. Commun. Nonlinear Sci. Numer. Simul., 11(3):413–423, 2006. [27] D. Karmed, M. Champion, and P. Bruel. Two-dimensional numerical modeling of a turbulent premixed flame stabilized in a stagnation flow. Combustion and Flame, 119(3):335–345, 1999. [28] Raphael Lardat, Bruno Koobus, Eric Schall, Alain Dervieux, and Charbel Farhat. Analysis of a possible coupling in a thrust inverter. 2000. [29] G. Lehnasch. Contribution ` a l’´ etude num´ eriques des jets supersoniques sous- d´

• etendus. PhD thesis, Universit´

e de Poitiers, Poitiers, 2005. Doctorat d’Universit´ e. [30] R. Luce and B.I. Wohlmuth. A local a posteriori error estimator based on equili- brated fluxes. SIAM J. Numer. Anal., 42(4):1394–1414, 2004. [31] Robert Luce and Sylvie Perez. Homogenization of the heterogeneous coefficients in an elliptic equation. Cruz Lopez de Silanes, Maria (ed.) et al., Actes des 6` emes journ´ ees Zaragoza-Pau de math´ ematiques appliqu´ ees et de statistiques. Pau: Pub- lications de Universit´ e de Pau et Pays de l’Adour, PUP. 401-408 (2001)., 2001. [32] E. Schall, D. Leservoisier, A. Dervieux, and B. Koobus. Mesh adaptation as a tool for certified computational aerodynamics.

• Int. J. Numer. Methods Fluids,

45(2):179–196, 2004. 51

GENERAL REFERENCES [33] Eric Schall, Raphael Lardat, Alain Dervieux, Bruno Koobus, and Charbel Farhat. Aeroelastic coupling between a thin divergent and high pressurs jets. 2000. [34] Eric Schall, Cecile Viozat, Bruno Koobus, and Alain Dervieux. Computation of low Mach thermical flows with implicit upwind methods. Int. J. Heat Mass Trans- fer, 46(20):3909–3926, 2003. [35] J.-M. Thomas and D. Trujillo. Finite volume variational formulation. Application to domain decomposition methods. Quarteroni, Alfio (ed.) et al., Domain decom- position methods in science and engineering. The sixth international conference on domain decomposition, Como, Italy, June 15-19, 1992. Providence, RI: American Mathematical Society. Contemp. Math. 157, 127-132 (1994)., 1994. [36] J.-M. Thomas and D. Trujillo. Mixed finite volume methods. Int. J. Numer. Meth-

• ds Eng., 46(9):1351–1366, 1999.

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[73] V. Girault and P.-A. Raviart. Finite Elements for the Navier Stokes Equations. Springer, Berlin, 1986. [74] V. Girault, B. Riviere, and M.F. Wheeler. A discontinuous Galerkin method with non-overlapping domain decomposition for the Stokes and Navier-Stokes prob-

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[75] E. Godlewski and P.-A. Raviart. Hyperbolic Systems of Conservation Laws. Math- ematics & Applications, Ellipses, Paris, 1991. [76] P. M. Gresho and R. L. Sani. Incompressible flow and the finite element method, Vol 1 and 2. Jossey-Bass, 1996. [77] J.-L. Guermond. Stabilization of Galerkin approximations of transport equations by subgrid modeling. Mod´

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mation of the Maxwell operator: non-stabilized formulation. 2005. [84] P. Houston, Chr. Schwab, and E. S¨

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58