CONCHA is a joint project of INRIA Futurs and LMA (Laboratoire de Mathématiques Appliquées– CNRS UMR 5142), UPPA (Université de Pau et des Pays de l'Adour).
The main objective of this project is the development of innovative algorithms and efficient software tools for the simulation of complex flow problems. On the one hand, our contributions concern modern discretization methods (highorder and adaptivity) and goaloriented simulation tools (prediction of physical quantities, numerical sensitivities, and inverse problems) on the other hand. Concrete applications originate from flows related to propulsion devices featuring important and challenging problems for numerical simulations in this field; see Section .
Reactive flow problems lead to very complex coupled systems of equations which are often very difficult to handle in routine way with industrial software. The difficulties are due to the physical complexity of the system of equations to be solved and the mathematical problems 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 twophase flows.
Our mediumterm goal is to develop flow solvers based on modern numerical methods such as highorder discretization in space and time and selfadaptive algorithms. Adaptivity based on a posteriori error estimators has become a new paradigm in scientific computing, first because of the objective to give rigorous error bounds, and second because of the possible speedup of simulation tools. A systematic approach to these questions requires an appropriate variational framework and the development of corresponding software tools.
It is our goal to study at hand of concrete applications the possible benefits and difficulties related to these numerical approaches in the context of complex fluid mechanics. Therefore, prototypical applications are chosen in order to represent important challenges in our field of application.
The main ingredients of our numerical approach are adaptive finite element discretizations combined with multilevel solvers and hierarchical modeling. In view of our applications described below in Section , it is natural to consider discontinuous and stabilized finite element methods, for example the socalled discontinuous Galerkin approach (DGFEM), since it generalizes classical finite volume methods and offers a unified framework for the development of adaptive higher order methods. The enhancements of these methods and their application to challenging physical problems induce numerous mathematical investigations.
Our longterm 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 different in nature as a diffusion coefficient and a velocity boundary condition. CONCHA has as longterm objective the development of the necessary computational framework and to contribute to the rational interaction between simulation and experiment.
Goals of the project are to evaluate the potential of recent numerical methods and to develop new approaches in the context of industrial CFD problems.
We intend to validate our software with respect to other commercial and research tools in the domain, such as Aéro3d (INRIASmash), AVBP (CERFACS), Cedre (ONERA), Fluent (ANSYS), FluidBox (INRIAScallaplix), OpenFoam (OpenCfd).
The development of CFD software benefits in an important measure from inhouse 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 an open data basis which serves for comparison with other simulation software and experiments. As outlined above, the project is based and generates different research topics. Especially, 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 inhouse experiments is available.
In the first phase we intend to develop highorder adaptive flow solvers. The availability of such tools is crucial for DNSbased 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. Testing of new numerical methods from the mathematical community at hand of concrete 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 adaptive 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 ?
Interaction with industry should profit from the different research networks listed above. The purposes 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 feedback 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 phdtheses.
CONCHA has been created as an 'equipe INRIA' in april 2007.
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 'highorder 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 highorder methods. Finally some perspectives are outlined.
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 physical model has to be represented. We propose to develop a systematic approach to these questions based on highorder and autoadaptive methods.
We note that most of the physical problems under consideration have a threedimensional character and involve the coupling of models. This makes the development of fast numerical methods a question of feasibility.
For the modeling of reactive flows see , . Numerical simulations of reactive flows have rapidly gained interest , , . For an overview on stateoftheart modeling of combustion and turbulent reactive flows see , .
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 Newtontype algorithms. A good speedup of the algorithms requires a clever combination of iteration and splitting techniques based on the structure of the concrete problem under consideration.
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, models with detailed chemistry are often prohibitive, and there exists a zoo of simplified equations, starting with flamesheettype models. The question of model reduction is of great interest for reacting flows, and different approaches have been developed , .
Although first attempts exist for generalization of a posteriori error estimators to model adaptation , , and , it remains a challenging question to develop numerical approaches using a hierarchy of models in a automatic way, especially combined with mesh adaptation.
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 incompressible 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 , , , whereas for the second one we cite exemplarily , , . The recent interest in questions related to lowMach number flows is for example documented in the proceedings , , .
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 the NavierStokes equations, and the corresponding idea that rational turbulence models might be based on adaptive techniques has gained much attention, see , and related work.
The discontinuous Galerkin finite element method (DGFEM) offers interesting perspectives, since it offers a framework for the combination of techniques developed in the incompressible finiteelement (wellfounded treatment of incompressibility constraints, pressure approximation, and stabilization for highReynoldsnumber flows) and the compressible finitevolume 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 (
hpmethods). At the same time, the wellestablished finiteelement knowledge for saddlepoint 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.
Overview articles concerning DGFEM and stabilized finite element methods are available, see for example , . 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.
The first discontinuous Galerkin method for hyperbolic equations is generally attributed to Reed and Hill
, where the authors propose a scheme for the neutron transport equation. The first analysis of the method has
been presented by Lesaint and Raviart
. Their result has been further improved and the analysis substantially broadened by Johnson, Pitkäranta and
Nävert
,
. We illustrate DGFEM for the linear firstorder scalar equations
·
u=
fin
with boundary conditions
u=
gon the inflow part of the boundary
^{}(defined by the condition
·
n
0with
nthe unit outer normal field at
). We seek an approximation
u_{h}which is allowed to be discontinuous across all interior edges of a given mesh
h. On a given cell
Kof the mesh, the restriction
u_{K}of
u_{h}to
Kis a polynomial. It is determined by the equation to be satisfied by all local test functions
v_{K}:
where
F(
u_{h},
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=
),
F(
u_{h},
n)equals for example
(
·
n)
gon
K^{}and
(
·
n)
u
_{K}on
K^{+}.
DGFEM became popular only ten years later when it was applied to nonlinear hyperbolic equations within the work of Cockburn and Shu , who combine a spatial DGFEM discretization with an explicit RungeKutta method in time. The breakthrough of DGFEM for the Euler equations was achieved by Bassi and Rebay . 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 .
Also around 1970, DGFEM for elliptic and parabolic equations were proposed. The different variants were generally called interior penalty (IP) methods 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
where a variational scheme for the socalled weak implementation of Dirichlet boundary conditions was developed
and analyzed. In order to solve the Dirichlet problem

u= 0in a bounded polygonal domain
with boundary condition
u=
gon
, Nitsche's method determines a finite element approximation
u_{h}to
uin a finite element space
V_{h}as the solution of the variational equation to be satisfied for all test functions
v_{h}V_{h}:
where and 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 ) 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 conditions; see also Wheeler for a collocation finite element method and the methods of Douglas and Dupont and Baker 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 convectiondiffusion, see for example . In order to deal with the NavierStokes equations, new ideas for the discretization of the diffusive terms have been developed in Bassi and Rebay and Cockburn and Shu . A unified framework of DGFEM methods for elliptic equations has been developed in .
The DGFEM framework offers many interesting possibilities. We mention here a robust scheme for the elasticity system
, the discretization of Maxwell's equation
,
, its use for the discretization on nonmatching meshes without Mortar spaces
, and
hpmethods
,
.
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 CrouzeixRaviart space); considering the system
matrix even leads to a more disadvantageous count. Concerning higherorder 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.
Standard finite element schemes do not lead to satisfactory schemes for convection dominated problems. Modifications of standard Galerkin methods by introducing additional socalled
stabilization terms in order to increase robustness have been introduced by Hughes and coworkers for the Stokes equations and convectiondominated problems
,
,
. In case of the linear transport equation
·
u=
fin
with boundary conditions
u= 0on the inflow part of the boundary
^{}a typical variational formulation reads: Find
u_{h}V_{h}such that for all test functions
v_{h}V_{h}there holds
The third term on the left of ( ) is the stabilization term which allows to control the streamline derivative of the discrete solution. The resulting socalled streamline diffusion method SDFEM (or sometimes Galerkin least squares or SUPG) has been analyzed in detail by Johnson and coworkers , . The method has been extended in many variants to the incompressible NavierStokes equations, see for example , .
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 (
) if
f=
f(
u)depends on
u. These difficulties are avoided by a new family of methods which replace the stabilization term in (
) by terms of the form
where
is a different approximation of
·
u, constructed for example with the help of an additional coarser mesh, see Guermond's twolevel scheme for the transport equation
and the local projection stabilization
,
. Another approach in this family is the stabilization based on the jumps of the derivatives of finite element
functions which goes back to
and has recently been developed in a systematic way for convectiondiffusion and incompressible flow problems
by Burman and Hansbo
,
.
There are many similarities between DGFEM and SDFEM (streamlinediffusion FEM) based on piecewise linear elements for the transport equation from a theoretical point of view, see for example the classical text book . Recently, attempts have been made to shed brighter light on the relations between these methods , . There is also a renewed interest in stabilizing nonconforming finite elements for the NavierStokes equations , , .
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.
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 higherorder spaces with special numerical integration in order to obtain fast computation of residuals and matrices ? How to stabilize such higherorder 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 such as mixed finite elements surprisingly lead to bad convergence behavior
. Standard conforming
Q^{p}finite elements are costly to compute and have disadvantages on anisotropic meshes.
Timediscretization:
The choice of DG timediscretization is natural in view of its good stability and conservation properties. However, the higherorder members of this family lead to coupled systems which have to be solved in each timestep.
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 tensorproduct structure of the spacetime mesh, it is still possible to organize a code in a timemarching procedure. The DG timediscretization leads to implicit RungeKutta schemes , 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, timestepping and specially tuned Newton algorithms.
Although direct solvers are often competitive in twodimensional computations, the complexity of threedimensional problems makes iterative solvers unavoidable; here, multilevel solvers are able to ensure optimal complexity for stationary problems. It should however be denoted, that timedependent problems lead to different situations, and the most efficient solution largely depends on the concrete application. The question of parallelization is intrinsically related and is of particular importance for the solution of the systems resulting from higherorder space or even time discretization.
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 methods which allow to locally adapt the approximation without necessarily adapting the mesh. Such a procedure has been proposed in and applied to fracture problems in solid mechanics in . We propose to develop a similar approach for the tracking of shock waves and combustion fronts.
There are many attempts to improve codes for compressible flows in order to deal with small Mach numbers, see for example , 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 physicaldifficulty in finding an appropriate model and a numericaldifficulty in finding a stable discretization. Concerning the numericaldifficulty, we know from the theory of numerics for incompressible flows that the pressure 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 modern 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 or by introduction of stabilization terms (or for more recent techniques , , , ).
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 tradition , and have recently regained attention , , . We remark here, that the discretization of secondorder terms as the diffusion terms has a sound theoretical background , , , .
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 conservativevariables and accurate shock approximation necessitates their use, diffusion and boundary conditions are expressed in primitivevariables.
How to solve the coupled system ?
Decoupling algorithms as the pressure correction schemes for transient incompressible flows are well understood. However, the generalization to higher Mach numbers, in particular for combustion, does not seem to be straightforward due to the different role of the pressure.
Concerning the physicaldifficulties, one has to be aware that the full system of flow equations comprises different physical phenomena with different timescales. One standard technique consists in developing the pressure with respect to the Mach number , . One is then able to construct numerical schemes for the resulting equations in the small Mach number limit , . However, in that case fast scales as acoustic 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.
The possible benefits of local mesh refinement for fluid dynamical problems is nowadays 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 autoadaptive methods as described above is a recent research topic, and only very limited theoretical results are known. Concerning the convergence of adaptive
methods for mesh refinement, only recently there has been made significant progress in the context of the Poisson problem
,
, based on twosided a posteriori error estimators
. The situation is completely open for
padaptivity, modeladaptivity or the DWR method
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.
A general approach towards the systematic a posteriori error estimation of modeling errors has been developed in , , and . 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.
The four 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 praxisoriented approach to error estimation is the DWR method, developed in ; see also the overview paper , application to laminar reacting flows in , and application to the Euler equations in . The idea of the DWR method is to consider a given, userdefined 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 meshrefinement for a wide range of model problems . Recently, it has been extended to the control of modeling errors . 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.
Numerical simulations generally involve parameters of different nature. Some parameters 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.
The generalization of the DWR method to parameter identification problems has been developed in , and for timedependent equations. The case of finitedimensional parameters, which is theoretically less challenging than the infinitedimensional 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
Iwhich are obtained from the approximated physical fields by additional computations, often termed
postprocessing. 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/
q_{i}where
q_{i}denotes a single parameter can be expected to be of practical and theoretical interest, either in order to improve the design of 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 .
Here, we list some possible research topics of interest for the longterm perspective of the project.
thermal, acoustic couplings
hierarchical turbulence modeling
other complex flows such as polymers
direct computation of turbulent flows
stability prediction
The targeted applications belong to the field of fluid mechanics with special emphasis on numerical simulation of complex flows. Of special interest are generic flow configurations related to the development of combustion chambers for jet or helicopter engines.
In the following, we shortly review the physics of combustion chambers. Then we present concrete example problems illustrating the challenging numerical difficulties described above. These examples serve as prototypical test problems.
As an introduction, we briefly remind hereafter the very basic aspects of airplane propulsion. 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 inlets. 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) or to entrain a propeller (turboshaft engine) or a rotor (helicopter). A schematic view of a diluted jet engine is given in Figure . 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 presents an overview of a recent jet engine (by RollsRoyce) 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 tendency 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 of pollutants emission.
In the following, we describe four major issues in the development of new combustion chambers: the environmental norms for pollution, the safety constraints, improved wall cooling efficiency, and combustion in microdevices.
The combustion chamber being at the source of most of the engine pollutants 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 chemical species, one can cite the various carbon oxides (which leads to ozone formation, greenhouse effect), the nitrogen oxides (provoking ozone formation locally and in the high atmosphere) and sulfur (which causes atmosphere acidity), unburned hydrocarbons (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 takeoff/flight/landing is thus defined in Annex 16  Volume II . In this framework, the monitored species are unburned hydrocarbons, soot, carbon monoxide, nitrogen oxides. Figure 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 of European research programs such as LOPOCOTEP and MOLECULES which group academic and industrial partners around this common objective. Many of the present device improvements rely on the recourse to Lean Premixed Prevaporized (LPP) combustion 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 negative impact of these phenomena. With this respect, the development of reliable and efficient CFD tools incorporating uptodate and extensively validated physical modeling is of paramount importance in order to supplement experiments.
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 or 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 threeminute impact of an highly underexpanded supersonic jet whose stagnation conditions are strictly imposed (see Figure ).
This threeminute 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 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 offer, 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 priorior 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 ), these jets are characterized by three main zones: i) the near field, whose “diamondlike” shock structure is a repetition 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 selfsimilar jet. In these cases, the jet core remains mainly supersonic so that the global features of the flow structure are reasonably predicted 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 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 or expansion zones, curvature and nonequilibrium effects on the shear layer growth. A comprehensive survey of the sensitivity of the flow structure to various parameters is given in . When such a flow is impacting a solid surface, the resulting flow structure 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 socalled 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 of a nonnormal 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 preliminary necessary step paving the way towards accurate simulations of such type of fluidsurface interaction, is the development of numerical tools able to simulate accurately 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 modeling, starting with the Euler equations (2D, 2Daxysemmetric, and 3D), then considering the NavierStokes equations before going to turbulence modeling.
The film cooling technique is now widely used in combustion chambers. The continuous 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 . Its goal is to study experimentally confined wall flows representative of those present in a real combustion chamber. This experimental rig is intended to be used in order to compare with computational results.
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 interest 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 related to these devices. A doctoral thesis on a related question with a UPPAONERA cosupervison is planned. Since the codes developed in the present project should have the capability of dealing with laminar and turbulent reacting flows, the numerical simulation of the flow in this type of microcombustion chamber is an interesting step to assess the capability of the developed tools to deal with reacting flows.
This kind of flow corresponds to the jets experimentally studied by Yüceïl and Ötügen and Ötügen et al. . Figure presents a schematic view of the flow configuration 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 (taken from ). 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 highorder 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.
This kind of flow corresponds to the experimental configuration investigated by Miron et al. . Figure presents a schematic view of the flow configuration.
The first objective is to simulate accurately the flow within the hole and the near field of the jet to crossflow interaction zone, in order to be able to assess the discharge coefficient sensitivity to the detailed shape of the hole which is in general noncircular. 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
selfadaptive methods for the efficient computation of this functional. We therefore intend to employ the DWRmethod
in order to perform automatic mesh selection. As a midterm goal we wish to develop a similar approach for
selfadaptive turbulence modeling. A first effort in this direction is the postdoctoral work of Phuc Nguyen Danh (supervised by P. Bruel and R. Becker), which is financed by the CDAPP
This kind of flow has been experimentally studied in the doctoral thesis of Dumand at ONERA (supervised by V. Sabel'nikov) . The flow geometry is sketched in Figure .
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 order 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 lowMach number flows.
Since the goals of the project are to evaluate the potential of recent numerical methods and to develop new approaches in the context of industrial CFD problems, it is important to possess flexible and extendable software which is able to integrate the methods under consideration such as local adaptive mesh refinement, anisotropic meshes, hierarchical meshes, hpmethods, and DGFEM. At the same time the codes have to be able to deal with the physics of complex reactive flow problems.
We have started the development of a toolkit Conchawhich serves as a basis for specialized solvers. 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 ), the software proposes different entrance levels. The development is made under the INRIA Gforge.
Based on the toolkit Concha, we have developped a code for the twodimensional Euler equations based on the lowestorder discontinuous finite element method. The objective is to compare the efficiency of different wellknown Riemann solvers, and to understand the computational bottlenecks in order to guide the further development concerning adaptive higher order methods.
Based on the toolkit Concha, we have developped a code for the twodimensional NavierStokes equations based on different stable and stabilized pairs of elements (P1P1, P2P2, P2P1, CR1P0). The objective is to compare the properties and efficiency of different stabilization methods for higher Reynolds numbers in order to guide the further development concerning adaptive higher order methods, especially in three dimensions.
The test facility MAVERIC (MAquette pour la Validation et lâ€™ExpeÌrimentation sur le Refroidissement par Injection Contrôlée has been installed and the particle image velocimetry (PIV) system has been fitted. The first velocity data were produced in June 2007. In Figure , a detailed view of the test section of MAVERIC is presented. The right image of Figure gives an example of profiles that have been extracted from the PIV data and compared to i) LDV measurements available in the literature and ii) RANS simulations carried out in parallel , . A first analysis of the sensitivity of the wall flow to the shape of the perforation hole has also been carried out .
Stabilized FEM for the Stokes problem are very popular in applications, since they allow for equalorder interpolation (same 'simple' FE space for both pressure and velocities) . In recent years, the trend is to move from residualbased schemes such as GLS to consistent penalty methods like LPS , . Especially the method of is advocated, since it avoid the explicit introduction of stabilization parameters. We have recently found an even simpler approach which only used the difference of the mass matrix and its lumped variant. The optimal convergence rate has been establishes, as well as a generalization to higher order elements.
The stabilization of higher Reynoldsnumber flow is the subject of . The approach is based on introduction of the vorticity as an additional unknown. The method has been analyzed, and computational comparisons have been made.
Adaptive finite element methods are becoming a standard tool in numerical simulations, and their application in CFD is one of the main topics of CONCHA. However, their theoretical analysis is very recent, and the known results have a lot of limitations with respect to the underlying PDE, which is supposed to be elliptic, but also with respect to important practical issues. In general, the mesh refinement is supposed to be done by the 'newest vertex algorithm', which generates interior nodes in the newly refined triangles. In addition the standard treatment of an unavoidable dataoscillation term either prohibits the control of the complexity (MNSalgorithm ), or leads to additional iteration . In order to overcome these difficulties, we have developed a new adaptive merking strategy, which allows us to proof geometric convergence and optimal complexity . We have extended this result to the case of the Courant FE with standard redgreen refinement , mixed FE , as well as nonconforming FE .
Turbomeca (SAFRAN Group)
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 combustion. Reactive flow problems and combustion are an important field of collaboration between UPPA and the manufacturer of turbines Turbomeca. Although numerical approaches have been part of it, the collaboration is at the moment mostly centered on experiments. Due to its experimental banks, the TURBOMECA site of PauBordes 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 (Advanced Combustion Initiative)
Numerical methods for complex fluid mechanical applications is a very huge subject. Here we focus on the most significant aspects related to reactive flow simulations.
Research network: Pôle de compétivité AESE
The themes of this project enter the 'pôle de compétitivité' AESE (aéronautique, espace et système embarqués) which brings together research activities in this domain and is located in the southwest of France.
Research network: INCA
Concerning the field of combustion, ONERA, CNRS, and the group SAFRAN (containing TURBOMECA), have created in 2002 the common project INCA 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 technology 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.
The LMA has proposed a new Master program starting in 2007, which is called MMS (Mathématiques, Modélisation et Simulation) and has a focus on analysis, modeling, and numerical computations
in PDEs. The core of this education is formed by lectures in four fields : PDEtheory, mechanics, numerical analysis, and simulation tools. Our software has a special part devoted to
educational purposes (library 'ConchaBase'
The second year of this master program includes lectures on physical applications, one of 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 defines a practical means of collaboration with our industrial partners such as CERFACS, ONERA, TOTAL, and Turbomeca.
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., Computing.
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, FLOCOME07, ICIAM07, ICFD07, MAMERN07, EuroMediterranean Conference on Biomathematics, Le Caire, 2629 juin 2007.
Finally we mention the onemonth visits of Shipeng Mao (Chinese academy of Sciences) and FrancescoJavier Sayas (University of Zaragoza).