Section: Scientific Foundations
Challenges related to numerical simulations of reacting flows
The modeling of reactive flows [37] , [68] and turbulent reactive flows [63] , [67] implies a number of difficulties.

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 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.

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, 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 [58] , [64] .
Although first attempts exist for generalization of a posteriori error estimators to model adaptation [38] , [47] , [48] and [61] , it remains a challenging question to develop numerical approaches using a hierarchy of models in a automatic way, especially combined with mesh adaptation.

AllMach regimes
The development of solvers able to deal with different Mach regimes simultaneously is a challenging subject. A robust and efficient methodology, which works for all the regimes, requires combination of the best techniques in the field of compressible and incompressible solvers.

Turbulence and chemistry
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, at least in the case of realistic geometries. Therefore, turbulence models such as RANS and LES are used in practice. We note that special difficulties are due to coupling of the flow with chemistry.