## Section: Overall Objectives

### Objectives

The main objective of this project is the development of innovative algorithms and efficient software tools for the simulation of complex flow problems. Our contributions concern modern discretization methods (high-order and adaptivity) and goal-oriented simulation tools (prediction of physical quantities, numerical sensitivities, and inverse problems). Concrete applications originate at the moment from three fields: combustion (see Section 4.1 ), turbulent flows (see Section 4.3 ), and viscoelastic flows (see Section 4.2 ).

Our short-term goal is to develop flow solvers based on modern numerical methods 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 scientific computing, first because of the necessity 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 a posteriori error estimates and adaptive algorithms, as well as sufficiently general software tools able to realize these algorithms. To this end we develop a single common library written in C++ and study at hand of concrete applications the possible benefits and difficulties related to these algorithms in the context of fluid mechanics. Prototypical applications are chosen in order to represent important challenges in our fields of application.

The main ingredients of our numerical approach are adaptive finite element discretizations combined with multilevel solvers and hierarchical modeling. We develop different kind of finite element methods, such as discontinuous (DGFEM) and stabilized finite element methods (SFEM), either based on continuous or non-conforming finite element spaces (NCFEM).

The availability of such tools is also a prerequisite for testing advanced physical models, concerning for example turbulence, compressibility effects, and realistic models of viscoelastic flows. In the case of polymer liquids, the numerical approximation of these flows is a challenging problem, due to the intrinsic physical properties (nonlinear viscoelastic behavior, high viscosity, low thermal viscosity) and due to the internal coupling between the viscoelasticity of the liquid and the flow, which is quantified by the dimensionless Weissenberg number We . The commercial codes are only able to deal with We up to 10, which is insufficient for many practical purposes.

Our long-term goals are described in the following. Having appropriate software tools at our disposal, we may tackle questions going beyond forward numerical simulations: parameter identification, design optimization, and questions related to interaction between numerical simulation and physical experiments.

Nowadays it appears that many questions 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. It is our long-term objective to develop the necessary computational framework and to contribute to the rational interaction between simulation and experiment.

The development of CFD software may benefit in an important measure from in-house experiments. To do so, 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 numerical simulations. The interdisciplinary collaboration between fluid mechanics, numerical analysis, and computer science 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 we develop collaborations with computer scientists.

The purposes of this project being to develop, analyze, and test new algorithms at hand of relevant configurations, collaboration with industrial partners is crucial. Technology transfer in form of integration of new methods into existing industrial codes is intended and could be the goal of future projects.