Research Area M
Coordination: Prof. Dr. I. Gasser
The Research area M covers projects that have a clear focus on modeling in specific application areas and combines the design of mathematical models with numerical algorithms for their (approximate) solution. Adequate numerical algorithms typically exploit properties of the mathematical models for gaining efficiency.
On the other hand numerical representations are required to conserve the physical properties of the modeled problem to ensure validity of computational solutions. Moreover, goal-oriented strategies directly adjust the discrete algorithm to the modeled question at hand.
Topics and PhD-themes in this research area address challenging mathematical research questions referring to accurate modeling and analysis of thermo-fluid dynamics (M1), hierarchical modeling (M2), and numerical modeling with adaptive discretization concepts (M3).
The team gathers researchers with comprehensive and complimentary skills in the analysis, modeling, and numerical simulation of complex problems. Scientific progress in this area will be facilitated through a strong collaboration between experts on the modeling and the algorithmic components.
M1 (Thermal Convection) - Heat Transport in thermal convection
M2 (Solar Plant Modeling) - Model-based design of solar thermal power plants
M3 (Adaptive Modeling) - Optimized refinement criteria for adaptive modeling of transient flows
M1: Thermal Convection
Principal Investigator: Camilla Nobili
The research is concerned with temperature driven Rayleigh-Benard (RBC) and Benard-Marangoni convection (BMC). Related mathematical models are important for meteorological and industrial flows ranging from electronic cooling to large-scale power plants [Chilla & Schumacher (2012), Incropera (1999), Chilla et al. (2004)]. For both models, our objective is to derive bounds on the average upward heat transport, the Nusselt number Nu, in terms of the non-dimensional parameters appearing in the models. In particular we aim at proving quantitative bounds which reproduce the predicted power-law scalings. The challenge here is at rigorously deriving the exponents and determining the transitions between scaling regimes which agree with scaling analysis, (direct) numerical simulation and experiments. In RBC the major goal is to produce upper bounds on Nu for low-viscosity fluids (i.e. low Pr-number fluids like liquid metals or engine oils), for which rigorous as well as experimental results are few and partial. Despite the widespread applications, heat transport properties of the shear-driven BMC have been less studied than the one of the buoyancy-driven RBC and our initial aim is to analyze a simplified model with negligible fluid inertia (i.e. Pr = 1). The novelty of the research refers to the employment of PDE-tools (e.g. regularity estimates) to derive predictions on a quantity of physical and engineering interest.
M2: Solar Plant Modeling
Principal Investigator: Ingenuin Gasser
There are many ideas to design and construct solar thermal power plants: From well-known parabolic troughs to exotic solar power or updraft towers, see e.g. [El Jai & Chalqi (2013),Zhai Rongrong et al. (2013),Richter (2017)] or [von Allwöorden et al. (2018)]. In almost all of these concepts solar power is used to heat a fluid (special liquid, salt, water, gas, liquid gas mixtures etc.), the increased inner energy of the fluid is transported to a heat storage or directly to the energy production unit where heat energy is transformed into mechanical energy which is used to power a generator for electrical energy production. Some of these concepts only exist on paper, some are still in a first phase of development and testing, others - such as the parabolic troughs - have been developed and used already over decades. At the moment there is no final conclusion on what the most promising technologies could be. In this phase of development it is natural to aim at optimization: to design and to operate a power plant in order to maximize e.g. the net. When operating such a power plant, the flow velocity in the pipes is a typical control parameter. Low flow velocities in the pipes guarantee low frictional losses but imply high temperatures (due to the long exposure time of the fluid) and thus high energy losses by radiation and heat transfer. On the other hand high flow velocities in the pipes induce high frictional losses but with low temperatures and thus low energy losses by radiation and heat transfer. Therefore we expect an optimal velocity somewhere in between with a related (maximal) net power output. This example refers to the operational phase of the power plant. Similar issues have to be addressed in the planning phase. The aim is to design the "optimal" power station, i.e., maximal net power output with respect to a series of design parameters under certain given constraints. As described above, the main physical concepts of many solar thermal power stations are very similar. Therefore, similar optimization issues can be formulated for other types of solar thermal power plants. The case of solar updraft towers is meanwhile well understood [von Allwörden et al. (2018)]. In the solar power tower case promising results were obtained in [Richter (2017)]. We will develop modelbased mathematical approaches to design and to operate solar thermal power plants with a focus on parabolic troughs. This requires to advance mathematical models for alternative energy power plants towards optimization, and leads to mathematical challenging optimization problems for transient, coupled thermo-fluid dynamic processes in a large 1D-network of pipes, whose analytical and numerical treatment demands the development of new solution methods.
M3: Adaptive Modeling
Principal Investigator: Jörn Behrens
Typical transient geoscientific fluid dynamics and wave propagation problems expose a large span of spatial and temporal scales and corresponding numerical methods need to resolve those. One approach to bridge the scale gap in transient geophysical fluid dynamics and wave propagation applications refers to adaptive meshing or order refinement [Becker & Rannacher (2001), Eriksson et al. (2017), Gerwing et al. (2017)]. While the general computational methods for adaptive simulation with appropriate numerical methods is well established [Behrens (2006)], an open issue is still to find efficient and robust criteria for determining the refinement. In this sub-project we focus on spatial adaptivity and seek to develop computationally efficient refinement indicators. A gold standard of physically motivated error estimation is the goal-oriented weighted residual adjoint method, originally proposed by [Becker & Rannacher (2001)]. It has been successfully applied for ocean modeling [Pflower et al. (2006)] and recently improvements for non-smooth situations as they appear in discontinuous Galerkin discretizations or along sharp interfaces have been studied in [Beckers (2017),Kröger et al. (2018)]. However, for nonlinear transient flow regimes such adjoint methods are extremely costly, requiring a forward-backward propagation of the equations, usually performed on coarse model grids or even with simplified model equations [Bauer et al. (2014)]. Therefore, this topic aims at developing novel error estimation techniques that achieve similar accuracy as sophisticated adjoint methods, but at lower computational cost. The strategies are based on discretization characteristics, e.g., the jumps at cell boundaries in flux-based methods, or the spectral coefficients in spectral methods [Vater et al. (2015)], and physics-induced error propagation. In particular Lagrangian flow paradigms as well as optimal transport methods are to be utilized in order to gain the necessary error propagation information for transient regimes. While efficient refinement strategies in adaptive mesh methods optimize the run-time behavior on a methodological or physical model level, efficiency is also required on an algorithmic level in order to enable solution of complex realistic problems. Therefore, parallelization of such possibly unstructured and dynamically changing problems is an important issue. Previous works [Kunst & Behrens (2014), Behrens & Bader (2009)] in this direction will need to be improved and extended, and developed refinement strategies be parallelized accordingly.