# Research Area M

**MODELING**

**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), andnumerical modeling with adaptive discretization concepts (M3) and modelling of dynamical systems (M4).

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.

### PhD Projects

## M1: Thermal Convection

**Principal Investigator: Camilla Nobili**

**PhD student: Fabian Bleitner**

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.

## M2.1: Solar Plant Modeling

**Principal Investigator: Ingenuin Gasser**

**PhD student: Hamzah Bakhti**

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.

## M2.2: Model-based design of power plants based on renewable energies

**Principal Investigator: Ingenuin Gasser**

**PhD student: Hanna Bartel**

There is a big variety of ideas and realisation on how to use the solar energy thermally, i.e. to produce directly heat energy, ideally in a highly efficient way. The most practiced and know realisations are parabolic trough power plants, more exotic and less efficient are the solar updraft towers.

There are many ideas to design and construct power plants based on renewable energies. Well known ideas are hydroelectric, wind energy, solar thermal (e,g, parabolic trough’s) and photovoltaics. Less known and more exotic ideas include pressure retarded osmosis (PRO) or reverse electro dialysis (RED).

The aim of this topic requires a detailed modeling of a complete power station with its various components. We plan to focus on realisations besides the parabolic trough's, such as solar power towers or more sophisticated solar updraft towers where codensation or even icing of water vapor is involved membrane based technologies in PRO, evolutions in PRO or RED.

Mathematically we have to model, analyze and optimize transient, coupled thermo-fluid dynamic processes for complex fluids.

## M3.1: Adaptive Modeling

**Principal Investigator: Jörn Behrens**

** Ezra Rozier**

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.

## M3.2: Model Adaptivity for 2D Shallow Water Flow

**Principal Investigator: Jörn Behrens**

**PhD student: Kemal Firdaus**

**Problem Formulation and Challenges:** In simulating wave propagation dispersive waves are often of great importance, however in significant approximations neglected. A prominent example is tsunami wave propagation, where a hydrostatic (shallow water) approximation to the wave propagation is very useful in many application cases (see e.g. [1]). However, in near shore wave amplification

and wave-induced currents non-hydrostatic effects play a significant role. In these situations Boussinesq-type equations, such as the Serre or Green-Naghdy equations, have been successfully applied.

In an earlier study we showed that the above mentioned Boussinesq-type equations are equivalent to a projection-based non-hydrostatic extension to the shallow water equations [2]. This projection method can be described as a predictor-corrector method, in which the hydrostatic (less expensive to solve) shallow water equations are solved in the predictor step, whereas these hydrostatic variables

are then used to set up an elliptic system to derive correction terms for the non-hydrostatic expansion of the predicted solution.

This projection method has been applied in a 1D Python code. One challenge is the computational effort necessary for solving the elliptic problem in each time step.

**Method Description and Milestones:** With the point-wise correction method one can restrict the correction locally, by this reducing the problem size of the elliptic problem. A PhD thesis and a master’s project were dedicated to derive methods for this model adaptive approach, where two different modeling approaches (i.e. the hydrostatic and the dispersive non-hydrostatic models) are

used adaptively [3, 4]. In particular for the 1D problem indicators for model refinement based on the quantities generated in the predictor step were derived empirically. Boundary conditions for interfacing the non-hydrostatic model to the hydrostatic environment were presented and tested. It could be shown that for certain test problems the model adaptivity achieves similar accuracy with

largely reduced computational effort.

## M4: Dynamics in fluid problems with continuous non-smooth forces

**Principal Investigator: Jens Rademacher**

**PhD student: Lütfiye Masur**

Continouous non-smooth terms arise in a variety of fluid models, in particular for nonlinear drag forces. Such terms pose challenges to study nonlinear effects since methods rely on Taylor expansion. Standard nonlinear friction drag force involves the term |u|u, where u is the velocity vector and |u| its length. Similarly, a variety of reduced models feature mixed component second order modulus terms of the form v|w|. Examples are rigid body mechanical models for ship manoeuvering [American Bureau of Shipping 2017] and box models for climate [Stommel 1961]. Such terms are mildly non-smooth since they are continuous, but details cannot be studied by standard methods that rely on direct Taylor expansion. Approximations by microscopic smoothening in general provide incorrect results on the relevant larger scale.

For the onset of oscillations via a Andronov-Hopf bifurcations an analytical method has been recently developed [Steinherr and Rademacher 2020, 2022]. This is partly based on invariant manifolds for Lipschitz vector fields [Aulbach and Wanner 1996]. An analogous approach has be used for bottom drag in certain shallow water equations [Prugger, Rademacher and Yang 2022 (preprint)].

Numerous challenges remain in this context, in particular the derivation of normal form coefficients in bifurcations of higher codimension, global bifurcations and constrained optimisation. An open problem in modelling is also an analytical derivation of the non-smooth terms.