Reducibility of Second Order Differential Operators with Rational Coefficients

This paper will provide several results for the reducibility of second order differential operators. More specifically, we will discuss second order operators that factor into two first order operators with either; one constant coefficient and one rational function coefficient, two polynomial coefficients, or two rational function coefficients with regular singularities. Furthermore, we will only consider operators with one or two regular singularities when considering rational function coefficients. The discussion will provide tests for reducibility of the operators as well

Instationäres aerodynamisches Verhalten einer bewegten Störklappe

In der vorliegenden Arbeit wurde untersucht, inwiefern numerische Verfahren zur Strömungssimulation auf Grundlage der Reynolds-gemittelten Navier-Stokes-Gleichungen in der Lage sind, das aerodynamische Antwortverhalten infolge einer bewegten Störklappe vorherzusagen. Neben der Erarbeitung einer geeigneten Simulationsstrategie stellt die Verifikation und Validierung des Simulationsansatzes einen wesentlichen Bestandteil dieser Arbeit dar. Hierzu wurde auf Daten aus einem eigens durchgeführten Windkanalversuch zurückgegriffen, in dem statische und zeitaufgelöste Oberflächendrücke an einem Profil bei statischen und dynamischen Ausschlägen einer Störklappe gemessen wurden. Bei schrittweise gesteigerter Modellierungstiefe konnte der Simulationsansatz im Niedergeschwindigkeitsbereich erfolgreich validiert werden. Darauf aufbauend wurde rein numerisch untersucht, inwiefern sich die unter Windkanalbedingungen gewonnenen Erkenntnisse beim Übergang zu flugrelevanten Mach- und Reynoldszahlen sowie bei noch schnelleren Stellvorgängen übertragen lassen. Dabei konnte bei ausreichend hohem Stellbereich der Störklappe gezeigt werden, dass eine kritische Stellrate existiert, deren Überschreitung dazu führt, dass die transienten Auftriebsüberschwinger zunächst einbrechen und schließlich in Gänze verschwinden. Die in dieser Arbeit gewonnenen Erkenntnisse sind insbesondere dann von Relevanz, wenn die Steuerfläche möglichst schnell ausschlagen soll, beispielsweise um Lasten infolge atmosphärischer Störungen wie Böen abzumindern

Towards the Investigation of Unsteady Control Surface Aerodynamics

Fostering the application range of CFD methods towards the simulation of entire flight manoeuvres still is a challenging task. It needs to be ensured that the aerodynamic effects evolving from the deflection of a control surface are properly resolved and that their impact on the overall aerodynamics is accurately predicted. Focussing on control surfaces CFD methods have traditionally been applied to investigate the performance of high-lift systems during take-off and landing. In recent years the interest in applying CFD to questions arising from other control surfaces gradually increased. The performance of a vertical tail with deflected rudder in engine-out conditions, for instance, or the efficiency of ailerons and spoilers are nowadays often determined using CFD. All these applications, however, have in common that they address static control surface deflections only. To close the gap towards manoeuvre flight simulations it is therefore indispensable to take the dynamic motion of a control surface and their transient aerodynamic behaviour into account. Spoilers, for example, are well known to exhibit a strongly nonlinear aerodynamic behaviour with significant time lags. Depending on their deflection rate spoilers may even yield a short-term lift augmentation. The investigation of spoiler aerodynamics is therefore hardly possible with low-fidelity tools. A profound assessment of the transient aerodynamic behaviour of a dynamic spoiler therefore requires an experimental data base with which the higher-fidelity CFD simulations can be validated. As DLR was lacking a wind tunnel model offering these functionalities it was decided to modify the DLR-F15 model with a dynamic control surface. The present paper highlights the constraints for the development of this new version of the DLR-F15 model and gives an overview on the background and the objective of the work planned

Numerical Simulation and Validation of Aerodynamics of Static and Dynamic Spoilers

Spoilers play a vital role in the flight control systems of modern transport aircraft. Because of their efficiency and fast deflection rates, they are widely used to assist in roll control or for gust load alleviation purposes. Simulating the aerodynamic behavior of spoilers still is a challenging task, as deflecting a spoiler always induces flow separation. The work presented in this paper therefore focused on extending the application range of DLR, German Aerospace Center’s in-house flow solver TAU by verifying and validating it for spoiler applications. In a first step, the work focused on steady and unsteady simulations of static and dynamic spoiler deflections in the low-speed regime. It was shown that the chosen numerical approach is well suited to reproduce the surface pressure distribution for static spoiler deflections of up to 50 deg. Above that, the simulated transient aerodynamic response was found to be in good agreement with experimental data for a variety of deflection times and deflection angles

Auf dem Weg zur simulationsbasierten Zertifizierung - Ein Überblick über das DLR-Projekt SimBaCon

Das DLR hat ab 2018 begonnen, sich verstärkt mit dem Thema 'Zulassung' und den Herausforderungen einer zukünftig verstärkt simulationsbasierten Nachweisführung zu befassen. Mit dem Ende 2021 abgeschlossenen Querschnittsprojekt 'Simulation based Certification' (kurz: SimBaCon) verfolgte das DLR das Ziel, Simulationsverfahren sukzessive für Zulassungsanwendungen in unterschiedlichen Bereichen weiterzuentwickeln. Weiterhin galt es, durch umfangreiche Verifikations- und Validierungsaktivitäten für industriell relevante Anwendungsfälle, einen Beitrag dazu zu leisten, das Vertrauen in diese Verfahren auszuweiten. Die Schwerpunkte bei den numerischen Aktivitäten bestanden in der Verbesserung der physikalischen Modellierung und der Modellierungstiefe sowie in der Demonstration dieser Verfahren für zertifizierungsrelevante Anwendungsfälle. Darüber hinaus wurden experimentelle Messmethoden weiterentwickelt und hochqualitative experimentelle Vergleichsdaten für die Validierung der Simulationen erzeugt. Im Vortrag wird ein Gesamtüberblick über das Projekt mit exemplarischen Anwendungen gegeben, sowie die Vernetzung zu weiteren internen und externen Aktivitäten vorgestellt

Validation of the Flow Topology Around Several Airdrop Cargo Configurations at Static Conditions

A numerical study was carried out to assess the interference effects between the wake of a transport aircraft and several generic cargo bodies during the early stage of an airdrop scenario. Based on experimental data and preceding numerical simulations distinct positions of the trajectory being subject to strong interference effects were stati- cally reproduced. The flow field around the bodies was experimentally investigated using stereoscopic Particle Image Velocimetry and compared to steady and unsteady Reynolds- averaged Navier-Stokes (RANS) computations. The latter were carried out using the unstructured DLR TAU code. The primary focus is to assess the suitability, accuracy and the limitations of RANS methods in such challenging flow conditions. Therefore, the influence of several turbulence models was investigated and compared to experimental field velocity data. Although deviations in the wake of the bodies were observed, the qualitative agreement between the steady simulations and experiment was very good. Quantitatively, however, the steady approach leaves room for further improvements. The gap to the experimental data could partially be reduced in applying unsteady RANS methods

REGULARITY FOR SOLUTIONS TO PARABOLIC SYSTEMS AND NONLOCAL MINIMIZATION PROBLEMS

The goal of this dissertation is to contribute to both the nonlocal and local settings of regularity within the calculus of variations. We provide analogues of higher differentiability results in the context of Besov spaces for minimizers of nonlocal functionals. We also establish the Holder continuity of solutions to a system of parabolic partial differential equations. Advisor: Mikil Fos

Development and Validation of a RANS-based Airdrop Simulation Approach

During the first few seconds of an airdrop the payload interacts with the flow field behind the aircraft. As most airdrop simulation tools rely on simplified aerodynamic models, e.g. ones with frozen wind field information, DLR has developed an alternative simulation approach. It computes the aerodynamic interference effects during the initial phase of an airdrop and the resulting trajectory. To achieve this the DLR TAU flow solver has been coupled with the multi-body simulation software SIMPACK. The major advantage of this approach, apart from the fact that it does not rely on simplified aerodynamic models, is its multi-body functionality. Thus, the relative motion of bodies, for example between payload and parachute, can easily be examined as kinematic constraints are considered. DLR has successfully demonstrated that its airdrop simulation approach is well-suited to predict the trajectories of different cargo and cargo--parachute configurations. An overview on the methodology, the verification and validation of this approach is presented in the following