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In this article, a concept of implicit methods for scalar conservation laws in one or more spatial dimensions allowing also for source terms of various types is presented. This material is a significant extension of previous work of the first author (Breuß SIAM J. Numer. Anal. 43(3), 970–986 2005). Implicit notions are developed that are centered around a monotonicity criterion. We demonstrate a connection between a numerical scheme and a discrete entropy inequality, which is based on a classical approach by Crandall and Majda. Additionally, three implicit methods are investigated using the developed notions. Next, we conduct a convergence proof which is not based on a classical compactness argument. Finally, the theoretical results are confirmed by various numerical tests.
The paper presents an aerodynamic investigation of 70 different streamlined bodies with fineness ratios ranging from 2 to 10. The bodies are chosen to idealize both unmanned and small manned aircraft fuselages and feature cross-sectional shapes that vary from circular to quadratic. The study focuses on friction and pressure drag in dependency of the individual body’s fineness ratio and cross section. The drag forces are normalized with the respective body’s wetted area to comply with an empirical drag estimation procedure. Although the friction drag coefficient then stays rather constant for all bodies, their pressure drag coefficients decrease with an increase in fineness ratio. Referring the pressure drag coefficient to the bodies’ cross-sectional areas shows a distinct pressure drag minimum at a fineness ratio of about three. The pressure drag of bodies with a quadratic cross section is generally higher than for bodies of revolution. The results are used to derive an improved form factor that can be employed in a classic empirical drag estimation method. The improved formulation takes both the fineness ratio and cross-sectional shape into account. It shows superior accuracy in estimating streamlined body drag when compared with experimental data and other form factor formulations of the literature.
The chemical imaging sensor is a semiconductor-based chemical sensor capable of visualizing pH and ion distributions. The spatial resolution depends on the lateral diffusion of photocarriers generated by illumination of the semiconductor substrate. In this study, two types of optical setups, one based on a bundle of optical fibers and the other based on a binocular tube head, were developed to project a hybrid illumination of a modulated light beam and a ring-shaped constant illumination onto the sensor plate. An improved spatial resolution was realized by the ring-shaped constant illumination, which suppressed lateral diffusion of photocarriers by enhanced recombination due to the increased carrier concentration.