Uniqueness of least area surfaces in the 3-torus

Given any -periodic metric g on and a plane through the origin, Bangert [4] shows that there exists a properly embedded surface homeomorphic to which is homotopically area-minimizing w.r.t. g, lies in a strip of bounded width around P, and does not have self-intersections when projected to the 3-torus . For the set of such surfaces, we show the following uniqueness theorems: If P is irrational, i.e., is not spanned by vectors in , the action of on by translations has a unique minimal set. If P is totally irrational, i.e., , then the surfaces in are pairwise disjoint. Received: 8 July 1999 / In final form: 14 February 2000 / Published online: 25 June 2001     

Detached eddy simulation of cyclic large scale fluctuations in a simplified engine setup

Computational Fluid Dynamics using RANS-based modelling approaches have become an important tool in the internal combustion engine development and optimization process. However, these models cannot resolve cycle to cycle variations, which are an important aspect in the design of new combustion systems. In this study the feasibility of using a Detached Eddy Simulation (DES) SST model, which is a hybrid RANS/LES model, to predict cycle to cycle variations is investigated. In the near wall region or in regions where the grid resolution is not sufficiently fine to resolve smaller structures, the two-equation RANS SST model is used. In the other regions with higher grid resolution an LES model is applied. The case considered is a geometrically simplified engine, for which detailed experimental data for the ensemble averaged and single cycle velocity field are available from Boreé et al. [Boreé, J., Maurel, S., Bazile, R., 2002. Disruption of a compressed vortex, Physics of Fluids 14 (7), 2543–2556]. The fluid flow shows a strong tumbling motion, which is a major characteristic for modern turbo-charged, direct-injection gasoline engines. The general flow structure is analyzed first and the extent of the LES region and the amount of resolved fluctuations are discussed. Multiple consecutive cycles are computed and turbulent statistics of DES SST, URANS and the measured velocity field are compared for different piston positions. Cycle to cycle variations of the velocity field are analyzed for both computation and experiment with a special emphasis on the useability of the DES SST model to predict cyclic variations.     

Photo-Trifluoromethylthiolation of Cyclopropyl Phenyl Propargyl Alcohol

Seven compounds are formed in various amounts when a solution of cyclopropopyl phenyl propargyl alcohol and trifluoromethylsulfenyl chloride in acetonitrile is exposed to UV light. The probable mechanism of formation and the mass spectral characterization of these compounds are described.     

Regioselective homolytic substitution of benzo[c][2,7]naphthyridines

Benzo[c][2,7]naphthyridines bearing electron-withdrawing substituents (bromo, acetyl) at C-4 undergo regioselective homolytic substitutions at C-5 with nucleophilic 1,3,5-trioxanyl and ethoxycarbonyl radicals under Minisci conditions. Surprisingly, mainly 5,6-dihydro derivatives are formed in these reactions. Rearomatization with manganese dioxide leads to 4,5-disubstituted benzo[c][2,7]naphthyridines, which should be attractive building blocks for the synthesis of pyridoacridine alkaloids. Homolytic methylation at C-5 takes place with methyl radicals generated from acetic acid and acetaldehyde, respectively.     

Measurement of instantaneous flow rates in periodically operating injection systems

Research and development work carried out to provide a method to measure accurately instantaneous flow rates in periodically operating injection systems is summarized. The instantaneous flow rate is reconstructed from axial velocity time series measured by a laser Doppler anemometer on the center-line of a capilary pipe flow. The theoretical background, on which the evaluation of the instantaneous flow rate is based is provided. It is shown that the axial mean velocity is sufficient to reconstruct the periodically varying flow rate and the pressure gradient. The application of the proposed method is described and an instrument is suggested that can be employed in many fields where fast, periodically varying flow rates occur and instantaneous information is needed.List of Symbols C _j complex Fourier coefficients - C.C. complex conjugate of different variables - D inner diameter of pipe - f ₀ repetition frequency of valve motion - i complex unit - J ₀ Bessel function of zeroth order - J _l Bessel function of first order - L distance between valve and measuring volume - m integrated fluid mass within one period - P/z mass flow rate - P/z pressure gradient along the pipe axis - p ₀,p _n normalized amplitude of the complex pressure gradient - r radial position - R pipe radius - Re Reynolds number - t time - T time of the period of the repetition valve opening - Ta Taylor number - U axial velocity - V integrated fluid volume within one period - instantaneous volumetric flow rate - viscosity - kinematic viscosity - fluid density - normalized open valve time - angular velocity of Fourier modes