Family Of Characteristic Curves Research Articles

Adjoint and Direct Characteristic Equations for Two-Dimensional Compressible Euler Flows

The method of characteristics is a classical method for gaining understanding in the solution of a partial differential equation. It has recently been applied to the adjoint equations of the 2D steady-state Euler equations and the first goal of this paper is to present a linear algebra analysis that greatly simplifies the discussion of the number of independent characteristic equations satisfied along a family of characteristic curves. This method may be applied for both the direct and the adjoint problem. Our second goal is to directly derive in conservative variables the characteristic equations of 2D compressible inviscid flows. Finally, the theoretical results are assessed for a nozzle flow with a classical scheme and its dual consistent discrete adjoint.

Weak shock wave interactions in isentropic Cargo‐LeRoux model of flux perturbation

In this paper, we explore the weak shock wave interactions in isentropic Cargo‐LeRoux model of flux perturbation which describes strictly hyperbolic quasilinear system of conservation laws. For the Riemann problem, we provide a global existence and uniqueness result of exact solution. Further, we establish the exact solution explicitly in terms of one parameter family of elementary wave curves. We devise a necessary and sufficient condition on initial data for which the solution of the Riemann problem consists of either a shock wave or a simple wave according to one and three family of characteristic curves. Lastly, we analyze the Von Neumann result related to collision of weak shock waves of same family.

A unified Fourier theory for time-of-flight PET data

Fully 3D time-of-flight (TOF) PET scanners offer the potential of previously unachievable image quality in clinical PET imaging. TOF measurements add another degree of redundancy for cylindrical PET scanners and make photon-limited TOF-PET imaging more robust than non-TOF PET imaging. The data space for 3D TOF-PET data is five-dimensional with two degrees of redundancy. Previously, consistency equations were used to characterize the redundancy of TOF-PET data. In this paper, we first derive two Fourier consistency equations and Fourier–John equation for 3D TOF PET based on the generalized projection-slice theorem; the three partial differential equations (PDEs) are the dual of the sinogram consistency equations and John’s equation. We then solve the three PDEs using the method of characteristics. The two degrees of entangled redundancy of the TOF-PET data can be explicitly elicited and exploited by the solutions of the PDEs along the characteristic curves, which gives a complete understanding of the rich structure of the 3D x-ray transform with TOF measurement. Fourier rebinning equations and other mapping equations among different types of PET data are special cases of the general solutions. We also obtain new Fourier rebinning and consistency equations (FORCEs) from other special cases of the general solutions, and thus we obtain a complete scheme to convert among different types of PET data: 3D TOF, 3D non-TOF, 2D TOF and 2D non-TOF data. The new FORCEs can be used as new Fourier-based rebinning algorithms for TOF-PET data reduction, inverse rebinnings for designing fast projectors, or consistency conditions for estimating missing data. Further, we give a geometric interpretation of the general solutions—the two families of characteristic curves can be obtained by respectively changing the azimuthal and co-polar angles of the biorthogonal coordinates in Fourier space. We conclude the unified Fourier theory by showing that the Fourier consistency equations are necessary and sufficient for 3D x-ray transform with TOF measurement. Finally, we give numerical examples of inverse rebinning for a 3D TOF PET and Fourier-based rebinning for a 2D TOF PET using the FORCEs to show the efficacy of the unified Fourier solutions.

Droplets wetting on filament rails: Surface energy and morphology transition

The morphology of liquid droplets wetting on filaments depends on the filament configuration, droplet volume, and contact angle. A stable morphology is the one that minimizes the potential energy of the droplet–filament system, while morphology transition may happen when an intermediate state exists which corresponds to a higher potential energy. This paper aims to explore such morphology transition of droplet wetting on filament rails made of two parallel identical microfilaments. Detailed numerical simulations were performed to extract the surface energy of the droplet–filament system at varying filament spacings, droplet volumes, and contact angles. Critical conditions of the morphology transition between two symmetrical wetting morphologies (i.e., liquid droplet bridge and barrel-shaped droplet) were determined. A family of characteristic curves in terms of the dimensionless droplet volume vs the filament spacing at varying contact angles was obtained, which can be used as a universal law to govern the morphology transition for such droplet–filament rail systems. The results and concepts presented in this work can be extended to broad wetting systems and utilized for the analysis and design of microfluidic devices and testers based on droplet–filament systems.

Technique for Comparing Operation of Alternative Interchange Types

Interchanges have been the focus of several recent studies that attempted to identify the conditions in which one interchange type is more efficient than another. However, the guidance offered is of such a general nature that it is difficult to apply with confidence in a specific situation. The objective of this study was to describe an interchange evaluation technique that is of sufficient precision to facilitate interchange-type selection or planning-level evaluations of interchange performance. This technique was based on the use of characteristic curves that define the relationship between interchange delay and the sum-of-critical-flow-ratios. This sum represents a unique parameter that combines the effect of signal phase sequence, traffic volume, number of lanes, and saturation flow rate. A family of characteristic curves was developed and used to illustrate the interchange evaluation technique. The curves describe the relationship between delay and the sum-of-critical-flow-ratios for the single-point urban interchange (SPUI) and the tight urban diamond interchange (TUDI), both serving frontage-road systems. For the range of conditions considered, it was concluded that the TUDI will generally operate with less delay than the SPUI.

Theory And Operating Characteristics Of The Thermionic Amplifier

The theory of operation of the three-electrode thermionic vacuum tube given in this paper is based on the fundamental equation of the family of characteristic curves for various plate and grid voltages. This equation, which may be written I= a( R_ EB + 2; E,+ C) where E b and E c , are the plate and grid voltages respectively, and α, μ 0 . and e are the structural parameters of the device, is obtained empirically with the help of the relation previously discovered by the author, which states that the voltage between filament and plate bears a linear relation to the effective voltage produced by it between the filament and a plane coincident with that of the grid. From this it follows that an electromotive force e impressed upon the grid circuit produces an electromotive force μ 0 e in the plate circuit. With the help of these fundamental relations the amplification equations of the tube are derived in terms of the structural parameters of the tube and the constants of the circuit. Methods are given for experimentally determining the constants of the tube, the two most important of which are the amplification constant μ 0 and the internal output impedance. Experiments are described which were performed to test these equations, and they indicate that the first order approximation made give results sufficiently accurate for amplification purposes.

Nonlinear transient analysis of coupled RLGC lines by the method of characteristics

A computationally efficient numerical technique for the evaluation of the time domain response of coupled lossy uniform and nonuniform transmission lines terminated in general nonlinear elements is presented. The technique is based on the method of characteristics where the original system of coupled transmission line equations is transformed into a system of ordinary differential equations which hold along a family of characteristic curves. An algorithm to solve these equations is presented. The procedure is applicable to both uniform and nonuniform coupled systems and examples of two and three line structures terminated in linear and nonlinear elements are included to demonstrate the versatility and the usefulness of the algorithm.

A Computer Code Using Exergy for Optimizing Thermal Plants

This paper presents an exergetic approach to reducing energy consumption in industrial plants. The minimum pinchpoint temperature difference is determined for several cases of heat exchange between hot and cold fluid streams with reduced thermal losses. The proposed method, designated TEXAS (Thermal and economic EXergetic Analysis of Systems), derives families of characteristic curves showing economic parameters versus exergetic efficiency in relation to factors such as heat transfer, surface, and scaling. As the method is highly affected by the economic parameter assumptions, the sensitivity of the variations in the parameters is also analyzed.

Photographic photometry of spectral lines in the convergence of a family of characteristic curves

Photographic photometry of spectral lines in the convergence of a family of characteristic curves

Photographic Sequence Exposure Experiment*

An instrument is described which has been built for the automatic recording of sequence exposures on 4-by-10-in. photographic plates. Twelve shutter slides are independently timed so that on one plate up to 384 4-by-5-mm rectangles can be uniformly exposed to high and low intensity radiation. The present design is for an intensity ratio of 100:1, where both intensities are below optimum. With minor modifications the instrument can be converted to operate at other intensity ratios. The densitometer readings are checked from calibration strip data placed adjacent to the sequence exposures. The corrected data, following conversion to Seidel values, are plotted as families of characteristic curves, with the sequence exposure ratio as the constant parameter. Isodense loop data are read from the curves at the intersection with a line drawn parallel to the exposure axis.A theoretical discussion of isodense loops is presented which is based on the hypothesis that for isodense exposures the probability of rendering the last grains developable must be equal. It is shown how this method should yield the limiting slope of the low intensity failure curve and the minimum number of quanta absorbed by the average grain to become developable. Preliminary results presented here for Eastman Kodak Emulsion Type 33 are within expectation.

The Characteristic Surface of the Triode

The paper outlines a method of representing triode characteristics by means of surfaces rather than by families of characteristic curves. The physical features of the surfaces are then described in considerable detail and the fundamental equation for the alternating component of the plate current is derived for an ideal surface. Finally the action of the triode both as a simple and as a regenerative amplifier is described with reference to its characteristic surface.