High-order compact schemes for incompressible flows: A simple and efficient method with quasi-spectral accuracy

In this paper, a finite difference code for Direct and Large Eddy Simulation (DNS/LES) of incompressible flows is presented. This code is an intermediate tool between fully spectral Navier–Stokes solvers (limited to academic geometry through Fourier or Chebyshev representation) and more versatile codes based on standard numerical schemes (typically only second-order accurate). The interest of high-order schemes is discussed in terms of implementation easiness, computational efficiency and accuracy improvement considered through simplified benchmark problems and practical calculations. The equivalence rules between operations in physical and spectral spaces are efficiently used to solve the Poisson equation introduced by the projection method. It is shown that for the pressure treatment, an accurate Fourier representation can be used for more flexible boundary conditions than periodicity or free-slip. Using the concept of the modified wave number, the incompressibility can be enforced up to the machine accuracy. The benefit offered by this alternative method is found to be very satisfactory, even when a formal second-order error is introduced locally by boundary conditions that are neither periodic nor symmetric. The usefulness of high-order schemes combined with an immersed boundary method (IBM) is also demonstrated despite the second-order accuracy introduced by this wall modelling strategy. In particular, the interest of a partially staggered mesh is exhibited in this specific context. Three-dimensional calculations of transitional and turbulent channel flows emphasize the ability of present high-order schemes to reduce the computational cost for a given accuracy. The main conclusion of this paper is that finite difference schemes with quasi-spectral accuracy can be very efficient for DNS/LES of incompressible flows, while allowing flexibility for the boundary conditions and easiness in the code development. Therefore, this compromise fits particularly well for very high-resolution simulations of turbulent flows with relatively complex geometries without requiring heavy numerical developments.     

Internal energy of high-density hydrogen: Analytic approximations compared with path integral Monte Carlo calculations

The internal energy of high-density hydrogen plasmas in the temperature range T=10000–50000 K is calculated by two different analytic approximation schemes—the method of an effective ion-ion interaction potential and the Padé approach within the chemical picture—and are compared with direct path integral Monte Carlo results. A reasonable agreement between the results obtained from the three independent calculations is found and the reasons for still existing differences are investigated. Interesting high-density phenomena such as the onset of ion crystallization are discussed.     

Finite volume treatment of dispersion-relation-preserving and optimized prefactored compact schemes for wave propagation

In developing suitable numerical techniques for computational aero-acoustics, the dispersion-relation-preserving (DRP) scheme by Tam and co-workers and the optimized prefactored compact (OPC) scheme by Ashcroft and Zhang have shown desirable properties of reducing both dissipative and dispersive errors. These schemes, originally based on the finite difference, attempt to optimize the coefficients for better resolution of short waves with respect to the computational grid while maintaining pre-determined formal orders of accuracy. In the present study, finite volume formulations of both schemes are presented to better handle the nonlinearity and complex geometry encountered in many engineering applications. Linear and nonlinear wave equations, with and without viscous dissipation, have been adopted as the test problems. Highlighting the principal characteristics of the schemes and utilizing linear and nonlinear wave equations with different wavelengths as the test cases, the performance of these approaches is documented. For the linear wave equation, there is no major difference between the DRP and OPC schemes. For the nonlinear wave equations, the finite volume version of both DRP and OPC schemes offers substantially better solutions in regions of high gradient or discontinuity.     

Radiation field calculations of pulsed ultrasonic transducers: Part 2: spherical disc- and ring-shaped transducers

The exact equation to describe the propagation of ultrasonic pressure waves in a lossless, homogeneous medium was evaluated assuming pulse-excited radiators.Acoustical field patterns were studied using the dimensions and shape of the transducer, the ultrasonic frequency and number of cycles within the pulse as parameters. The results emphasize the influence of the piston shape on interference phenomena within the near — and far-field and on beam-narrowing effects in the intermediate range between near- and far-field. The following transducer shapes are studied: spherical disc and spherical ring.     

A class of asymptotic-preserving schemes for kinetic equations and related problems with stiff sources

In this paper, we propose a general time-discrete framework to design asymptotic-preserving schemes for initial value problem of the Boltzmann kinetic and related equations. Numerically solving these equations are challenging due to the nonlinear stiff collision (source) terms induced by small mean free or relaxation time. We propose to penalize the nonlinear collision term by a BGK-type relaxation term, which can be solved explicitly even if discretized implicitly in time. Moreover, the BGK-type relaxation operator helps to drive the density distribution toward the local Maxwellian, thus naturally imposes an asymptotic-preserving scheme in the Euler limit. The scheme so designed does not need any nonlinear iterative solver or the use of Wild Sum. It is uniformly stable in terms of the (possibly small) Knudsen number, and can capture the macroscopic fluid dynamic (Euler) limit even if the small scale determined by the Knudsen number is not numerically resolved. It is also consistent to the compressible Navier–Stokes equations if the viscosity and heat conductivity are numerically resolved. The method is applicable to many other related problems, such as hyperbolic systems with stiff relaxation, and high order parabolic equations.     

Computation of the aberrations of concave diffraction gratings. Part 2. Aberration calculations and estimation of resolving power

The aberrations of a concave grating are derived by computation; the resolving power is estimated from the intersections of the rays with the image plane. It is shown that an aspheric grating can be used to advantage in a monochromator of Zeya-Namioka type.We are indebted to V. A. Efimov for compiling the computer program.     

Rotational spectra of isotopic species of silyl fluoride. Part I: Lamb-dip measurements and quantum-chemical calculations

The Lamb-dip technique has been employed for recording the rotational spectra of three isotopic species of silyl fluoride, namely ²⁸SiH₃F, ²⁹SiH₃F, and ³⁰SiH₃F, in order to improve the knowledge of their spectroscopic parameters as well as to try to resolve their hyperfine structure. High-level quantum-chemical computations using state-of-the-art coupled-cluster techniques together with core-polarized correlation-consistent basis sets have been employed to provide reliable reference values for the hyperfine parameters involved and have been used to guide the experimental investigation. Analysis of the experimental spectra allowed to improve the accuracy of the known spectroscopic parameters as well as to determine for the first time sextic and octic centrifugal-distortion constants.     

Artificial compressibility, characteristics-based schemes for variable-density, incompressible, multispecies flows: Part II. Multigrid implementation and numerical tests

The paper presents an investigation of the accuracy and efficiency of artificial compressibility, characteristics-based (CB) schemes for variable-density incompressible flows. The CB schemes have been implemented in conjunction with a multigrid method for accelerating numerical convergence and a fourth-order, explicit Runge–Kutta method for the integration of the governing equations in time. The implementation of the CB schemes is obtained in conjunction with first-, second- and third-order interpolation formulas for calculating the variables at the cell faces of the computational volume. The accuracy and efficiency of the schemes are examined against analytical and experimental results for diffusion broadening in two- and three-dimensional microfluidic channels, a problem that has motivated the development of the present methods. Moreover, unsteady, inviscid simulations have been performed for variable-density mixing layer. The computations revealed that accuracy and efficiency depend on the CB scheme design. The best multigrid convergence rates were exhibited by the conservative CB scheme, which is obtained by the fully conservative formulation of the variable-density, incompressible equations.     

Finite-aperture effects for Fourier transform systems with convergent illumination. Part I: 2-D system analysis

The Optical Fourier Transform (OFT) is one of the most fundamental operations in analogue Optical Signal Processing (OSP). There are many optical arrangements for implementing the OFT, however one which is particularly popular is the Scaled Optical Fourier Transform (SOFT) because it offers the user the ability to scale the output Fourier distribution. In this paper we study some of the practical limits introduced by using a converging spherical lens of finite aperture to produce the illuminating field in the implementation of the SOFT. By deriving simple rules of thumb, based on examining phase and intensity deviations from the ideal unapertured case, we define an area inside the geometric shadow, which we refer to as a sub-geometric shadow. Inside this sub-geometric shadow we show that the worst-case errors in the resulting SOFT, arising due to diffraction, can be quantified and avoided.     

Variational calculations of vibrational energy levels for XY4 molecules: 2. Bending states of methane

A variational method for calculating excited bending states of symmetric tetrahedral pentaatomic molecules is presented based on the use of Radau coordinates and Jacobi polynomials as the basis functions. Symmetry is used both to reduce the size of secular matrix to be diagonalized and to calculate potential energy matrix elements over a reduced grid of quadrature points. Methods of treating the redundant coordinate are investigated and fitting is found to be more effective than the use of Taylor expansions. Test results are presented for methane, for which stretch-bend coupling and the contribution due to the redundant coordinate are found to be significant. Converged results are obtained for bending states significantly higher than considered in previous calculations. These states will be used as a basis for bending motions in a fully coupled stretch-bend calculation.     

Design and development of compact readout electronics with silicon photomultiplier array for a compact imaging detector

This work aims at developing compact readout electronics for a compact imaging detector module with silicon photomultiplier (SPM) array. The detector module consists of a LYSO crystal array coupling with a SensL's 4×4 SPM array. A compact multiplexed readout based on a discretized positioning circuit (DPC) was developed to reduce the readout channels from 16 to 4 outputs. Different LYSO crystal arrays of 4×4, 8×8 and 12×12 with pixel sizes of 3.2, 1.6 and 1.0 mm respectively, have been tested with the compact readout board using a ¹³⁷Cs source. The initial results show that the compact imaging detector module with the compact multiplexed readout could clearly resolve 1 mm× 1 mm× 10 mm LYSO scintillation crystal array except those at the edges. The detector's intrinsic spatial resolution up to 1 mm can be achieved with the 3 mm×3 mm size SPMArray4 through light sharing and compact multiplexed readout. Our results indicate that this detector module is feasible for the development of high-resolution compact PET.     

Artificial compressibility, characteristics-based schemes for variable density, incompressible, multi-species flows. Part I. Derivation of different formulations and constant density limit

The paper presents various formulations of characteristics-based schemes in the framework of the artificial-compressibility method for variable-density incompressible flows. In contrast to constant-density incompressible flows, where the characteristics-based variables reconstruction leads to a single formulation, in the case of variable density flows three different schemes can be obtained henceforth labeled as: transport, conservative and hybrid schemes. The conservative scheme results in pseudo-compressibility terms in the (multi-species) density reconstruction. It is shown that in the limit of constant density, the transport scheme becomes the (original) characteristics-based scheme for incompressible flows, but the conservative and hybrid schemes lead to a new characteristics-based variant for constant density flows. The characteristics-based schemes are combined with second and third-order interpolation for increasing the computational accuracy locally at the cell faces of the control volume. Numerical experiments for constant density flows reveal that all the characteristics-based schemes result in the same flow solution, but they exhibit different convergence behavior. The multigrid implementation and numerical studies for variable density flows are presented in Part II of this study.     

Development of a compact CO2 sensor based on near-infrared laser technology for enological applications

This paper reports the development of an infrared laser spectrometer using commercial diode laser emitting at 2.68 μm. The instrument is designed to measure CO₂ concentrations above a glass poured with a sparkling liquid, such as beer or champagne in the present case. This spectrometer was developed in order to realize the cartography of CO₂ outgassing in the headspace above various glasses. We provide details of the instrument design and data processing. Absorption lines were carefully selected to minimize interferences from neighboring water vapor transitions. The instrument performance allows to measure ambient CO₂ concentrations so that one can be very confident in the CO₂ concentrations measurements above the glass. Some preliminary results on sparkling liquids such as beer and champagne are presented and compared to a model describing the flux of CO₂ discharging from glasses due to the contribution of bubbles.     

On the theoretical calculations of cross-sections for alkali metal atoms reacting with Cl2 and Br2 molecules

Summary An adiabatic capture mechanism is presented to calculate reactive cross-sections for the alkali+Cl₂ and alkali+Br₂ systems. The model is based on empirically constructed potential-energy surfaces and the results are found to be consistent with the experimental values. To speed up publication, the author of this paper has agreed to not receive the proofs for correction.     

Wavelet Analysis of Fractal Boundaries. Part 2: Multifractal Analysis

This second part deals with the global analysis of the boundary of domains . We develop methods for determining the dimensions of the sets where the local behaviors introduced in Part 1 occur. These methods are based on analogies with the thermodynamic formalism in statistical physics and lead to new classification tools for fractal domains.The first author is supported by the Institut Universitaire de France.This work was performed while the second author was at the Laboratoire d’Analyse et de Mathématiques Appliquées (University Paris XII) and at the Istituto di Matematica Applicata e Tecnologie Informatiche (Pavia, Italy) and partially supported by the Société de Secours des amis des Sciences and the TMR Research Network “Breaking Complexity”.     

Dispersion of interface waves in sediments with power-law shear speed profiles. II. Experimental observations and seismo-acoustic inversions.

The propagation of seismic interface waves is investigated in soft marine sediments in which the density is constant, the shear modulus is small, and the profile of shear speed c(s) versus depth z is of the power-law form c(s) (z) = c0z(v), in which c0 and v are constants (0< v < 1). Both the phase speed V and the group speed U of interface waves scale with frequency as f(v/(v -1)) and they obey the simple relation U= (1 - v) V. These relations are derived in a simple way using ray theory and the WKB method; a companion paper [O. A. Godin and D. M. F. Chapman, J. Acoust. Soc. Am. 110, 1890 (2001)] rigorously derives the same result from the solutions to the equations of motion. The frequency scaling is shown to exist in experimental data sets of interface wave phase speed and group speed. Approximate analytical formulas for the dispersion relations (phase and group speed versus frequency) enable direct inversion of the profile parameters c0 and v from the experimental data. In cases for which there is multi-mode dispersion data, the water-sediment density ratio can be determined as well. The theory applies to vertically polarized (P-SV) modes as well as to horizontally polarized (SH) modes (that is, Love waves).     

Magnetic resonance in systems with equivalent spin-1/2 nuclides. Part 2: energy values and spin states

In an earlier paper (Part 1), featuring group-theoretical analysis, it was shown that the isotropic EPR spectra of free radical (S=1/2) species XL(n), where the n equivalent nuclei also have spin 1/2, have a more complicated form than anticipated from the usual (first-order) oversimplified analysis. The nucleus of X is taken to be spin-less. The latter predicts n+1 lines with intensity ratios given by the coefficients of the binomial expansion; for systems with n=3, the EPR spectrum in fact consists of 6 lines. Analogous considerations hold for NMR spectroscopy of XL(n) non-radicals. For n > or = 3 systems, the degeneracy of the energy levels cannot be completely removed by the Zeeman electronic and nuclear interactions. Explicit solutions for n=3 (analytic, as well as computational) of the spin-hamiltonian for the energies and spin states have been obtained and are given in the present work. Discussion of the underlying theory, invoking exchange degeneracy, is included herein in some detail, focusing on line positions and relative spectral intensities.     

Comparative study of Mo_2Ga_2C with superconducting MAX phase Mo_2GaC: First-principles calculations

The structural, electronic, optical and thermodynamic properties of Mo_2Ga_2C are investigated using density functional theory(DFT) within the generalized gradient approximation(GGA). The optimized crystal structure is obtained and the lattice parameters are compared with available experimental data. The electronic density of states(DOS) is calculated and analyzed. The metallic behavior for the compound is confirmed and the value of DOS at Fermi level is 4.2 states per unit cell per e V. Technologically important optical parameters(e.g., dielectric function, refractive index, absorption coefficient, photo conductivity, reflectivity, and loss function) are calculated for the first time. The study of dielectric constant(ε1) indicates the Drude-like behavior. The absorption and conductivity spectra suggest that the compound is metallic.The reflectance spectrum shows that this compound has the potential to be used as a solar reflector. The thermodynamic properties such as the temperature and pressure dependent bulk modulus, Debye temperature, specific heats, and thermal expansion coefficient of Mo_2Ga_2C MAX phase are derived from the quasi-harmonic Debye model with phononic effect also for the first time. Analysis of T c expression using available parameter values(DOS, Debye temperature, atomic mass,etc.) suggests that the compound is less likely to be superconductor.     

Sixth order compact scheme combined with multigrid method and extrapolation technique for 2D poisson equation

We develop a sixth order finite difference discretization strategy to solve the two dimensional Poisson equation, which is based on the fourth order compact discretization, multigrid method, Richardson extrapolation technique, and an operator based interpolation scheme. We use multigrid V-Cycle procedure to build our multiscale multigrid algorithm, which is similar to the full multigrid method (FMG). The multigrid computation yields fourth order accurate solution on both the fine grid and the coarse grid. A sixth order accurate coarse grid solution is computed by using the Richardson extrapolation technique. Then we apply our operator based interpolation scheme to compute sixth order accurate solution on the fine grid. Numerical experiments are conducted to show the solution accuracy and the computational efficiency of our new method, compared to Sun–Zhang’s sixth order Richardson extrapolation compact (REC) discretization strategy using Alternating Direction Implicit (ADI) method and the standard fourth order compact difference (FOC) scheme using a multigrid method.