Large Wave Numbers Research Articles

Analysis of Dirichlet\u2013Robin Iterations for Solving the Cauchy Problem for Elliptic Equations

The Cauchy problem for general elliptic equations of second order is considered. In a previous paper (Berntsson et al. in Inverse Probl Sci Eng 26(7):1062–1078, 2018), it was suggested that the alternating iterative algorithm suggested by Kozlov and Maz’ya can be convergent, even for large wavenumbers k^2, in the Helmholtz equation, if the Neumann boundary conditions are replaced by Robin conditions. In this paper, we provide a proof that shows that the Dirichlet–Robin alternating algorithm is indeed convergent for general elliptic operators provided that the parameters in the Robin conditions are chosen appropriately. We also give numerical experiments intended to investigate the precise behaviour of the algorithm for different values of k^2 in the Helmholtz equation. In particular, we show how the speed of the convergence depends on the choice of Robin parameters.

Acoustic and shock wave dispersion in cylindrical ducts

An asymptotic approach is employed to investigate the effects of dispersion on acoustic disturbances propagating through cylindrical ducts, the walls of which may be impermeable or acoustically treated. The disturbances are assumed to be those generated by a rotor-locked pressure field. In the large azimuthal wavenumber (high-frequency) limit, m ≫ 1, linear potential perturbations are found to be weakly dispersive, with the leading order correction occurring at O(m−2/3). When a liner is present, its additional dispersive and dissipative effects appear at O(m−1). The effects of weak nonlinearity, in addition to those of dispersion are then studied. Specifically, on the assumption that the disturbance is comprised of a single radial mode, an evolution equation is derived for the pressure field. Numerical solutions of this equation are carried out for idealized perturbation patterns corresponding to tuned and detuned rotor blades, featuring respectively shocks of uniform and varying strength. In the tuned, untreated case, dispersive effects are found to reduce the dissipation across the shocks without substantially changing the character of the waveform in comparison with the non-dispersive case. The inclusion of the acoustic liner brings about more drastic changes: the periodic shock pattern is transformed into a series of humps with dispersive tails. With the detuned blades, it is found that the untreated duct scenario yields a waveform that differs significantly from the non-dispersive case. In particular, the shock merging that occurs in the latter instance does not transpire. Instead, the shocks exchange amplitudes even as they decay. The presence of the acoustic liner in the detuned case is similar to the tuned case, except with a richer spectrum being generated. Despite the relative simplicity of the models studied, these results exhibit many of the features of previous numerical simulations and point to the need for the inclusion of dispersive effects in designing liners for turbofan applications.

Stress analysis of large crude oil storage tank subjected to harmonic settlement

Differential settlement has a significant effect on the safe operation of tanks. In order to investigate the stress response of crude oil storage tank subjected to harmonic settlement. A numerical simulation model of the steel storage tank was developed in this study. Based on a validated finite element model, parametric analysis was conducted based on the main factors. The results show that the stress at the top of the tank wall is symmetrically distributed along the circumference of the tank. The stress result increases linearly with the increasing harmonic amplitude. The axial stress increases firstly, and then decreases for large wave number. Meanwhile, The stress value increases with the increment of the height-to-radius ratio, and decreases with the increment of the radius-to-thickness ratio. Especially, when the height-to-radius ratio ranging from 1.0 to 1.5 or the ratio r/t is more than 1500, the tendency of stress variation is slighter. This study could be referenced in strength or safety assessment of crude oil storage tank subjected to harmonic settlement.

Fixed-point proximity algorithms solving an incomplete Fourier transform model for seismic wavefield modeling

Seismic wavefield modeling is an important tool for the seismic interpretation. We consider modeling the wavefield in the frequency domain. This requires to solve a sequence of Helmholtz equations of wave numbers governed by the Nyquist sampling theorem. Inevitably, we have to solve Helmholtz equations of large wave numbers, which is a challenging task numerically. To address this issue, we develop two methods for modeling the wavefield in the frequency domain to obtain an alias-free result using lower frequencies of a number fewer than typically required by the Nyquist sampling theorem. Specifically, we introduce two ℓ1 regularization models to deal with incomplete Fourier transforms, which arise from seismic wavefield modeling in the frequency domain, and propose a new sampling technique to avoid solving the Helmholtz equations of large wave numbers. In terms of the fixed-point equation via the proximity operator of the ℓ1 norm, we characterize solutions of the two ℓ1 regularization models and develop fixed-point algorithms to solve these two models. Numerical experiments are conducted on seismic data to test the approximation accuracy and the computational efficiency of the proposed methods. Numerical results show that the proposed methods are accurate, robust and efficient in modeling seismic wavefield in the frequency domain with only a few low frequencies.

Centrifugal/elliptic instabilities in slowly varying channel flows

Abstract

Darcy–Brinkman Flow in a Corrugated Curved Channel

A theoretical analysis of a viscous flow in a corrugated curved channel enclosing a porous medium is carried out. The alignment of the corrugations of the outer and the inner curved walls is taken arbitrarily, and the corrugations are considered to be sinusoidal in nature with periodicity. The flow problem is described by Darcy–Brinkman model, derived in the curvilinear coordinates. The effects of the channel curvature, the wall corrugations and the medium permeability are studied through the boundary perturbation technique, for small corrugation amplitude. A substantial effect of the porous medium on the flow is observed when compared to that of the flow in a corrugated curved channel with clear conduit, especially for low permeability medium. Flow enhancement is found to take place for small corrugation wavenumbers, and maximum augmentation is realized for the completely out-of-phase alignment of the two corrugated curved walls. However, the flow reduces for large enough wavenumbers, and the alignment of corrugated curved walls eventually becomes irrelevant, with no influence on the flow. For low permeability medium, the results also show no effect of the wall alignment on the flow. In general, the effect of the channel curvature on the corrugated curved channel flow is discussed relative to a corrugated straight channel flow to demonstrate the implications of the wall geometry enclosing the porous medium.

A kind of operator regularization method for Cauchy problem of the Helmholtz equation in a multi-dimensional case

In this paper, a Cauchy problem of Helmholtz equation in a multi-dimensional case is investigated. This problem is severely ill-posed and small perturbations to measurement data can result in large changes in the solution. A kind of operator regularization method is proposed. The stable error estimates are obtained in the norm and norm under the conditions that m is even, md>x, mk>1, and suitable choice of regular parameters. Error estimates show that the regularized solution depends continuously on the perturbation noisy data and wave number. Our method makes up the limitation of small waves. Three numerical experiments show that our proposed method is effective and stable, especially in the case of a large wave number.

Analysis of transient wave propagation in inhomogeneous media using edge-based gradient smoothing technique and bathe time integration method

As is known to all, the classical finite element method (FEM) always fails to effectively solve the wave problems in the relatively large wave numbers due to the troublesome numerical error issue. In this paper, the standard FEM with edge-based gradient smoothing technique and Bathe time integration method is developed to analyze the transient wave propagation problems in inhomogeneous media. We explicitly show that the numerical error of the calculated solutions for transient wave propagation problems consists of two different parts, namely the spatial discretization error and the temporal discretization error. Due to the edge-based gradient smoothing technique and the appropriate numerical dissipation effects from Bathe time integration scheme, it is found that the total numerical error can be significantly suppressed and more accurate numerical solutions can be obtained. Several typical numerical examples have been conducted to examine the capacity of the proposed method in solving transient wave propagation problems in inhomogeneous media.

Statistical model for filamentary structures of molecular clouds

We propose a new statistical model that can reproduce the hierarchical nature of the ubiquitous filamentary structures of molecular clouds. This model is based on the multiplicative random cascade, which is designed to replicate the multifractal nature of intermittency in developed turbulence. We present a modified version of the multiplicative process where the spatial fluctuations as a function of scales are produced with the wavelet transforms of a fractional Brownian motion realisation. This simple approach produces naturally a log-normal distribution function and hierarchical coherent structures. Despite the highly contrasted aspect of these coherent structures against a smoother background, their Fourier power spectrum can be fitted by a single power law. As reported in previous works using the multiscale non-Gaussian segmentation (MnGSeg) technique, it is proven that the fit of a single power law reflects the inability of the Fourier power spectrum to detect the progressive non-Gaussian contributions that are at the origin of these structures across the inertial range of the power spectrum. The mutifractal nature of these coherent structures is discussed, and an extension of the MnGSeg technique is proposed to calculate the multifractal spectrum that is associated with them. Using directional wavelets, we show that filamentary structures can easily be produced without changing the general shape of the power spectrum. The cumulative effect of random multiplicative sequences succeeds in producing the general aspect of filamentary structures similar to those associated with star-forming regions. The filamentary structures are formed through the product of a large number of random-phase linear waves at different spatial wavelengths. Dynamically, this effect might be associated with the collection of compressive processes that occur in the interstellar medium.

Weyl's problem: A computational approach

The distribution of eigenvalues of the wave equation in a bounded domain is known as Weyl's problem. We describe several computational projects related to the cumulative state number, defined as the number of states having wavenumber up to a maximum value. This quantity and its derivative, the density of states, have important applications in nuclear physics, degenerate Fermi gases, blackbody radiation, Bose–Einstein condensation, and the Casimir effect. Weyl's theorem states that, in the limit of large wavenumbers, the cumulative state number depends only on the volume of the bounding domain and not on its shape. Corrections to this behavior are well known and depend on the surface area of the bounding domain, its curvature, and other features. We describe several projects that allow readers to investigate this dependence for three bounding domains—a rectangular box, a sphere, and a circular cylinder. Quasi-one- and two-dimensional systems can be analyzed by considering various limits. The projects have applications in statistical mechanics, but can also be integrated into quantum mechanics, nuclear physics, or computational physics courses.

Determination and evaluation of secondary structure content derived from calcium-induced conformational changes in wild-type and mutant mnemiopsin 2 by synchrotron-based Fourier-transform infrared spectroscopy

Mnemiopsin 2 from a luminous ctenophore with two functional EF-hand motifs is a calcium-regulated photoprotein that is responsible for emitting a bright blue bioluminescence upon reacting with coelenterazine and calcium ions in Mnemiopsis leidyi. Synchrotron radiation-based Fourier-transform infrared (SR-FTIR) spectroscopy was applied to analyze the distribution of secondary structures, the conformational changes resulting from calcium binding and the structural stabilities in wild-type mnemiopsin 2, as well as its mutant type that possesses three EF-hand motifs. The distribution of secondary structures of these proteins indicates that mutant apo-mnemiopsin 2 has a more stable secondary structure than the wild-type. Analyses of the SR-FTIR spectra revealed that the conformational changes at the secondary structures of both mnemiopsin 2 depend on the calcium concentrations, such that the most noticeable changes in structures of wild-type and mutant mnemiopsin 2 occur at optimum concentrations 6 and 2 mM of calcium chloride, respectively. The addition of calcium to both proteins increases the proportions of their secondary structures in the amide I and II regions. The major amide I bands in the IR spectra of both mnemiopsin‑calcium complexes shift towards smaller wavenumbers, whereas their main amide II bands are identified at larger wavenumbers.

Causality and stability conditions of a conformal charged fluid

In this paper, I study the conditions imposed on a normal charged fluid so that the causality and stability criteria hold for this fluid. I adopt the newly developed General Frame (GF) notion in the relativistic hydrodynamics framework which states that hydrodynamic frames have to be fixed after applying the stability and causality conditions. To do this, I take a charged conformal matter in the flat and 3 + 1 dimension to analyze better these conditions. The causality condition is applied by looking to the asymptotic velocity of sound hydro modes at the large wave number limit and stability conditions are imposed by looking to the imaginary parts of hydro modes as well as the Routh-Hurwitz criteria. By fixing some of the transports, the suitable spaces for other ones are derived. I observe that in a dense medium having a finite U(1) charge with chemical potential μ0, negative values for transports appear and the second law of thermodynamics has not ruled out the existence of such values. Sign of scalar transports are not limited by any constraints and just a combination of vector transports is limited by the second law of thermodynamic. Also numerically it is proved that the most favorable region for transports {tilde{upgamma}}_{1,2}, coefficients of the dissipative terms of the current, is of negative values.

Transient Wave Propagation Dynamics with Edge-Based Smoothed Finite Element Method and Bathe Time Integration Technique

In this work, the edge-based smoothed finite element method (ES-FEM) is incorporated with the Bathe time integration scheme to solve the transient wave propagation problems. The edge-based gradient smoothing technique (GST) can properly soften the “overly soft” system matrices from the standard finite element approach; then, the spatial numerical dispersion error of the calculated solutions for wave problems can be significantly reduced. To effectively solve the transient wave propagation problems, the Bathe time integration scheme is employed to perform the involved time integration. Due to the appropriate “numerical dissipation effects” from the Bathe time integration method, the spurious oscillations in the relatively large wave numbers (high frequencies) can be effectively suppressed; then, the temporal numerical dispersion error in the calculated solutions can also be notably controlled. A number of supporting numerical examples are considered to examine the capabilities of the present approach. The numerical results show that ES-FEM works very well with the Bathe time integration technique, and much more numerical solutions can be reached for solving transient wave propagation problems compared to the standard FEM.

Effect of the Ceramic Burning Temperature on the Characteristics of the Emission Spectrum (LIBS) of Al and Transmittance (FTIR) of Al-O on Ceramic Body of BL-1 Type

Aluminum (Al) contained in materials such as clay, kaolin is one of the elements that play an important role in the ceramic body. Al binds to other atoms like Si (quartz) to form a ceramic frame that is strong and not easily cracked. In addition, the bond with the group -OH plays an important role in relation to the ability of ceramics to absorb water. In this study, BL-1 type ceramic samples have been made with five variations of the burning temperature between 800-1000oC. To determine changes in the Al atomic content of ceramic samples were characterized by LIBS and FTIR spectrophotometers. The results of LIBS characterization of all samples showed two bands with peaks at wavelengths of 394.4 nm (Al-I) and 396.6 nm (Al-II) which are the characteristic peaks of Al atoms. The peaks with the highest intensity of Al-I and Al-II were resulted by samples that were burned at 900oC. Meanwhile, the results of FTIR characterization have shown a spectrum with bands at wavenumbers between 1030-1060 and 1150 cm-1 which are characteristic bands of stretching vibrations from Al-O. The peak of the band in the wavenumber between 1030-1060 cm-1 shifts towards a larger wavenumber with increasing the burning temperature.

How the circulation and axial velocity deficit in Batchelor vortices affect their disturbance growth?

In this study, inviscid linear stability theory is used to investigate the compressibility effects of Batchelor vortices. The growth rates are obtained from asymptotic analysis with large wavenumbers, and it is shown that the growth of disturbances is affected largely by both the Mach number and the axial velocity deficit. From the maximum growth rates for various base flow conditions, the growth properties of Batchelor vortices are summarized reasonably by using a convective Mach number based on two relative Mach numbers for three-dimensional disturbances. In addition, the ratio of the circulation and velocity deficit at the maximum growth rate is closely related to the growth rate as a function of the convective Mach number because the correlation between the ratio and the growth rate is high when the Mach number is less than unity. Therefore, the compressibility effects, which are expressed as the relation between the growth property and the convective Mach number, are estimated simply from only the base flows (circulation and velocity deficit) in Batchelor vortices.

Multiple Rayleigh waves guided by the planar surface of a continuously twisted structurally chiral material.

The Stroh formalism was adapted for Rayleigh-wave propagation guided by the planar traction-free surface of a continuously twisted structurally chiral material (CTSCM), which is an anisotropic solid that is periodically non-homogeneous in the direction normal to the planar surface. Numerical studies reveal that this surface can support either one or two Rayleigh waves at a fixed frequency, depending on the structural period and orientation of the CTSCM. In the case of two Rayleigh waves, each wave possesses a different wavenumber. The Rayleigh wave with the larger wavenumber is more localized to the surface and has a phase speed that changes less as the angular frequency varies in comparison with the Rayleigh wave with the smaller wavenumber.

Unique fermionic vacuum in de Sitter spacetime from hybrid quantum cosmology

In this work we show how the criterion of asymptotic Hamiltonian diagonalization originated in hybrid quantum cosmology serves to pick out a unique vacuum for the Dirac field in de Sitter, in the context of quantum field theory in curved spacetimes. This criterion is based on the dynamical definition of annihilation and creationlike variables for the fermionic field, which obey the linearized dynamics of a Hamiltonian that has been diagonalized in a way that is adapted to its local spatial structure. This leads to fermionic variables that possess a precise asymptotic expansion in the ultraviolet limit of large wavenumbers. We explicitly show that, when the cosmological background is fixed as a de Sitter solution, this expansion uniquely selects the choice of fermionic annihilation and creationlike variables for all spatial scales, and thus picks out a unique privileged Fock representation and vacuum state for the Dirac field in de Sitter. The explicit form of the basis of solutions to the Dirac equation associated with this vacuum is then computed.

Boltzmann equation and hydrodynamic equations: their equilibrium and non-equilibrium behaviour.

This short article summarizes the key features of equilibrium and non-equilibrium aspects of Boltzmann and hydrodynamic equations. Under equilibrium, the Boltzmann equation generates uncorrelated random velocity that corresponds to k2 energy spectrum for the Euler equation. The latter spectrum is produced using initial configuration with many Fourier modes of equal amplitudes but with random phases. However, for a large-scale vortex as an initial condition, earlier simulations exhibit a combination of k-5/3 (in the inertial range) and k2 (for large wavenumbers) spectra, with the range of k2 spectrum increasing with time. These simulations demonstrate an approach to equilibrium or thermalization of Euler turbulence. In addition, they also show how initial velocity field plays an important role in determining the behaviour of the Euler equation. In non-equilibrium scenario, both Boltzmann and Navier-Stokes equations produce similar flow behaviour, for example, Kolmogorov's k-5/3 spectrum in the inertial range. This article is part of the theme issue 'Fluid dynamics, soft matter and complex systems: recent results and new methods'.

Adaptive-multilevel BDDC algorithm for three-dimensional plane wave Helmholtz systems

In this paper, we are concerned with the weighted plane wave least-squares (PWLS) method for three-dimensional Helmholtz equations, and develop the multi-level adaptive BDDC algorithms for solving the resulting discrete system. In order to form the adaptive coarse components, the local generalized eigenvalue problems for each common face and each common edge are carefully designed. The condition number of the two-level adaptive BDDC preconditioned system is proved to be bounded above by a user-defined tolerance and a constant which is dependent on the maximum number of faces and edges per subdomain and the number of subdomains sharing a common edge. The efficiency of these algorithms is illustrated on a benchmark problem. The numerical results show the robustness of our two-level adaptive BDDC algorithms with respect to the wave number, the number of subdomains and the mesh size, and illustrate that our multi-level adaptive BDDC algorithm can reduce the scale of the coarse problem and can be used to solve large wave number problems efficiently.

Excitation of Negative Energy Surface Magnetohydrodynamic Waves in an Incompressible Cylindrical Plasma

Abstract Negative energy wave (NEW) phenomena may appear in shear flows in the presence of a wave decay mechanism and external energy supply. We study the appearance of negative energy surface waves in a plasma cylinder in the incompressible limit. The cylinder is surrounded by an axial magnetic field and by a plasma of different density. Considering flow inside and viscosity outside the flux tube, we derive dispersion relations and obtain analytical solutions for the phase speed and growth rate (increment) of the waves. It is found that the critical speed shear for the occurrence of the dissipative instability associated with NEWs and the threshold of Kelvin–Helmholtz instability (KHI) depend on the axial wavelength. The critical shear for the appearance of sausage NEW is lowest for the longest axial wavelengths, while for kink waves the minimum value of the critical shear is reached for the axial wavelength comparable to the diameter of the cylinder. The range between the critical speed of the dissipative instability and the KHI threshold is shown to depend on the difference of the Alfvén speeds inside and outside of the cylinder. For all axial wavenumbers, NEW appears for the shear flow speeds lower than the KHI threshold. It is easier to excite NEW in an underdense cylinder than in an overdense one. The negative energy surface waves can be effectively generated for an azimuthal number m = 0 with a large axial wavenumber and for higher modes (m > 0) with a small axial wavenumber.