Azimuthal Fourier Coefficients Research Articles

Superresolution and analysis of three-dimensional velocity fields of underexpanded jets in different screech modes

Time-resolved (TR), three-dimensional (3D) velocity fields of screeching, underexpanded jets are estimated using non-time-resolved particle image velocimetry and simultaneous TR microphone measurements. Specifically, we aim to reconstruct TR 3D velocity fluctuation fields associated with the A2, B, and C modes of a screeching jet using a linear regression model and to analyze screech dynamics of these modes. The linear regression model is constructed on the basis of a linear relationship between the velocity and acoustic fields. Three nozzle pressure ratios (NPRs) of 2.30, 2.97, and 3.40 are employed. The dominant azimuthal modes for three cases are investigated using azimuthal Fourier coefficients of the acoustic data obtained by the azimuthal array of eight microphones placed near the nozzle exit. The dominant azimuthal modes at NPRs of 2.30, 2.97, and 3.40 are m=0, 1, and 1, respectively. The first two proper orthogonal decomposition (POD) modes in these azimuthal modes are dominant at all NPRs and are associated with screech. 3D velocity fluctuation fields associated with screech are reconstructed from these leading POD modes of the acoustic data. The reconstructed 3D velocity fluctuation fields at NPRs of 2.97 and 3.40 exhibit two helical structures with opposite rotation directions. The present results demonstrate that, in the B mode, the flapping structure exhibits random clockwise and counterclockwise rotations over an extended time domain, while maintaining a consistent direction within short time domains. In addition, in the C mode, two helical structures with opposite rotation directions, as well as the flapping structure, are observed. Published by the American Physical Society 2024

Fracture detection, fluid identification and brittleness evaluation in the gas-bearing reservoir with tilted transverse isotropy via azimuthal Fourier coefficients

Seismic characterization of the sweet spot plays a crucial role in the exploration and exploitation of oil and gas reservoirs. Therefore, we focused on fluid discrimination, natural fracture detection and brittleness evaluation in the reservoir permeated by an inclined set of parallel fractures using pre-stacked seismic inversion method. To this end, we proved by numerical experiments that the suggested fluid indicator is available for identifying the gas-bearing reservoir due to its high sensitivity to the gaseous hydrocarbon and extremely weak dependences on the porosity and the fracture density of the gas-bearing formation. To reduce the cumulative errors induced by indirect calculations, we derived the reflection coefficient of the TTI (tilted transverse isotropy) rock in terms of Young's modulus, Poisson's ratio, the fluid indicator and the fracture parameters and then rewrote it in the form of the Fourier series. By comparing with the published results, the derived reflectivity equation was proven rational and of satisfactory accuracy. Furthermore, we proposed a hierarchical inversion method incorporating the discrete Fourier transformation (DFT) and the Bayesian inversion strategy, which consists of three steps. Firstly, the DFT method is utilized to decouple anisotropic seismic components from the isotropic seismic component. In the last two steps, the Bayesian inversion approach is taken to acquire the stable and reliable estimations of the fluid and brittleness indicators and the fracture parameters from the second-order and zeroth-order seismic components, respectively. The synthetic example proved that we can realize rational and reliable predictions of the unknown parameters using the hierarchical method even for the case with moderate noise. The field case demonstrated that the proposed method is feasible and stable, and the suggested sweet spot indicators worked well for gaseous hydrocarbon discrimination, natural fracture detection and brittleness evaluation.

Characteristics of collective modes in anisotropic plasmas

We use a quasi-particle picture and the hard-loop approximation to study collective modes in anisotropic plasmas. We study a general class of anisotropic distribution functions and derive and solve the dispersion equations. We focus on imaginary modes because of their potential influence on plasma dynamics, and we study their dependence on the parameters that characterize the anisotropy of the system. We show that the strength of the imaginary modes is related to the oblateness of the distribution, and the size of the azimuthal Fourier coefficients.

Estimation of in situ stresses from PP-wave azimuthal seismic data in fracture-induced anisotropic media

Horizontally transverse isotropy (HTI) induced by vertical or subvertical aligned fractures is common for unconventional fractured porous shale oil or gas reservoirs. Compared with the unfractured rocks, the seismic response characteristics of PP-wave azimuthal amplitudes are usually disturbed by the fractures and the in situ stresses. Knowledge of fracture properties, as well as in situ stresses, is required to optimize horizontal well planning and hydraulic fracturing during production, and the seismic inversion for in situ stresses from the PP-wave azimuthal amplitude data in fracture-induced anisotropic media is an essential step. Using the linear-slip theory and the effective stress law, we derive the fluid-saturated effective elastic stiffness tensors parameterized by background elastic moduli, the effective stress coefficient of isotropic host rocks, fluid modulus, porosity, and fracture parameters based on the anisotropic Gassmann’s fluid substitution equation. Combining the perturbations in saturated stiffness tensors and scattering theory, we formulate the reflection coefficient equation of PP-wave data as a function of background porosity-related stress parameter and two (i.e., normal and shear) fracture weakness parameters. Following Bayes’ rule, we estimate the porosity-related stress parameter and fracture weaknesses using the inversion method of azimuthal Fourier coefficients. Finally, we compute the effective horizontal and vertical in situ stresses using the estimated elastic moduli, porosity-related stress parameter, and fracture weaknesses. Synthetic and real data sets demonstrate that our proposed inversion approach based on the derived reflection equation provides us another way to obtain reasonable estimates of in situ stresses in a complex-fractured porous shale reservoir.

Gegenbauer expansions and addition theorems for a binomial and logarithmic fundamental solution of the even-dimensional Euclidean polyharmonic equation

On even-dimensional Euclidean space for integer powers of the positive Laplace operator greater than or equal to half the dimension, a fundamental solution of the polyharmonic equation has binomial and logarithmic behavior. Gegenbauer polynomial expansions of these fundamental solutions are obtained through a limit applied to Gegenbauer expansions of a power-law fundamental solution of the polyharmonic equation. This limit is accomplished through parameter differentiation. By combining these results with previously derived azimuthal Fourier series expansions for these binomial and logarithmic fundamental solutions, we are able to obtain addition theorems for the azimuthal Fourier coefficients. These logarithmic and binomial addition theorems are expressed in Vilenkin polyspherical geodesic polar coordinate systems and as well in generalized Hopf coordinates on hyperspheres in arbitrary even dimensions.

Flow Harmonics vn in pPb and PbPb Collisions

Previous CMS measurements have demonstrated the collective nature of multiparticle correlations in high-multiplicity pPb collisions at the LHC. This collectivity is consistent with a hydrodynamic flow origin. However, it can also be interpreted in terms of initial state effects arising from gluon saturation. The pseudorapidity dependence of the azimuthal Fourier coefficients (vn) is expected to be sensitive to the underlying mechanism with, in the hydrodynamic picture, the longer lifetime of the fireball on the Pb-going side expected to lead to a larger flow signal than found on the p-going side. To investigate the detailed properties of the observed collectivity, differential vn values in transverse momentum (pT) and pseudorapidity (η) are presented over the full range of the CMS tracker detector (−2.4<η<2.4) for pPb collisions at a nucleon-nucleon center-of-mass energy of 5.02 TeV. Results based on multiparticle analyses involving four or more particles are shown. An event plane analysis is presented where the influence of recently demonstrated event-plane decorrelation is considered. Comparisons are made with peripheral PbPb collisions measured at similar mid-rapidity particle multiplicities. The results will be discussed in the context of current models of the longitudinal dependence of the multiparticle correlations.

Interpreting azimuthal Fourier coefficients for anisotropic and fracture parameters

Amplitude variation with offset and azimuth (AVOAz) analysis can be separated into two separate parts: amplitude variation with offset (AVO) analysis and amplitude variation with azimuth (AVAz) analysis. Useful information about fractures and anisotropy can be obtained just by examining the AVAz. The AVAz can be described as a sum of sinusoids of different periodicities, each characterized by its magnitude and phase. This sum is mathematically equivalent to a Fourier series, and hence the coefficients describing the AVAz response are azimuthal Fourier coefficients (FCs). This FC parameterization is purely descriptive. The aim of this paper is to help the interpreter understand what these coefficients mean in terms of anisotropic and fracture parameters for the case of P-wave reflectivity using a linearized approximation. The FC representation is valid for general anisotropy. However, to gain insight into the significance of FCs, more restrictive assumptions about the anisotropy or facture system must be assumed. In the case of transverse isotropic media with a horizontal axis of symmetry, the P-wave reflectivity linearized approximation may be rewritten in terms of azimuthal FCs with the magnitude and phase of the different FCs corresponding to traditional AVAz attributes. Linear slip theory is used to show that the FCs can be interpreted similarly for the cases of a single set of parallel vertical fractures in isotropic media and in transverse isotropic media with a vertical axis of symmetry (VTI). The magnitude of the FCs depends on the fracture weakness parameters and the background media. For the case of vertical fractures in a VTI background, the AVOAz inverse problem is underdetermined, so extra information must be incorporated to determine how the weights are modified due to this background anisotropy. We evaluated this on a 3D data set from northwest Louisiana for which the main target was the Haynesville shale.

Fourier and Gegenbauer Expansions for a Fundamental Solution of Laplace's Equation in Hyperspherical Geometry

For a fundamental solution of Laplace's equation on the $R$-radius $d$-dimensional hypersphere, we compute the azimuthal Fourier coefficients in closed form in two and three dimensions. We also compute the Gegenbauer polynomial expansion for a fundamental solution of Laplace's equation in hyperspherical geometry in geodesic polar coordinates. From this expansion in three-dimensions, we derive an addition theorem for the azimuthal Fourier coefficients of a fundamental solution of Laplace's equation on the 3-sphere. Applications of our expansions are given, namely closed-form solutions to Poisson's equation with uniform density source distributions. The Newtonian potential is obtained for the 2-disc on the 2-sphere and 3-ball and circular curve segment on the 3-sphere. Applications are also given to the superintegrable Kepler-Coulomb and isotropic oscillator potentials.

Fourier, Gegenbauer and Jacobi Expansions for a Power-Law Fundamental Solution of the Polyharmonic Equation and Polyspherical Addition Theorems

We develop complex Jacobi, Gegenbauer and Chebyshev polynomial expansions for the kernels associated with power-law fundamental solutions of the polyharmonic equation on d-dimensional Euclidean space. From these series representations we derive Fourier expansions in certain rotationally-invariant coordinate systems and Gegenbauer polynomial expansions in Vilenkin's polyspherical coordinates. We compare both of these expansions to generate addition theorems for the azimuthal Fourier coefficients.

Object identification using correlated orbital angular momentum states.

Using spontaneous parametric down-conversion as a source of correlated photon pairs, correlations are measured between the orbital angular momentum (OAM) in a target beam (which contains an unknown object) and that in an empty reference beam. Unlike previous studies, the effects of the object on off-diagonal elements of the OAM correlation matrix are examined. Because of the presence of the object, terms appear in which the signal and idler OAM do not add up to that of the pump. Using these off-diagonal correlations, the potential for high-efficiency object identification by means of correlated OAM states is experimentally demonstrated for the first time. The higher-dimensional OAM Hilbert space enhances the information capacity of this approach, while the presence of the off-diagonal correlations allows for recognition of specific spatial signatures present in the object. In particular, this allows the detection of discrete rotational symmetries and the efficient evaluation of multiple azimuthal Fourier coefficients using fewer resources than in conventional pixel-by-pixel imaging. This represents a demonstration of sparse sensing using OAM states, as well as being the first correlated OAM experiment to measure properties of a real, stand-alone object, a necessary first step toward correlated OAM-based remote sensing.

Fourier and Gegenbauer expansions for a fundamental solution of the Laplacian in the hyperboloid model of hyperbolic geometry

Due to the isotropy of d-dimensional hyperbolic space, there exists a spherically symmetric fundamental solution for its corresponding Laplace–Beltrami operator. The R-radius hyperboloid model of hyperbolic geometry with R > 0 represents a Riemannian manifold with negative-constant sectional curvature. We obtain a spherically symmetric fundamental solution of Laplace’s equation on this manifold in terms of its geodesic radius. We give several matching expressions for this fundamental solution including a definite integral over reciprocal powers of the hyperbolic sine, finite summation expressions over hyperbolic functions, Gauss hypergeometric functions and in terms of the associated Legendre function of the second kind with order and degree given by d/2 − 1 with real argument greater than unity. We also demonstrate uniqueness for a fundamental solution of Laplace’s equation on this manifold in terms of a vanishing decay at infinity. In rotationally invariant coordinate systems, we compute the azimuthal Fourier coefficients for a fundamental solution of Laplace’s equation on the R-radius hyperboloid. For d ⩾ 2, we compute the Gegenbauer polynomial expansion in geodesic polar coordinates for a fundamental solution of Laplace’s equation on this negative-constant curvature Riemannian manifold. In three dimensions, an addition theorem for the azimuthal Fourier coefficients of a fundamental solution for Laplace’s equation is obtained through comparison with its corresponding Gegenbauer expansion.