Azimuthal Wave Research Articles

Characterization of micro-fractures in carbonate Sarvak reservoir, using petrophysical and geological data, SW Iran

In this paper, an integrated study of micro-fractures is carried out in Sarvak Formation. To achieve this goal the data from three wells (A, B and C) in one of the southwestern Iranian oil fields were used. In this study petrographic thin sections, porosity-permeability core data, image logs, Dipole Shear Sonic Imager (DSI) log, Velocity Deviation Log (VDL), Hydraulic Flow Units (HFU) concept, resistivity logs and drilling mud reports were used to identify and analyze micro-fractures in Sarvak Formation. The micro-fractures were first detected in thin sections and image logs directly, then the results of other methods in micro-fracture zones were investigated. The results show there is an abnormal trend of porosity-permeability in core data which is compatible with micro-fracture zones. Also, the relationship between micro-fractures and flow units was studied. Eight HFUs were recognized using porosity-permeability core data and FZI* (Flow Zone Indicator-Star) method. Later each of the HFUs was divided into two sub-HFUs; one of them is well-matched with the micro-fracture zones. The negative parts of VDL were in good agreement with the micro-fracture zones. The slowness anisotropy was mainly affected by the micro-fracture zones, but the energy and traveltime anisotropies were under the influence of the other phenomena such as solution seams, shale layers and borehole washouts. The fast shear wave azimuth was mainly aligned with the SHmax (maximum horizontal stress) direction, however, in micro-fracture zones, the direction suddenly changes. It was also observed that the micro-fractures do not have a considerable effect on Stoneley wave reflection, while the borehole washouts and shale layers have a great impact on this. The mud loss and mud invasion were very low in micro-fractured zones, which is related to the lack of lateral continuity of micro-fractures in the formation. The measured fracture apertures using thin section and image logs are often less than 0.1 mm, but the calculated values using Pezard-Anderson equations are very high.

Propagating helical waves as a building block of round turbulent jets

Turbulent jets are known to support large-scale vortical wave packets traveling downstream. We show that a propagating helical wave represents a common form of the optimal eigenfunction tracking these structures from the near to the far field of a round jet issuing from a pipe. Two first mirror-symmetric modes containing around 5% of the total turbulent kinetic energy capture all significant large-scale events and accurately replicate the full shear-layer dynamics of the azimuthal wave number m=1. A family of the most energy-containing traveling waves represents low wave numbers and is described in terms of empirical dispersion laws.

A New Analytical Procedure to Solve Two Phase Flow in Tubes

A new formulation for a proposed solution to the 3D Navier-Stokes Equations in cylindrical co-ordinates coupled to the continuity and level set convection equation is presented in terms of an additive solution of the three principle directions in the radial, azimuthal and z directions of flow and a connection between the level set function and composite velocity vector for the additive solution is shown. For the case of a vertical tube configuration with small inclination angle, results are obtained for the level set function defining the interface in both the radial and azimuthal directions. It is found that the curvature dependent part of the problem alone induces sinusoidal azimuthal interfacial waves whereas when the curvature together with the equation for the composite velocity is considered oscillating radial interfacial waves occur. The implications of two extremes indicate the importance of looking at the full equations including curvature.

E × B electron drift instability in Hall thrusters: Particle-in-cell simulations vs. theory

The E × B Electron Drift Instability (E × B EDI), also called Electron Cyclotron Drift Instability, has been observed in recent particle simulations of Hall thrusters and is a possible candidate to explain anomalous electron transport across the magnetic field in these devices. This instability is characterized by the development of an azimuthal wave with wavelength in the mm range and velocity on the order of the ion acoustic velocity, which enhances electron transport across the magnetic field. In this paper, we study the development and convection of the E × B EDI in the acceleration and near plume regions of a Hall thruster using a simplified 2D axial-azimuthal Particle-In-Cell simulation. The simulation is collisionless and the ionization profile is not-self-consistent but rather is given as an input parameter of the model. The aim is to study the development and properties of the instability for different values of the ionization rate (i.e., of the total ion production rate or current) and to compare the results with the theory. An important result is that the wavelength of the simulated azimuthal wave scales as the electron Debye length and that its frequency is on the order of the ion plasma frequency. This is consistent with the theory predicting destruction of electron cyclotron resonance of the E × B EDI in the non-linear regime resulting in the transition to an ion acoustic instability. The simulations also show that for plasma densities smaller than under nominal conditions of Hall thrusters the field fluctuations induced by the E × B EDI are no longer sufficient to significantly enhance electron transport across the magnetic field, and transit time instabilities develop in the axial direction. The conditions and results of the simulations are described in detail in this paper and they can serve as benchmarks for comparisons between different simulation codes. Such benchmarks would be very useful to study the role of numerical noise (numerical noise can also be responsible to the destruction of electron cyclotron resonance) or the influence of the period of the azimuthal domain, as well as to reach a better and consensual understanding of the physics.

Spatio-Temporal Koopman Decomposition

This paper deals with a new purely data-driven method, called the spatio-temporal Koopman decomposition, to approximate spatio-temporal data as a linear combination of (possibly growing or decaying exponentially) standing or traveling waves. The method combines (i) either standard singular value decomposition (SVD) or higher-order SVD and (ii) either standard dynamic mode decomposition (DMD) or an extension of this method by the authors, called higher-order DMD. In particular, for periodic or quasiperiodic attractors, the method gives the spatio-temporal pattern as a superposition of standing and/or traveling waves, which are identified in an efficient and robust way. Such superposition may give the whole pattern as a modulated, periodic or quasiperiodic, standing or traveling wave. The method is illustrated in some simple toy-model dynamics, and its performance is tested in the identification of standing and traveling waves in the Ginzburg–Landau equation and of azimuthal waves in a rotating spherical shell with thermal convection.

Far-wake meandering induced by atmospheric eddies in flow past a wind turbine

A novel algorithm is developed to calculate the nonlinear optimal boundary perturbations in three-dimensional incompressible flow. An optimal step length in the optimization loop is calculated without any additional calls to the Navier–Stokes equations. The algorithm is applied to compute the optimal inflow eddies for the flow around a wind turbine to clarify the mechanisms behind wake meandering, a phenomenon usually observed in wind farms. The turbine is modelled as an actuator disc using an immersed boundary method with the loading prescribed as a body force. At Reynolds number (based on free-stream velocity and turbine radius) $Re=1000$, the most energetic inflow perturbation has a frequency $\unicode[STIX]{x1D714}=0.8$–2, and is in the form of an azimuthal wave with wavenumber $m=1$ and the same radius as the actuator disc. The inflow perturbation is amplified by the strong shear downstream of the edge of the disc and then tilts the rolling-up vortex rings to induce wake meandering. This mechanism is verified by studying randomly perturbed flow at $Re\leqslant 8000$. At five turbine diameters downstream of the disc, the axial velocity oscillates at a magnitude of more than 60 % of the free-stream velocity when the magnitude of the inflow perturbation is 6 % of the free-stream wind speed. The dominant Strouhal number of the wake oscillation is 0.16 at $Re=3000$ and keeps approximately constant at higher $Re$. This Strouhal number agrees well with previous experimental findings. Overall the observations indicate that the well-observed stochastic wake meandering phenomenon appearing far downstream of wind turbines is induced by large-scale (the same order as the turbine rotor) and low-frequency free-stream eddies.

Global stability analysis of axisymmetric boundary layer over a circular cylinder

This paper presents a linear global stability analysis of the incompressible axisymmetric boundary layer on a circular cylinder. The base flow is parallel to the axis of the cylinder at inflow boundary. The pressure gradient is zero in the streamwise direction. The base flow velocity profile is fully non-parallel and non-similar in nature. The boundary layer grows continuously in the spatial directions. Linearized Navier–Stokes (LNS) equations are derived for the disturbance flow quantities in the cylindrical polar coordinates. The LNS equations along with homogeneous boundary conditions forms a generalized eigenvalues problem. Since the base flow is axisymmetric, the disturbances are periodic in azimuthal direction. Chebyshev spectral collocation method and Arnoldi’s iterative algorithm is used for the solution of the general eigenvalues problem. The global temporal modes are computed for the range of Reynolds numbers and different azimuthal wave numbers. The largest imaginary part of the computed eigenmodes is negative, and hence, the flow is temporally stable. The spatial structure of the eigenmodes shows that the disturbance amplitudes grow in size and magnitude while they are moving towards downstream. The global modes of axisymmetric boundary layer are more stable than that of 2D flat-plate boundary layer at low Reynolds number. However, at higher Reynolds number they approach 2D flat-plate boundary layer. Thus, the damping effect of transverse curvature is significant at low Reynolds number. The wave-like nature of the disturbance amplitudes is found in the streamwise direction for the least stable eigenmodes.

Excitation of higher radial modes of azimuthal surface waves in the electron cyclotron frequency range by rotating relativistic flow of electrons in cylindrical waveguides partially filled by plasmas

Azimuthal surface waves are electromagnetic eigenwaves of cylindrical plasma-dielectric waveguides which propagate azimuthally nearby the plasma-dielectric interface across an axial external stationary magnetic field. Their eigenfrequency in particular can belong to the electron cyclotron frequency range. Excitation of azimuthal surface waves by rotating relativistic electron flows was studied in detail recently in the case of the zeroth radial mode for which the waves' radial phase change within the layer where the electrons gyrate is small. In this case, just the plasma parameters cause the main influence on the waves' dispersion properties. In the case of the first and higher radial modes, the wave eigenfrequency is higher and the wavelength is shorter than in the case of the zeroth radial mode. This gain being of interest for practical applications can be achieved without any change in the device design. The possibility of effective excitation of the higher order radial modes of azimuthal surface waves is demonstrated here. Getting shorter wavelengths of the excited waves in the case of higher radial modes is shown to be accompanied by decreasing growth rates of the waves. The results obtained here are of interest for developing new sources of electromagnetic radiation, in nano-physics and in medical physics.

Two mechanisms of resonance overlapping in excitation of azimuthal surface waves by rotating relativistic electron beams

Azimuthal surface waves are electromagnetic eigenwaves of cylindrical plasma-filled metallic waveguides with a stationary axial magnetic field. These waves with extraordinary polarization can effectively interact with relativistic electron beams rotating along large Larmor orbits in the gap, which separates the plasma column from the waveguide wall. Both widening the layer and increasing the beam particle density are demonstrated to cause resonance overlapping seen from the perspective of the growth rate dependence on the effective wave number.

Wave number selection in the presence of noise: Experimental results.

In this study, we consider how the wave number selection in spherical Couette flow, in the transition to azimuthal waves after the first instability, occurs in the presence of noise. The outer sphere was held stationary, while the inner sphere rotational speed was increased linearly from a subcritical flow to a supercritical one. In a supercritical flow, one of two possible flow states, each with different azimuthal wave numbers, can appear depending upon the initial and final Reynolds numbers and the acceleration value. Noise perturbations were added by introducing small disturbances into the rotational speed signal. With an increasing noise amplitude, a change in the dominant wave number from m to m ± 1 was found to occur at the same initial and final Reynolds numbers and acceleration values. The flow velocity measurements were conducted by using laser Doppler anemometry. Using these results, the role of noise as well as the behaviour of the amplitudes of the competing modes in their stages of damping and growth were determined.

Transmission of a Magnetospheric Pc1 Wave Beam Through the Ionosphere to the Ground

AbstractA characteristic feature of the upper ionosphere is the occurrence of the ionospheric Alfvén resonator (IAR) and the ionospheric fast‐mode waveguide (IFW), which can trap electromagnetic wave energy in the Pc1 frequency range from fractions of a hertz to a few hertz. This wave trapping ensures the dependence of ionospheric transmission/reflective properties on frequency. We numerically model magnetospheric Alfvén wave transmission through the ionosphere to the ground based on solution of the magnetohydrodynamic full‐wave equations in a realistic ionosphere, whose parameters are reconstructed from the International Reference Ionosphere model. The spatial structure of an incident wave is modeled as a localized beam with a finite latitudinal scale λ⊥ and an azimuthally propagating wave. The IAR and IFW modes are coupled owing to the Hall conductivity and geomagnetic field inclination. Ground spatial and spectral structures of the Pc1 wave (0.1‐ to 6‐Hz band) have been calculated for summer day/night conditions at the Antarctic Halley observatory, though results may be qualitatively applied to any midlatitude site. An incident wave with azimuthal wave vector ky=1.7·10−3 km−1, and λ⊥=102 km has been considered. The model predicts that beneath the incident beam the ground magnetic response “duplicates” its structure after accounting for a π/2 rotation and some latitudinal shift/widening. The transmission has an oscillatory dependence on frequency, thus forming “transmission windows” at resonant frequencies of IAR and IFW modes. The spectra vary depending on distance from an incidence point. Interference between IFW modes is revealed in the nonmonotonic and frequency‐dependent character of latitudinal variations of wave amplitude.

Effect of dust on tilted electrostatic resistive instability in a Hall thruster

Effect of negatively charged dust on resistive instability corresponding to the electrostatic wave is investigated in a Hall thruster plasma when this purely azimuthal wave is tilted and strong axial component of wave vector is developed. Analytical calculations are done to obtain the relevant dispersion equation, which is solved numerically to investigate the growth rate of the instability. The magnitude of the growth rate in the plasma having dust particles is found to be much smaller than the case of pure plasma. However, the instability grows faster for the increasing dust density and the higher charge on the dust particles. The higher magnetic field is also found to support the instability.

Analysis and modeling of localized invariant solutions in pipe flow

Turbulent spots surrounded by laminar flow are a landmark of transitional shear flows, but the dependence of their kinematic properties on spatial structure is poorly understood. We here investigate this dependence in pipe flow for Reynolds numbers between 1500 and 5000. We compute spatially localized relative periodic orbits in long pipes and show that their upstream and downstream fronts decay exponentially towards the laminar profile. This allows to model the fronts by employing the linearized Navier-Stokes equations, and the resulting model yields the spatial decay rate and the front velocity profiles of the periodic orbits as a function of Reynolds number, azimuthal wave number and propagation speed. In addition, when applied to a localized turbulent puff, the model is shown to accurately approximate the spatial decay rate of its upstream and downstream tails. Our study provides insight into the relationship between the kinematics and spatial structure of localized turbulence and more generally into the physics of localization.

Central recirculation zones and instability waves in internal swirling flows with an annular entry

The characteristics of the central recirculation zone and the induced instability waves of a swirling flow in a cylindrical chamber with a slip head end have been numerically investigated using the Galerkin finite element method. The effects of Reynolds number as well as swirl level adjusted by the injection angle were examined systematically. The results indicate that at a high swirl level the flow is characterized by an axisymmetric central recirculation zone (CRZ). The fluid in the CRZ takes on a solid-body rotation driven by the outer main flow through a free shear layer. Both the solid-body rotating central flow and the free shear layer provide the potential for the development of instability waves. When the injection angle increases beyond a critical value, the basic axisymmetric flow loses stability, and instability waves develop. In the range of Reynolds numbers considered in this study, three kinds of instability were identified: inertial waves in the central flow, and azimuthal and longitudinal Kelvin-Helmholtz waves in the free shear layer. These three types of waves interact with each other and mix together. The mode selection of the azimuthal waves depends strongly on the injection angle, through the perimeter of the free shear layer. Compared with the injection angle, the Reynolds number plays a minor role in mode selection. The flow topologies and characteristics of different flow states are analyzed in detail, and the dependence of flow states on the injection angle and Reynolds number is summarized. Finally, a linear analysis of azimuthal instabilities is carried out; it confirms the mode selection mechanisms demonstrated by the numerical simulation.

Van Allen Probes Observations of Second Harmonic Poloidal Standing Alfvén Waves

AbstractLong‐lasting second‐harmonic poloidal standing Alfvén waves (P2 waves) were observed by the twin Van Allen Probes (Radiation Belt Storm Probes, or RBSP) spacecraft in the noon sector of the plasmasphere, when the spacecraft were close to the magnetic equator and had a small azimuthal separation. Oscillations of proton fluxes at the wave frequency (∼10 mHz) were also observed in the energy (W) range 50–300 keV. Using the unique RBSP orbital configuration, we determined the phase delay of magnetic field perturbations between the spacecraft with a 2nπ ambiguity. We then used finite gyroradius effects seen in the proton flux oscillations to remove the ambiguity and found that the waves were propagating westward with an azimuthal wave number (m) of ∼−200. The phase of the proton flux oscillations relative to the radial component of the wave magnetic field progresses with W, crossing 0 (northward moving protons) or 180° (southward moving protons) at W ∼ 120 keV. This feature is explained by drift‐bounce resonance (mωd ∼ ωb) of ∼120 keV protons with the waves, where ωd and ωb are the proton drift and bounce frequencies. At lower energies, the proton phase space density ( ) exhibits a bump‐on‐tail structure with occurring in the 1–10 keV energy range. This is unstable and can excite P2 waves through bounce resonance (ω ∼ ωb), where ω is the wave frequency.

Система соосных дисковых диэлектрических резонаторов с азимутальными колебаниями

The own q-factor of a disk dielectric resonator (DDR) with azimuthal oscillations is determined by the radiation losses from the disk surface and dissipation in the dielectric. A change of disk parameters reducing the influence of one of these factors invariably causes an increased influence of the other parameter, whereas the own q-factor does not exceed 1/tgδ of the disk material. In a resonant structure made in the form of two coaxially arranged disks, the q-factor may be several times higher than this value. That the effect is in principle possible is obvious: in a multilayer structure with different dissipation in layers, it is possible to localize the electromagnetic field in an area with low dissipation. Preliminary experimental studies have shown that the effect does take place and depends on the choice of the working type of azimuthal oscillations and system parameters. The aim of this study is to justify the choice of the type of azimuthal oscillations and the region of structural parameters with which the highest levels of the own q-factor are obtained. For calculating the characteristics of azimuthal waves of the HE- and EH types producing oscillations in the DDR system, the phenomenological method of effective dielectric permeability was used. A system of distributed coupled planar dielectric waveguides and a dielectric cylinder were considered as the method’s first and second models, respectively. The results from an analysis of dispersion and energy characteristics of the own types of azimuthal resonators have shown that the highest levels of own q-factors are characteristic of an even EH-type and can be several times higher than 1/tgδ.

Evolution and transition mechanisms of internal swirling flows with tangential entry

The characteristics and transition mechanisms of different states of swirling flow in a cylindrical chamber have been numerically investigated using the Galerkin finite element method. The effects of the Reynolds number and swirl level were examined, and a unified theory connecting different flow states was established. The development of each flow state is considered as a result of the interaction and competition between basic mechanisms: (1) the centrifugal effect, which drives an axisymmetric central recirculation zone (CRZ); (2) flow instabilities, which develop at the free shear layer and the central solid-body rotating flow; (3) the bouncing and restoring effects of the injected flow, which facilitate the convergence of flow on the centerline and the formation of bubble-type vortex breakdown; and (4) the damping effect of the end-induced flow, which suppresses the development of the instability waves. The results show that the CRZ, together with the free shear layer on its surface, composes the basic structure of swirling flow. The development of instability waves produces a number of discrete vortex cores enclosing the CRZ. The azimuthal wave number is primarily determined by the injection angle. Generally, the wave number is smaller at a higher injection angle, due to the reduction of the perimeter of the free shear layer. At the same time, the increase in the Reynolds number facilitates the growth of the wave number. The end-induced flow tends to reduce the wave number near the head end and causes a change in wave number from the head end to the downstream region. Spiral-type vortex breakdown can be considered as a limiting case at a high injection angle, with a wave number equal to 0 near the head end and equal to 1 downstream. At lower Reynolds numbers, the bouncing and restoring effect of the injected flow generates bubble-type vortex breakdown.

Effects of the Reynolds and Weissenberg’s Numbers on the Stability of Linear Pipe Flow of Viscous Fluid

A Fourier-Chebyshev Petrov-Galerkin spectral method is described for high accuracy computation of linearized dynamics for flow in a circular pipe. The code used here is based on solenoidal velocity variables and is written in FORTRAN. Systematic studies are presented of the dependence of eigenval-ues and other quantities on the axial and azimuthal wave numbers; the Reyn-olds’ number of up to 107 and the Weissenberg’s number that is considered lower here. The flow will be considered stable if all the real parts of the ei-genvalues obtained are negative and unstable if only one of these values is positive.

Azimuthal Ocean Wave Emerged in SAR Image Spectra Under Specific Condition

AbstractVelocity bunching causes nonlinear modulations in synthetic aperture radar (SAR) images of oceanic scenes. SAR images for the azimuth waves exhibit destructive wavy patterns because of azimuthal distortion caused by velocity bunching. Meanwhile, velocity bunching creates constructive wavy patterns for long waves in certain cases. This paper aims to provide a simple indication for practical interpretation of plausible wavy patterns. A numerical simulation had been performed to address how velocity bunching forms a clear wave‐like pattern caused by orbital motions of a long single swell under a certain condition. More specifically, when the azimuth shifts caused by the orbital motions of the azimuth wave concentrate in the wave trough or crest, the bunching causes a strong single peak in the SAR images. This paper explains the mechanism of displaying plausible wavy patterns from the viewpoint of the ocean wave spectra and this specific condition. First, the simulation was validated for irregular waves generated from ocean wave spectra matching actual buoy data from which SAR images were simultaneously obtained. The SAR image spectra showed longer waves than those of the ocean wave spectra estimated via buoy data. The simulated spectra exhibited a similar trend to the actual SAR image, thus validating the numerical simulation. Next, an additional numerical simulation was conducted for irregular waves under the specific condition. The simulated results suggest that if a significant wave height and wavelength match the specific condition, the long waves in SAR image spectra can be similar to the original ocean wave spectra.

The Effect of a Twisted Magnetic Field on the Nature of Kink MHD Waves

We consider a pressureless plasma in a thin magnetic-flux tube with a twisted magnetic field. We study the effect of twisted magnetic field on the nature of propagating kink waves. To do this, the restoring forces of oscillations in the linear ideal magnetohydrodynamics (MHD) were obtained. In the presence of a twisted magnetic field, the ratio of the magnetic-tension force to the gradient of the magnetic pressure increases for the mode with negative azimuthal wave number, but it decreases for the mode with positive azimuthal wave number. For the kink mode with positive azimuthal mode number, the ratio of the forces is more affected by the twisted magnetic field in dense loops. For the kink mode with negative azimuthal mode number, the perturbed magnetic pressure is negligible under some conditions. The magnetic twist increases (diminishes) the damping of the kink waves with positive (negative) azimuthal mode number due to resonant absorption. Our conclusion is that introducing a twisted magnetic field breaks the symmetry between the nature of the kink waves with positive and negative azimuthal wave number, and the wave can be a purely Alfvenic wave in the entire loop.