Azimuthal Mode Number Research Articles

Gain in efficiency of jet receptivity to acoustic disturbances with increasing nozzle-lip thickness

The gain in efficiency of the receptivity of jets to acoustic disturbances as the nozzle lip is thicker is investigated using numerical simulations. For that, axisymmetric acoustic pulses are introduced in jets with Blasius laminar boundary-layer profiles at Mach numbers $M=0.4$ , 0.6, 0.9 and 1.3 for nozzle-lip thicknesses between 1 % and 93 % of the nozzle radius. They are located on the jet axis or outside the jet with incidence angles $\varphi$ between $5^{\circ }$ and $90^{\circ }$ with respect to the downstream direction. Instability waves develop in the jet shear layer after the acoustic disturbances hit the nozzle. In all cases except for $\varphi \geq 75^{\circ }$ , their amplitudes and hence the efficiency of the jet receptivity to the disturbances increase with the nozzle-lip thickness. The gains in efficiency are greater for a pulse inside the jet, generating upstream-travelling pressure waves resembling guided jet waves, than for a pulse outside the jet, producing free-stream sound waves. In the second case, the gains are significant for $\varphi =5^{\circ }$ and decrease with the incidence angle, especially for $\varphi >30^{\circ }$ . Moreover, the gains are stronger for a higher Mach number, and roughly double between $M=0.4$ and $1.3$ , thus reaching, for a pulse inside the jet, values close to 6 between the thinnest and thickest lips. Finally, according to additional simulations for $M=0.9$ , the gains in receptivity efficiency do not change appreciably when different azimuthal mode numbers of the acoustic disturbances, widths of the pulse and shapes of the boundary-layer profile are considered.

Parametric oscillations of the sessile drop

Waves are formed on the surface of a sessile drop driven through substrate vibrations oriented at a slanting angle from the normal. A mathematical model is derived, which leads to an infinite system of coupled Mathieu equations governing the wave dynamics that are solved using Floquet theory. The spatial structure of the waves is described by the mode number pair $[\ell,m]$ , where $\ell$ and $m$ are the polar and azimuthal mode numbers, respectively. Limiting cases corresponding to horizontal and vertical vibrations are discussed with predictions agreeing well with prior literature. We focus our results on three drop motions – (1) harmonic $[1,1]$ rocking mode, (2) harmonic $[2,0]$ pumping mode, and (3) subharmonic rocking $[1,1]$ mode – as they depend upon the slanting angle, static contact angle, and contact-line conditions, which we assume to be either pinned or freely moving with fixed contact angle. New theoretical predictions are tested through experiments over a range of parameters, showing good agreement.

Compact high-Q Ka-band sapphire distributed Bragg resonator

In a class of high quality (Q-) factor dielectric resonators with low radiative losses, including popular whispering-gallery mode (WGM) resonators with high azimuthal mode numbers, due to high confinement of modal field in dielectric, the Q-factor is limited by the value of inverse dielectric loss tangent of dielectric material. Metal enclosures necessary for device integration only marginally affect the Q-factor while eliminating the residual radiative loss and allowing the optimization of input and output coupling. While very high Q-factors ∼200000 are available in sapphire WGM resonators in X-band, at millimeter wave frequencies increasing dielectric loss limits the Q-factor to much smaller values, e.g. ∼50000 and ∼25000 for quasi-TE and quasi-TM modes, correspondingly, at 36 GHz. The use of distributed Bragg reflection (DBR) principle allows to push modal energy outside dielectric while also isolating it from Joule losses in metallic enclosure walls. Very high Q∼600000>tgδ has been demonstrated in X-band [C. A. Flory and R. C. Taber, IEEE Trans. Ultrason., Ferroelectr., Freq. Control 44, 486–495 (1997).] at the expense of impractically large dimensions. In this work, we report on the assembly and testing of a compact Ka-band sapphire distributed Bragg reflector cavity characterized with Q-factor seven times larger than one predicted by the material’s dielectric loss at the frequency of interest. An intrinsic Q-factor of ∼200000 is demonstrated at 36 GHz for the lowest order TM-mode of a sapphire DBR. The resonator has 50 cm3 volume, smaller than previously demonstrated DBRs.

Statistical Study on the Azimuthal Mode Number of Pc5 ULF Wave in the Inner Magnetosphere

AbstractThe azimuthal mode number, m, of ultra‐low frequency (ULF) waves is a significant contributing factor for radiation belt electron energization, because it determines the conditions for resonant interaction between waves and particles. Based on multi‐point magnetic field measurements of GOES satellites from January to September of 2011, we statistically analyze the distributions of the characteristics of m of Pc5 ULF waves. In the dayside, the local peaks in the distributions of wave power spectra density locate at ∼10 and ∼13 MLT for m < 0 (westward propagation) and m > 0 (eastward propagation) waves respectively, suggesting the waves generally propagate anti‐sunward. In the nightside, the local peaks are at 22–23 MLT for both m < 0 and m > 0 waves, suggesting possible relation to substorm activities. Further investigation shows that, with increasing solar wind activities, the enhancements of dayside peaks are primarily contributed by |m| ≤ 3 waves, whereas the enhancements of nightside peak are contributed by both |m| ≤ 3 and |m| > 3 waves. With increasing AE index, the enhancements are more significant for the nightside peaks comparing to dayside peaks, and for |m| > 3 waves comparing to |m| ≤ 3 waves. The results of this study provide inputs for further investigation on the radial diffusion coefficient of radiation belt electrons with considering mode number information.

Receptivity of the rotating disk boundary layer to traveling disturbances

An adjoint approach is developed to undertake a receptivity study of the rotating disk boundary layer. The adjoint linearized Navier–Stokes equations are first derived in cylindrical coordinates. A receptivity formula is then formulated that specifies the response of stationary and traveling linear perturbations to an external force, including sources of momenta and mass and unsteady wall motion. Using the parallel flow approximation, in which the radial dependence of the undisturbed flow is ignored, receptivity characteristics are computed for a broad range of temporal frequencies, radial wavenumbers, azimuthal mode numbers, and Reynolds numbers. The type-I crossflow instability attains a maximum amplitude for external forces fixed near the wall-normal location of the critical layer (i.e., α¯rF+βG=ω), and the type-II Coriolis instability achieves larger amplitudes when external forces are located in the vicinity of a vanishing effective shear stress (i.e., α¯rF′+βG′=0). Sources of radial momenta fixed about these wall-normal locations establish larger-sized disturbances than equivalent-sized sources of azimuthal momenta, wall-normal momenta, and mass. At the disk surface, motion along the wall-normal direction induces a stronger receptivity response than wall motions acting along the radial and azimuthal directions. In general, the crossflow instability achieves larger-sized amplitudes than the Coriolis instability, with the peak response realized for Reynolds numbers near the critical conditions for linear instability.

Kerr microresonator dual-comb source with adjustable line-spacing.

Optical microresonators offer a highly-attractive new platform for the generation of optical frequency combs. Recently, several groups have been able to demonstrate the generation of dual-frequency combs in a single microresonator driven by two optical pumps. This opens the possibility for microresonator-based dual-comb systems suitable for measurement applications such as spectroscopy, ranging and imaging. Key to the performance of these systems are the parameters of the radio-frequency comb spectrum that arises from the interference of the two optical combs. In this work, we present a simple mechanism to enable the discrete fine-tuning of these parameters by driving the two optical combs with optical pumps with different azimuthal mode numbers. The mechanism consists of tuning the difference in azimuthal mode number between the two pumps by selection of the pumps' frequencies. We are able to implement this technique when the two counter-propagating pumps are set to drive resonances of the same spatial mode family, as well as different mode families. In each case, we experimentally observe ∼1 MHz of discrete tunability in the line-spacing of the radio-frequency comb as the frequency offset between the two pumps is scanned between 0 to 80 free-spectral-ranges.

Absolute and global instabilities driven by Cherenkov interaction between magnetized electron beam and spoof surface plasmon polaritons

In this study, we examine absolute and global instabilities driven by the Cherenkov interaction between a magnetized electron beam and spoof surface plasmon polaritons (SPPs) with an azimuthal mode number m. The absolute and global instabilities are induced in long and short lengths of the cylindrical corrugated waveguides (CCWs), respectively. The temporal and spatial growth rates have different dominant modes of spoof SPPs that, respectively, affect the absolute and global instabilities. In the experiment, the G-band radiation, which corresponds to the dominant mode in the spatial growth rate, is observed with the short length CCW. In the long CCW, the G-band radiation vanishes because the dominant mode in the temporal growth rate has lower frequency than the G-band cutoff frequency of the detecting system. Our results demonstrate that the instability and the multimode radiation are changed by the length of the CCW.

Transverse vibration modes analysis and acoustic response in optical fibers

Fiber optic sensors are often used as acoustic sensors to detect sound waves because of their apparent advantages, such as anti-electromagnetic interference and strong adaptation to the environment. The transverse vibration mode of the fiber caused by the acoustic wave can be obtained, and the principle of the optical fiber sensor to detect the acoustic wave signal was explored by using a simple model. It is found that the acoustic wave can effectively cause the change in birefringence of the fiber only when the number of azimuthal modes is 2, and the acoustic wave was detected by using a fiber sensor. It is found, by analyzing the detection mechanism, that the spectral width is proportional to the acoustic impedance of the surrounding medium, and the acoustic interaction between the TR22 mode and the surrounding medium is much weaker than that of the TR21 mode. This provides a theoretical basis for the detection of acoustic signals by fiber optic sensors.

Gregory-Laflamme encounters Superradiance

We investigate the effect of superradiant scattering of gravitational perturbations on the stability of rotating black strings, focusing on the six dimensional equal-spinning Myers-Perry black string. We find that rapidly rotating black strings are unstable to gravitational superradiant modes within a bounded range of string lengths. The instability occurs because momentum along the string direction creates a potential barrier that allows for the confinement of superradiant modes. Yet, five dimensional Myers-Perry black holes do not have stable particle orbits so, unlike other known superradiant systems, these black strings remain stable to perturbations with sufficiently high azimuthal mode number — this is a ‘finite-m’ superradiant instability. For some parameters, this instability competes with the Gregory-Laflamme instability, but otherwise exists independently. The onset of this instability is degenerate and branches to multiple steady-state solutions. This paper is the first of a trilogy: in the next two, we construct two distinct families of rotating strings emerging from the superradiant onset (the ‘black resonator strings’ and ‘helical black strings’). We argue that similar physics is present in 5-dimensional Kerr black strings, but not in D > 6 equal-spinning Myers-Perry black strings.

Vortex chaoticons in thermal nonlocal nonlinear media.

This paper numerically investigates the propagation of Laguerre-Gaussian vortex beams launched in nonlocal nonlinear media, such as lead glass. Our results show that the propagation properties depend on the selection of beam parameters m and p, which represent the azimuthal and radial mode numbers. When p=0, these profiles can be stable solitons for m≤2, or break up and then form a set of single-hump profiles for m≥3, which are unbounded states with scattered remnants of the energy. However, for p≥1, the broken beams can evolve into vortex chaoticons, which exhibit both chaotic and solitonlike properties. The chaotic properties are determined by the positive Lyapunov exponents and spatial decoherence, while the solitonlike properties are demonstrated by the invariance of beam width and the interaction of beams in the form of quasielastic collisions. In addition, the power and orbital angular momentum of unbounded beam states both decay in propagation, while those of the chaoticons maintain their values well.

Two-dimensional rogue wave clusters in self-focusing Kerr-media

In this paper we consider two-dimensional (2D) rogue waves that can be obtained from the approximate solution of the (2 + 1)-dimensional nonlinear Schrödinger (NLS) equation with Kerr nonlinearity. The approximate method proposed in this paper not only reduces complex operations needed for the solution of high-dimensional nonlinear partial differential equations, but also furnishes an effective mechanism for constructing stable high-dimensional rogue wave clusters, and also provides for more abundant local rogue wave excitations. Such an investigation is important in view of the known tendency for instability of localized solutions to the 2D and 3D NLS equations with focusing Kerr nonlinearity. We adopt the (2 + 1)-dimensional NLS equation in cylindrical coordinates as the basic model, and reduce the complexity of its solution by looking for the radially-symmetric ring-like solutions with the specific feature that the radius of the ring L is much greater than the ring thickness w. In this manner, we obtain an approximate analytical solution with the separated radial and angular functions which depend on the two integer parameters, the azimuthal mode number m and the rogue wave solution's order number n. With the two parameters L and w also included in the approximate solution, we show the profiles of the first-order and the second-order rogue waves, and discuss their characteristics. The method for finding approximate rogue wave solutions of the (2 + 1)-dimensional NLS equation proposed in this paper can be used as an effective way to obtain higher-order rogue wave excitations, and can be extended to other (2 + 1)-dimensional nonlinear systems.

Formation and breakup of twisting ligaments in a viscous swirling liquid jet

We analyze the successive steps of the breakup morphology of a swirling liquid jet. Three-dimensional numerical simulations are carried out using the Volume of Fluid method with adaptive mesh refinement for axial Reynolds numbers of 50 and swirl numbers of 0.50≤S≤1.55. We present fundamental flow features of the swirling jet in terms of time-averaged axial and azimuthal velocity profiles for the considered range of swirl numbers. The provision of a swirl induces helical disturbance at the interface of the jet, which exhibits an azimuthal mode number of m = 4. We identified that viscous forces are the most dominant force in the flow, which causes the suppression of Kelvin–Helmholtz instability at the interface. In contrast, we found the existence of centrifugal instability, which destabilizes the helical rim developing at the interface. As a result, centrifugally induced corrugations in the form of tiny protrusions develop along each of the helical rims, which triggers Rayleigh–Taylor instability. Subsequently, these tiny protrusions get stretched in the radially outward direction and transform into twisting ligaments that break into droplets. We have elucidated the mechanism for the twisting of ligaments and its further disintegration into first-generation droplets, which has not been reported in previous studies.

Decoherence of Higher Order Orbital Angular Momentum Entangled State in Non-Kolmogorov Turbulence

The decay of OAM entanglement in non-Kolmogorov turbulence has been numerically evaluated. In this work, we explore the evolution of OAM entanglement with higher-order OAM mode in the weak scintillation regime. In particular, the results of the numerical evaluation show that the OAM entanglement state with higher value of the azimuthal mode and larger radial quantum number survives over a longer distance. Meanwhile, the beam parameters and turbulence parameters usually have significant influences on OAM entanglement. In addition, it is demonstrated that the effect of turbulence on the OAM entanglement is the most serious when the generalized exponent is around 3.07.

Investigation of reflection bands of 1D annular photonic crystal containing double negative index material and non-magnetized plasma

In this paper, optical reflectance properties of an annular photonic crystal (APC) composed of alternate layers of double negative (DNG) material and a non-magnetized plasma (NMP) layer, immersed in free space, have been theoretically investigated and studied. The reflectance spectra of the annular PC have been obtained by employing the transfer matrix method (TMM) for the cylindrical waves in the case of TEpolarized wave only. It has been found that the spectral position and width of the reflection bands of APC are greatly influenced by the variation in the thickness of DNG metamaterial and NMP layer, respectively. Interestingly, it is observed that the presence of NMP layer causes the increase in photonic band gap (PBG) whereas the DNG layer reduces the PBG. Further, the effect of azimuthal mode number (m) on the reflectance spectra shows that for m > 0, splitting in the reflection bands occurs at the frequency corresponding to the zero permeability value of DNG metamaterial layer. Moreover, with the increase in azimuthal mode number one PBG is red-shifted and the second one is blue-shifted. Finally, the effect of change in the starting radius parameter of curved surface and plasma electron density on the reflectance spectra of APC has also been studied and very interesting results have been observed.

A Nonlocal Magneto-curvature Instability in a Differentially Rotating Disk

A global mode is shown to be unstable to nonaxisymmetric perturbations in a differentially rotating Keplerian disk containing either vertical or azimuthal magnetic fields. In an unstratified cylindrical disk model, using both global eigenvalue stability analysis and linear global initial-value simulations, it is demonstrated that this instability dominates at strong magnetic fields where local standard magnetorotational instability (MRI) becomes stable. Unlike the standard MRI mode, which is concentrated in the high flow shear region, these distinct global modes (with low azimuthal mode numbers) are extended in the global domain and are Alfvén-continuum-driven unstable modes. As its mode structure and relative dominance over MRI are inherently determined by the global spatial curvature as well as the flow shear in the presence of a magnetic field, we call it the magneto-curvature (magneto-spatial-curvature) instability. Consistent with the linear analysis, as the field strength is increased in the nonlinear simulations, a transition from MRI-driven turbulence to a state dominated by global nonaxisymmetric modes is obtained. This global instability could therefore be a source of nonlinear transport in accretion disks at a higher magnetic field than predicted by local models.

Ion-acoustic instability of the cylindrical inhomogeneous helicon discharge plasma with rotating electrons

The kinetic theory for the microinstabilities of a cylindrical plasma, produced by the cylindrical azimuthally symmetric (azimuthal mode number m0=0) helicon wave, based on the Vlasov–Poisson system of equations, is developed. The derived linear integral equation for the Fourier–Bessel transform of the electrostatic potential is the basic equation for the investigations of the parametric and the current-driven instabilities of the radially inhomogeneous cylindrical plasma in the radially inhomogeneous helicon wave. The short-wavelength solution of this equation for the electrostatic potential is derived in the form of the functional equation, which includes an infinite number of its satellites at a frequency separation equal to the helicon wave frequency. The analytical solution is derived for the high-frequency kinetic ion-acoustic instability of the cylindrical helicon plasma, driven by the coupled effect of the electron diamagnetic drift and of the steady azimuthal rotation of electrons relative to the ions with a radially inhomogeneous angular velocity.

Wave Transformation And Air Entrainment By A Harmonically Forced Plunging Jet

It is known that air may be entrained by a plunging jet even when that jet is relatively small and slow, so long as it has significant disturbances. In the current work, harmonic disturbances of controlled frequency and amplitude are used to produce air entrainment. At sufficient frequency and amplitude, the harmonic jet disturbances cause azimuthal subharmonic waves on the plunge pool surface to appear, oscillating at half of the forcing frequency. These subharmonic waves interact with the jet flow to entrain air at lower disturbance amplitudes and jet velocities than previously documented. The dependence of the wave shape and inception criteria on jet parameters, namely frequency and diameter, is investigated, giving results that agree well with past investigations of azimuthal waves driven by surface-piercing bodies. The number of azimuthal wave lobes may be understood as the ratio of the perimeter length of the forced jet and the wavelength of the resulting azimuthal water waves, given by nwaves = Ckf/2rj, where C is the ratio of the wave and jet diameters. In past experiments, C was approximately 1.2, but with much smaller liquid jets, we found that C can be as high as 1.4-2. The volumetric flow rate of entrained air was measured directly, by way of a capture hood, enabling investigation of air entrainment behavior, and it is concluded that the resultant air flow rate is dependent on several factors, including jet frequency, azimuthal wave mode number, jet velocity, and the jet disturbance amplitude. The influence of several of these factors was determined by the use of an effective bubble size parameter, hypothesizing that every wave trough during every wave cycle was equally likely to entrain air, an assumption that appears borne out in fact. This effective bubble size, free of the influences of forcing frequency and wave mode number, was found to depend on the forcing intensity in an approximately linear fashion, at least for forcing amplitudes close to those required for entrainment inception. Jet and wave measurements were conducted using high speed videography, enabling quantitative measurements of the nearjet wave amplitude, as well as the size and spectral nature of the jet disturbances.

Double cylinder target design for study of hydrodynamic instabilities in multi-shell ICF

Cylindrical implosions are used to study hydrodynamic instability growth for inertial confinement fusion (ICF) applications, as the cylindrical geometry allows for easier diagnostic access while retaining convergence effects. In this work, we use the established cylindrical implosion platform [Palaniyappan et al., Phys. Plasmas 27, 042708 (2020)] to inform the double shell ICF campaign [Montgomery et al., Phys. Plasmas 25, 092706 (2018)]. We present a design for a double cylindrical target as an analogue to the double shell ICF capsule in order to study hydrodynamic instability growth on the high-Z inner shell. Our design work is done with two-dimensional (2D) Eulerian radiation-hydrodynamics simulations, considering the axial uniformity of the implosion and feasibility of measuring the instability growth of pre-seeded single mode sinusoidal perturbations. We discuss in depth the design for a target to be directly driven at the OMEGA laser facility [Boehly et al., Opt. Commun. 133, 495 (1997)]. We evaluate the design for axial implosion symmetry and visibility of instability growth using synthetic radiographs constructed from the simulations, as the instability growth on the inner cylinder is experimentally measured using x-ray radiography of the implosion. We find that the seeded perturbation growth on the inner cylinder should be visible in an experiment, even with axial implosion asymmetry and preheat. We compare our 2D simulations with linear theory predictions for perturbation growth and show that a cylinder with lower azimuthal mode number (mode-20) perturbations compares more favorably with linear theory, while a cylinder with higher azimuthal mode number (mode-40) perturbations at the same starting amplitude saturates and is over-predicted by linear theory.

Simultaneous PIV and shadowgraph measurements of thermo-electrohydrodynamic convection in a differentially heated annulus

Abstract Convection in a silicone oil with a temperature dependant fluid property is investigated experimentally in a differentially heated cylindrical annulus. A convective flow is induced by terrestrial gravity in axial direction and combined with an electrical tension applied between both cylinders to induce thermo-electrohydrodynamic convection. To capture the evolving velocity and temperature fields a novel combination of simultaneous PIV and shadowgraph technique is utilized. Results reveal azimuthal modes with cold and hot jets from inner and outer cylinder that are referred to the electric tension. The lack of information in the shadowgraph pictures were recovered by the PIV technique providing a deeper understanding in the nature of the development of the azimuthal mode number and the surrounding axial flow of natural convection. In addition, the results provide a robust framework in the capability of combining both techniques to investigate complex flow patterns that are non-axisymmetric.

Thermoacoustics of multi-burner combustors with plenum and chamber cross-talk

This paper presents a thermoacoustic model in the frequency domain for multi-burner combustors composed of multiple plenum-nozzle-chamber modules with cross-talk at both plenums and chambers. The thermoacoustic model is given as a multi-input-multi-output transfer function matrix from heat fluctuation input vector to velocity fluctuation output vector. The discrete rotational symmetry of a multi-burner combustor is exploited to obtain a simple, diagonal transfer function matrix. The resonance equation, whose roots are resonances, is given as a set of scalar equations. The explicit form of the resonance equation reveals how combustor and operating parameters influence the resonances, and which parameters are significant for the longitudinal-azimuthal coupling and the plenum-chamber coupling. The diagonalization also allows us to show constructively that every mode shape is a Bloch wave and those mode shapes can characterize all resonances in terms of longitudinal and azimuthal mode numbers. At a low frequency, the resonance equation can be approximated as biquadratic and quadratic equations for a typical combustor geometry and operating conditions. A bunch of low-frequency resonances, peculiar to multi-burner combustors with cross-talk, are interpreted as azimuthal Helmholtz modes in which plenums/chambers and cross-talk ducts serve as the cavities and necks, respectively. Our combustor model is combined with a diagonal flame transfer function matrix for combustion instability analysis and we develop a set of scalar stability equations whose roots determine combustion instability. This stability equation reveals how combustion instability depends on the combustor geometry and operating parameters. Several numerical examples are presented to validate our thermoacoustic model and theoretical findings.