Azimuthal Mode Order Research Articles

A simple method improving acoustic mode identification capability based on genetic algorithms

This letter develops a simple approach of duct mode identification and reconstruction based on genetic algorithms, which can extend the azimuthal mode order range compared to the conventional method based on the (spatial) discrete Fourier transform. The underlying principle is reconstructing the dominant mode from the modal identification forward model through optimization by exploiting the sparsity of the mode amplitude vector. The performance is experimentally demonstrated for detections of one and two azimuthal modes under noisy conditions with nondominant modes. Overall, the proposed genetic-algorithm-based framework for solving acoustic inverse problems is beneficial to duct acoustic testing, particularly design evaluations of fan blades and acoustic liners for aeroengines.

Bridging the gap between annular and can-annular acoustic spectra.

In the literature on thermoacoustic instabilities in combustors, a distinction is typically made between annular and can-annular systems because these are the most common gas turbine architectures. In reality, however, annular combustors typically feature discretely symmetric elements, such as burner tubes, and can-annular combustors feature an azimuthally symmetric plenum at the turbine inlet. To better understand the general case in between the annular and can-annular extremes, we analyze the acoustic spectrum of an idealized can-annular combustion chamber with variable geometry, where the length of the axial gap distance beyond the ends of the cans-hence, the coupling strength-may be adjusted. For small gap sizes, the geometry approaches a set of isolated combustor cans, whereas for large gap sizes, it approaches that of pure annular systems. We present two theoretical models based on Bloch wave theory and validate them against finite-element simulations of the Helmholtz equation. We demonstrate that the azimuthal modes transform into the eigenmodes of an annular chamber as the gap is fully opened, and we show that below a certain frequency, all the modes in between the can-annular and annular limits can be classified by their axial and azimuthal mode orders.

Nonlinear interactions between the fundamental and its harmonics of self-excited azimuthal thermoacoustic instabilities

Thermoacoustic instabilities in combustors are inherently nonlinear processes, manifested by multiple peaks in the spectrum. Analyzing these signals is crucial to understand the mechanism of nonlinear limit cycle oscillations. The goal of this study is to investigate the acoustic modal dynamic relation between the fundamental of azimuthal instabilities and its first harmonic. The paper first demonstrates that the nonlinear interaction of the fundamental azimuthal mode produces not only higher order azimuthal modes but also an axial mode in the harmonic. This is confirmed by the experimental data showing a correlation between the axial mode in the harmonic and two counter-rotating modes in the fundamental. Furthermore, the study shows that neglecting this axial mode may result in inaccurate estimation of the maximum pressure in the chamber, potentially leading to safety concern. The combustor is operated under various conditions to examine the correlation between the fundamental and its harmonic in terms of dominant spinning direction, orientation of the acoustic structure, and the maximum pressure magnitude. It is found that the azimuthal components of the fundamental and its harmonic rotate in the same direction under a wide range of operating conditions. Additionally, the maximum pressure magnitudes are positively correlated with each other despite the difference in the order of magnitude between them. Specifically, the harmonic's magnitude shows quadratic dependence on the fundamental's, even when the mode transitions from a standing wave to a spinning wave. However, the anti-node locations of the fundamental and its harmonic exhibit no correlation, and their difference in location varies with operating conditions.

Tip flow on rotating instability on an axial compressor with different tip clearances

Rotating instability (RI) as an unsteady phenomenon usually occurs at off-design conditions in compressors, featuring a broadband hump in the pressure spectra below the blade passing frequency (BPF). RI phenomenon could not only cause deterioration to the performance, but also lead to tip clearance noise or non-synchronous vibration to the compressor. In this paper a detailed experiment was carried out in a low-pressure single-rotor compressor. By replacing the casing the relative tip clearance size was changed and the time-resolved wall pressure with a distinct RI signature was thus measured under several different conditions through the Kulite pressure transducers both circumferentially and chordwise installed. The temporal-spatial distribution of RI signature evolution with the flow coefficient is interrogated, and its propagation characteristics are also investigated through the cross-correlation method. Moreover, the quasi-instantaneous contour measured by the chordwise transducers shows no distinct flow structure which could be connected to the RI signature, i.e., the prestall disturbance. Therefore, a quasi-SFT method is developed to identify different disturbance counts and the corresponding relationship between the disturbance count per revolution and the azimuthal mode order could thus be strictly established. The prestall disturbance is found to originate from the mid-chord positions as RI initiates and travel upstream as the reduction of flow coefficient, and eventually reaches the leading edge as rotating stall occurs. Furthermore, the location of prestall disturbances is proven to be independent on the blade position. These findings describe the tip flow field when the RI mode orders are relatively high, and provide new insights for the shear layer instability theory as RI mechanism.

Laguerre-Gaussian Expansion of Step-Index Multimode Fiber Beams in Free-Space

We show that the complex conjugate of the overlap integrals between the fiber end-facet field and the Laguerre-Gaussian modes provides the expansion coefficients for the step-index multimode fiber beams. We also show that the fractional power in different azimuthal mode orders remains unchanged from fiber to free-space propagation.

Arbitrary higher-order optical spatial state generation by using spontaneously broken degeneracy modes in helically twisted ring-core hole assisted fibers.

A helically twisted ring-core hole assisted fiber (HAF) is proposed for an arbitrary higher-order spatial state generation. In usual circular core fibers, HE and EH modes are degenerate and have vector field distributions. We will show the degeneracy of these modes having specific azimuthal mode order is lifted by periodically placing air-holes around the core with specific symmetry while preserving their vector field distributions. It is called spontaneously broken degeneracy (SBD) in this paper. Azimuthal order of the SBD modes can be changed with the arrangement of surrounding air-holes. By using this SBD modes, it is shown that arbitrary higher-order spatial state generation, including orbital angular momentum (OAM) state, is possible based on a geometric phase existing in twisted fibers. Furthermore, by using periodically inverted twisting, efficient OAM state generation is possible by accumulating the geometric phase. The topological charge can be changed by changing the arrangement of surrounding air-holes. Since the mechanism does not depend on a phase matching, such as a long-period grating, but on a topological effect, the wavelength dependence is very small, leading to novel and efficient mechanism for the manipulation of the spatial state of the light.

Characteristics of Space, Leaky, and Creeping Waves in a Uniaxial Dielectric Rod

The excitation of space wave, leaky wave, and creeping wave in a cylindrical rod with uniaxial anisotropic permittivity and permeability is thoroughly characterized. The closed-form expressions of the space, leaky, and creeping waves in the uniaxial rod are derived for all source polarizations through the rigorous formulation, taking into account all higher order azimuthal harmonics and TE-TM mode coupling for the space and creeping waves. Conditions for obtaining a high space- and leaky-wave directivity are thoroughly examined for all source polarizations with variation in the vertical and horizontal permittivities/permeabilities and the rod radius. The leaky-wave phase and attenuation constants with variation in the rod parameters are investigated to identify the leaky-wave propagation regions of the uniaxial rod. Beam-scanning characteristics for the space and leaky waves are examined with variation in the uniaxial rod parameters. The complex pole loci for the azimuthally propagating creeping waves around the uniaxial rod are also investigated to identify the leaky, the creeping, and the trapped-mode regions.

Swirling mean flow effects on locally reacting interstage liner

Sound transmission through a finite-lined section in a rigid annular duct with swirling and sheared mean flow is analyzed with a new mode-matching method based on the conservation of the total enthalpy and the mass flow, which does not reduce to the conservation of the pressure and the axial velocity when the swirl is non-zero. It relies on a new projection method based on the property of the Chebyshev polynomials and on the scattering matrix formalism to yield transmission losses. This new method is first validated against a finite elements method tool in the uniform axial flow case, and then provides a parametric study of the effect of swirl. At low azimuthal mode order [Formula: see text], the swirl amplifies the attenuation of the contra-rotating modes and makes the attenuation of the co-rotating modes decrease with a trend of a general shift of the transmission loss curve toward contra-rotating modes. A small rotation of the transmission loss curves at low [Formula: see text] is also generally observed. The boundary condition in the lined section has a small effect on the transmission loss, except close to the cut-on thresholds. Finally, the duct boundary-layer thickness has a significant effect on the cut-on modes and the transmission loss but not its profile.

Experimental study on self-excited thermoacoustic instabilities and intermittent switching of azimuthal and longitudinal modes in an annular combustor

For annular combustors, the interactions between different modal structures (longitudinal, azimuthal spinning, standing, and slanted) remain a challenge in ongoing studies of thermoacoustic instabilities. The present work develops an annular combustor named TurboCombo, similar to the multiple-injector annular combustor named MICCA, but with a convergent turbine section connected to the combustor outlet. A Helmholtz solver provides a proper estimate of the modal frequencies and reveals the spatial structures for both two rigs. The experimental results present the consistencies of frequency domains between the acoustic pressure signals and the hydroxyl (OH*) radicals emission signal. The high-speed flame images are processed by the dynamic mode decomposition method, statistically showing longitudinal, azimuthal spinning, or standing nature in different cases. The stability maps of the two combustors show slight differences under atmospheric conditions. In the TurboCombo rig, there are two types of interaction behaviors between the 1/4 wave longitudinal mode of the chamber and the 1st order azimuthal mode of the plenum. Unlike the prevalent “dual-mode coexistence” characteristics, an interesting behavior of “intermittent switching” is revealed, which manifests as conspicuous two-way jumps between the two modes with different frequencies. Such intermittent switching occurs randomly between two stable limit cycles in a long-time observation without any external excitation, but is driven by the turbulence-induced stochastic process. Moreover, there still exist some weak azimuthal oscillations while the longitudinal mode is dominated. The standing or spinning bias of the azimuthal mode also features intermittent transition, corresponding to the occasions dominated by longitudinal mode or azimuthal mode, respectively.

Azimuthal Mode Characteristics of Rotating Instability in Axial Compressor Using Compressed Sensing Method

Abstract A signal reconstruction algorithm based on the compressed sensing (CS) theory with dual-uniform sampling point (DUSP) distribution is developed and applied to identify the azimuthal mode of axial compressor. A regular failure signal pattern is found, and the corresponding explanation is presented with validation. Azimuthal mode analysis is applied to both numerical and experimental pressure fluctuation signals of rotating instability (RI) in the axial compressor tip region. For numerical calculations, the signal in the azimuthal mode domain is reconstructed by the CS with random measurement points and DUSP, respectively. The success rates and reconstruction errors are discussed in detail. It is shown that the azimuthal mode reconstruction method based on CS combined with DUSP is capable of identifying the complex flow modes in the tip region of the axial compressor. For the experimental results, high azimuthal mode orders are reconstructed based on dynamic pressure signals measured by DUSP. Azimuthal mode analysis efficiency is thereby significantly improved. The time-resolved characteristics of the RI are discussed. Moreover, a robustness analysis is conducted, demonstrating the ability of the CS-based method with DUSP to address sensor failure problems.

Low-Order Modeling to Investigate Clusters of Intrinsic Thermoacoustic Modes in Annular Combustors

Abstract The intrinsic thermoacoustic (ITA) feedbackloop constitutes a coupling between flow, flame and acoustics that does not involve the natural acoustic modes of the system. One recent study showed that ITA modes in annular combustors come in significant number and with the peculiar behavior of clusters, i.e. several modes with close frequencies. In the present work, an analytical model of a typical annular combustor is derived via Riemann invariants and Bloch theory. The resulting formulation describes the full annular system as a longitudinal combustor with an outlet reflection coefficient that depends on frequency and the azimuthal mode order. The model explains the underlying mechanism of the clustering phenomena and the structure of the clusters associated with ITA modes of different azimuthal orders. In addition, a phasor analysis is proposed, which encloses the conditions for which the 1D model remains valid when describing the thermoacoustic behavior of an annular combustor.

Experimental and analytical investigation of contra-rotating multi-rotor UAV propeller noise

Contra-rotating propellers could increase the efficiency and lifting capacity of multi-rotor unmanned aerial vehicles (UAVs); however, contra-rotating propellers can produce significant noise levels. This work experimentally investigated the noise from static contra-rotating UAV propellers in an anechoic chamber. The effects of propeller diameter were studied by testing different configurations of 12″ and 15″ propellers. The effects of propeller spacing, rotational speed and blade number were also investigated; around 1400 propeller configurations were tested in total. It was observed that interaction tones were a dominant contributor to the overall noise level, which also contained rotor-alone tones and broadband noise. A theoretical model for the far-field interaction tone noise produced by a contra-rotating propeller is presented and used to explain the tonal noise levels observed in the experimental results. It was found that the tones with a zero azimuthal mode order dominate the noise spectrum close to the propeller axis. It was shown that the amplitude and number of these tones with zero azimuthal mode order could be reduced by using contra-rotating propellers with mismatched blade numbers. It was also experimentally shown that increased spacing between the propellers generally reduced the amplitude of the interaction tones. The rate of this reduction in amplitude with spacing between the propellers was dependent on frequency and spacing. High rates of decay were generally observed at small spacings and lower rates at larger spacings. However, the decay was neither monotonic nor at a uniform rate for all frequencies, which suggests that there could be a change in the noise generation mechanism as the spacing increases. Finally, the validity of the indoor static results was demonstrated by comparing the results to an in-flight test of a quadcopter with contra-rotating propellers.

Influence of Resonant Absorption on the Generation of the Kelvin-Helmholtz Instability

The inhomogeneous solar corona is continuously disturbed by transverse MHD waves. In the inhomogeneous environment of coronal flux tubes, these waves are subject to resonant absorption, a physical mechanism of mode conversion in which the wave energy is transferred to the transition boundary layers at the edge between these flux tubes and the ambient corona. Recently, transverse MHD waves have also been shown to trigger the Kelvin-Helmholtz instability (KHI) due to the velocity shear flows across the boundary layer. Also, continuous driving of kink modes in loops has been shown to lead to fully turbulent loops. It has been speculated that resonant absorption fuels the instability by amplifying the shear flows. In this work, we show that this is indeed the case by performing simulations of impulsively triggered transverse MHD waves in loops with and without an initially present boundary layer, and with and without enhanced viscosity that prevents the onset of KHI. In the absence of the boundary layer, the first unstable modes have high azimuthal wavenumber. A boundary layer is generated relatively late due to the mixing process of KHI vortices, which allows the late onset of resonant absorption. As the resonance grows, lower azimuthal wavenumbers become unstable, in what appears as an inverse energy cascade. Regardless of the thickness of the initial boundary layer, the velocity shear from the resonance also triggers higher order azimuthal unstable modes radially inwards inside the loop and a self-inducing process of KHI vortices occurs gradually deeper at a steady rate until basically all the loop is covered by small-scale vortices. We can therefore make the generalisation that all loops with transverse MHD waves become fully turbulent and that resonant absorption plays a key role in energising and spreading the transverse wave-induced KHI rolls all over the loop.

Acoustic theory of the many-bladed contra-rotating propeller: analysis of the effects of blade sweep on wake interaction noise

An analytical model is presented for the wake interaction tones produced by a contra-rotating propeller. We re-cast the usual far-field radiation formulae as a double integral over a nominal propeller source annulus. Assuming that the number of blades on both propellers is large, we evaluate the integral asymptotically in terms of its leading-order contributions from interior stationary or boundary critical points which represent the specific locations on the propeller annulus that dominate the sound radiation. The asymptotic approach is powerful producing results in the form of one-line algebraic formulae that contain no integrals or special functions yet remain accurate. The asymptotics show that sweep is not necessarily beneficial and can cause the blade design to become critical for particular tones and directions in terms of a continuum of interior points distributed along a line on the propeller source annulus producing a higher-order result and thus an enhanced radiated sound field. The paper also shows how the interior points are completely consistent with the sub- or super-critical gust response of a swept blade. Tones with low and zero azimuthal mode order are treated as special cases and the asymptotics show that, as the mode order reduces, the radiated sound becomes concentrated around the flight axis where even higher-order solutions are possible, including rings and annuli of stationary points around the propeller annulus. Full numerical calculations confirm the accuracy of the asymptotic approach.

Optimum Impedance in the Presence of an Inviscid Sheared Flow

In recent years, much effort has been devoted to find the “optimum impedance” (i.e., the impedance that results in the maximum modal decay rate in flow duct acoustics for a given frequency, Mach number, and azimuthal mode order). Although such analysis can be carried out by means of numerical simulations, analytical expressions can also be derived to predict the optimum impedance. Previous works have been concerned with the optimum impedance of higher-order modes in rectangular ducts with uniform flow. In this work, the analysis is expanded to circular ducts for both uniform and sheared inviscid flows. Focus is given to typical operating conditions found in turbofan engine intakes and vehicle exhaust systems. It is shown that, in certain conditions, the optimum impedance is affected even by the presence of a small boundary-layer thickness. It is also noted that, for low Helmholtz numbers, the optimum impedance may have a negative resistance.

In situ identification strategy of thermoacoustic stability in annular combustors

In annular combustion systems, thermoacoustic eigenmodes can manifest as standing waves, traveling waves or some form in between. Which dynamic solution appears in a combustor depends on details, regarding the flow field and (unintentional) breaking of the cylindrical symmetry of the annular combustion system. When these details are unknown, the specific behavior cannot be predicted from the characteristics of a single burner. Due to the (nearly) degenerate nature of the acoustic solution, annular eigenmodes come in pairs with practically the same eigenfrequency. In order to identify the thermoacoustic modes, conventional analysis of a spectral peak from a measurement does not suffice, because the peak is a superposition of the two eigenmodes. A method has been proposed to identify the two eigenmodes of given azimuthal mode order from multiple simultaneous measurements around the circumference of the combustion system. Using output-only identification on the acoustic signals, it is possible to estimate the individual mode shapes, frequencies and growth rates of the co-existing eigenmode pair. In this work, the strategy is applied to experimental data from an annular combustor. A split in the growth rate pair is observed during stable operation, depending on the equivalence ratio and flame-to-flame distance. It shows that in situ identification of annular thermoacoustics can reveal subtle dynamic effects, which is useful for testing and online monitoring of annular combustors. The moment when instability occurs can be foreseen under prevailing conditions, with simultaneous identification of the azimuthal mode structure.

Measurement and Visualization System of Eigenmode Shapes Considering Orthogonal Modes on Circular Membrane

This work proposes a visualization system of eigenmode shapes of a tapped circular membrane of membranophones. Under tapped membrane vibration, it is not enough to divide modes in frequency domain, because of the existence of eignmodes whose frequencies are close to each other including orthogonal modes. Therefore, we structured this visualization system of eigenmode shapes, which could divide the orthogonal modes appearing at slightly different frequencies. This system firstly measures the sound pressures emitted from the vibration on tapped circular membrane of membranophone with a circular microphone array placed in proximity to the membrane. Secondly, it extracts the circumferential shape of each eigenfrequency. In this system, dominant frequencies of estimated power spectrum density obtained by Yule-Walker method are considered as approximate eigenfrequences of this vibration. Multiple eigenmodes whose azimuthal mode order is different, are divided by Fourier series expansion of the circumferential shapes of each eigenfrequency. In addition, orthogonal modes, which have the same mode order and slightly different frequencies are divided by considering the orthogonality of mode shapes, sine and cosine functions. Lastly, the system estimates the eigenmode shape of the whole membrane using the Bessel function of the first kind correspond to each mode order as radial shape, and shows them to users. It is expected that user could recognize the unbalance of tension distribution intuitively using this system. It is also expected the system to be applied to the tuning assistive system of the tuning of circular membranophone.

Active chiral control of GHz acoustic whispering-gallery modes

We selectively generate chiral surface-acoustic whispering-gallery modes in the gigahertz range on a microscopic disk by means of an ultrafast time-domain technique incorporating a spatial light modulator. Active chiral control is achieved by making use of an optical pump spatial profile in the form of a semicircular arc, positioned on the sample to break the symmetry of clockwise- and counterclockwise-propagating modes. Spatiotemporal Fourier transforms of the interferometrically monitored two-dimensional acoustic fields measured to micron resolution allow individual chiral modes and their azimuthal mode order, both positive and negative, to be distinguished. In particular, for modes with 15-fold rotational symmetry, we demonstrate ultrafast chiral control of surface acoustic waves in a micro-acoustic system with picosecond temporal resolution. Applications include nondestructive testing and surface acoustic wave devices.

Higher-order modes of vacuum-clad ultrathin optical fibers

We present a systematic treatment of higher-order modes of vacuum-clad ultrathin optical fibers. We show that, for a given fiber, the higher-order modes have larger penetration lengths, larger effective mode radii, and larger fractional powers outside the fiber than the fundamental mode. We calculate, both analytically and numerically, the Poynting vector, propagating power, energy, angular momentum, and helicity (or chirality) of the guided light. The axial and azimuthal components of the Poynting vector can be negative with respect to the direction of propagation and the direction of phase circulation, respectively, depending on the position, the mode type, and the fiber parameters. The orbital and spin parts of the Poynting vector may also have opposite signs in some regions of space. We show that the angular momentum per photon decreases with increasing fiber radius and increases with increasing azimuthal mode order. The orbital part of angular momentum of guided light depends not only on the phase gradient but also on the field polarization, and is positive with respect to the direction of the phase circulation axis. Meanwhile, depending on the mode type, the spin and surface parts of angular momentum and the helicity of the field can be negative with respect to the direction of the phase circulation axis.

Experimental determination of the azimuthal and radial mode orders of a partially coherent LGpl beam (Invited Paper)

It is known that one can determine the mode orders (i.e., the azimuthal order and radial order) of a partially coherent LGpl beam (i.e., a partially coherent vortex beam) based on the measurement of the cross-correlation function (CCF) and the double correlation function (DCF) together. The technique for measuring the CCF is known. In this Letter, we propose a method for measuring the DCF. Based on the proposed method, the determination of the mode orders of a partially coherent LGpl beam is demonstrated experimentally.