Axisymmetric Modes Research Articles

Jet Installation Noise Modeling in Static and Flight Conditions Using Centerline Fluctuations

The present work is dedicated to the modeling of the low-frequency part of the jet installation noise (JIN) in flight conditions. It is known that the properties of JIN can be predicted based on the characteristics of the near field of an isolated jet. Unlike the static case, in the presence of a coflow, it is difficult to directly measure the structure of the pressure perturbations in the jet near field. To overcome this problem, we propose a technique based on hot-wire measurements on the jet axis, suitable both for static and flight conditions. The well-known agreement between a parabolized stability equations (PSEs) model and experimentally measured velocity fluctuations on the jet axis allows using the PSE approach for the reconstruction of the axisymmetric pressure fluctuations in the vicinity of the wing trailing-edge location. The first helical mode, which is also important for the jet installation noise prediction, is approximately reconstructed based on the fact that its properties are close to those of the axisymmetric mode. These pressure characteristics are then used as input in an analytical jet installation noise model. To confirm this approach, acoustic measurements of JIN in static and flight conditions are conducted for a laboratory subsonic jet installed near a flat plate simulating a wing. It is shown that the analytical model informed by the PSE-reconstructed pressure field is capable of capturing the main features of the low-frequency jet–plate interaction noise both in static conditions and in the presence of coflow.

Large Amplitude Axisymmetric Free Vibration of a Closed Two-Dimensional Spherical Pressure Vessel with a Constrained Volume

The nonlinear axisymmetric vibrational modes of a closed two-dimensional spherical pressure vessel with a constrained volume are presented using the fundamental forms of surfaces. The strain energy of the spherical pressure vessel is derived as a quadratic function of the initial Eulerian, added, and total Lagrangian strains. Two-dimensional bilinear elements described in spherical polar coordinates with the symmetric boundary condition along both sides of the meridian line are used for investigating the nonlinear axisymmetric mode shapes of a spherical pressure vessel. Numerical results were obtained by a nonlinear finite element approach and verified in the case of an empty spherical shell for both lower and upper branches. This study showed that the spherical pressure vessel gave a higher nonlinear axisymmetric natural frequency than the empty spherical shell. The elastic modulus has a large effect on the nonlinear axisymmetric frequencies of both the lower and upper branches for the spherical pressure vessel. Changing the thickness and initial internal pressure significantly affects only the lower branch of the spherical pressure vessel at high mode numbers. The results also indicate that the ratios of the nonlinear natural frequencies of the lower to upper branches decrease when the vibrational modes increase.

Closure mechanism of the A1 and A2 modes in jet screech

This paper explores the screech closure mechanism for different axisymmetric modes in shock-containing jets. While many of the discontinuities in tonal frequency exhibited by screeching jets can be associated with a change in the azimuthal mode, there has to date been no satisfactory explanation for the existence of multiple axisymmetric modes at different frequencies. This paper provides just such an explanation. As shown in previous works, specific wavenumbers arise from the interaction of waves in the flow with the shocks. This provides new paths for driving upstream-travelling waves that can potentially close the resonance loop. Predictions using locally parallel and spatially periodic linear stability analyses and the wavenumber spectrum of the shock-cell structure suggest that the A1 mode resonance is closed by a wave generated when the Kelvin–Helmholtz mode interacts with the leading wavenumber of the shock-cell structure. The A2 mode is closed by a wave that arises owing to the interaction between the Kelvin–Helmholtz wave and a secondary wavenumber peak, which arises from the spatial variation of the shock-cell wavelength. The predictions are shown to closely match experimental data, and possible justifications for the dominance of each mode are provided based on the growth rates of the absolute instability.

Dynamic identification of pretensile forces in a spider orb-web

In this paper we propose a constructive procedure to determine the radial pretension in an axially-symmetric orb-web, on the basis of one eigenfrequency and its corresponding vibration mode. The method can be applied both to axisymmetric modes, corresponding to the solution of a regular Sturm–Liouville problem, and to non-axisymmetric modes, corresponding to the solution of a singular Sturm–Liouville problem. In the absence of measurement errors of the eigenfunction, the identification of the pretension field is almost perfect. On the contrary, if the measurement presents some noise, then identification is imprecise. In this case, a regularization technique applied to the measurement of the eigenfunction, or to its derivative, is proposed, as well as a filtering process, which significantly improves the reconstruction of the pretension field. The fundamental mode of vibration, sampled with a suitable number of points, is the one that provides the lowest identification error.

Numerical simulation on the formation of a helical mode in the initial region of a round jets using synthetic jets

Axisymmetric mode (varicose mode) and asymmetric modes (helical modes) are involved in a round jet. Although there are many studies on the symmetric modes related to the formation of vortex rings, it seems to be a lack of studies on the helical mode. It is necessary to investigate the effect of the helical modes on the spatial development process of a jet. It has been found in previous research that helical vortex structures are formed from flow visualization in the initial region of a round jet by driving loudspeakers in sequence with phase differences. In this study, the conditions for the formation of helical modes by driving synthetic jets in sequence with phase differences are found by numerical simulations using OpenFOAM.

Effect of geometric irregularities induced during manufacturing of a crash-box on its crashworthiness performance

Imperfections associated with the geometric features of any component are inevitable during its fabrication and some of those can even have a drastic impact on its functionalities. One such case arises while fabricating one of the essential passive safety devices in an automobile called as 'thin-walled collapsible energy absorbers'. Thin-walled collapsible energy absorbers also known as crash-boxes are the tubular shell structures located in between the bumper and front side rails of an automobile. Crash-boxes collapse in a stable progressive manner to absorb kinetic energy of an external impact to protect the occupants during an accident. Tubular crash-boxes are generally manufactured by extrusion. Geometric imperfections in the form of irregularities in the shell thickness may be induced during extrusion of the crash-boxes. The present study aims to gauge the effects of irregularities present in the thickness of the extruded crash-box on it’s crashworthiness performance, numerically. Cylindrical aluminium alloy tubes are modelled as crash-boxes using ABAQUS/Explicit. The irregularities in the thickness of the tubular crash-box are modelled with the help of three different imperfect models. In every imperfect model, maximum variation in the thickness is kept within the tolerance value, provided by the standards DIN EN-755-9. The quasi-static axial collapse of these ‘imperfect’ cylindrical tubes is simulated and their crashworthiness performance indicators are compared with those of an ‘ideal’ circular crash-box. It is seen that the energy absorption and the mean crushing force reduced drastically due to these shell thickness irregularities in comparison to the ideal model. It is also observed that the mode of collapse of the circular tube changed from axi-symmetric concertina mode to the diamond mode due to irregularities in their thickness. Such significant change in the crashworthiness performance of crash-box may lead to a huge disaster in terms of human lives. Hence, crash-boxes should be fabricated carefully with more tight geometric and dimensional tolerances.

Ultrasonic transmission through small tubes such as rat tibias for axial Young’s modulus estimation: Discussion and recommendations

In the framework of in vitro osteoporosis investigation with ultrasound on rat model, we propose a study of axisymmetric modes guided in small cylindrical structures such as rat tibias. The main objective is to analyze how an ultrasonic pulse (with a frequency close to the MHz) transmitted through the cortical bone diaphysis can be used to obtain a rapid estimation of the axial Young's modulus. Thanks to a numerical approach and experimentations on bone phantoms we show that the velocity of the first arrival signal (VFAS) measured with a simple time of flight method corresponds to the maximum of the group velocity of the axisymmetric L0,2 mode. Considering the geometry of the cylinders analyzed, we show that it is possible to use this velocity to estimate the axial Young's modulus. The ratio (internal radius / thickness) is an important parameter for small tubes such as rat tibias and has to be considered for results analysis. In this framework, µCT measurements are of first interest. At the end of this communication, preliminary experiments on rat tibias are presented and discussed.

Prograde and retrograde precession of a fluid-filled cylinder

We numerically study precession driven flows in a cylindrical container whose nutation angle varies between 60 and 90 degrees for prograde and retrograde precession. For prograde precession we observe sharp transitions between a laminar and a turbulent flow state with low and high geostrophic axisymmetric flow components related with a centrifugal instability, while for retrograde precession a rather smooth transition between a low state and a high state occurs. At the same time prograde and perpendicular precession shows an abrupt breakdown of the flow directly excited by the forcing mechanism, which is not the case for retrograde motion. We characterize the corresponding flow states in terms of the directly driven, non-axisymmetric Kelvin mode, the axisymmetric geostrophic mode, and an axisymmetric poloidal flow which is promising for precession-driven dynamo action. The latter issue is discussed with particular view on an optimal parameter choice for the DRESDYN dynamo project.

Tones in the Acoustic Far Field of Jets in the Upstream Direction

The presence of tones in the acoustic far field of isothermal round jets for angles relative to the flow direction is investigated for Mach numbers between and 2 based on data from both experiments and large-eddy simulations. For all Mach numbers, as observed in the jet near-nozzle region in previous studies, peaks are found in the far-field pressure spectra regardless of the nozzle-exit boundary-layer properties. They emerge near the cutoff frequencies of the free-stream upstream-propagating guided jet modes predicted by a vortex sheet model, sharply for but more weakly for lower Mach numbers. The variations of their characteristics with the radiation angle are shown. As the angle increases, the peaks are more prominent, and the most apparent ones are related to lower azimuthal modes. Thus, peaks associated with the axisymmetric mode strongly predominate in the upstream direction.

Acoustic microstreaming produced by two interacting gas bubbles undergoing axisymmetric shape oscillations

An analytical theory is developed that describes acoustic microstreaming produced by two interacting bubbles. The bubbles are assumed to undergo axisymmetric oscillation modes, which can include radial oscillations, translation and shape modes. Analytical solutions are derived in terms of complex amplitudes of oscillation modes, which means that the modal amplitudes are assumed to be known and serve as input data when the velocity field of acoustic microstreaming is calculated. No restrictions are imposed on the ratio of the bubble radii to the viscous penetration depth and the distance between the bubbles. The interaction between the bubbles is considered both when the linear velocity field is calculated and when the second-order velocity field of acoustic microstreaming is calculated. Capabilities of the analytical theory are illustrated by computational examples.

Instability of coaxial viscoelastic jets under a radial electric field

In this paper we investigate temporal linear instability and absolute/convective instability of coaxial viscoelastic jets in the presence of a radial electric field. With a leaky dielectric model to describe the finite conductivity of the outer liquid, both axisymmetric instability and nonaxisymmetric instability are studied and compared to determine the predominant mode in the jet instability. For relatively small wavenumbers, the electric field suppresses the axisymmetric instability, while it enhances axisymmetric instability for relatively large wavenumbers. As the electric field increases, the instability mode tends to be a short wave. The elasticity of the inner and outer liquid is found to contribute to the coaxial jet instability, and the increase of the inner-liquid elasticity is discovered to allow the para-sinuous mode to predominate over the helical mode. Furthermore, the spatiotemporal instability analysis indicates that under weaker electric fields the axisymmetric mode is convectively unstable, but becomes absolutely unstable when the electric field is relatively strong, when the outer liquid has a certain degree of elasticity. While the nonaxisymmetric mode is found to be always absolutely unstable. A boundary line between absolute and convective instability is given on the parameter space of the Deborah number (relative relaxation time) of the outer liquid and the electrical Euler number. Our results in temporal and spatiotemporal instability analysis can be useful for understanding the origin of the different modes and desired mode control in coaxial electrospinning.

Absolute instability in shock-containing jets

We present an analysis of the linear stability characteristics of shock-containing jets. The flow is linearised around a spatially periodic mean, which acts as a surrogate for a mean flow with a shock-cell structure, leading to a set of partial differential equations with periodic coefficients in space. Disturbances are written using the Floquet ansatz and Fourier modes in the streamwise direction, leading to an eigenvalue problem for the Floquet exponent. The characteristics of the solution are directly compared with the locally parallel case, and some of the features are similar. The inclusion of periodicity induces minor changes in the growth rate and phase velocity of the relevant modes for small shock amplitudes. On the other hand, the eigenfunctions are now subject to modulation related to the periodicity of the flow. Analysis of the spatio-temporal growth rates led to the identification of a saddle point between the Kelvin–Helmholtz mode and the guided jet mode, characterising an absolute instability mechanism. Frequencies and mode shapes related to the saddle points for two conditions (associated with axisymmetric and helical modes) are compared with screech frequencies and the most energetic coherent structures of screeching jets, resulting in a good agreement for both. The analysis shows that a periodic shock-cell structure has an impulse response that grows upstream, leading to oscillator behaviour. The results suggest that screech can occur in the absence of a nozzle, and that the upstream reflection condition is not essential for screech frequency selection. Connections to previous models are also discussed.

Nano-optical trapping using an all-dielectric optical fiber supporting a TEM-like mode

Fiber optical tweezers benefit from compact structures and compatibility with fiber optic technology, however, trapping of nano-objects are rarely demonstrated. Here, we predict stable optical trapping of a 30 nm polystyrene particle using an all-dielectric coaxial optical fiber supporting an axisymmetric TEM-like mode. We demonstrate, via comprehensive finite-difference time-domain simulations, that the trapping behavior arises from a significant shift of the fiber-end-fire radiation directivity originated from the nanoparticle-induced symmetry breaking, rather than the gradient force which assumes an invariant optical field. Fabrication of the fiber involved is entirely feasible with existing techniques, such as thermal-drawn and electrospinning, and therefore can be mass-produced.

Electric and viscous correction for viscous potential flow analysis of electrohydrodynamic instability of an electrified leaky-dielectric jet

The electric and viscous correction of viscous potential flow (EVCVPF) is developed for analyzing the electrohydrodynamic instability of an electrified leaky-dielectric viscous jet. The EVCVPF model is based on the viscous potential flow (VPF) and the viscous correction of VPF (VCVPF), proposed by Joseph and Wang [“The dissipation approximation and viscous potential flow,” J. Fluid Mech. 505, 365–377 (2004)]. The purpose is to resolve the discrepancy between the non-zero irrotational viscous and the electric tangent stresses. The power of the pressure correction is introduced to compensate the neglected viscous dissipation in the flow bulk in VPF, which is equal to the average power of the irrotational viscous and the electric tangent stresses. The model has been validated by comparing it to the exact normal-mode solution of the linearized Navier–Stokes equations (fully viscous flow, FVF). The energy budget is also performed to assist in understanding underlying mechanisms. Results show that EVCVPF is accurate for charged jets with low and moderate viscosities, i.e., the Ohnesorge number approximately Oh ≤ 0.1. The inaccuracy for highly viscous jets are the limitations of VPF itself. The electric field has less influence compared to the fluid viscosity. To achieve more accurate approximations, VCVPF and VPF are chosen for axisymmetric and non-axisymmetric modes under weak electric fields. EVCVPF is in remarkably good agreement with FVF under moderate and strong electric fields. In general, as VCVPF extends the applicability in fluid viscosity of VPF, EVCVPF further improves the adequacy when studying the electrohydrodynamic instability.

Influence of nozzle external geometry on wavepackets in under-expanded supersonic impinging jets

In this study, large-eddy simulations are utilised to unravel the influence of the nozzle's external geometry on upstream-travelling waves in under-expanded supersonic impinging jets. Three configurations, a thin-lipped, a thin-lipped with a sponge and an infinite-lipped nozzle are considered with the other non-dimensionalised geometrical and flow variables identical for the three cases. Spectral proper orthogonal decomposition is applied to the Mack norm, i.e. the energy norm based on the stagnation energy, to obtain the spatial modes at their corresponding frequency. The spectral decomposition of the spatial modes at optimal and suboptimal frequencies is used to isolate the wavepackets into upstream- and downstream-propagating waves based on their phase velocity. It is found that the external geometry of the nozzle has a significant influence on the first-order statistics even though the governing non-dimensional parameters are the same for all three cases. Multiple peaks emerge in the energy spectra at distinct frequencies corresponding to axisymmetric azimuthal modes for each case. The downstream-propagating wavepackets have a high amplitude at the shear layer of the three jets with the mode shapes resembling Kelvin–Helmholtz instability waves, while the upstream-travelling wavepackets exist in the three regions of the near field, shear layer and inside of the jet. The barrel shock at the nozzle exit appears as a flexible shield, which prevents upstream-travelling waves from reaching the internal region of the nozzle, where the upstream-travelling waves travel obliquely with one side of the wavefront is crawling on the reflected shock while the other side is guided by the shear layer. These latter waves can reach the nozzle lip via inside of the jet. The spectral decomposition of the spatial modes at optimal and suboptimal frequencies show that all three forms of the near field, shear layer and inside jet upstream-travelling wavepackets contribute to the receptivity process while their contributions and strength are altered by the change of the external geometry of the nozzle.

Mean field dynamo action in shearing flows – II. Fluctuating kinetic helicity with zero mean

ABSTRACT Here we explore the role of temporal fluctuations in kinetic helicity on the generation of large-scale magnetic fields in the presence of a background linear shear flow. Key techniques involved here are same as in our earlier work, where we have used the renovating flow based model with shearing waves. Both the velocity and the helicity fields are treated as stochastic variables with finite correlation times, τ and τh, respectively. Growing solutions are obtained when τh > τ, even when this time-scale separation, characterized by m = τh/τ, remains below the threshold for causing the turbulent diffusion to turn negative. In regimes when turbulent diffusion remains positive, and τ is of the order of eddy turnover time T, the axisymmetric modes display non-monotonic behaviour with shear rate S: both, the growth rate γ and the wavenumber k* corresponding to the fastest growing mode, first increase, reach a maximum and then decrease with |S|, with k* being always smaller than eddy-wavenumber, thus boosting growth of magnetic fields at large length-scales. The cycle period Pcyc of growing dynamo wave is inversely proportional to |S| at small shear, exactly similar to the fixed kinetic helicity case of our earlier work. This dependence becomes shallower at larger shear. Interestingly enough, various curves corresponding to different choices of m collapse on top of each other in a plot of mPcyc with |S|.

Onset of Plasmoid Reconnection during Magnetorotational Instability

Abstract The evolution of current sheets in accretion flows undergoing magnetorotational instability (MRI) is examined through two- and three-dimensional numerical modeling of the resistive MHD equations in global cylindrical geometry. With an initial uniform magnetic field aligned in the vertical (z) direction, MRI produces radially extended toroidal (azimuthal) current sheets. In both 2D and 3D when axisymmetric modes dominate, these current sheets attract each other and merge in the poloidal (rz) plane, driving magnetic reconnection when the Lundquist number S > 3 × 102, making it a possible source of plasmoids (closed magnetic loops) in accretion disks. At high Lundquist numbers in the 2D regime, starting at S = 5 × 103, self-consistent MRI-generated current sheets become thin and subject to plasmoid instability, and therefore spontaneous magnetic reconnection. When nonaxisymmetric 3D modes dominate, turbulence makes the azimuthal current sheets more unstable and stretch vertically. Toroidally extended vertical current sheets in the inner region, as well as larger 3D magnetic islands in the outer regions of the disks are also formed. These findings have strong ramifications for astrophysical disks as potential sources of plasmoids that could cause local heating, particle acceleration, and high energy EM radiation.

Impact of magnetic X-points on the vertical stability of tokamak plasmas

The ideal-magneto-hydro-dynamic (MHD) theory of axisymmetric modes with toroidal mode number n = 0 in tokamak plasmas is developed. These modes are resonant at the magnetic X-points of the tokamak divertor separatrix. As a consequence, current sheets form along the separatrix, which profoundly affect the stability of vertical plasma displacements. In particular, current sheets at the magnetic separatrix lead to stabilization of n = 0 modes, at least on the ideal-MHD time scale, adding an important ingredient to the mechanism of passive feedback stabilization.

Impact of negative triangularity plasma shaping on the n = 0 resistive wall mode in a tokamak

The axisymmetric (n = 0) resistive wall mode instability is numerically investigated for the negative triangularity plasma shape, which has shown several benefits in terms of improved confinement time and fusion engineering. To evaluate the shape effects, we modified a MHD instability code AVSTAB (Axisymmetric Vertical STABility), which calculates the marginally controllable elongation in a simplified feedback capability parameter. The plasma characteristics (poloidal beta and internal inductance) as well as the geometric effects (wall shape and plasma location) are important to determine the instability. In contrast to positive triangularity, the higher poloidal beta provides more instability drive for the negative triangularity because of the higher Shafranov shift and the higher elongation of the inner flux surface of the MHD equilibrium. Non-conformal wall shapes to the plasmas (positive triangularity wall and negative triangularity plasma) are found to be rather helpful to stabilize the n = 0 mode, unless the plasma is too close to the walls at the nulls for the opposite triangularity.

Applicability and Limitations of Simplified Elastic Shell Theories for Vibration Modelling of Double-Walled Carbon Nanotubes

The applicability and limitations of simplified models of thin elastic circular cylindrical shells for linear vibrations of double-walled carbon nanotubes (DWCNTs) are considered. The simplified models, which are based on the assumptions of membrane and moment approximate thin-shell theories, are compared with the extended Sanders–Koiter shell theory. Actual discrete DWCNTs are modelled by means of couples of concentric equivalent continuous thin, circular cylindrical shells. Van der Waals interaction forces between the layers are taken into account by adopting He’s model. Simply supported and free–free boundary conditions are applied. The Rayleigh–Ritz method is considered to obtain approximate natural frequencies and mode shapes. Different aspect and thickness ratios, and numbers of waves along longitudinal and circumferential directions, are analysed. In the cases of axisymmetric and beam-like modes, it is proven that membrane shell theory, differently from moment shell theory, provides results with excellent agreement with the extended Sanders–Koiter shell theory. On the other hand, in the case of shell-like modes, it is found that both membrane and moment shell theories provide results reporting acceptable agreement with the extended Sanders–Koiter shell theory only for very limited ranges of geometries and wavenumbers. Conversely, for shell-like modes it is found that a newly developed, simplified shell model, based on the combination of membrane and semi-moment theories, provides results in satisfactory agreement with the extended Sanders–Koiter shell theory in all ranges.