Absolute Instability Research Articles

Numerical Simulation of a Two-Dimensional Helium Gas Jet with Side-Jets Formation

Side jets are generated when the jet density is smaller than that of the ambient gas. It has been suggested that the side jets are related to self-excited oscillations due to the local absolute instability that occurs when the density ratio is less than 0.7, and vorticity generation due to baroclinic torque caused by the cross product of density gradient and pressure gradient. In addition, it has been suggested that a side jet is also formed by a pair of streamwise vortices with different rotation directions. First of all, in order to confirm previous hypotheses, we have tried the formation of side jets in a helium gas jet by a numerical simulation using ANSYS Fluent. As a result, it is shown that a side jet is formed between a streamwise vortex pair.

Stability of an air–water mixing layer: focus on the confinement effect

The shear instability occurring at the interface between a slow water layer and a fast air stream is a complex phenomenon driven by momentum and viscosity differences across the interface, velocity gradients as well as by injector geometries. Simulating such an instability under experimental conditions is numerically challenging and few studies exist in the literature. This work aims at filling a part of this gap by presenting a study of the convergence between two-dimensional simulations, linear theory and experiments, in regimes where the instability is triggered by the confinement, i.e. finite thicknesses of gas and liquid streams. It is found that very good agreement between the three approaches is obtained. Moreover, using simulations and linear theory, we explore in detail the effects of confinement on the stability of the flow and on the transition between absolute and convective instability regimes, which is shown to depend on the length scale of the confinement as well as on the dynamic pressure ratio. In the absolute regime under study, the interfacial wave frequency is found to be inversely proportional to the smallest injector size (liquid or gas).

Transition in an infinite swept-wing boundary layer subject to surface roughness and free-stream turbulence

The instability of an incompressible boundary-layer flow over an infinite swept wing in the presence of disc-type roughness elements and free-stream turbulence (FST) has been investigated by means of direct numerical simulations. Our study corresponds to the experiments by Örlü et al. (Tech. Rep., KTH Royal Institute of Technology, 2021, http://urn.kb.se/resolve?urn=urn:nbn:se:kth:diva-291874). Here, different dimensions of the roughness elements and levels of FST have been considered. The aim of the present work is to investigate the experimentally observed sensitivity of the transition to the FST intensity. In the absence of FST, flow behind the roughness elements with a height above a certain value immediately undergoes transition to turbulence. Impulse–response analyses of the steady flow have been performed to identify the mechanism behind the observed flow instability. For subcritical roughness, the generated wave packet experiences a weak transient growth behind the roughness and then its amplitude decays as it is advected out of the computational domain. In the supercritical case, in which the flow transitions to turbulence, flow as expected exhibits an absolute instability. The presence of FST is found to have a significant impact on the transition behind the roughness, in particular in the case of a subcritical roughness height. For a height corresponding to a roughness Reynolds number $Re_{hh}=461$ , in the absence of FST the flow reaches a steady laminar state, while a very low FST intensity of $Tu =0.03\,\%$ causes the appearance of turbulence spots in the wake of the roughness. These randomly generated spots are advected out of the computational domain. For a higher FST level of $Tu=0.3\,\%$ , a turbulent wake is clearly visible behind the element, similar to that for the globally unstable case. The presented results confirm the experimental observations and explain the mechanisms behind the observed laminar–turbulent transition and its sensitivity to FST.

Absolute and convective stability of flow between closely spaced co-rotating disks with imposed throughflow

Different types of rotating disk boundary layers were previously shown to be absolutely unstable. The present study checks if this is also true for the flow between narrowly spaced co-rotating disks with merged boundary layers and imposed throughflow. The described method is able to reproduce the absolute instability of the von Kármán boundary layer and of radially outward flow between co-rotating disks with separated boundary layers. Nevertheless, no absolute instability is found for the case of narrow disk spacing with merged boundary layers for both radially inward and outward flow. The data does however show convective instability. Stability maps are provided for the analyzed parameter space.

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.

Theoretical analysis and numerical modeling of the evolution mechanism of polymer melt unstable flow in two‐dimensional ultrasonic‐assisted micro‐injection molding

AbstractMicro polymer parts are usually fabricated by ultrasonic‐assisted micro‐injection molding which is an effective and low‐cost method. This method solves some injection molding defects such as short shot by reducing the melt viscosity, and the clarification of its mechanism is a challenge. This article is aimed at studying the evolution mechanism of two‐dimensional ultrasonic vibration on the polymer melt unstable flow. A two‐dimensional ultrasonic vibration, whose displacements components in the‐ and‐directions are large, is applied to the polymer melt. The unstable flow is a flow field with perturbations. The formulas of perturbation stream function and velocity perturbations are established based on the theories of absolute instability and convective instability. The established formulas are verified by numerical analysis. The results show that the flow of polymer melt is convectively unstable and the vortices are formed under the combined actions of shear stress and transverse pressure. The distribution of vortices is bound to affect the mold filling quality. Accordingly, a two‐dimensional ultrasonic vibration with the phase difference of 90° is adopted to make strong vortices occur symmetrically, to improve the uniformity of viscosity field, and to obtain a better melt filling capability during injection molding.

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.

A fast multi-fidelity method with uncertainty quantification for complex data correlations: Application to vortex-induced vibrations of marine risers

We develop a fast multi-fidelity modeling method for very complex correlations between high- and low-fidelity data by working in modal space to extract the proper correlation function. We apply this method to infer the amplitude of motion of a flexible marine riser in cross-flow, subject to vortex-induced vibrations (VIV). VIV are driven by an absolute instability in the flow, which imposes a frequency (Strouhal) law that requires a matching with the impedance of the structure; this matching is easily achieved because of the rapid parametric variation of the added mass force. As a result, the wavenumber of the riser spatial response is within narrow bands of uncertainty. Hence, an error in wavenumber prediction can cause significant phase-related errors in the shape of the amplitude of response along the riser, rendering correlation between low- and high-fidelity data very complex. Working in modal space as outlined herein, dense data from low-fidelity data, provided by the semi-empirical computer code VIVA, can correlate in modal space with few high-fidelity data, obtained from experiments or fully-resolved CFD simulations, to correct both phase and amplitude and provide predictions that agree very well overall with the correct shape of the amplitude response. We also quantify the uncertainty in the prediction using Bayesian modeling and exploit this uncertainty to formulate an active learning strategy for the best possible location of the sensors providing the high fidelity measurements.

Instability in large bounded domains—branched versus unbranched resonances

We study transitions from convective to absolute instability near a trivial state in large bounded domains for prototypical model problems in the presence of transport and negative nonlinear feedback. We identify two generic scenarios, depending on the nature of the linear mechanism for instability, which both lead to different, universal bifurcation diagrams. In the first, classical case of a linear branched resonance the transition is hard, that is, small changes in a control parameter lead to a finite-size state. In the second, novel case of an unbranched resonance, the transition is gradual. In both cases, the bifurcation diagram is determined by interaction of the leading edge of an invasion front with upstream boundary conditions. Technically, we analyze this interaction in a heteroclinic gluing bifurcation analysis that uses geometric desingularization of the trivial state.

One-Dimensional Nonlinear Parametric Instability of Inhomogeneous Plasma: Time Domain Problem

A numerical one-dimensional model of convective parametric instability of inhomogeneous plasma is developed. By using this model, a numerical solution describing spatial and temporal characteristics of interacting waves is obtained. The results obtained are in a good agreement with known analytical models and substantially generalize them. In particular, an important advantage of the proposed model is the possibility of varying initial conditions, analyzing behavior of the system in the presence of incident wave fluctuations that is important for the future study of the absolute instability mode. The model is also provides possibility to simulate absolute parametric instability with a wide range of controllable parameters, as well as to study interacting wave transients.

Modal and absolute thermal instability in a vertical porous layer

The conduction regime in a vertical porous layer subject to a horizontal temperature gradient is studied. The boundaries are considered as isothermal, with different temperatures, and permeable to the external environment. The linear stability of this basic flow state is analysed by testing the dynamics of the normal modes of perturbation. The numerical solution of the stability eigenvalue problem leads to the determination of the neutral stability condition. Then, the evolution in time of localised wavepacket perturbations is investigated leading to the determination of the threshold to absolute instability.

Design and Simulation Investigations of W-Band Second Harmonic Periodically Dielectric-Loaded Gyro-TWT

ABSTRACT In this article, the complete design and beam-wave interaction studies of W-band gyro-TWT amplifier have been presented. The gyro-TWT has been designed to operate at second harmonic TE 02 mode to reduce the magnetic field requirement. A triode-type magnetron injection gun has been designed and optimized for low-velocity spread using E-GUN. A novel, mode selective input coupler with a gradually tapered slotted waveguide has been used to feed the desired TE 02 mode and suppress the reflection and spurious oscillations at the input end (upstream). To abridge the absolute instability of the operating mode and backward wave oscillations caused by spurious modes, a periodic lossy dielectric-loaded RF structure has been adopted. The particle-in-cell simulation of the amplifier predicted a peak RF output power of ∼500 kW with ∼3.6 GHz bandwidth, ∼20% efficiency, and ∼32.2 dB saturated gain for the DC input of 100 kV, ∼25 A with 4% axial velocity spread and velocity ratio of 1.22. Further, to extract the output power with minimum reflection, a double-disc window has been studied. An undepressed curved collector is also designed using E-GUN and its simulation predicted a maximum heat wall loading of ∼0.39 kW /cm2.

Generation of Short Microwave Pulses by Compression of Chirped Signals Obtained by Raman Backscattering on Electron Beam With Variable Energy

We suggest method of short microwave pulse generation by means of compression of chirped pulses obtained in the process of Stimulate Raman Backscattering (SRB) on electron beam with variable energy. We assume that in frequency modulator the pump wave, co-propagating with the electrons is scattered into counter propagating signal wave with frequency downshift. The absolute instability, developing in this system, allows for the smooth tuning of the signal frequency by varying the energy of electrons. The SRB modulator can produce a frequency-tunable signal with a bandwidth up to 10% and conversion efficiency up to 70%, when using a 50 kW, 35 GHz gyrotron as a source of pump wave. Simulation of subsequent compression with multi-fold helically corrugated waveguide (HCW) as dispersive line showed that optimization of energy temporal dependence in modulator together with compressor parameters provides the possibility to produce pulses with the peak power up to 3 MW which exceeds the power of driving gyrotron by the factor of 60.

Reversible Stream Drop Transition in a Microfluidic Coflow System via On Demand Exposure to Acoustic Standing Waves.

Transition between stream and droplet regimes in a coflow is typically achieved by adjusting the capillary numbers (Ca) of the phases. Remarkably, we experimentally evidence a reversible transition between the two regimes by controlling exposure of the system to acoustic standing waves, with Ca fixed. By satisfying the ratio of acoustic radiation force to the interfacial tension force, Ca_{ac}>1, experiments reveal a reversible stream drop transition for Ca<1, and stream relocation for Ca≥1. We explain the phenomenon in terms of the pinching, advection, and relocation timescales and a transition between convective and absolute instability from a linear stability analysis [P. Guillot et al., Phys. Rev. Lett. 99, 104502 (2007)PRLTAO0031-900710.1103/PhysRevLett.99.104502].

Predominance of pressure transport in spatial energy budget for a mixing layer approaching absolute instability

In this letter, we report the outcome of a spatial energy budget performed for the linear convective instability of the plane incompressible mixing layer within the inviscid framework. We find that, as the critical condition for the onset of absolute instability is approached, the integrated pressure-transport term becomes increasingly more prominent as compared to the integrated production term and it dominates the energy budget completely at the critical condition. This implies that, near the threshold of absolute instability, the growth of disturbances is almost entirely due to the pressure transport mechanism (rather than the production mechanism), which is a striking result. The part of the pressure-transport term that represents the work done by the fluctuating pressure forces is seen to be primarily responsible for the observed shift in the energy balance. These results can help us better understand the physical processes causing absolute instability in a mixing layer. In particular, the redistribution of disturbance energy in streamwise direction by fluctuating pressure, which is "non-local" in character for incompressible flows, seems to play a key role in this respect.

Hydrodynamics of granular gases of inelastic and rough hard disks or spheres. II. Stability analysis.

Conditions for the stability under linear perturbations around the homogeneous cooling state are studied for dilute granular gases of inelastic and rough hard disks or spheres with constant coefficients of normal (α) and tangential (β) restitution. After a formally exact linear stability analysis of the Navier-Stokes-Fourier hydrodynamic equations in terms of the translational (d_{t}) and rotational (d_{r}) degrees of freedom, the transport coefficients derived in the companion paper [A. Megías and A. Santos, "Hydrodynamics of granular gases of inelastic and rough hard disks or spheres. I. Transport coefficients" Phys. Rev. E 104, 034901 (2021)10.1103/PhysRevE.104.034901] are employed. Known results for hard spheres [Garzó, Santos, and Kremer, Phys. Rev. E 97, 052901 (2018)10.1103/PhysRevE.97.052901] are recovered by setting d_{t}=d_{r}=3, while novel results for hard disks (d_{t}=2,d_{r}=1) are obtained. In the latter case, a high-inelasticity peculiar region in the (α,β) parameter space is found, inside which the critical wave number associated with the longitudinal modes diverges. Comparison with event-driven molecular dynamics simulations for dilute systems of hard disks at α=0.2 shows that this theoretical region of absolute instability may be an artifact of the extrapolation to high inelasticity of the approximations made in the derivation of the transport coefficients, although it signals a shrinking of the conditions for stability. In the case of moderate inelasticity (α=0.7), however, a good agreement between the theoretical predictions and the simulation results is found.

Spatiotemporal instability of a shear-imposed viscous flow

We study the linear spatiotemporal instability of a two-dimensional gravity-driven viscous fluid flow where the fluid surface is subjected to an imposed shear stress. The fourth order Orr–Sommerfeld boundary value problem is derived and solved numerically up to moderate values of the Reynolds number. Numerical solution based on AUTO07p identifies four spatial branches, viz., I, II, III, and IV, where the spatial branches I, II, and IV lie in the upper half zone, while the spatial branch III lies in the lower half zone of the complex wavenumber plane. The spatial growth rate −ki corresponding to branch I becomes stronger as long as the imposed shear stress increases and ensures a destabilizing effect. Furthermore, the spatial branch I enters in the lower half zone of the complex wavenumber plane as soon as the temporal growth rate ωi decreases and may collide with other spatial branch lying in the lower half zone of the complex wavenumber plane. Moreover, a study of absolute and convective instabilities is carried out within the frameworks of saddle point technique and collision criterion. The saddle point technique provides only one unstable branch of the unstable wavepacket, while the collision criterion provides two unstable branches of the wavepacket. The unstable range of the wavepacket with ray velocity enhances in the presence of imposed shear stress. It is observed that the shear-imposed fluid flow is convectively unstable. In addition, the simplified second order two-equation model is developed for a shear-imposed flow in terms of the local fluid layer thickness and local flow rate, which in fact renders three spatial branches rather than four. However, the two-equation model recovers the physically relevant spatial branch I very well. Finally, nonlinear spatiotemporal simulation of the two-equation model displays a formation of the regular train of solitary waves downstream at low forcing frequency.

Experimental study on performance change in F-band surface-wave oscillator due to slow-wave structure length

A surface-wave oscillator (SWO) is a high-power terahertz-wave source based on a slow-wave structure (SWS). A cylindrical corrugated waveguide is used as the SWS. The length of SWS is an important factor in the oscillation. Experiments of the SWO with various length structures are conducted. An instability responsible for the SWO changes depending on the SWS length. For the 200-period SWS, the radiation is due to the absolute instability and depends on the beam current like the temporal growth rate. By contrast, for the 80-period SWS, the radiation is due to the global instability having the current dependence like the spatial growth rate. Compared to the 80-period SWS, the 200-period SWS can improve the radiation power more than seven times in the backward wave oscillation (BWO) region. The maximum BWO power reaches up to about 10 kW and the maximum electric efficiency of 1.5% is obtained.

Theoretical analysis of a 0.22 THz multistage confocal waveguide gyro-TWT with circle-sector-shaped electron beam

In order to improve the beam–wave interaction efficiency, a special circle-sector-shaped electron beam is designed and simulated in the confocal waveguide gyrotron traveling-wave tube (gyro-TWT). The linear and nonlinear characteristics of a 0.22 THz multistage confocal waveguide gyro-TWT are analyzed. Based on kinetic theory, the linear gain, critical current of absolute instability, and starting current of backward wave oscillation (BWO) are analyzed. A quasi-optical sever is utilized in the confocal waveguide to suppress the BWO while having a slight impact on the output power. Meanwhile, the impact of velocity spread on the output parameters is also taken into consideration. The nonlinear analyses indicate that the efficiency can be up to 34%, which is substantially higher than the traditional annular electron beam.

On Linear Instability of Atmospheric Quasi-hydrostatic Equations in Response to Small Shortwave Perturbations

A set of 3-dimensional atmospheric-dynamics equations with quasi-hydrostatic approximation is proposed and justified with the practical goal to optimize atmospheric modelling at scales ranging from meso meteorology to global climate. Sound waves are filtered by applying the quasi-hydrostatic approximation. In the closed system of hydro/thermodynamic equations, the inertial forces are negligibly small compared to gravity forces, and the asymptotically exact equation for vertical velocity is obtained. Investigation of the stability of solutions to this system in response to small shortwave perturbations has shown that solutions have the property of shortwave instability. There are situations when the increment of the perturbation amplitude tends to infinity, corresponding to absolute instability. It means that the Cauchy problem for such equations may be ill-posed. Its formulation can become conditionally correct if solutions are sought in a limited class of sufficiently smooth functions whose Fourier harmonics tend to zero reasonably quickly when the wavelengths of the perturbations approach zero. Thus, the numerical scheme for the quasi-hydrostatic equations using the finite-difference method requires an adequately selected pseudo-viscosity to eliminate the instability caused by perturbations with wavelengths of the order of the grid size. The result is useful for choosing appropriate vertical and horizontal grid sizes for modelling to avoid shortwave instability associated with the property of the system of equations. Implementation of pseudo-viscosities helps to smoothen or suppress the perturbations that occur during modelling.