scholarly journals Linear instability of Vlasov-Maxwell systems revisited-A Hamiltonian approach

2022 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Zhiwu Lin
Keyword(s):  
Linear Stability ◽  
Hamiltonian Systems ◽  
Linear Instability ◽  
Steady States ◽  
Hamiltonian Approach ◽  
Collisionless Plasmas ◽  
Exponential Trichotomy

<p style='text-indent:20px;'>We consider linear stability of steady states of 1<inline-formula><tex-math id="M1">\begin{document}$ \frac{1}{2} $\end{document}</tex-math></inline-formula> and 3DVlasov-Maxwell systems for collisionless plasmas. The linearized systems canbe written as separable Hamiltonian systems with constraints. By using ageneral theory for separable Hamiltonian systems, we recover the sharp linearstability criteria obtained previously by different approaches. Moreover, weobtain the exponential trichotomy estimates for the linearized Vlasov-Maxwellsystems in both relativistic and nonrelativistic cases.</p>

scholarly journals Linear stability of boundary layers under solitary waves

2014 ◽  
Vol 761 ◽  
pp. 62-104 ◽  
Author(s):  
Joris C. G. Verschaeve ◽  
Geir K. Pedersen
Keyword(s):  
Linear Stability ◽  
Solitary Waves ◽  
Reynolds Numbers ◽  
Quantitative Agreement ◽  
Linear Instability ◽  
Navier Stokes ◽  
Acceleration Region ◽  
Critical Reynolds Numbers ◽  
Tollmien Schlichting ◽  
The Stability

AbstractIn the present treatise, the stability of the boundary layer under solitary waves is analysed by means of the parabolized stability equation. We investigate both surface solitary waves and internal solitary waves. The main result is that the stability of the flow is not of parametric nature as has been assumed in the literature so far. Not only does linear stability analysis highlight this misunderstanding, it also gives an explanation why Sumer et al. (J. Fluid Mech., vol. 646, 2010, pp. 207–231), Vittori & Blondeaux (Coastal Engng, vol. 58, 2011, pp. 206–213) and Ozdemir et al. (J. Fluid Mech., vol. 731, 2013, pp. 545–578) each obtained different critical Reynolds numbers in their experiments and simulations. We find that linear instability is possible in the acceleration region of the flow, leading to the question of how this relates to the observation of transition in the acceleration region in the experiments by Sumer et al. or to the conjecture of a nonlinear instability mechanism in this region by Ozdemir et al. The key concept for assessment of instabilities is the integrated amplification which has not been employed for this kind of flow before. In addition, the present analysis is not based on a uniformization of the flow but instead uses a fully nonlinear description including non-parallel effects, weakly or fully. This allows for an analysis of the sensitivity with respect to these effects. Thanks to this thorough analysis, quantitative agreement between model results and direct numerical simulation has been obtained for the problem in question. The use of a high-order accurate Navier–Stokes solver is primordial in order to obtain agreement for the accumulated amplifications of the Tollmien–Schlichting waves as revealed in this analysis. An elaborate discussion on the effects of amplitudes and water depths on the stability of the flow is presented.

scholarly journals Decomposition of the temporal growth rate in linear instability of compressible gas flows

2015 ◽  
Vol 778 ◽  
pp. 120-132 ◽  
Author(s):  
Mario Weder ◽  
Michael Gloor ◽  
Leonhard Kleiser
Keyword(s):  
Growth Rate ◽  
Energy Balance ◽  
Linear Stability ◽  
Couette Flow ◽  
Stability Theory ◽  
Compressible Flows ◽  
Linear Instability ◽  
Linear Stability Theory ◽  
Gas Flows ◽  
Temporal Growth Rate

We present a decomposition of the temporal growth rate ${\it\omega}_{i}$ which characterises the evolution of wave-like disturbances in linear stability theory for compressible flows. The decomposition is based on the disturbance energy balance by Chu (Acta Mech., vol. 1 (3), 1965, pp. 215–234) and provides terms for production, dissipation and flux of energy as components of ${\it\omega}_{i}$. The inclusion of flux terms makes our formulation applicable to unconfined flows and flows with permeable or vibrating boundaries. The decomposition sheds light on the fundamental mechanisms determining temporal growth or decay of disturbances. The additional insights gained by the proposed approach are demonstrated by an investigation of two model flows, namely compressible Couette flow and a plane compressible jet.

Subcritical bifurcation and bistability in thermoacoustic systems

2013 ◽  
Vol 715 ◽  
pp. 210-238 ◽  
Author(s):  
Priya Subramanian ◽  
R. I. Sujith ◽  
P. Wahi
Keyword(s):  
Hopf Bifurcation ◽  
Linear Stability ◽  
Multiple Scales ◽  
Landau Equation ◽  
Linear Instability ◽  
Numerical Continuation ◽  
Rijke Tube ◽  
Subcritical Hopf Bifurcation ◽  
Fold Bifurcation ◽  
Parameter Values

AbstractThis paper analyses subcritical transition to instability, also known as triggering in thermoacoustic systems, with an example of a Rijke tube model with an explicit time delay. Linear stability analysis of the thermoacoustic system is performed to identify parameter values at the onset of linear instability via a Hopf bifurcation. We then use the method of multiple scales to recast the model of a general thermoacoustic system near the Hopf point into the Stuart–Landau equation. From the Stuart–Landau equation, the relation between the nonlinearity in the model and the criticality of the ensuing bifurcation is derived. The specific example of a model for a horizontal Rijke tube is shown to lose stability through a subcritical Hopf bifurcation as a consequence of the nonlinearity in the model for the unsteady heat release rate. Analytical estimates are obtained for the triggering amplitudes close to the critical values of the bifurcation parameter corresponding to loss of linear stability. The unstable limit cycles born from the subcritical Hopf bifurcation undergo a fold bifurcation to become stable and create a region of bistability or hysteresis. Estimates are obtained for the region of bistability by locating the fold points from a fully nonlinear analysis using the method of harmonic balance. These analytical estimates help to identify parameter regions where triggering is possible. Results obtained from analytical methods compare reasonably well with results obtained from both experiments and numerical continuation.

scholarly journals Parabolized stability analysis of jets from serrated nozzles

2016 ◽  
Vol 789 ◽  
pp. 36-63 ◽  
Author(s):  
Aniruddha Sinha ◽  
Kristján Gudmundsson ◽  
Hao Xia ◽  
Tim Colonius
Keyword(s):  
Linear Stability ◽  
Near Field ◽  
Pressure Fluctuations ◽  
Three Dimensional ◽  
Hydrodynamic Pressure ◽  
Linear Instability ◽  
Base Flow ◽  
Linear Stability Theory ◽  
Instability Wave ◽  
Mean Flow

We study the viscous spatial linear stability characteristics of the time-averaged flow in turbulent subsonic jets issuing from serrated (chevroned) nozzles, and compare them to analogous round jet results. Linear parabolized stability equations (PSE) are used in the calculations to account for the non-parallel base flow. By exploiting the symmetries of the mean flow due to the regular arrangement of serrations, we obtain a series of coupled two-dimensional PSE problems from the original three-dimensional problem. This reduces the solution cost and manifests the symmetries of the stability modes. In the parallel-flow linear stability theory (LST) calculations that are performed near the nozzle to initiate the PSE, we find that the serrated nozzle reduces the growth rates of the most unstable eigenmodes of the jet, but their phase speeds are approximately similar. We obtain encouraging validation of our linear PSE instability wave results vis-à-vis near-field hydrodynamic pressure data acquired on a phased microphone array in experiments, after filtering the latter with proper orthogonal decomposition (POD) to extract the energetically dominant coherent part. Additionally, a large-eddy simulation database of the same serrated jet is investigated, and its POD-filtered pressure field is found to compare favourably with the corresponding PSE solution within the jet plume. We conclude that the coherent hydrodynamic pressure fluctuations of jets from both round and serrated nozzles are reasonably consistent with the linear instability modes of the turbulent mean flow.

scholarly journals Instabilities of the flow around a cylinder and emission of vortex dipoles

2013 ◽  
Vol 730 ◽  
pp. 419-441 ◽  
Author(s):  
Ziv Kizner ◽  
Viacheslav Makarov ◽  
Leon Kamp ◽  
GertJan van Heijst
Keyword(s):  
Linear Stability ◽  
Linear Instability ◽  
Instability Region ◽  
Base Flow ◽  
Two Dimensional ◽  
Azimuthal Mode ◽  
Dimensional Parameter ◽  
Contour Dynamics ◽  
Instability Regions ◽  
Dynamics Simulations

AbstractInstabilities and long-term evolution of two-dimensional circular flows around a rigid circular cylinder (island) are studied analytically and numerically. For that we consider a base flow consisting of two concentric neighbouring rings of uniform but different vorticity, with the inner ring touching the cylinder. We first study the inviscid linear stability of such flows to perturbations of the free edges of the rings. For a given ratio of the vorticity in the rings, the governing parameters of the problem are the radii of the inner and outer rings scaled on the cylinder radius. In this two-dimensional parameter space, we determine analytically the regions of linear stability/instability of each azimuthal mode $m= 1, 2, \ldots . $ In the physically most meaningful case of zero net circulation, for each mode $m\gt 1$, two regions are identified: a regular instability region where mode $m$ is unstable along with some other modes, and a unique instability region where only mode $m$ is unstable. After the conditions of linear instability are established, inviscid contour-dynamics and high-Reynolds-number finite-element simulations are conducted. In the regular instability regions, simulations of both kinds typically result in the formation of vortical dipoles or multipoles. In the unique instability regions, where the inner vorticity ring is much thinner than the outer ring, the inviscid contour-dynamics simulations do not reveal dipole emission. In the viscous simulation, because viscosity has time to widen the inner ring, the instability develops in the same manner as in the regular instability regions.

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