Asymptotic approach to combustion instability

2005 ◽  
Vol 363 (1830) ◽  
pp. 1247-1259 ◽  
Author(s):  
Xuesong Wu
Keyword(s):  
Combustion Instability ◽  
Parametric Instability ◽  
Acoustic Mode ◽  
Sound Waves ◽  
Asymptotic Approach ◽  
Reaction Heat ◽  
Weakly Nonlinear ◽  
One Dimensional ◽  
Entire Flow ◽  
Asymptotic Formulation

This paper presents an asymptotic approach to combustion instability in premixed flames under the assumptions of large activation energy and small Mach number. The entire flow consists of four distinct yet fully interactive sub-regions, which accommodate the chemical reaction, heat transport, hydrodynamics and acoustics, respectively. A reduced system was derived to describe the intricate coupling between the flame and acoustics that underlies the combustion instability. The asymptotically reduced system was employed to study the weakly nonlinear interaction between the Darrieus–Landau instability and the longitudinal acoustic mode of the combustion chamber. The general asymptotic formulation includes the influence of enthalpy fluctuation in the oncoming mixture. It is shown that one-dimensional enthalpy fluctuation, through its interaction with flame, produces sound waves, and may cause parametric instability of the flame. The mutual coupling between the sound wave and parametric instability is analysed at the instability thresholds.

scholarly journals Linear and weakly nonlinear instability of a premixed curved flame under the influence of its spontaneous acoustic field

2014 ◽  
Vol 758 ◽  
pp. 180-220 ◽  
Author(s):  
Raphaël C. Assier ◽  
Xuesong Wu
Keyword(s):  
Acoustic Field ◽  
Numerical Solutions ◽  
Parametric Instability ◽  
Linear Instability ◽  
Weakly Nonlinear ◽  
Observed Behaviour ◽  
Hydrodynamic Field ◽  
The Stability ◽  
Sivashinsky Equation ◽  
Asymptotic Formulation

AbstractThe stability of premixed flames in a duct is investigated using an asymptotic formulation, which is derived from first principles and based on high-activation-energy and low-Mach-number assumptions (Wu et al., J. Fluid Mech., vol. 497, 2003, pp. 23–53). The present approach takes into account the dynamic coupling between the flame and its spontaneous acoustic field, as well as the interactions between the hydrodynamic field and the flame. The focus is on the fundamental mechanisms of combustion instability. To this end, a linear stability analysis of some steady curved flames is undertaken. These steady flames are known to be stable when the spontaneous acoustic perturbations are ignored. However, we demonstrate that they are actually unstable when the latter effect is included. In order to corroborate this result, and also to provide a relatively simple model guiding active control, we derived an extended Michelson–Sivashinsky equation, which governs the linear and weakly nonlinear evolution of a perturbed flame under the influence of its spontaneous sound. Numerical solutions to the initial-value problem confirm the linear instability result, and show how the flame evolves nonlinearly with time. They also indicate that in certain parameter regimes the spontaneous sound can induce a strong secondary subharmonic parametric instability. This behaviour is explained and justified mathematically by resorting to Floquet theory. Finally we compare our theoretical results with experimental observations, showing that our model captures some of the observed behaviour of propagating flames.

Nonlinear Spectral Properties of Elastic Waves Propagating Along a Pantographic Metamaterial With Local Inertia Amplifiers

2021 ◽  
Author(s):  
Valeria Settimi ◽  
Marco Lepidi ◽  
Andrea Bacigalupo
Keyword(s):  
Elastic Waves ◽  
Degrees Of Freedom ◽  
Invariant Manifolds ◽  
Nonlinear Effects ◽  
Harmonic Waves ◽  
Nonlinear Dispersion ◽  
Asymptotic Approach ◽  
Weakly Nonlinear ◽  
One Dimensional ◽  
Atomic Cell

Abstract Pantographic mechanisms can be introduced in the cellular periodic microstructure of architected metamaterials to achieve functional effects of local inertia amplification. The paper presents a one-dimensional pantographic metamaterial, characterized by an inertially amplified tetra-atomic cell. An internally constrained two-degrees-of-freedom model is formulated to describe the undamped free propagation of harmonic waves in the weakly nonlinear regime. A general asymptotic approach is employed to analytically determine the linear and nonlinear dispersion properties. Analytical, although asymptotically approximate, functions are obtained for the nonlinear wavefrequencies and waveforms, which show significant nonlinear effects including softening/hardening bending of the backbone curves and synclastic/anticlastic curvatures of the invariant manifolds.

Chaos in positive ion–negative ion magnetized plasmas

2020 ◽  
Vol 86 (6) ◽  
Author(s):  
Samiran Ghosh ◽  
Biplab Maity ◽  
Swarup Poria
Keyword(s):  
Nonlinear Wave ◽  
Low Frequency ◽  
Negative Ions ◽  
Dynamical Behaviour ◽  
Negative Ion ◽  
Sound Waves ◽  
Weakly Nonlinear ◽  
Positive Ions ◽  
Experimental Parameters ◽  
Map Analysis

The dynamical behaviour of weakly nonlinear, low-frequency sound waves are investigated in a plasma composed of only positive and negative ions incorporating the effects of a weak external uniform magnetic field. In the plasma model the mass (temperature) of the positive ions is smaller (larger) than that of the negative ions. The dynamics of the nonlinear wave is shown to be governed by a novel nonlinear equation. The stationary plane wave (analytical and numerical) nonlinear analysis on the basis of experimental parameters reveals that the nonlinear wave does have quasi-periodic and chaotic solutions. The Poincarè return map analysis confirms these observed complex structures.

ASYMPTOTICAL ANALYSIS OF ONE DIMENSIONAL GAS DYNAMICS EQUATIONS

2001 ◽  
Vol 6 (1) ◽  
pp. 117-128 ◽  
Author(s):  
A. Krylovas ◽  
R. Čiegis
Keyword(s):  
Gas Dynamics ◽  
Numerical Experiments ◽  
Method Of Averaging ◽  
Almost Periodic Solutions ◽  
Almost Periodic ◽  
Resonance Case ◽  
Gas Dynamics Equations ◽  
Weakly Nonlinear ◽  
One Dimensional ◽  
Asymptotical Analysis

A method of averaging is developed for constructing a uniformly valid asymptotic solution for weakly nonlinear one dimensional gas dynamics systems. Using this method we give the averaged system, which disintegrates into independent equations for the non‐resonance systems. Conditions of the resonance for periodic and almost periodic solutions are presented. In the resonance case the averaged system is solved numerically. Some results of numerical experiments are given.

scholarly journals On sound waves of finite amplitude

1953 ◽  
Vol 216 (1127) ◽  
pp. 539-547 ◽  
Keyword(s):  
Integral Equation ◽  
Analytical Solution ◽  
Arbitrary Function ◽  
Initial Conditions ◽  
Finite Amplitude ◽  
Sound Waves ◽  
One Dimensional ◽  
The One ◽  
Complex Integral ◽  
Gas Cloud

An analytical solution of Riemann’s equations for the one-dimensional propagation of sound waves of finite amplitude in a gas obeying the adiabatic law p = k ρ γ is obtained for any value of the parameter γ. The solution is in the form of a complex integral involving an arbitrary function which is found from the initial conditions by solving a generalization of Abel’s integral equation. The results are applied to the problem of the expansion of a gas cloud into a vacuum.

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