Bistability and Triggering Analysis of Thermoacoustic Instability in a Horizontal Rijke Tube

水平Rijke管热声不稳定的双稳态和触发分析

Science and Technology on Combustion, Internal Flow and Thermo-Structure Laboratory, Northwestern Polytechnical University, Xi’an 710072, China

Received: 9 December 2017

Abstract

Dynamical systems theory has been often employed to study nonlinear flow and flame dynamics in combustion systems. However, the corresponding studies using nonlinear dynamics to analyze the Rijke tube thermoacoustic system are still occasional. Little study has been performed to elucidate the characteristics of triggering phenomenon in the bistable region of the thermoacoustic system. In this regard, the main objectives of the present research are to contribute analysis for the lack of literature in these areas, especially to study the bistability and triggering properties of a thermoacoustic system. The thermoacoustic model of a horizontal Rijke tube is firstly established. The governing equations are expanded and solved by using Galerkin method. The analysis of the system is carried out by using nonlinear dynamics theory. Linear and nonlinear stability boundaries for the variation of non-dimensional heater power, damping coefficient and the relative heater location are obtained for different values of non-dimensional time lag in the system. Regions of global stability, global instability and bistability are characterized. The bistable region in the relative heater location is distributed symmetrically with x_f=0.25. It is observed that the bistable region in the relative heater location firstly decreases with an increase in the non-dimensional time lag, reaching a minimum value at a certain critical value of τ, then increases. The situation for the bistable region in the damping coefficient presents a reverse variation, And the bistable region reach the maximum at τ=0.5. The triggering phenomenon and limit cycle of the system in the bistable region are studied. The critical triggering values are determined with the changes of the non-dimensional heater power, the damping coefficient and the relative heater location. The critical triggering value of velocity perturbation decreases with the increase of non-dimensional heater power, whereas an increasing trend is observed with the increase of damping coefficient. Interestingly, the critical triggering value firstly decreases and then increases with the increase of the relative heater location. The variation of critical triggering value for pressure perturbation is found to correspond with velocity perturbation. In the bistable region, the amplitude and frequency of the steady limit cycle oscillation of the system are independent of the initial perturbation values, but the perturbation value has strong effect on the duration needed to achieve the steady limit cycle, and the time required for the system to reach the limit cycle under the perturbation of U₁=0.4 is about 3 times longer than that of U₁=0.8.

摘要

建立了水平Rijke管热声模型，并利用Galerkin方法对控制方程进行展开，实现数值求解。利用非线性动力学理论对系统进行分析，得到系统的全局稳定区域、全局不稳定区域以及双稳态区域。获得了无量纲加热功率K、热源相对位置x_f、阻尼系数c₁与无量纲时间延迟τ之间的稳定区域图谱。发现热源相对位置x_f的稳定性区域关于x_f=0.25近似呈对称分布，阻尼系数c₁的双稳态区域在τ=0.5时达到最大。研究了系统在双稳态区域内的触发和极限环振荡现象，获得无量纲加热功率K、阻尼系数c₁和热源相对位置x_f等参数变化时的临界触发值。发现系统的临界触发值P₁与U₁具有一致的变化规律，其随无量纲加热功率K的增大而减小，但随阻尼系数c₁的增大呈现增大趋势。特别的，临界触发值随热源相对位置x_f的增大呈现先减小后增大的趋势。在双稳态区域内，系统稳定极限环振荡的振幅和频率与初始扰动值无关，但扰动值会影响系统达到稳定极限环的时间，系统在U₁=0.4扰动下达到极限环所需时间比U₁=0.8延长约3倍。