Identification of nonlinear characteristics of thermoacoustic oscillations in helium piping systems

Abstract

Insufficient dynamic information on helium piping systems limits our ability to understand and identify thermoacoustic oscillation (TAO) phenomena in these systems. In this study, we investigate the associated oscillation behaviours and the effects of the temperature distribution steepness, S, and the inflection point position, x₀, on the TAO characteristics through numerical simulations. The results indicate that x₀ has a greater impact on the oscillation behaviour and is a decisive factor for oscillation-free operation. It is found that oscillation-free operation can be achieved when x₀ = 0.9 m. To identify the nonlinear dynamic characteristics related to TAO, the original pressure time series are unfolded in phase portraits and Poincaré maps. In contrast to the period-1 and period-2 limit circle oscillations at other positions, the trajectory at x₀ = 0.9 m displays a transition from limit circles to a fixed point (zero amplitude), corresponding to a series of points including the point (0,0) in the Poincaré map. Therefore, an approach of adjusting the inflection point position can be used to control the state of the TAO, and the transition in the nonlinear dynamic characteristics can be identified, which can also provide guidelines for designers to mitigate unfavourable TAO.

Introduction

Self-excited thermoacoustic oscillation (TAO), which is also known as thermoacoustic instability, causes consistent problems such as Taconis oscillations in cryogenic piping systems [1]. Taconis oscillations are a phenomenon that occur in tubes that open to the cryogenic environment and expose their closed end to the ambient environment [2,3]. Large-amplitude pressure oscillations are the main characteristic of instability, which often results in undesired noise, energy dissipation, and equipment vibration. However, in practical applications, stability is an essential requirement for many cryogenic systems, e.g. temperature and pressure control in single-pressure refractive-index gas thermometry (SPRIGT) systems [[4], [5], [6]] and the commissioning stage of a superconducting linear accelerator (linac) [7]. Therefore, it is necessary to establish the operating conditions corresponding to the thermoacoustic instability state to predict the realisation of an oscillation-free cryogenic system.

To estimate the occurrence of TAO in cryogenic systems, Rott [8] proposed the stability curves criterion. Experimental verification of these stability curves has been performed by many researchers [9,10], who have concluded that TAO can be predicted based on the Rott stability curves. Based on the parameter Y_c at the junction of the two branches of the Rott stability curve, Prabhat [11] designed oscillation-free systems. Because the angular frequency and sound speed depend on the physical properties of helium and the temperature distribution, the value of λ_c in the Rott mathematical formulation is an estimated value. Studies employing the Rott stability curves have typically focused on linear stability analyses, whereas nonlinear interaction mechanisms have rarely been reported. However, the design and control parameters of helium piping systems are sensitive to the existence of nonlinear dynamic behaviour. Successful simulation of the TAO process with computational fluid dynamics (CFD) [12] can facilitate the realisation of nonlinear state capture. Hence, the dynamic pressure time-series of the self-excited oscillation in helium piping systems will be obtained using the CFD method in the current study.

To represent and analyse the dynamic behaviour and evolution process of the TAO system, nonlinear time series (NTS) analysis is used, which is a rich field involving many methods adapted from dynamic systems [13,14]. Gotoda et al. [15] characterised the nonlinear dynamic behaviour of combustion instability and showed that NTS was capable of characterising complexities in thermoacoustic instability. In another study [16], recurrent quantification analysis was used to reveal the phase space corresponding to the pressure fluctuation. The existence of high-dimensionality and the pseudo-periodic nature of the dynamic behaviour of thermoacoustic instability was revealed using cyclic networks and phase-space networks from the pressure time-series [17]. To reduce the thermoacoustic amplitude, Guan et al. [[18], [19], [20]] studied the control of self-excited TAO in a tube combustor system subjected to various forcing types, including periodic forcing, transient forcing, hysteresis, and model switching. Sujith et al. [21] demonstrated that the nonlinear behaviour was highly sensitive to many parameters, which were sensitive to changes in the system design. They [22] further discussed various prognosis and mitigation strategies for thermoacoustic state-based complex system theory. Ni et al. [23] found that the generation of nonlinear pulsating oscillation depends on the mass flow, and variations in the temperature can decrease the amplitude of the pulsating oscillations. However, in cryogenic helium piping systems, oscillation state switching due to variations in a control parameter has not been previously reported. Crucially, it has yet to be shown how state switching can be integrated into an oscillation control strategy in such a way that TAO will be weakened. Therefore, this study will focus on identification of the nonlinear characteristics of TAO in helium piping systems using NTS analysis.

The pressure time-series of the self-excited oscillation in the system is defined as P_t = [ P₁, P₂, P₃, ⋯, P_n], which is usually regarded as a one-dimensional measurement of a dynamic system. According to chaos theory, a set of variables is employed in a phase-space reconstruction (PSR) diagram to demonstrate the dynamic system. Each point in the PSR diagram represents the dynamic behaviour state of the system corresponding to a specific time period. The hidden dynamic information in the system can be extracted from the pressure time-series by the PSR. Embedding the delay time in the PSR diagram will map the signal to a high dimension, where the embedding emphasises iteration. The embedding dimension, m, and the delay time, τ, are two important parameters for reconstructing the phase space based on the NTS analysis. When reconstructing a proper phase space, the parameter τ must be selected before the minimum embedding dimension because the minimum embedding dimension depends on the delay time. To select the appropriate delay time, the average mutual information (AMI) function [24] is applied because it can account for both linear and nonlinear correlations, and each component of the reconstructed vectors is independent.

The purpose of the embedding dimension is to embed the data in a time-delayed space with a suitable dimension. An appropriate dimension can unfold the full dynamics of the system. If the embedding dimension is too small, distant points on the attractor will overlap in the embedding space like the two-dimensional shadow of a three-dimensional object; otherwise, the embedding space may be poorly reconstructed and more calculations will be needed. As such, it is important to choose a minimum embedding dimension to construct a phase space in which no false neighbours exist. The correlation dimension (CD) [25,26] and false nearest neighbours (FNN) [27] are popular methods for determining the minimum embedding dimension. Whether a neighbour is “false” depends on the distance between a point and the reference point. When the fraction of FNN is equal to zero, the minimum embedding dimension is defined. In the FNN algorithm, two criteria [28] are used to designate which nearest neighbour is false by comparing R_m+1²(n) to R_m²(n) when they are not independent of each other. Therefore, the typical problem with the FNN method is that the data are intensive and certainly subjective. To identify the dynamic behaviour in a helium piping system, this study adopts the CD method to determine the minimum embedding dimension. Chaotic analysis of the helium piping system can indicate the fractal dimension needed to illustrate a dynamic system by estimating the CD. It can be simplified to provide an explanation of a complex TAO system, thereby providing state estimation for nonlinear dynamic models.

Our aim in this study is to select an appropriate temperature distribution to accomplish oscillation-free operation of the cryogenic helium piping system. Theoretical analysis and numerical simulations are introduced to investigate the oscillation states of the system. As determinant factors of the temperature distribution, the effects of the steepness, S, and the inflection point position, x₀, on the TAO characteristics are revealed. Then, the nonlinear dynamic characteristics of the pressure oscillations at different inflection point positions are revealed through NTS analysis. The original pressure time series are unfolded in the embedding space and displayed in phase portraits and Poincaré maps. The results show that the nonlinear dynamics related to the TAO can be identified, which can also provide guidelines for designers to mitigate unfavourable TAO.