Estimation of limit cycle amplitude after onset threshold of thermoacoustic Stirling engine

Empirical Modeling of Subcritical Hopf Bifurcation of Thermoacoustic Stirling Engine

“…Note that Equation (5) can also be determined using MATLAB's built-in function "gradient". This experimental data analysis provides insight into the oscillation's growth rate, ω I , realizing from the Landau amplitude Equation (3). By plotting d(ln|A(t)|)/dt against |A(t)| 2 , as shown in the latter, we can ascertain the Landau constants from Equation (3) through least-squares fitting of polynomial curves.…”

“…It is important to note that we preprocess the measured data by applying band filtering to p ′ (t) to reduce noise. We isolated measured p ′ (t) within the 22 to 26 Hz range, determined by the engine's spontaneous oscillation frequency, roughly estimated at 24.1 Hz using a/(L + L b ) × 1/4 [3], where a is the adiabatic sound speed at room temperature.…”

“…The certain values of parameters at which these changes occur are known as bifurcation points. Understanding bifurcations is essential for gaining insights into the dynamic behavior of thermoacoustic engines and crucial for predicting and managing thermoacoustic instability Within thermoacoustic engines, both supercritical and subcritical Hopf bifurcations are observed [3]. An increase in the temperature gradient results in a supercritical Hopf bifurcation, leading to a gradual increase in oscillation amplitude, whereas a subcritical Hopf bifurcation causes an abrupt amplitude jump at the Hopf point, where the system's stable steady-state becomes unstable and a stable limit cycle emerges.…”

“…An increase in the temperature gradient results in a supercritical Hopf bifurcation, leading to a gradual increase in oscillation amplitude, whereas a subcritical Hopf bifurcation causes an abrupt amplitude jump at the Hopf point, where the system's stable steady-state becomes unstable and a stable limit cycle emerges. Subcritical cases have been extensively documented across various thermoacoustic engine configurations, including standing wave, traveling wave, phase change, and free-piston Stirling types [3][4][5][6][7][8][9][10][11]. Furthermore, a decrease in the temperature gradient reveals hysteresis in oscillation amplitude with subcritical Hopf bifurcations, eventually ceasing through a saddle-node bifurcation at a distinct fold point [2], which signifies the minimum temperature gradient necessary for engine operation.…”

“…While computational fluid dynamics (CFD) methods effectively capture these nonlinear phenomena [15], they do come with significant computational costs. Recently, we have refined Rott's model by integrating empirically derived nonlinear flow resistance [3]. This addition temporarily characterizes the reduction of the linear growth rate, enhancing Rott's frequency theory to better describe the evolution of amplitude saturation from onset to limit cycles.…”