In the present work, we formulate a new method to represent a given Flame Describing Function by analytical expressions. The underlying idea is motivated by the observation that different types of perturbations in a burner travel with different speeds and that the arrival of a perturbation at the flame is spread out over time. We develop an analytical model for the Flame Describing Function, which consists of a superposition of several Gaussians, each characterised by three amplitude-dependent quantities: central time-lag, peak value and standard deviation. These quantities are treated as fitting parameters, and they are deduced from the original Flame Describing Function by using error minimisation and nonlinear optimisation techniques. The amplitude-dependence of the fitting parameters is also represented analytically (by linear or quadratic functions). We test our method by using it to make stability predictions for a burner with well-documented stability behaviour (Noiray's matrix burner). This is done in the time-domain with a tailored Green's function approach.

中文翻译：

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具有幅度相关时滞分布的非线性分析火焰模型

在当前的工作中，我们制定了一种新方法，通过解析表达式来表示给定的火焰描述函数。潜在的想法是由以下观察得出的：燃烧器中不同类型的扰动以不同的速度行进，并且扰动到达火焰的时间随时间扩展。我们为火焰描述函数开发了一个分析模型，该模型包含几个高斯的叠加，每个高斯的特征在于三个与振幅有关的量：中心时滞，峰值和标准偏差。这些量被视为拟合参数，并通过使用误差最小化和非线性优化技术从原始的火焰描述函数推导得出。拟合参数的幅度依赖性也可以解析地表示（通过线性或二次函数）。我们通过使用它对具有良好记录的稳定性行为的燃烧器（Noiray矩阵燃烧器）进行稳定性预测来测试我们的方法。这是在时域中使用量身定制的格林函数方法完成的。