This work performs a theoretical and numerical analysis of the communication between one-dimensional acoustic and entropy waves in a duct with a mean temperature gradient. Such a situation is highly relevant to combustor flows where the mean temperature drops axially due to heat losses. A duct containing a compact heating element followed by an axial temperature gradient and choked end is considered. The proposed jump conditions linking acoustic and entropy waves on either side of the flame show that the generated entropy wave is generally proportional to the mean temperature ratio across the flame and the ratio ( F - 1 ) , where F is the flame transfer function. It is inversely proportional to the Mach number immediately downstream of the flame M2. The acoustic and entropy fields in the region of axial mean temperature gradient are calculated using four approaches: (1) using the full three linearised Euler equations as the reference; (2) using two linearised Euler equations in which the acoustic and entropy waves are assumed independent (thus allowing the extent of communication between the acoustic and entropy wave to be evaluated); (3) using a Helmholtz solver which neglects mean flow effects and (4) using a recently developed analytical solution. It is found that the communication between the acoustic and entropy waves is small at low Mach numbers; it rises with increasing Mach number and cannot be neglected when the mean Mach number downstream of the heating element exceeds 0.1. Predictions from the analytical method generally match those from the full three linearised Euler equations, and the Helmholtz solver accurately determines the acoustic field when M 2 ≤ 0 . 1 .

中文翻译：

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轴向平均温度梯度对一维声波和熵波之间通信的影响

这项工作对管道中具有平均温度梯度的一维声波和熵波之间的通信进行了理论和数值分析。这种情况与燃烧器流量高度相关，在燃烧器流量中，平均温度由于热量损失而轴向下降。考虑一种管道，该管道包含紧凑的加热元件，其后是轴向温度梯度和阻塞的末端。提出的将火焰两侧的声波和熵波联系起来的跳跃条件表明，所产生的熵波通常与整个火焰的平均温度比和比率（F-1）成正比，其中F是火焰传递函数。它与紧邻火焰M2下游的马赫数成反比。使用四种方法计算轴向平均温度梯度区域中的声场和熵场：（1）使用全部三个线性化的Euler方程作为参考；（2）使用两个线性化的欧拉方程，其中假设声波和熵波是独立的（因此可以评估声波和熵波之间的连通程度）；（3）使用Helmholtz求解器忽略了平均流量影响，（4）使用最新开发的分析解决方案。发现在低马赫数下，声波和熵波之间的通信很小。它随着马赫数的增加而增加，并且当加热元件下游的平均马赫数超过0.1时不能忽略。分析方法的预测通常与全部三个线性化的Euler方程的预测匹配，当M 2≤0时，亥姆霍兹求解器可以精确地确定声场。1。