Analysis and application of relationship between Reynolds number index and Reynolds number ratio
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摘要:
雷诺数指数（RNI）与雷诺数比（RNR）作为常用的雷诺数相关问题重要无量纲数广泛地用于航空发动机研发过程中，然而前者多用于工程研制阶段，后者多用于研究初期阶段，两者长期处于应用层级割裂状态。为了厘清RNI与RNR之间的关系，本文首先用
Π 定理和无量纲数代数推导两种方法推导出了RNI，结果揭示了RNI的物理含义为考虑马赫数修正的RNR，代表了雷诺数强相似性原理；RNR仅基于表速相似，代表了雷诺数弱相似性原理。其次，比较了雷诺数指数与雷诺数比之间的关系，在两者的相对差仅为温度的函数，在工况温度比处于0.94～1.06的范围时，RNI与RNR两者相差±3%以内，认为两者可以互换；因为工况间温度比差距过大目前以RNR为自变量的雷诺数修正公式在实际使用中误差过大的原因之一。最后，给出了一种基于RNI=1.0、保证雷诺数强相似性的冷热态工作点模化方法，作为RNI的应用范例，并给出了两组冷热态换算结果，计算得到冷热态工况的Π 函数是一致的，因此可以认为冷热态模化满足相似性原理。本文分析并解释了RNI与RNR的关系，可作为航空发动机研制各阶段中雷诺数相关问题无量纲参数选取的依据。Abstract:The Reynolds number index (RNI) and Reynolds number ratio (RNR) are commonly used as important dimensionless numbers for Reynolds number related problems in the development of aero-engine. However, the former is mostly used in the engineering development stage and the latter mostly used in the early stage of research, and the two have long been in a fragmented state at the application level. In order to clarify the relationship between RNI and RNR, RNI was firstly derived from two perspectives:
$ \varPi $ theorem and algebraic derivation of dimensionless numbers, whose physical meaning was RNR considering Mach number correction, which represented the Reynolds number strong similarity principle. Secondly, the relationship between RNI and RNR was compared, in which the relative difference between these two was only a function of temperature; and when the difference between RNI and RNR was within the working temperature ratio range of 0.94—1.06, these two were considered interchangeable; given the temperature ratio gap between working conditions was too large, this was one of the reasons why the current Reynolds number correction formula with RNR as the independent variable had too much errors in practice. Finally, the operation point at cold and hot states based on RNI of 1.0 was given as one of the applications of RNI, which guaranteed the strong Reynolds number similarity; and the results of two sets of cold and hot state conversions were given, and the$ \varPi $ functions of cold and hot state conditions were calculated to be consistent, and the cold and hot state modelling was considered to satisfy the similarity principle. The relationship between RNI and RNR explained and analyzed herein can be used as a basis for selection of dimensionless parameters for Reynolds number related problems in all stages of aero-engine development. -
图 1 0～20 km ISA大气环境气体物性及RNR比较
Figure 1. Comparison of atmospheric gas properties and RNR for 0—20 km （ISA condition）
图 4 冷热态工况点换算流程
Figure 4. Working point conversion process from hot state to cold state
表 1 [M−L−T]系统下叶轮机械表征参数
Table 1. Characterization parameters of turbomachinery under [M−L−T] system
特性类型 参数 符号 量纲 机械特征 特征直径/尺寸 $ D $ $ [L] $ 转速 $ N $ $ [{T^{ - 1}}] $ 工质物性 等熵指数 $ \kappa $ $ [1] $ 气体常数 $ R $ $ [1] $ 动力黏度 $ \,\mu $ $ [M{L^{ - 1}}{T^{ - 1}}] $ 流动状态 进口总压 $ p_1^* $ $ [M{L^{ - 1}}{T^{ - 2}}] $ 进口总温 $ T_1^* $ $ [{L^2}{T^{ - 2}}] $ 出口总压 $ p_2^* $ $ [M{L^{ - 1}}{T^{ - 2}}] $ 质量流量 $ \dot m $ $ [M{T^{ - 1}}] $ 特性参数 比焓升/降 $ \Delta h $ $ [{L^2}{T^{ - 2}}] $ 效率 $ \eta $ $ [1] $ 
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表 2 某低压涡轮冷热态模化结果
Table 2. Results of simularity under hot and cold conditions for a certain low pressure turbine
项目 热态[25] 冷态 给定转速 给定温度 进口质量流量/（kg/s） 4.200 2.333 2.558 进口总压/MPa 0.300 0.0926 0.111 进口总温/K 1078.000 332.716 400.000 膨胀比$ \pi （ {{\varPi _{p_2^*}}} ） $ 2.764 2.764 2.764 转子转速/（r/min） 21600.000 12000.000 13157.530 $ \,\mu $/10−5 （Pa·s） 5.110 2.840 3.112 $ {\varPi _N} $ 657.876 657.876 657.876 $ {\varPi _{\dot m}} $ 459.661 459.661 459.661 $ {\varPi _{\mu} } $ 178.824 178.824 178.824
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