Analysis and application of relationship between Reynolds number index and Reynolds number ratio

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.