Abstract
The elementary task is to calculate the growth rates of disturbances when the eN method in transition prediction is performed. However, there is no unified knowledge to determine the growth rates of disturbances in three-dimensional (3D) flows. In this paper, we study the relation among the wave parameters of the disturbance in boundary layers in which the imaginary parts of wave parameters are far smaller than the real parts. The generalized growth rate (GGR) in the direction of group velocity is introduced, and the conservation relation of GGR is strictly deduced in theory. This conservation relation manifests that the GGR only depends on the real parts of wave parameters instead of the imaginary parts. Numerical validations for GGR conservation are also provided in the cases of first/second modes and crossflow modes. The application of GGR to the eN method in 3D flows is discussed, and the puzzle of determining growth rates in 3D flows is clarified. A convenient method is also proposed to calculate growth rates of disturbances in 3D flows. Good agreement between this convenient method and existing methods is found except the condition that the angle between the group velocity direction and the x-direction is close to 90° which can be easily avoided in practical application.
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Acknowledgements
The authors are grateful to Ph. D. candidate Dongdong XU of Tianjin University for valuable discussion and suggestions.
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Project supported by the National Natural Science Foundation of China (Nos. 11672351 and 11332007), the National Key R&D Plan (No. 2016YFA0401200), and the FengLei Youth Innovation Fund of China Aerodynamics Research and Development Center (No.KT-FLJJ-201803)
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Song, R., Zhao, L. & Huang, Z. Conservation relation of generalized growth rate in boundary layers. Appl. Math. Mech.-Engl. Ed. 39, 1755–1768 (2018). https://doi.org/10.1007/s10483-018-2394-9
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DOI: https://doi.org/10.1007/s10483-018-2394-9