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Conservation relation of generalized growth rate in boundary layers

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Applied Mathematics and Mechanics(English Edition) 2018年第12期39卷 1755-1768页

作者： Runjie SONG Lei ZHAO Zhangfeng HUANG Department of Mechanics Tianjin University Hypervelocity Aerodynamics Institute China Aerodynamics Research and Development Center

The elementary task is to calculate the growth rates of disturbances when the e;method in transition prediction is performed. However, there is no unified knowledge to determine the growth rates of disturbances in three-dimensional（3 D） 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 3 D flows is discussed, and the puzzle of determining growth rates in 3 D flows is clarified. A convenient method is also proposed to calculate growth rates of disturbances in 3 D 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.

关键词： generalized growth rate(GGR) boundary layer stability e^N method

NONLINEAR EVOLUTION ANALYSIS OF T-S DISTURBANCE WAVE AT FINITE AMPLITUDE IN NONPARALLEL boundary layerS

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Applied Mathematics and Mechanics(English Edition) 2002年第6期23卷 660-669页

作者： TANG Deng-bin(唐登斌) XIA Hao(夏浩) Department of Aerodynamics Nanjing University of Aeronautics and Astronautics Nanjing 210016 P R China Department of Aerodynamics Nanjing University of Aeronautics and Astronautics Nanjing 210016 P R China

The nonlinear evolution problem in nonparallel boundary layer stability was studied. The relative parabolized stability equations of nonlinear nonparallel boundary layer were derived. The developed numerical method, which is very effective, was used to study the nonlinear evolution. of T-S disturbance wave at finite amplitudes. Solving nonlinear equations of different modes by using predictor-corrector and iterative approach, which is uncoupled between modes, improving computational accuracy by using high order compact differential scheme, satisfying normalization condition I determining tables of nonlinear terms at different modes, and implementing stably the spatial marching, were included in this method. With different initial amplitudes, the nonlinear evolution of T-S wave was studied. The nonlinear nonparallel results of examples compare with data of direct numerical simulations (DNS) using full Navier-Stokes equations.

关键词： boundary layer stability nonlinear evolution nonparallelism T-S disturbance wave compact scheme spatial mode parabolized stability equation

On Nonlinear Evolution of C-type Instability in Nonparallel boundary layers

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Chinese Journal of Aeronautics 2007年第4期20卷 313-319页

作者： Liu Jixue Tang Dengbin YangYingzhao Department of Aerodynamics Nanjing University of Aeronautics and Astronautics Nanjing 210016 China Chengdu Aircraft Design Institute Chengdu 610041 China

The process of evolution, especially that of nonlinear evolution, of C-type instability of laminar-turbulent flow transition in nonparallel boundary layers are studied by means of a newly developed method called parabolic stability equations （PSE）. Initial conditions, which are very important for the nonlinear problem, are investigated by computing initial solution of the harmonic waves, modifying the mean-flow-distortion, and giving initial value of TS wave and its subharmonic waves at initial station by solving linear PSE. A numerical method with high-order accuracy are developed in the text, the key normalization conditions in the PSE are satisfied, and nonlinear PSE are solved efficiently and implemented stably by the spatial marching. It has been shown that the computed process of nonlinear evolution of C-type instability in Blasius boundary layer is in good agreement with the experimental results.

关键词： C-type instability nonlinear evolution nonparallelism boundary layer stability parabolic stability equations