Conservation relation of generalized growth rate in boundary layers

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.

Improved nonlinear parabolized stability equations approach for hypersonic boundary layers

The nonlinear parabolized stability equations(NPSEs)approach is widely used to study the evolution of disturbances in hypersonic boundary layers owing to its high computational efficiency.However,divergence of the NPSEs will occur when disturbances imposed at the inlet no longer play a leading role or when the nonlinear effect becomes very strong.Two major improvements are proposed here to deal with the divergence of the NPSEs.First,all disturbances are divided into two types:dominant waves and non-dominant waves.Disturbances imposed at the inlet or playing a leading role are defined as dominant waves,with all others being defined as non-dominant waves.Second,the streamwise wavenumbers of the non-dominant waves are obtained using the phase-locked method,while those of the dominant waves are obtained using an iterative method.Two reference wavenumbers are introduced in the phase-locked method,and methods for calculating them for different numbers of dominant waves are discussed.Direct numerical simulation(DNS)is performed to verify and validate the predictions of the improved NPSEs in a hypersonic boundary layer on an isothermal swept blunt plate.The results from the improved NPSEs approach are in good agreement with those of DNS,whereas the traditional NPSEs approach is subject to divergence,indicating that the improved NPSEs approach exhibits greater robustness.

Instability analysis and drag coefficient prediction on a swept RAE2822 wing with constant lift coefficient

Swept wing is widely used in civil aircraft,whose airfoil is chosen,designed and optimized to increase the cruise speed and decrease the drag coefficient.The parameters of swept wing,such as sweep angle and angle of attack,are determined according to the cruise lift coefficient requirement,and the drag coefficient is expected to be predicted accurately,which involves the instability characteristics and transition position of the flow.The pressure coefficient of the RAE2822 wing with given constant lift coefficient is obtained by solving the three-dimensional Navier-Stokes equation numerically,and then the mean flow is calculated by solving the boundary layer（BL） equation with spectral method.The cross-flow instability characteristic of boundary layer of swept wing in the windward and leeward is analyzed by linear stability theory（LST）,and the transition position is predicted by eNmethod.The drag coefficient is numerically predicted by introducing a laminar/turbulent indicator.A simple approach to calculate the lift coefficient of swept wing is proposed.It is found that there is a quantitative relationship between the angle of attack and sweep angle when the lift coefficient keeps constant;when the angle of attack is small,the flow on the leeward of the wing is stable.when the angle of attack is larger than 3°,the flow becomes unstable quickly;with the increase of sweep angle or angle of attack the disturbance on the windward becomes more unstable,leading to the moving forward of the transition position to the leading edge of the wing;the drag coefficient has two significant jumping growth due to the successive occurrence of transition in the windward and the leeward;the optimal range of sweep angle for civil aircraft is suggested.