Influence of leading-edge curvature on excited unsteady cross-flow vortices in three-dimensional boundary-layer

Abstract

Three-dimensional boundary-layer receptivity is the first stage of the laminar-turbulent transition in a three-dimensional boundary layer, and also a key issue for predicting and controlling the laminar-turbulent transition in the three-dimensional boundary layer. At a high turbulence level, the three-dimensional boundary-layer instability in the transition is caused mainly by the unsteady cross-flow vortices. And the leading-edge curvature has a significant influence on three-dimensional boundary-layer receptivity. In view of this, the direct numerical simulation is utilized in this paper to study the mechanism of receptivity to exciting unsteady cross-flow vortices in the three-dimensional (swept-plate) boundary layer with various elliptic leading edges. In order to solve the Navier-Stokes equation numerically, a modified fourth-order Runge-Kutta scheme is introduced for discretization in time; high-order compact finite difference schemes are utilized for discretization in the x-and y-direction; and Fourier transform is used in the z-direction. The pressure Helmholtz equation is solved by a fourth-order iterative scheme. Additionally, the numerical calculation is performed in the curvilinear coordinate system via Jaccobi transform. And the elliptic equation technique is used to gene-rate the body-fitted mesh. The effect of leading-edge curvature on the propagation speed and direction, distribution and receptivity coefficient of the excited unsteady cross-flow vortex wave packet, and the amplitude, dispersion relation and growth rate of the extracted unsteady cross-flow vortex are revealed. In addition, the inner link among the receptivity to unsteady cross-flow vortex, intensity, and direction of free-stream turbulence is established. Furthermore, the receptivity to anisotropic free-stream turbulence is also analyzed in detail. The numerical results indicate that the more intense receptivity to the unsteady cross-flow vortex wave packets is triggered with a smaller leading-edge curvature; whereas, the less intense receptivity is triggered with a greater leading-edge curvature. The receptivity to the unsteady cross-flow vortex wave packets in different curvatures are also found to vary with the angle of free-stream turbulence. Moreover, the anisotropic degree of free-stream turbulence can affect the excitation of the unsteady cross-flow vortex obviously. Through the above study, a further step can be taken to understand the prediction and control of laminar-turbulent transition in the three-dimensional boundary layer and also improve the theory of the hydrodynamic stability.