Effect of leading-edge curvature variation on flat-plate boundary-layer receptivity

Abstract

Boundary-layer receptivity is the initial stage of the laminar-turbulent transition, which is the key step to implement the prediction and control of laminar-turbulent transition in the boundary layer. Current studies show that under the action of acoustic wave or vortical disturbance, the variation of leading-edge curvature significantly affects the boundary-layer receptivity. Additionally, the free-stream turbulence is universal in nature. Therefore, direct numerical simulation is performed in this paper to study the receptivity to free-stream turbulence in the flat-plate boundary layer with an elliptic leading edge. To discretize the Navier-Stokes equation, a modified fourth-order Runge-Kutta scheme is introduced for the temporal discretization; the high-order compact finite difference scheme is utilized for the x- and y-direction spatial discretization; the Fourier transform is conducted in the z-direction. The pressure Helmholtz equation is solved by iterating a fourth-order finite difference scheme. In addition, the Jaccobi transform is introduced to convert the curvilinear coordinate system into Cartesian coordinate system. And elliptic equation technique is adopted to generate the body-fitted mesh. Then the effect of elliptic leading-edge curvature on the receptivity mechanism and the propagation speed of the excited Tollmien-Schlichting (T-S) wave packet in the flat-plate boundary layer are revealed. Subsequently, a group of multi-frequency T-S waves is extracted from the T-S wave packets by temporal fast Fourier transform. The influences of different leading-edge curvatures on the amplitudes, dispersion relations, growth rates, phases and shape functions of the excited T-S waves are analyzed in detail. Finally, the position occupied by leading-edge curvature in the boundary-layer receptivity process for the excitation of T-S wave is also confirmed. The numerical results show that the more intensive receptivity is triggered in the smaller leading-edge curvature; on the contrary, the less intensive receptivity is triggered in the greater leading-edge curvature. But in different leading-edge curvatures, the structures of the excited T-S wave packets are almost identical, and the group velocity is close to constant, which is approximate to one-third of the free-stream velocity. Similarly, the greater amplitude of the excited T-S wave can be induced with the smaller leading-edge curvature; whereas the smaller amplitude of the excited T-S wave can be induced with the greater leading-edge curvature. Moreover, the dispersion relations, growth rates, phases and shape function of the excited T-S waves in the boundary layer are found to be nearly invariable in different leading-edge curvatures. Through the above study, a further step can be made to understand the boundary-layer leading-edge receptivity and also improve the theory of the hydrodynamic stability.