Introduction

Understanding and controlling the transition from laminar to turbulent flow is necessary for optimal designs in aeronautical industries where the lower drag that characterizes the laminar flows is sought. The process of transition in boundary layers starts from the moment an external disturbance enter the boundary layer and generates instabilities. This process is commonly known as Receptivity. In the case where the disturbance is an acoustic wave, energy is transferred from the acoustic wave to Tollmien-Schlichting (TS) instabilities through wavelength scattering such as in the presence of surface inhomogeneity with a finite height where the boundary layer is forced to make a rapid change in order to adjust itself. This paper deals with the numerical simulation of the receptivity of a subsonic boundary layer to an external acoustic perturbation.

 Methodology

The numerical simulation is largely inspired from the experimental work of (Saric et al. 1991) The roughness is located at 0.46m of the leading edge which has a Reynolds number 1015 based on the BL displacement thickness and Frequency parameter of 49.33 10-6. The Navier-Stokes equations are expressed in conservative form and are solved using a time accurate Navier Stokes code . Viscous terms are approximated using central differencing while the convective terms are discretized using a third order MUSCL TVD scheme in conjunction with a Riemann solver. The inflow boundary condition is a first order non-reflective boundary based on perturbations around the Blasius velocity profile. The boundary condition on the top sets u and v velocity gradients to zero and with the pressure calculated from the appropriate characteristic compatibility relations. A non-reflective outflow boundary condition was imposed on exit zone.

Results

A converged steady state without any acoustic source is obtained using the implicit approximately factored algorithm and then the calculation is restarted using an explicit Runge-Kutta method. This converged steady state will provide a baseline against which to compare the unsteady solution. During the second stage of the explicit computation, the acoustic source is triggered. The obtained solution is a blend of the base flow; the unsteady stokes component as well as the TS component. The Stokes solution is obtained when using the acoustic source that interacts with the boundary layer flow of a homogeneous flat plate (without roughness) . A sample of the u'TS at a fixed value of y/Lref =0.00078 for different x location is shown in figure 1. The exponential amplification of the u'TS component is clear from plot. Likewise, a plot of the u'TS versus the y elevation is presented in figure 2. These two plots exhibit the same behavior and compare favorably to the largely published results. The most unstable TS obtained from figure 2 is equal to 0.054 which is similar to the analytical value based on the linear stability theory.