Numerical Simulations of Unsteady Low-Reynolds-Number Flows Over the APEX Airfoil

Laminar and transitional separation bubbles are an important feature of low-Reynolds- number flows over airfoils. The separation bubbles are unsteady and have a significant impact on the aerodynamic properties of the airfoils. In this paper unsteady low-Reynolds- number separated flows over the APEX airfoil are cal- culated using a Navier-Stokes solver. The numerical re- sults show the presence of unsteady separation bubbles in the flowfield. An analysis of the numerical results shows that flowfield disturbances are amplified signifi- cantly in the separation bubble, leading to periodic vor- tex shedding. A linear stability analysis of the separated boundary layer-is performed and the results show that the dominant wavenumber and frequency in the numer- ical simulations agree with the most unstable wavenum- ber and frequency from the linear stability analysis. The numerical results also show the growth and inter- action of disturbance waves in the separation bubble. For transonic flows over the APEX airfoil, the calcula- tions show that the presence of shocks causes significant changes in the separation location and consequently, the . overall flowfield. dient. The separated boundary layer is unsteady, and usually becomes unstable and rapidly undergoes transi- tion. The turbulent flow reattaches, and the region be- tween the separation point and the reattachment point is called the separation bubble. The separated region is usually highly unstable and characterized by signifi- cant growth of disturbance waves. This is shown in Fig. 1 (b) which illustrates the transition process as the linear growth of disturbances corresponding to the instability of the separated flow, followed by nonlinear interaction of disturbances leading to turbulent flow. The strong influence of the separation bubbles on the performance of airfoils in low-Reynolds- number flows has been shown in many experimental studies. t3'4-7J The structure of the unsteady separation bubbles has been studied through experimental investigations of the growth of disturbance waves and the transition pro- cess. Leblanc, Blackwelder, and Liebeck ^5' measured the dominant frequencies in the velocity spectra in the separated region. They showed that the peak frequen- cies measured in the velocity spectra for the instabil- ity region match the most amplified wavenumber and frequency scaling calculated by linear stability theory. Dovgal, Kozlov, and Michalke ^ also confirmed the linear growth of disturbances in the separated region. They also showed that the subsequent nonlinear interac- tions of the disturbance waves led to transition. An im- portant observation was that the mean flow is changed by the presence of disturbance waves in the flowfield. Drela and Giles I8'12! used a viscous-inviscid approach to compute transonic low-Reynolds-number flows over airfoils. Their calculations show the strong influence of separation bubbles on the overall flowfield, and pre- dicted the rapid degradation in the performance of the airfoils with decreasing Reynolds-numbers. Lin and Pauley '-11-' used unsteady Navier-Stokes simulations to compute low-Reynolds-number flows over airfoils. Their results show that the separation bubble is unsteady with an associated periodic vortex shedding. The dominant frequency was shown to be in agreement with the most amplified frequency from the linear stability analysis, of a mixing layer corresponding to the separated bound- ary layer. The instability of the separated flow and the growth of disturbances in the separation bubbles have been previously studied.'-13-' Hildings ^ studied tran- sitional separation bubbles on a flat plate using direct numerical simulations. His results also show that the growth of small disturbances in the separated region