The leading-edge stall of airfoils with blunt noses at low to moderately high Reynolds numbers

Abstract

The inception of leading-edge stall on stationary, smooth thin airfoils with various blunt nose shapes of the form $y=\pm k(ax)^{\frac{1}{a}}$ (where $a \geq 2$ and $k$ is a constant) at low to moderately high Reynolds numbers ($Re$) is studied. A reduced-order, multi-scale model problem is developed for relatively low $Re$ flows and is complimented by numerical computations using a Reynolds-Averaged Navier-Stokes (RANS) flow solver for moderately high $Re$. The asymptotic theory demonstrates that a flow about a thin airfoil may be described in terms of an outer region, around most of the airfoil's chord, and an inner region, around the nose, that asymptotically match each other. The flow in the outer region is dominated by the classical thin airfoil theory. Scaled (magnified) coordinates and a modified (smaller) Reynolds number ($Re_M$) are used to correctly account for the nonlinear behavior and acute velocity changes in the inner region, where both the near-stagnation and high-suction areas appear. The far-field of the inner region is described by a symmetric effect due to nose shape and an asymmetric effect with a lumped circulation parameter ($\tilde{A}$) due to angle of attack and camber.

The inner flow problem is solved numerically using a transformation from the physical domain to a computational domain and a second-order finite-difference scheme for integrating the vorticity and stream function. The computed results demonstrate numerical convergence with mesh refinement. The inner region solutions reveal, for various values of $a$, the nature of the flow around the nose and the inception of global separation and stall as $\tilde{A}$ increases above a certain critical value, $\tilde{A}_s$, at fixed $a$ and $Re_M$. For $a \geq 2$, the value of $\tilde{A}_s$ decreases with $Re_M$ up to a limit value, $Re_{M,lim}$, above which unsteady effects increase $\tilde{A}_s$ and delay the onset of stall. For airfoils with the same thickness ratio and position of maximum thickness, global stall is delayed to higher angles of attack as $a$ is increased above 2. The results of the RANS computations for various $a$ show matching with the asymptotic results in a certain region of $Re_M$ values, as well as extend the stall predictions of $\tilde{A}_s$ to higher $Re_M$. Parametric studies provide data for the design of novel airfoils with blunt noses and higher stall angles of attack at various $Re$.