Stall of airfoils with blunt noses at low to moderately high Reynolds numbers

The onset of leading-edge stall on stationary, smooth, thin, two-dimensional airfoils with various blunt nose shapes of the form $y=\pm k{\left(ax\right)}^{\frac{1}{a}}$ (where $a\ge 2$ and $k$ is a constant) at moderately high chord Reynolds numbers (Re) is studied. A reduced-order, multiple-scale model problem is developed and is complimented by direct numerical simulations for low Re and numerical computations using a Reynolds-averaged Navier-Stokes (RANS) flow solver for moderately high Re. The asymptotic theory results in a description of the flow around a thin airfoil composed of an outer region about a majority of the airfoil's chord, and an inner region, surrounding the nose, that match each other. The classical thin airfoil theory dominates the outer region. The coordinates in the inner region are scaled with respect to a characteristic length of the nose and the Reynolds number is modified $\left({\text{Re}}_M\right)$ in order to account for the acute velocity changes in the inner region, where both 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 $\left(\tilde{A}\right)$ 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 ${\text{Re}}_M$. For $a\ge 2$, the value of ${\tilde{A}}_s$ decreases with ${\text{Re}}_M$ up to a limit value, ${\text{Re}}_{M,\lim }$, 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 ${\text{Re}}_M$ values, as well as extend the stall predictions of ${\tilde{A}}_s$ to higher ${\text{Re}}_M$. Parametric studies provide data for the design of novel airfoils with blunt noses and higher stall angles of attack at various Re.

Figure 1

The physical model and various regions studied.

Figure 2

(a) Computational domain with $\mathrm{\Psi }$ contours and (b) physical domain for the case of $a=2$ at ${\text{Re}}_M=100$ and $\tilde{A}=1.5$. The bottom of panel (a), the $\mu$ axis, represents the upper (positive) and lower (negative) surfaces of the canonic nose. The top of panel (a) represents the far field away from the nose toward infinity. The left and right sides of panel (a), the $\eta$ axis, represent the far field away from the nose. The boundaries of panel (b) are the nose wall and the inflow and outflow surfaces.

Figure 3

Results of mesh convergence studies for the cases with $a=2.5$ (a ${\left(2.5x^{*}\right)}^{\frac{2}{5}}$ nose), ${\text{Re}}_M=100,$ and (a) $\tilde{A}=1.75$, (b) $\tilde{A}=1.8$.

Figure 4

Comparison of $\mathrm{\Psi }$ contour lines for the case with $a=2.5$ (a ${\left(2.5x^{*}\right)}^{\frac{2}{5}}$ nose), ${\text{Re}}_M=100,$ and $\tilde{A}=1.8$. The red lines are results from the mesh with $2M=200$ by $N=200$. The black lines are results from the mesh with $2M=200$ by $N=400$.

Figure 5

Results of mesh convergence studies for the cases with $a=2.5$ (a ${\left(2.5x^{*}\right)}^{\frac{2}{5}}$ nose), ${\text{Re}}_M=900,$ and $\tilde{A}=1$.

Figure 6

Comparison of $\mathrm{\Psi }$ contour lines for the case with $a=2.5$ (a ${\left(2.5x^{*}\right)}^{\frac{2}{5}}$ nose), ${\text{Re}}_M=900,$ and $\tilde{A}=1$. The red lines are results from the mesh with $2M=200$ by $N=200$. The black lines are results from the mesh with $2M=200$ by $N=400$.

Figure 7

The $\mathrm{\Psi }$ contour lines for the case with $a=2.5$ (a ${\left(2.5x^{*}\right)}^{\frac{2}{5}}$ nose), ${\text{Re}}_M=900,$ and $\tilde{A}=1.5$ computed with the mesh $2M=200$ by $N=400$ at $t=500$.

Figure 8

Mesh refinement study of FFT results at long time flow behavior at ${\text{Re}}_M=900$ and $\tilde{A}=1.5$ during the time interval $500\le t\le 600$ from three meshes, $2M=200$ by $N=200,2M=400$ by $N=200$, and $2M=400$ by $N=400$ at points (a) $\mu =2.6,\eta =1.05$ and (b) $\mu =6.6,\eta =1.15$.

Figure 9

Long-term states with $a=2$ (a ${\left(2x^{*}\right)}^{\frac{1}{2}}$ nose), ${\text{Re}}_M=100,$ and (a) $\tilde{A}=0$, (b) $\tilde{A}=1.3$, (c) $\tilde{A}=1.4$, (d) $\tilde{A}=1.77$.

Figure 10

Long-term states with $a=2.5$ (a ${\left(2.5x^{*}\right)}^{\frac{2}{5}}$ nose), ${\text{Re}}_M=100,$ and (a) $\tilde{A}=1.5$, (b) $\tilde{A}=1.6$, (c) $\tilde{A}=1.65$, (d) $\tilde{A}=1.7$, (e) $\tilde{A}=1.75$, (f) $\tilde{A}=1.8$.

Figure 11

Long-term states with $a=2.5$ (a ${\left(2.5x^{*}\right)}^{\frac{2}{5}}$ nose), ${\text{Re}}_M=500,$ and (a) $\tilde{A}=1.1$, (b) $\tilde{A}=1.2$, (c) $\tilde{A}=1.3$, (d) $\tilde{A}=1.4$.

Figure 12

Separated, unsteady flow state that is on the verge of global stall for a nose with $a=2.5$ at ${\text{Re}}_M=1000$ and $\tilde{A}=1.7$.

Figure 13

Long-term states with $a=3$ (a ${\left(3x^{*}\right)}^{\frac{1}{3}}$ nose), ${\text{Re}}_M=500,$ and (a) $\tilde{A}=0.9$, (b) $\tilde{A}=1$, (c) $\tilde{A}=1.1$, (d) $\tilde{A}=1.2$.

Figure 14

Plot of ${\tilde{A}}_s$ for various values of ${\text{Re}}_M$ and $a$.

Figure 15

(a) Example of symmetric airfoils with various noses, a common tail, and the same thickness ratio (0.12) with $x_t/c=0.133$. (b) Magnified view of various nose geometries in the range $0<x/c<x_t/c$.

Figure 16

The distribution of pressure coefficient, $c_p$, as a function of $x/c$ for various angles of attack, $\alpha =8^{\circ },9^{\circ },{10}^{\circ },{11}^{\circ }$ for an airfoil with $a=2$ nose with maximum thickness at $x_t/c=0.190$ and $\text{Re}=150000$. The stall of the airfoil is depicted as $\alpha$ increases form ${10}^{\circ }$ to ${11}^{\circ }$.

Figure 17

The fields of velocity magnitude around the airfoil with $a=2,x_t/c=0.190,\text{Re}=150000$ at various angles of attack: (a) $\alpha =8^{\circ }$; at this angle of attack, the flow is attached to both the lower and upper surfaces of the airfoil. (b) $\alpha =9^{\circ }$; flow around most of the airfoil is attached but there is local flow separation near the trailing edge. (c) $\alpha ={10}^{\circ }$; flow around most of the airfoil is attached but there is local flow separation near the trailing edge. (d) $\alpha ={11}^{\circ }$; leading-edge stall state. (e) Velocity scale for parts (a) through (d) in meters per second.

Figure 18

The distribution of pressure coefficient, $c_p$, as a function of $x/c$ for various angles of attack, $\alpha =9^{\circ },{10}^{\circ },{11}^{\circ },{12}^{\circ }$ for an airfoil with $a=2.5$ nose with maximum thickness at $x_t/c=0.190$ and $\text{Re}=150000$. The stall of the airfoil is depicted as $\alpha$ increases form ${11}^{\circ }$ to ${12}^{\circ }$.

Figure 19

The fields of velocity magnitude around the airfoil with $a=2.5,x_t/c=0.190,\text{Re}=150000$ at various angles of attack: (a) $\alpha =9^{\circ }$; at this angle of attack, the flow is slightly separated near the trailing edge. (b) $\alpha ={10}^{\circ }$; flow around most of the airfoil is attached but there is local flow separation near the trailing edge. (c) $\alpha ={11}^{\circ }$; flow around most of the airfoil is attached but there is local flow separation near the trailing edge. (d) Velocity scale for parts (a) through (c) in meters per second.

Figure 20

${\tilde{A}}_s$ as a function of ${\text{Re}}_M$ for an airfoil with a nose shape parameter of $a=2$. The solid line represents the results of the direct numerical simulations. The dashed line represents the results extrapolated from the existing direct numerical simulation results. The dotted lines represent the results of the RANS computations for various values of $x_t/c$. Note that the RANS computations represent a continuation of the results of the direct numerical simulation to higher values of ${\text{Re}}_M$. Also note that the value of $x_t/c$ does not have an impact on ${\tilde{A}}_s\left({\text{Re}}_M\right)$.

Figure 21

${\tilde{A}}_s$ as a function of ${\text{Re}}_M$ for an airfoil with a nose shape parameter of $a=2.5$. The solid line represents the results of the direct numerical simulations. The dashed line represents the results extrapolated from the existing direct numerical simulation results. The dotted line represents the results of the RANS computations. Note that the RANS computations represent a continuation of the results of the direct numerical simulation to higher values of ${\text{Re}}_M$.

Figure 22

${\tilde{A}}_s$ as a function of ${\text{Re}}_M$ for an airfoil with a nose shape parameter of $a=3$. The solid line represents the results of the direct numerical simulations. The dashed line represents the results extrapolated from the existing direct numerical simulation results. The dotted line represents the results of the RANS computations. Note that the RANS computations represent a continuation of the results of the direct numerical simulation to higher values of ${\text{Re}}_M$.

Figure 23

Leading-edge stall angle vs Reynolds number for an airfoil with a nose shape parameter of $a=2.5$ and various values of $x_t/c$. Note that as $x_t/c$ is increased, the angle of stall decreases. The solid lines represent the results of direct numerical simulations. The dashed lines are from extrapolated results of the direct numerical simulations. The dotted lines represent predictions from RANS computations. The circles represent the RANS results.

Figure 24

Leading-edge stall angle vs Reynolds number for an airfoil with a fixed value of $x_t/c=0.190$ and various $a$. Note that as $a$ is increased, the angle of stall increases. The solid lines represent the results of direct numerical simulations. The dashed lines are from extrapolated results of the direct numerical simulations. The dotted lines represent predictions from RANS computations. The circles represent the RANS results.

Figure 25

Comparison of theoretical predictions with RANS computations of the leading-edge stall angle of attack for (a) the NACA 0012 airfoil and (b) a special transonic airfoil design.