Simulation and characterization of the laminar separation bubble over a NACA-0012 airfoil as a function of angle of attack

This study examines the effects of angle of attack on the characteristics of the laminar separation bubble (LSB), its associated low-frequency flow oscillation (LFO), and the flow field about a NACA-0012 airfoil at Reynolds number of $5\times {10}^4$ and $9\times {10}^4$, Mach number of 0.4, and several angles of attack near stall. In the range of the investigated angles of attack, statistics of the flow field suggest the existence of three distinct angle-of-attack regimes. At angles of attack lower than the stall angle of attack, the mean flow field is attached, a short bubble is formed, and the flow field is not much affected by the LFO. At angles of attack higher than the stall angle of attack and lower than the angle of attack of maximum LFO, the flow field undergoes a transition process in which the LFO develops until the flow field reaches a quasiperiodic switching between separated and attached flow and the LSB switches between short and long bubble. At angles of attack higher than the angle of attack of maximum LFO, the mean flow field is massively separated, an open bubble is formed, and the LFO gradually loses momentum and becomes unable to reattach the flow until the airfoil approaches the angle of a full stall. These angle-of-attack regimes have, to the best of the authors' knowledge, not been reported in the literature before.

Figure 1

Time history of the lift coefficient at ${\mathrm{Re}}_c=9\times {10}^4$ and the angle of attack of $11.0^{\circ }$. The dashed, solid, and dash-dot horizontal lines show the high-lift, mean-lift, and low-lift time average, respectively.

Figure 2

The conditional phase-averaging process. Left: One sinusoidal cycle. Right: Time history of the lift coefficient at ${\mathrm{Re}}_c=9\times {10}^4$ and the angle of attack of $11.0^{\circ }$.

Figure 3

Plots of the computational domain, grid distribution, and the LSB.

Figure 4

Instantaneous ${\lambda }_2$-criterion isosurface (${\lambda }_2=-200$) colored by the total energy per unit mass ($E$).

Figure 5

Locally time-averaged pressure and skin-friction coefficients plotted versus the distance from the airfoil leading edge $x_c/c$. The arrows indicate the direction in which the angle of attack, $\alpha$, increases in ascending order.

Figure 6

Profiles of the scaled mean ($U/U_{\infty }$) of the streamwise velocity plotted versus vertical distance from the airfoil surface ($\mathrm{\Delta }y/c$) at eight locations ($x_c/c$) measured from the leading edge along the airfoil chord at ${\mathrm{Re}}_c=5\times {10}^4$ and $\alpha =9.0^{\circ }$. Solid line: LES data; circles: incompressible experimental data of Ohtake and Motohashi [42] and Ohtake [43] at ${\mathrm{Re}}_c=5\times {10}^4$.

Figure 7

Profiles of the mean pressure and skin-friction coefficients plotted versus the distance from the airfoil leading edge computed along the curvilinear coordinate on the airfoil suction side $x_s/c$ with $x_s/c=0$ at the stagnation point at ${\mathrm{Re}}_c=9\times {10}^4$ and $\alpha =10.{25}^{\circ }$.

Figure 8

Time histories of the lift and the drag coefficients.

Figure 9

Mean lift and drag coefficients plotted versus the angle of attack $\alpha$. Circles, upward triangles, and downward triangles display the mean-lift, high-lift, and low-lift time average of the lift and drag coefficients, respectively. The filled black squares indicate the incompressible data of Ohtake et al. [44] at ${\mathrm{Re}}_c=5\times {10}^4$. The $\times$'s display the LES data of Almutairi and Alqadi [27].

Figure 10

Mean moment and skin-friction coefficients plotted versus the angle of attack $\alpha$.

Figure 11

Spectra of the streamwise (black line), wall-normal (red line), and spanwise (blue line) velocities for the angles of attack of $9.0^{\circ }$, $9.8^{\circ }$, and $10.5^{\circ }$ at ${\mathrm{Re}}_c=5\times {10}^4$ (top), and the angles of attack of $10.{25}^{\circ }$, $11.0^{\circ }$, and $11.2^{\circ }$ at ${\mathrm{Re}}_c=9\times {10}^4$ (bottom).

Figure 12

Spectra of the lift coefficient for the angles of attack $\alpha =9.{25}^{\circ }$–$10.5^{\circ }$ at ${\mathrm{Re}}_c=5\times {10}^4$ (top), and $\alpha =10.{25}^{\circ }$–$11.2^{\circ }$ at ${\mathrm{Re}}_c=9\times {10}^4$ (bottom).

Figure 13

Low-frequency Strouhal number $\left(\text{St}\right)$ for the lift coefficient plotted versus the angle of attack $\alpha$. Circles: LES data, and $\times$'s: the LES data of Almutairi and Alqadi [27].

Figure 14

Streamlines patterns of the time-averaged flow field superimposed on color maps of the time-averaged streamwise velocity component at the stall angle of attack [${\alpha }_s=9.0^{\circ }$ at ${\mathrm{Re}}_c=5\times {10}^4$ (left), and ${\alpha }_s=10.{25}^{\circ }$ at ${\mathrm{Re}}_c=9\times {10}^4$ (right)].

Figure 15

Streamlines patterns of the time-averaged flow field superimposed on color maps of the time-averaged streamwise velocity component at the angle of attack of maximum LFO [${\alpha }_{LFO}=9.8^{\circ }$ at ${\mathrm{Re}}_c=5\times {10}^4$ (left), and ${\alpha }_{LFO}=11.0^{\circ }$ at ${\mathrm{Re}}_c=9\times {10}^4$ (right)].

Figure 16

Streamline patterns of the time-averaged flow field superimposed on color maps of the time-averaged streamwise velocity component for $\alpha =10.5^{\circ }$ at ${\mathrm{Re}}_c=5\times {10}^4$ (left), and $\alpha =11.2^{\circ }$ at ${\mathrm{Re}}_c=9\times {10}^4$ (right).

Figure 17

The length of the Laminar Separation Bubble (LSB) and the Trailing-Edge Bubble (TEB) plotted versus the angle of attack $\alpha$.

Figure 18

The height of the Laminar Separation Bubble (LSB) and the Trailing-Edge Bubble (TEB) plotted versus the angle of attack $\alpha$.

Figure 19

The aspect ratio of the LSB plotted versus the angle of attack $\alpha$ (left). The bursting parameter $P_h$ plotted versus the Reynolds number ${\text{Re}}_h$. The arrows indicate the direction in which the angle of attack, $\alpha$, increases in ascending order (right).

Figure 20

Maximum reverse velocity ($\mathrm{MRV}$) plotted versus the angle of attack $\alpha$.

Figure 21

Color map of the variance of the streamwise velocity component arising from the total fluctuations ($\overline{{\breve{u}}_1^2}$) at ${\mathrm{Re}}_c=5\times {10}^4$ (left) and ${\mathrm{Re}}_c=9\times {10}^4$ (right).

Figure 22

Color map of the variance of the wall-normal velocity component arising from the total fluctuations ($\overline{{\breve{u}}_2^2}$) at ${\mathrm{Re}}_c=5\times {10}^4$ (left) and ${\mathrm{Re}}_c=9\times {10}^4$ (right).

Figure 23

Color map of the variance of the spanwise velocity component arising from the total fluctuations ($\overline{{\breve{u}}_3^2}$) at ${\mathrm{Re}}_c=5\times {10}^4$ (left) and ${\mathrm{Re}}_c=9\times {10}^4$ (right).

Figure 24

Maximum values of the variance of the streamwise velocity component arising from the periodic fluctuations ($\overline{{u_1^{″}}^2}$) (left) and from the turbulent fluctuations ($\overline{{u_1^{\prime }}^2}$) (right) plotted versus the angle of attack $\alpha$.

Figure 25

Maximum values of the variance of the wall-normal velocity component arising from the periodic fluctuations ($\overline{{u_2^{″}}^2}$) (left) and from the turbulent fluctuations ($\overline{{u_2^{\prime }}^2}$) (right) plotted versus the angle of attack $\alpha$.

Figure 26

Maximum values of the variance of the spanwise velocity component arising from the turbulent fluctuations ($\overline{{u_3^{\prime }}^2}$) (left) and maximum Turbulent Kinetic Energy (TKE) (right) plotted versus the angle of attack $\alpha$.

Figure 27

Color map of the turbulence production arising from the total fluctuations ($-\overline{{\breve{u}}_1{\breve{u}}_2}\frac{\partial \overline{u_1}}{\partial y}$) at ${\mathrm{Re}}_c=5\times {10}^4$ (left) and ${\mathrm{Re}}_c=9\times {10}^4$ (right).

Figure 28

Maximum values of the turbulence production arising from the total fluctuations ($-\overline{{\breve{u}}_1{\breve{u}}_2}\frac{\partial \overline{u_1}}{\partial y}$) (left) and the distance of the maximum turbulence production measured from the airfoil leading edge ($\mathrm{\Delta }x/c$) (right) plotted versus the angle of attack $\alpha$.

Figure 29

Color map of the variance of the pressure arising from the total fluctuations ($\overline{{\breve{p}}^2}$) at ${\mathrm{Re}}_c=5\times {10}^4$ (left) and ${\mathrm{Re}}_c=9\times {10}^4$ (right).

Figure 30

Maximum values of the variance of the pressure arising from the periodic fluctuations ($\overline{{p^{″}}^2}$) (left) and from the turbulent fluctuations ($\overline{{p^{\prime }}^2}$) (right) plotted versus the angle of attack $\alpha$.