Experiments on the unsteady massive separation over an aerofoil

We have experimentally investigated an unsteady separating flow over a NACA-0012 aerofoil. Both the surface pressure measurements and the whole flow field measurements using the time-resolved particle image velocimetry (TR-PIV) technique were carried out at different angles of attack for a chord-based Reynolds number, ${\text{Re}}_c=5\times {10}^4$. The instantaneous flow fields were measured at the stall angle of attack ($\alpha ={10}^{\circ }$) and at different post-stall angles of attack ($\alpha ={11}^{\circ },{12}^{\circ },{13}^{\circ }$) using the TR-PIV system. Although the mean flow data, as expected, show that the flow is fully/massively separated for $\alpha >{10}^{\circ }$ over the suction surface of the aerofoil, the time sequence of the instantaneous flow fields reveals that the massive separation is intermittent in nature. A small separated region grows with time, eventually leading to the massive flow separation and then the massively separated region decays with time leading to a small separated region. In this process, the vortex shedding from the separated shear layer is also observed. We find that an interplay exists between the temporal growth of the separated shear layer and the vortex shedding during the intermittent massive separation. The mechanism for the intermittent massive flow separation has been explained using the time sequence of the instantaneous flow fields. This has further been supported using the proper orthogonal decomposition analysis. Furthermore, the spatiotemporal stability analysis of the local velocity profile shows that when the small separated region goes to a fully separated state, the velocity profiles become absolutely unstable from a convectively unstable state, and vice versa. In addition, these results are found to be consistent with the absolute instability criteria proposed in Alam and Sandham [J. Fluid Mech. 410, 1 (2000)] and Avanci et al. [Phys. Fluids 31, 014103 (2019)].

Figure 1

A schematic diagram of the experimental setup in a close circuit wind tunnel.

Figure 2

(a) Variations of $C_l$ and $C_d$ with angle of attack, $\alpha$. Description of symbols: $C_l$($•$) and $C_d$ ($\blacksquare$) for the present experiment; colored symbols correspond to the selected angles of attack for which pressure distributions are shown in panel (b); $C_l$($◯$) and $C_d$($\square$) correspond the data reproduced from the study of Ohtake et al. [33], for comparison. (b) Variation of the mean surface pressure coefficient ($C_p$) for the selected $\alpha$, as shown in panel (a).

Figure 3

Comparison of the mean flow velocity vectors overlaid with the $u_{\text{rms}}$ contour for the fixed and surface attached coordinate systems. (a)–(d) Fixed coordinate system, and (e)–(g) Surface attached coordinate system. The white line corresponds to the zero streamwise mean velocity.

Figure 4

Streamwise variation of the mean flow parameters at different AOA. (a) The boundary layer edge velocity variation. (b) Displacement thickness variation. (c) Maximum reverse flow velocity in percentage (considering the maximum MRV value for a corresponding chordwise position). A dashed line corresponds to the limit above which the separated shear layer becomes absolutely unstable.

Figure 5

Variation of the instantaneous displacement thickness at $x/c=0.3$; open and solid symbols indicate the minimum and maximum values of the displacement thickness, respectively.

Figure 6

Instantaneous streamlines superimposed with the instantaneous vorticity contours and the $U=0$ line. The first column of the figure represents the realizations corresponding to the time instants marked by the open symbols in Fig. 5. The second column of the figure represents the realizations corresponding to the time instants marked by the closed symbols in Fig. 5. The roller structure is marked by a red colored box, and it is zoomed-in for better visualization.

Figure 7

Comparison of the backflow intermittency factor ($\gamma$) for $\alpha ={10}^{\circ }$, ${11}^{\circ }$, ${12}^{\circ }$, and ${13}^{\circ }$.

Figure 8

Comparison of the $u$ and $v$ fluctuating velocity signals extracted at the wall-normal location close to $u_{\text{rms},\text{max}}$ and $v_{\text{rms},\text{max}}$, respectively. First and second signal in each panel from the top corresponds to the $u$ and $v$ fluctuating velocity, respectively. The mean value of the signal is represented by a dashed line.

Figure 9

Comparison of the power spectra of the $u$ and $v$ fluctuating velocity signals shown in Fig. 8 at different streamwise locations. (a)–(d) For $u$ fluctuating velocity signal. (e)–(h) For $v$ fluctuating velocity signals. Four power spectrum curves (from bottom to top) correspond to the streamwise locations $x/c=0.40$, 0.55, 0.70, and 0.85.

Figure 10

Variations of the relative and cumulative energy levels of the POD modes (i.e., the $uv$-POD modes) for $\alpha ={10}^{\circ }$, ${11}^{\circ }$, ${12}^{\circ }$, and ${13}^{\circ }$. Open symbols correspond to the relative energy ($E_r$) level, and closed symbols correspond to the cumulative energy ($E_c$) level.

Figure 11

Two-component POD modes (i.e., the $uv$-POD modes) superimposed with the contours of its wall-normal component (i.e., ${\varphi }_v$). (a) $\alpha ={10}^{\circ }$, (b) $\alpha ={11}^{\circ }$, (c) $\alpha ={12}^{\circ }$, (d) $\alpha ={13}^{\circ }$.

Figure 12

Comparison of the time coefficients of the first six POD modes for $\alpha ={10}^{\circ },{11}^{\circ },{12}^{\circ },{13}^{\circ }$.

Figure 13

Comparison of the frequency spectra of the first six POD modes for $\alpha ={10}^{\circ },{11}^{\circ },{12}^{\circ },{13}^{\circ }$.

Figure 14

One oscillation cycle of the ${\delta }_I^{*}$ from the Fig. 5, along with the ${\delta }_I^{*}$ estimated from the reconstructed POD modes for the cases of $\alpha ={11}^{\circ }$ and $\alpha ={13}^{\circ }$. Filled and open symbols correspond to the selected frames for further investigation on increasing and decreasing ${\delta }_I^{*}$ curve, respectively.

Figure 15

Comparison of the instantaneous and the reconstructed streamlines using first 25 POD modes for the case of $\alpha ={11}^{\circ }$. (a), (b) Instantaneous and its reconstructed streamlines corresponding to the time instants shown by filled symbols in Fig. 14. (c), (d) Instantaneous and its reconstructed streamlines corresponding to the time instants shown by open symbols in Fig. 14. Description of blue line: (a), (c) the isoline of $U=0$ estimated from the instantaneous velocity field; (b), (d) the isoline of $U=0$ estimated from the reconstructed velocity field using first POD mode.

Figure 16

Comparison of the instantaneous and its reconstructed streamlines using first 25 POD modes for the case of $\alpha ={13}^{\circ }$. (a), (b) Instantaneous and its reconstructed streamlines corresponding to the time instants shown by open symbols in Fig. 14. (c), (d) Instantaneous and its reconstructed streamlines corresponding to the time instants shown by filled symbols in Fig. 14. Description of blue line: (a), (c) the isoline of $U=0$ estimated from the instantaneous velocity field; (b), (d) the isoline of $U=0$ estimated from the reconstructed velocity field using first POD mode.

Figure 17

Variation of the reconstructed instantaneous velocity profiles extracted at $x/c=0.3$, using the first POD mode, for $\alpha ={11}^{\circ }$. (a) Reconstructed velocity profiles that correspond to TR-PIV realizations shown in Fig. 15. (b) Reconstructed velocity profiles that correspond to TR-PIV realizations shown in Fig. 15.

Figure 18

Variation of the reconstructed instantaneous velocity profiles extracted at $x/c=0.3$, using the first POD mode, for $\alpha ={13}^{\circ }$. (a) Reconstructed velocity profiles that correspond to TR-PIV realizations shown in Fig. 16. (b) Reconstructed velocity profiles that correspond to TR-PIV realizations shown in Fig. 16.

Figure 19

Space-time variation of $y_{\text{in}}/y_d$ calculated from the reconstructed velocity profiles using the first POD mode. (a) Corresponding to the TR-PIV realizations shown in Fig. 15. (b) Corresponding to the TR-PIV realizations shown in Fig. 15. (c) Corresponding to the TR-PIV realizations shown in Fig. 16. (d) Corresponding to the TR-PIV realizations shown in Fig. 16. The shaded region represents the absolutely unstable region according to the criterion of Avanci et al. [62]. The direction of the arrow indicates increasing time in panels (a) and (b).

Figure 20

Space-time variation of MRV calculated from the reconstructed velocity profiles using the first POD mode. (a) Corresponding to the TR-PIV realizations shown in Fig. 15. (b) Corresponding to the TR-PIV realizations shown in Fig. 15. (c) Corresponding to the TR-PIV realizations shown in Fig. 16. (d) Corresponding to the TR-PIV realizations shown in Fig. 16. The shaded region represents the absolutely unstable region according to the criterion of Alam and Sandham [23]. The direction of the arrow indicates increasing time in panels (a) and (b).

Figure 21

Validation of the spatiotemporal stability analysis results. (a) Velocity profile corresponding to the profile $P3$ mentioned in Avanci et al. [62]. (b) Contours of temporal growth rate ${\omega }_i$ in the complex $k$ plane. (c) Temporal growth rate ${\omega }_i$ versus group velocity $C_g=\partial \omega /\partial k$.

Figure 22

Space-time variation of temporal growth rate estimated from the reconstructed velocity profiles using the first POD mode. (a) Corresponding to the TR-PIV realizations shown in Fig. 15. (b) Corresponding to the TR-PIV realizations shown in Fig. 15. (c) Corresponding to the TR-PIV realizations shown in Fig. 16. (d) Corresponding to the TR-PIV realizations shown in Fig. 16. The shaded region represents an absolutely unstable region according to the local stability analysis. The direction of the arrow indicates increasing time in panels (a) and (b), for ${\omega }_{i,0}$.

Figure 23

Space-time variation of most amplified frequency estimated from the reconstructed velocity profiles using the first POD mode. (a) Corresponding to the TR-PIV realizations shown in Fig. 15. (b) Corresponding to the TR-PIV realizations shown in Fig. 15. (c) Corresponding to the TR-PIV realizations shown in Fig. 16. (d) Corresponding to the TR-PIV realizations shown in Fig. 16. The shaded region in panels (a), (b) and (c), (d) represents the experimentally measured peak-frequency band from the Figs. 9 and 9, respectively.

Figure 24

The temporal variation of the eigenmodes for $\alpha ={11}^{\circ }$, at $x/c=0.3$. (a) Corresponding to the velocity profiles shown in Fig. 17. (b) Corresponding to the velocity profiles shown in Fig. 17.

Figure 25

The temporal variation of the Eigenmodes for $\alpha ={13}^{\circ }$, at $x/c=0.3$. (a) Corresponding to the velocity profiles shown in Fig. 18. (b) Corresponding to the velocity profiles shown in Fig. 18.

Figure 26

A schematic to illustrate the mechanism of intermittent massive separation. (a) Attached to separated state, (b) separated to attached state. The temporal evolution of the separation bubble is shown by a blue colored line.

Figure 27

Variation of the relative and cumulative energy levels of the $v$-POD modes for $\alpha ={10}^{\circ }$, ${11}^{\circ }$, ${12}^{\circ }$, and ${13}^{\circ }$. Open symbols correspond to the relative energy ($E_r$) level and closed symbols correspond to the cumulative energy ($E_c$) level.

Figure 28

First two energetic $v$-POD modes. (a) $\alpha ={10}^{\circ }$, (b) $\alpha ={11}^{\circ }$, (c) $\alpha ={12}^{\circ }$, (d) $\alpha ={13}^{\circ }$.

Figure 29

Comparison of the time coefficients of the first two $v$-POD modes and their power spectra. $v$-POD time coefficients, (a) $\alpha ={10}^{\circ }$, (b) $\alpha ={11}^{\circ }$, (c) $\alpha ={12}^{\circ }$, (d) $\alpha ={13}^{\circ }$; $v$-POD frequency spectra, (e) $\alpha ={10}^{\circ }$, (f) $\alpha ={11}^{\circ }$, (g) $\alpha ={12}^{\circ }$, (h) $\alpha ={13}^{\circ }$.