Study of the wall pressure variations on the stall inception of a thick cambered profile at high Reynolds number

We present an experimental study of the aerodynamic forces on a thick and cambered airfoil at a high Reynolds number (Re) ($3.6\times {10}^6$), which is of direct relevance for wind turbine design. Unlike thin airfoils at low chord-based Reynolds numbers, no consistent description currently exists for the stall process on such airfoils. We consider two chordwise rows of instantaneous wall pressure measurements, taken simultaneously at two spanwise locations over a range of angles of attack. We show that around maximum lift conditions, a strong asymmetry is observed in the statistics of the normal force on each chord. In this range of angles of attacks, the pressure fluctuations are largest in the adverse pressure gradient region, and the fluctuation peak along the chord is systematically located directly upstream of the mean steady separation point, indicating intermittent flow separation. Moreover, the fluctuations are characterized by bistability in both space and time: For each spanwise location, large excursions of the local wall pressure between two different levels can be observed in time (jumps), and these excursions are highly anticorrelated between the two spanwise locations (spatial switches). The characteristic timescale for the switches is found to be well correlated with the amplitudes of the fluctuations. Application of proper orthogonal decomposition (POD) analysis to each row of sensors confirms that the flow separation is an inherently local, three-dimensional, and unsteady process that occurs in a continuous manner when the angle of attack increases. The correlation between the dominant POD mode amplitudes is found to be a good indicator of bistability. For all angles of attack, most of the fluctuations can be captured with the two most energetic POD modes. This suggests that force fluctuations near the maximal lift could be modelled by a low-order approach for monitoring and control purposes.

Figure 1

(a) The CSTB wind tunnel and (b) schematics of the 2D blade installation in the aerodynamic test section with the two rows of 78 pressure taps each in the chordwise direction at two different spanwise locations (red and blue corresponds to respectively $Y^{+}$ and $Y^{-}$ locations). (c) The inflow is measured 2.65 m upstream of the profile by a cobra probe. From Neunaber et al. [34].

Figure 2

Normal force coefficient $C_N$ in solid lines and standard deviation of $C_N$ in dashed lines from both spanwise locations, $Y^{+}$ and $Y^{-}$.

Figure 3

Asymmetry coefficient $\mathrm{\Delta }{C}^{\mathrm{rel}}=2[C_N\left(Y^{+}\right)-C_N\left(Y^{-}\right)]/[C_N\left(Y^{+}\right)+C_N\left(Y^{+}\right)]$ for the complete data set (solid points). Gray areas represent the results when only 25% of the data set length are used.

Figure 4

Pressure coefficient ($C_p$) distribution in chordwise direction $x/c$ for both spanwise locations ($Y^{+}$ in dotted lines and $Y^{-}$ in solid lines) for different angles of attack: (a) from $6^{\circ }$ to ${12}^{\circ }$ and (b) from ${12}^{\circ }$ to ${24}^{\circ }$. The markers indicate the pressure measurement locations.

Figure 5

Standard deviation of the wall pressure coefficient, $\mathrm{Std}\left(Cp\right)$, along the chord $x/c$ for both spanwise locations ($Y^{+}$ in dotted lines and $Y^{-}$ in solid lines) for different angles of attack: (a) from $6^{\circ }$ to ${12}^{\circ }$ and (b) from ${12}^{\circ }$ to ${24}^{\circ }$. The markers indicate the pressure measurement locations. Note the difference of the scale for the $y$ axis between the low angles of attack (a) and high angles of attack (b) to highlight the apparitions of local maxima at ${10}^{\circ }$. Local maxima are respectively denoted LEM, leading-edge maximum; ISP, intermittent separation point; TEM, trailing-edge maximum.

Figure 6

Evolution of the separation point with the angle of attack for both spanwise locations. Error bars represent the accuracy in the evaluation of the separation point due to the pressure tap resolution.

Figure 7

Sketch of the flow physics corresponding to the spatial distribution of the pressure fluctuations. The red arrows along the blade surface show the intermittent flow separation associated with a large pressure standard deviation at the ISP, while the red arrows above the blade represent the fluctuations of the shear layer associated with the TEM.

Figure 8

Pressure coefficients $C_p$ (in red), gradient of pressure $d{C}_p/dx$ (in black), and standard deviation of the pressures $\mathrm{Std}\left(Cp\right)$ (in blue) versus the chordwise direction, $x/c$, at the $Y^{-}$ row for different angles of attack (from ${10}^{\circ }$ to ${24}^{\circ }$). The ISP and the SSP locations are indicated in the figures.

Figure 9

Left: Spatiotemporal evolution of the wall pressure fluctuations at the spanwise location $Y^{+}$ (only suction side) and for $\mathrm{AoA}={12}^{\circ }$. Right: Wall pressure standard deviation distribution for $\mathrm{AoA}={12}^{\circ }$. For both plots, the red line represents the maximum location of the standard deviation peak in the recovery region and the blue lines correspond to the two local fluctuation minima delimiting the region of strong fluctuations.

Figure 10

Coherence $C_{x_M}(x,f)$ between the pressure coefficient at location on the suction side and the intermittent separation point $x^{\mathrm{ISP}}$ (indicated with a red line) on the chord $Y=Y^{+}$. The frequency is indicated in hertz. The blue lines correspond to local minima of the pressure fluctuations.

Figure 11

Left: $\mathrm{AoA}={12}^{\circ }$; instantaneous pressure signal ($Y^{+}$) at $x/c=0.52$ at the intermittent separation point. Center: Histogram of the pressure signal. Right: Averaged pressure coefficient distribution based on full and conditional averages: high values (red), low values (green), and all values (orange).

Figure 12

Instantaneous pressure signals ($Y^{+}$) for different angle of attacks at four chordwise locations, $x/c=0.52$, 0.39, 0.36, and 0.30, corresponding to the intermittent separation points at respectively ${12}^{\circ }$, ${14}^{\circ }$, ${16}^{\circ }$, and ${20}^{\circ }$. The associated histograms are plotted on the right-hand side of each figure.

Figure 13

Spatial pressure switch between $Y^{+}$ in red to $Y^{-}$ in blue for different chordwise locations corresponding to the intermittent separation point of the considered angle of attack.

Figure 14

Spectra of the pressure signal at the intermittent separation point for different angles of attack (the spectra have been rescaled for easier comparison).

Figure 15

Evolution of the bistability timescale with the angle of attack for different values of the time threshold $s$. The open symbols correspond to the different intervals measured for $s{U}_c/c=32$. The variance of the pressure fluctuations $p_{\mathrm{rms}}^2$ is also represented on the plot for comparison as star markers.

Figure 16

Evolution with the angle of attack of the first two POD eigenvalues ${\lambda }_1$ (squares) and ${\lambda }_2$ (triangles), referring to the energy of the two first modes and of the total variance (solid circles), for each row of pressure $Y^{+}$ (in red) and $Y^{-}$ (in blue).

Figure 17

First two POD modes of the pressure coefficient ${\mathrm{\Phi }}_n{\lambda }_n^{1/2}$ on the suction side for $Y^{-}$ (solid lines) and $Y^{+}$ (dashed lines). The red solid and dashed lines respectively correspond to the steady separation point at $Y^{+}$ and $Y^{-}$.

Figure 18

POD-based reconstructions of the pressure coefficient on the suction side for different angles of attack using the two first modes (the modes represented are those determined at $Y_{+}$; the same trends were observed for those obtained at $Y_{-}$).

Figure 19

Histogram of the first two POD amplitudes; the black dotted line indicates a reference Gaussian distribution.

Figure 20

POD-based reconstructions of the pressure coefficient on the suction side for different angles of attack using the dominant mode and the most frequent value of the amplitude (a rescaling factor of 2 has been applied to the fluctuation for easier visualization).

Figure 21

Autocorrelation of the dominant (normalized) POD amplitude for different angles of attack. Cross-correlation of the dominant POD amplitude for different angles of attack.