Resolvent analysis of separated and attached flows around an airfoil at transitional Reynolds number

We analyze the resolvent operator in three flows around a nominal NACA-0012 airfoil at ${\text{Re}}_C=50,000$ and $5^{\circ }$ angle of attack. In particular, we study two naturally developing flows (around the airfoil with straight and blunt trailing edges) and one tripped flow. The naturally developing flows exhibit laminar separation, transition and turbulent reattachment, while the tripped flow remains attached in the suction side. For all cases, the time-averaged flow fields are computed from separate DNS simulations. The resolvent analysis can identify the areas of maximum receptivity of the flow field as well as the most amplified modes (optimal response). The former was located close to the leading edge, and in the case of naturally developing flows, also in the free-stream. The spatial distribution of optimal forcing was dominated by structures tilted against the mean flow, in agreement with other studies of boundary layer flows. The optimal response of the two naturally developing flows were different. For the NACA-0012 airfoil with straight trailing edge, the response was maximized at the natural frequency of the separating shear layer, and the shape matched closely with the corresponding DMD mode obtained from processing the DNS results. For the blunt airfoil, the receptivity of the separating shear layer was suppressed in the region of natural frequency, while it was maximized at the frequency of vortex shedding from the trailing edge. This is attributed to the lock-in mechanism between the shedding and the separating shear layer observed in the DNS simulations; the lock-in makes the separating shear layer less sensitive to forcing. In the tripped flow, the amplification of the perturbations is significantly diminished, and only by restricting the resolvent analysis to a region close to the suction side can we get a dominant frequency that matches with DNS at that region. The dominant resolvent modes were used to get estimations of the velocity spectra based only on the mean flow and the signal of one velocity component at discrete calibration points. Very good approximation is observed with only one or two points (depending on the flow examined), provided that they are located in energetic regions of the flow that contain sufficient spectral content at the relevant frequencies.

Figure 1

Mean streamwise velocity field for the naturally developing and tripped flows around the straight trailing edge airfoil.

Figure 2

Mean streamwise velocity field for the naturally developing flow around the blunt trailing edge airfoil.

Figure 3

Ratio of the local characteristic cell size to the Kolmogorov length scale.

Figure 4

Spectrum of the resolvent operator for the naturally developing flow around a NACA-0012 with straight trailing edge. The first five gain values of the operator are plotted against the angular frequency $\omega$.

Figure 5

Streamwise component of the optimal forcing ${\widehat{f}}_x$ at frequency ${\omega }_P$ at the leading edge with the displacement thickness of the boundary layer superimposed. The contours range is $\left[-72,72\right]$.

Figure 6

Cross-stream component of the optimal response at frequency ${\omega }_P$ at the suction side of the airfoil. The solid black line shows the displacement thickness of the boundary layer while the solid brown line is the zero-streamwise mean velocity contour. The contours range is $\left[-4.17,4.17\right]$.

Figure 7

Absolute value of the maximum reverse flow (blue line-left axis) and velocities ratio $R$ (red line-right axis), over the airfoil's suction side.

Figure 8

Streamwise component of the optimal response ${\widehat{u}}_x$ at the wake shedding frequency ${\omega }_1$. The contours range is $\left[-4.1,4.1\right]$.

Figure 9

Resolvent spectra of the straight and blunt trailing edge cases.

Figure 10

Cross-stream component of the optimal response ${\widehat{u}}_y$ at the vortex shedding frequency ${\omega }_{\text{SH}}$. The contours range is $\left[-3.59,3.59\right]$.

Figure 11

Cross-stream component of the optimal response at the suction side of the airfoil at frequency ${\omega }_{\text{SH}}$. The solid black line shows the displacement thickness of the boundary layer. The contours range is $\left[-0.8,0.8\right]$.

Figure 12

Probability density function of $C_f$ and $C_P$ for the naturally developing (left column) and tripped (right column) flows. The solid line in each plot denotes the mean value, while the dashed lines bracket the $\pm 1\sigma$ and $\pm 2\sigma$ ranges around the mean. The vertical dashed lines in Figs. 12 and 12 indicate the thin strip where tripping was applied.

Figure 13

Tangential velocity profiles at the airfoil's surface (shown up to $Y/C=0.01)$. The dashed blue arrows indicate increasing $x/c$.

Figure 14

Cross-stream velocity spectra at two locations on the suction' side boundary layer.

Figure 15

Instantaneous streamwise velocity field. The red curves delineate large scale structures.

Figure 16

Resolvent spectrum for the tripped straight trailing edge airfoil.

Figure 17

Optimal forcing in the cross stream direction at frequency ${\omega }_{P,\text{Trip}}$. The contours range is $\left[-69.2,69.2\right]$

Figure 18

Cross-stream components of responses at frequencies ${\omega }_{P,\text{Trip}}$ and ${\omega }_{2,\text{Trip}}$. The contours in Fig. 18 range between $\left[-3.6,3.6\right]$ while in Fig. 18 $\left[-9.8,9.8\right]$.

Figure 19

Optimization domain defined on the suction side of the airfoil.

Figure 20

Resolvent spectra with different optimization domains. The gray lines correspond to optimization over the whole domain, while the colored lines correspond to a domain on the suction side of the airfoil.

Figure 21

Cross-stream component of the optimal response at the frequency ${\omega }_{2,\text{Trip}}$ when the domain shown in Fig. 19 is used in the optimization process. The contours range is $\left[-7.7,7.7\right]$.

Figure 22

Comparison between DNS and model spectra (calibration point $x/c=0.62,y/c=0.022)$.

Figure 23

Model spectrum at the wake point $\left(X/C=2.0,Y/C=-0.17\right)$ using two calibration points to estimate $\mathrm{\Psi }\left(\omega \right)$.

Figure 24

PSD spectrum as calculated from the model (calibration at $X/C=0.62,Y/C=0.022)$.