Resolvent analysis of an airfoil laminar separation bubble at $\text{Re}=500000$

We perform a resolvent analysis to examine the perturbation dynamics over the laminar separation bubble (LSB) that forms near the leading edge of a NACA 0012 airfoil at a chord-based Reynolds number (Re) of 500 000 and an angle of attack of 8 degrees. While we focus on the LSB residing over $6\%$ of the chord length, the resolvent operator is constructed about the global mean flow over the airfoil, avoiding numerical issues arising from domain truncation. Moreover, randomized singular value decomposition is adopted in the present analysis to relieve the computational cost associated with the high-Re global base flow. To examine the local physics over the LSB, we consider the use of exponential discounting to limit the time horizon that allows for the instability to develop with respect to the base flow. With discounting, the gain distribution over frequency accurately captures the spectral content over the LSB obtained from flow simulation. The peak-gain frequency also agrees with previous flow control results on suppressing dynamic stall over a pitching airfoil. According to the gain distribution and the modal structures, we conclude that the dominant energy-amplification mechanism is the Kelvin-Helmholtz instability. In addition to discounting, we also examine the use of spatial windows for both the forcing and response. From the response-windowed analysis, we find that the LSB serves the main role of energy amplifier, with the amplification saturating at the reattachment point. The input window imposes the constraint of surface forcing, and the results show that the optimal actuator location is slightly upstream of the separation point. The surface-forcing mode also suggests the optimal momentum forcing in the surface-tangent direction, with strong unidirectionality that is ideal for synthetic-jet-type actuators. This study demonstrates the strength of randomized resolvent analysis in tackling high-Reynolds-number base flows, and it calls attention to the care needed for base-flow instabilities. The physical insights provided by the resolvent analysis can also support flow control studies that target the LSB for suppressing flow separation or dynamic stall.

Figure 1

The grid setups for the linear stability analysis (left) and biglobal resolvent analysis (right). The mean flow obtained on the LES mesh is interpolated onto a separate grid to conduct the resolvent analysis. Every eight grid lines are shown. Contour lines are shown for the time- and spanwise-averaged streamwise velocity.

Figure 2

The discounted resolvent analysis for unstable ${\mathcal{A}}_{\overline{\mathit{q}}}$ can be related to the shift of the vertical integral path in the inverse Laplace transform (7) such that the unstable eigenvalues are enclosed in the Bromwich contour.

Figure 3

Discounted resolvent analysis of a simple system: (a) Finite-time energy amplification for the forced unstable system and discounted stable system with forcing frequency $\omega =1.2$; (b) comparison of the finite-time gain from the unstable system (evaluated at $t=30$ and rescaled as $e^{-\beta t}{\left|\right|\mathit{q}\left|\right|}_2$) to the gains obtained from discounted and nondiscounted resolvent analyses. Finite-time gain of the harmonically forced unstable system (evaluated at $t=30$ and rescaled as $e^{-\text{Re}\left({\lambda }_1\right)t}{\left|\right|\mathit{q}\left|\right|}_2$) also shows agreement with the gain profile obtained from the discounted resolvent analysis. Inserted in (b) are the eigenvalues (magenta dots) and pseudospectrum (contour levels) of ${\mathcal{A}}_{\text{u}}$. The nondiscounted analysis is performed over the path of $s=i\omega$ and the discounted analysis is performed over $s=\beta +i\omega$.

Figure 4

The spatial window $\mathcal{M}(x,y)=\left\{(x,y)\right|x\in (-\infty ,x_b],y\in (-\infty ,\infty )\}$ that defines the output $\widehat{\mathit{y}}=\mathit{C}\widehat{\mathit{q}}$. We perform a series of response-windowed resolvent analyses by moving the right-boundary $x_b$ of this window over the region of the LSB, which is highlighted by the magenta dashed line for the zero-streamwise velocity contour (${\overline{v}}_x=0$).

Figure 5

Comparison of the randomized SVD results to those from full SVD using arpack. Agreements are found with respect to (a) gain profile and (b) cosine similarity between the response modes $\langle {\tilde{\mathit{y}}}_1,{\widehat{\mathit{y}}}_1\rangle$ over the range of examined frequency. Here, we use $k_z{L}_c=0$ and $\beta L_c/\left(2\pi v_{\infty }\right)=20$, without spatial windowing, i.e., $\mathit{B}=\mathit{C}=\mathit{I}$. Inserted in (b) are the ${\widehat{v}}_y$-components of the representative response mode at $\text{St}=20$, highlighted with $•$ on the cosine similarity profile.

Figure 6

(a) Instantaneous flow-field visualization with the isosurface of $Q$-criterion colored by the streamwise velocity. The extent of the laminar separation bubble is represented by the zero time-averaged streamwise velocity contour ${\overline{v}}_x=0$, highlighted by the magenta dashed line in (b). Time-resolved measurements are taken along the time-averaged shear layer, as highlighted with the magenta dashed line in (c). The vertical scale in (b) and (c) is stretched by two times to visualize the LSB.

Figure 7

Results of quasiparallel linear stability analysis. Shown are (a) the contour of the local amplification rate $k_{x,i}$, (b) representative boundary-layer profiles, and (c) the corresponding eigenfunctions. Eigenfunctions in (c) correspond to $\text{St}=200$ at $x/L_c=0.009$ (first station of adverse pressure gradient) and $\text{St}=120$ at $x/L_c=0.022$ (separation point) and 0.048 (upstream of absolute instability). The dotted black line in (a) tracks the most amplified frequency at a given streamwise location.

Figure 8

Amplification profile using Eq. (2), according to the quasiparallel instability analysis.

Figure 9

Global stability analysis for determining the minimal discounting parameter. (a) Eigenvalues of ${\mathcal{A}}_{\overline{\mathit{q}}}\left(k_z\right)$ appearing in the frequency range of the LSB. (b) Modal structures of two representative global modes (${\widehat{\mathit{v}}}_x$-component, $k_z=0$) circled in (a).

Figure 10

(a) Isosurfaces of the leading gain over the parameter space of $(\text{St},k_z,t_{\beta })$; (b) gain profiles for $k_z=0$ for different $t_{\beta }$; (c) effect of $t_{\beta }$ on the peak frequency.

Figure 11

The ${\widehat{\mathit{v}}}_x$-components of forcing (green-yellow) and response (red-blue) modes for representative frequencies and discounting parameters. Here the resolvent modes of $k_z{L}_c=0$ are shown. The lowest magnitude marked by the contour lines is $1\%$ of the modal maximum.

Figure 12

(a) Gain profiles of $t_{\beta }{v}_{\infty }/L_c=0.05$ (black line) and $\beta =max({\lambda }_r\left({\mathcal{A}}_{\overline{\mathit{q}}}\right))+\epsilon$ (blue line), where $\epsilon ={10}^{-4}$; (b) velocity spectra in terms of probability density function, and (c) in terms of power spectral density.

Figure 13

Resolvent analysis with discounting and/or spatial windowing. (a) Gain profiles for $k_z{L}_c=0$; (b)–(d) representative response modes at $\text{St}=100$.

Figure 14

The leading gain against $x_b$ of the spatial window. (a) $k_z{L}_c=0$; (b) $k_z{L}_c=200\pi$. For all frequencies, the increase in gain with $x_b$ saturates at $x_3/L_c=0.08$, the turbulent reattachment point.

Figure 15

(a) The leading gain profile against the windowing $x_b$ for $\text{St}=110$ and $k_z{L}_c=0$. The energy amplification according to the linear stability analysis using Eq. (2) is also shown for comparison, where the dashed part of the curve is obtained by interpolating for $k_{x,i}$ in the regions of absolute instability. (b) The rate of change in the leading gain with respect to $x_b$. (c) Base flow streamwise velocity profiles at three representative locations.

Figure 16

Comparison of the gain profiles from the input-window analysis with that without windowing ($k_z=0$).

Figure 17

(a) The surface-forcing modes (in their magnitudes $|\widehat{\mathit{f}}|$) and the ${\widehat{\mathit{v}}}_x$-components of response modes for representative frequencies ($k_z=0$). The magenta dashed lines mark $x_{\text{act}}\equiv {argmax}_x(\left|\widehat{\mathit{f}}\left(x\right)\right|)$ as the optimal actuator locations. (b) The orbit of the momentum-based forcing at $x_{\text{act}}$ computed by rotating the phase of ${[{\widehat{v}}_{f_x},{\widehat{v}}_{f_y}]}_{x_{\text{act}}}$ over a period.